Two-machine flow-shop scheduling to minimize total late work

(1)

This article was downloaded by: [National Chiao Tung University 國立交通大學] On: 26 April 2014, At: 02:16

Publisher: Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Engineering Optimization

Publication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/geno20

Two-machine flow-shop scheduling to

minimize total late work

B. M. T. Lin a , F. C. Lin b & R. C. T. Lee c

a

Department of Information and Finance Management , Institute of Information Management, National Chiao Tung University , Taiwan, 300, R.O.C

b

Department of Computer Science and Information Engineering , National Chi Nan University , Taiwan, 545, R.O.C

c

Department of Information Management , National Chi Nan University , Taiwan, 545, R.O.C

Published online: 25 Jan 2007.

To cite this article: B. M. T. Lin , F. C. Lin & R. C. T. Lee (2006) Two-machine flow-shop scheduling to minimize total late work, Engineering Optimization, 38:04, 501-509, DOI: 10.1080/03052150500420439

To link to this article: http://dx.doi.org/10.1080/03052150500420439

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

(2)

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

(3)

Engineering Optimization

Vol. 38, No. 4, June 2006, 501–509

Two-machine flow-shop scheduling to minimize

total late work

B. M. T. LIN*†, F. C. LIN‡ and R. C. T. LEE§

†Department of Information and Finance Management, Institute of Information Management, National Chiao Tung University, Taiwan 300, R.O.C.

‡Department of Computer Science and Information Engineering, National Chi Nan University, Taiwan 545, R.O.C.

§Department of Information Management, National Chi Nan University, Taiwan 545, R.O.C.

(Received 31 January 2005; revised 8 April 2005; in final form 28 April 2005)

This article considers a two-machine flow-shop scheduling problem of minimizing total late work. Unlike tardiness, which is based upon the difference between the job completion time and the due date, the late work of a job is defined as the amount of work not completed by its due date. This article first shows that the problem remains non-deterministic polynomial time (NP) hard even if all jobs share a common due date. A lower bound and a dominance property are developed to design branch-and-bound algorithms. Computational experiments are conducted to assess the performance of the proposed algorithms. Numerical results demonstrate that the lower bound and dominance rule can help to reduce the computational efforts required by exploring the enumeration tree. The average deviation between the solution found by tabu search and the proposed lower bound is less than 3%, suggesting that the proposed lower bound is close to the optimal solution.

Keywords: Late work; Flow shop; Complexity; Branch-and-bound algorithm

1. Introduction

Together with the rapid growth of applications of scheduling theory, new manufacturing models as well as measures have been proposed and studied in the open literature (Pinedo and Chao 1999). This article studies a scheduling problem of minimizing a new penalty measure, namely the total penalty for unfinished parts of the jobs. Consider a set of jobs N = {1, 2, . . . , n} available from time zero onwards for processing in a two-machine flow shop, which was first motivated by the study by Johnson (1954). Each job must be first processed on machine one and then transferred to machine two for second-stage processing. Operations on both machines are processed in the same sequence; i.e. only permutation schedules are considered. The processing time of job i on machine k (k= 1 or 2) is denoted by pi,k. A specific due

date di is assigned to each individual job i to define the time before which it is expected to

*Corresponding author. Email: [email protected]

Engineering Optimization

DOI: 10.1080/03052150500420439

(4)

502 B. M. T. Lin et al.

be completed. If a job cannot be completely finished by its due date, then it is called late and a penalty proportional to the amount of remaining unfinished part will be incurred. Let Ci,1and Ci,2be the completion times of operation one and operation two of job i on the two

machines respectively. The late work of job i is defined by Yi = Yi,1+ Yi,2 = min{max{Ci,1− di,0}, pi,1} + min{max{Ci,2− di,0}, pi,2}. If the two operations of a job are completed by its

due date, then no penalty is incurred; otherwise, it will be penalized in proportion to the amount of unfinished parts of operation one and operation two. The goal of the studied problem seeks to construct a job sequence such that the sum of late work penalties over all jobs is minimum. Following the standard three-field notation (Graham et al. 1979), this article uses F2||Yito

denote the studied problem, where F2 dictates the two-machine flow shop and the third field specifies the objective function to minimize.

The late-work concept was first introduced by Blazewicz (1984) in a parallel-machine production environment. The total late-work measure has practical significance in such real-world applications as data collection (Blazewicz 1984, Blazewicz and Finke 1987) and corn harvesting (Potts and Van Wassenhove 1992a, 1992b). The study was later extended to single-machine (Hariri et al. 1995, Kethley and Alidaee 2002, Potts and Van Wassenhove 1992a, 1992b), flow-shop (Blazewicz et al. 2005) and open-shop environments (Blazewicz et al. 2004). The known results concerning total late-work criteria are summarized in table 1. The research studies that are most relevant to this study are by Blazewicz et al. (2005), Hariri et al. (1995), Lin and Hsu (2005) and Potts and Van Wassenhove (1992a). Hariri et al. (1995) con-sidered the single-machine environment with the objective of minimizing total weighted late work. They presented a proof to show that 1||wiYi is non-deterministic polynomial time

(NP) hard and developed a pseudo-polynomial time dynamic programming algorithm. Lin and Hsu (2005) considered 1|ri|

Yi, in which release dates are introduced. They showed that

1|ri, di = d|

Yi and 1|ri, pmt n|

Yi are polynomially solvable and developed a

branch-and-bound algorithm for the general case. Study on late-work criteria in flow shops first appeared in an article by Blazewicz et al. (2005) so as to minimize weighted late work. A reduction from PARTITIONwas given to establish the NP-hardness of F2|di = d|

wiYi.

They also developed a pseudo-polynomial time dynamic program. This article shows that the unweighted case F2|di = d|

Yi remains NP-hard and uses the results obtained by

Hariri et al. (1995) and Lin and Hsu (2005) to develop a branch-and-bound algorithm for F2||Yi.

Table 1. Known results about the total late-work criterion.

Problem Complexity Source

1Yi NP-hard Potts and Wassenhove (1992a)

1|pmtn|Yi O(n log n) Potts and Wassenhove (1992a)

1|di= d|Yi O(n) Potts and Wassenhove (1992a)

1|pi= p|Yi O(n log n) Potts and Wassenhove (1992a)

1|pmtn|wiYi O(n log n) Hariri et al. (1995)

1|di= d|wiYi O(n) Hariri et al. (1995)

1|pi= p|wiYi O(n3) Hariri et al. (1995)

1|ri, di= d|Yi O(n log n) Lin and Hsu (2005)

1|ri, pmtn|Yi O(n log n) Lin and Hsu (2005)

P|ri|wiYi NP-hard Blazewicz (1984)

P|ri, pmtn|wiYi O(n7log n) Blazewicz and Finke (1987)

Pk|ri, pmtn|wiYi O(k3n7log kn) Blazewicz and Finke (1987)

F2|di= d|wiYi NP-hard Blazewicz et al. (2005)

O|ri, pmtn|wiYi O(n3) Blazewicz et al. (2004)

O2|di= d|Yi O(n) Blazewicz et al. (2004)

O2|di= d|wiYi NP-hard Blazewicz et al. (2004)

(5)

Two-machine flow-shop scheduling 503

The rest of this article is organized as follows. Section 2 is dedicated to the NP-hardness of a special case where a common due date is assumed. Section 3 introduces a lower bound and a dominance property to develop branch-and-bound algorithms. In section 4, computational experiments are conducted to assess the effectiveness of the proposed properties. Finally, some concluding remarks are given in section 5.

2. Non-deterministic polynomial time hardness result

This section mainly includes the proof that the studied problem remains NP-hard even if there is only one due date common to all jobs. Denote this special case by F2|di = d|

Yi.As

mentioned above, the weighted case F2|di = d|

wiYi problem was shown to be NP-hard

by Blazewicz et al. (2005). The reduction in this article is also based upon PARTITION, which is NP-complete in the ordinary sense (Garey and Johnson 1979).

PARTITION: Given an integer B and a set of A of s positive integers{x1, x2, . . . , xs} such

thats_i₌₁ xi = 2B, does there exist a partition A1and A2 of set A such that

xi∈A1 xi=

xi∈A2 xi = B?

THEOREM1 The decision version of the F2|di= d|

Yiproblem is NP-complete.

Proof It is clear that the decision counterpart of F2|di = d|

Yibelongs to NP. Given an

instance of PARTITION, an instance of the F2|di = d|

Yi problem consisting of n= s + 1

jobs is constructed as follows:

ordinary jobs: pi,1= xiω, pi,2= xi, 1 i s, ω is an integer greater than B, say

ω= B + 1;

enforcer job: ps+1,1= 0, ps+1,2= ωB;

common due date d= ωB + B.

The following will establish the claim that there is a specified partition for PARTITIONif and

only if there is a schedule for F2|di= d|

Yiwhose total late work is exactly ωB.

⇒) Let subsets A1 and A2 be a partition as specified for set A. Schedule enforcer job s+ 1 as the first job in the flow shop. The jobs corresponding to the elements of A1 then

follow in arbitrary order. The remaining jobs are scheduled arbitrarily at the rear part of the schedule. See figure 1 for an illustration. Note that the late machine-two operations can be postponed and processed consecutively at the rear part of the schedule without sacrificing the solution quality. In the figure, each late machine-two operation is processed immediately on completion of its corresponding machine-one operation so as to emphasize the sequencing order. It is not difficult to verify that the enforcer job and the jobs corresponding to the elements of A1 are all early. Moreover, the first of the remaining jobs has an early part of length B

Figure 1. Gantt chart of the schedule corresponding to the specified partition.

(6)

on machine-one. Therefore, the total late work of the remaining jobs on both machines is

xi∈A2 xiω− B +

xi∈A2 xi = ωB − B + B = ωB.

⇐) Suppose that there is a schedule for F2|di = d|

Yiwhose total late work is no more

than B. Note that the total processing length of all jobs on the two machines is 2ωB+ ωB + 2B. Because d = ωB + B, to ensure that the total late work is no greater than B, on both machines the time span before the due date must be fully filled; i.e. no idle time is allowed on either machine before the due date. This observation leads to the fact that the enforcer job must be scheduled first. Let N1be the set of ordinary jobs that are completely early on

machine-one. First the situation when_i_∈N₁ xi is strictly smaller than B is considered. In

this case, the completion time on machine-two of the last job in N1 is strictly earlier than ωB+ B. Because on machine-one the processing of the immediate successor following N₁is not completed by or at the due date d= ωB + B, a non-zero idle time will be introduced into its machine-two processing before the due date. As a sequel,_i_∈N₁ xi B must hold. On the

other hand, ifi_∈N1 xi > B, then

i_∈N1 xi B + 1. Therefore, on machine-one the largest

completion time of the jobs in N1 is

i_∈N1 pi,1=

i_∈N1 ωxi ωB + ω, which is greater

than d = ωB + B. It is a contraction to the assumption that all jobs of N1 are completely

early on machine-one. Therefore,i∈N1 ximust equal B exactly, and thus a desired partition

is attained. From the above reduction, the proof is readily complete.

The result implies that it is very unlikely to design polynomial time algorithms for producing optimal solutions to F2|di= d|

Yi. Because Blacewicz et al. (2005) have

proposed a pseudo-polynomial algorithm for the weighted version F2|di = d|

wiYi,the

F2|di = d|

Yi problem is ordinary NP-hard. In the next section, some useful properties

will be developed and deployed to reduce the solution-finding time.

3. Branch-and-bound algorithm

This section is dedicated to the development of a branch-and-bound algorithm, which is one of the most commonly adopted approaches to tackling hard combinatorial optimization problems. The efficiency of a branch-and-bound algorithm mainly depends on such structural properties as lower bounds and dominance rules, which can help to trim off unnecessary branches in the enumeration tree that corresponds to the solution space. In most of the objective functions for flow-shop scheduling problems, the objective value is dependent on the completion of machine-two operations. For the total late-work criterion, it is, however, necessary to consider the performance measure on both machines. Therefore, it is viable to find a lower bound for the sequence of operations on each machine and then to combine two lower bounds into an aggregate lower bound for the original problem. In addition to lower bounds, a dominance rule will also be used to reduce the time required by exploring the solution space.

To develop a lower bound, the machine-two operations are ignored and only the operations on the first machine are considered. The problem then becomes the single-machine problem 1||Yi,using{pi,1, pi,2, . . . , pi,n}.Although the single-machine 1||

Yiproblem is still

NP-hard, Potts and Wassenhove (1992a) proposed a pseudo-polynomial dynamic programming algorithm to solve it optimally. Therefore, given a partial schedule σ for the original problem, the machine-one operations of the unscheduled jobs can be collected and the single-machine case is solved by appealing to this dynamic program.

The discussion now proceeds to when the variant involves only the machine-two operations. Given a partial schedule σ of the original F2||Yi problem, let t1(σ ) and t2(σ )be the

completion times of the last job scheduled in σ on machine-one and machine-two respectively. Note that any machine-two operation of job i∈ N\{σ } cannot start its processing earlier than

(7)

max{t1(σ )+ pi,1, t2(σ )}. Following this line of reasoning, max{t1(σ )+ pi,1, t2(σ )} may be

treated as the release time for machine-two operation of unscheduled job i. As a result, this becomes the single-machine schedule problem of the machine-two operations of the remaining unscheduled jobs under release-date constraints, 1|ri|

Yi. Similarly, this problem remains

NP-hard. Lin and Hsu (2005) presented an O(n log n) algorithm for the case 1|pmtn, ri|

Yi,

where preemption is allowed. The optimal solution value of 1|pmtn, ri|

Yiis a lower bound

on the optimal solution value 1|ri|

Yi; i.e. there is a lower bound for the problem involving

only the machine-two operations. In an optimal solution to the original F2||Yiproblem, the

total late work contributed by the machine-one operations cannot be smaller than the solution derived using the dynamic program, and similarly the total late work contributed by the machine-two operations cannot be smaller than the solution derived by using the preemption relaxation. Therefore, by combining these two values, an aggregate lower bound on the optimal solution value of F2||Yican be obtained.

To cut off unnecessary branches further, the relationship between any two jobs that are arranged consecutively is examined. For partial schedule σ and two unscheduled jobs i and j, we consider two (partial) schedules σ= σij and σ= σji. If

(a) Cj,2(σ) Ci,2(σ)and

(b) Yi(σ)+ Yj(σ) Yj(σ)+ Yi(σ),

then

(i) σand σhave the same completion time on machine-one,

(ii) on machine-two, the completion time of σis no later than that of σand (iii) the total late work of σis no greater than that of σ.

The three consequence parts jointly imply that the best solution value fulfilled from σ can-not be smaller than that fulfilled from σ; i.e. the branch corresponding to partial schedule σ is fathomed. If conditions (a) and (b) are both satisfied with equalities, then in implementation it is possible that σand σwill trim off each other. Therefore, another condition

(c) i < j is added.

The discussion is summarized in the following property.

Dominance property: Let σ= σij and σ= σji be defined from the partial schedule σ and two unscheduled jobs i and j. If

(a) Cj,2(σ) Ci,2(σ),

(b) Yi(σ)+ Yj(σ) Yj(σ)+ Yi(σ)and

(c) i < j

are satisfied, then the subtree rooted at σcan be trimmed off without further exploration.

4. Computational evaluation

To test the performance of the properties when they are incorporated into the design of branch-and-bound algorithms, a series of computational experiments was performed. The experiments consist of two parts. For small-scale problems (n 30), the experiments are aimed at studying the actual improvement that the lower bound and dominance property can make. For medium-scale problems (n 100), the focus is mainly on the gap between the lower bound and

(8)

the solution produced by a tabu search algorithm. The gap is defined as[(TS − LB)/LB] × 100%, where TS denotes the solution value produced by tabu search and LB is the lower bound.

The algorithms were implemented in C++ and a personal computer with an Intel Pen-tium 4, 1.7 GHz central processing unit and 256 MB random-access memory was used as the platform. Following the convention adopted in the generation of test data for flow shops, the processing times pi,kwere randomly generated from a discrete uniform distribution over

[1, 10]. Moreover, to allow potential discrepancy among jobs, another set of test cases were generated with processing times over a wider interval [1, 100]. To generate due dates with different levels of tightness, first the total processing length pi,1+ pi,2of each job i is

com-puted, and then the aggregate processing lengths are sorted in non-decreasing order. Denote the ith-largest aggregate length by p_i. The due date of the job corresponding to p_iwas ran-domly generated from the interval (p_i, p_i+i_k₌₁ p_n_−k+1/β], where β is a parameter used to control the tightness of the due dates. When the value of β is large, the due dates are tight and most of the jobs will be late. On the other hand, if the value of β is small, most of the jobs will be early. To avoid these two extreme cases, preliminary tests were conducted and the value of β set to 3, 5 or 7. Moreover, a limit of 1800 s was used as a threshold for the execution of branch-and-bound algorithms; i.e., if the algorithm cannot finish exploring the solution space, it aborts with a failure. The limit was chosen based on the fact that the maximum value of the system variable used to store the clock ticks in Linux is equivalent to 1800 s. To avoid problems caused by resetting the ticks counter, this limit was adopted. For each problem size nand degree of tightness, β, ten sets of jobs were generated.

Table 2 shows the numerical results for the branch-and-bound algorithm. Columns LB correspond to the algorithm equipped with the lower bound only. LB_Dom indicates the incorporation of both the lower bound and the dominance rule. Three types of information were kept as follows: Avg_Node, average number of nodes explored for the instances that have been successfully solved; Avg_Time, average execution time required by the successfully solved instances; #Opt, number of instances that have been successfully solved. Exploring a search tree of size n! using crude enumeration usually can solve instances with 12 or 13 jobs. With the lower bound and the dominance rule, the algorithm can solve most instances with up to 30 jobs. The statistics first show that incorporating two properties simultaneously can provide better performances with respect to the number of visited nodes and the elapsed execution times. It is also interesting to see that the application of the dominance property does not incur too much computational load at each node. For example, consider the scenarios with n= 25 and β = 5 for LB and LB_Dom. The ratio of the number of nodes explored by LB_Dom to that explored by LB is 3 611 820/11 763 800= 0.307. Similarly, the ratio of their execution times is 92.086/288.442= 0.319. The statistics also suggest that the performance of the algorithm is better when the job-processing times are generated from the shorter interval [1, 10]. A second observation is made on the effects of due date tightness. When the due dates are tighter, i.e. β is larger, the algorithm will visit more nodes and thus take a longer execution time.

Table 3 summarizes the numerical results of the second part of the experiments. It appears that different interval lengths from which processing times were generated do not have sig-nificant impact on the gaps between the approximate solutions provided by tabu search and the lower bound values. However, the tightness of due dates still plays a significant role. Gaps become larger when the given due dates are relatively larger. As a general observation, it may be said that the average gaps are less than 3% between the 240 test cases. This observation in the mean time demonstrates the tightness of our proposed lower bound.

(9)

T wo-mac hine flow-shop sc heduling 507

Table 2. Numerical results for the branch-and-bound algorithm.

pi∈ [1, 10] pi∈ [1, 100]

LB LB−Dom LB LB−Dom

n β Avg₋Node Avg₋time #Opt Avg₋Node Avg₋time #Opt Avg₋Node Avg₋time #Opt Avg₋Node Avg₋time #Opt

10 3 477 0.006 10 381 0.000 10 452 0.008 10 355 0.006 10 5 1 222 0.008 10 798 0.006 10 1 722 0.033 10 1 069 0.019 10 7 2 454 0.019 10 1 340 0.010 10 2 484 0.051 10 1 446 0.031 10 12 3 903 0.010 10 734 0.005 10 551 0.060 10 1 653 0.042 10 5 2 128 0.039 10 1 331 0.014 10 5 651 0.134 10 3 096 0.081 10 7 8 503 0.098 10 2 888 0.041 10 4 482 0.169 10 2 804 0.111 10 14 3 3 713 0.044 10 2 451 0.031 10 11 409 0.336 10 5 532 0.173 10 5 6 237 0.077 10 4 080 0.052 10 7 360 0.282 10 4 377 0.166 10 7 13 273 0.184 10 5 838 0.086 10 33 149 1.358 10 15 596 0.700 10 16 3 6 451 0.086 10 3 514 0.053 10 29 511 1.309 10 14 012 0.648 10 5 27 344 0.366 10 11 112 0.158 10 30 544 1.269 10 15 268 0.670 10 7 68 624 0.952 10 25 181 0.389 10 773 582 31.578 10 147 228 6.981 10 18 3 20 912 0.342 10 10 755 0.172 10 121 519 5.326 10 57 280 2.622 10 5 136 351 2.552 10 49 093 0.925 10 68 255 3.197 10 26 674 1.338 10 7 1 910 370 32.000 10 309 733 5.586 10 1 162 890 61.764 10 304 443 17.438 10 20 3 114 870 2.231 10 45 899 0.878 10 81 759 3.938 10 35 743 1.878 10 5 160 031 3.152 10 69 007 1.411 10 918 015 47.996 10 205 962 11.614 10 7 1 972 320 41.322 10 407 924 9.053 10 4 680 728 315.445 10 870 196 60.777 10 25 3 3 699 560 101.491 10 673 347 18.405 10 5 744 130 220.252 10 1 349 170 51.864 10 5 11 763 800 288.442 9 3 611 820 92.086 10 6 013 670 246.296 7 3 281 320 131.233 10 7 7 506 770 208.495 10 1 016 960 32.263 10 1 312 040 105.806 3 5 980 120 403.266 9 30 3 4 256 164 132.816 9 3 744 051 115.826 10 7 585 302 724.504 4 6 618 136 594.965 9 5 18 090 878 583.491 5 13 894 589 432.300 9 11 451 931 1034.640 1 2 753 549 264.187 1 7 12 852 938 563.106 4 10 292 675 385.578 4 – – – – – –

(10)

Table 3. Average gap between the tabu search solution and the lower bound.

n β pi∈ [1, 10](%) pi∈ [1, 100](%) 40 3 2.069 2.162 5 2.540 2.356 7 3.036 3.496 50 3 1.374 1.492 5 2.292 2.598 7 3.600 3.106 70 3 0.857 1.065 5 1.240 1.654 7 2.729 3.092 100 3 0.978 0.881 5 0.994 1.300 7 1.702 1.707 5. Concluding remarks

This article has considered the minimization of total late work in a two-machine flow shop. A lower bound and a dominance property have been proposed. Branch-and-bound algorithms incorporating these two properties were tested through a computational study. Statistics have showed the significance of the lower bound and the dominance rule in reducing the computa-tional efforts required by exploring the enumeration tree. Comparisons between lower bound values and approximate solution values also suggest that the proposed lower bound is fairly close to the optimal solution.

As mentioned above, although the late-work criterion has practical implications as well as theoretical challenges, it has not been extensively studied in the literature. There could be many interesting problems to study. For example, it is of research interest to consider such relationship as precedence constraints or release dates.

Acknowledgement

The authors are partially supported by the National Science Council of the R.O.C. under contracts NSC-93-2416-H-009-026 and NSC-93-2213-E-260-009.

References

Blazewicz, J., Scheduling preemptible tasks on parallel processors with information loss. Technique Sci. Inf., 1984, 3(6), 415–420.

Blazewicz, J. and Finke, G., Minimizing mean weighted execution time loss on identical and uniform processors. Inf. Processing Lett., 1987, 24, 259–263.

Blazewicz, J., Pesch, E., Sterna, M. and Werner, F., Open shop scheduling problems with late work criteria. Discrete Appl. Math., 2004, 134, 1–24.

Blazewicz, J., Pesch, E., Sterna, M. and Werner, F., The two-machine flow-shop problem with weighted late work criterion and common due date. Eur. J. Operational Res., 2005, 165(2), 408–415.

Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-completeness, 1979 (Freeman: San Francisco, CA).

Graham, R.L., Lawler, E.L., Lenstra, J.K. and Rinnoy Kan, A.H.G., Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math., 1979, 5, 287–326.

Hariri, A.M.A., Potts, C.N. and Wassenhove, L.N., Single machine scheduling to minimize total weighted late work. ORSA J. Comput., 1995, 7, 232–242.

Johnson, S.M., Optimal two- and three-stage production schedules with setup times included. Nav. Res. Logistics Q., 1954, 1(1), 61–68.

Kethley, R.B. and Alidaee, B., Single machine scheduling to minimize total weighted late work: a comparison of scheduling rules and search algorithms. Comput. Ind. Engng., 2002, 43, 509–528.

(11)

Lin, B.M.T. and Hsu, S.W., Minimizing total late work on a single machine with release and due dates, in 2005 SIAM Conference on Computational Science and Engineering, Orlando, Florida, USA, February 2005.

Pinedo, M. and Chao, X., Operations Scheduling with Applications in Manufacturing and Services, 1999 (McGraw-Hill: Singapore).

Potts, C.N. and Van Wassenhove, L.N., Single machine scheduling to minimize total late work. Operations Res., 1992a, 40, 586–595.

Potts, C.N. and Van Wassenhove, L.N., Approximation algorithms for scheduling a single machine to minimize total late work. Operations Res. Lett., 1992b, 11, 261–266.