1

A Reference Point based Application to Financial Planning Problems with

Multiple Linear Fractional Objectives

João Paulo Costa – jpaulo@fe.uc.pt ; João Lourenço – jlourenc@fe.uc.pt

Fac. de Economia da Univ. de Coimbra; Av. Dias da Silva, 165; 3004-512 Coimbra; Portugal INESC – Coimbra; Phone: +351 239 790 586 Fax: + 351 239 403 511

JEL Codes: C61; C63; G31

ABSTRACT

In this paper we present a new technique to compute non-dominated solutions, with an

error that can be made as low as the user wants, in multiple objective linear fractional

programming (MOLFP), using reference points. The basic idea is to divide, by the

approximate ‘middle’, the non-dominate region in two sub-regions and to analyze each of

them in order to try to discard one. The process is repeated with the remaining region and it

ends when the regions are so little that the differences among their non-dominated solutions

are lower than a predefined error. The results of several tests to the technique performance are

reported.

We also report a financial planning model with multiple linear fractional objectives, and

propose a straightforward computational tool implementing the above mentioned technique to

help the decision maker (DM) finding the most preferred non-dominated solution. The

financial planning model includes three objective functions: (1) the maximization of the

company net present value; (2) the maximization of the average interest cover ratio during the

planning period; and (3) the maximization of a productivity ratio. The management of the

company has to evaluate a set of investment projects, simultaneously considering a series of

financing decisions like: the balance of funds, dividend distribution policy, debt and equity

issuing limitations.

We tested the model with a set of 10 projects over a 6 year planning period resulting on

a problem with 45 decision variables and 38 constraints. The computation technique proved to

be fast enough to deal with this model in an interactive way, using a desktop computer.

Keywords: Financial planning; Multiple objective linear fractional programming; Reference points.

2

1. INTRODUCTION

The main reason for the interest and importance of fractional programming stems from

the fact that various problems consider the optimization of a ratio between physical and/or

economic linear functions. For example, in many linear applications the programming models

could fit better the ‘real-problem’ if we use linear fractional goals – see Kornbluth and Steuer

(1981), Schaible (1981) and Stancu-Minasian (1997), among others. The explicit use of

several criteria or objectives can be more suitable for most decision situations where there are

several conflicting, or at least different, view points to cope with.

Reference point (Wierzbicki, 1980) methods and techniques can be considered as

‘generalized goal programming’. Their essence is that a reference point is a goal, but the sense

of ‘coming close’ to it does not mean the minimization of a distance, but the minimization of

an achievement scalarising function (ASF). The characteristics of this function enable it to

provide non-dominated solutions, even if the goals are settled below them (that is, ‘better’

solutions than the settled goals if they are feasible). We developed a technique that computes

non-dominated solutions, based on reference points, in multiple objective linear fractional

programming (MOLFP), with an error that can be made as low as the user wants. We only

acknowledge the existence of fairly complex non-linear methods to compute local optima that

can be non-dominated solutions, namely the one of Metev and Gueorguieva (2000).

The basic idea of the new technique (presented in section 3) is to divide, by the

approximate ‘middle’, the non-dominate region in two sub-regions and to analyze each of

them in order to discard one, if it can be proved that the minimum of the ASF is in the other.

The process is repeated with the remaining region. It is not always possible to discard one of

the regions and so the process must be repeated for both, building a search tree. In most

problems, it is only after a certain level of the search tree that we can start to discard regions.

The process will end when the remaining regions are so little that the differences among their

non-dominated solutions are lower than a predefined error. As we demonstrate (section 3.1),

one region can be discarded when the value of the ASF for its ideal point is worst than the

value of the ASF for a non-dominated solution belonging to another region, not yet discarded.

The idea behind the new technique stems from the algorithm presented by Costa and

Lourenço (2001), in the context of the computation of the maximum of a weighted sum. The

results of several tests to the technique performance and some conclusions are presented in

section 4.

3

We define financial planning in a company, like Goedhart and Spronk (1995), as “... a

structured process of identification and selection of present and future capital investment

projects, taking account the financing of these projects over time”. To support this process we

developed a model that can also be included in the field of financial modelling in the sense of

Spronk and Hallerbach (1997). They settled the three main roles of financial planning:

(1) to get insight into the feasibility of different strategies and possibly also to generate

new alternatives;

(2) to clarify the relations between decision alternatives and the (potential) results of

these alternatives;

(3) to help in finding a stream (or a set of streams) of decision alternatives.

In our opinion, to fulfil these roles it is necessary to formulate a substantive model, to

analyse it in detail, in order to gain some insight into its validity in the contingent situation,

and, finally, to explore how it generates some suitable streams of decisions according to the

preference system of the decision makers.

Several models for financial planning have been proposed in the literature. Section 5

presents a collection of such models and problems. Although different, all these models are

similar. The idea behind them is to state one or more objectives, subjected to several kinds of

constraints, while capturing the management politics and preferences, and some limitations of

the companies or their environments.

Concerning multiple objective models, most of the proposed exploration techniques fall

under goal programming. The main advantages of goal programming are related to the

psychologically appealing idea that we should set a goal and try to come close to it

(Wierzbicki, 1998). The basic disadvantage relates to the fact that this idea, without additional

assumptions, cannot guarantee efficient or non-dominated results.

Park and Sharp-Bette (1990, Chapter 8) and Wilkes (1977) organize good reviews of

financial planning models and decision approaches. An interesting feature of financial

planning problems, apart from having multiple and possibly conflicting goals, is the fact that

many of those goals have the form of ratios, making them a good application for MOLFP. In

this paper, we claim that MOLFP is better suited for this kind of problems, and present a

financial planning model constructed to demonstrate it. We tried to include some features of

financial planning which are rather usual in practise. Nevertheless, we are conscious that this

model can be no more than an example, and most surely do not apply to all financial planning

problems.

4

Goedhart and Spronk (1995) present an interactive approach to deal with such problems

using multiple linear fractional goal variables. Through the interactive procedure, constraints

on the values of the goal variables are formulated and then changed one by one and interaction

to interaction. Kornbluth and Steuer (1981b) extend goal programming in order to include

linear fractional criteria, resulting in the core mechanism of an interactive MOLFP method.

This mechanism resorts to the algorithm of Kornbluth and Steuer (1981a), which consists on

computing a set of weakly efficient points to present to the DM. By selecting a weakly

efficient point, the DM selects an indifference region in the weight space. However, this

method is not easy to apply.

The financial planning model proposed in this paper includes three objective functions:

(1) the maximization of the company net present value;

(2) the maximization of the average interest cover ratio during the planning period; and

(3) the maximization of a productivity ratio.

The company management has to evaluate a set of investment projects, simultaneously

considering a series of financing decisions like: the balance of funds, dividend distribution

policy, debt and equity issuing limitations.

The remainder of the paper is as follows. We start by presenting some concepts of

multiobjective linear fractional programming. On section 3, we define the computational

technique giving its validity proof. The results of several performed tests to the technique

performance are presented on section 4. We report a literature collection of models and

programs related to the proposed financial planning model on section 5 and after we detail it.

On section 7, we illustrate the use of the model and the technique with a numerical planning

problem. Finally, we present a brief summary and some conclusions.

2. MULTIPLE OBJECTIVE LINEAR FRACTIONAL PROGRAMMING

We formulate the MOLFP problem in the following way:

{ }mn

pp

pp

p

RbxbAxRxSxts

xdxc

zxdxcz

∈≥=∈=∈

++

=

++

=

,0,|:..

maxmax1

11

1

1 βα

βα

(2.1)

Where ck, dk ∈ Rn, and αk, βk ∈ R, k = 1, ..., p and ∀k, x ∈ S: dk x + βk > 0.

We will differentiate between weakly non-dominated solutions – a point x’ ∈ S is

weakly non-dominated if and only if there does not exist another point x ∈ S such that zk(x) ≥

5

zk(x’), for all k =1, ..., p – and non-dominated solutions – a point x’ ∈ S is non-dominated if

and only if there does not exist another point x ∈ S such that zk(x) ≥ zk(x’), for k = 1, ..., p,

and zk(x) > zk(x’) for at least one k. In this paper we will say that z(x) is non-dominated if and

only if x is non-dominated.

The ideal point, z*, is the point of the objective function space whose coordinates are

equal to the maximum that can be achieved by each objective function in the feasible region,

S. z* is computed through the determination of the pay-off table, that is, computing zk = z

(x*k); k = 1, …, p; where x*k is non-dominated and optimizes the program:

Sxtsxzk ∈:..)(max (2.2)

There is a variable change technique (Charnes and Cooper, 1962), that turns a linear

fractional problem (only with one criterion), into a plain linear program. We will use this

technique to compute the solutions of the problems of steps 2 and 5 of the algorithm presented

in section 3.1. The technique is as follows:

Consider p = 1 in the problem of Eq. 2.1. We define the new variables:

( )β+= dxt 1 and y = x t. (2.3)

Making the variables substitution (Eq. 2.3) we arrive to the following plain linear

program:

{ }0,,0,

1;0:..max

≥∈≥∈

=+=−+=tRtyRy

tdybtAytstcyzn

βα

(2.4)

We will use the following achievement scalarizing function (ASF):

( ) ( ) ∑==

−−=p

kkkkpk

zzzzzs1,...,1

max, ε (2.5)

where z ∈ Rp is a reference point in the criterion space and is defined according to the

DM’s preferences, the parameter ε ∈ R+ is used to prevent the result from being a weakly non-

-dominated solution and must be small enough (to allow for the possibility of determining all

the interesting non-dominated solutions).

The solution of the following scalarized problem is non-dominated:

( ) Sxtszzzp

kkkkpk

∈

−− ∑==

:..maxmin1,...,1

ε (2.6)

The problem stated in Eq. 2.6 can be rewritten in the following way:

6

RSx

pkxdxc

ztsxdxc

kk

kk

k

p

k kk

kk

∈∈

=

++

−≥

++

− ∑=

δβα

δβα

εδ

;

;,,1,:..min1

(2.7)

In Eq. 2.7 we can observe that the objective function is non-linear. Moreover (even if

we consider ε = 0, disregarding the weakly non-dominated solutions issue), the auxiliary

constraints, corresponding to the objective functions, are non-linear, because there are some

terms where the variable δ is multiplied by x. Thus, the scalarized problem, Eq. 2.6 or 2.7, is

very difficult to solve. Metev and Gueorguieva (2000) show several characteristics of the

scalarizing function and of the auxiliary problem that allows the use of complex non-linear

algorithms giving local optima.

3. COMPUTING NON-DOMINATED SOLUTIONS

Step 1 – Initialization

To define the error: Error. To define the reference point: z .

To initialize: Q = ∅ - Set of sub-regions the range of which is inferior to Error; M = {0} - Set

of sub-regions that can be further sub-divided; n = 0 - Identification of the current region; g =

0 - Sub-region counter. zI = (+∞, +∞, ..., +∞) - Incumbent solution; nI = n - Incumbent region.

S(n) is the feasible region n.

Step 2 - Analysis of the first region

For k = 1, ..., p , compute zkn = max zk (x), s.t.: x ∈ S(0).

( )pnp

nnn zzzz ,...,,* 22

11= is the ideal point of region n (n = 0).

Note that at this step zkn, k = 1, ..., p, can be weakly non-dominated.

For j = 1, ..., p do: If s( z ,zjn) < s ( z , zI) then zI ← zjn .

Step 3 - To choose the sub-region to proceed

If M ≠ ∅ then ( )m

Mmzzsmn *,min:

∈= ; M ← M \ {n};

Else (The better non-dominated solution is chosen and the algorithm stops.)

For all q ∈ Q compute zkq = z(x*k), k = 1, ...,p,

where x*k is non-dominated and optimizes the program:

max zk (x), s.t.: x ∈ S(q)

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The non-dominated solution corresponding to the reference point, z , is the solution

that minimizes: ( )jq

pjQqzzs ,min

,...,1; =∈

The algorithm stops.

End of If.

Step 4 – To sub-divide a region

The index of the objective function that will be constrained, indicated by r, in order to

sub-divide the region n, corresponds to the objective function that has the biggest range in the

pay-off table:

{ }

−=∆=∆

≠=

jnkjk

knk

nkpk

nr zzzz minmax

,...,1

(3.1)

The new regions are:

g ← g + 1; ( ) ( ) ( ){ }nr

rnrr

n zzxzRxnSgS ∆−≥∈∩= 21| ;

g ← g + 1; ( ) ( ) ( ){ }nr

rnrr

n zzxzRxnSgS ∆−≤∈∩= 21| ;

Step 5 – Analysis of the two new regions

For k=1,...,p do:

If ( ) nr

rnr

knr zzxz ∆−≥ 2

1 then

kngk zz =− )1( ; )(max xzz kkg = s.t.: ( )gSx∈

Else

knkg zz = ; )(max)1( xzz kgk =− s.t.: ( )1−∈ gSx

End of If

End of For

For t = g-1, g do:

If s ( z , z*t) < s ( z , zI) then

For j = 1, ..., p do: If s ( z , zjt) < s ( z , zI) then zI ← zjt ; nI ← t;

If ( )Errorzzpjpk jtk

t

kjk >−==∃ *|,...,1;,...,1, then M ← M + {t}

Else Q ← Q + {t};

End of If s ( z , z*t) < s ( z , zI)

End of For t.

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Step 6 – Discarding regions

If ( nI = g ) or ( nI = (g-1) ) then

For all m ∈ M do: If s ( z , z*m) > s ( z , zI) then M ← M \ {m};

For all q ∈ Q do: If s ( z , z*q) > s ( z , zI) then Q ← Q \ {q};

End of If.

Return to Step 3.

3.1. Condition to discarding a region (Step 6)

Theorem – Consider that z* is the ideal point of region A, from S, and z1 is one

solution of region B from S.

( ) ( )

−−>

−− ∑∑

====

p

kkkkpk

p

k kkkpkzzzzzz

1

11

,...,11,...,1max**max εε

(3.2)

If condition from Eq. 3.2 holds, then the non-dominated solution, +z , minimizing s ( z ,

z), x ∈ S, does not belong to region A.

Proof:

Consider that +z belongs to A, then if +z minimizes s ( z , z) over S:

( ) ( ) ( ) ∑∑∑=====

++

=−−<−−≤−−

p

k kkkpk

p

kkkkpk

p

kkkkpk

zzzzzzzzz1,...,11

11

,...,11,...,1**maxmaxmax εεε

(3.3)

That is:

( ) ( ) ∑∑===

++

=−−<−−

p

k kkkpk

p

kkkkpk

zzzzzz1,...,11,...,1

**maxmax εε (3.4)

( ) ( ) ( ) 0**maxmax1,...,1,...,1

<−+−−− ∑=

+

=

+

=

p

kkkkkpkkkpk

zzzzzz ε (3.5)

Consider i, without loss of generality, such as pkzzzz kkii ,...,1, =−≥− ++ and

remember that pkzz kk,...,1;* =≥ + , because z* is the ideal point of A. Then

pkzzzzkkii ,...,1,* =−≥− + , and ( ) ( )

kkpkkkpkzzzz *maxmax

,...,1,...,1−≥−

=

+

=. Concluding:

( ) ( ) 0*maxmax,...,1,...,1

≥−−−=

+

= kkpkkkpkzzzz (3.6)

Due to Eq. 3.6, to verify Eq. 3.5 it is necessary that:

9

( ) 0*1

<−∑=

+p

kkk

zzε (3.7)

Eq. 3.7 implies that there is at least one k, lets say k’, for which '' *

kk zz >+ and so z* could

not be the ideal point of A. This contradiction proofs the theorem.

4. TESTING THE PERFORMANCE OF THE TECHNIQUE

The technique was implemented with Delphi Pascal 5.0 for Microsoft Windows and a

PC Pentium III 500MHz, with 256Mb of memory, was used for running the tests. The simplex

code for solving the linear problems involved in the method was obtained through URL

http://www.netcologne.de/~nc-weidenma/readme.htm, and adapted to the present case.

The tests were done using random generated problems where the parameters followed

an uniform distribution in the range ]0,1]. Each measure was obtained through the average of

20 runs, ignoring the two worst and better values.

We studied the performance of the technique according to several parameters, but we

will only present the results we found more relevant, that is the performance according to the

error and according to the number of the objective functions. We also tested the performance

of the technique with other criteria to sub-divide the regions (step 4 of the algorithm), but we

reached the conclusion that the one presented on section 3.1 was the best.

We used two performance evaluation measures: the elapsed time from the start to the

end of the method and the number of generated sub-regions. The time is an absolute and

effective measure, but it is significantly affected by the quality of the machine that was used to

perform the tests. Figs. 4.1 and 4.2 show the performance of the technique according to the

elapsed time obtained on the tests. The problems were generated with 10 decision variables

and 10 constraints.

The number of sub-regions is a measure that indicates a more correct idea of the

growing complexity of the computations with the different parameters. This measure only

depends on the structure of the method it-self. On Fig. 4.3, the elapsed time ranged from 0.271

seconds for 2 objective functions, Error = 0.1 to 443.344 seconds for 4 objective functions,

Error = 0.0001. On Fig. 4.4 the elapsed time ranged from 0.271 seconds for 2 objective

functions to 1 104,864 seconds for 7 objective functions.

All the reported tests where done for what we found to be the computation worst case,

that is: when the used reference point coordinates are all equal among them.

10

0,1

1

10

100

1000

0,1 0,01 0,001 0,0001

Error

Tim

e [s

] 2 ob. functions

3 ob. functions

4 ob. functions

Fig. 4.1. Performance of the technique according to time [seconds]. Both axes have a logarithmic scale.

0

200

400600

800

1000

1200

2 3 4 5 6 7

No. of objective functions

Tim

e [s

]

Fig. 4.2. Performance of the technique according to time [seconds]. Error = 0.1 for all measures.

1

10

100

1000

0,1 0,01 0,001 0,0001

Error

No.

of s

ub-r

egio

ns

2 ob. functions

3 ob. functions

4 ob. functions

Fig. 4.3. Performance of the technique according to the No. of sub-regions. Both axes have a logarithmic scale.

11

0100200300400500600700

2 3 4 5 6 7

No. of objective functions

No.

of s

ub-r

egio

ns

Fig. 4.4. Performance of the technique according to the No. of sub-regions. Error = 0.1 for all measures.

The performance of the technique proved to be much better than the known ones, which

are complex non-linear techniques. The reported tests show it is reasonable to solve problems

with a maximum of 7 objective functions. The technique reveals some performance sensitivity

to the error (precision).

5. PREVIOUSLY PROPOSED FINANCIAL PLANNING MODELS

In the past, several financial planning models and decision approaches have been

proposed. We will focus on mathematical programming models, some of them presented by

Wilkes (1977) and Park and Sharp-Bette (1990, chapter 8).

Lorie and Savage (1955) were pioneers on presenting the problem of capital allocation

among projects taking into account their Net Present Value. Weingartner (1963) proposes a

model emphasizing the accumulated value at the final of the planning period and

incorporating the possibility of several different kinds of debt. Baumol and Quandt (1965)

attempt to solve the inconsistency underlying the use of the net present value, constraints on

the amount of the available capital and discount rates only determined by external (to the

company) issues.

The study of different dividends policies and their impact over the profit is the bigger

contribution of the model presented by Chambers (1967). The model of Bernhard (1969) also

incorporates some issues concerning the dividend policy, achieving it by loosing the objective

function linearity. The model of Hayes (1984), not concerned with the dividend policy, allows

the explicit consideration of the impacts of holding cash from one period to another and

maximizes (as a proxy for the company value) the amount of cash to be hold in the last

planning period.

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Charnes et al. (1959) formulated a similar problem: to maximize a profit function,

taking in to consideration several economic/financial issues (e.g. cash flows and amount of

per period needed capital) and physical constraints (e.g. stocks capacity), and defining the

amounts to buy and sell on each period. This formulation, combined with the models

described, was the basis of several reported models that are very complex (due to their

extension and interdependence among variables), and try to encompass a lot of aspects of a

company. The models of Jaaskelainen (1966) and Deam et al. (1975) can be given as

examples. These models incorporate not only financial planning aspects, but also production

management policies and accounting equations.

Another similar problem is the constitution of a portfolio of securities. Markowitz

(1952) was a pioneer in the field through his well-known quadratic programming formulation,

which includes a proxy for the risk and starts to introduce the idea of multicriteria. Konno

(1990) and Yamakazi and Konno (1991) propose an alternative formulation where the

quadratic function is substituted by a linear one. They use, as a proxy for risk, the ‘mean

absolute deviation’ (which is an L1 metrics), instead of the variance (an L2 metrics). Young

(1998) introduces a new principle, based on historical returns data: the chosen portfolio

minimizes the maximum loss over all past observation periods, for a given return level. This

is basically an L∞ metrics and leads to a linear programming problem. The application of

some ideas of portfolio optimization to the capital allocation to investment projects was done

by several authors, like Yoon and Sadrian (1992), Hoadley et al. (1993) and Aloysius and

Rosenthal (1999).

Some authors have made the explicit use of multiobjective programming formulations

and associated solving techniques. Spronk (1981) presents one of the earliest formulation,

going through the advantages of explicitly considering several criteria in financial planning.

Costa and Clímaco (1995) propose a multiobjective linear programming model and propose

the use of reference points with a local search technique to explore the potential solutions.

Goedhart and Spronk (1995) introduce some linear fractional objective functions and a

convergent interactive technique to cope with them. Mukherjee and Bera (1995) present an

application example, using goal programming techniques.

13

6. THE FINANCIAL PLANNING MODEL

The financial planning model that we constructed (Lourenço, 2002; Lourenço and

Costa, 2003) was inspired by Spronk (1981), Goedhart and Spronk (1995) and Costa et al.

(submitted).

In this model, covering a H year planning period, the management of a company has to

evaluate a set of N investment projects, simultaneously considering a series of financing

decisions. We consider that the impact of the dependencies among investment projects, and

between investment projects and all other assets of the company, being accounted for, as

thoroughly as possible, in the project cash flows, can be negleted.

Projects, with a fixed duration, tD (where Nt ,...,1= ), are mainly characterized by: their

annual investment cash flows, itCFI ; their annual operating cash flows, itCFE (always

positive, where Ni ,...,1= and tDt ≤≤0 ); their contribution, itE , to the annual earnings

before interest and taxes (where Ni ,...,1= and tDt ≤≤0 ); their discount rate, ir , associated

with systematic risk (where Ni ,...,1= ); and, iw , their contribution to the company total

employment. The model accounts for cash flows generated after the planning period.

On the other hand, the company is described by the tax rate, cT , and the interest rate for

debt, dr . As for the operating environment of the company (operating in a quasi-perfect free

market), it is characterized by the short-term interest rate, sr , the long-term interest rate, Lr ,

and the cost, c, of issuing equity (a cost proportional to the amount of equity issued). Interest

rates, as well as tax rates, are fixed over the planning period, and independent from the issued

amounts.

The management strategy towards dividend distribution is modelled through: the target

dividend payout policy, TD , that is the proportion of the re-invested and to distribute earnings;

the adjustment rate of dividend distribution, AR ; and the cost, tCD , of not following the

management policy at a certain year t (where Ht ,...,1,0= ).

The instrumental variables are:

ix , the amount of investment units on project i (where 10 ≤≤ ix for every Ni ,...,1= );

ty , the amount of debt issued over year t (where Ht ,...,1,0= );

tSI , the amount of equity issued in year t (where Ht ,...,1,0= );

tL the annual cash held (where 1,...,1 += Ht );

tDIV , the amount of dividends distributed at the end of year t (where Ht ,...,1,0= );

14

tPD , the penalties for not following company policy towards dividend distribution at the

end of year t (where Ht ,...,1,0= ).

All debt issues can be seen as one period loans, borrowed at the end of year, t, and

repaid at the end of year, t + 1. 0L , the amount of cash held in year 0, is given.

We have included three objective functions. Generally the most important is assumed to

be the maximization of the net present value of the company. It can be defined as the sum of

the net present value of the investment projects, plus the present value of tax savings resulting

from debt financing, minus the costs of issuing equity, and minus the opportunity cost of

issuing stocks. Following Costa et al. (submitted), we also subtract the penalties for not

following the management policy in dividend distribution. The first objective function is a

linear:

( ) ( )

( ) ( )( )( )

−

++

−+

−+

−

−+

++−

=

∑∑∑

∑∑∑

=

−

=+++

=

===

H

ttt

H

tt

L

tst

L

tH

tt

L

t

H

tt

d

tdcD

tt

i

ititN

ii

PDCDr

Lrr

Lr

cSI

ryrT

rCFICFE

xz

Max

i

0

1

0111

0

0011

11

11

11

(6.1)

Another objective function consists in the maximization of the average interest cover

ratio during the planning period, enforcing company policy towards limiting the risk of being

unable to pay the interest on debt out of company earnings. According to Goedhart and

Spronk (1995), some financial planning models handle this risk by formulating chance

constraints on debt level. The model is deterministic and the uncertainty that is inherent to

financial planning is a great extent exogenously accounted for. Yet, the risk of bankruptcy in

case of more debt financing is endogenously represented. Therefore, we chose to deal with

this matter by including an objective function in the form of the maximization of a ratio of

accounting earnings to interest charges. This objective function is linear fractional:

=∑

∑∑

=

= =H

ttd

H

t

N

iiit

yr

xEzMax

0

0 12

(6.2)

Finally, the third objective function is the maximization of a productivity ratio, as an

example of an organizational factor that may influence the company investment decisions.

This objective function is defined as the ratio of accounting earnings to the contribution to the

company total employment, and therefore is linear fractional:

15

=∑

∑∑

=

= =N

iii

H

t

N

iiit

xw

xEzMax

1

0 13

(6.3)

The financial planning problem is modelled through four sets of constraints. The first

set bounds the variables - all the variables must be bigger or equal to zero, and the amount on

investment units on project i (xi, i = 1,..., N) must also be lower or equal to one.

The second set of constraints, Eq. 6.4, includes the equations for the annual balance of

funds. We assume that interest over debt is due in the beginning of each period.

( )( ) ( ) ( ) ( ) ;,...,1,011 111

HtDIVLLrSIcyryyxCFICFE tttsttdtt

N

iiitit ==−−++−+−−+− +−

=∑

( )( ) ( ) .0,00100100001==−−+−+−+−∑

=

tDIVLLSIcydryixiCFIiCFEN

i

(6.4)

The third set of constraints, Eq. 6.5, models the management strategy towards the

distribution of dividends (mainly trying to stabilize them).

;,...,1,0111

HtPDDIVDIVDIVxETDAR tttt

N

iiit =≤−−+

− −−

=∑

( ) .0,01 011

0 =≤−−+−∑∑==

tPDDIVtHtDIV t

H

t

H

t

(6.5)

Finally, the model has a set of constraints, Eq. 6.6, for bounding the debt and equity

issued each year to the investment amount in that year:

;,...,1,11

HtyySIxCFI ttt

N

iiit =−+≥ −

=∑

.0,001

0 =+≥∑=

tySIxCFIN

iii

(6.6)

7. FINANCIAL MODEL EXAMPLE

We built an example in order to test both the financial model and the computation

technique. This example shows the benefits of using MOLFP and reference points approaches

in dealing with financial panning problems.

7.1. Data of the example

In this section we present the data used on the example. We considered a planning

period of 6 years, H = 6, and a set of 10 (N = 10) potential investment projects. Table 7.1

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gives the per period investment cash flow, itCFI , and the discount rate, ir , for each project.

Table 7.2 shows the annual operating cash flow, itCFE . Note that these values are considered

for the total duration of each project, tD , and not only for the planning period. This is

necessary to take into account what happens after the planning horizon.

Projects Periods

ir 0 1 2 3 4

1 300 200 10% 2 350 260 10% 3 300 200 10% 4 350 260 10% 5 3500 9% 6 5500 7% 7 5000 11% 8 5000 11% 9 2500 500 12%

10 2000 6%

Table 7.1. Per period investment cash flow, itCFI , in 103 Euros and the discount rate, ir , for each project.

Project

s Periods

1 2 3 4 5 6 7 8 9 10 11 12 13 1 60 100 100 100 100 100 100 100 100 100 2 70 110 110 110 110 110 110 110 110 110 3 60 100 100 100 100 100 100 100 100 100 4 70 110 110 110 110 110 110 110 110 110 5 700 700 700 700 700 700 700 700 6 660 660 660 660 660 660 660 660 660 660 660 660 1853 7 2500 2500 2500 8 2500 2500 2500 9 770 840 910 980 1050

10 400 400 400 400

Table 7.2. Annual operating cash flow, itCFE , in 103 Euros.

Projects Periods

iw 1 2 3 4 5 6

1 45 60 60 60 60 60 5 2 50 65 65 65 65 8 3 45 60 60 60 2 4 50 65 65 5 5 490 490 490 490 490 490 115 6 160 160 160 160 160 160 23 7 350 500 750 45 8 350 500 750 97 9 110 230 300 350 500 45

10 250 250 28

Table 7.3. Project contribution to earnings, itE , in 103 Euros and to employment, iw .

17

Table 7.3 presents the project contribution to the annual earnings before interest and

taxes, itE , and the contribution of each project for the company total employment, iw .

Finally, Table 7.4 gives the remaining data, which are necessary to characterize the

company, its operating environment and the management strategy towards dividend

distribution.

The model has 45 decision variables and 38 constraints, with these data.

cT 40% CD0 2.00%

dr 6% CD1 1.93%

sr 1% CD2 1.85%

Lr 4% CD3 1.78% c 2% CD4 1.71% 0L 100 103 Euros CD5 1.64%

TD 40% CD6 1.58% AR 50%

Table 7.4. Remaining data.

7.2. Testing the example

The DM started by computing the pay-off table, that is, the non-dominated solutions that

respectively maximize each objective function. Table 7.5 presents the pay-off table. In this

table we can note the range of the values of the 3 objective functions, and we can obtain the

ideal point of the problem (bold numbers).

z1 z2 z3 Max obj. function 1 2 842,8 2,0 27,5 Max obj. function 2 836,0 7 974,0 25,1 Max obj. function 3 120,3 1,6 75,0

Table 7.5. Pay-off table of the example.

None of the non-dominated solutions of the pay-off table is balanced because they

correspond to the extreme points of the non-dominated region. So the DM computed the non-

-dominated solution that corresponds to an approximation of the ideal point. Table 7.6 a)

presents the result and the algorithm performance. This is a more balanced solution but the

DM considers that the first objective function (the net present value of the investments) is too

far from its maximum. Therefore, s/he tries to improve the first objective function by using a

reference point with a bigger value in the first coordinate. Table 7.6 b) shows the computed

18

non-dominated solution. More solutions were computed using different reference points, in

order to explore the non-dominated solution. The DM was satisfied (in what concerns our

example) with solution e) presented in table 7.6.

Reference point Non-dominated solution Algorithm performance z1 z2 z3 z1 z2 z3 Time [s] No. Regions

a) 3 000 8 000 80 1 650,1 6 650,1 26,5 2,1 81 b) 6 000 8 000 80 1 944,7 4 164,0 26,0 2,8 99 c) 8 000 4 000 100 2 842,8 2,0 27,5 10,7 173 d) 8 000 6 000 300 2 030,0 30,0 26,0 2,2 75 e) 8 000 6 000 600 2 027,4 27,4 27,4 19,2 259

Table 7.6. Reference points and corresponding non-dominated solutions.

The algorithm performance was good enough to explore the non-dominated region in an

interactive way. All the computations were done with an error of 1%.

In table 7.7 we present the values of the decision variables for the non-dominated

solution e), which has the following relevant characteristics:

. not to invest on projects 5, 6 and 10;

. to issue some debt on the first and third year;

. to only issue equity on the beginning of the planning period;

. to only hold some cash on the third period;

. to have some variation on the distribution of dividends, with a severe drop from the

second to the third year;

. to have a dividend penalty on the third year, which corresponds to the mentioned drop

on the distributed amount of dividends from the second to the third years.

Variable Value Variable Value Variable Value

x1 1,0 y5 0,0 L7 0,0 x2 1,0 y6 0,0 DIV0 0,0 x3 1,0 SI0 5 000,0 DIV1 74,8 x4 1,0 SI1 0,0 DIV2 1 109,8 x5 0,0 SI2 0,0 DIV3 509,8 x6 0,0 SI3 0,0 DIV4 425,4 x7 0,9 SI4 0,0 DIV5 3 700,8 x8 1,0 SI5 0,0 DIV6 3 760,9 x9 0,9 SI6 0,0 PD0 0,0

x10 0,0 L1 0,0 PD1 0,0 y0 0,0 L2 0,0 PD2 0,0 y1 382,8 L3 38,4 PD3 260,5 y2 0,0 L4 0,0 PD4 0,0 y3 2 915,3 L5 0,0 PD5 0,0 y4 0,0 L6 0,0 PD6 0,0

Table 7.7. Values (103 Euros) of the decision variables of the non-dominated solution of table 7.6 e).

19

8. CONCLUSIONS

The explicit use of linear fractional objectives in financial planning can make multi-

objective models to fit better ‘real world’ problems. In this paper we presented one of such

MOLFP models, and advocated the use of reference points to help to deal with it. We also

defined one computational technique that makes possible to work with such problems, greatly

improving the potential application of MOLFP.

The performance of the technique proved to be much better than the known ones, which

are complex non-linear techniques. The reported tests show that it is reasonable to solve

problems with a maximum of 7 objective functions. The technique also revealed some

performance sensitivity to the error (precision).

The financial planning model includes three objective functions: (1) the maximization

of the company net present value; (2) the maximization of the average interest cover ratio

during the planning period; and (3) the maximization of a productivity ratio. The company

management has to evaluate a set of investment projects, simultaneously considering a series

of financing decisions like: the balance of funds, dividend distribution policy, debt and equity

issuing limitations. We tested the model with a set of 10 projects over a 6 year planning

period, leading to a problem with 45 decision variables and 38 constraints. The computation

technique proved to be fast enough to deal with this model in an interactive way, using a

desktop computer.

Acknowledgement: This work was supported by the ‘Fundação para a Ciência e Tecnologia’ and FEDER, project POCTI/32405/GES/2000.

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