Minimizing makespan subject to minimum total flow-time on identical parallel machines

Theory and Methodology

Minimizing makespan subject to minimum total ¯ow-time onidentical parallel machines

Jatinder N.D. Gupta a,*, Alex J. Ruiz-Torres b

a Department of Management, College of Business, Ball State University, Muncie, IN 47306-0350, USAb Department of Information and Decision Sciences, College of Business Administration, The University of Texas at El Paso, El Paso,

TX 79968-0544, USA

Received 17 September 1998; accepted 1 May 1999

Abstract

This paper considers the identical parallel-machine scheduling problem of minimizing makespan subject to minimum

total ¯ow-time. In view of the NP-hard nature of the problem, lower bounds on the optimal makespan value and

e�cient heuristic algorithms for ®nding optimal schedules are described and empirically evaluated as to their e�ec-

tiveness in ®nding optimal schedules. Ó 2000 Elsevier Science B.V. All rights reserved.

Keywords: Parallel-machine scheduling; Hierarchical criteria; Makespan; ¯ow-time; Heuristic algorithms; Empirical

results

1. Introduction

The need to consider multiple criteria inscheduling is widely recognized. Either a simulta-neous or a hierarchical approach can be adopted.For simultaneous optimization, there are two ap-proaches. First, all e�cient schedules can be gen-erated, where an e�cient schedule is one in whichany improvement to the performance with respectto one of the criteria causes a deterioration withrespect to one of the other criteria. Second, a single

objective function can be constructed, for exampleby forming a linear combination of the variouscriteria, which is then optimized. Under a hierar-chical approach, the criteria are ranked in order ofimportance; the ®rst criterion is optimized ®rst, thesecond criterion is then optimized, subject toachieving the optimum with respect to the ®rstcriterion, and so on. Surveys of algorithms andcomplexity results in this area are given by Chenand Bul®n [3], Lee and Vairaktarakis [13] andNagar et al. [14].

In their survey of multi-machine multi-criteriascheduling problems, Nagar et al. [14] observedthat only a few papers have considered multiplemachine settings, although this last assumption isa more appropriate depiction of most shop ¯oors.

European Journal of Operational Research 125 (2000) 370±380www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +1-765-285-5301; fax: +1-765-

285-8024.

E-mail addresses: [email protected] (J.N.D. Gupta), aruiz-

[email protected] (A.J. Ruiz-Torres).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 3 8 6 - 0

The authors suggest the reason for the lack of re-search in this area may be the complex nature ofthese problems. Recently, Bernardo and Lin [1]developed solution procedures for the bi-criteriaminimization of the total tardiness and set-up coston parallel machines.

This paper considers problem of scheduling njobs on m identical machines to minimize make-span (maximum completion time) given that thetotal completion (¯ow) time is minimum. Clearly,the above described problem is one of hierarchicalmulti-criteria scheduling. Following the three-®eldnotation of a scheduling problem and the discus-sion in [13], we will designate the identical parallel-machine problem to minimize makespan subject tominimum total ¯ow-time as a P jjFh�Cmax=

PCi�

problem, where P designates the identical parallelmachines, Cmax denotes the maximum completiontime (makespan),

PCi represents the total ¯ow-

time, and the functional notation Fh�Cmax=P

Ci�designates that we hierarchically minimize make-span subject to minimum total ¯ow-time.

The P jjFh�Cmax=P

Ci� problem has been shownto be NP-hard by Bruno et al. [2]. This complexitystatus of the problem justi®es the development ofheuristics that can provide close to optimal solu-tions in polynomial time. Heuristic algorithms fortheir solution are developed by Co�man and Sethi[5]. Eck and Pinedo [7] improved the results ofCo�man and Sethi [5] and proposed two heuristicalgorithms for solving the P jjFh�Cmax=

PCi�

problem. They also developed a heuristic proce-dure for the P2jjFh�Cmax=

PCi� problem that gives

a minimum ¯ow-time schedule with makespan thatis guaranteed to be no more than 3.7037% abovethe makespan of the optimal schedule. Gupta andHo [10] developed an optimization algorithm forthe P2jjFh�Cmax=

PCi� problem and empirically

compared the e�ciency and e�ectiveness of Eckand Pinedo's algorithm [7] and a modi®ed versionof the MULTIFIT algorithm with their proposedalgorithm.

The rest of the paper is organized as follows.Section 2 discusses the P jjFh�Cmax=

PCi� problem

and describes lower bounds on the optimalmakespan. Existing and proposed heuristic algo-rithms are described in Section 3 and illustratedwith a numerical example. Section 4 discusses the

results of the simulation experiments designed toevaluate the e�ectiveness of the heuristics in ®nd-ing the optimal schedule. Finally, Section 5 con-cludes the paper with a summary and some fruitfuldirections for future research.

2. Lower bounds

Without loss of generality, we assume that thetotal number of jobs, n is an integer multiple of m,the total number of identical machines. If this isnot the case, enough jobs with zero processingtimes are added so that this assumption is satis®ed(i.e., we assume that s � n=m is an integer). LetN � f1; 2; . . . ; ng be the set of all jobs. The pro-cessing time of each job i 2 N , pi is independent ofthe machine and includes setup time if needed.Preemption and division of jobs is not allowed,and machines can only process one job at a time.All jobs arrive at time zero and are consideredequally important. The completion time of jobi 2 N is represented by Ci.

Let r � r�1�; r�2�; . . . ; r�n�� be a shortestprocessing time (SPT) order of n jobs wherepr�i�6 pr�i�1� for 16 i6 nÿ 1. Further, for eachk6 s, let qk � �qk�1�; . . . ; qk�m�� be a subscheduleof r such that for each j6m, qk�j� � r�m�k ÿ 1��j�. Jobs in qk are said to have a rank (position) k.Using these de®nitions, Conway et al. [6] showedthat the P jjPCi problem is optimally solved as-signing m-jobs-at-a-time rule where m jobs of rank1 are assigned to di�erent machines to be pro-cessed ®rst, followed by m jobs of rank 2 assignedto di�erent machines to be processed second, . . .,and ®nally m jobs of rank s assigned to di�erentmachines to be processed last. Notice that the jobsin the ®rst set can be assigned to the ®rst positionof any machine, that jobs in the second set can beassigned to the second position of any machine,and so on. As the job±machine assignments arenot speci®ed, the m-jobs-at-a-time procedure re-sults in multiple schedules, all with an optimaltotal ¯ow-time.

In order to evaluate the proposed heuristic al-gorithms, the optimal makespan given an optimaltotal ¯ow-time C�max=

PC�i is needed. However,

without knowing an optimal schedule, this value

J.N.D. Gupta, A.J. Ruiz-Torres / European Journal of Operational Research 125 (2000) 370±380 371

cannot be found. Therefore, we describe a lowerbound on the value of C�max=

PC�i . To do so, we

notice that the minimum possible makespan of thesingle criterion problem, P jjCmax, LB1, is a lowerbound for the P jjFh�Cmax=

PCi� problem where

LB1 � max max16 i6 n

pj;Xn

j�1

pj=m

( ): �1�

The above lower bound can be improved bynoticing that in the P jjFh�Cmax=

PCi� problem,

each machine is assigned s jobs. The optimalmakespan of the P jjFh�Cmax=

PCi� problem,

therefore, cannot be less than the sum of thesmallest jobs for the ®rst sÿ 1 positions and thejob with the longest processing time of all n jobs.In other words, the longest job must be started assoon as possible in order to minimize the make-span. In order to keep the m-jobs-at-a-time struc-ture that guarantees the optimal total ¯ow-time,the smallest job from each position in the scheduleis assigned in front of the longest job. Using thede®nitions of the job ranks described earlier,therefore, the second lower bound, LB2 is calcu-lated as shown below:

LB2 � max pqs�m�

"(�Xsÿ1

k�1

pqk�1�

#;Xn

j�1

pj=m

): �2�

In the case of integer processing times, the secondvalue in Eqs. (1) and (2) above is changed todPn

j�1 pj=me where dX e is the smallest integergreater than or equal to X.

It is readily seen that LB2 P LB1. Therefore, wewill assume that the lower bound for theP jjFh�Cmax=

PCi� problem, LB � LB2.

3. Heuristic algorithms

Several polynomial bounded heuristic algo-rithms have been proposed and evaluated for theP jjCmax problem. Graham's [9] well-known LPTrule has been shown to perform well for themakespan single criterion problem [11]. Both SPTand LPT rules are part of the family of algorithmsknown as list-scheduling, which iteratively assignsa list of jobs to the ®rst available machine. The

m-jobs-at-a-time assignment procedure used togenerate all the possible optimal schedules for theP jjPCi problem is also an application of thegeneralized SPT algorithm.

For the P jjCmax problem, the MULTIFIT al-gorithm proposed by Co�man et al. [4] for themakespan problem utilizes the close relation be-tween the bin-packing problem and the maximumcompletion time problem. Although MULTIFITis not guaranteed to perform better than LPT, ithas been shown to have a worst case bound thatis better than LPT [8]. Lee and Massey [12]proposed the algorithm COMBINE which utilizesthe LPT result as an incumbent for the MUL-TIFIT algorithm. In this algorithm, the error isnever worse than MULTIFIT, and its perfor-mance is at least as good as the best of eitherLPT or MULTIFIT.

3.1. Existing heuristics

As stated before, the P jjFh�Cmax=P

Ci� problemis NP-Hard [2]. Therefore, heuristics have beenproposed for this problem, two by Co�man andSethi [5] and two by Eck and Pinedo [7].

The ®rst heuristic by Co�man and Sethi [5],called LI, positions (ranks) jobs by increasingorder of the largest pj. From that position, jobs areassigned largest ®rst onto the machines as theybecome available. The second heuristic proposed,called LD, works in the following manner: assignpositions (ranks) in decreasing order of the largestpj. From that position, jobs are assigned largest®rst onto the machines as they become available.After all the jobs have been assigned, jobs in eachmachine must be re-sequenced by increasing orderof processing times (optimal ¯ow-time).

Eck and Pinedo [7] propose two heuristicscalled LPT � and LPT �� for the m-machine case al-though their paper focuses on the case wherem � 2. Both heuristics are based on the solution ofa reduced and constrained P jjCmax problem ob-tained by de®ning the processing time of job qk�j�in rank k � 1; 2; . . . ; s as the di�erence dqk�j� be-tween its original processing time and the pro-cessing time of the ®rst (minimum processing time)job in rank k. Thus, dqk�j� is de®ned as follows:

372 J.N.D. Gupta, A.J. Ruiz-Torres / European Journal of Operational Research 125 (2000) 370±380

dqk�j� � pqk�j� ÿ pqk�1�: �3�

The makespan of the reduced and constrainedP jjCmax problem obtained will di�er from themakespan of the original problem by an amount Xgiven by

X �Xs

k�1

pqk�1�: �4�

In addition, let LB0 be the lower bound on themakespan when the processing time for each jobi 2 N is replaced by di given in Eq. (3). Since theprocessing times in rank k are decreased by pqk�1�,LB0 can be obtained by calculating LB using Eq. (2)as follows:

LB0 � LBÿ X; �5�

where X is calculated using Eq. (4).The LPT � heuristic is similar to Co�man and

Sethi's LD heuristic and works in the followingmanner: assign positions (ranks) in decreasingorder of the largest dj. From that position, jobs areassigned largest ®rst onto the machines as theybecome available. The LPT �� heuristic selects jobsby decreasing order of dj where only one job fromeach position can be assigned to each machine. Inboth heuristics after all jobs have been assigned,jobs on each machine must be re-sequenced byincreasing order of processing times.

While Co�man and Sethi [5] and Eck andPinedo [7] provide some worst case performancebounds for these heuristics, no empirical resultsare available to compare their average case per-formance.

3.2. Proposed heuristic procedure

This section presents a heuristic procedure tosolve the P jjFh�Cmax=

PCi� problem. The pro-

posed heuristic algorithm combines a list-schedul-ing heuristic proposed by Ruiz-Torres et al. [15]for the P jjFh�Cmax;

PCi� problem, a modi®ed as-

signment procedure coupled with a backtrackingprocedure in the MULTIFIT algorithm [4]).Further, as done by Eck and Pinedo [7], we solve

the reduced and constrained P jjCmax problem de-®ned by processing times given by Eq. (3).

The proposed algorithm is composed of twophases. As in Lee and Massey's [12] COMBINEalgorithm, the ®rst phase of the proposed algo-rithm generates an initial solution as an upperbound for the P jjFh�Cmax=

PCi� problem while the

second phase searches for improved solutions. The®rst phase combines a heuristic developed by Ruiz-Torres et al. [15] for the P jjF �Cmax;

PCi� problem

with Eck and Pinedo's approach for heuristicLPT ��, where each machine can only have one jobfrom each position (rank). The second phasecombines a modi®ed assignment and a backtrack-ing procedure with the MULTIFIT algorithm.

In the proposed algorithm, an equivalentmakespan problem is created with n jobs where theprocessing time of job j is given by Eq. (3). A list ofjobs is created by combining two sub-lists, A andB. The jobs assigned to the two sub-lists changesiteratively, as well as the ordering method for eachsub-list. Jobs in list A and B are scheduled in anSPT or LPT order where the processing time ofjob j is dj. Similar to LPT ��, only one job from eachposition can be assigned to each machine. Alsolike LPT ��, once all the jobs have been assigned toone of the m machines, the jobs assigned to eachmachine are re-sequenced by increasing order oftheir original processing times.

Let pr and /r be the ordering of jobs in listsA and B respectively according to an ordering ruler where r � 1 implies SPT order and r � 2 meansan LPT order. Further, recall that r � r�1�;�r�2�; . . . ; r�n�� is an SPT order of n jobs wherepr�i�6 pr�i�1� for 16 i6 nÿ 1. For each job r�j�de®ne Rr�j� � dj=me as the rank of job r�j� wherethe symbol dX e represents the smallest integergreater than or equal to X. For each machine i6mand each rank k6 s, let I�i; k� � 1 if a job of rank khas been assigned to machine i and 0 otherwise.Then, the steps of Phase I of the proposedalgorithm, called Algorithm U , are a follows:

Algorithm U (An upper bounding procedure).Input: n, m, processing times pj for j � 1; . . . ; n.Step 0. Let r � �r�1�; r�2�; . . . ; r�n�� wherepr�i�6 pr�i�1� for 16 i6 nÿ 1, s � n=m and


Rr�j� � dj=me for j � 1; . . . ; n. For each j �1; 2; . . . ; n, let the reduced processing time ofjob r�j� be dr�j� � pr�j� ÿ pr�x� where x �m�Rr�j� ÿ 1� � 1 is the ®rst job in rank Rr�j� ofjob r�j�. For each machine i � 1; 2; . . . ;m, letui � ;. Set A � f1; . . . ; ng; B � ;, UB � 1,r � 1, and q � 2. Using Eqs. (2)±(5) calculateLB0 and enter Step 1.Step 1. Set I�i; k� � 0 for i6m; k6 s. Let pr and/q be the ordered lists of jobs in lists A and B ar-ranged by orderings r and q, respectively usingthe reduced processing times dj. Let b ��b�1�; b�2�; . . . ; b�n�� /qpr. Set T1 � � � � �Tm � 0, j � 1, and enter Step 2.Step 2. Among the machines with I�i;Rb�j�� 0,select machine h such that Th is as small as pos-sible. Schedule job b�j� on machine h, setTh � Th � db�j�. Set I�h;Rb�j�� 1, uh � uhb�j�and enter Step 3.Step 3. If j < n, set j � j� 1 and return to Step2; otherwise enter Step 4.Step 4. If Cmax � maxfT1; . . . ; Tmg < UB, setUB � Cmax, Si � ui for i � 1; 2; . . . ;m, and enterStep 5; otherwise enter Step 6.Step 5. If UB � LB0, go to Step 9; otherwise en-ter Step 6.Step 6. If pr � ;, enter Step 7, otherwise removelast job of pr and place it into job list B. Updatejob lists A and B, and return to Step 1.Step 7. Set A � f1; . . . ; ng; B � ;. If r < 2, setr � r � 1 and return to Step 1; otherwise, enterStep 8.Step 8. If q > 1, set q � qÿ 1, r � 1, and returnto Step 1; otherwise, enter Step 9.Step 9. STOP. UB is the feasible upper boundfor the problem where jobs in schedule Si areprocessed on machine i � 1; 2; . . . ;m in theSPT order of their processing times.

The complexity of Algorithm U isO�n log n� nm� since after sorting all jobs, it re-quires assignments for each list.

As an illustration of the proposed Algorithm U ,consider the 12-job example where the processingtimes are given in Table 1 and the number ofmachines m � 3.

Since jobs are arranged in the SPT order, Table 2shows the ranks, SPT order of jobs in each rank,and the processing times of all jobs for the trans-formed and constrained P jjCmax problem obtainedby using Eq. (3) with X � 91.

Initially job list A has all the jobs while job list Bis empty. We set r � 1 and q � 2. Using SPT(r � 1) ordering, p1 � f1; 4; 7; 10; 8; 2; 5; 11; 9; 6;12; 3g and /2 � ;. Hence, b � /qpr � �1; 4; 7; 10;8; 2; 5; 11; 9; 6; 12; 3�. Using Steps 2±4 of AlgorithmU , utilizing the least loaded machine allocationand the constraint that only one job from eachposition can be assigned to a machine, we obtainthe following assignments: u1 � �1; 4; 7; 10�, u2 ��8; 5; 12; 3�, and u3 � �2; 11; 9; 6� with Cmax � 21.Hence, we set UB � 21, S1 � u1 � �1; 4; 7; 10�,S2 � u2 � �8; 5; 12; 3�, and S3 � u3 � �2; 11; 9; 6�.

Then, Step 4 moves the last job in set A to set B,thus A � f1; 2; 4; 5; 6; 7; 8:9; 10; 11; 12g, B � f3g.Step 1 results in p1 � f1; 4; 7; 10; 8; 2; 5; 11; 9; 6; 12gand /2 � �3� giving b � �3; 1; 4; 7; 10; 8; 2; 5; 11; 9;6; 12�. Steps 2±4 result in the following assign-ments: u1 � �3; 9; 6; 12�, u2 � �1; 4; 7; 10�, andu3 � �8; 2; 5; 11�. Since Cmax � 29, no change ismade in UB. The next iteration results in A �f1; 2; 4; 5; 6; 7; 8; 9; 10; 11g, B � f12; 3g. Step 1 re-sults in p1 � f1; 4; 7; 10; 8; 2; 5; 11; 9; 6g and/2 � �3; 12� giving b � �3; 12; 1; 4; 7; 10; 8; 2; 5; 11;9; 6� and machine assignments: u1 � �3; 5; 11; 9�,u2 � �12; 8; 2; 6�, and u3 � �1; 4; 7; 10�, and Cmax �21, thus no change in UB � 21.

The process continues and a schedule with amakespan of 14 is found by Algorithm U . Thisschedule is generated when job list A � f1; 4;7; 8; 10g is ordered by SPT and job listB � f2; 3; 5; 6; 9; 11; 12g is ordered by LPT. Step 1results in p1 � f1; 4; 7; 10; 8g and /2 � �3; 12; 6;9; 11; 5; 2� to give b��3;12;6;9;11;5;2;1;4;7;10;8�.

Table 1

Processing times for the example problem

i 1 2 3 4 5 6 7 8 9 10 11 12

pi 2 4 12 20 22 26 27 28 32 42 46 50


Steps 2±4 result in the following assignments:S1�u1��3;11;4;7�, S2�u2��12;5;2;8�, S3�u3��6;9;1;10� with Cmax�14. Thus UB forAlgorithm U is 14. By comparison, algorithmsLPT � and LPT �� ®nd schedules with a makespanof 15.

The Phase II of the proposed algorithm com-bines MULTIFIT's moving bound with a back-tracking procedure. Similar to the upper boundingprocedure in Phase I, only one job from each po-sition (rank) can be assigned to each machine. Thestrategy of Phase II is to load machines one at atime so they are as close as possible to the bound C,but not greater than it. Each machine is ®rst loadedwith the largest possible job in each position. If thatresults in a machine makespan greater than thebound C, a backtracking procedure is used to as-sign jobs by either the best ®t (makespan less thanC and smallest slack) or by the smallest job stillavailable for that position. The forward pass isthen continued to assign jobs so that the machinemakespan is no greater than C. This process is re-peated for each machine. If this process does notlead to a feasible schedule whose makespan is atmost C, the lower bound is updated, otherwiseupper bound is updated. The process is continuedfor a speci®ed number of iterations.

To describe the steps of Phase II, letdi � �di�1�; . . . ; di�k�� be the jobs assigned to ma-chine i. Further, let D�di� �

Px2di

dx. Then, thesteps of Phase II of the proposed algorithm are asfollows:

Algorithm M (Modified MULTIFIT Algorithm).Input: n, m, dj for j � 1; . . . ; n; K, UB, LB0, and,from Algorithm U , Si for each machinei � 1; 2; . . . ;m.Step 0. Set a � 1, s � n=m, and b � 0. For eachmachine i � 1; 2; . . . ;m, let di � ; and go toStep 9.

Step 1. If b P K, go to step 10; otherwise setb � b� 1, C � �UB� LB0�=2, di � ; for i �1; . . . ;m, pk � qk, and sk � s, for k � 1; . . . ; s.Set k � 1, r � sk; i � 1, and enter Step 2.Step 2. Let x � pk�r�. If D�di� � dx6C, enterStep 3; otherwise enter Step 4.Step 3. Let di � �di;x�, pk � pk ÿ fxg, andsk � sk ÿ 1. If k < s, set k � k � 1, r � sk, andreturn to Step 2; otherwise go to Step 5.Step 4. If r � 1, go to Step 6, otherwise setr � r ÿ 1, and return to Step 2.Step 5. If i � mÿ 1, let dm � �p1; . . . ; ps� and en-ter Step 8; otherwise, set i � i� 1, k � 1, r � sk,and return to Step 2.Step 6. If k > 1, set k � k ÿ 1 and enter Step 7;otherwise set LB0 � C and return to step 1.Step 7. Let pk � fpk; di�k�g, sk � sk � 1, rear-range jobs in pk in SPT order, ®nd r such thatpk�r� � di�k�; set di � di ÿ fdi�k�g and returnto Step 4.Step 8. If Cmax > C, set LB0 � C and returnto Step 1; otherwise set UB � Cmax and enterStep 9.Step 9. If a > UB, set a � UB, Si � di for eachi � 1; . . . ;m. If a � LB0, enter Step 10, otherwisereturn to Step 1.Step 10. STOP. Processing jobs in schedule Si

on machine i � 1; 2; . . . ;m in the SPT order oftheir original processing times is the heuristicsolution of the problem.

The complexity of Algorithm M is similar to thecomplexity of the MULTIFIT algorithm given a®xed number of iterations K: O�Kn3=m�. In ourimplementation of Algorithm M , we used K � 7.

As an illustration of Algorithm M , we solve theproblem of Table 1 where the output from Algo-rithm U is used as input. Thus, UB � 14. UsingEqs. (2) and (4), LB0 � 13 (since processing timesare integers) giving C � 13:5. We start with the®rst machine, i.e., i � 1. The last job from the ®rst

Table 2

Processing times for the equivalent problem

q q1 q2 q3 q4

j 1 2 3 4 5 6 7 8 9 10 11 12

dj 0 2 10 0 2 6 0 1 5 0 4 8

Rj 1s 1 1 2 2 2 3 3 3 4 4 4


rank (x � 3) results in a load D�di� � dx�6C, thusjob 3 is assigned to machine i � 1. The last jobfrom second rank results in a load for machinei � 1 greater than C, thus we attempt to assign thenext largest job (x � 5) in rank 2 and ®nd thatD�di� � dx�6C, thus job 5 is assigned to machinei � 1. The last job from rank 3, job 9 results in aload greater than C so we attempt to assign job 8to machine 1. This process is repeated for thefourth rank jobs where job 10 is selected. The re-sult from steps 1 through 7 for the ®rst machine isthe schedule: d1 � �3; 5; 8; 10�. The process startsfor the next machine (i � 2); and the largest jobs inranks 1, 2, and 3 are selected: jobs 2, 6, and 9 re-spectively. From rank 4, no job satis®es the loadrequirement, thus we backtrack to rank 3, removejob 9 from machine 2 and attempt to assign thelargest processing time job (x � 7) to machine 2.Since D�di� � dx�6C, we assign job x � 7 tomachine 2. From rank 4, job 11 is selected andassigned to machine 2 to give d2 � �2; 6; 7; 11�.Finally Step 5 is used for the last machine (i � 3)to get d3 � �1; 4; 9; 12� with Cmax � 13. UB � 13.Since UB � LB � 13, we stop as the solution ob-tained is optimal. Reverting back to originalproblem data in Table 1, we ®nd that the heuristicschedule obtained has a makespan equal toCmax � X � 13� 91 � 104.

While each of the proposed Algorithms U andM can be used independently to solve a problem,our proposed algorithm is a combination ofAlgorithms U and M and we represent it asAlgorithm UM . The overall complexity for theproposed heuristic Algorithm UM , therefore, isO�n3=m� n log�n��.

4. Computational results

A simulation experiment was conducted to testthe e�ectiveness of the heuristics, their relativeperformance, and the e�ect of three factors onperformance: the number of machines (m), thenumber of jobs (n) related to problem complexity,and the range of processing times (pgen) an attrib-ute of the jobs to be scheduled. The LPT � andLPT �� heuristics from Eck and Pinedo [7] are usedfor comparison as they outperformed LI and LD

by a large margin. In addition, we used AlgorithmsU and M (where initial UB was found using theSPT rule with the m-jobs-at-a-time assignment)separately as well as Algorithms LPT �M andLPT ��M , combinations of Algorithms LPT �,LPT ��M , and M . The number of machines wasinvestigated at four levels: 5, 10, 15, and 20 and thenumber of jobs at four levels: 3m, 5m, 7m, 9m(s � n=m � 3; 5; 7; and 9). Pilot experiments withrational values of n=m, for example n � 7m� 2resulted in no signi®cant di�erence in relativeheuristic performance from the case where the n=mratio was an integer, as for example n � 7m. Theprocessing times were generated from uniformdistributions with three ranges: U�1; 25�, U�1; 50�,and U�1; 100�.

One hundred replications are taken at eachexperimental point for these problems, which re-sults in 4800 problem instances. Each of theseproblems was solved by the proposed algorithm aswell as the LPT � and LPT �� algorithms. For eachheuristic Algorithm h, the ratio of its makespan toits lower bound (for the original problem data),Rh � Ch

max=LB was calculated and averaged acrossall 100 problems.

An analysis of variance was carried out to de-termine the e�ects of the factors and the heuristics,while multiple-comparison t-tests were conductedin order to rank the proposed and existing heu-ristics. The use of the analysis of variance is ap-propriate for this experiment given equal samplesand normality in data (large number of replica-tions). Finally Levene's test, described by Sheskin[16] was used to test the assumption of homoge-neous variances for the data.

4.1. Empirical results

For each heuristic Algorithm h and eachproblem size used in the experiments, average Rh

values are calculated. Tables 3±5 show these valuesfor di�erent ranges of the processing times used.For each of the 48 factor combinations, AlgorithmU outperformed (average of 100 replications) LPT �

and LPT ��. Generally, LPT �� outperformed LPT �.Further, Algorithm UM outperformed AlgorithmsLPT �, LPT ��, and LPT �M in all factor combina-


tions, and outperformed Algorithm LPT ��M in 43combinations, whereas for the 5 other cases, therewas no di�erence in performance. These resultsdemonstrate that the performance of the proposedheuristic Algorithms U and UM is superior to that

of LPT � and LPT �� and their augmentation byAlgorithm M . However, Algorithm M by itself isinferior to all the other algorithms.

Table 6 shows the average Rh values for eachheuristic algorithm according to the machine

Table 3

Computational results with pgen � U�1; 25�m n LPT � LPT �� U M LPT �M LPT ��M UM

5 3m 1.0238 1.0237 1.0207 1.0410 1.0195 1.0204 1.0195

5m 1.0044 1.0034 1.0008 1.0426 1.0026 1.0029 1.0008

7m 1.0007 1.0007 1.0003 1.0281 1.0003 1.0005 1.0003

9m 1.0001 1.0006 1.0001 1.0168 1.0000 1.0004 1.0000

10 3m 1.0280 1.0211 1.0173 1.0343 1.0175 1.0155 1.0134

5m 1.0071 1.0049 1.0017 1.0446 1.0052 1.0046 1.0017

7m 1.0023 1.0022 1.0005 1.0300 1.0018 1.0018 1.0004

9m 1.0006 1.0017 1.0001 1.0220 1.0003 1.0014 1.0001

15 3m 1.0314 1.0196 1.0136 1.0308 1.0159 1.0135 1.0108

5m 1.0077 1.0045 1.0018 1.0360 1.0039 1.0030 1.0014

7m 1.0018 1.0017 1.0006 1.0335 1.0011 1.0014 1.0005

9m 1.0008 1.0019 1.0002 1.0311 1.0007 1.0018 1.0002

20 3m 1.0369 1.0204 1.0127 1.0271 1.0156 1.0137 1.0100

5m 1.0105 1.0076 1.0024 1.0402 1.0073 1.0057 1.0021

7m 1.0039 1.0030 1.0010 1.0276 1.0021 1.0024 1.0010

9m 1.0018 1.0022 1.0008 1.0226 1.0007 1.0016 1.0003

Table 4


5 3m 1.0314 1.0319 1.0297 1.0521 1.0279 1.0297 1.0279

5m 1.0060 1.0049 1.0034 1.0389 1.0048 1.0043 1.0031

7m 1.0023 1.0014 1.0007 1.0234 1.0018 1.0013 1.0006

9m 1.0010 1.0008 1.0004 1.0185 1.0010 1.0007 1.0004

10 3m 1.0338 1.0253 1.0215 1.0364 1.0200 1.0187 1.0170

5m 1.0095 1.0046 1.0027 1.0410 1.0079 1.0043 1.0026

7m 1.0036 1.0022 1.0007 1.0225 1.0026 1.0020 1.0007

9m 1.0022 1.0015 1.0006 1.0176 1.0016 1.0012 1.0005

15 3m 1.0376 1.0264 1.0227 1.0339 1.0205 1.0189 1.0169

5m 1.0115 1.0060 1.0034 1.0376 1.0091 1.0053 1.0030

7m 1.0043 1.0021 1.0007 1.0351 1.0036 1.0020 1.0006

9m 1.0023 1.0015 1.0005 1.0414 1.0021 1.0015 1.0005

20 3m 1.0454 1.0237 1.0200 1.0297 1.0211 1.0172 1.0160

5m 1.0111 1.0058 1.0038 1.0418 1.0090 1.0052 1.0036

7m 1.0051 1.0025 1.0009 1.0194 1.0033 1.0020 1.0009

9m 1.0026 1.0014 1.0006 1.0150 1.0016 1.0011 1.0006


level, the number of jobs level, and the process-ing times range level. The multiple-comparisont-test analysis demonstrated that UM signi®cantlyoutperformed all other algorithms, and thatLPT �� signi®cantly outperformed LPT �. It is im-portant to remember that, similar to previousexperiments [11,12], even when the ratio Rh isgreater than 1.00, the heuristic solution may still

be optimal given the comparison is made to alower bound.

The analysis of variance results showed that themain e�ects: heuristic (h) and number of jobs (n)were most signi®cant. The processing time range,the number of machines (m), and the interactionsh� n and h� m were also signi®cant at the 0.05level. These results indicate that the performance

Table 5


5 3m 1.0291 1.0277 1.0263 1.0425 1.0234 1.0243 1.0234

5m 1.0075 1.0059 1.0046 1.0435 1.0065 1.0056 1.0043

7m 1.0031 1.0023 1.0015 1.0243 1.0027 1.0021 1.0014

9m 1.0013 1.0008 1.0003 1.0191 1.0012 1.0007 1.0003

10 3m 1.0336 1.0280 1.0256 1.0378 1.0228 1.0229 1.0217

5m 1.0099 1.0066 1.0047 1.0413 1.0083 1.0060 1.0043

7m 1.0044 1.0023 1.0014 1.0211 1.0035 1.0022 1.0013

9m 1.0023 1.0013 1.0006 1.0158 1.0019 1.0012 1.0006

15 3m 1.0395 1.0272 1.0249 1.0360 1.0228 1.0216 1.0204

5m 1.0117 1.0064 1.0045 1.0458 1.0095 1.0061 1.0043

7m 1.0053 1.0027 1.0016 1.0293 1.0046 1.0026 1.0015

9m 1.0031 1.0014 1.0008 1.0412 1.0027 1.0014 1.0008

20 3m 1.0456 1.0255 1.0220 1.0281 1.0225 1.0186 1.0172

5m 1.0128 1.0059 1.0040 1.0371 1.0104 1.0055 1.0038

7m 1.0062 1.0022 1.0013 1.0204 1.0048 1.0020 1.0013

9m 1.0037 1.0012 1.0007 1.0116 1.0026 1.0011 1.0006

Table 6

Summary of results by experimental variable

Factor Value LPT � LPT �� U M LPT �M LPT ��M UM

n 3m 1.0347 1.0250 1.0214 1.0358 1.0208 1.0196 1.0179

5m 1.0091 1.0055 1.0032 1.0409 1.0070 1.0049 1.0029

7m 1.0036 1.0021 1.0009 1.0262 1.0027 1.0019 1.0009

9m 1.0018 1.0014 1.0005 1.0227 1.0014 1.0012 1.0004

m 5 1.0092 1.0087 1.0074 1.0326 1.0076 1.0077 1.0068

10 1.0114 1.0085 1.0065 1.0304 1.0078 1.0068 1.0054

15 1.0131 1.0085 1.0063 1.0360 1.0080 1.0066 1.0051

20 1.0155 1.0085 1.0059 1.0267 1.0084 1.0063 1.0048

Pgen U�1; 25� 1.0101 1.0075 1.0047 1.0318 1.0059 1.0057 1.0039

U�1; 50� 1.0131 1.0089 1.0070 1.0315 1.0086 1.0072 1.0059

U�1; 100� 1.0137 1.0092 1.0078 1.0309 1.0094 1.0077 1.0067

Overall 1.0123 1.0085 1.0065 1.0314 1.0080 1.0069 1.0055


of the heuristics is dependent primarily on theshop environment (problem complexity) and willbe somewhat a�ected by the range of the jobprocessing times. However, in all cases, AlgorithmUM outperformed all others, thus showing that itsrelative performance is not a�ected by the shopenvironment or the processing time range.

The average percentage deviations shown inTable 6 are plotted in Figs. 1±3 and show that achange in factors m, n, and pgen changed the per-formance of various algorithms. Fig. 1 illustratesthat as n increased, the performance of each al-gorithm improved. This last outcome is similar tothat observed in the results from Lee and Massey[12], and Kedia [11] for the P jjCmax problem, whereas n increases, the ratio Rh approached 1.0. Whilethe di�erence between LPT � LPT �, U , and UM is

very notable when n � 3m, the gap quickly nar-rows as n increases, thus Algorithms U , and UMare highly preferable to LPT � when the schedulingwindow is small (small number of positions s).

Fig. 2 illustrates that as the number of machines(m) increases the performance of the LPT � andLPT �M heuristics worsen, and the performanceof the LPT �� heuristic remains stable. However, asm increases, the performance of Algorithms U ,LPT ��M , and UM improves.

Fig. 3 shows that an increase in processing timevariability, measured by the range of the uniformdistribution (Pgen�, causes a decrease in each heu-ristic's performance in ®nding an optimal schedule(except for M).

Based on the above computational results,the proposed heuristic Algorithms U and UMare quite e�ective, and superior to the existingheuristics, in solving the P2jjFh�Cmax=

PCi�

problem.

5. Conclusions

This paper proposed a new heuristic for theproblem of scheduling n jobs on m identical ma-chines to minimize the maximum completion timesubject to an optimal total ¯ow-time. The heuristicwas empirically compared to existing heuristicsand their adaptations and was found to outper-form both of the existing ones. The performance ofthe proposed algorithm was not a�ected by anincrease in the number of machines, and waspositively a�ected by an increase in the number

Fig. 1. E�ect of the number of jobs (n) on the heuristic per-

formance.

Fig. 2. E�ect of the number of machines (m) on heuristic

performance.

Fig. 3. E�ect of the processing time range (pgen) on heuristic

performance.


jobs. However, the range of the jobs processingtime did not have a slight negative e�ect on theperformance of any of the heuristics. Overall, UMheuristic outperformed existing heuristics on allfactor combinations, making it a better solutionmethodology for the P jjFh�Cmax=

PCi� problem

when average performance is the measure ofinterest.

Several issues are worthy of future investiga-tions. First, the derivation of improved lowerbounds that would enable us to ®nd optimal so-lutions of larger problems is an interesting researchtopic. Second, the development of a suitable tie-breaking rule in the assignment of jobs in Algo-rithms U and M may improve their performance.Third, the development of the worse-case perfor-mance bounds for the existing and proposedalgorithms will be useful. Fourth, the solution ofthe problem with the reverse optimality criteria,namely, the identical parallel-machine schedulingproblem where the primary criterion is the mini-mization of maximum completion time (make-span) and the secondary criterion is theminimization of the total ¯ow-time is both inter-esting and useful. Fifth, the development of algo-rithms for other secondary criteria such as thetotal tardiness subject to the minimum makespan isa fruitful area of research. Finally, extension ofour results to more complex machine envi-ronments, such as multi-stage ¯ow-shop and job-shop problems, is important for application inindustry.

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Minimizing makespan subject to minimum total flow-time on identical parallel machines