Single-machine scheduling simultaneous with ...

Applied Mathematical Modelling 35 (2011) 2068–2074

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Single-machine scheduling simultaneous with position-based and sum-of-processing-times-based learning considerations under group technology assumption Suh-Jenq Yang a,⇑, Dar-Li Yang b a b

Department of Industrial Engineering and Management, Nan Kai University of Technology, Nan-Tou 54210, Taiwan Department of Information Management, National Formosa University, Yun-Lin 63201, Taiwan

a r t i c l e

i n f o

Article history: Received 8 February 2010 Received in revised form 6 October 2010 Accepted 15 November 2010 Available online 21 November 2010 Keywords: Group technology Group availability Learning effect Makespan Total completion time

a b s t r a c t This paper considers the problems of scheduling with the effect of learning on a singlemachine under group technology assumption. We propose a new learning model where the job actual processing time is linear combinations of the scheduled position of the job and the sum of the normal processing time of jobs already processed. We show that the makespan minimization problem is polynomially solvable. We also prove that the total completion time minimization problem with the group availability assumption remains polynomially solvable under agreeable conditions. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Recently, scheduling problems with the effect of learning have attracted particular attention in management science, since it was perceived as worthwhile to be taken into consideration during production planning as it can increase the production efficiency. Generally, in scheduling with the learning effect, the actual processing time of a job is modeled as a decreasing function if it is scheduled later in a sequence. Cheng and Kovalyov [1] were among the pioneers who brought the concept of the learning effect into the field of scheduling. Biskup [2] proposed a position-dependent learning model where the actual time processing time of a job is a decreasing function of its position in a sequence. Cheng and Wang [3] considered a single-machine scheduling problem with a volume-dependent piecewise linear processing time to model the effect of learning. Mosheiov and Sidney [4] extended Biskup’s learning model to a job-dependent learning model by the introduction of different learning factors for jobs. Kuo and Yang [5] proposed a time-dependent learning model in which the actual time required to perform a job is a function of the sum of the normal processing time of jobs already processed. Janiak and Rudek [6] provided an extensive study of a single-machine scheduling with an experience-based learning model, where job processing times are described by S-shaped functions that are dependent on the experience of the machine. Wu and Liu [7] studied a single-machine problem with the learning effect and release times where the objective is to minimize the makespan. They proposed a branch-and-bound algorithm and three two-stage heuristic algorithms for the problem. For a complete list of studies, the reader may refer to the comprehensive surveys by Bachman and Janiak [8], Janiak and Kovalyov [9], Biskup [10], and Janiak and Rudek [6]. ⇑ Corresponding author. Fax: +886 49 2565842. E-mail address: [email protected] (S.-J. Yang). 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.11.041

S.-J. Yang, D.-L. Yang / Applied Mathematical Modelling 35 (2011) 2068–2074

2069

It is well-known that groups similar products into families helps increase the efficiency of production and decrease the requirement of facilities. This concept is known as group technology in the literature [11]. In the domain of the group technology on scheduling problem, jobs are classified into groups or families by the commonality or similarity of their designs and/or production processes, and a setup time is incurred whenever a machine transfers job processing from a group to another group. The essence that the learning effect always accompanies a series of jobs with similar manufacturing requirements highlights the rationality of incorporating learning effect into the group scheduling. Hence, the group scheduling with the learning effect results in a new stream of research. Although different models of learning have been studied extensively in various areas, it has rarely been studied in the context of group scheduling. Kuo and Yang [12] studied a single-machine group scheduling with a time-dependent learning model to minimize the makespan and the total completion time of all jobs. They proved that these problems are polynomially solvable. Wang et al. [13] considered single-machine group scheduling problems with the effect of learning. They showed that all the studied problems are polynomially solvable where the objectives are to minimize the makespan and the total completion time of all jobs. Lee and Wu [14] proposed a group learning model on a single-machine where the learning effect not only depends on the job position, but also depends on the group position. They showed that the makespan and the total completion time minimization problems remain polynomially solvable under the proposed model. Yang and Yang [15] proposed other agreeable conditions for the problem studied by Lee and Wu [14] and showed that the total completion time minimization problem remains polynomially solvable under the agreeable conditions. Yang and Yang [16] investigated single-machine group scheduling problems with simultaneous considerations of deterioration and learning effects to minimize the makespan and the total completion time. They derived polynomial time optimal solutions for these problems with or without the presence of certain conditions. On the other hand, several recent papers have been conducted to address the group technology on scheduling problems with jobs deterioration, including Guo and Wang [17], Xu et al. [18], Wu et al. [19], Wu and Lee [20], and Wang and Sun [21]. Furthermore, Biskup [10] classified the learning models into two types, namely the position-based learning model and the sum-of-processing-times-based learning model. Yin et al. [22] developed a general learning model where the job actual processing time is not only a function of the sum of the normal processing times of jobs already processed, but also a function of the job’s scheduled position. In this paper we consider a new learning model where the job actual processing time is linear combinations of the scheduled position of the job and the sum of the normal processing time of jobs already processed. We investigate the single-machine group scheduling with the learning effect such that the makespan is minimized. We also aim to study the total completion time minimization problem under the group availability assumption [23–25], in which jobs are completed only when their entire group has completed processing. For example, this situation occurs if the jobs in a group are placed on a container, and the container is only moved from the machine when all of these jobs are processed. The remainder of this paper is organized as follows. In Section 2 we formulate the problem. The solution procedure for the makespan minimization problem is discussed in Section 3. The total completion time minimization problem under group availability assumption is presented in Section 4. The paper concludes with a summary of the results and directions for future research. 2. Notation and problem formulation

m Gi ni n

We use the following notation throughout the paper and will introduce additional notation when needed: the number of groups (m P 2) the ith group, i = 1, 2, . . . , m the number of jobs of group Gi, i = 1, 2, . . . , m P the total number of jobs (i.e., n ¼ m i¼1 ni Þ

si s[i] sik c d u, v Jij pij pi[r] pijr ai bi a, b Ci

Wi

the normal setup time of group Gi, i = 1, 2, . . . , m the normal setup time of a group scheduled in the ith position, i = 1, 2, . . . , m the actual setup time of group Gi scheduled in the kth position, i, k = 1, 2, . . . , m the common time-based learning factor of group setup time, c 6 0 the common position-based learning factor of group setup time, d 6 0 the group setup time weighted factors, 0 6 u 6 1, 0 6 v 6 1, and u + v = 1 the jth job of group Gi, i = 1, 2, . . . , m and j = 1, 2, . . . , ni the normal processing time of job Jij, i = 1, 2, . . . , m and j = 1, 2, . . . , ni the normal processing time of a job scheduled in the rth position of group Gi, i = 1, 2, . . . , m and r = 1, 2, . . . , ni the actual processing time of job Jij scheduled in the rth position of group Gi, i = 1, 2, . . . , m and j, r = 1, 2, . . . , ni the time-based learning factor of job in group Gi, ai 6 0, i = 1, 2, . . . , m the position-based learning factor of job in group Gi, bi 6 0, i = 1, 2, . . . , m the job processing time weighted factors, 0 6 a 6 1, 0 6 b 6 1, and a + b = 1 the completion time of group Gi, i = 1, 2, . . . , m the sum of the actual processing time of group Gi when jobs within group Gi are arranged in an optimal sequence, i = 1, 2, . . . , m.

The problem under study can be formally described as follows: There are n jobs classified into m groups ready to be processed on a single-machine. All the jobs are non-resumable and available for processing at time zero. We assume that jobs in

2070


the same group are processed consecutively and a group setup time is incurred if the machine transfers job processing from one group to another. For a given schedule, we denote by Cmax the makespan of all jobs and GATC the total completion time under group availability assumption. Due to the effect of learning, the actual group setup time of group Gi when scheduled in the kth position in a group sequence is given by:

sik ¼ si u 1 þ

k1 X

!c s½q

! d

þ vk

i; k ¼ 1; 2; . . . ; m;

;

ð1Þ

q¼1

where 0 6 u 6 1, 0 6 v 6 1, and u + v = 1, and the actual job processing time of job Jij when scheduled in the rth position of group Gi is defined by:

pijr ¼ pij a 1 þ

r1 X

!ai pi½q

! þ brbi ;

i ¼ 1; 2; . . . ; m;

j; r ¼ 1; 2; . . . ; ni ;

ð2Þ

q¼1

where 0 6 a 6 1, 0 6 b 6 1, and a + b = 1. The objectives are to find jointly the optimal job sequence in each group and the optimal group sequence to minimize the makespan of all jobs and the total completion time under group availability assumption. Let G indicate that the problem is a group scheduling problem. Using the conventional notation [26], the makespan and the total completion time minimization problems are respectively denoted as 1/G, sik, pijr/Cmax and 1/G, sik, pijr/GATC. 3. Problem 1/G, sik, pijr/Cmax In this section we study the single-machine group scheduling simultaneous with setup and job processing times under the learning effect considerations to minimize the makespan of all jobs. First, we provide two useful lemmas that help find the optimal schedule for the problem. Lemma 1. g(z) = q((1 (x + z)a) + az(x + kz)a1) + r(yb (y + 1)b) P 0 for q P 0, r P 0, k P 1, x P 1, y P 1, a 6 0, b 6 0 and z P 0.

Proof. Taking the first derivative of g(z) with respect to z, we have:

g 0 ðzÞ ¼ aqðx þ zÞa1 þ aqðx þ kzÞa1 þ aða 1Þqzðx þ kzÞa2 :

ð3Þ

If q P 0, r P 0, k P 1, x P 1, y P 1, a 6 0, b 6 0 and z P 0, then g(0) P 0 and g0 (z) P 0. Hence g(z) is a non-decreasing function of z. h Lemma 2. f(k) = q(k(1 (x + z)a) (1 (x + kz)a) + r (k 1)(yb (y + 1)b) P 0 for q P 0, r P 0, k P 1, x P 1, y P 1, a 6 0, b 6 0 and z P 0. Proof. Taking the first and second derivatives of f(k) with respect to k, we have:

f 0 ðkÞ ¼ qðð1 ðx þ zÞa Þ þ azðx þ kzÞa1 Þ þ rðyb ðy þ 1Þb Þ

ð4Þ

f 00 ðkÞ ¼ qaða 1Þz2 ðx þ kzÞa2 :

ð5Þ

and

If q P 0, k P 1, x P 1, a 6 0, and z P 0, then f00 (k) P 0. Hence, we have f0 (k) is a non-decreasing function of k. Moreover, from Lemma 1, we have f0 (k) P 0 for q P 0, r P 0, k P 1, x P 1, y P 1, a 6 0, b 6 0 and z P 0. Hence, f(k) is a non-decreasing function. Thus, f(k) P 0 for k P 1 since f(1) = 0. h In the following, we will show that the 1/G, sik, pijr/Cmax problem is polynomially solvable. Theorem 1. For the 1/G, sik, pijr/Cmax problem, the optimal schedule satisfies (a) the jobs in each group are arranged in nondecreasing order of their normal processing times (i.e., the SPT rule), and (b) the groups are arranged in non-decreasing order of their normal group setup times.

Proof. We prove the result by contradiction. First, we show that an optimal job sequence within a group is sequencing the jobs according to the SPT rule. Suppose Si1 = (pi1, Jik, Jil, pi2) is an optimal job sequence in group Gi such that pik P pil, where job Jik is scheduled in the rth position, job Jil is scheduled in the (r + 1)th position, and pi1 and pi2 respectively denote the partial job sequences of Si1. Let Si2 denote the same job sequence in group Gi with jobs Jik and Jil in opposite positions, i.e., Si2 = (pi1, Jil, Jik, pi2). Note that pi1 and pi2 may be empty.

2071


Let Cip(Si1) and Cip(Si2) respectively denote the completion times of job Jip in sequences Si1 and Si2, respectively. We denote by t the completion time of the last job of pi1. Then the completion times of jobs Jil in sequence Si1 and job Jik in sequence Si2 are respectively given by:

"

r1 X

C il ðSi1 Þ ¼ t þ pik a 1 þ

!a i

#

"

þ br bi þ pil a 1 þ

pi½q

q¼1

r1 X

!ai

# þ bðr þ 1Þbi

pi½q þ pik

ð6Þ

q¼1

and

"

r1 X

C ik ðSi2 Þ ¼ t þ pil a 1 þ

!a i pi½q

# þ br

bi

" þ pik a 1 þ

r1 X

q¼1

!ai

#

pi½q þ pil

þ bðr þ 1Þ

bi

ð7Þ

:

q¼1

Then the difference between Cil(Si1) and Cik(Si2) is

C il ðSi1 Þ C ik ðSi2 Þ ¼ aðpik pil Þ 1 þ

r1 X

!ai þ apil 1 þ

pi½q

q¼1

r1 X

!ai apik 1 þ

pi½q þ pik

q¼1

r1 X

!a i pi½q þ pil

þ bðpik pil Þ

q¼1

ðrbi ðr þ 1Þbi Þ: Again,

Pr1

ð8Þ

q¼1 pi½q

Let k ¼

pik pil

¼ 0 if r = 1. P ; x ¼ ð1 þ r1 q¼1 pi½q Þ P 1; z ¼ pil > 0 and y = r P 1. If k P 1, then, by Lemma 2, we have that:

h i C il ðSi1 Þ C ik ðSi2 Þ ¼ pil a kð1 ðx þ zÞai Þ ð1 ðx þ kzÞai Þ þ bðk 1Þ ybi ðy þ 1Þbi P 0:

ð9Þ

Consequently, we obtain that Cik(Si2) 6 Cil(Si1). This contradicts the optimality of Si1. Therefore, we have that the optimal job sequence within a group is sequencing the jobs in the SPT rule. Next, we show that an optimal group sequence is arranged the groups in non-decreasing order of their normal setup times. Suppose Q1 = (p1, Gk, Gl, p2) is an optimal group sequence such that sk P sl, where group Gk is scheduled in the hth position, group Gl is scheduled in the (h + 1) th position, and p1 and p2 respectively denote the partial job sequences of Q1. Note that p1 and p2 may be empty. Let Q2 denote the same group sequence with groups Gk and Gl in opposite positions, i.e., Q2 = (p1, Gl, Gk, p2). Let Cpj(Q1) and Cpj(Q2) denote the completion times of job Jpj in sequences Q1 and Q2, respectively. Moreover, we denote by t the completion time of the last job of p1. Then the completion times of group Gl in sequence Q1 and group Gk in sequence Q2 are respectively given by:

"

h1 X

C lnl ðQ 1 Þ ¼ t þ sk u 1 þ

!c

# d

þ vh

s½q

þ

q¼1

" þ sl u 1 þ

h1 X

nk X

" pk½j a 1 þ #

þ v ðh þ 1Þ

!a k

# bk

pk½q

þ bj

q¼1

j¼1

!c s½q þ sk

j1 X

d

þ

nl X

q¼1

" pl½j a 1 þ

j1 X

!a l pl½q

# þ bj

bl

ð10Þ

q¼1

j¼1

and

" C knk ðQ 2 Þ ¼ t þ sl u 1 þ

h1 X

!c s½q

# þ vh

d

þ

q¼1

" þ sk u 1 þ

h1 X

nl X

" pl½j a 1 þ

!c

# þ v ðh þ 1Þd þ

q¼1

!al pl½q

nk X

" pk½j a 1 þ

q¼1

!c

s½q

þ usl 1 þ

j1 X

!a k pk½q

# þ bj

bk

ð11Þ

:

q¼1

Then the difference between C lnl ðQ 1 Þ and C knk ðQ 2 Þ is

C lnl ðQ 1 Þ C knk ðQ 2 Þ ¼ uðsk sl Þ 1 þ

þ bj

j¼1

h1 X

# bl

q¼1

j¼1

s½q þ sl

j1 X

h1 X

!c s½q þ sk

q¼1

usk ð1 þ

h1 X

d

s½q þ sl Þc þ v ðsk sl Þðh ðh þ 1Þd Þ:

q¼1

ð12Þ Again,

Ph1

q¼1 s½q

¼ 0 if h = 1.

Similar to the above analysis, we obtain that C knk ðQ 2 Þ 6 C lnl ðQ 1 Þ since sk P sl. This contradicts the optimality of Q1. Therefore, we have that the optimal group sequence is sequencing the groups in non-decreasing order of their normal group setup times. h For the 1/G, sik, pijr/Cmax problem, we propose an algorithm to solve it.

2072


Algorithm 1 Step 1. Arrange the jobs of each group in the SPT rule. Step 2. Arrange the groups in non-decreasing order of their normal group setup time. The complexity of obtaining the optimal job sequence within a certain group Gi is O(ni log ni). Hence, the complexity of Step 1 P in Algorithm 1 is m i¼1 Oðni log ni Þ 6 Oðn log nÞ. In addition, the complexity of Step 2 in Algorithm 1 is O(m log m). Therefore, the complexity of Algorithm 1 is at most O(n log n).

4. Problem 1/G, sik, pijr/GATC In this section, we introduce the concept of group availability into group scheduling with the learning effect, where a job is completed at the completion time of its entire group. First, we will show that an optimal job sequence within a group remains satisfied the SPT rule. Theorem 2. For the 1/G, sik, pijr/GATC problem, the optimal job schedule in each group is arranged the jobs in the SPT rule. Proof. We prove the result by contradiction. Suppose Si1 = (pi1, Jik, Jil, pi2) is an optimal job sequence in group Gi such that pik P pil, where job Jik is scheduled in the rth position, job Jil is scheduled in the (r + 1)th position, and pi1 and pi2 denote the partial job sequences of Si1, respectively. Let Si2 denote the same job sequence in group Gi with jobs Jik and Jil in opposite positions, i.e., Si2 = (pi1, Jil, Jik, pi2). Note that pi1 and pi2 may be empty. Let Cip(Si1) and Cip(Si2) denote the completion times of job Jip in sequences Si1 and Si2, respectively. We denote by t the completion time of the last job of pi1. Then the completion times of jobs Jik and Jil in sequence Si1 are respectively given by:

" C ik ðSi1 Þ ¼ t þ pik a 1 þ

r1 X

!ai pi½q

# þ brbi

ð13Þ

q¼1

and

" C il ðSi1 Þ ¼ t þ pik a 1 þ

r1 X

!a i pi½q

# þ br

bi

" þ pil a 1 þ

q¼1

r1 X

!ai pi½q þ pik

# þ bðr þ 1Þ

bi

ð14Þ

:

q¼1

Similarly, the completion times of jobs Jil and Jik in sequence Si2 are respectively given by:

" C il ðSi2 Þ ¼ t þ pil a 1 þ

r1 X

!a i

# þ br bi

pi½q

ð15Þ

q¼1

and

" C ik ðSi2 Þ ¼ t þ pil a 1 þ

r1 X

!a i pi½q

# þ br

bi

" þ pik a 1 þ

r1 X

q¼1

!ai pi½q þ pil

# þ bðr þ 1Þ

bi

ð16Þ

:

q¼1

After taking the difference between the sum of Eqs. (13) and (14) and that of Eqs. (15) and (16), we have that:

ðC ik ðSi1 Þ þ C il ðSi1 ÞÞ ðC il ðSi2 Þ þ C ik ðSi2 ÞÞ ¼ aðpik pil Þ 1 þ

r1 X

!a i

apik 1 þ

r1 X

pil Þ 1 þ

r1 X

pi½q

!a i pi½q þ pik

q¼1

!a i

pi½q þ pil

q¼1

þ apil 1 þ

pi½q

q¼1

r1 X

þ bðpik pil Þðrbi ðr þ 1Þbi Þ þ aðpik

!a i þ br bi ðpik pil Þ:

ð17Þ

q¼1

Since pik P pil, the last two terms of the right side of Eq. (17) is non-negative. Moreover, we can see that the rest of the terms of the right side of Eq. (17) are the same as Eq. (8). As a result, we obtain that ðC ik ðSi1 Þ þ C il ðSi1 ÞÞ P ðC il ðSi2 Þ þ C ik ðSi2 ÞÞ. This contradicts the optimality of Si1 and proves that the jobs within the same group are optimally ordered in the SPT rule. h By Theorem 2, we have that the optimal job sequence in each group is arranged the jobs in the SPT rule. We denote by Wi the sum of the job actual processing times of group Gi when the jobs within group Gi are arranged in the SPT rule. That is, h ai i Pi P Wi ¼ nr¼1 pi½r a 1 þ r1 þ brbi and pi½1 6 pi½2 6 6 pi½ni , where pi[r] denotes the normal processing time of the job q¼1 pi½q scheduled in the rth position of group Gi, i = 1, 2, . . . , m and r = 1, 2, . . . , ni. In addition, we assume that ssl P nnk P 1 and Wl P Wk k l for any two groups Gk and Gl, k, l = 1, 2, . . . , m. Then, the following theorem provides the optimal group sequence for the 1/ G, sik, pijr/GATC problem under the agreeable conditions.

2073


Theorem 3. For the 1/G, sik, pijr/GATC problem, the optimal group sequence is arranged the groups in non-decreasing order of Wi/ni, i = 1, 2, . . . , m, if sskl P nnkl P 1 and Wl P Wk, for any two groups Gk and Gl, k, l = 1, 2, . . . , m. Proof. Without loss of generality, we denote that a group sequence Q1 = (p1, Gk, Gl, p2), where group Gk is scheduled in the hth position, group Gl is scheduled in the (h + 1)th position, and p1 and p2 respectively denote the partial job sequences of Q1. Note that p1 and p2 may be empty. Let Q2 denote the same group sequence with groups Gk and Gl in opposite positions, i.e., Q2 = (p1, Gl, Gk, p2). For a certain group sequence, let Cpj(Q1) and Cpj(Q2) denote the completion times of job Jpj in sequences Q1 and Q2, respectively. Moreover, we denote by t the completion time of the last job of p1. Then the completion times of groups Gk and Gl in sequence Q1 are respectively given by:

"

C k ðQ 1 Þ ¼ t þ sk u 1 þ

h1 X

!c

#

þ vh

s½q

d

þ

q¼1

and

" C l ðQ 1 Þ ¼ t þ sk u 1 þ

h1 X

þ sl u 1 þ

h1 X

pk½j a 1 þ

!c

# þ vh

s½q

d

þ

nk X

" pk½j a 1 þ

!ak

#

pk½q

þ bj

bk

ð18Þ

j1 X

!ak pk½q

# þ bj

bk

q¼1

j¼1

!c

# d

þ v ðh þ 1Þ

s½q þ sk

j1 X q¼1

j¼1

q¼1

"

"

nk X

þ

q¼1

nl X

" pl½j a 1 þ

j1 X

!al pl½q

# þ bj

bl

ð19Þ

:

q¼1

j¼1

Similarly, the completion times of groups Gl and Gk in sequence Q2 are respectively given by:

"

h1 X

C l ðQ 2 Þ ¼ t þ sl u 1 þ

!c

#

þ vh

s½q

d

þ

q¼1

nl X

"

pl½j a 1 þ

j1 X

!a l

#

pl½q

þ bj

bl

ð20Þ

q¼1

j¼1

and

" C k ðQ 2 Þ ¼ t þ sl u 1 þ

h1 X

!c s½q

# þ vh

q¼1

" þ sk u 1 þ

h1 X q¼1

þ

nl X

" pl½j a 1 þ

j1 X

!a l pl½q

# þ bj

bl

q¼1

j¼1

!c s½q þ sl

d

# þ v ðh þ 1Þd þ

nk X

"

j1 X

pk½j a 1 þ

!a k pk½q

# þ bj

bk

ð21Þ

:

q¼1

j¼1

Under group availability assumption, the difference of the sum of the completion time of groups Gk and Gl between sequences Q1 and Q2 is given by:

"

ðnk C k ðQ 1 Þ þ nl C l ðQ 1 ÞÞ ðnl C l ðQ 2 Þ þ nk C k ðQ 2 ÞÞ ¼ ðnk sk nl sl Þ u 1 þ " þ nl sk u 1 þ " nk sl u 1 þ

h1 X

!c

s½q

#

þ vh

q¼1 h1 X

!c

q¼1 h1 X q¼1

d

!c s½q

þ nl Wk nk Wl

! þ vh

s½q

d

þ sl u 1 þ

h1 X q¼1

! þ vh

d

!c

þ sk u 1 þ

h1 X

!# þ v ðh þ 1Þd

s½q þ sk !c s½q þ sl

!# þ v ðh þ 1Þd

:

q¼1

ð22Þ Clearly, based on the analysis of part (b) of Theorem 1, we have that the last two terms of the right side of Eq. (22) is nonpositive if nk P nl and sl P sk. In addition, if ssl P nnk P 1 and Wl P Wk, then we obtain that (nkCk(Q1) + nlCl(Q1)) 6 (nlCl(Q2) + k l nkCk(Q2)). As a result, it is optimal to process group Gk before group Gl. Therefore, repeating this interchange argument for all groups which are not sequenced in non-decreasing order of Wi/ni, i = 1, 2, . . . , m, yields Theorem 3. h Based on Theorems 2 and 3, we provide an algorithm to solve the 1/G, sik, pijr/GATC problem. Algorithm 2 Step 1. Arrange the jobs of each group in the SPT rule. Pni pi½r rai , i = 1, 2, . . . , m. Step 2. Calculate Wi ¼ r¼1 sl nk Step 3. If s P n P 1 and Wl P Wk, for any two groups Gk and Gl, k, l = 1, 2, . . . , m, arrange the groups in non-decreasing order k l of Wi/ni, i = 1, 2, . . . , m. Similarly, the complexity of Algorithm 2 is at most O(n log n).

2074


5. Conclusions This paper proposed a new learning model where the job actual processing time is linear combinations of the scheduled position of the job and the sum of the normal processing time of jobs already processed. We investigated the single-machine group scheduling problem for minimizing the makespan of all jobs. We also investigated the total completion time minimization problem under group availability assumption. We first provided a polynomial time solution for the makespan minimization problem. We then showed that the total completion time minimization problem remains polynomially solvable under certain conditions. Future research may investigate problems concerning more general learning models, under other shop environments, and involving other performance measures. Acknowledgment We thank the Editor and two anonymous reviewers for their helpful comments on an earlier version of our paper. References [1] T.C.E. Cheng, M.Y. Kovalyov, Scheduling with Learning Effects on Job Processing Times, Working Paper, No. 06/94, Faculty of Business and Information Systems, The Hong Kong Polytechnic University, 1994. [2] D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178. [3] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Ann. Oper. Res. 98 (2000) 273–290. [4] G. Mosheiov, J.B. Sidney, Scheduling with general job-dependent learning curves, Eur. J. Oper. Res. 147 (2003) 665–670. [5] W.-H. Kuo, D.-L. Yang, Minimizing the total completion time in a single-machine scheduling problem with a time-dependent learning effect, Eur. J. Oper. Res. 174 (2006) 1184–1190. [6] A. Janiak, R. Rudek, Experience based approach to scheduling problems with the learning effect, IEEE Trans. Syst. Man Cybern. Part A 39 (2009) 344– 357. [7] C.C. Wu, C.-L. Liu, Minimizing the makespan on a single machine with learning and unequal release times, Comput. Ind. Eng. 59 (2010) 419–424. [8] A. Bachman, A. Janiak, Scheduling jobs with position-dependent processing times, J. Oper. Res. Soc. 55 (2004) 257–264. [9] A. Janiak, M.Y. Kovalyov, Scheduling problems with position dependent job processing times, in: A. Janiak (Ed.), Scheduling in Computer and Manufacturing Systems, WKL, Poland: Warszawa, 2006, pp. 26–32. [10] D. Biskup, A state-of-the-art review on scheduling with learning effects, Eur. J. Oper. Res. 188 (2008) 315–329. [11] I. Ham, K. Hitomi, T. Yoshida, Group Technology: Applications to Production Management, Kluwer-Nijhoff, Boston, 1985. [12] W.-H. Kuo, D.-L. Yang, Single-machine group scheduling with a time-dependent learning effect, Comput. Oper. Res. 33 (2006) 2099–2112. [13] J.-B. Wang, A.-X. Guo, F. Shan, B. Jiang, L.-Y. Wang, Single machine group scheduling under decreasing linear deterioration, J. Appl. Math. Comput. 24 (2007) 283–293. [14] W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (2009) 2159– 2163. [15] S.-J. Yang, D.-L. Yang, Note on A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 34 (2010) 4306–4308. [16] S.-J. Yang, D.-L. Yang, Single-machine group scheduling problems under the effects of deterioration and learning, Comput. Ind. Eng. 58 (2010) 754–758. [17] A.-X. Guo, J.-B. Wang, Single machine scheduling with deteriorating jobs under the group technology assumption, Int. J. Pure Appl. Math. 18 (2005) 225–231. [18] F. Xu, A.-X. Guo, J.-B. Wang, F. Shan, Single machine scheduling problem with linear deterioration under group technology, Int. J. Pure Appl. Math. 28 (2006) 401–406. [19] C.-C. Wu, Y.-R. Shiau, W.-C. Lee, Single-machine group scheduling problems with deterioration consideration, Comput. Oper. Res. 35 (2008) 1652– 1659. [20] C.-C. Wu, W.-C. Lee, Single-machine group-scheduling problems with deteriorating setup times and job-processing times, Int. J. Prod. Econ. 115 (2009) 128–133. [21] J.-B. Wang, L. Sun, Single-machine group scheduling with linearly decreasing time-dependent setup times and job processing times, Int. J. Adv. Manuf. Technol. 49 (2010) 65–772. [22] Y. Yin, D. Xu, K. Sun, H. Li, Some scheduling problems with general position-dependent and time-dependent learning effects, Inf. Sci. 179 (2009) 2416– 2425. [23] A. Allahverdi, C.T. Ng, T.C.E. Cheng, M.Y. Kovalyov, A survey of scheduling problems with setup times or costs, Eur. J. Oper. Res. 187 (2008) 985–1032. [24] G. Mosheiov, D. Oron, A single machine batch scheduling problem with bounded batch size, Eur. J. Oper. Res. 187 (2008) 1069–1079. [25] C.N. Potts, M.Y. Kovalyov, Scheduling with batching: A review, Eur. J. Oper. Res. 120 (2000) 228–249. [26] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Ann. Discrete Math. 5 (1979) 287–326.