Single machine scheduling of deteriorating jobs to minimize total absolute differences in completion times

ARTICLE IN PRESS Int. J. Production Economics 118 (2009) 424–429

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Single machine scheduling of deteriorating jobs to minimize total absolute differences in completion times Yongqiang Li a,b,, Gang Li a, Linyan Sun a, Zhiyong Xu a a b

The Management School of Xi’an Jiaotong University, The State Key Lab for Manufacturing Systems Engineering, Xi’an 710049, China School of Economics and Management, Xidian University, Xi’an 710071, China

a r t i c l e i n f o

abstract

Article history: Received 27 December 2007 Accepted 12 November 2008 Available online 24 December 2008

This paper investigates a single machine scheduling problem with deteriorating jobs. By a deteriorating job, we mean that the processing time is an increasing function of its execution starting time. Job deterioration is described by a function which is proportional to a linear function of time. The objective is to find a schedule that minimizes total absolute differences in completion times (TADC). We show that the optimal schedule is V-shaped, i.e., jobs are arranged in descending order of their deterioration rates if they are placed before the job with the smallest deterioration rate, but in ascending order of their deterioration rates if placed after it. We also prove some other properties of an optimal schedule, and propose two heuristic algorithms that are tested against a lower bound. We also provide computational results to evaluate the performance of the heuristic algorithms. & 2009 Elsevier B.V. All rights reserved.

Keywords: Scheduling Single machine Deteriorating jobs The total absolute deviation of completion times (TADC)

1. Introduction In many scheduling environments, it is assumed that the processing times of jobs are an increasing function of their starting times. This phenomenon, known as deteriorating jobs, has been extensively studied in the last decade in different scheduling models and problems (see recent reviews Alidaee and Womer, 1998; Cheng et al., 2004a). Scheduling deteriorating jobs was first considered by Browne and Yechiali (1990) who assumed that the job processing time is a non-decreasing, start-time dependent linear function. They showed that the single machine expected makespan minimization problem could be

 Corresponding author at: The State Key Lab for Manufacturing Systems Engineering, The Management School of Xi’an Jiaotong University, Xi’an 710049, China. Tel.: +86 13319292665; fax: +86 29 82664643. E-mail addresses: [email protected] (Y. Li), [email protected] (G. Li), [email protected] (L. Sun), [email protected] (Z. Xu).

0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2008.11.011

solved in polynomial time. Mosheiov (1991) considered the problem that all the jobs are characterized by a common positive basic processing time. Using this basic assumption, Mosheiov proved that the optimal schedule to minimize flowtime is symmetric and has a V-shaped property with respect to the increasing rates. Mosheiov (1994) considered the following objective functions: makespan, total flow time, sum of weighted completion times, total lateness, maximum lateness and maximum tardiness, and the number of tardy jobs. When the values of the basic processing time equal zero, all these problems can be solved polynomially. Wu and Lee (2008) investigated two single machine group-scheduling problems where the group setup times and the job processing times are both increasing functions of their starting times. They proved that the makespan minimization problem remains polynomially solvable when the deterioration is present, and the sum of completion times problem is polynomially solvable when the numbers of jobs in each group are equal. Sundararaghavan and Kunnathur (1994) considered the single machine scheduling problem in which the processing time is a binary function of a common start

ARTICLE IN PRESS Y. Li et al. / Int. J. Production Economics 118 (2009) 424–429

time due date. The jobs have processing time penalties for starting after the due date, and the objective was to minimize the sum of the weighted completion times. Three special cases of this problem can be solved optimally. Bachman and Janiak (2000) showed that the maximum lateness minimization problem under the linear deterioration assumption is NP-hard, and two heuristic algorithms were presented as a consequence. Bachman et al. (2002) considered the problem of minimizing the total weighted completion time with general linear deterioration. They proved that the problem is NPhard. Wang and Xia (2005) considered the scheduling problems under a special type of linear decreasing deterioration. They presented optimal algorithms for single machine scheduling of minimizing the makespan, maximum lateness, maximum cost and number of late jobs, respectively. For the two-machine flow shop scheduling problem to minimize the makespan, they proved that the optimal schedule can be obtained by Johnson’s rule. If the processing times of the operations are equal for each job, they proved that the flow shop scheduling problems could be transformed into single machine scheduling problems. Wang et al. (2008) considered the single machine scheduling problems with deteriorating jobs, and the jobs are related by a series-parallel graph. They proved that for the general linear problem to minimize the makespan, polynomial algorithms exist. They also proved that for the proportional linear problem to minimize the total weighted completion time, polynomial algorithms exist, too. Wu et al. (2007) considered the problem of minimizing the total weighted completion time introduced by Bachman et al. (2002). They gave three heuristic algorithms and a branch-and-bound algorithm that incorporates two lower bounds. They also studied the effects of normal processing times and deterioration rates. Chen (1996), Hsieh and Bricker (1997), and Mosheiov (1998) considered scheduling deteriorating jobs in a multi-machine setting. They assumed that the jobs deteriorate linearly and they are processed on parallel identical machines. Chen (1996) considered minimizing the flow time, while Hsieh and Bricker (1997), and Mosheiov (1998) studied makespan minimization. The reader is referred to the survey by Cheng et al. (2004a) for more details on single machine and multi-machine scheduling with deteriorating jobs. Kononov and Gawiejnowicz (2001) considered the makespan minimization problem. They showed that under linear deterioration the two-machine flow shop problem is strongly NP-hard, and the two-machine open shop problem is ordinary NP-hard. They also showed that in the three-machine flow shop with simple deterioration, there does not exist a polynomial time approximation algorithm with a worst-case ratio bounded by a constant. Finally, they proved that the three-machine open shop problem with simple linear deterioration is ordinary NP-hard. Kang and Ng (2007) studied the NP-hard problem of scheduling n deteriorating jobs on m identical parallel machines to minimize the makespan. They assumed that each job’s processing time is a linear nondecreasing function of its start time and proposed a fully

425

polynomial time approximation scheme for the problem. Mosheiov (2002) considered the computational complexity of flow shop, open shop and job shop problems with simple linear deterioration of minimizing the makespan. He introduced a polynomial time algorithm for the two-machine flow shop and two-machine open shop problems, respectively. He also proved that the threemachine flow shop, the three-machine open shop and the two-machine job shop problems are all NP-hard. Wang and Xia (2006a, b) considered flow shop scheduling problems with job processing times dependent on their starting times. In these problems there are some dominating relationships between the machines. They showed that for the problems to minimize makespan or minimize weighted sum of completion time, polynomial algorithms still exist. However, when the objective is to minimize maximum lateness, the solutions of a classical version may not hold. Wang et al. (2006) considered a twomachine flow shop scheduling problem with a simple linear deterioration. Several dominance conditions and two lower bounds for the problem to minimize total completion time were implemented in the proposed branch-and-bound algorithm to search for the optimal solution. They also provided a heuristic algorithm to overcome the inefficiency of the branch-and-bound algorithm for large-sized problems. Most of the scheduling of the frontal papers with deteriorating jobs literature examines regular measures of the performance, which are non-decreasing functions of job completion times. Yet in certain situations one is more interested in performance measures that are non-regular. To the best of our knowledge, there exist only a few research results on scheduling models considering nonregular performance measures. Cheng et al. (2004b, 2005) considered single machine scheduling problem with linear job-independent increasing (decreasing) deterioration jobs. The problem was to determine an optimal combination of the due date and schedule so as to minimize the sum of due date, earliness and tardiness penalties. They gave a polynomial time algorithm to solve this problem. Oron (2008) considered a single machine scheduling with simple linear deterioration. The objective function was to minimize the total absolute deviation of completion times (TADC). They proved some properties of an optimal schedule, and introduced two heuristic algorithms to solve this problem. In this paper we consider a single machine scheduling problem with proportional deterioration to minimize a non-regular performance measure, i.e., the total absolute differences in completion times (TADC). This objective was proposed by Kanet (1981) as an alternative measure of completion time variation. The rest of this paper is organized as follows. In the next section we give the problem description. In Section 3 we introduce the V-type and some other properties. In Section 4 we analyse the complexity of the problem, and offer two efficient polynomial time heuristic algorithms and a lower bound. In Section 5 we present computational experiments to evaluate the performance of the heuristic algorithms. Concluding remarks are given in the last section.

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Y. Li et al. / Int. J. Production Economics 118 (2009) 424–429

2. Formulation of the problem

we have

We now formulate the problem of single machine scheduling with deteriorating jobs to minimize the total absolute deviation of completion times (TADC). We are given a set of n jobs N ¼ fJ 1 ; J 2 ; . . . ; Jn g that have to be processed on a single machine. All the jobs are available at time t0X0. The machine can handle one job at a time and preemption is not allowed. The machine is assumed to be continuously available from time zero onwards only. The problem is to schedule the jobs in such a way that the variation in flow time (time-in-system) is minimal. This type of problem has applications in any service or manufacturing setting whenever it is deemed desirable to provide jobs (customers) the same treatment; i.e., each customer spends approximately the same time in the system or waits for service the same time as every other customer (Kanet, 1981). As in Kononov and Gawiejnowicz (2001), we assume that the actual processing time pj of job Jj if its starting time is Sj is given by pj ¼ bj(a+bSj), where bj denotes the deterioration rate of job Jj. For a given schedule p ¼ [J1,J2,y,Jn], Cj ¼ Cj(p) represents the completion time of job Jj. The objective is to find a schedule that minimizes the total absolute deviation of P P completion times (TADC): TADC ¼ nj¼1 ni¼j jC j  C i j, where |CiCj| is the absolute difference of pair-wise comparisons of job completion times, and the n jobs are numbered in order of completion. Using the three-field notation for scheduling problem classification, the problem can be denoted as 1|pj ¼ bj(a+bSj)|TADC.

TADC ¼

3. Properties of the optimal schedule

n X ðj  1Þðn  j þ 1Þpj j¼1

j1 n  Y a X ¼ b t0 þ ðj  1Þðn  j þ 1Þbj ð1 þ bbi Þ. b j¼1 i¼1

(2)

Similar to Oron (2008), we have Propositions 1–5 and Theorem 1. Proposition 1. For problem 1|pj ¼ bj(a+bSj)|TADC, let l ¼ arg minj2N fbj ; j ¼ 1; 2; . . . ; ng. If nX3, then, within the job set N, Jl is scheduled neither first nor the last in the optimal schedule. Proof. For problem 1|pj ¼ bj(a+bSj)|TADC, consider any schedule with job Jl placed first. For convenience, let p1 ¼ [Jl,J[2],J[3],y,J[n]], where J[i] denotes the job scheduled in the ith position (C[i] and b[i] are defined accordingly). p2 is the schedule obtained by interchanging the first two jobs, i.e., p2 ¼ [J[2],Jl,J[3],y,J[n]]. Then  a (3) TADCðp1 Þ  TADCðp2 Þ ¼ b t 0 þ ðn  1Þðb½2  bl Þ. b Since b[2]Xbl, (3) is non-negative and therefore p2 is a better policy. Similarly, let p01 ¼ ½J ½1 ; J½2 ; . . . ; J ½n1 ; J l . p02 is the schedule obtained by interchanging the last two jobs, i.e., p02 ¼ ½J½1 ; J½2 ; . . . ; Jl ; J½n1 . Then TADCðp01 Þ  TADCðp02 Þ n 2  Y a ¼ b t 0 þ ðn  3Þðb½n1  bl Þ ð1 þ bb½i Þ. b i¼1

(4)

Since b[n1]Xbl and nX3, (4) is non-negative and therefore p02 is a better policy. This completes the proof. & Lemma 1. For a given schedule p ¼ [J1,J2,y,Jn] of 1|pj ¼ bj(a+bSj)|TADC, if all jobs start at time t0X0, then the actual processing time pj of job Jj is equal to j1  a Y pj ¼ b t 0 þ bj ð1 þ bbi Þ, b i¼1

where

Q0

i¼1 ð1

(1)

þ bbi Þ ¼ 1.

Proof. For a given schedule p ¼ [J1,J2,y,Jn], the starting Q time Sj of job Jj is Sj ¼ C j1 ¼ ðt 0 þ a=bÞ j¼1 i¼1 ð1 þ bbi Þ  a=b (Kononov and Gawiejnowicz, 2001), hence, pj ¼ bj ða þ bSj Þ ¼ bj ða þ bC j1 Þ ¼ bj a þ b



a Y a ð1 þ bbi Þ  b i¼1 b j1

t0 þ

!!

j1  a Y ð1 þ bbi Þ ¼ b t 0 þ bj b i¼1

This completes the lemma.

&

For a given schedule p ¼ [J1,J2,y,Jn], from TADC ¼ Pn j¼1 ðj  1Þðn  j þ 1Þpj (Kanet, 1981), and Lemma 1,

Proposition 2. A schedule containing three consecutive jobs, Ji1, Ji and Ji+1, such that bi4bi1 and bi4bi+1 is not optimal. Proof. We show that an interchange between Ji and Ji1 or between Ji and Ji+1 reduces the value of TADC. Let p1 ¼ ½J ½1 ; J ½2 ; . . . ; J ½i2 ; J i1 ; J i ; J iþ1 ; J ½iþ2 ; . . . ; J ½n , p2 ¼ ½J½1 ; J ½2 ; . . . ; J½i2 ; ; J i ; J i1 ; J iþ1 ; J ½iþ2 ; . . . ; J½n  and p3 ¼ ½J ½1 ; J½2 ; . . . ; J½i2 ; Ji1 ; J iþ1 ; Ji ; J½iþ2 ; . . . ; J n . Then TADCðp1 Þ  TADCðp2 Þ i2  Y a ¼ b t 0 þ ðn þ 3  2iÞðbi  bi1 Þ ð1 þ bb½j Þ, b j¼1 TADCðp1 Þ  TADCðp3 Þ  a ¼ b t 0 þ ð2i  n  1Þðbi  biþ1 Þð1 þ bbi1 Þ b i2 Y  ð1 þ bb½j Þ.

(5)

(6)

j¼1

We show that at least one of the schedules is better than the original one, i.e., that either TADC(p1)TADC(p2)40 or TADC(p1)TADC(p3)40.

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Under the condition that bi4bi1 and bi4bi+1, if the term n+32i in (5) is negative, we have TADC(p1) TADC(p2)o0; it implies that i4(n+3)/2. Under the same condition, TADC(p1)TADC(p3)o0 only if the term 2in1 in (6) is negative, it implies that io(n+1)/2, a contradiction. We conclude that the schedule p1 is never optimal. & Theorem 1. (V-shape). The optimal schedule has a V-shape, i.e., jobs are arranged in descending order of their deterioration rates if they are placed before the job with the smallest bl, but in ascending order of their deterioration rates if placed after it. Proof. It is straightforward from Propositions 1 and 2.

&

In the following we give some propositions about the position of the smallest b-value job in an optimal schedule. Proposition 3. If the number of jobs n is even, then an optimal schedule consists of sequencing the smallest b-value job, Jl, in the n/2+1 position. Proof. Using schedules p1, p2 and p3 in Proposition 2, if biobi1 and biobi+1, then Eqs. (5) and (6) have to be nonpositive. This follows from the fact that the optimal schedule has a V-shape. From (5) p0, we have n+32iX0, hence ip(n+3)/2. Similarly, from (6) p0, we have 2in1 and iX(n+1)/2. Thus, ðn þ 1Þ=2pipðn þ 3Þ=2. Because the number of jobs n is even, we have i ¼ n/2+1. &

 b½ðnþ3Þ=2

ðnþ1Þ=21 Y j¼1

427

   a n þ 3 1 ð1 þ bb½j Þ  b t 0 þ b 2

  ðnþ1Þ=21 Y nþ3  n þ 1 b½ðnþ1Þ=2 ð1 þ bb½ðnþ3Þ=2 Þ ð1 þ bb½j Þ 2 j¼1     a n þ 1 nþ1 ¼ b t0 þ 1 n þ1 b 2 2    nþ3 nþ3  1 n þ1 2 2  ðb½ðnþ1Þ=2  b½ðnþ3Þ=2 Þ

ðn1Þ=2 Y

ð1 þ bb½j Þ

j¼1

Since ((n+1/21)(n(n+1)/2+1)((n+3)/21)(n(n+3)/2+ 1) ¼ 0, hence TADC(p1)TADC(p2) ¼ 0. We can obtain that the smallest b-value job can be scheduled in position (n+1)/2 or (n+3)/2. & For convenience, we choose to schedule the smallest bvalue job in position (n+3)/2 if n is odd. From Proposition 4, we know that when the number of jobs is odd, the optimal solution is not unique. Proposition 5. In an optimal schedule, the job scheduled i positions after the smallest b-value job will have a larger bvalue than the job scheduled i positions prior to the smallest b-value job, i ¼ 1; 2; . . . ; bn=2  1c. Proof. Consider p1 to be an optimal schedule of job set {J1,J2,y,Jn} and p2 (p3) to be the schedule obtained from p1 by interchanging jobs J[n/2+1i] (J[(n+3)/2i]) and J[n/2+1+i] (J[(n+3)/2+i]) when n is even (odd). When n is even, we have TADCðp1 Þ  TADCðp2 Þ

Proposition 4. If the number of jobs n is odd, then an optimal schedule consists of sequencing the smallest b-value job, Jl, in position (n+1)/2 or (n+3)/2. Proof. Using the same conditions on Eqs. (5) and (6) we obtain that Jl should be scheduled in the ith position where ðn þ 1Þ=2pipðn þ 3Þ=2. Consequently, Jl can be scheduled in one of two positions (n+1)/2 and (n+3)/2. Note that an interchange of jobs scheduled in these two positions is no change in the objective function TADC. Let p1 ¼ ½J½1 ; J½2 ; . . . ; J½ðnþ1Þ=2 ; J½ðnþ3Þ=2 ; . . . ; J½n , p2 ¼ ½J½1 ; J½2 ; . . . ; J½ðnþ3Þ=2 ; J½ðnþ1Þ=2 ; . . . ; J½n ; then TADCðp1 Þ  TADCðp2 Þ     a n þ 1 nþ1 1 n þ1 ¼ b t0 þ b 2 2   ðnþ1Þ=21  Y a n þ 3  b½ðnþ1Þ=2 1 ð1 þ bb½j Þ þ b t 0 þ b 2 j¼1   ðnþ1Þ=21 Y nþ3 þ 1 b½ðnþ3Þ= ð1 þ bb½ðnþ1Þ=2 Þ  n ð1 þ bb½j Þ 2 j¼1     a n þ 1 nþ1  b t0 þ 1 n þ1 b 2 2

n=2i   n  Y a  n ð1 þ bb½j Þ i þ i b½n=2iþ1 ¼ b t0 þ b 2 2 j¼1       a n n iþ1 þi1 þ b t0 þ b 2 2 n=2i  Y a ð1 þ bb½j Þ þ    þ b t 0 þ  b½n=2iþ2 ð1 þ bb½n=2iþ1 Þ b j¼1



n

n=2þi n=2i  n  Y Y ð1 þ bb½k Þ ð1 þ bb½j Þ þi  i b½n=2þiþ1 2 2 j¼1 k¼n=2iþ1

n=2i   n  Y a  n ð1 þ bb½j Þ i þ i b½n=2þiþ1  b t0 þ b 2 2 j¼1       a n n  b t0 þ iþ1 þi1 b 2 2 n=2i  Y a ð1 þ bb½j Þ      b t 0 þ  b½n=2iþ2 ð1 þ bb½n=2þiþ1 Þ b j¼1 n  n  þi  i b½n=2iþ1 ð1 þ bb½n=2þiþ1 Þ  2 2 n=2þi n=2i Y Y ð1 þ bb½k Þ ð1 þ bb½j Þ  k¼n=2iþ2

j¼1

n=2i  Y a ð1 þ bb½j Þ ¼ b t 0 þ ðb½n=2þ1i  b½n=2þ1þi Þ b j¼1

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2 n=2þi n  n  X 4 i ðk  1Þðn  k þ 1Þbb½k þi þ 2 2 k¼n=2iþ2

n=2þ1 j1  Y a X LB2 ¼ b t 0 þ ðj  1Þðn  j þ 1Þbn=2þ2j ð1 þ bbn=2þ2i Þ b j¼1 i¼1

3  n  n=2þi Y ð1 þ bb½j Þ  ð1 þ bb½k Þ5. þi i  2 2 j¼n=2iþ2 k¼n=2iþ2 n

k1 Y

(7) Obviously, the term bðt 0 þ a=bÞ positive, and

Qn=2i j¼1

ð1 þ bbj Þ is always

n=2þi n  X i þi þ ðk  1Þðn  k þ 1Þbb½k 2 2 k¼n=2iþ2 k 1 Y

ð1 þ bb½j Þ 

j¼n=2iþ2

n 2

n  þi i ð1 þ bb½k Þ 2 k¼n=2iþ2 n=2þi Y

0 n=2þi k 1 n  n  Y X þ i @1 þ bb½k ð1 þ bb½j Þ 4 i 2 2 j¼n=2iþ2 k¼n=2iþ2 1 n=2þi Y  ð1 þ bb½k ÞA ¼ 0.

(9)

where b1 pb2 p    pbn . &

Proposition 9. For the problem 1|pj ¼ bj(a+bSj)|TADC, when n is odd, a lower bound is equal to ðnþ3Þ=2  a X LB2 ¼ b t 0 þ ðj  1Þðn  j þ 1Þbðnþ3Þ=2þ1j b j¼1 j1 Y ð1 þ bbðnþ3Þ=2þ1i Þ i¼1 n  a X þ b t0 þ ðj  1Þðn  j þ 1Þb2jn2 b j¼ðnþ5Þ=2

Thus, under the assumption that p1 is an optimal schedule, (7) is non-positive if and only if b½n=2þ1i p b½n=2þ1þi . When n is old, the proof is similar to the above. &



j1 Y ð1 þ bbi Þ,

Similar to Oron (2008), from Propositions 3–5, we can obtain the complexity of the problem 1|pj ¼ bj(a+bSj)|TADC. Proposition 6. For the problem 1|pj ¼ bj(a+bSj)|TADC, the computational time required to find the optimal schedule is O((n)3/22n). Proof. From the proof of Oron (2008), the result can be easily obtained. & From Proposition 6, we know that the computational time of the problem 1|pj ¼ bj(a+bSj)|TADC remains exponential in the number of jobs. Thus, the heuristic algorithms are the main tool for solving the problem containing a large number of jobs. Therefore, in order to evaluate any proposed heuristic algorithms, we first introduce a lower bound LB. Proposition 7. For the problem 1|pj ¼ bj(a+bSj)|TADC, when n is even, a lower bound is equal to n=2 j  Y a X ð2j  n  1Þ ð1 þ bbnþ22j Þ LB1 ¼ b t 0 þ b j¼1 i¼1

(8)

where b1 pb2 p    pbn . &

Proposition 8. For the problem 1|pj ¼ bj(a+bSj)|TADC, when n is even, a lower bound is equal to

(10)

i¼1

where b1 pb2 p    pbn . Proof. Similar to the proof of Oron (2008).

4. Complexity analysis and heuristic algorithms

Proof. Similar to the proof of Oron (2008).

j1 Y ð1 þ bbi Þ, i¼1



k¼n=2iþ2

j n  Y a X þ b t0 þ ð2j  n  1Þ ð1 þ bbi Þ, b j¼n=2þ1 i¼1



Proof. Similar to the proof of Oron (2008).

n



n  a X þ b t0 þ ðj  1Þðn  j þ 1Þb2jn1 b j¼n=2þ2

&

In order to make the lower bound tighter, we choose the maximum value of Eqs. (8), (9) or (10) as a lower bound for the problem 1|pj ¼ bj(a+bSj)|TADC. That is, LB ¼ maxfLB1 ; LB2 g. As in Oron (2008), we introduce two simple heuristic algorithms to solve the problem 1|pj ¼ bj(a+bSj)|TADC. As previously, we assume that the job set N ¼ {J1,J2,y,Jn} is ordered in non-decreasing order of bj, i.e., b1 pb2 p    pbn . Heuristic 1 (H1). If n is even proceed to Step 1 and if n is odd proceed to Step 10. Step 1: Assign job Jn/2+2i to position i, i ¼ 1,2,y,n/2+1. Step 2: Assign job Ji to position i, i ¼ n=2 þ 2; n=2 þ 3; . . . ; n. The resulting permutation is of the form ½Jn=2þ1 ; Jn=2 ; . . . ; J 3 ; J 2 ; J1 ; Jn=2þ2 ; Jn=2þ3 ; . . . ; J n . Step 10 : Assign job Jðnþ5Þ=2i to position i, i ¼ 1,2, y,(n+3)/2. Step 20 : Assign job Ji to position i, i ¼ ðn þ 5Þ=2; ðn þ 7Þ=2; . . . ; n. The resulting permutation is of the form ½Jðnþ3Þ=2 ; J ðnþ1Þ=2 ; . . . ; J3 ; J2 ; J 1 ; J ðnþ5Þ=2 ; Jðnþ7Þ=2 ; . . . ; J n . Heuristic 2 (H2). If n is even proceed to Step 1 and if n is odd proceed to Step 10. Step 1: Assign job Jn+22i to position i, i ¼ 1,2,y,n/2. Step 2: Assign job J2i1 to position n/2+i, i ¼ 1,2,y,n/2. The resulting permutation is of the form ½Jn ; J n2 ; J n4 ; . . . ; J 2 ; J 1 ; J 3 ; . . . ; J n3 ; Jn1 . Step 10 : Assign job Jn+22i to position i, i ¼ 1,2,y,n+1/2. Step 20 : Assign job J2i to position (n+1)/2+i, i ¼ 1,2, y,n1/2. The resulting permutation is of the form ½Jn ; Jn2 ; . . . ; J5 ; J 3 ; J 1 ; J 2 ; J 4 ; . . . ; Jn3 ; J n1 . Obviously, the running time of both heuristic algorithms is O(n log n).

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Table 1 Optimality gaps for heuristic algorithms H1 and H2. Problem

H1 Mean (%)

20 jobs 50 jobs 100 jobs

1.169 8.91 103 6.61 107

H2 Best case (%) 2

7.53  10 7.76  104 2.63  108

Worst case (%)

Mean (%)

Best case (%)

Worst case (%)

6.663 0.154 2.68  105

5.613 2.214 1.375

1.993 0.723 0.281

14.638 7.879 4.771

5. Computational experiments Computational experiments were conducted to evaluate the effectiveness of the heuristic algorithms. The heuristic algorithms were coded in VC++ 6.0 and the computational experiments were run on a Pentium 4 personal computer with a RAM size of 512 MB. The deterioration rates were generated from a uniform distribution over (0.05, 1), i.e., bj(0.05,1). The heuristic algorithms were tested for three sizes of the problem (n ¼ 20, 50 and 100). As a consequence, 3 experimental conditions were examined and 200 instances randomly generated for each condition. A total of 600 problems were tested. For all the tests, the values t0 ¼ 0, a ¼ 1 and b ¼ 0.1 were used. For the heuristic algorithm, the worst, average and best percentage deviations of the heuristic solution from the lower bound, i.e., (Heuristic-LB)/LB, are reported. From Table 1 we see that the performance of the heuristic algorithms is effective in obtaining near-optimal solutions for large problems. 6. Conclusions In this paper we considered a single machine scheduling problem with proportional deterioration. We considered a non-regular objective function, i.e., minimizing the total deviation of job completion time (TADC). Several important properties of an optimal schedule were proved: the optimal schedule has a V-shape with respect to the deterioration rates, the positions of the smallest b-value were found, and a relationship between jobs scheduled prior and after the smallest b-value was introduced. Two heuristic algorithms were also proposed, which was shown by computational experiments to be effective and efficient in obtaining near-optimal solutions. Further research may focus on determining the complexity of the problem, considering the general deteriorating model, or studying the other non-regular objective functions.

Acknowledgements The authors wish to thank the anonymous referees for their valuable suggestions. References Alidaee, B., Womer, N.K., 1998. Scheduling with time dependent processing times: review and extensions. Journal of the Operational Research Society 50 (5), 711–720.

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