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Supian SUDRADJAT
editura universitaţii bucureşti
Referenţi ştiinţifici: Prof. univ. dr. Vasile PREDA
Prof. univ. dr. Ion VĂDUVA
@ editura universităţii din bucureşti Şos. Panduri, 90-92, Bucureşti-050663;Telefon/Fax: 410.23.84
E-mail:editura_unibuc@yahoo.com Internet:www.editura.unibuc.ro
Descrierea CIP a Bibliotecii Naţionae a Romaniei SUDRADJAT, SUPIAN Mathematical programming models for portfolio selection / Supian Sudradjat – Bucureşti: Editura Universittăţii din Bucureşti, 2007 ISBN 978-973-737-351-9 51-7:336.71+336.717
This work would not have been possible without the advice and help of many people. Foremost, I wish to express my deep gratitude to: - Professor Vasile PREDA, - Prof. univ. dr. Ion VĂDUVA I would also like to thank all the people who helped me during the course of my studies. Above all, - Rector of Bucharest University Romania, - Rector of Padjadjaran University Bandung Indonesia, - H.E. Nuni Turnijati Djoko,(the Ambasador of the Republic of Indonesia in Bucharest Romania), - Purno Wirawan - Islah Abdullah Dedication
To: - my dear parents,
Halimah and the late Ojon SUPIAN - my wife Deti SUDIARTI, and - my childrens
Sudradjat ISMAIL HASBULLAH, Sudradjat MUHAMMAD IKHSAN, and Sudradjat FITRIYANTI
Preface
Gratitude to the Almighty, the only God, for completeness of this book
entitled “Mathematical Programming Models For Portfolio Selection” so it can be publish
as planed. The subject of this book is in close connection to some mathematical techniques
applications in financial modeling. More specifically, multicriteria portfolio optimization
started with the Markowitz mean-variance model. Basically, Harry Markowitz introduced the
theory of modern portfolios, which originates in a quadratic programming problem applied
for evaluating a portfolio of assets. The resulting model, namely the mean-variance model, is
one of the most used quadratic programming models. Then, Markowitz’s model was
extended in various directions. Recently, some authors implemented dynamic investments
models in order to study long-term effect and improve the performance.
Constructing a dynamic financial model consists of three basic components: 1) a
stochastic differential system of equations for describing the model’s relevant random
quantities development (alternative scenarios are therefore generated); 2) a decision simulator
for finding investor position at each moment and 3) a dynamic optimization model.
In the classical approach of portfolios selection, expected utility theory is applied
based on a set of axioms related to investor’s behavior and on order relation between
deterministic and random events from the set of possible choices. The specific characteristics
of axioms characterizing the utility function take into account the assumption that a
probability measure could be defined on random results. If, in addition, one assumes that the
origins of these random results are not very well known, then the probability theory proves
itself inadequate due to the lack of experimental information. In these situations, the decision
problem could be addressed on uncertainty basis, using different mathematical instruments.
Furthermore, the preferences function describing investor’s utility could be modified with
respect to uncertainty degree.
The portfolio selection problem on uncertainty assumption could be transformed into
a decision problem in fuzzy environment. Fuzzy theory was intensively used from 1960 for
solving many problems, including financial risk management problems. The concept of fuzzy
random variable is a proper extension of classical random variable. Using fuzzy approach, the
experts’ knowledge and subjective opinions of investors could be easier fit in a portfolio
selection model.
The main goal of this book is to examine the methods for solving statistical problems
involving fuzzy element in the random experiment and it aims to be a starting point in
constructing a portfolio selection model of Markowitz type. There are presented models
which involve stochastic dominance constraints on the returns of portfolios and necessary
conditions for possible constraints programming, which are solved by transforming them into
multi-objective linear programming problems.
In the first chapter there is underlined the importance of the topic proposed in this
book, and then, some important results from the literature are presented. Also, in this chapter
are slightly detailed the other chapters of the book, and some results are highlighted.
In the second chapter, “Some classes of stochastic problems”, the relationships
between efficiency sets for some multi-objective determinist programming problems are
presented. These results will be used later in analyzing the concept of efficient solution for a
multi-objective stochastic programming problem. We have to note here the results obtained
in Sections 2.4, 2.6. and 2.7, which extend the results of Cabalero, Cerda, Munoz, Rez,
Stancu-Minasian and White.
In third chapter, “Portofolio optimization with stochastic dominance constraints, it is
considered the construction of a portfolio with finite assets whose returns are described by a
discrete distribution. A portfolio optimization model with stochastic dominance constraints
on the returns is presented. Optimality and duality of these models are studied and, also,
equivalent optimization models are constructed using utility functions.
In forth chapter, “The dominance-constrained portfolio”. We remark the results from
Sections 4.3, 4.4 and 3.6, extending the results of Dentcheva, Ruszczynski, Rothschild,
Stiglitz and Ogrzczak.
In fifth chapter, “Portfolio optimization using fuzzy decision”. In this chapter we
introduce with fuzzy linear programming models and interactive fuzzy linear programming.
Also represents a generalization of Chapter 4. Here optimization problems with stochastic
dominance constraints, using fuzzy decisions. The fuzzy linear programming problems and
fuzzy multi-objective programming problems are thoroughly treated. We remark again the
important results of Sections 5.4, 5.5, 5.6, 5.7 and 5.8. and the extensions of some results
belonging to Markowitz, Klirr, Zuan, Gasimov, Lai and Hwang, and in Section 5.9, we
studied about multiobjective fractional programming problem under fuzziness.
In sixth chapter, “A possibilistic aprroach for portfolio selection problem“ there is
considered a programming problem with possible constraints, which will be solved by
transforming it into a multi-objective programming problem. The results from Sections 6.22,
6.3.3, 6.4 and 6.5, extend some results given by Chen, Inuiguchi, Ramik, Majlender, Yhou
and Li.
In seventh chapter, “Atzbergerţ’s extension of Markowitz portfolio selection”,
represent one basic manner by which Markowitz’s theory for portfolio selection can be
extended to account for non-gaussian distributed returns. We then discuss how a model
incorporating information about the performance of the assets in different market regimes
over the holding period can be developed.
Most of the original results presented in this book were presented in very important
conferences and workshops. Also, we have to note the large list of references considered
elaborating this book.
I wish to acknowledge the teachers, colleagues, and reviewers who contributed to
earlier editions of this book and further to extend my appreciation for the guidance and
suggestions donated during its revision.
Gratitude is particularly due to Prof. DR. Vasile PREDA, Prof. DR. Ion VĂDUVA,
Acad. Marius IOSIFESCU, Dr. Ion M. STANCU-MINASIAN, DR. Miruna BELDIMAN,
DR. Roxana CIUMARA. I would also like to thank all the people who helped me. Above
all, Rector of Bucharest University Romania, Rector of Padjadjaran University Bandung
Indonesia, H.E. Nuni Turnijati Djoko (Ambasador of the Republic of Indonesia in Bucharest
Romania), Islah Abdullah, Purno Wirawan, Sam E. Marentek, Hary Irawan, Pratiwi
Amperawati, Dedin M. Nurdin.
Supian SUDRADJAT
CONTENTS Preface Chapter 1 Introduction …………..……………………….. ……….... 1 Chapter 2 Some classes of stochastic problems …………………....... 7
2.1 Introduction …………………………………………………… 7
2.2 Efficient solution concepts ………………………………… 10
2.3 Relations between the efficient sets of several deterministic
multiobjective programming problems ………………………... 13
2.4 Some relations between expected-value efficient solution,
minimum-variance efficient solutions and expected-value
standar-deviation efficient solutions ………………………... 19
2.5 Multicriteria problems …………………………………………. 20
2.6 Relations between classes of solutions for (P1), (P2) and (P3)….. 21
2.7 White’s approach multiobjective weighting factors auxiliary
optimization problem for (P1), (P2) and (P3) ……………….. 27
2.7.1 Introduction …………………………………………. 28
2.7.2 Transformations and auxiliary optimization problem
associated to (P1), (P2) and (P3) ……………….. 29
2.7.3 Non-convex auxiliary optimization problem 32
Chapter 3 Stochastic dominance …..………….…………………….. 40
3.1 Introduction …………………………………………………… 40
3.2 Stochastic dominance …………………………………………. 42
3.3 The portfolio problem ………………………………………... 44
3.4 Consistency with stochastic dominance ………………………. 45
Chapter 4 The dominance-constrained portfolio problem ………... 52
4.1 Introducere ………………………………………………….. 52
4.2 Dominance-constrained . …….………………………………… 53
4.3 Optimality and duality ………………………………………… 56
4.5 Spliting ….………………………………………………….. 60
4.6 Decomposition …….…………………………………………. 64
Chapter 5 A fuzzy approach to portfolio optimization ……………. 68
5.1 Introduction ………………………………………………….. 68
5.2 Fuzzy linear programming models ..………………………... 69
5.3 Interactive fuzzy programming ……………………………….. 76
5.3.1 Interactive fuzzy linear programming algorithm ……… 78
5.4 Portfolio problem .…………………………………………. 80
5.5 Case of fuzzy technological coefficient and fuzzy
right-hand side numbers ………………………………… 83
5.5.1 Case of fuzzy technological coefficients …………… 83
5.5.2 Portfolio problems with fuzzy technological
coefficients and fuzzy right-hand-side numbers ………. 88
5.6 The modified subgradient method …………………………….. 93
5.7 Defuzzification and solution of defuzzificated problem ………. 96
5.7.1 A modified subgradient method to fuzzy
linear programming ………………………………… 96
5.7.2 Fuzzy decisive set method ..……………………....... 98
5.8 Portfolio problem with fuzzy multiple objective …………….. 110
5.9 Multiobjective fractional programming problem under fuzziness.. 115
5.9.1 Problem formulation and the solution concept ………… 116
5.9.2 Solution algorithm ………………………………….. 122
5.9.3 Basic stability nations for problem (FMOFP) ………. 125
5.9.4 Utilization of Kuhn-Tucker conditions corresponding
to problem ……………………..……………..... 125 )( λP
Chapter 6 A possibilistic approach for a portfolio selection problems .. 128
6.1 Introduction ………………………………………………….. 128
6.2 Mean VaR portfolio selection multiobjective model
with transaction costs ..………………………………………. 129
6.2.1 Case of downside-risk ..…………………………….. 129
6.2.2 Case of proportional transaction costs model ………... 131
6.3 A possibilistic mean Var portfolio selection model …………... 131
6.3.1 Possibilistic theory. Some preliminaries ……………. 132
6.3.2 Triangular and trapezoidal fuzzy numbers …………... 133
6.3.3 Construction of efficient portfolios .……………….. 135
6.4 A weighted possibilistic mean value approach ……………….. 138
6.5 A weighted possibilistic mean variance and covariance
of fuzzy numbers …………………………………………... 142
Chapter 7 An extention of Markowitz portfolio selection ……….. 146
7.1 Introduction ………………………………………………….. 146
7.2 Gaussian mixture distribution ………………………………… 148
7.3 An extention of the Markowitz portfolio theory ……………….. 151
7.4 Portfolio selection problem (GM-PoS) ………………………... 152
Bibliography ….……………………………….……………… 154 Apendix Notations ………………………………………………………. 172
Acronyms & Abbreviations …………………………… 174 Index ………………………………………………….. 175
1
CHAPTER 1
INTRODUCTION
The problem of optimizing a portfolio of finitely many assets is a classical problem in
theoretical and computational finance. Since the seminal work of Markowitz [112] it is
generally agreed that portfolio performance should be measured in two distinct
dimensions: the mean describing the expected return, and the risk which measures the
uncertainty of the return. In the mean–risk approach, we select from the universe of all
possible portfolios those that are efficient: for a given value of the mean they minimize
the risk or, equivalently, for a given value of risk they maximize the mean. This
approach allows one to formulate the problem as a parametric optimization problem, and
it facilitates the trade-off analysis between mean and risk.
In the classical approach to portfolio selection, one often applies the theory of
expected utility that is derived from a set of axioms concerning investor behaviour as
regards the ordering relationship for deterministic and random events in the choice set.
The specific nature of the axioms that characterize the utility function is based on the
assumption that a probability measure can be defined on the random outcomes.
However, if we assume that the origins of these random events are not well known, then
the theory of probability proves inadequate because of a lack of experimental
information. In such instances, one has to approach the decision theory problem under
uncertainty using different mathematical tools. Further, the preference function that
describes the utility of the investor may itself be changing with the degree of uncertainty.
Moreover, one could postulate that the investor has multiple preference functions each of
which corresponds to a particular view on various factors that influence the future state
of the economy and the confidence with which it is held. Under these conditions, the
existing literature in the field of economic theory does not provide the investor with
sufficient tools to address the portfolio selection problem. The discussion above
highlighted potential difficulties one would encounter when addressing the portfolio
selection problem under uncertainty. It was postulated that under uncertainty the investor
2
would be confronted with multiple utility functions. Each one of these utility functions
may be attributed to a particular market view being held and can be broadly described as
capturing the investor’s level of satisfaction if it turns out to be true. For instance, a fund
manager structuring a fixed-income portfolio may have only vague views regarding
future interest rate scenarios and these can broadly be described as being “bullish”,
“bearish” or “neutral”. Such views may arise out of the subjective and/or intuitive
opinion of the decision-maker on the basis of information available at the given point in
time. Under these circumstances, one might try to characterize the range of acceptable
solutions to the portfolio selection problem as a fuzzy set (see Bellman and Zadeh [9]).
In simple terms, a fuzzy set is a class of objects in which there is no clear distinction
between those objects that belong to the class and those that do not. Further, associated
with each object is a membership function that defines the degree of membership of the
object in the set. In this respect, fuzzy set theory provides a framework to deal with
problems in which the source of imprecision is the absence of sharply defined criteria of
class membership rather than the presence of random variables. This provides the point
of departure from probability theory, where the uncertainty arises from the random
nature of the environment rather than from any vagueness of human reasoning. In the
context of choosing optimal portfolios that target returns above the risk-free rate for
certain market scenarios while at the same time guaranteeing a minimum rate of return,
fuzzy decision theory provides an excellent framework for analysis. This is because the
nature of the problem requires one to examine various market scenarios, and each such
scenario will in turn give rise to an objective function. In the face of uncertainty, one will
not be able to assign a numerical value to the probability of these scenarios occurring.
Under this constraint, it is not clear how a suitable weighting vector can be determined to
solve the multi-objective optimization problem. One way to overcome this difficulty is to
use the membership function that arises in fuzzy decision theory to serve as a suitable
preference function for finding an ordering relation for the uncertain events. In fact, one
can describe the membership function as the fuzzy utility of the investor, which
describes the behaviour of indifference, preference or aversion towards uncertainty,
Mathieu-Nicot [115]. The advantage of using the membership function is that it does not
rely necessarily on the existence of a probability measure but rather on the existence of
relative preference between the uncertain events.
3
The above arguments show how the portfolio selection problem under uncertainty can be
transformed into a problem of decision-making in a fuzzy environment Bellman and
Zadeh [9]. To do this, one has to model the aspirations of the investor on the basis of the
strength of the views held on various market scenarios through suitable membership
functions of a fuzzy set. For instance, a fund manager structuring a fixed-income
portfolio may have aspiration levels as to what the portfolio’s acceptable excess return
over the risk-free rate should be for those scenarios he/she considers more likely. The
concepts of fuzzy sets, fuzzy goals and fuzzy decision will be introduced and a fuzzy
multi-criteria optimization problem will be formulated.
As stated by Markowitz in [112,114), “The expected utility maxim appears reasonable
offhand. But so did the expected return maxim. Perhaps there is some equally strong
reason for decisively rejecting the expected utility maxim as well”.
The classical Markowitz model is
[ ])()( xRarx V=ρ ,
where )(xρ is the variance of the return, and )(xR is total return.
The mean–risk portfolio optimization problem is formulated as follows:
[ ])()(max xxx
λρμ −∈X
.
where R and X are defined in section 3.3.
Here, λ is a nonnegative parameter representing our desirable exchange rate of mean
for risk. If 0=λ , the risk has no value and the problem reduces to the problem of
maximizing the mean. If 0>λ we look for a compromise between the mean and the
risk. The general question of constructing mean–risk models which are in harmony with
the stochastic dominance relations has been the subject of the analysis of the recent
papers Dentcheva and Ruszcynski [41,42], Rothschild and Stiglitz [155], Ogryczak and
Ruszczynski [127, 128].
Portfolio selection is generally based on a trade-off between expected return and risk,
and requires a choice for the risk measure to be implemented. Usually, the risk is
evaluated by the conditional second-order moment, i.e., conditional variance or
volatility. This leads to the determination of the mean-variance efficient portfolio
introduced by Markowitz [114]. It can also be based on a safety-first criterion
4
(probability of failure), initially proposed by Roy [149] and then implemented by Levy
and Sarnat [100]. The efficient portfolio is one for which there does not exist another
portfolio that has higher mean and no higher variance, and/or has less variance and no
less mean at the terminal time T . In other words, an efficient portfolio is one that is
Pareto optimal.
Notwithstanding its popularity, mean variance approach has also been subject to a lot of
criticism. Alternative approaches attempt to conform the fundamental assumptions to
reality by dismissing the normality hypothesis in order to account for the fat-tailedness
and the asymmetry of the asset returns. Consequently, other measures of risk, such as
Value at Risk (VaR), expected shortfall, mean absolute deviation, semi-variance and so
on are used.
Another theoretical approach to the portfolio selection problem
- Stochastic dominance (Mosler and Scarsini, [121]), the concept of stochastic dominance
is related to models of risk-averse preferences Fishburn [52]. It originated from the
theory of majorization Hardly, Littlewood and Poya [70] for the discrete case, was later
extended to general distributions Quirk and Saposnik[146]; Hadar and Russel [66];
Hanoch and Levy [68]; Rothschild and Stielits [155], and is now widely used in
economics and finance (Levy [99]).
- The usual (first order) definition of stochastic dominance gives a partial order in the
space of real random variables. More important from the portfolio point of view is the
notion of second-order dominance, which is also defined as a partial order. It is
equivalent to this statement: a random variable R dominates the random variable Y if
)]([)]([ YuERuE ≥ for all non-decreasing concave functions u(·) for which these
expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio
with return Y over a portfolio with return R.
- The stochastic optimization model with stochastic dominance constraints Dentcheva and
Ruszcynsk [42, 44], can be used for risk-averse portfolio optimization. We add to the
portfolio problem the condition that the portfolio return stochastically dominates a
reference return, for example, the return of an index. We identify concave non-
decreasing utility functions which correspond to dominance constraints. Maximizing the
5
expected return modified by these utility functions, guarantees that the optimal portfolio
return will dominate the given reference return.
- Fuzzy set theory, since 1960s, has been widely used to solve many problems including
financial risk management. The concept of a fuzzy random variable is a reasonable
extension of the concept of a usual random variable in the many practical applications of
random experiments, where the implicit assumption of data precision seems to be an
inappropriate simplification rather than an adequate modeling of the real physical
conditions. By using fuzzy approaches, the experts’ knowledge and the investors’
subjective opinions can be better integrated into a portfolio selection model. Bellman and
Zadeh [9] proposed the fuzzy decision theory. Ramaswamy [14] presented a bond
portfolio selection model based on the fuzzy decision theory, Sudradjat and Preda [188]
presented on portfolio optimization using fuzzy decisions. The notion of a fuzzy random
variable (see for example, Kwakernaak [91], Puri and Ralescu [145], Kruse and Meyer
[89] provides a valuable model that is manageable in a probabilistic framework. Also,
the concept of fuzzy information presented by Zadeh [216] can formalize either the
experimental data or the events involving fuzziness. The concept of a fuzzy random
variable Puri and Ralescu [145] was defined as a tool for establishing relationships
between the outcomes of a random experiment and inexact data, Ostermark [128]
proposed a dynamic portfolio management model. Watada [201] presented another type
of portfolio selection model based on the fuzzy decision principle. The model is directly
related to the mean-variance model, where the goal rate for an expected return and the
corresponding risk described by logistic membership functions.
- In standard portfolio models uncertainty is equated with randomness, which actually
combines both objectively observable and testable random events with subjective
judgments of the decision maker into probability assessments. A purist on theory
would accept the use of probability theory to deal with observable random events, but
would frown upon the transformation of subjective judgments to probabilities. Tanaka
et al [194] give a special formulation of fuzzy decision problems by the probability
events. Carlsson et al [26] studied the portfolio selection model in which the rate of
return of security follows the possibility distribution. Sudradjat, Popescu and Ghica
[187] studied on possibilistic approach a portfolio selection problem. Applying
possibilistic distribution may have two advantages: (1) the knowledge of the expert can
6
be easily introduced to the estimation of the return rates and (2) the reduced problem is
more tractable than that of the stochastic programming approach. Korner [86] pointed
out that the variability is given by two kinds of uncertainties: randomness (stochastic
variability) and imprecision (vagueness). Randomness models the stochastic variability
of all possible outcomes of an experiment. Fuzziness describes the vagueness of the
given or realized outcome. Kwakernaak [91] presented another explanation for the
difference between randomness and fuzziness. He pointed out that when we consider
an opinion poll in which a number of people are questioned, randomness occurs
because it is not known which response may be expected from any given individual.
Once the response is available, there still is uncertainty about the precise meaning of
the response.
The aim of this book is to examine methods for handling statistical problems
involving fuzziness in the elements of the random experiment, and serves as a point from
which to derive the Markowitz portfolio model in the presence of efficient solution
concepts for a stochastic multi-objective programming, develop portfolio optimization
model involving stochastic dominance constraints on the portfolio return and necessary
and sufficient conditions of optimality and duality, we develop portfolio optimization
using fuzzy decisions in concentrate on fuzzy linear programming, and finally we
consider a mathematical programming model with possibilistic constraint and we it solve
by transforming into multi-objective linear programming problem.
7
CHAPTER 2
SOME CLASES OF STOCHASTIC PROBLEMS
2.1 Introduction Stochastic programming deals with a class of optimization models and algorithms in
which some of the data may be subject to significant uncertainty. Such models are
appropriate when data evolve over time and decisions need to be made prior to observing
the entire data stream. For instance, investment decisions in portfolio planning problems
must be implemented before stock performance can be observed. Similarly, utilities must
plan power generation before the demand for electricity is realized. Such inherent
uncertainty is amplified by technological innovation and market forces. As an example,
consider the electric power industry. Deregulation of the electric power market, and the
possibility of personal electricity generators (e.g. gas turbines) are some of the causes of
uncertainty in the industry. Under these circumstances it pays to develop models in
which plans are evaluated against a variety of future scenarios that represent alternative
outcomes of data. Such models yield plans that are better able to hedge against losses
and catastrophic failures. Because of these properties, stochastic programming models
have been developed for a variety of applications, including electric power generation
(Murphy [124]), financial planning (Carino et al [23]), telecommunications network
planning (Sen et al [170]), and supply chain management (Fisher et al [51]), to mention a
few.
The widespread applicability of stochastic programming models has attracted
considerable attention from the OR/MS community, resulting in several recent books
(Kall and Wallace [77], Birge and Louveaux [16], Prekopa [138, 139]) and survey
articles (Birge [15], Sen and Higle [169]). Nevertheless, stochastic programming models
remain one of the more challenging optimization problems.
While stochastic programming grew out of the need to incorporate uncertainty in linear
and other optimization models (Dantzig [39], Beale [8], Charnes and Cooper [30]), it has
close connections with other paradigms for decision making under uncertainty. For
8
instance, decision analysis, dynamic programming and stochastic control, all address
similar problems, and each is effective in certain domains. Decision analysis is usually
restricted to problems in which discrete choices are evaluated in view of sequential
observations of discrete random variables. One of the main strengths of the decision
analytic approach is that it allows the decision maker to use very general preference
functions in comparing alternative courses of action. Thus, both single and multi-
objectives are incorporated in the decision analytic framework. Unfortunately, the need
to enumerate all choices (decisions) as well as outcomes (of random variables) limits this
approach to decision making problems in which only a few strategic alternatives are
considered.
These limitations are similar to methods based on dynamic programming, which also
require finite action (decision) and state spaces. Under Markovian assumptions the
dynamic programming approach can also be used to devise optimal (stationary) policies
for infinite horizon problems of stochastic control (see also Neuro-Dynamic
Programming by Bertsekas and Tsitsiklis [13]). However, DP-based control remains
wedded to Markovian Decision Problems, whereas path dependence is significant in a
variety of emerging applications. Stochastic programming provides a general framework
to model path dependence of the stochastic process within an optimization model.
Furthermore, it permits uncountably many states and actions, together with constraints,
time-lags etc. One of the important distinctions that should be highlighted is that unlike
dynamic programming, stochastic programming separates the model formulation activity
from the solution algorithm. One advantage of this separation is that it is not necessary
for stochastic programming models to all the same mathematical assumptions. This leads
to a rich class of models for which a variety of algorithms can be developed. On the
downside of the ledger, stochastic programming formulations can lead to very large scale
problems, and methods based on approximation and decomposition become paramount.
A whole series of production processes, economic system of different types, and
technical objective is described by mathematical models which are multi-criteria
optimization problems (Steuer [177], Chankong and Haimes [29] and Stancu-Minasian
[175]) . This situation is quite usual, because frequently it is necessary to take into
9
account simultaneously the influence of a number of contradictory external factors on
the system.
The most intensive development of the theory and the methods of detailed bibliographic
description of which is given in Zeleny [217] and Urli and Nadeau [196], are linear and
non linear multi-criteria optimization problems. Some classifications of the methods of
this type, oriented to the specific user, and multi-criteria optimization problems with
contradictory constraints were explored in are given (Salukavadze and Topchishvili
[166]). Very interesting results generalized into the general domination cone for different
classes of solutions of multi-criteria problem are given (Salukavadze and Topchishvili
[166]).
Now, one of the widely developing fields in multi-criteria optimization is its qualitative
theory; the most important results are given (Salukavadze and Topchishvili [166]). Well-
known algorithms can be modified and new theoritical results.
The objective of this chapter is to examine some properties of different classes of multi-
criteria optimization problem solutions.
Most real-life engineering optimization problems require simultaneous optimization of
more than one objective function. In these cases, it is unlikely that the same values of
design variables will results in the best optimal values for all the objectives. Hence, some
trade-off between the objectives is needed to ensure a satisfactory design.
As the system efficiency indices can be different (and mutually contradictory), it is
reasonable to use the multi-objective approach to optimize the overall efficiency. This
can be done mathematically correctly only when some optimality principle is used. We
use Pareto optimality principle, the essence of which is following. The multi-objective
optimization problem solution is considered to be Pareto-optimal if there are no other
solutions that are better in satisfying all of the objectives simultaneously. That is, there
can be other solutions that are better in satisfying one or several objectives, but they
must be worse than the Pareto-optimal solution in satisfying the remaining objectives.
10
2.2 Efficient solution concepts Consider a model in which the design/decision associated with a system is specified via
vector x. Under uncertainty, the system operates in an environment in which there are
uncontrollable parameters which are modeled using random variables. Consequently, the
performance of such a system can also be viewed as a random variable. Accordingly,
stochastic programming models provide a framework in which designs (x) can be chosen
to optimize some measure of the performance (random variable). It is therefore natural to
consider measures such as the worst case performance, expectation and other moments
of performance, or even the probability of attaining a predetermined performance goal.
Let us consider the stochastic multi-objective programming problem Caballero, et al [21]
( ))~,(),...,~,(min 1 cxzcxz qDx∈, (2.1)
where the following notations and assumptions
• there is a compact set nD R⊆ of feasible actions;
• nx R∈ is thevector of decision variables of the problem and c~ is a random
vector whose components are random continous variables, defined on the set nRE∈ . We assume given the family F of events (that is, subset of E ) and the
distribution of probability P defined on F so that, for any subset of E , E⊂A ,
F∈A , the probability P(A) is known. Also, we assume that the distribution of
probability P is independent of the decision variables nxx ,...,1 ;
• there are q objective functions )( ⋅kf with +∈R)(xfk for all Dx∈ and
c~ is a random vector whose components are random continuous variable;
• it is required to find members of the efficient (vector minimal) set E of D with
respect to the order relation ≤ on qR , where, by definition,
)()()()(,: xfyfxfyfDyDxE =→≤∈∈= (2.2)
Let )(xzk is the expected value of the kth objective function, and let )(xkσ be its
standard deviation, ,...,1 qk ∈ . Let us assume that, for every ,...,1 qk ∈ and for
every feasible vector x of the stochastic multi-objective programming problem, the
standard deviation )(xkσ is finite. In this section we will shows the definitions and
11
relations between expected value standard deviation efficient solution and efficient
solutions.
Next the following definitions by Caballero, et al [21],
Definition 2.1 [21] Expected-Value Efficient Solution. The point Dx∈ is an expected-
value efficient solution of the stochastic multi-objective problem if it is Pareto efficient to
the following problem :
( ))(),...,(min:)( 1 xzxzPE qDx∈.
Let PEE be the set of expected-value efficient solution of the stochastic multi-objective
problem.
Definition 2.2 [21] Minimum-Variance Efficient Solution. The point Dx∈ is a
minimum-variance efficient solution for the stochastic multi-objective problem if it is a
Pareto efficient solution for the problem :
( ))(),...,(min:)( 221
2 xxP qDxσσσ
∈.
Let 2σPE be the set of efficient solutions of the problem )( 2σP .
Definition 2.3 [21] Expected-Value Standard-Deviation Efficient Solution or σE
Efficient Solution. The point Dx∈ is an expected-value standard-deviation efficient
solution for the stochastic multi-objective programming problem if it is a Pareto efficient
solution to the problem
( ))(),...,(),(),...,(min:)( 11 xxxzxzPE qqDxσσσ
∈.
Let σPEE be the set of expected-value standard-deviation efficient solutions of the
stochastic multi-objective programming problem (2.1).
Now, we give the concepts of efficiency for two criteria of maximum probability. As we
will see next, in order to define these two concepts, the minimum-risk criterion (concept
of minimum-risk efficiency) and Kataoka criterion (efficiency in probability) are applied
respectively to each stochastic objective.
Definition 2.4 [21] Minimum-Risk Efficient Solution for the Levels quu ,...,1 . See
Stancu-Minasian and Tigan (180). The point Dx∈ is a minimum risk vectorial solution
for levels quu ,...,1 if it is a Pareto efficient solution to the problem:
12
( )))~,(~(,...,)~,(~(max:))(( 11 qqDxucxzPucxzPuPRM ≤≤
∈,
Let )(uPRME be the set of efficient solution for the problem (PMR(u)).
Definition 2.5 [21] Efficient Solution with Probabilities qββ ,...,1 or β -Efficient
Solution. The point Dx∈ is an efficient solution with probabilities qββ ,...,1 if there
exist qu R∈ such that ttt ux ),( is a Pareto efficient solution to problem:
))(( βPP
( )
DxqkucxzP
uu
kkk
qux
∈=≥≤ ,,1,)~,(~
,...,min 1,
β
Let )(βPPE nR⊂ be the set of efficient solutions with probabilities qββ ,...,1 for the
stochastic multi-objective programming problem (2.1).
It may be noted that these definitions of efficient solution are obtained by applying the
same transformation criterion to each one of the objectives separately (expected value,
minimum variance, etc.), and by building after word the resulting deterministic
multiobjective problem. In this sense, it is necessary to the following results.
Remark 2.1 The concepts of expected value, minimum variance, etc., weak and properly
efficient solution can be defined in a natural way.
Remark 2.2 The concepts of minimum-risk efficiency and β -efficiency require setting a
priori a vector of aspiration levels u or a probability vector β . This implies that, in both
cases, the efficient set obtained depends on the predetermined vectors in such a way that,
in general, different level and proba bility vector give rise to different efficient sets,
).()(
),()(''
''
ββββ PPPP
PRMPRM
EE
uEuEuu
≠⇒≠
≠⇒≠
Remark 2.3 The concept of expected standard-deviation efficient solution is an
extention to multiobjective case of the concept of the mean-variance efficient solution
that Markowitz [114] defines for the stochastic single objective problem of portfolio
selection. In this way, we have the two statistical moments corresponding to each
stochastic objective in the same measuring units. Since the square root function is
13
strictly increasing, the set of efficient solutions does not vary in problem if we substitute
standard deviation for variance, White [209].
Remark 2.46 The efficiency in probability criterion is a generalization of the one
presented by Goicoechea, Hansen, and Duckstein [63], who define the same concept
taking the same probability β for all the stochastic objectives and with the probabilistic
equality constraints taking the form
β=≤ )~,(~ kk ucxzP .
This notion was introduced by Stancu-Minasian [179], considering the Kataoka problem
in the case of multiple criteria.
2.3 Relations between the efficient sets of several of deterministic
multiobjective programming problems
We present some relations between the efficient sets of several problems of deterministic
multi-objective programming problems. These results will be used later for analysis of
concepts of efficient solutions for multi-objective stochastic problems.
Considered f and g be vectorial functions defined on the same set nH R⊂ with
nHf R⊂: qR→ and nHg R⊂: qR→ and let γα , be nonnull vectors with q
real components, that is, qR∈γα , and 0, ≠γα . Let us consider the following
multiobjective problems:
(PD1) ( )))(()),...,((),(),...,(min 111 xgxgxfxf qqqDxγγ
∈ (2.3)
(PD2) ( ))(),...,(min 1 xfxf qDx∈ (2.4)
(PD3) ( ))))(()),...,((min 2211 xgxg qqDx
γγ∈
(2.5)
with, qR∈γ , 0≠γ . Let 321 ,, EEE be the sets of weakly efficient, efficient, and
proper efficient points of problem )( iPD , respectively. The following theorem relates
these problems (PD1), (PD2) and (PD3) problems to each other.
14
Theorem 2.1 We assume that 0>g for every Dx∈ ,. Then:
i1 132 EEE ⊂∩
i2 wEEE 132 ⊂∪
i3 www EEE 132 ⊂∪
Proof:
( 1i ) 32 EEx ∩∈
Let us show that 1Ex∈ by reductio ad absurdum. We assume that 1Ex∉ . Then, there
exist an Dx ∈* such that )()( * xfxf kk ≤ and ))(())(( * xgxg kkkk γγ ≤ , for every
,...,1 qk ∈ , there being an ,...,1 qs∈ for which the inequality is strict,
)()( * xfxf ss < or ))(())(( * xgxg ssss γγ < .
Therefore, 2Ex∉ or 3Ex∉ , since ))(())(( * xgxg kkkk γγ ≤ , implies ≤))(( *0 xg skkγ
))(( 0 xg skkγ , contrary to 32 EEx ∩∈ .
(i2 ) wEEE 132 ⊂∪
Let 32 EEx ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that
wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates x and so verifies
)()( * xfxf kk < and ))(())(( * xgxg kkkk γγ < , for every ,...,1 qk = . Thus, 2Ex∉
and, since ))(())(( * xgxg kkkk γγ < , implies <))(( *0 xg skkγ ))(( 0 xg s
kkγ , 3Ex∉ ,
contrary to 32 EEx ∪∈ .
(i3) www EEE 132 ⊂∪
Let ww EEx 32 ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that
wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates the vector x and
therefore verifies that )()( * xfxf kk < and ))(())(( * xgxg kkkk γγ < , for every
,...,1 qk ∈ . Thus, wEx 2∉ and, since ))(())(( * xgxg kkkk γγ < , implies
<))(( *0 xg skkγ ))(( 0 xg s
kkλ , wEx 3∉ , contrary to ww EEx 32 ∪∈ .
15
Thus, (i2) can be deduced from (i3)
Now we consider the following problem
( )))(()()),...,(()(min 111 xgxfxgxf qqqDxαα ++
∈ (2.6)
where q1 :),...,( RR →= +qααα .
Let )(4 αE and )(4 αGE denote the efficient solutions set and the properly efficient
solutions set respectively for problem (2.6). We will now present some relations between
these sets and the set of efficient solutions and properly efficient solutions for problem
(PD1).
Theorem 2.2 [21] For qq RR →= +:),...,( 1 γγγ , q
1 :),...,( RR →= +qααα , with
0, ≠kk γα and qksignsign kk ,1),()( == γα , the following relation holds :
14 )( EE ⊂α .
Proof: Let )(4 αEx∈ . We assume that 1Ex∉ . In this case, there is a solution *x that
dominates the solution x, that is,
)()( * xfxf kk ≤ and ))(())(( * xgxg kkkk γγ ≤ , for every ,...,1 qk ∈ , and there
exist at least one ,...,1 qs∈ for which the inequality is strict, that is,
)()( * xfxf ss < or ))(())(( * xgxg ssss γγ <
From this point onward, since
)()( * xfxf kk ≤ , ))(())(( * xgxg kkkk γγ ≤ , implies ≤))(( *xgkkα ))(( xgkkα ,
the following inequalities are verified:
))(()())(()( *** xgxfxgxf kkkkkk λα +≤+ , for every ,...,1 qk ∈ , (2.7)
))(()())(()( * xgxfxgxf kkkkkk αα +≤+ , for every ,...,1 qk ∈ . (2.8)
From (2.7) and (2.8), we obtain
))(()())(()( ** xgxfxgxf kkkkkk αα +≤+ , for every ,...,1 qk ∈ .
In particular, for sk = , we have the results bellow:
(a) if )()( * xfxf ss < ,
))(()())(()( *** xgxfxgxf ssssss αα +<+ ,
16
and the following inequality is obtained from (2.8):
))(()())(()( ** xgxfxgxf ssssss αα +<+ ;
(b) if ))(())(( * xgxg ssss αα < ,
))(()())(()( *** xgxfxgxf ssssss αα +<+ ,
and since )()( * xfxf ss ≤ , we obtain
))(()())(()( ** xgxfxgxf ssssss αα +<+ .
Therefore, for every ,...,1 qk ∈ ,
))(()())(()( ** xgxfxgxf kkkkkk αα +≤+ ,
and there is at least a subscript ,...,1 qs∈ for which
))(()())(()( ** xgxfxgxf ssssss αα +<+ ,
which implies that the solution *x dominates the solution x; therefore, we reach a
contradiction with the hypothesis of *x being the efficient solution to problem (2.6).
Next, we prove that, in some conditions, this relationship is hold for the set of properly
efficient solution. For this purpose, we define problems ),(, μλγgfP and )(ξαP ,
obtained by applying the weighting method to problems (2.3)-(2.6) respectively as
follows:
∑=
∈+
q
kkkk
t
Dxgf xgxfP1
, )()(min:)),(( γμλμλγ ,
))()((min:))((1
xgxfP kkk
q
kkDx
αξξα +∑=
∈.
We use the results available in the literature about the relationships between the
optimal solution to the weighting problem and the efficient solutions to the multi-
objective problem. Some results, see Chankong and Haimes [29], applied to problem
(2.3) and its associated weighted problem ),(, μλγgfP , are as follow :
17
(a) If f and tqq gg ),...,( 11 γγ are convex functions, D is convex, and *x is a properly
efficient solution for the multi-objective problem (2.3), there exist some weight
vector μλ, with strictly positive components such that *x is the optimal solution
for weighted problem ),(, μλγgfP .
(b) For each vector of weights with strictly positive components, the optimal solution to
the weighted problem ),(, μλγgfP is properly efficient for the multi-objective
problem (P1).
Proposition 2.1 If f and ))(),...,(( 11 qq gg γγ are convex functions, D is a convex set
and there exist qq RR →= +:),...,( 1 ααα , )()( kk signsign γα = , for every
,...,1 qk ∈ then GG EE 14 )( ⊂α .
Proof: If f and ))(),...,(( 11 qq gg γγ are convex functions and if D is a convex set, then
the set of properly efficient solutions to problems (PD1) and (2.6) are obtained from the
associated weighted problems for strictly positive weight vectors. We will prove that any
solutions to the optimization problem )(ξαP , with 0>ξ , is a solution to problem
),(, μλγgfP for some vector 0),( >μλ .
Let )(4 αGEx∈ . Then, given the established hypotheses, there exist a vector 0>ξ for
which x is the solution for problem )(ξαP . Let us assume that, for every
0,,,...,1 ≠∈ kkqk γα . Then, we take
kk ξλ = , 0,,/ >= kkkkkk μλγαξμ ,
Since 0>ξ , we obtain that x is an optimal solution to problem ),(, μλλgfP . For some
,...,1 qi∈ if 0== ii γα , then the proof would be the same, since in problem (2.3)
the function ig is not involved and since in problem (2.6) the function ith objective
would be if .
In general, the inverse inclution does not hold, as it’s shown by the following example.
18
Example 2.1. Let us consider the following problem:
,0,
,19./
),,(max
22
,
≥≤+
yxyxts
yxyx
with 1,),(,),( === uyyxgxyxf .
The set of efficient points for this problem is 0,,14/),( 222 >=+∈ yxyxyx t R
and is represented in Fig. 2.1.
We outline the solution of the problem
,0,
,19./
,max
22
,
≥≤+
+
yxyxts
yxyx
α
with 0>α . For each fixed 0>α , the optimal solution of the resulting problem is one
of property efficient solutions to the original becriterion problem.
y
1 ε
D
3 x
Figura 2.1
Proposition 2.2 If f and ))(),...,(( 11 qq gg γγ are convex functions, then
UΩ∈
⊂α
α )(41GG EE ,
with ,1),()(:),...,( 1 qksignsignR kkq
q ==→==Ω + γαααα R .
Proof: As the previous case, the proof of the proposition is carried out by demonstrating
that any solution to the problem ),(, μλγgfP is a solution to the problem )(ξαP for
some vector nR∈α , with ,...,1),()( qksignsign kk ∈= γα , and for some 0>ξ .
19
Consider GEx 1∈ . Since f and tqq gg ),...,( 11 γγ are convex functions, there exist vector
0, >μλ such that x is a solution to problem ),( μλfugP . Because 0, >μξ we put
kk λξ = , k
kkk ξ
γμα = ,
since 0, >μξ , therefore we obtain that x is also a solution to the problem )(ξαP .
From Proposition 2.1 and Proposition 2.2, if f and tqq gg ),...,( 11 γγ are convex
functions and if )()( kk signsign γα = , 0)().( >tt kk γα , for every ,...,1 qk ∈ , the
sets of properly efficient solutions to problem (2.3) and (2.6) verify the following
properties:
a. Every properly efficient solution to problem (2.6) is properly efficient for problem
(2.3);
b. Setting qR∈γ , with nonnull components, the set of properly efficient solutions to
problem (2.3) is a subset of the union in α of the set of properly efficient solutions
for problem (2.6).
2.4 Some relation between expected-value efficient solution, minimum-variance efficient solution and expected-value standard deviation efficient solution
Consider a problem (2.1) and sets efficient solution expected value ( PEE ) minimum
variance ( 2σPEE ), and expected value standard deviation ( σPEE ) associated with the
problem. Let wPE
wPE
wPE EEE σσ
,, 2 be the sets of weakly efficient solutions associated with
the problems in Definitions 2.1-2.3, respectively.
If we consider
)()(),()( xxgxzxf kkkk σ==
And if we choose 1=γ , given that, for ,...,1 qk ∈ , its verified that +→ RR n:σ ,
then the relations between these efficient sets are deduced directly from Theorem
20
2.5 Multi-criteria problems
Consider the following model of a multi-criteria optimization problem:
( ))(),...,(min 1 xFxF q (2.8)
Dx∈ (2.9)
where D is a nonempty set of all feasible solution, mD R⊂ ; R→DFF q :,...,1 . Stated
briefly, a multi-criteria optimization problem consists in the choice of a particular
solution Dx ∈* for which all of the utility functions qkxFk ,1),( = , simultaneously
approach bigger values or at least do not decrease.
Let us recall some concepts of multi-criteria optimization problem solutions; (Zeleny
[217] and Urli and Nadeau [196], Salukavadze and Topchishvili [166]).
Definition 2.6 The solution DxP ∈ is called Pareto-optimal (or efficient) for the
problem (2.8)-(2.9) if and only if, for every Dx∈ , the system of inequalities
)()( Pkk xFxF < , qk ,1= ,where at least one inequality is strict, is inconsistent.
Definition 2.7 The solution Dxw ∈ , is called weakly efficient (or Slater-optimal) for the
problem (2.8)-(2.9) if and only if, for every Dx∈ , the system of strict inequalities
)()( wkk xFxF < , qk ,1= , is inconsistent.
Definition 2.8 The solution DxG ∈ , is called proper efficient (or Geoffrion-optimal)
for the problem (2.8)-(2.9) if and only if it is a Pareto-optimal solution for the problem
(2.8)-(2.9) and there exists a positive number 0>θ such that, for each pk ,1= , we
have
θ≤−− )]()(/[)]()([ xFxFxFxF jG
jG
kk ,
for some j such that )()( Gjj xFxF > where Dx∈ and )()( G
kk xFxF < qk ,1= , is
inconsistent.
Let ,wjE ,E G
jE denoted the sets of weakly-efficient, efficient, and proper efficient
solutions, respectively, for the problem (2.8)-(2.9). It is obvious that
GjE ⊂ E ⊂ w
jE .
21
Next we will studied some relations between the efficient sets of several problems of
deterministic multi-objective programming.
Let f and g be vectorial functions defined on the same set nH R⊆ , with
qRHf →: and qRHg +→: . Let us consider the following multi-objective problems:
(P1) ( )))(()),...,((),(),...,(min 111 xguxguxfxf qqqDx∈ (2.10)
(P2) ( ))(),...,(min 1 xfxf qDx∈ (2.11)
(P3) ( ))))(()),...,((min 0011 xguxgu s
qqs
Dx∈ (2.12)
with, HD ⊆ , qu RR →+: , ),...,( 1 quuu = and 00 >s a real number.
2.6 Relations between classes of solutions for (P1), (P2) and (P3)
We present some relations between the efficient sets of above considered deterministic
multi-objective programming problems. These results will be used later for analysis of
concepts of efficient solutions for multi-objective stochastic problems. These results
extend Section 2.4.
For 3,2,1=i , let Gii
wi EEE ,, be the sets of weakly efficient, efficient, and proper
efficient points of problem )( iP , respectively. The following theorem relates these
problems (P1), (P2) and (P3) problems to each other.
Theorem 2.3 We assume that 0>g for every Dx∈ , and for 0, 21 >tt and qk ,1=
we have )()()( 21 tutu kk <≤ implies that )()()( 0021s
ks
k tutu <≤ . Then:
(i) 132 EEE ⊂∩
(ii) wEEE 132 ⊂∪
(iii) www EEE 132 ⊂∪
Proof:
(i) 32 EEx ∩∈
22
Let us show that 1Ex∈ by reductio ad absurdum. We assume that 1Ex∉ . Then, there
exist an Dx ∈* such that
)()( * xfxf kk ≤ and ))(())(( * xguxgu kkkk ≤ , for every ,...,1 qk ∈ , there
being an ,...,1 qs∈ for which the inequality is strict,
)()( * xfxf ss < or ))(())(( * xguxgu ssss < .
Therefore, 2Ex∉ or 3Ex∉ , since ))(())(( * xguxgu kkkk ≤ , implies ≤))(( *0 xgu skk
))(( 0 xgu skk , contrary to 32 EEx ∩∈ .
(ii) wEEE 132 ⊂∪
Let 32 EEx ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that
wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates x and so verifies
)()( * xfxf kk < and ))(())(( * xguxgu kkkk < , for every ,...,1 qk = . Thus, 2Ex∉
and, since ))(())(( * xguxgu kkkk < , implies <))(( *0 xgu skk ))(( 0 xgu s
kk , 3Ex∉ ,
contrary to 32 EEx ∪∈ .
(iii) www EEE 132 ⊂∪
Let ww EEx 32 ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that
wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates the vector x and
therefore verifies that )()( * xfxf kk < and ))(())(( * xguxgu kkkk < , for every
,...,1 qk ∈ . Thus, wEx 2∉ and, since ))(())(( * xguxgu kkkk < , implies
<))(( *0 xgu skk ))(( 0 xgu s
kk , wEx 3∉ , contrary to ww EEx 32 ∪∈ .
Thus, (ii) can be deduced from (iii)
Remark 2.5 Also we see that (ii) can be deduced from (iii). It is obvious that
www EEE 132 ⊂∩
is also verified. Futhermore, as
wEE 22 ⊂ and wEE 33 ⊂
23
then
ww EEEE 3131 ∪⊂∪ .
Remark 2.6 We note that RR →+:u with ttu λ=)( or 0,)( >= pttu pλ and
0>λ satisfy condition from Theorem 2.3.
Now we consider the following problem
( )))((~)()),...,((~)(min 111 xguxfxguxf qqqDx++
∈ (2.13)
where q1 :)~,...,~(~ RR →= +quuu .
Let )~(4 uE and )~(4 uEG denote the efficient solutions set and the properly efficient
solutions set respectively for problem (2.13). We will now present some relations
between these sets and the set of efficient solutions and properly efficient solutions for
problem (P1).
Theorem 2.4 For qquuu RR →= +:),...,( 1 and q
1 :)~,...,~(~ RR →= +quuu , such that
)()()( 21 tutu kk <≤ implies )(~)()(~21 tutu kk <≤ , qk ,1= , the following relation holds :
14 )~( EuE ⊂ .
Proof: Let )~(4 uEx∈ . We assume that 1Ex∉ . In this case, there is a solution *x that
dominates the solution x, that is,
)()( * xfxf kk ≤ and ))(())(( * xguxgu kkkk ≤ , for every ,...,1 qk ∈ , and there
exist at least one ,...,1 qs∈ for which the inequality is strict, that is,
)()( * xfxf ss < or ))(())(( * xguxgu ssss <
From this point onward, since
)()( * xfxf kk ≤ , ))(())(( * xguxgu kkkk ≤ , implies ≤))((~ *xgu kk ))((~ xgu kk ,
the following inequalities are verified:
))((~)())((~)( *** xguxfxguxf kkkkkk +≤+ , for every ,...,1 qk ∈ , (2.14)
))((~)())((~)( * xguxfxguxf kkkkkk +≤+ , for every ,...,1 qk ∈ . (2.15)
From (2.14) and (2.14), we obtain
24
))((~)())((~)( ** xguxfxguxf kkkkkk +≤+ , for every ,...,1 qk ∈ .
In particular, for sk = , we have the results bellow:
(c) if )()( * xfxf ss < ,
))((~)())((~)( *** xguxfxguxf ssssss +<+ ,
and the following inequality is obtained from (2.15):
))((~)())((~)( ** xguxfxguxf ssssss +<+ ;
(d) if ))((~))((~ * xguxgu ssss < ,
))((~)())((~)( *** xguxfxguxf ssssss +<+ ,
and since )()( * xfxf ss ≤ , we obtain
))((~)())((~)( ** xguxfxguxf ssssss +<+ .
Therefore, for every ,...,1 qk ∈ ,
))((~)())((~)( ** xguxfxguxf kkkkkk +≤+ ,
and there is at least a subscript ,...,1 qs∈ for which
))((~)())((~)( ** xguxfxguxf ssssss +<+ ,
which implies that the solution *x dominates the solution x; therefore, we reach a
contradiction with the hypothesis of *x being the efficient solution to problem
(2.13).
Next, we prove that, in some conditions, this relationship is hold for the set of properly
efficient solution. For this purpose, we define problems ),( μλfugP and )(~ ξuP , obtained
by applying the weighting method to problems (2.10) and (2.13) respectively as follows:
( )∑=
∈+
q
kkkkkkDxfug xguxfP
1))(()(min:)),(( μλμλ ,
)))((~)((min:))((1
~ xguxfP kkk
q
kkDxu +∑
=∈
ξξ .
We use the results available in the literature about the relationships between the
optimal solution to the weighting problem and the efficient solutions to the multi-
25
objective problem. Some results, see Chankong and Haimes [29], applied to problem
(2.10) and its associated weighted problem ),( μλfugP , are as follow :
(c) If f and ))(),...,(( 11 qq gugu are convex functions, D is convex, and *x is a properly
efficient solution for the multi-objective problem (2.10), there exist some weight
vector μλ, with strictly positive components such that *x is the optimal solution
for weighted problem ),( μλfugP .
(d) For each vector of weights with strictly positive components, the optimal solution to
the weighted problem ),( μλfugP is properly efficient for the multi-objective
problem (P1).
Proposition 2.3 If f and ))(),...,(( 11 qq gugu are convex functions, D is a convex set
and there exist qquuu RR →= +:)~,...,~(~
1 with 0)(~).( >tutu kk , for every 0≥t and
,...,1 qk ∈ then GG EuE 14 )~( ⊂ .
Proof: If f and ))(),...,(( 11 qq gugu are convex functions and if D is a convex set, then
the set of properly efficient solutions to problems (P1) and (2.13) are obtained from the
associated weighted problems for strictly positive weight vectors. We will prove that any
solutions to the optimization problem )(~ ξuP , with 0>ξ , is a solution to problem
),( μλfugP for some vector 0),( >μλ .
Let )~(4 uEx G∈ . Then, given the established hypotheses, there exist a vector 0>ξ for
which x is the solution for problem )(~ ξuP . Let us assume that, for every
0,,...,1 ≠∈ kuqk , then we take
kk ξλ = , u~ and kμ such that )()(~ tutu kkkk μξ = for ,...,1 qk ∈ .
Since 0>ξ , we obtain that x is an optimal solution to problem ),( μλfugP . For some
,...,1 qk ∈ if 0=ku , then the proof would be the same, since in problem (2.10) the
function ig is not involved and since in problem (2.13) the kth objective would be kf .
Proposition 2.4 If f and ))(),...,(( 11 qq gugu are convex functions, then
26
UΩ∈
⊂u
GG uEE~
41 )~( ,
with ,,0)(.)(~:)~,...,~(~ 1 tktutuRuuu kkq
q ∀>→==Ω + R .
Proof: Using the Proposition 2.3, we see that the proof of this proposition is carried out
by demonstrating that any solution to the problem ),( μλfugP is a solution to the
problem )(~ ξuP with qquuu RR →= +:)~,...,~(~
1 , 0)().(~ >tutu kk , tk,∀ and for some
0>ξ .
Consider GEx 1∈ . Since f and ))(),...,(( 11 qq gugu are convex functions, there exist
vector 0, >μλ such that x is a solution to problem ),( μλfugP . Because 0, >μξ we
put
kk λξ = , k
kkk
tutuξ
μ )(.)(~ = , for +∈∈ Rtqk ,,...,1 .
Therefore we obtain that x is also a solution to the problem )(~ ξuP .
From Proposition 2.3 and Proposition 2.4, if f and ))(),...,(( 11 qq gugu are convex
functions and if tktutuuuu kkq
q ,,0)().(~,:)~,...,~(~1 ∀>→= + RR , then the sets of
properly efficient solutions to problem (2.10) and (2.13) verify the following properties:
a. Every properly efficient solution to problem (2.13) is properly efficient for
problem (2.10);
b. Setting 0\: qu RR →+ , the set of properly efficient solutions to problem (2.3)
is a subset of the union in u~ of the set of properly efficient solutions for problem
(2.6).
By combining both results, the following corollary:
Corollary 2.4 If f and ))(),...,(( 11 qq gugu are convex functions, then
UΩ∈
=u
GG uEE~
41 )~( .
Corollary 2.5 If f , and ))(),...,(( 11 qq gugu are convex functions and
0)(~).( >tutu kk , for every 0≥t and ,...,1 qk ∈ then GG EuE 14 )~( ⊂ .
27
Corollary 2.6 (Caballero et al [21]) If 0\qR∈α , ),...,( 1 qααα = , and
ttu kk .)( α= , then problem (2.13) to reduced from problem (4) in Caballero et al [21].
Corollary 2.7 If 0\qR∈γ , ),...,( 1 qγγγ = , 20 =s , ttu kk .)( γ= , qk ,1= , then
(P1)-(P3) to reduced from (1)-(3) in Caballero et al [21].
In this case we obtaint true consequence from Theorem 3.1 and Theorem 3.2 in
Caballero et al [21]. Further if we have in view Corollary 2.6, we see that we get a
consequently Proposition 3.1, Proposition 3.2 and Corollary 3.1 in Caballero et al [21].
For example, using Chankong and Haimes [29], we can applied to problem (P1) and its
associated weighted with problem ),( μλfugP . Thus we have:
Proposition 2.5 If f and ))(),...,(( 11 qq gugu are convex function, D is a convex set and
10 Ex ∈ , then there exist qq μμλλ ,...,,,..., 11 nonnegative real numbers such that 0x is
optimal solution for problem ),( μλfugP .
Proposition 2.6 Let qq μμλλ ,...,,,..., 11 nonnegative real numbers and 0x is an of
optimal solution for problem ),( μλfugP then 10 Ex ∈ .
2.7 White’s approach multiobjective weighting factor auxiliary
optimization problem for P1, P2 and P3
We consider the generation of efficient stochastic multi-objective solution using
weighting factor, q th-power approach for some non-convex auxiliary function
optimization problem. We will introduce our class of auxiliary function and give some
known standard results and we will show tree classes of non-convex auxiliary
optimization problems, giving a concavity-preserving transformation for the q th power
of concave function for range solution for auxiliary optimization problem.
2.7.1 Introduction
Consider the following class of multi-objective problems:
28
(i) there is a compact set nD R⊆ of feasible actions;
(ii) there are q objective functions )( ⋅kF with +∈R)(xFk for all Dx∈ ;
(iii) it is required to find members of the efficient (vector minimal) set E of D, where,
by definition
)()()()(,: xFyFxFyFDyDxE =→≤∈∈= , (2.16)
which ),...,( 1 qFFF = .
The most common of these auxiliary forms is the positively weighted form given by the
class of auxiliary functions
∑=
=⋅q
kkk FF
1),;( δδν , (i.e., ∑
=
=q
kkk xFFx
1)(),;( δδν ), (2.17)
with
⎭⎬⎫
⎩⎨⎧
=∈=Δ∈ ∑=
++ 1:1
q
kk
q δδδ R
or
0: >Δ∈=Δ∈ +++ δδδ .
We define ),;(minarg),( FxFMDx
δνδ∈
= .
Then using well known results in multi-objective programming (see, for example Karlin
[82]) we have the following proposition.
Proposition 2.7
a) EFM ⊆++Δ∈
),(δδU ;
b) If F is convex vector function and D is convex, then (Karlin[82])
U++Δ∈
⊆δ
δ ),( FME .
According to Karlin [82] there are two central issues arising from this class of auxiliary
functions, namely
a. this class of auxiliary functions may not be capable of generating enough points in
E,
29
b. if F is not convex vector function or D is not convex, there may be no current
method for finding ),( FM δ .
2.7.2 Transformations and auxiliary optimization problems associated to (P1), (P2),
and (P3)
We consider the transformation given by
qq ++ →= RR:),...,( 1 θθθ
Relative to (P1), (P2), and (P3), we consider the following transformated problems (2.18),
(2.19), and (2.20), respectively.
(PS1) ⎟⎟⎠
⎞⎜⎜⎝
⎛+∑∑
==∈
)))((())((min11
xguxf kkk
q
kkkk
q
kkDx
υυ θμθλ (2.18)
(PS2) ⎟⎟⎠
⎞⎜⎜⎝
⎛∑=
∈))((min
1
xf kk
q
kkDx
υθλ (2.19)
(PS3) 0,)))(((min 01
0 >⎟⎟⎠
⎞⎜⎜⎝
⎛∑=
∈sxgu s
kkk
q
kkDx
υθμ (2.20)
with 0>υ , qquuu RR →= +:),...,( 1 .
We now define for problem (2.18) an auxiliary optimization problem ),,,(1 θμλqAP as
follows for 1),(,0\ ++ Δ∈∈ μλZq ,
where ⎭⎬⎫
⎩⎨⎧
=+×∈=Δ ∑∑==
+++ 1/),(11
1q
ii
q
ii
qq μλμλ RR .
Find ),,,,(minarg),,,( 11 θμλψθμλ qxqMDx∈
= the set of optimal solutions for (2.18),
where +→ RDqx :),,,;(1 θμλψ is given by
)))((())((),,,;(11
1 xguxfqx kkqk
q
kkk
qk
q
kk θμθλθμλψ ∑∑
==
+= (2.21)
For ∞=q relation (2.21) is replaced by
⎭⎬⎫
⎩⎨⎧=∞ )))(((max)),((maxmax),,,;(1 xguxfx kkkkkkkk
kθμθλθμλψ (2.22)
30
By using the lines of White [207, 209], Bowman [19], Karlin [82], relative to the
problem (2.18) we obtain the following results:
Theorem 2.5
)1a If ∞≠q , then 11 ),,,( EqM ⊆θμλ if 1),( ++Δ∈μλ ;
)2a If a certain uniform dominance condition hold (Bowman [19]), then
11 ),,,( EM ⊆∞ θμλ for 1),( ++Δ∈μλ ;
)3a If ∞≠q , ))(( ⋅kqk fθ , qk ,1= are all convex on D and D is convex, then
U1),(
11 ),,,(++Δ∈
⊆μλ
θμλqME ;
)4a U1),(
11 ),,,(++Δ∈
∞⊆μλ
θμλME ;
)5a If D is finite, then there exists a 0\*+∈ Zq , such that
U1),(
11 ),,,(++Δ∈
=μλ
θμλqME *qq ≥∀ .
For problem (2.19) an auxiliary optimization problem ),,(2 θλqAP as follows for
2,0\ ++ Δ∈∈ λZq ,
where ⎭⎬⎫
⎩⎨⎧
=∈=Δ ∑=
++ 1/1
2q
ii
q λλ R .
Find ),,,(minarg),,( 22 θλψθλ qxqMDx∈
= , where +→ RDqx :),,;(2 θλψ is given
by
))((),,;(1
2 xfqx kqk
q
kkθλθλψ ∑
=
= . (2.23)
For ∞=q equation (2.23) is replaced by
))((max),,;(2 xfx kkkk
θλθλψ =∞ (2.25)
By using the lines of White [207, 209], Bowman [19], Karlin [82], relative to the
problem (2.19) we obtain the following results:
Theorem 2.6
)1a If ∞≠q , then 22 ),,( EqM ⊆θλ if 2++Δ∈λ ;
31
)2a If a certain uniform dominance condition hold (Bowman [19]), then
22 ),,( EM ⊆∞ θλ if 2++Δ∈λ ;
)3a If ∞≠q , ))(( ⋅kqk fθ , qk ,1= are all convex on D and D is convex, then
U2
),,(22++Δ∈
⊆λ
θλqME ;
)4a U2
),,(22++Δ∈
∞⊆λ
θλME ;
)5a If D is finite, then there exists a 0\*+∈ Zq , such that
U2
),,(22++Δ∈
=λ
θλqME *qq ≥∀ .
For problem (2.20) an auxiliary optimization problem ),,(3 θμqAP as follows for
2,0\ ++ Δ∈∈ μZq ,
Find ),,,(minarg),,( 33 θμψθμ qxqMDx∈
= , where +→ RDqx :),,;(3 θμψ is given by
∑= ))((),,;( 03
skk
qkk guqx θμθμψ (2.25)
For ∞=q equation (2.25) is replaced by
)))(((max),,,( 03 xgux s
kkkkkθμθμψ =∞ (2.19)
Finally for problem (2.20) we obtain the following results:
Theorem 2.7
)1a If ∞≠q , then 33 ),,( EqM ⊆θμ if 2++Δ∈μ
)2a If a certain uniform dominance condition hold (Bowman [19]), then
33 ),,( EM ⊆∞ θλ if 2++Δ∈μ
)3a If ∞≠q , ))(( ⋅kqk fθ , qk ,1= are all convex on D and D is convex, then
U2
),,(33++Δ∈
⊆μ
θμqME
)4a U2
),,(33++Δ∈
∞⊆μ
θμME
32
)5a If D is finite, then there exists a 0\*+∈ Zq , such that
U2
),,(33++Δ∈
=μ
θμqME *qq ≥∀ .
Remark 2.7 Some proofs of these results are given in Sudradjat and Preda [189].
2.7.3 Non-convex auxiliary optimization problem
The problem is to finding points in D which are in E or close to points in E. We also
wish to use convexity and concavity properties. For these to be meaningful, we need
appropriate convex sets within which to embed our analysis. nR is too large, because
we have stipulate merely that ∈)(xF q+R for all Dx∈ and not for all ∈x nR . When
D is convex, all that we need state is that F is defined on *D , with ∈)(xF q+R for all
*Dx∈ . In the following we use lines given by White [149].
a) Case of concave ∞≠⋅⋅ qguf kkk )),((),(
We assume that the kf are all concave on *D and look at the choice of )( ⋅kθ and
q and associated algorithms for auxiliary optimization problems ),,,(1 θμλqAP ,
),,(2 θλqAP and ),,(3 θμqAP .
Generally even if ))(( ⋅kk fθ is concave on *D it is not necessarily true that
))(( ⋅kqk fθ is concave on *D . We need to choose ))(( ⋅kk fθ so that, at least for some
q , ))(( ⋅kqk fθ are all concave on *D . The following form of )( ⋅kθ will provide an
instance which will do what is required, namely
qkatt kk ,1),)(log())(( =+= ϕϕθ
over the range
,1)( −−≥ katϕ qk ,1= , where R⊆ ka
Lemma 2.1 If )( ⋅kf are concave on *D , then ))(( ⋅kqk fθ are all concave on *D
for all 0\+∈ Zq and ),min( 21 qqq = such that
33
)]1)))(([log(minmin
)]1))([log(minmin
,12
,11
++≤
++≤
∈=
∈=
kkkDxqk
kkDxqk
axguq
axfq (2.27)
provided that
)]([min xfa kDxk ∈≥ , qk ,1= (2.28)
Proof: For 1,,1 −−≥∈≥ kazRzq ,
2
2
2
2
)()]log()1)[((log)(
k
kkq
kqk
azazqazq
dzazd
++−−+
=+ −θ
.
Thus for any given ,1−−≥ kaz
0)(2
2
≤dz
zd qkθ ,
if )log(1 kazq ++≤ .
Replacing z by )(xfk and ))(( xgu kk , we see that )(⋅qkθ is concave on *D provided
that (2.26) and (2.27) hold.
b) Case of convex )( ⋅kf , ))(( ⋅kk gu and finite D
If )( ⋅kθ are all convex on +R and )( ⋅kf and ))(( ⋅kk gu are all convex on *D ,
then ))(( ⋅kqk fθ are all convex on *D . This applies, for example, when
+∈∀= R)()())(( tttk ϕϕϕθ , qk ≤≤1 .
In this case the auxiliary optimization problem ),,,(1 θμλqAP become one of
minimizing a convex function over a finite set D, also for ),,(2 θλqAP and
),,(3 θμqAP .
This also hold in the case of ∞=q .
Let us now assume that )(xψ is ),,,;(1 θμλψ qx , ),,;(2 θλψ qx or ),,;(3 θμψ qx
respectively, and that )(xψ is convex on *D . Then consider the following algorithm,
given the qualifier ,,, θμλq , ,, θλq or ,, θμq respectively for ease of
34
exposition : )( txψ∂ is the subdifferential of ψ at txx = and φψ ≠∂ )( tx
(Rockafellar, [150]); tS is any finite non-empty subset of )( txψ∂ obtained by some
specified method; 1x is the first component of x.
Algorithm 2.1
(i) Select 0\+∈ Rε .
(ii) Set DDt = .
(iii) Set t = 1.
(iv) Assume that we have derived tD .
(v) Find tS and
)(minarg xxtDx
t ψ∈
∈
(vi) Set tttt SyxxyDxD ∈∀−≤−∈=+ ε)(:1
(vii) If φ=+1tD , set
)(minarg],...,[ 1
xxtxxx
A ψ∈
∈ ,
and stop.
(viii) If φ≠+1tD , go to step (v).
We have the following theorem.
Theorem 2.8
(i) Algorithm 2.1 terminates in a finite number of iterations.
(ii) If *ψ is the minimal value of ψ on D, then
εψψψ +≤≤ ** )( Ax , where ,, 321 ψψψψ ∈ .
Proof: (i) Let us suppose that the algorithm, is not finite.
Because D is finite, there exist a set 0\ +⊆ Zr such that
10 ≥∀= rxx rt
where 0x is some member of D and
35
11 ≥∀∈+ rDx rr tt ,
then
1),()( 01 ≥∂∈∀−≤−+ rxyxxy rr tt ψε .
Thus
1),()( 01 ≥∂∈∀−≤−+ rxyrxxy rr tt ψε .
This is possible.
(ii) Let φ=+1tD and
,\ 1+= sss DDY ts ≤≤1 ,
then
Ut
ssYD
1== .
Let sYx∈ . Then 1+∉ sDx and
ε−≥− )( sxxy for some )( sxy ψ∂∈ .
Because ,, 321 ψψψψ ∈ are convex,
εψψ −≥−≥− )()()( ss xxyxx .
Hence
εψψψ +≤≤ ** )( Ax .
Remark 2.8 Step (v) really only requires finding a feasible solution in tD . The use of
1x as an objective function is merely to facilitate this.
Remark 2.9 Step (v) is a subproblem of minimizing a linear function over those
solutions defined by a polytope, say tZ , generated by the subgradient constraints, which
are also in D. When all function are differentiable, tD is singleton gradient vector for
ψ at 1xx = . In the general case )( txψ∂ and hence tD may be found in term of the
subdifferentials of the functions ))(( ⋅kk fθ , (Rockafellar, [150]).
Remark 2.10 If D is the vertex set of a polytope *D , then any solution from tD is also
a vertex of tZD ∩* . Thus a vertex search subalgorithm such as that of Murty [124] can
36
be developed. Other procedures for enumerating the vertices of a polytope may be
adaptable for this problem (Matheiss and Rubin, [116]).
Remark 2.11 If D is integral set of a polytope, the subproblem in step (v) is an integer
linear programming problem, for which a range of algorithms exist.
Whatever method is use to solve the auxiliary problem, the convexity of ψ is helpful in
providing lower bounds, because if sx is any set of solutions generated, then
)]()([maxmin)(,
* ss
xysDxxxyx
s−+≥
∂∈∈ψψ
ψ.
Remark 2.12 If D is a subset of a polytope *D , we note that
)(min*
* xDxψψ
∈≥ . (2.22)
The determination of right-hand side of (2.22) is a convex programming problem. Lower
bounds may be useful in determining how close the best solution to date is to an optimal
solution, so that computations may be termined early if wished.
c) Case of mixture of concave and convex ∞≠⋅⋅ qguf kkk )),((),(
From a global optimization point of view this is the hardest problem of all.
Consider 1M , 2M , '1M and '
2M are non-empty subsets of ,...,1 q such that
,...,1'2
'121 qMMMM =∪=∪ , φ=∩ 21 MM , and φ=∩ '
2'1 MM ; ))(( ⋅k
qk fθ be
concave on *D for ))((;1 ⋅∈ kqk fMk θ be convex on *D for 2Mk ∈ , also that
)))((( ⋅kkqk guθ be concave on *D for '
1Mk ∈ and )))((( ⋅kkqk guθ be convex on *D
for 2Mk ∈ ;
∑∑∈∈
⋅+⋅=⋅'11
)))((())((),,,;(1Mk
kkqkkk
qk
Mkk gufq θμθλθμλψ
)))((())((),,,;('22
2 ⋅+⋅=⋅ ∑∑∈∈
kkqk
Mkkk
qk
Mkk gufq θμθλθμλψ
Then, dropping the qualifiers ,,, θμλq for ease of exposition, we have
)()()( 21 ⋅+⋅=⋅ ψψψ
where )(1 ⋅ψ and )(2 ⋅ψ are respectively concave and convex on *D .
37
The following algorithm is an extension of an algorithm of White ([206]). tS2 is any
finite subset of the sub-differential )(2txψ∂ of )(2 ⋅ψ at txx = obtained by some
specified method.
Algorithm 2.2
(i) Select 0\+∈ Rε .
(ii) Set DD =1 .
(iii) Set t = 1.
(iv) Assume that we have derived tD .
(v) Find tS and
)(minarg xxtDx
t ψ∈
∈
(vi) Set tttt SyxxyDxD ∈∀−≤−∈=+ ε)(:1
(vii) If φ=+1tD , set
)(minarg],...,[ 1
xxtxxx
A ψ∈
∈
and stop
(viii) If φ≠+1tD , go to step (v).
We have the following theorem, where )(2 xψ∂ is the subdifferential of )(2 ⋅ψ at point
x.
Theorem 2.9
(i) If U Dxx
∈∂ )(2ψ is compact, then Algorithm 2.2 terminates in a finite number of
iterations.
(ii) If *ψ is the minimal value of )(⋅ψ on D, then
εψψψ +≤≤ ** )( Bx
Proof: (i) Let us suppose that the algorithm, is not finite.
Because of the assumptions about the sub-differentials and the compactness of D, there
exist a ny R∈0 and a set 0\ +⊆ Zr such that
38
12/)( 10 ≥∀−≤−+ rxxy rr tt ε
Thus
12/)( 10 ≥∀−≤−+ rrxxy rr tt ε ,
Inequalities which is not possible.
(ii) Let φ=+1tD and ,\ 1+= sss DDY ts −=1 , then Ut
ssYD
1== .
Let sYx∈ . Then 1+∉ sDx and ε−≥− )( sxxy for some )(2txψ∂ .
Because )(2 ⋅ψ is convex,
εψψ −≥−≥− )()()( 22ss xxyxx .
Also,
)()( 11sxx ψψ ≥ .
Thus
εψψψ +≤≤ ** )( Bx .
Remark 2.13 Step (v) is a subproblem involving the minimization of a concave function
over tD . When D is a polytope, so is tD , and algorithms exist for solving such
problems (e.g. Glover and Klingman, [62]; Falk and Hoffman, [49]; Carino, [23]). We
note that in this case step (vi) adds cutting constraints which exclude the current solution
and for which the dual simplex method is useful (Hadley, [65]).
Remark 2.14 If D is the integral set of a polytope *D , then the subproblem takes an
integer programming form, for which algorithm exist.
Remark 2.15 If D is the vertices of a polytope *D , then except in special cases (e.g.
when vertices are integral as in the assignment problem) some new algorithm is
required.
39
CHAPTER 3
STOCHASTIC DOMINANCE 3.1 Introduction. The relation of stochastic dominance is a fundamental concept of decision theory and
economics , Dentcheva and Ruszczynski [41, 42], Hanoch and Levy [68], Quirk and
Saposnik [146] and Rothschild [155].
A random variable X dominates another random variable Y in the second order, which
we write as YX )2(f , if )]([)]([ YuXu EE ≥ for every concave nondecreasing function
)(⋅u , for which these expected values are finite.
A basic model of stochastic optimization can be formulated as follows:
)],([max ωϕ zZz
E∈
. (3.1)
In this formulation ω denotes an elementary event in a probability space ),,( PFΩ , z is a
decision vector in an appropriate space Z , and R→Ω×Z:ϕ . The set Z⊂Z is
defined either explicitly, or via some constraints that may involve the elementary event ω
and must hold with some prescribed probability.
The first stochastic optimization models with expected values were introduced by
Lehmann [96] and Hanoch and Levy [68]. Mathematical theory of expectation models
involving two-stage and multistage decisions has been developed by Wets [204, 205]
and Birge [16].
Models involving constraints on probability were introduced by Charnes and Cooper
[31], Prekopa [139], Dentcheva and Ruszczynski [42] discusses in detail the theory and
numerical methods for linear models with one probabilistic constraint on finitely many
inequalities.
Another way to look at problem (3.1) is to consider the set C of random variables X such
that, for some Zz∈ , one has ),()( ωϕω zX ≤ a.s. Then we can write the model as
][max XCX
E∈
.
40
In practice, however, it is almost impossible to elicit the utility function of a decision
maker explicitly. Additional difficulties arise when there is a group of decision makers
with different utility functions who have to come to a consensus.
In some applications a reference outcome Y in ),,(1 PFL Ω is available. It may have the
form Ω∈= ωωϕω ),,()( zY , for some policy z . Our intention is to have the new
outcome, X, preferable over Y. Therefore, we consider the following optimization
problem:
)(max Xf (3.2)
subject to YX )2(f , (3.3)
CX ∈ . (3.4)
Here Y is a random variable in ),,(1 PFL Ω , the set ),,(C 1 PFL Ω⊂ is convex and
closed, and R→Cf : is a concave continuous functional. Constraints (3.3) guarantees
that for any decision maker, whose utility function )(⋅u is concave and nondecreasing,
the solution X of the problem will satisfy the relation )]([)]([ YuXu EE > .
Another class of models that recently attracted much attention are mean-risk models. In
our notation they take the form
)(][max XXCX
λρ−∈
E .
In this problem λ > 0 and )(⋅ρ is a risk functional which depends on the entire
distribution of X and assigns to it a scalar measure of its variability. For example, the
expected shortfall below the mean,
[ ]+−= )][()( XXEEXρ ,
may be used as the risk functional. Here ),0max()( XX =+ . Mean-risk models are also
closely related to stochastic dominance relations. If we use an appropriate risk measure
ρ and the parameter λ is within a certain range, then the optimal outcome X is not
stochastically dominated by any other feasible outcome Dentcheva and
Ruszczynski [42], Orgryczak and Ruszczynski [127, 128, 129].
41
Other stochastic optimization models involving general risk functional were considered
by Dentcheva and Ruszczynski [42], Rockafellar and Uryasev [152]. Model (3.2)–(3.4)
correspond to a new approach in stochastic optimization problem.
3.2 Stochastic dominance
In the stochastic dominance approach, random returns are compared by a point-wise
comparison of some performance functions constructed from their distribution functions.
For a real random variable V , its first performance function is defined as the right-
continuous cumulative distribution function of V :
),( ηη ≤= VVF P for R∈η .
A random return V is said stochastically dominate another random return S to the first
order (Dentcheva and Ruszczynski [42], Lehmann [96] and Quirk and Saposnik [146]),
denoted SV FSDf , if
);();( ηη SFVF ≤ for all R∈η .
Define the function );(2 ⋅VF as
ααηη
dVFVF ∫ ∞−= );();(2 for R∈η , (3.5)
as an integral of a nondecreasing function, it is a convex function of η and defines the
weak relation of the second-order stochastic dominance (SSD). That is, the random return
V stochastically dominates S to the second order, denoted SV SSDf , if
);();( 22 ηη SFVF ≤ for all R∈η .
The corresponding strict dominance relations for FSDf and SSDf are defined in the
usual way: SV f if and only if SV f , VS f/ .
Furthermore, for ),,( PV FLm Ω∈ we can define recursively the functions
∫ ∞− −=η
ααη dVFVF kk ),();( 1 for 1,3k +=∈ mR,η . (3.6)
Furthermore, for ),,( PV FLm Ω∈ we can define recursively the functions
42
∫ ∞− −=η
ααη dVFVF kk ),();( 1 for 1,3k +=∈ mR,η . (3.6)
Figure 3.3 First order dominance
RE ∈−== +∞−∫ ηηααη
ηforXdXFXF )();();( 12
Figure 3.2 Scond-order dominnce
They are also convex and nondecreasing functions of the second argument.
Definition 3.1. A random variable ),,(1-k PX FL Ω∈ dominates in the kth order
another random variable ),,(1-k PY FL Ω∈ if
);();( ηη YFXF kk ≤ for all R∈η . (3.7)
We shall denote relation (3.7) as
YX k )(f (3.8)
and the set of X satisfying this relation as
:),,()( )(1 YXPXYA k
kk fFL Ω∈= − . (3.9)
43
By changing the order of integration we can express the function );(2 ⋅VF as the
expected shortfall (Rockafellar and Uryasev [152]): for each target value η we have
[ ]+−= )();(2 VEVF ηη , (3.10)
where )0,max()( VV −=− + ηη . The function );(2 ⋅VF is continuous, convex,
nonnegative and nondecreasing. It is well defined for all random variables V with finite
expected value.
3.3 The portfolio problem
Let nRR ,...,1 be random returns of n assets. We assume that ∞<][ jRE for all
nj ,1= . Our aim is to invest our capital in these assets in order to obtain some desirable
characteristics of the total return on the investment. Denoting by nxx ,...,1 the fractions of
the initial capital invested in assets n,...,1 respectively we can easily derive the formula
for the total return:
nn xRxRxR ++= ...)( 11 . (3.11)
Clearly, the set of possible asset allocations can be defined as follows:
X njxxxx jnn ,1,0,1...: 1 =≥=++∈= R ,
where ,..., 1 nxxx = .
In some applications one may introduce the possibility of short positions, i.e., allow some
jx ’s to become negative. Other restrictions may limit the exposure to particular assets or
their groups, by imposing upper bounds on the jx ’s or on their partial sums. One can
also limit the absolute differences between the jx ’s and some reference investments jx ,
which may represent the existing portfolio, etc. Our analysis will not depend on the
detailed way this set is defined; we shall only use the fact that it is a convex polyhedron.
All modifications discussed above define some convex polyhedral feasible sets, and are,
therefore, covered by our approach.
44
The main difficulty in formulating a meaningful portfolio optimization problem is the
definition of the preference structure among feasible portfolios. If we use only the mean
return
[ ])()( xRx E=μ ,
then the resulting optimization problem has a trivial and meaningless solution: invest
everything in assets that have the maximum expected return. For these reasons the
practice of portfolio optimization resorts usually to two approaches.
In the first approach we associate with portfolio x some risk measure )(xρ representing
the variability of the return )(xR . In the classical Markowitz model )(xρ is the variance
of the return,
[ ])()( xRarx V=ρ ,
but many other measures are possible here as well.
The mean–risk portfolio optimization problem is formulated as follows:
)]()([max xxx
λρμ −∈X
(3.12)
Here, λ is a nonnegative parameter representing our desirable exchange rate of mean for
risk. If 0=λ , the risk has no value and the problem reduces to the problem of
maximizing the mean. If 0>λ we look for a compromise between the mean and the
risk. The general question of constructing mean–risk models which are in harmony with
the stochastic dominance relations has been the subject of the analysis of the recent
papers Dentcheva and Ruszczynski [41, 42], Rothschild and Stiglitz [155], Ogryczak and
Ruszczynski [127, 128].
We have identified there several primal risk measures, most notably central semi-
deviations, and dual risk measures, based on the Lorenz curve, which are consistent with
the stochastic dominance relations.
The second approach is to select a certain utility function RR →:u and to formulate
the following optimization problem
( )[ ])(max xRuEx X∈
(3.13)
It is usually required that the function u(·) is concave and nondecreasing, thus
representing preferences of a risk-averse decision maker. The challenge here is to select
45
the appropriate utility function that represents well our preferences and whose application
leads to non-trivial and meaningful solutions of (3.13).
3.4 Consistency with stochastic dominance
The concept of stochastic dominance is related to an axiomatic model of risk-averse
preferences Fishburn [52]. It originated from the theory of majorization (Hardy,
Litltewood and Polya [70], Marshall and Olkin [109]) for the discrete case and was later
extended to general distributions (Quirk and Saposnik [146], Hadar and Russell [66],
Hanoch and Levy [68], Rothschild and Stiglitz [155]. It is nowadays widely used in
economics and finance Bawa[7], Levy [99].
In the stochastic dominance approach, random returns are compared by a point-wise
comparison of some performance functions constructed from their distribution functions.
Figure 3.3. Mean–risk analysis. Portfolio x is better than portfolio y in the mean–risk sense, but none of them is efficient.
For a real random variable V, its first performance function is defined as the
rightcontinuous cumulative distribution function of V :
RP ∈≤= ηηη forVFV ,)( .
A random return V is said (Lehmann[94], Quirk and Saposnik [144]) to stochastically
dominate another random return S to the first order, denoted SV FSDf , if
R∈= ηηη allforFF SV ),()( .
The second performance function 2F is given by areas below the distribution function F,
∫ ∞−=
ηααη dVFVF );();(2 for R∈η (3.14)
46
and defines the weak relation of the second-order stochastic dominance (SSD). That is,
random return V stochastically dominates S to the second order, denoted SV SSDf , if
);();( 22 ηη SFVF ≤ for R∈η . (3.15)
(see Hadar and Russell [66], Hanoch and Levy [99]). The corresponding strict
dominance relations FSDf and SSDp are defined in the usual way
SV FSDf ⇔ SV SSDp . and SV SSDf (3.16)
For portfolios, the random variables in question are the returns defined by (3.11). To void
placing the decision vector, x, in a subscript expression, we shall simply write It will not
lead to any confusion, we believe. Thus, we say that portfolio x dominates portfolio y
under the FSD rules, if );();( yFxF ηη = for all R∈η , where at least one strict
inequality holds. Similarly, we say that x dominates y under the SSD rules
))()(( yRxR SSDf , if );();( )2()2( yFxF ηη = for all R∈η , with at least one inequality
strict.
)();( )( ηη xFxF R= and )();( )(22 ηη xFxF R= .
Figure 3.4. The expected shortfall function.
Stochastic dominance relations are of crucial importance for decision theory. It is known
that )()( yRxR FSDf if and only if ))](([))](([ yRUxRU EE ≥ for any nondecreasing
function U(·) for which these expected values are finite. Also, ))()(( yRxR SSDf if and
only if ))](([))](([ yRUxRU EE ≥ for every nondecreasing and concave U(·) for which
these expected values are finite (see, e.g., Levy [97]).
47
For a set P of portfolios, a portfolio P∈x is called SSD-efficient (or FSD-efficient) in
P if there is no P∈y such that )()( xRyR SSDf (or )()( xRyR FSDf ).
We shall focus our attention on the SSD relation, because of its consistency with risk-
averse preferences: if ))()(( yRxR SSDf , then portfolio x is preferred to y by all risk-
averse decision makers. By changing the order of integration we can express the function
);(2 xF ⋅ as the expected shortfall (Orgyczak ang Rusczynski [129)]: for each target value
η we have
)]0),([max();(2 xRExF −= ηη . (3.17)
The function );()2( xF ⋅ ) is continuous, convex, nonnegative and nondecreasing. Its graph
is illustrated in Figure 3.4.
Following [42, 43], we introduce the following definition.
Definition 2.1 Ruszczy´nski and Vanderbei [156] The mean-risk model ),( ρμ is
consistent with SSD with coefficient 0>α , if the following relation is true
)()()()()()( yyxxyRxR SSD λρμλρμ −≥−⇒f for all αλ ≤≤0 .
for all αλ ≤≤0
In fact, as we shall see in the proof below, it is sufficient to have the above inequality
satisfied for α ; its validity for all αλ ≤≤0 follows from that.
The concept of consistency turns out to be fruitful. In [129] we have proved the following
result.
Theorem 3.1. The mean–risk model in which the risk is defined as the absolute
semideviation,
)0),()(max()( xRxx −= μδ E , (3.18)
is consistent with the second-order stochastic dominance relation with coefficient 1.
We provide an easy alternative proof here.
Proof. First, it is clear from (3.17) that the line )(xμη − is the asymptote of );()2( xF ⋅
for ∞→η . Therefore )()( yRxR SSDf implies that
)()( yx μμ ≥ . (3.19)
48
Secondly, setting )(xμη = in (4) we obtain
))()(,0max()( yRxx −≤ μδ E .
Since 0)()( ≥− yx μμ , we have
))()(,0max()()())()()()(,0max())()(,0max(
yRxyxyRyyxyRx
−+−≤−+−=−
μμμμμμμ
Taking the expected value of both sides and combining with the preceding inequality we
get
)()()()( yyxx δμμδ +−= ,
which can be rewritten as
)()()()( yyxx δμδμ +≥− . (3.20)
Combining inequalities (3.19) and (3.20) with coefficients λ−1 and λ , where
]1,0[∈λ , we obtain the required result.
An identical result (under the condition of finite second moments) has been obtained in
Ogryczak and Ruszczynski [128] for the standard semideviation, and further extended in
Ogryczak and Ruszczynski [129] to central semideviations of higher orders and
stochastic dominance relations of higher orders (see also Gotoh and Konno [64]).
Elementary calculations show that for any distribution
)(21)( xx δδ = ,
where )(xδ is the mean absolute deviation from the mean:
)()()( xxRx μδ −= E . (3.21)
Thus, )(xδ is a consistent risk measure with the coefficient 21
=α . The mean–absolute
deviation model has been introduced as a convenient linear programming mean–risk
model by Konno and Yamazaki [86].
Another useful class of risk measures can be obtained by using quantiles of the
distribution of the return R(x). Let )(xqp denote the p-th quantile (In the financial
literature, the quantity )(xqp W, where W is the initial investment, is sometimes called
the Value at Risk ) of the distribution of the return R(x), i.e.,
49
)(xδ .
We may define the risk measure
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−= zxRxRz
ppxp )()),((1max)( Eρ . (3.22)
In the special case of 21
=p the measure above represents the mean absolute deviation
from the median. For small p, deviations to the left of the p-th quantile are penalized in a
much more severe way than deviations to the right.
Although the p-th quantile )(xqp might not be uniquely defined, the risk measure
)(xpρ is a well defined quantity. Indeed, it is the optimal value of a certain optimization
problem:
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−= zxRxRz
ppxp )()),((1maxmin)( E
Xρ . (3.23)
It is well known that the optimizing z will be one of the p-th quantiles of R(x) (see, e.g.,
Bloomfield [17]). In Ogryczak and Ruszczynski [127] we have proved the following
result.
Theorem 2. The mean–risk model with the risk defined as )(xpρ is consistent with the
second-order stochastic dominance relation with coefficient 1, for all )1,0(∈p ).
Again, we provide here an alternative proof.
Proof. Let us consider the composite objective in our mean–risk model (scaled by p):
)()();( xpxpxpG pρμ −= . (3.24)
If follows from (3.23) that we can represent it as an optimal value:
( )[ ][ ]))(()),()(1(maxsup);( zxRpxRzppxpG −−−−= EμX
. (3.25)
Clearly, we have the identity
( ) ( ) ( )zxRpxRzzxRpxRzp −+−=−−− )()(,0max))(()),()(1(max .
Using this in (3.25) we obtain
[ ]),(sup[);( )2( xzFpzxpG −=X
. (3.26)
50
Figure 3.6 The absolute Lorenz curve
Therefore, the function );( xG ⋅ is the Fenchel conjugate of );(2 xF ⋅ (see Fenchel [50]
and Rockafellar[148]). Consequently, the second-order dominance )()( yRxR SSDf
implies that
);();( ypGxpG ≥
for all ]1,0[∈p . Recalling (13) we conclude that
)()()()( yyxx pp ρμρμ −≥− .
Since we also have (8), Definition 1 is satisfied with all ]1,0[∈λ .
Interestingly, the function );( xG ⋅ can also be expressed as the integral:
∫=η
α α0
)(),( dxqxpG (3.27)
(non-uniqueness of the quantile does not matter here). Indeed, it follows from (3.26) that
the quantile )(xq p , which is the maximizer in (3.26), is a subgradient of );( xG ⋅ at p (see
Fenchel [50] and Rockafellar[150]). The integral in (3.27) is called the absolute Lorenz
curve and is frequently used (for nonnegative variables and in a normalized form) in
income inequality studies (see Arnold [2], Gastwirth [59], Lorenz [106], Hardy,
Litlewood and Polya [70] and the references therein). It is illustrated in Figure 3.6.
51
CHAPTER 4
THE DOMINANCE-CONSTRAINED PORTFOLIO PROBLEM
4.1 Introduction
The problem of optimizing a portfolio of finitely many assets is a classical problem in
theoretical and computational finance. Since the seminal work of Markowitz [112, 114,
114] it is generally agreed that portfolio performance should be measured in two distinct
dimensions: the mean describing the expected return, and the risk which measures the
uncertainty of the return. In the mean–risk approach, we select from the universe of all
possible portfolios those that are efficient: for a given value of the mean they minimize
the risk or, equivalently, for a given value of risk they maximize the mean. This
approach allows one to formulate the problem as a parametric optimization problem, and
it facilitates the trade-off analysis between mean and risk.
Another theoretical approach to the portfolio selection problem is that of stochastic
dominance (see Mosler and Scarsini [121], Whitmore and Findlay [210], and Levy
[99]). The concept of stochastic dominance is related to models of risk averse
preferences, Fishburn [52]. It originated from the theory of majorization, Hardy and
Litlewood [70], and Marshall and Olkin [109] for the discrete case, was later extended to
general distributions (Quirk and Saposnik [146], Hadar and Russell [66], Hanoch and
Levy [68], Rothschild and Stiglitz [155], and is now widely used in conomics and
finance Levy [99].
The usual (first order) definition of stochastic dominance gives a partial order in the
space of real random variables. More important from the portfolio point of view is the
notion of second-order dominance, which is also defined as a partial order. It is
equivalent to this statement: a random variable R dominates the random variable Y if
)]([)]([ yuERuE ≥ for all nondecreasing concave functions u(·) for which these
52
expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio
with return Y over a portfolio with return R.
Introduced a new stochastic optimization model with stochastic dominance constraints
Dencheva and Ruszczynski [41,42]. In this chapte we show how this theory can be used
for risk-averse portfolio optimization. We add to the portfolio problem the condition that
the portfolio return stochastically dominates a reference return, for example, the return of
an index. We identify concave nondecreasing utility functions which correspond to
dominance constraints. Maximizing the expected return modified by these utility
functions, guarantees that the optimal portfolio return will dominate the given reference
return.
4.2. Dominance-constrained
Consider stochastic dominance relations between random returns defined by (3.11).
Thus, we say that portfolio x dominates portfolio y under the FSD rules, if
));(());(( ηη yRFxRF ≤ for all R∈η ,
where at least one strict inequality holds. Similarly, we say that x dominates y under
the SSD rules ))()(( yRxR SSDf , if
));(());(( 22 ηη yRFxRF ≤ for all R∈η
with at least one inequality strict. Recall that the individual returns kjR have finite
expected values and thus the function ));((2 ηxRF is well defined.
Stochastic dominance relations are of crucial importance for decision theory. It is known
that )()( yRxR FSDf if and only if
))](([))](([ yRuExRuE ≥ , (4.1)
for any nondecreasing function u(·) for which these expected values are finite.
Furthermore, )()( yRxR SSDf if and only if (4.1) holds true for every nondecreasing and
concave u(·) for which these expected values are finite (see, Dentcheva and Ruszczynski
[41, 42] and Levy [99]).
A portfolio x is called SSD-efficient (or FSD-efficient) in a set of portfolios X if there
is no ∈y X such that ))()(( xRyR SSDf (or FSDyR f)( ))(xR .
53
We shall the focus to the SSD relation, because of its consistency with risk-averse
preferences: if )()( xRyR SSDf , then portfolio x is preferred to y by all risk-averse
decision makers.
The starting point consider the assumption that a reference random return Y having a
finite expected value is available. It may have the form )(zR , for some reference
portfolio z . It may be an index or our current portfolio. Our intention is to have the
return of the new portfolio, )(xR , preferable over Y . We introduce the following
optimization problem :
)(max xf (4.2)
Subject to kSSD
k YxR f)( , υ,1=k (4.3)
X∈x , (4.4)
where nkn
kk xRxRxR ++= ...)( 11 , υ,1=k .
Here R→X:f is a concave continuous functional. In particular for 0>kw , υ,1=k ,
we may use
)]([)(1
xREwxf k
kk∑
=
=υ
,
and this will still lead to nontrivial solutions, due to the presence of the dominance
constraints.
Using the line of Dentcheva and Ruszczynski [41] we obtain the following proposition.
Proposition 4.1 Assume that kY , υ,1=k has a discrete distribution with realizations
kiy , mi ,1= , υ,1=k . Then relation (3.16) is equivalent to
υ,1,,1],)[(]))([( ==∀−≤− ++ kmiYyxRy ki
ki EE . (4.5)
Proof. If relation (4.3) is true, then the equivalent representation (3.10) implies (4.5).
It is sufficient to prove that (4.5) imply that
);());(( 22 ηη kk YFxRF ≤ for all R∈η , υ,1=k .
With no loss of generality we may assume that km
k yy << ...1 , mi ,1= , υ,1=k . The
distribution function );( ⋅kYF is piecewise constant with jumps at kiy , mi ,1= ,
54
υ,1=k . Therefore, the function );( ⋅kYF is piecewise linear and has break points at
kiy , mi ,1= , υ,1=k . Let us consider three cases, depending on the value of η .
Case 4.1: If ky1≤η we have
0);());(());((0 12122 =≤≤≤ kkkkk yYFyxRFxRF η
Therefore the required relation holds as an equality.
Case 4.2: Let ],[ 1ki
ki yy +∈η for some i. Since, for any random return )(xRk , the
function ));((2 ⋅xRF k is convex, inequalities (4.5) for i and 1+i imply that for all
],[ 1ki
ki yy +∈η one has
));(()1());(());(( 1222ki
kki
kk yxRFyxRFxRF +−+≤ λλη
);();()1();( 2122 ηλλ kki
kki
k YFyYFyYF =−+≤ +
where kii
ki
yyy
−−
=+1
ηλ The last equality follows from the linearity of );(2 ⋅kYF in the
interval ],[ 1ki
ki yy + .
Case 4.3: For kmy∈η the function );(2 ηkYF is affine with slope 1, and therefore
km
km
kk yyYFYF −+= ηη );();( 22
));(());(());(( 22 ηααη
xRFdxRFyxRF k
y
kkm
kkm
=+≥ ∫ ,
as required.
Let us assume now that the returns have a discrete joint distribution with realizations
njTtr kjt ,1,,1, == and υ,1=k , attained with probabilities Ttp k
t ,1, = . The
formulation of the stochastic dominance relation (4.3) resp. (4.5) simplifies even further.
Introducing variables kits representing shortfall of )(xRk below k
iy in realization,
mit ,1, = Tt ,1= and υ,1=k we obtain the following result.
Proposition 4.2 Assume that njR kj ,1, = , υ,1=k and kY have discrete distributions.
Then problem (4.2)–(4.4) is equivalent to the problem
55
)(max xf (4.6)
Subject to ki
kit
kjt
n
jj ysrx ≥+∑
=1, mi ,1= , Tt ,1= , υ,1=k , (4.7)
),(21
ki
kkit
T
t
kt yYFsp ≤∑
=
, mi ,1= , υ,1=k , (4.8)
0≥kits mi ,1= , Tt ,1= , υ,1=k , (4.9)
X∈x . (4.10) Proof. If nx R∈ is a feasible point of (4.2)–(4.4), then we can set
+=⎟⎟⎠
⎞⎜⎜⎝
⎛−= ∑ k
jt
n
jj
ki
kit rxys
1, mi ,1= , Tt ,1= , υ,1=k .
The pair ),( sx is feasible for (4.7)–(4.10).
On the other hand, for any pair ),( sx , which is feasible for (4.7)–(4.10), we get
⎟⎟⎠
⎞⎜⎜⎝
⎛−≥ ∑
=
kjt
n
jj
ki
kit rxys
1
, mi ,1= , Tt ,1= , υ,1=k .
Taking the expected value of both sides and using (3.21) we obtain
),,());(( 22ki
kki
k yYFyxRF ≤ mi ,1= , υ,1=k .
Proposition 4.1 implies that x is feasible for problem (4.2)–(4.4).
4.3 Optimality and duality
From now on we shall assume that the probability distributions of the returns and of the
reference outcome kY are discrete with finitely many realizations. We also assume that
the realizations of kY are ordered: km
k yy << ...1 , υ,1=k . The probabilities of the
realizations are denoted by miki ,1, =π .
We define the set U of functions RR →:u satisfying the following conditions:
)(⋅u is concave and nondecreasing;
)(⋅u is piecewise linear with break points kiy , mi ,1= , υ,1=k ;
0)( =tu for all kmyt ≥ , υ,1=k .
It is evident that U is a convex cone.
56
Let us define the Lagrangian of (4.2)–(4.4), RR →× υUnL : , as follows
( )∑=
−+=υ
υ
1
)( ))(())(()(),(k
kkkk YuExREuxfuxL . (4.11)
where ),...,( 1)( υυ uuu = .
It is well defined, because for every U∈ku and every nx R∈ the expected value
))](([ xRu kkE exists and is finite.
Theorem 4.4 If x is an optimal solution of (4.2)–(4.4) then there exists a function υυ U∈)(u , such that
)ˆ,(max)ˆ,ˆ( )()( υυ uxLuxLx X∈
= (4.12)
and
)](ˆ[))]ˆ((ˆ[ kkkk YuxRu EE = , υ,1=k , (4.13)
where )ˆ,...,ˆ(ˆ 1)( υυ uuu = .
Conversely, if for some function υυ U∈)(u an optimal solution x of (4.12) satisfies
(4.3) and (4.13), then x is an optimal solution of (4.11)–(4.13).
Proof. By Proposition 4.2 problem (4.2)–(4.4) is equivalent to problem (4.6)–(4.10). We
associate Lagrange multipliers mR∈μ with constraints (4.8) and we formulate the
Lagrangian Λ : RRRR →×× mmTn υ)( as follows:
⎟⎠
⎞⎜⎝
⎛−+=Λ ∑∑∑
== =
T
t
kit
kt
ki
k
k
m
i
ki spyYFxfsx
12
1 1
)()( ),()(),,(υ
υυ μμ
where ),...,( 1)( υυ sss = , ),...,( 1)( υυ μμμ = and ),...,( 1km
kk μμμ = .
Let us define the set
⎭⎬⎫
⎩⎨⎧
===≥+×∈= ∑=
+ υυυ ,1,,1,,1,:)(),(1
)( kTtmiysrxRsxZ ki
n
j
kit
kjtj
mTX .
Since Z is a convex closed polyhedral set, the constraints (4.8) are linear, and the
objective function is concave, if the point )ˆ,ˆ( )(υsx is an optimal solution of problem
57
(4.2)–(4.4), then the following Karush-Kuhn-Tucker optimality conditions hold true.
There exists a vector of multipliers 0ˆ ≥μ such that:
)ˆ,,(max)ˆ,ˆ,ˆ( )()(
),(
)()()(
υυυυ μμυ
sxsxZsxΛ=Λ
∈ (4.14)
and
υμ ,1,,1,0),(ˆ1
2 ===⎟⎠
⎞⎜⎝
⎛−∑
=
kmispyYFT
t
kit
kt
ki
kki . (4.15)
We can transform the Lagrangian Λ as follows:
∑∑∑∑∑= = == =
−+=Λυυ
υυ μμμ1 1 1
21 1
)()( ),()(),,(k
kit
ki
m
i
T
t
ki
ki
k
k
m
i
ki spyYFxfsx
∑∑∑∑∑== == =
−+=m
i
kit
ki
k
T
t
kt
ki
k
k
m
i
ki spyYFxf
11 12
1 1),()( μμ
υυ
.
For any fixed x the maximization with respect to )(υs such that Zsx ∈),( )(υ yields
+=⎟⎟⎠
⎞⎜⎜⎝
⎛−= ∑
n
j
kjtj
ki
kit rxys
1
( ) ,,1,,1,,1,)]([ υ===−= + kTtmixRy tkk
i
where tk xR )]([ is the t-th realization of the portfolio return. Define the functions
RR →:kiu , mi ,1= , υ,1=k by
+−−= )()( ηη ki
ki yu ,
and let
)()(1
ημημ ∑
=
=m
i
ki
ki
k uu k , ),...,( 1km
kk μμμ = .
Let us observe that υμ
U∈kku . We can rewrite the result of maximization of the
Lagrangian Λ with respect to s as follows:
( )∑∑∑∑ ∑== === =
++=Λm
it
kki
ki
k
T
t
kt
ki
k
k
m
i
ki
sxRupyYFxfsx
11 12
11 1
)()( )]([),()(),,(max)(
μμμυυ
υυυ
( )∑∑∑∑= == =
++=υ
μ
υ
μ1 1
21 1
)]([),()(k
tkk
T
t
kt
ki
k
k
m
i
ki xRupyYFxf k .
58
(4.16)
Furthermore, we can obtain a similar expression for the sum involving kY :
+== == =∑∑∑∑∑ −= )(),(
11 12
1 1
m
l
kl
ki
kl
k
m
i
ki
ki
k
k
m
i
ki yyyYF πμμ
υυ
+== =∑∑∑ −= )(
11 1
m
i
kl
ki
ki
k
m
l
kl yyμπ
υ
.
)(1 1
kl
km
i
m
l
kl yu kμ
π∑∑= =
−=
Substituting into (4.16), we obtain
[ ]∑=
−+=Λυ
μμυυ μ
υ1
)()( )]([))](([)(),,(max)(
k
kkkk
sYuxRuxfsx kk EE ),( )(υ
μuxL= ,
(4.17)
Setting kkkuu
μ=ˆ , υ,1=k we conclude that the conditions (4.14) imply (4.12), as
required. Furthermore, adding the complementarity conditions (4.15) over mi ,1= , and
using the same transformation we get (4.13).
To prove the converse, let us observe that for every υυ U∈)(u we can define
),()ˆ()()ˆ(ˆ '' ki
kki
kki yuyu +− −=μ mi ,1= , υ,1=k .
with ')ˆ( ku− and ')ˆ( ku+ denoting the left and right derivatives of ku :
t
tuuukk
t
k
−−
=↑− η
ηηη
)(ˆ)(ˆlim)()ˆ( ' and
ttuuu
kk
t
k
−−
=↓+ η
ηηη
)(ˆ)(ˆlim)()ˆ( .
Since ku is concave, 0ˆ ≥kμ . Using the elementary functions +−−= )()( ηη ki
ki yu we
can represent ku as follows:
∑=
=m
i
ki
ki
k uu1
)(ˆ)(ˆ ημη , υ,1=k .
Consequently, correspondence (4.17) holds true for kμ , and )(ˆ υu . Therefore, if x is the
maximizer of (4.12), then the pair )ˆ,ˆ( )(υsx , with )ˆ,...,ˆ(ˆ )(1)( υυ sss = ,
+=⎟⎟⎠
⎞⎜⎜⎝
⎛−= ∑
n
j
kjtj
kiit rxys
1
)( ˆˆ υ ,
59
is the maximizer of )ˆ,,( )()( υυ μsxΛ , over Zsx ∈),( )(υ . Our result follows then from
standard sufficient conditions for problem (4.6)–(4.10) (see,e.g.,Rockafellar [150,
Theorem. 28.1]).
We can also develop duality relations for our problem. With the Lagrangian (4.11) we
can associate the dual function
),(max)( )()( υυ uxLuDx X∈
= .
We are allowed to write the maximization operation here, because the set X is compact
and ),( )(υuxL is continuous.
The dual problem has the form
⎪⎩
⎪⎨⎧
∈∈
.
),(min)()(
)()(
υυ
υυυ
UU
u
uDu (4.18)
The set υU is a closed convex cone and )(⋅D is a convex functional, so (4.18) is a
convex optimization problem.
Theorem 4.5 Assume that (4.11)–(4.13) has an optimal solution. Then problem (4.18)
has an optimal solution and the optimal values of both problems coincide. Furthermore,
the set of optimal solutions of (3.31) is the set of functions U∈u satisfying (4.11)–
(4.13) for an optimal solution x of (4.11)–(4.13).
Proof. The theorem is consequence of Theorem 4.4 and general duality relations in
convex non-linear programming (see Beale [10, Theorem. 2.165]). Note that all
constraints of our problem are linear or convex polyhedral, and therefore we do not need
any constraints qualification conditions here.
4.4 Splitting
Let us now consider the special form of problem (4.11)–(4.13), with
∑=
=υ
1)]([)(
k
kk xREwxf , 0>kw for υ,1=k .
Recall that the random returns υ,1,,1,R == knjkj , have discrete distributions with
realizations υ,1,,1, == kTtr kjt , attained with probabilities k
tp .
60
In order to facilitate numerical solution of problem (4.11)–(4.13), it is convenient to
consider its split-variable form:
⎥⎦
⎤⎢⎣
⎡∑=
υ
1)(max
k
kk xRwE (4.19)
subject to kk VxR ≥)( , υ,1=k a.s., (4.20)
kk YV )2(f , υ,1=k (4.21)
X∈x . (4.22)
In this problem, kV is a random variable having realizations ktv attained with
probabilities ktp , υ,1,,1 == kTt , and relation (3.33) is understood almost surely. In
the case of finitely many realizations it simply means that
kt
kj
n
j
kjt vxr ≥∑∑
= =
υ
1 1 , υ,1,,1 == kTt . (4.23)
We shall consider two groups of Lagrange multipliers: a utility function υυ U∈)(u , and
vector 0, ≥∈ kTk θθ R . The utility functions )()( ⋅υu will correspond to the dominance
constraints (4.21), as in the preceding section. The multipliers kt
ktp θ , υ,1,,1 == kTt ,
will correspond to the inequalities (4.23). The Lagrangian takes on the form
+⎟⎟⎠
⎞⎜⎜⎝
⎛−+= ∑∑∑∑∑∑
== === =
kt
n
jj
kjt
k
T
t
kt
kt
n
jj
kjt
k
T
t
ktk vxrpxrpwuVx
11 111 1
)()()( ),,,(υυ
υυυ θθL
∑∑∑∑= == =
−+υυ
π1 11 1
)()(k
kl
km
l
kl
kt
k
k
T
t
kt yuvup , (4.24)
where the random variable kV is identified by its realization kT
k vv ,...,1 . (Thus we put
),...,,( 21υT
kkk vvvV = and ),...,( 1)( υυ VVV = ).
The optimality conditions can be formulated as follows.
Theorem 4.6 If )ˆ,ˆ( )(υVx is an optimal solution of (4.19)–(4.22), then there exist
U∈u and a nonnegative vector υυθ )(ˆ )( TR∈ , such that
)ˆ,ˆ,,(max)ˆ,ˆ,ˆ,ˆ( )()()(
)(),(
)()()()(
υυυυυυ θθυυ
uVxuVxTVx
LLX R×∈
= , (4.25)
61
0)(ˆ)ˆ(ˆ11
=−∑∑==
m
l
kl
kkl
kt
kT
t
kt yuvup π , υ,1=∀k (4.26)
0)ˆˆ(ˆ1
=−∑=
n
jj
kjt
kt
kt xrvθ , υ,1,,1 == kTt . (4.27)
Conversely, if for some function υυ U∈)(u and nonnegative vector υυθ )(ˆ )( TR∈ , an
optimal solution ( ))(ˆ,ˆ υVx of (4.25) satisfies (4.20)–(4.22) and (4.26)–(4.27), then
( ))(ˆ,ˆ υVx is an optimal solution of (4.19)–(4.22).
Proof. By Proposition 4.1, the dominance constraints (4.21) is equivalent to finitely
many inequalities
υ,1,,1],)[(]))([( ==−≤− ++ kmiYyxRy kki
kki EE .
Problem (4.19)–(4.22) takes on the form:
⎥⎦
⎤⎢⎣
⎡∑=
υ
1)(max
kkk xRwE
subject to ∑=
≥n
j
ktj
kjt vxr
1, υ,1,,1 == kTt
υ,1,,1],)[(]))([( ==−≤− ++ kmiYyxRy kki
kki EE
.X∈x
Let us introduce Lagrange multipliers miki ,1, =μ , υ,1=k associated with the
dominance constraints. The standard Lagrangian takes on the form:
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=Λ ∑∑∑∑∑∑
== === =
n
j
ktj
kjt
kt
k
T
t
ktj
n
j
kjt
k
T
t
ktk vxrpxrpwVx
11 111 1
)()()( ),,,( θθμυυ
υυυ
∑∑∑∑ ∑∑∑=
+= == =
+= =
−+−−m
l
kl
ki
kl
k
m
i
ki
T
t
n
jj
kjt
kit
k
m
i
ki yyxryp
11 11 11 1][][ πμμ
υυ
.
Rearranging the last two sums, exactly as in the proof of Theorem 4.4, we obtain the
following key relation. For every 0≥kμ , setting
+=
−−= ∑ )()(1
ημημ
ki
m
i
ki
k yu k
62
we have
),,,(),,,( )()()()()()( υυμ
υυυυ θθμ kuVxVx L=Λ ,
where ),...,( )(1)( υμμ
υμ kkk uuu = .
The remaining part of the proof is the same as the proof of Theorem 4.4.
The dual function associated with the split-variable problem has the form
),,,(sup),( )()()(
)(,
)()(
)(
υυυυυ θθυυ
uVxuTVx
LDX R∈∈
=
and the dual problem is, as usual,
),(min )()(
0,)(
υυ
θθ
υυu
uD
U ≥∈, (4.28)
The corresponding duality theorem is an immediate consequence of Theorem 4.5 and
standard duality relations in convex programming. Note that all constraints of our
problem (4.19)–(4.22) are linear or convex polyhedral, and therefore here we do not
need additional constraints qualification conditions.
Theorem 3.4 Assume that (4.19)–(4.22) has an optimal solution. Then the dual problem
(4.28) has an optimal solution and the optimal values of both problems coincide.
Furthermore, the set of optimal solutions of (4.28) is the set of functions υυ U∈)(u and
vectors 0)( ≥υθ satisfying (4.25)–(4.27) for an optimal solution )ˆ,ˆ( )(υVx of (4.19)–
(4.22).
We can analysis in more detail the structure of the dual function:
∑ ∑∑∑∑∑= =====∈∈ ⎭
⎬⎫
⎩⎨⎧
+−+=υ
θθ1 11111,
)()(sup),,(k
T
t
kt
kt
ktj
n
j
kjt
T
t
kt
ktj
n
j
kjt
T
t
kt
VxvupvxrpxrpvuD
TRX
)(1 1
kl
k
m
l
kl yu∑∑
= =
−υ
π
∑ ∑∑∑∑= ==∈= =
∈ ⎭⎬⎫
⎩⎨⎧
−−++=υ
πθθ1 111 1
)(])([sup)1(maxk
n
j
kl
kl
T
t
kt
kt
kt
kt
RVj
kj
m
j
kt
T
t
ktx
yuvvupxrpTX
.
63
In the last equation we have used the fact that X is a simplex and therefore
the maximum of a linear form is attained at one of its vertices. It follows that the
dual function can be expressed as the sum
∑∑=
+=
++=T
t
kT
kt
kt
kt
kuDvuDpDvuD
11
10 )(),,()(),,( θθθ
υ
(4.29)
with
∑∑= =
≤≤−=
υ
θθ1 110 )1(max)(
k
kjt
kt
k
t
ktnj
rpD , (4.30)
])([sup),( kt
kt
kt
v
kt
kt vvuuD
t
θθ −= , υ,1,,1 == kTt , (4.31)
and
∑∑= =
+ −=m
k
kl
m
l
kl
kT yuvuD
1 11 ))(),( π , υ,1=k . (4.32)
If the set X is a general convex polyhedron, the calculation of 0D involves a
linear programming problem with n variables.
To determine the domain of the dual function, observe that if kt
kyu θ<− )( 1' , then
+∞=−∞→
])([lim kt
kt
kt
vvvu
kt
θ ,
and thus the supremum in (4.22) is equal to ∞+ . On the other hand, if kt
kyu θ≥− )( 1' , then the function k
tkt
kt vvu θ−)( has a nonnegative slope for kk
t yv 1≤
and nonpositive slope ktθ− for kk
t yv 1≥ . It is piecewise linear and it achieves its
maximum at one of the break points.
Therefore
)(:),( 1' k
tkk
tkt yuRUuDdom θθ ≥×∈= −+ .
At any point of the domain,
])([max),(1
kl
kl
klmk
kt
kt yyuuD θθ −=
≤≤. (4.33)
The domain of kD0 is the entire space TR .
64
4.5 Decomposition
It follows that the dual functional can be expressed as a weighted sum of 2+T
functions (4.30)–(4.32).
In order to analyze their properties and to develop a numerical method we need to
find a proper representation of the utility function u . We represent the function
u by its slopes between break points. Let us denote the values of u at its break
points by
)( kl
kl yuu = , υ,1,,1 == kml .
We introduce the slope variables
)(' kl
kl yu−=β , υ,1,,1 == kml
The vectors ,....),...,,,...,( 221
111 mm βββββ = is nonnegative, because u is
nondecreasing. As u is concave, kl
kl 1+≥ ββ , υ,1,1,1 =−= kml . We can represent
the values of u at break points as follows
∑>
−−−=ll
kl
kl
kl
kl yyu
'
)( 1β , 1,1 −= ml .
The function (4.24) takes on the form
⎥⎦
⎤⎢⎣
⎡−−−=−= ∑
>−≤≤≤≤
ll
kl
kt
kl
kl
klml
kl
kl
klml
kt
kt yyyyuuD
'
)(max][max),( 111θβθθ .
In this way we have expressed ),( kt
kt uD θ as a functions of the slope vector
mR∈β and of +∈Rktθ . We denote
⎥⎦
⎤⎢⎣
⎡−−−= ∑
>−≤≤
ll
kl
kt
kl
kl
klml
kt
kt yyyB
'
)(max),( 11θβθβ . (4.34)
Observe that ktB is the maximum of finitely many linear functions in its domain.
The domain is a convex polyhedron defined by
kl
kt βθ ≤≤0 .
65
Consequently, ktB is a convex polyhedral function. Therefore its subgradient at a
point ),( ktθβ of the domain can be calculated as the gradient of the linear
function at which the maximum in (4.34) is attained. Let *l be the index of this
linear function. Denoting by 'lδ the 'l th unit vector in mR we obtain that:
⎟⎟⎠
⎞⎜⎜⎝
⎛−−− ∑
>−
*'
),( 1ll
kl
kl
kll yyyδ is the subgradient of ),( k
tktB θβ .
Similarly, function (3.32) can be represented as a function kTB 1+ of the slope
vectors β :
∑ ∑= >
−+ −=m
l ll
kl
kl
kl
kl
kT yyB
111
'
)()( βπβ .
It is linear in β and its gradient has the form
)( 11 ''
'kl
kl
ll
kl
n
ll yy −
<=
−∑∑ πδ .
Finally, denoting by *j the index at which the maximum in (4.30) is attained, we
see that the vector with coordinates
ktj
kt rp * , υ,1,,1 == kTt , (4.35)
is a subgradient of kD0 .
Summing up, with our representation of the utility function by its slopes, the dual
function is a sum of T + 2 convex polyhedral functions with known domains.
Moreover, their subgradients are readily available. Therefore the dual problem
can be solved by nonsmooth optimization methods (see Dentcheva and
Ruszczynski [41], Beale [8] and Bonnans and Shapiro [18]). We have developed
a specialized version of the regularized decomposition method described in
Dentcheva and Ruszczy´nski [41] and Ogryezak and Ruszczy’nski [127]. This
approach is particularly suitable, because the dual function is a sum of very many
polyhedral functions.
66
After the dual problem is solved, we obtain not only the optimal dual solution
)ˆ,ˆ( θβ , but also a collection of active cutting planes for each component of the
dual function.
Let us denote by kj0 the collection of active cuts for kD0 . Each cutting plane for
kD0 provides a subgradient (4.35) at the optimal dual solution. A convex
combination of these subgradients provides the subgradient of kD0 that enters the
optimality conditions for the dual problem. The coefficients of this convex
combination are also identified by the regularized decomposition method. Let kg0
denote this subgradient and let 0, Jjvkj ∈ the corresponding coefficients. Then
∑∑∈=
=0Jj1
0kj
kjt
kt
T
t
kt
k vrpg δ ,
where
0≥kjv , ∑
∈
=0Jj
1kjv .
For each t the subgradient of ktB with respect to k
tθ entering the optimality
conditions is
)34.4(inmaximizeris:ˆ ** lyconvv k
lkt ∈ .
Therefore
0ˆ1
0 =−∑=
Tkt
kt
k vpgl
,
Using these relations we can verify that v are the vector of optimal portfolio
returns in scenarios Tt ,1= . Thus the optimal portfolio has the weights
⎩⎨⎧
∉=∈=
.,0ˆ,,ˆ
0
0
JjxJjvx
j
jj
67
68
CHAPTER 5
A FUZZY APPROACH TO PORTFOLIO OPTIMIZATION
5.1 Introduction
In general optimization problem in which the objective function(s) and the constraints in
the space of the decision variables are linear is said to be a linear programming problem.
In addition, if there are multiple objectives, then the optimization problem will be called a
multi-objective linear programming problem.
In a recent paper Sakawa and Yana [165] very nicely demonstrate the state of the art
when we want to deal with multiple criteria problems which are - and cannot be - well-
defined. Their model is a multiple objective linear fractional programming model, with
fuzzy parameters and an uncertain goal for the objective function. The uncertainty is of
two types: (i) an uncertainty of the satisfaction with the value of an objective function;
and (ii) an uncertainty of the possibility to generate the wanted value of the objective
function. The Sakawa-Yana method handles both types of uncertainties and reduces the
problem to an ordinary multi-objective programming problem.
Another approach has been developed by Kacprzyk and Yager [74], in which they use
fuzzy logic with linguistic qualifiers to bring human consistency to multiobjective
decision making. They use rather a nonconventional solution concept, which is based on
searching for some optimal option which ”best satisfies most of 2 the important
objectives”; this differs from the traditional notion to try to find an optimal option which
best satisfies ”all the objectives”.
Both the Sakawa-Yana and the Kacprzyk-Yager papers seem to support the idea that
traditional multi-criteria decision models (MCDM), and their underlying notion of an
optimal solution, are much to o limited for actual, real-world problem-solving with
MCDM methods. The reason for this is simple: when the solution derived from a well-
formulated mathematical MCDM-model is applied to an actual problem there are some
major problems to consider (Roy [149]): (i) the set of feasible decision alternatives is
69
fuzzy, and this set changes during the problem solving process; (ii) the DM does not exist
as an active entity, and the preferences consist of badly formulated beliefs, which are
riddled with conflicts and contradictions; (iii) data on preferences are imprecise, and (iv)
a decision should be good or bad not only in relation to some model, but in relation to the
actual context. These problems have initiated active and fast-growing research on the use
of fuzzy set theory in solving multiple criteria decision problems (Carlsson [28, 29],
Takeda [191] and Zimmermann [222]).
The problem of optimizing a portfolio of finitely many assets is a classical problem in
theoretical and computational finance. Since the seminal work of Markowitz [112] it is
generally agreed that portfolio performance should be measured in two distinct
dimensions: the mean describing the expected return, and the risk which measures the
uncertainty of the return. As one of theoretical approach to the portfolio selection
problem is that of stochastic dominance (see Rockafellar and Uryasev [152].
In the context of choosing optimal portfolios that target returns above the risk-free rate
for certain market scenarios while at the same time guaranteeing a minimum rate of
return, fuzzy decisions theory provides an excellent framework for analysis. This is
because the nature of the problem requires one to examine various market scenarios, and
each such scenario will in turn give rise to an objective function. In the last section, we
will describe a multi-objective linear programming problem formulation where the
objective functions are considered to be fuzzy.
5.2 Fuzzy linear programming models
Empirical surveys reveal that linear programming (LP) is one of most frequently applied
operations research techniques is real-word problems. However, given the power of LP
one could have expected oven more applications. This might be due to the fact that LP
requiresmunch well-defined and precise data which involves high-information cost. In
real-word applications certainty, reliability and precision of data is often illusory.
Furthermore the optimal solution of LP only depends on a limited number of constraints
and, thus, much of the information collected has little inpact on the solution. Fuzz linear
programming, propused by Bellman and Zadeh, is an extention of LP with both objective
function(s) and constraints represented by fuzzy sets.
70
Now we can defined a LP problem with crisp of fuzzy resource constraints, and a crisp or
fuzzy objective as:
subject to
,0
,1,~,~max
1
≥
=≤
=
∑=
X
pibXa
XcZ
ij
m
jij
T
(5.1)
where fuzzy resources ibi ∀,~
have the same form of membership function. We may also
consider the following fuzzy inequality constraints:
subject to
,0
,1,~~
,~max
1
≥
=≤
=
∑=
X
pibXa
XcZ
ij
m
jij
T
(5.2)
even though (5.1) and (5.2) are defferent in some points of view, we can use the same
approach to handle them under the pre-assumtion of the membership functions of the
fuzzy available resources and fuzzy inequality constraints.
The difference between crisp and fuzzy constraints is that in case of crisp constraints the
decision maker can strictly differentiate between feasibility and infeasibility; in case of
fuzzy constraints he wants to consider a certain degree of feasibility in the interval
(Werners, 1987).
Now, we consider some approaches for fuzzy linear programming models.
Correspondingly we could build portfolio models as sections 5.5, 5.6 and 57.
First approach: The resources can be determined precisely, a traditional LP problem is
consider as :
subject to
,0
,~,max
1
≥
∀≤
=
∑=
X
ibXa
XcZ
ij
m
jij
T
(5.3)
where ibac iij ∀,and, are precisely given. The optimal solution of (5.3) ia a unique
optimal solution.
71
Second approach Chanas and Verdegay: A decision maker wishes to make a
postoptimization analysis. Thus, a parametric programming problem is formulated as:
subject to
,0
],1,0[,~,max
1
≥
∈+≤
=
∑=
X
pbXa
XcZ
ij
m
jij
T
θθ (5.4)
where ipbac iiij ∀,and,, are precisely given and θ is a parameter, ipi ∀, are
maximum tolerances which are always positive. The solution )(* θZ of (5.4) are function
of θ . That is, for each θ we can obtain an optimal solution.
On the other hand, the available resources may be fuzzy. Then the LP problem with fuzzy
resources becomes:
subject to
,0
,~,max
1
≥
∀≤
=
∑=
X
ibXa
XcZ
ij
m
jij
T
(5.5)
It is possible to determine the maximum tolerance ip of the fuzzy resources ibi ∀, . Then
we can construct the membership functions iμ assumed linear for each fuzzy constraints,
as follows:
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
+>
+≤≤−
−
≤
=
∑
∑∑
∑
=
=
=
=
.if0
,if1
,if1
1
1
1
1
ii
m
jjij
ii
m
jjiji
i
i
m
jjij
i
m
jjij
i
pbXa
pbXabp
bXa
bXa
μ (5.6)
Verdegay and Chanas, propuse that (5.5) and (5.6), however, are equivalent to (5.4), a
parametric LP where ipbac iiij ∀,and,, are given, by use of the λ -level cut concept.
72
For each λ -level cut of the fuzzy constraint set (5.5) becomes a traditional LP problem.
That is,
subject to
].1,0[,0,,,
,max
∈≥∀≥=
∈=
λλμλ
λ
XandiXXXX
XcZ
i
T
(5.7)
and equivalent to:
subject to
,0and]1,0[
,,)1(
,~max
1
≥∈
∀−+≤
=
∑=
X
ipbXa
XcZ
ii
m
jjij
T
λ
λ (5.8)
where ipbac iiij ∀,and,, are precisely given. Now, if we set θλ −=1 , then equation
given by (5.8) will be the same as (5.4). Then a solution table is presented to the decision
maker to determine the satisfying solution. )(* θZ , ]1,0[∈θ is the fuzzy solution
corresponding to Verdegay’s approach.
Third approach (Weners’s approach): A decision maker may want to solve a FLP
problem with a fuzzy objective and fuzzy constraints, while the goal 0b , is not given.
That is:
subject to
,0
,,~
,~max
1
≥
∀≤
=
∑=
X
ibXa
XcZ
i
m
jjij
T
(5.9)
which is equivalent to:
subject to
,0and]1,0[
,,~
,~max
1
≥∈
∀+≤
=
∑=
X
ipbXa
XcZ
ii
n
jjij
T
λ
θ (5.10)
where ipbac iiij ∀,and,, are given, but the goal of the fuzzy objective is not given.
To solve (5.10) by use of Werners’s approach, let us first define 0Z and 1Z as follows:
73
),0()maxinf( *0 ===∈
θZXcZ T
X X (5.11)
)1()maxsup(( *1 ===∈
θZXcZ T
X X (5.12)
where .0and],1,0[,,1
≥∈∀+≤= ∑=
XipbXaX ii
n
jjij θθX
Then, we can obtain Werners’s membership function 0μ of the fuzzy objective. That is:
⎪⎪⎩
⎪⎪⎨
⎧
<
≤≤−−
−
>
=
.if0
,if1
,if1
0
1001
1
1
0
ZXc
ZXcZZZ
XcZZXc
T
TT
T
μ (5.13)
The membership functions ii ∀,μ , of the fuzzy constraints are defined as (5.6). By use of
the min-operator proposed by Bellman and Zadeh, we can obtain the decision space D
which is defined by its membership function Dμ where,
),...,min( 0 pD μμμ = . (5.14)
It is reasonable to choose the decision where Dμ is maximal as the optimal solution of
(5.9). Therfore, (5.9) is equivalent to:
λmax
,0],1,0[and,
,,
0
0
≥∀∈
≥≥
Xii
i
μμλλμλμ
(5.15)
where ipbac iiij ∀,and,, are given, and ),...,min( 0 mD μμμλ == .
Let θλ −=1 . Then the problem given by (5.15) will be equivalent to:
θmax
subject to
,0and]1,0[
,,)(),( 011
≥∈
∀+≤−−≥
X
ipbXaZZZXc
iiiij
T
θ
θθ
(5.16)
74
where ipbac iiij ∀,and,, are given and θ is fraction of )( 01 ZZ − for the first
constraint and a fraction of maximum tolerance for others. The solution is a unique
optimal solution.
Forth approach (Zimmermann’s approach): A decision maker may want to solve a FLP
problem with a fuzzy objective and fuzzy constraints, when the goal 0b of the fuzzy
objective and its minimum tolerance are given. That is,
subject to
,0
,,~
,~max
1
≥
∀+≤
=
∑=
X
ipbXa
XcZ
ii
n
jjij
T
θ (5.17)
where iij bpbac ,,,, 00 and ipi ∀, are given. The problem given by (5.17) is actually
equivalent to:
Find X,
subject to
,0
,,
,~max
1
≥
∀+≤
=
∑=
X
ipbXa
XcZ
ii
m
jjij
T
θ (5.18)
with the membership function of the fuzzy constraints as previously described in (5.6)
and the membership function of the fuzzy objective 0μ as follows:
⎪⎪
⎩
⎪⎪
⎨
⎧
−<
≤≤−−−
−
>
=
.if0
,if1
,if1
00
000010
0
0
pbXc
bXcpbZZ
Xcb
bXc
T
TT
T
μ (5.19)
Thus by use of the maximum concept, (5.18) is actually equivalent to:
λmax
subject to
,0],1,0[and,,,and
0
i0
≥∀∈
∀≥
Xi
i
iμμλλμμ
(5.20)
75
where iij bpbac ,,,, 00 and ipi ∀, are given. Let θλ −=1 . Then (5.15) will be
equivalent to:
subject to
,0and]1,0[
,,
,,max
1
00
≥∈
∀+≤
+≥
∑=
X
ipbXa
pbXc
ii
m
jjij
iT
θ
θ
θ
θ
(5.21)
where iij bpbac ,,,, 00 and ipi ∀, are given and θ is a fraction of the maximum
tolerances. The optimal solution (5.21) is unique.
When a fuzzy objective is assumed, what Zimmermann and Werner’s approaches are
asumming is essentially a performance function on the objective,
]1,0[)()( ∈= XcFXf T .
Thn, in all cases, if ]1,0[),(* ∈θθZ , is the fuzzy solution to the problem, the
corresponding point solution for each fuzzy objective (performance function associated to
Zimmermann’s or Werner’s approach) to be considered, can be obtained by solving the
point-fix equation. It is showed that in the application.
Fifth approach: A decision maker may want to solve a FLP problem with a fuzzy
objective and fuzzy constraints, while only the goal 0b of the fuzzy objective is given, but
its tolerance 0p is not given. That is,
subject to
,0
,,~
,x~ma~max
1
≥
∀≤
=
∑=
X
ibXa
XcZ
i
n
jjij
T
(5.22)
where iij bbac ,,,, 0 and ipi ∀, are given, but 0p is not given. While 0p is not given, we
do know that 0p should be in between 0 and 00 Zb − . For each ],0[ 0
00 Zbp −∈ , we
can obtain the membership function of the fuzzy objective as (5.19). Sice in a high-
productivity system the objective value should be larger then 0Z at 0=θ , there is no
meaning to given a positive grade of membership for those which are less than 0Z .
76
The difference between this problem and Zimmermann is that 0p is not initially given in
this problem. Therefore, we may assume a set of sp0 , where ],0[ 000 Zbp −∈ . Then, the
problem of each 0p given is a Zimmermann problem.
The decision maker may choose a refined 0p among the solution for this given set of
sp0 . Then a Zimmermann problem with the decision maker’s refined 0p is solved. This
solution will be the final optimal solution for (5.22).
5.3 Interactive fuzzy linear programming
Decision processes are better described and solved using fuzzy sets theory, rather
than precise approaches. However, the decision maker himself always plays the
most important role in using fuzzy sets theory. Therefore, an interactive process
between the decision maker and the decision process is necessary to solve our
problem. That is actually a user-dependent fuzzy LP technique.
Furthermore, a problem-oriented concept is also a vitally important concept in
solving practical problems, as noted by Simon.
By use of fuzzy sets theory, and user dependent (interactive) and problem,
oriented concepts, the flexibility and robustness of LP techniques are improved.
An IFLP approach which is a symmetric integration of Zimmermann’s, Werner’s,
Verdegay’s and Chanas’s FLP approaches is developed and additionally it
provides a decision support system for solving a specific domain of a rel-world
LP system Lai and Hwang [92]. Lai and Hwang suggested “expert decision
support system” that give an aggregate solution to all possible cases.
The system determines fuzzy-efficient extreme solution and a fuzzy efficient
compromise solution. They are judged by the decision maker and he decides
whether modification are necessary. In the latter case the decision maker change
membership functions assisted by the system, Werners [202].
The application of FLP implies that the problems will be solved in an interactive
way. In the first step, the fuzzy system is modeled by using only the information
77
which the decision maker can provide without any expensive additional
information acquisition. Knowing a first “compromise solution” the decision
maker can perceive which further information should be obtained and he is able to
justify the decision by comparing carefully additional advantages and arising
costs. In doing so, step by step the compromise solution are improved. The
procedure obviously offer the possibility to limit the acquisition and processing
information to the relevant components and therefore information costs will be
distinctly reduced, Rommefanger [154].
The most important element that affects solutions of FLP problems is
parameters which are used reflecting fuzziness of model. How these parameters
define fuzzy geometry is the most sensitive point. Because the success of solution
depends on the success of reflecting the system of model.
Moreover, the interactive concept provides a learning process about the
system and makes allowance for psychological convergence for the decision
maker, whereby, (s)he learns recognize good solutions, the relative importance of
factors in the system and then design a high-productivity system, instead of
optimizing a given system.
This IFLP system provides integration-oriented, adaption and learning
features by considering all possibilities of specific domain of LP problems which
are integrated in logical order using an IF-THEN rule.
IFLP methods have been studied, since 1980. Typical works are Baptistella and
Ollero, Fabian, Cibiobanu and Stoica, Ollero, Aracil and Camacho, Sea and
Sakawa, Slowinski, Werners and Zimmermann, Zimmermann described some
general concepts and modeling methods of decision support system and expert
system in a fuzzy enviroment. Others developed interactive approaches to solve
multiple criteria decision making problem, Lai and Hwang [92].
With the aim of solutions for the models like these, there are many studies
on LP models. However the studies of Zimmermann, Chanas, Werners and
78
Vedegay have become quite efficient for improving LP models with decision
support to solve real-world problem.
5.3.1 Interactive fuzzy linear programming algorithm
Step 1 Solve a traditional LP problem of (5.3) by use of the simplex method.
The unique optimal solution with its corresponding consumed resources is
presented to the decision maker.
Step 2 Does this solution satisfy the decision maker ? Consider the following
cases.
1.If solution is satisfied the print out results an top
2.If resource i, for some i are idle then reduce available ib and go to Step 1.
3. If available resources are not precise and some tolerances are possible then
make a parametric analysis with and go to Step 3.
Step 3 Solve a parametric LP problem of (5.4). Then results are depicted in a
table. At the same time, let us identify )0(*0 == θZZ and )1(*1 == θZZ .
Step 4 Do any of these solutions shown in table satisfy the decision maker?
Consider the following cases:
1. If solution is satisfied then print out results ad stop.
2. If resource i, for some i are idle then decrease ib (and change ip ) and
then go to Step 1.
3. If tolerance i, for same i are not acceptable then change ip as desired and
go to Step 3.
4. If the objective should be considered as imprecise then to Step 5.
Step 5 After reffering to first table, the decision maker is then asked for his
subjective goal 0b and its tolerance 0p for solving a symmetric FLP problem.
If the decision maker does not like to give his goal for the fuzzy objective, to to
Step 6. If 0b is given, go to Step 8.
79
Step 5 Solve problem of (15). A unique Werners’s solution is the provided.
Step 7 Is the solution of (5.16) satisfying ? Consider the following cases:
1. If the solution is satisfied then print out results and stop.
2. If the user has realized his/her goal then give the goal 0b and go to Step 8.
3. If resource i, for some i are idle then decrease ib (and change ip ) and then
go to Step 1.
4. If tolerance i, for same i are not acceptable the change ip as desired and go
to Step 3.
Step 8 Is 0p determined by the decision maker ? If the decision maker would like
to specify 0p , we should provide a table to help the decision maker. Then go to
Step 9. If 0p is not given, then go to Step 11.
Step 9 Solve problem of (5.21). A unique Zimmermann’s solution is obtained.
Step 10 Is the solution of (5.21) satisfying ?
1. If solution is satisfied then print out results and stop.
2. If the user has realized beter his/her goal (and its tolerance) then give the
goal 0b ( and 0p ) and go to Step 8.
3. If resource i, for some i are idle then decrease ib (and change ip ) and then
go to Step 1.
4. If tolerance i, for same i are not acceptable the change ip as desired and go
to Step 3.
Step 11 Solve last problem. That is, call Step 9 to solve problem of (5.21) for a set
of sp0 . Then the solutions are depicted in a table.
Step 12 Are the solution satisfying ? If yes, print out the solution and then
terminate the solution procedure. Otherwise, go to Step 13.
80
Step 13 Ask the decision maker to specify the refined 0p , and then go to Step 0.
It is rather reasonable to ask the decision maker 0p at this step, because he has a
good idea about 0p now figure 5.1.
For implementing the above IFLP, we need only two solution-finding techniques,
the simplex method and parametric method. Therefore, the IFLP approach
proposed here can be easily programmed in a PC system for its simplicity, Lai and
Hwang [92].
5.4 Portfolio problem
The process of selecting a portfolio may be divided into two stages. The first stage starts
with observation and experience and end with beliefs about the future performance of
available securities. The second stage starts with the relevant beliefs about future
performances and ends with the choice of portfolio by Markowitz [114]. This chapter is
concerned with the second stage.
The problem of standard portfolio selection is as follows. Assume
(a) n securities,
(b) an initial sum of money to be invested,
(c) the beginning of a holding period,
(d) the end of the holding period,
and let x1,…,xn be the investors investment proportion weights. These are the proportions
of the initial sum to be invested in the n securities at the beginning of the holding period
that define the portfolio to be held fixed until the end of the holding period. The standard
view is that there is only one purpose in portfolio selection, and that is to maximize
portfolio return, the percent return earned by the portfolio over the course of the holding
period.
Now we consider the problem (4.11)-(4.12) given in Chapter 4, for 1=υ . Thus we have
)(max xf (5.23)
subject to YxR SSDf)( , (5.24)
X∈x . (5.25)
81
Here R→X:f is a concave continuous functional. Also in particular, we may
use
)]([)( xRxf E=
and this will still lead to nontrivial solutions, due to the presence of the dominance
constraint.
Yes
No
Yes
No
Figure 5.1 Flow chart decision support system (Werner’s, 1987)
Using the Chapter 4, for 1=υ , we get the following proposition.
Proposition 5.1 Assume that Y has a discrete distribution with realizations miyi ,1, = .
Then relation (5.24) is equivalent to
])[(]))([( ++ −≤− YyxRy ii EE , mi ,1=∀ . (5.26)
Model formulation
Efficient Extremesolution
Compromise Solution Local Information
Solution Acceptable ?“Best”
Compromise STOP
Modification of membership functionstion
Local consequences ?
82
Let us assume now that the returns have a discrete joint distribution with realizations jtr ,
Tt ,1= , nj ,1= , attained with probabilities tp , Tt ,1= . The formulation of the
stochastic dominance relation (5.24) respectively (5.26) simplifies event further.
Introducing variables its representing shortfall of R(x) below yi in realization t, mi ,1=
and Tt ,1= , we obtain the following proposition.
Proposition 5.2 The problem (5.23)-(5.25) is equivalent to the problem
)(max xf (5.27)
subject to iitj
n
jjt ysxr −≤−−∑
=1, mi ,1= , Tt ,1= , (5.28)
);(21
iit
T
tt yYFsp ≤∑
=
, mi ,1= (5.29)
0≥its mi ,1= , Tt ,1= , (5.30)
∑=
≤n
jjx
11, (5.31)
∑=
−≤−n
jjx
11 , (5.32)
0≥jx , nj ,1= , (5.33)
and problem (5.27)-(5.33) can be written as
)(max Xϕ = ∑=
n
jjj Xc
1, (5.34)
subject to: i
mTn
jjij bXa ≤∑
+
=1, 2,1 ++= mmTi , (5.35)
0≥jX , mTnj += ,1 , (5.36)
where, ),...,,...,,...,,,...,,,...,( 12211111 mTmTTn ssssssxxX = .
83
⎪⎩
⎪⎨
⎧
+−++=−=++=−−=++==−
=otherwise
iTnnjandTKmKKmiTKmKKminjr
aij
ij
,01)1(,1,)1(,0,)1(,1,1
)1(,0,)1(,1,,1,
⎩⎨⎧ +==
=otherwise
mTinjaij ,01,,1,1
⎩⎨⎧ +==−
=otherwise
mTinjaij ,02,,1,1
⎩⎨⎧ +++==++−+=
= −−−
otherwisemmTmTimKTKnKTnjpa KTnj
ij ,02,3,,1,,1)1(,)1(
In the next section we extended this result to fuzzy decisions theory.
5.5 Case of fuzzy technological coefficients and fuzzy right-hand side numbers
5.5.1 Case of fuzzy technological coefficients
In this section presents an approach to portfolio selection using fuzzy decisions theory.
We consider the problem (5.34) – (5.36) with fuzzy technological coefficients Gasimov
[57].
)(max Xϕ =∑=
n
jjj Xc
1 (5.37)
subject to i
mTn
jjij bXa ≤∑
+
=1
~ , 2,1 ++= mmTi , (5.38)
0≥jX , mTnj += ,1 . (5.39)
Assumption 5.1. ija~ is a fuzzy number for any i and j.
In this case we consider the following membership functions:
(i) 1. For )1(,0,)1(,1 −=++= TKmKKmi and nj ,1=
⎪⎩
⎪⎨
⎧
+−≥+−<≤−−+−
−<=
.0,/)(
,1)(
ijij
ijijijijijij
ji
a
drtifdrtrifdtdr
rtift
ijμ
84
2. For )1(,0,)1(,1 −=++= TKmKKmi and j=n+T(i-Km-1)+K+1
⎪⎩
⎪⎨
⎧
+−≥+−<≤−−+−
−<=
,10,11/)1(
11)(
ij
ijijija
dtifdtifdtd
tift
ijμ
(ii) For 2,3 +++= mmTmTi , mK ,1= and TKnKTnj ++−+= ,1)1(
⎪⎩
⎪⎨
⎧
+≥+<≤−+
<=
−−−
−−−−−−
−−−
−−−
,
)1(
)1(
)1()1(
)1(
0,/)(
,1)(
ij
ijijijKTnja
dptifdptpifdtdp
ptift
KTnj
KTnjKTnj
KTnj
ijμ
where Rt ∈ and 0>ijd for all 2,1 ++= mmTi , )1(,0 −= TK and mTnj += ,1 .
For defuzzification of this problem, we first fuzzify the objective function. This is done
by calculating the lower and upper bound of the optimal values first. The bounds of the
optimal values lz and uz are obtained by solving the standard linear programming
problems
)(max1 Xz ϕ= (5.40)
subject to ij
mTn
jij bXa ≤∑
+
=1, 2,1 ++= mmTi , (5.41)
0≥jX , mTnj += ,1 , (5.42)
and
)(max2 Xz ϕ= (5.43)
subject to ij
mTn
jij bXa ≤∑
+
=1
ˆ , 2,1 ++= mmTi , (5.44)
0≥jX , mTnj += ,1 , (5.45)
where
85
⎪⎩
⎪⎨
⎧
−=+−++=++=+−−=++==+−
=otherwised
TKandiTnnjmKmKmidTKandmKKminjdr
a
ij
ij
ijij
ij
,)1(,0,1)1(,1,)1(,1,1
)1(,0)1(,1,,1,ˆ
⎪⎩
⎪⎨⎧ +==+
=otherwised
mTinjda
ij
ijij ,
1,,,1,1ˆ
⎪⎩
⎪⎨⎧ +==+−
=otherwised
mTinjda
ij
ijij ,
2,,1,1ˆ
⎪⎩
⎪⎨⎧ +++==++−+=+
= −−−
otherwisedmmTmTiandmKTKnKTnjdp
aij
ijKTnjij ,
2,3,,1,,1)1(,ˆ )1(
The objective function takes values between 1z and 2z while technological
coefficients vary between ija and ijij da + . Let ),min( 21 zzz =l and ),max( 21 zzz =u .
Then lz and uz are called the lower and upper bounds of the optimal values,
respectively.
Assumption 5.2. The linear crisp problems (5.40)-( 5..42) and (5.43)-(5.45) have finite
optimal values.
In this case the fuzzy set of optimal values, G, which is subset of mTnR + , is defined as
Klir and Yuan [84 ]
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
≤≤−−
<
=
∑
∑∑
∑
=
==
=
n
jjj
n
jjj
n
jjj
n
jjj
G
zXcif
zXczifzzzXc
zXcif
X
1
11
1
1
)/()(
0
)(
u
ullul
l
μ (5.46)
The fuzzy set of the ith constraint, iC , which is a subset of mTnR + , is defined by
(i) 1. For mKKmi )1(,1 ++= and )1(,0 −= TK
86
)(XiCμ =
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+−≥
+−<≤−+
−<
∑
∑ ∑∑ ∑
∑
=
= == =
=
n
jjijjii
n
j
n
jjijijijij
n
j
n
jjijjiji
n
jjjii
Xdrb
XdrbXrXdXrb
Xrb
1
1 11 1
1
)(,1
)(,/)(
,0
(5.47)
2. For mKKmi )1(,1 ++= and )1(,0 −= TK
=)(XiCμ
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+−≥
+−<≤−+
−<
∑
∑ ∑∑ ∑
∑
+=
+= +=+= =
+=
),(
1
),(
1
),(
1
),(
1 1
),(
1
,)1(,1
,)1(,/)(
,,0
Kin
njjiji
Kin
nj
Kin
njjijij
Kin
nj
n
jjijji
Kin
njji
Xdb
XdbXXdXb
Xb
(5.48)
where n(i,K)=n+T(i-Km-1)+K+1
(ii) For ,2,3 +++= mmTmTi and mK ,1=
)(XiCμ =
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+≥
+<≤−
<
∑
∑ ∑∑ ∑
∑
+
−+=−−−
+
−+=
+
−+=−−−−−−
+
−+=
+
−+=−−−
+
−+=−−−
TKn
KTnjjijKTnji
TKn
KTnj
TKn
KTnjjijKTnjijKTnj
TKn
KTnj
TKn
KTnjjijjKTnji
TKn
KTnjjKTnji
Xdpb
XdpbXpXdXpb
Xpb
)1()1(
)1( )1()1()1(
)1( )1()1(
)1()1(
.)(,1
,)(,/)(
,,0
(5.49)
By using the definition of the fuzzy decisions proposed by Bellman and Zadeh
[9], we have
))((min),(min()( XXXjCjGD μμμ = .
i.e.
))((min),(min(max))((max00
XXXjCjGXDX
μμμ≥≥
=
87
Consequently, the problem (5.37)-(5.39) can be written as
λmax (5.50)
,)( λμ ≥XG (5.51)
2,1,)( ++=≥ mmTiXiC λμ , (5.52)
0≥jX , 10 ≤≤ λ , mTj ,1= . (5.53)
By using (5.46) and (5.47)-(5.53), we obtain the following theorem.
Theorem 5.1 The portfolio problem with fuzzy technological coefficient can be reduced
to the following problem
λmax (5.54)
0)( 21
21 ≤+−− ∑=
zXczzn
jjjλ , (5.55)
∑+
=
≤−mTn
jijij bXa
10)(ˆ λ , 2,1 ++= mmTi , (5.56)
0≥jX , 10 ≤≤ λ , mTnj += ,1 . (4.57)
where
⎪⎩
⎪⎨
⎧
+−+=−=++=+−−=++==+−
=otherwise,,
,1)1(,1)1(,0,)1(,1,1,)1(,0,)1(,1,,1,
)(ˆ
ij
ij
ijij
ij
diTnjandTKmKKmid
TKandmKKminjdra
λλλ
λ
⎪⎩
⎪⎨⎧ +==+
=,otherwise,
,1,,1,1)(ˆ
ij
ijij d
mTinjda
λλ
λ
⎪⎩
⎪⎨⎧ +==+−
=otherwise,,
,2,,1,1)(ˆ
ij
ijij d
mTinjda
λλ
λ
⎪⎩
⎪⎨⎧ +++=++−+=+
= −−−
. otherwise,2,3,,1)1(,
)(ˆ )1(
ij
ijKTnjij d
mmTmTiTKnKTnjdpa
λλ
λ
Notice that, the constraints in problem (5.54)-(5.57) containing the cross product term
jXλ are not convex. Therefore the solution of this problem requires the special approach
88
adopted for solving general nonconvex optimization problem (Rockafellar and Wets
[153] and White [209]).
5.5.2 Portfolio problems with fuzzy technological coefficients and fuzzy right-
hand-side numbers
We consider the linear programming problem (5.34)-(5.36) with fuzzy technological
coefficients and fuzzy right-hand-side numbers
)(max Xϕ =∑=
n
jjj Xc
1 (5.58)
subject to i
mTn
jjij bXa ~~
1≤∑
+
=
, 2,1 ++= mmTi , (5.59)
0≥jX , mTnj += ,1 . (5.60)
Assumption 5.3. ija~ and ib~ are fuzzy numbers for any i and j.
In this case we consider the following linear membership functions:
i) 1. For )1(,0,)1(,1 −=++= TKmKKmi and nj ,1= ,
⎪⎩
⎪⎨
⎧
+−≥+−<≤−−+−
−<=
+
,
)1(
0,/)(
,1)(
ijij
ijijijijijij
jk
a
drtifdrtrifdtdr
rtift
ijμ
2. For )1(,0,)1(,1 −=++= TKmKKmi and j=n+T(i-Km-1)+K+1,
⎪⎩
⎪⎨
⎧
+−≥+−<≤−−+−
−<=
,10,11/)1(
11)(
ij
ijijija
dtifdtifdtd
tift
ijμ
(ii) For mKmmTmTi ,1,2,3 =+++= and TKnKTnj +−+= ),1( ,
⎪⎩
⎪⎨
⎧
+≥+<≤−+
<=
,0,/)(
,1)(
ijij
ijijijijijij
ij
a
dptifdptpifdtdp
ptift
ijμ
and
89
⎪⎩
⎪⎨
⎧
+≥+<≤−+
<=
ii
iiiiii
i
b
pbtifpbtbifptpb
btift
i
0,/)(
,1)(μ
where Rt ∈ and 0>ijd for all 2,1,,1 ++== mmTinj . For defuzzification of this
problem (5.58)-(5.60), we first calculate the lower and upper bounds of the optimal
values. The optimal values lz and uz can be defined by solving the following standard
linear programming problems, for which we assume that all they have the finite optimal
values.
Now defuzzification of this problem (5.58)-(5.60). first we fuzzify the objective function.
This is done by calculating the lower and upper bound of the optimal values first. The
bounds of the optimal values lz and uz are obtained by solving the standard linear
programming problems
)(max1 Xz ϕ= (5.61)
subject to ijij
n
jij bXdr ≤+−∑
=
)(1
, 2,1 ++= mmTi (5.62)
0≥jX , nj ,1= , (5.63)
and
)(max2 Xz ϕ= (5.64)
subject to iij
n
jij pbXr +≤−∑
=1, (5.65)
0≥jX , nj ,1= , (5.66)
and
)(max3 Xz ϕ= (5.67)
subject to iij
n
jijij pbXdr +≤+−∑
=1)( , (5.68)
0≥jX , nj ,1= , (5.69)
and
90
)(max4 Xz ϕ= (5.70)
subject to: ij
n
jij bXr ≤−∑
=1, (5.71)
0≥jX , nj ,1= . (5.72)
Let ),,,min( 4321 zzzzz =l and ),,,max( 4321 zzzzzu = . The objective function takes
values between lz and uz while technological coefficients take values between ijr− and
ijij dr +− and the right-hand side numbers take values between ib and ii pb + .
Then, the fuzzy set of optimal values, G, which is a subset of mTnR + , is defined by
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
≤≤−−
<
=
∑
∑∑
∑
=
==
=
n
jjj
n
jjj
n
jjj
n
jjj
G
zXcif
zXczifzzzXc
zXcif
X
1
11
1
.,1
,,)/()(
,,0
)(
u
ullul
l
μ (5.73)
The fuzzy set of the ith constraint, iC , which is a subset of mTnR + , is defined by:
(i) 1. For mKKmi )1(,1 ++= and )1(,0 −= TK
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
++−≥
++−<≤−++
−<
=
∑
∑ ∑∑ ∑
∑
=
= == =
=
n
jijijiji
n
j
n
jijijijijij
n
j
n
jijijjiji
n
jjiji
C
pXdrb
pXdrbXrpXdXrb
Xrb
Xi
1
1 11 1
1
.)(,1
,)(,)(/()(
,,0
)(μ
(5.74)
2. For mKKmi )1(,1 ++= and )1(,0 −= TK
91
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
++−≥
++−<≤−++
−<
=
∑
∑ ∑∑ ∑
∑
+=
+= +=+= =
+=
),(
1
),(
1
),(
1
),((
1 1
),(
1
.)1(,1
,)1(,)(/()(
,,0
)(
Kin
njijiji
Kin
nj
Kin
njijijij
Kin
nj
n
jijijji
Kin
njji
C
pXdb
pXdbXpXdXb
Xb
Xi
μ
(5.75)
where 1)1(),( ++−−+= KKmiTnKin .
(ii) For 2,3 +++= mmTmTi and mK ,1=
)(XiCμ =
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
++≥
++<≤+−
<
∑
∑ ∑∑ ∑
∑
+
−+=
+
−+=
+
−+=
+
−+=
+
−+=
+
−+=
TKn
KTnjijijiji
TKn
KTnj
TKn
KTnjijijijijij
TKn
KTnj
TKn
KTnjijijjiji
TKn
KTnjjiji
pXdpb
pXdpbXppXdXpb
Xpb
)1(
)1( )1(,
)1( )1(
)1(
.)(,1
)(,)(/)(
,,0
(5.76)
By using the definition of the fuzzy decisions proposed by Bellman and Zadeh [9], we
have
)))((min),(min()( XXXjCjGD μμμ = .
In this case the an optimal fuzzy decision is a solution of the problem
))((min),(min(max))((max00
XXXjCjGXDX
μμμ≥≥
= ).
Consequently, the problem (5.58)-(5.60) can be written as to the following optimization
problem
λmax (5.77)
λμ ≥)(XG (5.78)
2,1,)( ++=≥ mmTiXiC λμ (5.79)
0≥X , 10 ≤≤ λ . (5.80)
92
By using the method of defuzzification as for the problem (5.50)-(5.53), we get the
following theorem.
Theorem 5.2 The problem (5.58)-(4.60) is reduced to one of the following crisp
problems :
λmax (5.81)
0)( 11
12 ≤+−− ∑=
zXczzn
jjjλ , (5.82)
∑=
≤−++−n
jiijijij bpXdr
10)( λλ , mKKmi )1(,1 ++= and )1(,0 −= TK (5.83)
0≥jX , nj ,1= , 10 ≤≤ λ ; (5.84)
λmax (5.85)
0)( 11
12 ≤+−− ∑=
zXczzn
jjjλ , (5.86)
∑+=
≤−++−),(
10)1(
Kin
njiijij bpXd λλ , mKKmi )1(,1 ++= and )1(,0 −= TK (5.87)
0≥jX , ),(,1 Kinnj += , 10 ≤≤ λ , (5.88)
where n(i,K)=n+T(i-Km-1)+K+1;
λmax (5.89)
0)( 11
12 ≤+−− ∑=
zXczzn
jjjλ , (5.90)
∑+
−+=
≤−++TKn
KTnjiijijij bpXdp
)1(0)( λ , 2,3 +++= mmTmTi and mK ,1= (5.91)
0≥jX , TKnKTnj +++= ),1( , 10 ≤≤ λ . (5.92)
Notice that, the problem given in this theorem are also nonconvex programming
problems, similar for the problem (5.77)-(5.80).
93
5.6 The modified subgradient method
In this section, we briefly present an algorithm of the modified subgradient method
suggested by Gasimov [57] which can be applied for solving a large class of nonconvex
and nonsmooth constrained optimization problems. This method is based on the
construction of dual problems by using sharp Lagrangian functions and has some
advantages Azimov and Gasimov[6], Gasimov [58], Rockafellar [150]. Some of them are
the following:
- The zero duality gap property is proved for suffciently large class of problems;
- The value of dual function strongly increases at each iteration;
- The method does not use any penalty parameters;
- The presented method has a natural stopping criterion.
Now, we give the general principles of the modified subgradient method. Let 0X be any
topological linear space, 0XS ⊂ be a certain subset of 00 ,YX be a real normed space
and *0Y be its dual. Consider the primal mathematical programming problem defined as
(P) 0)(
)(infinf
=
=∈
xgtosubject
xfPSx
where f is a real-valued function defined on S and g is a mapping of S into 0Y :
For every 0Xx∈ and 0Yy∈ let
⎩⎨⎧
∞+=∈
=Φ.,
)(),(),(
otherwiseyxgandSxifxf
yx (5.93)
We define the augmented Lagrange function associated with problem (P) in the following
form: (see Azimov and Gasimov [6] and Rockafellar and Wets [153),
),(),(inf),,( uyycyxcuxLYy
−+Φ=∈
for 0Xx∈ , *0Yu∈ and 0≥c . By using (5.93) we concretize the augmented Lagrangian
associated with (P):
))),((()()(),,( uxgxgcxfcuxL −+= , (5.94)
where 0Xx∈ , *0Yu∈ and 0≥c .
94
It is easy to show that,
),,(supinfinf*),(
cuxLPYcuSx
+×∈∈=
R
The dual function H is defined as
),,(inf),( cuxLcuHSx∈
= (5.95)
for *0Yu∈ and 0≥c . Then, a dual problem of (P) is given by
),(,sup*
0),(
* cuHPSupYcu +×∈
=R
.
Any element +×∈ R*
0),( Ycu with ),(* cuHPSup = is termed a solution of *P .
Proofs of the following three theorems can be found in Gasimov [58].
Theorem 5.3. Suppose in (P) that f and g are continuous, S is compact and a feasible
solution exists. Then *PSupInfP = and there exists a solution to (P). Furthermore, in
this case, the function H in ( *P ) is concave and finite everywhere on +×R*
0Y , so this
maximization problem is efficiently unconstrained.
Theorem 5.4. Let *supinf PP = and for some +×∈ R*
0),( Ycu ,
)),(()()(),,(inf uxgxgcxfcuxLSx
−+=∈
. (5.96)
Then x is a solution to (P) and ),( cu is a solution to ( *P ) if and only if
g(x) = 0. (5.97)
When the assumptions of the theorems, mentioned above, are satisfied, the maximization
of the dual function H by using the subgradient method will give us the optimal value of
the primal problem.
It will be convenient to introduce the following set :
)()()(minimizes),( SxoverxguxgcxfxxcuS ∈−+=
95
Theorem 5.5 Let S be a nonempty compact set in nR and let f and g be continuous so
that for any ),(,),( cuScu mkk+×∈ RR is not empty. If ),( cuSx ∈ , then
))(),(( xgxg− is a subgradient of H at ),( cu .
Now we are able to present the algorithm of the modified subgradient method.
Algorithm
Initialization Step. Choose a vector ),( 11 cu with 01 ≥c let k = 1, and go to main step.
Main Step.
Step 1. Given ),( kk cu . Solve the following subproblem :
( )
.)),(()()(min
Sxtosubjectuxgxgcxf kk
∈
−+
Let kx be any solution. If 0)( =kxg , then stop; ),( kk cu is a solution to dual problem
( *P ), kx is a solution to primal problem (P). Otherwise, go to Step 2.
Step 2. Let
)()(
)(1
1
kkkkk
kkkk
xgscc
xgsuu
ε++=
−=+
+
(5.76)
where ks and kε are positive scalar stepsizes, replace k by k + 1; and repeat Step 1.
One of the stepsize formulas which can be used is
3)(5)),((
k
kkkkk
xgcuHH
s−
=α
where kH is an approximation to the optimal dual value, 20 << kα and kk s<< ε0 .
The following theorem shows that in contrast with the subgradient methods developedfor
dual problems formulated by using ordinary Lagrangians, the new iterate improves the
cost for all values of the stepsizes ks and kε .
Theorem 5.6. Suppose that the pair +×∈ RR mkk cu ),( is not a solution to the dual
problem and ),( kkk cuSx ∈ . Then for a new iterate ),( 11 ++ kk cu calculated from (5.76)
for all positive scalar stepsizes ks and kε we have
96
211 )()2(),(),(0 k
kkkkkk xgscuHcuH ε+≤−< ++ .
5.7 Defuzzification and solution of defuzzificated problem
In this section, we present the modified subgradient method (Gasimov [57]) and
use it for solving the defuzzificated problems (5.55)-(5.57) for nonconvex constrained
problems and can be applied for solving a large class of such problems.
Notice that, the constraints in problem (5.55)-(5.57) is generally not convex. These
problems may be solved either by the fuzzy decisive set method, which is presented by
Sakawa and Yana [165], or by the linearization method of Kettani and Oral [83].
5.7.1 A modified subgradient method to fuzzy linear programming
For applying the subgradient method ( Gasimov [57]) to the problem (5.54)-(5.57),
we first formulate it with equality constraints by using slack variables 0y and iy ,
2,1 ++= mmTi . Then, we can be written as
λmax , (5.99)
0)(),,( 021
210 =++−−= ∑=
yzXczzyXgn
jjjλλ , (5.100)
=),,( λyXgi ∑+
=
=+−mTn
jiijij ybXa
10)(ˆ λ , 2,1 ++= mmTi , (5.101)
0≥jX , 0,0 ≥iyy , 10 ≤≤ λ , mTnj += ,1 , 2,1 ++= mmTi . (5.102)
where ),...,( 0 nyyy =
For this problem we define the set S as
10,0,0),,( ≤≤≥≥= λλ yXyXS .
Since )min(max λλ −−= and ),...,( 20 ++= mmTggg the augmented Lagrangian
associated with the problem (5.99)-(5.102) can be written in the form
97
.2
1)(ˆ021
)21(0
21
22
1)(ˆ
2
021
)21(),,(
1
1
∑++
=⎟⎠⎞
⎜⎝⎛ +−∑−⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+∑
=+−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∑++
=⎟⎠⎞
⎜⎝⎛ +−∑+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++∑=
−−+−=
+
=
+
=
mmT
iybXauyz
n
j jXjczz
mmT
iybXayz
n
j jXjczzccuXL
iij
mTn
jiji
iij
mTn
jij
λλμ
λλλ
The modified subgradient method may be applied to the problem (5.99)-(5.102) in the
following way:
Initialization Step. Choose a vector ),,...,,( 112
11
10 cuuu mmT ++ with 01 ≥c . Let 1=k , and
go to main step.
Main Step.
Step 1 . Given ),,...,,( 210kk
mmTkk cuuu ++ ; solve the following subproblem :
,2
1)(
1ˆ021
)21(0
21
22
1 1)(ˆ
2
021
)21(min
∑++
= ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+−∑
+
=−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+∑
=+−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∑++
= ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+−∑
+
=+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++∑=
−−+−
mmT
i iyibjXmTn
j ijaiuyzn
j jXjczzu
mmT
i iyibmTn
j jXijayzn
j jXjczzc
λλ
λλλ
.),,( SyX ∈λ
Let ),,( kkk yX λ be a solution. If 0),,( =kkk yXg λ , then stop; ),,...,,( 210kk
mmTkk cuuu ++
is a solution to dual problem, ),,( kkk yX λ is a solution to problem (5.54)-(5.57).
Otherwise, go to Step 2.
Step 2 . Let
⎟⎟⎠
⎞⎜⎜⎝
⎛++−−−= ∑
=
+02
1210
10 )( yzXczzhuu
n
jjj
kkk λ
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−= ∑
+
=
+mTn
jiijij
kki
ki ybXahuu
1
1 )(ˆ λ , 2,1 ++= mmTi
),,()(1 kkkkkkk yXghcc λε++=+
98
where kh and kε are positive scalar stepsizes and 0>> kkh ε , replace k by k + 1; and
repeat Step 1.
5.7.2 Fuzzy decisive set method
For a fixed value of λ , the problem (5.54)-(5.57) is a linear programming
problem. Thus obtaining the optimal solution *λ to the problem (5.54)-(5.57) is
equivalent to determining the maximum value of λ so that the feasible set is nonempty.
Bellow is presented the algorithm (Gasimov [57]), of this method for the problem (5.54)-
(5.57).
Algorithm
Step 1. Set λ = 1 and test whether a feasible set satisfying the constraints of the problem
(5.54)-(5.57) exists or not using phase one of the simplex method. If a feasible set exists,
set λ = 1: Otherwise, set 0=Lλ and 1=Rλ and go to the next step.
Step 2. For the value of 2/)( RL λλλ += ; update the value of Lλ and Rλ using the
bisection method as follows :
λλ =L if feasible set is nonempty for λ ;
λλ =R if feasible set is empty for λ .
Hence, for each λ , we test whether a feasible set of the problem (5.54)-(5.57)
exists or not using phase one of the Simplex method and determine the maximum value *λ satisfying the constraints of the problem (5.54)-(5.57).
Example 5.1.
Solve the optimization problem, see Gasimov [40]
,0,61~3~42~1~
32max
21
21
21
21
≥≤+
≤+
+
xxxx
xx
xx
(5.103)
which take fuzzy parameters as )2,3(3~),3,2(2~),1,1(1~ LLL === and )3,1(1~ L= as
used by Shaocheng [171]. That is,
99
⎥⎦
⎤⎢⎣
⎡=
1321
)( ija , ⎥⎦
⎤⎢⎣
⎡=
3231
)( ijd ⇒ ⎥⎦
⎤⎢⎣
⎡=+
4552
)( ijij da
For solving this problem we must solve the folowing two subproblems:
,0,6342
32max
21
21
21
211
≥≤+≤+
+=
xxxxxx
xxz
and
,0,645452
32max
21
21
21
212
≥≤+≤+
+=
xxxxxx
xxz
Optimal solutions of these subproblems are
2.16.18.6
2
1
1
===
xxz
and
,82.047.006.3
2
1
2
===
xxz
respectively. By using these optimal values, problem (5.103) can be reduced to the
following equivalent non-linear programming problem:
0,10
3236
32406.38.6
06.332max
21
21
21
21
21
21
≥≤≤
≥+−−
≥+−−
≥−
−+
xx
xxxx
xxxx
xx
λ
λ
λ
λ
λ
that is
100
0,10
6)31()23(4)32()1(
74.306.332max
21
21
21
21
≥≤≤
≤+++≤+++
+≥+
xx
xxxx
xx
λλλλλ
λλ
(5.104)
Let's solve problem (5.104) by using the fuzzy decisive set method.
For 1=λ , the problem can be written as
0,6454528.632
21
21
21
21
≥≤+≤+≥+
xxxxxx
xx
and since the feasible set is empty, by taking 0=Lλ and Rλ =1; the new value of
21)
210( =+=λ is tried.
For 21
=λ , the problem can be written as
,0,
6254
427
23
9294.432
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 0=Lλ and 21
=Rλ , the new value of
412/)
210( =+=λ is tried.
For 41
=λ , the problem can be written as
101
,0,
647
27
44
1145
9941.332
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 41
=Lλ and 21
=Rλ , the new value of
832/)
21
41( =+=λ is tried.
For 83
=λ , the problem can be written as
,0,
68
174
15
4825
811
4618.432
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is nonempty, by taking 83
=Lλ and 21
=Rλ , the new value
of 1672/)
21
83( =+=λ is tried.
For 4375.0167==λ , the problem can be written as
,0,
61637
831
41653
1623
6956.432
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is nonempty, by taking 83
=Lλ and 167
=Rλ , the new value
102
of 32132/)
167
83( =+=λ is tried.
For 40625.03213
==λ , the problem can be written as
,0,
63271
32122
432
1033245
5787.432
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 83
=Lλ and 3213
=Rλ , the new value of
64252/)
3213
83( =+=λ is tried.
For 390625.06425
==λ , the problem can be written as
,0,
664
13964242
464203
6489
5202.432
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 6425
=Lλ and 3213
=Rλ , the new value of
128512/)
3213
6425( =+=λ is tried.
For 398475.012851
==λ , the problem can be written as
103
,0,
6128281
128486
4128409
128179
5494.432
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 6425
=Lλ and 12851
=Rλ , the new value of
2561012/)
12851
6425( =+=λ is tried.
The following value of λ are obtained in the next thirteen iterations:
256/101=λ = 0.39453125
512/1203=λ = 0.396484325
1024/407=λ = 0.396972656
4096/1627=λ = 0.397216796
8192/3255=λ = 0.397338867
16384/6511=λ = 0.397399902
32768/13021=λ = 0.397369384
65536/26043=λ = 0.397384643
131072/52085=λ = 0.397377014
262144/104169=λ = 0.3973731
524288/208337=λ = 0.3973733
1048576/416675* =λ = 0.3973723
Consequently, we obtain the optimal value of λ at the twenty first iteration by using the
fuzzy decisive set method.
Now, let's solve the same problem by using the modified subgradient method. Before
solving the problem, we first formulate it in the form
)min(max λλ −−=
104
,0,,0,10
06)31()23(04)32()1(006.33274.3
210
21
221
121
021
≥≥≤≤
=+−+++=+−+++=++−−
pppxx
pxxpxx
pxx
λλλλλ
λ
where 10 , pp and 2p are slack variables. The augmented Lagrangian function for this
problem is
).6)31()23(()4)32()1(()06.33274.3(])6)31()23((
)4)32()1(()06.33274.3[(),,(
22121211
021021
2221
2121
2021
pxxupxxupxxupxx
pxxpxxccuxL
+−+++−+−+++−++−−−+−++++
+−++++++−−+−=
λλλλλλλ
λλλλ
Let the initial vector is ),,,( 112
11
10 cuuu = (0; 0; 0; 0) and let's solve the following
subproblem
.2.174.06.182.0
10)0,0,(min
2
1
≤≤≤≤
≤≤
xx
xLλ
The optimal solutions of subproblem are obtained as
.1),,(
2),,(8.4),,(
101
1113
1112
1111
2
1
−=
−=
=
===
λ
λ
λ
λ
pxg
pxgpxg
xx
Since 0),,( 111 ≠λpxg , we calculate the new values of Lagrangemultipliers
),,( 222
21
20 cuuu by using Step 2 of the modified subgradient method. The solutions of the
second iteration are obtained as
105
62223
62222
62221
*2
1
1031.2),,(108.3),,(
109),,(3973723.0
75147.01475877,1
−
−
−
×=
×−=
×=
=
==
λ
λ
λ
λ
pxgpxg
pxg
xx
Since )(xg is quite small, by Theorem 5.4 75147.0,1475877.1 *2
*1 == xx and =*λ
3973723.0 are optimal solutions to the problem (5.100). This means that, the vector
),( *2
*1 xx is a solution to the problem (5.99) which has the best membership grade *λ .
Note that, the optimal value of λ found at the second iteration of the modified
subgradient method is approximately equal to the optimal value of λ calculated at the
twenty first iteration of the fuzzy decisive set method.
Example 5.2.
Solve the optimization problem, see Gasimov [40]
,0,4~3~2~3~2~1~
max
21
21
21
21
≥≤+
≤+
+
xxxx
xx
xx
(5.105)
which take fuzzy parameters as
)2,3(3~),2,3(3~),2,2(2~),1,2(2~),1,1(1~ LbLLLL ====== and
)3,4(4~2 Lb == as used by Shaocheng [171]. That is,
⎥⎦
⎤⎢⎣
⎡=
3221
)( ija , ⎥⎦
⎤⎢⎣
⎡=
2211
)( ijd ⇒ ⎥⎦
⎤⎢⎣
⎡=+
5432
)( ijij da
⎥⎦
⎤⎢⎣
⎡=
43
)( ib , ⎥⎦
⎤⎢⎣
⎡=
32
)( ip ⇒ ⎥⎦
⎤⎢⎣
⎡=+
75
)( ii pb .
To solving this problem, first, we must solve the folowing two subproblems:
106
,0,454332
max
21
21
21
211
≥≤+≤++=
xxxxxx
xxz
and
,0,732
52max
21
21
21
212
≥≤+≤+
+=
xxxx
xxxxz
Optimal solutions of these subproblems are
011
2
1
1
===
xxz
and
,05.35.3
2
1
2
===
xxz
respectively. By using these optimal values, problem (5.105) can be reduced to the
following equivalent non-linear programming problem:
0,10
22324
2315.3
1max
21
21
21
21
21
21
≥≤≤
≥+−−
≥+−−
≥−−+
xx
xxxx
xxxx
xx
λ
λ
λ
λ
λ
that is
0,10
6)31()23(4)32()1(
74.306.332max
21
21
21
21
≥≤≤
≤+++≤+++
+≥+
xx
xxxx
xx
λλλλλ
λλ
(5.106)
Let's solve problem (5.106) by using the fuzzy decisive set method.
107
For 1=λ , the problem can be written as
0,154132
5.3
21
21
21
21
≥≤+≤+
≥+
xxxxxx
xx
and since the feasible set is empty, by taking 0=Lλ and Rλ =1; the new value of
212/)10( =+=λ is tried.
For 21
=λ , the problem can be written as
,0,2543
225
23
25.2
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 0=Lλ and 21
=Rλ , the new value of
412/)
210( =+=λ is tried.
For 41
=λ , the problem can be written as
,0,4
1327
25
25
49
45
625.1
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is empty, by taking 41
=Lλ and 41
=Rλ , the new value of
812/)
410( =+=λ is tried.
108
For 81
=λ , the problem can be written as
,0,829
413
49
822
817
89
3125.1
21
21
21
21
≥
≤+
≤+
≥+
xx
xx
xx
xx
and since the feasible set is nonempty, by taking 81
=Lλ and 41
=Rλ , the new value
of 1632/)
41
81( =+=λ is tried.
The following value of λ are obtained in the next thirteen iterations:
16/3=λ = 0.1875
32/5=λ = 0.15625
64/11=λ = 0.171875
128/23=λ = 0.1796875
256/47=λ = 0.18359375
512/93=λ = 0.181640625
1024/187=λ = 0.182617187
2048/375=λ = 0.183349609
4096/751=λ = 0.183166503
8192/1501=λ = 0.397377014
16384/3001=λ = 0.183166503
32768/6003=λ = 0.183197021
65536/12007=λ = 0.18321228
131072/24015=λ = 0.183219909
62144/48029=λ = 0.183216095
524288/96057=λ = 0.183214187
1048576/192115=λ = 0.183215141
2097152/383231=λ = 0.183215618
109
4194304/786463=λ = 0.183215856
8388608/1536927=λ = 0.183215975
16777216/3073853* =λ =0.183215916
Hence,, we obtain the optimal value of λ at the twenty fifth iteration by using the fuzzy
decisive set method.
By using the modified subgradient we solve the same problem method. Before solving
the problem, we first formulate it in the form
)min(max λλ −−=
,0,,0,10
04)23()22(03)2()1(
015.2
210
21
221
121
021
≥≥≤≤
=+−+++=+−+++
=++−−
pppxx
pxxpxx
pxx
λλλ
λλλ
where 10 , pp and 2p are slack variables. The augmented Lagrangian function for this
problem is
).4)23()22(()3)2()1(()15.2(])4)23()22((
)3)2()1(()15,2[(),,(
22121211
021021
2221
2121
2021
pxxupxxupxxupxx
pxxpxxccuxL
+−+++−+−+++−++−−−+−++++
+−++++++−−+−=
λλλλλλλ
λλλλ
Let the initial vector is ),,,( 112
11
10 cuuu = (0; 0; 0; 0) and let's solve the following
subproblem
.005.31
10)0,0,(min
2
1
≤≤≤≤≤≤
xx
xLλ
The optimal solutions of subproblem are obtained as
110
3),,(
1),,(5.2),,(
101
1113
1112
1111
2
1
=
=
=
===
λ
λ
λ
λ
pxg
pxgpxg
xx
Since 0),,( 111 ≠λpxg , we calculate the new values of Lagrangemultipliers
),,( 222
21
20 cuuu by using Step 2 of the modified subgradient method. The solutions of the
second iteration are obtained as
82223
82222
72221
*
82
1
1083.7),,(102.8),,(
1028.3),,(1832159.0
108.7
45804,1
−
−
−
−
×−=
×=
×=
=
×=
=
λ
λ
λ
λ
pxgpxg
pxg
x
x
Since )(xg is quite small, by Theorem 5.4, 0108.7,45804.1 8*2
*1 ≅×== xx and
=*λ 0.1832159 are optimal solutions to the problem (5.106). This means that, the vector
),( *2
*1 xx is a solution to the problem (5.105) which has the best membership grade *λ .
Note that, the optimal value of λ found at the second iteration of the modified
subgradient method is approximately equal to the optimal value of λ calculated at the
twenty first iteration of the fuzzy decisive set method.
5.8 Portfolio problem with fuzzy multi-objective
The Fuzzy Multiple Objective Decision Model (FMODM) studied by Lai and Hwang
[93] states that the effectiveness of a decision makers’ performance in a decision process
can be improved as a result of the high quality of analytic information supplied by a
computer. They propose an Interactive Fuzzy Multiple Objective Decision Model
(IFMODM) to solve a specific domain of Multiple Objective Decision Model (MODM).
111
In this section we consider this approach for (4.2)-(4.4) portfolio model. Thus we have
the following problem
))(),...,(( xfxfMax qi (5.107)
subject to
itjt
n
jji srxy ≤−∑
=1, Ttmi ,1,,1 == , (5.108)
);(21
iit
T
tt yYFsp ≤∑
=
, ,,1 mi = (5.109)
0≥its , Ttmi ,1,,1 == , (5.110)
0≥x , X∈x . (5.111)
where qkxcxfn
iiikk ,1,)(
1
==∑=
.
Let us now consider the case of a decision-maker who has a fuzzy goal such as “the
objective function )(xf k should be much greater than minkp ”. Further, let us assume that
the degree of satisfaction of the decision-maker with respect to achieving the objective
does not change beyond the level maxkp . Then the corresponding linear membership
function that characterises the fuzzy goal of the decision-maker is given by:
⎪⎪⎩
⎪⎪⎨
⎧
≤
<<−−
≤
=
.)(;1
,;)(,;0
)]([max
maxminminmax
min
minmin
kk
kkkkk
kk
kk
kk
pxf
pfppppxf
pf
xfμ (5.112)
Given the membership functions for the various objectives of the decision-maker, the
maximizing decision can be computed by solving the following optimization problem:
)]([minmaximize,1
xfkkqkμ
= (5.113)
subject to itjt
n
jji srxy ≤−∑
=1, mi ,1= , (5.114)
);(21
iit
T
tt yYFsp ≤∑
=
, mi ,1= (5.115)
112
0≥its , Ttmi ,1,,1 == (5.116)
0≥x , X∈x . (5.117)
By introducing the auxiliary variable λ , the above optimization problem can be reduced
to the following conventional linear programming problem :
λMaximize (5.118)
subject to qkxfkk ,1,)]([ =≥ λμ , (5.119)
itjt
n
jji srxy ≤−∑
=1, Ttmi ,1,,1 == , (5.120)
);(21
iit
T
tt yYFsp ≤∑
=
, mi ,1= (5.121)
0≥its , Ttmi ,1,,1 == (5.122)
0≥x , X∈x , R∈λ , 10 ≤≤ λ . (5.123)
Let us consider the case of a fund manager who has to choose a structured portfolio from
an investment universe of n assets with jl and jl , nj ,1= being the minimum and
maximum weight of the ith asset in the portfolio. In order to select the structured
portfolio, the fund manager may examine k potential market scenarios, and for each of
these scenarios the decision maker may wish to maximize the portfolio return. To achieve
the return objective the fund manager could formulate the following optimization
problem:
maximize ( )(),...,(1 xRxR q ) (5.124)
subject to itjt
n
jji srxy ≤−∑
=1, Ttmi ,1,,1 == , (5.125)
);(21
iit
T
tt yYFsp ≤∑
=
, ,,1 mi = (5.126)
11
=∑=
m
iix , (5.127)
jjj lxl ≤≤ , nj ,1= , (5.128)
113
0≥its , Ttmi ,1,,1 == , (5.129)
0≥x , X∈x . (5.130)
where qkxRxRn
jj
kj
k ,1,)(1
==∑=
In equation (5.76), kjr denotes the return from the jth asset for the kth market scenario at
the end of the investment period and )(xRk the portfolio return for the kth scenario.
Since the above optimization problem has multiple objective functions, one has to
compute a Pareto optimal solution for the problem (see Sakawa [164]). Also we can use
the model of Chapter 2 for instance, one could characterize the set of Pareto optimal
solutions using the weighted minimax method and select one solution from this set. The
set of Pareto optimal solutions to the above optimization problem is characterized by:
Maximize λ (5.131)
subject to λ≥)(xRw kk , pk ,1= , (5.132)
itjt
n
jji srxy ≤−∑
=1, Ttmi ,1,,1 == , (5.133)
);(21
iit
T
tt yYFsp ≤∑
=
, mi ,1= (5.134)
11
=∑=
n
jjx , (5.135)
jjj lxl ≤≤ , ,,1 nj = (5.136)
0≥its , Ttmi ,1,,1 == (5.137)
0≥x , X∈x , R∈λ . (5.138)
In above relation, λ is an auxiliary variable and qkwk ,1, = are any arbitrarily chosen
nonnegative weights. Given any suitable weighting vector, one can determine the Pareto
optimal solution. Here, we assume without loss of generality that ,0)( >xRk
jjj lxl ≤≤ , nj ,1= . If this is not the case, the objective functions can be rewritten as
qkCRxR kk ,1),()(ˆ == , (5.139)
114
where C is a suitable constant that ensures kxRk ∀> ,0)(ˆ . Incorporating this change in
equation (5.132), one can compute the Pareto optimal solution.
The optimization problem formulated above is a linear programming problem and can be
easily solved using standard algorithms. However, finding a satisfactory Pareto optimal
solution requires one to define the a priori probabilities of various scenarios that
incorporate the market views. In the face of uncertainty these a priori probabilities are not
computable, and hence it is difficult to compute a Pareto optimal solution that can be
characterised as being satisfactory. Moreover, the fund manager may like to structure the
portfolio such that the return targets are different for each market scenario, for instance
with those scenarios that he/she considers more likely to occur (although no experimental
evidence is available) being targeted to achieve greater return. Transforming such goals
into suitable weights qkwk ,1,0 => for the various scenarios is not obvious from the
fund manager’s perspective.
Let us now consider a fund manager structuring a portfolio based on p potential
market scenarios. For each such scenario, the fund manager may have a target range for
the expected return over the investment period. We will denote by minkp and max
kp the
minimum and maximum expected return for the jth market scenario. Note that it is quite
easy for the fund manager to provide information on the expected target range of return
for various scenarios rather than to define the a priori probabilities for different scenarios.
Using the linear membership function given in equation (5.112) it is possible to compute
the degree of satisfaction ))(( xRkkμ for any given portfolio x for the kth market
scenario. Given that the degree of satisfaction to the fund manager for the kth market
scenario is ))(( xRkkμ , the structured portfolio can be computed by solving the
following optimization problems, for qk ,1=
λMaximize (5.140)
subject to λμ ≥)(xRkk , (5.141)
itjt
n
jji srxy ≤−∑
=1, Ttmi ,1,,1 == , (5.142)
115
);(21
iit
n
jt yYFsp ≤∑
=
, mi ,1= (5.143)
11
=∑=
n
jjx , (5.144)
jjj lxl ≤≤ , ,,1 mi = (5.145)
0≥its , Ttmi ,1,,1 == (5.146)
0≥x , X∈x , R∈λ . (5.147)
It is easy to show that the solution to the above optimization problem (if one exists) will
be Pareto optimal, Sakawa [164]. It is again useful to remind that we can interpret the
membership function ))(( xRkkμ for the kth market scenario in (5.136)) as modelling the
fuzzy utility of the investor for the given scenario. In this case, the structured portfolio
computed by solving the above optimization problem maximizes the fuzzy utility of the
investor.
5.9. Multiobjective fractional programming problems under fuzziness
Fractional programming has attracted the attention of many researchers in the
past. The main reason for interest in fractional programming stems from the fact that
linear fractional objective functions occur frequently as measures of performance in a
variety of circumstances such as when satisfying objectives under uncertainty. In some
real world decision-making situations, when formulating fractional objectives, some or all
of the parameters of the optimization problem are described by fuzzy or stochastic
variables.
Saad [160 ] presented a solution procedure for solving linear fraction programs
having fuzzy parameters in the right-hand side of the constraints. These parameters have
been characteristized by fuzzy numbers and the concept of α -optimality has been
introduced. On the other hand, Bicriterion integer nonlinear fractional programs
(BINOLFP) involving fuzzy parameters in the objective functions have been studied by
Saad and Abdelkader [161] .
116
Moreover, a solution algorithm has been described to solve the (BINOLFP). Furthermore,
a solution algorithm has been proposed by Saad and Abd-Rabo [162] was based upon the
chance-constrained programming technique Seppala [168] along with the branch-and-
bound method Ammar (1988). Recently, Saad and Sharif developed a solution method to
solve integer linear fractional program with chance-constraints and having statistically
independent random parameters Dutta (1992). Pareto-optimality for multiobjective linear
fractional programming problems with fuzzy parameter has been discussed by Sakawa
and Yano [165]. Programming with linear fractional functions was introduced into the
literature by Charnes and Cooper [30]. Since we can use a fuzzy multiobjective fractional
portfolio models, in this section we give some recently results on fuzzy multiobjective
fractional programming problem.
5.9.1 Problem formulation and the solution concept
The problem to be considered in this paper is the following fuzzy multiobjective
fractional programming problem:
(FMOFP) ,)()(
,...,)(
()()(max
1
1
⎥⎥⎦
⎤
⎢⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡xgxf
xgxf
xgxf
p
p
subject to 0,~)~,( ≥≤∈=∈ xbAxRxbAXx n ,
where A is an )( nm× -matrix b~ is an m-vector of fuzzy parameters and we suppose that
they are given by fuzzy numbers, estimated from the information provided by the
decision maker. Moreover ),1(,0)( pixgi => for all x in the feasible region of problem
(FMOFP).
Definition 5.1 (Dubois and Prade [44]) It is apropiate to recall that a real fuzzy number
a~ is a continuous fuzzy subset from the real R whose membership function )(~ aaμ is
defined by:
(1) A continuous mapping function R to the closed interval [0,1],
(2) 0)(~ =aaμ for all ],( 1aa −∞∈ ,
(3) )(~ aaμ is strictly increasing on ],[ 21 aa ,
117
(4) 1)(~ =aaμ for all ],( 32 aaa∈ ,
(5) )(~ aaμ is strictly decreasing on ],( 43 aaa∈ ,
(6) 0)(~ =aaμ for all ),[ 4 ∞∈ aa ,
Figure 5.2 Illustrates the graph of possible shape of a membership function of a fuzzy
number a~ .
Here, the vector of fuzzy parameters b~ involved in problem (FMOFP) is a vector of
fuzzy numbers whose membership function is )(~ bbμ .
)(~ aaμ 1 0 1a 2a 3a 4a a
Fig. 5.2 Membership function of a fuzzy number a~ .
In what follows, we give the definition of the α -level set or α -cut of the fuzzy vector
]~,...,~[~1 mbbb = .
Definition 5.2 [44] The α -level set of the vector of fuzzy parameters b~ in problem
(FMOFP) is defined as the ordinary set )()~( ~ αμα ≥∈= bRbbL bm .
For a certain degree *αα = in [0,1], estimated by the decision maker, the (FMOFP) can
be understood as the following nonfuzzy α -multiobjective fractional programming
problem (α -MOFP):
(α -MOFP):
).~(,0,)~,(
,)()(
,...,)(
()()(max
1
1
bLbxbAXRxbAXxtosubject
xgxf
xgxf
xgxf
n
p
p
α∈≥≤∈=∈
⎥⎥⎦
⎤
⎢⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
It should be emphasized here in the (α -MOFP) above that the vector of parameters
b is treated as a vector of decision variables rather than constants.
Problem (α -MOFP) can be reformulated in the following form:
118
(P) ))(),...,(()(max 1 xFxFxF p= ,
subject to ),,( bAXx∈
where )()()()( * xgxxfxF iiii θ−= and ),1(,)()(,0 *
*** pi
xgxf
r
rri ==≥ θθ are fixed
parameters and for their specification, Singh and Hanson [174].
Based on Definition 2 of the α -level set of the vector of fuzzy numbers b~ , we introduce
the concept of α -efficient solution of problem (P) above as follows:
Definition 5.3 (Sakawa and Yano [165]) A point ),(* bAXx ∈ is said to be an α -
efficient solution of problem (P) if only if there exists no other )~(),,(* bLbbAXx α∈∈
such that ),1();()( * pixFxF ii =≤ with strictly inequality holding for at least one i,
where the corresponding values of parameters ),1(* mrbr = are called the α -level
optimal parameters.
Now, consider λ is a p-dimensional strictly positive fixed vector, then problem (P) can
be written again in a problem of scalar single-objective function )( λP in the following
form:
)( λP :
).,(
),(max1
bAXxtosubject
xFp
iii
∈
∑=
λ
Let ),( bAX denote the set of feasible solutions of problem (α -MOFP) or (P) or )( λP .
We assume that 0)( ≥xf , 0)( >xg , for all ),( bAXx∈ . We further assume that f,-g
are concave functions and ),( bAX is a convex set. It follows that F is concave (Singh
and Hanson [174]).
Problem )( λP can be solved at 1* == ii λλ with the corresponding fixed parameters
),1(,* piii ==θθ using any available nonlinear programming package, for example,
GINO (Lieberman, et al [102], to find the α -optimal solution *x together with the
optimal parameters ),1(* mrbr = .
119
It should be noted from Singh and Hanson [174] that *x is an α -efficient solution to
problem (α -MOFP) or problem (P) with the corresponding α -level optimal parameters
),1(* mrbr = if there exists 0* ≥λ such that *x solves problem )( λP and either one of
the following conditions holds:
i. 0* >= ii λλ for all ),1( pi = .
ii. *x is the unique maximizer of problem )( λP .
Definition 5.4 (Geofrion [60]) Consider the multiobjective programming problem
))(),...,(()(max 1 xxx kφφφ = ,
subject to nRSx ⊆∈ .
We say that Sx ∈0 is efficient if ond only if there exists no Sx∈ such that
)()( 0 xx φφ ≤ .
Definition 5.5 (Geofrion [60]) For the multiobjective programming problem in
Definition 4, we say that an efficient solution 0x is properly efficient if only if for each i
and Sx∈ , there exists a positive real number M and a j such that
0)()( 0 >− xx jj φφ and )))()((()()( 00 xxMxx jjii φφφφ −≤− ,
whenever 0)()( 0 >− xx ii φφ .
Before we go any further, the reader is reminded that for multiobjective linear
fractional programming, when the emphasis is on finding efficient solution, there is no
general method for finding all the efficient solutions but Choo and Atkins [34] have
developed an algorithm, using row parameters, for solving the bicriterion linear fractional
programming problem (BLFP). Choo [33] has also shown that if 0x is an efficient
solution to (BLFP) then 0x is properly efficient Geofrion [60].
The nonnegativity of *iθ is needed to establish part (b) of Theorem 5.7 bellow.
Theorem 5.7
(a) If *x is an α -optimal solution of )( λP , then *x is properly an α -efficient for )(P .
(b) If f and -g are concave and *x is properly an α -efficient for (P), then it is an α -
optimal for )( λP .
120
To prove Theorem 5.7 above, the reader is referred to Geofrion [60].
Theorem 5.8 The point ),(* bAXx ∈ is an α -efficient solution of (α -MOFP) if it is an
α -efficient of (P) with 0)( * =xF .
Proof Suppose ),(* bAXx ∈ is an α -efficient solution of (α -MOFP). Then by
Definition4, there is no ),(* bAXx ∈ such that
pixgxf
xgxf
i
i
i
i ,1,)()(
)()(
*
*
=∀≤ .
Letting )()(
*
**
xgxf
i
ii =θ for pi ,1= , we see from the above inequality that there does not
exists an ),( bAXx∈ such that
pixFxgxf iiii ,1),()()(0 * =∀=−≤ θ
Since pixFxgxf iiii ,1),()()(0 ** ==−= θ , we see that there exists no x in ),( bAX
such that pixFxF ii ,1),()( * =≤ . Therefore, *x is an α -efficient of (P) with
0)( * =xF .
Conversely, suppose that *x is an α -efficient solution of (P) with
)()(0)( **** xgxfxF θ−== . That means, by Definition 5.4, there exists no
),( bAXx∈ such that
pixgxfxFxF iii ,1),()()()(0 ** =∀−=≤= θ .
That is, there exists no ),( bAXx∈ such that
pixgxf
xgxf
i
ii
i
i ,1,)()(
)()( *
*
*
=∀≤= θ
Hence, *x is an α -efficient solution of (α -MOFP).
For the development that follows, we assume that there exists real numbers 0,0 >> Kk
such that Kxgk i << )( for all i. Applying Definition 5.5 of the proper efficiency to
121
problem (α -MOFP), we note that an α -efficient solution *x of problem (α -MOFP) is
properly α -efficient if there exists a real number 0>M such that for each i, we have
)](/)()(/)([)(/)()(/)( *** xgxfxgxfMxgxfxgxf jjjjiiii −≤−
for some j such that )(/)()(/)( ** xgxfxgxf jjjj < whenever ),( bAXx∈ and
)(/)()(/)( ** xgxfxgxf iiii > . Or, rewriting these inequalities slightly differently, we
say an α -efficient solution *x of problem (α -MOFP) is properly α -efficient if there
exists a real number 0>M such that for each i, we have
)(/)](/)()()([)(/)]()()()([ ****** xgxgxfxgxfMxgxgxfxgxf jjjjjiiiii −≤− ,
(5.148)
where kMKM /= for some j such that
)()(/)()( ** xfxgxgxf jjj < (5.149)
Whenever ),( bAXx∈ and
0)]()()()([ ** >− xgxfxgxf iiii . (5.150)
To link proper α -efficiently of problem (α -MOFP) and (P), we prove the following
theorem.
Theorem 5.9 The point ),(* bAXx ∈ is a properly α - efficiently solution of problem
(α -MOFP) if and only if it is a properly α - efficiently solution on (P) with 0)( * =xF .
Proof. Supose *x is a properly α - efficiently solution of problem (α -MOFP). Then by
Theorem 5.8, we know its an α - efficiently solution on (P) with 0)( * =xF . Now *x is
a properly α - efficiently solution of problem (P) if there exists a positive real number M
such that for each i,
))()(()()( ** xFxFMxFxF jjii −≤− . (5.151)
for some j such that
. 0)()( * <− xFxF jj (5.152)
whenever ),( bAXx∈ and
0)()( * >− xFxF ii . (5.153)
122
Or [in view of the fact that 0)( * =xF for all i and )()()( * xgxfxF iiii θ−= with
)(/)( *** xgxf iii =θ for pi ,1= ], the result holds if and only if there exists an 0>M
such for each i,
)(/)](/)()()([
)](/)]()()(/)([***
***
xgxgxfxgxfM
xgxgxfxgxf
jjjjj
iiiii
−
≤− (5.154)
for some j such that
0)()()()( ** <− xgxfxgxf jjjj (5.155)
whenever ),( bAXx∈ and
0)()()()( ** >− xgxfxgxf iiii (5.156)
Relation (5.154)-(5.156) hold by (5.148)-(5.150) with MM = . Conversely, suppose *x
is a properly α -efficient solution of (P) with 0)( * =xF . Then by Definition 5.5,
relation (5.151)-(5.153) hold for some M and eacj i and ),( bAXx∈ . From this it follows
that (5.154)-(5.156) hold which are (5.148)-(5.150) with MM = .
5.9.2. Solution algorithm
A solution algorithm to solve fuzzy multiobjective fractional programming problem
(FMOFP) is described in a series of steps. The suggested algorithm can be summarized in
the following manner : Saad [159]
Step 1. Start with an initial level set 0* ==αα .
Step 2. Determine point ),,,( 4321 bbbb for the vector of fuzzy parameters b~ in problem
(FMOFP) to elicit a membership function )(~ bbμ satisfying assumptions (5.148)-(5.153)
in Definition 5.1.
Step 3. Convert problem (FMOFP) into its nonfuzzy version )( MOFP−α .
Step 4. Rewrite problem )( MOFP−α in the form of problem )( λP of single-objective
function.
123
Step 5. Choose 0* >= ii λλ and 11
* =∑=
p
iiλ with fixed values of ),1(,* piii ==θθ and
use GINO software package Lieberman, et al [102] to find the α -optimal solution *x of
prolem )( λP .
Step 6. Set ]1,0[)( * ∈+= stepαα and go to step 1.
Step 7. Repeat again the above procedure until the interval [0,1] is fully exhausted. Then,
stop.
Example. (Saad [160]) In what follows we provide a numerical example to clarify the
solution algorithm suggested above.
Let
222211
211 2)(,2)(,21)(,1)( xxgxfxxgxxf −==+=−=
So
.2
2)()()(,
211
)()()(
22
222
1
21
1
11 xxg
xfxFx
xxgxfxF
−==
+−
==
Consider the followings fuzzy bicriterion practionl programming problem (FBFP)
)),(),(()(max 21 xFxFxF =
subject to .0,
~
21
22
21
≥≤+
xxbxx
where b~ is a fuzzy parameter and is characterized by the following fuzzy
numbers:
)5,3,1,0(~=b
Assume that the membership function of these fuzzy numbers in the following
form:
124
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
≥
≤≤⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−
≤≤
≤≤⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−
≤
=
4
43
2
34
3
32
21
2
21
2
1
~
,0
,1
,1
,1
,0
)(
bb
bbbbbbb
bbb
bbbbbbb
bb
bbμ
Let 19.0=α , for example, then we get:
8.41.0 ≤≤ b
Choosing 1=b , then non-fuzzy α -bicriterion fractional programming problem
(α -BFP)becomes:
)),(),(()(max 21 xFxFxF =
subject to .0,1
21
22
21
≥≤+
xxxx
observe that point )0,0(* =x is an α -efficient solution of problem (α -BFP) since,
for each feasible x and then we have
02121
211)()( 2
2
22
21
22
21*
11 ≤++
−=−+−
=−xxx
xxxFxF ,
and
02
12
2)()(2
2
2
*22 ≥
−=−
−=−
xx
xxFxF ,
and there is no other feasible point for which
)1,1())(),(()( 21 ≥= xFxFxF .
We now consider the case when 1,2 == ji in the definition of a properly efficient
solution and therefore it can be seen that )0,0(* =x is also properly α -efficient solution.
When
)()( *22 xFxF − we have 0
2 2
2 >− xx
; that is, 02 >x .
125
Then
0212)()( 2
2
22
21
1*
1 >++
=−xxxxFxF and 0)()( *
22 >− xFxF .
0)2)(2(
)21(22
212
222 >+−
+=
xxxxxM .
We have
))()(()()( 1*
1*
22 xFxFMxFxF −≤− .
So that point )0,0(* =x is properly α -efficient solution for problem (α -BFP) with the
corresponding α -level set equals 0.19.
5.9.3. Basic stability notions for problem (FMOFP)
Based on definition of the set of feasible parameters; the solvability set and the stability
set of the first kind (SSK1) of problem (FMOFP) via problem (α -MOFP).
Let
⎭⎬⎫
⎩⎨⎧
=∈= ∑∑=
∈=
p
iiibAXx
p
iii
n xFxFRxE1),(1
**** )(max)()( λλλ
be the set of α -optimal solutions of problem )( *λP .
Definition 5.6 The set of feasible parameter of problem (α -MOFP), which is denoted b,
which is denoted by U, is defined by:
φα ≠=∈∈= ),(and,1),~( bAXmrbLbbU rrmR .
Definition 5.7 The solvability set of problem (α -MOFP), which is denoted by V, is
defined by:
)(where,solutionefficientanhas)(Problem *** λαα ExxMOFPUbV ∈−−∈= .
Definition 5.8 (The stability set of the first kind). Supose that Vb ∈* with the
corresponding α -efficient solution *x of problem (α -MOFP) such that )( ** λEx ∈ ,
then the stability set of the first kind(SSK1) of problem (α -MOFP), which is denoted by
)( *xS , is defined by:
126
)( problem ofsolution efficientn-an is)( ** MOFPxVbxS −∈= αα .
5.9.4. Utilization of Kuhn-Tucker conditions corresponding to problem )( λP
Problem )( λP can be written in the followings form:
)( λP : ,)(max1∑=
p
iii xFλ
subject to
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=≥
=≤≤
=≤=∑=
.,1,0
,,1,
,,1,),(1
njx
mrHbh
mrbxabx
j
rrr
n
jrjrjrrψ
It is clear that the constraint )~(bLb α∈ in the problem (P) has been replaced by the
equivalent constraint mrHbh rrr ,1, =≤≤ in problem )( λP , where rh and rH , are
lower and upper bounds on rb , respectively.
Therefor, the Kuhn-Tucker necessary optimality conditions corresponding to the
maximization problem )( λP we have the following form:
∑ ∑ ∑∑
∑ ∑ ∑
= = ==
= = =
=+−+∂
∂
=+∂
∂−
∂∂
m
r
m
r
m
rrr
m
rr
j
rrr
p
i
m
r
n
jj
j
rrr
j
ii
xbx
xbx
xxF
1 1 11
1 1 1
,0),(
,0),()(
ηγξψξ
βψξλ
rrr bbx ≤),(ψ ,
,rr bh ≤
,rr Hb ≤
,0≥jx
,0]),([ =− rrrr bbxψξ
,0)( =− rrr bhγ
127
,0)( =− rrr Hbη
0=jj xβ
0≥rξ ,
,0≥jβ
,0≥rγ
,0≥rη
where ,...,1 mIr =∈ and ,...,1 nJj =∈ . In addition, all the expressions of Kuhn-
Tucker conditions are evaluated at the α -optimal solution *x of problem )( *λP .
Furthermore, rrjr ηγβξ ,,, are the Lagrange multipliers.
The first two together with the last four relations of the above system of the Kuhn-Tucker
conditions represent a Polytope in ξβγη -space for which its vertices can be determined
using any algorithm based upon the simplex method, for example, Balinski (1961).
According to whether any of the variables ,...,1,,,, mIrrrjr =∈ηγβξ and
,...,1, nJjj =∈β are zero positive, then the set of parameters mrbr ,1, = for which
the α -efficient solution *x for one vector of parameters Vb ∈* rests efficient for all
parameters Vb∈ .
128
CHAPTER 6
A POSSIBILISTIC APPROACH FOR A PORTFOLIO SELECTION PROBLEM
6.1. Introduction
A half century ago, H. Markowitz pioneered the modern finance theory by his
meanvariance portfolio selection model. Although his work is perhaps technically simple
in today’s view, his idea still inspires the work in finance.
Zhou and Li [218] explored Markowitz’s work in a complete continuous-time financial
market. Variance has been commonly taken as a measure of risk. Meanwhile there are a
lot of researches on how to measure the risk of an investment, such as semivariance
advised by Markowitz. In the mean-risk framework, only mean-variance was widely
accepted in the discrete-time market.
The portfolio selection model of Markowitz [112, 113] consists of two interrelated
modules:
- a nonlinear programming problem where risk-averse investors solve a utility
maximization problem involving the risk and the expected rate of return of any
portfolio, subject to the constraint of an efficiency frontier. The latter is defined
pointwise, as a sequence of solutions to a quadratic programming problem which
minimizes the risk associated with each possible portfolios expected rate of return
subject to the constraint that the elements of the portfolio be non-negative and sum to
unity, and
- a parametric stochastic returns-generating process by which, in each period, the
investors determine the requisite vector of expectations and the variance-covariance
matrix of the investors anticipated rates of return on all risky assets.
The managers are constantly faced with the dilemma of guessing the direction of
market moves in order to meet the return target for assets under management. Given the
uncertainty inherent in financial markets, the managers are very cautious in expressing
their market views. The information content in such cautious views can be best
129
described as being “fuzzy” or vague, in terms of both the direction and the size of
market moves. Nevertheless, such fuzzy views are the ones needed to structure
portfolios so that the target return, which is assumed to be higher than the risk-free
theory to select optimal portfolios that target returns above the risk-free rate by taking
only market risk.
6.2 A Mean VaR portfolio selection multi-objective model
with transaction costs
A value-at-risk (VaR) model measures market risk by determining how much the value of
a portfolio could decline over a given period of time with a given probability as a result of
changes in market prices or rates. The two most important components of VaR models are
the length of time over which market risk is to be measured and the confidence level at
which market risk is measured. The choice of these components by risk managers greatly
affects the nature of the value-at-risk model.
We begin by using the rates of return of the risky securities in the economic have a
multivariate normal distribution. In practice, this is a popular assumption when
computing a portfolio’s VaR ( see Hull and White [74]).
In this section, we will study a possibilistic mean VaR multi-objective model with
transaction costs.
6.2.1 Case of Mean downside-risk
In this section we extended Chen et al [31], Inuiguchi and Ramik [75] for n
assets. In practice investors are concerned about the risk that their portfolio value falls
below a certain level. That is the reason why different measures of downside-risk are
considered in the multi asset allocation problem. Denoted the random variable iν ,
qi ,1= the future portfolio value, i.e., the value of the portfolio by the end of the
planning period, then the probability
))(( ii VaRP <ν , qi ,1=
130
that iν the portfolio value falls below the iVaR)( level, is called the shortfall probability.
The conditional mean value of the portfolio given that the portfolio value has fallen
below (VaR)i , called the expected shortfall, is defined as
))(( iii VaRE <νν .
Other risk measures used in practice are the mean absolute deviation
)())(( iiii EEE νννν <− ,
and the semi-variance
))())((( 2iiii EEE νννν <− ,
where we consider only the negative deviations from the mean.
Let ),1( njx j = represents the proportion of the total amount of money devoted to
security j and jM1 and jM 2 represent the minimum and maximum proportion of the
total amount of money devoted to security j , respectively. For nj ,1= , qi ,1= let jir
be a random variable which is the rate of the i return of security j. Then we have
∑=
=n
jjjii xr
1ν .
Assume that an investor wants to allocate his/her wealth among n risky securities. If the
risk profile of the investor is determined in terms of (VaR)i, qi ,1= , a mean-VaR
efficient portfolio will be a solution of the following .
Multi-objective optimization problem
[ ])(,),(max 1 qRx
EEn
νν L∈
(6.1)
qiVaRtosubject iii ,1,)(Pr =≤≤ βν , (6.2)
∑=
=n
jjx
11, (6.3)
njMxM jjj ,1,21 =≤≤ . (6.4)
In this model, the investor is trying to maximize the future value of portfolio,
which requires the probability that the future value of his portfolio falls below (VaR)i not
to be greater than iβ , qi ,1= .
131
6.2.2. Case of the proportional transaction costs model
The introduction of transaction costs adds considerable complexity to the optimal
portfolio selection problem. The problem is simplified if one assumes that the transaction
costs are proportional to the amount of the risky asset traded, and there are no transaction
costs on trades in the riskless asset. Transaction cost is one of the main sources of concern
to managers (see Arnott and Wagner [1], Zhou and Li [218] are found that ignoring
transaction costs would result in efficient portfolio and some conclusion.
Assume the rate of transaction cost of security j ( nj ,1= ) and allocation of i, qi ,1=
assets is jic , thus the transaction cost of security j and allocation of i assets is jji xc . The
transaction cost of portfolio ),...,( 1 nxxx = is qixcn
jjji ,1,
1=∑
=
. Considering the
proportional transaction cost and the shortfall probability constraint, we purpose the
following mean VaR portfolio selection model with transaction costs:
⎥⎦
⎤⎢⎣
⎡−− ∑∑
==∈
n
jjjkk
n
jjj
RxxcvExcvEMax
n11
11 )(....,,)( (6.5)
iii VaRvtosubject β≤< )(Pr , qi ,1= , (6.6)
∑=
=n
jjx
11, (6.7)
njMxM jjj ,1,21 =≤≤ . (6.8)
6.3 Possibilistic mean Var portfolio selection model.
In this section we introduce the concepts of possibilistic mean VaR portfolio selection
model and with assume that the rates of return on securities are modeled by possibility
distributions rather than probability distributions. Applying possibilistic distribution may
have two advantages (Hull and White [74]): (1) the knowledge of the expert can be easily
introduced to the estimation of the return rates and (2) the reduced problem is more
tractable than that of the stochastic programming approach. Possibility theory may be
132
quantitative or qualitative (Dubois and Prade, [44]) according to the range of these
measures which may be the real interval [0, 1], or a finite linearly ordered scale as well.
6.3.1 Possibilistic theory. Some preliminari
We consider the possibilistic theory proposed by Zadeh [216]. Let a~ and b~ be
two fuzzy numbers with membership functions a~μ and b~μ respectively. The possibility
operator (Pos) is defined as follows (Dubois and Prade [44]).
⎪⎩
⎪⎨
⎧
∈==<∈=<≤∈=≤
.))(),(supmin()~~(,,))(),(supmin()~~(
,,))(),(supmin()~~(
~~
~~
~~
RR,R,
xxxbaPosyxyxyxbaPosyxyxyxbaPos
ba
ba
ba
μμμμμμ
(6.9)
In particular, when b~ is a crisp number b, we have
( )
⎪⎩
⎪⎨
⎧
==<∈=<≤∈=≤
).()~(,,)(sup)~(
,,)(sup~
~
~
~
bbaPosbxxxbaPosbxxxbaPos
a
a
a
μμμ
RR
(6.10)
Let RRR →×:f be a binary operation over real numbers. Then it can be
extended to the operation over the set of fuzzy numbers. If we denoted for the fuzzy
numbers ba ~,~ the numbers )~,~(~ bafc = , then the membership function c~μ is obtained
from the membership function a~μ and b~μ by
),(,,))(),(supmin()( ~~~ yxfzyxyxz bac =∈= Rμμμ (6.11)
for R∈z . That is, the possibility that the fuzzy number )~,~(~ bafc = achives value
R∈z is as great as the most possibility combination of real numbers x,y such that z =
f(x,y), where the value of a~ and b~ are x and y respectively.
133
6.3.2 Triangular and trapezoidal fuzzy numbers
Let the rate of return on security given by a trapezoidal fuzzy number ),,,(~4321 rrrrr =
where 4321 rrrr <≤< . Then the membership function of the fuzzy number r~ can be
denoted by:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≤≤−−
≤≤
≤≤−−
=
.,0
,
,,1
,,
)(,43
43
4
32
2112
1
otherwise
rxrrrrx
rxr
rxrrrrx
xμ (6.12)
We mention that trapezoidal fuzzy number is triangular fuzzy number if 32 rr = .
)(~ xbμ )(~ xrμ 1 δ 0 b1 b2 r1 b3 xδ r2 r3 b4 r4
Figura 6.1: Two trapezoidal fuzzy number r~ and b~ .
Let us consider two trapezoidal fuzzy numbers ),,,(~
4321 rrrrr = and =b~
),,,( 4321 bbbb , as shown in Figure 6.1.
If 32 br ≤ , then we have
( ) yxyxbrPos br ≤=≤ )(),(minsup~~~~ μμ
,11,1min)(),(min 3~2~ ==≥ br br μμ
which implies that 1)~~( =≤ brPos . If 32 br ≥ and 41 br ≤ then the supremum is
achieved at point of intersection xδ of the two membership function )(~ xrμ and )(~ xbμ .
A simple computation shows that
134
( ))()(
~~1234
14
rrbbrbbrPos
−+−−
==≤ δ
and
δδ )( 121 rrrx −+= .
If 41 br > , then for any yx < , at least one of the equalities
0)(,0)( ~~ == yx br μμ
hold. Thus we have ( ) 0~~ =≤ brPos . Now we summarize the above results as
( )⎪⎩
⎪⎨
⎧
≥≤≥
≤=≤
.,0,,,
,,1~~
41
4132
32
brbrbr
brbrPos δ (6.13)
Especially, when b~ is the crisp number 0, then we have
( )⎪⎩
⎪⎨
⎧
≥≤≤
≤=≤
0,00,
0,10~
1
21
2
rrr
rrPos δ (6.14)
where
21
1
rrr−
=δ . (6.15)
We now turn our attention the following lemma.
Lemma 6.1 (Dobois and Prade [42]) Let ( )4321 ,,,~ rrrrr = be a trapezoidal fuzzy
number. Then for any given confidence level α with ( ) αα ≥≤≤≤ 0~,10 rPos if and
only if 1)1( rα− + 02 ≤rα .
The λ level set of a fuzzy number ( )4321 ,,,~ rrrrr = is a crisp subset of R and denoted
by ,)(]~[ Rxxxr ∈≥= λμλ , then according to Carlsson et al [26], we have
)](),([,)(]~[ 344121 rrrrrrRxxxr −−−+=∈≥= λλλμλ .
Given )](),([]~[ 21 λλλ aar = , the crisp possibilistic mean value of ( )4321 ,,,~ rrrrr =
is
∫ +=1
0 21 ))()(()~(~ λλλλ daarE .
135
where E~ denotes fuzzy mean operator.
We can see that if ( )4321 ,,,~ rrrrr = is a trapezoidal fuzzy number then
63
))()(()~(~ 41321
0 344121rrrrdrrrrrrrE +
++
=−−+−+= ∫ λλλλ . (6.16)
6.3.3 Construction efficient portfolios
Let jx the proportional of the total amount of money devoted to security j,
jM1 and jM 2 represent the minimum and maximum proportion respectively of the total
amount of money devoted to security j . The trapezoidal fuzzy number of jir is
( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where 4)(3)(2)(1)( jijijiji rrrr <≤< . In addition, we denote the
(VaR)i level by the fuzzy number trapezoidal ( )4321 ,,,~iiiii bbbbb = , qi ,1= .
Using this approach we see that the model given by (6.5)-(6.8) reduces itself to the form
from the following theorem.
Theorem 6.1 The possibilistic mean VaR portfolio selection for the vector mean VaR
efficient portfolio model (6.5)-(6.8) is
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛∑∑∑∑====∈
n
jjjq
n
jjjq
n
jjj
n
jjj
RxxcxrExcxrE
n111
11
1~~,...,~~max (6.17)
i
n
jijji bxrPostosubject β≤⎟⎟⎠
⎞⎜⎜⎝
⎛<∑
=1
~~ , qi ,1= , (6.18)
∑=
=n
jjx
11, (6.19)
njMxM jjj ,1,21 =≤≤ . (6.20)
In the following using Section 6.3 we obtain the efficient portfolios given by the Theorem
6.1.
Theorem 6.2. If qii ,1,0 =>λ , then an efficient portfolio for possibilistic model is an
optimal solution of the following problem:
136
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛∑∑∑===∈
n
jjji
n
jjji
q
ii
RxxcxrE
n111
~~max λ (6.21)
i
n
jijji bxrPostosubject β≤⎟⎟⎠
⎞⎜⎜⎝
⎛<∑
=1
~~ , qi ,1= , (6.22)
∑=
=n
jjx
11, (6.23)
njMxM jjj ,1,21 =≤≤ . (6.24)
Using the fact that rate of return on security ),1( njj = is given by trapezoidal fuzzy
number, then we get the following results.
Theorem 6.3 Let rate of return on security ),1( njj = by the trapezoidal fuzzy number
( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where 4)(3)(2)(1)( jijijiji rrrr <≤< and addition
( )4321 ,,,~iiiii bbbbb = is trapezoidal fuzzy number for VaR level and 0>iλ ,with qi ,1= .
Then using the possibilistic mean VaR portfolio selection model an efficient portfolio is
an optimal solution for the following problem:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−+
++
∑∑∑∑∑
∑=
====
=∈
n
jjji
n
jjji
n
jjji
n
jjji
n
jjjiq
ii
Rxxc
xrxrxrxr
n1
14)(
11)(
13)(
12)(
1 63max λ (6.25)
:. toubjects
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= , (6.26)
∑=
=n
jjx
11, (6.27)
njMxM jjj ,1,21 =≤≤ . (6.28)
Proof : Really, from the equation (6.16), we have
137
63
~~ 14)(
11)(
13)(
12)(
1
∑∑∑∑∑ ====
=
++
+=⎟⎟
⎠
⎞⎜⎜⎝
⎛
n
jjji
n
jjji
n
jjji
n
jjjin
jjji
xrxrxrxrxrE , pi ,1= .
From Lemma 6.1, we have that, for any qi ,1= ,
i
n
jijji bxrPos β≤⎟⎟⎠
⎞⎜⎜⎝
⎛<∑
=1
~~ , is equivalent with
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ .
Furthermore, from (6.25)-(6.28) given by Theorem 6.2, we get that this problem is of the
following form :
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−+
++
∑∑∑∑∑
∑=
====
=∈
n
jjji
n
jjji
n
jjji
n
jjji
n
jjjiq
ii
Rxxc
xrxrxrxr
n1
14)(
11)(
13)(
12)(
1 63max λ (6.29)
tosubject
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= (6.30)
∑=
=n
jjx
11, (6.31)
njMxM jjj ,1,21 =≤≤ . (6.32)
This completes the proof.
Problem (6.29)-(6.32) is a standard multi-objective linear programming problem. For
optimal solution we can used some algorithms of multiobjective programming (Slowinski
and Teghem [175] and White [209]).
138
6.4 A Weighted possibilistic mean value approach
In this section introducing a weighting function measuring the importance of λ -
level sets of fuzzy numbers we shall define the weighted lower possibilistic and upper
possibilistic mean values, crisp possibilistic mean value of fuzzy numbers, which is
consistent with the extension principle and with the well-known definitions of expectation
in probability theory. We shall also show that the weighted interval-valued possibilistic
mean is always a subset (moreover a proper subset excluding some special cases) of the
interval-valued probabilistic mean for any weighting function.
A trapezoidal fuzzy number ),,,(~4321 rrrrr = is a fuzzy set of the real line R with a
normal, fuzzy convex and continuous membership function of bounded support. The
family of fuzzy numbers will be denoted by F. A λ -level set of a fuzzy number
),,,(~4321 rrrrr = is defined by ,)(]~[ R∈≥= xxxr λμλ , then
)](),([,)(]~[ 414121 rrrrrrxxxr −−−+=∈≥= λλλμλ R ,
if 0>λ and 0)(]~[ ≥∈= xxclr μλ R (the closure of the support of r~ ) if 0=λ . It is
well-known that if r~ is a fuzzy number then λ]~[r is a compact subset of R for all
]1,0[∈λ .
Definition 6.1 (Majlender [110]) Let F∈r~ be fuzzy number with =λ]~[r
)],(),([ 21 λλ aa ]1,0[∈λ . A function ]1,0[:w → R is said to be a weighting function if
wis non-negative, monoton increasing and satisfies the following normalization
condition
∫ =1
01)( λλ dw . (6.33)
We define the w-weighted possibilistic mean (or expected) value of fuzzy number r~ as
∫+
=1
021 )(
2)()()~(~ λλλλ dwaarEw . (6.34)
It should be noted that if ]1,0[,2)( ∈= λλλw then
∫ +=1
0 21 .)]()([)~(~ λλλλ daarEw
139
That is the w-weighted possibilistic mean value defined by (6.24) can be
considered as a generalization of possibilistic mean value in Chen [32]. From the
definition of a weighting function it can be seen that )(λw might be zero for certain
(unimportant) λ -level sets of r~ . So by introducing different weighting functions we can
give different (case-dependent) important to λ -levels sets of fuzzy numbers.
Let ),,,(~21 βαrrr = be a fuzzy number of trapezoidal form and with peak ],[ 21 rr , left-
with 0>α and right-with 0>β and let
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−−=
−
1)1()12()( 21γλγλw ,
where 1≥γ . It’s clear that w is weighting function with 0)0( =w and
∞=−→
)(lim01
λλ
w .
Then the w-weighted lower and upper possibilistic mean values of r~ are computed by
( )[ ]∫ −−−−−= −− 1
0
2/11 11)12]()1([)~(~ λλγαλ γ drrEw
)14(2)12(
1 −−
−=γγαr ,
and
( )[ ]∫ −−−−−= −+ 1
0
2/11 11)12]()1([)~(~ λλγβλ γ drrEw
)14(2)12(
2 −−
+=γγβr
and therefore
⎥⎦
⎤⎢⎣
⎡−−
+−−
−=)14(2)12(,
)14(2)12()~(~
21 γγβ
γγα rrrEw
)14(4
))(12(2
)~(~ 21
−−−
++
=γ
αβγrrrEw . (6.35)
This observation along with Theorem 6.1 as in Section 6.3.3 leads to the
following theorem.
140
Theorem 6.4 The mean VaR efficient portfolio model is
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛∑∑∑===∈
n
jjji
n
jjjiw
q
ii
RxxcxrE
n111
~~max λ (6.36)
i
n
jijji bxrPostosubject β≤⎟⎟⎠
⎞⎜⎜⎝
⎛<∑
=1
~~ , qi ,1= , (6.37)
∑=
=n
jjx
11, (6.38)
njMxM jjj ,1,21 =≤≤ . (6.39)
In the next theorem we extend Theorem 6.3 to the case weighted possibility mean value
approach with a special weighted )(λw .
Theorem 6.5 Let ⎥⎥⎦
⎤
⎢⎢⎣
⎡−−−=
−
1)1()12()( 21γλγλw , 1≥γ the weighted possibility mean
of the trapezoidal fuzzy number ( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where ≤< 2)(1)( jiji rr
4)(3)( jiji rr < and addition ( )4321 ,,,~iiiii bbbbb = is a trapezoidal fuzzy number for (VaR)i
level, qi ,1= . Then the possibilistic mean VaR portfolio selection model is
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
++
∑∑∑∑∑
∑=
====
=∈
n
jjji
n
jjji
n
jjji
n
jjji
n
jjjiq
ii
Rxxc
xrxrxrxr
n1
13)(
14)(
12)(
11)(
1 )14(4
)12(
2max
γ
γλ (6.40)
tosubject
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= , (6.41)
∑=
=n
jjx
11, (6.42)
njMxM jjj ,1,21 =≤≤ . (6.43)
141
Proof : From the equation (6.16), we have
)14(4
)12(
2~~ 1
3)(1
4)(1
2)(1
1)(
1 −
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
++
=⎟⎟⎠
⎞⎜⎜⎝
⎛ ∑∑∑∑∑ ====
= γ
γn
jjji
n
jjji
n
jjji
n
jjjin
jjjiw
xrxrxrxrxrE , 1≥γ .
From Lemma 6.1, we have that
i
n
jijji bxrPos β≤⎟⎟⎠
⎞⎜⎜⎝
⎛<∑
=1
~~ , qi ,1= is equivalent with
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= .
Furthermore, from (6.40)-(6.43) given by Theorem 6.4, we get the following form :
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
++
∑∑∑∑∑
∑=
====
=∈
n
jjji
n
jjji
n
jjji
n
jjji
n
jjjiq
ii
Rxxc
xrxrxrxr
n1
11)(
14)(
12)(
11)(
1 )14(4
)12(
2max
γ
γλ (6.44)
tosubject
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= (6.45)
∑=
=n
jjx
11, (6.46)
njMxM jjj ,1,21 =≤≤ . (6.47)
This completes the proof.
Problem (6.44)-(6.47) is a standard multi-objective linear programming problem. Also we
can obtain an optimal solution by using some algorithms of multi-objective programming
(Kacprzyk and Yager [76] and Stanley and Li [177]).
For ∞→γ we see that =∞→
)~(~lim rEwγ 824321 rrrr −
++
. Thus we get
Corollary 6.1 For +∞→γ , the weighted possibilistic mean VaR efficient portfolio
selection model can be reduce to the following linear programming problem:
142
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
++
∑∑∑∑∑
∑=
====
=∈
n
jjji
n
jjji
n
jjji
n
jjji
n
jjjiq
ii
Rxxc
xrxrxrxr
n1
11)(
14)(
12)(
11)(
1 82max λ
subject to
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= ,
∑=
=n
jjx
11,
njMxMjj j ,1,21 =≤≤ .
6.5 A weighted possibilistic mean variance and covariance of
fuzzy numbers The classical mean-variance portfolio selection problem uses the variance as the
measure for risk, which puts the same weight on the down side and upside of the return.
In this section, we study the “weighted” possibilistic mean-variance and covariance
portfolio selection model.
Definition 6.2 (Fuller and Majlender, [53]) Let F∈r~ be a fuzzy number with
)](),([]~[ 21 λλλ rrr = , ]1,0[∈λ . The w -weighted possibilistic variance of r~ is
∫ ⎟⎠⎞
⎜⎝⎛ −
=1
0
212 )(
2)()()~( λλλλ dwrrrVarw
λλλλ
λλλλ dwrrrrrr )(
2)()()()(
2)()(
211
0
221
2
2
121∫ ⎟
⎟⎠
⎞⎜⎜⎝
⎛⎥⎦⎤
⎢⎣⎡ +
−+⎥⎦⎤
⎢⎣⎡ −
+=
where weighting function is non-decreasing and satisfies
∫ =1
01)( λλ dw . (6.48)
We note that the weighted possibilistic variance of r~ is defined as the expected value of
the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e.
the lower possibility-weighted average of the squared distance between the left-hand
endpoint and the arithmetic mean of the endpoints of its level sets plus the upper
143
possibility weighted average of the squared distance between the right-hand endpoint and
the arithmetic mean of the endpoints their of its level sets.
The standard deviation of r~ is defined by
)~(~ rVarr =σ (6.49)
Let r~ fuzzy number and w be a weighting function, we define the weighted possibilistic
variance of r~ by
∫ ⎟⎠⎞
⎜⎝⎛ −
=1
0
212 )(
2)()(
)~( λλλλ
dwrr
rVarw
and the weighted covariance of r~ and b~ is defined as
∫ ⎟⎠⎞
⎜⎝⎛ −−
=1
01212 )(
2)()(.
2)()(),~( λλ
λλλλ dwbbrrbrCovw .
If ]1,0[,2)( ∈= λλλw
∫ −=1
0 12 ))()((21)~( λλλλ drrrVarw , (6.50)
and
∫ −−=1
0 1212 2))()()(()((21),~( λλλλλλ dbbrrbrCovw . (6.51)
Let ),,,(~4321 rrrrr = and ),,,(~
4321 bbbbb = be fuzzy numbers of trapezoidal form.
Let
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−=
−
1)1()12()( 21γλγλw ,
where 1≥γ , be a weighting function then the power-weighted variance and covariance
r~ and b~ are computed by
λλλλ
γ γ drrrVarw ∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟
⎠⎞
⎜⎝⎛ −
−=−1
0
212
12 1)1(2
)()()12()~(
⎥⎦
⎤⎢⎣
⎡−
++−
+−+−
−−
=318
1)(28
1))((212
1)(4
)12( 2434312
212 γγγ
γ rrrrrrrr
144
⎥⎦
⎤⎢⎣
⎡−
++
−+−
+−
−−=
)16(3)(
)14(2))((2
12)(
4)12( 2
4343122
12
γγγγ rrrrrrrr
∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛−−
+−−+−−−=
−1
0
21
43124312 1)1(2
))(1)((.
2))(1)((
)12()~,~( λλλλ
γ γ dbbbbrrrr
brCovw
= ⎥⎦
⎤⎢⎣
⎡−++
+−
++−+
−−−−
)16(3))((
)14(2))()((
12))((
4)12( 43434343121212
γγγγ bbrrrrbbrrbbrr
Theorem 6.6 The mean-variance efficient portfolio model is
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛∑∑∑===∈
n
jjji
n
jjjiw
q
ii
RxxcxrE
n111
~~max λ (6.52)
tosubject i
n
jijji bxrPos β≤⎟⎟⎠
⎞⎜⎜⎝
⎛<∑
=1
~~ , qi ,1= , (6.53)
∑=
=n
jjx
11, (6.54)
njMxM jjj ,1,21 =≤≤ . (6.55)
In the next theorem we extend Theorem 6.3 to the case weighted possibility mean-
variance approach with a special weighted )(λw .
Theorem 6.7 Let ⎥⎥⎦
⎤
⎢⎢⎣
⎡−−−=
−
1)1()12()( 21γλγλw , 1≥γ the weighted possibility mean
variance of the trapezoidal number ( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where
4)(3)(2)(1)( jijijiji rrrr <≤< and addition ( )4321 ,,,~iiiii bbbbb = is a trapezoidal number for
(VaR)i level, qi ,1= . For qii ,1,0 =>λ , then the possibilistic mean variance portfolio
selection model is
145
⎢⎢⎢⎢⎢
⎣
⎡
−
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
+−
⎟⎟⎠
⎞⎜⎜⎝
⎛−− ∑∑∑∑ ∑
∑ ==== =
=∈ )14(4
)12(
)12(4
)12(max 1
4)(1
3)(1)(1
2)(
2
1 11)(2)(
1 γ
γ
γ
γλ
n
jjjij
n
jjijji
n
jjji
n
j
n
jjjijjiq
ii
Rx
xrxrxrxrxrxr
n
+
⎥⎥⎥⎥⎥
⎦
⎤
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
∑∑∑
=
==n
jjji
n
jjji
n
jjji
xcxrxr
1
2
14)(
13)(
)16(12
)12(
γ
γ (6.56)
tosubject
( ) 011
32)(1
41)( ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−− ∑∑
==
n
jijjii
n
jijjii bxrbxr ββ , qi ,1= , (6.57)
∑=
=n
jjx
1
1, (6.58)
njMxM jjj ,1,21 =≤≤ . (6.59)
Proof: The proof is the on the line of Theorem 6.5.
146
CHAPTER 7
ATZBERGER’S EXTENSION OF MARKOWITZ PORTFOLIO SELECTION
7.1 Introduction
Recently Atzberger [3] we represent one basic manner by which Markowitz’s theory for
portfolio selection can be extended to account for non-gaussian distributed returns. Thus
we discuss how a model incorporating information about the performance of the assets in
different market regimes over the holding period can be developed. This basic extension
follows the work of Buckley, Comezana, Djerrond, and Seco [20].
When attempting to apply Markowitz’s theory for portfolio selection in practice a
number of well known challenges arise. First, in the theory the mean returns μ and
covariance V must somehow be estimated. One approach is to use historic data, but this
has its obvious limitations in predicting the future performance of assets. In practice, the
historic data is often used in combination with securities analysis to construct forecasts
incorporating information about the fundamentals of the security and the opinions of
financial experts. Second, in the theory the variance is used as the primary indicator of
risk. However, the variance does not capture important information about the risk of an
investment if the returns are non-Gaussian with a multimodal or strongly skewed
distribution, see figures 7.1–7.3. Third, the covariance structure and realized returns for
assets may be drastically altered if the market conditions suddenly change, such as a
rally with investors having bullish expectations changing to a sell-out with bearish
expectations because of the announcement of important news, such as the results of an
election, oil production adjustments, or interest rate changes. In particular, the character
of the asset returns may change drastically depending on which important economic
conditions prevail over the holding period of the asset, leading to a break-down of the
assumptions usually justifying a Gaussian distribution. To better enable the key insights
147
of the Markowitz theory to be adapted in this case, we shall discuss a basic extension
with the following features.
- Returns will be modeled by Gaussian Mixture Distributions, discussed in more
detail below.
- We shall have a separate mean and covariance structure to model returns for
each of the market scenarios anticipated over the holding period (for example:
election results, oil production adjustments, interest rate changes) and
probabilities assigned for these different scenarios to occur.
- Since the variance may no longer be a good indication of the uncertainty (risk)
of the asset returns, we shall introduce a new quantification of risk. In particular,
we shall consider the probability that the returns fall short of a desired return.
As we mentioned above we seek to model the asset returns for different market
conditions which may prevail over the holding period of the portfolio. One useful
way to extend the model, while retaining much of the computational convenience of
the Markowitz Theory, is to describe the asset returns across the different market
scenarios using Gaussian Mixture Distributions, which are distributions derived from
linear combinations of Gaussian distributions. One basic motivation for this
approach, is that the returns of the assets for each market scenario may still retain
their Gaussian behavior, while averaging over the different scenarios leads to
multimodal or skewed distributions. The mixture coefficients then correspond to the
probability of each scenario. We now give the mathematical details of this extension.
Figure 7.1: Asset return having a Gaussian distribution, well characterized by the mean and variance.
148
As we mentioned above we seek to model the asset returns for different market
conditions which may prevail over the holding period of the portfolio. One useful way to
extend the model, while retaining much of the computational convenience of the
Markowitz Theory, is to describe the asset returns across the different market scenarios
using Gaussian Mixture Distributions, which are distributions derived from linear
combinations of Gaussian distributions. One basic motivation for this approach, is that
the returns of the assets for each market scenario may still retain their Gaussian behavior,
while averaging over the different scenarios leads to multimodal or skewed distributions.
The mixture coefficients then correspond to the probability of each scenario. We now
give the mathematical details of this extension.
Figure 7.2: Asset return having a non-Gaussian distribution, not well characterized by the mean and variance as consequence of skew.
7.2 Gaussian Mixture Distributions
Definition 7.3 [3] A scalar random variable Z has the univariate Gaussian Mixture
(GM) distribution if its probability density has the form:
∑ ∑= =
⎟⎟⎠
⎞⎜⎜⎝
⎛ −==
n
i
n
i i
iiXi
zzfzf
i1 1
)()(σμ
φαα
where ),(~ 2iiiX σμη are Gaussian distributed random variables with mean iμ and
2iσ variance. The density of the standard Gaussian is denoted by
149
2
2
21)(
x
ex−
=π
φ .
The additional condition 11
=∑ =
n
i iα is imposed to ensure the resulting density
describes a probability distribution. We now extend this definition to the vector-valued
case.
Definition 7.3: [3] A vector-valued random variable Z has the multivariate GM
distribution if its probability density is of the form:
)(,)()(1
)(
1)()( zvzfzf
n
i
in
iXiz ii ∑∑
==
==μ
φα
Figure 7.3: Asset return having a non-Gaussian distribution, not well characterized by the mean and variance as a consequence of biomodality.
where )(iX are multivariate Gaussian random variables with component means given by
the vector )(iμ and covariances given by the matrix )(..( )()()( ia
ia
i XEeiV =μ and
)),cov( )()()(,
jb
ia
iba XXV = . The multivariate Gaussian random variables with the specified
means and variances is given by the density:
)()()(21
21
)(2/,
)(1)()(
)()(
)()2(
1)(iiTi
ii
zVz
iV e
DetVz
μμ
πμ
πφ
−−− −
= ,
We remark that by definition we shall assume 0,cov( )()( =jb
ia XX We now show how
moments can be computed for these random variables.
150
Proposition 7.1 [3] Let Z be a vector-valued GM random variable with mean )(iμ ,
covariance )(iV , and mixture weights iα , then for any function g we have
∑=
=n
i
ii XgEZgE
1
)( ))(())(( α .
where )(iX are the Gaussian random variables defined above.
Proof: This follows immediately from the definition of the GM random variable with a
density which is a linear combination of the densities of )(iX . We leave the details as an
exercise .
A particular consequence of this proposition is that ∑ ==
n
ii
iZE1
)()( μα .
Proposition 7.2 [3] Let Z be as in the previous proposition, then the ath component Za =
[Z]a has a variance which can be expressed as
( )
2
1
)()(2)(
1
1
)()(
1
)(
)()(
)()(var()var(
∑∑∑
∑∑∑
= <=
= <=
−+=
−+=
n
i
ja
ia
ijji
ia
n
ii
n
i
ia
ia
ijji
n
i
iaia XEXEXZ
μμαασα
ααα
Proof:
( )
( )∑ ∫
∫ ∑
∑ ∑∫
=
=
= =
−+−=
−==
===
n
iXa
ia
iai
n
iXiaa
ia
ia
i
n
iiXiaa
dzzz
dzzzZ
dzzzZE
ia
ia
ia
1
2)()(
1
22)(
)(1
1 1
)(
)()()var(
)()(
)(
)(
)(
φμμμα
φαμσ
μαφαμ
( ) ( )
( ) ( )∑ ∫
∑ ∫∑ ∫
=
==
−−+
−+−=
n
iXa
ia
iai
n
iXa
iai
n
iX
iai
dzzz
dzzdzzz
ia
ia
ia
1
)(2)(
1
2)(
1
2)(
)(2
)()(
)(
)()(
φμμμα
φμμαφμα
( ) ( )∑∑==
−+=n
i
iaai
n
i
iai X
1
2)(
1
)(var μμαα .
Now using ∑ ==
n
ii
aia 1)(μαμ and )(2)(22)( 2)()( i
aai
aai
aa μμμμμμ ++=− , we have
151
( ) ( )
( ) ( )( )
( )∑∑
∑
∑ ∑ ∑∑
= <
=
= = = =
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
−=−
n
i
ia
ia
ijji
n
ji
ja
ia
ja
iaji
n
i
n
i
ja
n
i
n
j
iaji
iai
iaai
1
2)()(
1,
)()(2)(2)(
1 1
)(
1 1
)(2)(2)(
221
μμαα
μμμμαα
μμααμαμμα
and substituting this above proves the proposition .
Proposition 7.3 [3] Let Z be defined as above then the covariance between the two
components of a and b is given by
( )( )∑∑∑− <=
−−+=n
i
jb
ib
ja
ia
ijji
iba
n
iiba VZZ
1
)()()()()(,
1),cov( μμμμααα
Proof: The proof is similar to the last proposition.
Remark 7.1: Let V denote the covariance matrix of Z then )),cov(( , baba ZZV =
( )( )Tn
i
jiji
ijji
in
iiVV ∑∑∑
− <=
−−+=1
)()()()()(
1μμμμααα .
7.3 An Extension of the Markowitz Portfolio Theory
We now discuss some results for the linear combinations of GM distributed random
variables, which will be useful in describing the return of a portfolio of assets.
Proposition 7.4 [3] Let Z be a vector-valued GM distributed random variable with m
components and let
∑=
=m
aaaZwY
1,
then
∑=
=n
iiY yyf
ii1
,)()( 2σμ
φα .
In the notation, ∑ ==
m
ai
aai w1
)(μμ and ∑ ∑= ==
m
aibab
m
b ai Vww1
)(,1
σ , where φ is the
density of a Gaussian as defined above.
Proof: The proof is similar to the previous propositions by making use of the fact that
the density is a linear combination of Gaussian densities.
152
From these propositions we see that the statistics of the GM distribution random
variables can be computed as readily as for Gaussian random variables, yet we can
construct a richer set of distributions. In principle, any distribution can be approximated
by a Gaussian mixture model provided a sufficient number are Gaussians are used.
As we discussed in the introduction, the variance may be a poor indicator of the risk of
an asset return. Instead, we shall consider a different approach to characterizing the risk
of a portfolio. An alternative criteria which can be used to model risk is to consider the
probability that a portfolio falls short of an intended (desired) return Bμ . We shall refer
to this measure of risk as the Probability of Shortfall (PoS). We remark that this is
similar in spirit, but distinct, from other measures used to evaluate risk, such as Value at
Risk (VaR).
Definition 7.4 [3] Probability of Short-fall (PoS) is defined for a desired return Bμ by
Pr BYPoS μ≤=
where Y is the portfolio return defined for a GM model of the returns Z given by
∑=
=m
aaaZwY
1
From the propositions above we have the following.
Proposition 7.5 [3] For asset returns modeled by a GM distribution the Probability of
Shortfall (PoS) is given by
∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −Φ=
n
iiT
iTB
iBwVw
wwF1
)(
)(
),( μμαμ
Proof: Follows from similar calculations as the proofs above .
7.4 Portfolio Selection Problem (GM-PoS)
Using the PoS to quantify the risk associated with a portfolio and using the GM model
for the asset returns, the portfolio selection problem for obtaining a target return Pμ but
not falling below the return Bμ is:
∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −Φ
n
iiT
iTB
iwVw
w1
)(
)(
min μμα
153
tosubject
PT
m
i
T
w
w
μμ ≥
=∑=
11
where ∑==
n
ii
i1)(μαμ and dyex
xy
∫ ∞−
−=Φ 2
2
21)(π
.
The objective function to be minimized is no longer quadratic as in Markowitz’s Theory
so in general a numerical optimization will have to be performed. Similarly, the lack of
analytic expressions for the solution requires that the efficient frontier be analyzed
numerically.
154
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NOTATIONS
0\+R : positive real number sets +R : the set of nonnegative real numbers −R the set of nonpositive real numbers m+R the set of m-dimension real vector with
nonnegative components m−R the set of m-dimension real vector with
nonpositive components ))(,,,(1 ⋅θμλqAP : auxilitiy optimization problem 1
11 ),,,( EqM ⊆θμλ : set of optimal solutions nD R⊆ : a compact set of feasible actions
FSDf : First-order stochastic dominance
SSDf : second-order stochastic dominance (Ω,F, P) : probability space Z : A decision vector in an appropriate space Z
][XE : mean X
n,1 : 1,2,…,n
nRR ,...,1 : random returns of assets n,1 )(xμ : mean return
[ ])()( xRarx V=ρ : the variance of the return
))()((max xxXx
λρμ −∈
: the mean–risk portfolio optimization problem
),( kt
kt uD θ : a function of the slope vector
+− )( vη : )0,max( v−η
ija : technology coefficient
B : right-hand side vector of the constraint
ija~ : fuzzy number
)(XGμ : the fuzzy set of optimal values, G
iC : the fuzzy set of the ith constrain
),,( pXg λ : augmented Lagrangian
),,,(~4321 rrrrr = : Trapezoidal
172
minkp : the minimum expected return for the kth
market scenario maxkp : the maximum expected return for the kth
market scenario il : the minimum weight of the ith asset in the
portfolio il : the maximum weight of the ith asset in the
portfolio; r~σ : the standard deviation of r~
)~(rVarw : weighted possibilistic variance of r~ u : υuu ,...,1
)ˆ,...,ˆ(ˆ 1)( υυ uuu = : )ˆ,...,ˆ( 1 υuu ),,,(~
4321 rrrrr = : E
F : the family of fuzzy numbers E : set of efficient solutions
wE : sets of weakly efficient solutions GE : Set Geoffrion/proper efficient solutions
PKE : the set of efficient solutions
173
ACRONYMS & ABBREVIATIONS
ALM : assets liability management a.s : almost surely BFP bicriterion fractional programming BINOLFP bicriterion integer nonlinear fraction programs BSDE backward stochastic differential equation Covr : covariance dom : domain F : he family of fuzzy numbers FLP : fuzzy linear programming FMODM : fuzzy multiple objective decision model FMOFP Fuzzy multiobjective fractional programming IFLP : interactive fuzzy linear programming IFMODM : interactive fuzzy multiple objective decision
model LQ linear quadratic MOFP multiobjective fractional programming MODM : multiple objective decision model ODE ordinary differential equation Pos : possibilistic; Pr : probability resp : respectively SRE stochastic riccati equation SSK1 the stability set of the first kind VaR : value at risk Var : variance
174
INDEX
Arnott,131 Wagner, 131
auxiliary, 27 optimization, 27 algorithm, 32, 34, 78, 95, 137, 141 asset return, 147 Bellman-Zaded, 3 BINOLFP, 151 concave, 28,32,36
convex, 36 Nondecreasing 40 continuous, 41, 54, 60, 94 functional, 81 function, 38
concavity, 28,32 convex, 28
function, 17,,18,19, 25,34 convexity, 32, 36 concavity, 26 cone 70 nonconvex, 87, 93 programming, 92 combination, 83 polyhedral 73 programming, 36, 93 set 17, 32,57
continuous, 116 fuzzy, 116 mapping, 116
classes, 9, 21
nonconvex, 9,28 auxiliary, 32 multiobjective, 9 solution, 21 covariance, 97, 98, 142 continuous,41, 44 cumulative, 42 functional, 41, 54 variable, 10 structur, 146 matrix, 143 defuzzification, 84, 95 Dentcheva, 3, 4, 41, 42, 66
Ruszcynski, 3, 4, 41, 42,54 66,79
dual, 45 Risk, 45 functions, 93, 94 problems, 60, 92, 94 ,110
solution, 80 duality, 56, relation, 60 function, 60 gap, 93 distributed, 146 return, 146 efficient 21
portfolio, 131,135 solution, 10, 21 sets, 10, 20 SSD/FSD, 26
175
stochastic,27 expected value, 4, 10, 11
efficient, 10 standard deviation, 10, 11 minimum, 13
fuzzy fuzzy numbers, 60, 88, 123, 140, , 134, 86, 88, 91 number trapezoidal,135 approach, 5, 68, 81 fuzzy decision, 2, 5,103 constraints, 70 efficient, 76 compromise, 76 bicriterion, 123 emviroment, 3,77 decision, 5, 91 decesive, 98 function, 82 geometry, 77 goal, 111 linear, 5, 69, 70 mean-operator, 135 parameter, 98, 115, 122, 123 multiobjective, 110, 122 fractional, 116 objective, 72, 75 Resources, 70, 71 system, 77 utility, 2, 114 technological coefficient, 88 right-hand, 88 convex, 138 Fuzzy mean operator, 135 multicriteria, 3 random, 5
nonfuzzy,117, 122 Geoffrion, 13, 119 Gaussian, 119, distribution, 146 mixture distribution, 147 investment, 7, 112, 113, 128 investor, 80, 115 interactive approach, 77 interactive fuzzy, 76, 110, linear, 78 multiobjective, 110 Karush-Kuhn-Tucker, 58 multipliers, 58 linear programming, 76 linear fractional, 115 nonlinear fractional, 115 nonlinear programming, 115 nonconvex, 28 auxiliary 32, 108 programming, 92 markowitz, 4, 60, 83
model, 4 mathieu-Nicot, 2 mean-variance, 142 portfolio, 142 return, 55, 113, 128
assets, 44 objective, 112 total return, 3, 44 security, 130, 136 rate, 131 target, 114, 128 scurity, 136
pareto-optimal, 9 portfolio retun, 112, 113 possibilistic, 7, 8 , 131
constraint, 8 distribution, 131 theory, 132
176
model, 135 mean VaR, 131 mean variance, 131, 142 mean covariance, 143
proper efficient, 21 random return, 55 splitting, 76 right-hand-side, 88 shortfall, 130
probability, 82, 130 stochastic contol, 8
dominance, 40, 48 multi-objective, 1, 11, 30, problem, 21, 40 programming, 7 optimization, 53 return, 128 variable, 115
sakawa-Yana method, 116 SSD/FSD efficient, 26, slater-optimal, 20 subgradient, 93, 94, 1044 method, 93
transaction, 128, cost, 129
security, 131 triangular, 133, fuzzy, 136 trapezoidal, 133, 143 fuzzy, 134 ,140 value
at Risk, 11, 129, portfolio, 140, 141 random, 149
variance, 128 covariance, 128,142 matrix, 128
semivariance, 128 of return, 45
weakly, 19, 37 efficient, 19,21 dominant, 22 weighted, 17
problem, 17 possibilistic, 142, 138
variance, 142 covariance, 143
177
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