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Scheduling problems with multiple due windows assignment and controllable processing times on a single machine
Int. J. Production Economics 150 (2014) 96–103
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Scheduling problems with multiple due windows assignment and controllable processing times on a single machine Dar-Li Yang a, Chien-Jung Lai b, Suh-Jenq Yang b,c,n a
Department of Information Management, National Formosa University, Taiwan Department of Distribution Management, National Chin-Yi University of Technology, Taiwan c Department of Industrial Management, Nan Kai University of Technology, No. 568, Zhongzheng Rd., Caotun Township, Nantou County 54243, Taiwan b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2012 Accepted 13 December 2013 Available online 22 December 2013
This paper deals with multiple due windows assignment scheduling problems and controllable processing times on a single machine. We assume that the actual processing time of a job can be controlled by the introduction of additional resource and any due window is not allowed to contain another due window as a proper subset. The objective is to determine the optimal due window positions and sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimal schedule to minimize a total cost function, which consists of the earliness, the tardiness, the processing time compressions, and the due windows related costs. We show that for the case when the number of jobs assigned to each due window is given in advance, the problem is polynomially solvable in Oðn3 Þ time, where n is the total number of jobs; while if the number of jobs assigned to each due window is unknown, the problem can be optimally solved in Oðnm þ 2 Þ time, where m is the number of due windows. Furthermore, we extend the problem by incorporating with the aging effect and prove that it remains polynomially solvable. & 2013 Elsevier B.V. All rights reserved.
Keywords: Scheduling Multiple due windows Controllable processing time Aging effect
1. Introduction Traditionally manufacturers have forecasted demand for their products into the future and then have attempted to smooth out production to satisfy that forecasted demand. Unfortunately, this approach has a number of major drawbacks, such as large inventories, long production times, production obsolescence, inability to meet delivery schedules, and high costs. On the other hand, Just-inTime (JIT) is a pull system of production, in which products are produced only as needed to meet the actual customer demand. It is based on planned elimination of all waste and continuous improvement of productivity. To cope with intensified global competition and escalating customer demand for superior service, JIT production has become a competitive strategy for world-class company. According to the principle of JIT production, a company producing orders (or jobs) early, as well as late, is discouraged. It incurs several costs when an order is produced early, for example, costs caused by the extra investment in its finished products inventory, costs involved in extra storage facilities and product spoilage cost. In practice, order completions can be accepted without penalty within an interval in time. This time interval is often called the due n Corresponding author at: Department of Industrial Management, Nan Kai University of Technology, No. 568, Zhongzheng Rd., Caotun Township, Nantou County 54243, Taiwan. Fax: þ886 49 2565842. E-mail address: [email protected] (S.-J. Yang).
0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.12.021
window. Applications of the due window problem in real-life situations can readily be found. For example, the due window might reflect an assembly environment in which the components of the product should be ready at a time interval in order to avoid staging delays, or a shop where several jobs constitute a single customer’s order. It is clear that a wide due window increases the supplier’s production and delivery flexibility. However, a large due window and delaying job completion reduce the supplier’s competitiveness and customer service level. The classical JIT scheduling with due windows assignment aims to determine the optimal due window locations and sizes and the optimal schedule to minimize a total cost function. Sidney (1977) was among the pioneers, who studied a single machine scheduling problem with due windows assignment. The objective was to find a schedule that minimizes the total costs of earliness and tardiness. He assumed that each job has its due window and no job’s due window is allowed to contain the due window of another job. He proved that this problem is solvable in polynomial time and developed an optimal procedure to determine the actual job start times to minimize the earliness and tardiness penalties. Liman et al. (1996) explored a variation of the problem where a common due window size is given but the earliest due date is not. Liman et al. (1998) further generalized their result to the problem where both the earliest due date and the due window size are to be determined. They proposed an Oðn log nÞ algorithm to solve the problem, where n is the total number of jobs. Recently, Mosheiov
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
and Sarig (2008) extended the problem proposed by Liman et al. (1998) to the case of position-dependent job processing times. They assumed that the processing time of a job is a function of its position in a sequence. They provided an Oðn3 Þ time solution for this case. Janiak et al. (2009) considered various models of due window assignment scheduling problems on a single processor such that the objective function containing the maximum or total earliness and tardiness and the due window parameters is minimized. They derived several properties of the solutions and constructed polynomial time algorithms for the considered problems. Wang and Wang (2011) studied a single machine common due window scheduling problem simultaneously with the learning effect and deteriorating jobs considerations. They introduced an Oðn log nÞ time algorithm to solve the problem. Su and Tien (2011) considered a problem of scheduling jobs on a single machine to minimize the mean absolute deviation of the job completion time about a large common due window subject to the maximum tardiness constraint. They proposed a branch and bound algorithm and a heuristic to solve the problem under study. Cheng et al. (2012) explored a common due window assignment scheduling problem with linear time-dependent deteriorating jobs and a deteriorating maintenance on a single machine setting. They showed that the proposed model is polynomially solvable. Yin et al., 2013 considered a batch delivery single machine scheduling problem where the jobs have an assignable common due window. They showed that the problem can be optimally solved in Oðn8 Þ time under a reasonable assumption on the relationships among the cost parameters. For new trends in scheduling with due window, we refer the reader to Mor and Mosheiov (2012), Huang et al. (2013), Chen et al. (2013), Ji et al. (2013), and Yin et al. (2014). On the other hand, scheduling problems with controllable processing times have received considerable attention by many researchers. In scheduling problems, the actual processing time of a job can be controlled by the allocation of additional resources. The concept of controllable processing time is from the area of project planning and control. Comprehensive surveys on this line of scheduling research can be found in Nowicki and Zdrzalka (1990), Chudzik et al. (2006), and Shabtay and Steiner (2007). For new trends in scheduling with controllable processing times, we refer the reader to Rudek and Rudek (2012), Shabtay et al. (2012), Yang et al. (2013), Oron (2014), and Kayvanfar et al. (2014). Furthermore, scheduling problem with controllable processing times to include the due window aspect has received relatively little attention in the literature. Liman et al. (1997) investigated single machine scheduling problems with a common due window and controllable processing times. The objective was to find the optimal common due window position and size, the optimal job compressions, and the optimal job sequence to minimize a total cost function including earliness, tardiness, due window location and size, and processing time reduction. They showed that the problem can be formulated as an assignment problem and thus can be solved in polynomial time. Wan (2007) studied a single machine common due window assignment scheduling problem where the job processing times are controllable with linear costs and the due window was variable. The objective was to find a job sequence, a processing time for each job, and a position of the common due window to minimize the total costs of weighted earliness and tardiness and the processing time compression. He proposed some properties of the optimal solution and provided a polynomial time algorithm to solve the problem. Although the due window assignment in JIT scheduling has been extensively studied in the literature, the multiple due windows assignment scheduling problem with controllable processing times has never been explored. In practice, n jobs may have distinct m due windows, where 1 r m r n. In this paper the
97
concept of single due window is extended to allow for a predetermined number of due windows when scheduling a single machine. This extension allows for greater flexibility in modeling real-life problems. For example, in an order picking operation process, the number of orders to be completed may be too great to realistically justify measurement from a single due window for a single customer. By viewing the order picking operation process as being composed of several discrete segments, each group of orders could be made uniform around its own due window. The orders should be ready at their due window in order to avoid staging delays. Moreover, a higher cost in the form of transship fee generally accompanies a later due window. Another example can be found in an assembly process where many components constitute a product. We consider single machine scheduling problems with multiple due windows assignment and controlling processing times. Similar to Liman et al. (1997), the objective is to determine the optimal due window positions and sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimal job sequence such that the cost function containing the earliness, the tardiness, the processing time compressions, and the due windows related costs is minimized. We also extend the proposed problem by the introduction of the aging effect. The considered issues are relatively new in the scheduling theory. We show that the considered problems are polynomially solvable and propose two efficient algorithms to solve them. This paper is divided into six sections. In Section 2 we formulate the problem. In Section3 we provide some properties of the optimal schedule. In Section 4 we present the optimal solution for the problem. We extend the problem to incorporate with the aging effect in Section 5. We conclude the paper and suggest some topics for future research in the last section.
2. Notation and problem formulation The following notation will be used throughout the paper and we will introduce additional notation when needed. n m Jj ½r di wi Di Ii ni pj xj xj pj Cj Ej Tj Gj
α β
the total number of jobs; the number of due windows, 1 r m rn; job j, j ¼ 1; 2; …; n; the job scheduled in the rth position of a job sequence; the due window starting time of the ith due window, i ¼ 1; 2; …; m; the due window finishing time of the ith due window, i ¼ 1; 2; …; m; the size of the ith due window, i ¼ 1; 2; …; m, i.e., D i ¼ w i di ; the set of jobs assigned to the ith due window, i ¼ 1; 2; …; m; the number of jobs assigned to the ith due window, i ¼ 1; 2; …; m, i.e., jI i j ¼ ni and n ¼ ∑m i ¼ 1 ni ; the normal processing time of job J j , j ¼ 1; 2; …; n; the amount of reduction in processing time of job J j , j ¼ 1; 2; …; n; the upper bound on the amount of reduction in processing time of job J j , j ¼ 1; 2; …; n; the actual processing time of job J j , j ¼ 1; 2; …; n; the completion time of job J j , j ¼ 1; 2; …; n; the earliness of job J j ¼ max 0; di C j , for J j A I i , j ¼ 1; 2; …; n and i ¼ 1; 2; …; m; the tardiness for job J j ¼ max 0; C j wi , for J j A I i , j ¼ 1; 2; …; n and i ¼ 1; 2; …; m; the unit cost of compression for job J j , j ¼ 1; 2; …; n; the per unit time earliness penalty, α 4 0; the per unit time tardiness penalty, β 4 0;
98
γ δ
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
the per unit time due window starting time penalty, γ 4 0. the per unit time due window size penalty, δ 4 0.
There are n independent jobs J ¼ J 1 ; J 2 ; …; J n to be processed on a single machine. All the jobs are ready for processing at time zero and no preemption is allowed. The machine can handle at most one job at a time. We consider the problem of multiple due windows assignment scheduling with controllable processing times simultaneously. Under the proposed model, job J j has a normal processing time pj , which can be controlled by an additional resource xj Z 0, j ¼ 1; 2; …; n. For each job J j , its compression has an upper bound xj , j ¼ 1; 2; …; n. Then, the actual processing time of job J j is given by pj ¼ pj xj ; 0 r xj rxj o pj ; j ¼ 1; 2; …; n
ð1Þ
The condition xj o pj is necessary to ensure that the actual processing time of a job is greater than zero. In practice, n jobs may have distinct m due windows, where 1 r m rn. We assume that the number of due windows m to be assigned to the jobs is given in advance, for 1 rm r n. We further assume that any due window is not allowed to contain another due window as a proper subset. Let di ð Z0Þ and wi ð Z di Þ denote the starting time and finishing time of the ith due window, respectively, for i ¼ 1; 2; …; m, and Di ¼ wi di denotes the size of the ith due window. Note that both di and wi are decision variables. If m ¼ 1, it means that all the jobs have one common due window; if m ¼ n, it indicates that there exists n distinct due windows for n jobs. Let I i denote the set of jobs assigned to the ith due window, for i ¼ 1; 2; :::; m. Then the earliness and the tardiness of job J j are Ej ¼ max f0; di C j g and T j ¼ max f0; C j wi g, respectively, for J j A I i . Similar to Liman et al. (1997), we aim to determine the set of the due window starting times d ¼ fd1 ; d2 ; :::; dm g, the set of the due window sizes D ¼ fD1 ; D2 ; :::; Dm g, the set of jobs assigned to each due window I ¼ fI 1 ; I 2 ; :::; I m g, the set of the job compressions x ¼ fx1 ; x2 ; :::; xn g, and the job sequence π such that the total cost function Zðd; D; I; x; π Þ m
Zðd; D; I; x; π Þ ¼ ∑ ∑ ðαEj þ β T j þ γ di þ δDi þ Gj xj Þ
ð2Þ
i ¼ 1 j A Ii
is minimized.
3. Preliminary analysis The following lemmas are useful to find the optimal solution for the problem under study. Lemma 1. In an optimal schedule, there exists no idle time between consecutive jobs on the machine and the first job starts at time zero.
3 indicates that for an optimal sequence, the jobs assigned to different due windows are mutually disjoint, that is, there is an optimal solution such that ni consecutive jobs (in positions Ni 1 þ 1 to Ni ) in a job sequence π are assigned to the ith due window. Lemma 3. For any given d, D, x, and π, there exists an optimal I such that I i ¼ ðJ ½Ni 1 þ 1 ; J ½Ni 1 þ 2 ; …; J ½Ni Þ, for i ¼ 1; 2; …; m, where J ½r is the job scheduled in position r in the job sequence π. Proof. For any given d, D, x, and π, without loss of generality, we assume that in a schedule S1 ¼ ðπ 1 ; J k ; J l ; π 2 Þ job J k immediately precedes job J l , while in a schedule S2 ¼ ðπ 1 ; J l ; J k ; π 2 Þ jobs J k and J l are mutually replaced, where π 1 and π 2 denote partial sequences and jobs J k and J l are, respectively, scheduled in the rth position and ðr þ1Þth position in the schedule S1 . In addition, we assume that for both schedules S1 and S2 , J k is early for the ði þ1Þth due window and J l is tardy for the ith due window, where J k A I i þ 1 and Jl A Ii . Let C j ðS1 Þ and C j ðS2 Þ be the completion times of job J j in schedules S1 and S2 , respectively. By definition, the completion times of jobs J k and J l in S1 are r 1
C k ðS1 Þ ¼ ∑ p½j þ pk
ð3Þ
j¼1
and r1
C l ðS1 Þ ¼ ∑ p½r þ pk þ pl
ð4Þ
j¼1
Similarly, r1
C l ðS2 Þ ¼ ∑ p½j þ pl
ð5Þ
j¼1
and r 1
C k ðS2 Þ ¼ ∑ p½r þ pl þ pk
ð6Þ
j¼1
Then, αðdi þ 1 C k ðS1 ÞÞ þ βðC l ðS1 Þ wi Þ αðdi þ 1 C k ðS2 ÞÞ þ βðC l ðS2 Þ wi Þ ¼ αpl þ βpk 40
ð7Þ
Clearly, α di þ 1 C k ðS1 Þ þ βðC l ðS1 Þ wi Þ 4 α di þ 1 C k ðS2 Þ þ βðC l ðS2 Þ wi Þ
Proof. The proof is obvious and omitted. □ Lemma 2. For any specified sequence π, there exists optimal common due windows with the property that the due window starting time di and finishing time wi coincide with some jobs’ completion times, for i ¼ 1; 2; …; m. Proof. When all the jobs have a common due window (i.e., m ¼ 1), Liman et al. (1998) and Mosheiov and Sarig (2008) showed that there exists an optimal common due window such that the due window starting time and finishing time coincide with some jobs’ completion times. Because their proofs are independent of the job distribution on the time axis, the result can immediately be generalized to the proposed problem. □ Let N i ¼ ∑ik ¼ 1 nk be the total number of jobs assigned to the first i due windows, for i ¼ 1; 2; …; m, and N 0 ¼ 0. Lemma
Similarly, the total cost decreases as repeating this interchange argument for the jobs which assigned to the same due window are not sequenced consecutively. Therefore, we conclude that for any given d, D, x, and π, the jobs assigned to the same due window are arranged consecutively. □ Furthermore, for the case that all the jobs have a common due window, Mosheiov and Sarig (2008) showed that if γ 4 min β; δ , an optimal schedule exists in which the due window starts at time zero and if β o min γ ; δ , there exists an optimal schedule in which the due window is reduced to a due date that starts at time zero. We assume δ 4 γ and β Z δ throughout the remainder of the paper. Given any real number x, the ceiling of x, denoted ⌈x⌉, is the smallest integer greater than or equal to x. Lemma 4 provides the optimal locations of common due windows.
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Lemma 4. For any given I, x, and π, there exists optimal due windows such that di ¼ C ½ki and wi ¼ C ½ki þ hi , where
n ðδ γ Þ ki ¼ N i 1 þ ⌈ i ⌉
ð8Þ
α
and n ðβ δÞ ⌉ ki þ hi ¼ N i 1 þ ⌈ i
ð9Þ
β
for i ¼ 1; 2; …; m. Proof. When all the jobs share a common due window (i.e., m ¼ 1 and n1 ¼ n), consider an optimal schedule and an optimal due window such that C ½k1 ¼ d1 and C ½k1 þ h1 ¼ w1 . Using the small perturbation technique (Mosheiov and Sarig, 2009), we have n1 ðδ γ Þ k1 ¼ ⌈ ⌉
ð10Þ
α
and ðk1 þ h1 Þ ¼ ⌈
n1 ðβ δ Þ
β
⌉
ð11Þ
Since any due window is not allowed to contain another due window as a proper subset and both (10) and (11) are obtained independent of the job distribution on the time axis, the results can immediately be generalized to the proposed model. Consequently, we have (8) and (9). □ Consequently, by Lemmas 2–4, we obtain that for the ith due window, the first ðki Ni 1 1Þ jobs in I i are early and ðN i ki hi Þ jobs in I i are tardy. 4. Optimal solution In this section we will find the set of jobs assigned to each due window, the optimal job compressions, and the optimal job sequence to minimize the objective function.
a negative position weight ur should be its normal processing time p½r , and the processing time of a job in a position with a positive weight ur should be its normal processing time p½r minus the upper bound of its compression x½r . If ur ¼ 0, then the optimal processing time of the job in this position may be any value between p½r x½r . Notice that if x½j ¼ 0, it implies that no additional resource is needed to compress the processing time of the job. Thus, after simplifying, we have the following result: ( p½r if ur r 0; n p½r ¼ right: ð14Þ p½r x½r if ur 4 0; where pn½r denotes the optimal processing time of the job in position r, for r ¼ 1; 2; …; n. As a result, the optimal job compression xn½r can be obtained by xn½r ¼ p½r pn½r , for r ¼ 1; 2; …; n. Therefore, we have the following theorem. Theorem 1. For a given job sequence, the optimal job compressions can be determined as follows: The compression of the job in a nonpositive weight position is zero; the compression of the job in a positive weight position is its maximum reduction in the processing time. That is, ( 0 if ur r 0; xn½r ¼ right: ð15Þ x½r if ur 4 0; Proof. The proof follows from the above analysis. □ In order to obtain the optimal job sequence, we formulate the problem as an assignment problem. We may ignore the term of ∑nr ¼ 1 G½r p½r in (12), as it represents a constant in the objective function. Let yjr be a 0/1 variable such that yjr ¼ 1 if job J j is scheduled in the rth position to be processed on the machine and yjr ¼ 0 otherwise. Then, we can formulate the sequencing problem as the following assignment problem: minimize n
m
Zðd; D; I; x; π Þ ¼ ∑ ∑ ðαEj þ βT j þ γ di þ δDi þGj xj Þ i ¼ 1 j A Ii
∑
i ¼ 1 j ¼ Ni 1 þ 1
ðαE½j þ β T ½j þ γ di þ δDi þ G½j x½j Þ
(
ni γ C ½ki þ ni δðC ½ki þ hi C ½ki Þ þ Ni
þ
∑
j ¼ ki þ hi þ 1
βðC ½j C ½ki þ hi Þ þ
n
n
r¼1
r¼1
ki
∑
j ¼ Ni 1 þ 1
Ni
∑
j ¼ Ni 1 þ 1
αðC ½ki C ½j Þ )
∑ yjr ¼ 1;
r ¼ 1; 2; …; n
ð17Þ
∑ yjr ¼ 1;
j ¼ 1; 2; …; n
ð18Þ
yjr ¼ 1 or 0;
j ¼ 1; 2; …; n
j¼1 n
r¼1
and
ð12Þ
r ¼ 1; 2; …; n
ð19Þ
i
½r
r ¼ N i 1 þ 1; N i 1 þ 2; …; ki ; r ¼ ki þ 1; ki þ 2; …; ki þ hi ; r ¼ ki þ hi þ 1; ki þ hi þ 2; …; N i ;
ð13Þ ∑nr ¼ 1 G½r p½r
i
i
j
j ¼ 1; 2; …; n; r ¼ N i 1 þ 1; N i 1 þ 2; …; ki ; j ¼ 1; 2; …; n; r ¼ ki þ 1; ki þ 2; …; ki þ hi ; j ¼ 1; 2; …; n; r ¼ ki þ hi þ 1; ki þ hi þ 2; …; N i ;
ð20Þ
pjr ¼
(
pj ;
if ujr r 0;
pj xj ;
if ujr 4 0;
ð21Þ
for i ¼ 1; 2; …; m. For a given vector ðn1 ; n2 ; …; nm Þ, the problem can be optimally solved by the following algorithm.
where 8 αðr 1 Ni 1 Þ þ γ ðn Ni 1 Þ G½r ; > < ur ¼ δðN i Ni 1 Þ þ γ ðn N i Þ G½r ; > : βðN r þ 1Þ þ γ ðn N Þ G ;
8 αðr 1 N i 1 Þ þ γ ðn N i 1 Þ Gj ; > < ujr ¼ δðN i Ni 1 Þþ γ ðn N i Þ Gj ; > : βðN r þ 1Þ þ γ ðn N Þ G ;
and
G½j x½j
¼ ∑ ur p½r þ ∑ G½r p½r
i
subject to n
where
Ni
m
¼ ∑
i¼1
ð16Þ
j¼1r ¼1
This subsection assumes that the vector ðn1 ; n2 ; …; nm Þ of m positive integers such that n ¼ ∑m and jI i j ¼ ni , for i ¼ 1 ni i ¼ 1; 2; …; m, is externally specified. Note that while jI i j ¼ ni is assumed to be known in advance, the specific jobs in I i are unknown and to be determined. Using Lemmas 3 and 4 and substituting C ½j ¼ ∑jr ¼ 1 p½r and x½j ¼ p½j p½j into (2), we have
m
n
∑ ∑ ujr pjr yjr
4.1. jI i j ¼ ni known
¼ ∑
99
and i ¼ 1; 2; …; m. Since is a constant, for any sequence, the optimal processing time of a job in a position with
Algorithm 1. Step 1. By Lemma 4, calculate ki ¼ N i 1 þ⌈ni ðδ γ Þ=α⌉ and ki þ hi ¼ Ni 1 þ⌈ni ðβ δÞ=β ⌉, for i ¼ 1; 2; …; m, where Ni ¼ ∑ik ¼ 1 nk and N 0 ¼ 0.
100
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
Step 2. Calculate the αjr and pjr values by using (20) and (21), for j ¼ 1; 2; …; n and r ¼ 1; 2; …; n. Step 3. Solve the assignment problem (16)-(19) to determine the optimal job sequence. Step 4. Calculate the optimal job compressions by using (13) and (15). Step 5. Calculate the optimal job processing times by using (14). Step 6. Calculate the position of the due window di ¼ C ½ki ¼ ∑kr i¼ 1 p½r and wi ¼ C ½ki þ hi ¼ ∑kr i¼þ 1hi p½r , for i ¼ 1; 2; …; m. Step 7. Calculate the size of the due window Di ¼ wi di , for i ¼ 1; 2; …; m. Theorem 2. For a given constant m, if jI i j ¼ ni is known in advance, for i ¼ 1; 2; …; m, the proposed model can be solved in Oðn3 Þ time. Proof.. In Algorithm 1, the time complexity of Step 2 is Oðn2 Þ and Step 3 is Oðn3 Þ time; Steps 1, 4, 5, 6, and 7 can be performed in linear time. Thus the overall time complexity of the algorithm is Oðn3 Þ. □ The following example illustrates applying Algorithm 1 to find the optimal solution of an 11 jobs instance. We solve the assignment problem of the example using the commercial software package LINGO version 11.0 on a personal computer. Example 1. There are n ¼ 11 jobs. The number of due windows is m ¼ 3 and n1 ¼ 3, n2 ¼ 5, and n3 ¼ 3 are given. The set of job parameters is presented in Table 1. The unit earliness, tardiness, due window starting time and due window size penalties are α ¼ 3, β ¼ 15, γ ¼ 5 and δ ¼ 6, respectively. Solution. By Lemma 4, we obtain that k1 ¼ 1, k1 þ h1 ¼ 2, k2 ¼ 5, k2 þh2 ¼ 6, k3 ¼ 9, and k3 þ h3 ¼ 10. We solve the assignment problem (16)–(19) to obtain the optimal job sequence and thus obtain the optimal compressions and the actual processing times of jobs. The results of the example are summarized in Table 2. From Table 2, we see that the optimal solution for the example consists of the following: (i) the optimal job sequences for the three due windows are I 1 ¼ ðJ 7 ; J 1 ; J 3 Þ, I 2 ¼ ðJ 10 ; J 5 ; J 11 ; J 9 ; J 4 Þ, and I 3 ¼ ðJ 6 ; J 2 ; J 8 Þ, respectively; (ii) the actual processing times of jobs J 1 , J 2 , J 3 , J 4 , J 5 , J 7 , J 9 , J 10 , and J 11 are compressed by additional resources, (iii) the first due window is starting at time 4.5 and finishing at time 7.5; the second due window is starting at time 31.0 and finishing at time 37.4; and the third due window is starting at time 72.7 and finishing at time 84.7; (iv) the sizes of the due windows are D1 ¼ 7:5 4:5 ¼ 3:0, D2 ¼ 37:4 31:0 ¼ 6:4, and D3 ¼ 84:7 72:7 ¼ 12:0, respectively; (v) job J 10 is early and jobs J 3 , J 4 , J 8 , and J 9 are tardy; (vi) the total cost is Z ¼ 3843:8. 4.2. jI i j ¼ ni unknown In this subsection, we consider that the vector ðn1 ; n2 ; …; nm Þ is unknown. d, D, I, x, and π are decision variables. Following Yang and Yang (2010b), we denote by Pðn; mÞ ¼ ðn1 ; n2 ; …; nm Þ the allocation vector of number of jobs assigned to each due window, where ni Z 1 is the number of jobs assigned to the ith due window and n ¼ ∑m i ¼ 1 ni . Note that the number of jobs assigned to a due window may be 1; 2; …; n m þ 1. So if we know the numbers of Table 1 Job parameters for Example 1. Job
J1
J2
J3
J4
J5
J6
J7
J8
J9
J 10
J 11
pj xj Gj
8 5 21
16 4 12
7 2.5 23
14 3.7 14
12 3.2 35
17 1.5 26
9 4.5 17
15 3.5 28
11 3 39
13 2.8 20
10 3.6 31
Table 2 The optimal solution for Example 1. r
1
2
3
4
5
6
7
8
9
10
11
J ½r xn½r
J7 4.5
J1 5.0
J3 2.5
J 10 2.8
J5 3.2
J 11 3.6
J9 3.0
J4 3.7
J6 0.0
J2 4.0
J8 0.0
p½r C ½r
4.5 4.5
3.0 7.5
4.5 12.0
10.2 22.2
8.8 31.0
6.4 37.4
8.0 45.4
10.3 55.7
17.0 72.7
12.0 84.7
15.0 99.7
Table 3 Job parameters for Example 2. Job
J1
J2
J3
J4
J5
J6
J7
pj xj Gj
8 5 21
16 4 12
7 2.5 23
14 3.7 14
12 3.2 35
17 1.5 26
9 4.5 17
jobs on the first m 1 due windows, the number of jobs processed on the last due window is then determined uniquely due to the fact that n ¼ ∑m i ¼ 1 ni . Therefore, the upper bound on Pðn; mÞ vector is ðn m þ 1Þm 1 r nm 1 . Based on the analysis in the previous section, if ðn1 ; n2 ; …; nm Þ is given, then we can obtain the optimal job sequence and the optimal job compressions by using (16)–(19) and (13) and (15), respectively. Consequently, we can determine the actual processing time of jobs, the optimal positions of the common due windows, and the size of each due window. Theorem 3. For a given constant m, if jI i j ¼ ni is unknown, for i ¼ 1; 2; …; m, the proposed model can be solved in Oðnm þ 2 Þ time. Proof. From Algorithm 1, for a given vector ðn1 ; n2 ; …; nm Þ, the proposed problem can be solved in Oðn3 Þ time. In addition, the upper bound on Pðn; mÞ vector is ðn m þ 1Þm 1 r nm 1 . Therefore, if jI i j ¼ ni is unknown, for i ¼ 1; 2; :::; m, the proposed problem can optimally be solved in Oðnm þ 2 Þ time. □ Clearly, for a given vector Pðn; mÞ ¼ ðn1 ; n2 ; …; nm Þ, we can determine the local optimal solution of the problem by Algorithm 1. The global optimal solution for the problem is the one with the minimum total cost for all the possible vectors Pðn; mÞ ¼ ðn1 ; n2 ; …; nm Þ. Example 2. There are n ¼ 7 jobs. The number of due windows is m ¼ 3. The set of job parameters is presented in Table 3. The unit earliness, tardiness, due window starting time and due window size penalties are α ¼ 3, β ¼ 15, γ ¼ 5 and δ ¼ 6, respectively. Solution. For given the number of jobs on each due window, i.e., ðn1 ; n2 ; n3 Þ, we solve the corresponding problem using Algorithm 1. Table 4 shows the optimal position of due windows and the total cost Z for all the possible vectors ðn1 ; n2 ; n3 Þ. From Table 4, we see that the minimum total cost for this example is obtained when ðn1 ; n2 ; n3 Þ ¼ ð3; 2; 2Þ. The optimal solution for the example is showed in Table 5. From Table 5, we know that: (i) the optimal job sequences for the three due windows are I 1 ¼ ðJ 3 ; J 1 ; J 7 Þ, I 2 ¼ ðJ 2 ; J 4 Þ, and I 3 ¼ ðJ 6 ; J 5 Þ, respectively; (ii) the actual processing times of jobs J 1 , J 2 , J 3 , J 4 , and J 7 are compressed by additional resources, (iii) the first due window is starting at time 4.5 and finishing at time 7.5; the second due window is starting at time 24.0 and finishing at time 34.3; and the third due window is starting at time 51.3 and finishing at time 63.3; (iv) the sizes of the due windows are D1 ¼ 7:5 4:5 ¼ 3:0, D2 ¼ 34:3 24:0 ¼ 10:3, and D3 ¼ 63:3 51:3 ¼ 12:0, respectively; (v) job J 7 is tardy; (vi) the total cost is Z ¼ 1548:4.
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
Table 4 The optimal due window positions and total cost for all the possible vectors ðn1 ; n2 ; n3 Þ of Example 2.
k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 5 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 2, k1 þ h1 ¼ 3, k3 þ h3 ¼ 7 k1 ¼ 2, k1 þ h1 ¼ 3, k3 þ h3 ¼ 7 k1 ¼ 2, k1 þ h1 ¼ 3, k3 þ h3 ¼ 7
ð1; 2; 4Þ ð1; 3; 3Þ ð1; 4; 2Þ ð1; 5; 1Þ ð2; 1; 4Þ ð2; 2; 3Þ ð2; 3; 2Þ ð2; 4; 1Þ ð3; 1; 3Þ ð3; 2; 2Þ ð3; 3; 1Þ ð4; 1; 2Þ ð4; 2; 1Þ ð5; 1; 1Þ
a
x½j ¼ p½j ja p½j into (2), we obtain m
Zðd; D; I; x; π Þ ¼ ∑ ∑ ðαEj þ β T j þ γ di þ δDi þ Gj xj Þ i ¼ 1 j A Ii
ðn1 ; n2 ; n3 Þ Optimal position of due windows ð1; 1; 5Þ
101
Total cost Z
k2 ¼ 2, k2 þ h2 ¼ 2, k3 ¼ 4,
1913.8
k2 ¼ 2, k2 þ h2 ¼ 3, k3 ¼ 5,
1741.0
Ni
m
¼ ∑
∑
i ¼ 1 j ¼ Ni 1 þ 1 m
¼ ∑
i¼1
k2 ¼ 2, k2 þ h2 ¼ 3, k3 ¼ 5,
ðαE½j þ βT ½j þ γ di þ δDi þ G½j x½j Þ
(
ni γ C ½ki þ ni δðC ½ki þ hi C ½ki Þ þ
ki
∑
j ¼ Ni 1 þ 1
αðC ½ki C ½j Þ )
1687.3
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 5,
1702.3
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 7,
1783.7
k2 ¼ 3, k2 þ h2 ¼ 3, k3 ¼ 5,
1738.0
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 5,
1637.3
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 6,
1604.8
k2 ¼ 4, k2 þ h2 ¼ 5, k3 ¼ 7,
1651.2
k2 ¼ 4, k2 þ h2 ¼ 4, k3 ¼ 5,
1624.8
k2 ¼ 4, k2 þ h2 ¼ 5, k3 ¼ 6,
1548.4
k2 ¼ 4, k2 þ h2 ¼ 5, k3 ¼ 7,
1569.7
k2 ¼ 5, k2 þ h2 ¼ 5, k3 ¼ 6,
1587.3
k2 ¼ 5, k2 þ h2 ¼ 6, k3 ¼ 7,
1562.3
k2 ¼ 6, k2 þ h2 ¼ 6, k3 ¼ 7,
1667.3
Ni
þ
∑
j ¼ ki þ hi þ 1
βðC ½j C ½ki þ hi Þ þ
n
n
r¼1
r¼1
Ni
∑
j ¼ Ni 1 þ 1
G½j x½j
¼ ∑ θr p½r r a þ ∑ ðG½r θr Þx½r where 8 > < αðr 1 Ni 1 Þ þ γ ðn Ni 1 Þ; θr ¼ δðNi Ni 1 Þ þ γ ðn N i Þ; > : βðN r þ 1Þ þ γ ðn N Þ; i
i
ð23Þ
r ¼ N i 1 þ 1; Ni 1 þ 2; …; ki ; r ¼ ki þ 1; ki þ 2; …; ki þ hi ; r ¼ ki þ hi þ 1; ki þ hi þ 2; …; Ni ; ð24Þ
N i ¼ ∑ik ¼ 1 nk
a
The minimum total cost.
Table 5 The optimal solution for Example 2. r
1
2
3
4
5
6
7
J ½r xn½r
J3 2.5
J1 5.0
J7 4.5
J2 4.0
J4 3.7
J6 0.0
J5 0.0
p½r C ½r
4.5 4.5
3.0 7.5
4.5 12.0
12.0 24.0
10.3 34.3
17.0 51.3
12.0 63.3
denotes the total number and i ¼ 1; 2; …; m. Recall that of jobs assigned to the first i due windows, for i ¼ 1; 2; …; m, and N0 ¼ 0. From (23), for any job sequence, we can find that the optimal compression of a job in a position with a negative G½r θr should be its maximum reduction in the processing time, and the optimal compression of a job in a position with a positive G½r θr should be 0. If G½r θr ¼ 0, then the optimal compression of the job in this position may be any value between 0 and x½j . After simplifying, for a given job sequence, the optimal job compressions can be determined as follows: ( x½r if G½r θr o 0; n x½r ¼ ð25Þ 0 if G½r θr Z 0; for r ¼ 1; 2; …; n. Then, we can formulate the sequencing problem as the following assignment problem: minimize n
n
∑ ∑ λjr yjr
ð26Þ
j¼1r ¼1
subject to n
∑ yjr ¼ 1;
j¼1
5. Extension
n
In this section, the proposed model is extended by the introduction of the effect of aging. The aging effect occurs when the production facility becomes less efficiency (e.g., wearing or tiredness) and the production rate deteriorates (Janiak and Rudek, 2006). Under the aging effect environment, the later a given job is scheduled in the sequence, the longer its processing time is. Several recent papers have studied scheduling problems with the aging effect in different machine environments, including Kuo and Yang (2008), Janiak and Rudek (2010), Zhao and Tang (2010), and Yang and Yang (2010a, 2010b, 2010c). We will show that the problem with incorporating the aging effect remains polynomially solvable. In this model, if job J j is scheduled in the rth position in a job sequence, its actual processing time is given by a
a
pj ¼ pj r xj ; 0 r xj r xj o pj n ;
j ¼ 1; 2; …; n
ð22Þ
where a Z 0 is the common aging factor of jobs. Clearly, Lemmas 1–4 still hold when the aging effect is examined. Similar to the above analysis, if the vector ðn1 ; n2 ; …; nm Þ is given in advance, using Lemmas 3 and 4 and substituting C ½j ¼ ∑jr ¼ 1 p½r and
∑ yjr ¼ 1;
r¼1
r ¼ 1; 2; …; n
ð27Þ
j ¼ 1; 2; …; n
ð28Þ
yjr ¼ 1 or 0; where (
λjr ¼
j ¼ 1; 2; …; n;
and
r ¼ 1; 2; …; n
θ r pj r a if Gj θr Z0; θr pj ra þ ðGj θr Þxj if Gj θr o0
ð29Þ
ð30Þ
For a given vector ðn1 ; n2 ; …; nm Þ, we propose a polynomial time algorithm to solve the problem. Algorithm 2. Step 1. By Lemma 4, calculate ki ¼ N i 1 þ⌈ni ðδ γ Þ=α⌉ and ki þ hi ¼ N i 1 þ ⌈ni ðβ δÞ=β⌉, for i ¼ 1; 2; …; m, where Ni ¼ ∑ik ¼ 1 nk and N 0 ¼ 0. Step 2. Calculate the λjr values by using (30), for j ¼ 1; 2; …; n and r ¼ 1; 2; …; n. Step 3. Solve the assignment problem (26)-(29) to determine the optimal job sequence.
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Table 6 The optimal solution for Example 3. r
1
2
3
4
5
6
7
8
J ½r xn½r
J4 3.7
J7 4.5
J1 5.0
J 10 2.8
J9 3.0
J 11 3.6
J3 2.5
J5 0.0
0.0
4.0
0.0
p½r C ½r
10.3 10.3
6.58 16.88
6.12 23.0
16.9 39.9
14.83 54.73
13.52 68.25
10.05 78.3
22.39 100.69
32.86 133.55
27.92 161.47
30.8 192.27
Step 4. Calculate the optimal job compressions by using (25). Step 5. Calculate the optimal job processing times by using (22). Step 6. Calculate the position of the due window di ¼ C ½ki ¼ ∑kr i¼ 1 p½r and wi ¼ C ½ki þ hi ¼ ∑kr i¼þ 1hi p½r , for i ¼ 1; 2; …; m. Step 7. Calculate the size of the due window Di ¼ wi di , for i ¼ 1; 2; …; m. Clearly, the overall time complexity of Algorithm 2 is Oðn3 Þ and thus the following theorem holds. Theorem 4. For a given constant m, if jI i j ¼ ni is known in advance, for i ¼ 1; 2; …; m, the proposed model with the aging effect can be solved in Oðn3 Þ time. Example 3 illustrates applying Algorithm 2 to find the optimal solution of an 11 job instance.
9 J6
10
11
J2
J8
processing time and an amount of reduction. We showed that the problem is polynomially solvable and presented an efficient algorithm to solve it. We also extended the proposed model by the introduction of the aging effect. We proved that even the introduction of the effect of aging, the problem remains polynomially solvable. Further research might be to consider the problem with other models of resource allocation or in multi-machine settings.
Acknowledgments The authors thank the Editor and anonymous reviewers for their helpful comments and suggestions on an earlier version of the paper. This research was supported in part by the National Science Council of Taiwan, Republic of China, under grant number NSC 102–2221-E-252-010-MY2.
Example 3. The same data in Example 1 is used and the common aging factor is a ¼ 0:3.
References
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In view of the analysis in the section 4.2, we have the following theorem. Theorem 5. For a given constant m, if jI i j ¼ ni is unknown, for i ¼ 1; 2; …; m, the proposed model with the aging effect can be solved in Oðnm þ 2 Þ time. 6. Conclusions We considered single machine scheduling problem with multiple due windows assignment and controllable processing times where the objective is to determine the optimal due window locations and sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimal job sequence to minimize the total penalty based on job earliness, tardiness, due window, and processing time compression. We assumed that the actual processing time of a job is a function of its normal
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Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Scheduling problems with multiple due windows assignment and controllable processing times on a single machine Dar-Li Yang a, Chien-Jung Lai b, Suh-Jenq Yang b,c,n a
Department of Information Management, National Formosa University, Taiwan Department of Distribution Management, National Chin-Yi University of Technology, Taiwan c Department of Industrial Management, Nan Kai University of Technology, No. 568, Zhongzheng Rd., Caotun Township, Nantou County 54243, Taiwan b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2012 Accepted 13 December 2013 Available online 22 December 2013
This paper deals with multiple due windows assignment scheduling problems and controllable processing times on a single machine. We assume that the actual processing time of a job can be controlled by the introduction of additional resource and any due window is not allowed to contain another due window as a proper subset. The objective is to determine the optimal due window positions and sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimal schedule to minimize a total cost function, which consists of the earliness, the tardiness, the processing time compressions, and the due windows related costs. We show that for the case when the number of jobs assigned to each due window is given in advance, the problem is polynomially solvable in Oðn3 Þ time, where n is the total number of jobs; while if the number of jobs assigned to each due window is unknown, the problem can be optimally solved in Oðnm þ 2 Þ time, where m is the number of due windows. Furthermore, we extend the problem by incorporating with the aging effect and prove that it remains polynomially solvable. & 2013 Elsevier B.V. All rights reserved.
Keywords: Scheduling Multiple due windows Controllable processing time Aging effect
1. Introduction Traditionally manufacturers have forecasted demand for their products into the future and then have attempted to smooth out production to satisfy that forecasted demand. Unfortunately, this approach has a number of major drawbacks, such as large inventories, long production times, production obsolescence, inability to meet delivery schedules, and high costs. On the other hand, Just-inTime (JIT) is a pull system of production, in which products are produced only as needed to meet the actual customer demand. It is based on planned elimination of all waste and continuous improvement of productivity. To cope with intensified global competition and escalating customer demand for superior service, JIT production has become a competitive strategy for world-class company. According to the principle of JIT production, a company producing orders (or jobs) early, as well as late, is discouraged. It incurs several costs when an order is produced early, for example, costs caused by the extra investment in its finished products inventory, costs involved in extra storage facilities and product spoilage cost. In practice, order completions can be accepted without penalty within an interval in time. This time interval is often called the due n Corresponding author at: Department of Industrial Management, Nan Kai University of Technology, No. 568, Zhongzheng Rd., Caotun Township, Nantou County 54243, Taiwan. Fax: þ886 49 2565842. E-mail address: [email protected] (S.-J. Yang).
0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.12.021
window. Applications of the due window problem in real-life situations can readily be found. For example, the due window might reflect an assembly environment in which the components of the product should be ready at a time interval in order to avoid staging delays, or a shop where several jobs constitute a single customer’s order. It is clear that a wide due window increases the supplier’s production and delivery flexibility. However, a large due window and delaying job completion reduce the supplier’s competitiveness and customer service level. The classical JIT scheduling with due windows assignment aims to determine the optimal due window locations and sizes and the optimal schedule to minimize a total cost function. Sidney (1977) was among the pioneers, who studied a single machine scheduling problem with due windows assignment. The objective was to find a schedule that minimizes the total costs of earliness and tardiness. He assumed that each job has its due window and no job’s due window is allowed to contain the due window of another job. He proved that this problem is solvable in polynomial time and developed an optimal procedure to determine the actual job start times to minimize the earliness and tardiness penalties. Liman et al. (1996) explored a variation of the problem where a common due window size is given but the earliest due date is not. Liman et al. (1998) further generalized their result to the problem where both the earliest due date and the due window size are to be determined. They proposed an Oðn log nÞ algorithm to solve the problem, where n is the total number of jobs. Recently, Mosheiov
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
and Sarig (2008) extended the problem proposed by Liman et al. (1998) to the case of position-dependent job processing times. They assumed that the processing time of a job is a function of its position in a sequence. They provided an Oðn3 Þ time solution for this case. Janiak et al. (2009) considered various models of due window assignment scheduling problems on a single processor such that the objective function containing the maximum or total earliness and tardiness and the due window parameters is minimized. They derived several properties of the solutions and constructed polynomial time algorithms for the considered problems. Wang and Wang (2011) studied a single machine common due window scheduling problem simultaneously with the learning effect and deteriorating jobs considerations. They introduced an Oðn log nÞ time algorithm to solve the problem. Su and Tien (2011) considered a problem of scheduling jobs on a single machine to minimize the mean absolute deviation of the job completion time about a large common due window subject to the maximum tardiness constraint. They proposed a branch and bound algorithm and a heuristic to solve the problem under study. Cheng et al. (2012) explored a common due window assignment scheduling problem with linear time-dependent deteriorating jobs and a deteriorating maintenance on a single machine setting. They showed that the proposed model is polynomially solvable. Yin et al., 2013 considered a batch delivery single machine scheduling problem where the jobs have an assignable common due window. They showed that the problem can be optimally solved in Oðn8 Þ time under a reasonable assumption on the relationships among the cost parameters. For new trends in scheduling with due window, we refer the reader to Mor and Mosheiov (2012), Huang et al. (2013), Chen et al. (2013), Ji et al. (2013), and Yin et al. (2014). On the other hand, scheduling problems with controllable processing times have received considerable attention by many researchers. In scheduling problems, the actual processing time of a job can be controlled by the allocation of additional resources. The concept of controllable processing time is from the area of project planning and control. Comprehensive surveys on this line of scheduling research can be found in Nowicki and Zdrzalka (1990), Chudzik et al. (2006), and Shabtay and Steiner (2007). For new trends in scheduling with controllable processing times, we refer the reader to Rudek and Rudek (2012), Shabtay et al. (2012), Yang et al. (2013), Oron (2014), and Kayvanfar et al. (2014). Furthermore, scheduling problem with controllable processing times to include the due window aspect has received relatively little attention in the literature. Liman et al. (1997) investigated single machine scheduling problems with a common due window and controllable processing times. The objective was to find the optimal common due window position and size, the optimal job compressions, and the optimal job sequence to minimize a total cost function including earliness, tardiness, due window location and size, and processing time reduction. They showed that the problem can be formulated as an assignment problem and thus can be solved in polynomial time. Wan (2007) studied a single machine common due window assignment scheduling problem where the job processing times are controllable with linear costs and the due window was variable. The objective was to find a job sequence, a processing time for each job, and a position of the common due window to minimize the total costs of weighted earliness and tardiness and the processing time compression. He proposed some properties of the optimal solution and provided a polynomial time algorithm to solve the problem. Although the due window assignment in JIT scheduling has been extensively studied in the literature, the multiple due windows assignment scheduling problem with controllable processing times has never been explored. In practice, n jobs may have distinct m due windows, where 1 r m r n. In this paper the
97
concept of single due window is extended to allow for a predetermined number of due windows when scheduling a single machine. This extension allows for greater flexibility in modeling real-life problems. For example, in an order picking operation process, the number of orders to be completed may be too great to realistically justify measurement from a single due window for a single customer. By viewing the order picking operation process as being composed of several discrete segments, each group of orders could be made uniform around its own due window. The orders should be ready at their due window in order to avoid staging delays. Moreover, a higher cost in the form of transship fee generally accompanies a later due window. Another example can be found in an assembly process where many components constitute a product. We consider single machine scheduling problems with multiple due windows assignment and controlling processing times. Similar to Liman et al. (1997), the objective is to determine the optimal due window positions and sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimal job sequence such that the cost function containing the earliness, the tardiness, the processing time compressions, and the due windows related costs is minimized. We also extend the proposed problem by the introduction of the aging effect. The considered issues are relatively new in the scheduling theory. We show that the considered problems are polynomially solvable and propose two efficient algorithms to solve them. This paper is divided into six sections. In Section 2 we formulate the problem. In Section3 we provide some properties of the optimal schedule. In Section 4 we present the optimal solution for the problem. We extend the problem to incorporate with the aging effect in Section 5. We conclude the paper and suggest some topics for future research in the last section.
2. Notation and problem formulation The following notation will be used throughout the paper and we will introduce additional notation when needed. n m Jj ½r di wi Di Ii ni pj xj xj pj Cj Ej Tj Gj
α β
the total number of jobs; the number of due windows, 1 r m rn; job j, j ¼ 1; 2; …; n; the job scheduled in the rth position of a job sequence; the due window starting time of the ith due window, i ¼ 1; 2; …; m; the due window finishing time of the ith due window, i ¼ 1; 2; …; m; the size of the ith due window, i ¼ 1; 2; …; m, i.e., D i ¼ w i di ; the set of jobs assigned to the ith due window, i ¼ 1; 2; …; m; the number of jobs assigned to the ith due window, i ¼ 1; 2; …; m, i.e., jI i j ¼ ni and n ¼ ∑m i ¼ 1 ni ; the normal processing time of job J j , j ¼ 1; 2; …; n; the amount of reduction in processing time of job J j , j ¼ 1; 2; …; n; the upper bound on the amount of reduction in processing time of job J j , j ¼ 1; 2; …; n; the actual processing time of job J j , j ¼ 1; 2; …; n; the completion time of job J j , j ¼ 1; 2; …; n; the earliness of job J j ¼ max 0; di C j , for J j A I i , j ¼ 1; 2; …; n and i ¼ 1; 2; …; m; the tardiness for job J j ¼ max 0; C j wi , for J j A I i , j ¼ 1; 2; …; n and i ¼ 1; 2; …; m; the unit cost of compression for job J j , j ¼ 1; 2; …; n; the per unit time earliness penalty, α 4 0; the per unit time tardiness penalty, β 4 0;
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γ δ
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
the per unit time due window starting time penalty, γ 4 0. the per unit time due window size penalty, δ 4 0.
There are n independent jobs J ¼ J 1 ; J 2 ; …; J n to be processed on a single machine. All the jobs are ready for processing at time zero and no preemption is allowed. The machine can handle at most one job at a time. We consider the problem of multiple due windows assignment scheduling with controllable processing times simultaneously. Under the proposed model, job J j has a normal processing time pj , which can be controlled by an additional resource xj Z 0, j ¼ 1; 2; …; n. For each job J j , its compression has an upper bound xj , j ¼ 1; 2; …; n. Then, the actual processing time of job J j is given by pj ¼ pj xj ; 0 r xj rxj o pj ; j ¼ 1; 2; …; n
ð1Þ
The condition xj o pj is necessary to ensure that the actual processing time of a job is greater than zero. In practice, n jobs may have distinct m due windows, where 1 r m rn. We assume that the number of due windows m to be assigned to the jobs is given in advance, for 1 rm r n. We further assume that any due window is not allowed to contain another due window as a proper subset. Let di ð Z0Þ and wi ð Z di Þ denote the starting time and finishing time of the ith due window, respectively, for i ¼ 1; 2; …; m, and Di ¼ wi di denotes the size of the ith due window. Note that both di and wi are decision variables. If m ¼ 1, it means that all the jobs have one common due window; if m ¼ n, it indicates that there exists n distinct due windows for n jobs. Let I i denote the set of jobs assigned to the ith due window, for i ¼ 1; 2; :::; m. Then the earliness and the tardiness of job J j are Ej ¼ max f0; di C j g and T j ¼ max f0; C j wi g, respectively, for J j A I i . Similar to Liman et al. (1997), we aim to determine the set of the due window starting times d ¼ fd1 ; d2 ; :::; dm g, the set of the due window sizes D ¼ fD1 ; D2 ; :::; Dm g, the set of jobs assigned to each due window I ¼ fI 1 ; I 2 ; :::; I m g, the set of the job compressions x ¼ fx1 ; x2 ; :::; xn g, and the job sequence π such that the total cost function Zðd; D; I; x; π Þ m
Zðd; D; I; x; π Þ ¼ ∑ ∑ ðαEj þ β T j þ γ di þ δDi þ Gj xj Þ
ð2Þ
i ¼ 1 j A Ii
is minimized.
3. Preliminary analysis The following lemmas are useful to find the optimal solution for the problem under study. Lemma 1. In an optimal schedule, there exists no idle time between consecutive jobs on the machine and the first job starts at time zero.
3 indicates that for an optimal sequence, the jobs assigned to different due windows are mutually disjoint, that is, there is an optimal solution such that ni consecutive jobs (in positions Ni 1 þ 1 to Ni ) in a job sequence π are assigned to the ith due window. Lemma 3. For any given d, D, x, and π, there exists an optimal I such that I i ¼ ðJ ½Ni 1 þ 1 ; J ½Ni 1 þ 2 ; …; J ½Ni Þ, for i ¼ 1; 2; …; m, where J ½r is the job scheduled in position r in the job sequence π. Proof. For any given d, D, x, and π, without loss of generality, we assume that in a schedule S1 ¼ ðπ 1 ; J k ; J l ; π 2 Þ job J k immediately precedes job J l , while in a schedule S2 ¼ ðπ 1 ; J l ; J k ; π 2 Þ jobs J k and J l are mutually replaced, where π 1 and π 2 denote partial sequences and jobs J k and J l are, respectively, scheduled in the rth position and ðr þ1Þth position in the schedule S1 . In addition, we assume that for both schedules S1 and S2 , J k is early for the ði þ1Þth due window and J l is tardy for the ith due window, where J k A I i þ 1 and Jl A Ii . Let C j ðS1 Þ and C j ðS2 Þ be the completion times of job J j in schedules S1 and S2 , respectively. By definition, the completion times of jobs J k and J l in S1 are r 1
C k ðS1 Þ ¼ ∑ p½j þ pk
ð3Þ
j¼1
and r1
C l ðS1 Þ ¼ ∑ p½r þ pk þ pl
ð4Þ
j¼1
Similarly, r1
C l ðS2 Þ ¼ ∑ p½j þ pl
ð5Þ
j¼1
and r 1
C k ðS2 Þ ¼ ∑ p½r þ pl þ pk
ð6Þ
j¼1
Then, αðdi þ 1 C k ðS1 ÞÞ þ βðC l ðS1 Þ wi Þ αðdi þ 1 C k ðS2 ÞÞ þ βðC l ðS2 Þ wi Þ ¼ αpl þ βpk 40
ð7Þ
Clearly, α di þ 1 C k ðS1 Þ þ βðC l ðS1 Þ wi Þ 4 α di þ 1 C k ðS2 Þ þ βðC l ðS2 Þ wi Þ
Proof. The proof is obvious and omitted. □ Lemma 2. For any specified sequence π, there exists optimal common due windows with the property that the due window starting time di and finishing time wi coincide with some jobs’ completion times, for i ¼ 1; 2; …; m. Proof. When all the jobs have a common due window (i.e., m ¼ 1), Liman et al. (1998) and Mosheiov and Sarig (2008) showed that there exists an optimal common due window such that the due window starting time and finishing time coincide with some jobs’ completion times. Because their proofs are independent of the job distribution on the time axis, the result can immediately be generalized to the proposed problem. □ Let N i ¼ ∑ik ¼ 1 nk be the total number of jobs assigned to the first i due windows, for i ¼ 1; 2; …; m, and N 0 ¼ 0. Lemma
Similarly, the total cost decreases as repeating this interchange argument for the jobs which assigned to the same due window are not sequenced consecutively. Therefore, we conclude that for any given d, D, x, and π, the jobs assigned to the same due window are arranged consecutively. □ Furthermore, for the case that all the jobs have a common due window, Mosheiov and Sarig (2008) showed that if γ 4 min β; δ , an optimal schedule exists in which the due window starts at time zero and if β o min γ ; δ , there exists an optimal schedule in which the due window is reduced to a due date that starts at time zero. We assume δ 4 γ and β Z δ throughout the remainder of the paper. Given any real number x, the ceiling of x, denoted ⌈x⌉, is the smallest integer greater than or equal to x. Lemma 4 provides the optimal locations of common due windows.
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
Lemma 4. For any given I, x, and π, there exists optimal due windows such that di ¼ C ½ki and wi ¼ C ½ki þ hi , where
n ðδ γ Þ ki ¼ N i 1 þ ⌈ i ⌉
ð8Þ
α
and n ðβ δÞ ⌉ ki þ hi ¼ N i 1 þ ⌈ i
ð9Þ
β
for i ¼ 1; 2; …; m. Proof. When all the jobs share a common due window (i.e., m ¼ 1 and n1 ¼ n), consider an optimal schedule and an optimal due window such that C ½k1 ¼ d1 and C ½k1 þ h1 ¼ w1 . Using the small perturbation technique (Mosheiov and Sarig, 2009), we have n1 ðδ γ Þ k1 ¼ ⌈ ⌉
ð10Þ
α
and ðk1 þ h1 Þ ¼ ⌈
n1 ðβ δ Þ
β
⌉
ð11Þ
Since any due window is not allowed to contain another due window as a proper subset and both (10) and (11) are obtained independent of the job distribution on the time axis, the results can immediately be generalized to the proposed model. Consequently, we have (8) and (9). □ Consequently, by Lemmas 2–4, we obtain that for the ith due window, the first ðki Ni 1 1Þ jobs in I i are early and ðN i ki hi Þ jobs in I i are tardy. 4. Optimal solution In this section we will find the set of jobs assigned to each due window, the optimal job compressions, and the optimal job sequence to minimize the objective function.
a negative position weight ur should be its normal processing time p½r , and the processing time of a job in a position with a positive weight ur should be its normal processing time p½r minus the upper bound of its compression x½r . If ur ¼ 0, then the optimal processing time of the job in this position may be any value between p½r x½r . Notice that if x½j ¼ 0, it implies that no additional resource is needed to compress the processing time of the job. Thus, after simplifying, we have the following result: ( p½r if ur r 0; n p½r ¼ right: ð14Þ p½r x½r if ur 4 0; where pn½r denotes the optimal processing time of the job in position r, for r ¼ 1; 2; …; n. As a result, the optimal job compression xn½r can be obtained by xn½r ¼ p½r pn½r , for r ¼ 1; 2; …; n. Therefore, we have the following theorem. Theorem 1. For a given job sequence, the optimal job compressions can be determined as follows: The compression of the job in a nonpositive weight position is zero; the compression of the job in a positive weight position is its maximum reduction in the processing time. That is, ( 0 if ur r 0; xn½r ¼ right: ð15Þ x½r if ur 4 0; Proof. The proof follows from the above analysis. □ In order to obtain the optimal job sequence, we formulate the problem as an assignment problem. We may ignore the term of ∑nr ¼ 1 G½r p½r in (12), as it represents a constant in the objective function. Let yjr be a 0/1 variable such that yjr ¼ 1 if job J j is scheduled in the rth position to be processed on the machine and yjr ¼ 0 otherwise. Then, we can formulate the sequencing problem as the following assignment problem: minimize n
m
Zðd; D; I; x; π Þ ¼ ∑ ∑ ðαEj þ βT j þ γ di þ δDi þGj xj Þ i ¼ 1 j A Ii
∑
i ¼ 1 j ¼ Ni 1 þ 1
ðαE½j þ β T ½j þ γ di þ δDi þ G½j x½j Þ
(
ni γ C ½ki þ ni δðC ½ki þ hi C ½ki Þ þ Ni
þ
∑
j ¼ ki þ hi þ 1
βðC ½j C ½ki þ hi Þ þ
n
n
r¼1
r¼1
ki
∑
j ¼ Ni 1 þ 1
Ni
∑
j ¼ Ni 1 þ 1
αðC ½ki C ½j Þ )
∑ yjr ¼ 1;
r ¼ 1; 2; …; n
ð17Þ
∑ yjr ¼ 1;
j ¼ 1; 2; …; n
ð18Þ
yjr ¼ 1 or 0;
j ¼ 1; 2; …; n
j¼1 n
r¼1
and
ð12Þ
r ¼ 1; 2; …; n
ð19Þ
i
½r
r ¼ N i 1 þ 1; N i 1 þ 2; …; ki ; r ¼ ki þ 1; ki þ 2; …; ki þ hi ; r ¼ ki þ hi þ 1; ki þ hi þ 2; …; N i ;
ð13Þ ∑nr ¼ 1 G½r p½r
i
i
j
j ¼ 1; 2; …; n; r ¼ N i 1 þ 1; N i 1 þ 2; …; ki ; j ¼ 1; 2; …; n; r ¼ ki þ 1; ki þ 2; …; ki þ hi ; j ¼ 1; 2; …; n; r ¼ ki þ hi þ 1; ki þ hi þ 2; …; N i ;
ð20Þ
pjr ¼
(
pj ;
if ujr r 0;
pj xj ;
if ujr 4 0;
ð21Þ
for i ¼ 1; 2; …; m. For a given vector ðn1 ; n2 ; …; nm Þ, the problem can be optimally solved by the following algorithm.
where 8 αðr 1 Ni 1 Þ þ γ ðn Ni 1 Þ G½r ; > < ur ¼ δðN i Ni 1 Þ þ γ ðn N i Þ G½r ; > : βðN r þ 1Þ þ γ ðn N Þ G ;
8 αðr 1 N i 1 Þ þ γ ðn N i 1 Þ Gj ; > < ujr ¼ δðN i Ni 1 Þþ γ ðn N i Þ Gj ; > : βðN r þ 1Þ þ γ ðn N Þ G ;
and
G½j x½j
¼ ∑ ur p½r þ ∑ G½r p½r
i
subject to n
where
Ni
m
¼ ∑
i¼1
ð16Þ
j¼1r ¼1
This subsection assumes that the vector ðn1 ; n2 ; …; nm Þ of m positive integers such that n ¼ ∑m and jI i j ¼ ni , for i ¼ 1 ni i ¼ 1; 2; …; m, is externally specified. Note that while jI i j ¼ ni is assumed to be known in advance, the specific jobs in I i are unknown and to be determined. Using Lemmas 3 and 4 and substituting C ½j ¼ ∑jr ¼ 1 p½r and x½j ¼ p½j p½j into (2), we have
m
n
∑ ∑ ujr pjr yjr
4.1. jI i j ¼ ni known
¼ ∑
99
and i ¼ 1; 2; …; m. Since is a constant, for any sequence, the optimal processing time of a job in a position with
Algorithm 1. Step 1. By Lemma 4, calculate ki ¼ N i 1 þ⌈ni ðδ γ Þ=α⌉ and ki þ hi ¼ Ni 1 þ⌈ni ðβ δÞ=β ⌉, for i ¼ 1; 2; …; m, where Ni ¼ ∑ik ¼ 1 nk and N 0 ¼ 0.
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D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
Step 2. Calculate the αjr and pjr values by using (20) and (21), for j ¼ 1; 2; …; n and r ¼ 1; 2; …; n. Step 3. Solve the assignment problem (16)-(19) to determine the optimal job sequence. Step 4. Calculate the optimal job compressions by using (13) and (15). Step 5. Calculate the optimal job processing times by using (14). Step 6. Calculate the position of the due window di ¼ C ½ki ¼ ∑kr i¼ 1 p½r and wi ¼ C ½ki þ hi ¼ ∑kr i¼þ 1hi p½r , for i ¼ 1; 2; …; m. Step 7. Calculate the size of the due window Di ¼ wi di , for i ¼ 1; 2; …; m. Theorem 2. For a given constant m, if jI i j ¼ ni is known in advance, for i ¼ 1; 2; …; m, the proposed model can be solved in Oðn3 Þ time. Proof.. In Algorithm 1, the time complexity of Step 2 is Oðn2 Þ and Step 3 is Oðn3 Þ time; Steps 1, 4, 5, 6, and 7 can be performed in linear time. Thus the overall time complexity of the algorithm is Oðn3 Þ. □ The following example illustrates applying Algorithm 1 to find the optimal solution of an 11 jobs instance. We solve the assignment problem of the example using the commercial software package LINGO version 11.0 on a personal computer. Example 1. There are n ¼ 11 jobs. The number of due windows is m ¼ 3 and n1 ¼ 3, n2 ¼ 5, and n3 ¼ 3 are given. The set of job parameters is presented in Table 1. The unit earliness, tardiness, due window starting time and due window size penalties are α ¼ 3, β ¼ 15, γ ¼ 5 and δ ¼ 6, respectively. Solution. By Lemma 4, we obtain that k1 ¼ 1, k1 þ h1 ¼ 2, k2 ¼ 5, k2 þh2 ¼ 6, k3 ¼ 9, and k3 þ h3 ¼ 10. We solve the assignment problem (16)–(19) to obtain the optimal job sequence and thus obtain the optimal compressions and the actual processing times of jobs. The results of the example are summarized in Table 2. From Table 2, we see that the optimal solution for the example consists of the following: (i) the optimal job sequences for the three due windows are I 1 ¼ ðJ 7 ; J 1 ; J 3 Þ, I 2 ¼ ðJ 10 ; J 5 ; J 11 ; J 9 ; J 4 Þ, and I 3 ¼ ðJ 6 ; J 2 ; J 8 Þ, respectively; (ii) the actual processing times of jobs J 1 , J 2 , J 3 , J 4 , J 5 , J 7 , J 9 , J 10 , and J 11 are compressed by additional resources, (iii) the first due window is starting at time 4.5 and finishing at time 7.5; the second due window is starting at time 31.0 and finishing at time 37.4; and the third due window is starting at time 72.7 and finishing at time 84.7; (iv) the sizes of the due windows are D1 ¼ 7:5 4:5 ¼ 3:0, D2 ¼ 37:4 31:0 ¼ 6:4, and D3 ¼ 84:7 72:7 ¼ 12:0, respectively; (v) job J 10 is early and jobs J 3 , J 4 , J 8 , and J 9 are tardy; (vi) the total cost is Z ¼ 3843:8. 4.2. jI i j ¼ ni unknown In this subsection, we consider that the vector ðn1 ; n2 ; …; nm Þ is unknown. d, D, I, x, and π are decision variables. Following Yang and Yang (2010b), we denote by Pðn; mÞ ¼ ðn1 ; n2 ; …; nm Þ the allocation vector of number of jobs assigned to each due window, where ni Z 1 is the number of jobs assigned to the ith due window and n ¼ ∑m i ¼ 1 ni . Note that the number of jobs assigned to a due window may be 1; 2; …; n m þ 1. So if we know the numbers of Table 1 Job parameters for Example 1. Job
J1
J2
J3
J4
J5
J6
J7
J8
J9
J 10
J 11
pj xj Gj
8 5 21
16 4 12
7 2.5 23
14 3.7 14
12 3.2 35
17 1.5 26
9 4.5 17
15 3.5 28
11 3 39
13 2.8 20
10 3.6 31
Table 2 The optimal solution for Example 1. r
1
2
3
4
5
6
7
8
9
10
11
J ½r xn½r
J7 4.5
J1 5.0
J3 2.5
J 10 2.8
J5 3.2
J 11 3.6
J9 3.0
J4 3.7
J6 0.0
J2 4.0
J8 0.0
p½r C ½r
4.5 4.5
3.0 7.5
4.5 12.0
10.2 22.2
8.8 31.0
6.4 37.4
8.0 45.4
10.3 55.7
17.0 72.7
12.0 84.7
15.0 99.7
Table 3 Job parameters for Example 2. Job
J1
J2
J3
J4
J5
J6
J7
pj xj Gj
8 5 21
16 4 12
7 2.5 23
14 3.7 14
12 3.2 35
17 1.5 26
9 4.5 17
jobs on the first m 1 due windows, the number of jobs processed on the last due window is then determined uniquely due to the fact that n ¼ ∑m i ¼ 1 ni . Therefore, the upper bound on Pðn; mÞ vector is ðn m þ 1Þm 1 r nm 1 . Based on the analysis in the previous section, if ðn1 ; n2 ; …; nm Þ is given, then we can obtain the optimal job sequence and the optimal job compressions by using (16)–(19) and (13) and (15), respectively. Consequently, we can determine the actual processing time of jobs, the optimal positions of the common due windows, and the size of each due window. Theorem 3. For a given constant m, if jI i j ¼ ni is unknown, for i ¼ 1; 2; …; m, the proposed model can be solved in Oðnm þ 2 Þ time. Proof. From Algorithm 1, for a given vector ðn1 ; n2 ; …; nm Þ, the proposed problem can be solved in Oðn3 Þ time. In addition, the upper bound on Pðn; mÞ vector is ðn m þ 1Þm 1 r nm 1 . Therefore, if jI i j ¼ ni is unknown, for i ¼ 1; 2; :::; m, the proposed problem can optimally be solved in Oðnm þ 2 Þ time. □ Clearly, for a given vector Pðn; mÞ ¼ ðn1 ; n2 ; …; nm Þ, we can determine the local optimal solution of the problem by Algorithm 1. The global optimal solution for the problem is the one with the minimum total cost for all the possible vectors Pðn; mÞ ¼ ðn1 ; n2 ; …; nm Þ. Example 2. There are n ¼ 7 jobs. The number of due windows is m ¼ 3. The set of job parameters is presented in Table 3. The unit earliness, tardiness, due window starting time and due window size penalties are α ¼ 3, β ¼ 15, γ ¼ 5 and δ ¼ 6, respectively. Solution. For given the number of jobs on each due window, i.e., ðn1 ; n2 ; n3 Þ, we solve the corresponding problem using Algorithm 1. Table 4 shows the optimal position of due windows and the total cost Z for all the possible vectors ðn1 ; n2 ; n3 Þ. From Table 4, we see that the minimum total cost for this example is obtained when ðn1 ; n2 ; n3 Þ ¼ ð3; 2; 2Þ. The optimal solution for the example is showed in Table 5. From Table 5, we know that: (i) the optimal job sequences for the three due windows are I 1 ¼ ðJ 3 ; J 1 ; J 7 Þ, I 2 ¼ ðJ 2 ; J 4 Þ, and I 3 ¼ ðJ 6 ; J 5 Þ, respectively; (ii) the actual processing times of jobs J 1 , J 2 , J 3 , J 4 , and J 7 are compressed by additional resources, (iii) the first due window is starting at time 4.5 and finishing at time 7.5; the second due window is starting at time 24.0 and finishing at time 34.3; and the third due window is starting at time 51.3 and finishing at time 63.3; (iv) the sizes of the due windows are D1 ¼ 7:5 4:5 ¼ 3:0, D2 ¼ 34:3 24:0 ¼ 10:3, and D3 ¼ 63:3 51:3 ¼ 12:0, respectively; (v) job J 7 is tardy; (vi) the total cost is Z ¼ 1548:4.
D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103
Table 4 The optimal due window positions and total cost for all the possible vectors ðn1 ; n2 ; n3 Þ of Example 2.
k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 5 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 1, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 6 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 1, k1 þ h1 ¼ 2, k3 þ h3 ¼ 7 k1 ¼ 2, k1 þ h1 ¼ 3, k3 þ h3 ¼ 7 k1 ¼ 2, k1 þ h1 ¼ 3, k3 þ h3 ¼ 7 k1 ¼ 2, k1 þ h1 ¼ 3, k3 þ h3 ¼ 7
ð1; 2; 4Þ ð1; 3; 3Þ ð1; 4; 2Þ ð1; 5; 1Þ ð2; 1; 4Þ ð2; 2; 3Þ ð2; 3; 2Þ ð2; 4; 1Þ ð3; 1; 3Þ ð3; 2; 2Þ ð3; 3; 1Þ ð4; 1; 2Þ ð4; 2; 1Þ ð5; 1; 1Þ
a
x½j ¼ p½j ja p½j into (2), we obtain m
Zðd; D; I; x; π Þ ¼ ∑ ∑ ðαEj þ β T j þ γ di þ δDi þ Gj xj Þ i ¼ 1 j A Ii
ðn1 ; n2 ; n3 Þ Optimal position of due windows ð1; 1; 5Þ
101
Total cost Z
k2 ¼ 2, k2 þ h2 ¼ 2, k3 ¼ 4,
1913.8
k2 ¼ 2, k2 þ h2 ¼ 3, k3 ¼ 5,
1741.0
Ni
m
¼ ∑
∑
i ¼ 1 j ¼ Ni 1 þ 1 m
¼ ∑
i¼1
k2 ¼ 2, k2 þ h2 ¼ 3, k3 ¼ 5,
ðαE½j þ βT ½j þ γ di þ δDi þ G½j x½j Þ
(
ni γ C ½ki þ ni δðC ½ki þ hi C ½ki Þ þ
ki
∑
j ¼ Ni 1 þ 1
αðC ½ki C ½j Þ )
1687.3
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 5,
1702.3
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 7,
1783.7
k2 ¼ 3, k2 þ h2 ¼ 3, k3 ¼ 5,
1738.0
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 5,
1637.3
k2 ¼ 3, k2 þ h2 ¼ 4, k3 ¼ 6,
1604.8
k2 ¼ 4, k2 þ h2 ¼ 5, k3 ¼ 7,
1651.2
k2 ¼ 4, k2 þ h2 ¼ 4, k3 ¼ 5,
1624.8
k2 ¼ 4, k2 þ h2 ¼ 5, k3 ¼ 6,
1548.4
k2 ¼ 4, k2 þ h2 ¼ 5, k3 ¼ 7,
1569.7
k2 ¼ 5, k2 þ h2 ¼ 5, k3 ¼ 6,
1587.3
k2 ¼ 5, k2 þ h2 ¼ 6, k3 ¼ 7,
1562.3
k2 ¼ 6, k2 þ h2 ¼ 6, k3 ¼ 7,
1667.3
Ni
þ
∑
j ¼ ki þ hi þ 1
βðC ½j C ½ki þ hi Þ þ
n
n
r¼1
r¼1
Ni
∑
j ¼ Ni 1 þ 1
G½j x½j
¼ ∑ θr p½r r a þ ∑ ðG½r θr Þx½r where 8 > < αðr 1 Ni 1 Þ þ γ ðn Ni 1 Þ; θr ¼ δðNi Ni 1 Þ þ γ ðn N i Þ; > : βðN r þ 1Þ þ γ ðn N Þ; i
i
ð23Þ
r ¼ N i 1 þ 1; Ni 1 þ 2; …; ki ; r ¼ ki þ 1; ki þ 2; …; ki þ hi ; r ¼ ki þ hi þ 1; ki þ hi þ 2; …; Ni ; ð24Þ
N i ¼ ∑ik ¼ 1 nk
a
The minimum total cost.
Table 5 The optimal solution for Example 2. r
1
2
3
4
5
6
7
J ½r xn½r
J3 2.5
J1 5.0
J7 4.5
J2 4.0
J4 3.7
J6 0.0
J5 0.0
p½r C ½r
4.5 4.5
3.0 7.5
4.5 12.0
12.0 24.0
10.3 34.3
17.0 51.3
12.0 63.3
denotes the total number and i ¼ 1; 2; …; m. Recall that of jobs assigned to the first i due windows, for i ¼ 1; 2; …; m, and N0 ¼ 0. From (23), for any job sequence, we can find that the optimal compression of a job in a position with a negative G½r θr should be its maximum reduction in the processing time, and the optimal compression of a job in a position with a positive G½r θr should be 0. If G½r θr ¼ 0, then the optimal compression of the job in this position may be any value between 0 and x½j . After simplifying, for a given job sequence, the optimal job compressions can be determined as follows: ( x½r if G½r θr o 0; n x½r ¼ ð25Þ 0 if G½r θr Z 0; for r ¼ 1; 2; …; n. Then, we can formulate the sequencing problem as the following assignment problem: minimize n
n
∑ ∑ λjr yjr
ð26Þ
j¼1r ¼1
subject to n
∑ yjr ¼ 1;
j¼1
5. Extension
n
In this section, the proposed model is extended by the introduction of the effect of aging. The aging effect occurs when the production facility becomes less efficiency (e.g., wearing or tiredness) and the production rate deteriorates (Janiak and Rudek, 2006). Under the aging effect environment, the later a given job is scheduled in the sequence, the longer its processing time is. Several recent papers have studied scheduling problems with the aging effect in different machine environments, including Kuo and Yang (2008), Janiak and Rudek (2010), Zhao and Tang (2010), and Yang and Yang (2010a, 2010b, 2010c). We will show that the problem with incorporating the aging effect remains polynomially solvable. In this model, if job J j is scheduled in the rth position in a job sequence, its actual processing time is given by a
a
pj ¼ pj r xj ; 0 r xj r xj o pj n ;
j ¼ 1; 2; …; n
ð22Þ
where a Z 0 is the common aging factor of jobs. Clearly, Lemmas 1–4 still hold when the aging effect is examined. Similar to the above analysis, if the vector ðn1 ; n2 ; …; nm Þ is given in advance, using Lemmas 3 and 4 and substituting C ½j ¼ ∑jr ¼ 1 p½r and
∑ yjr ¼ 1;
r¼1
r ¼ 1; 2; …; n
ð27Þ
j ¼ 1; 2; …; n
ð28Þ
yjr ¼ 1 or 0; where (
λjr ¼
j ¼ 1; 2; …; n;
and
r ¼ 1; 2; …; n
θ r pj r a if Gj θr Z0; θr pj ra þ ðGj θr Þxj if Gj θr o0
ð29Þ
ð30Þ
For a given vector ðn1 ; n2 ; …; nm Þ, we propose a polynomial time algorithm to solve the problem. Algorithm 2. Step 1. By Lemma 4, calculate ki ¼ N i 1 þ⌈ni ðδ γ Þ=α⌉ and ki þ hi ¼ N i 1 þ ⌈ni ðβ δÞ=β⌉, for i ¼ 1; 2; …; m, where Ni ¼ ∑ik ¼ 1 nk and N 0 ¼ 0. Step 2. Calculate the λjr values by using (30), for j ¼ 1; 2; …; n and r ¼ 1; 2; …; n. Step 3. Solve the assignment problem (26)-(29) to determine the optimal job sequence.
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Table 6 The optimal solution for Example 3. r
1
2
3
4
5
6
7
8
J ½r xn½r
J4 3.7
J7 4.5
J1 5.0
J 10 2.8
J9 3.0
J 11 3.6
J3 2.5
J5 0.0
0.0
4.0
0.0
p½r C ½r
10.3 10.3
6.58 16.88
6.12 23.0
16.9 39.9
14.83 54.73
13.52 68.25
10.05 78.3
22.39 100.69
32.86 133.55
27.92 161.47
30.8 192.27
Step 4. Calculate the optimal job compressions by using (25). Step 5. Calculate the optimal job processing times by using (22). Step 6. Calculate the position of the due window di ¼ C ½ki ¼ ∑kr i¼ 1 p½r and wi ¼ C ½ki þ hi ¼ ∑kr i¼þ 1hi p½r , for i ¼ 1; 2; …; m. Step 7. Calculate the size of the due window Di ¼ wi di , for i ¼ 1; 2; …; m. Clearly, the overall time complexity of Algorithm 2 is Oðn3 Þ and thus the following theorem holds. Theorem 4. For a given constant m, if jI i j ¼ ni is known in advance, for i ¼ 1; 2; …; m, the proposed model with the aging effect can be solved in Oðn3 Þ time. Example 3 illustrates applying Algorithm 2 to find the optimal solution of an 11 job instance.
9 J6
10
11
J2
J8
processing time and an amount of reduction. We showed that the problem is polynomially solvable and presented an efficient algorithm to solve it. We also extended the proposed model by the introduction of the aging effect. We proved that even the introduction of the effect of aging, the problem remains polynomially solvable. Further research might be to consider the problem with other models of resource allocation or in multi-machine settings.
Acknowledgments The authors thank the Editor and anonymous reviewers for their helpful comments and suggestions on an earlier version of the paper. This research was supported in part by the National Science Council of Taiwan, Republic of China, under grant number NSC 102–2221-E-252-010-MY2.
Example 3. The same data in Example 1 is used and the common aging factor is a ¼ 0:3.
References
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