Single-machine Sequencing
2.4 Problems with Due Dates: Elementary Results
2.4.3 Minimizing Total Tardiness
The performance objective of“meeting job due dates”is one of the scheduling criteria most frequently encountered in practical problems. While meeting due dates is only a qualitative goal, it usually implies that time-dependent penalties are assessed on late jobs but that no benefits derive from completing jobs early.
This interpretation leads naturally to the tardiness measure as a quantification of the scheduling objective, and a fundamental sequencing problem is the min- imization of total tardiness. The difficulty of dealing with this measure, and with most other tardiness-based performance measures, arises from the fact that tardiness is not a linear function of completion time. This means that finding optimal solutions often requires that we draw on general techniques of combi- natorial optimization. Furthermore, because of the complexities of combinato- rial methods, there is apt to be more attention paid to efficient heuristic techniques. In the next chapter, we shall discuss general-purpose combinatorial optimization techniques and demonstrate their application to the total tardiness
B B B B A A A
Late jobs Early jobs
Figure 2.6 The form of a sequence that minimizesU.
2 Single-machine Sequencing 26
criterion. Here, we examine how much progress we can make with simpler techniques.
A logical first approach to the tardiness problem is to analyze an adjacent pair- wise interchange. Consider a scheduleS, in which jobsiandjare adjacent in sequence, and the schedule S that is identical to S except that jobs i and j are interchanged (see Figure 2.4). We seek conditions that will tell us which job should appear earlier in the sequence. Rather than comparingTfor both sequences, it suffices to compare the contributions toTthat come from jobs iandj, because the total contributions of the other jobs are the same in both sequences. Thus let
Tij=Ti S +Tj S = max p B +pi–di, 0 + max p B +pi+pj–dj, 0 and
Tji=Tj S +Ti S = max p B +pj–dj, 0 + max p B +pi+pj–di, 0 where, as before,p(B) denotes the time at which jobior jobjcan be started. To begin, let us assume thatpi≥pjanddi≥dj. When the processing times and due dates of jobsiandjare ordered similarly, as in this assumption, we say that the processing time and due date parameters areagreeable. (Formally, two sets of parameters,ujandvj, are agreeable ifui<ujimpliesvi≤vj.) For the time being, we shall refer to the case of agreeable processing times and due dates as Case 1.
Case 1.1 p(B) +pi≤di
Tij= max p B +pi+pj–dj, 0
Tji= max p B +pj–dj, 0 + max p B +pj+pi–di, 0
Notice thatTijis at least as large as the first maximum inTji(becausepi≥0) and at least as large as the second (becausedi≥dj). Therefore, if one or both of the maxima inTjiare zero, we will haveTij≥Tji. Now suppose that neither term in Tjiis zero. Then
Tij–Tji= p B +pi+pj–dj – p B +pj–dj – p B +pj+pi–di
Tij–Tji=–p B –pj+di≥–p B –pi+di≥0
Therefore, Case 1.1 yieldsTij≥Tji, so it is preferable to have jobjprecede jobi.
Case 1.2 di<p B +pi
Tij=p B +pi–di+p B +pi+pj–dj
Tji= max p B +pj–dj, 0 +p B +pj+pi–di Tij–Tji=p B +pi–dj–max p B +pj–dj, 0
2.4 Problems with Due Dates: Elementary Results 27
If the maximum in the last term is zero, then the condition specifying Case 1.2 implies thatTij≥Tji; and if the maximum in the last term is positive,
Tij–Tji=p B +pi–dj– p B +pj–dj =pi–pj≥0
Therefore, Case 1.2 yieldsTij≥Tji, so it is preferable to have jobjprecede jobi.
These two cases reveal that when the processing times and the due dates are agreeable, the shorter job (or, equivalently, the job with the earlier due date) should come first. We state this partial result more formally as follows.
∎Theorem 2.8 If processing times and due dates are agreeable for all pairs of jobs, then total tardiness (T) is minimized by SPT sequencing with ties broken by EDD (or, equivalently, by EDD with ties broken by SPT).
Proof. The proof follows directly from adjacent pairwise interchange analysis, with the same interpretation as in the proof of Theorem 2.6. □
Furthermore, although Theorem 2.8 assumes thatallpairs of jobs have agree- able parameters, it can be shown that if anytwojobs are agreeable, then they should be sequenced by EDD/SPT even if some other jobs are sequenced between them. Now we turn to the more complicated situation, where the para- meters are not agreeable. Letpi≥pjanddi<dj.
Case 2.1 p B +pi≤di
Tij= max p B +pi+pj–dj, 0 Tji= max p B +pi+pj–di, 0 ≥Tij
Therefore, Case 2.1 yieldsTji≥Tij, so it is preferable to have jobi(the job with the earlier due date) precede jobj.
Case 2.2 di<p(B) +pi
Case 2.2.1 p B +pi+pj≤dj Tij=p B +pi–di
Tji=p B +pj+pi–di≥Tij
Therefore, Case 2.2.1 yieldsTji≥Tij, so it is preferable to have jobi(the job with the earlier due date) precede jobj.
Case 2.2.2 p B +pj≤dj<p B +pi+pj
Tij=p B +pi–di+p B +pi+pj–dj Tji=p B +pj+pi–di
Tij–Tji=p B +pi–dj
2 Single-machine Sequencing 28
Therefore, Case 2.2.2 yields the result that it is preferable to have jobi(the job with the earlier due date) precede jobjunlessp(B) +pi>dj, in which case jobj (the shorter job) may precede jobi.
Case 2.2.3 dj<p B +pj
Tij=p B +pi–di+p B +pi+pj–dj
Tji=p B +pj–dj+p B +pj+pi–di Tij–Tji=pi–pj≥0
Therefore, Case 2.2.3 yieldsTij≥Tji, so it is preferable to have jobj(the shorter job) precede jobi.
We can now combine the various subcases and conclude that, for Case 2, jobi may come first except when
p B +pi>dj
in which case jobjshould come first. In fact, we can combine Case 2 with Case 1 and restate the result as follows.
∎Theorem 2.9 If jobsiandjare the candidates to begin at timet, then the job with the earlier due date should come first, except if
t+ max {pi,pj} > max {di,dj},
in which case the shorter job should come first.
This decision rule is specific–it provides a choice between any pair of can- didate jobs–but the outcome may depend ont. That is, the rule could choose jobiin favor of jobjearly in the schedule but jobjin favor of jobilate in the schedule. More importantly, the rule does not tell us whether jobsiandj should come early in the schedule or late in the schedule. Thus, the decision rule is a weaker result than those in Theorems 2.3–2.7 because it does not sequence the jobs unambiguously.
We can look at this result from another perspective. Suppose we define the modified due date(MDD) of jobjat timetto be
dj= max dj,t+pj
In words, MDD is either the original due date or else the earliest time at which the job could possibly be completed, whichever is later. It is a dynamic quantity, because it may change as time passes. Therefore, if we give priority to the job with the earliest MDD, then the choice between jobsiandjmay be different early in the schedule than it is late in the schedule. The MDD priority rule is consistent with the prescriptions of Cases 1 and 2: if jobsiandjare the candi- dates to begin at timet, then the job with the earlier MDD should come first.
2.4 Problems with Due Dates: Elementary Results 29
Again, the MDD rule is weaker than such rules as SPT and SWPT. It tells us that if we examined an optimal sequence, we would find that each pair of jobs is sequenced consistently with MDD; however, starting at time zero and sequen- cing the jobs by MDD may not produce an optimal schedule. To put it another way, the MDD rule represents a necessary condition for optimality, but it is not a sufficient condition.
We conclude our treatment of theT-problem with some specialized results concerning optimal sequences:
•
If the EDD sequence produces no more than one tardy job, it yields the min- imum value ofT.•
If all jobs have the same due date, thenTis minimized by SPT sequencing.•
If it is impossible for any job to be on time in any sequence, thenTis mini- mized by SPT sequencing.•
If SPT sequencing yields no jobs on time, then it minimizesT.The weighted version of the total tardiness problem is even more difficult to solve than theT-problem, which itself is NP-hard, and we postpone its discus- sion until we examine more general methods of solution in Chapter 3.