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JORC (2013) 1:239252DOI 10.1007/s40305-013-0014-y

Uniform Parallel-Machine Scheduling with TimeDependent Processing Times

Juan Zou Yuzhong Zhang Cuixia Miao

Received: 16 January 2013 / Revised: 9 April 2013 / Accepted: 18 April 2013 /Published online: 24 May 2013 Operations Research Society of China, Periodicals Agency of Shanghai University, andSpringer-Verlag Berlin Heidelberg 2013

Abstract We consider several uniform parallel-machine scheduling problems inwhich the processing time of a job is a linear increasing function of its starting time.The objectives are to minimize the total completion time of all jobs and the totalload on all machines. We show that the problems are polynomially solvable whenthe increasing rates are identical for all jobs; we propose a fully polynomial-timeapproximation scheme for the standard linear deteriorating function, where the ob-jective function is to minimize the total load on all machines. We also consider theproblem in which the processing time of a job is a simple linear increasing functionof its starting time and each job has a delivery time. The objective is to find a sched-ule which minimizes the time by which all jobs are delivered, and we propose a fullypolynomial-time approximation scheme to solve this problem.

Keywords Scheduling Uniform machine Linear deterioration Fully polynomialtime approximation scheme

1 Introduction

For most scheduling problems, the processing times of jobs are considered to beconstant and independent of their starting time. However, this assumption is not ap-propriate for the modeling of many modern industrial processes, we often encounter

This work was supported by the National Natural Science Foundation of China (Nos. 11071142,11201259), the Natural Science Foundation of Shan Dong Province (No. ZR2010AM034) and theDoctoral Fund of the Ministry of Education (No. 20123705120001).J. Zou () Y. ZhangSchool of Management, Qufu Normal University, Rizhao, Shandong, Chinae-mail: zoujuanjn@163.com

J. Zou C. MiaoSchool of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

240 J. Zou et al.

situations in which processing time increases over time, when the machines gradu-ally lose efficiency. Such problems are generally known as scheduling with deteriora-tion effects. Scheduling with linear deterioration was first considered by Browne andYechiali [1] who assumed that the processing times of jobs are nondecreasing, starttime dependent linear functions. They provided the optimal solution for the singlemachine when the objective is to minimize the makespan. In addition, they solveda special case when the objective function is to minimize the total weighted com-pletion time. Mosheiov [2] considered simple linear deterioration where jobs have afixed job-dependent growth rate but no basic processing time. He showed that mostcommonly applied performance criteria, such as the makespan, the total flow time,the total lateness, the sum of weighted completion time, the maximum lateness, themaximum tardiness, and the number of tardy jobs, remain polynomially solvable.

In the standard form of linear models, the actual processing time of job Jj is givenby pj = aj + bj tj , where aj , bj , and tj denote the basic processing time, the dete-riorating rate, and the starting time of job Jj , respectively. Kuo and Yang [3] studiedseveral parallel-machine scheduling problems in which the actual processing time ofjob Jj is given by pj = aj + btj or pj = aj btj . The objectives are to minimizethe total completion time of all jobs and the total load on all machines. They showedthat the problems are polynomially solvable. Kononov [4] established the ordinaryNP-completeness of P2|pj = bj t, rj = t0|Cmax and P2|pj = bj t, rj = t0|Cj .Mosheiov [5] studied multi-machine makespan minimization and total load mini-mization with linear deterioration, he proved that the two problems are NP-hard evenfor two machines. Liu et al. [6] considered m uniform machine scheduling with lin-ear deterioration to minimize the makespan, they proposed a fully polynomial-timeapproximation scheme for this problem. An extensive survey of different models andproblems was provided by Alidaee and Womer [7]. Cheng et al. [8] presented an up-dated survey of the results on scheduling problems with time-dependent processingtimes. In addition, the concept of the learning effect and deteriorating jobs has beenextensively studied. Lee [9] considered single-machine scheduling problem with de-teriorating jobs and the learning effect, the objective was to minimize the makespanand the total completion time. He introduced the polynomial solutions for schedulingproblems with the effects of learning and deterioration. But in the literature, there areonly a few studies dealing with uniform parallel-machine scheduling problems underlinear deterioration.

In this paper, we consider uniform-machine scheduling problems in which theactual processing time of job Jj is given by pj = aj + btj . The objectives are tominimize the total completion time of all jobs and the total load on all machines. Fol-lowing the three-field notation introduced by Graham et al. [10], the correspondingproblems are denoted by Qm|pj = aj + btj |Cj and Qm|pj = aj + btj |Cimax.We show that the two problems are polynomially solvable. For the standard linear de-teriorating function, we denote the problem by Qm|pj = aj +bj tj |Cimax. We pro-pose a fully polynomial-time approximation scheme for this problem. We also con-sider the scheduling problem with simple linear deteriorating jobs on uniform parallelmachine, each job Jj has a delivery time qj . The objective is to find a schedule whichminimizes the time by which all jobs are delivered. Let Cj denote the completion timeof job Jj , thus the delivery completion time of a job Jj is Cj +qj , we will continue to

Uniform Parallel-Machine Scheduling with Time Dependent 241

use Lj to represent Cj +qj . The problem is denoted by Qm|pj = bj tj , rj = t0|Lmax,we present a fully polynomial-time approximation scheme for it.

2 Problem Description

There are n independent jobs J = {J1, J2, , Jn} to be processed on m uniform par-allel machines and each machine can process at most one job at a time. Preemption isnot allowed. For the problems Qm|pj = aj +btj |Cj , Qm|pj = aj +btj |Cimaxand Qm|pj = aj + bj tj |Cimax, we assume that all jobs are simultaneously avail-able at time 0 and each job Jj has a positive deteriorating rate b or bj . For the problemQm|pj = bj tj , rj = t0|Lmax, all the jobs are simultaneously available at time t0 > 0.Each job Jj has a positive deterioration rate bj and has a subsequent nonnegativedelivery time qj .

Let Mi and si denote the ith machine in the system and its speed factor, respec-tively. Given a schedule, we use ni to denote the number of jobs scheduled on ma-chine Mi , J[i,r] denotes a job when it is scheduled in position r on machine Mi ina sequence. Let C[i,r], Cimax, and L[i,r] denote the completion time of job J[i,r], thetotal load on machine Mi , and the delivery completion time of job J[i,r], respectively,for i = 1,2, ,m, r = 1,2, , ni , and mi=1 ni = n.

For Qm|pj = aj + btj |Cj and Qm|pj = aj + btj |Cimax, the completiontime of each job scheduled on Mi can be expressed as follows:

C[i,1] = a[i,1]si

,

C[i,2] = C[i,1] + a[i,2] + bC[i,1]si

= a[i,2]si

+(

1 + bsi

)a[i,1]si

,

C[i,3] = C[i,2] + a[i,3] + bC[i,2]si

= a[i,3]si

+(

1 + bsi

)a[i,2]si

+(

1 + bsi

)2a[i,1]si

,

...

C[i,r] = C[i,r1] + a[i,r] + bC[i,r1]si

= a[i,r]si

+(

1 + bsi

)a[i,r1]

si+ +

(

1 + bsi

)r1a[i,1]si

,

Cimax = C[i,ni ] =a[i,ni ]

si+

(

1 + bsi

)a[i,ni1]

si+ +

(

1 + bsi

)ni1 a[i,1]si

.

The total completion time and the total load can be written as follows:

m

i=1

ni

r=1C[i,r] =

m

i=1

ni

r=1

1si

(

1 +(

1 + bsi

)

+(

1 + bsi

)2+ +

(

1 + bsi

)nir)a[i,r],

242 J. Zou et al.

m

i=1Cimax =

m

i=1

ni

r=1

1si

(

1 + bsi

)nira[i,r].

For the problem Qm|pj = aj + bj tj |Cimax, the total load on Mi can be ex-pressed as follows:

Cimax = C[i,ni ] =a[i,ni ]

si+

ni1

j=1

(

1 + b[i,ni ]si

)(

1 + b[i,ni1]si

)

(

1 + b[i,j+1]si

)a[i,j ]si

.

For the problem Qm|pj = bj tj , rj = t0|Lmax, the delivery completion time of jobJ[i,r] can be denoted by L[i,r] = C[i,r] + q[i,r] = t0 rj=1(1 + b[i,j ]si ) + q[i,r].

3 Minimizing Total Completion Time and Minimizing Total Load

3.1 Polynomial Algorithms

Lemma 1 Let xi and yi be two sequences of numbers, then the sum

i xiyi of prod-ucts of the corresponding elements is the least if the sequences are monotonic in theopposite sense.

Proof The proof is obtained from [11].

Then, according to Lemma 1 and the expression form of the total completion time,we can construct a polynomial algorithm for the problem Qm|pj = aj + btj |Cj .

Algorithm 1

Step 1 Select n smallest numbers from the set { 1si, 1

si+ 1

si(1 + b

si), 1

si+ 1

si(1 + b

si)+

1si(1 + b

si)2, 1

si+ 1

si(1 + b

si) + 1

si(1 + b

si)2 + 1

si(1 + b

si)3, | i = 1,2, ,m} and

place these numbers in a nondecreasing sequence.Step 2 Sort the jobs in the nonincreasing order of normal processing aj (i.e., an an1 a1).

Step 3 Form a correspondence between these numbers and the jobs, if Jj corre-sponds to 1

si+ 1

si(1 + b

si) + + 1

si(1 + b

si)k1, then schedule job Jj on machine

Mi as the kth last position.

Theorem 1 Algorithm 1 gives an optimal schedule for the problem Qm|pj = aj +btj |Cj in O(n logn) time.

Proof Suppose that is an optimal schedule which is different from the schedule obtained by Algorithm 1. Let k be the largest index such that job Jk is scheduledon different processors in and . Let job Jk be scheduled on machine Mi in and ri denote the number of jobs scheduled to be processed after and including jobJk on machine Mi in . Let job Jk be scheduled on machine Mj in and rj denote

Uniform Parallel-Machine Scheduling with Time Dependent 243

the number of jobs scheduled to be processed after and including job Jk on machineMj in . Since and are identical for job indices greater than k and Algorithm 1assigned job Jk to machine Mj , it follows that

1sj

+ 1sj

(

1+ bsj

)

+ + 1sj

(

1+ bsj

)rj1 1

si+ 1

si

(

1+ bsi

)

+ + 1si

(

1+ bsi

)ri1.

Let Jl be the job at position rj in , by interchanging the position of job Jk andJl in , the objective function of changes by

(rj1

r=0

1sj

(

1 + bsj

)r)

ak +(

ri1

r=0

1si

(

1 + bsi

)r)

al

(rj1

r=0

1sj

(

1 + bsj

)r)

al (

ri1

r=0

1si

(

1 + bsi

)r)

ak

=(rj1

r=0

1sj

(

1 + bsj

)r

ri1

r=0

1si

(

1 + bsi

)r)

(ak al) 0.

Based on the definition of k and the ordering of jobs normal processing time,it follows that ak al 0. Then, a finite number of repetitions of this argumentestablishes that there exists an optimal schedule in which the jobs are sequenced byAlgorithm 1.

Similarly, we give an optimal algorithm for the problem Qm|pj = aj +btj |Cimax.

Algorithm 2

Step 1 Select n smallest numbers from the set { 1si, 1

si(1 + b

si), 1

si(1 + b

si)2, | i =

1,2, ,m} and place these numbers in a nondecreasing sequence.Step 2 Sort the jobs in the nonincreasing order of normal processing aj .Step 3 Form a correspondence between these numbers and the jobs, if Jj corre-

sponds to 1si(1 + b

si)k1, then schedule job Jj on machine Mi as the kth last posi-

tion.

Theorem 2 Algorithm 2 gives an optimal schedule for the problem Qm|pj = aj +btj |Cimax, which can be solved in O(n logn) time.

Proof The proof is similar to that of Theorem 1.

3.2 An FPTAS

An algorithm A is a -approximation ( 1) algorithm for a minimization problemif it produces a solution that is at most times that of the optimal one ( is alsoreferred to as the worst-case ratio). A family of algorithms {A : > 0} is called afully polynomial-time approximation scheme (FPTAS) if for each > 0 the algorithm

244 J. Zou et al.

A is a (1 + )-approximation algorithm running in polynomial time in the input sizeand 1/. In the sequel, we assume 0 < 1.

Since the problem P2|pj = bj tj , rj = t0|Cimax has been proved to be ordinar-ily NP-hard by Mosheiov [5], Qm|pj = aj + bj tj |Cimax is at least NP-hard inthe ordinary sense. In this section, we propose a fully polynomial-time approxima-tion scheme for the scheduling problem Qm|pj = aj + bj tj |Cimax. In order to usethe procedure Partition proposed in Kovalyov and Kubiak [12] which requires that afunction used within it be a nonnegative integer function, we first modify the originalobjective function Cimax to satisfy this restriction without affecting the sequence.The transformation method from the original objective function to a nonnegative in-teger function is given by the following scale procedure:

For any i {1, ,m} and j {1, , n}, define 1 = min{ ajsi }, 2 = min{1+bjsi

}.For simplicity, we suppose that 1 and 2 are finite decimals. Indeed, data are alwaysobtained within some error range in industrial production. If the error range is set tobe infinitesimal, the values will be real numbers. Find integers {l1, l2} N+ such that10l11 N+ and 10l22 N+. Since

C[i,r]a[i,r]si

+r1

j=1

(

1 + b[i,r]si

)(

1 + b[i,r1]si

)

(

1 + b[i,j+1]si

)a[i,j ]si

,

10l1+(r1)l2C[i,r] can be verified as an integer. Then define L = 10l1+nl2 , where nis the total number of jobs, the transformed objective function can be expressedas L

Cimax. We use the new objective function instead of the original one in

the following. Now our concerned problem can be denoted by Qm|pj = aj +bj tj |LCimax.

Gupta and Gupta [13] proved that the problem 1|pj = aj + bj tj |Cmax is solvedby sequencing the jobs in nondecreasing order of aj /bj , which leads to the followinglemma.

Lemma 2 For Qm|pj = aj + bj tj |LCimax, on each machine Mi (i {1, ,m})in an optimal solution, all the jobs are sequenced in the non-decreasing order of aj

bj.

Based on Lemma 2, it is natural to consider the jobs in nondecreasing order ofaj /bj . So we index the jobs such that a1/b1 a2/b2 an/bn. We intro-duce variables xj , j = 1,2, , n, where xj = i if job Jj is scheduled on ma-chine Mi , i = 1,2, ,m. Let X be the set of all vectors x = (x1, x2, , xn) withxj {1,2, ,m}, and j = 1,2, , n. We define the following initial and recursivefunctions on X:

Initial function: F i0(x) = 0, i = 1, ,m.Recursive function, for j = 1, , n,

F ij (x) =aj

si10l1+j l2 + F ij1(x)

(

1 + bjsi

)

10l2, i = xj ,

F ij (x) = F ij1(x) 10l2, i = xj ,

Uniform Parallel-Machine Scheduling with Time Dependent 245

Gj(x) =m

i=1F ij (x),

where F ij (x) is the magnified workload of machine Mi for the jobs amongJ1, J2, , Jj .Note that in the recursion functions, the workload of each machine is enlarged

by a scale factor exactly 10l1+nl2 . Therefore,m

i=1 F in(x) is our desired value for agiven x; the problem Qm|pj = aj + bj tj |LCimax reduces to the following mini-mization problem:

minGn(x), x X.First, we present the procedure Partition(A,h, ) proposed by Kovalyov and Ku-

biak [12], where A X, h is a nonnegative integer function on X, and 0 < 1.This procedure partitions A into disjoint subsets Ah1,Ah2, ,Ahkh such that |h(x) h(x)| min...