Parallel machine scheduling with batch delivery costs

Transcription

1 Int. J. Production Economics 68 (2000) 177}183 Parallel machine scheduling with batch delivery costs Guoqing Wang*, T.C. Edwin Cheng Department of Business Administration, Jinan University, Guangzhou, People's Republic of China Department of Management, The Hong Kong Polytechnic University, Kowloon, Hong Kong Ozce of the Vice President (Research and Postgraduate Studies), The Hong Kong Polytechnic University, Kowloon, Hong Kong Received 12 May 1998; accepted 8 July 1999 Abstract We consider a scheduling problem in which n independent and simultaneously available jobs are to be processed on m parallel machines. The jobs are delivered in batches and the delivery date of a batch is equal to the completion time of the last job in the batch. The delivery cost depends on the number of deliveries. The objective is to minimize the sum of the total #ow time and delivery cost. We "rst show that the problem is NP-complete in the ordinary sense even when m"2, and NP-complete in the strong sense when m is arbitrary. Then we develop a dynamic programming algorithm to solve the problem. The algorithm is pseudopolynomial when m is constant and the number of batches has a "xed upper bound. Finally, we identify two polynomially solvable cases by introducing their corresponding solution methods Elsevier Science B.V. All rights reserved. Keywords: Parallel machine scheduling; Batch delivery cost 1. Introduction Batch scheduling, as combinations of sequencing and partitioning, has attracted much attention of researchers in recent years. Most of the results in batch scheduling area fall into the following three categories: (i) item availability family scheduling problems, (ii) batch availability scheduling problems, and (iii) batch processing scheduling problems. The interested reader is referred to the recent reviews by Webster and Baker [1], Liaee and Emmons [2], and Cheng et al. [3] In this paper, we study a problem which falls into a di!erent category of batch scheduling problems, * Correspondence address. Department of Management, The Hong Kong Polytechnic University, Kowloon, Hong Kong. namely batch delivery problems. Batch delivery problems were "rst introduced by Cheng and Kahlbacher [4]. In [4], they studied single machine batch delivery scheduling to minimize the sum of the total weighted earliness and delivery costs. The earliness of a job is de"ned as the di!erence between its delivery date and completion time. They showed that the problem is NP-complete in the ordinary sense, and the equal weight case is polynomially solvable. Cheng and Gordon [5] provided a dynamic programming algorithm to solve the general problem. The algorithm is pseudopolynomial when the number of batches has a "xed upper bound. They also provided a polynomial algorithm to solve the common processing time case. Cheng et al. [6] further showed that this problem can be formulated as a classical parallel machine scheduling problem, thus the complexity /00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S ( 9 9 ) X

2 178 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177}183 results and algorithms for the corresponding parallel machine scheduling problem can be easily extended to the problem. Cheng et al. [7] studied the single machine batch delivery problem to minimize the sum of the total weighted earliness and mean batch delivery time. Hermann and Lee [8] considered a single machine batch delivery problem where all jobs have a common due date and the objective is to minimize the sum of the earliness and tardiness penalties and delivery costs of the tardy jobs. They provided a pseudopolynomial dynamic programming algorithm to solve the problem. Chen [9] showed that, when the common due date is a decision variable, the problem can be solved polynomially. Yuan [10] showed that the single machine batch delivery problem to minimize the sum of the total weighted earliness and tardiness and delivery costs of the tardy jobs is NP-complete in the strong sense. While all prior batch delivery scheduling research focuses on the single machine environment, we study a parallel machine batch delivery scheduling problem in this paper. The problem can be formally stated as follows (see Fig. 1). There are n independent nonpreemptive jobs to be sequenced for processing on m parallel identical machines, and partitioned into several batches for delivery. All jobs are available for processing at time zero. All jobs in a batch are delivered to the customer together. The batch delivery date is equal to the completion time of the last job in a batch. Thus the #ow time of a job is equal to the batch delivery date on which it is delivered. The delivery cost is a nondecreasing function of the number of batch deliveries. The objective is to sequence and partition the jobs to minimize the sum of the total #ow time and delivery cost. Batch delivery is characteristic of many practical systems in which jobs are transported and ultimately delivered in containers such as boxes or carts. For such systems, an important performance criterion is to minimize the work-in-process (WIP) inventories which are related to the total #ow time. As there are always some costs associated with each delivery, we obtain a situation which can be modelled as the above batch delivery problem. The rest of the paper is organized as follows. In Section 2, we introduce the notation to be used. In Section 3, we show that the problem under study is NP-complete and provide a dynamic programming algorithm to solve the problem optimally. In Section 4, we identify some polynomially solvable cases. Finally, we present our conclusions in the last section. 2. Notation In this section, we introduce the notation to be used in the paper: n : number of jobs; m : number of machines; J"J,2, J : job set to be processed; p : processing time of job J ; R : number of batch deliveries; α(r) : delivery cost function, a nondecreasing function of R; Fig. 1. Problem descriptions.

3 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177} B : batch l; b : number of jobs in B ; D : delivery date of B ; C : completion time of job J ; F : #ow tome of J, which is equal to the batch delivery date on which J is delivered; π"b,2, B : a schedule; C "maxc : makespan of a schedule; G(π)" F #α(r) : total penalty of π. Adopting the three-"eld notation introduced by Graham et al. [11], we denote our problem as Pm/bd/(F #α(r)). 3. NP-completeness and dynamic programming In this section, we consider the complexity issues of the problem. First, it is interesting to note that, when the delivery cost is negligible, Pm/bd/(F #α(r)) simply reduces to the classical parallel machine scheduling problem Pm//C.Itis well known that Pm//C is solved by the generalized shortest processing time (SPT) rule: schedule the jobs in the order of nondecreasing processing times, and assign each job to the earliest available machine [12]. Next, we give a simple proof for the NP-completeness for the problem under study. Assume that the batch delivery cost is so large that all jobs must be delivered in one batch. i.e. R"1. Then Pm/bd/(F #α(r)) is equivalent to the classical parallel machine scheduling problem Pm//C. Since Pm//C has been shown to be NP-complete in the ordinary sense when m is "xed, and NP-complete in the strong sense when m is arbitrary [13], we have the following theorem. Theorem 1. Even when R"1, Pm/bd/(F #α(r)) is NP-complete in the ordinary sense when m is xxed and m*2, and NP-complete in the strong sense when m is arbitrary. The following lemma establishes several properties for an optimal schedule for the problem. Lemma 1. There exists an optimal schedule πh"b,2, B for Pm/bd, R);/(F #α(r)) in which (i) there is no idle time before each job; (ii) all jobs assigned to the same machine are scheduled in the SPT order; (iii) B contains all jobs which xnish processing in the time interval (D, D ], l"1,2, R. Proof. (i) Trivial. (ii) Assume that jobs J and J are assigned to the same machine and J follows J immediately such that p *p in πh. Let π be a schedule obtained by swapping J and J. It is easy to show that G(πH)*G(π), regardless of whether J and J are delivered in the same batch or not. (iii) Let us number all jobs in the order of their completion time in πh. Assume that the batches are numbered in accordance with the numbers of their last jobs. It is clear that, without loss of generality, we can assume D (2(D. Let J be the "rst job in B. If there are any jobs between J and J (the last job in B ) which are assigned to other batches, then there must be at least one batch, say B, such that C )D (D. Let π be a schedule obtained by simply assigning J to B. It is obvious that G(πH)*G(π), a contradiction. Following the same argument with the jobs in batch B and so on, we can show that there exists an optimal schedule in which all batches consist of a number of jobs which "nish processing contiguously. Since all jobs which processing at D, l"1,2, R, can all be assigned to B in any optimal schedule, we have shown that B contains all jobs which "nish processing in the time interval (D, D ]. Based on Lemma 1, we can develop a dynamic programming algorithm to solve the problem. Let ; be an upper bound for the number of batch deliveries, and P" p. The algorithm is formally described as follows. Algorithm PMBD-1: (a) Renumber the jobs in the SPT order, i.e. p )p 2)p. (b) De"ne H (j, t,2, t, D,2, D ) as the minimum total #ow time if we have scheduled jobs J up to J such that the total processing time

4 180 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177}183 of the jobs assigned to machine u is t, u"1,2, m, and the delivery date is D for batch B, l"1,2, R. (c) Recursive relations: For j"0,2, n, t "0,2, P, u " 1,2, m, D " D # 1,2, P, l"1,2, R, D "0, and R"1,2, ;, H ( j, t,2, t, D,2, D )" min X, (1) where X "H ( j!1, t,2, t!p,2, t, D,2, D ) # F ; (2) F "D D (t )D, D "0, l"1,2, R. (3) (d) Initial conditions: For each t "0,2, P, u " 1,2, m, D " D # 1,2, P, l"1,2, R, D "0, and R"1,2, ;, H ( j, t,2, t, D,2, D ) " if j"0, t 0 "t " 2"t "0, R otherwise. (e) Optimal solution: GH"minH (n, t,2, t, D,2, D )#α(r) over all t "0,2, P, u"1,2, m, D "D #1,2, P, l"1,2, R, D "0, and R"1,2, ;. Lemma 2. Algorithm PMBD-1 solves the problem Pm/bd, R);/(F #α(r)) in O(nm;P) time. Proof. Due to Lemma 1, there exists an optimal schedule with jobs assigned to each machine in the SPT order, and each job in assigned to the "rst batch after the completion of the job. If J is assigned to machine u, then C "t, and if D (t )D, then J is assigned to B, and so F "D. This justi"es Eqs. (2) and (3). Since H (j, t,2, t, D,2, D ) is determined by the minimum assignment by de"nition, we have justi- "ed the validity of the recursive relations. So the algorithm PMBD-1 solves the problem Pm/bd, R);/(F #α(r)). The time complexity of the algorithm can be established as follows. Since only m!1 of the values t,2, t are independent, the number of di!erent states of the recursive relations is at most np for R"1,2, ;. For each state, the righthand side of Eq. (1) can be calculated in O(m;) time. Thus, the overall computational complexity of Algorithm PMBD-1 is O(nm;P). Lemma 2 implies that the problem Pm/bd, R);/(F #α(r)) is not strongly NP-complete for any constant m and ;. But it is not clear whether Pm/bd, R);/(F #α(r)) is strongly NP-complete or pseudopolynomically solvable for a constant m and an arbitrary ;. 4. Polynomially solvable cases In this section, we "rst consider a special case where the job assignment is predetermined. It is evident that the problem reduces to an optimal batching problem in this case. This special case characterizes the practical scenario where each machine is dedicated to a special group of jobs. According to Lemma 1, we can provide a backward dynamic programming algorithm to solve the optimal batching problem as follows. Algorithm PMBD-2: (a) Schedule the jobs on each machine in the SPT order, and then renumber all the jobs in accordance with the job completion times. (b) De"ne H (j, l) as the minimum total completion time of the jobs J,2, J when they are assigned to the delivery batches B,2, B. (c) Recursive relations: For R"1,2, n, l"r,2, 1, and j"n,2, l, H ( j, l)" min H (k, l#1)#(k!j)c. (4) (d) Initial conditions: For each j"1,2, n l"1,2, R, and R"1,2, n, H (j, l)" 0 R if j"n#1, and l"r#1; otherwise. (e) Optimal solution: G(πH)" min H (1, 1)#α(R). While the optimality of the algorithm can be easily justi"ed, it is also not di$cult to see that the time complexity of the algorithm is O(n).

5 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177} H (1, 1)"H (2, 2)#C "31. Thus G(πH)"7R#H (1, 1)"52. Fig. 2. Example 1. Now we present a numerical example for the special case to demonstrate the optimality of the algorithm. Example 1. Consider the instance with J" J,2, J, m"2, and α(r)"7r. Assume that J and J are assigned on M, J and J are assigned on M, and all jobs are sequenced in the SPT order on each machine (as shown in Fig. 2). Now using algorithm PMBD-2 to solve the instance, we have the following results: When R"1, we have H (4, 1)"H (5, 2)#C "12, H (3, 1)"H (5, 2)#2C "24, H (2, 1)"H (5, 2)#3C "36, H (1, 1)"H (5, 2)#4C "48, and so G(πH)"7R#H (1, 1)"55. When R"2, we have H (4, 2)"H (5, 3)#C "12, H (3, 2)"H (5, 3)#2C "24, H (2, 2)"H (5, 3)#3C "36, H (1, 1)"minH (4, 2)#3C, H (3, 2)#2C, H (2, 2)#C "32, and so G(πH)"7R#H (1, 1)"46. When R"3, we have H (4, 3)"H (5, 4)#C "12, H (3, 3)"H (5, 4)#2C "24, H (2, 2)"minH (4, 3)#2C, H (3, 3)#C "28, When R"4, we have H (4, 4)"H (5, 5)#C "12, H (3, 3)"H (4, 4)#C "20, H (2, 2)"H (3, 3)#C "24, H (1, 1)"H (2, 2)#C "27, and so G(πH )"7R#H (1, 1)"55. Hence, we obtained an optimal schedule πh" B, B with B "J, J and B "J, J. We now consider the special case with identical processing times, Pm/bd, p "p/(f #α(r)). Let n "n/m, g"n m!n. Let n be the number of jobs processed on machine u under a speci"c schedule. Then we have the following lemma. Lemma 3. There exists an optimal schedule for Pm/bd, p "p/(f #α(r)) in which n!1) n )n,u"1,2, m. Proof. Suppose there exists an optimal schedule πh in which the condition is not satis"ed. According to Lemma 1, we can assume that there is no inserted idle time in πh. Then there must be a pair of machines u and v such that n *n #2 and the last job on machine u is also the last job of the last batch delivery. It is clear that moving the last job on machine u to the last position on machine v will not increase the total penalty. Repeating this process, we can obtain a desired optimal schedule. Let πh"b,2, B be an optimal schedule with R batch deliveries for the problem Pm/bd, p "p/(f #α(r)). Let bm "b /m, l"1, 2, R!1. We have Lemma 4. There exists an optimal schedule with R batch deliveries for Pm/bd, p "p/(f #α(r)) in which bm, l"1, 2, R!1, is integral.

6 182 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177}183 Proof. According to Lemma 1, we can assume that in πh there is no inserted idle time and B contains all jobs which "nish processing in the time interval (D, D ], l"1,2, R!1. It is evident that (D!D )/p is integral, l"1,2, R. Since b "(D!D )m/p, l"1,2, R!1, bm is also integral. Now assume that πh satis"ed Lemmas 3 and 4. Let bm "b /m. We have Lemma 5. There exists an optimal schedule with R batch deliveries for Pm/bd, p "p/(f #α(r)) in which bm!bm )1 for any pair of k and l, where k"1,2, R, l"1,2, R. Proof. We "rst show that changing the sequence of B, l"1,2, R!1, will not cause any increase in the total penalty. It is not di$cult to see that the sequencing of batch deliveries B, l"1,2, R!1, is equivalent to the classical single machine total weighted #ow time scheduling problem, denoted as 1//w C, with w "bm m and p "bm p. It is well known that the total weighted #ow time is minimized by sequencing the jobs in the weighted shortest processing time (WSPT) order [14]. Since p /w "2"p /w "p/m, B, l"1,2, R!1, can be sequenced in an arbitrary order. Now, we can prove the lemma by showing bm!bm )1, l"2, 2, R. (5) We assume that bm '1, l"2, 2, R. Byde"nition, we have D b #D b )(D!p)(b!m) # (D #p)(b #m), (6) D b #D b )(D #p)(b #m) # (D!p)(b!m). (7) Then we can easily obtain the desired results. Now, assume that there are some batches such that bm "1. Note that b may be less than m in this case. Since it is trivial when bm "1 (or bm "1), we suppose bm '1 (or bm '1). From (6) (or (7)), we can easily show that bm )2or(bM )2), and thus (5) holds again. This completes the proof. Based on these results, we can easily construct an optimal schedule with R batch deliveries πh"b,2, B such that bm "b M!1, l"1, 2, h, bm "b M, l"h#1, 2, R, where bm "n /R and h"bm R!n. The associated total penalty can be calculated as G(πH)"α(R)# lmp(bm!1) #h(bm!1)p ) bm (R!h)m # lmpbm!gn p ) "α(r)#(rbm!h#bm )(RbM!h)mp/2! h(bm!1)mp/2!gn p. Now, we can construct a simple algorithm to solve the problem as follows. Algorithm PMBD-3: 1. n :"n/m; g :"n m!n; RH :"0; GH :"R; 2. for R"1 ton do begin bm :"n /R; h :"b M R!n ; G(πH):"α(R)#(RbM!h#b M ) (RbM!h)mp/2!h(b M!1)mp/2!gn p; if GH'G(πH) then RH :"R; GH :"G(πH); end end It is clear that Algorithm PMBD-3 solves the problem Pm/bd, p "p/(f #α(r)) in O(n/m) time. It should be pointed out that, the algorithm, although e$cient, is not actually polynomial when each batch delivery has an equal delivery cost c, i.e.α(r)"cr. The following numerical example demonstrates the optimality of the algorithm. Example 2. Consider the instance with J"J, 2, J, m"2, α(r)"7r, p "p"2, i"1,2, 7. From Lemma 3, we know that there exists an optimal schedule in which all jobs are sequenced as shown in Fig. 3. It is clear that n "4, g"1. Using algorithm PMBD-3 to solve the instance, we have the following results. When R"1, we have bm G(πH)"7R#28p"63. "4 and h"0, and so

7 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177} Acknowledgements Fig. 3. Example 2. When R"2, we have bm "2 and h"0, and so G(πH)"7R#20p"54. When R"3, we have bm "2 and h"2, and so G(πH)"7R#18p"57. When R"4, we have bm "1 and h"0, and so G(πH)"7R#16p"60. Now we can obtain an optimal schedule πh"b, B with B "J, J, J, J, and B "J, J, J. 5. Conclusions In this paper, we have studied the parallel machine scheduling with batch delivery costs. We have shown that the problem to minimize the sum of the total #ow time and delivery cost is NP-complete in the strong sense. We have then provided a dynamic programming algorithm to solve the problem. The algorithm is pseudopolynomial when the number of machines is constant and the number of batches has a "xed upper bound. We also have provided two polynomial time algorithms to solve the special cases where the job assignment is given or the job processing times are equal. There are a number of issues which are of interest for further research. First, it is interesting to investigate the open problem posed in the paper, i.e. whether it is pseudopolynomially solvable or strongly NP-complete when the number of machines is constant and the number of batches is arbitrary. It is also interesting to investigate polynomial time algorithms for the special case where the job processing times are equal and the batch delivery cost function is linear. Another interesting issue is to develop e!ective heuristics to solve the general problem, and it is evident that a viable strategy is to combine the list scheduling procedure for the classical parallel machine scheduling problems [15] with the optimal batching algorithm proposed in this paper. This research was supported in part by The Hong Kong Polytechnic University under grant number 350/239. References [1] S. Webster, K.R. Baker, Scheduling groups of jobs on a single machine, Operations Research 43 (1995) 692}703. [2] M.M. Liaee, H. Emmons, Scheduling families of jobs with setup times, International Journal of Production Economics 51 (1997) 165}176. [3] T.C.E. Cheng, J.N.D. Gupta, G. Wang, A review of #owshop scheduling research with setup times, Production and Operations Management (1999), to appear. [4] T.C.E. Cheng, H.G. Kahlbacher, Scheduling with delivery and earliness penalties, Asia-Paci"c Journal of Operational Research 10 (1993) 145}152. [5] T.C.E. Cheng, V.S. Gordon, Batch delivery scheduling on a single machine, Journal of the Operational Research Society 45 (1994) 1211}1215. [6] T.C.E. Cheng, V.S. Gordon, M.Y. Kovalyov, Single machine scheduling with batch delivery, European Journal of Operational Research 94 (1996) 277}283. [7] T.C.E. Cheng, M.Y. Kovalyov, B.M.T. Lin, Single machine scheduling to minimize batch delivery and job earliness penalties, SIAM Journal on Optimization 7 (1997) 547}559. [8] J.W. Hermann, C.-Y. Lee, On scheduling to minimize earliness}tardiness and batch delivery costs with a common due date, European Journal of Operational Research 70 (1993) 272}288. [9] Z.-L. Chen, Scheduling and common due date assignment with earliness}tardiness penalties and batch delivery costs, European Journal of Operational Research 93 (1996) 49}60. [10] J. Yuan, A note on the complexity of single-machine scheduling with a common due date, earliness}tardiness, and batch delivery costs, European Journal of Operational Research 94 (1996) 203}205. [11] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Annals of Discrete Mathematics 5 (1979) 287}326. [12] R.W. Conway, W.L. Maxwell, L.W. Miller, Theory of Scheduling, Addison-Wesley, Reading, MA, [13] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, New York, [14] W.E. Smith, Various optimizers for single-stage production, Naval Research Logistics Quarterly 3 (1956) 59}66. [15] R.L. Graham, Bounds for certain multiprocessing anomalies, Bell System Technical Journal 45 (1996) 1563}1583.