integrated scheduling of flexible manufacturing system using evolutionary algorithms

ORIGINAL ARTICLE

Integrated scheduling of flexible manufacturing systemusing evolutionary algorithms

P. Udhayakumar & S. Kumanan

Received: 3 June 2010 /Accepted: 24 October 2011 /Published online: 24 November 2011# Springer-Verlag London Limited 2011

Abstract Effective sequencing and scheduling of thematerial handling system (MHS) have an impact on theproductivity of the flexible manufacturing system (FMS).The MHS cannot be neglected while scheduling theproduction tasks. It is necessary to take into account theinteraction between machines and MHS. This paper high-lights the importance of integration between productionschedule and MHS schedule in FMS. The Giffler andThompson algorithm with different priority dispatchingrules is developed to minimize the makespan in the FMSproduction schedule. Its output is used for MHS schedulingwhere the distance traveled and the number of back-trackings of the automated-guided vehicles are minimizedusing an evolutionary algorithms such as an ant colonyoptimization algorithm and particle swarm optimization(PSO) algorithm. The proposed evolutionary algorithms arevalidated with benchmark problems. The results availablefor the existing algorithms are compared with resultsobtained by the proposed evolutionary algorithms. Theanalysis reveals that PSO algorithm provides better solutionwith reasonable computational time.

Keywords FMS . Production scheduling .

MHS scheduling . ACO algorithm . PSO algorithm

1 Introduction

Manufacturing companies are to manage increasing productcomplexities, shorter time to market, newer technologies,

threats of global competition, and rapidly changing envi-ronment. To cope up the manufacturing competition,flexible manufacturing system (FMS) is established. FMSis an integrated manufacturing system that consists ofmultifunctional numerically controlled machine toolsconnected with an automated material handling system(MHS). The objective of FMS is flexibility in productionwithout compromising the quality of products. The FMSflexibility depends on the flexibility of computer numericalcontrol machines, automated material handling devices, andcontrol softwares. There are number of problems facedduring the life cycle of an FMS. These problems areclassified into design, planning, scheduling, and controlproblems. During the implementation and operation phaseof the FMS, the user requires adjusting and fine-tuning ofthe FMS to the best operating conditions. Researchers arecontinuously putting efforts to solve design and operationalproblems of FMS. The dynamic planning of FMS operationneeds attention.

The strategic roles of a MHS in an FMS as given byBedworth and Bailey [1] are given in Fig. 1. The mainfunction of an MHS is to supply the right materials at theright locations and at the right time. The cost of materialhandling is a significant portion of total cost of production.Eynan and Rosenblatt [2] indicated that the materialhandling cost is about 30% of the total production cost.This makes the subject of material handling increasinglyimportant.

Automated-guided vehicles (AGV) are used in manymaterial handling situations involving large, heavy loadsover flexible routes. AGVs are the most flexible of thefloor cart systems and follow electromagnetic impulsestransmitted from a guide wire embedded in the plantfloor. Two advantages of the AGV systems, in an FMSare that it can accommodate work stations and commu-

P. Udhayakumar (*) : S. KumananDepartment of Production Engineering, National Institute ofTechnology,Tiruchirappalli 620 015 Tamilnadu, Indiae-mail: [email protected]

Int J Adv Manuf Technol (2012) 61:621–635DOI 10.1007/s00170-011-3727-2

nicate with a computer-controlled system on a real-timebasis [3]. AGV systems continue to play a significant rolein low- to medium-flow manufacturing operations, includ-ing FMS and other applications. An AGV is a driverlessvehicle that accomplishes the material handling tasksflexibly and so is considered appropriate for an FMSenvironment [4]. The production demands a flexible MHSto move the parts to various processing stations during aproduction run. The integration of the machines into asystem, achieved by automated MHS and by the overallcomputer control, result in FMS characterized by flexibil-ity, high productivity, and low cost per unit produced. Therole and operation of MHS in a FMS is important as it hasto conform to the production schedule. Hence, the majorissue for the operational executives of any FMS is todevelop an efficient scheduling mechanism for the FMS.The development of effective and efficient FMS schedul-ing strategies remains an important and active researcharea.

This paper is organized as follows: Section 2includes the literature review and Section 3 includes theproblem descriptions of the integrated scheduling ofFMS. In Section 4, elements of the ant colony optimiza-tion (ACO) and particle swarm optimization (PSO)algorithm based on the integrated scheduling problemsof FMS are discussed. Numerical examples of integratedscheduling of FMS are discussed in Section 5. Theresults and discussion by the application of evolutionaryalgorithms are discussed in Section 6. Section 7 sums upconclusions.

2 Literature review

Giffler and Thompson [5] developed an enumerativeprocedure to generate all active schedules for the general“n” jobs and “m” machine problem. Stecke [6] defines anddescribes the design, planning, scheduling, and controlproblems of flexible manufacturing systems. Buzacott andYao [7] presented a comprehensive review of the analyticalmodels developed for the design and scheduling of FMS.Kimemia and Gershwin [8] reported on an optimizationproblem that optimizes the routing of the parts in an FMSwith the objective of maximizing the flow while keepingthe average in-process inventory below a fixed level. Themachines in the cell have different processing times for anoperation. Jaikumar and Solomon [9] studied a manufac-turing system, integrated with a central warehouse bymeans of AGVs. The jobs are returned to the centralwarehouse after each manufacturing operation. The job andAGV scheduling problem in a two-level hierarchicaloptimization is considered. Sabuncuoglu and Hommertzheim[10] proposed a dispatching algorithm that considers theavailability of both machine and AGV to select a job forloading. The performance of the proposed methodology iscompared with a few simple dispatching rules by using themean flow time and mean tardiness criteria through asimulation test.

Karabtlik and Sabuncuoglu [11] introduced a beamsearch-based algorithm for the simultaneous scheduling ofmachines and AGVs. The assumptions made are vehiclesalways return to the load/unload station after transferring aload. Rachamadugu and Stecke [12] classified andreviewed FMS scheduling procedures. Classificationscheme is provided that is based on key factors such asthe FMS type, the mode of system operation, the nature ofthe demands placed on the system, the scheduling environ-ment, and the responsiveness of the system to disturbances.The choice of appropriate scheduling criteria is discussed.

Spanno et al. [13] reviewed the work done on the designof FMS in the areas of facility design, MHS design, controlsystem design, and scheduling. In AGV scheduling, theoptimal design of the physical layout is one of theimportant issues that must be resolved in the early stagesof the FMS design [14]. Ulsoy and Bilge [15] pointed outthe coordination and integration of material handlingsystem with machine scheduling during the schedulingphase of the FMS have not received much attention. Anattempt is to make scheduling of AGVs an integral part ofthe overall scheduling activity of FMS environment. Leeand DiCesare [16] formulated the integrated production andmaterial handling scheduling in an offline job shop context.Two shortest path routes (in opposite directions) existsbetween every pair of machines. Routes are constrainingresources. Sawik [17] presented a multilevel decision

Forecast Management Decisions Environment

Capacity Planning Design

Purchase /

Vendor

MRP (Inventory)

Computer Aided Process Planning

Computerized FMS

Scheduling Control of MHS, AS / RS, etc

Testing / Inspection

Shop floor capture system

Fig. 1 Production planning and control in CIM

622 Int J Adv Manuf Technol (2012) 61:621–635

model for simultaneous scheduling of machine and vehiclein an FMS. Dorigo and Gamberdella [18] proposed an antcolonies algorithm for the traveling salesman problem. TheACO algorithm performs better in problems such as thequadratic assignment problem [19], job shop schedulingproblem [20], and the vehicle routing problem [21]. Smithet al. [22] considered not only material handling activitiesbut also explicitly loading and unloading activities foroffline scheduling. Kim and Hwang [23] proposed a newdispatching algorithm for an efficient operation of AGVS.It utilizes the information of work-in-process in buffers andtravel times of AGVs. The performance of the algorithm iscompared with some well-known dispatching rules in termsof the system throughput through simulation. Lee [24]presented a contribution where the material handlingactivities that influence the schedule take place betweenthe load/unload station and the machine. Dispatchingstrategies are proposed and evaluated for rail guidedvehicles in a loading/unloading zone of an FMS.

Noorul Haq et al. [25] discussed the multilevel sched-uling decisions of a FMS to generate realistic schedules forthe efficient operation of the FMS. Abdelmaguid et al. [26]has presented a new hybrid genetic algorithm for thesimultaneous scheduling problem for the makespan mini-mization objective. The hybrid GA is composed of GA anda heuristic. The GA is used to address the first part of theproblem that is theoretically similar to the job shopscheduling problem and the vehicle assignment is handledby a heuristic called vehicle assignment algorithm. Jerald etal. [27] proposed the two approaches such as geneticalgorithm and adaptive genetic algorithm used for schedul-ing both parts and AGVs simultaneously in an FMSenvironment. Vis [28] described the research perspectivesin the design and control of AGV systems in distribution,transshipment, and transportation systems. Reddy and Rao[29] addressed the simultaneous scheduling problem as amulti-objective problem in scheduling with conflictingobjectives and solved by nondominated sorting evolution-ary algorithms. Xia and Wu [30] proposed a hybrid PSOalgorithm for the problem of finding the minimum make-span in the job-shop scheduling environment. Huang andLiao [31] presented a hybrid algorithm combining antcolony optimization algorithm with the taboo searchalgorithm for the classical job shop scheduling problem.Ponnambalam and Kiat [32] proposed a PSO algorithm tosolve machine loading problem in a flexible manufacturingsystem with bicriterion objectives of minimizing systemunbalance and maximizing system throughput in theoccurrence of technological constraints such as availablemachining time and tool slots. Gnanavelbabu et al. [33]discussed the artificial immune system for multi-objectivescheduling of jobs, AGVs and AS/RS in FMS. Kashan andKarimi [34] proposed a discrete PSO algorithm to tackle the

problem of optimal assignment of jobs to machine tominimize the makespan time. Subbaiah et al. [35]addressed the problem of simultaneous scheduling ofmachines and two identical automated-guided vehicles ina FMS so as to minimize makespan and mean tardinessusing sheep flock heredity algorithm. Krishnan et al. [36]discussed the scatter search technique to locate theoptimum arrangement of machines with the minimumdistance traveled by the MHS. Gnanavelbabu [37]addressed the differential evolution algorithm for simulta-neous scheduling machines and AGV’s in FMS environ-ment. Fauadi and Murata [38] addressed binary particleswarm optimization algorithm to optimize simultaneousmachines and AGVs scheduling process with makespanminimization function.

In MHS schedule, research has been progressed forminimizing the distance traveled by an AGV but lessattention has been paid to considering the objective ofminimization of the number of backtracking movements.The application of evolutionary algorithms for integratedscheduling problem of FMS needs attention. In this paper,evolutionary algorithms such as an ant colony optimizationalgorithm and particle swarm optimization algorithm havebeen proposed and developed to solve the integratedscheduling of FMS problems.

3 Problem descriptions

Given a set of M machines, a set of P part types, theoperations are sequenced for each part Pi processed bymachine Mi to allocate machines and parts into a layout tominimize the distance traveled by AGV inside the system.Given the processing requirements (operation/job matrix) of“P” parts on “M” machines, the best routing of the parts forits routing flexibility is determined by applying prioritydispatching rules (PDRs) using Giffler and Thompson (GT)algorithm. The PDRs are listed as follows:

& TWORK—greatest total work& SRPT—select the job having the smallest remaining

processing time& LRPT—select the job having the longest remaining

processing time& SIO—select the job with the smallest ratio obtained by

dividing the processing time of the imminent operationby the total processing time for the part

& LIO—select the job with the largest ratio obtained bydividing the processing time of the imminent operationby the total processing time for the part

This optimum part routing is taken as an input toschedule the MHS. In MHS, the distance traveled and thenumber of backtrackings of the AGV is to be minimized.

Int J Adv Manuf Technol (2012) 61:621–635 623

The objective function is stated as in Eq. 1.

Minimize Z ¼ w1Dþ w2B ð1Þwhere

D ¼XP

i¼1

XM

j¼1

XM

k¼1

dijk ;

B ¼XP

i¼1

XM

j¼1

XM

k¼1

bijk ;

w1 ¼ 0:6 and w2 ¼ 0:4

Let dijk distance traveled by AGV in moving part ifrom machine j to machine k.

bijk number of backtrackings occurred in movingpart i from machine j to machine k.

D total distance traveled by AGV in moving allparts P to be processed for completion.

B total number of backtrackings occurred inmoving parts P to complete its processing.

w1, w2 The normalized weights assigned to eachobjective.

4 Proposed methodologies

The general description of the proposed solution method-ologies is dealt.

4.1 ACO algorithm

An individual ant constructs candidate solutions by startingwith an empty solution and then iteratively adding solutioncomponents until a complete candidate solution is generated.After the solution construction is completed, the ants givefeedback on the solutions as they have constructed bydepositing pheromone on solution component. In addition,old pheromone evaporates to a certain extent.

Typically, solution components which are part of bettersolution will hold a higher amount of pheromone, and hence,such solution components are more likely to be used by theants in future iterations of the ACO algorithm. Pheromoneevaporation is the process by means of which the pheromonetrail intensity on the components decreases over time. From apractical point of view, pheromone evaporation is needed toavoid a too rapid convergence of the algorithm towards a sub-optimal region. It implements a useful form of forgetting,favoring the exploration of new areas of the search space. Theprobability for choosing the next path by an ant will bedirectly proportional to the amount of pheromone on that path.

State transition rule Starting from the initial node, each antchooses the next node in its path according to the statetransition rule [20] by using probability of transition. Let S bethe set of nodes at a decision point i. The transition from thenode i to a node k by an ant is calculated as given in Eq. 2.

k ¼ maxjeS

t i; jð Þ½ �a hðjÞ½ �bn o

if q � q0

0 otherwiseð2Þ

τ(i, j) is the quantity of the pheromone on the edgebetween the node i and the node j. η(j) is the heuristicinformation stored on the node j. α and β tune the relativeimportance in probability of the amount of the pheromoneversus the operation time. q is a random number generatedbetween 0 and 1 at each time of the selection of a node. q0is a constant given as input within a range from 0 to 1 at thebeginning of the algorithm. k is a random node selectedaccording to Eq. 3.

p i; kð Þ ¼ t i; kð Þ½ �a hðkÞ½ �bPj2S

t i; jð Þ½ �a hðjÞ½ �b if k e S

0 otherwise

ð3Þ

Global updating rule Global updating is intended to rewardedges belonging to the shortest path. Once all the ants havearrived at their destination, the amount of pheromone on theedge (i, j) belonging to the shortest path at a time t ismodified by applying the global updating rule given in Eq. 4.

t i; jð Þt ¼ 1� rð Þt i; jð Þt�1 þ Q=Lk ð4Þρ is the coefficient representing pheromone evaporation

(note: 0<ρ<1). Q is a constant whose value is chosenaccording to the size of the problem. Lk is the minimumtour length traveled by an ant among all the ants in theiteration.

Local updating rule The evaporation phase is substitutedby a local updating rule of the pheromone applied duringthe construction of the solution. The pheromone associatedwith an edge is modified each time the ant moves fromnode i to node j according to Eq. 5.

t i; jð Þt ¼ 1� rð Þt i; jð Þt�1 þ r � t0 ð5Þ

τ0 is the initial pheromone value and is defined as τ0=n/L, where L is the tour length produced by the execution offirst iteration without pheromone information. The localupdating rule is equivalent to the trail evaporation in thereal ants. The procedural steps of the proposed ACOalgorithm for integrated scheduling of FMS are explainedas below and its flowchart is shown in Fig. 2.


Initial solution An initial solution of 200 randomlygenerates solutions.

Sort the regions according to the objective function The200 solutions are then sorted in ascending order withrespect to the objective function. The regions pertaining tominimum objective function value are referred to assuperior solutions, while regions pertaining to maximumobjective function value are referred to as inferior solutions.

Distribution of ants The total number of ants is 100, whichis half of 200 and is distributed as 80 for global (G) and 20for local search (L).

Global search Using global search, global ants create newregions by replacing the inferior solutions of the existingsolutions. It consists of the following operations: (a)crossover, (b) mutation, and (c) trial diffusion

Crossover Parents are selected from superior solution (N–G) and randomly selected from 90% of the inferiorsolutions from the crossover. A representation of N–Gparents (superior solutions) and G children (inferiorsolutions) is shown in Fig. 3.

Mutation Each variable of the solution is randomly addedor subtracted by a value of 0.005, with a probability equalto or greater than the mutation probability 0.6.

Trial diffusion Two parents are selected at random from theparents regions. The child can have (1) the value of thecorresponding variable of the first parent, (2) the value ofthe corresponding variable of the second parent, and (3) acombination arrived from the weighted average of theabove X childð Þ ¼ ax parent 1ð Þ þ 1� a½ �x parent 2ð Þ, whereis a uniform random number in the range [0,1]. Theprobability of selecting the third option is set equal tothe mutation probability 0.6, and the probability ofselecting the first and second options is allotted aprobability of 0.2.

After completion of these global search operations, theobjective function of the new G regions is compared withprevious G regions. The G region which has the minimumobjective function is selected for further operation.

Updating of pheromone trail value of new solution inglobal search After the global search, the pheromone trailvalue of the new solutions is updated proportionally to theimprovement in the fitness value.

Local search With a local search, the L local ants select Lregions from N regions move in search of better fitness.Here L is 20, and L solutions are selected as per the currentpheromone trial value. In the selected regions, the variablesare changed by a finite random increment (0.005) in thepositive or negative direction. If the fitness is improved, thenew solutions are updated to the current region.

Stop

Yes

Create solutions

Sort regions according the objective function

Send 90% of global ants for crossover and mutation

Send 10% of global ants for trial diffusion

Update trial value of weakest solution

Select regions and send local ants

If new function value is improved?

Move ant to new solution

Update trial value of weakest solution

Evaporation of trail value

Sort according to objective function value

Print best objective function value

Check for termination criteria?

No

Yes

No

Fig. 2 Flowchart for integrated scheduling of FMS using ACOalgorithm

1 2 3 …….. 120 121 ……. 199 200

Superior solutions Inferior solutions

N-G parents G children

Fig. 3 Representation of N–Gparents and G children


Updating the pheromone trail value of new solution in localsearch If there is a decrease the objective function value inthe local search procedure, the regions position is updatedto the current position. The pheromone trail value is thenupdated proportional to the decrease in the objectivefunction value.

Sort the regions according to the objective functionvalue New solutions will be obtained after the global andlocal search. The solutions will also have the new pheromonetrail values. The solutions are sorted in ascending order of theobjective values and the best objective value is stored. Theprocess is repeated for a specified number of iterations.

4.2 PSO algorithm

PSO algorithm is an optimization tool provides a population-based search procedure in which individuals called particleschange their position (state) with time. In a PSO algorithm,swarm is initiated randomly with particles and evaluated tocompute fitness’s together with finding the particle best (bestvalue of each individual so far) and global best (best particle inthe whole swarm). Initially, each individual with its dimen-sions and fitness value is assigned to its particle best. The best

individual among particle best swarm, with its dimension andfitness value is, on the other hand, assigned to the global best.Then a loop starts to converge to an optimum solution. In theloop, particle and global bests are determined to update thevelocity first. Then the current position of each particle isupdated with the current velocity. Evaluation is againperformed to compute the fitness of the particles in the swarm.This loop is terminated with a stopping criterion predeter-mined in advance.

The flowchart for integrated scheduling of FMS usingPSO algorithm is shown in Fig. 4. The procedural steps ofthe proposed PSO algorithm are explained as below:

Step 1: Swarm initialization and velocityInput the number of parts, number of machines,

number of AGV, number of operation in each joband number of iterations. Randomly generate aninitial swarm. Initialize the velocity of particles aslist of moves.

Step 2: Evaluation of objective functionThe objective function (x) such as total distance

traveled by the AGV and the number of back-tracking movement in MHS is evaluated.

Step 3: pbest selectionIdentify the particle best position (pbest).

Initialize swarm and velocity

For each particle

Evaluate objective function (x)

If objective function (x) > objective function (gbest) then gbest = x

If objective function (x) > objective function (pbest) then pbest = x

Next particle

Update position pi = pi + vi

Update velocity vi = vi + c1R1(pi, best – pi) + c2R2(gi, best – pi)

If termination criteria

satisfied?

Print gbest solution

Stop

Start

Yes

No

Fig. 4 Flowchart for integratedscheduling of FMS using PSOalgorithm


Step 4: gbest selectionIdentify the gbest. If objective function (x)>

objective function (gbest) then gbest=x. If objec-tive function (x)>objective function (pbest) thenpbest=x; print the gbest values and then stop.

Step 5: Velocity updateThe velocity is updated by using Eq. 6.

vi ¼ vi þ C1R1 pi; best � pi� �þ C2R2 gi; best � pi

� �

ð6Þwhere,

vi is the particle velocitypi is the current particle (solution)R1 and R2 is a random number between 0 and 1C1, C2 are learning factors. Usually, C1 equals to

C2 and ranges from 0 to 4.

Step 6: Position updateThe position is updated by using Eq. 7.

pi¼pi þ vi ð7ÞStep 7: Termination checking

The algorithm repeats steps 2–6 until termina-tion conditions are met. Once terminated, thealgorithm reports the values of gbest as itssolution.

5 Numerical examples

Example 1: To illustrate the production scheduling, anFMS with the configuration is given inTable 1 and the layout of FMS [25] is shownin Fig. 5.

Let there be two parts to be processedon nine different machines. The part–machine incidence matrix indicating theoperation sequence number of two partswith alternative routes is given in Table 2.The processing time of the parts in eachmachine is given in Table 3.

The FMS process the parts P1 and P2 inany of the four possible combinations asshown in Table 4. The PDRs used inliterature [25] are listed as follows:

& SPT—select the job with the shortest processingtime

& LPT—select the job with the longest processing time& LOR—select the job having the least number of

operations remaining

Table 1 Configuration of FMS (numerical example 1)

Layout type No. ofmachines

No. ofparts

Load / unloadstations

No. ofAGV

U loop 9 2 2 1

Slot 4 Slot 3 Slot 2 Slot 1

Slot 6 Slot 7 Slot 8 Slot 9

Slot 5

Load

Unload

AGV

Fig. 5 Layout of FMS [25]

Table 2 Part-machine incidence matrix (numerical example 1)

Part types Operation sequence

1 2 3 4 5 6 7 8 9

P1 M2 M5 M3 M1 M8 M4 M6 M9 M7

M2 M6 M3 M5 M1 M4 M9 M7 M8

P2 M6 M2 M1 M7 M8 M3 M4 M9 M5

M6 M9 M5 M3 M1 M4 M7 M8 M2

Table 3 Processing times of parts in machines (numerical example 1)

Part types Processing times

M1 M2 M3 M4 M5 M6 M7 M8 M9

P1 62 9 89 87 10 66 95 49 82

P2 30 95 58 61 58 60 17 8 5


& MOR—select the job having the most number ofoperations remaining

& LWR—select the job having the least amount ofwork remaining

& MWR—select the job having themost work remaining

Using GT algorithm, the best route selected in this studyis the one for which most of the parts give the minimummakespan. From Table 4, a second combination gives theminimum makespan for five parts namely LPT, LOR,MOR, MWR, SRPT, and SIO.

So the best route is

P1:M2-M5-M3-M1-M8-M4-M6-M9-M7

P2:M6-M9-M5-M3-M1-M4-M7-M8-M2

This route is given as an input for AGV scheduling,where the objective is to arrive at a layout, whichminimizes the distance traveled and the number of back-trackings.

The integration of the machines into a system, achievedby the automated material handling and by the overallcomputer control, can result in manufacturing systemscharacterized by flexibility, high productivity and low costper unit produced. Hence the role of MHS in FMS isimportant and further, the operation of MHS requires a co-coordinated effort (between machines, MHS, and comput-er), as it has to conform to the production schedule. Theinput to the AGV schedule is an optimal routing of partsobtained from the production schedule that excluded thetransfer activities of the FMS (i.e., the production scheduleobtained with zero transportation time). Here, the transferactivity is included. The transportation time depends on thedistance between machines, the mode of operation and thespeed of the AGV. The AGV moves in the bidirectionalmode.

The application of the ACO and PSO algorithms areused to find an optimum solution for scheduling the ULoop layout in the FMS. The layout has fixed slots intowhich machines can be arranged. The path to be traversedby a part is obtained as an input from the productionschedule. The distance matrix between machines given inTable 5 and the load/unload distance matrix given inTable 6 is considered for illustration.

The optimum route attained by the production scheduleshown in Table 7 is considered for AGV scheduling toarrange the machines in a U loop to optimize the objectivefunction.

Example 2: Configuration of FMS is given in Table 8 istaken from the literature [36]. The layout ofFMS is shown in Fig. 6.

Let there be four parts to be processed onseven different machines. The part-machineincidence matrix indicating the operationsequence number of two parts with alternativeroutes is tabulated in Table 9. The distancematrix between machines is tabulated inTable 10. The load/unload distance matrixbetween machines is given in Table 11.

The numerical illustration of proposedparticle swarm optimization algorithm fornumerical example 2 [36] is described below:

The initial swarm is generated randomlyand objective function is calculated. Togenerate new value from the present, thefollowing method is employed.

New sequence ¼ present sequenceþ velocity

where, velocity=C1R1(pi,best − pi) +C2R2

(gi,best − pi)

Table 4 Optimum production schedule with makespan obtained using various PDRs (numerical example 1)

S. no. Part routes Makespan

Literature [25] Proposed PDRs

SPT LPT LOR MOR LWR MWR TWORK SRPT LRPT SIO LIO

1 P1: M2-M5-M3-M1-M8-M4-M6-M9-M7 549 626 676 626 659 549 615 641 549 587 607P2: M6-M2-M1-M7-M8-M3-M4-M9-M5





Table 7 Part machine incidence matrix (best route taken fromproduction scheduling numerical example 1)


1 2 3 4 5 6 7 8 9

P1 M2 M5 M3 M1 M8 M4 M6 M9 M7

P2 M6 M9 M5 M3 M1 M4 M7 M8 M2

Table 8 Configuration of FMS (numerical example 2)

Layout type No. ofmachines

No. ofparts

Load / unloadstations

No. ofAGV

U Loop 7 4 2 1

Table 9 Part-machine incidence matrix (numerical example 2)


1 2 3 4

P1 M1 M4 M2 M7

P2 M1 M6 M3 M4

P3 M2 M6 M3 M5

P4 M3 M4 M2 M7

Table 5 The distance matrix between machines (numerical example 1)

Machines M1 M2 M3 M4 M5 M6 M7 M8 M9

M1 0 6 8 12 14 14 12 8 6

M2 6 0 6 10 12 14 12 10 8

M3 8 6 0 8 10 12 10 9 7

M4 12 10 8 0 8 11 12 9 10

M5 14 12 10 8 0 7 9 12 6

M6 14 14 12 11 7 0 12 14 12

M7 12 12 10 12 9 12 0 4 6

M8 8 10 9 9 12 14 4 0 4

M9 6 8 7 10 6 12 6 4 0

Table 6 The load / unload distance matrix between machines (numericalexample 1)

Machines M1 M2 M3 M4 M5 M6 M7 M8 M9

Load 2 4 6 7 12 12 12 11 14

Unload 3 4 7 8 13 14 10 15 16

Slot 3 Slot 2 Slot 1

Slot 5 Slot 6 Slot 7

Slot 4 Load

Unload

AGV

Fig. 6 Layout of FMS [36]

Table 10 The distance matrix between machines (numerical example 2)

Machines M1 M2 M3 M4 M5 M6 M7

M1 0 2 4 8 12 12 10

M2 2 0 2 6 10 12 12

M3 4 2 0 4 8 10 12

M4 8 6 4 0 4 6 8

M5 12 10 8 4 0 2 4

M6 12 12 10 6 2 0 2

M7 10 12 12 8 4 2 0

Table 11 The load/unload distance matrix between machines(numerical example 2)

Machines M1 M2 M3 M4 M5 M6 M7

Load 4 6 8 12 10 8 6

Unload 6 8 10 12 8 6 4


Let the particle sequence at iteration ‘n’ is:

present n½ � : 5 4 3 1 2 6 7where, n is the iteration number

Let the particles best (pbest[]) sequencebe:

pbest½� : 5 6 1 3 2 4 7

and the global best (gbest[]) sequence till thepresent iteration be:

gbest½� : 2 1 5 4 6 3 7

The velocity can be found using therelation

velocity ¼ C1R1 pi; best � pi� �þ C2R2 gi; best � pi

� �

where, C1=C2=0.5

R1 is generated randomly as 0.6833R2 is generated randomly as 0.2513

pbest½� � present½� ¼ 5 6 1 3 2 4 7½ � � 5 4 3 1 2 6 7½ �

The differences in the sequences are taken to be thevelocity. The difference in the sequence may be calculatedas the changes need to be made by swapping theindividuals of a string to get the other. Hence, to get pbest[5 6 1 3 2 4 7], firstly 6 and 4 should be swapped in presentsequence and making it as [5 6 3 1 2 4 7]. Then, 3 and 1must be swapped to get [5 6 1 3 2 4 7]. Hence, pbest[]−present[] is termed as (6,4) (3,1).

Similarly, gbest[]−present[]

gbest½� : 2 1 5 4 6 3 7

present½� : 5 4 3 1 2 6 7

Following the same procedure as followed for pbest[]−present[], and it is given as (2,5) (1,4) (5,3) and (6,3).

Hence velocity ¼ 0:5� 0:6833� 6; 4ð Þ 3; 1ð Þ½ � þ 0:5

� 0:2513 2; 5ð Þ 1; 4ð Þ 5; 3ð Þ; 6; 3ð Þ½ �

Taking 0.34165 [(6,4) (3,1)] is not possible, hence aminimum part of difference is considered and the firstchange (6,4) is taken.

Taking 0.12565 of [(2,5) (1,4), (5,3), (6,3)] which is12.565% of the sequence is not possible as only 4changes are to be made. Hence (2,5) is taken(25% ofthe change).

Hence the velocity becomes (6,4) (2,5)

New sequence nþ 1½ � ¼ present n½ � þ velocity

¼ 5 4 3 1 2 6 7½ � þ 6; 4ð Þ 2; 5ð Þ½ �¼ 2 6 3 1 5 4 7½ �

Hence the new sequence is [2 6 3 1 5 4 7] and it isevaluated for the objective function.

6 Results and discussion

If the standard set of test problems is accessible, theperformance of metaheuristics can be compared on exactlythe same set of test problems. For this reason, problem istaken from the literature [25] and [36] as the test problem ofthis paper. The performance of the proposed evolutionaryalgorithms is compared with existing algorithms such asGA, SA, and scatter search algorithm used in the literature.A comparison of optimum performance parameters by theproposed evolutionary algorithms for numerical example 1is given in Table 12.

Comparison of distance traveled by AGV by theproposed evolutionary algorithms for numerical example1 is given in Fig. 7. Comparison of number of back-trackings by the proposed evolutionary algorithms fornumerical example 1 is given in Fig. 8. Comparison ofobjective function by the proposed evolutionary algorithmsfor numerical example 1 is given in Fig. 9. Comparison ofcomputational time by the proposed evolutionary algorithmsfor numerical example 1 is given in Fig. 10. The numericalresults of the ACO and PSO algorithm employed to obtainthe optimum parameters for numerical example 1 are shownin Table 13.

Table 13 shows that new better solutions have beenobtained from PSO algorithm. The application of PSOalgorithm has proved as the optimal values (best from 150iterations) of the objective function are very close to thevalues obtained with the existing algorithms.

Comparison of optimum performance parameters by theproposed evolutionary algorithms for numerical example 2is given in Table 14. The numerical results of the ACO andPSO algorithm employed to obtain the optimum parametersfor numerical example 2 are shown in Table 15. Compar-ison of distance traveled by AGV by the proposedevolutionary algorithms for numerical example 2 is givenin Fig. 11. Comparison of number of backtrackings by theproposed evolutionary algorithms for numerical example 2is given in Fig. 12. Comparison of objective function by theproposed evolutionary algorithms for numerical example 2is given in Fig. 13.


Comments on the feasibility of the application of ACOand PSO algorithms to this problem are discussed in thissection. The solution of the ACO algorithm is a collectiveoutcome of the solution found by all the ants. Thepheromone trial is updated after all the ants have foundout their respective solutions. The pheromone updating canbe explored to enhance the capability of the proposed ACOalgorithm. The parameters of the state transition rule,determine the convergence rate of the algorithm as well asthe quality of the obtained solution. To allow the algorithmto converge to a satisfactory solution, the evaporationconstant has to be well tuned so as to guide the search intofavored regions in the search space and at the same timeprevent searching in small neighborhoods of local optima.Once the parameters are properly tuned, the algorithmconverges satisfactory, thus accomplishing the stated goalof this work.

The advantages of the PSO algorithm are very fewparameters to deal with and the large number ofprocessing elements, so called dimensions, which enableto fly around the solution space effectively. PSO doesnot have genetic operators such as crossover andmutation. Particles update themselves with the internalvelocity; they also have a memory that is important tothe algorithm. In PSO algorithm, only the ‘best’ particlegives out the information to others. It is a one-wayinformation sharing mechanism, the evolution onlylooks for the best solution. In general, the searchprocess of a PSO algorithm should be a process

Table 12 Comparison of optimum parameters by the proposed evolutionary algorithms (numerical example 1)

Parameters Literature [25] Proposed evolutionary algorithms

GA SAA ACO PSO

Objective function Z* 96.99 103 98.66 94.23

Distance D* 147 157 149 135

No. of backtrackings B* 3 3 3 3

Optimum sequence 1 6 3 4 2 5 7 9 8 1 4 2 3 7 6 8 5 9 3 2 9 1 6 4 5 8 7 6 5 2 7 4 1 9 3 8

0

20

40

60

80

100

120

140

160

1 40 80 120

160

200

240

280

Number of Iterations

Ob

ject

ive

Fu

nct

ion

GA

SAA

ACO

PSO

Fig. 9 Comparison of objective function by the proposedevolutionary algorithms (numerical example 1)

0123456789

10

1 30 60 90 120

150

180

210

240

270

300

Number of iterations

Nu

mb

er o

f B

ackt

rack

ing

s

GA

SAA

ACO

PSO

Fig. 8 Comparison of number of backtrackings by the proposedevolutionary algorithms (numerical example 1)

0

20

40

60

80

100

120

140

160

180

200

1 30 60 90 120

150

180

210

240

270

300

Number of Iterations

To

tal D

ista

nce

Tra

vele

d b

y A

GV

GA

SAA

ACO

PSO

Fig. 7 Comparison of distance traveled by AGV by the proposedevolutionary algorithms (numerical example 1)

Fig. 10 Comparison of computational time by the proposed evolu-tionary algorithms (numerical example 1)


Table 13 Comparison of numerical results of the proposed evolutionary algorithms (numerical example 1)

Iterations Literature [25] Proposed evolutionary algorithms

GA SAA ACO PSO

Distance(D*)

No. of backtracking(B*)

Objectivefunction(Z*)

Distance(D*)



Distance(D*)



Distance(D*)



1 176 8 137.6 186 7 137.59 180 9 137.12 174 7 136.76

10 183 6 133.8 174 9 135.59 178 8 134.27 172 6 131.22

20 172 6 127.2 172 9 132 174 8 129.68 171 7 124.37

30 182 5 126.8 189 5 130.59 185 6 129.42 179 5 123.55

40 175 6 121.4 180 6 129.99 178 6 126.59 170 6 118.91

50 164 7 120 175 7 128.59 171 8 124.98 162 6 117.86

60 170 6 118.9 183 5 126.99 174 7 122 165 5 114.73

70 159 6 117 162 8 126.39 159 7 119.84 154 6 113.14

80 168 4 114.8 168 7 124 170 7 116.36 162 5 112

90 158 5 112.8 153 9 121.79 160 8 114.73 149 5 110.57

100 156 5 110.4 172 5 119.99 164 6 112 151 5 109.42

110 165 3 109.8 163 6 118.19 169 6 114.56 156 4 108.67

120 161 4 108.6 151 8 116.59 158 7 113.71 142 4 104.64

130 156 4 106.4 161 5 113.39 156 6 111.37 148 4 98.78

140 148 5 105.6 164 4 110.39 150 5 108.88 139 3 95.36

150 147 3 96.99 150 5 104.39 152 5 102.32 135 3 94.23

160 147 3 96.99 150 4 104.1 149 5 101.55 135 3 94.23

170 147 3 96.99 157 3 103 149 4 99.76 135 3 94.23

180 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

190 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

200 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

210 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

220 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

230 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

240 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

250 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

260 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

270 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

280 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

290 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

300 147 3 96.99 157 3 103 149 3 98.66 135 3 94.23

Table 14 Comparison ofoptimum parameters by theproposed evolutionaryalgorithms (numericalexample 2)

Parameters Literature [36] Proposed evolutionary algorithms

SS ACO PSO

Distance D* 136 142 129

No. of backtrackings B* 6 4 3

Objective function Z* 84 88.2 79.2

Optimum sequence 2 4 1 7 6 3 5 3 2 6 4 7 1 5 2 1 7 4 3 6 5


consisted of both contraction and expansion so that itcould have the ability to escape from local minima, andeventually find good enough solutions.

The application of PSO algorithm to this problem hasproved fruitful as the optimal values of the objective functionare very close to the values obtained with the other algorithms.The PSO algorithm is able to provide better results than otheralgorithms many times. It is observed that the utilization ofAGV is more than 80%. This makes it evident that PSOalgorithm is an efficient method to solve integrated schedulingof FMS environment.

7 Conclusions

The production scheduling conforming to the materialhandling system scheduling of flexible manufacturingsystem is addressed. Giffler and Thompson algorithm isused with different priority dispatching rules to attainthe objective of the minimization of makespan forproduction scheduling. Ant colony optimization andParticle swarm optimization algorithm have been pro-posed and developed to evaluate the objective function

Table 15 Comparison of numerical results of the proposed evolutionary algorithms (numerical example 2)

Iterations Literature [36] Proposed evolutionary algorithms

Scatter search ACO PSO

Distance(D*)

No. of backtracking (B*)

Objectivefunction (Z*)

Distance(D*)



Distance(D*)



1 168 5 102.8 170 6 108.8 164 4 94.2

100 150 4 91.6 158 5 95.6 144 4 86.6

200 142 4 86.8 149 5 92.4 135 4 82.8

300 136 6 84 142 4 88.2 129 3 79.2

Fig. 13 Comparison of objective function by the proposed evolution-ary algorithms (numerical example 2)

Fig. 11 Comparison of distance traveled by AGV by the proposedevolutionary algorithms (numerical example 2)

Fig. 12 Comparison of number of backtrackings by the proposedevolutionary algorithms (numerical example 2)


of minimization of the distance traveled and the numberof backtrackings for material handling system schedul-ing. The performances of the proposed evolutionaryalgorithms are evaluated with benchmark problems. Theresults of the proposed evolutionary algorithms arecompared with existing algorithms such as geneticalgorithm, simulated annealing algorithm and scattersearch algorithm. The PSO algorithm provides bettersolution with reasonable computational time.

7.1 Scope for future work

The proposed evolutionary algorithms can be extendedand analyzed for multi AGV systems with differentlayouts. The objective in a MHS schedule can beextended to minimize the multiple objectives by therelocation costs of machines including the constraintthat some of the machines may be immovable. Othermetaheuristics like sheep flock heredity algorithm,artificial immune system can also be attempted.

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