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Earliness-tardiness scheduling problems with a common delivery window
ELSEVIER
Operations Research Letters 15 (1994) 195-203
Earliness-tardiness scheduling problems with a common delivery window S u r y a D . L i m a n *'a, S a n j a y R a m a s w a m y
b
aDepartment of Industrial Engineering, Texas Teeh University Lubbock, TX 79409 USA b Department qf Industrial and Management Systems, Penn State University, Unit,ersity Park, PA 16802 USA
(Received 11 January 1993; revised 1 December 1993)
Abstract This paper deals with the scheduling of n jobs on a single machine to minimize the sum of weighted earliness and weighted number of tardy jobs given a delivery window. Penalties are not incurred if jobs are completed within the delivery window. The length of this delivery window (which corresponds to the time period within which the customer is willing to take deliveries) is a given constant. We consider two cases of the problem; one where the position of the delivery window is internally determined (unrestricted window case) and the other where the position is an externally specified parameter (restricted window case). We present some optimal properties, prove that the problem (even for the unrestricted window case) is NP-complete, and present dynamic programming algorithms for both cases. K e y words." Scheduling; Earliness-tardiness penalty; Delivery window; Dynamic programming
1. Introduction The problem considered is closely related to a variety of practical problems in scheduling. There are n jobs to be processed on a single machine. All jobs are assumed to have originated from a single customer and hence they have a c o m m o n delivery window. The length of the window is prespecified by the customer. Typically, this corresponds to the time frame during which the customer is most willing to take delivery of the jobs, and thus we have an earliest and a latest due date. Jobs that are completed within the delivery window incur no costs. However, if a job is completed before the earliest due date, it has to be held in inventory until such time the customer is ready to receive the job (i.e., the earliest due date). Consequently, this job incurs an earliness penalty that depends on its value and the length of time it is held in inventory. This earliness cost can be likened to such costs as capital, insurance, and deterioration costs. On the other hand, if a job is completed after the latest due date, it incurs a fixed tardiness cost that is independent of how tardy the job is. We can draw a parallel here to a situation involving the production of seasonable goods, where off-season
* Corresponding author. 0167-6377/94/07.00 (c~ 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 3 7 7 ( 9 4 ) 0 0 0 0 8 - 7
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S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
deliveries, while are still accepted by the customer, have to be sold at a discounted price. The objective is to minimize the total weighted earliness and weighted number of tardy jobs penalties incurred. This objective function is a nonregular measure of performance, as opposed to regular measures which are nondecreasing functions of job completion times [6], and is also in line with current developments in just-in-time (JIT) management [17]. With the recent increase in awareness of JIT concept, there has been a rapid growth of literature in the area of earliness-tardiness scheduling (the reader is referred to [18, 3] for comprehensive reviews of works in this area). Hall and Posner [8] examined the symmetric weights problem (where the per unit time penalties of earliness and tardiness are equal) and proved it to be NP-complete in the ordinary sense (see [7] for the theory of NP-completeness). The case of nonsymmetric weights, where all earliness as well as tardiness weights are job independent, was investigated by Bagchi et al. [2], among others. Lee et al. [14] discussed the more general case of agreeable earliness and tardiness weights with weighted number of tardy jobs. In their paper they provided pseudo-polynomial dynamic programming algorithms for both the unrestricted (due date is internally determined) and restricted (externally specified due date) cases of the problem. The restricted due date case is already NP-complete [9] even when we have unit weights. For this problem, Liman and Lee [15] proposed a heuristic which was proven to have a tight worst case relative error bound of 0.5. Panwalkar et al. [16] investigated the single machine problem of minimizing the (constant weighted) sum of due date, earliness, and tardiness penalties. They proposed an O(n log n) algorithm that solved the problem using positional weights concept. Cheng and Kahlbacher ['5] studied a similar problem where instead of minimizing the weighted tardiness penalty, they minimized the weighted number of tardy jobs. An O(n log n) algorithm was also proposed. One of the first papers to discuss the due date window problem was by Anger et al. [1], who considered the problem of minimizing the number of jobs finishing outside the window. Cheng [4] discussed the common due date problem with the objective of minimizing the total earliness-tardiness penalty, where a job finishing within a prespecified tolerance from the due date did not incur any penalty. The tolerance, however, was assumed to be relatively small compared to the job processing times (such that at most one job can avoid the penalty). Moreover, the earliness (or tardiness) of a job was calculated from the due date and not from the ends of the tolerance interval. Kr~imer and Lee [12] investigated the due window problem to minimize the total earliness and total tardiness costs with the penalties calculated from the end points of the window. In the paper, they studied two cases of the problem: one with the window position as a decision variable and one with the position as a given parameter. For the first case, they proposed a polynomial time algorithm for the constant weights problem and pseudo-polynomial algorithm for the more general agreeable weights problem. For the second case, they provided a pseudo-polynomial algorithm for the unweighted case. Hoogeveen [10] and Lee [13] had also studied scheduling with due date windows. Their bicriteria objectives were to find schedules that minimize some functions of maximum earliness and maximum lateness. They provided polynomial algorithms to solve special cases of the problem. In this paper, we consider the problem with an arbitrary due date or delivery window, per unit cost of earliness, and a fixed tardiness penalty. The motivation for such consideration stems from the many parallels that can be drawn between the theoretical problem under study and the practical applications in scheduling jobs for delivery within an externally specified due date window. We study both the restricted and unrestricted cases of the problem described. In the restricted case, the scheduler has the freedom to choose the delivery dates (although the window size is specified). A study of this case is useful in establishing the optimal properties of the problem and in providing insights into the basic problem structure. The more practically relevant restricted case of the problem (where the position of the due date window is an external parameter) is then examined and a solution is proposed. We do not explicitly consider the ordering of tardy jobs and note that such ordering may be used for a secondary criterion.
S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195 203
197
The remainder of the paper is organized as follows. In the next section, the problem definition and notations used are introduced. Optimal properties of the problem under study are presented next. The unrestricted window case is then studied. We provide a proof that this version of the problem is NP-complete and propose a dynamic programming algorithm for the same. Finally, the restricted version is discussed along with a dynamic programming algorithm to solve it.
2. Notations and problem definition The following notations are used throughout the paper: Ji dl d2 d Pi
= job i = common earliest delivery date for all jobs = common latest delivery date for all jobs = d2 - dl = delivery window size = processing time of Ji
M S - - ~ Pi i=1
Ci = :¢~ ~> 0 = ~,;/> 0 = E -W-T --
completion time of J~ earliness penalty of J~ per time unit early fixed penalty incurred if Ji is tardy set of all jobs finishing on or before dl (Ci ~ dl ) set of all jobs finishing between dl and d2(dl ~ Ci ~ d2) set of all tardy jobs (Ci > d2)
Ui = {10
ifJ~eT, otherwise.
The earliness-tardiness window problem ETW is defined by n jobs with known integer per unit earliness costs ~,:¢z . . . . . :t,, known integer per unit tardiness penalties 7~,72 ..... 7,, and an externally specified delivery window size d. Preemption is not allowed and all jobs are assumed to be available at time zero. There is one continuously available machine that can handle only one job at a time. The objective of ETW is to find a schedule that minimizes the following total penalty function: ~ [ ~ i m a x {0,dl - Ci} + 7iUi]. i=l
We will consider two cases of the problem ETW. In the first case, d~ (or d2) is a decision variable. We call this case the unrestricted (internally determined) window case. In the second case, d~ (and hence d2) is specified as a problem parameter. We call this case the restricted (externally determined) window case. We will now state some properties of optimal schedules for both the restricted and the unrestricted cases.
3. Optimal properties Property 1. There exists an optimal schedule to ETW where all jobs are processed without any inserted idle time.
Proof. Omitted due to simplicity.
Property 2. There exists an optimal solution to ETW where some job starts {or completes) at d 2.
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S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
Proof. Assume there exists an optimal schedule tr* where no job completes at d2. Let Ji be the job that starts before and completes after d2. Since Ji completes after d2, it will incur a fixed penalty cost 7i, irrespective of how much later than d2 it is completed. Therefore, ifJi is moved to the right and made to start exactly at time d2, the tardiness penalty of J~ will remain unchanged while the earliness penalties of jobs processed before Ji, if there are any, will decrease. Hence, we can find another schedule with the same total penalties (if there are no jobs before J~), or one with smaller total penalties (if there are jobs before Ji), by letting Ji start at time d2. Henceforth, we will only consider schedules where there exists a job that starts (or completes) at dE. Property 3. There exists an optimal schedule where jobs in set E are in nonincreasing order of pJaj. Proof. The proof follows from the well-known V-shaped property (see, e.g. [14]). Property 4. For a given set W and a given set T, the orderings of jobs within these sets are irrelevant as they do not affect the total penalties. Note that, from Property 2, the first job in set W starts on or after dl and the last job completes at d2 and the first job in set T starts at d2. The ordering can, however, be of use in optimizing a secondary criterion.
Proof. Omitted due to simplicity. In the sections that follow, we will present an NP-completeness proof and a dynamic programming algorithm for the unrestricted case followed by a dynamic programming algorithm for the restricted case. From now on, we will assume that jobs are numbered in nondecreasing order of p~/ej (i.e. Pl/~1 <<.... <<-P,/e,).
4. Unrestricted window problem In this section, we will study the unrestricted window case of ETW problem which we will denote by ETW U. In this case, we are given d and we can either assume that dl is very large (so as not to constrain the schedule) or that dl is a decision variable. We will first prove that ETW U is NP-complete by reducing the partition problem, which is known to be NP-complete, in polynomial time to the ETW U problem. 4.1. An NP-completeness proof for the unrestricted window case The partition problem (P) Given a finite set A of 2n integers a l, az .... , a2n, where each ai~ Z +, does there exist a subset A ' _ A such that ~.,~a' ai = Y~.,~A- .r ai? The corresponding earliness-tardiness window problem instance ETWI can be constructed as follows. Instance ETWI Number of jobs:
2n+ 1
Window size:
d=½E
Processing times:
Pi=ai
Earliness weights:
• ~=S
fori=l,2
O~2n+ 1 ~
$2
2n
S
a,=~
2n
w h e r e S = ~ a,
i=1
i=1
f o r i = 1,2,...,2n / 2 , "~2 P2n+ I = ~,,i~=lPi ) .... 2n
199
S.D. Liman. S. Ramaswamy / Operations Research Letters 15 (1994) 195 203
L
2d
.1.
set E
T'a"
[
d
.L.
[...I
set W
d~
[
~
d
...I
set T
A
I
d~ Fig. 1
Tardiness weights:
7i = a~ for i = 1,2 .... 2n ~2n + 1 ~ $ 2
Total earliness-tardiness penalty = y = d. The recognition version of the problem ETW v for the instance ETWI is Does there exist a nonpreemptive, single machine schedule with total earliness-tardiness penalty no larger than y? Lemma 1. I f there exists a partition o f set A such that ~a,~A ai = ~ a , ~ A - A ' a i = ly~2n ~ ai, then there exists a schedule a for ETWI with total penalty y. Proof. By assumption, the partition of set A exists. Define a set W of jobs corresponding to set A' (i.e., p~ = ag for all Ji E W and a~ cA') and a set T of jobs corresponding to set A - A' (i.e., Pi = a~ for all Ji E T and al~ d - A'). Therefore ~s,~w P~ = ~J,~r P~ = d. The delivery window d can therefore hold exactly jobs in set W, while other jobs are assigned to set T. Figure 1 shows the corresponding schedule. J2, +1 has a very large earliness and tardiness penalty and is therefore placed to the left of the delivery window to finish at time d~. The total penalty incurred is due to jobs in set T and is equal to ~s,~rTg = S/2 = d = y (since ;~ = pg = al). Lemma 2. I f there exists a solution to ETWI with total penalty <<.y, then the following must be true: 1. Jz,+~ must be processed so as to finish at dl. 2. ~ j,~w Pi = ~ J,~T Pl : d.
Proof: Assume J 2 n + l does not complete at dl. If J 2 n + l finishes before dl it will incur an earliness cost of S 2 > y. On the other hand, if it finishes beyond d2, a tardiness penalty of S 2 > y will be incurred. Therefore, J 2 n + l has to finish between dl and d2 (in set W). Now assume that J2,+ 1 completes after time d~ but before time d2. Again, there should not be any jobs in set E f o r otherwise the total penalty would exceed y. Therefore, all other jobs should be processed immediately after J2, +1. This implies that the total processing times of jobs after d2 are greater than d. Since the total tardiness penalty is at least equal to the sum of processing times of jobs in set T, the total penalty in this case will also exceed y. Hence, J2,+1 has to complete exactly at dl. To prove part two, we only need to consider the first 2n jobs and that the first of these jobs (the first job in set W) will start processing at time dl immediately after J2n+ 1. Assume that YJ,~wPi # ~ J ~ r P i (note that by definition, ~J,~w Pi < d and hence ~s,~r Pi > d). However, we can see that since the total tardiness penalty is at least equal to the sum of processing times of jobs in set T, the total penalty in this case will also exceed y. Hence ~ J , e w P i = ~J~eTPi = d. This implies that there exists a solution to (P). Theorem 1. Problem ETW U is NP-complete. Proof. From Lemmas I and 2.
S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
200
~=0
I QIDII set E
~" "T"
set W
d,
.a.. T
1
set T
,,A
set T
,d
"1
d2
~-- 6>0 --~ DOO
set E
~ T
set W
d,
-]
d2
•
set E
~x, T
ha" T
set W
d,
I
.x. T
""1
set T
I -1
d2 Fig. 2
4.2. A dynamic programming algorithm for the unrestricted window case In this section we will provide a dynamic programming algorithm to find the optimal solution to problem ETW v. From Property 2 we know that there exists an optimal schedule where some job finishes at time dE. Property 3 tells us that jobs in set E are processed in nonincreasing order of pi/cti, and from Property 4 we know that the ordering of jobs in set T and set W are of no consequence. While an optimal schedule has a job finishing at time dE, w e have two possibilities with regard to dl (see Fig. 2). 1. There exists Jk that either starts or finishes at time dl. 2. There exists Jk that straddles (completes before and finishes after) dl. To take all cases into account, we would first remove Jk (k -- 1.... n) from the set of n jobs and schedule it at all possible locations (starting with JR completing at time dl and ending with either Jk completing at time (dl + Pk) or (dl + d) whichever is smaller). In other words, if we let 6 = Ck -- dl, then 6 = 0 . . . . . min {pk,d}. The remaining (n - l)jobs would be scheduled around Jk. We would begin with J1 (the job with the smallest ratio of Pi/~i). We will now present the dynamic programming algorithm. Algorithm 1. Let fk(j, t b t 2 ) = the minimum total penalty if we have already scheduled the first j jobs, assuming that we have already removed Jk and have renumbered the remaining jobs from J x to J,_ l (still in nondecreasing order of p~/~j), t 1 is the difference between the start time of the first job in the sequence and dl, and t2 is the difference between the completion time of last job in the window and dl (see Fig. 3); k = 1,2,..., n;j = 0,1,..., n - l; tl = 0 , 1 . . . . . MS; t 2 = 0,1,...,d.
Boundary conditions t0. fk(O, tl, t2) ----- k - - O0
if t l = p k - - 6 otherwise
and t z = 6 where 6 = 0 ..... min {pk,d}t
f (j, tbt2)= +oo f o r k = l , . . . , n ; j = l k
"
..... n - 1 ; t x < O o r t 2 < O .
for k = 1, .-.,n,
(1)
201
S.D. Liman, S. Ramaswarny / Operations Research Letters 15 (1994) 195-203 L.
.L
[
4
"
set E
t2
• ..
J
l --fO
...1.-~
set W
d,
.J "1
••
t.. F
set T
d2 Fig. 3
Recursive relations f k ( j __ 1, q -- pj, t2) + C~j(tl -- pfl, fk(j, tDt2) = m i n i f k ( j
l,tt, t2 -- pfl,
I fk(j
1,Q,t2) + 72"
(2)
F o r k = 1,2 . . . . . n;j = 1,2 . . . . . n - 1; t~ = 0, 1. . . . . MS; te = 0, 1. . . . . d.
Optimal solution Z = min{fk(n -- 1, tl, re);
k = 1. . . . . n;
tl = 0, 1. . . . . MS;
t2 = 0, 1. . . . . d}.
(3)
Remark. Algorithm 1 is a p s e u d o - p o l y n o m i a l time algorithm with complexity O(n 2 {MS}d). Hence, problem E T W v is N P - c o m p l e t e in the ordinary sense only. Justifications of Algorithm 1. We assume that Jk(k = 1,..., n) is placed first before all other jobs. Correspondingly, fk(o, t l , t z ) = 0 only if tl = P k - 6 and t2 = 6 for 6 = 0 .... , min{pk, d}. Otherwise, the initial placement is infeasible and f k ( O , h , t 2 ) = +0o. Similarly, since neither t~ nor t2 can be negative, .fk(j,&,t2)= +0O if either t~ < 0 or t2 < 0 . Hence, the b o u n d a r y conditions are given by Eq. (1). T h e first equation in the recursive relations, given by Eq. (2), is for placing Jj in set E. The second and the third are for placing Jr in set W and set T, respectively. If Jr is placed in set E, the additional cost will be the earliness penalty of the j o b which is equal to ~i(t~ - pfl. Placing Jr in set W incurs no additional cost while placing it in set T incurs an additional cost equals to the fixed tardiness cost of the j o b ( = 7j). And hence we have the three equations in the recursive relations. The optimal solution to the algorithm is obtained after we have scheduled all jobs. That is, after we have scheduled the remaining n - 1 jobs (in addition to j o b k). We have to consider all possibilities of k, tl, and t2 in obtaining the m i n i m u m earliness-tardiness cost given by Eq. (3).
5. Restricted window problem
In this section, we will look at the restricted window case of E T W which we will denote by E T W R. In this case, both dl and d (and hence dz) are given p r o b l e m parameters. It is easy to see that E T W R is also
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S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
NP-Complete since ETW u is NP-complete. Also, Karp [11] had shown that a special case of ETW R, where c~i = 0, for all i, a n d d = d2, is N P - c o m p l e t e . Hence, we will state T h e o r e m 2 w i t h o u t proof. T h e o r e m 2. Problems E T W R is NP-complete. W e will n o w m o d i f y A l g o r i t h m 1 to h a n d l e this case.
5.1. A dynamic programming algorithm f o r the restricted window case W e observe t h a t the restricted case can be solved b y a d y n a m i c p r o g r a m m i n g a l g o r i t h m similar to the one p r o v i d e d for the unrestricted case, except for the fact that the range of tl a n d possibly 6 are n o w restricted by an externally specified dl. Hence, we only need to m o d i f y the range of ta a n d 6. A l g o r i t h m 2, S a m e as A l g o r i t h m 1 except t h a t in Eq. (1), 6 = m a x { 0 , p ~ - da} . . . . , min {pk, d}, a n d in e q u a t i o n s (2) a n d (3), ta = 0, 1 , . . . , min {d~, MS}. T h e overall c o m p l e x i t y of this a l g o r i t h m remains unc h a n g e d at O(n 2 { M S } d). R e m a r k . A l g o r i t h m 2 is a p s e u d o - p o l y n o m i a l time a l g o r i t h m . Hence, p r o b l e m E T W R is N P - c o m p l e t e in the o r d i n a r y sense only.
Acknowledgement The authors would like to acknowledge the associate editor and the anonymous referee for the helpful comments and suggestions.
References [1] F.D. Anger, C.-Y. Lee and L.A. Martin-Vega (1986), "Single machine scheduling with tight windows", Research Report 86-16, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida. [2] U. Bagchi, Y.L Chang and R.S. Sullivan (1987), "'Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date", Nar,al Res. Logist. 34, 739-751. [3] K.R. Baker and G.D. Scudder (1990), "Sequencing with earliness and tardiness penalties: A review", Oper. Res. 38, 22 36. [4] T.C.E Cheng (1988), "Optimal common due-date with limited completion times deviation", Comput, Oper. Res. 15, 91-96. [5] T.C.E. Cheng and H.G. Kahlbacher (1991), "Single-machine scheduling to minimize earliness and number of tardy jobs", Preprint No. 179, Universitat Kaiserslautern, FB Mathematik, Kaiserslautern, Germany. [6] R.W. Conway, W.L. Maxwell and L.W. Miller (1967), Theory of Scheduling, Addison-Wesley, MA. [7] M.R. Garey and D.S. Johnson (1979), Computers and Intractability: A guide to the theory of NP-Completeness, Freeman, California. [8] N.G. Hall and M.E. Posner (1991), "Earliness tardiness scheduling problems, I: Weighted deviation of completion times about a common due date", Oper. Res. 39, 836-846. [9] N.G. Hall, W. Kubiak and S.P. Sethi (1991), "Earliness tardiness scheduling problems, I1: Deviation of completion times about a restrictive common due date", Oper. Res. 39, 847-856. [10] J.A. Hoogeveen (1992), "Minimizing maximum promptness and maximum lateness on a single machine", in Single-Machine Bicriteria Scheduling, Ph.D. Dissertation, Technische Universiteit Eindhoven, Amsterdam. [11] R.M. Karp (1992), "Reducibility among combinatorial problems", in: R.E. Miller and J.W. Thatcher (eds.),Complexity of Computer Computations, Plenum Press, New York, 85-103. [12] F.-J. Kr~imer and C.-Y. Lee (1992), "Common due window scheduling", Research Report 92-5, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida. [13] C.-Y, Lee (1991) "Earliness-tardiness scheduling problems with constant size of due date window", Research Report 91-17, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida.
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[14] C.-Y. Lee, S.L, Danusaputro and C.-S. Lin, (1991), "Minimizing weighted number of tardy jobs and weighted earliness-tardiness penalties about a common due date", Comput. Oper. Res. 18, 379-389. [I 5] S.D. Liman and C.-Y. Lee (1992), "Error bound for a heuristic on the common due date scheduling problem", ORSA J. Comput., to appear. [16] S.S. Panwalkar, M.L. Smith and A. Seidmann (1982), "Common due date assignment to minimize total penalty for the one machine sequencing problem", Oper. Res. 30, 391 399. [17] E.L. Porteus (1985), "Investing in reduced setups in the EOQ model", Mymt. Sci. 31,998-1010. [18] M. Raghavachari (1988), "'Scheduling problems with non-regular penalty functions - a review", Opsearch, 25, 144 164.
Operations Research Letters 15 (1994) 195-203
Earliness-tardiness scheduling problems with a common delivery window S u r y a D . L i m a n *'a, S a n j a y R a m a s w a m y
b
aDepartment of Industrial Engineering, Texas Teeh University Lubbock, TX 79409 USA b Department qf Industrial and Management Systems, Penn State University, Unit,ersity Park, PA 16802 USA
(Received 11 January 1993; revised 1 December 1993)
Abstract This paper deals with the scheduling of n jobs on a single machine to minimize the sum of weighted earliness and weighted number of tardy jobs given a delivery window. Penalties are not incurred if jobs are completed within the delivery window. The length of this delivery window (which corresponds to the time period within which the customer is willing to take deliveries) is a given constant. We consider two cases of the problem; one where the position of the delivery window is internally determined (unrestricted window case) and the other where the position is an externally specified parameter (restricted window case). We present some optimal properties, prove that the problem (even for the unrestricted window case) is NP-complete, and present dynamic programming algorithms for both cases. K e y words." Scheduling; Earliness-tardiness penalty; Delivery window; Dynamic programming
1. Introduction The problem considered is closely related to a variety of practical problems in scheduling. There are n jobs to be processed on a single machine. All jobs are assumed to have originated from a single customer and hence they have a c o m m o n delivery window. The length of the window is prespecified by the customer. Typically, this corresponds to the time frame during which the customer is most willing to take delivery of the jobs, and thus we have an earliest and a latest due date. Jobs that are completed within the delivery window incur no costs. However, if a job is completed before the earliest due date, it has to be held in inventory until such time the customer is ready to receive the job (i.e., the earliest due date). Consequently, this job incurs an earliness penalty that depends on its value and the length of time it is held in inventory. This earliness cost can be likened to such costs as capital, insurance, and deterioration costs. On the other hand, if a job is completed after the latest due date, it incurs a fixed tardiness cost that is independent of how tardy the job is. We can draw a parallel here to a situation involving the production of seasonable goods, where off-season
* Corresponding author. 0167-6377/94/07.00 (c~ 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 3 7 7 ( 9 4 ) 0 0 0 0 8 - 7
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S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
deliveries, while are still accepted by the customer, have to be sold at a discounted price. The objective is to minimize the total weighted earliness and weighted number of tardy jobs penalties incurred. This objective function is a nonregular measure of performance, as opposed to regular measures which are nondecreasing functions of job completion times [6], and is also in line with current developments in just-in-time (JIT) management [17]. With the recent increase in awareness of JIT concept, there has been a rapid growth of literature in the area of earliness-tardiness scheduling (the reader is referred to [18, 3] for comprehensive reviews of works in this area). Hall and Posner [8] examined the symmetric weights problem (where the per unit time penalties of earliness and tardiness are equal) and proved it to be NP-complete in the ordinary sense (see [7] for the theory of NP-completeness). The case of nonsymmetric weights, where all earliness as well as tardiness weights are job independent, was investigated by Bagchi et al. [2], among others. Lee et al. [14] discussed the more general case of agreeable earliness and tardiness weights with weighted number of tardy jobs. In their paper they provided pseudo-polynomial dynamic programming algorithms for both the unrestricted (due date is internally determined) and restricted (externally specified due date) cases of the problem. The restricted due date case is already NP-complete [9] even when we have unit weights. For this problem, Liman and Lee [15] proposed a heuristic which was proven to have a tight worst case relative error bound of 0.5. Panwalkar et al. [16] investigated the single machine problem of minimizing the (constant weighted) sum of due date, earliness, and tardiness penalties. They proposed an O(n log n) algorithm that solved the problem using positional weights concept. Cheng and Kahlbacher ['5] studied a similar problem where instead of minimizing the weighted tardiness penalty, they minimized the weighted number of tardy jobs. An O(n log n) algorithm was also proposed. One of the first papers to discuss the due date window problem was by Anger et al. [1], who considered the problem of minimizing the number of jobs finishing outside the window. Cheng [4] discussed the common due date problem with the objective of minimizing the total earliness-tardiness penalty, where a job finishing within a prespecified tolerance from the due date did not incur any penalty. The tolerance, however, was assumed to be relatively small compared to the job processing times (such that at most one job can avoid the penalty). Moreover, the earliness (or tardiness) of a job was calculated from the due date and not from the ends of the tolerance interval. Kr~imer and Lee [12] investigated the due window problem to minimize the total earliness and total tardiness costs with the penalties calculated from the end points of the window. In the paper, they studied two cases of the problem: one with the window position as a decision variable and one with the position as a given parameter. For the first case, they proposed a polynomial time algorithm for the constant weights problem and pseudo-polynomial algorithm for the more general agreeable weights problem. For the second case, they provided a pseudo-polynomial algorithm for the unweighted case. Hoogeveen [10] and Lee [13] had also studied scheduling with due date windows. Their bicriteria objectives were to find schedules that minimize some functions of maximum earliness and maximum lateness. They provided polynomial algorithms to solve special cases of the problem. In this paper, we consider the problem with an arbitrary due date or delivery window, per unit cost of earliness, and a fixed tardiness penalty. The motivation for such consideration stems from the many parallels that can be drawn between the theoretical problem under study and the practical applications in scheduling jobs for delivery within an externally specified due date window. We study both the restricted and unrestricted cases of the problem described. In the restricted case, the scheduler has the freedom to choose the delivery dates (although the window size is specified). A study of this case is useful in establishing the optimal properties of the problem and in providing insights into the basic problem structure. The more practically relevant restricted case of the problem (where the position of the due date window is an external parameter) is then examined and a solution is proposed. We do not explicitly consider the ordering of tardy jobs and note that such ordering may be used for a secondary criterion.
S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195 203
197
The remainder of the paper is organized as follows. In the next section, the problem definition and notations used are introduced. Optimal properties of the problem under study are presented next. The unrestricted window case is then studied. We provide a proof that this version of the problem is NP-complete and propose a dynamic programming algorithm for the same. Finally, the restricted version is discussed along with a dynamic programming algorithm to solve it.
2. Notations and problem definition The following notations are used throughout the paper: Ji dl d2 d Pi
= job i = common earliest delivery date for all jobs = common latest delivery date for all jobs = d2 - dl = delivery window size = processing time of Ji
M S - - ~ Pi i=1
Ci = :¢~ ~> 0 = ~,;/> 0 = E -W-T --
completion time of J~ earliness penalty of J~ per time unit early fixed penalty incurred if Ji is tardy set of all jobs finishing on or before dl (Ci ~ dl ) set of all jobs finishing between dl and d2(dl ~ Ci ~ d2) set of all tardy jobs (Ci > d2)
Ui = {10
ifJ~eT, otherwise.
The earliness-tardiness window problem ETW is defined by n jobs with known integer per unit earliness costs ~,:¢z . . . . . :t,, known integer per unit tardiness penalties 7~,72 ..... 7,, and an externally specified delivery window size d. Preemption is not allowed and all jobs are assumed to be available at time zero. There is one continuously available machine that can handle only one job at a time. The objective of ETW is to find a schedule that minimizes the following total penalty function: ~ [ ~ i m a x {0,dl - Ci} + 7iUi]. i=l
We will consider two cases of the problem ETW. In the first case, d~ (or d2) is a decision variable. We call this case the unrestricted (internally determined) window case. In the second case, d~ (and hence d2) is specified as a problem parameter. We call this case the restricted (externally determined) window case. We will now state some properties of optimal schedules for both the restricted and the unrestricted cases.
3. Optimal properties Property 1. There exists an optimal schedule to ETW where all jobs are processed without any inserted idle time.
Proof. Omitted due to simplicity.
Property 2. There exists an optimal solution to ETW where some job starts {or completes) at d 2.
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Proof. Assume there exists an optimal schedule tr* where no job completes at d2. Let Ji be the job that starts before and completes after d2. Since Ji completes after d2, it will incur a fixed penalty cost 7i, irrespective of how much later than d2 it is completed. Therefore, ifJi is moved to the right and made to start exactly at time d2, the tardiness penalty of J~ will remain unchanged while the earliness penalties of jobs processed before Ji, if there are any, will decrease. Hence, we can find another schedule with the same total penalties (if there are no jobs before J~), or one with smaller total penalties (if there are jobs before Ji), by letting Ji start at time d2. Henceforth, we will only consider schedules where there exists a job that starts (or completes) at dE. Property 3. There exists an optimal schedule where jobs in set E are in nonincreasing order of pJaj. Proof. The proof follows from the well-known V-shaped property (see, e.g. [14]). Property 4. For a given set W and a given set T, the orderings of jobs within these sets are irrelevant as they do not affect the total penalties. Note that, from Property 2, the first job in set W starts on or after dl and the last job completes at d2 and the first job in set T starts at d2. The ordering can, however, be of use in optimizing a secondary criterion.
Proof. Omitted due to simplicity. In the sections that follow, we will present an NP-completeness proof and a dynamic programming algorithm for the unrestricted case followed by a dynamic programming algorithm for the restricted case. From now on, we will assume that jobs are numbered in nondecreasing order of p~/ej (i.e. Pl/~1 <<.... <<-P,/e,).
4. Unrestricted window problem In this section, we will study the unrestricted window case of ETW problem which we will denote by ETW U. In this case, we are given d and we can either assume that dl is very large (so as not to constrain the schedule) or that dl is a decision variable. We will first prove that ETW U is NP-complete by reducing the partition problem, which is known to be NP-complete, in polynomial time to the ETW U problem. 4.1. An NP-completeness proof for the unrestricted window case The partition problem (P) Given a finite set A of 2n integers a l, az .... , a2n, where each ai~ Z +, does there exist a subset A ' _ A such that ~.,~a' ai = Y~.,~A- .r ai? The corresponding earliness-tardiness window problem instance ETWI can be constructed as follows. Instance ETWI Number of jobs:
2n+ 1
Window size:
d=½E
Processing times:
Pi=ai
Earliness weights:
• ~=S
fori=l,2
O~2n+ 1 ~
$2
2n
S
a,=~
2n
w h e r e S = ~ a,
i=1
i=1
f o r i = 1,2,...,2n / 2 , "~2 P2n+ I = ~,,i~=lPi ) .... 2n
199
S.D. Liman. S. Ramaswamy / Operations Research Letters 15 (1994) 195 203
L
2d
.1.
set E
T'a"
[
d
.L.
[...I
set W
d~
[
~
d
...I
set T
A
I
d~ Fig. 1
Tardiness weights:
7i = a~ for i = 1,2 .... 2n ~2n + 1 ~ $ 2
Total earliness-tardiness penalty = y = d. The recognition version of the problem ETW v for the instance ETWI is Does there exist a nonpreemptive, single machine schedule with total earliness-tardiness penalty no larger than y? Lemma 1. I f there exists a partition o f set A such that ~a,~A ai = ~ a , ~ A - A ' a i = ly~2n ~ ai, then there exists a schedule a for ETWI with total penalty y. Proof. By assumption, the partition of set A exists. Define a set W of jobs corresponding to set A' (i.e., p~ = ag for all Ji E W and a~ cA') and a set T of jobs corresponding to set A - A' (i.e., Pi = a~ for all Ji E T and al~ d - A'). Therefore ~s,~w P~ = ~J,~r P~ = d. The delivery window d can therefore hold exactly jobs in set W, while other jobs are assigned to set T. Figure 1 shows the corresponding schedule. J2, +1 has a very large earliness and tardiness penalty and is therefore placed to the left of the delivery window to finish at time d~. The total penalty incurred is due to jobs in set T and is equal to ~s,~rTg = S/2 = d = y (since ;~ = pg = al). Lemma 2. I f there exists a solution to ETWI with total penalty <<.y, then the following must be true: 1. Jz,+~ must be processed so as to finish at dl. 2. ~ j,~w Pi = ~ J,~T Pl : d.
Proof: Assume J 2 n + l does not complete at dl. If J 2 n + l finishes before dl it will incur an earliness cost of S 2 > y. On the other hand, if it finishes beyond d2, a tardiness penalty of S 2 > y will be incurred. Therefore, J 2 n + l has to finish between dl and d2 (in set W). Now assume that J2,+ 1 completes after time d~ but before time d2. Again, there should not be any jobs in set E f o r otherwise the total penalty would exceed y. Therefore, all other jobs should be processed immediately after J2, +1. This implies that the total processing times of jobs after d2 are greater than d. Since the total tardiness penalty is at least equal to the sum of processing times of jobs in set T, the total penalty in this case will also exceed y. Hence, J2,+1 has to complete exactly at dl. To prove part two, we only need to consider the first 2n jobs and that the first of these jobs (the first job in set W) will start processing at time dl immediately after J2n+ 1. Assume that YJ,~wPi # ~ J ~ r P i (note that by definition, ~J,~w Pi < d and hence ~s,~r Pi > d). However, we can see that since the total tardiness penalty is at least equal to the sum of processing times of jobs in set T, the total penalty in this case will also exceed y. Hence ~ J , e w P i = ~J~eTPi = d. This implies that there exists a solution to (P). Theorem 1. Problem ETW U is NP-complete. Proof. From Lemmas I and 2.
S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
200
~=0
I QIDII set E
~" "T"
set W
d,
.a.. T
1
set T
,,A
set T
,d
"1
d2
~-- 6>0 --~ DOO
set E
~ T
set W
d,
-]
d2
•
set E
~x, T
ha" T
set W
d,
I
.x. T
""1
set T
I -1
d2 Fig. 2
4.2. A dynamic programming algorithm for the unrestricted window case In this section we will provide a dynamic programming algorithm to find the optimal solution to problem ETW v. From Property 2 we know that there exists an optimal schedule where some job finishes at time dE. Property 3 tells us that jobs in set E are processed in nonincreasing order of pi/cti, and from Property 4 we know that the ordering of jobs in set T and set W are of no consequence. While an optimal schedule has a job finishing at time dE, w e have two possibilities with regard to dl (see Fig. 2). 1. There exists Jk that either starts or finishes at time dl. 2. There exists Jk that straddles (completes before and finishes after) dl. To take all cases into account, we would first remove Jk (k -- 1.... n) from the set of n jobs and schedule it at all possible locations (starting with JR completing at time dl and ending with either Jk completing at time (dl + Pk) or (dl + d) whichever is smaller). In other words, if we let 6 = Ck -- dl, then 6 = 0 . . . . . min {pk,d}. The remaining (n - l)jobs would be scheduled around Jk. We would begin with J1 (the job with the smallest ratio of Pi/~i). We will now present the dynamic programming algorithm. Algorithm 1. Let fk(j, t b t 2 ) = the minimum total penalty if we have already scheduled the first j jobs, assuming that we have already removed Jk and have renumbered the remaining jobs from J x to J,_ l (still in nondecreasing order of p~/~j), t 1 is the difference between the start time of the first job in the sequence and dl, and t2 is the difference between the completion time of last job in the window and dl (see Fig. 3); k = 1,2,..., n;j = 0,1,..., n - l; tl = 0 , 1 . . . . . MS; t 2 = 0,1,...,d.
Boundary conditions t0. fk(O, tl, t2) ----- k - - O0
if t l = p k - - 6 otherwise
and t z = 6 where 6 = 0 ..... min {pk,d}t
f (j, tbt2)= +oo f o r k = l , . . . , n ; j = l k
"
..... n - 1 ; t x < O o r t 2 < O .
for k = 1, .-.,n,
(1)
201
S.D. Liman, S. Ramaswarny / Operations Research Letters 15 (1994) 195-203 L.
.L
[
4
"
set E
t2
• ..
J
l --fO
...1.-~
set W
d,
.J "1
••
t.. F
set T
d2 Fig. 3
Recursive relations f k ( j __ 1, q -- pj, t2) + C~j(tl -- pfl, fk(j, tDt2) = m i n i f k ( j
l,tt, t2 -- pfl,
I fk(j
1,Q,t2) + 72"
(2)
F o r k = 1,2 . . . . . n;j = 1,2 . . . . . n - 1; t~ = 0, 1. . . . . MS; te = 0, 1. . . . . d.
Optimal solution Z = min{fk(n -- 1, tl, re);
k = 1. . . . . n;
tl = 0, 1. . . . . MS;
t2 = 0, 1. . . . . d}.
(3)
Remark. Algorithm 1 is a p s e u d o - p o l y n o m i a l time algorithm with complexity O(n 2 {MS}d). Hence, problem E T W v is N P - c o m p l e t e in the ordinary sense only. Justifications of Algorithm 1. We assume that Jk(k = 1,..., n) is placed first before all other jobs. Correspondingly, fk(o, t l , t z ) = 0 only if tl = P k - 6 and t2 = 6 for 6 = 0 .... , min{pk, d}. Otherwise, the initial placement is infeasible and f k ( O , h , t 2 ) = +0o. Similarly, since neither t~ nor t2 can be negative, .fk(j,&,t2)= +0O if either t~ < 0 or t2 < 0 . Hence, the b o u n d a r y conditions are given by Eq. (1). T h e first equation in the recursive relations, given by Eq. (2), is for placing Jj in set E. The second and the third are for placing Jr in set W and set T, respectively. If Jr is placed in set E, the additional cost will be the earliness penalty of the j o b which is equal to ~i(t~ - pfl. Placing Jr in set W incurs no additional cost while placing it in set T incurs an additional cost equals to the fixed tardiness cost of the j o b ( = 7j). And hence we have the three equations in the recursive relations. The optimal solution to the algorithm is obtained after we have scheduled all jobs. That is, after we have scheduled the remaining n - 1 jobs (in addition to j o b k). We have to consider all possibilities of k, tl, and t2 in obtaining the m i n i m u m earliness-tardiness cost given by Eq. (3).
5. Restricted window problem
In this section, we will look at the restricted window case of E T W which we will denote by E T W R. In this case, both dl and d (and hence dz) are given p r o b l e m parameters. It is easy to see that E T W R is also
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S.D. Liman, S. Ramaswamy / Operations Research Letters 15 (1994) 195-203
NP-Complete since ETW u is NP-complete. Also, Karp [11] had shown that a special case of ETW R, where c~i = 0, for all i, a n d d = d2, is N P - c o m p l e t e . Hence, we will state T h e o r e m 2 w i t h o u t proof. T h e o r e m 2. Problems E T W R is NP-complete. W e will n o w m o d i f y A l g o r i t h m 1 to h a n d l e this case.
5.1. A dynamic programming algorithm f o r the restricted window case W e observe t h a t the restricted case can be solved b y a d y n a m i c p r o g r a m m i n g a l g o r i t h m similar to the one p r o v i d e d for the unrestricted case, except for the fact that the range of tl a n d possibly 6 are n o w restricted by an externally specified dl. Hence, we only need to m o d i f y the range of ta a n d 6. A l g o r i t h m 2, S a m e as A l g o r i t h m 1 except t h a t in Eq. (1), 6 = m a x { 0 , p ~ - da} . . . . , min {pk, d}, a n d in e q u a t i o n s (2) a n d (3), ta = 0, 1 , . . . , min {d~, MS}. T h e overall c o m p l e x i t y of this a l g o r i t h m remains unc h a n g e d at O(n 2 { M S } d). R e m a r k . A l g o r i t h m 2 is a p s e u d o - p o l y n o m i a l time a l g o r i t h m . Hence, p r o b l e m E T W R is N P - c o m p l e t e in the o r d i n a r y sense only.
Acknowledgement The authors would like to acknowledge the associate editor and the anonymous referee for the helpful comments and suggestions.
References [1] F.D. Anger, C.-Y. Lee and L.A. Martin-Vega (1986), "Single machine scheduling with tight windows", Research Report 86-16, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida. [2] U. Bagchi, Y.L Chang and R.S. Sullivan (1987), "'Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date", Nar,al Res. Logist. 34, 739-751. [3] K.R. Baker and G.D. Scudder (1990), "Sequencing with earliness and tardiness penalties: A review", Oper. Res. 38, 22 36. [4] T.C.E Cheng (1988), "Optimal common due-date with limited completion times deviation", Comput, Oper. Res. 15, 91-96. [5] T.C.E. Cheng and H.G. Kahlbacher (1991), "Single-machine scheduling to minimize earliness and number of tardy jobs", Preprint No. 179, Universitat Kaiserslautern, FB Mathematik, Kaiserslautern, Germany. [6] R.W. Conway, W.L. Maxwell and L.W. Miller (1967), Theory of Scheduling, Addison-Wesley, MA. [7] M.R. Garey and D.S. Johnson (1979), Computers and Intractability: A guide to the theory of NP-Completeness, Freeman, California. [8] N.G. Hall and M.E. Posner (1991), "Earliness tardiness scheduling problems, I: Weighted deviation of completion times about a common due date", Oper. Res. 39, 836-846. [9] N.G. Hall, W. Kubiak and S.P. Sethi (1991), "Earliness tardiness scheduling problems, I1: Deviation of completion times about a restrictive common due date", Oper. Res. 39, 847-856. [10] J.A. Hoogeveen (1992), "Minimizing maximum promptness and maximum lateness on a single machine", in Single-Machine Bicriteria Scheduling, Ph.D. Dissertation, Technische Universiteit Eindhoven, Amsterdam. [11] R.M. Karp (1992), "Reducibility among combinatorial problems", in: R.E. Miller and J.W. Thatcher (eds.),Complexity of Computer Computations, Plenum Press, New York, 85-103. [12] F.-J. Kr~imer and C.-Y. Lee (1992), "Common due window scheduling", Research Report 92-5, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida. [13] C.-Y, Lee (1991) "Earliness-tardiness scheduling problems with constant size of due date window", Research Report 91-17, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida.
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[14] C.-Y. Lee, S.L, Danusaputro and C.-S. Lin, (1991), "Minimizing weighted number of tardy jobs and weighted earliness-tardiness penalties about a common due date", Comput. Oper. Res. 18, 379-389. [I 5] S.D. Liman and C.-Y. Lee (1992), "Error bound for a heuristic on the common due date scheduling problem", ORSA J. Comput., to appear. [16] S.S. Panwalkar, M.L. Smith and A. Seidmann (1982), "Common due date assignment to minimize total penalty for the one machine sequencing problem", Oper. Res. 30, 391 399. [17] E.L. Porteus (1985), "Investing in reduced setups in the EOQ model", Mymt. Sci. 31,998-1010. [18] M. Raghavachari (1988), "'Scheduling problems with non-regular penalty functions - a review", Opsearch, 25, 144 164.