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time, total absolute differences in completion times and total resource cost; minimizing acost function containing makespan, total waiting time, total absolute differences in waitingtimes and total resource cost. We show that the problems remain polynomially solvable

is as

lems with time-dependent job processing times. Janiak and Kovalyov [6] considered the problem of scheduling jobs exe-cuted by a human in a contaminated area. Wu and Lee [7] considered the two-machine ow shop total completion timeminimization problem with deteriorating jobs. They derived several dominance properties and two lower bounds to facil-itate the search for the optimal solution in the branch-and-bound algorithm. Gawiejnowicz [8] considered two single-ma-chine makespan minimization scheduling problems with proportionally deteriorating jobs. In the rst problem, the

0307-904X/$ - see front matter 2011 Elsevier Inc. All rights reserved.

Corresponding author.E-mail addresses: wcm1234460@sohu.com (C.-M. Wei), wangjibo75@yahoo.com.cn (J.-B. Wang), mfpji@inet.polyu.edu.hk (P. Ji).

Applied Mathematical Modelling 36 (2012) 792798

Contents lists available at ScienceDirect

Applied Mathematical Modellingdoi:10.1016/j.apm.2011.07.005however, we often encounter settings in which job processing times may be subject to change due to the phenomenonof deterioration. Job deterioration appears, for instance, in the steel production where the temperature of an ingot dropsbelow a certain level while waiting to enter a rolling machine, which requires reheating of the ingot before rolling. Similarsituations will also occur in scheduling maintenance tasks, national defense or cleaning assignments, where any delay inprocessing a job is penalized by incurring additional time for accomplishing the job. Extensive surveys of different sched-uling models and problems involving deteriorating jobs can be found in Alidaee and Womer [2], and Cheng et al. [3]. Morerecent papers that have considered scheduling problems with deteriorating jobs include Wang and Xia [4], Gawiejnowiczet al. [5], Janiak and Kovalyov [6], Wu and Lee [7], Gawiejnowicz [8], Wang et al. [9], Lee et al. [10, 11], Li et al. [12], Tangand Liu [13], Ng et al. [14], Yang [15], Yang and Wang [16], Huang and Wang [17], Wang and Wang [18], and Wang et al.[19]. Wang and Xia [4] considered various single-machine and ow-shop scheduling problems with decreasing lineardeterioration of job processing times. Gawiejnowicz et al. [5] considered two single-machine bicriterion scheduling prob-Keywords:SchedulingSingle-machineDeteriorating jobsResource allocation

1. Introduction

In classical scheduling theory, itunder the proposed model. 2011 Elsevier Inc. All rights reserved.

sumed that the job processing times xed and constant values [1]. In practice,Single-machine scheduling with time-and-resource-dependentprocessing times

Cai-Min Wei a,, Ji-Bo Wang b, Ping Ji caDepartment of Mathematics, Shantou University, Shantou 515063, Chinab School of Science, Shenyang Aerospace University, Shenyang 110136, ChinacDepartment of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

a r t i c l e i n f o

Article history:Received 5 September 2009Received in revised form 24 May 2011Accepted 1 July 2011Available online 23 July 2011

a b s t r a c t

We consider single-machine scheduling problems in which the processing time of a job is afunction of its starting time and its resource allocation. The objective is to nd the optimalsequence of jobs and the optimal resource allocation separately. We concentrate on twogoals separately, namely, minimizing a cost function containingmakespan, total completion

journal homepage: www.elsevier .com/locate /apm

machine is not continuously available for processing but the number of non-availability periods, and the start time and

C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792798 793the end time of each period are known in advance. In the second problem, the machine is available all the time butfor each job a ready time and a deadline are dened. He showed that decision versions of these two problems are NP-com-plete in the ordinary sense or in the strong sense, depending on the number of non-availability periods or the number ofdistinct ready times and deadlines. Wang et al. [9] considered single-machine scheduling with deteriorating jobs in whichthe jobs are constrained by a seriesparallel graph constraint. They proved that the problem can be solved in polynomialtime. Lee et al. [10] considered the same model of Wu and Lee [7]. But with a makespan objective function. Lee et al. [11]considered the permutation ow shop scheduling with machine-dependent job deterioration rates. The objective functionis to minimize the total completion time. They proposed a dominance rule and an efcient lower bound to speed up thesearching for the optimal solution. Li et al. [12] considered single machine scheduling of deteriorating jobs to minimizetotal absolute differences in completion times. Tang and Liu [13] considered two scheduling problems for a two-machineowshop where a single machine is followed by a batching machine. The rst problem is that there is a transporter tocarry the jobs between machines. The second problem is that there are deteriorating jobs to be processed on the singlemachine. For the rst problem with minimizing the makespan, they formulate it as a mixed integer programming modeland then proved that it is strongly NP-hard. A heuristic algorithm is proposed for solving this problem and its worst caseperformance is analyzed. For the second problem, they derived the optimal algorithms with polynomial time for minimiz-ing the makespan, the total completion time and the maximum lateness, respectively. Ng et al. [14] considered a two-ma-chine ow shop scheduling problem to minimize the total completion time with proportional linear deterioration. Theyderived several dominance properties, some lower bounds, and an initial upper bound and applied them in a proposedbranch-and-bound algorithm to search for the optimal solution. Yang [15] considered some single-machine schedulingproblems with both start-time dependent learning and position dependent aging effects under deteriorating maintenanceconsideration. Yang and Wang [16] considered a two-machine ow shop scheduling problem with simple linear deterio-ration and total eighted completion time criterion. They derived several dominance properties, some lower bounds, and aninitial upper bound and applied them in a proposed branch-and-bound algorithm to search for the optimal solution.Huang and Wang [17] considered parallel identical machines scheduling problems with deteriorating jobs. They showedthat two scheduling problems remains polynomially solvable under the proposed model. Wang and Wang [18] consideredthe single-machine scheduling problems with nonlinear deterioration. They showed that even with the introduction ofnonlinear deterioration to job processing times, single machine makespan minimization problem remains polynomiallysolvable. Wang et al. [19] considered a single machine scheduling problem with simple linear deterioration. For the jobswith chain precedence constraints, they proved that the weighted sum of squared completion times minimization problemwith strong chains and weak chains can be solved in polynomial time, respectively. We refer the reader to review Gaw-iejnowicz [20] for more details on single-machine, parallel-machine and dedicated-machine scheduling problems withtime-dependent processing times.

On the other hand, the problems with xed job processing times dependent on resources have been considered by Ja-niak [21], Nowicki and Zdrzalka [22], Panwalkar and Rajagopalan [23], Cheng and Janiak [24], Blazewicz et al. [25], Wangand Xia [26], and Tseng et al. [27]. However, to the best of our knowledge, there exist only a few research results onscheduling models considering the resource allocation and deteriorating jobs at the same time. The phenomena of re-source allocation and deteriorating jobs occurring simultaneously can be found in many real-life situations. For example,in steel production, more precisely, in the process of preheating ingots by gas to prepare them for hot rolling on theblooming mill. Before the ingots can be hot rolled, they have to achieve the required temperature. However, the preheat-ing time of the ingots depends on their starting temperature, i.e., the longer ingots wait for the start of the preheatingprocess, the lower goes their temperature and therefore the longer lasts the preheating process. The preheating timecan be shortened by the increase of the gas ow intensity, i.e., the more gas is consumed, the shorter lasts the preheatingprocess. Thus, the ingot preheating time depends on the starting moment of the preheating process and the amount of gasconsumed during it [28]. Bachman and Janiak [28] rst considered single-machine scheduling with job processing timesdependent on the starting moments of job execution and on the amounts of resource allocation to the jobs. They provedthat the makespan minimization problem is NP-hard. They also gave some properties of the optimal resource allocation.Janiak and Iwanowski [29] considered the single machine scheduling problems with time and resource dependent pro-cessing times. They considered the following criteria: the makespan and the total completion time subject to a given con-straint on the total resource consumption and the total resource consumption criterion subject to a given constraint eitheron the makespan or on the total completion time, respectively. For a given schedule of jobs, they proved that an optimalresource allocation vector can be constructed in polynomial time. We proved that they operate in polynomial time. Zhaoet al. [30,31] considered single-machine scheduling with deteriorating jobs where the release times of the jobs depend onthe amounts of resource allocation. For two resource constrained scheduling problems, they gave optimal algorithms tond the optimal resource allocations.

In this paper, we consider single machine scheduling problems with time and resource dependent processing times at thesame time. The rest of this paper is organized as follows. Notations and assumptions are given in Section 2. In Sections 3 and4, we show that the problems can be formulated as an assignment problem, respectively. In Section 5, a test example is gi-ven. In Section 6, conclusions are presented.

2. Problem formulation

We consider the problem of scheduling n jobs J1, J2, . . . , Jn on a continuously available machine. All the jobs are available forprocessing at some time 0. The machine can handle one job at a time and job preemption is not allowed. Let p be the actual

whereis its supper

ForCmax;

functi

X

whereand Gconsid

3. Pro

Let

And t

where

794 C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792798For the model (2), if we substitute, pj aj huj b l11 b al hul, Eq. (2) can be rewritten as

xj = d1 + d2(n + 1 j) + d3(j 1)(n j + 1). Pj1 j1ll1

For the model (2), if we substitute, Cj Pj

l1pl;Cmax Pn

j1pj;TC Pn

j1Cj and TADC Pn

j1j 1n j 1pj [33]into (2) and simplify, we have

f p;u d1Xnj1

pj d2Xnj1

n j 1pj d3Xnj1

j 1n j 1pj d4Xnj1

Gjuj

Xnj1

d1 d2n 1 j d3j 1n j 1pj d4Xnj1

Gjuj Xnj1xjpj d4

Xnj1

Gjuj;Cn Pnj1

1 bnjaj huj:

he actual processing time of job J[r] can be expressed as follows:

pj aj bC j1 huj aj huj bXj1

1 bj1lal hul: 5r in a sequence, respectively. Then the completion times of jobs be expressed as follows:

C1 a1 hu1;C2 a1 hu1 a2 ba1 hu1 hu2 a2 hu2 1 ba1 hu1;C3 C 2 a3 bC2 hu3 a3 hu3 1 ba2 hu2 1 b2a1 hu1;

. . .

Cj Pjl1

1 bjlal hul;. . .

4f p;u d1Cmax d2TW d3 TADW d4j1

Gjuj; 3

weights d1P 0, d2P 0, d3P 0 and d4P 0 are given constants (the decision-maker selects the weights d1, d2, d3, d4)j is the per time unit cost associated with the resource allocation. In the remaining part of the paper, all the problemsered will be denoted using the three-eld notation schema introduced by Graham et al. [32].

blem 1jpj aj bt hujjd1Cmax d2TC d3 TADC d4+nj1Gjuj

p[r] and a[r] denote the actual processing time and the normal processing time of a job when it is scheduled in positionj1nons be optimal:

f p;u d1Cmax d2TC d3 TADC d4Xn

Gjuj; 2the total completion times, the total waiting times, the total absolute differences in completion times, and the total absolutedifferences in waiting times, where Wj = Cj pj be the waiting time of job Jj. The objective is to determine the optimal re-source allocations and the optimal sequence of jobs in the machine so that the corresponding value of the following costajP 0 is the normal (basic) processing time of the job Jj, bP 0 is the common deterioration rate for all the jobs, tP 0tart time, hP 0 and uj is the amount of a non-renewable resource allocated to job Jj, with 0 6 uj 6 mj 6

ajh andmj is the

bound on the amount of resource that can be allocated to job Jj.a given sequence p = [J1, J2, . . . , Jn], Cj = Cj(p) represents the completion time for job Jj. Let

TC Pnj1Cj;TW Pnj1Wj; TADC Pni1PnjijCi Cjj and TADW Pni1PnjijWi Wjj be the makespan of all jobs,processing time of job Jj. In this paper, we consider the following time-and-resource dependent processing times model

pj aj bt huj; 1j

n2

where

X x bx b1 bx b1 b x ;

Inassign

Furtheand R

Rechence

4. Pro

AsTADW

where

C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792798 795rmore, let zij be a 0/1 variable such that zij = 1 if job Ji is scheduled in position j, and zij = 0, otherwise. As in Panwalkarajagopalan [23], the optimal matching of jobs to positions requires a solution for the following assignment problem:

minPni1

Pnj1

kijzij

subject toPni1

zij 1; i 1;2; . . . ;n;Pnj1

zij 1; j 1;2; . . . ; n;

zij 0 or 1; i; j 1;2; . . . ;n:

7

all that solving an assignment problem of size n requires an effort of O(n3) (using the well-known Hungarian method),the optimal solution can be found in polynomial time.

blem 1jpj aj bt hujjd1Cmax d2TW d3 TADW d4+nj1Gjuj

in Section 3, for the model (3), if we substitute, W j Pj1

l1pl;Cmax Pn

j1pj; TW Pn

j1W j andPnj1jn jpj [34] into (3) and simplify, we havef p;u d1

Xnj1

pj d2Xnj1

n jpj d3Xnj1

jn jpj d4Xnj1

Gjuj Xnj1mjpj d4

Xnj1

Gjuj;

mj = d1 + d2(n j) + d3j(n j).kij Xjai d4Gi hXjmi; if d4Gi hXj < 0:

63 3 4 5 n

. . .

Xn1 xn1 bxn;Xn xn:order to obtain the optimal resource allocations and the optimal sequence of jobs, we formulate the model (2) as anment problem, respectively. Let

Xjai; if d4Gi hXj P 0;X2 x2 bx3 b1 bx4 b1 bn3xn;n4X1 x1 bx2 b1 bx3 b1 bn2xn;j1

Xjaj j1

d4Gj hXjuj; x1 bx2 b1 bx3 b1 b xn a1 hu1

x2 bx3 b1 bx4 b1 bn3xn

a2 hu2

x3 bx4 b1 bx5 b1 bn4xn

a3 hu3 xn1 bxnan1 hun1 xnan

hun d4Xnj1

Gjuj

Xn Xnx4a4 hu4 ba3 hu3 1 ba2 hu2 1 b2a1 hu1 xn1an1 hun1 ban2 hun2 1 ban3 hun3 1 bn4a2 hu2 1 bn3a1...