Single Machine Models (Deterministic)
Algorithm 3.5.1 FPTAS for Minimizing Total Tardiness) Step 1
3.7 Discussion
that are close to the boundaries separating NP-hard problems from polynomial time problems, e.g., 1|rj |
Cj.
Most of the problems described in this chapter can be formulated as Mixed Integer Programs (MIPs). Mixed Integer Programming formulations of several single machine scheduling problems are presented in Appendix A. This appendix gives also an overview of the techniques that can be applied to MIPs.
This chapter does not exhibit all the possible procedures and techniques that can be brought to bear on single machine scheduling problems. One important class of solution procedures is often referred to as constraint programming.
Appendix C gives a detailed description of this class of procedures and Chapter 15 provides an example of a constraint programming procedure that can be applied to 1|rj|
wjUj.
Many heuristic procedures have been developed that can be applied to single machine scheduling problems. These procedures include the so-called composite dispatching rules as well as local search techniques. Chapter 14 provides an in- depth overview of these techniques and their applications to single machine problems.
The next chapter considers more general and more complicated single ma- chine problems. It focuses on problems with non-regular objective functions and on problems with multiple objective functions.
Exercises (Computational)
3.1. Consider 1||
wjCj with the following weights and processing times.
jobs 1 2 3 4 5 6 7
wj 0 18 12 8 8 17 16
pj 3 6 6 5 4 8 9
(a) Findall optimal sequences.
(b) Determine the effect of a change in p2 from6 to 7 on the optimal sequence(s).
(c) Determine the effect of the change under (b) on the value of the objec- tive.
3.2. Consider 1 | chains |
wjCj with the same set of jobs as in Exercise 3.1.(a). The jobs are now subject to precedence constraints which take the form of chains:
1 → 2 3 → 4 → 5
6 → 7 Find all optimal sequences.
Exercises 63 3.3. Consider 1||
wj(1−e−rCj) with the same set of jobs as in Exercise 3.1.
(a) Assume the discount rate r is 0.05. Find the optimal sequence. Is it unique?
(b) Assume the discount rateris 0.5. Does the optimal sequence change?
3.4. Find all optimal sequences for the instance of 1||hmaxwith the following jobs.
jobs 1 2 3 4 5 6 7
pj 4 8 12 7 6 9 9
hj(Cj) 3C1 77 C32 1.5C4 70 +√
C5 1.6C6 1.4C7
3.5. Consider 1|prec|hmax with the same set of jobs as in Exercise 3.4 and the following precedence constraints.
1 → 7 → 6 5 → 7 5 → 4 Find the optimal sequence.
3.6. Solve by branch-and-bound the following instance of the 1 | rj | Lmax problem.
jobs 1 2 3 4 5 6 7 pj 6 18 12 10 10 17 16 rj 0 0 0 14 25 25 50 dj 8 42 44 24 90 85 68
3.7. Consider the same problem as in the previous exercise. However, now the jobs are subject to the following precedence constraints.
2 → 1 → 4 6 → 7 Find the optimal sequence.
3.8. Find the optimal sequence for the following instance of the 1 ||
Tj
problem.
jobs 1 2 3 4 5 6 7 8
pj 6 18 12 10 10 11 5 7 dj 8 42 44 24 26 26 70 75
Hint: Before applying the dynamic programming algorithm, consider first the elimination criterion in Lemma 3.4.1.
3.9. Consider a single machine and 6 jobs.
jobs 1 2 3 4 5 6
pj 1190 810 1565 719 1290 482 dj 1996 2000 2660 3360 3370 3375
Apply the FPTAS described in Section 3.5 to this instance with4= 0.02. Are all sequences that are optimal for the rescaled data set also optimal for the original data set?
3.10. Find the optimal sequence for the following instance of the 1||
wjTj
problem.
jobs 1 2 3 4 5 6 7 pj 6 18 12 10 10 17 16
wj 1 5 2 4 1 4 2
dj 8 42 44 24 90 85 68
Exercises (Theory)
3.11. Consider 1||
wj(1−e−rCj). Assume thatwj/pj=wk/pk for alljand k. Show that forrsufficiently close to zero the optimal sequence is WSPT.
3.12. Show that if all jobs have equal weights, i.e.,wj = 1 for allj, the WDSPT rule is equivalent to the Shortest Processing Time first (SPT) rule for anyr, 0< r <1.
3.13. Consider the problem1| prmp |
hj(Cj). Show that if the functions hj are nondecreasing there exists an optimal schedule that is nonpreemptive.
Does the result continue to hold for arbitrary functionshj? 3.14. Consider the problem1|rj|
Cj.
(a) Show through a counterexample that the nonpreemptive rule that se- lects, whenever a machine is freed, the shortest job among those available for processing is not always optimal. In part (b) and (c) this rule is referred to as SPT∗.
(b) Performa worst case analysis of the SPT∗ rule, i.e., determine the maximum possible value of the ratio
Cj(SP T∗)/
Cj(OP T).
Exercises 65 (c) Design a heuristic for 1|rj|Cj that performs better than SPT∗.
3.15. Consider the problem1 | rj, prmp |
Cj. Show that the preemptive Shortest Remaining Processing Time first (SRPT) rule is optimal.
3.16. Consider the problem1 | prmp |
Cj with the additional restriction that jobj has tobe completed by a hard deadline ¯dj. Assuming that there are feasible schedules, give an algorithm that minimizes the total completion time and prove that it leads to optimality.
3.17. Consider the following preemptive version of the WSPT rule: if pj(t) denotes the remaining processing time of job j at time t, then a preemptive version of the WSPT rule puts at every point in time the job with the highest wj/pj(t) ratio on the machine. Show, through a counterexample, that this rule is not necessarily optimal for 1|rj, prmp|
wjCj. 3.18. Give an algorithmfor 1|intree|
wjCj and prove that it leads to an optimal schedule (recall that in an intree each job has at most one successor).
3.19. Give an algorithmfor 1|outtree|
wjCj and show that it leads to an optimal schedule (recall that in an outtree each job has at most one predecessor).
3.20. Consider the problem1 || Lmax. The Minimum Slackfirst (MS) rule selects at timet, when a machine is freed, among the remaining jobs the job with the minimum slack max(dj−pj−t,0). Show through a counterexample that this rule is not necessarily optimal.
3.21. Performan Adjacent Sequence Interchange for the weighted discounted flow time cost function. That is, state and prove a result similar to Lemma 3.1.2.
3.22. Consider the problem1|chains|
wj(1−e−rCj). Describe the algo- rithmthat solves this problemand prove that it results in an optimal sequence.
3.23. Consider the problem1|prec| max(h1(S1), . . . , hn(Sn)), whereSj de- notes the starting time of jobj. The cost functionhj,j= 1, . . . , nisdecreasing.
Unforced idleness of the machine isnot allowed. Describe a dynamic program- ming type algorithmfor this problemsimilar to the one in Section 3.2. Why does one have to use here forward dynamic programming instead of backward dynamic programming?
3.24. Consider the problem1|rj, prmp|Lmax. Determine the optimal sched- ule and prove its optimality.
3.25. Show that
(a) SPT is optimal for 1|brkdwn| Cj,
(b) Algorithm3.3.1 is optimal for 1|brkdwn| Uj, (c) WSPT is not necessarily optimal for 1|brkdwn|
wjCj.
3.26. Consider 1||
wjTj. Prove or disprove the following statement: If wj/pj> wk/pk,
pj < pk, and
dj < dk,
then there exists an optimal sequence in which jobj appears before jobk.
3.27. Complete the first step of the proof of Theorem 3.3.2.
Comments and References
The optimality of the WSPT rule for1||
wjCj appears in the seminal paper by W.E. Smith (1956). Lawler (1978), Monma and Sidney (1979, 1987), M¨ohring and Radermacher (1985a) and Sidney and Steiner (1986) all present very elegant results for1|prec|
wjCj; the classes of precedence constraints considered in these papers is fairly general and includes chains as well as intrees and outtrees.
The 1 | rj, prmp |
Cj problemhas been analyzed by Schrage (1968). The complexity proof for1|rj, prmp|
wjCjis due to Labetoulle, Lawler, Lenstra and Rinnooy Kan (1984). Rothkopf (1966a, 1966b) and Rothkopf and Smith (1984) analyze1||
wj(1−e−rCj).
The EDD rule is due to Jackson (1955) and the algorithmfor1|prec|hmaxis due to Lawler (1973). The complexity proof for1|rj|Lmaxappears in Lenstra, Rinnooy Kan and Brucker (1977). Many researchers have worked on branch- and-bound methods for1|rj|Lmax; see, for example, McMahon and Florian (1975), Carlier (1982) and Nowicki and Zdrzalka (1986). Potts (1980) analyzes a heuristic for1|rj|Lmax.
Algorithm 3.3.1, which minimizes the number of late jobs, is from Moore (1968). Kise, Ibaraki and Mine (1978) consider the 1 | rj |
Uj problem.
The NP-hardness of 1 ||
wjUj (i.e., the knapsack problem) is established in the classic paper by Karp (1972) on computational complexity. Sahni (1976) presents a pseudopolynomial time algorithm for this problem and Gens and Levner (1981) and Ibarra and Kim(1978) provide fast approximation algo- rithms. Potts and Van Wassenhove (1988) give a very efficient algorithm for a Linear Programming relaxation of the Knapsack problem. Van den Akker and Hoogeveen (2004) give an in-depth overview of scheduling problems with the wjUj objective. (A problemrelated to the knapsack problemis the so-called due date assignment problem. This problem has received a lot of attention as well; see Panwalkar, Smith and Seidmann (1982) and Cheng and Gupta (1989).) The dominance condition in Lemma 3.4.1 is due to Emmons (1969) and the pseudo-polynomial time Algorithm 3.4.4 is from Lawler (1977). The NP- hardness of1 ||
Tj is shown by Du and Leung (1990). For additional work
Comments and References 67 on dynamic programming and other approaches for this problem, see Potts and van Wassenhove (1982, 1987).
An enormous amount of work has been done on Polynomial Time Approxima- tion Schemes (PTAS) and on Fully Polynomial Time Approximation Schemes (FPTAS). The algorithmdescribed in Section 3.5 is one of the very first schemes developed for scheduling problems. This section is based entirely on the paper by Lawler (1982). A significant amount of work has been done on approximation algorithms for1|rj|
Cj; see, for example, Chekuri, Motwani, Natarajan, and Stein (1997). There are many other interesting applications of PTAS to schedul- ing problems; see, for example, Hochbaum and Shmoys (1987), Sevastianov and Woeginger (1998), and Alon, Azar, Woeginger and Yadid (1998). Schuurman and Woeginger (1999) present in their paper ten open problems concerning PTAS; their paper also contains an extensive reference list of PTAS papers. For a general overview of approximation techniques (covering more than just PTAS and FPTAS), see Chen, Potts and Woeginger (1998).
The complexity of 1||
wjTj is established in the Lawler (1977) paper as well as in the Lenstra, Rinnooy Kan and Brucker (1977) paper. Branch-and- bound methods using bounding techniques based on relaxations to transporta- tion problems are discussed in Gelders and Kleindorfer (1974, 1975). Many other approaches have been suggested for1 ||
wjTj, see for example Fisher (1976, 1981), Potts and van Wassenhove (1985) and Rachamadugu (1987).