Customer order scheduling to minimize the number of late jobs

Short Communication

Customer order scheduling to minimize

the number of late jobs

q

B.M.T. Lin

, A.V. Kononov

a

Department of Finance and Information Management, Institute of Information Management, National Chiao Tung University, Taiwan

b

Sobolev Institute of Mathematics, Novosibirsk, Russia Received 12 January 2006; accepted 13 October 2006

Available online 13 December 2006

Abstract

In the order scheduling problem, every job (order) consists of several tasks (product items), each of which will be pro-cessed on a dedicated machine. The completion time of a job is defined as the time at which all its tasks are finished. Min-imizing the number of late jobs was known to be strongly NP-hard. In this note, we show that no FPTAS exists for the two-machine, common due date case, unless P = NP. We design a heuristic algorithm and analyze its performance ratio for the unweighted case. An LP-based approximation algorithm is presented for the general multicover problem. The algo-rithm can be applied to the weighted version of the order scheduling problem with a common due date.

Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Order scheduling; Number of late jobs; Approximability; Approximation algorithm; Multicover problem

1. Introduction

Order management is one of the crucial issues in the manufacturing industry. In this study, we con-sider order scheduling to minimize the number of late orders. For consistency with the scheduling literature we use jobs instead of orders hereafter. Consider a set of jobs N = {J1, J2, . . . , Jn} available from time zero

onwards for processing. Each job Ji2 N consists of

m operations Oik, 1 6 k 6 m, to be processed on m

independent dedicated machines M1, M2, . . . , Mm

for which operation Oik,1 6 k 6 m, can be processed

on machine Mkonly and has a processing time pik.

The m operations of a job are independent and there-fore can be processed simultaneously by their specific machines. At any time, each machine can process at most one operation. No preemption is allowed. The completion time of operation Oikon machine Mkis

denoted by Cik. A job is completed only if its

opera-tions are all finished. Therefore, the completion time of job Jiis defined as Ci= max16k6m{Cik}. Each job

Ji2 N is associated with a due date dithat specifies

the time it is expected to be completed. Binary vari-able Ui dictates whether job Ji is late or not, i.e.

0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.10.021

q

This research is supported by Taiwan–Russia joint grant under contract numbers NSC94-2416-H-009-013 and RP05H01(05-06-90606-HHCa).

* _{Corresponding author. Tel.: +886 3 5131472; fax: +886 3}

5729915.

E-mail addresses: bmtlin@iim.nctu.edu.tw (B.M.T. Lin),

alvenko@math.nsc.ru(A.V. Kononov).

Ui= 1 if Ci> di; 0, otherwise. In this paper, we want

to schedule the jobs so as to minimize the number of late jobs. Following the three-field notation used in Leung et al. (2006), the studied problem is denoted by PDkPUi.

As indicated by Roemer (2006), the first paper concerning order scheduling may be due toAhmadi and Bagchi (1990). Order scheduling can also be regarded as a degenerate case of the two-stage assembly-type flowshop (Lee et al., 1993) by ignor-ing the second-stage assembly operation. Wagneur and Sriskandarajah (1993) later studied the order scheduling problem from a different aspect ‘‘open shop scheduling with job overlaps’’. They investi-gated several standard objectives, including make-span, total completion time, maximum lateness, total tardiness, and number of late jobs. Roemer (2006)summarized several different but independent lines of order scheduling research that have appeared in the literature. Because this note focuses on only the objective of number of late jobs PUi,

we do not review the known results on other objec-tives. The reader is referred toRoemer (2006)for a thorough classification. To minimize the number of late jobs, Wagneur and Sriskandarajah (1993) showed that the problem is NP-hard even if there are only two machines. Ahmadi and Bagchi (1997) and Cheng et al. (2006) independently developed a pseudo-polynomial time algorithm for the weighted case PDmkPwiUi, where m is a fixed number of

machines. Leung et al. (2006) showed that the PDkPUi problem is strongly NP-hard, even if a

common due date is assumed. They introduced a Revised Hodgson–Moore (RHM) algorithm to solve the case with agreeable conditions.Ng et al. (2003) presented a negative approximability result that the PDjdi¼ d; pik2 f0; 1gj

P_U

i problem has no

c Æ ln m-approximation algorithm for some constant c > 0, unless P = NP. Further, an LP-rounding algo-rithm with performance ratio d + 1 was designed for the weighted case PDjdi¼ d; pik2 f0; 1gj

P_w

iUi.

The rest of this study proceeds as follows. In Sec-tion2, we introduce a proof for the non-existence of fully polynomial time approximation schemes for the two-machine, common due date case. Section 3is dedicated to the development of a heuristic algo-rithm that exhibits a worst-case performance of m. In Section 4, we design and analyze an LP-based approximation algorithm for the multicover prob-lem. The result is applied to the common due date case of minimizing the weighted number of late jobs. Section 5gives some concluding remarks.

2. Non-approximability about FPTAS

In this section, we prove that there is no fully polynomial time approximation scheme (FPTAS) unless P = NP for the PDkPUiproblem even when

there are only two machines and a common due date.

Theorem 1. No FPTAS exists for PD2jdi¼ djPUi

unless P = NP.

Proof. To prove the theorem, we use the NP-hard Equal-Size-Partition problem [SP12] (Garey and Johnson, 1979): The input instance consists of integer W and 2t positive integers S = {x1, . . . , x2t} that sum

up to 2W. The problem is to determine if there is a subset of items S0 _{with jS}0_{j = t and} P

xi2S0xi¼

P

xi62S0xi¼ W . Let X = 1 + W. Consider the

follow-ing instance of the PD2jdi¼ djPUi problem with

2t jobs: For i = 1, . . . ,2t, job Ji is created with

pi1= xi+ X and pi2¼2Wt  xiþ xmax, where xmax¼

maxxi2Sfxig. The common due date is d = W + tX.

Suppose that PD2jdi¼ djPUi does permit the

existence of an FPTAS. We set ¼1

2t. Assume that

the FPTAS returns a schedule with at most t late jobs. So we have at least t jobs completed before time d. Note that for any t + 1 jobs, the sum of their processing times on machine one is greater than (t + 1)X > W + tX. It follows that there must be exactly t tardy jobs. Denote the set of non-tardy jobs by N0_{. With this definition, we must have}

P Ji2N0pi1 6_{d. It follows that} P Ji2N0ðxiþ X Þ 6 W þ tX . Therefore, X Ji2N0 xj6W : ð1Þ

Also, we have P_Ji2N0p_i26d. It follows that

P Ji2N0 2W t  xiþ X  ₆ W þ tX . Therefore, W 6 X Ji2N0 xi: ð2Þ

Inequality(1)and Inequality(2)together imply that P

Ji2N0xi¼ W . Consequently, the elements

corre-sponding to the jobs in set N0_{constitute a solution}

to Equal-Size-Partition.

Next we assume that the Equal-Size-Partition problem has a solution S0 _{with jS}0_{j = t and}

P

i2S0x_i¼ W . We schedule the jobs corresponding

to the elements in S0_{first. Then, we have t jobs that}

are early or on-time. By the choice of , the approximate objective value reported by the FPTAS must be at most (1 + )t < t + 1. Since the objective

function takes only integer values, the approximate objective value must be at most t

Therefore, we come up with the fact that the FPTAS of PD2jdi¼ djPUi would find in

polyno-mial time a solution with an objective value at most t if and only if Equal-Size-Partition has a solution. h

3. Approximation algorithm for the unweighted case In this section, we develop a lower bound and a heuristic with performance analysis. We start from the development of a lower bound of the general problem. An algorithm due to Hodgson and Moore finds an optimal solution of the single machine problem 1kPUiin O(n log n) time. We can

decom-pose the m-machine order scheduling problem to m independent single machine problems. Let Gkbe the

number of late jobs in the optimal solution of the single machine problem on machine Mk. It is clear

that Gk6OPT, where OPT stands for the optimal

solution value in the order scheduling problem. Thus we have the following lower bound by com-bining all these independent bounds: LB = maxk=1, . . . , m{Gk} 6 OPT. Note that the lower

bound can be obtained in O(mn log n) time. To generate a feasible schedule, it is sufficient to find a set S of jobs which are early or on-time. We will construct set S by the following steps. Add the jobs to S one by one in nondecreasing order of due dates. If job Jjis completed after djon some machine Mkwhen

added into S, we call Jja critical job. A job with the

largest processing time on this machine is marked to be late and removed from S. Let tkdenote the

cur-rent schedule time on machine Mkin the running

ses-sion of the algorithm. The algorithm is outlined as: Algorithm A

1. Re-index the jobs such that d16d26   6 dn.

2. S : =;; tk: = 0 for all k = 1, . . . ,m. 3. For j : = 1 to n do: S : = S[ {Jj}; For k : = 1 to m do: tk: = tk+ pjk; For k : = 1 to m do: If tk> djthen

Find job Jiin S with the largest pik;

S : = Sn{Ji};

For k : = 1 to m do: tk: = tk pik.

4. Stop.

Denote the number of late jobs in the schedule constructed byAlgorithm AbyPUiðAÞ.

Theorem 2. For the PDkPUi problem, the

perfor-mance ratio P

UiðAÞ

LB 6m.

Proof. Let Nkbe the set of jobs that were removed

from S when the algorithm encountered due-date violation on machine Mk. Then PUiðAÞ ¼

Pm

k¼1jNkj. Consider machine Mkthat has the largest

value of jNkj, i.e. jNkj = max1 6 j 6 m{jNjj}. Let

N0= S[ Nk. On machine Mk, consider the classical

1kPUiproblem on the set of jobs N0with

process-ing times pjkof jobs Jj2 N0. Let G0k be the optimal

solution value of the set N0_{. Since N}0_{ N, we know}

that G0_k6_{Gk, the optimal value of 1k}P_{Uion set N.} But Algorithm A on set N0 _{constructs the same}

schedule as Hodgson–Moore algorithm, i.e our algorithm produces an optimal solution to the 1kPUi problem. It follows that PUiðAÞ ¼

Pm

k¼1jNkj 6 mG0k6mGk6mLB and that

Algo-rithm A is an m-approximation algorithm for PDmkPUi. h

To establish the tightness of the performance ratio m, we consider the following instance of m + 1 jobs. For jobs Ji,1 6 i 6 m, pik= 2 if k = i; 0,

otherwise. Job Jm+1 has pm+1,k= 1 on all k. All

jobs have a common due date d = 2. An optimal solution to the instance successfully schedules jobs J1,J2, . . . ,Jm. Algorithm A reports a schedule with

only job Jm+1 scheduled early. Therefore,

P

UiðAÞ

LB ¼ m.

The agreeable conditions specify that for any Ji,Jj2 N, either pik6pjk for all k,1 6 k 6 m or

pikP pjk for all k,1 6 k 6 m. The RHM algorihtm

proposed by Leung et al. (2006) works in the following way to optimally solve the agreeable case: If an order is scheduled and the maximum of its completion time over all m machines is larger than its due date, then the order in the partial schedule that has the longest processing time on each one of the machine is deleted. Although Algorithm A provides approximate solutions to the general case, it can solve the agreeable case to optimality. The RHM however cannot deal with the general case.

4. Approximation algorithm for the weighted case In this section, we consider the order scheduling problem of minimizing the weighted number of late jobs, PDkPwiUi. Now, we assume that job Ji has

weight wiP0. Let Pk ¼

Pn

m, and Pmax= max1 6 k 6 m{Pk}. As noted in Ng

et al. (2003) the weighted version with common due date d can be formulated as the following inte-ger linear program:

ILPPDMinimize Xn i¼1 wixi ð3Þ subject to Xn i¼1 p_ikxiP Pk d; 1 6 k 6 m; ð4Þ xi2 f0; 1g; 1 6 i 6 n: ð5Þ

Here, job Jiis late if xi= 1.

Note that ILPPD (3)–(5) is a special case of the

multicover problem (Hochbaum, 1997). ILPMCMinimize Xn i¼1 wixi ð6Þ subject to Xn i¼1 p_ikxiP bk; 1 6 k 6 m; ð7Þ xi2 f0; 1g; 1 6 i 6 n: ð8Þ

For the multicover problem, three approxima-tion algorithms were proposed: LP-algorithm, rounding algorithm (Hochbaum, 1997), and dual-feasible algorithm (Hall and Hochbaum, 1986). The three approaches are all Pmax-approximate

algorithms, but the heuristic solution value of the dual-feasible algorithm does not exceed maxk¼1; ... ;mPJi2Nlatepik times the value of optimum,

where Nlateis the set of late jobs. For instances with

large due dates, which correspond to small values of bi, i = 1, . . . ,n, this quantity could be much smaller

than Pmax.

Another approach is considered by Ng et al. (2003) for the case of common due date and 0–1 processing times, PDjdi¼ d; pik 2 f0; 1gj

P wiUi.

They presented a d + 1-approximate algorithm where d is the common due date. In the following, we show that a similar algorithm can be used for the multicover problem and thereby for the problem PDjdi¼ djPwiUi. Let LP be a linear program

obtained from ILP (6)–(8) by relaxing the integer constraints (8), i.e. 0 6 xik61 for i = 1, . . . ,n.

Denote the value of optimal solution of LP by LBLP. It is obvious that LBLP is a lower bound

of the multicover problem. Let dk= Pk bk,

1 6 k 6 m and dmax= maxk=1, . . . ,m{dk}. The

follow-ing LP-based algorithm can provide new approxi-mation results for the multicover problem.

Algorithm LP:

1. Solve the LP and obtain an optimal solution x_{¼ ðx}

1; x2; :::; xnÞ.

2. Output a subsetfijx

i P1=ðdmaxþ 1Þg.

Theorem 3. For the multicover problem, Algorithm LP has a performance ratio

P_w

iUiðLPÞ

LBLP

6_{dmaxþ 1.}

Proof. It is clear that after the rounding step,Pwixi

is no greater than (dmax+ 1)LBLP. So we have to

prove that the derived solution is feasible for ILP (6)–(8). Let constraints (7) fail for some k, i.e. P ijx iP 1 dmaxþ1 f gpik 6Pk dk 1. It follows that X ijx i< 1 dmax þ1 f g p_ik P dkþ 1: ð9Þ Then, we have Xn k¼1 p_ikx_i ¼ X ijx iPdmax þ11 f g p_ikx_i þ X ijx i<dmaxþ11 f g p_ikx_i < X ijx iPdmax þ11 f g p_ikþ 1 dmaxþ 1 X ijx i<dmaxþ11 f g p_ik ¼ Pk dmax dmaxþ 1 X ijx i<dmax þ11 f g p_ik:

Inequality(9)further implies Xn i¼1 p_ikx_i < Pk dmaxðdkþ 1Þ dmaxþ 1 6_{Pk dk:}

The last inequality follows from dk6dmax.

There-fore, we come to a contradiction to the assumption Pn

i¼1pikxi P Pk dk. h

Assume dk= d for all 1 6 k 6 m. The following

corollary readily follows.

Corrollary 1. For the PDjdi¼ djPwiUi problem,

Algorithm LP has a performance ratio P_w

iUiðLPÞ

LBLP

6 dþ 1.

The performance ratio is the same as that given in Ng et al. (2003) for PDjdi¼ d; pik 2

f0; 1gjPwiUi. However, our result is given for a

more general case PDjdi¼ djPwiUi, in which no

constraint is assumed for processing times. Note that in contrast to the dual-feasible algorithm, Algo-rithm LP will produce good approximate solutions for the case with small due dates.

5. Conclusion

This study has investigated the order scheduling problem of minimizing the (weighted) number of late jobs. A reduction from Equal-Size-Partition was con-ducted to establish the fact that there cannot exist a fully polynomial time approximation scheme unless P = NP. A heuristic algorithm with performance ratio m was designed. We have also examined the gen-eral multicover problem and proposed a new approx-imation algorithm. The results in the mean time lead to a (d + 1)-approximation algorithm for the order scheduling problem with a common due date.

As aforementioned, order scheduling problems are not yet extensively studied. There is considerable room for further research involving such constraints as release dates or precedence relationships. Acknowledgements

The authors are grateful to the anonymous refer-ees for indicating typos and providing information on some papers that have addressed the studied problem.

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