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Scheduling about a common due date with kob-dependent asymmetric earliness and tardiness penalties
EUROPEAN
JOURNAL OF OPERATIONAL RESEARCH
ELSEVIER
European Journal of Operational Research 98 (1997) 154-168
Theory and Methodology
Scheduling about a common due date with job-dependent asymmetric earliness and tardiness penalties X. Cai *, V.Y.S. Lum, J.M.T. Chan Department of Systems Engineering and Engineering Management, The Chinese Universityof Hong Kong, Shatin, N.T., Hong Kong Received October 1994; revised August 1995
Abstract This paper is concerned with the problem of scheduling n jobs with a common due date on a single machine so as to minimize the total cost arising from earliness and tardiness. A general model is examined, in which earliness penalty and tardiness penalty are, respectively, arbitrary non-decreasing functions. Moreover, the model includes two important features that commonly appear in practical problems, namely, 1) earliness and tardiness are penalized with different weights which are job-dependent, and 2) the earliness (or tardiness) penalty consists of two parts, one is a variable cost dependent on the length of earliness (or tardiness), while the other is a fixed cost incurred when a job is early (or tardy). This model provides a general and flexible performance measure for earliness/tardiness scheduling, which has not been addressed before. We establish a number of results on the characterizations of optimal and sub-optimal solutions, and propose two algorithms based on these results. The first algorithm can find, under an agreeable weight condition, an optimum in time O(n2P,,), and the second algorithm can generate a sub-optimum in time O(nP,,), where Pn is the sum of the processing times. Further, we derive an upper bound on the relative error of the sub-optimal solution and show that, under certain conditions, the error tends to zero as n increases. Computational results are also reported to demonstrate the effectiveness of the algorithms proposed. © 1997 Elsevier Science B.V.
Keywords: General earliness and tardiness costs; Manufacturing; Scheduling/sequencing; Due date assignment; Exact and approximate algorithms
1. Introduction C o n s i d e r the scheduling situation w h e r e a single m a c h i n e is to process a set o f n i n d e p e n d e n t j o b s under a c o m m o n due date d. A s s o c i a t e d with each j o b i, i = 1, 2 . . . . . n, are a p r o c e s s i n g time Pi, an earliness w e i g h t el, a tardiness w e i g h t t i, a fixed earliness penalty a i, and a fixed tardiness p e n a l t y / 3 i. It is a s s u m e d that Pi, i = 1, 2 . . . . . n, are positive integers w h e r e a s all other parameters are n o n n e g a -
tive numbers. All j o b s are r e a d y for processing at time zero. The m a c h i n e is available continuously f r o m time zero and can process at m o s t one j o b at a time. J o b splitting and p r e e m p t i o n are not allowed. T h e p r o b l e m is to find a schedule A to process the n j o b s so as to m i n i m i z e the total cost arising f r o m earliness and tardiness, e x p r e s s e d as
~_, [ a i + e i f (L;)]
J(A)=
+ * Corresponding author. 0377-2217/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 5 ) 0 0 2 5 3 - 7
Y'~ [ 1 9 i + t i g ( L i ) ] ,
(1.1)
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
where L i = I Ci - d l denotes the absolute lateness of job i from the due date, C~ is the completion time of job i under A, and E(A)={i:C i d} represent, respectively, the sets of early jobs and tardy jobs under h, and f ( . ) and g ( . ) are general non-decreasing real-valued functions. For ease of reference, let us denote the model formulated above as P1. This model espouses the concept of Just-In-Time production in advanced manufacturing systems. Suppose a company (a machine) plans to process n jobs under a common due date d. A tardiness penalty is incurred if a job is completed after: d. This occurs due to contract penalty as well as other intangible losses like customer dissatisfaction, loss of reputation, etc. On the other hand, completion of a job before the due date also incurs a cost, since an early job takes up scarce inventory space, ties up capital, and requires additional effort to manage. Thus an important problem for the company is to find an optimal schedule to process the jobs so as to minimize the total cost due to tardiness and earliness. In practice, the tardiness penalty of a job is generally not identical with the earliness penalty of the job and these are usually job-dependent, which thus justifies the asymmetric structure of penalties on earliness and tardiness as formulated b y (1.1). In addition to manufacturing systems, the model may have applications in any other situations where a number of tasks have to be scheduled about a certain event. Examples include the scheduling of a number of astronomical experiments about an external event like the passing of a comet. Earliness/tardiness scheduling problems have been extensively investigated in the literature. The majority of prior researches, however, usually dealt with problems under the assumptions: 1) Earliness and tardiness weights are all equal, namely, e~ = t; for i = 1, 2 . . . . . n, 2) Earliness and tardiness penalty functions are linear or quadratic functions, namely, g ( L ~ ) = f ( L i) = L i or g ( L i) = f ( L i) = L~, a n d / o r 3) fixed penalties a i and /3~ are zero. For a comprehensive review, see Baker and Scudder (1990) and Cheng and Gupta (1989). Problems
155
with asymmetric earliness and tardiness weights have received some attention in recent years. The first work that explicitly investigates such a model was reported by Bagchi, Chang and Sullivan (1987), in which they addressed a quadratic penalty function. De, Ghosh and Wells (1993) considered a problem with g ( L ~ ) = f ( L i) = L r, where r is a positive integer. Federgruen and Mosheiov (1993) and Kahlbacher (1993) examined respectively more general penalty functions (quasiconvex or unimodal functions). While these researches have provided significant insights on the investigation of asymmetric earliness and tardiness penalties, there is still a common restriction imposed on these models, which assumes that all jobs should have the same earliness weight and the same tardiness weight, namely, e i = ej and t i = t j for i , j = l , 2 . . . . . n. On the problem involving job-dependent asymmetric earliness and tardiness weights, the work of Lee, Danusaputro and Lin (1991) should be noted. They have derived some excellent results including a pseudopolynomial algorithm under certain conditions like the ratio of processing times and weights is agreeable. Nevertheless, their model is concerned only with a linear penalty function. Besides the restriction that all jobs should have a common earliness weight and a common tardiness weight, another assumption commonly made in prior researches is that the cost for the tardiness/earliness of a job should approach zero if the deviation of its completion time from the due date tends to zero. Noting that a fixed cost is incurred once a job is late no matter how late it is, Lee, Danusaputro and Lin (1991) have proposed to incorporate into the objective function a fixed penalty, which depends upon the jobs which are tardy, Nevertheless, their model is concerned with fixed tardiness penalty only. Notice that there also exists a fixed earliness cost once a job is completed early. This includes, for example, the basic cost for removing an early job to a designated warehouse and other fixed administrative costs. In this paper we study the problem P1. This is a general model, which subsumes most single-machine. common due-date problems having been studied in the literature and provides a more general and flexible performance measure to better reflect the need of decision-making in practice. To the best of our knowledge, such a general model has not been
X. Cai et a l . / European Journal of Operational Research 98 (1997) 154-168
156
addressed before. To find an optimal solution for this problem is difficult. Many special instances of the model have been shown to be NP-complete. For example, if e i = t i = 1 , of i = J~i ~-" 0 , g ( Z i) : f ( L i) - L i, and d >~ P,( = ET= t Pi), the problem is known to be equivalent to the completion time variance (CTV) problem, which has been shown to be NPcomplete by Kubiak (1993). We conjecture that the general problem P1 is NP-complete in the strong sense and thus unlikely to be solved even by a pseudopolynomial algorithm. Considering the inherent difficulty in finding an exact optimal solution, a primary objective of this paper will be to investigate the characterizations of optimal and sub-optimal solutions so that algorithms can be developed which can find an optimal solution under certain conditions or a sub-optimal solution efficiently. Our main results include a W-shape property for an exact optimal schedule and a V-shape property for a sub-optimal schedule. The concept of V-shaped schedule was firstly introduced by Eilon and Chowdhury (1977) in the context of the CTV problem. The existence of a V-shaped optimum has become an important optimality property in many earliness/ tardiness problems and served as the basis for development of effective (pseudopolynomial) solution methods, see Baker and Scudder (1990), Raghavachad (1986), Krieger and Raghavachari (1992), Lee, Danusaputro and Lin (1991), Hall and Posner (1991), Hall, Kubiak and Sethi (1991), De, Ghosh and Wells (1992), etc. The structure of a W-shaped schedule is similar to a V-shape, except that the job straddling the due date (or the tip of the V-shape) is unclear; see Section 2 below for a definition. Hoogeveen and van de Velde (1991) have found that an optimum for an earliness/tardiness problem with a linear penalty function possesses a W-shaped structure and, consequently, applied the optimality property to devise a solution algorithm with pseudopolynomial time complexity. We will show in the sequel that the general problem P1 will have a W-shaped optimum when it is agreeably weighted, in the sense that Pi > Pj implies e i <~ ej and t i <~ (j, V i, j. A motivation for this agreeable condition is, for instance, the common practice that a smaller job is charged at a higher rate, which in turn makes it relatively more important and thus the penalty of missing its due date become heavier. Agreeable conditions of this nature have _
2
often been used in many scheduling studies in the literature, see Lawler (1976, 1977), Kise, Ibaraki and Mine (1978), Pinedo (1983), Lee, Danusaputro and Lin (1991), Cai (1995), etc. We will also show, in an example, that some problem instances of P1 have no V-shaped optimum even under the agreeable condition. Thus it is impossible to establish a V-shape property for optimal schedules. To facilitate the development of more efficient solution algorithms, we will turn to examining a V-shape property for schedules that are sub-optimal. We will show that there exists at least one V-shaped schedule, the relative error of which approaches zero as n increases under some conditions. Specifically, the conditions are satisfied when ]e i ej[f(pj), I t i - t j l g ( p j ) and I ( t x i - a j ) - ( / 3 i /32) 1, Vi, j, are bounded above. Note that these conditions are naturally satisfied under the assumption that all jobs have the same earliness weights and the same tardiness weights (including the fixed penalties), which are usually made in prior researches as we reviewed above. Practically, the conditions are reasonable since the problem parameters are usually bounded above in a practical situation. On the basis of the W-shape property and the V-shape property, we will develop two algorithms to solve the problem. Under the agreeable condition, the first algorithm can find an optimal solution in time O(n2P,). The second algorithm, which is n times faster than the first one, can generate a sub-optimal schedule with relative error approaching zero as n grows. Both algorithms will be shown to be applicable to problems with d as a fixed constant or a decision variable. In the general case without the agreeable weight condition, we suggest to extend the second algorithm as a heuristic procedure to find approximate solutions. The algorithms proposed will be evaluated by computational experiments, for both cases with and without the agreeable condition.
2. Characterizations of solutions
We n o w describe some important results concerning the basic characterizations of optimal and suboptimal schedules. The proof for the result below is straightforward, since if a schedule has a non-zero idle time between two consecutive jobs, then the gap
X. Cai et al./ European Journal of Operational Research 98 (1997) 154-168
can be closed towards the due date to generate a schedule without the idle time, which is not worse than the schedule with the idle time (or even better if f and g are strictly increasing functions). T h e o r e m 1. There exists an optimal schedule f o r P1 which has no idle time between any two consecutive jobs. The result below is on the W-shape property of the problem. To be more precise, let us introduce the following definition first, in which S~ denotes the starting time of job i, namely, Si=Ci-Pi
for i = 1 , 2 . . . . . n.
Definition 1. A schedule h is said to be W-shaped with respect to the processing times if pj >I-Pk when Cj < C k < d, and Pj < Pk when d <~Sj < S k. Notice that in a W-shaped schedule as defined above, it is possible to have a job m with S m < d <~ C m, called straddling job, such that P m > P,,, and Pm > P r o " , where jobs m' and m" are, respectively, the jobs sequenced immediately before and after job m.
T h e o r e m 2. There exists an optimal schedule f o r P1 which is W-shaped with respect to the processing times if p i < p: implies e i >1 e: and t i >~ tj, V i, j. Proof. Suppose that A* is an optimal schedule. Clearly if it is not W-shaped, it must have three consecutive jobs i, j and k such that pj > Pi, Pj > Pk, and c? + pj= Cf < d or
d
S; = s ; - p .
where Cq and Sq are, respectively, the completion time and the starting time of job q under A*. When C j < d, we may construct a new schedule /X' by interchanging jobs i and j in h*. In this case, E(X) = E( A* ) and T(A') = T(A* ). Thus, letting z = d - C j , we can see that
=J(x) -J(A °)
=
157
e i f ( Z ) + e j f ( Z + Pi) -- e i f ( Z + pj)
-- e J ( Z) ~ O,
(2.1)
since f ( z + p ) > ~ f ( z + p i ) > ~ f ( z ) and ei>~ ej because of the agreeable condition. Similarly, when d ~< S f , we can construct h" by interchanging jobs j and k and show A ( h " ) ~ 0. If ~' or X' is still not W-shaped, we may further interchange a pair of consecutive jobs similar to the above. Clearly a series of interchanging operations will finally result in a W-shaped schedule h ¢ with A(h¢) ~< 0, which indicates either a contradiction with the fact that h* is optimal (if A(A/) < 0) or the optimality of h f (if a(,V) = o).
[]
Observing that the agreeable condition is satisfied when all jobs have the same earliness weight and the same tardiness weight, we have: Corollary 1. There exists a W-shaped optimal schedule f o r PI if e i = ej and t i = tj, V i , j. The results above show that P1 has an optimal schedule which is W-shaped when the problem is agreeably weighted or has an equal earliness weight and an equal tardiness weight. Since V-shape is a special case of W-shape, the optimal schedules for some instances of P1 can actually be V-shaped. However, on the other hand, it may also be shown that there do exist some instances which do not have any V-shaped optimal schedule, e v e n if they m e e t the agreeable condition (see an example given in Section 4.1.1). Considering that the problem in general may have no V-shaped optimum, we now turn to examining the sub-optimality of the best V-shaped schedule. We will first describe, in the theorem below, an upper bound on the relative error of the best V-shaped schedule. Corollaries will be then deduced from the theorem, which show that, under certain conditions, the relative error tends to zero or even equals zero. To be more precise, let us lay down the following definition first, Definition 2. A schedule h is said to be V-shaped with respect to the processing times if pi >~Pk when C j < C k~
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
158
Let A* be a W-shaped optimum for P1 and h v be the best schedule in ~-, the set of all V-shaped schedules. Further, let O = m a x i,j{ 1(o~ i - ogj) - ( ]3 i - ~ j ) 1} +max/d{[ e i -- ejl f ( p j ) } + m a x i o { ] t i -- tj ] g( pj)}. F = E~= l[min{ai, /3i}] - m a x i { m i n { a i, /~i}}. w~ = The i-th smallest element in the set 7 f = {e l, e 2 . . . . . e,, t I, t 2 . . . . . tn}. q~= The i-th smallest element in the set ~ = {f(Pmin), f(2pmin) . . . . ; f ( n P m i . ) , g(Pmi~)' g(2Pmin) . . . . . g(nprnin)}, where P~in is the minimum processing time. The proof for the following theorem is given in Appendix A.
Theorem 3. There exists a V-shaped schedule h ° f o r P1 such that
R(v) =
J(a°)-J(a J(a*)
*)
o n--2
1~+ E i = l W i q n _ l _ i
Letting O= maxi,j{ l( oli-- otj) -- ( t~i -- t~j) l }, we have:
Corollary 3. If e;---ei=e and t i = t j = t ,
V i , j,
where e and t are constant, then 0
R( h ~) ~<
.
(2.3)
F-}- E72~Wiqn_l_i Furthermore, if e > O, t > O, f and g are increasing functions, and 0 is b o u n d e d above, then P1 has a V-shaped schedule which is asymptotically optimal in the sense that R(A °) ~ 0 as n increases.
It can be seen that R(A v) = 0 if ofi : a j and /3i =/3j in addition to e~ = ej and t~ = tj. Hence, we have:
Corollary 4. There exists a V-shaped schedule which
(2.2)
is exactly optimal f o r P1 i f e i = ej, t i = tj, a i = aj, and ~i = ~j, V i, j.
The corollaries below follow directly from Theorem 3.
The following corollary comes from the observation that 0 = 0 when the fixed earliness penalty for each job is equal to its fixed tardiness penalty, namely, a i = fli, V i.
if p i
implies e i >~ ej and t i > tj, V i , j.
Corollary 2. Under the agreeable condition, R(A v) 0 as n increases i f f and g are increasing f u n c tions, w i > O , f o r i = 1 , 2 . . . . . n - 2 , and 6 ) < . M , where M is a constant independent o f n.
Corollary 5. A s s u m e e i = ej and t i = tj, V i , j. Then a V-shaped schedule is exactly optimal f o r P1 i f Oli ":" ~i' V i .
R e m a r k . Note that the conditions that f and g are increasing functions and w[are positive numbers are satisfied by most of the earliness/tardiness models investigated in the literature. Examples include the cases f = g = [Li[, f = g =L2i, e i = t i = 1, etc. The condition that (9 ~< M is not a tight one in a practical sense, a s in practical problems the values of the parameters involved are usually bounded above. Corollary 2 actually indicates that the best V-shaped schedule Av asymptotically approaches the optimal schedule A* if the variations in the parameters of any pair Of jobs do not dominate the sum of the fixed and variable penalties of other jobs when the number of jobs increases. This result is particularly useful for large problems.
In general, the strucfure of optimal schedules for P1 without the agreeable weight condition remains to be an open question. When e; and t i can take arbitrary values, it is not hard to devise an example which does not have any W-shaped optimum (such an example can be constructed by assigning very large weights to certain jobs with large processing times).
3. Algorithms According to Theorem 1, we know that an optimal schedule for P1 should arrange the jobs about the due date d without inserting any idle time be-
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3) R( A”) is bounded above by (2.3) if ei = ej and ti = tj, Vi, j; 4) h” is asymptotically optimal in the sense that R( h”) -+ 0 as n increases if the conditions of Corollaries 2 and 3 are satisfied; and 5) h” is exactly optimal if the conditions of Corollaries 4 and 5 are satisfied. Remarks. 1) Algorithms 1 and 2 described above are proposed to solve probelms in which the due date d is a given constant. Nevertheless, they are also applicable to problems in which d is a decision variable. Such a situation may occur during sales negotiations of a company with its customers, when the company may hope to locate an optimal due date so as to minimize its overall cost. 2) When d is decision variable to be determined, we may apply the following procedure to determine an optimal or sub-optima1 d: Let d = P,, -and apply Algorithm 1 or 2 to find A* or h”. Accordingly, let the optimal or sub-optimal starting time be r& or r&,. (ii) Rearrange h * or h” through shifting the starting times of all jobs to the left by a constant r& or r,&. Let the optimal (sub-op timal) due date be d * = P, - riin (d” = P, rGin). This procedure is justifiable since J(h) actually only depends on the relative distance of job completion times from d and d = P, has given us sufficient freedom to schedule the jobs to minimize J(h). Step ii) above chooses the due date that corresponds to the schedule starting at time zero. This will result in a shortest due date without changing J( h * ) or J( h”). 3) The conclusions in Theorems 4 and 5 remain unchanged when the procedure above applies. 3.2. Heuristic for PI without the agreeable
condition
Theorems 2 and 3 are the basis of the algorithms developed above, which indicate that an optimal (or sub-optimal) solution will be W-shaped (or V-shaped) when the problem is agreeably weighted. A W-shaped (or V-shaped) structure makes it possible for us to devise a dynamic programming procedure to search for the required solution. However, for a general problem Pl without the agreeable condition, we are
unable to know whether or not it has a W-shaped optimum or a V-shaped sub-optimum (note that the agreeable condition in Theorems 2 and 3 is a sufficient condition), so in a strict sense these algorithms are not applicable considering they have no guarantee on the optimality of the solutions obtained. Nevertheless, if in practical situations one only aims to find some approximate solutions, we can still use these algorithms as heuristic procedures. Considering that even if a best W-shaped schedule might not be optimal in the general case, we suggest to generalize Algorithm 2, which is IZ times faster than Algorithm 1, as a heuristic for solving the general problem Pl. Moreover, considering that pi/( e, + ti> represents, in some sense, the relative importance of job i and that a more important job should be scheduled nearer the due date (this is essentially what the V-shape property suggests), we propose to label the jobs so that Pl
-<
-< el + tl
P2
...
Q-
e2 + t2
PTl e, + t,
*
(3.4)
Algorithm 2 can now be applied to generate a solution after the jobs are sorted according to this criterion. Theoretically we have no guarantee on the optimality of the solution found by this heuristic, unless the problem solved happens to have an optimum which is V-shaped with respect to the weighted processing times { pi/(ei + ti)} (in this case the solution found will be an exact optimum). Nevertheless, the time complexity of the heuristic remains to be O(nPJ.
4. Computational results To evaluate the effectiveness of the algorithms proposed above, a number of computational experiments have been carried out, in which problems with the following cost structure were solved:
f(x)
=
g(x)
=
b, + c,x
if x> A,,
b,-t-c,x
ifO
&As
x2
4,
(4.1)
if x&A,, ifO
(4.2)
160
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
Furthermore, let pro(r) be the contribution of the jobs in Ni m to the overall objective function (1.1), subject to r being the starting time of these jobs. Clearly, to ensure job m to be the straddling job, the possible values that r can take will be integers contained in [ ai, bi] with a i = max{0, d - P i } and b i = d, where P~ = ~ j ~ N:"Pj" According to Theorem 2, job i, if i v~ m, must be sequenced either the first or the last among all jobs in Nim. Thus, we have a dynamic programming algorithm for solving P1 in which d is a given constant; see Table 1. Theorem 4. The schedule A* found by Algorithm 1 is optimal for P1 under the agreeable condition and the time complexity required by Algorithm 1 is bounded above by O(n2p,).
Proof. The optimality of A* follows directly from Theorem 2 and the principle of optimality of dynamic programming. As for the time complexity, it is obvious that (b i - a i ) < ~ P, steps are needed to calculate p~"(r) for each i and m. Since we need to enumerate i = 1 , 2 . . . . . n and m = l , 2 . . . . . n, the time requirement to compute p ~ ( r ) is bounded above by O(n2p,). This dominates the time complexity of the algorithm. []
shortest job) is not necessarily the straddling job. This is different from a W-shaped schedule in which the straddling job must be the job at the tip (the peak) of the W-shape. Algorithm 1 has made use of this fact in setting the range [ai, b i] for r. To develop an algorithm to search for the best V-shaped schedule Av in the set ~ ' , we use the following result: L e m m a 1. Under the agreeable condition, A" sequences job 1 as: 1) the straddling job; 2) the last job among all the early jobs; or 3) the first job among all the tardy jobs that start no earlier than d. The proof follows from a job interchange argument. From this result we know that the completion time of job 1 will not go beyond the range [ d - p., d + P . + P l ] - Thus, letting N~ be the set of jobs 1, 2 . . . . . and i, we can see that r may only take integer values in the interval [a;, b;], where a i = max(0, d -- Pi - P,}, b i = d + p,~, e i = ]~ pj.
(3.3)
j~N i
3.1.2. An approximate algorithm ,It follows from Theorem 3 that, if one only aims to find an approximate solution, then searching in the set ~- of V-shaped schedules will be enough. This will reduce the time complexity of Algorithm 1 as it is no longer needed to enumerate the straddling job m.
First we have to note that, in a V-shaped schedule, the job lying at the tip of the V-shape (the
We propose the algorithm presented in Table 2. In light of Theorem 3, Lemma 1, and the proof for Theorem 4, we have: Theorem 5. 1) Algorithm 2 requires at most O(nP.) time to find a schedule a"; 2) R(A") is bounded above by (2.2) under the agreeable condition;
Table 2 Algorithm 2 • Given m = 1, compute OT'(r) according to Eq. (3.1) for i = 1, 2 . . . . . n and r E [ai, bi]. where a i and b i are calculated in accordance with Eq. (3.3), subject to the following boundary condition: , +e,f(d-r-p~)
Io
[ill+tlg(r+p,--d) I !, + 0% •
if d - p , < ~ r + p l
if r + p l = d , ifd
Compute: J(A") = minr~ta., b,.l{p],(r)}, then construct A" by a backtracking procedure after the best r is found.
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X. Cai et al./ European Journal of Operutional Research 98 (1997) 154-168
3) R( A”) is bounded above by (2.3) if ei = ej and ti = tj, Vi, j; 4) h” is asymptotically optimal in the sense that R( h”) -+ 0 as n increases if the conditions of Corollaries 2 and 3 are satisfied; and 5) h” is exactly optimal if the conditions of Corollaries 4 and 5 are satisfied. Remarks. 1) Algorithms 1 and 2 described above are proposed to solve probelms in which the due date d is a given constant. Nevertheless, they are also applicable to problems in which d is a decision variable. Such a situation may occur during sales negotiations of a company with its customers, when the company may hope to locate an optimal due date so as to minimize its overall cost. 2) When d is decision variable to be determined, we may apply the following procedure to determine an optimal or sub-optima1 d: Let d = P,, -and apply Algorithm 1 or 2 to find A* or h”. Accordingly, let the optimal or sub-optimal starting time be r& or r&,. (ii) Rearrange h * or h” through shifting the starting times of all jobs to the left by a constant r& or r,&. Let the optimal (sub-op timal) due date be d * = P, - riin (d” = P, rGin). This procedure is justifiable since J(h) actually only depends on the relative distance of job completion times from d and d = P, has given us sufficient freedom to schedule the jobs to minimize J(h). Step ii) above chooses the due date that corresponds to the schedule starting at time zero. This will result in a shortest due date without changing J( h * ) or J( h”). 3) The conclusions in Theorems 4 and 5 remain unchanged when the procedure above applies. 3.2. Heuristic for PI without the agreeable
condition
Theorems 2 and 3 are the basis of the algorithms developed above, which indicate that an optimal (or sub-optimal) solution will be W-shaped (or V-shaped) when the problem is agreeably weighted. A W-shaped (or V-shaped) structure makes it possible for us to devise a dynamic programming procedure to search for the required solution. However, for a general problem Pl without the agreeable condition, we are
unable to know whether or not it has a W-shaped optimum or a V-shaped sub-optimum (note that the agreeable condition in Theorems 2 and 3 is a sufficient condition), so in a strict sense these algorithms are not applicable considering they have no guarantee on the optimality of the solutions obtained. Nevertheless, if in practical situations one only aims to find some approximate solutions, we can still use these algorithms as heuristic procedures. Considering that even if a best W-shaped schedule might not be optimal in the general case, we suggest to generalize Algorithm 2, which is IZ times faster than Algorithm 1, as a heuristic for solving the general problem Pl. Moreover, considering that pi/( e, + ti> represents, in some sense, the relative importance of job i and that a more important job should be scheduled nearer the due date (this is essentially what the V-shape property suggests), we propose to label the jobs so that Pl
-<
-< el + tl
P2
...
Q-
e2 + t2
PTl e, + t,
*
(3.4)
Algorithm 2 can now be applied to generate a solution after the jobs are sorted according to this criterion. Theoretically we have no guarantee on the optimality of the solution found by this heuristic, unless the problem solved happens to have an optimum which is V-shaped with respect to the weighted processing times { pi/(ei + ti)} (in this case the solution found will be an exact optimum). Nevertheless, the time complexity of the heuristic remains to be O(nPJ.
4. Computational results To evaluate the effectiveness of the algorithms proposed above, a number of computational experiments have been carried out, in which problems with the following cost structure were solved:
f(x)
=
g(x)
=
b, + c,x
if x> A,,
b,-t-c,x
ifO
&As
x2
4,
(4.1)
if x&A,, ifO
(4.2)
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X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
Table 3 Pl = 1, a I = 0, /31 = 30,
P2 = 1, a 2 = 0, /32 = 30,
P3 = 1, a 3 = 30, /33 = 0,
f(xp
A1
0
x
g(x)
/ / J r
0
A2
x
Fig. 1. The earliness function f ( x ) a n d the tardiness function
g(x).
P4 = 1, a 4 = 30, /34 = 0,
P5 = 10, a 5 = 50, /35 = 50.
The functions f(x) and g(x) are illustrated by Fig. 1. In our experiments, we restricted 0 ~< c~ ~< c 2 in the earliness function f ( x ) . This corresponds to, for instance, the situation where two types of places are available for storage of finished jobs before the due date d. The use of the first one is charged at a lower rate, which however requires a minimum storage time A~ and thus can only be for jobs completed before d - A ~ . The other place is available for short-term use, which is thus charged at a higher rate. Those jobs completed after d - A l can only be stored in this place. Technically, we chose c 1 = l/f3 and c 2 = 1. Accordingly, in our experiments we chose b 2 = 0 and b 1 = A I ( 1 - 1/7~-). These selections made the function f ( x ) be continuous at the point x = A l and take zero value when x = 0. The tardiness function g(x) consists of two parts. If a job is late but is not too late (completed before a threshold d + A2), then it will be penalized by a quadratic cost function. However, if a job is later than the threshold, it will be penalized by an exponential cost function, g(x) reflects the structure of tardiness penalties of problems where large lateness is highly undesirable. The coefficient c 3 was selected to be (2 log A 2 ) / A 2 in our experiments which made g(x) be continuous at the point x = A 2.
Table 4 C o m p u t a t i o n a l results o f A l g o r i t h m 1 ( A 1 ) a n d A l g o r i t h m 2 ( A 2 ) when d - ~ P , n
10 20 30 40 50 60 70 80 90 100 200
Ave. J (X:1012)
Ave. C P U (sec.) AI
A2
A1
A2
0.88 7.68 25.24 62.36 125.14 215.06 346.90 515.94 730.96 988.76 8193.58
0.19 0.42 0.86 1.56 2.46 3.54 4.98 6.58 8.08 9.86 41.80
0.081 0.855 3.324 8.753 23.581 47.693 98.372 174.710 283.581 447.211 12951.654
0.081 0.855 3.324 8.753 23.581 47.693 98.372 174.710 283.581 447.211 12951.654
X. Cai et al./ European Journal of Operational Research 98 (1997) 154-168
163
Table 5 Computational results of Algorithm 1 (A1) and Algorithm 2 (A2) when d = 3 Pn n
10 20 30 40 50 60 70 80 90 100 200
Ave. CPU (sec.)
Ave. J (X 107)
A1
A2
A1
A2
1.20 10.16 33.66 81.40 164.32 278.04 461.44 678.10 946.68 1313.30 11873.85
0.12 0.56 1.16 2.06 3.22 4.66 6.56 8.42 10.56 13.32 54.52
0.093 0.415 1.077 1.954 3.712 5.971 9.517 14.174 19.763 25.367 216.431
0.093 0.415 1.077 1.954 3.712 5.971 9.517 14.174 19.763 25.367 216.431
Other parameters were selected as follows: p~ were uniformly sampled from an integer distribution in the range [1,200], e i and t i in the range [1, 30], 1
2
a i and /3 i in [0, 500], a n d A l = ~Pn a n d A 2 = ~Pn. F o r e a c h n, f i v e p r o b l e m i n s t a n c e s w e r e g e n e r a t e d and s o l v e d , a n d the a v e r a g e c o m p u t a t i o n a l results are r e p o r t e d h e r e . T h e a l g o r i t h m s w e r e c o d e d in S U N F O R T R A N a n d run o n a S U N S P A R C M P 6 7 0 . In the f o l l o w i n g w e c l a s s i f y our c o m p u t a t i o n a l r e s u l t s into t w o g r o u p s , o n e f o r a g r e e a b l e p r o b l e m s w h e r e a s the o t h e r f o r g e n e r a l p r o b l e m s w i t h o u t the a g r e e a b l e condition. 4.1. A g r e e a b l y w e i g h t e d p r o b l e m s
Note that Algorithm 1 can find the exact optima for these problems. Hence, the solutions obtained by
Algorithm 1 can be used as a basis to evaluate the performance of Algorithm 2. Results on problems with d as a fixed constant and a decision variable are reported below respectively. 4.1.1. R e s u l t s o n p r o b l e m s w i t h d a s a f i x e d Constant 1 3 Problems with d = ~Pn and d = 3Pn were solved respectively. The results are summarized in Tables 4 and 5, from which we can observe that, for all the problem instances tested, the solutions found by Algorithm 2 are exactly optimal. In fact, in addition to the results reported here, we also solved a large number of problems with different d and different parameter configurations, and surprisingly found that Algorithm 2 never failed to find an exact optimum for any problem instance we randomly generated. This justifies the theoretical result that the solution
Table 6 Computational results of Algorithm 1 (A1) and Algorithm 2 (A2) when d is decision variable n
10 20 30 40 50 60 70 80 90 100 200
Ave. CPU (sec.)
Ave. J (X 104)
AI
A2
A1
A2
1.46 12.50 41.36 100.76 274.52 419.64 554.20 825.38 I 156.22 1595.96 12729.00
0.16 0.64 1.38 2.50 3.98 5.66 9.70 10.24 13.16 16.02 65.68
0.027 0.108 0.211 0.384 0.597 0.809 1.101 1.430 1.770 2.161 8.968
0.027 0.108 0.211 0.384 0.597 0.809 1.101 1.430 1.770 2.161 8.968
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Table 7 Computational results of Algorithm 2 (A2) and Complete Enumeration (C.E.) when d = 3 P. n
3 4 5 6 7 8 9 10 a
Ave. CPU (sec.)
Ave. J (X 104)
Ave. R (%)
A2
C.E.
A2
C.E.
0 0.02 0.04 0.04 0.06 0.08 0.10 0.10
0.02 0.12 0.90 6.70 81.78 794.56 8629.36 116078.50
17.612 27.038 60.137 48.284 89.165 139.094 114.492 120.629
17.612 26.885 53.727 45.918 85.469 132.025 113.256 106.243
0 0.57 11.93 5.15 4.32 5.35 1.09 13.54
a Computational results of one instance only.
actually uses large fixed penalties to force the optimal schedule to be W-shaped.
found by Algorithm 2 is nearly optimal or even exactly optimal (when the problem solved has a V-shaped optimum). Unfortunately, as we discussed in previous sections, there do exist some problems for which there is no V-shaped optimal schedule, and thus Algorithm 2 cannot secure any optimal solution in these cases. Consider an example, in which n--5, d = 20, e i = 1, t i = 2, V i , and Pi, °li and ~i are given by Table 3. Note that this problem satisfies the agreeable condition. Under the cost functions given by (4.1) and (4.2), it is not hard to see that any optimal schedule for the problem must have job 5 finished at the due date, jobs 1 and 2 completed early, and jobs 3 and 4 late. This is a typical W-shaped schedule, which has an overall cost J ( A * ) = 2 4 . 4 9 1 2 . The best V-shaped schedule Av of this problem will schedule job 5 early. This alone has resulted a fixed earliness cost greater than J(A*). This example
4.1.2. Results on problems with d as a decision variable
Table 6 summarizes the results obtained by Algorithms 1 and 2 when d is a decision variable. Again, we can see that Algorithm 2 had actually obtained the exact optimal solutions for all the problem instances tested. Furthermore, Algorithm 2 was computationally efficient, which was able to generate optimal solutions for problems with up to 200 jobs quickly. 4.2. Results on general problems
As discussed in Section 3.2, Algorithm 2 can be used as a heuristic to solve the general problem P1 without the agreeable condition. To evaluate the
Table 8 Computational results of Algorithm 2 (A2) and Complete Enumeration (C.E.) when d is decision variable n
3 4 5 6 7 8 9 I0 a
Ave. CPU (sec.)
Ave. J (X 104)
Ave. R (%)
A2
C.E.
A2
C.E.
0 0.02 0.04 0.04 0.08 0.10 0.12 0.20
0.02 0.16 1.36 9.80 118.92 1155.24 12521.50 169903.50
0.181 0.563 0.765 0.787 2.265 2.041 2.925 3.536
0.156 0.545 0.733 0.742 2.179 1.808 2.697 3.321
a Computational results of one instance only.
16.03 3.30 4,37 6.06 3.95 12,89 8,45 6,47
X. Cai et al./ European Journal of Operational Research 98 (1997) 154-168
quality of the solutions obtained by this heuristic, the approach of complete enumeration was used to generate the optimal solutions for comparison. The results obtained are reported in Tables 7 and 8, from which the following observations can be made: (a) Algorithm 2 is efficient - the computing time it required was almost negligible for problems with small n. (b) The solutions found by Algorithm 2 are reasonably good. We believe, based on Corollaries 2 and 3, that the solutions found by Algorithm 2 could become better and better as n grows. (c) The time requirement of complete enumeration increased rapidly - it became almost impossible to use it for problems with n > 10.
5. Concluding remarks This paper examines the problem of scheduling n jobs on a single machine to minimize the total earliness and tardiness penalty under a common due date. We have addressed a general model in which earliness penalty and tardiness penalty are arbitrary non-decreasing functions. In addition, the model includes two important features that commonly appear in realistic problems: 1) earliness and tardiness are penalized with different weights which are j o b dependent; and 2) the earliness (or tardiness) penalty consists of two parts, one being a variable cost while the other being a fixed charge once a job is early (or tardy). The model provides a flexible performance measure to better reflect the need of decision making in practice. We have shown that an optimal schedule is Wshaped when the problem is agreeably weighted. Moreover, we have shown that there exists at least one V-shaped schedule whose relative error of performance approaches zero as n increases under some conditions on the bounds of parameters. We have highlighted certain cases satisfying these conditions, including the one in which all jobs have the same earliness weight and the same tardiness weight. The conditions are not hard to satisfy in a practical sense since the parameters in real-world problems usually have an upper bound. Two algorithms have been provided to solve the problem. Under the agreeable
165
condition, the algorithms can generate an optimal schedule in time O(nZPn), or a sub-optimal solution in time O(nPn). For problems without the agreeable weight condition, we have suggested to extend the O(nPn) algorithm as a heuristic procedure to find approximate solutions. Numerical results are reported, from which we see that the O(nPn) algorithm is very effective in terms of computational efficiency as well as the quality of solutions. Note that it is easy to further generalize the model to include a general due date assignment cost h(d) (when d is a decision variable) a n d / o r a flow time cost in the performance measure. The main results we have obtained will remain basically unchanged in these generalized models. A nontrivial future work is to consider the case where multiple machines are available to process the jobs. Another important work is to consider the problem of finding efficient schedules when d is a decision variable, which are good in terms of minimizing both total earliness/tardiness cost and due date assignment cost (cf. Bagchi 1989; De, Ghosh, and Wells 1992). Since our algorithms are based on dynamic programming which actually give the optimal schedules corresponding to all due date values, this bi-criteria problem can be tackled without much difficulty. Theoretical analysis to further relax the agreeable condition is also interesting. In general, it has been known that there exist some problem instances for which no W-shaped schedules (and therefore no V-shaped schedules) are optimal. Nevertheless, it may be possible to derive some conditions thal are looser than the agreeable condition. Besides, the heuristic we have proposed for solving general problems without the agreeable condition should be further evaluated, provided that some kind of approach can be devised to evaluate the optimality of solutions for large problems.
Acknowledgements We wish to thank the referees for their helpful comments and suggestions. We are particularly indebted to an anonymous referee who indicated the possibility of generalizing the results in an earlier version of the paper.
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Appendix A P r o o f of Theorem 3. Let Si* and Ci* be respectively the starting time and completion time of job i, i = 1, 2 . . . . . n, under h*. If the problem has a Vshaped optimum, namely, h ° = h *, then R(A ~) = 0 and Eq. (2.2) holds. It is thus evident that we only need to consider the situation that the optimal schedule A* is exactly W-shaped, in which there are three consecutive jobs i, j and k such that pj > Pi and Pj > Pk and job j is the straddling job, namely, Ci = S j < d a n d S~ = C j ~>d. We will now show that J ( A °) < J ( A * ) + O,
(A.1)
through examining the following cases respectively, in which we let z = Cf - d: Case 1. If Cf - p ~ < d and C f >2t, we interchange jobs i and j in A* to construct a new schedule h (al~, in which jobs i and j will become tardy and early respectively. Thus, one can see that j(/~(al)) __ j ( A,~ )
-- OLi -- e i f ( p j -- Z) -- ~ j -- t j g ( Z) -
-
J(A (bl)) - J ( h * )
= tSk + tkg( z - - p j + pk) + ~j + tjg( z + pk) -- ~j-- t i e ( z ) -- ~k-- tkg( z + Pk) = (tkg ( z - p j +(tj-
(
+ ejf( P i - z) -- e i f ( pj -- z) + (t i -- t j ) g ( Z ) <~ ( a j - ai) - ( ~ j - ~i) + ( t i - t j ) g ( z)
tk)g ( z +Pk)
<~ ( t k g ( z) - tjg( z ) ) + ( t j - t k ) g ( z + Pk) ~< 0 ~< 69.
(A.3)
If h (bl) is still not V-shaped, job j can be further interchanged with the job sequenced immediately after it to get a new schedule h (b2~. Similarly, this procedure can be continued until a V-shaped schedule i b with J(A b)~< --- ~
a' = J ( X )
- J(;~*)
= t~i + tjg( z-- pi) + t~i + tig( z)
(since ej ~< e i and f ( . ) is nondecreasing) ~< O.
+Pk) - t j g ( z ) )
is derived. Again, this implies (A.1) as h ° is the best V-shaped schedule. Case 3. If C f - Pi > d, Sf + Pk < d and C f > d, then the interchange of jobs i and j will yield a schedule h' in which jobs i and j are all tardy. Thus, it can be seen that
= Otj + ejf( P i - Z) + fii + tigi(Z)
= ( .j
sequence h (b~), in which jobs j and k are all tardy. Again, bearing in mind that pj > Pk, tj <~t k, and g(. ) is a non-decreasing function, we can see that
(A.2)
Moreover, after interchanging jobs i and j, if the processing time of job j is still greater than that of the job immediately before it, we further interchange job j and that job to construct a new schedule h (a2). Clearly J(A (a2)) ~ J(A (al)) (see the proof for Theorem 2). Such a procedure can be continued until a V-shaped schedule A~ is obtained, which satisfies
J( h a) ~< "-- ~ d and Cj* > d, we interchange jobs j and k in A* to construct a new
-- a i -- e i f ( pj -- Z) -- [~j -- tjg( Z) = ( ~ i - ai) + ( t i - t j ) g ( z ) + tjg( Z--Pi) -- eif( P j " Z).
(A.4)
Moreover, the interchange of jobs j and k will yield a schedule X' with job j tardy and job k early. This gives us a" = J ( X ' ) - J ( h * )
= a k + ekf (pj-
z-pk)
+ ¢3j + t j g ( z +Pk)
-- ~j-- tjg( z) - ~k-- t , g ( z + Pk) =
( ak --
~k) + (tj -- tk)g ( Z +Pk)
+ekf(pj--z--pk
) - - t i g ( z ).
(A.5)
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
Combining (A.4) and (A.5), we have
since f is a non-decreasing function. Similarly, if job i is mith tardy job after the job j~, then
A' + A "
= ( ~ i - ~k)-- (OLi- OLk) ~- (ti-- tj)g(z) +
g( Lj) = g ( C f - d) >1 g(mjPmin) , since g is a non-decreasing function. Hence,
tj(g( z -Pi) - g(z))
JV(fi*) =
+ ( e k f ( pj -- z -- pk) - e l f ( pj - z ) )
~_~ e i f ( Li) + iEE(A*)
+(tj--tk)g(z+Pk)
>/
( J~i- ~k) -- (OLi- Ogk) %- ( t i - t j ) g ( Z ) +(ekf( pj- z-Pk)
167
E
tjg( Lj)
j ~ T(A*)
E
eif{miPmin}
iE {E(A* ) - {J's}}
+
- e,f( p j - z ) )
E
tjg(mjPmin)
j ~ {r(x* )-- {Js}}
(since tj ~< t k andg( z - Pi) <~g ( z ) )
= Ew'q,
= (t~i--~k) -- ( a i - - ak) + (ti-- t j ) g ( z )
where w and q are, respectively, a certain element in the set ~ and ~ as defined in Section 2. (~w- q) is the sum of the product of two sequences, each having n - 2 elements (the jobs Js and ~ are excluded) from the set ~ or ~, According to Hardy, Littlewood and Polya (1934, p.261), it is evident that
+(e k- ei)f(pj-- z--pk) + ei( f ( Pj -- z -- P, ) -- f ( Ps -- Z ) ) <~ ( ~ i - ~Sk) - ( % - - ak) + (ti-- t j ) g ( z) + ( e, -- e j ) f ( pj -- z --Pk),
n--2
JV(a*)>~ ~.,wiq,-l-i.
(since f ( pj -- z --Pk) ~
i=l
and e i > %) ~< O,
(A.6)
which implies either A'~< O or A"~< O, namely J(X)
E a,+ E ~i >~r. iEE(~*) i~r(a*)
(A.7)
NOW examine J~(A*). Let j~ denote the straddling job and j{ denote the job sequenced immediately before the job j~, under A*. Clearly, if job i is mi-th early job before the job f~, then f ( Li) = f ( d -
Ci* ) >/ f ( miPmin),
(A.8)
This together with (A.1) and (A.7) proves the theorem. []
References Bagchi, U. (1989), "Simultaneous minimization of mean and variation of flow time and waiting time in single machine systems", Operations Research 37, 118-125. Bagchi, U., Chang, Y.L., and Sullivan, R.S. (1987), "'Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date", Naval Research Logistics 34, 739-751. Baker, K.R., and Scudder, G.D. (1990), "Sequencing with earliness and tardiness penalties: A review", Operations Research 38, 22-36. Cai, X. (1995), "Minimization of agreeably weighted variance in single machine systems", European Journal of Operational Research 85, 576-592. Cheng, T.C.E., and Gupta, M. (1989), "Survey of scheduling research involving due date determination decisions", European Journal of Operational Research 38, 156-166. De, P., Ghosh, J.B., and Wells, C.E. (1992), "On the minimization of completion time variance with a bi-criteria extension", Operations Research 40, 1148-1155. De, P., Ghosh, J.B., and Wells, C.E. (1993), "On the general solution for a class of early/tardy problems", Computers & Operational Research 20, 141-149.
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Eilon, S., and Chowdhury, I.G. (1977), "Minimizing waiting variance in the single machine problem", Management Science 23, 567-575. Federgruen, A., and Mosheiov, G. (1993), "Simultaneous optimization of efficiency and performance balance measures in single-machine scheduling problems", Naval Research Logistics 40, 951-970. Hall, N., and Posner, M.E. (1991), "Earliness-tardiness scheduling problems, I: Weighted deviation of completion times about a common due date", Operations Research 39, 836-846. Hall, N., Kubiak, W., and Sethi, S.P. (1991), "Earliness-tardiness scheduling problems, II: Deviation of completion times about a restrictive common due date", Operations Research 39, 847-856. Hardy, G.H., Littlewood, J.E., and Polya, G. (1934) Inequalities, Cambridge University Press, London. Hoogeveen, J.A., and van de Velde, S.L. (1991), "Scheduling around a small common due date", European Journal of Operational Research 55, 237-242. Kahlbacher, H.G. (1993), "Scheduling with monotonous earliness and tardiness penalties", European Journal of Operational Research 64, 258-277. Kise, H., Ibaraki, T., and Mine, H. (1978), " A solvable case of the one-machine scheduling problem with ready and due times", Operations Research 26, 121-126.
Krieger, A.M., and Raghavachari, M. (1992), "V-shape property for optimal schedules with monotone penalty functions", Computers & Operations Research 19, 533-534. Kubiak, W. (1993), "Completion time variance minimization on a single machine is difficult", Operations Research Letters 14, 49-59. Lawler, E.L. (1976), "Sequencing to minimize the weighted number of tardy jobs", Rev. Automat. InJbrmat. Rec. Opnl. 10, 27-33. Lawler, E.L. (1977), " A pseudopotynomial algorithm for sequencing jobs to minimize total tardiness", Annals of Discrete Mathematics 1,331-342. Lawler, E.L., and Moore, J. (1969), "A functional equation and its applications to resource allocation and sequencing problems", Management Science 16, 77-84. Lee, C.-Y., Danusaputro, S.L., and Lin, C.-S. (1991), "Minimizing weighted number of tardy jobs and weighted earlinesstardiness penalties about a common due date", Computers & Operations Research 18, 379-389. Pinedo, M. (1983), "Stochastic scheduling with release dates and due dates", Operations Research 31,559-572. Raghavachari, M. (1986), "A V-shape property for optimal schedule of jobs about a common due date", European Journal of Operational Research 23, 401-402.
JOURNAL OF OPERATIONAL RESEARCH
ELSEVIER
European Journal of Operational Research 98 (1997) 154-168
Theory and Methodology
Scheduling about a common due date with job-dependent asymmetric earliness and tardiness penalties X. Cai *, V.Y.S. Lum, J.M.T. Chan Department of Systems Engineering and Engineering Management, The Chinese Universityof Hong Kong, Shatin, N.T., Hong Kong Received October 1994; revised August 1995
Abstract This paper is concerned with the problem of scheduling n jobs with a common due date on a single machine so as to minimize the total cost arising from earliness and tardiness. A general model is examined, in which earliness penalty and tardiness penalty are, respectively, arbitrary non-decreasing functions. Moreover, the model includes two important features that commonly appear in practical problems, namely, 1) earliness and tardiness are penalized with different weights which are job-dependent, and 2) the earliness (or tardiness) penalty consists of two parts, one is a variable cost dependent on the length of earliness (or tardiness), while the other is a fixed cost incurred when a job is early (or tardy). This model provides a general and flexible performance measure for earliness/tardiness scheduling, which has not been addressed before. We establish a number of results on the characterizations of optimal and sub-optimal solutions, and propose two algorithms based on these results. The first algorithm can find, under an agreeable weight condition, an optimum in time O(n2P,,), and the second algorithm can generate a sub-optimum in time O(nP,,), where Pn is the sum of the processing times. Further, we derive an upper bound on the relative error of the sub-optimal solution and show that, under certain conditions, the error tends to zero as n increases. Computational results are also reported to demonstrate the effectiveness of the algorithms proposed. © 1997 Elsevier Science B.V.
Keywords: General earliness and tardiness costs; Manufacturing; Scheduling/sequencing; Due date assignment; Exact and approximate algorithms
1. Introduction C o n s i d e r the scheduling situation w h e r e a single m a c h i n e is to process a set o f n i n d e p e n d e n t j o b s under a c o m m o n due date d. A s s o c i a t e d with each j o b i, i = 1, 2 . . . . . n, are a p r o c e s s i n g time Pi, an earliness w e i g h t el, a tardiness w e i g h t t i, a fixed earliness penalty a i, and a fixed tardiness p e n a l t y / 3 i. It is a s s u m e d that Pi, i = 1, 2 . . . . . n, are positive integers w h e r e a s all other parameters are n o n n e g a -
tive numbers. All j o b s are r e a d y for processing at time zero. The m a c h i n e is available continuously f r o m time zero and can process at m o s t one j o b at a time. J o b splitting and p r e e m p t i o n are not allowed. T h e p r o b l e m is to find a schedule A to process the n j o b s so as to m i n i m i z e the total cost arising f r o m earliness and tardiness, e x p r e s s e d as
~_, [ a i + e i f (L;)]
J(A)=
+ * Corresponding author. 0377-2217/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 5 ) 0 0 2 5 3 - 7
Y'~ [ 1 9 i + t i g ( L i ) ] ,
(1.1)
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
where L i = I Ci - d l denotes the absolute lateness of job i from the due date, C~ is the completion time of job i under A, and E(A)={i:C i
155
with asymmetric earliness and tardiness weights have received some attention in recent years. The first work that explicitly investigates such a model was reported by Bagchi, Chang and Sullivan (1987), in which they addressed a quadratic penalty function. De, Ghosh and Wells (1993) considered a problem with g ( L ~ ) = f ( L i) = L r, where r is a positive integer. Federgruen and Mosheiov (1993) and Kahlbacher (1993) examined respectively more general penalty functions (quasiconvex or unimodal functions). While these researches have provided significant insights on the investigation of asymmetric earliness and tardiness penalties, there is still a common restriction imposed on these models, which assumes that all jobs should have the same earliness weight and the same tardiness weight, namely, e i = ej and t i = t j for i , j = l , 2 . . . . . n. On the problem involving job-dependent asymmetric earliness and tardiness weights, the work of Lee, Danusaputro and Lin (1991) should be noted. They have derived some excellent results including a pseudopolynomial algorithm under certain conditions like the ratio of processing times and weights is agreeable. Nevertheless, their model is concerned only with a linear penalty function. Besides the restriction that all jobs should have a common earliness weight and a common tardiness weight, another assumption commonly made in prior researches is that the cost for the tardiness/earliness of a job should approach zero if the deviation of its completion time from the due date tends to zero. Noting that a fixed cost is incurred once a job is late no matter how late it is, Lee, Danusaputro and Lin (1991) have proposed to incorporate into the objective function a fixed penalty, which depends upon the jobs which are tardy, Nevertheless, their model is concerned with fixed tardiness penalty only. Notice that there also exists a fixed earliness cost once a job is completed early. This includes, for example, the basic cost for removing an early job to a designated warehouse and other fixed administrative costs. In this paper we study the problem P1. This is a general model, which subsumes most single-machine. common due-date problems having been studied in the literature and provides a more general and flexible performance measure to better reflect the need of decision-making in practice. To the best of our knowledge, such a general model has not been
X. Cai et a l . / European Journal of Operational Research 98 (1997) 154-168
156
addressed before. To find an optimal solution for this problem is difficult. Many special instances of the model have been shown to be NP-complete. For example, if e i = t i = 1 , of i = J~i ~-" 0 , g ( Z i) : f ( L i) - L i, and d >~ P,( = ET= t Pi), the problem is known to be equivalent to the completion time variance (CTV) problem, which has been shown to be NPcomplete by Kubiak (1993). We conjecture that the general problem P1 is NP-complete in the strong sense and thus unlikely to be solved even by a pseudopolynomial algorithm. Considering the inherent difficulty in finding an exact optimal solution, a primary objective of this paper will be to investigate the characterizations of optimal and sub-optimal solutions so that algorithms can be developed which can find an optimal solution under certain conditions or a sub-optimal solution efficiently. Our main results include a W-shape property for an exact optimal schedule and a V-shape property for a sub-optimal schedule. The concept of V-shaped schedule was firstly introduced by Eilon and Chowdhury (1977) in the context of the CTV problem. The existence of a V-shaped optimum has become an important optimality property in many earliness/ tardiness problems and served as the basis for development of effective (pseudopolynomial) solution methods, see Baker and Scudder (1990), Raghavachad (1986), Krieger and Raghavachari (1992), Lee, Danusaputro and Lin (1991), Hall and Posner (1991), Hall, Kubiak and Sethi (1991), De, Ghosh and Wells (1992), etc. The structure of a W-shaped schedule is similar to a V-shape, except that the job straddling the due date (or the tip of the V-shape) is unclear; see Section 2 below for a definition. Hoogeveen and van de Velde (1991) have found that an optimum for an earliness/tardiness problem with a linear penalty function possesses a W-shaped structure and, consequently, applied the optimality property to devise a solution algorithm with pseudopolynomial time complexity. We will show in the sequel that the general problem P1 will have a W-shaped optimum when it is agreeably weighted, in the sense that Pi > Pj implies e i <~ ej and t i <~ (j, V i, j. A motivation for this agreeable condition is, for instance, the common practice that a smaller job is charged at a higher rate, which in turn makes it relatively more important and thus the penalty of missing its due date become heavier. Agreeable conditions of this nature have _
2
often been used in many scheduling studies in the literature, see Lawler (1976, 1977), Kise, Ibaraki and Mine (1978), Pinedo (1983), Lee, Danusaputro and Lin (1991), Cai (1995), etc. We will also show, in an example, that some problem instances of P1 have no V-shaped optimum even under the agreeable condition. Thus it is impossible to establish a V-shape property for optimal schedules. To facilitate the development of more efficient solution algorithms, we will turn to examining a V-shape property for schedules that are sub-optimal. We will show that there exists at least one V-shaped schedule, the relative error of which approaches zero as n increases under some conditions. Specifically, the conditions are satisfied when ]e i ej[f(pj), I t i - t j l g ( p j ) and I ( t x i - a j ) - ( / 3 i /32) 1, Vi, j, are bounded above. Note that these conditions are naturally satisfied under the assumption that all jobs have the same earliness weights and the same tardiness weights (including the fixed penalties), which are usually made in prior researches as we reviewed above. Practically, the conditions are reasonable since the problem parameters are usually bounded above in a practical situation. On the basis of the W-shape property and the V-shape property, we will develop two algorithms to solve the problem. Under the agreeable condition, the first algorithm can find an optimal solution in time O(n2P,). The second algorithm, which is n times faster than the first one, can generate a sub-optimal schedule with relative error approaching zero as n grows. Both algorithms will be shown to be applicable to problems with d as a fixed constant or a decision variable. In the general case without the agreeable weight condition, we suggest to extend the second algorithm as a heuristic procedure to find approximate solutions. The algorithms proposed will be evaluated by computational experiments, for both cases with and without the agreeable condition.
2. Characterizations of solutions
We n o w describe some important results concerning the basic characterizations of optimal and suboptimal schedules. The proof for the result below is straightforward, since if a schedule has a non-zero idle time between two consecutive jobs, then the gap
X. Cai et al./ European Journal of Operational Research 98 (1997) 154-168
can be closed towards the due date to generate a schedule without the idle time, which is not worse than the schedule with the idle time (or even better if f and g are strictly increasing functions). T h e o r e m 1. There exists an optimal schedule f o r P1 which has no idle time between any two consecutive jobs. The result below is on the W-shape property of the problem. To be more precise, let us introduce the following definition first, in which S~ denotes the starting time of job i, namely, Si=Ci-Pi
for i = 1 , 2 . . . . . n.
Definition 1. A schedule h is said to be W-shaped with respect to the processing times if pj >I-Pk when Cj < C k < d, and Pj < Pk when d <~Sj < S k. Notice that in a W-shaped schedule as defined above, it is possible to have a job m with S m < d <~ C m, called straddling job, such that P m > P,,, and Pm > P r o " , where jobs m' and m" are, respectively, the jobs sequenced immediately before and after job m.
T h e o r e m 2. There exists an optimal schedule f o r P1 which is W-shaped with respect to the processing times if p i < p: implies e i >1 e: and t i >~ tj, V i, j. Proof. Suppose that A* is an optimal schedule. Clearly if it is not W-shaped, it must have three consecutive jobs i, j and k such that pj > Pi, Pj > Pk, and c? + pj= Cf < d or
d
S; = s ; - p .
where Cq and Sq are, respectively, the completion time and the starting time of job q under A*. When C j < d, we may construct a new schedule /X' by interchanging jobs i and j in h*. In this case, E(X) = E( A* ) and T(A') = T(A* ). Thus, letting z = d - C j , we can see that
=J(x) -J(A °)
=
157
e i f ( Z ) + e j f ( Z + Pi) -- e i f ( Z + pj)
-- e J ( Z) ~ O,
(2.1)
since f ( z + p ) > ~ f ( z + p i ) > ~ f ( z ) and ei>~ ej because of the agreeable condition. Similarly, when d ~< S f , we can construct h" by interchanging jobs j and k and show A ( h " ) ~ 0. If ~' or X' is still not W-shaped, we may further interchange a pair of consecutive jobs similar to the above. Clearly a series of interchanging operations will finally result in a W-shaped schedule h ¢ with A(h¢) ~< 0, which indicates either a contradiction with the fact that h* is optimal (if A(A/) < 0) or the optimality of h f (if a(,V) = o).
[]
Observing that the agreeable condition is satisfied when all jobs have the same earliness weight and the same tardiness weight, we have: Corollary 1. There exists a W-shaped optimal schedule f o r PI if e i = ej and t i = tj, V i , j. The results above show that P1 has an optimal schedule which is W-shaped when the problem is agreeably weighted or has an equal earliness weight and an equal tardiness weight. Since V-shape is a special case of W-shape, the optimal schedules for some instances of P1 can actually be V-shaped. However, on the other hand, it may also be shown that there do exist some instances which do not have any V-shaped optimal schedule, e v e n if they m e e t the agreeable condition (see an example given in Section 4.1.1). Considering that the problem in general may have no V-shaped optimum, we now turn to examining the sub-optimality of the best V-shaped schedule. We will first describe, in the theorem below, an upper bound on the relative error of the best V-shaped schedule. Corollaries will be then deduced from the theorem, which show that, under certain conditions, the relative error tends to zero or even equals zero. To be more precise, let us lay down the following definition first, Definition 2. A schedule h is said to be V-shaped with respect to the processing times if pi >~Pk when C j < C k~
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
158
Let A* be a W-shaped optimum for P1 and h v be the best schedule in ~-, the set of all V-shaped schedules. Further, let O = m a x i,j{ 1(o~ i - ogj) - ( ]3 i - ~ j ) 1} +max/d{[ e i -- ejl f ( p j ) } + m a x i o { ] t i -- tj ] g( pj)}. F = E~= l[min{ai, /3i}] - m a x i { m i n { a i, /~i}}. w~ = The i-th smallest element in the set 7 f = {e l, e 2 . . . . . e,, t I, t 2 . . . . . tn}. q~= The i-th smallest element in the set ~ = {f(Pmin), f(2pmin) . . . . ; f ( n P m i . ) , g(Pmi~)' g(2Pmin) . . . . . g(nprnin)}, where P~in is the minimum processing time. The proof for the following theorem is given in Appendix A.
Theorem 3. There exists a V-shaped schedule h ° f o r P1 such that
R(v) =
J(a°)-J(a J(a*)
*)
o n--2
1~+ E i = l W i q n _ l _ i
Letting O= maxi,j{ l( oli-- otj) -- ( t~i -- t~j) l }, we have:
Corollary 3. If e;---ei=e and t i = t j = t ,
V i , j,
where e and t are constant, then 0
R( h ~) ~<
.
(2.3)
F-}- E72~Wiqn_l_i Furthermore, if e > O, t > O, f and g are increasing functions, and 0 is b o u n d e d above, then P1 has a V-shaped schedule which is asymptotically optimal in the sense that R(A °) ~ 0 as n increases.
It can be seen that R(A v) = 0 if ofi : a j and /3i =/3j in addition to e~ = ej and t~ = tj. Hence, we have:
Corollary 4. There exists a V-shaped schedule which
(2.2)
is exactly optimal f o r P1 i f e i = ej, t i = tj, a i = aj, and ~i = ~j, V i, j.
The corollaries below follow directly from Theorem 3.
The following corollary comes from the observation that 0 = 0 when the fixed earliness penalty for each job is equal to its fixed tardiness penalty, namely, a i = fli, V i.
if p i
implies e i >~ ej and t i > tj, V i , j.
Corollary 2. Under the agreeable condition, R(A v) 0 as n increases i f f and g are increasing f u n c tions, w i > O , f o r i = 1 , 2 . . . . . n - 2 , and 6 ) < . M , where M is a constant independent o f n.
Corollary 5. A s s u m e e i = ej and t i = tj, V i , j. Then a V-shaped schedule is exactly optimal f o r P1 i f Oli ":" ~i' V i .
R e m a r k . Note that the conditions that f and g are increasing functions and w[are positive numbers are satisfied by most of the earliness/tardiness models investigated in the literature. Examples include the cases f = g = [Li[, f = g =L2i, e i = t i = 1, etc. The condition that (9 ~< M is not a tight one in a practical sense, a s in practical problems the values of the parameters involved are usually bounded above. Corollary 2 actually indicates that the best V-shaped schedule Av asymptotically approaches the optimal schedule A* if the variations in the parameters of any pair Of jobs do not dominate the sum of the fixed and variable penalties of other jobs when the number of jobs increases. This result is particularly useful for large problems.
In general, the strucfure of optimal schedules for P1 without the agreeable weight condition remains to be an open question. When e; and t i can take arbitrary values, it is not hard to devise an example which does not have any W-shaped optimum (such an example can be constructed by assigning very large weights to certain jobs with large processing times).
3. Algorithms According to Theorem 1, we know that an optimal schedule for P1 should arrange the jobs about the due date d without inserting any idle time be-
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3) R( A”) is bounded above by (2.3) if ei = ej and ti = tj, Vi, j; 4) h” is asymptotically optimal in the sense that R( h”) -+ 0 as n increases if the conditions of Corollaries 2 and 3 are satisfied; and 5) h” is exactly optimal if the conditions of Corollaries 4 and 5 are satisfied. Remarks. 1) Algorithms 1 and 2 described above are proposed to solve probelms in which the due date d is a given constant. Nevertheless, they are also applicable to problems in which d is a decision variable. Such a situation may occur during sales negotiations of a company with its customers, when the company may hope to locate an optimal due date so as to minimize its overall cost. 2) When d is decision variable to be determined, we may apply the following procedure to determine an optimal or sub-optima1 d: Let d = P,, -and apply Algorithm 1 or 2 to find A* or h”. Accordingly, let the optimal or sub-optimal starting time be r& or r&,. (ii) Rearrange h * or h” through shifting the starting times of all jobs to the left by a constant r& or r,&. Let the optimal (sub-op timal) due date be d * = P, - riin (d” = P, rGin). This procedure is justifiable since J(h) actually only depends on the relative distance of job completion times from d and d = P, has given us sufficient freedom to schedule the jobs to minimize J(h). Step ii) above chooses the due date that corresponds to the schedule starting at time zero. This will result in a shortest due date without changing J( h * ) or J( h”). 3) The conclusions in Theorems 4 and 5 remain unchanged when the procedure above applies. 3.2. Heuristic for PI without the agreeable
condition
Theorems 2 and 3 are the basis of the algorithms developed above, which indicate that an optimal (or sub-optimal) solution will be W-shaped (or V-shaped) when the problem is agreeably weighted. A W-shaped (or V-shaped) structure makes it possible for us to devise a dynamic programming procedure to search for the required solution. However, for a general problem Pl without the agreeable condition, we are
unable to know whether or not it has a W-shaped optimum or a V-shaped sub-optimum (note that the agreeable condition in Theorems 2 and 3 is a sufficient condition), so in a strict sense these algorithms are not applicable considering they have no guarantee on the optimality of the solutions obtained. Nevertheless, if in practical situations one only aims to find some approximate solutions, we can still use these algorithms as heuristic procedures. Considering that even if a best W-shaped schedule might not be optimal in the general case, we suggest to generalize Algorithm 2, which is IZ times faster than Algorithm 1, as a heuristic for solving the general problem Pl. Moreover, considering that pi/( e, + ti> represents, in some sense, the relative importance of job i and that a more important job should be scheduled nearer the due date (this is essentially what the V-shape property suggests), we propose to label the jobs so that Pl
-<
-< el + tl
P2
...
Q-
e2 + t2
PTl e, + t,
*
(3.4)
Algorithm 2 can now be applied to generate a solution after the jobs are sorted according to this criterion. Theoretically we have no guarantee on the optimality of the solution found by this heuristic, unless the problem solved happens to have an optimum which is V-shaped with respect to the weighted processing times { pi/(ei + ti)} (in this case the solution found will be an exact optimum). Nevertheless, the time complexity of the heuristic remains to be O(nPJ.
4. Computational results To evaluate the effectiveness of the algorithms proposed above, a number of computational experiments have been carried out, in which problems with the following cost structure were solved:
f(x)
=
g(x)
=
b, + c,x
if x> A,,
b,-t-c,x
ifO
&As
x2
4,
(4.1)
if x&A,, ifO
(4.2)
160
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Furthermore, let pro(r) be the contribution of the jobs in Ni m to the overall objective function (1.1), subject to r being the starting time of these jobs. Clearly, to ensure job m to be the straddling job, the possible values that r can take will be integers contained in [ ai, bi] with a i = max{0, d - P i } and b i = d, where P~ = ~ j ~ N:"Pj" According to Theorem 2, job i, if i v~ m, must be sequenced either the first or the last among all jobs in Nim. Thus, we have a dynamic programming algorithm for solving P1 in which d is a given constant; see Table 1. Theorem 4. The schedule A* found by Algorithm 1 is optimal for P1 under the agreeable condition and the time complexity required by Algorithm 1 is bounded above by O(n2p,).
Proof. The optimality of A* follows directly from Theorem 2 and the principle of optimality of dynamic programming. As for the time complexity, it is obvious that (b i - a i ) < ~ P, steps are needed to calculate p~"(r) for each i and m. Since we need to enumerate i = 1 , 2 . . . . . n and m = l , 2 . . . . . n, the time requirement to compute p ~ ( r ) is bounded above by O(n2p,). This dominates the time complexity of the algorithm. []
shortest job) is not necessarily the straddling job. This is different from a W-shaped schedule in which the straddling job must be the job at the tip (the peak) of the W-shape. Algorithm 1 has made use of this fact in setting the range [ai, b i] for r. To develop an algorithm to search for the best V-shaped schedule Av in the set ~ ' , we use the following result: L e m m a 1. Under the agreeable condition, A" sequences job 1 as: 1) the straddling job; 2) the last job among all the early jobs; or 3) the first job among all the tardy jobs that start no earlier than d. The proof follows from a job interchange argument. From this result we know that the completion time of job 1 will not go beyond the range [ d - p., d + P . + P l ] - Thus, letting N~ be the set of jobs 1, 2 . . . . . and i, we can see that r may only take integer values in the interval [a;, b;], where a i = max(0, d -- Pi - P,}, b i = d + p,~, e i = ]~ pj.
(3.3)
j~N i
3.1.2. An approximate algorithm ,It follows from Theorem 3 that, if one only aims to find an approximate solution, then searching in the set ~- of V-shaped schedules will be enough. This will reduce the time complexity of Algorithm 1 as it is no longer needed to enumerate the straddling job m.
First we have to note that, in a V-shaped schedule, the job lying at the tip of the V-shape (the
We propose the algorithm presented in Table 2. In light of Theorem 3, Lemma 1, and the proof for Theorem 4, we have: Theorem 5. 1) Algorithm 2 requires at most O(nP.) time to find a schedule a"; 2) R(A") is bounded above by (2.2) under the agreeable condition;
Table 2 Algorithm 2 • Given m = 1, compute OT'(r) according to Eq. (3.1) for i = 1, 2 . . . . . n and r E [ai, bi]. where a i and b i are calculated in accordance with Eq. (3.3), subject to the following boundary condition: , +e,f(d-r-p~)
Io
[ill+tlg(r+p,--d) I !, + 0% •
if d - p , < ~ r + p l
if r + p l = d , ifd
Compute: J(A") = minr~ta., b,.l{p],(r)}, then construct A" by a backtracking procedure after the best r is found.
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X. Cai et al./ European Journal of Operutional Research 98 (1997) 154-168
3) R( A”) is bounded above by (2.3) if ei = ej and ti = tj, Vi, j; 4) h” is asymptotically optimal in the sense that R( h”) -+ 0 as n increases if the conditions of Corollaries 2 and 3 are satisfied; and 5) h” is exactly optimal if the conditions of Corollaries 4 and 5 are satisfied. Remarks. 1) Algorithms 1 and 2 described above are proposed to solve probelms in which the due date d is a given constant. Nevertheless, they are also applicable to problems in which d is a decision variable. Such a situation may occur during sales negotiations of a company with its customers, when the company may hope to locate an optimal due date so as to minimize its overall cost. 2) When d is decision variable to be determined, we may apply the following procedure to determine an optimal or sub-optima1 d: Let d = P,, -and apply Algorithm 1 or 2 to find A* or h”. Accordingly, let the optimal or sub-optimal starting time be r& or r&,. (ii) Rearrange h * or h” through shifting the starting times of all jobs to the left by a constant r& or r,&. Let the optimal (sub-op timal) due date be d * = P, - riin (d” = P, rGin). This procedure is justifiable since J(h) actually only depends on the relative distance of job completion times from d and d = P, has given us sufficient freedom to schedule the jobs to minimize J(h). Step ii) above chooses the due date that corresponds to the schedule starting at time zero. This will result in a shortest due date without changing J( h * ) or J( h”). 3) The conclusions in Theorems 4 and 5 remain unchanged when the procedure above applies. 3.2. Heuristic for PI without the agreeable
condition
Theorems 2 and 3 are the basis of the algorithms developed above, which indicate that an optimal (or sub-optimal) solution will be W-shaped (or V-shaped) when the problem is agreeably weighted. A W-shaped (or V-shaped) structure makes it possible for us to devise a dynamic programming procedure to search for the required solution. However, for a general problem Pl without the agreeable condition, we are
unable to know whether or not it has a W-shaped optimum or a V-shaped sub-optimum (note that the agreeable condition in Theorems 2 and 3 is a sufficient condition), so in a strict sense these algorithms are not applicable considering they have no guarantee on the optimality of the solutions obtained. Nevertheless, if in practical situations one only aims to find some approximate solutions, we can still use these algorithms as heuristic procedures. Considering that even if a best W-shaped schedule might not be optimal in the general case, we suggest to generalize Algorithm 2, which is IZ times faster than Algorithm 1, as a heuristic for solving the general problem Pl. Moreover, considering that pi/( e, + ti> represents, in some sense, the relative importance of job i and that a more important job should be scheduled nearer the due date (this is essentially what the V-shape property suggests), we propose to label the jobs so that Pl
-<
-< el + tl
P2
...
Q-
e2 + t2
PTl e, + t,
*
(3.4)
Algorithm 2 can now be applied to generate a solution after the jobs are sorted according to this criterion. Theoretically we have no guarantee on the optimality of the solution found by this heuristic, unless the problem solved happens to have an optimum which is V-shaped with respect to the weighted processing times { pi/(ei + ti)} (in this case the solution found will be an exact optimum). Nevertheless, the time complexity of the heuristic remains to be O(nPJ.
4. Computational results To evaluate the effectiveness of the algorithms proposed above, a number of computational experiments have been carried out, in which problems with the following cost structure were solved:
f(x)
=
g(x)
=
b, + c,x
if x> A,,
b,-t-c,x
ifO
&As
x2
4,
(4.1)
if x&A,, ifO
(4.2)
162
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
Table 3 Pl = 1, a I = 0, /31 = 30,
P2 = 1, a 2 = 0, /32 = 30,
P3 = 1, a 3 = 30, /33 = 0,
f(xp
A1
0
x
g(x)
/ / J r
0
A2
x
Fig. 1. The earliness function f ( x ) a n d the tardiness function
g(x).
P4 = 1, a 4 = 30, /34 = 0,
P5 = 10, a 5 = 50, /35 = 50.
The functions f(x) and g(x) are illustrated by Fig. 1. In our experiments, we restricted 0 ~< c~ ~< c 2 in the earliness function f ( x ) . This corresponds to, for instance, the situation where two types of places are available for storage of finished jobs before the due date d. The use of the first one is charged at a lower rate, which however requires a minimum storage time A~ and thus can only be for jobs completed before d - A ~ . The other place is available for short-term use, which is thus charged at a higher rate. Those jobs completed after d - A l can only be stored in this place. Technically, we chose c 1 = l/f3 and c 2 = 1. Accordingly, in our experiments we chose b 2 = 0 and b 1 = A I ( 1 - 1/7~-). These selections made the function f ( x ) be continuous at the point x = A l and take zero value when x = 0. The tardiness function g(x) consists of two parts. If a job is late but is not too late (completed before a threshold d + A2), then it will be penalized by a quadratic cost function. However, if a job is later than the threshold, it will be penalized by an exponential cost function, g(x) reflects the structure of tardiness penalties of problems where large lateness is highly undesirable. The coefficient c 3 was selected to be (2 log A 2 ) / A 2 in our experiments which made g(x) be continuous at the point x = A 2.
Table 4 C o m p u t a t i o n a l results o f A l g o r i t h m 1 ( A 1 ) a n d A l g o r i t h m 2 ( A 2 ) when d - ~ P , n
10 20 30 40 50 60 70 80 90 100 200
Ave. J (X:1012)
Ave. C P U (sec.) AI
A2
A1
A2
0.88 7.68 25.24 62.36 125.14 215.06 346.90 515.94 730.96 988.76 8193.58
0.19 0.42 0.86 1.56 2.46 3.54 4.98 6.58 8.08 9.86 41.80
0.081 0.855 3.324 8.753 23.581 47.693 98.372 174.710 283.581 447.211 12951.654
0.081 0.855 3.324 8.753 23.581 47.693 98.372 174.710 283.581 447.211 12951.654
X. Cai et al./ European Journal of Operational Research 98 (1997) 154-168
163
Table 5 Computational results of Algorithm 1 (A1) and Algorithm 2 (A2) when d = 3 Pn n
10 20 30 40 50 60 70 80 90 100 200
Ave. CPU (sec.)
Ave. J (X 107)
A1
A2
A1
A2
1.20 10.16 33.66 81.40 164.32 278.04 461.44 678.10 946.68 1313.30 11873.85
0.12 0.56 1.16 2.06 3.22 4.66 6.56 8.42 10.56 13.32 54.52
0.093 0.415 1.077 1.954 3.712 5.971 9.517 14.174 19.763 25.367 216.431
0.093 0.415 1.077 1.954 3.712 5.971 9.517 14.174 19.763 25.367 216.431
Other parameters were selected as follows: p~ were uniformly sampled from an integer distribution in the range [1,200], e i and t i in the range [1, 30], 1
2
a i and /3 i in [0, 500], a n d A l = ~Pn a n d A 2 = ~Pn. F o r e a c h n, f i v e p r o b l e m i n s t a n c e s w e r e g e n e r a t e d and s o l v e d , a n d the a v e r a g e c o m p u t a t i o n a l results are r e p o r t e d h e r e . T h e a l g o r i t h m s w e r e c o d e d in S U N F O R T R A N a n d run o n a S U N S P A R C M P 6 7 0 . In the f o l l o w i n g w e c l a s s i f y our c o m p u t a t i o n a l r e s u l t s into t w o g r o u p s , o n e f o r a g r e e a b l e p r o b l e m s w h e r e a s the o t h e r f o r g e n e r a l p r o b l e m s w i t h o u t the a g r e e a b l e condition. 4.1. A g r e e a b l y w e i g h t e d p r o b l e m s
Note that Algorithm 1 can find the exact optima for these problems. Hence, the solutions obtained by
Algorithm 1 can be used as a basis to evaluate the performance of Algorithm 2. Results on problems with d as a fixed constant and a decision variable are reported below respectively. 4.1.1. R e s u l t s o n p r o b l e m s w i t h d a s a f i x e d Constant 1 3 Problems with d = ~Pn and d = 3Pn were solved respectively. The results are summarized in Tables 4 and 5, from which we can observe that, for all the problem instances tested, the solutions found by Algorithm 2 are exactly optimal. In fact, in addition to the results reported here, we also solved a large number of problems with different d and different parameter configurations, and surprisingly found that Algorithm 2 never failed to find an exact optimum for any problem instance we randomly generated. This justifies the theoretical result that the solution
Table 6 Computational results of Algorithm 1 (A1) and Algorithm 2 (A2) when d is decision variable n
10 20 30 40 50 60 70 80 90 100 200
Ave. CPU (sec.)
Ave. J (X 104)
AI
A2
A1
A2
1.46 12.50 41.36 100.76 274.52 419.64 554.20 825.38 I 156.22 1595.96 12729.00
0.16 0.64 1.38 2.50 3.98 5.66 9.70 10.24 13.16 16.02 65.68
0.027 0.108 0.211 0.384 0.597 0.809 1.101 1.430 1.770 2.161 8.968
0.027 0.108 0.211 0.384 0.597 0.809 1.101 1.430 1.770 2.161 8.968
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Table 7 Computational results of Algorithm 2 (A2) and Complete Enumeration (C.E.) when d = 3 P. n
3 4 5 6 7 8 9 10 a
Ave. CPU (sec.)
Ave. J (X 104)
Ave. R (%)
A2
C.E.
A2
C.E.
0 0.02 0.04 0.04 0.06 0.08 0.10 0.10
0.02 0.12 0.90 6.70 81.78 794.56 8629.36 116078.50
17.612 27.038 60.137 48.284 89.165 139.094 114.492 120.629
17.612 26.885 53.727 45.918 85.469 132.025 113.256 106.243
0 0.57 11.93 5.15 4.32 5.35 1.09 13.54
a Computational results of one instance only.
actually uses large fixed penalties to force the optimal schedule to be W-shaped.
found by Algorithm 2 is nearly optimal or even exactly optimal (when the problem solved has a V-shaped optimum). Unfortunately, as we discussed in previous sections, there do exist some problems for which there is no V-shaped optimal schedule, and thus Algorithm 2 cannot secure any optimal solution in these cases. Consider an example, in which n--5, d = 20, e i = 1, t i = 2, V i , and Pi, °li and ~i are given by Table 3. Note that this problem satisfies the agreeable condition. Under the cost functions given by (4.1) and (4.2), it is not hard to see that any optimal schedule for the problem must have job 5 finished at the due date, jobs 1 and 2 completed early, and jobs 3 and 4 late. This is a typical W-shaped schedule, which has an overall cost J ( A * ) = 2 4 . 4 9 1 2 . The best V-shaped schedule Av of this problem will schedule job 5 early. This alone has resulted a fixed earliness cost greater than J(A*). This example
4.1.2. Results on problems with d as a decision variable
Table 6 summarizes the results obtained by Algorithms 1 and 2 when d is a decision variable. Again, we can see that Algorithm 2 had actually obtained the exact optimal solutions for all the problem instances tested. Furthermore, Algorithm 2 was computationally efficient, which was able to generate optimal solutions for problems with up to 200 jobs quickly. 4.2. Results on general problems
As discussed in Section 3.2, Algorithm 2 can be used as a heuristic to solve the general problem P1 without the agreeable condition. To evaluate the
Table 8 Computational results of Algorithm 2 (A2) and Complete Enumeration (C.E.) when d is decision variable n
3 4 5 6 7 8 9 I0 a
Ave. CPU (sec.)
Ave. J (X 104)
Ave. R (%)
A2
C.E.
A2
C.E.
0 0.02 0.04 0.04 0.08 0.10 0.12 0.20
0.02 0.16 1.36 9.80 118.92 1155.24 12521.50 169903.50
0.181 0.563 0.765 0.787 2.265 2.041 2.925 3.536
0.156 0.545 0.733 0.742 2.179 1.808 2.697 3.321
a Computational results of one instance only.
16.03 3.30 4,37 6.06 3.95 12,89 8,45 6,47
X. Cai et al./ European Journal of Operational Research 98 (1997) 154-168
quality of the solutions obtained by this heuristic, the approach of complete enumeration was used to generate the optimal solutions for comparison. The results obtained are reported in Tables 7 and 8, from which the following observations can be made: (a) Algorithm 2 is efficient - the computing time it required was almost negligible for problems with small n. (b) The solutions found by Algorithm 2 are reasonably good. We believe, based on Corollaries 2 and 3, that the solutions found by Algorithm 2 could become better and better as n grows. (c) The time requirement of complete enumeration increased rapidly - it became almost impossible to use it for problems with n > 10.
5. Concluding remarks This paper examines the problem of scheduling n jobs on a single machine to minimize the total earliness and tardiness penalty under a common due date. We have addressed a general model in which earliness penalty and tardiness penalty are arbitrary non-decreasing functions. In addition, the model includes two important features that commonly appear in realistic problems: 1) earliness and tardiness are penalized with different weights which are j o b dependent; and 2) the earliness (or tardiness) penalty consists of two parts, one being a variable cost while the other being a fixed charge once a job is early (or tardy). The model provides a flexible performance measure to better reflect the need of decision making in practice. We have shown that an optimal schedule is Wshaped when the problem is agreeably weighted. Moreover, we have shown that there exists at least one V-shaped schedule whose relative error of performance approaches zero as n increases under some conditions on the bounds of parameters. We have highlighted certain cases satisfying these conditions, including the one in which all jobs have the same earliness weight and the same tardiness weight. The conditions are not hard to satisfy in a practical sense since the parameters in real-world problems usually have an upper bound. Two algorithms have been provided to solve the problem. Under the agreeable
165
condition, the algorithms can generate an optimal schedule in time O(nZPn), or a sub-optimal solution in time O(nPn). For problems without the agreeable weight condition, we have suggested to extend the O(nPn) algorithm as a heuristic procedure to find approximate solutions. Numerical results are reported, from which we see that the O(nPn) algorithm is very effective in terms of computational efficiency as well as the quality of solutions. Note that it is easy to further generalize the model to include a general due date assignment cost h(d) (when d is a decision variable) a n d / o r a flow time cost in the performance measure. The main results we have obtained will remain basically unchanged in these generalized models. A nontrivial future work is to consider the case where multiple machines are available to process the jobs. Another important work is to consider the problem of finding efficient schedules when d is a decision variable, which are good in terms of minimizing both total earliness/tardiness cost and due date assignment cost (cf. Bagchi 1989; De, Ghosh, and Wells 1992). Since our algorithms are based on dynamic programming which actually give the optimal schedules corresponding to all due date values, this bi-criteria problem can be tackled without much difficulty. Theoretical analysis to further relax the agreeable condition is also interesting. In general, it has been known that there exist some problem instances for which no W-shaped schedules (and therefore no V-shaped schedules) are optimal. Nevertheless, it may be possible to derive some conditions thal are looser than the agreeable condition. Besides, the heuristic we have proposed for solving general problems without the agreeable condition should be further evaluated, provided that some kind of approach can be devised to evaluate the optimality of solutions for large problems.
Acknowledgements We wish to thank the referees for their helpful comments and suggestions. We are particularly indebted to an anonymous referee who indicated the possibility of generalizing the results in an earlier version of the paper.
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X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
Appendix A P r o o f of Theorem 3. Let Si* and Ci* be respectively the starting time and completion time of job i, i = 1, 2 . . . . . n, under h*. If the problem has a Vshaped optimum, namely, h ° = h *, then R(A ~) = 0 and Eq. (2.2) holds. It is thus evident that we only need to consider the situation that the optimal schedule A* is exactly W-shaped, in which there are three consecutive jobs i, j and k such that pj > Pi and Pj > Pk and job j is the straddling job, namely, Ci = S j < d a n d S~ = C j ~>d. We will now show that J ( A °) < J ( A * ) + O,
(A.1)
through examining the following cases respectively, in which we let z = Cf - d: Case 1. If Cf - p ~ < d and C f >2t, we interchange jobs i and j in A* to construct a new schedule h (al~, in which jobs i and j will become tardy and early respectively. Thus, one can see that j(/~(al)) __ j ( A,~ )
-- OLi -- e i f ( p j -- Z) -- ~ j -- t j g ( Z) -
-
J(A (bl)) - J ( h * )
= tSk + tkg( z - - p j + pk) + ~j + tjg( z + pk) -- ~j-- t i e ( z ) -- ~k-- tkg( z + Pk) = (tkg ( z - p j +(tj-
(
+ ejf( P i - z) -- e i f ( pj -- z) + (t i -- t j ) g ( Z ) <~ ( a j - ai) - ( ~ j - ~i) + ( t i - t j ) g ( z)
tk)g ( z +Pk)
<~ ( t k g ( z) - tjg( z ) ) + ( t j - t k ) g ( z + Pk) ~< 0 ~< 69.
(A.3)
If h (bl) is still not V-shaped, job j can be further interchanged with the job sequenced immediately after it to get a new schedule h (b2~. Similarly, this procedure can be continued until a V-shaped schedule i b with J(A b)~< --- ~
a' = J ( X )
- J(;~*)
= t~i + tjg( z-- pi) + t~i + tig( z)
(since ej ~< e i and f ( . ) is nondecreasing) ~< O.
+Pk) - t j g ( z ) )
is derived. Again, this implies (A.1) as h ° is the best V-shaped schedule. Case 3. If C f - Pi > d, Sf + Pk < d and C f > d, then the interchange of jobs i and j will yield a schedule h' in which jobs i and j are all tardy. Thus, it can be seen that
= Otj + ejf( P i - Z) + fii + tigi(Z)
= ( .j
sequence h (b~), in which jobs j and k are all tardy. Again, bearing in mind that pj > Pk, tj <~t k, and g(. ) is a non-decreasing function, we can see that
(A.2)
Moreover, after interchanging jobs i and j, if the processing time of job j is still greater than that of the job immediately before it, we further interchange job j and that job to construct a new schedule h (a2). Clearly J(A (a2)) ~ J(A (al)) (see the proof for Theorem 2). Such a procedure can be continued until a V-shaped schedule A~ is obtained, which satisfies
J( h a) ~< "-- ~
-- a i -- e i f ( pj -- Z) -- [~j -- tjg( Z) = ( ~ i - ai) + ( t i - t j ) g ( z ) + tjg( Z--Pi) -- eif( P j " Z).
(A.4)
Moreover, the interchange of jobs j and k will yield a schedule X' with job j tardy and job k early. This gives us a" = J ( X ' ) - J ( h * )
= a k + ekf (pj-
z-pk)
+ ¢3j + t j g ( z +Pk)
-- ~j-- tjg( z) - ~k-- t , g ( z + Pk) =
( ak --
~k) + (tj -- tk)g ( Z +Pk)
+ekf(pj--z--pk
) - - t i g ( z ).
(A.5)
X. Cai et al. / European Journal of Operational Research 98 (1997) 154-168
Combining (A.4) and (A.5), we have
since f is a non-decreasing function. Similarly, if job i is mith tardy job after the job j~, then
A' + A "
= ( ~ i - ~k)-- (OLi- OLk) ~- (ti-- tj)g(z) +
g( Lj) = g ( C f - d) >1 g(mjPmin) , since g is a non-decreasing function. Hence,
tj(g( z -Pi) - g(z))
JV(fi*) =
+ ( e k f ( pj -- z -- pk) - e l f ( pj - z ) )
~_~ e i f ( Li) + iEE(A*)
+(tj--tk)g(z+Pk)
>/
( J~i- ~k) -- (OLi- Ogk) %- ( t i - t j ) g ( Z ) +(ekf( pj- z-Pk)
167
E
tjg( Lj)
j ~ T(A*)
E
eif{miPmin}
iE {E(A* ) - {J's}}
+
- e,f( p j - z ) )
E
tjg(mjPmin)
j ~ {r(x* )-- {Js}}
(since tj ~< t k andg( z - Pi) <~g ( z ) )
= Ew'q,
= (t~i--~k) -- ( a i - - ak) + (ti-- t j ) g ( z )
where w and q are, respectively, a certain element in the set ~ and ~ as defined in Section 2. (~w- q) is the sum of the product of two sequences, each having n - 2 elements (the jobs Js and ~ are excluded) from the set ~ or ~, According to Hardy, Littlewood and Polya (1934, p.261), it is evident that
+(e k- ei)f(pj-- z--pk) + ei( f ( Pj -- z -- P, ) -- f ( Ps -- Z ) ) <~ ( ~ i - ~Sk) - ( % - - ak) + (ti-- t j ) g ( z) + ( e, -- e j ) f ( pj -- z --Pk),
n--2
JV(a*)>~ ~.,wiq,-l-i.
(since f ( pj -- z --Pk) ~
i=l
and e i > %) ~< O,
(A.6)
which implies either A'~< O or A"~< O, namely J(X)
E a,+ E ~i >~r. iEE(~*) i~r(a*)
(A.7)
NOW examine J~(A*). Let j~ denote the straddling job and j{ denote the job sequenced immediately before the job j~, under A*. Clearly, if job i is mi-th early job before the job f~, then f ( Li) = f ( d -
Ci* ) >/ f ( miPmin),
(A.8)
This together with (A.1) and (A.7) proves the theorem. []
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