algorithm abc

7/31/2019 Algorithm ABC

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Solving the Assignment problem using Genetic Algorithm and

Simulated Annealing

Anshuman Sahu, Rudrajit Tapadar.

AbstractThe paper attempts to solve the generalized Assignment problem

through genetic algorithm and simulated annealing. The generalizedassignment problem is basically the N men- N jobs problem where a single

job can be assigned to only one person in such a way that the overall cost of

assignment is minimized. While solving this problem through genetic

algorithm (GA), a unique encoding scheme is used together with Partially

Matched Crossover (PMX). The population size can also be varied in each

iteration. In simulated annealing (SA) method, an exponential cooling

schedule based on Newtonian cooling process is employed and

experimentation is done on choosing the number of iterations (m) at each step.

The source codes for the above have been developed in C language and

compiled in GCC. Several test cases have been taken and the results obtained

from both the methods have been tabulated and compared against the results

obtained by coding in AMPL.

Index TermsAssignment problem, Genetic Algorithm, Newtonian cooling

schedule, Partially Matched Crossover (PMX), Simulated Annealing.

I. INTRODUCTION

The Assignment model, as discussed in different text-books of

Operations Research, can be paraphrased as: Given N men and N

machines, we have to assign each single machine to a single man in

such a manner that the overall cost of assignment is minimized.

To put it mathematically, let us define the following symbols:

i row number denoting ith man i [1, N]

j column number denoting jth machine j [1, N]

[][]Cij cost of assigning jth machine to ith man

[][]Xij= 1 if jth machine is assigned to ith man

= 0 otherwise

The problem can be formulated as:

Minimize the total cost function 11[][][][]NNijCijXij==Subject to the following constraints:

Manuscript received March 7, 2006.

Anshuman Sahu is a senior undergraduate student with a major in

Production and Industrial Engineering in Motilal Nehru National


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Institute of Technology, Allahabad, India. (e-mail:

[email protected]).

Rudrajit Tapadar is a junior undergraduate student with a major in

Computer Science and Engineering in Motilal Nehru National

Institute of Technology, Allahabad, India. (e-mail:[email protected]).

1[][]NiXij= = 1j=1, 2 N (1)

= 1 i=1, 2 N (2) 1[][]NjXij=

[][]Xij=1 or 0 (3)

The Hungarian mathematician D.Knig proved an essential theorem

for the development of the Hungarian method to solve this

model. The problem can also be formulated as an integer-

programming model and solved by techniques such as Branch-and-Bound technique. Reference [1] states that the Hungarian

algorithm for solving the assignment model is more efficient than

branch-and-bound algorithms. This paper attempts to solve the same

model using two non-traditional techniques: Genetic Algorithm and

Simulated Annealing. It is basically an experimental investigation

into the various parameters affecting these two algorithms and

adapting them to our own problem. These two approaches are

discussed one by one.

II. GENETIC ALGORITHM APPROACHGenetic algorithms (GA) are computerized search and optimization

algorithms based on the mechanics of natural genetics and natural

selection. They were first envisioned by John Holland and were

subsequently developed by various researchers. Each potential

solution is encoded in the form of a string and a population of

strings is created which is further processed by three operators:

Reproduction, Crossover, and Mutation. Reproduction is a process

in which individual strings are copied according to their fitnessfunction (Here the fitness function is taken to be the total cost

function). Crossover is the process of swapping the content of two

strings at some point(s) with a probability. Finally, Mutation is

the process of flipping the value at a particular location in a


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string with a very low probability. A more comprehensive treatment

of GA can be found in [2], [3], [4].

Now, for adapting GA to our problem, it is necessary that we

develop an encoding scheme. Consider the case when N=3 and let us

presume that machine M1 is assigned to man m1, machine M2 to manm2, and machine M3 to man m3 as shown:

.

IAENG International Journal of Applied Mathematics, 36:1, IJAM_36_1_7

__________________________________________________________________

____________________

(Advance online publication: 1 February 2007)

Machine

M1 M2 M3

m1 1 0 0Man m2 0 1 0

m3 0 0 1

Figure 1. A sample assignment.

Consider the first column: 100 which is equivalent to 4 in base 10

representation. Similarly the other two columns decode to 2 and 1

respectively. Hence, the above assignment can be encoded as . A quicker insight leads us to the observation that the each

permutation of i.e., , , , ,, and is a possible solution. As the total number

of solutions possible to this particular problem are 3! =6, we can

easily conjecture that in case of N men, N machine, the total

number of solutions possible is N! and our task is to select the

best string (the one with minimum total cost). As our encoding

scheme also generates N! strings, therefore it is correct and

there is one to one correspondence between each possible solution

and each string. It is also evident that each component (value at

each position) in the string can be uniquely expressed as 2r

where r is a positive whole number varying from 0 to N-1. As the

powers of 2 increase rapidly, a more compact way of encoding would

be to express the component 2r simply as r. This is easier to

write and saves space when N is high.


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After encoding of the string, the population selection for

crossover is done by Binary tournament selection method. Here

s=2 strings are randomly chosen and compared, the best one being

selected for parenthood. This is repeated M times where M is the

size of the population. Reference [4] also cites a method forgenerating the parent strings which are then ready for crossover.

Here simple crossover will not work; instead we choose the method

of Partially Matched Crossover (PMX) which was initially developed

for tackling the Traveling Salesman Problem [2]. The concept

of PMX can be understood by considering an example:

Suppose we want to have crossover between two permutations of the

string i.e., and . Two random

numbers are generated between 1 and L where L is the length of thestring (L=5 in this case). Suppose the crossover points have been

choosen as shown below:

1 3 4 2 5

2 1 3 5 4

Where the dashed positions show the chosen points. Now PMX defines

the following scheme for interchangeability:

3 1 4 3 25 implies 14 and 25

Now the portion between the selected crossover points is swappedand the rest of the values are changed according to the above rule

(this means 1 in the portion outside the two crossover points is

replaced by 4 and 2 in the portion outside the two crossover

points is replaced by 5). So the two children strings generated

are:

4 1 3 5 2

5 3 4 2 1

Which are again valid permutations of . AfterCrossover, we have a family of parent population and children

population out of which we are to select the population for next

iteration. Here we have a choice of altering the population size

at each iteration. We must maintain the diversity in population or

else it may lead to premature convergence to a solution which may


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not be optimal. One method of selecting the population may be to

arrange the entire population in ascending order of their

objective function value (the string that decodes to lowest total

cost of assignment will have the highest objective function value)

and choose a predetermined number of individual strings from eachcategory i.e., from those that are above average, from those

around the average, and from those below the average. This

threshold can be set by using the concept of mean and standard

deviation applied to the population. For instance, if we assume

the string values to be normally distributed with mean value

and standard deviation , we divide the population into four

categories: those having values above + 3*, those having

values between + 3* and , those having values between and -3*, and those having values less than -3*. In

this way the diversity in population is maintained. Another aspect

is that the string with the best objective function value at each

iteration is stored in a separate array and subsequently compared

with the best string of the population at next iteration. In this

way, the best string cannot escape. Also note that we are not

using mutation but a slight variant of it (Inversion) by choosing

two random spots in a string and swapping the corresponding values

at that position. Inversion is allowed only when the sum of thecosts at these positions before swapping is greater than the sum

of costs associated with these positions after swapping.

Thus, if we want to swap in the string at say second and

third positions, it will only be allowed if cost of is

greater than cost associated with .

The program was developed for the test problem given in [1]. Two

cases were implemented: one in which Inversion was used and

another in which Inversion was not used. In both the cases, theanswer converged to the final optimum value. On an average, there

was not much difference in the number of iterations required to

reach the final value in both the cases. The observations are

plotted in the table as shown below:

750800850900950100010501100123456789generation numbercost (


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function) valuewithInversionwithoutInversionFigure 2. Graph showing

convergence to the global minimum in case of both inversion and

without inversion.

On an average, the time taken was 0.01s measured on a standard

desktop with processor Intel Pentium 4, 2.40 GHz. The populationsize at each generation was kept equal to 20.

While using this algorithm in our case, we represented each

possible solution by the string as developed previously in the

case of genetic algorithm. E refers to the function value (the

total cost of assignment for a particular string). We have

employed the scheme of Newtonian cooling wherein the temperature

at each generation is determined according to the law: T

III. SIMULATED ANNEALING APPROACHSimulated Annealing is another non-traditional method which was

originally developed by S. Kirkpatrik, C.D. Gelatt, Jr., and M.P.

Vecchi [5]. The simulated annealing procedure simulates the

process of slow cooling of molten metal to achieve the minimum

function value in a minimization problem. It is a point-by-point

method. The algorithm begins with an initial point and a high

temperature T. A second point is taken at random in the vicinity

of the initial point and the difference in the function values

(E) at these two points is calculated. The second point ischosen according to the Metropolis algorithm which states that if

the second point has a smaller function value, the point is

accepted; otherwise the point is accepted with a probability exp

(-E / T). This completes one iteration of the simulated

annealing procedure. In the next generation, another point is

created at random in the neighborhood of the current point and the

Metropolis algorithm is used to accept or reject the point. In

order to simulate the thermal equilibrium at every temperature, anumber of points (m) is usually tested at a particular temperature

before reducing the temperature. The algorithm is terminated when

a sufficiently small temperature is obtained or a small enough

change in the function values is found. A detailed description of

this can be found in [4].


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i = T0 * exp (-) where T i is the temperature at ith generation,

T0 is the initial temperature and is a suitable constant ( is

initially taken 0 when temperature equals T0 and is incremented by

a factor increment at each stage). Now consider the task of

randomly generated valid strings: two techniques are beingemployed. Suppose we have the string and we want to

produce another random permutation of these 4 numbers. The first

method is to slide each number by a random number generated

between 1 and L (L not included) where L is the length of the

string. Thus, assuming that the random number generated is 2, the

string gets transformed to . The second method

of generating a valid permutation is two choose two positions at

random in the string and swap the values at those points. Wesearch for the potential solution in two regions: first we search

in the region of strings created by the above first method. When

the answer converges to a particular value, we store the

corresponding string in a separate array. Then we proceed with our

search again in the region of strings created by the second

method. Once again, we converge to another string and this string

is compared with the string which was initially stored in a

separate array. The minimum of these two (the one with lesser

function value) is selected as the final answer.The important parameters affecting simulated annealing are the

number of iterations (m) at each step and the cooling schedule.

The total number of iterations is proportional to m as well as the

rate of change of temperature. The cooling schedule is based on

Newtons law of cooling. This model of cooling can be compared to

the discharge of an initially charged capacitor in a RC circuit as

they both follow exponential decay law. For all practical

purposes, it is assumed that the capacitor is fully discharged att=5*RC. Hence, in our schedule we also ran our program from

Tmax=700 to Tmin around 700*exp (-5), keeping the number of

iterations m fixed (=50). Tmax is generally computed by

calculating the average of function values at several points. The

program was run on a standard desktop with processor Intel Pentium


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4, 2.40 GHz and the test case considered was the one given in [1].

The results obtained have been plotted as shown in figure 3 as

shown below:800850900950100013579111315 iteration number (taken at

an interval of 200 iterations)cost (function)

valueTmax=700,Tmin=4.5Tmax=700,Tmin=5Tmax=700,Tmin=7 Figure 3.Program run for various schedules (m constant)

The average time taken was 0.051s. Now m at each step was changed,

decreasing it from an initial value of 100 till a minimum value

(=20 in our case) was reached. It was observed that the program

converged to the minimum value at lesser total number of

iterations. This is shown below:

800850900950100012345678iteration number (taken at interval of 200

iterations)cost (function)

valueTmax=700,Tmin=4.5Tmax=700,Tmin=5Tmax=700,Tmin=7 Figure 4.

Program run for various schedules (m varying)

Reference [1] reports to have solved the above test problem

in 0.09s of IBM 370/168 time. The problem was also coded in AMPL

with MINOS 5.5 as the solver and it took 0.03125s on the standard

desktop mentioned earlier. While solving the problem using Genetic

Algorithm, the average time taken was 0.01s while the time taken

for solving it using Simulated Annealing was 0.05s (The time was

noted on a standard desktop with processor Intel Pentium 4, 2.40GHz).

IV. CONCLUSION

An experimental investigation into solving the Assignment model

using Genetic Algorithm and Simulated Annealing is presented.

Various parameters affecting the algorithms are studied and their

influence on convergence to the final optimum solution is shown.

ACKNOWLEDGMENT

The Authors would like to thank Dr. Sanjeev Sinha, Asst.

professor, Dept. of Mechanical Engineering, MNNIT, India for his

invaluable advice and guidance. The Authors would also like to

thank their friends with whom they discussed their ideas which

sometimes led to many new insights.

REFERENCES


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[1] Billy E. Gillett,Introduction to Operations Research A Computer-

Oriented Algorithmic Approach.Tata McGraw-Hill Publishing Company

Limited, New Delhi (Copyright 1976 by McGraw-Hill, Inc., New

York), TMH EDITION 1979, ch. 3, ch.4.

[2] Goldberg, D.E.,Genetic Algorithms in Search, Optimization, and MachineLearning.Reading, Mass.: Addison-Wesley, 1989.

[3] Kalyanmoy Deb,Optimization for Engineering Design Algorithms and

Examples.Prentice-Hall of India Private Limited, New Delhi, 1995,

ch. 6.

[4] Fred Glover, Gary A. Kochenberger, Ed.HANDBOOK OF

METAHEURISTICS.2003 Kluwer Academic Publishers New York,

Boston, Dordrecht, London, Moscow (Print ISBN: 1-4020-7263-5,

ebook ISBN: 0-306-48056-5).[5] Kirkpatrick, S., Gelatt, Jr., C.D. and Vecchi, M.P. (1983)

Optimization by simulated annealing,Science, 220, 671-680.


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AbstractSelecting the routes and the assignment of link flow

in a computer communication networks are extremely complex

combinatorial optimization problems. Metaheuristics, such as genetic

or simulated annealing algorithms, are widely applicable heuristic

optimization strategies that have shown encouraging results for a

large number of difficult combinatorial optimization problems. This

paper considers the route selection and hence the flow assignment

problem. A genetic algorithm and simulated annealing algorithm are

used to solve this problem. A new hybrid algorithm combining the

genetic with the simulated annealing algorithm is introduced. A

modification of the genetic algorithm is also introduced.

Computational experiments with sample networks are reported. The

results show that the proposed modified genetic algorithm is efficient

in finding good solutions of the flow assignment problem compared

with other techniques.

KeywordsGenetic Algorithms, Flow Assignment, Routing,

Computer network, Simulated Annealing.

I. INTRODUCTION

ETWORK design is a fundamental problem with a large

scope of applications that have given rise to many

different models and solution approaches [6], [10]. The

general network design problem involves the minimization of


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a cost objective function over a lot of design variables, such

as link capacities, flow assignment, network topology, and

node locations. This problem belongs to the class of

combinatorial problems. Efficient solutions to this problem

are much sought after because such solutions could lead to

better utilization of the networks. The traditional Lagrange

relaxation and sub-gradient optimization methods can be used

for tackling this problem. The results generated by these

methods are locally optimal instead of globally optimal [7].

The flow assignment (FA) problem focuses on assigning the

traffic requirements on the best routes used by nodes in the

network in order to ensure an acceptable performance level at

a minimum cost.

For solving the FA problem, we consider different

approaches. The first one uses the genetic algorithm (GA).

The second approach uses the simulated annealing algorithm

(SA).

Tarek M. Mahmoud is with the Computer Science Dept., Faculty of

Science, Minia University, El-Minia, Egypt (e-mail: [email protected]).

The third approach combines GA with simulated annealing

(SA) to improve the performance of the GA. In the fourth

approach, a new modification of the GA is introduced.


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GAs have gained considerable attention in recent years for

solving various combinatorial optimization problems. It is an

evolutionary technique that simulates the process of natural

evolution and applies genetic search operators like

recombination, mutation, and selection to a sequence of

alleles. The sequence of alleles is the equivalent of a

chromosome in nature. GA can be used to solve variety of

different problems using a survival of the fittest idea [4], [8].

SA is a metaheuristic algorithm derived from thermodynamic

principles. It has recently turned out to be one of the most

powerful tools for solving hard combinatorial problems [5],

[9].

The remainder of the paper is organized as follows. In

section 2, the FA problem is formulated. Section 3 gives a

review of both the GA and SA. The implementation of the

GA and SA for solving the FA problem is given in section 4

and 5. The hybrid genetic-simulated annealing algorithm to

solve the FA problem is given in section 6. In section 7, a

modification of the GA is introduced. The results of

computational experiments are presented in section 8. Section

9 concludes and summarizes the main results obtained in this

paper.


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II. PROBLEM DESCRIPTION AND FORMULATION

A computer network can be modeled as an undirected

graph G = (N, L), in which the sets of nodes N and links L

represent computer sites and communication cables,

respectively. There are communication demands between n

different nodes. The demands are specified by an n x n

demand matrix M = (mij), where mij is the amount of traffic

required between ni and nj. The arrival rate on each link (i, j)

is denoted by ij and is expressed in the same units as

capacity Cij.

The flow assignment problem (FA) can be described as

follows: given the network topology, the traffic requirement

(OD matrix), and the link capacities, minimize the average

time delay of messages by selecting the route such that traffic

requirements are satisfied. Once the route is decided, the

flows are sent along this route from origin node (O) to

A Genetic and Simulated Annealing Based

Algorithms for Solving the Flow Assignment

Problem in Computer Networks

Tarek M. Mahmoud

N

World Academy of Science, Engineering and Technology 27 2007


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360

destination node (D). In this paper, as in most previous work

in the literature [7], [10], and [11] we make several

simplifying assumptions. We assume that nodes have

practically unlimited buffers to store messages and the arrival

process of messages to the network follows a Poisson

distribution. The computer network can be modeled as a

network of independent M/M/1 queue in which links are

treated as servers with service rates proportional to the link

capacities [1]. The queuing and transmission delay in link ij is

given by:

ij ij

ij C

T 1

= (1)

where ij C is the capacity of link ij in bits per second, ij is

the arrival rate of messages to link ij, and1 is the expected

message length. Using the above notation, the expected

network delay is given by:


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=

ij L

r R

r

r

ij ij ij

r R

r

r

ij ij

C x

x

T 1

(2)

where


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is the total arrival rate of messages in the network,

L is the index set of links in the network,

R is the index set of candidate routes,

r

ij is an indicator function, which is 1 if link ij is used in

route r and is 0 otherwise,

r x is a decision variable which is 1 if route r is selected for

message routing and 0 otherwise.

The FA problem is then to assign the traffic demand on a

route r from R for each nodes pair, which can satisfy the

following condition:

= =


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ij L

r R

r

r

ij ij ij

r R

r

r

ij ij

C x

x

D min T 1

(3)

subject to:

x , ij L C 1

r R

r

r


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ij ij ij

(4)

=x 1, P r

r SP

(5)

= r x {0,1},rR (6)wherep S is the index set of candidate routes for commodity

p, and is the index set of communicating

source/destination pairs in the network. The first constraint

ensures that the flow on each link does not exceed its

capacity, while selection of exactly one route per commodity

is ensured by the second and third constraints.

III. REVIEW OF GENETIC AND SIMULATED ANNEALING

ALGORITHMS

In recent years, genetic algorithms (GAs), which based on

the idea of natural selection and survival of the fittest, have

been applied with a high degree of success to a variety of

problems [3], [7]. GAs are search techniques for global


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optimization in a complex search space. Search space in GA

is composed of possible solutions to the problem. A solution

in the search space can be represented by a sequence of 0s

and 1s, integers, or any other type from which a specific

solution can be deduced. This solution string is referred to as

the chromosome in the search space. Each chromosome has

an associated objective value called the fitness value. The

fitness of a chromosome corresponds to its ability to survive

and reproduce offspring. A good chromosome (that has good

chance to survive) is the one that has a high/low fitness value

depending upon the problem (maximization/ minimization). A

set of chromosomes and associated fitness values are called

the population. This population at a given stage of the GA is

referred to as generation. The general GA has the following

elements:

A solution encoding (representation).

A mechanism to generate initial solutions

(population) from where the iterative search will

proceed.

An evaluation function (fitness function) to rate

solutions from a current population.

Perturbation operators to create new solutions from a


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current population.

Assignment to the parameters of the algorithm, and

Stopping criteria.

Creating new population from a given current population

can be achieved by shuffling two randomly selected

chromosomes. This process is called crossover. Sometimes

one or more bits of a chromosome are complemented to

generate a new offspring. This process of complementation is

called mutation. The GA can be summarized as follows:

Step 1: Initialize population

Step 2: Evaluate population

Step 3: Repeat Steps 4, 5, and 6 while termination criterion

not reached

Step 4: Select parent chromosomes for next population

Step 5: Perform crossover and mutation

Step 6: Evaluate population

Simulated annealing algorithm (SA) is a general-purpose

optimization technique and has been applied to many

combinatorial optimization problems [5], [9]. The main idea

behind SA is an analogy with the way in which liquids freeze

and crystallize. When liquids are at a high temperature their

molecules can move freely in relation to each other. As the


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liquid's temperature is lowered, this freedom of movement is

lost and the liquid begins to solidify. If the liquid is cooled

slowly enough, the molecules may become arranged in a

crystallize structure. The molecules making up the crystallize


361

structure will be in a minimum energy state. If the liquid is

cooled very rapidly it does not form such a crystallize

structure, but instead forms a solid whose molecules will not

be in a minimum energy state. The fundamental idea of SA is

therefore that the moves made by an iterative improvement

algorithm are like the re-arrangement of the molecules in a

liquid that occur as it is cooled and that the energy of those

molecules corresponds to the cost function which is being

optimized by the iterative improvement algorithm. Thus, the

SA aims to achieve a global optimum by slowly convergence

to a final solution, making downwards moves with occasional

upwards moves and thus hopefully ending up in a global

optimum. SA can be described formally as follows: start from

a random solution. Given a solution Sc, select a neighboring

solution Sn and compute the difference in the objective

function values,f f(S ) - f(S ). n c = If the objective


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function is improved (f< 0 ), then replace the current

solution by the new one. Iff0 , then accept a move with

probability p(f) = exp(-f/T) , where Tis the control

parameter or temperature. This probabilistic acceptance is

achieved by generating a random number in [0, 1] and

comparing it against the threshold exp(-f/T). If

exp(-f/T) is greater than the generated random number

then replace the current solution by the new one. The

procedure is repeated until a stopping condition is satisfied.

The SA can be summarized as follows:

Step 1: Initialize temperature T at random, and set a cooling

rate.

Step 2: Initialize initial solution Sc at random.

Step 3: Evaluatef(Sc).

Step 4: Repeat steps 5, 6, 7, and 8 while termination condition

not satisfied.

Step 5: Select a solution Sn in the neighborhood ofSc at

random.

Step 6: Set ( n ) ( c )f=f Sf S {Compare the change in

objective function}

Step 7: Iff 0 then


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ScSn {Sn replaces Sc}

Else

If exp(-(f)/T) > random (0, 1) then

ScSn

Step 8: Update (T) using the relation T = T, where is the

cooling rate {Cooling step}

IV. IMPLEMENTING THE GA FOR SOLVING THE FLOW

ASSIGNMENT PROBLEM

This section presents an implementation of the GA for

identifying the link flows of the computer network that

satisfies the traffic requirements while satisfying the problem

constraints. Firstly, we describe how the main components of

the GA are implemented to solve the FA problem. Then we

give the overall algorithm used to solve this problem.

A. Genetic Representation

In route selection, and hence flow assignment, each

chromosome represents a routing table which includes a path

list P1, P2, , PN(N-1)/2 that represents the entire network. Each

path Pk is a particular route between two nodes i, j. A path

(route) is encoded as list of integers by listing the nodes from

its source to its destination based on the network topology [4].

If a path cannot be realized on the network, it cannot be


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encoded into a chromosome, which means that each step in a

path must pass through a physical link in the network. The

first gene on the chromosome represents the source node; the

last gene represents the destination node, while other genes

represent intermediate nodes along that path from the source

to the destination.

B. Population Initialization

The initial process is used to compose the routing tables for

all chromosomes in the current generation. Each chromosome

includes a random routing table for a given topology.

C. Evaluation

To evaluate the solution quality of the flow assignment

problem, equation (3) is used as the objective function to

determine the ability of a chromosome to survive and produce

offspring. In our implementation, a route with less average

time delay is frequently employed in sending packets.

D. Selection

The selection (reproduction) operator is intended to

improve the average quality of the population by giving the

high fitness chromosomes (less average time delay) a better

chance to get multiple copies into the next generation,

whereas chromosomes with low fitness (high average time


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delay) have fewer copies or even none at all. Many types of

selection scheme can be used [3]. One of the selection

methods is based on spinning the roulette wheel pop_size

times; each time a single chromosome is selected as a new

offspring, where pop_size denotes the population size. The

steps of this selection method are as follows [8]:

Step 1: Calculate the fitness valuef(vi) for each chromosome

vi (i=1, , pop_size).

Step 2: Find the total fitness of the population

=

=

pop_size

i 1

i Ff(v )

Step 3: Calculate the probability of the selection pi for each

chromosome vi where

p (v ) / F i i =f, i = 1, 2, , pop_size

Step 4: Calculate a cumulative probability qi for each

chromosome vi where

=

=

i


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j 1

i j q p , i = 1, 2, , pop_size

Step 5: Generate a random number r from the range (0, 1).

Step 6: If r


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should have the same source and destination nodes. Crossing

sites for the path crossover operator are limited to nodes

contained in both paths. A node is randomly selected as a

crossing site from the potential crossing sites and exchanges

subroutes. When applying the crossover operator to a pair of

paths P1 and P2, the operation proceeds according to the

following algorithm [4]:

Step 1: List up a set of nodes Nc included in both P1 and P2

(excluding source and destination nodes) as potential

crossing sites.

Step 2: Select a node i as a crossing site from the Nc.

Step 3: Crossover the paths by exchanging all nodes after the

crossing site i.

Fig. 1 shows an overview of the crossover operator

applying for a pair of paths P1 and P2 from node 1 to node 4.

Their potential crossing sites is node 3. When we select node

3 as a crossing site, the new offspring are generated by

exchanging the subroutes as shown in the figure below.

Path P1 1 5 3 4

Path P2 1 3 2 4

Child P1 1 5 3 2 4

Child P2 1 3 4


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Fig. 1 Example of crossover

F. Mutation Operator

The path mutation is another genetic operator, which is

applied to a randomly selected single solution (chromosome)

from the population with a certain probability. It makes small

random changes in the solution. These random changes will

gradually add some characteristics to the population, which

could not be supplied by the crossover operator. Similar to the

crossover operator, the application of the mutation operator is

governed by a mutation probability Pm. In each generation,

the mutation operator is also tried pop-size times and thus, the

expected number of mutated chromosomes is Pm* pop-size.

To perform a mutation in the FA problem, a node is

randomly selected from the path, which is called a mutation

node. Then another node is randomly selected from the nodes

directly connected to the mutation node. If any duplication of

nodes exists in the offspring path, then this path can be

discarded. The path mutation operator can be described as

follows [4]:

Step 1: Select mutation node i randomly from all nodes in

parent V.

Step 2: Select a node j from the neighbors of the mutation


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node i.

Step 3: Generate a random path from the source to node j, and

another random path from node j to the destination node.

Step 4: If any duplication of nodes exists in the offspring

path, discard the routes and do not perform mutation.

Otherwise the routes are connected to make up a mutated

path.

G. Overall Algorithm

The GA algorithm used to solve the flow assignment

problem can be described as follows:

Step 1: Input all kinds of data of the problem (network

topology, OD matrix, and link capacities) and the

controlling parameters of the algorithm (crossover

and mutation probability, population size, and

generation number).

Step 2: Randomly generate initial population, where each

chromosome in the population represents a routing

table for the given network.

Step 3: For each OD pair, assign the flow between the origin

O and the destination D on the route connecting

them, then compute the fitness of every chromosome

in the current population using equation (3). Save the


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best chromosome.

Step 4: Select the best chromosomes (routing tables) using

the roulette wheel method.

Step 5: Perform the crossover and mutation operations to get

a new population.

Step 6: Repeat steps 3 5 until the termination condition is

met.

Note that, the termination condition is met either after a

specified number of generations or no improvement occurs on

the best solution for successive generations. In our

implementation, a specified number of generations are used as

a termination condition.

V. IMPLEMENTING THE SA FOR SOLVING THE FLOW

ASSIGNMENT PROBLEM

Using SA for solving the FA problem requires the

determination of an initial feasible solution. In our

implementation, we used the population initialization step of

the GA, discussed in section III, to get an initial feasible

solution. Thus, the initial solution includes a random routing

table for each source-destination pair of a given network. The

selecting of a solution in the neighborhood of the initial

solution can be obtained by changing randomly one or more


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links of the route from the source to the destination. As in the


363

implementation of GA for solving the FA problem, equation 3

is used to compute the value of each candidate solution for the

problem.

VI. HYBRID GENETIC-SIMULATED ANNEALING ALGORITHM

APPROACH (HGSA)

The HGSA algorithm combines both the GA and SA

algorithms to solve the FA problem. This combination occurs

in the selection process of the GA algorithm. A SA-selection

[2] is used to choose a single candidate solution between the

best of the two parents, offspring, and best solution of the

generations. The selection function SA-selection(offspring[i],

parent[i], best solution, T[i]) is defined, which applies the

traditional SA function SA(a, b, T) multiple times to identify

the single surviving candidate, where i in the SA-selection

function indicates the index of the new individual. The

function SA(a, b, T) calculates the acceptance probability

P = exp(- (f(a) - f(b))/T), where f(x) is the cost function of a

candidate x. As mentioned in section 3, if f(a) f(b), then

candidate a will be selected. If f(a)>f(b) and P > random


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number in [0, 1), then candidate a will be selected too. Except

these cases, candidate b will be selected. All possible cases of

this selection process are represented in Table 1. In this SAselection

function, offspring[i] and parent[i] are compared

with each other, then with best solution. To illustrate the

selection cases given in Table I, consider case 1, where

offspring[i] is accepted over both parent[i] and best solution,

but parent[i] is accepted over best solution. Hence,

offspring[i] is accepted and returned.

The HGSA algorithm for solving the FA problem has the

following steps:

Step 1: Input all kinds of data of the problem (network

topology, OD matrix, and link capacities) and the controlling

parameters of the algorithm (crossover and mutation

probability, population size, and generation number, and

cooling rate).

Step 2: Randomly generate initial population, where each

chromosome in the population represents a routing table for

the given network.

Step 3: For each individual i randomly generate an initial

temperature T[i].

Step 4: For each OD pair, assign the flow between the origin


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O and the destination D on the route connecting them, then

compute the fitness of every chromosome in the current

population.

Step 5: Save the current population as the parent population.

Step 6: Perform the crossover and mutation operations to get

the offsprings.

Step 7: Find the best routing table among the parents,

offsprings, and current best solution, and then update the best

solution.

Step 8: For each individual of the population do

Find the i-th chromosome between parent, offspring,

and best solution according to the processes given in

Table I.

i-th chromosome = SA-selection(offspring[i],

parent[i], best solution, T[i])

Update the fitness function of the i-th chromosome.

Set T[i] = T[i] x cooling rate

End for

Step 9: Repeat steps 4 8 until termination condition met.

VII. A MODIFIED GENETIC ALGORITHM APPROACH (MGAA)

As mentioned earlier, the selection operator is intended to

improve the average quality of the population and many types


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of selection scheme can be used. Calling the selection process

for each generation increases the algorithm complexity. We

can ignore this process and perform the reproduction process

directly on the current population. In the MGAA, we generate

the initial population randomly. In each generation, the new

population consists of offsprings produced from mating

individuals from the current population and possibly some

individuals from the current population. The parents are

selected for crossover and mutation according to the

crossover and mutation probability. If the fitness value of the

offspring has a better value than one or both of its parents

then this offspring is accepted in the new population.

Otherwise, mate the best of the two parents with another

parent. In the later case, we can change the crossover point

instead of replacing one of the parents. This step guarantees

that the new population contains always the best

chromosomes. Applying the traditional crossover process to

generate new offspring is not always guaranteed to produce

good chromosomes.

The MGAA can be summarized as follows:

Step 1: Initialize population

Step 2: Evaluate population and save the best chromosome.


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Step 3: Repeat Steps 4 and 5 while termination criterion not

reached

Step 4: Perform crossover and mutation

Step 5: Evaluate population and update the best chromosome.

It is clear that; the selection step in the genetic algorithm

given in section 4 is eliminated. This modification speeds up

the GA and guarantees that the reproduction step always

produces good offsprings.

VIII. EXPERIMENTAL RESULTS AND DISCUSSION

In this section, we present a comparison between the

performance of the four approaches GA, SA, HGSA, and

MGAA. These algorithms were implemented in C++. The

results presented in this section are obtained from simulations

on 2 sample networks. The topologies of these networks are

as shown in Fig. 2 and 3.

Fig. 2 A network example with 10 nodes and 36 links

1

2

3

4

5

6


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7

8

9 10


364

Fig. 3 A network example with 15 nodes and 38 links

In the network topology given in Fig. 2, node 10 is a

geosynchronous satellite. All links are full duplex. The

capacity of each terrestrial link is 38.4 kb/s and that of each

satellite link is 50 kb/s. Each of the six satellite links is

assumed to introduce a propagation delay of 125 ms. All link

capacities in the second network topology are 50 kb/s. To

handle the situation of overflow, the terms of the objective

function given in equation 3 are replaced with the following

terms [12]:

+

+


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=

( 0.99 ) .T (0.99 ) otherwise

2

1

T (0.99 ) ( 0.99 ).T (0.99 )

T ( ) if 0 0.99

T

ij ij

2

ij ij

ij ij ij ij ij ij

ij ij ij ij

ij

where ij ij T and T are the first and second derivatives of the

objective function respectively.

The traffic matrix for the nodes of the first network

example with 10 nodes and the second network example with

15 nodes is given in Table II and III.


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The SA algorithm was applied using the following

parameters:

The initial temperature of the process selected

randomly in the range (0..1)

The cooling coefficient was 0.99

The genetic parameters are chosen as follows:

The population size is 10.

The crossover probability is 0.35.

The mutation probability is 0.01.

The computer simulation results generated by the GA, SA,

HGSA, and MGAA for the two network topologies given in

Figs. 2, 3 are shown in Figs. 4, 5. The figures show the

relation between the number of generations and the fitness

(average time delay) values for the FA problem. As can be

seen from Fig. 4 and 5, MGAA is better than GA, HGSA, and

SA algorithms.

According to the simulation results, the hybridization

between SA and GA improves the performance of the GA in

solving the FA problem. It is clear also that SA is better than

GA in solving the FA problem. As mentioned above, the

MGAA has another advantage than the traditional GA. The

time used in the selection process either using the roulette


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wheel discussed in section IV.D or any other selection

method is saved. Our implementation of the MGAA applies in

the reproduction step one-cut point crossover.

0,001

0,021

0,041

0,061

0,081

0,101

100 200 300 400 500 600 700 800 900 1000

Generation number

F itn e s s

GA

HGSA

SA

MGAA

Fig. 4 Comparison chart for fitness values using GA, SA, HGSA and

MGAA for the first network example

0

0,04

0,08

0,12


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0,16

0,2

100 200 300 400 500 600 700 800 900 1000

Generation number

F itn e ss

GA

HGSA

SA

MGAA

Fig. 5 Comparison chart for fitness values using GA, SA, HGSA and

MGAA for the second network example

IX. CONCLUSION

In this paper, we studied the route selection and flow

assignment in computer networks, and the use of a genetic

algorithm and simulated annealing algorithm for solving the

problem. To improve the performance of the genetic

algorithm, hybridization between the genetic and the

simulated annealing algorithm is introduced. A modification

of the genetic algorithm is also introduced.

We have compared the performance of the four algorithms

in solving the route selection and flow assignment problem.

Our experimental results showed that the proposed modified


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genetic algorithm provided better solutions than the

traditional genetic, simulated annealing, and hybrid geneticsimulated

annealing algorithms.

For the same problem, the simulated annealing and its

hybridization with the genetic algorithm have better results

than the traditional genetic algorithm. The modified genetic

algorithm can be used in any problems to which the genetic

algorithm approach is applicable. Further modifications of the

initial population step can improve the performance of the

genetic algorithm. Instead of generating the initial population

randomly, we can use any method to generate only better

chromosomes.

REFERENCES

*1+ D. Berttsekas, and R. Gallager. Data Networks, Second Edition,

Prentice Hall, Englwood Cliffs, New Jersey (1992).

[2] H. Chang, and P. Jung, "SA-selection-based Genetic Algorithm for the

Design of Fuzzy Controller", International Journal of Control,

Automation, and Systems, Vol. 3, no. 2, pp. 236-243, June (2005).

[3] W. Chang and R. S. Ramakrishna, "A genetic algorithm for shortest path

routing problem and the sizing of populations", IEEE Transaction on

Evolutionary Computation, Vol.6, No. 6, December (2002).

[4] M. Gen and R. Cheng, "Genetic algorithms and engineering


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optimization", John Wiley&Sons, Inc., (2000).

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


365

[5] Y.,Habib, M. Sait and H. Adiche, "Evolutionary algorithms, simulated

annealing and tabu search: a comparative study", Engineering

Applications of Artificial Intelligence 14, pp. 167-181, (2001).

[6] K. Ko, K. Tang, C. Chan, K. Man and S. Kwong, "Using genetic


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algorithms to design mesh networks", IEEE Computer, pp 56-61,

(1997).

[7] X. Lin, Y. Kwok and V. Lau, A genetic algorithm based approach to

route selection and capacity flow assignment, Computer

Communications 26, pp 961-974, (2003).

[8] Z. Michalewicz, "Genetic algorithms + data structure = evolution

programs", 3rd edition, Springer Verlag, New York, USA, (1996).

[9] P. Mills, E. Tsang, Q. Zhang and J. Ford, "A survey of AI-based metaheuristics

for dealing with local optima in local search", Technical

Report Series, Report No. CSM-416, September (2004).

[10] J. Shen, F. Xu and P. Zheng,"A tabu search algorithm for routing and

capacity assignment problem in computer networks", Computers &

Operations Research 32, pp 2785 2800, (2005).

[11] K. Walkowiak, "Ant algorithm for flow assignment in connectionoriented

networks", International Journal of Applied Mathematics and

Computer Science, 15, pp 205-220, (2005).

[12] Z. Wang, D. Browning, "An Optimal Distributed Routing Algorithm",

IEEE Transaction on Communication, vol.39, no. 9, (1991).

TABLE II

TRAFFIC REQUIREMENT OF THE FIRST NETWORK EXAMPLE WITH 10 NODES

1 2 3 4 5 6 7 8 9 10

1 0 5 5 0 0 5 0 10 0 0


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2 5 0 10 5 5 3 0 10 0 0

3 5 10 0 5 0 3 5 5 20 0

4 0 5 5 0 5 0 5 5 5 0

5 0 5 0 5 0 5 0 0 5 0

6 5 3 3 0 5 0 0 0 5 0

7 0 0 5 5 0 0 0 5 0 0

8 10 10 5 5 0 0 5 0 5 0

9 0 0 20 5 5 5 0 5 0 0

10 0 0 0 0 0 0 0 0 0 0

TABLE III

TRAFFIC REQUIREMENT OF THE SECOND NETWORK EXAMPLE WITH 15 NODES

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 0 1

0 3 0 0 5 0 8 0 3 0 6 0 4 0

2 1

0 0 8 5 0 4 10 0 0 5 0 7 0 0 9

3 3 8 0 0 6 8 0 2 0 5 0 8 0 5 0

4 0 5 0 0 10 9 0 3 0 6 0 4 0 0 2

5 0 0 6 10 0 0 5 6 0 3 0 6 8 10 0

6 5 4 8 9 0 0 5 0 7 7 4 3 0 0 5

7 0 1

0 0 0 5 5 0 5 2 4 0 7 0 8 0


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8 8 0 2 3 6 0 5 0 11 12 0 5 0 8 0

9 0 0 0 0 0 7 2 11 0 10 0 10 0 4 0

10 3 5 5 6 3 7 4 12 10 0 5 0 5 5 0

11 0 0 0 0 0 4 0 0 0 0 0 4 0 7 0

12 6 7 8 4 6 3 7 5 10 0 2 0 5 2 5

13 0 0 0 0 8 0 0 0 0 5 0 5 0 5 5

14 4 0 5 0 10 0 8 8 4 5 7 2 5 0 8

15 0 9 0 2 0 5 0 0 0 0 0 5 5 8 0

TABLE I

SA-SELECTION FUNCTION FOR CHOOSING THE SURVIVING SOLUTION AMONG

OFFSPRING, BEST SOLUTION, AND PARENT

SA-selection(offspring[i], parent[i], best solution, T[i])

Case

SA(offspring[i],

parent[i], T[i])

SA(offspring[i],

best solution, T[i])

SA(parent[i],

best solution, T[i])

return

1 offspring[i] offspring[i] parent[i] offspring[i]

2 offspring[i] offspring[i] best solution offspring[i]


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3 offspring[i] best solution parent[i] offspring[i]

4 offspring[i] best solution best solution best solution

5 parent[i] offspring[i] parent[i] parent[i]

6 parent[i] offspring[i] best solution offspring[i]

7 parent[i] best solution parent[i] parent[i]

8 parent[i] best solution best solution best solution


366


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53

Genetic Algorithmic Approach to Generalized

Assignment Problems in Conjunction with

Supply Chain Optimization : A Survey

One of the fundamental Combinatorial Optimisation Problems is the Generalised

Assignment Problem

(GAP). Several Operations Research experts, researchers, academicians,

industrialists tried to optimise

the cost related functions with source and destination in various assignment

problems like Machine

Learning Problem, Machine Assignment Problem, Traveling Salesman Problem,

Supplier Assignment

Problem, Scheduling Problem etc. In the recent years, Genetic Algorithms are

used in most of the fields of

Manufacturing Management / Supply Chain Management and Optimisation.

Several researchers studied

and analysed the GAP through the classical Lexi Search, Branch and Bound

approaches and later by

algorithms like Ant Colony, Particle Swarm, Bee Colony, Simulated Annealing, and

heuristics methods of

optimisation. It was followed by the study of Genetic Algorithm approach in

solving the GAP.

In this paper, an attempt is made with an analytic review of the literature on the

Genetic Algorithmic


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approach to GAP, which is proved to be convenient and efficient in deriving the

required solutions. The

outcome of this survey proves to be very handy for the researchers / practitioners

that the tailor made

Genetic Algorithm is of potential use to simulate the possible solutions and for

manufacturing segments

like Single Product Manufacturer, Multi Product Manufacturer, Multi Objective

Optimisation,

Outsourcing, Multiple Supplier, Partner Selection and so on, in terms of Supplier

Assignment Cost,

Manufacturing Cost, Time Bounded Assignment which leads to Supply Chain

Optimisation.

Key Words: Generalised Assignment Problem, Genetic Algorithm, Supply Chain

Management,

Optimisation

Introduction care of the logistics, distribution of various products

at multi product manufacturer level and the last

In the present global market scenario the module, 4. The customer module, which

reflects the

manufacturer is facing many challenges and one performance of the above

modules in chain like

such is meeting the demand which is uncertain and fashion which should result in

the customer delight

even after meting this requirement, subsequent and satisfaction.

supply of spare parts is once again an uncertain


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commodity. To overcome this, the concept called Mishra[25] defines the concept

of Supply Chain

Supply Chain Management (SCM) was evolved to Management as a business

process as depicted in the

the rescue of many manufacturers with its figure 1. The very essence right from

the

outstanding solutions at every link level in the manufacturer to customer delight

every stage of the

supply chain. Basically the Supply Chain defined supply chain is to be optimised to

its

Management can be viewed in four different corresponding constraints so as to

establish a

modules vis., 1.Supplier Management, which prcised processes and which leads

to the Supply

includes various aspects like supplier relation, Chain Optimisation concept which

was discussed

supplier selection, and material selection at different by Romero Morales [30].

stages etc., 2. The producer module, which discusses

about the order, inventory and manufacturing In Supply Chain Management

utmost care is to be

managements (including the assembly line exercised between the suppliers

manufacturer tie up

balancing), 3. The wholesaler module, which takes with regard to meeting the

required specifications. It

V.Kalyan Chakravarthy, Dr.V.V.Venkata Ramana & Dr. C.Umashankar


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54

is at this stage assigning the supplier to a of orders, any supplier can be assigned

to perform

manufacturer based on the requirement is the any order in the supplier

manufacturer module of

management problem in the sense to assign the best SCM, incurring some cost

that may vary depending

of the supplier to a manufacturer(s) which gives rise on the supplier order

assignment. It is required to

to an exiting Assignment Problem through Supply perform all orders by assigning

exactly one supplier

Chain, visualising this concept the Generalised to each order in such a way that

the total cost of the

Assignment criteria has been evolved to Supply assignment is minimum.

Chain Management, which leads the analyst to

suggest novel procedures in optimising the supplier To minimise the total cost,

the GAPis considered to

assignments to different manufacturers is the study at supplier manufacturer

module of SCM in

concept of application of assignment problems in detail through Genetic

Algorithmic approach.

Supply Chain Management. It is at this juncture felt

necessary to establish the supplier producer link

level relation [24] which is briefly explained in


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figure 2. Generally the initial consideration of Genetic Algorithms [40] were

invented to mimic

Supply Chain Management is to choose the best of some of the processes

observed in the natural

the available suppliers considering its evolution. Many people, biologists included,

are

corresponding cost relate to the manufacturer, lead astonished that life at the

level of complexity that

time reduction, shipping, delivery date we observe could have evolved in the

relatively

commitment, customer satisfaction, supply in right short time suggested by the

fossil record. The idea

time, right quality with right quantity at the right with GA is to use this power of

evolution to solve

location. This necessitated the researcher to study optimisation problem. The

father of the original GA

and establish different modeling aspects of Supply is John Holland, who invented

in the early 1970s.

Chain Management in choosing the best of the

supplier pave way for different types of Assignment Genetic Algorithms are

Adaptive Heuristic Search

Problems like Classical, Continuous, Quadratic and Algorithms, based on the

evolutionary ideas of

Discrete which can be used in optimising the natural selection and genetics. As

such they

supplier producer assignment. represent an intelligent exploitation of a random


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search used to solve optimisation problems.

The information on the application of algorithmic Although randomised, Genetic

Algorithms are by

aspects revealed fact that initially the Lexi Search, no means random, instead they

exploit historical

Branch and Bound, Simulated Annealing, information to direct the search into the

region of

Heuristics etc., are applied and later by the Genetic better performance within the

search space. The

Algorithms. Compared to the different algorithmic basic techniques of the Genetic

Algorithms are

aspects of application, for certain reasons the designed to simulate processes in

natural systems

Genetic Algorithmic approach is proved to be an necessary for evolution;

especially those follow the

efficient method, for which reason the author is principles first laid down byCharles Darwin of

fascinated to study, probe and to report the Supply Survival of the Fittest. Since,

in nature,

Chain Optimisation problem solving methods competition among individuals for

scanty resources

through the application of GAP with Genetic results in the fittest individuals

dominating over the

Algorithmic approach which adds to the literature. weaker ones.

The Assignment Problem is one of the fundamental A Genetic Algorithm [13] is an

optimisation


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Combinatorial Optimisation Problems in the branch technique as it can be used to

find the minimum or

of optimisation or Operations Research in maximum of some arbitrary function.

While there

Mathematics. It consists of finding a maximum are a large number of

mathematical techniques for

weight matching in a weighted bipartite graph. accomplishing this, both in general

and for specific

When there are a number of suppliers and a number circumstances, a GA is

unique. It is a stochastic

History and Genesis of Genetic Algorithm (GA)

Gap in Relation to SCM Importance

55

method, and it will find a global minimum, neither programming techniques in the

process of

property being singular. The approach is optimisation. The paper tries to probe

natural

remarkable because it is based on the way that a algorithms in particular with

Genetic Algorithm

population of living organisms grows and evolves, approach on Generalised

Assignment Problems

fitting into their ecological niche better with each towards Supply Chain

Optimisation. The author

generation. tries to focus the pioneering research papers in the

case of applications of algorithms like Branch and


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GAs is different from more normal optimisation and Bound, Lexi search approach,

Tabu Search,

search procedures in four ways: Simulated Annealing for GAP prior to GA. Ross

and Soland *35+ describes the Generalised

1. GAs work with a coding of the parameter set, Assignment Problem as a

generalisation of the

not the parameters themselves. ordinary Assignment Problem of linear

2. GAs search from a population of points, not a programming in which multiple

assignments of

single point. tasks to agents are limited by some resource

3. GAs use pay off (objective function) available to the agents. A Branch and

Bound

information, not derivatives or other auxiliary algorithm is developed that solves

the GAP by

knowledge. solving a series of binary knapsack problems to

4. GAs use probabilistic transition rules, not determine the bounds.

Computational results are

deterministic rules. cited for problems with up to 40000 1 variables and

comparisons are made with other algorithms.

One of the best natural algorithms suits for several Martello and Toth [22]

proposed a typical algorithm

applications of the industry and market and even in for the GAPwhich is

considered as a basic paper by

the analysis of customer delight and satisfaction. many authors. Subsequent his

work, Venkata


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Another form of sensing the Genetic Algorithms Ramana and Umashankar [36]

discussed in detail

[29] is an optimisation and search technique based about the several cases of

Assignment Problems

on the principles of genetics and natural selection. A using LexiSearch algorithm.

Cerny [3] developed a

GA allows a population composed of many Thermo dynamical approach to the

traveling

individuals to evolve under specified selection rules salesman problem with the

help of an efficient

to a state that maximises the fitness (i.e., Simulation algorithm. Branch and

Price algorithm

minimises the cost function). for the GAP developed by Savelsberg [23]

considering the column generation and Branch and

GA Operators: The following operators based on Bound to attain optimal integer

solution. Similarly

natural selection [29] the solution method for solving the k-best Traveling

Reproduction Selects good strings in a Salesman Problem reported by Poort

[11] for

population and forms a mating pool. finding k-best solutions for one of the most

Crossover New strings are created by notorious NP-hard problems, namely

the Traveling

exchanging information among strings of the Salesman Problem(TSP). Osman [26]

in his paper

mating pool. suggested novel heuristic procedures for the GAP


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Mutation does a local search around the and also presented the GAP with

Simulated Annealing and Tabu Search algorithms by

current solution to create a better string demonstrating the different merits and

demerits of

hopefully. Also used to maintain diversity in the the suggested algorithms. Abada

[1] solving a time

population. table problem and presented the solution using Tabu

Search algorithm.

Several researchers / practitioners and Operation

Cattrysse and Wassenhove [10] had conducted a Research experts used variety of

natural method of

Survey of Algorithms on Generalised Assignment algorithms in the process of

optimisation which

Problems, specially based on Branch and Bound lead to satisfying the supplier and

producer module

technique. The authors opined there is a tremendous relation in a Supply Chain.

Of all the algorithmic

change in the applications of several mathematical aspects application to GAP

with particular relation

Survey of Algorithms

56

to SCM, it is observed that still there is a lack of proves that GA solution is a

quality solution. The

mathematical expertise while using different Guided Genetic Algorithm (GGA) is a

hybrid of


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algorithmic approaches. The mathematical Genetic Algorithm (GA) and Meta-

Heuristic

treatment in the application of algorithms is felt by Search Algorithm, Guided

Local Search (GLS). It

many researchers, and at this juncture the reported builds on the framework and

robustness of GA, and

Genetic Algorithm is proved to be very handy for integrating GLS's conceptual

simplicity and

many. Experiencing the novel merits of GA many effectiveness to arrive at a

flexible algorithm well

researchers started the GA application initially to meant for constraint

optimisation problems. Lau,

GAP and as the author focusing this article the Tsang [19] reported on GGA and its

successful

applications of GAP to SCM through GA this application to the GAP.

review is presented which will be useful

information for the practitioners in studying this A constructive Genetic Algorithm

for the GAP is

aspect. An effort is made to present the scanty described by Luiz et.al [20] as a

problem of

available literature on GAusage in conjunction with assigning n items to m

knapsacks, n>m such that

Supply Chain Optimisation in specific. each item is assigned to exactly one

knapsack to

evolve the binary representation and assignment


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An initial survey of GAP through Branch and heuristic is presented through GA

which allocates

Bound technique is reported by Cattrysse and items to knapsacks and many

instances are taken

Wassenhove [10] in finding the minimum cost from the literature by the authors.

assignment of jobs to agents so that each job is

assigned exactly once and agents are not Stephen Chen [34] presented the

traditional

overloaded. The author in his paper opines still advantages and the efficient

based coded

there is lack of representativeness of mathematical algorithmic procedures and

compared to other

treatment in the reported heuristics. Ribeiro et.al algorithmic aspects to conclude

that GA is a new

[18] presented the Genetic Algorithm programming perspective for the

practitioners which can improve

environments into application oriented system, upon the solution methodologies.

Jiyin Liu, Lixin

algorithmic oriented systems and tool kits by Tang [16] proposes a modified

Genetic Algorithm

providing case studies with particular focus to for the single machine scheduling

problem with

biological environment, and confess that GA is the ready times. This algorithm

improves the simple

best of algorithmic approaches. Genetic Algorithm by introducing two new steps:


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(1) a filtering step to filter out the worst solutions in

Genetic Algorithms provide an alternative to each generation and fill in their

positions with the

traditional optimisation techniques by using best solutions of previous

generations; and (2) a

directed random searches to locate optimal selective cultivation step to cultivate

the most

solutions in complex landscapes. Srinivas and promising individual when no

improvement is

Patnaik [33] introduce the art and science of Genetic made for certain

generations. Improvement is also

Algorithms and survey current issues in GAtheory made on the crossover

operation for the problem.

and practice. A new GA for GAP is successfully Computational experiments are

carried out,

evolved by Wilson [37] on a class of set covering comparing the performance ofthe proposed

problems. He has suggested an alternative to algorithm, the simple Genetic

Algorithm and

geometric procedure in improving the feasible special purpose heuristics.

solutions through GA. This GA is also applied to a

set of assignment problems of substantial sise and A Genetic Algorithm method

for one dimensional

compared to an exact integer programming machine location problems is

suggested by Dijin


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approach. Gong et al [9] proposing the design of a generalised

flow line to minimise the backtracking of the jobs

A GA based heuristic for solving the GAP is formulating as a quadratic assignment

problem. He

discussed by Chu and Beasley [5] while presenting also discussed the specific

advantage of application

the problem the author compares the GA with the of GA approach when

compared to different

other existing heuristic algorithms and analytically heuristics.

57

Very interesting teacher assignment problems to procedures and compared the

results from these two

assign various courses to the teachers were dealt by procedures while confirming

the coded GA

the principle of multiple constraints to be solved by procedure is worthwhile than

the other. To solve a

a heuristic. This application problem is very well GAP with imprecise cost (s) /

times (s) instead of

reported and solved using the GAby Yen Zen Wang precise one, a Elitist GA is

applied and many

[39] which results are uniformly accepted by the extensive comparative

computational studies are

teachers of the universities. Application of a GA to performed on different

parameters while


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improve the existing GAP solution is proposed by developing the GA and is

validated through

Juell et.al [28] and his approach proved to be an illustrative procedures is

demonstrated by

optimum solution method and this is being Majumdar and Bunia [21].

demonstrated through numerical illustration.

An Immune Genetic Algorithm (IGA) is used to

An improved Hybrid Genetic Algorithm for the solve Weapon-Target Assignment

Problem (WTA).

GAP to find minimum cost assignment for a set of The used immune system

serves as a local search

jobs to a set of agents is proposed by Feltl et.al mechanism for Genetic Algorithm.

Besides, in our

[15].In this procedure the author suggests the two implementation, a new

crossover operator is

alternative initialisation of heuristics, a modified proposed to preserve good

information contained in

selection and replacement scheme for handling the chromosome. A comparison

of the proposed

infeasible solutions more appropriately, a heuristic algorithm with several existing

search approaches

mutation operator. Further the author authenticates shows that the IGA

outperforms its competitors on

that for the largest and most difficult instances, all tested WTA problems will

presented by Shang


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however, the proposed GA yields on average the et.al [12]. The coordination of

just-in-time

best results in shortest time. The concern of production and transportation in a

network of

allocation of projects to students through GA partially independent facilities to

guarantee timely

proving its very best advantages in finding a delivery to distributed customers is

one of the most

feasible project assignment, facilitating the challenging aspects of Supply Chain

Management.

discussion on the merits of various allocations and From a theoretical perspective,

the timely

supporting multi-objective decision making is well production/distribution can be

viewed as a hybrid

presented by Harper et.al [27]. combination of planning, scheduling and routing

problems, each notoriously affected by nearly

Over the last decade, there has been a rapid growth prohibitive combinatorial

complexity. David et.al

of the use of Genetic Algorithms in the various areas [8] proposed a novel meta-

heuristic approach based

of Production and Operations Management. on a Hybrid Genetic Algorithm

combined with

Chaudhry and Luo [4] provides a review of Genetic constructive heuristics. A

detailed case study

Algorithms research published in twenty-one major derived from industrial data is

used to illustrate the


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production and Operations Management journals potential of the proposed

approach.

from 1990-2001. More specifically, it identifies

research trends and publication outlets of Genetic Salik Yadav et.al [32]

introduces the algorithm

Algorithms applications. Our findings show that portfolio concept to solve a

combinatorial

there are only a handful of production and optimisation problem pertaining to a

Supply Chain.

operations management areas to which Genetic The Supply Chain problem is

modeled with

Algorithms have been applied as the solution capacity constraints and demand

variations over

approach. Furthermore, recognising and discussing different time periods to

minimise the total supply

potential research areas and outlets in which chain configuration cost. Thealgorithm portfolio is

researchers may target their work as well as the need implemented over various

problem instances to

for top ranked POM journals to consider publishing inspect and alleviate the

computational

Genetic Algorithms related papers. expensiveness of a solution strategy. Abunch

of five

algorithms are utilised hereby viz. AIS, GA,

Anshuman Sahu, RudrajitTadapar [2] discussed the Endosymbiotic Optimisation,

PSO and


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GAP with GA and Simulated Annealing heuristic Psychoclonal algorithm. The

observations reflect

58

the appropriateness and effect of algorithm

portfolios over the adopted supply chain, and

viability over other optimisation problems.

Chen et.al [6] studied an application of Genetic

Algorithms for flow shop problems with make span

as the criterion generating a Genetic Algorithm

(GA) based heuristic for these problems, and

compare the computational results of the heuristic

with the results of some existing heuristics. The

conclusions show that the GA based heuristic can

always give the best results in a short time on a SUN

workstation.

AJob scheduling problem with distinct due dates in

a single machine is considered by Lee and Choi [7]

with general penalty weights which are not

necessarily proportional to the processing times are Figure 2: Extended supplychain

applied to jobs either early or tardy. An optimal

timing algorithm is presented which decides the


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optimal starting time of each job in a given job

sequence. The near optimality of the solutions by

the GAis also illustrated by the comparison with an Abada, H. and EI-Darzi, (1996),

E-Solving the

exact algorithm. Berning et.al [14] consider a Timetable Problem Using Tabu

Search, Technical

complex scheduling problem in the chemical Report, University of Westminister.

process industry involving batch production. They

address three distinct aspects: (i) a scheduling Sahu, A. and Tadapar, R., (2007),

Solving the

solution obtained from a Genetic Algorithm based Assignment Problem Using

Genetic Algorithm and

optimiser, (ii) a mechanism for collaborative Simulated Annealing, International

Journal of

planning among the involved plants, and (iii) a tool Applied Mathematics, Vol. 36,

No. 1, pp. 1-7.

for manual updates and schedule changes and an

integrated system solution for optimising the supply Cerny, V., (1985),

Thermodynamical Approach to

chain process. the Traveling Salesman Problem: An Efficient Simulation

Algorithm, Journal Optimisation

In large-scale computer communication networks Theory Applications, Vol. 45,

No. 1, pp. 41-51.

(e.g. the nowadays Internet), the assignment of link


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capacity and the selection of routes (or the Chaudhry, S. S. and Luo, W.,(2005),

Application

assignment of flows) are extremely complex of Genetic Algorithms in Production

and

network optimisation problems. Efficient solutions Operations Management: A

Review International

to these problems are much sought after because Journal of Production Research,

Vol 43, Issue 19,

such solutions could lead to considerable monetary pp. 4083-4101.

savings and better utilisation of the networks. With

novel formulation and genetic modeling, the Chu, P. C. H. and Beasley, J. E.,

(1997), AGenetic

proposed algorithm by Hui et.al [38] generates Algorithm for the Generalised

Assignment

much better solutions than two well known efficient Problem, Computers &

Operations Research, 24,

methods in our simulation studies. pp. 17-23.

Chen, C. L., Vempati, S. V. and Aljaber, N., (1995),

An Application of Genetic Algorithms for Flow Shop

Problems, European Journal of Operational

The literature has been surveyed and an exclusive Research Vol 80, Issue 2, pp.

389-396.

survey on GAapproaches to GAP with reference to

SCM has been presented. Charles Lee Y. and Choi, J. Y., (2000). A Genetic


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