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Scheduling Jobs with Piecewise LinearDecreasing Processing Times

T.C. Edwin Cheng,1 Qing Ding,1 Mikhail Y. Kovalyov,2Aleksander Bachman,3 Adam Janiak3

1 Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon,Hong Kong SAR, Peoples Republic of China

2 Institute of Engineering Cybernetics, National Academy of Sciences of Belarus, Surganova6, 220012 Minsk, Belarus

3 Institute of Engineering Cybernetics, Wroclaw University of Technology, Janiszewskiego11/17, 50-372 Wroclaw, Poland

Received 9 January 2001; revised 11 April 2002; accepted 25 November 2002

10.1002/nav.10073

Abstract: We study the problems of scheduling a set of nonpreemptive jobs on a single ormultiple machines without idle times where the processing time of a job is a piecewise linearnonincreasing function of its start time. The objectives are the minimization of makespan andminimization of total job completion time. The single machine problems are proved to beNP-hard, and some properties of their optimal solutions are established. A pseudopolynomialtime algorithm is constructed for makespan minimization. Several heuristics are derived for bothtotal completion time and makespan minimization. Computational experiments are conducted toevaluate their efciency. NP-hardness proofs and polynomial time algorithms are presented forsome special cases of the parallel machine problems. 2003 Wiley Periodicals, Inc. Naval ResearchLogistics 50: 531554, 2003.

Keywords: machine scheduling; start time dependent processing times; computational com-plexity

1. INTRODUCTION

The following scheduling problem with start time dependent job processing times is studied.There are n independent nonpreemptive jobs, which are simultaneously available, to bescheduled for processing on m parallel machines. Each job can be completely processed by anymachine. Each machine can handle at most one job at a time and cannot stand idle until the last

Correspondence to: T.C.E. Cheng

2003 Wiley Periodicals, Inc.

job assigned to it has nished processing. A schedule is characterized by the sequences of jobsarranged in order of processing on the machines. The processing time of job j scheduled onmachine l depends on its start time t in the following way:

plj alj, if t y,alj bljt y, if y t Y,alj bljY y, if t Y.

On machine l, each job j thus has a normal processing time alj, a common initial decreasingdate y, after which the processing time starts to decrease linearly with a decreasing rate blj anda common nal decreasing date Y (Y y), after which it decreases no further. It is assumedthat 0 blj 1 and alj blj(min{i1n ali alj, Y} y) hold for each job j and machinel. The rst condition ensures that the decrease of each job processing time is less than one unitper every unit of delay in its starting moment. The second condition guarantees positive jobprocessing times. These conditions are natural for real-life applications (see Ho, Leung, and Wei[14]). Given a schedule, the job completion times Cj, j 1, . . . , n, are easily determined. Twoclassical scheduling criteria are considered: makespan, Cmax max{Cjj 1, . . . , n}, andtotal completion time, j1n Cj. These criteria are related to throughput time and total work-in-process inventories of a production system, respectively.

Two variants of the above problem are distinguished. If it is known that all jobs can becompleted by Y for any schedule, then the corresponding problem is called unbounded;otherwise, it is called bounded. It is assumed that all alj, y, and Y are integral and all blj arerational so that, blj vlj/L, where vlj are integers and L is a natural number, j 1, . . . , n,l 1, . . . , m. For the case of a single machine or identical machines, the index l is omittedin the corresponding notation.

We adopt the three-eld notation of Graham et al. [12] for describing traditional schedulingproblems, //, to denote our problem and its special cases. We add the description plj alj blj(t y) in the second eld to indicate the presence of start time dependent job processingtimes and the description Y to signify the unbounded problems. Some other descriptionsare added to the second eld, which are easily understood.

Scheduling models with start time dependent job processing times began to attract theattention of the research community only in the last two decades. Most fundamental results areobtained for the situation where there is a single machine, the job processing time is a linearincreasing function of the job start time, i.e., pj aj bj(t y) for each j, y 0, Y ,and the objective is to minimize makespan.

Assuming y 0 and pj aj bjt, Gupta and Gupta [13] prove that the unbounded problemis solved by sequencing the jobs in nonincreasing order of bj/aj. Browne and Yechiali [4]consider a stochastic version of this problem. They show that the expected makespan isminimized when the jobs are sequenced in nondecreasing order of E(aj)/bj, where E(aj) is themean of aj. Glazebrook [10, 11] extends the latter model by incorporating precedence relationsbetween the jobs and allowing preemptions. Kunnathur and Gupta [20] address the unboundedproblem with individual initial increasing dates yj for the jobs. They develop a branch-and-bound algorithm and a dynamic programming algorithm for such a problem.

Mosheiov [23] addresses the unbounded problem of minimizing total completion time wherey 0 and pj 1 bjt for all j. He proves that the optimal job sequences are V-shaped sothat the jobs appearing before the job with the smallest bj are sequenced in nonincreasing orderof bj, and the remaining jobs are sequenced in nondecreasing order of bj. This property provides

532 Naval Research Logistics, Vol. 50 (2003)

a basis for an O(n2n3) enumerative algorithm and some heuristics. Mosheiov [24] also derivespolynomial algorithms for the unbounded problems of minimizing makespan, total completiontime, and number of tardy jobs if y 0 and pj bjt for all j, where it is assumed that theprocessing of the rst job starts at time 0 for a given small .

Kubiak and van de Velde [19] prove that the unbounded problem of minimizing makespan isNP-hard in the ordinary sense if y 0 and pj aj bj(t y) for all j. They also providepseudopolynomial algorithms for both unbounded and bounded cases. Kovalyov and Kubiak[18] develop a fully polynomial time approximation scheme for the bounded case.

There are many applications of the model where the job processing time is an increasingfunction of the job start time. These include the control of queues and communication systems,shops with deteriorating machines, and/or delay of maintenance or cleaning, re ghting, andhospital emergency wards, scheduling steel rolling mills, etc. ([4], [20], [23], [19]).

The single machine model with pj aj bjt has recently been suggested by Ho, Leung, andWei [14]. They show that the sequence in nonincreasing order of aj/bj is optimal for theunbounded problem of minimizing maximum lateness if y 0 and the jobs have a common duedate. This problem is motivated by military applications, where the task is to destroy an aerialthreat and its execution time decreases with time as the longer the action is delayed, the closerthe threat gets. Chen [6] gives an O(n2) algorithm for the problem where the objective is tominimize number of tardy jobs.

Another application of the model with decreasing job processing times can be found inenvironments where actual job processing times decrease due to the learning effect. As anexample, consider a machine operator who receives an order to produce a batch of enginecomponents that are similar in processing requirements but different in sizes. During the initialproduction period, his productivity is low as he needs to work out the proper operationalprocedures and work method through trial-and-error to produce the components. His produc-tivity will gradually increase as a result of learning through practice and experience gained fromrepeating similar operations and procedures over time.

A review of the results on scheduling with time dependent processing times has recently beenprovided by Alidaee and Womer [1]. Recent contributions to this eld, not mentioned in theabove review, have been made by Bachman [2], Bachman and Janiak [3], Gawiejnowicz [9], andKononov [17].

We introduce a model more general than that suggested by Ho, Leung, and Wei [14]. Thismodel can provide a better approximation for some real-life situations where job processingtimes decrease with their start times. For example, in environments where learning effect takesplace, the productivity of an operator is at a low level initially, and it gradually increases to astable level after some time because of physical and safety limitations.

We prove that the bounded problems 1/pj aj bj(t y), y 0/Cmax and 1/pj aj bj(t y), y 0/ Cj are both NP-hard in the ordinary sense. We further establish someproperties for the optimal solutions of the single machine problems and construct a pseudopoly-nomial time algorithm for the problem 1/pj aj bj(t y)/Cmax. It follows that this problemis only NP-hard in the ordinary sense. The strong NP-hardness of the problem 1/pj aj bj(t y), y 0/ Cj remains an open question. Derivation of a pseudopolynomial timealgorithm for this problem and for its extension to an arbitrary y could be possible if such analgorithm would exist for the problem with y 0 and Y . The latter problem was studiedby Ng et al. [25]. The authors show that sequencing jobs in nondecreasing order of aj is optimalfor the cases bj b, j 1, . . . , n, and bj kaj, j 1, . . . , n. They derive apseudopolynomial time algorithm for the case aj a, j 1, . . . , n, but fail to derive suchan algorithm for the problem in its general setting.

533Cheng et al.: Scheduling Jobs with Decreasing Processing Times

For practical purposes, we derive several heuristics to solve the NP-hard single machineproblems and conduct computational experiments to evaluate their efciency.

We show that the unbounded problems with identical machines, P2/pj aj bj(t y),bj b, y 0, Y /Cmax, and P/pj aj bj(t y), bj b, y 0, Y /Cmax areNP-hard and strongly NP-hard, respectively, and the unbounded problem with unrelatedmachines, R/plj alj blj(t y), y 0, blj b, Y / Cj, is solvable in O(n3) timeby a transformation to a weighted bipartite matching problem. If the machines are uniform, thelatter time complexity can be further reduced to O(n log n).

We would like to contrast our results with those of Kubiak and van de Velde [19]. The presentstudy of problems with piecewise linear decreasing job processing times is a continuation of thework reported in Cheng and Kovalyov [7]. We have attempted to adapt the NP-hardness proofand pseudopolynomial algorithms provided by Kubiak and van de Velde for the Cmax-problemwith piecewise linear increasing job processing times. However, the effort was unsuccessfulbecause inversion of the sign in the formula for processing times does not lead to a mirrorproblem that could then be similarly analyzed. The cause can be due to the fact that processingtimes and, therefore, the Cmax value depend on the job sequence. In this case, a known inversionapproach like that for the classical problems 1/rj/Cmax and 1/ /Lmax (see Lawler et al. [22])does not work.

We have exploited the specic characteristics of our problem to construct the NP-hardnessproof, which is based on different ideas from those of Kubiak and van de Velde. Moreover, theydid not study the Cj criterion. Our dynamic programming algorithm has some similarities withthat of Kubiak and van de Velde. It is based on specic properties of the problem withdecreasing job processing times, which are different from those established by Kubiak and vande Velde for the problem with increasing job processing times. As the main goal of our paperis to provide classication of computational complexities, the derivation of a pseudopolynomialalgorithm is important as its existence, coupled with the NP-hardness proof, establishes that aproblem is only NP-hard in the ordinary sense.

2. SINGLE MACHINE

In this section, we prove that the bounded single machine problems are NP-hard in theordinary sense, establish some properties of their optimal solutions, and show that makespan canbe minimized in pseudopolynomial time. We further describe several heuristics for theseproblems and conduct computational experiments to evaluate their efciency.

2.1. NP-Hardness Proofs

THEOREM 1: The problem 1/pj aj bj(t y), y 0/Cmax is NP-hard.

PROOF: We show that the decision version of the above problem is NP-complete by atransformation from the known NP-complete problem PARTITION (Garey and Johnson [8]): Givenpositive integers h1, . . . , hr, is there a set S {1, . . . , r} such that jS hj H, where2H j1r hj?

Given any instance I of PARTITION, we dene V (r!)(2r)3r6H2, A V4, 1/V20, 1/V22, and construct the following instance II for our problem. There are 2r jobs: (1, j) and (2,j) with normal processing times

534 Naval Research Logistics, Vol. 50 (2003)

ai, j A ai, j, i 1, 2, where a1, j 2jH hj, a2, j 2jH,

and decreasing rates

bi, j ai, j bi, j, i 1, 2, where b1, j 2rjHhj

r j 1 , b2,j 2rjH.

It can easily be seen that the decreasing rates can be represented in the form vi, j/L, where vi, jare some natural numbers and L (r!)V22.

The common initial decreasing date is y 0 and the common nal decreasing date is Y rA 2r1H H. Dene

E 3Y2/2 rA2/2 A2r1H H and

F j1

r r j 12rrj1H rhjr j 1 H.

The threshold value for Cmax is G 2Y E AF 2V.We prove that instance I of PARTITION has a solution if and only if there exists a solution for

the constructed instance II of our problem.It is easy to see that the construction of II can be done in time polynomial in the length of

I.From the denition of V, A, , and , we have

V 1V21

1V20 V

1V19 A

1V16 A

21V12 . (1)

Consider a job sequence which is a solution for instance II. This solution is called a canonicalsolution if it is of the form ((kr, r), (kr1, r 1), . . . , (k1, 1), (3 kr, r), (3 kr1, r 1), . . . , (3 k1, 1)), where kj {1, 2}, j 1, . . . , r. We will show that if II has a solution,then there exists a canonical solution.

Let a job sequence be a solution for II. Denote by [ j] the job in the position j of thissequence and denote by s[ j] the start time of this job. Dene j1r a[ j] Y and j2r( j 1)b[ j] r jr12r b[ j] F. The proof of the theorem is based on the following threelemmas.

LEMMA 1: If is a solution for II, then 0.

PROOF: If 0, then the (r 1)st job starts after Y. Then we have

Cmax j1

r

aj bjsj jr1

2r

aj bjY

j1

2r

aj j2

r

bj i1

j1

ai Y jr1

2r

bj

535Cheng et al.: Scheduling Jobs with Decreasing Processing Times

2Y j2

r i1

j1

aj bjai Y jr1

2r

aj bj 2Y

j2

r i1

j1

ajai Y jr1

2r

aj. (2)Dene X1 j2r i1j1 a[ j]a[i] Y jr12r a[ j]. Since j1r a[ j] Y and a[ j] A

a[ j], we have

X112 j1

r

aj 2 12 j1r

aj2 Y2Y

j1

r

aj

12 Y

212 j1

r

A2 2Aaj aj2 YY

12 Y...