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Complexity of single machine, multi-criteria scheduling problems
European Journal of Operational Research 70 (1993) 115-125 North-Holland
115
Theory and Methodology
Complexity of single machine, multi-criteria scheduling problems Chuen-Lung Chen Department of Industrial Engineering, Mississippi State University, Mississippi State, MS 39762, USA R o b e r t L. B u l f i n
Department of Industrial Engineering, Auburn University, Auburn, AL 36849, USA Received March 1991; revised September 1991
Abstract: We examine the complexity of scheduling problems when more than one measure of perfor-
mance is appropriate. The criteria we study are maximal tardiness, flowtime, number of tardy jobs, tardiness and the weighted counterparts of the last three measures. The machine environment is restricted to a single machine. Complexity results are given for secondary criterion, bicriteria and weighted criteria approaches for all combinations of measures. Of the problems examined, only six remain open. Keywords: Complexity; Multiple criteria; Scheduling; Single machine 1. Introduction
In 1973, a survey by Panwalkar, Dudek and Smith (1973) pointed out the importance of considering multiple measures of performance in scheduling problems. Almost two decades later, there are still quite a few open problems in scheduling with multiple criteria. We consider the complexity of some of these problems in this paper. Not surprisingly, almost all are NP-hard. While other researchers have developed heuristics or enumerative algorithms for multi-criteria scheduling problems, we will not discuss their work unless it deals with problem complexity. We start with a brief background discussion of scheduling, multicriteria optimization and complexity. We then discuss single machine results for secondary, multiple criteria, and weighted criteria approaches for each combination of the various criteria. A brief discussion of the results concludes the paper. 2. Background
This section will introduce the notation and define the environment for the remainder of the paper. It will be divided into three parts, the scheduling problems addressed, the types of multiple criteria problem considered, and complexity. Correspondence to: R.L. Bulfin, Department of Industrial Engineering, College of Engineering, Auburn University, 207 Dunstan Hail, Auburn, AL 36849-5346, USA.
0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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C.-L. Chen, R.L. Bulfin / Single machine multi-criteria scheduling
2.1. Scheduling environment
To define a scheduling problem, let n be the number of jobs to be processed and Pi, ri, di, and w i be the processing time, release data, due date, and weight (or 'value') of job i, respectively. Given a schedule, for each job i, we can define Ci to be its completion time, F~ = C~ - r~ its flowtime, L~ = C i - di its lateness, and T,. = max{0, L~} its tardiness. Let the maximum tardiness be Tmax = max i {T/}. Finally, if job i is tardy (i.e. T/> 0), let U/ be one, and zero if job i is completed on or before its due date (T/< 0). Traditional measures of performance for machine scheduling include minimizing total flowtime, total tardiness, maximum tardiness or the number of tardy jobs, which we will denote by F, T, Tmax and U, respectively. These measures (except for Tmax) are simply the sum over all jobs of the respective quantity for each job. If all jobs are not equally important, an equivalent weighted measure can be calculated by multiplying the quantity by the appropriate job weight. These will be denoted by WF, W T and w v respectively. For a more detailed discussion of measures of performance used in scheduling, see Baker (1974) or French (1986). Assumptions made for most scheduling problems are that data are known with certainty, setup times are independent of the order of processing, all jobs are immediately available (r i = 0), no precedence exists between jobs and once processing of a job starts, it is not interrupted. Graham et al. (1979) have defined a three-field notation, t~/13/3", to describe scheduling problems where the first field corresponds to the machine environment, the second to deviations from normal assumptions, and the third denotes the criteria or measure of performance to be used. For this work, standard assumptions will be in force. The criteria will be as previously defined, and hence 3' ~ F = {Tmax, F, WF, U, w v, T, WT}. Because one subgroup of these criteria occur frequently, we define F h = {wv, T, wr}. The machine environment will consist of a single machine, i.e. a = 1. As an example 1 / /Tma x represents minimizing maximal tardiness on a single machine under the standard assumptions. 2.2. Multicriteria scheduling Although few researchers have addressed multicriteria scheduling problems, much work has been done on general multicriteria optimization problems; for details, see Goicoechea et al. (1982) or Steuer (1986). We will discuss three general multicriteria approaches that are applicable to scheduling problems: secondary criterion, efficient set generation and weighting of criteria. We present results in terms of two criteria since it greatly simplifies discussion. However, in most cases the generalization to three or more criteria is straightforward. The secondary criterion approach designates one criterion as primary and the other as secondary. The objective is to find the best schedule for the secondary criterion among all alternative optimal schedules for the primary criterion. In an obvious extension to the notation of Graham et al. (1979), we will denote these problems 1 / /3'213'1 where 3'1 is the primary criterion and 3'2 is the secondary criterion. Smith (1956) was the first to look at a single machine secondary criteria scheduling problem, minimizing flowtime with no tardy jobs ( 1 / / F I Tma~ = 0). Another approach is to generate all efficient (nondominated, Pareto optimal) schedules for a problem, and then allow the decision maker to make explicit trade-offs between these schedules. When there are two criteria, we call this a bicriteria approach, and denote it by 1 / /3'1,3"2, where the two criteria of interest are 3'1 and 3"2- 1 / /F,Tma x would denote the problem to generate all nondominated schedules considering flowtime and maximum tardiness simultaneously. A final approach is to use a weighting function. Here the decision maker expresses a tradeoff which, once specified, allows the problem to be solved with a single criterion. We will consider only functions which depend on the previously defined measures. A scheduling problem with two criteria, say 3'1 and 3'2, and a given weighting function f is denoted by 1 / / f ( 3 ' 1 , 3"2). For scheduling problems, it seems t~ make sense to restrict f to linear combinations of the completion times of the criteria, e.g. f(3'1, 3"2) = A13'1 + A23"2.
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We will not consider Cm~x as a criterion, since it is constant for a single machine. Neither will we consider a measure and its weighted counterpart, e.g. F and w F. 2.3. Complexity The complexity of 1 / /3' for any 3" previously defined is well known. Lageweg et al. (1981) provide the results for all measures except tardiness. This gap in single machine complexity results was recently closed by Du and Leung (1990), who showed 1 / / T is NP-hard. The secondary criteria and weighting approaches result in a single schedule, and complexity analysis is straightforward. This is not the case for the multicriteria approach, since the number of nondominated schedules could be exponential in the problem size. For our discussion, we will say that a multicriteria problem is polynomially solvable if we can generate each nondominated schedule in polynomial time. If the number of nondominated schedules is bounded by a polynomial in the problem size, we can generate them all in polynomial time. Unfortunately, unless there are special restrictions, such as unit execution time jobs, there seems to be no way to guarantee this. Since almost none of the multicriteria problems can generate a single nondominated schedule in polynomial time, it is a moot point anyway. We will now give some theorems which, while quite obvious, prove to be very powerful. The first relates the complexity of single criterion problems to multiple criteria problems. Theorem 1. / f 1 /
//3,1 /S NP-hard, then 1///3'213,1, 1//3,1,3,2 and 1 / //f(3,1' 3,2) are NP-hard.
Proof. We will show that the existence of a polynomial algorithm for 1 / //3/213,1, 1 / /3'1, 3,2 and
1// //f(3,1, 3,2 ) implies the existence of a polynomial algorithm for 1 / //3,1, thus proving the result. Clearly, any schedule solving 1// //3'213'1 must be optimal for 1 / //3"1, giving the first result. Since 1//3"1,3"2 generates all nondominated schedules and a solution to 1///3'1 is a nondominated schedule for 1 / //3/1,3,2, the second result is at hand. Finally, a solution to 1 / //f(3,1, 3,2 ) with )t 1 = 1 and A2 = 0 provides a solution to 1 / / Y r [] A second theorem relates the complexity of secondary criterion problems to bicriteria problems. Theorem 2. If 1 / //3/213'1 /s NP-hard, 1 /
//3"1,3,2 /S NP-hard.
Proof. Again we will show that the existence of a polynomial algorithm for 1///3'1,3"2 implies 1 / //3"2 I 3,1 can be solved in polynomial time. Since the schedule which solves 1 / //3,213,1 is a nondominated schedule for 1 / //3/1,3,2, it is produced as a consequence of solving 1// //3,1,3,2- [] Finally, we show that finding a solution to the weighted problem is just as hard as the bicriteria problem. Theorem 3. / f 1 / /3'1,3/2 /S NP-hard, 1 / /f(3"1, 3"2) is NP-hard.
Proof. For the proper choice of weights, the weighting method can be used to generate nondominated solutions (Goicoechea et al., 1982). Therefore, a polynomial algorithm for 1 / / f ( 3 , 1 , 3'2) can generate a nondominated schedule in polynomial time, which gives the result. [] These results will prove useful in showing various problems are NP-hard. We will examine the complexity of secondary criterion, bicriteria and weighted criteria for the single machine problem. We begin with secondary criterion problems.
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3. Secondary criterion problems In this section we examine the 36 secondary criterion problems 1//3"21 3'1, where 3'1 and 3,2 belong to F. Recall that problems with a criterion and its weighted counterpart, such as 1 / / F l w v , will not be considered. We will examine these problems in order of their primary criterion. 3.1. Problems with m a x i m u m tardiness as the primary criterion
The first secondary criterion problem studied was 1 / / F I Tmax. Smith (1956) developed a polynomial algorithm for this problem. The weighted flowtime problem, 1 / /wvlTm~x, can easily be shown to be NP-hard. This is an obvious extension of a result by Lenstra et al. (1977) which showed 1 / / w F I Tmax = 0 is NP-hard. Now consider 1//3'21Tmax where 3,2 ~ Fh. We have
Theorem 4. 1 / / w v l T m a x, 1 / / T I T m a x and 1 / / w r l T m ~ x are NP-hard. Proof. We will show that any algorithm for 1//3,21Tmax will solve 1 / / 3 ' 2 . Since 1 / / 3 ' 2 , 3"2 ~ Fh, are all NP-hard, the result follows. Consider an instance of 1 / / w v with n jobs having processing times Pi, due dates d i and weights wr Assume d 1 < d 2 < • .. _
d , , + l = d , , + 1 and
Wn+l=0.
Any schedule which minimizes Tmax must have job n + 1 last. The order of the other jobs has no effect on Tmax, and since wn+ t = 0, job n + 1 has no effect on w v. Applying an algorithm for 1 / / w t r l T m a x to this instance would result in a schedule which solves 1 / / w U for the original instance. Since 1 / / w v is NP-hard, then so is 1 / / w v I Tmax. A similar argument holds for 3'2 ~ {T, wr}. [] These results leave 1 / / U
I Tm~x as an open problem.
3.2. Problems with flowtime as the primary criterion
Recall that 1 / / F is solved by sequencing the jobs in nondecreasing order of their processing times (SPT rule). In order for there to be alternate optimal schedules, there must be ties in the SPT sequence. For any jobs with identical processing times, we can use the results of Chen and Bulfin (1990) to solve the secondary criterion problem. For 3"2 = Tm~x or 3'2 = T, simply break ties by earliest due date (EDD) first. For U, simply apply Hodgson's algorithm to each set of tied jobs. For w u use Lawler's (1976) algorithm for 1 / U E T / w v for the tied jobs. Unfortunately, for 3'2 = w r , we must solve an assignment problem over the tied jobs; details can be found in Chen and Bulfin (1990). Therefore we see that 1 / / 3 " 2 1 F can be solved in polynomial time for all 3"2 ~ F. 3.3. Problems with weighted flowtime as the primary criterion
Ordering the jobs with the first job having the smallest ratio of processing time to weight, the second job the next smallest, and so on, i.e. WSPT, solves 1 / / w F. As there must be ties in SPT for 1 / / 3 , 2 1 F , there must be ties in the processing time to weight ratio to have alternative optimal schedules 1//3'2 [wF. For 3'2 = Tm~, simply break ties by placing the job with the earliest due date first. When 3'2 = U, Hodgson's algorithm can be used on the set of tied jobs. However, secondary criterion problems with 3"2 ~ Fh are NP-hard, as shown by the following theorem.
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Theorem 5. 1 / /3"21WF is NP-hard for 3"2 ~ Fh.
Proof. Consider an instance of 1 / / w v with n jobs and Pi, di and w~ the processing time, due date and weight of job i, respectively. Further, let w~ =Pi for all i. Create an instance of 1 / / w v l w t from the same data. Since the ratios of processing time to weight are all one, every schedule has the same weighted flowtime, and any solution to 1 / / w v l w F is also a solution to 1 / / w U. Lenstra et al. (1977) show that 1 / / w e is NP-hard even if the weights are equal to the processing times. Therefore 1 / /WFI w F is NP-hard. Similar arguments can be made for 3'2 = T and 3"2 = w~r. []
3.4. Problems with number of tardy jobs as the primary criterion Surprisingly, this is probably the most intriguing group of problems in multiple criteria scheduling. Two of the five problems are open ( 1 / /Tma,, t U and 1 / / T I U ) and the proof showing another ( 1 / / F I U) is NP-hard is very complex. One of the open problems, 1 / / T m = I U, seems to be equivalent to 1 / / U I T m = , which is also open. If 3'2 = wF or 3'2 = wr, we can easily show the problems are NP-hard. Formally, for wF, we have Theorem 6. 1 / / w F I U is NP-hard.
Proof. Clearly, any algorithm for 1 / / w F I U would solve 1 / / w e l For
wT
Tmax = 0, which is NP-hard.
[]
we have
Theorem 7. 1 / / w r [U is NP-hard.
Proof. The proof of this theorem mimes the proof which shows knapsack is reducible to 1 / / w r , and thus NP-hard. For an instance of knapsack, al, a 2. . . . . a n, b, create an instance of 1 / / w T - I U by setting
Pi=wi=ai,
di=O
,
i = 1, 2 . . . . . n,
and add a new job, n + 1, with Pn+l = 1,
wn+ 1 = 2 and dn+ 1 = b + 1.
To minimize U, n + 1 must be completed before time b + 1, and hence it acts as the enforcer. From this point the proof is exactly the same as the one presented in Rinnooy Kan (1976) for 1 / / w T. [] To resolve the complexity of 1 / ~ F L U , we need to define Equal Partition (EP) as follows: Given g integers a 1 < a 2 < . . . < a g , with g even, let G = {1, 2 . . . . . g} and A = Y ' . g = l a i . Does there exits a subset of G, say G 1, with [ G 1 [ = l g and Ey ~ amaj Y'.j~c_61ay = 2A.19 Garey and Johnson (1979) show EP is NP-hard. For notational convenience, let h = ~g 1 represent half the elements of EP. Given an instance of EP, define an instance of 1 / / F I U, to be called I(FI U), by creating a 'real' job from each element of EP, with processing time and due date Pi and di. For each i, let Qi ([ Qil = ni) be a set of 'follower' jobs with Pi,j and di, j the processing time and due date for the j-th job in the set. Determine n i so that the decrease in flowtime resulting from placing job i at the end of the E D D sequence of I ( F I U) is Ai=
[ ag
-
a i +
1] X
where X = agn s.
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Let ng = ¼g[h - 1] and 8g = O. We can compute n i and 8i, i = g - 1, g - 2 . . . . ,1, recursively by
/a,-
E nk-(g-i).
k=i+l
k=i+l
The parameter 8~ will be used to determine the processing time of the last job in the follower set. Note 8 i < a;. For each i = 1, 2 . . . . . g let Pi = A + a i and Pi,j = A ,
j = 1, n i - 1,
Pi,,i = A
"1-~i"
Also, let i
i - 1 nk
Epk+ ~ EPk,~
di= [
k=l
for i = 1 , 2 . . . . . h,
k=l j=l i-I nk
i
Y'.
k=l+i-h
pk+ E E pI,,j
for i = h + 1, h + 2 . . . . . g.
k=l j=l
Furthermore, let
di, j = di+ 1 for i = 1, 2 . . . . . g - 1 and j = 1, 2 . . . . . ni, and g
dgj=
g
~'~pk + E k=l
nk
~"~Pk,v--½(g+l) A
forj=l,E,...,ng.
k=l v=l
Figures 1 and 2 show the E D D sequence of I ( F I U). From Figures 1 and 2, we see that I ( F I U ) has some important properties. First, it is constructed so that for each i, 3i < ai, which implies Pi,,~
There are many schedules for I ( F I U) which have exactly h tardy jobs. The next lemma characterizes some of them. [
nI
n2
I I:,+ 1
1,1
1,2
...
1,n~
[
ng
[,? la +l 2
2,1
2,2
...
2,n 2
...
g
I g,1
g,2
...
Figure 1. Real jobs and follower jobs of I ( F I U ) dh
dh +1
dh +2
dg
I
I
I
]
PP~I
I
...
[ I
Pl +P2
I
l
...
]P l
+
"" " +
Ph
I
h+2
Figure 2. Due dates for jobs h through g for
..
I(FIU)
g
...
g,ng
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121
L e m m a 2. I f the EDD schedule for I( F I U) is modified by moving any h real jobs whose processing times sum to at least ½(g + 1)A to the end o f the schedule, then all other jobs will be on time.
Proof. In the E D D schedule, jobs 1, 2 , . . . , h are on time, as are all of their followers. By the way the instance was constructed, the tardiness of job h + i is Pl + P 2 + " ' " +P~, for i = 1, 2 . . . . , h. Jobs in Oh +i- 1 have the same due date as job h + i, and since they are scheduled immediately before job h + i, their completion times must be at least Ph+i less than the completion time of job h + i. Therefore the tardiness of the jobs in Qh+i-1 is no greater than p~ + P 2 + " " " +P~ --Ph+2 for i = 2, 3 . . . . . h - 1. This also implies jobs in Qh a r e on time. Finally, the tardiness of jobs in Qg is at most ½(g + 1)A. At least one job, say k, 1 < k < h + 1, must be moved to the end if no more than h jobs are tardy. If it is not job h + 1, job h + 1 will be moved forward by Pk and job h + 1 will be on time, since its tardiness was Pl, and Pxp~ + P 2 --Ph+2 and hence Pl + P 2 --Ph+2 --Pk is less than or equal to zero. If job h + 1 is moved to the end, its followers will be on time since
Ph+l >Pk" In general, i real jobs between 1 and h + i must be moved to get h real jobs at the end. If job h + i is not moved, its completion time will be reduced by at least Pl + P 2 + " " " +P~, making it on time. The same reduction for its followers will make them on time also. If h + i is moved, the completion time for its followers will be reduced by at least p~ +P2 + "" " +Pi-1 +Ph+i, again making them on time. This argument takes care of all jobs except the followers of g, whose tardiness is, at most, ½(g + 1)A. If we require the sum of processing times of the real jobs moved to the end to be at least this amount, they too will be on time. Thus the result follows. [] From L e m m a 2 we know there are at least g choose h schedules for I ( F I U) with only h tardy jobs; i.e. more than 2 h. F r o m all schedules with h tardy jobs, we seek the one with smallest flowtime. While we cannot easily determine the best flowtime schedule for I ( F I U), we give a bound on how much it differs from the schedule with early jobs in E D D and the tardy jobs at the end in any order. I ~ t Ai be the reduction in flowtime gained by moving job i to the end of the E D D sequence for I ( F I U). Recall that X = agng. We now have L e m m a 3. A i > Ai, j + X , i = 1, 2 . . . . . g.
Proof. From E m m o n s (1975a), job i + 1 dominates follower job i,j for any i and j, so Ai+ 1 > Ai,j. By definition, Ai+ 1 = [ a g - a i + 1 + 1]X and A i = [ a g - a i + 1]X. Now A i > Ai+ 1 + X , since ai+ 1 > a i + 1. Therefore Ai>__Ai+l-~-X> Ai,j"~-S. [] To make this schedule better, we know from E m m o n s (1975a) that the tardy jobs should be in SPT order, and this can reduce the flowtime. Another decrease in flowtime could result when a real job is tardy and moved to the end of the schedule, and a follower job moves in front of one or more real jobs and the follower job immediately preceding them. This leads to the following lemma. L e m m a 4. The schedule with the early jobs in EDD followed by the tardy jobs in any order has flowtime no more than X greater than the optimal. Proof. Let A D J be a bound on the reduction of flowtime. The h tardy jobs can be reordered in SPT in at
most l g ( h - 1) movements with each movement reducing flowtime no more than ag, giving a reduction bounded by l g ( h - 1)ag. For each real job ' r e m o v e d ' one follower job may move in front of at most h - 1 real jobs, and for each, the last follower of the previous set. At most h followers can be advanced. Each follower job so moved could reduce flowtime by at most ag for each real job and each preceding follower, giving a maximum reduction of 2ag for each move. Advancing followers gives a reduction bounded by h(h - 1)2ag. Adding these two components together gives
A D J < { ( g [ h - 1])ag = X .
[]
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122
We are now ready for the result: 1 / ~FlU is NP-hard with respect to the id-encoding. Recall the id-encoding represents identical jobs by one copy of their data and the number of jobs with the same characteristics (Rinnooy Kan, 1976).
Theorem 8.
1//F
IU is NP-hard with respect to the M-encoding.
Proof. Let Y = F ( E D D ) - [ h ( a g + 1 ) - 1 A ] X . To prove 1 / / F I U is NP-hard with respect to the id-encoding, we have to show that EP reduces to 1 / / F I U. First we show that if EP has a partition, then I(F IU) has a schedule with h tardy jobs and total flowtime less than or equal to Y. Then we show that if I(F IU) has a schedule with h tardy jobs and total flowtime less than or equal to Y, then the tardy job set is a partition of EP. =~ : If EP has a partition, then I(F I U) has a schedule with h tardy jobs and total flowtime less than or equal to Y. Let L be the set of tardy jobs be given by the partition of EP, i.e. L = G 1, and let S be the optimal schedule for I(FIU) given this tardy set, with F(S) its total flowtime. From Lemma 2, we know that placing the jobs in L at the end results in a schedule with exactly h tardy jobs. Also, since ~ i ~ 1.ai = 1A, moving the jobs in L to the end of the E D D schedule will reduce F ( E D D ) by at least
~7. [ag-a i+ 1]X= [h(ag + 1)- ½A]X. i~L
So, F ( S ) < F ( E D D ) - [h(ag+ 1) - ½A]X< Y. Therefore I(F I U) has a solution at least as good as Y. : If I(F IU) has a schedule with h tardy jobs and total flowtime less than or equal to Y, then the tardy job set is a partition of EP. Let S be a schedule for I(FIU) with tardy set L, ILl = h , and F(S) < Y. To prove that the jobs in L are a partition of EP, we first show that all jobs in L are real jobs, and then show the sum of the processing times of the jobs in L is exactly equal to l ( g + 1)A. (a) We will show all the jobs in L are real jobs by contradiction. Assume L contains both real jobs and follower jobs. Let L = L U L 2, where L 1 contains real jobs and L 2 follower jobs. Of course, Y~'i E L l a i + ~ i j E L2~i j >-- 21A, else the number of tardy jobs will be larger than h. The reduction of F ( E D D ) resulting from moving jobs in L 1 and L 2 to the end is Ei ~/~,/[i and Ei,j ~ L2/[i~, respectively. So,
F( S) = F(EDD) - Y'~ Ai i~L1
:F(EDD)-
Y'~ /[ii- ADJ i,j~L2
Y'~ [ a g - a i + l ] X i~Ll
E
/[ij - A D J .
i,j~L2
Recall from Lemma 3,/[i~
F(S) >F(EDD) - E [ag-ai+ 1]X- E [ A i - - g ] - A D J iGL 1
i~L'2
:F(EDD)- • [ ( a g - a i + l ) ] X i~L 1
]~ [ ( a g - a i + l ) X - X ] - A D J i~L'2
[i l ai*
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123
Clearly, Ei~l:ai > Ei~L,2~i , SO (~,i~Llai + Y'.i~L, ai)> 1A, and recall that ADJ
F(S) > F ( E D D ) - [ h ( a g
+ 1 ) ] X + ½AX+ I L ~ I X - A D J
> F ( E D D ) - [h(ag + 1 ) ] X + ITAX= Y, and we have a contradiction. (b) Now we show that ~,i~Lai = ½A. Note that from the construction of I(FIU), this is equivalent to E i ~ L P i = ½(g + 1)A. From Lemma 2, for no more than h jobs to be tardy, we have Ei~ LP,- > ½(g + 1)A. To show equality of the sum, we assume the opposite and a contradiction results. Suppose ~,i~ LPi > ½(g + 1)A, or equivalently, ~,i~Lai > ½h. By definition, the total decrease in flowtime resulting from moving the jobs in L to the end of the EDD schedule is Ei~L[ag- a i + 1]g. So,
F(S)=F(EDD)-
Y'~ [ a g - - a i +
1]X-ADJ.
i~L
Since ~'i ~ Lai ~ ½A and ADJ < X, it follows that
[h(ag + 1 ) - ½ A ] X = Y.
F(S) >F(EDD)-
which is a contradiction. Therefore, Ei~Lai = ½A.
[]
3.5. Problems with weighted number of tardy jobs, tardiness or weighted tardiness as the primary criterion The fifteen problems with Yl ~/-,h and 3"2 ~/~ are all NP-hard. This is a direct consequence of Theorem 1.
3.6. Conclusions A summary of complexity results for single machine secondary criterion problems is given in Table 1. Only three of the 36 problems remain open, with 25 NP-hard and eight polynomially solvable. Seven of the eight polynomially solvable problems have flowtime or weighted flowtime as the primary criterion. Consider a tertiary criterion problem (1//Y313'213'1). This requires finding the best schedule for 3'3 out of all schedules that are best for 3'2 out of all alternative optimal schedules for 3"1-It seems unlikely that for a single machine there will be more than one or two schedules to choose from when we consider 3"3" However, it is clear that if 1 / / 3 " 2 [ 3'1 is NP-hard, so is 1 / //3"3 [3"2 13"1" Therefore, it seems unlikely that any tertiary problem will be polynomially solvable unless 3"1 = F or 3'1 = wr-
Table 1 C o m p l e x i t y o f 1 / //72171 Primary
Secondary criterion
criterion
Tmax
EF
Ew F
EU
Ew v
ET
Ew r
Tma~
-
P
NP-h
0
NP-h
NP-h
NP-h
EF
P
-
-
P
P
P
P
Y'.wF
P
-
-
P
NP-h
NP-h
NP-h
~-U
0
NP-h a
NP-h
-
-
0
NP-h
~w U
NP-h
NP-h
NP-h
-
-
NP-h
NP-h
ET
NP-h
NP-h
NP-h
NP-h
NP-h
-
-
~w r
NP-h
NP-h
NP-h
NP-h
NP-h
-
a W i t h r e s p e c t to t h e i d - e n c o d i n g .
-
124
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4. Bicriteria problems Since the 'order' of a criterion does not matter for bicriteria problems, there are only eighteen problems to consider.
4.1. Problems with maximum tardiness Van Wassenhove and Gelders (1980) give a polynomial algorithm for 1 / /Tm~I,F. It is based on Smith's work on 1 / /F[Tm~. Theorem 2 can be used to show that 1 / /Tmax,W v is NP-hard, since 1 / /wFlTma x is NP-hard. As in the case for its secondary criteria counterparts, 1 / /Tmax,U remains open. Theorem 1 can be used to show 1//Tm~x,Wv, 1//Tma~,T and 1//Tmax,W r are all NP-hard.
4.2. Problems with flowtime or weighted flowtime Theorems 2 and 7 can be used to show 1 / / F , U is NP-hard with respect to the id-encoding. Theorem 1 proves 1 / /F,T 2 is NP-hard for 3'2 ~ / , h . The same reasoning used for 1 / / F , T 2 shows 1 / //WF,Y 2 is NP-hard for Y2 = U and T2 E/-'h.
4.3. Problems with number of tardy jobs or weighted number of tardy jobs The problems 1 / / U , T ,
1 / / U , w r, 1 / / w v , T
and 1 / / w v , w r are all NP-hard as a result of Theorem
1.
4.4. Conclusions All bicriteria problems are NP-hard except 1 / /Tmax,F, which can be solved in polynomial time, and 1 / /Tmax,U, which is open. If we consider three or more criteria, everything is NP-hard.
5. Weighted criteria From Theorem 3, we see that except for 1 / / f ( T m ~ , F ) and 1 / /f(Tm~,U), all weighted criteria problems must be NP-hard. These two remain open. Again, if we consider three or more criteria, all problems are NP-hard.
6. Discussion We see from the previous results that polynomial algorithms are unlikely to exist for most single machine, multiple criteria scheduling problems. For secondary criterion problems there are three open problems, eight polynomially solvable problems and twenty-five NP-hard problems. All of the polynomially solvable problems have either F or w F as one of the criteria, while all three open problems have U as one of the criteria. With more than two criteria, unless the primary criterion is F or we, the problems appear to be NP-hard. For bicriteria problems we have one open, one polynomially solvable and sixteen NP-hard problems. All problems with three criteria considered simultaneously are NP-hard. For weighted two-criteria problems, there are two open problems, no polynomially solvable problems and sixteen NP-hard problems. Again, considering three or more criteria implies the problem is NP-hard. The first three theorems are very helpful in providing results. It should be noted that they are valid for multiple machine environments. Most multiple machine, multi-criteria problems are readily seen to be NP-hard, but some cases are not so obvious. Further work on these problems is needed.
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References Baker, K.R. (1974), Introduction To Sequencing And Scheduling, Wiley, New York. Chen, C.-L., and Bulfin, R.L. (1989), "Scheduling unit processing time jobs on a single machine with multiple criteria", Computers & Operations Research 17, 1-17. Du, J., and Leung, J.Y.-T. (1990), "Minimizing total tardiness on one machine is NP-hard", Mathematics of Operations Research 15, 483-493. Emmons, H. (1975a), "One machine sequencing to minimize mean flow time with minimum number tardy", Naval Research Logistics Quarterly 22, 585-592. Emmons, H. (1975b), "A note on a scheduling problem with dual criteria", Naval Research Logistics Quarterly 22, 615-616. French, S. (1986), Sequencing and Scheduling, Wiley, New York. Garey, M.R., and Johnson, D.S. (1979), Computers and Intractability, Freeman, San Francisco, CA. Goicoechea, A., Hansen, D.R., and Duckstein, L. (1982), Multiobjective Decision Analysis with Engineering and Business Applications, Wiley, New York. Graham, R.L., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G. (1979), "Optimization and approximation in deterministic sequencing and scheduling", Annals of Discrete Mathematics 5, 287-326. Lageweg, B.J., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G. (1981), "Computer aided complexity classification of deterministic scheduling problems", Stichting Mathematisch Centrum, Amsterdam, Netherlands. Lawler, E.L. (1976), "Sequencing to minimize the weighted number of tardy jobs", Recherche 01~rationnelle 10, 17-33. Lenstra, J.K., Rinnooy Kan, A.H.G., and Brucker, P. (1977), "Complexity of machine scheduling problems", Annals of Discrete Mathematics 1, 343-362. Panwalkar, S.S., Dudek, R.K., and Smith, M.L. (1973), "Sequencing research and the industrial scheduling problem", in: S.E. Elmaghraby (ed.), Symposium on the Theory of Scheduling and its Application, Springer-Verlag, New York, 29-38. Rinnooy Kan, A.H.G. (1976), Machine Scheduling Problems: Classification, Complexity and Computations, Martinus Nijhoff, The Hague, Holland. Smith, W.E. (1956), "Various optimizers for single stage production", Naval Research Logistics Quarterly 3, 59-66. Steuer, R.E. (1986), Multiple Criteria Optimization: Theory, Computation, and Applications, Wiley, New York. Van Wassenhove, L.N., and Gelders, L.F. (1980), "Solving a bicriteria scheduling problem", European Journal of Operational Research 4, 42-48.
115
Theory and Methodology
Complexity of single machine, multi-criteria scheduling problems Chuen-Lung Chen Department of Industrial Engineering, Mississippi State University, Mississippi State, MS 39762, USA R o b e r t L. B u l f i n
Department of Industrial Engineering, Auburn University, Auburn, AL 36849, USA Received March 1991; revised September 1991
Abstract: We examine the complexity of scheduling problems when more than one measure of perfor-
mance is appropriate. The criteria we study are maximal tardiness, flowtime, number of tardy jobs, tardiness and the weighted counterparts of the last three measures. The machine environment is restricted to a single machine. Complexity results are given for secondary criterion, bicriteria and weighted criteria approaches for all combinations of measures. Of the problems examined, only six remain open. Keywords: Complexity; Multiple criteria; Scheduling; Single machine 1. Introduction
In 1973, a survey by Panwalkar, Dudek and Smith (1973) pointed out the importance of considering multiple measures of performance in scheduling problems. Almost two decades later, there are still quite a few open problems in scheduling with multiple criteria. We consider the complexity of some of these problems in this paper. Not surprisingly, almost all are NP-hard. While other researchers have developed heuristics or enumerative algorithms for multi-criteria scheduling problems, we will not discuss their work unless it deals with problem complexity. We start with a brief background discussion of scheduling, multicriteria optimization and complexity. We then discuss single machine results for secondary, multiple criteria, and weighted criteria approaches for each combination of the various criteria. A brief discussion of the results concludes the paper. 2. Background
This section will introduce the notation and define the environment for the remainder of the paper. It will be divided into three parts, the scheduling problems addressed, the types of multiple criteria problem considered, and complexity. Correspondence to: R.L. Bulfin, Department of Industrial Engineering, College of Engineering, Auburn University, 207 Dunstan Hail, Auburn, AL 36849-5346, USA.
0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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2.1. Scheduling environment
To define a scheduling problem, let n be the number of jobs to be processed and Pi, ri, di, and w i be the processing time, release data, due date, and weight (or 'value') of job i, respectively. Given a schedule, for each job i, we can define Ci to be its completion time, F~ = C~ - r~ its flowtime, L~ = C i - di its lateness, and T,. = max{0, L~} its tardiness. Let the maximum tardiness be Tmax = max i {T/}. Finally, if job i is tardy (i.e. T/> 0), let U/ be one, and zero if job i is completed on or before its due date (T/< 0). Traditional measures of performance for machine scheduling include minimizing total flowtime, total tardiness, maximum tardiness or the number of tardy jobs, which we will denote by F, T, Tmax and U, respectively. These measures (except for Tmax) are simply the sum over all jobs of the respective quantity for each job. If all jobs are not equally important, an equivalent weighted measure can be calculated by multiplying the quantity by the appropriate job weight. These will be denoted by WF, W T and w v respectively. For a more detailed discussion of measures of performance used in scheduling, see Baker (1974) or French (1986). Assumptions made for most scheduling problems are that data are known with certainty, setup times are independent of the order of processing, all jobs are immediately available (r i = 0), no precedence exists between jobs and once processing of a job starts, it is not interrupted. Graham et al. (1979) have defined a three-field notation, t~/13/3", to describe scheduling problems where the first field corresponds to the machine environment, the second to deviations from normal assumptions, and the third denotes the criteria or measure of performance to be used. For this work, standard assumptions will be in force. The criteria will be as previously defined, and hence 3' ~ F = {Tmax, F, WF, U, w v, T, WT}. Because one subgroup of these criteria occur frequently, we define F h = {wv, T, wr}. The machine environment will consist of a single machine, i.e. a = 1. As an example 1 / /Tma x represents minimizing maximal tardiness on a single machine under the standard assumptions. 2.2. Multicriteria scheduling Although few researchers have addressed multicriteria scheduling problems, much work has been done on general multicriteria optimization problems; for details, see Goicoechea et al. (1982) or Steuer (1986). We will discuss three general multicriteria approaches that are applicable to scheduling problems: secondary criterion, efficient set generation and weighting of criteria. We present results in terms of two criteria since it greatly simplifies discussion. However, in most cases the generalization to three or more criteria is straightforward. The secondary criterion approach designates one criterion as primary and the other as secondary. The objective is to find the best schedule for the secondary criterion among all alternative optimal schedules for the primary criterion. In an obvious extension to the notation of Graham et al. (1979), we will denote these problems 1 / /3'213'1 where 3'1 is the primary criterion and 3'2 is the secondary criterion. Smith (1956) was the first to look at a single machine secondary criteria scheduling problem, minimizing flowtime with no tardy jobs ( 1 / / F I Tma~ = 0). Another approach is to generate all efficient (nondominated, Pareto optimal) schedules for a problem, and then allow the decision maker to make explicit trade-offs between these schedules. When there are two criteria, we call this a bicriteria approach, and denote it by 1 / /3'1,3"2, where the two criteria of interest are 3'1 and 3"2- 1 / /F,Tma x would denote the problem to generate all nondominated schedules considering flowtime and maximum tardiness simultaneously. A final approach is to use a weighting function. Here the decision maker expresses a tradeoff which, once specified, allows the problem to be solved with a single criterion. We will consider only functions which depend on the previously defined measures. A scheduling problem with two criteria, say 3'1 and 3'2, and a given weighting function f is denoted by 1 / / f ( 3 ' 1 , 3"2). For scheduling problems, it seems t~ make sense to restrict f to linear combinations of the completion times of the criteria, e.g. f(3'1, 3"2) = A13'1 + A23"2.
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We will not consider Cm~x as a criterion, since it is constant for a single machine. Neither will we consider a measure and its weighted counterpart, e.g. F and w F. 2.3. Complexity The complexity of 1 / /3' for any 3" previously defined is well known. Lageweg et al. (1981) provide the results for all measures except tardiness. This gap in single machine complexity results was recently closed by Du and Leung (1990), who showed 1 / / T is NP-hard. The secondary criteria and weighting approaches result in a single schedule, and complexity analysis is straightforward. This is not the case for the multicriteria approach, since the number of nondominated schedules could be exponential in the problem size. For our discussion, we will say that a multicriteria problem is polynomially solvable if we can generate each nondominated schedule in polynomial time. If the number of nondominated schedules is bounded by a polynomial in the problem size, we can generate them all in polynomial time. Unfortunately, unless there are special restrictions, such as unit execution time jobs, there seems to be no way to guarantee this. Since almost none of the multicriteria problems can generate a single nondominated schedule in polynomial time, it is a moot point anyway. We will now give some theorems which, while quite obvious, prove to be very powerful. The first relates the complexity of single criterion problems to multiple criteria problems. Theorem 1. / f 1 /
//3,1 /S NP-hard, then 1///3'213,1, 1//3,1,3,2 and 1 / //f(3,1' 3,2) are NP-hard.
Proof. We will show that the existence of a polynomial algorithm for 1 / //3/213,1, 1 / /3'1, 3,2 and
1// //f(3,1, 3,2 ) implies the existence of a polynomial algorithm for 1 / //3,1, thus proving the result. Clearly, any schedule solving 1// //3'213'1 must be optimal for 1 / //3"1, giving the first result. Since 1//3"1,3"2 generates all nondominated schedules and a solution to 1///3'1 is a nondominated schedule for 1 / //3/1,3,2, the second result is at hand. Finally, a solution to 1 / //f(3,1, 3,2 ) with )t 1 = 1 and A2 = 0 provides a solution to 1 / / Y r [] A second theorem relates the complexity of secondary criterion problems to bicriteria problems. Theorem 2. If 1 / //3/213'1 /s NP-hard, 1 /
//3"1,3,2 /S NP-hard.
Proof. Again we will show that the existence of a polynomial algorithm for 1///3'1,3"2 implies 1 / //3"2 I 3,1 can be solved in polynomial time. Since the schedule which solves 1 / //3,213,1 is a nondominated schedule for 1 / //3/1,3,2, it is produced as a consequence of solving 1// //3,1,3,2- [] Finally, we show that finding a solution to the weighted problem is just as hard as the bicriteria problem. Theorem 3. / f 1 / /3'1,3/2 /S NP-hard, 1 / /f(3"1, 3"2) is NP-hard.
Proof. For the proper choice of weights, the weighting method can be used to generate nondominated solutions (Goicoechea et al., 1982). Therefore, a polynomial algorithm for 1 / / f ( 3 , 1 , 3'2) can generate a nondominated schedule in polynomial time, which gives the result. [] These results will prove useful in showing various problems are NP-hard. We will examine the complexity of secondary criterion, bicriteria and weighted criteria for the single machine problem. We begin with secondary criterion problems.
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3. Secondary criterion problems In this section we examine the 36 secondary criterion problems 1//3"21 3'1, where 3'1 and 3,2 belong to F. Recall that problems with a criterion and its weighted counterpart, such as 1 / / F l w v , will not be considered. We will examine these problems in order of their primary criterion. 3.1. Problems with m a x i m u m tardiness as the primary criterion
The first secondary criterion problem studied was 1 / / F I Tmax. Smith (1956) developed a polynomial algorithm for this problem. The weighted flowtime problem, 1 / /wvlTm~x, can easily be shown to be NP-hard. This is an obvious extension of a result by Lenstra et al. (1977) which showed 1 / / w F I Tmax = 0 is NP-hard. Now consider 1//3'21Tmax where 3,2 ~ Fh. We have
Theorem 4. 1 / / w v l T m a x, 1 / / T I T m a x and 1 / / w r l T m ~ x are NP-hard. Proof. We will show that any algorithm for 1//3,21Tmax will solve 1 / / 3 ' 2 . Since 1 / / 3 ' 2 , 3"2 ~ Fh, are all NP-hard, the result follows. Consider an instance of 1 / / w v with n jobs having processing times Pi, due dates d i and weights wr Assume d 1 < d 2 < • .. _
d , , + l = d , , + 1 and
Wn+l=0.
Any schedule which minimizes Tmax must have job n + 1 last. The order of the other jobs has no effect on Tmax, and since wn+ t = 0, job n + 1 has no effect on w v. Applying an algorithm for 1 / / w t r l T m a x to this instance would result in a schedule which solves 1 / / w U for the original instance. Since 1 / / w v is NP-hard, then so is 1 / / w v I Tmax. A similar argument holds for 3'2 ~ {T, wr}. [] These results leave 1 / / U
I Tm~x as an open problem.
3.2. Problems with flowtime as the primary criterion
Recall that 1 / / F is solved by sequencing the jobs in nondecreasing order of their processing times (SPT rule). In order for there to be alternate optimal schedules, there must be ties in the SPT sequence. For any jobs with identical processing times, we can use the results of Chen and Bulfin (1990) to solve the secondary criterion problem. For 3"2 = Tm~x or 3'2 = T, simply break ties by earliest due date (EDD) first. For U, simply apply Hodgson's algorithm to each set of tied jobs. For w u use Lawler's (1976) algorithm for 1 / U E T / w v for the tied jobs. Unfortunately, for 3'2 = w r , we must solve an assignment problem over the tied jobs; details can be found in Chen and Bulfin (1990). Therefore we see that 1 / / 3 " 2 1 F can be solved in polynomial time for all 3"2 ~ F. 3.3. Problems with weighted flowtime as the primary criterion
Ordering the jobs with the first job having the smallest ratio of processing time to weight, the second job the next smallest, and so on, i.e. WSPT, solves 1 / / w F. As there must be ties in SPT for 1 / / 3 , 2 1 F , there must be ties in the processing time to weight ratio to have alternative optimal schedules 1//3'2 [wF. For 3'2 = Tm~, simply break ties by placing the job with the earliest due date first. When 3'2 = U, Hodgson's algorithm can be used on the set of tied jobs. However, secondary criterion problems with 3"2 ~ Fh are NP-hard, as shown by the following theorem.
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119
Theorem 5. 1 / /3"21WF is NP-hard for 3"2 ~ Fh.
Proof. Consider an instance of 1 / / w v with n jobs and Pi, di and w~ the processing time, due date and weight of job i, respectively. Further, let w~ =Pi for all i. Create an instance of 1 / / w v l w t from the same data. Since the ratios of processing time to weight are all one, every schedule has the same weighted flowtime, and any solution to 1 / / w v l w F is also a solution to 1 / / w U. Lenstra et al. (1977) show that 1 / / w e is NP-hard even if the weights are equal to the processing times. Therefore 1 / /WFI w F is NP-hard. Similar arguments can be made for 3'2 = T and 3"2 = w~r. []
3.4. Problems with number of tardy jobs as the primary criterion Surprisingly, this is probably the most intriguing group of problems in multiple criteria scheduling. Two of the five problems are open ( 1 / /Tma,, t U and 1 / / T I U ) and the proof showing another ( 1 / / F I U) is NP-hard is very complex. One of the open problems, 1 / / T m = I U, seems to be equivalent to 1 / / U I T m = , which is also open. If 3'2 = wF or 3'2 = wr, we can easily show the problems are NP-hard. Formally, for wF, we have Theorem 6. 1 / / w F I U is NP-hard.
Proof. Clearly, any algorithm for 1 / / w F I U would solve 1 / / w e l For
wT
Tmax = 0, which is NP-hard.
[]
we have
Theorem 7. 1 / / w r [U is NP-hard.
Proof. The proof of this theorem mimes the proof which shows knapsack is reducible to 1 / / w r , and thus NP-hard. For an instance of knapsack, al, a 2. . . . . a n, b, create an instance of 1 / / w T - I U by setting
Pi=wi=ai,
di=O
,
i = 1, 2 . . . . . n,
and add a new job, n + 1, with Pn+l = 1,
wn+ 1 = 2 and dn+ 1 = b + 1.
To minimize U, n + 1 must be completed before time b + 1, and hence it acts as the enforcer. From this point the proof is exactly the same as the one presented in Rinnooy Kan (1976) for 1 / / w T. [] To resolve the complexity of 1 / ~ F L U , we need to define Equal Partition (EP) as follows: Given g integers a 1 < a 2 < . . . < a g , with g even, let G = {1, 2 . . . . . g} and A = Y ' . g = l a i . Does there exits a subset of G, say G 1, with [ G 1 [ = l g and Ey ~ amaj Y'.j~c_61ay = 2A.19 Garey and Johnson (1979) show EP is NP-hard. For notational convenience, let h = ~g 1 represent half the elements of EP. Given an instance of EP, define an instance of 1 / / F I U, to be called I(FI U), by creating a 'real' job from each element of EP, with processing time and due date Pi and di. For each i, let Qi ([ Qil = ni) be a set of 'follower' jobs with Pi,j and di, j the processing time and due date for the j-th job in the set. Determine n i so that the decrease in flowtime resulting from placing job i at the end of the E D D sequence of I ( F I U) is Ai=
[ ag
-
a i +
1] X
where X = agn s.
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120
Let ng = ¼g[h - 1] and 8g = O. We can compute n i and 8i, i = g - 1, g - 2 . . . . ,1, recursively by
/a,-
E nk-(g-i).
k=i+l
k=i+l
The parameter 8~ will be used to determine the processing time of the last job in the follower set. Note 8 i < a;. For each i = 1, 2 . . . . . g let Pi = A + a i and Pi,j = A ,
j = 1, n i - 1,
Pi,,i = A
"1-~i"
Also, let i
i - 1 nk
Epk+ ~ EPk,~
di= [
k=l
for i = 1 , 2 . . . . . h,
k=l j=l i-I nk
i
Y'.
k=l+i-h
pk+ E E pI,,j
for i = h + 1, h + 2 . . . . . g.
k=l j=l
Furthermore, let
di, j = di+ 1 for i = 1, 2 . . . . . g - 1 and j = 1, 2 . . . . . ni, and g
dgj=
g
~'~pk + E k=l
nk
~"~Pk,v--½(g+l) A
forj=l,E,...,ng.
k=l v=l
Figures 1 and 2 show the E D D sequence of I ( F I U). From Figures 1 and 2, we see that I ( F I U ) has some important properties. First, it is constructed so that for each i, 3i < ai, which implies Pi,,~
There are many schedules for I ( F I U) which have exactly h tardy jobs. The next lemma characterizes some of them. [
nI
n2
I I:,+ 1
1,1
1,2
...
1,n~
[
ng
[,? la +l 2
2,1
2,2
...
2,n 2
...
g
I g,1
g,2
...
Figure 1. Real jobs and follower jobs of I ( F I U ) dh
dh +1
dh +2
dg
I
I
I
]
PP~I
I
...
[ I
Pl +P2
I
l
...
]P l
+
"" " +
Ph
I
h+2
Figure 2. Due dates for jobs h through g for
..
I(FIU)
g
...
g,ng
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121
L e m m a 2. I f the EDD schedule for I( F I U) is modified by moving any h real jobs whose processing times sum to at least ½(g + 1)A to the end o f the schedule, then all other jobs will be on time.
Proof. In the E D D schedule, jobs 1, 2 , . . . , h are on time, as are all of their followers. By the way the instance was constructed, the tardiness of job h + i is Pl + P 2 + " ' " +P~, for i = 1, 2 . . . . , h. Jobs in Oh +i- 1 have the same due date as job h + i, and since they are scheduled immediately before job h + i, their completion times must be at least Ph+i less than the completion time of job h + i. Therefore the tardiness of the jobs in Qh+i-1 is no greater than p~ + P 2 + " " " +P~ --Ph+2 for i = 2, 3 . . . . . h - 1. This also implies jobs in Qh a r e on time. Finally, the tardiness of jobs in Qg is at most ½(g + 1)A. At least one job, say k, 1 < k < h + 1, must be moved to the end if no more than h jobs are tardy. If it is not job h + 1, job h + 1 will be moved forward by Pk and job h + 1 will be on time, since its tardiness was Pl, and Px
Ph+l >Pk" In general, i real jobs between 1 and h + i must be moved to get h real jobs at the end. If job h + i is not moved, its completion time will be reduced by at least Pl + P 2 + " " " +P~, making it on time. The same reduction for its followers will make them on time also. If h + i is moved, the completion time for its followers will be reduced by at least p~ +P2 + "" " +Pi-1 +Ph+i, again making them on time. This argument takes care of all jobs except the followers of g, whose tardiness is, at most, ½(g + 1)A. If we require the sum of processing times of the real jobs moved to the end to be at least this amount, they too will be on time. Thus the result follows. [] From L e m m a 2 we know there are at least g choose h schedules for I ( F I U) with only h tardy jobs; i.e. more than 2 h. F r o m all schedules with h tardy jobs, we seek the one with smallest flowtime. While we cannot easily determine the best flowtime schedule for I ( F I U), we give a bound on how much it differs from the schedule with early jobs in E D D and the tardy jobs at the end in any order. I ~ t Ai be the reduction in flowtime gained by moving job i to the end of the E D D sequence for I ( F I U). Recall that X = agng. We now have L e m m a 3. A i > Ai, j + X , i = 1, 2 . . . . . g.
Proof. From E m m o n s (1975a), job i + 1 dominates follower job i,j for any i and j, so Ai+ 1 > Ai,j. By definition, Ai+ 1 = [ a g - a i + 1 + 1]X and A i = [ a g - a i + 1]X. Now A i > Ai+ 1 + X , since ai+ 1 > a i + 1. Therefore Ai>__Ai+l-~-X> Ai,j"~-S. [] To make this schedule better, we know from E m m o n s (1975a) that the tardy jobs should be in SPT order, and this can reduce the flowtime. Another decrease in flowtime could result when a real job is tardy and moved to the end of the schedule, and a follower job moves in front of one or more real jobs and the follower job immediately preceding them. This leads to the following lemma. L e m m a 4. The schedule with the early jobs in EDD followed by the tardy jobs in any order has flowtime no more than X greater than the optimal. Proof. Let A D J be a bound on the reduction of flowtime. The h tardy jobs can be reordered in SPT in at
most l g ( h - 1) movements with each movement reducing flowtime no more than ag, giving a reduction bounded by l g ( h - 1)ag. For each real job ' r e m o v e d ' one follower job may move in front of at most h - 1 real jobs, and for each, the last follower of the previous set. At most h followers can be advanced. Each follower job so moved could reduce flowtime by at most ag for each real job and each preceding follower, giving a maximum reduction of 2ag for each move. Advancing followers gives a reduction bounded by h(h - 1)2ag. Adding these two components together gives
A D J < { ( g [ h - 1])ag = X .
[]
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We are now ready for the result: 1 / ~FlU is NP-hard with respect to the id-encoding. Recall the id-encoding represents identical jobs by one copy of their data and the number of jobs with the same characteristics (Rinnooy Kan, 1976).
Theorem 8.
1//F
IU is NP-hard with respect to the M-encoding.
Proof. Let Y = F ( E D D ) - [ h ( a g + 1 ) - 1 A ] X . To prove 1 / / F I U is NP-hard with respect to the id-encoding, we have to show that EP reduces to 1 / / F I U. First we show that if EP has a partition, then I(F IU) has a schedule with h tardy jobs and total flowtime less than or equal to Y. Then we show that if I(F IU) has a schedule with h tardy jobs and total flowtime less than or equal to Y, then the tardy job set is a partition of EP. =~ : If EP has a partition, then I(F I U) has a schedule with h tardy jobs and total flowtime less than or equal to Y. Let L be the set of tardy jobs be given by the partition of EP, i.e. L = G 1, and let S be the optimal schedule for I(FIU) given this tardy set, with F(S) its total flowtime. From Lemma 2, we know that placing the jobs in L at the end results in a schedule with exactly h tardy jobs. Also, since ~ i ~ 1.ai = 1A, moving the jobs in L to the end of the E D D schedule will reduce F ( E D D ) by at least
~7. [ag-a i+ 1]X= [h(ag + 1)- ½A]X. i~L
So, F ( S ) < F ( E D D ) - [h(ag+ 1) - ½A]X< Y. Therefore I(F I U) has a solution at least as good as Y. : If I(F IU) has a schedule with h tardy jobs and total flowtime less than or equal to Y, then the tardy job set is a partition of EP. Let S be a schedule for I(FIU) with tardy set L, ILl = h , and F(S) < Y. To prove that the jobs in L are a partition of EP, we first show that all jobs in L are real jobs, and then show the sum of the processing times of the jobs in L is exactly equal to l ( g + 1)A. (a) We will show all the jobs in L are real jobs by contradiction. Assume L contains both real jobs and follower jobs. Let L = L U L 2, where L 1 contains real jobs and L 2 follower jobs. Of course, Y~'i E L l a i + ~ i j E L2~i j >-- 21A, else the number of tardy jobs will be larger than h. The reduction of F ( E D D ) resulting from moving jobs in L 1 and L 2 to the end is Ei ~/~,/[i and Ei,j ~ L2/[i~, respectively. So,
F( S) = F(EDD) - Y'~ Ai i~L1
:F(EDD)-
Y'~ /[ii- ADJ i,j~L2
Y'~ [ a g - a i + l ] X i~Ll
E
/[ij - A D J .
i,j~L2
Recall from Lemma 3,/[i~
F(S) >F(EDD) - E [ag-ai+ 1]X- E [ A i - - g ] - A D J iGL 1
i~L'2
:F(EDD)- • [ ( a g - a i + l ) ] X i~L 1
]~ [ ( a g - a i + l ) X - X ] - A D J i~L'2
[i l ai*
C.-L. Chen, R.L. Bulf'm / Single machine multi-criteria scheduling
123
Clearly, Ei~l:ai > Ei~L,2~i , SO (~,i~Llai + Y'.i~L, ai)> 1A, and recall that ADJ
F(S) > F ( E D D ) - [ h ( a g
+ 1 ) ] X + ½AX+ I L ~ I X - A D J
> F ( E D D ) - [h(ag + 1 ) ] X + ITAX= Y, and we have a contradiction. (b) Now we show that ~,i~Lai = ½A. Note that from the construction of I(FIU), this is equivalent to E i ~ L P i = ½(g + 1)A. From Lemma 2, for no more than h jobs to be tardy, we have Ei~ LP,- > ½(g + 1)A. To show equality of the sum, we assume the opposite and a contradiction results. Suppose ~,i~ LPi > ½(g + 1)A, or equivalently, ~,i~Lai > ½h. By definition, the total decrease in flowtime resulting from moving the jobs in L to the end of the EDD schedule is Ei~L[ag- a i + 1]g. So,
F(S)=F(EDD)-
Y'~ [ a g - - a i +
1]X-ADJ.
i~L
Since ~'i ~ Lai ~ ½A and ADJ < X, it follows that
[h(ag + 1 ) - ½ A ] X = Y.
F(S) >F(EDD)-
which is a contradiction. Therefore, Ei~Lai = ½A.
[]
3.5. Problems with weighted number of tardy jobs, tardiness or weighted tardiness as the primary criterion The fifteen problems with Yl ~/-,h and 3"2 ~/~ are all NP-hard. This is a direct consequence of Theorem 1.
3.6. Conclusions A summary of complexity results for single machine secondary criterion problems is given in Table 1. Only three of the 36 problems remain open, with 25 NP-hard and eight polynomially solvable. Seven of the eight polynomially solvable problems have flowtime or weighted flowtime as the primary criterion. Consider a tertiary criterion problem (1//Y313'213'1). This requires finding the best schedule for 3'3 out of all schedules that are best for 3'2 out of all alternative optimal schedules for 3"1-It seems unlikely that for a single machine there will be more than one or two schedules to choose from when we consider 3"3" However, it is clear that if 1 / / 3 " 2 [ 3'1 is NP-hard, so is 1 / //3"3 [3"2 13"1" Therefore, it seems unlikely that any tertiary problem will be polynomially solvable unless 3"1 = F or 3'1 = wr-
Table 1 C o m p l e x i t y o f 1 / //72171 Primary
Secondary criterion
criterion
Tmax
EF
Ew F
EU
Ew v
ET
Ew r
Tma~
-
P
NP-h
0
NP-h
NP-h
NP-h
EF
P
-
-
P
P
P
P
Y'.wF
P
-
-
P
NP-h
NP-h
NP-h
~-U
0
NP-h a
NP-h
-
-
0
NP-h
~w U
NP-h
NP-h
NP-h
-
-
NP-h
NP-h
ET
NP-h
NP-h
NP-h
NP-h
NP-h
-
-
~w r
NP-h
NP-h
NP-h
NP-h
NP-h
-
a W i t h r e s p e c t to t h e i d - e n c o d i n g .
-
124
C.-L. Chen, R.L. Bulfin / Singlemachinemulti-criteriascheduling
4. Bicriteria problems Since the 'order' of a criterion does not matter for bicriteria problems, there are only eighteen problems to consider.
4.1. Problems with maximum tardiness Van Wassenhove and Gelders (1980) give a polynomial algorithm for 1 / /Tm~I,F. It is based on Smith's work on 1 / /F[Tm~. Theorem 2 can be used to show that 1 / /Tmax,W v is NP-hard, since 1 / /wFlTma x is NP-hard. As in the case for its secondary criteria counterparts, 1 / /Tmax,U remains open. Theorem 1 can be used to show 1//Tm~x,Wv, 1//Tma~,T and 1//Tmax,W r are all NP-hard.
4.2. Problems with flowtime or weighted flowtime Theorems 2 and 7 can be used to show 1 / / F , U is NP-hard with respect to the id-encoding. Theorem 1 proves 1 / /F,T 2 is NP-hard for 3'2 ~ / , h . The same reasoning used for 1 / / F , T 2 shows 1 / //WF,Y 2 is NP-hard for Y2 = U and T2 E/-'h.
4.3. Problems with number of tardy jobs or weighted number of tardy jobs The problems 1 / / U , T ,
1 / / U , w r, 1 / / w v , T
and 1 / / w v , w r are all NP-hard as a result of Theorem
1.
4.4. Conclusions All bicriteria problems are NP-hard except 1 / /Tmax,F, which can be solved in polynomial time, and 1 / /Tmax,U, which is open. If we consider three or more criteria, everything is NP-hard.
5. Weighted criteria From Theorem 3, we see that except for 1 / / f ( T m ~ , F ) and 1 / /f(Tm~,U), all weighted criteria problems must be NP-hard. These two remain open. Again, if we consider three or more criteria, all problems are NP-hard.
6. Discussion We see from the previous results that polynomial algorithms are unlikely to exist for most single machine, multiple criteria scheduling problems. For secondary criterion problems there are three open problems, eight polynomially solvable problems and twenty-five NP-hard problems. All of the polynomially solvable problems have either F or w F as one of the criteria, while all three open problems have U as one of the criteria. With more than two criteria, unless the primary criterion is F or we, the problems appear to be NP-hard. For bicriteria problems we have one open, one polynomially solvable and sixteen NP-hard problems. All problems with three criteria considered simultaneously are NP-hard. For weighted two-criteria problems, there are two open problems, no polynomially solvable problems and sixteen NP-hard problems. Again, considering three or more criteria implies the problem is NP-hard. The first three theorems are very helpful in providing results. It should be noted that they are valid for multiple machine environments. Most multiple machine, multi-criteria problems are readily seen to be NP-hard, but some cases are not so obvious. Further work on these problems is needed.
C.-L. Chen, R.L. Bulfin / Single machine multi-criteria scheduling
125
References Baker, K.R. (1974), Introduction To Sequencing And Scheduling, Wiley, New York. Chen, C.-L., and Bulfin, R.L. (1989), "Scheduling unit processing time jobs on a single machine with multiple criteria", Computers & Operations Research 17, 1-17. Du, J., and Leung, J.Y.-T. (1990), "Minimizing total tardiness on one machine is NP-hard", Mathematics of Operations Research 15, 483-493. Emmons, H. (1975a), "One machine sequencing to minimize mean flow time with minimum number tardy", Naval Research Logistics Quarterly 22, 585-592. Emmons, H. (1975b), "A note on a scheduling problem with dual criteria", Naval Research Logistics Quarterly 22, 615-616. French, S. (1986), Sequencing and Scheduling, Wiley, New York. Garey, M.R., and Johnson, D.S. (1979), Computers and Intractability, Freeman, San Francisco, CA. Goicoechea, A., Hansen, D.R., and Duckstein, L. (1982), Multiobjective Decision Analysis with Engineering and Business Applications, Wiley, New York. Graham, R.L., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G. (1979), "Optimization and approximation in deterministic sequencing and scheduling", Annals of Discrete Mathematics 5, 287-326. Lageweg, B.J., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G. (1981), "Computer aided complexity classification of deterministic scheduling problems", Stichting Mathematisch Centrum, Amsterdam, Netherlands. Lawler, E.L. (1976), "Sequencing to minimize the weighted number of tardy jobs", Recherche 01~rationnelle 10, 17-33. Lenstra, J.K., Rinnooy Kan, A.H.G., and Brucker, P. (1977), "Complexity of machine scheduling problems", Annals of Discrete Mathematics 1, 343-362. Panwalkar, S.S., Dudek, R.K., and Smith, M.L. (1973), "Sequencing research and the industrial scheduling problem", in: S.E. Elmaghraby (ed.), Symposium on the Theory of Scheduling and its Application, Springer-Verlag, New York, 29-38. Rinnooy Kan, A.H.G. (1976), Machine Scheduling Problems: Classification, Complexity and Computations, Martinus Nijhoff, The Hague, Holland. Smith, W.E. (1956), "Various optimizers for single stage production", Naval Research Logistics Quarterly 3, 59-66. Steuer, R.E. (1986), Multiple Criteria Optimization: Theory, Computation, and Applications, Wiley, New York. Van Wassenhove, L.N., and Gelders, L.F. (1980), "Solving a bicriteria scheduling problem", European Journal of Operational Research 4, 42-48.