Solving machine-loading problem of a ﬂexible manufacturing ...€¦ · Production, Manufacturing...

European Journal of Operational Research 175 (2006) 1043–1069

www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Solving machine-loading problem of a flexiblemanufacturing system with constraint-based

genetic algorithm

Akhilesh Kumar a, Prakash b, M.K. Tiwari a, Ravi Shankar c, Alok Baveja d,*

a Department of Manufacturing Engineering, National Institute of Foundry and Forge Technology (NIFFT),

Ranchi 834003, Indiab Department of Metallurgy and Materials Engineering, National Institute of Foundry and Forge Technology (NIFFT),

Ranchi 834003, Indiac Department of Management studies, Indian Institute of Technology Delhi, New Delhi 110016, India

d School of Business, Rutgers, The State University of New Jersey, Camden, NJ 08102, USA

Received 12 January 2004; accepted 6 June 2005Available online 31 August 2005

Abstract

Machine-loading problem of a flexible manufacturing system is known for its complexity. This problem encompassesvarious types of flexibility aspects pertaining to part selection and operation assignments along with constraints rangingfrom simple algebraic to potentially very complex conditional constraints. From the literature, it has been seen thatsimple genetic-algorithm-based heuristics for this problem lead to constraint violations and large number of genera-tions. This paper extends the simple genetic algorithm and proposes a new methodology, constraint-based genetic algo-rithm (CBGA) to handle a complex variety of variables and constraints in a typical FMS-loading problem. To achievethis aim, three new genetic operators—constraint based: initialization, crossover, and mutation are introduced. Themethodology developed here helps avoid getting trapped at local minima. The application of the algorithm is testedon standard data sets and its superiority is demonstrated. The solution approach is illustrated by a simple exampleand the robustness of the algorithm is tested on five well-known functions.� 2005 Elsevier B.V. All rights reserved.

Keywords: Genetic algorithm; Flexible manufacturing system; Machine loading

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reservdoi:10.1016/j.ejor.2005.06.025

* Corresponding author. Tel.: +1 856 225 6694; fax: +1 856225 6231.

E-mail address: [email protected] (A. Baveja).

1. Introduction

�Time to market� is now accepted as a pivotalstrategy to increasing and/or retaining a firm�s

ed.

mailto:[email protected]

1044 A. Kumar et al. / European Journal of Operational Research 175 (2006) 1043–1069

market share in the face of global competition. Formany industries, it is an accepted norm that areduction in time to market of even one week canaccrue significant revenues. Currently, firms oftenapply flexible manufacturing system (FMS), to re-duce the time to market for their products. FMSensures quality product at lowest cost while main-taining small lead-time. So, firms adopt the FMSas a means for meeting mounting requirementsof customized production. One of the main pur-poses of the FMS is to achieve efficiency of awell-balanced transfer line while retaining the flex-ibility of the job shop (Stecke, 1983).

The FMS is a set of flexible machine tools hav-ing general-purpose NC machines and an auto-mated material handling system, all under acentral computer control. As FMSs are veryexpensive, it is essential to manage them effectivelyto achieve desired goals with less investment risk;and this largely depends on how the decision prob-lems of FMS are tackled. Stecke (1983) mentionsfour stages of decision problems for the FMS:designing, planning, scheduling, and control. Theauthor deals with problems where designing deci-sions are already made and hence efforts are fo-cused on the planning problem. FMS planningconsists of pre-release and post-release decisions.The pre-release decision problems consider pre-arrangement of parts and tools before the mainprocesses of an FMS. FMS scheduling or post-re-lease decision considers sequencing and routing ofpart types, when the system is in progress. Stecke(1983), Sarin and Chen (1987) discuss the follow-ing post-release decision problems: (a) machinegrouping, (b) part-type selection, (c) batching ofpart type, (d) production rate determination, (e)resource allocation, and (f) loading.

Hwang (1986) investigates the production-plan-ning problem and finds that the two sub-prob-lems—part selection and machine loading—arecrucially important. Decisions pertaining to load-ing problems receive their inputs from the preced-ing decision levels (such as grouping of resources,selection of part mixes, aggregate planning) thatgenerate inputs to the succeeding decisions relatedto scheduling resources, dynamic operations plan-ning, and control. Hence, it is clear that the load-ing decision is acting as an important link between

strategic and operational level decisions in manu-facturing. Van Looveren et al. (1986), Kusiak(1985), Singhal (1978), Whitney and Gaul (1984)discuss the inter-relationships of various decisionsand their hierarchies in a flexible manufacturingenvironment. Liang and Dutta (1993) address thepart selection and machine-loading problem simul-taneously, which had been addressed separately byprior researchers. Part selection, machine loading,and tool configuration are three different, butinterlinked, problems with shared restrictions suchas tool magazine capacity, part-tool machine com-patibility and available machine time. Still, mostpast research has treated part selection, machineloading, and tool configuration separately primar-ily due to the complexity in solving themsimultaneously.

In this paper, a machine-loading problem in anFMS environment is considered. The machine-loading problem can be described as ‘‘. . . given aset of part types to be produced, set of tools thatare needed for processing the parts on a set of ma-chines, and using a set of resources such as mate-rial handling systems, pallets and fixtures, howshould the parts be assigned and tool allocatedso that some measure of productivity is opti-mized’’. In fact, Stecke (1983) studied machine-loading problems in detail and described its sixmain objectives:

1. Balancing the machine processing time;2. Minimizing the number of movements;3. Balancing the workload per machine for a sys-

tem of groups of pooled machines of equalsizes;

4. Unbalancing the workload per machine for asystem of groups of pooled machines ofunequal sizes;

5. Filling the tool magazines as densely aspossible;

6. Maximizing the sum of operations priorities.

It is clear from the above literature that the ma-chine-loading problem involves multiple objectives(Stecke, 1983; Shanker and Srinivasulu, 1989).Ammons et al. (1985) resolves the loading problemconsidering a bi-criteria objective of balancingworkload and minimizing work stations visits,

A. Kumar et al. / European Journal of Operational Research 175 (2006) 1043–1069 1045

whereas Shanker and Tzen (1985) consider balanc-ing workload and meeting due-date of part types.Tiwari et al. (1997) and Mukhopadhyay et al.(1992) tackle machine-loading problem using heu-ristic approaches with an objective of minimizingsystem unbalance and maximizing throughput.Contributions of several researchers pertaining tothe loading problem have been summarized suc-cinctly in a detailed survey of literature by Griecoet al. (2001). Extending this survey, we present inthis article an updated list that incorporates someof the recent contributions (Tables 1a and 1b).

Objective functions used in this research are theminimization of system unbalance and the maxi-mization of throughput for the following reasons:

(1) Minimization of system idle time leads tohigher machine utilization,

(2) One of the most important goals of any load-ing policy is enhancing total system output,which is the same as throughput,

(3) Kim and Yano (1997) have found thatthroughput maximization by balancing theworkloads on the machine often results inlimiting the tardiness.

Past research efforts have focused on solving theproblem with several pre-determined sequencingrules such as: shortest processing time (SPT), lastin first out (LIFO), first in first out (FIFO), earliestdue-date (EDD). Machine-loading problem can beaddressed mainly by two approaches: (a) heuristicoriented (Mukhopadhyay et al., 1991, 1992, 1998;Shanker and Srinivasulu, 1989; Tiwari et al., 1997;Tiwari and Vidyarthi, 2000; Mukhopadhyay andTiwari, 1995; Sarma et al., 2002; Swarnkar and Ti-wari, 2004) and (b) optimization-based methods(Stecke, 1983; Sawik, 1990; Sarin and Chen,1987; Shanker and Tzen, 1985; Nayak and Ach-arya, 1998). Mukhopadhyay et al. (1998) solvethe machine-loading problem using the simulatedannealing approach. Tiwari et al. (1997) proposea heuristic approach and a petrinet model for solv-ing the machine-loading problem of an FMS. Toaddress the same problem Tiwari and Vidyarthi(2000) have proposed a genetic-algorithm-basedheuristic approach. Sarma et al. (2002) and Swarn-kar and Tiwari (2004) employ a tabu search based

heuristic to solve the machine-loading FMSproblem.

Heuristic approaches are largely based uponrules and rely on empirical experiences. Therefore,one of the limitations of a heuristic approach is inits difficulty to estimate results in a new or com-pletely changed environment. While optimization-based methods such as—integer programming,dynamic programming, branch and bound, etc.—are robust in applicability, they tend to becomeimpractical when problem size increases. In fact,the machine-loading problem involving simulta-neous determination of job sequence, systemunbalance, and system throughput by satisfyingvarious technological constraints (such as limitedtools slots, machine time, etc.) lies under the broadcategory of NP-hard problems (Srinivas et al.,2004). As Goldberg et al. (1989) puts it—‘‘Eventhe most touted enumerative scheme, dynamic pro-gramming breaks-down on problems of moderatesize and complexity’’. In summary, the machine-loading problems pertaining to automated manu-facturing system belong to the category of NP-hardproblems where the computational solution timesare non-polynomial in the size of the problem(Mukhopadhyay et al., 1991, 1992, 1998; Morenoand Ding, 1993; Shanker and Srinivasulu, 1989;Shanker and Tzen, 1985).

It is evident from the aforementioned literaturethat both the heuristic techniques and the mathe-matical-programming-based methodologies arehandicapped by their limitations in producinggood quality optimal/near-optimal solutions tomachine-loading FMS problems. This limitationhas prompted researchers to investigate the use ofnew and innovative search techniques for solvingthe FMS machine-loading problem. One such po-tential approach is genetic algorithm (GA) (Hol-land, 1975), which is an intelligent probabilisticsearch algorithm. Several manufacturing problemssuch as—scheduling, layout, process planning, andmechanical design—have been solved using geneticalgorithms. Tiwari and Vidyarthi (2000) have pro-posed a GA-based heuristic that performs well andis superior to other methods on similar problemsand under the same assumptions. Liaw (2000) con-siders the minimization of the makespan in an openshop using a GA-based heuristic. GA developed by

Table 1aContributions of authors vis-a-vis objective functions for the machine-loading problem

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

Stecke (1983)p p p p

Shanker and Tzen (1985)p p

Kusiak (1985)p

Berrada and Stecke (1986)p

Rajagopalan (1986)p

Stecke (1986)p p p

Stecke and Kim (1986)p

Sarin and Chen (1987)p

Lashkari et al. (1987)p p

Shanker and Srinivasulu (1989)p

Wilson (1989)p

Han et al. (1989)p

Lee and Jung (1989)p p

Shanker and Rajamarthandan (1989)p

Moreno and Ding (1993)p p

Chung and Doong (1989)p

Ram et al. (1990)p

Chen and Askin (1990)p p p p

Kumar et al. (1990)p p

Sawik (1990)p

Brethauer and Venkataramanan (1990)p

Co et al. (1990)p

De Werra and Widmer (1991)p p

Chen and Chung (1991)p

Liang and Taboun (1992)p

Bernado and Mohamed (1992)p p

Mukhopadhyay et al. (1992)p

Stecke (1992)p

De Vecchi et al. (1993)p

Kim and Yano (1993a,b)p

Kiravak and Dincer (1993)p

Liang (1993)p p

Liang and Dutta (1993)p p

Moreno and Ding (1993)p p

Liang (1994)p p

Sodhi et al. (1994b)p p

Katayama (1994)p

Modi and Shankar (1994)p

Sodhi et al. (1994a)p p p p

D�Alfonso and Venture (1995)p

Grassi et al. (1995)p

Kuhn (1995)p

Song et al. (1995)p

Hsu and De Matta (1997)p

Roh and Kim (1997)p

Tiwari et al. (1997)p

Atmani and Lashkari (1998)p

Colosimo et al. (1998)p

Mukhopadhyay et al. (1998)p

Kumar and Shankar (2000)p

Sawik (2000)p

Tiwari and Vidyarthi (2000)p


Table 1bObjective function representation used in Table 1a

No. in Table 1a Objective function implied in Table 1a

1 Minimization of manufacturing costs2 Minimization of inventory costs3 Maximization of the total profit (difference between income and costs)4 Minimization of flow time and minimization of WIP5 Minimization of the number of tardy jobs6 Minimization of the total (weighted) tardiness7 Minimization of makespan or maximization of system production rate or maximization of system saturation8 Maximization of the differences of load among the machines9 Minimization of total overload and underload of the machines10 Minimization of load of workpiece transport system11 Minimization of load of tool transport system12 Minimization of load of re-fixturing stations13 Maximization of number of alternative routings14 Minimization of number of tool magazine configuration changes15 Minimization of system unbalance and maximization of throughput


Cai and Li (2000) minimizes the cost of staffing,with secondary objectives of maximizing staffingsurplus and reducing the variation of this staff sur-plus over different time periods. Mori and Tseng(1997) propose a GA for multi-mode resource-con-strained project scheduling problem. The GAdeveloped by them performs well on large prob-lems with the caveat that its computational require-ments were significantly more. Eatson and Mansour(1999) designed a distributed genetic algorithm (orDGA) for the deterministic and stochastic laborscheduling problem. Here the GA operators workin parallel on a network and good solutions mi-grate among the populations. Bortfeldt and Geh-ring (2001) proposed a hybrid GA to solve thecontainer-loading problem (three-dimensionalknapspack problem). Guvenir and Erel (1998) ap-plied GAs to solve multi-criteria ABC inventoryclassification problems. Gravel et al. (1998) de-signed a double-loop genetic algorithm, whereouter loop assigns machines to cells, while theinner loop determines the optimal part routing.

Clearly genetic algorithms have a great potentialand one of the major research thrusts currentlyunderway is in devising strategies for handlingproblem constraints within the GA framework.The current work pursues research in which GAis amalgamated with logic-based constraints tosolve the machine-loading problems for an FMS.This work is motivated by the work of Cormier

et al. (1998) where they develop a similar methodol-ogy to solve design problems in concurrent engi-neering. In fact, they first proposed an algorithmthat accommodates a mixture of algebraic, condi-tional, and database constraints with numeric aswell as symbolic variables. They put forward theconcepts of constraint-based initialization, cross-over, and mutation. Although, Cormier et al.(1998) use the term constraint-based initialization,they seed the initial population randomly. Thishas motivated our work whereby we devise anextension that we call the constraint-based geneticalgorithm (CBGA). The CBGA is a semi-automaticstochastic search algorithm, which combines thefeatures of logic-based constraint modeling withthe strengths of a traditional genetic algorithm.The constraint-based genetic algorithm (CBGA)is transformed as per the needs of machine-loadingproblem and tested on 10 moderate-size test prob-lems adopted from Mukhopadhyay et al. (1992).The results are compared with the solutions ob-tained by Tiwari et al. (1997), Shanker and Srini-vasulu (1989), Mukhopadhyay et al. (1992), andTiwari and Vidyarthi (2000). In this work we findthat the CBGA-based heuristic performs well, infewer number of generations, and with computa-tional superiority over existing approaches.

The remainder of the paper is organized as fol-lows: Section 2 provides the description of theproblem. The mathematical modeling is illustrated


in Section 3. Section 4 provides a background ofGA, along with difficulties associated with adapt-ing the algorithm in an FMS environment. TheCBGA was specifically developed for overcomingthese limitations. The proposed algorithm is pre-sented in Section 5. For explaining the steps ofthe algorithm, an illustrative example is discussedin Section 6. The numerical simulation is providedin Section 7. Finally conclusions with scope for fu-ture research are included in Section 8.

2. Problem description

To analyze a machine-loading problem for arandom FMS, this work considers multiple ma-chines and a fixed number of tool slots. In a givenplanning horizon, part types arrive randomly andtheir operation times and tool slot requirementsare well known (Shanker and Tzen, 1985; Mukho-padhyay et al., 1992; Tiwari et al., 1997; Tiwariand Vidyarthi, 2000). The random FMS, consid-ered here, is capable of performing operations thatmay be either essential or optional. Essential oper-ations are those operations that can be done onlyon specific machines using specific tool slotswhereas optional operations are flexible and canbe carried out on one or more machines inter-changeably. Therefore, it makes intuitive sense topreserve the optional operations as long as possi-ble while considering all possible routes. It isinstructive to note that flexibility lies in the selec-tion of a machine for processing the optional oper-ations for a particular part. The FMS underconsideration is able to elicit the flexibility pertain-ing to selection of a machine, processing of anoperation, selection of part-type sequence, etc.

In a given planning horizon, machine-loadingproblem deals with choosing a part type from apool of part types, and allocates its operation toappropriate machines achieving desired systemperformance measures. This is done taking into ac-count the technological and capacity constraints.In order to understand the complexities of the ma-chine-loading problem for a random FMS, let usconsider an example FMS in which four part typesare to be processed on four machines, each havingfive tool slots and different processing times for

every operation. Each part type consists of fouroperations, which can be performed, on any ofthe machines without altering the sequence ofoperations. The adaptability of each machine toperform many different operations allows severaloperation assignment possibilities generating alter-native part routes. Thus, there can be a fairly largenumber of combinations in which operations ofthe part type can be assigned on the different ma-chines while satisfying all the technological andcapacity constraints. Further consideration of flex-ibilities such as: tooling flexibilities, part move-ment flexibilities, etc., along with the constraintsof the system configuration and operational feasi-bility make the problem even more complex.

These operation–machines allocation combina-tions are evaluated using two common perfor-mance measures: system unbalance andthroughput. System unbalance is the sum of unuti-lized or over-utilized time on all machines avail-able in the system. Maximization of machineutilization is identical to minimization of systemunbalance, whereas �throughput� refers to the unitsof part types produced. In the worst case, to arriveat an optimal or a near-optimal solution for themachine-loading problem, it may be necessary toexplore each combinatorial allocation with respectto a given objective function (minimization of sys-tem unbalance or maximizing throughput), andsimultaneously satisfying all the constraints. Asstated earlier, the number of possible allocationsto be explored increases exponentially as the sizeof the problem increases. This NP-hard nature ofthe problem, illustrated in Table 2, can be ex-plained as follows. Considering the problem pro-vided in Table 2—there are eight jobs with 8! jobsequences. For one such ordering there exist:(1) Æ (2) Æ (2) Æ (1) Æ (2) Æ (6) Æ (9) Æ (6) = 2,592 operation/machine-allocation combinations. In turn, for 8!Job orderings, total number of possible allocationswill be (8!) Æ (2592) = 104,509,440. Therefore, forany realistic problem an exhaustive search in thesolution space will be practically intractable.

Early researchers have tackled the above prob-lem by generating pre-determined part sequenc-ing-based heuristics, which do not guaranteeoptimal/near-optimal solutions. More recently,the application of local intelligent search tech-

Table 2Description of problem no. 1 (adopted from Shanker and Srinivasulu, 1989)

Part type Operation no. Batch size Unit processing time Machine no. Tool slots needed

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 22 25 2 1

6 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 2 1 113 3 113 1 1

3 23 4 18 1 13 25 1 1

25 2 125 3 1

2 7 2 11 1 1

3 24 1 3


niques, such as genetic algorithm (GA) (Tiwariand Vidyarthi, 2000), tabu search (TS) (Sarmaet al., 2002) and simulated annealing (SA)(Mukhopadhyay et al., 1998) have been used byresearchers to solve such computationally complexoptimization problems. In this paper further exten-sion of GA, i.e., CBGA (constraint-based geneticalgorithm) is applied to solve machine-loadingproblem to obtain optimal or near-optimal combi-nation of operation–machine allocations of parttypes. The above problem is addressed consideringthe following objective functions:

(1) minimization of system unbalance alone;(2) maximization of throughput alone;

(3) multiple objectives: a combination of mini-mization of system unbalance and maximiza-tion of throughput.

In this work, the proposed algorithm is testedon 10 standard problems illustrated in Mukhopad-hyay et al. (1992), and Shanker and Srinivasulu(1989). Further, Problem no. 1 is taken up (fromMukhopadhyay et al. (1992), Shanker and Srini-vasulu (1989)) for an exhaustive study (see Table2).

To investigate the generic behavior of our algo-rithm, it is tested on five functions from Bandho-padhyay and Pal (1998) (see below). These fivefunctions are:


1. Two max function:This function is of the form:

f ðxÞ ¼ j18n� 8lj;where n is the number 1�s in l bit string repre-senting x. There is one global maxima with va-lue 10l (when x is composed of all 1s, i.e. n = l),and one local maxima with value 8l (when x iscomposed of all zeros, i.e. n = 0). The boundarybetween two peaks occurs at 4l/9.

2. Trap function:This function, with one global and one local

maxima, deals with a situation where the collect-ing area of the local maxima is much larger thanthe collecting area of the global maxima. Thefunction is defined as follows:For z = [(3/4)l],

f ðxÞ ¼ 8lðz� nÞ=z for n 6 z;

¼ 10lðn� zÞðl� zÞ for n > z.

This function has a global maxima with value 10l

when x is composed of all 1s and local maximawith a value of 8l when x is composed of all 0s.

3. Plateau function:This function includes large plateau regions

that are areas in the solution space with sameobjective function value (Ackley, 1987). Thefunction can be defined as follows:The l bits are divided into four equal sized groups.If group contains all 1s, a score of 2.5l is assig-ned and if it contains at least one 0 then its scoreis 0.5l. Sum of all scores of the four groups isthe value of objective function for each chromo-some. It is easy to see that the only possible valuesof the objective function are 2l, 4l, 6l, 8l, and 10l.

4. Exponential function:The function can be expressed as:

f ðxÞ ¼ 2þ expðx� 10Þ cosð10� xÞ x 6 10:0;

¼ 2þ expð10� xÞ cosðx� 10Þ x > 10:0.

For l = 22, x is allowed to vary in the range[0, 20]. Global maxima exists at x = 10.00,where objective function takes a value of 3.0.

5. Sine square function:

f ðx1; x2Þ ¼ 0:5�sin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 þ x22

p� �2

� 0:5

½1þ 0:001ðx21 þ x2

2Þ�2

.

This is a function of two variables (x1, x2). Eachnumber is encoded using l = 44 and is allowedto vary in range of �100 to 100. Global maximawith value 1 occurs when x1 = x2 = 0.0.

3. Mathematical model

Next we provide a formal mathematical modelfor the problem including notations, decision vari-ables, various characteristics, objective functionsand the underlying constraints. To facilitate read-er�s understanding, an intuitive explanation of theconstraints is also provided.

Notations

m Machine, m = 1, . . ., M

t Tool type, t = 1, . . ., T

i Part type, i = 1, . . ., N

j Operation type, j = 1, . . ., O

bi Batch size of part type ia Processing alternative(i, j) Operation j for part type i

(m, t) Tool type t on machine m

s Characteristic type (i, j, m, t)c Chromosome

Decision variables

ai ¼1 if part type i is selected

0 otherwise

bm;t ¼1 if tool type t is allocated on machine m

0 otherwise

ci;j;m;t ¼1 if operation ði; jÞ is assigned to

processing alternative ðm; tÞ0 otherwise

diðbÞ ¼1 if part type i is included in batch b

0 otherwise

Characteristics

i: characteristics pertaining to part type:di demand for part type i

o(i) number of operations for part type i


j: characteristics pertaining to operation type:aij number of processing alternative for oper-

ation (i, j)uptij processing time needed to process unit

operation j for part type imijr volume of total operation (i, j) to be pro-

cessed using route r

Rij Set of routes r for processing operation(i, j)

m: characteristics pertaining to machine type:MCm capacity of tool magazine on machine m

wm workload on machine m

T Total planning horizon (8 hours) for allthe problems

TAm available time on machine m

TTm set of tool types congenial for machinem

TUm under-utilized time on machine mTOm over-utilized time on machine m

Tp,o,r,m time remaining on machine m before allo-cation of (i, j)

Tp,o,a,m time remaining on machine m after alloca-tion of (i, j)

UNSU set of part types unassigned due to nega-tive system unbalance

t: characteristics pertaining to tool type:Nt number of tools of types t availableQt number of slots required by tool type t for

(i, j)CMt set of machine congenial for tool type t

TLt life of tool type tUTSC set of part types unassigned due to tool

slot constraints

Thus, the objective functions for this researchcan be formulated as:

(1) First objective function is to minimize systemunbalance leading to maximizing system uti-lization. Thus the first objective function canbe formulated as:

Minimize f 1¼M �T �

PMm¼1ðTU mþTOmÞM �T

( ).

ð1Þ

(2) Second objective function is to maximizethroughput, leading to maximization of sys-tem efficiency. So the second objective func-tion can be stated as:

Maximize f 2 ¼PN

i¼1biaiPNi¼1bi

( ). ð2Þ

(3) Third objective function is a combination ofboth of the above objective functions:

Maximize f 3 ¼W 1 � ½f1� þ W 2 � ½f2�

W 1 þ W 2

; ð3Þ

where
W1 is the weight assigned to objective 1 (min-imization of the system unbalance),W2 is the weight assigned to objective 2(maximization of throughput),W1 = W2 = 1 (for simplicity, we have con-sidered equal weights).
For calculation of these objective functions, a flowchart is shown in Fig. 1.

All these functions are subject to the followingsystem constraints:Xði;jÞ2CMt

cijmt 6 di 8i; o; ð4Þ

cijmt 6 bmt 8i; j;m; t 2 Rij; ð5ÞXij

Xr

diðbÞmijruptij 6 TLtCMt; ð6Þ

Xij

Xt

diuptijcijmt ¼ wm 8m; t 2 TT m; ð7Þ

Xt2TTm

Qtbmt 6 MCm 8m; ð8ÞX

m

bmt 6 Nt 8t;m 2 CMt; ð9ÞX

jr

mijr ¼ di 8i; j; r 2 Rij; ð10Þ

XM

m¼1

ðTU m þ TOmÞP 0; ð11ÞX

mt

cijmt ¼ aij; ð12Þ

XN

t¼1

XM

m¼1

diðbÞbmt ¼ aibi. ð13Þ

Start

For each part type calculate total processing time (o(i) × uptij) and arrange them in increasing order

Initially there is no part type selected

Select next part type

Calculate CMt entertaining (i,j).CMt= MCm= Nt.

Is (Nt >0)

Yes

Reject part type

No

Allocate a part type on machine m having maximum remaining processing time.

uptij= Tp,o,a,m= Tp,o,r,m

SU = Tp,o,r,m. Is (SU>0) ?

Are all part types arranged in given sequence?

Yes

No

Calculate TH for all suitable part types

Stop

Yes

No

Fig. 1. Flow chart for calculating the objective functions.



Let us consider each constraint sequentially. Con-straint (4) is for operations assignment where theassignment depends on the selection of its parttype. Constraint (5) is for an assignment of anoperation, based on appropriate tool allocation.Constraint (6) mandates that, if an operation is as-signed to a machine, the required number of toolslots also be available on that machine. Constraint(7) ensures capacity of the machines workloadcannot be more than the time available on it.Constraint (8) takes care that slots needed by toolson machine must be within its tool magazinecapacity. Constraint (9) concerns with tool typeavailability and ensures that the total numbers ofallocated copies are not more than the availablecopies. Constraint (10) and constraint (12) requirethat for any part type, all the operations must becompleted (unique part-type routing). Constraint(11) take cares of positive system unbalanceand when the system unbalance becomes nega-tive the part type will be unassigned. Constraint(13) requires that a new part type be consid-ered only if all operations of a part type arecompleted.

4. Background of GA

Simulating the natural evolutionary process re-sults in a stochastic optimization technique calledevolutionary algorithm. Genetic algorithms arestochastic search techniques that rely on the pro-cess of natural selection (Goldberg et al., 1989).At any given iteration, the genetic algorithm oper-ates on a pool of solutions rather than a singlesolution. In GA, search starts with an initial setof random solutions known as population. Eachchromosome of population is evaluated usingsome measure of fitness function. Based on thevalue of the fitness functions, a set of chromo-somes is selected for breeding. In order to simulatea new generation, genetic operators such as cross-over and mutation are applied. According to thefitness value, parents and offsprings are selected,while rejecting some of them so as to keep thepopulation size constant for new generation. Thecycle of evaluation–selection–reproduction is con-tinued until an optimal or a near-optimal solution

is found (Goldberg et al., 1989; Michaelwicz,1992).

Holland (1975) introduced GA and later ap-plied this technique on a wide variety of problems.Process of GA is commonly called simple geneticalgorithm (SGA). Fig. 2 illustrates the workingof the typical SGA.

Although GA is a global search technique, itspractical usefulness depends on how well it canhandle problem constraints. Therefore, a numberof techniques have been developed for handlingconstraints. Michaelwicz and Nazhiyath (1995)has surveyed the several constraint-handling meth-ods for GA. The most common approach is to adda penalty function to the objective function inorder to transform a constrained problem intoan unconstrained one. In a penalty function ap-proach, the basic rule is to add a zero penaltywhen all constraints are satisfied, and a non-zeropenalty otherwise. There can be two types of pen-alty functions: static or dynamic. In static penalty(Homaifiar et al. (1994)), the penalties increase insteps. Dynamic penalty (Joines and Houck,1994), on the other hand, ascertains the magnitudeof the penalty based on the amount of constraintviolation. However quantifying constraint viola-tion is not always feasible especially when we aredealing with non-numeric variables. Thus, imple-menting penalty functions directly within anFMS framework is a tedious task.

Another approach to handle constraints wasdevised by Richardson et al. (1989) and Powelland Skolnick (1993), with key guiding phrases like‘‘the worst feasible solution is better than bestinfeasible solution’’. Generally speaking, the prob-ability of generating feasible solution is very small.Hence, this type of penalty functions will takemassive amounts of time to discover even a smallset of feasible solution.

GenoCopIII (Michaelwicz and Nazhiyath(1995)) is another alternative and innovativeapproach for handling constraints in GA. Thisapproach, however, is only compatible with prob-lems involving continuous and convex constraints.Paredis (1993) tried a different repair techniquethat he called genetic state space search (GSSS).GSSS is very effective but, unfortunately, satisfac-tion of constraints has been historically hampered

New Population

Y

N

START

Encoding

Initial population Generation

Crossover Mutation

Evaluation DecodingSelection

Solutions

Fitness Computation

Termination?

STOP

Fig. 2. Flowchart illustrating the process of a simple genetic algorithm.


by slow performance. As mentioned earlier, Cor-mier et al. (1998) have introduced constraint-basedgenetic algorithm to solve problems in concurrentengineering. In our current work this concept, withsome modifications, is employed to solve FMSproblems. For a search strategy like CBGA tohandle an FMS problem, it is necessary that itaccommodate the inherent features of the FMSproblems, such as: mixture of algebraic, condi-tional database, and database constraints with nu-meric as well as symbolic variables. The proposedCBGA for an FMS environment discussed in thenext section specifically describes how geneticoperators interact and work with a set ofconstraints.

The idea of constraint satisfaction within a ge-netic algorithm is a promising one. In CBGA, un-like its traditional counterpart, the geneticalgorithm incorporates constraint satisfactionwhile focusing on performance measures. Thus,the idea of constraint-based initialization, con-straint-based crossover and constraint-basedmutation operators are built into this enhancedprocedure.

5. Proposed constraint-based genetic algorithm

As mentioned above, strength of the CBGAover a simple genetic algorithm is in its use of con-straint-based initialization, whereby it does notgenerate the initial population randomly. Charac-teristic values of constraints are considered whilegenerating the initial solution. Furthermore, con-straint-based crossover and constraint-basedmutation is used to generate new population off-springs, making the search space wider.

5.1. Nomenclature

Dc,s(t) domain for characteristic s of chromo-some c at generation t

Ac,s(t) value of characteristics s for chromosomec at generation t

T_P(t) = {P1(t), . . ., Pn(t)} Total population atgeneration t

Pc(t) = {Cc,1(t), . . ., Cc,m(t)} Set of m character-istic values for chromosome c at genera-tion t


Disseminate (Cc,s(t)) Function that disseminatescharacteristic s of chromosome c at gener-ation t

Revise (R) Function that revises the set of charac-teristics without values (R)

5.2. Constraint-based initialization

Constraint-based initialization operator gener-ates an initial population set, which provides the‘‘gene pool’’ of characteristic values for subsequentgenerations. A flowchart illustrating workingof constraint-based initialization is shown inFig. 3.

It can be seen from the flowchart that this oper-ator assigns a value for each chromosome in a gen-eration ensuring that constraints are not violated.The problem constraints have the potential tochange feasible values that may be assigned to

ForPc (

S = m, m R = {1, 2, …, m}

Ac,s(t) =

Dissemi

R

R

Fig. 3. Flowchart illustrating con

the remaining unknown characteristics. Dissemi-nation is proposed for performing several opera-tions. It involves inserting the new values into allrelated constraints. If only one unknown variableremains in any of these constraints, that variable�svalue is obtained by solving the constraints. A newvalue is then inserted into all related constraints,and the process repeats itself until no new valuescan be disseminated. Following local constraintsdissemination, the system must also revise the do-main of feasible values that can be arranged forthe remaining variables without values. Hence, Re-

vise is a function that revises the set of characteris-tics without assigned values.

5.3. Evaluation

After initialization, each sequence of the initialpopulation is evaluated according to problem spe-cific objective function. As the FMS involves

Yes

No

START

each t) ∈ T_P (t)

∈R

x, x = Dc,s(t)

nate (A c,s(t))

evise (R)

If ≠ { }

STOP

straint-based initialization.


multiple objective functions, the model must en-able user to specify several objectives. In the pro-posed model the decision-maker supplies relativeweighted averages of the individual objectives.Regardless of how the objective is formulated,the constraint-satisfaction tools used in the CBGAare highly effective for maintaining solutionfeasibility.

5.4. Selection

Following the evaluation, a sub-set of the pop-ulation is stochastically selected for survival.According to Neo-Darwinism, the process of selec-tion can be sub-divided into three categories—(a)stabilizing selection, (b) directional selection, and(c) disruptive selection.

Stabilizing solution is also called normalizingselection, as it tends to eliminate chromosomeswith extreme values. Directional selection has theeffect of either increasing or decreasing the meanvalue of the population. In this paper, we use a sta-bilizing selection (which provides stability to thesolutions space) and also disruptive selection thattends to eliminate chromosomes with moderatevalues. In this method, the chromosomes areranked from best to the worst according to theirraw fitness values. It is found that the probabilityof survival increases as fitness of a chromosome in-creases. The survival probability is proportional tothe chromosome�s fitness or the chromosome�srank (best to worst). It is important to note thata combination of these three selection strategiesbalances variation and stability, allowing a richsurviving population that includes the optimalsolution.

5.5. Constraint-based crossover

Following the above steps, the discarded chro-mosomes are combined together to form newchromosomes. As the reader may recall, the goalof constraint-based initialization is to seed thecharacteristic values in the initial chromosome bycarrying out dynamic domain calculations andlocal constraint dissemination. The constraint-based crossover assigns characteristic valuesdirectly from parents to the child by satisfying

the constraints. Initially two parents are selectedrandomly from surviving chromosomes. Then,characteristic values are assigned from parents tothe offsprings. It may be possible that only oneor neither of parents can contribute any feasiblecharacteristic values to the offsprings. In order toovercome this limitation, a check is made to deter-mine which of the parents can contribute a feasiblevalue to the offsprings. When offspring cannot in-herit any of the feasible characteristic value fromthe parents, a value is randomly selected fromcharacteristic�s domain in the same manner as pro-posed by constraint-based initialization. Finally,the new characteristic value is disseminated in asimilar fashion as the constraint-based initializa-tion. A flow chart discussing steps involved in con-straint-based crossover is shown in Fig. 4.

5.6. Constraint-based mutation

To introduce diversity into the population, thecharacteristic value can be modified using con-straint-based mutation. Constraint-based muta-tion is a semi-intelligent mutation technique thatchecks constraint violation during the process ofmutation. Adoption of this operator in a con-straint environment can help prevent a previouslyfeasible option from becoming infeasible. A majorcomplication of the constraint-based mutation canbe clearly observed from Fig. 5. A NULL value isassigned to the characteristic and disseminatedthroughout the network; this dissemination isneeded as the mutated characteristics may belinked to other characteristics by virtue of problemconstraints. Dissemination of this NULL valuealso retracts values of other characteristics. It canbe seen that mutation loop is similar to that ofconstraint-based initialization, as new characteris-tic values are selected from each characteristic do-main and checked for constraint violations. But,this leads to more processing time during muta-tion. This limitation can, however, be overcomeby decreasing the magnitude of mutations as thegenerations pass. When population begins to con-verge in later generations this step has a desirableeffect of fine-tuning the solution. A flowchart illus-trating constraint-based mutation is shown inFig. 5.

YES

NO

YES

NO

YESNO

YES

NO

For C = survivors+1 to n

R = {1, 2, …, m}

P=P1

P=P2

S = n, where n R

Select two parents, P1 P2

P = set of available Parent (1,2…,survival)

If R { }

AP1,s(t) Dc,s(t) AP2,s(t) Dc,s(t)

Ac,s(t) = x

Ac,s(t) =AP1,s

AP1,s(t) Dc,s(t) AP2,s(t) Dc,s(t)

Ac,s(t) =AP2,s Ac,s(t) = x

Disseminate (Ac,s(t))

STOP

If R { }

Revise (R)

START

≠

≠

∈

∈

≠

∈

∉

∉

Fig. 4. Flowchart illustrating constraint-based crossover.


YES

P= 1,2,…,n

K= y P

S = n, where n A

For K = 1 to M (For each mutation to be performed)

Ac,s(t)=NULL


Revise (R)

S = n, where n R

Ac,s(t)= x, x Dc,s(t)


Revise (R)

STOP

If R {}

A = {1, 2, …, m}

START

≠

∈

∈

∈

∈

Fig. 5. Flowchart illustrating constraint-based mutation.


6. An illustrative example

Here the proposed constraint-based geneticalgorithm is applied to solve a well-known prob-

lem described by Shanker and Srinivasulu (1989)and used by others (Mukhopadhyay et al., 1992;Tiwari et al., 1997; Tiwari and Vidyarthi, 2000)for a random flexible manufacturing system. We

Fig. 6. Block-diagram representation of solution procedure for the illustrative example.



use this specific example for two reasons—(a) toease comparative understanding of the CBGAvis-a-vis existing algorithms where this examplehas already been used by past researchers, (b) thecomplexity built into this illustrative example ishelpful in explaining some of the intricacies ofthe CBGA methodology. A detailed descriptionof the test problem is presented in Table 2.

The nomenclature used in the algorithm is dis-cussed below in context of the underlying ma-chine-loading problem:

• Dc,s(t): Set of characteristics s {i, j, m, t} (dis-cussed in Section 3).

• Ac,s(t): Corresponding values of chromosome c

{i, j, m, t}.• Cc,s(t): Chromosome c at generation t.• Disseminate (Cc,s(t)): For any sequence of parts,

dissemination is performed to check the feasi-bility of the part allocation based on the variousconstraints discussed in Section 3.

• Revise (R): At the end of each generation, func-tion Revise (R) gets a new list of characteristics;e.g., if some part type is allocated on machine m

then after its allocation the remaining time lefton the machine m is revised using functionRevise (R).

A block diagram detailing the various steps of thesolution procedure for the illustrative exampleabove is presented in Fig. 6. Based on severalpre-runs, the parameters are chosen to achieve

Table 3Comparison of CBGA-based heuristic solutions with other-heuristic

Problemnumber

Total numberof part types

Shanker andSrinivasulu (1989)

Muet a

SU TH SU

1 8 253 39 1222 6 388 51 2023 5 288 63 2864 5 819 51 8195 6 467 62 3646 6 548 51 3657 6 189 54 1478 7 459 36 4599 7 462 79 31510 6 518 44 320

near-optimal solutions. These empirical valuesfor the parameters are provided in Step 1 of theblock diagram in Fig. 6.

7. Numerical simulation

Past research related to GA applications, espe-cially in planning and scheduling problems ofFMS, has shown that the number of generations/iterations required to achieve near-optimal solu-tions is substantially high with further disadvan-tage of constraint violations. Therefore, it isdesirable, indeed necessary, to develop an algo-rithm that can solve large-sized combinatorialproblems in fewer number of generations utilizingless CPU time. As stated earlier, keeping in mindthese requirements, in this research three newoperators are proposed: (1) Constraint-based ini-tialization, (2) constraint-based crossover, and (3)constraint-based mutation. The resulting algo-rithm, constraint-based genetic algorithm (CBGA),introduces high level of diversity in solving the ma-chine-loading problem of a random-type FMS.This enables rejection of infeasible solutions, fasterinformation interchange among chromosomes, inturn, resulting in a quicker convergence of thealgorithm. In addition, several modifications areincorporated so that known computational hur-dles are resolved efficiently.

By dealing with constraints in the CBGA, weare able to achieve near-optimal solutions rela-

solutions for the problem set

khopadhyayl. (1992)

Tiwari et al.(1997)

CBGA-basedheuristic

TH SU TH SU TH

42 76 42 0 3663 234 63 18 4679 152 69 72 6951 819 51 819 5176 264 61 187 5362 314 63 28 6166 996 48 165 5436 158 43 63 4888 309 88 309 8856 166 55 122 56


tively quickly. The convergence of results startsmanifesting typically in only a few generations.The computational burden and CPU time havebeen considerably reduced as compared to theGA-based heuristic proposed in Tiwari and Vid-yarthi (2000).

To judge the quality of solutions the CBGA em-ploys three objectives or criteria, namely: systemunbalance alone, throughput alone and a combi-nation of system unbalance and throughput.Minimization of system unbalance leads to maxi-

Table 4Results of problems adopted from Shanker and Srinivasulu (1989)

Problem no. Objectivefunction

Parameter settingof CBGA

Part-typesequence

1 1 5,0.5,0.3,30 4,7,2,5,6,3,1,82 5,0.5,0.3,30 5,7,4,3,2,6,8,13 5,0.5,0.3,30 3,7,4,1,2,5,6,8

2 1 5.0.6,0.2,30 5,2,4,6,1,32 5,0.5,0.2,30 5,3,1,4,2,6 or

3,4,5,1,6,23 5,0.5,0.2,30 5,4,3,1,2,6 or

5,3,1,4,2,6

3 1 5,0.5,0.2,30 4,5,2,3,12 5,0.5,0.2,30 2,3,1,5,43 5,0.5,0.2,30 5,3,2,4,1 or

5,1,2,3,4

4 – Any sequence –

5 1 5,0.75,0.2,30 3,1,5,4,6,22 5,0.85,0.2,30 6,2,5,1,3,43 5,0.80,0.2,30 2,1,4,6,3,5

6 1 5,0.5,0.2,30 1,5,2,4,6,32 5,0.5.0.2,30 4,3,1,5,6,23 5,0.5,0.2,30 1,4,6,2,5,3

7 1 5,0.5,0.2,30 4,1,2,5,6,32 5,0.5,0.2,30 3,6,1,4,5,2 and

3,1,2,5,6,43 5,0.5,0.2,30 1,5,2,3,6,4

8 1 5,0.5,0.2,30 7,5,4,6,3,1,22 5,0.5,0.2,30 4,5,2,7,3,1,6 an

5,2,7,1,6,4,33 5,0.5,0.2,30 3,2,5,7,1,6,4

9 – – Any sequence

10 1 5,0.5,0.2,30 5,2,6,4,1,32 5,0.5,0.2,30 6,1,3,5,4,23 5,0.5,0.2,30 1,6,5,2,3,4

mization of capacity utilization of the system.The CBGA heuristic, with the objective of mini-mizing system unbalance and maximizing through-put, yields a system unbalance of 14 minutes and athroughput of 48 units in 30 generations. These re-sults are a significant improvement over Tiwariand Vidyarthi (2000) where the system unbalanceof 14 minutes and a throughput of 48 units wasachieved in 50 generations. For the same problem,Tiwari et al. (1997), Mukhopadhyay et al. (1992)and Shanker and Srinivasulu (1989) reported

Part-typeunassigned

Value ofobjective function

SU TH

NSU TSC

6,3,1,8 – 1.000 0 362,6 8 0.640 14 482,6 8 0.796 14 48

3, 1 – 0.919 18 462 – 0.630 154 63– 22 – 0.891 154 63– 2

1 – 0.963 72 694 – 0.924 128 734 4 0.928 128 73– –

– – – 819 51

6,2 – 0.903 187 534 – 0.815 479 623 – 0.829 276 61

3 – 0.985 28 612 – 0.863 314 633 – 0.910 28 61

6 3 0.914 165 542 – 0.807 486 634 – 231 634 – 0.843 231 63

3,1,2 – 0.993 13 44d 1,6 – 0.800 63 48

6,3 288 481,6 – 0.883 63 48

– – – 309 88

1 – 0.957 82 54– 4 0.835 122 56– 4 0.886 122 56


(system unbalance, throughput) of (76 minutes, 42units), (122 minutes, 42 units), and (253 minutes,39 units), respectively. A detailed comparativestudy is presented in Table 3.

It is important to recognize that the above citedresearch articles have the same technological con-straints (i.e. availability of machining time andtools slots) making the comparative frameworkfair and equitable. Similarly, for the other prob-lems generated by Mukhopadhyay et al. (1992),detailed results are provided in Table 4. Differentparameters for all 10 problems such as choice ofobjective function, GA control parameters (popu-lation size, crossover probability, mutation proba-bility and maximum generation), part-typesequences, fitness function value, correspondingsystem unbalance and throughput, and the set ofpart unassigned along with causes of rejection—are described in Table 4. Apart from showing sub-stantial improvement in the solution quality,CBGA also presents diversity due to the numberof constraints applied within the proposedheuristic.

Shanker and Srinivasulu (1989) consideredunutilized time in the system unbalance, whereasMukhopadhyay et al. (1992) and Tiwari et al.(1997) have considered the unutilized and over-uti-lized machine time in their definition of systemunbalance. The heuristic of Tiwari and Vidyarthi(2000) considered the allocation of next part type(from the set of unassigned part types) by remov-

Table 5Comparison of CBGA-based heuristic with pre-determined sequencin

Problem number SPT sequence LPT sequence

SU TH SU TH

1 76 42 51 382 236 63 18 463 152 69 132 484 819 51 819 515 264 61 187 536 314 63 139 497 996 48 165 548 158 43 133 299 309 88 309 8810 166 55 184 49

Average 349 58.3 213.7 50.5

ing the last assigned part type that caused theactual system unbalance to increase. Mukhopad-hyay et al. (1992) have adopted the max–max rulefor the allocation of unassigned part type in antic-ipation that system unbalance improves further.However the CBGA takes care of above situationby proposing a choice among different objectivefunctions (f1, f2, f3). On the known data sets, wetest the efficacy of CBGA heuristic vis-a-vis otherfixed sequencing rule heuristics such as shortestprocessing time (SPT), longest processing time(LPT), last in first out (LIFO), first in first out(FIFO). The results listed in Table 5 show thatthe proposed heuristic, almost always, does betterthan these procedures.

7.1. Parameter setting

This section deals with selecting the geneticparameters for solving the machine-loading prob-lem of a random-type FMS. De Jong (1975) firststudied the significance of the various parametersof GA such as population size, number of genera-tions, crossover probability, mutation probability,and replacement strategies, etc. It is now fairly wellestablished that the optimal values for theseparameters vary from problem to problem, envi-ronment to environment, and constraints to con-straints (Schaffer et al. (1989)). Pakath andZaveri (1993) proposed a decision-support-systemapproach to systematically determine appropriate

g rules

LIFO sequence FIFO sequence CBGA-basedsequence

SU TH SU TH SU TH

92 35 35 45 0 36154 63 500 57 18 4672 69 152 69 72 69

819 51 819 51 819 51211 53 207 52 187 53314 63 500 57 28 61996 48 189 54 165 5413 44 158 43 63 48

309 88 309 88 309 8882 54 166 55 122 56

306.1 56.8 303.5 57.1 178.3 57.4


parameter values. Gupta et al. (1993a,b) proposedan experimental design approach for evaluatingthe settings of several GA parameters.

These parameters cannot be determined inde-pendently as it involves solving complex non-linearoptimization problems. Thus, it is important torecognize that the control parameters cannot beset until the interactions among the genetic opera-tions are considered in detail. Our investigationshave found the following important observationsfor control parameter selection of GA are alsoapplicable to CBGA. These are:

CBC:PMXUOX:EER:OX: OCX: C

0.85

0.875

0.9

0.925

0.95

0.975

1

0 4 8 12 16 20 2

Generatio

Ave

rage

val

ue o

f ob

ject

ive

func

tion

f1

Fig. 7. Performance of c

• Whenever the crossover probability increases, itleads to an increase in the recombination of thebuilding blocks. However, it is accompanied bydisruption of good strings.

• In case where the mutation probabilityincreases, it leads to a transformation of thegenetic search into random search, but it is alsoaccompanied by a reproduction of large geneticsearch material.

• Increasing the population size leads to anincrease in its diversity and reduces the pro-bability that the GA will prematurely converge

Constraint-based Crossover : Partially-mapped Crossover Uniform-order-based Crossover Enhance-edge-recombination Crossover rder Crossover yclic Crossover

4 28 32 36 40

ns

CBC

UOX

OX

CX

EER

PMX

rossover operators.

CBM: Constraint-based mutation DIS: Displacement RE: Reciprocal exchange INS: Insertion INV: Inversion

0.7

0.75

0.8

0.85

0.9

0.95

1

1 6 11 21 26 31 36 41

Generations

Ave

rage

val

ue o

f obj

ectiv

e fu

nctio

n f1

CBM

DIS

RE

INS

INV

Fig. 8. Performance of mutation operators.

CBC-CBM: Combination of Constraint-based Crossover and Constraint-based Mutation

EER-INS: Combination of Enhance-edge-recombination Crossover and Insertion

PMX-RE: Combination of Partially-mapped Crossover and Reciprocal Exchange

EER-RE: Combination of Enhance-edge Recombination and Reciprocal Exchange

CX-DIS: Combination of Cyclic Crossover and Displacement

UOX-INV: Combination of Uniform-order-based Crossover and Inversion

0.85

0.9

0.95

1

0 4 8 12 16 20 24 28 32 36 40Generations

Ave

rage

val

ue o

f ob

ject

ive

func

tion

f1

EER-INS

PMX-RE

EER-RE

CBC-CBM

UOX-INV

CX-DIS

Fig. 9. Performance of combination of operators.



to local optimal results; however, it alsoincreases the time to converge to the optimalsolution.

Therefore, these observations serve as a guide in asystematic empirical selection of the parametervalues.

Ten test problems, considered in this research,consist of 5–8 part types. Hence the maximumnumber of genes varies from 5 to 8. Several trialswere performed with population size varying from5 to 15 (in steps of 1) and 15 to 30 (in steps of 2).Similarly, the crossover probability and mutationprobability were varied from 0 to 1 in steps of 0.1.

To capture the diversity of the objective functionvalues, these problems were also solved by settingthe mutation probability to 0 and varying cross-over probability in steps of 0.1 and vice versa.

200210220230240250

40 140 240 340 440 540 640 740Iterations

Ave

rage

fitn

ess

CBGA

GA

0

50

100

150

200

40 140 240 340 440 540 640 740Iterations

Ave

rage

fitn

ess

GACBGA

0.50.60.70.80.9

1

800

900

1000

1100

1200

1300

14

Itera

Ave

arge

fitn

ess

a

c

e

Fig. 10. Performance of CBGA vs GA for the two max function (a),and sine square function (e).

The maximum number of generations was variedfrom 20 to 40 in steps of 2. The global optimal ornear-optimal part sequences are found at a gen_max (maximum generation) of 30. This appearsto be the fewest number of generations requiredfor solving machine-loading problem (Tiwari andVidyarthi, 2000). Various crossover operators,which are used for sequencing and schedulingproblem were tested on given data set and it wasfound that the constraint-based crossover operatoremployed in the CBGA outperforms most opera-tors. The comparative results are shown in Fig. 7.

Similarly, performance of constraint-basedmutation operator is compared with various muta-tion operators. Fig. 8 shows the superiority of theconstraint-based mutation over other methods.

A combination of constraint-based crossoveroperator and constraint-based mutation operator

0

50

100

150

200

40 140 240 340 440 540 640 740Iterations

Ave

rage

fitn

ess

GACBGA

050

100150200250300350

40 80 120

160

200

240

280

320

360

Iterations

Ave

rage

fitn

ess

GACBGA

0015

0016

0017

0018

0019

00

tions

GACBGA

b

d

trap function (b), plateau function (c), exponential function (d),


does better than their separate use. A joint use ofconstraint-based crossover operator and mutationoperator is recommended to obtain optimal ornear-optimal part type sequences in a small searchspace. The performance of a combination of vari-ous crossover and mutation operators is shown inFig. 9. It can be inferred from Fig. 9 that CBC-CBM operator outperforms the EER-INS, CX-DIS, and UOX-INV operators. With respect toEER-RE and PMX-RE operators, the perfor-mance of CBC-CBM operator gradually augmentsas generations pass.

To establish the generic efficacy of the CBGA, itwas also tested on five standard problems reportedby Bandhopadhyay and Pal (1998). For theseproblems, near-optimal solutions have beenachieved in less number of generations, comparedto the standard GA procedure. Fig. 10(a)–(e),shows the performance of CBGA vis-a-vis GAfor these five functions.

The CBGA was coded in C++ programminglanguage and the program was run on an IBMPC with a 1.9 GHz Pentium processor. The sourcecode for this algorithm is available at: http://www.geocities.com/gurukul007/cbga_source.txt.

8. Conclusions

FMS machine-loading problem is characterizedby a model that involves grappling with a widevariety of constraints and objectives. This paperpresents a new integrated approach to concur-rently address the machine loading and the toolallocation problems in an FMS environment.The proposed CBGA heuristic enhances the capa-bilities of the traditional GA procedure throughthe use of specialized operators. The operatorsproposed in this paper allow exhaustive explora-tions that exploit the search space during thegrowth mechanism of the algorithm. Thus, itsimultaneously prevents premature convergenceand helps in rejecting the infeasible solutions. Thisresults in decreased CPU time for obtainingoptimal/near-optimal solutions. In this work, acomparative study shows that the proposedCBGA-based heuristic outperforms the existingheuristics on standard data sets.

While solving the problem, emphasis is placedon the following aspects of the CBGA heuristic:

(1) The representation scheme of the CBGA is insync with the part-type sequencing represen-tation of machine-loading problem for a ran-dom-type FMS.

(2) Attributes are selected such that they coverall aspects of the given problem.

(3) A proper fitness function is formulated toinclude the more realistic multiple objectivecases.

(4) The proposed operators are applied with aprudent choice of attributes.

(5) Control parameters are one of the crucialfacets that govern the solution quality. Thishelps achieve near-optimal results, maintain-ing reasonable search space while utilizingless computational time.

Preliminary testing of the CBGA-based heuris-tic for determining the optimal assembly/disassem-bly sequences under varying geometric constraintsshows promising results. Therefore, a comprehen-sive exploration of this procedure in context of theoptimal assembly/disassembly sequence problemcould be a useful future research endeavor.

We sincerely hope that this research can be ex-panded to various problems that encompass thesequencing or allocation of resources (such asmaterial handing, pallets, and fixture) in an FMSenvironment. Also, this research can be exploitedto solve the multi-objective loading and schedulingproblems by introducing more flexible attributes.

Acknowledgement

We gratefully acknowledge the numerousthoughtful suggestions of the anonymous refereesand the editor that significantly helped us inimproving the content and delivery of this article.

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