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Part selection and operation-machine assignmentin a flexible manufacturing system environment:a genetic algorithm with chromosomedifferentiation-based methodologyA K Choudhary2, M K Tiwari1,and J A Harding2*

1Department of Manufacturing Engineering, National Institute of Foundry and Forge Technology, Ranchi, India2Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, UK

The manuscript was received on 6 December 2004 and was accepted after revision for publication on 8 November 2005.

DOI: 10.1243/09544054JEM207

Abstract: Production planning of a flexible manufacturing system (FMS) is plagued by twointerrelated problems, i.e. part type selection and operation allocation on machines. Thecombination of these problems is termed the machine-loading problem, which is a well-known complex puzzle and treated as a strongly NP-hard problem. In this research, amachine-loading problem has been modelled, taking into consideration several technologicalconstraints related to the flexibility of machines, availability of machining time, tool slots,etc., while aiming to satisfy the objectives of minimizing the system unbalance, maximizingthroughput, and achieving very good overall FMS utilization. The solution of such problems,even for moderate numbers of part types and machines, is marked by excessive computationcomplexities and therefore advanced random search and optimization techniques are

needed to resolve them. In this paper, a new kind of genetic algorithm, termed a geneticalgorithm with chromosome differentiation, has been used to address a well-knownmachine-loading problem. The proposed algorithm overcomes the drawbacks of the simplegenetic algorithm and the methodology reported here is capable of achieving abetter balance between exploration and exploitation and of escaping from local minima.The proposed algorithm has been tested on ten standard test problems adopted fromliterature and extensive computational experiments have revealed its superiority over earlierapproaches.

Keywords:flexible manufacturing system, machine loading, genetic algorithm, chromosomedifferentiation

1 INTRODUCTION

Flexible manufacturing systems (FMSs) aim to com-bine the productivity of flow lines with the flexibilityof job shops, to attain very versatile manufacturingunits achieving high operational efficiencies. Theyare particularly designed for low-volume high-variety manufacturing, and good decision makingandmanagement are crucial to maximize the benefits

that they offer. Stecke and Solberg [1] mentionedfour decision stages for FMS, i.e. design, planning,scheduling, and control. This paper will focus onthe planning stage.

Stecke and Solberg [1] also defined five sub-problems in FMS planning:

(a) part type selection;(b) machine grouping;(c) production ratio determination;(d) palletfixture allocation;

(e) machine loading.If there is no grouping of machines, the palletfixtureallocation is ignored, and the production volume is

*Corresponding author: Wolfson School of Mechanical and

Manufacturing Engineering, Loughborough University, Ashby

Road, Loughborough LE11 3TU, UK. email: j.a.harding@

lboro.ac.uk

JEM207 IMechE 2006 Proc. IMechE Vol. 220 Part B: J. Engineering Manufacture

677


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fixed as non-splittable lot sizes. Planning then onlyneeds to address the following two problems.

1. Part type selection. Choose a subset of part typefor immediate production.

2. Machine loading. Allocate the operation andrequired tools of the selected part types amongthe machine groups subject to the technologicaland capacity constraints of the FMS.

These have been called the two indispensableprerelease problems and they have received consid-erable attention during the first stage of research inthis area, resulting in the emergence of a series ofarticles [2]. Rajagopalan [3] formulated a mixed-integer programming (MIP) solution, which solvedthe problem of part grouping and part-toolallocation and at the same time eliminated somenon-linearity of the model given by Stecke [4].Berrada and Stecke [5] pointed out that the lineariza-tion process carried out by Stecke [4] might notbe always viable and they proposed a new MIPformulation. Lashkariet al. [6] extended the problemof operation allocations discussed by Stecke [4]by introducing the concepts of refixturing andlimitations on the number of available tools. Thenon-linear model proposed by Lashkari et al. [6]

was subsequently reconsidered and simplified byWilson [7].

The second generation of articles that emerged

tended to redefine the characteristics of the loadingproblem and was more concerned with the problemof tool management, since several workers recog-nized that tools are expensive and therefore shouldbe considered as a scarce resource [812]. In thiscontext, Han et al. [13] proposed a model in whicha tool transport device is applied with the objectiveof minimizing tool traffic. Lee and Jung [14]discussed the application of goal programming formulti-objective problems which are to be handledsimultaneously during the production phase of anFMS. Another interesting tendency of this second

phase of literature was that different problems wereconsidered together, leading to the proposal of newhierarchies for the whole production-planningproblem. Joint solutions to the grouping and loadingproblems were tried in references [15] an d [16].Sawik [17] proposed a hierarchy for the produc-tion-planning task based on part type selection,machine loading, part input sequence, and opera-tional scheduling. Loading and routing problems

were considered [18]; Hsu and Matta [19] pointedout that feedback among the various models ofthe hierarchy is very important and proposed amodel to evaluate the feasibility of the loading

problem.Loading decisions act as an important link

between strategic- and operational-level decisions

in manufacturing, and the interrelationships of var-ious decisions and hierarchies in flexible manufac-turing environments have been discussed [20, 21].The formulation and solution methodologies for

various scenarios and combinations of parametersrelating to the loading problem in FMS have attractedthe attention of numerous researchers, resulting inmany publications including references [22] to [29].

Ammons et al. [30] described the bicriterionobjectives for the machine-loading problem, i.e. bal-ancing workloads and minimizing workstation visits.Bicriterion objectives for a loading problem, whichincludes balancing workloads and meeting duedate of part types, were also discussed in reference[28]. They suggested that a mathematical modelapproach is impractical because of the large compu-tational time requirement even for a moderate-sizetest problem. This problem is still complex despitethe advances in computing over the past 20 yearsand major hurdles exist in producing software thatcan achieve the requisite search time. Kumar andShanker [31] solved the part type selection andmachine-loading problem by formulating an MIPmodel to make their solution, which was based ona genetic algorithm (GA), more meaningful and lesscomputationally intensive. Yang and Wu [32] alsoapplied a GA-based integrated approach to solvethe FMS part type selection and machine-loadingproblem, where they introduced the concept of a vir-

tual job and virtual operation while implementingthe encoding scheme of their algorithm.

Contrary to the common practices of groupingmachines with identical tooling arrangements, Leeand Kim [33] discussed a scheduling problem thatensures that a loading plan is achieved for theconfiguration of partially grouped machines, whereeach machine has different tools, but multiplemachines can be arranged for each operation. Intheir study, Lee and Kim decomposed the schedul-ing problem into a two-shop problem, i.e. a routeselection problem and a job shop scheduling.

Therefore, they also discussed two types of solutionalgorithm in satisfying the decomposition problem.Mukhopadhyay and Tiwari [25] and Tiwari et al.

[23] attempted to solve the machine-loading pro-blem using heuristic procedures with the objectiveof minimizing the system unbalance and maximiz-ing throughput. Tiwari and Vidyarthi [34] applied aGA-based heuristic to solve the machine-loadingproblem. In order to exploit fully the effectivenessof the heuristic proposed by Tiwari et al. [23] andto ensure that optimal and near-optimal solutionsare obtained in each case, it is not only desirablebut also inevitable that the part type sequence

for practical problems be determined by someevolutionary random search rather than by adoptingfixed predetermined sequencing rules. In fact,

678 A K Choudhary, M K Tiwari, and J A Harding

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simultaneous determination of a part type sequence,system unbalance, throughput, and overall FMSutilization, by satisfying the technologicalconstraints (limited tool slot, planning horizon,

etc.) are key components of a machine-loadingproblem and hence makes it NP-hard in nature.

In the current research, a new kind of GA based onthe principle of chromosome differentiation (GACD)is applied and a heuristic has been developed tosolve a moderate-size machine-loading problempertaining to random FMSs. In addition, theproposed heuristic has also been applied to a well-known function optimization problem to show theeffectiveness of the algorithm. The GACD has thefollowing advantages over the GA.

1. GACD appears to strike a better balance between

exploration and exploitation, which is crucialfor any adaptive system, hence offering an edgeover GA.

2. Increasing diversity in population and restrictedcrossover result in faster information interchangebetween the chromosomes and thereby fasterconvergence than a GA.

3. It is evident from the analysis of the GACD, espe-cially with respect to schema sampling asrevealed by Bandhopadhyay and Pal [35], thatthe lower bond of number of instances offeredby schema sampled by GACD offers better values

than that of GA.The organization of the present paper is as follows.

The next section discusses the various aspects of theGACD and this is followed by a description of theproblem environment in section 3, which is furtherdivided into three, namely the loading problem inFMSs, the mathematical model, and the constraints.Section 4 deals with the solution methodology andalso describes details related to encoding, popula-tion initialization, fitness computation, selectionscaling, crossover, mutation, and the selection ofvarious GACD parameters. After that, an illustrative

example is taken and solved with proposed GACD-based heuristic in section 5. Results and discussionsare illustrated in section 6 to validate claims aboutthe performance of the proposed heuristic. Variousfunction optimization problems are also discussedin this section. Finally the paper concludes witha note about the future scope of this approach insection 7.

2 GENETIC ALGORITHM WITH CHROMOSOMEDIFFERENTIATION

2.1 Overview of the GAA GA is an intelligent probabilistic evolutionarysearch and optimization algorithm that simulates

the process of maturation by taking a collection ofchromosomes called a population of solutions andapplying various biologically inspired genetic opera-tors such as selection, crossover, and mutation that

occur naturally in reproduction. Each member ofthe population is evaluated according to some fit-ness measures, and fitter solutions are used forreproduction. New offsprings are generated andpoor solutions are replaced. The cycle of evaluation,selection, and reproduction is continued until asatisfactory and near-optimal solution is found.Pioneering work in developing GAs was carried outin references [36] to [38] and GAs have now beenapplied to a wide variety of problems. Elmarghyet al. [39] used a GA to address a dual-resourcescheduling problem in a manufacturing system con-strained by both machines and workers. Sinrich andSawaka [40] applied a genetic approach to developan efficient heuristic for the pickupdelivery locationproblem to get rid of the existing, computationallyprohibitive (01) programming model used earlierin the design procedure of a segmented-flowtopology-based material-handling system.

GAs are typically used in areas as diverse as imageprocessing, function optimization, pattern classifica-tion, neural network design and optimization, jobscheduling, and classifier systems [4146].

2.2 GA incorporating chromosome

differentiation

In the proposed algorithm, sexual differentiation isapplied so that genotypes (chromosomes) aredivided into two classes, i.e. male (M) and female(F), thereby giving rise to two populations, namelya male population (MP) and a female population(FP). In addition, these populations are madedissimilar artificially, and both populations aregenerated in a way that maximizes the Hammingdistance (HD) between the two classes. Crossover isonly allowed between individuals belonging to twodistinct populations [35], and this introduces a

greater degree of diversity and simultaneously leadsto greater exploration in the search space. Selectionis applied over the entire population, which servesto exploit the information gained so far. Thus theGACD accomplishes greater equilibrium betweenexploration and exploitation, which is one of themain features for any adaptive system, therebymaking the GACD superior to the simple geneticalgorithm (SGA).

2.3 Description of the GA with chromosomedifferentiation

The basic steps of the GA are shown in Fig. 1 andmodified versions of these processes are alsofollowed in GACD.

Part selection and operation-machine assignment in FMS environment 679



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Therefore subsections 2.3.1 to 2.3.6 will focus ondiscussing the various parameters of the GACD.

2.3.1 Population initialization

The construction of a chromosome in GACD isdelineated in Fig. 2. The data bits are representedas a individual bits here and are used to encode the

parameters of the problem. In addition, the firsttwo bits of a chromosome are called class bits asthese are used to indicate whether the class of thechromosome is M or F. Two separate populations,one (MP) consisting of M chromosomes and theother (FP) containing the F chromosomes, are main-tained over each generation. The total population(TP) equals MP FP, and initially MP FP TP/2,but the sizes of these two populations, MP and FP,may vary in different generations. The MP is initia-lized first; the data bits for each M chromosome aregenerated randomly and the two class bits arerandomly initialized to either 01 or 10. The FP is

then initialized, and data bits are generated foreach F chromosome, by maximizing the HDbetween the two populations. Both the class bits

of each F chromosome are set to 0. For twochromosomes g1 and g2; g1, g22t, where t is theinitial population, HD (g1,g2) is defined as the num-ber of bit positions at which the two chromosomesare dissimilar. The HD between the two populationsMP and FP is denoted as

HDMP,FP X

i

Xj

HD gigj

8

gi2 MPgj2 FP

1

Reproduction

Chromosome

Evaluation

Chromosome

Alteration

Population of

chromosomes

Dustbin

Offspring

Initial Chromosome

Evaluated Offspring

Discard

Chromosome

Selection

Fig. 1 Description of the SGA

Chromosomes

Class

bits

Data bits

01/

10

Data bits 00 Data bits

Male chromosome Female chromosome

F.class bit 00

M.class bit 01,10Fig. 2 Framework of chromosomes in the GACD




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For two chromosomes, the HD may be defined as

HD gi,gj

Xi;j2t

bibj

n

where bi2 gi,bj2 gj

2FP, the number of F chromosomes, is generated bysatisfying the previously mentioned constraints,

while permitting a certain element of randomness.

2.3.2 The pseudo code

The pseudo code for the generation of FP is asfollows:

{For i0 to MP doFor j0 toa-1 do // Initialization

Check (i, j) 0For i0 to FP doFor j0 toa-1 dok random (MP); // Return an integer in the range of0 to MP-1

While check (k, j) 0 //M (k, j) not to be chosenbeforeCheck (k, j) 1; // M (k, j) now chosen not to bechosen again

F (i, j) complement (M (k, j))}

Maximizing the HD between the two populationsresults in a greater degree of diversity in the totalpopulation.

2.3.3 Fitness evaluation

Only the a data bits of each chromosome are usedfor the fitness computation. Class bits do not contri-bute in the calculation of the fitness function andtherefore do not affect selection.

2.3.4 Selection

Members of the next generation are selected from allchromosomes in the enlarged sampling space, based

on their fitness evaluation. The best chromosomesare therefore preserved, and the characteristics ofelitism are therefore incorporated in both the Mand the F chromosomes.

2.3.5 Crossover

Crossover is applied with probabilityxcbetween theparticipating M and F chromosomes. Each parentcontributes one class bit to the child, but clearlythe F parent can only contribute a 0. The class of theoffspring is therefore primarily determined by themale parent who can contribute either 1 or 0. In

the present context, various types of crossoveroperators are applied on the data bits. Crossover isapplied until the following conditions are satisfied.

1. There are no traces of chromosomes remaining inthe mating pool

2. There are either only M or only F chromosomesremaining in the mating pool.

In case 1 the crossover procedure terminates, andin case 2 the remaining M or F chromosomes aremated with the best F or M chromosome. If, at theinitial stage, the mating pool contains chromosomesof only one class, the crossover procedure isdiscontinued.

2.3.6 Mutation

Various types of mutation operator are applied ondata bits with mutation probability mp. Mutationoperators are not applied on class bits.

3 PROBLEM ENVIRONMENT

3.1 FMS loading problem

The loading decision is concerned with the alloca-tion of operations and required tools to themachines or workstations that are subject to theresource and technological (direct and indirect)constraints of the system. Loading is one of themost critical decisions in the FMS planning. Eachoperation is assigned to only one machine; as a con-sequence, each part type has only one route throughthe system. This manufacturing policy is referred toas fixed routing since the machining centres in theFMS are flexible and most machines are capable ofprocessing more than one operation, when equipped

with appropriate tools. Thus a part can have morethan one route through the system compared withidentical machines existing in a group. Multiple-operation assignment and tool loading provideeconomical ways to alternate part routing practices.

In order to minimize the complexity of analysingthe loading problem of the system the followingassumptions have been made.

1. Initially, all the part types and machines are avail-able simultaneously.2. For all the part types, the processing time

required to complete all the operations is known.3. Non-splitting of part types is to be observed.4. An operation on a part type, once commenced, is

continued until it is completed.5. The transportation time between the machines is

neglected.6. The sharing and duplication of tool slots are not

allowed.

3.2 Proposed modelIn this research, the loading problem considered forthe analysis is for a random FMS equipped with M




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machines. Each machine has a fixed number of toolslots. Part types are arriving randomly in a givenplanning period and their operation times and toolslot requirements are well known. In the recent

past, several researchers have already addressedthis type of problem. The random FMS consideredhere is capable of performing operations that maybe essential or optional. Essential operations arethose that can be carried out on a particular machineusing particular tool slots while optional operationscan be performed on more than one machine.Thus, the FMS under consideration is able to deline-ate the flexibility pertaining to machine selection,operation processing, part type selection, etc. It isknown that the machine-loading problem deals

with selecting a subset of part types from a set ofpart types and assigning their operation to theappropriate machines in a given planning horizonin order to achieve certain performance measuresof the system by taking into account the technolog-ical constraints of the system. System unbalance,throughput and overall FMS utilization are themost commonly used performance measures in thecontext of the loading problem, whereas commontechnological constraints encountered include theavailability of machining time and tool slots onmachines.

In order to comprehend the complexities of aloading problem of a random FMS, consider an

example that consists of four machines, each havingfive tool slots, and the processing times for carryingout the different operations of the part types areknown. Each part type consists of four operations,

which can be performed on any of the machinesbut the sequence of operations remains unaltered.Different operations of part types can be performedon different machine with unequal machining timesand different tool slots. The versatility of eachmachine and its capability of performing many dif-ferent operations enable several operation assign-ments to be duplicated to generate alternative part

routes. A fairly large number of combinations there-fore exist in which operations of part type can beassigned to the different machines while satisfyingthe system constraints. The complexity of the pro-blem increases exponentially if other flexibilitiessuch as tooling flexibilities or part movement flex-ibilities are considered with the constraints of thesystem configuration and operational feasibility.The above operationmachine allocation combina-tion is to be evaluated using three common yard-sticks: system unbalance, throughput, and overallFMS utilization. System unbalance can be definedas the sum of over-utilized and non-utilized time

on all the machines available in the system. Minimi-zation of system unbalance is the same as maximiza-tion of system utilization. Throughput is referred to

as the units of part types produced and the ratio ofthroughput to maximum throughput gives the sys-tem efficiencies. Overall the FMS utilization may bedefined as the weighted average of the utilization of

all machines.It is very difficult to evaluate all the possible com-

binations of operationmachine allocation in orderto achieve minimum system unbalance and maxi-mum throughput. This is because it takes a largesearch space as well as a huge computational time.Earlier researchers have addressed the problems bydeveloping predetermined part-sequence-basedheuristic solutions that do not guarantee optimal ornear-optimal solutions. So heuristic solutions weresuggested to resolve these types of problem. Sincethe 1980s, a few intelligent heuristic techniques,namely GA, tabu search, and simulated annealing,have been extensively used by researchers to solvecomputationally complex optimization problems.In simple GAs, no restriction is placed upon theselection of mating pairs for the crossover operationand often chromosomes with similar characteristicsare mated, which wastes computational resourcesas no significant new information is gained out ofthe process because of trapping in local minima.The advanced GACD has been applied to minimizecomputational complexities, and to obtain anoptimal or near-optimal combination of opera-tionmachine allocation of given part types, with

tool slots and machine availability as constraints.Thus, the problem has been modelled keepingin mind the following multi-criteria objectivefunctions:

(a) minimization of system unbalance or maximi-zation of system utilization alone;

(b) maximization of throughput or maximization ofsystem efficiency alone;

(c) maximization of multi-objectives that arecombinations of (a) and (b);

(d) maximization of overall FMS utilization.

3.3 Notation

For easy recall and portability, the notation used isgiven in the Appendix at the end of the paper.

3.4 Integrality of decision variables

Several decisions can be characterized using binaryor 01 integer values, according to

Xp 1, if part typep is selected

0, otherwise

Ymt 1, if tool slott is allocated to machine m

0, otherwise




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zpomt

1, if operationp, o is arranged to theprocessing alternative m,t 2 SMTp,o

0, otherwise

8 0 9

Constraint C7: part mix constraint. The mix of thevarious part or product styles produced by thesystem is defined byjpand may be stated as

Xpp1

jpXp 1 10

3.6 Formulation of objective function

The problem described earlier is formulated as onewith multicriterion objectives; it can be expressedas follows.

1. The first objective function is to minimize the sys-tem unbalance, which is equivalent to maximiz-ing the system utilization according to

f1 SUseq SUminSUmax SUmin

Pm

m1 Pp

p1 Po

o1

PTpomtXpZ

poMT

Mjmax 11

2. The second objective function is to maximizethroughput or equivalently to maximize systemefficiency according to

f2

Ppp1

bpXp

Ppp1

bp

THseq THminTHmax THmin

12

3. The overall objective function is

F

m1f1m2f2m1m2 13

wherem1and m2are weighted parameters. For thepresent case, m1 m2 1.

4. The third objective function is the maximizationof overall FMS utilization. It may be defined asthe weighted average of mean utilization of eachmachine and stated as

However for the present case, the value of theweighted parameter for all the machines aretaken as 1.

UFMSPM

m1mUmPMm1m

14

4 GACD-BASED SOLUTION METHODOLOGY

This section defines the relevant terminologies anddiscusses the various design issues (e.g. encoding,population initialization, fitness function evaluation,crossover, mutation, selection, and scaling) relatedto the GACD with reference to its application in sol-ving the machine-loading problems of an FMS. The

major advantages of the GACD are its flexibility andability to adapt to the changing optimization criteriaand constraints. Factors such as the representation




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of individuals, encoding methods, initial M and Fpopulations, selection and scaling approaches, andchoice of genetic operators have a tremendousinfluence on the performance of the proposed

heuristic. In the next few sections, these factors arediscussed in detail.

4.1 Encoding

Encoding the chromosome of a solution and itsrepresentation is a key issue in a GAs implementa-tion. Encoding the GA includes encoding of the classbit and encoding of the data bits. Holland [38]applied a binary string encoding scheme, but suchschemes are not normally appropriate to real-worldproblems. During the past ten years, various non-

string encoding techniques have been evolved forparticular problems, e.g. real encoding to solve con-strained optimization problems and integer codingfor combinatorial optimization problems. In addi-tion to these, adjacency, permutation, and matrix-based encoding are also used. A sequence-orientedencoding scheme for data bits and binary stringcoding for the class bits have been adopted for thisresearch. For example, in a situation where eightparts are to be processed on machine, then it canbe encoded as

0 1 8 5 4 3 1 6 2 7

Here the class bit 01 ensures an M chromosomeand the data bits represent part types in sequence.

4.2 Population initialization

The construction of chromosomes in the GACD wasdiscussed generally in section 2.3, considering asequencing problem having eight-part types to beprocessed. The MP and FP are represented as

Initial populationMale population Female population0 1 4 5 7 3 1 6 2 8 0 0 5 4 2 6 8 3 7 10 1 5 6 1 4 2 8 7 3 0 0 4 3 8 5 7 1 2 61 0 7 2 6 4 5 1 3 8 0 0 2 7 3 5 4 8 6 11 0 3 1 5 2 4 7 8 6 0 0 6 8 4 7 5 2 1 3

4.3 Fitness computation

During each generation, chromosomes are evaluatedusing some measures of fitness. In this research,M POP and F POP are generated by the heuristicmentioned in section 2 and their fitness values arecalculated according to a part type sequence, while

also taking into account all the technologicalconstraints. The mathematical models of fitnessfunctions are shown in equations (11) to (14).

Various steps are required to evaluate a part typesequence using an objective function, as illustratedin Fig. 3.

4.4 Selection

Selection is performed over all the population(M POP F POP) disregarding class information.

A scaling approach helps to sustain the steadyselective pressure in the population and preventsthe premature convergence of the population to asuboptimal solution. In this research, variousscaling approaches have been tested, includingdynamic linear scaling, sigma truncation, power lawscaling, logarithmic scaling, windowing, normal-izing, and Boltzmann selection. Computational

experiments reveal that the proportional-selection-based roulette wheel strategy embedded with theBoltzmann selection scheme works more effec-tively than others. For chromosome K and fitness

fk, the scaling function may be defined as

f0k efk=T 15

Selection pressure is low when the controlparameter T is high. Now the selection probabilitycan be interpreted as

Pk

f0kPPOP SIZEk1

fk16

Ck POP SIZE Pk, where Ck is the improvedroulette wheel count (IRWC).

The best individual always survives into the nextgeneration so as to enable the GACD to convergemore rapidly. For example, if the population consistsof two M POP and two F POP, the above proceduremay be applied as shown in Table 1. Thus the nextgeneration consists of copies of the first, third, andfourth sequences and two copies of the second

sequence.

4.5 Crossover

In the GACD, crossover is applied until the conditionas mentioned above is satisfied. Crossover is aprocess in which one M and one F chromosomerecombine to produce two chromosomes (M or For both). Various types of standard crossover opera-tor are most commonly used in solving sequencingand scheduling problems. These include heuristiccrossover, partially mapped crossover (PMX),

enhanced edge recombination (EER), order cross-over (OX), uniform-order-based crossover (UOX),and cycle crossover (CX). In this research, PMX is




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used and results obtained have been comparedwith others.

4.6 Procedure for the partially mapped crossover

Step 1. Two positions are selected at random fromalong the strings; the substring formed in this way

is known as the mapping section.Step 2. Exchange two substrings between parents toproduce a proto-child.

Step 3. Establish the mapping relationship betweenthe two mapping sections.Step 4. Legalize offsprings with the mapping relation.

The procedure for partially mapped crossover is asshown in Fig. 4.

4.7 MutationAfter crossover, the strings are normally subjected tomutation. This introduces some extra variability,

Fig. 3 Flow chart describing the calculation of objective functions




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provides and maintains diversity in a population,and explores the search space. In this research,mutation is applied on a data bit. The mutationprobability is the rate with which a bit will beflipped. During the past decade, several mutationoperators have been proposed for permutationrepresentation, such as inversion, insertion,displacement, and reciprocal exchange mutation.In this research, a heuristic-based method is

applied, using the neighbourhood technique toproduce an improved offspring. The mechanism ofmutation operator is discussed.

4.8 Procedure: heuristic mutation

Step 1. Pick upn genes at random.Step 2. Generate the neighbourhood offsprings byconsidering all possible permutations of theselected genes.Step 3. Evaluate all the neighbours and select thebest as the offspring.

The proposed heuristic-based mutation procedureis illustrated in Fig. 5.

4.9 SELECTION OF THE GACD PARAMETERS

The parameter setting is a time-consuming issue inGAs. The parameters are the population sizePOP SIZE, the number of generations MAX GEN,the crossover probabilityxp, and the mutation prob-

ability mp. Michalewicz [37] mentioned that selec-tion of the parameters of the GA still remains an artrather than science.

4.9.1 Population size factor

GA experts have recommended that the populationsize varies as a multiple of chromosome length (CL)according to

POP SIZE PSF CL (levels considered 1, 2)

where PSF in the population size factor. In this

research, POP SIZE INIM POP INIF POP.

4.9.2 Crossover and mutation probability

Moderately large values of xc (0.40.9) and smallvalues ofmp (0.050.2) are commonly employed ina GA. Increasing values of xp and mp promoteexploration at the expense of exploitation. However,in this research, intensive experiments have beenconducted to achieve this trade-off between explora-tion and exploitation by varying xp, mp, MAX GEN,and PSF to obtain optimal or near-optimal solutions,and the results obtained are within the range men-

tioned above. The exact values of these parametershave been specified in the following illustrativeexample.

Table 1 Description of the IRWC

Class bit Data bit (sequence) fk fk0 pk Ck IRWC

0 1 3 4 6 2 1 5 338 1.34 0.1719 0.6876 10 1 3 4 2 1 5 6 1152 2.71 0.3474 1.39 20 0 4 3 1 5 6 2 617 1.71 0.2190 0.876 10 0 4 3 1 5 6 2 822 2.04 0.2625 1.046 1Total 7.80

Step 1:-

Male Parent Female Parent

0 1 3 5 6 4 2 1 7 8 0 0 6 4 3 5 7 8

Step 2:-

0 1 3 5 5 7 8 1 7 8 0 0 6 4 3 6 4 2

Step 3:-

Step 4:-

Male Child Female Child

0 1 3 6 5 7 8 1 4 2 0 0 5 7 3 6 4 2 8

5 7 8

6 4 2

Fig. 4 PMX applied to M and F chromosomes

Select three positions at random

0 1 6 5 2 8 3 4 1 7

The neighbors formed with data-bits chosen

0 1 6 8 2 1 3 4 5 7

0 1 6 1 2 5 3 4 8 7

0 1 6 1 2 8 3 4 5 70 1 6 8 2 5 3 4 1 7

0 1 6 5 2 1 3 4 8 7

Fig. 5 Description of heuristic-based mutation




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5 AN ILLUSTRATIVE EXAMPLE

To demonstrate the effectiveness of the proposedmodel and the quality of the proposed GACD solu-

tion approach, a slightly modified version of theillustrative example considered in references [23] to[25] for a random type FMS is now solved. Table 2shows the detailed description of the test problemfor the given FMS type. Nine more problems gener-ated by Mukhopadhyay and Tiwari [25] have alsobeen solved to show the effectiveness of the pro-posed algorithm. Machine data are given in Table 3.

The following steps describe the application of theabove GACD-based heuristic approach for solvingthe machine-loading problem.

Step 1. Set the initial and global parameters:INIM POP INIF POP 5, xp 0.5, mp 0.1,MAX GEN 30, and f1 0.Step 2. The total number of part types,pmax 8, andpart type sequence is generated as per initializationand initial population mentioned in section 4.Step 3. Choose the objective functionsf1, i.e. minimi-zation of the system unbalance.Step 4. Initialize GEN 0 1, and recall the parttype sequence generated in step 2. It comes out to be

[0 1 5 4 3 7 1 6 8 2]Step 5. The value of f1 is calculated as per the flowchart in Fig. 2. Only data bits are used for calculationof the objective function.

Step 6. For the M chromosome the data bits of thepart type sequence are [5 4 3 7 1 6 8 2], correspond-ing to the objective function 1, i.e. minimization ofthe system unbalance: SU 55, TH 45,JRNSU (1,8), JRTSC (2), and f1 0.971, where SUis the system unbalance, TH is the throughput, JRNSUis the part rejected owing to negative system unba-lance, and JRTSC is the part rejected owing to toolslot constraints. Now, the selection strategy isapplied on all M POP and F POP.Step 7. PMX is applied on selected chromosome asmentioned in section 4.Step 8. Heuristic-based mutation is applied on databits related to the part type sequence.Step 9. Initial population selected for reproductionconsists of crossover offspring and mutationoffspring and, hence, constructs the extendedpopulation.Step 10. Then the extended population is evaluatedusing the objective function f1and the selection pro-cess is carried out according to the methoddescribed in section 4.Step 11. The terminal test is as follows: if GEN MAX GEN, terminate and print the output.Otherwise, GEN GEN 1 and go to step 4.

Therefore after executing all the steps of algorithmthe following part type sequence is obtained: parttype sequence [5 4 3 7 6 1 8 2] and, correspondingto this sequence, the objective functions are asfollows: SU 0 min, TH 36 units, f1 1.0,JRNSU {1,3,6,8}, and JRTSC {}. The overall FMSutilization equals 76 per cent.

6 RESULTS AND DISCUSSION

Research contributions in the broad area of GA

applications show that in SGA applications the num-ber of generations required to achieve near-optimalsolutions are substantially high. Therefore it isessential that new GAs be identified that can solvelarge-size combinatorial problems in fewer genera-tions and also take less central processing unit(CPU) time. With these primary requirements in

Table 2 Description problem 1 (adopted from Shanker

and Srinivasula [27])Parttype

Operationnumber

Batchsize

Unitprocessing time

Machinenumber

Tool slotneeded

1 1 8 18 3 12 1 9 25 1 1

25 4 12 24 4 13 22 2 1

3 1 13 26 4 226 1 2

2 11 3 34 1 6 14 3 1

2 19 4 15 1 9 22 2 2

22 3 2

2 25 2 16 1 10 16 4 12 7 4 1

7 2 17 3 1

3 21 2 121 1 1

7 1 12 19 3 119 2 119 4 1

2 13 1 113 3 113 1 1

3 23 4 18 1 13 25 1 1

25 2 125 3 1

2 7 2 17 1 1

3 24 1 3

Table 3 Data pertaining to machines

Machinenumber

Planninghorizon

Number of toolslots available

M1 480 5

M2 480 5M3 480 5M4 480 5




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mind, a new kind of GA, known as the GACD, hasbeen coded. A high level of population diversity isintroduced and subsequently maintained in thissystem, resulting in faster information interchangeamong chromosomes, and therefore faster conver-gence of the algorithm. In addition, several modifi-cations are performed to make it suitable as far as acomputationally complex problem is concerned ina lower number of generations and modest CPU time.

For all the objectives considered, the GACD is ableto achieve the best solution, thereby proving its abil-ity as a powerful randomized search optimizationtool. It uses a modified population generation con-cept, new crossover operator, and heuristic-based

mutation mechanism, enabling it to obtain near-optimal results in substantially fewer generations,for the given loading problem. After a very few gen-erations, the algorithm shows the trace of conver-gence of solutions, making it more effective thanthe simple GA used by Kumar and Shanker [31],

Yang and Wu [32], and Tiwari and Vidyarthi [34].The proposed GACD-based heuristic employs

four different objective functions namely systemunbalance alone, throughput alone, combination ofthroughput and system unbalance, and overall FMSutilization while keeping in view the system con-

straints (available machining time, tool slot capacity,etc.). It is important to highlight here that the con-cepts of overloading and underloading of machineshave already been taken into account with systemunbalance in this research. In this context, it isimportant to mention that a few of the relatedresearches have used slightly different definitions.For example, Shanker and Srinivasulu [27] havetaken only non-utilized machine time in the systemunbalance, whereas Mukhopadhyay and Tiwari[25], Tiwari et al. [23], and Tiwari and Vidyarthi[34] have considered the non-utilized machine timeas well as the over-utilized machine time in their

system unbalance. Mukhopadhyay and Tiwari haveadopted the maxmax rule for the allocation of anunassigned part type in anticipation that the system

unbalance improves further. The performance of theproposed GACD-based heuristic is superior when itis compared with a few predetermined sequencingrules, such as the longest processing time (LPT),last-in first-out (LIFO), first-in first-out (FIFO), andalso with other approaches given in the literature.Figure 6 reveals one such comparison in detail andfound that the proposed heuristic outperforms theresults obtained by Tiwariet al. [23].

To confirm the performance of the proposedheuristic, nine more problems generated byMukhopadhyay et al. [4] have been solved and acomparative study of results from Mukhopadhyayet al. [4], Shanker and Srinivasulu [27], and Tiwariet al. [23] are shown in Fig. 7. It is evident from thefigure that, in these cases, the proposed heuristicoutperforms the others. A detailed description ofthe parameter setting for different objective func-tions, part type sequence fitness function values,corresponding system unbalance and throughput,overall FMS utilization, and unassigned part type(along with cause of rejection) are given in Table 4.

As mentioned in section 4, the selection of thecontrol parameters of the GACD is a time-consumingand frustrating issue. In this research, intensive

Fig. 6 Comparison of the GACD-based heuristic with the heuristic proposed by Tiwariet al. [23] (using different sequen-cing rules)

[27] [4]

[23]

Fig. 7 Comparison of proposed GACD-based heuristic

with the heuristics proposed by variousresearchers




14/19

experiments have been carried out to determine theparameters governing the execution of the GACDalgorithm and these parameters are the initial malepopulation, initial female population, crossover rate,mutation rate, crossover type, mutation type, selec-tion strategies, etc. In most applications of GAs, thetuning of the parameters is based on some trials onknown problems. Here it is found that near-optimal

part type selection are obtained at a MAX GEN of30. Various crossover operators which are used forsequencing and scheduling problems were tested onsample problem and found that the PMX applied inthe GACD outperforms other crossover operators.The comparative results are shown in Fig. 8.

A heuristic-based mutation operator is proposedand its performances are compared with othermutation operators. Figure 9 reveals the superiorityof heuristic-based mutation over others. Numerousselection scaling methods were examined while sol-ving these problems. It is evident from Fig. 10 thatthe proportional-selection-based roulette wheel

strategy embedded with the Boltzmann selectiondominates others. The performances of severalcombinations of crossover and mutation operators

and their influences on the outcome of the loadingproblem are given in Fig. 11. Other problemsadopted from Mukhopadhyayet al. [4] were solvedand the results are reported on the basis of thesetwo types of operator.

The performance of the GACD has been comparedwith that of the GA on the test problem reported byTiwari and Vidyarthi [34]. In Fig. 12, with the help

of graphical aids, a comparison has been madebetween the GACD and the GA for the previouslymentioned problem. From this comparison it isaxiomatic that the best result obtained for thepopulation remains the same. However, averagevalues of the population for objective function 1,i.e. system unbalance, has a lower magnitude thanthat obtained by the SGA.

To validate the proposed GACD-based heuristic,four different types of multi-modal function optimi-zation problem have been considered with variouscomplexities, as described in the literature, to checkthe robustness of the solution quality. Here, these

functions [47] have been chosen because they havebeen widely used by other researchers to show theefficacy of their solution methodologies.

Table 4 Details of the parameters and results obtained using the proposed GACD-based heuristic

Problemnumber

Objectivefunction

Parametersetting ofGACD*

Part typesequence

Part type unassigned Value ofobjectivefunction

Systemunbalance

Throughput Overall FMSutilization(%)

Negative systemunbalance Tool slotconstraints

1 1 5,0.5,0.2,30 5,7,4,2,6,3,1,8 6,3,1,8 1.000 0 36 882 5,0.5,0.1,30 4,7,3,1,2,6,8,5 2,6 8 0.640 14 48 753 5,0.5,0.1,30 4,7,3,2,1,5,6,8 2,6 8 0.796 14 48 75

2 1 5,0.5,0.2,30 6,2,4,5,3,1 3, 1 0.919 18 46 722 5,0.5,0.1,30 5,3,1,4,2,6 or 2 0.630 84

3,4,5,1,6,2 2 154 633 5,0.5,0.1,30 5,4,3,1,2,6 or 2 0.891 84

5,3,1,4,2,6 2 154 633 1 5,0.5,0.2,30 4,5,2,3,1 1 0.963 72 69 88

2 5,0.5,0.1,30 2,3,1,5,4 4 0.924 128 73 81.63 5,0.5,0.1,30 5,3,2,4,1 or 4 4 0.928 81.6

5,1,2,3,4 128 734 5,0.5,0.2,30 819 51 1005 1 5,0.5,0.2,30 3,1,5,4,6,2 6,2 0.903 187 53 76

2 5,0.5,0.1,30 6,2,5,1,3,4 4 0.815 479 62 723 5,0.5,0.1,30 2,1,4,6,3,5 3 0.829 76 61 70.2

6 1 5,0.5,0.2,30 1,5,2,4,6,3 3 0.985 28 61 862 5,0.5,0.1,30 4,3,1,5,6,2 2 0.863 314 63 783 5,0.5,0.1,30 1,4,6,2,5,3 3 0.910 28 61 80.4

7 1 5,0.5,0.2,30 4,1,2,5,6,3 6 3 0.914 165 54 782 5,0.5,0.1,30 3,6,1,4,5,2 and 2 0.807 486 63 68

3,1,2,5,6,4 4 2313 5,0.5,0.1,30 1,5,2,3,6,4 4 0.843 231 63 72.4

8 1 5,0.5,0.2,30 7,5,4,6,3,1,2 3,1,2 0.993 13 44 922 5,0.5,0.1,30 4,5,2,7,3,1,6 and 1,6 0.800 63 81.6

5,2,7,1,6,4,3 6,3 288 483 5,0.5,0.1,30 3,2,5,7,1,6,4 1,6 0.883 63 48 81.6

9 Any sequence 309 88 10010 1 5,0.5,0.2,30 2,3,5,4,1,6 1 0.957 82 54 81.3

2 5,0.5,0.1,30 3,5,1,2,6,4 4 0.887 122 56 77.23 5,0.5,0.1,30 5,1,3,6,2,4 4 0.860 122 56 77.2

*Parameter setting: (INIM POP, crossover probability, mutation probability, maximum generation).




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6.1 Dejongs functionf5

This is a spiky function (also known as Shekels fox-holes) with 25 sharp spikes of various heights. Thefunction has two variables and the solution isencoded using binary digit. The task of the GACD isto locate the highest peak. The expression for f5is

f5 0:002X25j1

1

jP2i1

xiaij6


This is a rapidly varying multi-modal function of twovariable. The variables are encoded using binarydigits each and assume values in range [100, 100].

f6 has been employed earlier [48] for comparativestudies where it is referred to as the sine envelopesine wavefunction and is expressed as

f6 0:5sin2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix21x22

q 0:5

10:001x21x22

2


This function is also similar to f6but has the barrierheight between adjacent minima approaching zeroas the global optimum is approached. It is given by

f7 x21x

22

0:25sin250x21x22

0:1 1:0

6.4 Ackleys function

This function is obtained by modulating an expo-nential function with a cosine wave of moderateamplitude. The complexities of this function areshown in Fig. 13, and can be expressed as

minfx1,x2 c1exp c2

ffiffiffi1

2

rX2j1

x2j

!

exp 1

2

X2j1

cosc3xj

" #c1e,

5 6xj6 5, j 1,2

Table 5 shows the comparative result of the bestvalues obtained after 100 generations. It can beeasily understood from Table 5 that the GACD out-performs the SGA for all these functions.

The chromosome differentiation can therefore beextremely beneficial for challenging combinatorialproblems. In addition, it has been shown to outper-form traditional GAs in function optimization pro-blems. Therefore, the proposed GACD-basedapproach may be applicable to resolve other similarcombinatorial complex problems related to manu-facturing systems.

The proposed GACD-based heuristic has beencoded in C and the program was run on an IBMpersonal computer with a Pentium 4 CPU at 1.2 GHz.

0.7

0.75

0.8

0.85

0.9

0.95

1

1 6 11 21 26 31 36 41 46 51 56

Genarations

A

veragevalueofobjective

functionf1

DIS

RE

INS

INV

HEU

DIS: Displacement

RE: reciprocal exchange

INS: insertionINV: inversionHEU: Heuristic

Fig. 9 Performance of mutation operators

PMX:Partially mapped crossoverUOX:Uniform Order-based crossoverEER:Enhance edge recombination crossoverCB: Constraints based crossoverOX: Order crossoverCX: Cyclic crossover

Performance of Crossover operators (Problem no.1)

0.8

0.85

0.9

0.95

1

1 5 9 13 17 21 25 29 33 37 41 45

Generations

Averagevalueof

objectivefunctionf1 PMX+GA

UOX+GA

EER+GA

PMX+GACD

OX+GA

CX+GA

Fig. 8 Comparison of various crossover operators




16/19

7 CONCLUSIONS

In this paper, a machine-loading problem with four

objective functions and two technological con-straints has been considered for analysis. One ofthe key issues tackled in this research was to deter-mine the number and sequence of part types to beprocessed on machines by satisfying technologicalconstraints to achieve the minimum systemunbalance and maximum throughput. From theliterature, it was known that the underlyingloading problem is computationally complex andmathematically intractable to solve. A different GAhas been proposed, employing the concept of chro-mosome differentiation to enhance the capabilityof the existing SGA. The resulting heuristic solution

has been tested to address a well-known part typesequence and operation allocation problem in theFMS environment. Here it is significant to mention

that, while coding the algorithm, binary coding isapplied to the class bit, and real coding is used forthe data bit, to minimize the chromosome length

and to maintain the level of infeasiblity. In additionto the above modifications of the GA, a strategy ofmaximizing the HD has been used to incorporatechromosome differentiation in the GACD for theFMS machine-loading problem. Restricted cross-over, resulting in faster information interchange,advanced mutation, and an evolutionary selectionscheme have led to a better balance betweenexploration and exploitation. Application of theproposed approach has been shown on various testproblems and the results achieved demonstratethat the algorithm performs well in terms of boththe solution quality and the CPU time. Extensive

computational experience of various types ofcrossover, mutation, and selection scaling aredemonstrated. The proposed algorithm has been

0.8

0.85

0.9

0.95

1

5 10 15 20 25 30 35 40 45 50

Generations

Averagevalu

eof

objectivefunctionf1 EER-RE

PMX-RE

EER-INS

CX-DIS

UOX-INV

PMX-HEU

EER-RE: combinations of enhance edge recombination and reciprocal exchangePMX-RE: combination of partially mapped crossover and reciprocal exchangeEER-INS: combination of enhance edge recombination crossover and insertionCX-DIS: combination of cyclic crossover and displacementUOX-INV: combination of uniform order-based crossover and inversionPMX-HEU: combination of partially mapped crossover and heuristic based mutation

Fig. 11 Performance of combination of operators (problem 1)

Dy-Lin: Dynamic Linear ScalingSIG-TRU: Sigma TruncationPL: Power Law ScalingLOG: Logarithmic ScalingNORMAL: Normal SelectionRWS+BOLTZ:Roulette Wheel Embedded with Boltzmann Selection

Fig. 10 Comparison of various scaling approaches with the proposed approach for the objective function combining sys-tem unbalance and throughput (objective function f3)




17/19

tested on several complex constrained and non-constrained optimization functions, and the resultsobtained have shown a significant improvementover SGA.

The research possesses enough scope forextension to encompass several other allocationsof resources, such as pallets, fixtures, and auto-

mated guided vehicles. The addition of apenalty and a few more objective functions, e.g.minimization of part movement, tool change

Fig. 13 Plot representing complexity of Ackleys function

P e r f o r m a n c e o f G A A N D G A C D ( P r o b l e m n o . 1 )

0 .6

0 . 6 5

0 .7

0 . 7 5

0 .80 . 8 5

0 .9

0 . 9 5

1

1 5 9 1 3 1 7 2 1 2 5 2 9 3 3 7 4

N u m b e r o f G e n e r a tio n s

G A

G A C

( a )

( b )

Fig. 12 Comparisons of (a) the best value of the objective function and (b) the average value of the population densityobtained by the GA and the GACD

Table 5 Comparative results of GA and GACD forfunction optimization problems

Solutionnumber

Best value obtained

Function Optimal GACD GA

1 Dejongsf5 1 0.987 324 4 0.910 832 2

2 Dejongsf6 1 0.970 266 2 0.911 959 13 Dejongsf7 1 0.995 027 1 0.956 223 14 Ackleys 0 0.000 462 8 0.082 354 9




18/19

together with the measure of flexibility pert-aining to the machine, and material handling,

will be the topics for future exploration. Inconclusion, the proposed GACD-based heuristic

approach offers a better solution in a lower numberof generations when tested on well-knownmachine-loading problem of an FMS and also onother functions.

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APPENDIX

Notation

a processing alternativesJRNSU part rejected owing to negative system

unbalanceJRTSC part rejected owing to tool slot constraintsm machine number 1, 2, 3, . . .,Mo operation numberOp number of operations on part typepp part type numberSU system unbalanceSUmax maximum system unbalanceSUmin minimum system unbalanceSUseq system unbalance corresponding to a

particular part sequenceT(a,m)(p,o) time available on machine m before allo-

cation of operation o of part type pT(c,m)

(p,o) time required by machine m foroperation o of part type p

T(r,m)(p,o) time remaining on machine m after

allocation of operation o of part type pTHmax maximum throughputTHmin minimum throughputTHseq throughput corresponding to a particular

part sequenceUm mean utilization of each machine

wm/w

w* workload at bottleneck stationwm workload on machinem

(r,o) operation of part type r

jmax length of planning horizonjp part mix of part typept(a,m)

(p,o) tool slot available on machinemallocatedto operation o on machine m of parttypep

t(c,m)(p,o) tool slot required by machine m for

operation o of part type p

t(r,m)(p,o)

tool slot remaining on machine m afterallocation of operationo of part typep

tm tool slot capacity on machinem


Part Selection

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