Minimizing makespan subject to minimum flowtime on two identical parallel machines

Computers & Operations Research 28 (2001) 705}717

Minimizing makespan subject to minimum #owtime on twoidentical parallel machines

Jatinder N.D. Gupta��*, Johnny C. Ho�

�Department of Management, College of Business, Ball State University, Muncie, IN 47306, USA�Abbott Turner College of Business, Columbus State University, Columbus, GA 31907-5645, USA

Received 1 July 1998; received in revised form 1 May 1999

Abstract

We consider the problem of scheduling jobs on two parallel identical machines where an optimal scheduleis de"ned as one that gives the smallest makespan (the completion time of the last job) among the set ofschedules with optimal total #owtime (the sum of the completion times of all jobs). We propose an algorithmto determine optimal schedules for the problem, and describe a modi"ed multi"t algorithm to "nd anapproximate solution to the problem in polynomial computational time. Results of a computational study tocompare the performance of the proposed algorithms with a known heuristic shows that the proposedheuristic and optimization algorithms are quite e!ective and e$cient in solving the problem. � 2001Elsevier Science Ltd. All rights reserved.

Scope and purpose

Multiple objective optimization problems are quite common in practice. However, while solving schedul-ing problems, optimization algorithms often consider only a single objective function. Consideration ofmultiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore,enumerative optimization techniques and heuristic solution procedures are required to solve multi-objectivescheduling problems. This paper illustrates the development of an optimization algorithm and polynomiallybounded heuristic solution procedures for the scheduling jobs on two identical parallel machines tohierarchically minimize the makespan subject to the optimality of the total #owtime.

Keywords: Parallel machine scheduling; Hierarchical criteria; Makespan minimization; Flowtime minimization; Empiri-cal results

*Corresponding author. Tel.: #765-285-5300; fax: #765-285-8024.E-mail address: [email protected] (J.N.D. Gupta).

0305-0548/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 8 3 - 0

1. Introduction

Consider the following scheduling problem: a setN"�1, 2,2, n� of n jobs available at time zerois to be processed on m identical parallel machines. Each job i3N is to be processed withoutinterruption on one of the mmachines with processing time p

�. Each machine can process only one

job at a time and no job may be processed by more than one machine. Setup time, if any, is includedin the processing time. It is desired to minimize the total #ow-time as the primary objective andminimize makespan (maximum completion time) as the secondary objective. Thus, it is required to"nd a schedule for which the maximum completion time (makespan) is minimized, subject to theconstraint that no reduction in the total #ow-time is possible. Both makespan and #owtimeperformance measures have signi"cant impact on a schedule's cost, since the former generallyrepresents the amount of resources tied to a set of jobs; while the latter is a useful indicator of theamount of work-in-process inventory [1]. With the current trend to minimize work-in-processinventory, and the desire to maximize production rate, "nding a minimum makespan scheduleamong those that minimize total #owtime is a useful criterion in practice. Parallel-machinescheduling problems arise often in practice as in the scheduling of jobs to a number of computerprocessors or the scheduling of jobs to a set of identical lathes.The need to consider multiple criteria in scheduling is widely recognized. Either a simultaneous

or a hierarchical approach can be adopted. For simultaneous optimization, there are twoapproaches. First, all ezcient schedules can be generated, where an e$cient schedule isone in which any improvement to the performance with respect to one of the criteriacauses a deterioration with respect to one of the other criteria. Second, a single objective functioncan be constructed, for example by forming a linear combination of the various criteria,which is then optimized. Under a hierarchical approach, the criteria are ranked in order ofimportance; the "rst criterion is optimized "rst, the second criterion is then optimized, subject toachieving the optimum with respect to the "rst criterion, and so on. Surveys of algorithms andcomplexity results in this area are given by Chen and Bul"n [2], Lee and Vairaktarakis [3] andNagar et al. [4]. Clearly, the above described problem is one of the hierarchical multi-criteriascheduling.Following the three-"eld notation of scheduling problems, we will designate the identical parallel

machine problem to minimize makespan subject to minimum total #owtime as a P��F�(C

��/�C

�)

problem where P designates the identical parallel machines, C��

denotes the maximum comple-tion time (makespan),�C

�represents the total #owtime, and the functional notation F

�(C

��/�C

�)

designates that we hierarchically minimize makespan subject to minimum total #owtime. Thisproblem has been shown to be NP-hard by Bruno et al. [5]. Heuristic algorithms for their solutionare developed by Co!man and Sethi [6]. Eck and Pinedo [7] improved the results of Co!man andSethi [6] and proposed a heuristic method for solving the P2��F

�(C

��/�C

�) problem that gives

a minimum #owtime schedule with makespan that is guaranteed to be no more than 3.7037%above the makespan of the optimal schedule. Leung and Young [8] considered the preemptive caseof the problem and developed an algorithm for its solution. To our knowledge, no algorithm isavailable to optimally solve the P2��F

�(C

��/�C

�) problem.

This paper proposes an optimization algorithm to solve the P2��F�(C

��/�C

�) problem which,

while exponential in its computational complexity, is quite e$cient in solving large-sized probleminstances. In view of the NP-hard nature of the problem, we also propose a modi"cation of the

706 J.N.D. Gupta, J.C. Ho / Computers & Operations Research 28 (2001) 705}717

multi"t algorithm to "nd an approximate solution to the P2��F�(C

��/�C

�) problem and compare

its performance to the LPT-based heuristic developed by Eck and Pinedo [7].The rest of the paper is organized as follows. Sections 2 and 3 review the available results for

minimizing total #owtime and makespan individually and describe the optimization and heuristicalgorithms speci"cally tailored to the two-machine case. Section 4 discusses the transformation ofthe P2��F

�(C

��/�C

�) problem to an equivalent P2��C

��problem which can be solved using the

algorithms in Section 3. The simulation experiment and computational results are given in Section5. Finally, we conclude the paper in Section 6 and provide some fruitful directions for futureresearch.

2. Minimizing total 6owtime

Conway et al. [9] describe a simple extension of the SPT rule to optimally solve the P��C�

problem. An optimal schedule is obtained as follows: at any stage of the assignment of jobs tovarious machines, m shortest processing time jobs to di!erent machines. At the end, if the numberof jobs left, k, is less than m, (i.e. k(m), they are assigned to k di!erent machines. For the P2��C

�problem, this procedure simpli"es as follows:

Algorithm S: shortest processing time procedureInput: p

�for i"1,2, n; �"(�(1),2, �(n)) such that p��)p��)2)p�� .

Step 1: Let S�"(�(1), �(3),2, �(k)) where k is the largest odd integer less than or equal to n.

Similarly, S�"(�(2), �(4),2, �(j)) where j is the largest even integer less than or equal to n. Enter

Step 2.Step 2: The schedule where jobs in S

�are processed on machine 1 and jobs in S

�are processed on

machine 2 is an optimal solution of the P2��C�problem.

The computational e!ort required to solve the problem through algorithm S is O(n log n).

3. Minimizing makespan

The P��C��

problem is known to be NP-hard in the ordinary sense [10]. Hence, most of theresearch to solve the P��C

��problem aims at providing a near-optimal solution in a polynomially

bounded computational time. In this section, we describe an optimization algorithm and twoheuristic algorithms to solve the P2��C

��problem.

3.1. An optimization algorithm

Utilizing the concept of lexicographic search and speci"c properties of the P2��C��

problem, Hoand Wong [11] developed an algorithm to optimally solve the P2��C

��problem. To describe this

algorithm, let the processing time of job i be t�for i"1,2, swhere s is the total number of jobs. Let

�"(�(1),2, �(k)) be the schedule of k jobs scheduled on machine 1 where t��*t��for

J.N.D. Gupta, J.C. Ho / Computers & Operations Research 28 (2001) 705}717 707

1)i)k!1. Further, let P�"��

t�� and B"��

t�. Let � be the makespan of a known

schedule. Let � be the smallest integer greater than or equal to B/2. Then, the lower bound on themakespan, � can be found as follows:

�"max�max��

t�, ��. (1)

Then, the algorithm to "nd optimal solution to the P2��C��

problem is described as follows:

Algorithm T: two-machine optimization procedureInput: p

�for i"1,2, s; �"(�(1),2, �(s)) such that t��*t��*2*t�� ; k"1; �"�; and

�"(�(1),2, �(k))"(�(1),2, �(k)).Step 1: If max�P� ,B!P��"�, let S

�"�, �"�, and go to step 5; otherwise enter step 2.

Step 2: If max�P� ,B!P��(�, set �"max�P� ,B!P��, and S�"�. Find q such that

�(q)"�(k). If P�(�, set k"k#1. Enter step 3.Step 3: If q(s, set �(k)"�(q#1) and return to step 1, otherwise enter step 4.Step 4: If k"1, enter step 5; otherwise, set k"k!1, "nd q such that �(q)"�(k) and return to

step 3.Step 5: The schedule where jobs in S

�are processed on machine 1 and remaining jobs processed

on machine 2 is an optimal solution of the P2��C��

problem with makespan �.

Even though the computational complexity of algorithm ¹ is O(2��), computational results ofHo and Wong [11] show that the algorithm is quite e$cient in solving problem instances witha large number of jobs.

3.2. Heuristic algorithms

In view of the NP-hard nature of the P2��C��

problem, it may be appropriate to use a poly-nomially bounded heuristic to "nd an approximate solution to the problem. Two of the mostpopular heuristics are Graham's [12] LPT algorithm and the Multi"t heuristic algorithmdeveloped by Co!man et al. [13]. Each of these is described below.The longest processing time (LPT) algorithm assigns an available job with maximum processing

time to the "rst available machine. The steps of this algorithm are as follows:

Algorithm L: longest processing time procedureInput: t

�for i"1,2, s; �"(�(1),2, �(s)) such that t��*t��*2*t�� ; k"1;

¹�"¹

�"0; and �"(�(1),2, �(k))"(�(1),2, �(k)).

Step 1: Select machineM�so that¹

�is as small as possible. Schedule job �(k) on machineM

�, set

¹�"¹

�#t�� . If h"1, set �"��(k).

Step 2: If k(s, then set k"k#1 and return to Step 1. Otherwise, set S�"�, �"max�¹

�,¹

��

and STOP. The schedule where jobs in S�are processed on machine 1 and remaining jobs

processed on machine 2 is the heuristic solution of the P2��C��


The computational e!ort required to solve the problem through algorithm L is O(s log s).


By doing some more computations, it may be possible to improve the makespan. To do this, themulti-"t algorithm assigns a subset of available jobs to single machine to pack as much of thecapacity C of that machine as possible. In doing so, it attempts to pack the jobs with longestprocessing time "rst. If all jobs can be processed within capacity C, then, an attempt is made toreduce the makespan by decreasing capacity C. To start with, capacity C is found as the average ofsome known lower and upper bounds. The initial lower bound (LB) used is the same as in Eq. (1)written as follows:

¸B"max�max��

t�; ��. (2)

The initial upper bound (UB) can be either the value of a known schedule or can be obtained by thefollowing equation:

;B"B. (3)

Algorithm M: the multi5t procedureInput: t

�for i"1,2, s; �"(�(1),2, �(s)) such that t��*t��*2*t�� ; ¸B,;B,¸, and

j"1.Step 1: LetC"(;B#¸B)/2, k"1; �"(�(1),2, �(k))"(�(1),2, �(k)), Q"0 and enter Step 2.Step 2: If Q#t��)C, set Q"Q#t�� and �"��(k). Enter step 3.Step 3: If k(s, set k"k#1 and return to step 2; otherwise let C

��"max��t�� ; B!�t��

and enter Step 4.Step 4: If C

��)C, set ;B"C, S

�"�, �"C

��and enter Step 5; otherwise set ¸B"C and

enter Step 6.Step 5: If C

��"¸B, enter step 7; otherwise, enter Step 6.

Step 6: If j(¸, set j"j#1, and return to step 1; otherwise, enter step 7.Step 7: The schedule where jobs in S

�are processed on machine 1 and remaining jobs processed

on machine 2 is the heuristic solution of the P2��C��


Algorithm M requires O(s log s#s¸) computational steps where ¸ is the number of iterationsused in the multi"t algorithm. In our implementation of algorithm M, we "xed the number ofiterations, ¸"15.

4. Minimizing makespan subject to minimum 6owtime

In this section, we show that the P2��F�(C

��/�C

�) problem can be formulated as an integer

programwhich can be transformed to an equivalent P2��C��

roblem and hence solved by using thesolution approaches described in Section 3 above. To do so, without the loss of generality, weassume that the total number of jobs n is even. If n is odd, an arti"cial job with zero processing timeis added to the set of jobs. Further, let the schedule �"(�(1),2, �(n)) be such thatp��*p��*2*p�� . From algorithm S in Section 2, it is clear that jobs �(2i!1) and �(2i) areassigned to di!erent machines. Therefore, we treat these jobs in pairs, remembering that if one ofthem is assigned to machine j, the other one is not assigned to machine j. Also, once the job


assignment to both machines is known, an optimal total #owtime schedule is found by arrangingjobs at each machine in the SPT order. For formulating the P2��F

�(C

��/�C

�) problem as an

integer program, de"ne a 0!1 variable x�as follows:

x�"1 if job �(2i!1) is assigned to machine 1 and job �(2i) is assigned to machine 2.

x�"0 if job �(2i!1) is assigned to machine 2 and job �(2i) is assigned to machine 1.

Then, the following integer program represents the P2��F�(C

��/�C

�) problem:

min C��

(4)

subject to:

C��

!

��

x�[p��

!p��]*��

p�� , (5)

C��

#

��

x�[p��

!p��]*��

p��. (6)

Now, for each i"1, 2,2, n/2, de"ne t�as follows:

t�"p��

!p�� . (7)

Further, let

C��

"C��

!��

p�� . (8)

Then, the above integer program represented by Eqs. (4)}(6) can be written as follows:

min C��

(9)

subject to:

C��

!

��

x�t�*0, (10)

C��

#

��

x�t�*

��

t�. (11)

A closer look at the above integer program in Eqs. (9)}(11) shows that it represents a P2��C��

problem with s"n/2 jobs where processing time of each job i is t�. Therefore, as noted by Eck and

Pinedo [7], the P2��F�(C

��/�C

�) problem can be transformed to an equivalent P2��C

��problem

shown above.Based on the above discussion, we propose the following optimization algorithm to solve the

problem.

Algorithm H: hierarchical criteria algorithmInput: n even, p

�for i"1,2, n;

Step 1: De"ne an alternate P2��C��

problem with s jobs where the processing time of job i ist�given by Eq. (7).


Table 1Processing times for the example problem

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

p�

30 8 15 25 12 41 8 13 29 18 21 0

Table 2Processing times for the alternate problem

i 1 2 3 4 5 6

t�

11 4 3 2 4 8

Step 2: Use AlgorithmX, �X3T, L, M�) to obtain �"(�(1),2, �(k)) as the assignment of jobs tomachine 1 and "((1),2, (q)) as the assignment of jobs to machine 2.Step 3: Let S

�"��

�� r)k; �

�� r)q� and S�"��

�� r)k; ��

� r)q�. EnterStep 4.Step 4: Processing jobs in S

�on machine 1 and S

�on machine 2 in the SPT order is the solution

to the problem, which is optimal if algorithm ¹ is used in step 2 above. Thus, we have describedthree algorithms to solve the problem. Algorithm HT will yield an optimal solution, but has anexponential computational complexity bound. AlgorithmsHL and HM are heuristics that requiresO(n log n) and O(n¸#n log n) computational time, respectively, but do not always provide anoptimal solution. In the worst case, the makespan of the schedule obtained from algorithm HL isno more than 28/27 times its optimal value [7].

As an illustration of the proposed algorithms, consider the 11-job example problem of Table 1.Since n was odd, we added a job with zero processing times to get n"12. The LPT schedule

�"(6, 1, 9, 4, 11, 10, 3, 8, 5, 2, 7, 12). In step 1, we create an alternative six-job P��C��

problemwith processing times given in Table 2.Solving the P��C

��problem of Table 2 using algorithm T results in schedules �"(1, 3, 4) and

"(2, 5, 6), where the corresponding minimum makespan schedule for the original problem isS�"(6, 11, 3, 4, 2, 12) and S

�"(1, 10, 8, 9, 5, 7), with a makespan of 110. Finally, we arrange

the jobs in each machine S�and S

�in the SPT order and obtain an optimal schedule:

S�"(12, 2, 3, 11, 4, 6) and S

�"(7, 5, 8, 10, 9, 1). For this example problem, use of algorithmHL

or HM, would have produced the schedule S�"(12, 10, 7, 6, 3, 1) and S

�"(11, 9, 8, 5, 4, 2) with

a makespan of 111, which is 0.91% above the optimal makespan.

5. Experimental results

A simulation experiment is performed to evaluate the e!ectiveness of the two heuristic algo-rithms in "nding an optimal solution. In addition, the optimization algorithmHT is evaluated as toits e$ciency in solving problems involving large number of jobs.


We consider two factors in this experimental study: the variability of processing time and thenumber of jobs. Problem hardness is likely to depend on the range and distribution of theprocessing times. Therefore, the processing times are generated from a uniform distribution,;(a, b),and set at three di!erent levels: ;(1, 2000), ;(1, 10 000), and ;(1, 50 000). The uniform distribu-tion is chosen because it is commonly used in the literature and often represents the hardestproblems to solve. The large range of processing times were selected because of two reasons: (1)preliminary computational results indicated that most problems with small processing time rangeare optimized by all three algorithms beyond about 30 jobs; and (2) problems with large range ofprocessing times have been shown to generate hard problems for the P��C

��problem [14] and the

subset-sum problem [15].In our initial experiments, we solve problems involving up to 7500 jobs. However, we found that

problems involving about 1000 or more jobs become easier to optimize by each of the threealgorithms, perhaps because of the asymptotic optimality property of the ¸P¹ rule. Therefore, wereport the results for problems with the number of jobs n)2000. Thus, the computational resultsinclude problems where the number of jobs is set at 17 di!erent levels: 9, 10, 11, 12, 13, 14, 15, 20, 25,30, 40, 50, 100, 250, 500, 1000, and 2000. The smaller problems were included to examine thee!ectiveness of the heuristics in "nding optimal solutions. The combination of these two factorsgive a total of 51 sets of problems. For each set of problems, 200 replications are made. Hence, wereport the computational results for a total of 10 200 problems are solved. All three algorithms wereprogrammed in Microsoft FORTRAN running on an Pentium 166-based microcomputer.Each problem was solved using each of the three algorithms. For each problem, we determine

the percentage deviations of HL and HM from the optimum (for both the original and alternativeproblems). We also recorded the CPU time for algorithms HT, HL, HM.For the original P2��F

�(C

��/�C

�) problem, Table 3 gives the mean and maximum percent

deviations (rounded o! to three decimal places) from the optimal makespan for heuristics HL andHM. The results in Table 3 show that the e!ectiveness of the heuristics is relatively insensitive tothe changes in the range of the processing times. For both algorithms, the maximum percentdeviation tends to reduce as the number of jobs increases; when the number of jobs reaches 20 ormore, both the mean and maximum percent deviations tend to decrease.Table 3 also shows that the largest percent deviation of algorithm HL was found be 1.488%,

which is much lower than the worst case bound of 3.7037% reported by Eck and Pinedo [7]. Ingeneral, algorithm HM outperforms algorithm HL since the mean percent deviation of algorithmHM, for each set, is smaller than or equal to that of algorithmHL. However, the maximum percentdeviation of algorithm HM is larger than that of algorithm HL when n"20 for all ranges ofprocessing times.Table 4 provides the mean and maximum percent deviations (rounded o! to three decimal

places) above the optimal makespan for the equivalent P2��C��

problem. For both algorithms HLand HM, mean and maximum deviations in Table 4 are signi"cantly larger than those in Table 3.This is expected since, as shown in Eq. (8), the constant K"��

��p�� , (where �"(�(1),2, �(s))

such that p��*p��*2*p��) is subtracted from the makespan of each of the three algo-rithms for the original problem. The largest mean deviations for HL and HM algorithms are 1.036and 0.590%, respectively, which are found when n"15 and b"50 000; while the largest maximumdeviations for HL and HM algorithms are 14.413 and 8.975%, respectively, when n"10 andb"50 000 for HL, and when n"14 and b"50 000 for HM.


Table 3Mean and maximum percent deviations (original problem)

n ;(1, 2000) ;(1, 10 000) ;(1, 50 000)

Mean Max Mean Max Mean Max

HL HM HL HM HL HM HL HM HL HM HL HM

9 0.037 0.000 1.488 0.000 0.037 0.000 1.473 0.000 0.037 0.000 1.469 0.00010 0.045 0.000 1.012 0.000 0.045 0.000 1.036 0.000 0.045 0.000 1.037 0.00011 0.047 0.023 0.858 0.821 0.047 0.023 0.862 0.837 0.047 0.023 0.861 0.83712 0.056 0.011 0.937 0.269 0.056 0.011 0.951 0.260 0.056 0.011 0.950 0.25913 0.059 0.032 0.500 0.423 0.060 0.032 0.500 0.416 0.060 0.032 0.497 0.41614 0.069 0.036 0.770 0.460 0.068 0.036 0.787 0.473 0.068 0.036 0.792 0.47315 0.070 0.039 0.746 0.457 0.071 0.040 0.751 0.465 0.071 0.040 0.749 0.46720 0.033 0.025 0.389 0.440 0.033 0.025 0.389 0.434 0.033 0.025 0.391 0.43625 0.020 0.012 0.211 0.080 0.021 0.013 0.210 0.066 0.021 0.013 0.213 0.06630 0.009 0.008 0.053 0.053 0.010 0.009 0.051 0.049 0.010 0.009 0.050 0.04940 0.004 0.002 0.023 0.020 0.005 0.003 0.024 0.022 0.005 0.004 0.024 0.02350 0.002 0.001 0.015 0.008 0.002 0.002 0.013 0.011 0.002 0.002 0.012 0.012100 0.000 0.000 0.002 0.002 0.000 0.000 0.002 0.001 0.000 0.000 0.002 0.001250 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000500 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0001000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0002000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Unlike the results of the original problem, the mean and maximum deviations and processingtime range show a positive correlation for both algorithms. Moreover, when the number of jobs isless than 20, the mean deviation increases as n increases; and when the number of jobs is 20 ormore, the mean deviation decreases as n increases. This indicates that, in general, hard instances forthe P2��C

��problem may not be hard instances for the corresponding P2��F

�(C

��/�C

�) problem

since the number of jobs in the alternate P2��C��

problem to be solved for the later problem arehalf of those for the original problem. In addition, because of the use of Eq. (8) in de"ning theprocessing times of the alternate P2��C

��problem, the range of the processing times is likely to be

less than that for the original P2��F�(C

��/�C

�) problem.

Table 5 reports the percentage of problems that could not be optimized by algorithm HL andalgorithm HM. These results show that as the processing time range increases, the percentage ofnon-optimal solutions increases signi"cantly. For example, the mean percentage of non-optimalsolutions increases from 30.1 to 48.5% for algorithmHL and from 21.1 to 42.4% for algorithmHMwhen b increases from 2000 to 50 000. The larger percentages are concentrated in sets with mediumvalues of n (25}250) and largest value of b (50 000). The largest percentage of non-optimal solutionsfor HL andHM as shown in Table 4 are 99.5 and 97%, respectively. These values occur in the set ofproblems with b"50 000.Table 6 gives the set CPU time for each of the three algorithms. Algorithm HM requires

approximately twice the CPU time as that for algorithm HT while algorithm HL requires a littleless CPU time than that for algorithm HT; nonetheless, all three algorithms are very fast.


Table 4Mean and maximum percent deviations (alternate problem)

n ;(1, 2000) ;(1, 10 000) ;(1, 50 000)

Mean Max Mean Max Mean Max

HL HM HL HM HL HM HL HM HL HM HL HM

9 0.298 0.000 13.322 0.000 0.296 0.000 13.183 0.000 0.295 0.000 13.154 0.00010 0.452 0.000 14.052 0.000 0.456 0.000 14.398 0.000 0.455 0.000 14.413 0.00011 0.512 0.258 7.312 8.124 0.510 0.258 7.137 8.304 0.511 0.257 7.127 8.30412 0.732 0.152 12.821 3.721 0.739 0.152 12.610 3.671 0.737 0.153 12.504 3.64213 0.790 0.404 6.701 5.542 0.801 0.403 6.918 5.451 0.802 0.402 6.947 5.45214 0.992 0.528 9.272 8.883 0.986 0.527 9.400 8.950 0.986 0.528 9.381 8.97515 1.025 0.589 9.864 6.157 1.035 0.590 9.922 6.135 1.036 0.590 9.891 6.16220 0.655 0.503 6.041 6.836 0.657 0.512 6.037 6.737 0.654 0.510 6.063 6.76225 0.478 0.309 4.276 1.772 0.503 0.334 4.235 1.805 0.508 0.335 4.303 1.80630 0.284 0.245 1.774 1.774 0.324 0.282 1.479 1.639 0.323 0.287 1.471 1.64040 0.155 0.086 0.730 0.780 0.186 0.143 0.835 0.857 0.196 0.153 0.784 0.89650 0.093 0.045 0.688 0.394 0.121 0.082 0.589 0.545 0.126 0.091 0.582 0.596100 0.007 0.002 0.210 0.209 0.028 0.017 0.180 0.130 0.032 0.024 0.205 0.125250 0.000 0.000 0.000 0.000 0.003 0.001 0.047 0.044 0.010 0.007 0.041 0.041500 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.017 0.0151000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.008 0.0082000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

From the results in Table 6, it is clear that the processing time range does not appear to havea major impact on the CPU time. However, for the number of jobs n'100, the CPU times increasesigni"cantly for each algorithm. Since our computer codes may be ine$cient, these results showthat there is not much di!erence in the CPU time required to solve a problem by algorithms HLand HT with algorithm HM requiring somewhat more CPU time than that for algorithm HL orHT. Further, as was stated earlier, we were able to solve problems involving up to 7500 jobs tooptimality with about the same CPU times as those required for algorithmHL. These results showthat use of the proposed algorithm HT is practical to optimally solve the P2��F

�(C

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problem with a very large number of jobs. While the asymptotic property of the ¸P¹ rule mayprovide optimal results while using algorithm HL, use of algorithm HT guarantees an optimalsolution for large problems without any appreciable increase in CPU time.

6. Conclusions

This paper considered the two-identical-parallel-machine problem to minimize makespan sub-ject to minimum total #owtime and proposed an optimization algorithm and a heuristic algorithmfor its solution. Computational results of a simulation study with randomly generated problemsshow that the proposed optimization algorithm HT is quite e$cient in optimizing large-sized


Table 5Percentage of problems the proposed algorithm returns a smaller makespan

n ;(1, 2000) ;(1, 10 000) ;(1, 50 000)

HL HM HL HM HL HM

9 8.0 0.0 8.0 0.0 8.0 0.010 10.0 0.0 10.0 0.0 10.0 0.011 19.5 12.0 18.5 13.5 18.5 13.512 22.0 8.0 22.5 8.0 22.5 8.013 37.5 23.5 39.5 25.0 39.0 25.014 45.0 31.5 45.5 31.5 45.0 32.015 59.0 43.0 63.0 48.0 64.0 47.520 73.5 63.0 81.0 72.0 84.0 75.025 77.5 64.5 93.5 87.0 95.5 92.530 65.0 57.0 90.5 85.5 96.0 97.040 53.0 35.0 87.5 83.0 94.5 96.550 38.5 21.0 84.5 74.0 99.5 96.0100 3.5 1.0 45.0 31.5 79.0 80.5250 0.0 0.0 8.5 3.5 59.5 48.0500 0.0 0.0 0.0 0.0 9.5 8.51000 0.0 0.0 0.0 0.0 0.5 1.02000 0.0 0.0 0.0 0.0 0.0 0.0Mean 30.1 21.1 41.0 33.1 48.5 42.4

Table 6Set CPU time (in seconds)

n ;(1, 2000) ;(1, 10 000) ;(1, 50 000)

HL HM HT HL HM HT HL HM HT

9 0.15 0.25 0.20 0.10 0.10 0.05 0.05 0.15 0.1010 0.10 0.15 0.15 0.05 0.20 0.05 0.10 0.05 0.0011 0.20 0.30 0.30 0.10 0.10 0.15 0.00 0.10 0.0012 0.00 0.20 0.05 0.10 0.10 0.05 0.10 0.10 0.0513 0.15 0.20 0.20 0.30 0.25 0.20 0.05 0.10 0.1514 0.10 0.20 0.10 0.05 0.25 0.15 0.15 0.10 0.1515 0.25 0.30 0.25 0.05 0.35 0.05 0.15 0.20 0.0520 0.10 0.25 0.25 0.15 0.25 0.25 0.05 0.30 0.2025 0.15 0.25 0.15 0.05 0.25 0.40 0.15 0.30 0.7030 0.05 0.40 0.10 0.10 0.25 0.50 0.15 0.40 1.8540 0.20 0.45 0.05 0.25 0.60 0.35 0.25 0.45 0.6050 0.10 0.65 0.15 0.20 0.60 0.30 0.25 0.75 1.81100 0.35 1.05 0.25 0.35 1.20 0.65 0.45 1.35 0.85250 1.20 2.75 1.10 1.15 2.90 1.40 1.35 2.50 1.70500 2.45 5.80 2.50 2.20 6.10 2.50 2.35 6.25 2.901000 5.10 11.85 5.30 4.95 12.35 5.55 4.90 12.65 6.012000 10.95 22.95 11.30 10.60 24.65 11.60 10.80 25.60 12.10

Mean 1.27 2.82 1.32 1.22 2.97 1.42 1.25 3.02 1.72


problems. Further, the heuristic algorithm HM is relatively more e!ective in "nding optimalsolutions than the existing algorithm HL. The computational results indicate that the proposedHT algorithm, which "nds optimal makespan, outperforms algorithm HL and algorithm HM inmost of the cases. Furthermore, the algorithm HT "nds a smaller makespan than those ofalgorithms HL and HM in 39.9 and 32.2% of the 10 200 test problems, respectively. In terms ofCPU time, all three algorithms are very e$cient, HL requiring the least CPU time, followed byalgorithm HT which requires a little more CPU time than algorithm HL. Algorithm HM requiresabout twice the CPU time than that for algorithm HT.Several issues are worthy of future investigations. First, the development of the worse-case

performance bounds for the HM algorithm will be useful. Second, extension of results to them-identical-parallel-machine case (m'2) will be worthwhile. Third, the solution of the problemwith the reverse hierarchical optimality criteria, namely, the identical-parallel-machine schedulingproblem where the primary criterion is the minimization of maximum completion time (makespan)and the secondary criterion is the minimization of the total #ow-time is both interesting and useful.Fourth, the development of algorithms for other secondary criteria such as the total tardinesssubject to the minimum makespan is a fruitful area of research. Finally, extension of our results tomore complex machine environments, such as multi-stage #owshop and job}shop problems, isimportant for application in industry.

References

[1] Pinedo M, Chao X. Operations scheduling with applications in manufacturing and services. Irwin/McGraw Hill,Homewood, IL/New York: 1991.

[2] Chen C, Bul"n RL. Complexity of single machine, multi-criteria scheduling problems. European Journal ofOperational Research 1993;70:115}25.

[3] Lee CY, Vairaktarakis GL. Complexity of single machine hierarchical scheduling: a survey, In: Pardalos PM,Editor. Complexity in numerical optimization. Singapore: World Scienti"c, 1993. p. 269}98.

[4] Nagar A, Haddock J, Heragu S. Multiple and bicriteria scheduling: a literature survey. European Journal ofOperational Research 1995;81:88}104.

[5] Bruno J, Co!man EG, Sethi R. Scheduling algorithms to minimize mean #ow time. Proceedings of the IFIPCongress 74, Amsterdam, North-Holland, 1974. p. 504}10.

[6] Co!man EG, Sethi R. Scheduling algorithms minimize mean #ow time: schedule length properties. Acta Infor-matica 1976;6:1}14.

[7] Eck BT, Pinedo M. On the minimization of the makespan subject to #owtime optimality. Operations Research1993;41:797}800.

[8] Leung JYT, Young GH. Minimizing schedule length subject to minimum #ow time. SIAM Journal of Computing1989;18:314}26.

[9] Conway RW, Maxwell WL, Miller LW. Theory of scheduling, Reading, MA: Addison-Wesley, 1967.[10] Garey MR, Johnson DS. Computers and intractability: a Guide to the theory of NP-completeness. San Francisco,

CA: Freeman, 1979.[11] Ho JC, Wong JS. Makespan minimization for parallel identical processors. Naval Research Logistics

1995;42:935}48.[12] Graham RL. Boundaries on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics

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[14] Dell'Amico M, Martello S. Optimal scheduling of tasks on identical parallel processors. ORSA Journal ofComputing 1995;7:191}200.

[15] Martello S, Toth P. Approximation schemes for the subset-sum problem: survey and experimental analysis.European Journal of Operational Research 1985;22:56}69.

Jatinder N.D. Gupta is Professor of Management, Information and Communication Sciences, and Industry andTechnology at the Ball State University, Muncie, Indiana, USA. He holds a Ph.D. in Industrial Engineering (withspecialization in Production Management and Information Systems) from Texas Tech University. Coauthor of a text-book in Operations Research, Dr. Gupta serves on the editorial boards of several national and international journals. Hehas published numerous research and technical papers in such journals as International Journal of Information Manage-ment, Journal of Management Information Systems, Operations Research, IIE Transactions, Naval Research Logistics,European Journal of Operational Research, etc. His current research interests include information technology, scheduling,planning and control, organizational learning and e!ectiveness, systems education, and knowledge management.Johnny C. Ho is Associate Professor of Operations Management at Columbus State University. He holds a Ph.D. in

Management from Georgia Institute of Technology. Dr. Ho has published articles such journals as Naval Research ofLogistics, Annals of Operations Research, European Journal of Operational Research, Production Planning and Control, andInternational Journal of Production Economics. His current research interests include scheduling, planning and control,and technology justi"cation.


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Minimizing makespan subject to minimum flowtime on two identical parallel machines