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eniv

an 7

Scheduling

atioa stimfun

and

time algorithm to nd the optimal job sequence and resource allocation. 2010 Elsevier Ltd. All rights reserved.

assuminedo,hich joenome

tion r in a sequence, its actual processing time is

pj pjra;where pj is the normal processing time of job Jj, a 6 0 is a constantlearning effect. He proved that single machine scheduling problems

total tardiness. To solve this scheduling problem, they developeda mathematical programming model. Eren and Guner (2007b) con-sidered the single machine total tardiness problem with a learningeffect. They developed an integer programming model for theproblem. Cheng et al. (2008a, 2008b) considered some schedulingproblems with the actual job processing time is a function of jobsalready processed. Toksar and Guner (2008) considered the parallelmachine scheduling with learning effects and deteriorating jobs.They introduced a mixed nonlinear integer programming formula-tion for parallel machine earliness/tardiness (ET) scheduling

q This manuscript was processed by Area Editor T.C. Edwin Cheng.* Corresponding author at: Operations Research and Cybernetics Institute,

School of Science, Shenyang Aerospace University, Shenyang 110136, China.Tel./fax: +86 24 89723548.

Computers & Industrial Engineering 59 (2010) 458462

Contents lists availab

Computers & Indus

.eE-mail address: wangjibo75@yahoo.com.cn (J.-B. Wang).controllable processing time (resource allocation). Job learningappears in, e.g., repeated processing of similar tasks improvesworker skills; workers are able to perform setup, to deal with ma-chine operations and software, or to handle raw materials andcomponents at a greater pace (Biskup, 1999). Job-processing timesare controllable by allocating a common limited resource suchas nancial budget, overtime, energy, fuel, subcontracting ormanpower.

Biskup (1999) and Cheng and Wang (2000) were among thepioneers that brought the concept of learning into the eld ofscheduling. Biskup (1999) assumed that the processing time of ajob is a log-linear learning curve, i.e., if job Jj is scheduled in posi-

minimize the maximum lateness. They showed that the problemis NP-hard in the strong sense and then identied two special casesthat are polynomially solvable. They also proposed two heuristicsand analysed their worst-case performance. A survey on this lineof the scheduling problems with learning effects could be foundin Biskup (2008). More recent papers that have considered schedul-ing problems with learning effects include Eren and Guner (2007a,2007b), Cheng, Wu, and Lee (2008a, 2008b), Toksar and Guner(2008), Wu and Lee (2009), Lee and Wu (2009), Eren (2009a,2009b) and Zhu et al. (2010). Eren and Guner (2007a) considereda bicriteria single machine scheduling problem with a learningeffect to minimize a weighted sum of total completion time andSingle-machineLearning effectResource allocation

1. Introduction

In classical scheduling theory, it ising times xed and constant values (Pever, we often encounter settings inwbe subject to change due to the ph0360-8352/$ - see front matter 2010 Elsevier Ltd. Adoi:10.1016/j.cie.2010.06.002ed that the job process-2002). In practice, how-b processing timesmaynon of learning and/or

to minimize the sum of job ow times and the total deviations ofjob completion times from a common due date are polynomial solv-able. Cheng and Wang (2000) considered a single machine schedul-ing problem in which the job processing times decrease as a resultof learning. A volume-dependent piecewise linear processing timefunction was used to model the learning effect. The objective is toKeywords:waiting time, total absolute differences in waiting times and total resource cost. We analyse the problemwith two different processing time functions. For each combination of these, we provide a polynomialSingle-machine scheduling with learningprocessing timesq

Dan Wang a, Ming-Zheng Wang b, Ji-Bo Wang a,c,*aOperations Research and Cybernetics Institute, School of Science, Shenyang Aerospace Ub School of Management, Dalian University of Technology, Dalian 116024, ChinacKnowledge Management and Innovation Research Centre, Xian Jiaotong University, Xi

a r t i c l e i n f o

Article history:Received 6 January 2010Received in revised form 2 June 2010Accepted 3 June 2010Available online 11 June 2010

a b s t r a c t

We consider resource allocfunction of its position insequence of jobs and the opnamely, minimizing a costences in completion times

journal homepage: wwwll rights reserved.ffect and resource-dependent

ersity, Shenyang 110136, China

10049, China

n scheduling with learning effect in which the processing time of a job is aequence and its resource allocation. The objective is to nd the optimalal resource allocation separately. We concentrate on two goals separately,ction containing makespan, total completion time, total absolute differ-total resource cost; minimizing a cost function containing makespan, total

le at ScienceDirect

trial Engineering

lsev ier .com/ locate/caie

striascheduling problems. Wu and Lee (2009) and Lee and Wu (2009)some single machine and owshop scheduling with a general learn-ing effect. Eren (2009a) considered a bicriteria parallel machinescheduling problems with a learning effect of setup times and re-moval times. The objective function of the problem is minimizationof the weighted sum of total completion time and total tardiness.He developed a mathematical programming model for the problemwhich belongs to NP-hard class. Eren (2009b) considered them-par-allel machine with maximum lateness consideration of the learningeffect. He developed a mathematical programming model for theproblem which belongs to NP-hard classes. Zhu, Sun, Sun, and Li(2010) considered two single machine scheduling problems withproportional linear deterioration of job processing times. For theseresource constrained problems, i.e., the total resource resource con-sumption minimization problem under the constraint that themakespan does not exceed a given limit, and the makespan minimi-zation problem under the constraint that the total resource con-sumption does not exceed a given limit, they proved that theseproblems can be solved in polynomial time.

On the other hand, sequencing problems with controllable pro-cessing times have been studied extensively by researchers since1980 (e.g., Alidaee & Ahmadian, 1993; Biskup & Jahnke, 2001;Cheng, Ogaz, & Qi, 1996; Daniels, 1990; Hoogeveen & Woeginger,2002; Janiak, 1987; Ng, Cheng, Kovalyov, & Lam, 2003; Panwalkar& Rajagopalan, 1992; Shabtay & Steiner, 2008; Van Wassenhove &Baker, 1982; Vickson, 1980; Wan, Yen, & Li, 2001; Wang & Cheng,2005; Wang & Xia, 2007Yedidsion et al., Yedidsion, Shabtay, &Moshe, 2007 ). A survey of results up to 1990 can be found inNowicki and Zdrzalka (1990). A more recent survey was given byShabtay and Steiner (2007). In most of the above mentioned stud-ies of scheduling with controllable processing times, researchersassumed that the job processing time is a bounded linear functionof the amount of resource allocated to the processing of the job, i.e.,the resource consumption function is of the form

pj pj ajuj; j 1;2; . . . ;n; 0 6 uj 6 uj 0;

where k is a positive constant. This resource consumption functionhas been used extensively in continuous resource allocation theory(e.g., Armstrong, Gu, & Lei, 1995, 1997; Monma et al., 1990; Scott &Jefferson, 1995; Shabtay, 2004; Shabtay & Kaspi, 2004; Shabtay &Steiner, 2008). In fact, Monma et al. (1990) were among the rstto proposed this model, He also pointed out that k = 1 correspondsto many actual government and industrial operations and thek = 0.5 case arises from VLSI (very large scale integration) circuit de-

D. Wang et al. / Computers & Indusign, where the product of the silicon area (resource) and the squareof time spent equals a constant value (the workload) for an individ-ual job.However, it is surprising that the effects of learning and re-source allocation have never been considered concurrently. Thephenomena of resource allocation and learning effects occurringsimultaneously can be found in many real-life situations. Forexample, in the chemical industry, the processing time of a chem-ical compound can be changed by increasing the amount of cata-lysts, which entails some extra costs (Wang & Cheng, 2005).Clearly, compressing jobs would be rational and possible only ifthe additional cost is compensated by the gains from job comple-tion at an earlier time. The scheduling problem with controllableprocessing times is concerned with determining not only the jobsequence but also the amount of compression for each job so asto minimize the total cost. On the other hand, the learning effectsreect that the workers become more skilled to operate the ma-chines through experience accumulation. For this situation, consid-ering these the job learning effects and resource allocation in jobscheduling is both necessary and reasonable. In this paper, westudy single machine scheduling problem with position and re-source-dependent processing times at the same time.

The rest of this paper is organized as follows. In Section 2 wewill give a formal description of the model under study. In Sections3, we obtain optimal resource allocation for any given sequence. InSection 4, we show that the problem can be formulated as anassignment problem. In Section 5, conclusions are presented.

2. Problem formulation

The focus of this paper is to study the learning effects and re-source allocation simultaneously. The new model is described asfollows.

There are given a single machine and a set J = {J1, J2, . . ., Jn} of nindependent and non-preemptive jobs immediately available forprocessing. The machine can handle one job at a time and job pre-emption is not allowed. Let pj be the actual processing time of job Jj.In this paper, we consider the following models:

A linear resource consumption function:

pj pjra ajuj; 1where pj is the basic (normal) processing time of job Jj, r is the posi-tion of job Jj is scheduled in a sequence, a 6 0 is a learning index,and uj is the amount of resource that can be allocated to job Jj, with

0 6 uj 6 uj 0; 2

where k is a positive constant.For a given sequence p = [J1, J2, . . ., Jn], Cj = Cj(p) represents

the completion time for job Jj. Let Cmax maxfCjjj 1;2; . . . ;ng;TC Pnj1Cj, TWPnj1Wj; TADCPni1PnjijCiCjj and TADW Pn

i1Pn

jijWi Wjj be the makespan of all jobs, the total comple-tion times, the total waiting times, the total absolute differencesin completion times, and the total absolute differences in waitingtimes, whereWj = Cj pj be the waiting time of job Jj. The objectiveis to determine the optimal resource allocations and the optimalsequence of jobs in the machine so that the corresponding valueof the following cost functions be optimal:

f p;u d1Cmax d2TC d3TADC d4Xnj1

Gjuj; 3

l Engineering 59 (2010) 458462 459f p;u d1Cmax d2TW d3TADW d4Xnj1

Gjuj; 4

hij m p ja d G m a u ; if d G m a < 0: 11

z 1; j 1;2; . . . ;n;

striawhere weights d1P 0,d2P 0, d3P 0, and d4P 0 are given con-stants (the decision-maker selects the weights d1, d2, d3, d4), andGj is the per time unit cost associated with the resource allocation.In the remaining part of the paper, all the problems considered willbe denoted using the three-eld notation schema introduced byGraham, Lawler, Lenstra, and Rinnooy Kan (1979).

3. Solution with a linear resource consumption function

In this section, we will show that with a linear resource con-sumption function the optimal schedule for both these objectivefunctions can be obtained in O(n3) time.

For the model (3), if we substitute, Cj Pj

l1pl; Cmax Pn

j1pj,TC Pnj1Cj; TADC Pnj1j 1n j 1pj (Kanet, 1981) andpj pjra ajuj into (3) and simplify, we have

f p;u d1Xnj1

pj d2Xnj1

n j 1pj d3Xnj1

j 1n j1pj

d4Xnj1

Gjuj;

Xnj1

d1 d2n1 j d3j 1n j 1pj d4Xnj1

Gjuj;

Xnj1

xjpj d4Xnj1

Gjuj;

Xnj1

xjpjja ajuj d4Xnj1

Gjuj;

Xnj1

xjpjja Xnj1

d4Gj xjajuj; 5

where xj = d1 + d2 (n + 1 j) + d3(j 1)(n j + 1).From (5), for any sequence, the optimal resource allocation of a

job in a position with a negative d4G[j] wja[j] should be its upperbound on the amount of resource uj, and the optimal resourceallocation of a job in a position with a positive d4G[j] wja[j] shouldbe 0. If d4G[j] wj a[j] = 0, then the optimal resource allocation ofthe job in this position may be any value between 0 and uj. Thesecan be written in the notational form as follows:

uj uj; if d4Gj wjaj < 0;uj 2 0; uj; if d4Gj wjaj 0;0; if d4Gj wjaj > 0;

8>: 6

where uj;1 6 j 6 n, represents the optimal resource allocation ofthe job in position j.

For the model (4), if we substitute, W j Pj1

l1pl; Cmax Pn

j1pj, TW

Pnj1W j; TADW

Pnj1jn jpj (Bagchi, 1989) and pj

pjra ajuj into (4) and simplify, we have

f p;u d1Xnj1

pj d2Xnj1

n jpj d3Xnj1

jn jpj d4Xnj1

Gjuj;

Xnj1

mjpj d4Xnj1

Gjuj;

Xnj1

mjpjja Xnj1

d4Gj mjajuj; 7

where mj = d1 + d2(n j) + d3j(n j).From (7), for any sequence, the optimal resource allocation of a

job in a position with a negative d4G[j] mja[j] should be its upperbound on the amount of resource uj, and the optimal resourceallocation of a job in a position with a positive d4G[j] mja[j] should

460 D. Wang et al. / Computers & Indube 0. If d4G[j] mja[j] = 0, then the optimal resource allocation of thejob in this position may be any value between 0 and uj. These canbe written in the notational form as follows:j i 4 i j i i 4 i j i

The optimal sequence is obtained, as the following assignmentproblem

minXni1

Xnj1

hijzij; 12

s:t: Xni1

zij 1; i 1;2; . . . ;n;Xnu^j u; if d4Gj mjaj < 0;uj 2 0; uj; if d4Gj mjaj 0;0; if d4Gj mjaj > 0;

8>: 8

where u^j; 1 6 j 6 n, represents the optimal resource allocation ofthe job in position j.

Lemma 1. Given a sequence, for the model (3), the optimal resourceallocation can be determined by (6), and for the model (4), the optimalresource allocation can be determined by (8).

Proof. For the model (3), substituting (1) for p[j] into (3) and takingthe derivative by u[j], we have

df p;uuj

d4Gj xjaj for j = 1,2,. . .,n.Hence, if d4G[j] wja[j] > 0, we should not allocate any resource tojob J[j]; if d4G[j] wja[j] < 0, we will allocate the maximal feasibleamount of resource to job J[j]; and if d4G[j] wja[j] = 0, any feasibleresource allocation can be optimal. For the model (4), the result canbe obtained by the similar method. h

Now we discuss the determination of optimal sequences for thetwo models. In view of the analysis in the previous sections, wherewe provided the expressions for computing the optimal optimalresource allocation for any given optimal sequence, the problemreduces to a pure sequencing problem. In order to obtain the opti-mal sequence, we formulate the models (3) and (4) as an assign-ment problem, respectively.

For the model (3), let

kij xjpij

a; if d4Gi xjai P 0;

xjpija d4Gi xjaiui; if d4Gi xjai < 0:

(9

Furthermore, let zij be a 0/1 variable such that zij = 1 if job Ji isscheduled in position j, and zij = 0, ot...