A set of n jobs has to be scheduled on a single machine which can handle only one job at a time. Each job requires a given positive uninterrupted processing time and has a positive weight. The problem is to find a schedule that minimizes the sum of weighted deviations of the job completion times from a given common due date d, which is smaller than the sum of the processing times. We prove that this problem is NP-hard even if all job weights are equal. In addition, we present a pseudopolynomial algorithm that requires O(n2d) time and O(nd) space

Hoogeveen, J.A. (Han)

Velde, S.L. (Steef) van de

Erasmus University Digital Repository

European Journal of Operational Research 55 (1991) 237-242 237 North-Holland Theory and Methodology Scheduling around a small common due date J .A. Hoogeveen and  S.L. van  de Ve lde  Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, Netherlands Received May 1989; revised April 1990 Abstract: A set of n jobs has to be scheduled on a single machine which can handle only one job at a time. Each job requires a given positive uninterrupted processing time and has a positive weight. The problem is to find a schedule that minimizes the sum of weighted eviations of the job completion times from a given common due date d, which is smaller than the sum of the processing times. We prove that this problem is NP-hard even if all job weights are equal. In addition, we present a pseudopolynomial lgorithm that requires O(n2d) time and O(nd) space. Keywords: Single machine scheduling, NP-hardness, pseudopolynomial algorithm I. Introduction Recently, we have seen a growing interest in just-in-time manufacturing. This concept decrees that products should be completed as close to their due dates as possible in order to avoid both storage costs as a result of early completions and penalty costs inflicted on account of late de- liveries. This might induce the following types of problems for the single-machine job shop. A set of n independent jobs has to be sched- uled on a single machine, which can handle only one job at a time. The machine is assumed to be continuously available from time 0 onwards. Job Ji (i = 1 . . . . .  n) has a given positive uninterrupted processing time Pi and should ideally be com- pleted at a given due date d~. Without loss of generality, we assume that the processing times and the due dates are integral. A schedule defines for each job ~ a completion time G such that the jobs do not overlap in their execution. Given a schedule S, the earliness and tardiness of job Ji are defined as E i = max ( d i - C i, 0} and T i = max {C i -  di, 0}, respectively. The just-in-time philosophy is reflected in the objective function f (S )  = ~ (aiE ~ + fliT,). ~=1 For a review on problems with this type of objec- tive function, see Baker and Scudder (1990). An important subclass contains the set of prob- lems that deal with a common due date d for all jobs. The common due date is either specified as part of the problem instance, or is a decision variable that has to be optimized with the job sequence simultaneously. As the first job may start later than time 0, the optimal schedule is identical for both problems unless the common due date d is restrictively small (d < ~Pi). There- fore, the first variant is referred to as the restricted problem and the second variant as the unrestricted problem. We will call the earliness and tardiness penalty weights symmetric if a~ = fli for each i = 1 . . . . .  n. For the case of nonsymmetric weights, only one problem type has been investigated, namely the 0377-2217/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved 238 J.A. Hoogeveen, S.L. van de Velde / Scheduling around asmall common due date case in which all a i are equal and all fl~ are equal. Bagchi, Chang and Sullivan (1987) and Emmons (1987) present an O(n log n) algorithm for the unrestricted variant, while Bagchi et al. (1987) propose a branch-and-bound algorithm for the restricted problem. If the earliness and tardiness penalty weights are symmetric, then the problem reduces to find- ing a schedule S that minimizes the weighted sum of the deviations of the completion times from the common due date: f (S )  = ~ we l f , -d l .  i=1 There are two notable results for the case that d >/EPv Kanet (1981) gives and O(n log n) time algorithm to find an optimal schedule, if all weights are equal. Hall and Posner (1989) show that the problem with symmetric weights is NP-hard. In contrast, we focus our attention on the case that d < Epv In Section 2 we prove some proper- ties of an optimal schedule. In Section 3 we estab- lish NP-hardness of the problem, even for the case that all job weights are equal. We note that Hall, Kubiak and Sethi (1989) independently obtained this result by a slightly more complicated proof. This result justifies the development of enumera- tive algorithms by Bagchi, Sullivan and Chang (1986) and by Szwarc (1989) for minimizing E" i=~ IC~- d l subject to a common due date d < Ep~. In contrast, we present a pseudopolynomial algorithm in Section 4 for 1 IIEw~ I C~- d I, which requires O(n2d) time and O(nd) space. Our al- gorithm is applicable to a more general problem type than the pseudopolynomial algorithm of Hall et al. (1989), which can only handle equal job weights. In Section 5 we present some well-solva- ble cases. 2. Basic concepts It is straightforward to verify that no optimal solution has any idle time between the execution of jobs. In case there were idle time, the schedul- ing cost could be reduced by closing the gap. The next two theorems further characterize any opti- mal solution. Theorem 1. In any optimal schedule S, the jobs J~ that are completed before or at the common due date d are scheduled in order of nondecreasing values of wi/pi, and the jobs that are started at or after d are scheduled in order of nonincreasing values of w i/pi.  Proof. This follows immediately from Smith's ratio rule (Smith, 1956). [] Theorem 2. In each optimal schedule S, either the first job starts at time 0 or the due date d coincides with the start time or completion time of the job with the largest ratio wi/Pi. Proof. For a given schedule S, let B(S)  denote the set of jobs that are completed before or at the common due date and A(S)  the set of jobs com- pleted after the due date. Define A = E j, ~ s(s)Wi - 5~j, ~ ms)W, We consider the cases in which A < 0 and A >t 0 separately. Suppose first A < 0. If S starts at time T > 0, determine t = rain ( T, rninj, ~ A(s)C~ - d }. If the entire schedule is put t time units earlier, then the reduction in cost equals - tA  > 0. In the new situation either schedule S starts at time T = 0 or one job has moved from A(S)  to B(S).  If still T > 0 and A < 0, we repeat he procedure until we arrive at a situation in which T = 0 or A >/0, and no further improvement is possible. The latter case implies that the due date coincides with the completion time of one job and the start time of another. Because of Theorem 1, one of these jobs must be the job with the largest ratio wJp ,  On the other hand, in the case of A >/0, reverse arguments can be applied to show that the due date coincides with the completion or start time of the job with the largest ratio wypv [] Note that Theorem 1 does not impose any restric- tions on a job that is started before and completed after the due date. Consider the following instance with n=3,  p ]=8,  p2= 10, p3=4,  w 1 =5,  w2= 7, w 3 = 3, and d= 15. The optimal solution is shown in Figure 1 and demonstrates that such a job can exist, and that it can even have the smal- lest ratio wi/pi. I 0 d=15 Figure IJ.A. Hoogeveen, S.L. van de Velde / Scheduling around a small common due date 239 3. Scheduling around a small common due date is NP-hard In this section we prove that this problem is NP-hard even if w~ = 1 for each job J,, by showing that the corresponding decision problem is NP- complete. The reduction is from Even-Odd parti- tion. Even-Odd partition (Garey, Tarjan and Wilfong, 1988): Given a set of 2n positive integers B = (b 1 . . . . .  b2, ) such that bi>bi+ 1 for each i=  1 . . . . .  2n -  1, is there a partit ion of B into two subsets B 1 and B 2 such that Y~bi~ B,b, = Eb,~ B2b, = A and such that B 1 contains exactly one of (b2, 1, b2i} for each i= l  . . . . .  n? We start by describing a reduction from the Even-Odd partit ion problem to the small common due date problem with w~ = 1 for all J,. Let B = (b  I . . . . .  b2n } be an arbitrary instance of the Even- Odd partition problem, with A = ~.bJ2. Con- struct the following set of jobs: 2n 'part i t ion'  jobs, Ji with processing times p~ = bi + nA for each i = 1 . . . . .  2n, an additional job J0 with p0 = 3(n2+ 1)A, weights wi= 1 for i=0  . . . . .  2n, and a com- mon due date d = (n 2 + 1)A. In addition, we de- fine a threshold value Y0 = Y~7=l[( i + 1)(P2i- i  + P2,)] + d on the scheduling cost. Consider a partit ioning of the set of partit ion jobs (J1 . . . . .  J2~) into the sets B1 = {Jll, J21 . . . . .  J~l } and B 2 ----- ( . /12 ,  "]22 . . . . .  Jn2 }, where ( J,1, Ji2 } = ( J2i- 1, J2i } for each i = 1 . . . . .  n. Lemma 1. I f  the partitioning into the sets B 1 and B 2 corresponds to a solution of the Even-Odd partition problem, then the cost of schedule S O constructed as shown in Figure 2 equals the threshold value Yo. Proof. Note that the jobs in B 1 and B 2 are  sched- uled as indicated in Theorem 1. The verification then only requires straightforward computations. [] r I t [ 0 d F igure  2. Schedu le  S O We now prove that, conversely, any schedule S with f(S)<~ Yo must have the same structure as So, and that the subsets B 1 and B e must corre- spond to a solution of the Even-Odd Partition problem. Proposition 1. Suppose S is a schedule with schedul- ing cost flS) <<. Yo. Then S has the following proper- ties." (1) At most n jobs can be completed before the due date d. (2) The first job must start at time O. (3) The additional job Jo is scheduled last. (4) At least n - 1 jobs must be completed before the due date d. Proof. (1) This is due to the choice of the processing times. (2) This follows immediately from the first property and the proof  of Theorem 2. (3) Suppose J0 is not scheduled last. Then, because of Theorem 1, J0 must start before the common due date d. Since at most n jobs can be scheduled before job Jo, for at least n + 1 jobs in S we have C i - d >/P0 - d = 2d. This implies that f (S )  >1 2(n + 1)d>~ (n + 4)d. However, as each of the multipliers of Pl . . . . .  Pn in Y0 is at most ½(n + 3), while ZT=l(i + 1) = ½n(n + 3), we have the following inequality: t/ Yo = E [ ( i+ l ) (p2 ,  I+P2, ) ]  +d i=1  2n  < ½(n+ 3) ~'~p~+d i=1  =(n+a)d<~f(S) ,  which contradicts the assumption. (4) This follows immediately from the first three properties and the choice of the processing times. [] Lemma 2. Suppose S & an optimal schedule with f(S) <~ Yo. Then the due date d must coincide with the completion time of the n-th job in the schedule S, the schedule S must have the same structure as the schedule So, and provide an affirmative answer to the Even-Odd partition problem. Proof. Assume that s(i) denotes the index of the job that is scheduled on position i in schedule S. 240 J.A. Hoogeveen, S.L. van de Velde /Schedul ing around a small common due date We compute the scheduling cost relative to the imaginary due date k=p~(1 ) + .-- +p~(,). Then we have 2n ~ IC,.-kl i=O /I = Y'. [ ( i -  1)p~(i)] i=1 2n + E [ (2n+ 2-i)ps(i)  ] + 3d i=n+l  = ~ [(i + l)p,(i) ] i=l 2n + y" [ (2n+2- i )p , ( i ) ]+3d-2k  i=n+l  : ~[ ( i+ I )P , (o  l i=1 + ~ [ ( i+ 1)p,(2,+l_i) ] + 3d-2k  i= l  >~ ~ [ ( i+  1)(p2i_ 1+p2i)] i=1 + 3d-  2k= Y0 + 2d-  2k. The true scheduling cost f (S)  can be written as 2n f (S )= ~ IG -d l  i=0 2n = ~_, IG -k l+(d -k )  i=O × (card(B(S)) - card(A (S)))  where card denotes the cardinality function. Be- cause of Proposition 1, we have only three cases to consider: - if d= k, then f (S )  >~Yo, - if d> k, then card(B(S)) = n, and therefore f(S)>~ yo + d -k> yo, - if d < k, then card(B(S)) = n - 1, and hence f (S )~ yo + k -d> yo. This implies that if f (  S)) <~Yo, then Cs(/1 ) = d, that is, the completion time of the n-th job in S must coincide with the due date. Furthermore, f (S)  <~Yo implies ( J s ( i ) ,  J s (2n- l+ i )}  = (J2i-1, J2,} for i = 1 . . . . .  n. Therefore, the schedule S has the same structure as the schedule SO depicted in Figure 2. This means that the original Even- Odd partition problem has an affirmative answer. [] Theorem 3. Given a set of jobs and a nonnegative integer y, the problem of deciding whether there exists a schedule S O with f(So) <~ y is NP-complete. Proof. The decision problem is clearly in NP. For any given instance of the Even-Odd Partition problem, we construct a set of jobs as described above and set y =Y0- This reduction requires polynomial time. Theorem 3 now follows from Lemmas 1 and 2. [] 4. A dynamic  programming a lgor i thm Theorem 3 implies that, unless P = NP, no polynomial algorithm exists for solving the small common due date problem. We present a pseudo- polynomial algorithm that requires O(n2d) time and O(nd) space, for which Theorems 1 and 2 provide the basis. According to Theorem 2 we must consider two cases: one in which the job with the largest weight to processing time ratio is scheduled such that either its completion or its start time coincides with the due date, and one in which all the jobs are scheduled in the interval [0, EPA. For the first option, we renumber the jobs according to nonincreasing weight to processing time ratios. Let ~( t )  denote the optimal objective value for the first j jobs subject o the condition that the interval [d - t ,  d+Y~{=lpi-t  ] is oc- cupied by the first j jobs. Then the initialization is F:(t) = (0  fo r t=0,  j=0  oo otherwise, and the recursion for j = 1 . . . . .  n is given by Fj( t ) = rnJn{ Fj_l( t -  pj ) + wj( t - pj), Fj_I( t ) + wj p~ - t for 0 ~< t ~< d. i In the second case, all jobs are scheduled in the interval [0, Y'.Pi]. In such a situation it might occur that one of the jobs is started before and yet completed after the due date (see Figure 1). To J.A. Hoogeveen, S.L van de Velde /Scheduling around a small common due date 241 allow for this possibility, we leave one job out of the recursion, and repeat the recursion n times, once for each job. Since the cost of the schedule can now only be computed relative to the end- points of the interval, it is assumed that the jobs have been renumbered according to nondereasing values of wi/p~. Consequently, we know that the first job either starts at time 0 or finishes at time EPi. Assume that Jh is the job that will be scheduled around the due date. Let Gh(t) denote the optimal cost for the first j jobs subject to the condition that the intervals [0, t] and [En=j+lPi + t, Ep~] are occupied by the first j jobs. The initialization is {0  for t=O,  j=O G~(t) = otherwise, and the recursion for j = 1 . . . . .  n, is Theorem 4. The dynamic programming algorithm solves the problem in O(n2d) time and O(nd) space. Note that the dynamic programming algorithm can be modified to cope with any common due date problem with nonsymmetric earliness and tardiness penalty weights that allow for a pre- specified processing order of the jobs that are completed before or started after the common due date. This includes the problem with all a~ equal and all /3, equal, for which Bagchi et al. (1987) presented a branch-and-bound algorithm. In ad- dition, the weighted tardiness problem with a common due data possesses this property. 5. Polynomially solvable cases c ; ( t )  = G~_l(t ) i f j=h ,  if d-pj<~t<~d, Gf , ( t -p j )+%(d- t )  if Y" pi < d -  t, j+ l  n a? ÷ w,(d- t)} otherwise. The recursion leaves the interval [t, t + Ph] idle, and it is here that we insert the job Jh and compute the resulting cost as { G.h(t) + Wh(t + Ph -- d) Gh(t) = i fd-ph<~t<~d , o¢ otherwise. The optimal solution is then found as min0_<,_< d<(t )} ,  by which we have established the following result. 5.1. Identical jobs If the jobs are identical, we have Pi =P  for each job J Since the processing times and due date are assumed to be integral, this situation is more general than the one in which pi = 1 for all Pi. Suppose the jobs have been renumbered according to nonincreasing weights. If d>~p[½n], then it is easy to show that Em- mons' matching approach (Emmons, 1987) gener- ates an optimal schedule S by partitioning the jobs into sets A(S)=( Jz ,  l i= I  .. . . .  [½n]} and B(S) = ( J2i- l l i  = 1 . . . . .  [½n]}, where the first job in B(S) starts at time t=d-E j ,~B(s )p~=d - p[½n l. In this notation, [½nl denotes the largest integer smaller than or equal to ½n and [½n] denotes the smallest integer greater than or equal to ½n. Conversely, if d<p[½n], then there are two options: either the first job starts at time 0 or the last job in B(S) is completed at time d. It is easy to see that in both cases Emmons' matching ap- proach generates optimal schedules, and the prob- lem is solved by choosing the better one. 5.2. Jobs with equal weight o processing time ratios Theorem 5. In the event hat p, = w~ for each job J~, there is an optimal schedule for any value of d in which the jobs are scheduled according to nonin- creasing processing times. 242 J.A. Hoogeveen, S.L. van de Velde /Scheduling around a small common due date Proof. Consider two adjacent jobs that are not scheduled according to the indicated order. If both jobs are completed before or started after the common due date, then these jobs can be inter- changed without affecting the cost of the schedule S, unless the due date lies in the interval between the start time of the first and the completion time of the other job. We prove that even in that case, an interchange of these two jobs does not increase the scheduling cost. Without loss of generality, let J1 and J2 be the two jobs that have to be interchanged, with p~ >/ P2, and let J2 start at time t. We have to investi- gate the following three situations: (1) t ~< d ~< t + P2. Then the interchange l ads to a schedule with the same cost. (2) t +p2 < d ~< t +pl .  Then the interchange lowers the cost by 2p2(d-p2  ) >10. (3) t +p l  < d ~< t +p l  +p2. Then the inter- change decreases the cost by 2dp2 - 2dpl + 2p~ - 2P 2 = 2(Pl +P2 - d)(p l  -P2)  >/0. [] Assume that the jobs have been renumbered inorder of nonincreasing processing times. Suppose r is the smallest index for which Y.r=lp ~ >/ ET=r+1P~" Theorem 5 then implies that, if d>~ 5"r=lPi, the problem is solved by putting B(S)= (J, I i= l  . . . . .  r} and A(S)=( J ,  l i=r+ 1 . . . . .  n }. If d < Er=ap, the first job needs to start at time 0, and the jobs are processed in order of nondecreasing processing times. Acknowledgement The authors would like to thank Lenstra for his helpful comments. Jan Karel References Bagchi, U., Sullivan, R.S., and Chang, Y.L. (1986), "Minimiz- ing mean absolute deviation of completion times about a common due date," Naval Research Logistics Quarterly 33, 227-240. Bagchi, U., Chang, Y.L., and Sullivan, R.S. (1987), "Minimiz- ing absolute and squared deviations of completion times with different earliness and tardiness penalties and a com- mon due date", Naval Research Logistics 34, 739-751. Baker, K., and Scudder, G. (1990), "Sequencing with earliness and tardiness penalties: A review, "'Operations Research 38, 22-36. Emmons, H. (1987), "Scheduling to a common due date on parallel uniform processors", Naval Research Logistics 34, 803-810. Garey, M.R., Tarjan, R.E., and Wilfong, G.T. (1988), "One- processor scheduling with earliness and tardiness penalties", Mathematics of Operations Research 13, 330-348. Hall, N.G., Kubiak, W., and Sethi, S.P. (1991), "Earliness- tardiness cheduling problems, II: Deviation of completion times about a restrictive common due date", Operations Research 39, to appear. Hall, N.G., and Posner, M.E. (1991), "Earliness-Tardiness scheduling problems, Operations Research 39, to appear. Kanet, J.J. (1981), "Minimizing the average deviation of job completion times about a common due date', Naval Re- search Logistics Quarterly 28, 643-651. Smith, W.E. (1956), "Various optimizers for single-stage pro- duction", Naval Research Logistics Quarterly 3, 59-66. Szwarc, W. (1989), "Single-machine scheduling to minimize absolute deviation of completion times from a common due date", Naval Research Logistics 36, 663-673. 

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