Discussion

Advanced Single Machine Models (Deterministic)

Algorithm 4.6.6 Minimizing Makespan – Batch Size Finite) Step 1. (Initialization)

4.7 Discussion

Over the last decade problems with earliness and tardiness penalties have re- ceived a significant amount of attention. Even more general problems than those considered in this chapter have been studied. For example, some research has focused on problems with jobs that are subject to penalty functions such as the one presented in Figure 4.8.

Because of the importance of multiple objectives in practice a considerable amount of research has been done on problems with multiple objectives. Of course, these problems are harder than the problems with just a single objective.

So, most problems with two objectives are NP-hard. These types of problems may attract in the near future the attention of investigators who specialize in PTAS and FPTAS.

The makespan minimization problem when the jobs are subject to sequence dependent setup times turns out to be equivalent to the Travelling Salesman Problem. Many combinatorial problems inspired by real world settings are equivalent to Travelling Salesman Problems. Another scheduling problem that is discussed in Chapter 6 is also equivalent to the particular Travelling Salesman Problemdescribed in Section 4.4.

The models in the section focusing on job families are at times also referred to as batch scheduling models. Every time the machine has to be set up for a new family it is said that a batch of a particular family is about to start. This batch of jobs fromthat family are processedsequentially. This is in contrast to the setting in the last section where a batch of jobs is processed inparallel on the batch processing machine.

Exercises (Computational)

4.1. Consider the following instance with 6 jobs andd= 156.

jobs 1 2 3 4 5 6

pj 4 18 25 93 102 114

Apply Algorithm4.1.4 to find a sequence. Is the sequence generated by the heuristic optimal?

4.2. Consider the following instance with 7 jobs. For each jobw_j =w_j =wj. However,wj is not necessarily equal towk.

jobs 1 2 3 4 5 6 7 pj 4 7 5 9 12 2 6 wj 4 7 5 9 12 2 6

All seven jobs have the same due dated= 26. Find all optimal sequences.

4.3. Consider again the instance of the previous exercise with 7 jobs. Again, for each jobw_j =w_j =w_j. However,w_j is not necessarily equal to w_k. However, now the jobs have different due dates.

jobs 1 2 3 4 5 6 7

pj 4 7 5 9 12 2 6

wj 4 7 5 9 12 2 6

dj 6 12 24 28 35 37 42

Find the optimal job sequence.

4.4. Give a numerical example of an instance with at most five jobs for which Algorithm4.1.4 does not yield an optimal solution.

4.5. Consider the following instance of the 1||

w_jC_j⁽¹⁾, L⁽²⁾_max problem.

jobs 1 2 3 4 5

wj 4 6 2 4 20

pj 4 6 2 4 10

dj 14 18 18 22 0

108 4 Advanced Single Machine Models (Deterministic) Find all optimal schedules.

4.6. Consider the following instance of the 1||L⁽¹⁾_max,

w_jC_j⁽²⁾ problem.

jobs 1 2 3 4 5

w_j 4 6 2 4 20

pj 4 6 2 4 10

dj 14 18 18 22 0 Find all optimal schedules.

4.7. Apply Algorithm4.3.2 to the following instance with 5 jobs and generate the entire trade-off curve.

jobs 1 2 3 4 5 p_j 4 6 2 4 2 dj 2 4 6 10 10

4.8. Consider the instance of 1 || θ₁L_max+θ₂

Cj in Example 4.3.3. Find the ranges ofθ₁ andθ₂ (assumingθ₁+θ₂= 1) for which each Pareto-optimal schedule minimizesθ₁L_max+θ₂

Cj.

4.9. Consider an instance of the 1 | sjk | C_max problemwith the sequence dependent setup times being of the formsjk =| ak−bj |. The parametersaj

andb_k are in the table below. Find the optimal sequence.

cities 0 1 2 3 4 5 6 b_j 39 20 2 30 17 6 27 a_j 19 44 8 34 16 7 23

4.10. Consider the following instance of 1|fmls, sgh |

wjCj withF = 2.

The sequence dependent setup times between the two families ares₁₂=s₂₁= 2 ands₀₁ =s₀₂= 0. There are two jobs in family 1 and three jobs in family 2, i.e.,n₁= 2 andn₂= 3. The processing times are in the table below:

jobs (1,1) (2,1) (1,2) (2,2) (3,2)

p_jg 3 1 1 1 3

w_jg 27 2 30 1 1

Apply Algorithm4.5.2 to find the optimal schedule.

Exercises (Theory)

4.11. Show that in an optimal schedule for an instance of 1|dj =d| Ej+ Tj there is no unforced idleness in between any two consecutive jobs.

4.12. Prove Lemma 4.1.1.

4.13. Consider the single machine scheduling problem with objective wE_j+ wT_jand all jobs having the same due date, i.e.,d_j=d. Note that the weight of the earliness penaltyw is different fromthe weight of the tardiness penalty w, but the penalty structure is the same for each job. Consider an instance where the due datedis so far out that the machine will not start processing any job at time zero. Describe an algorithm that yields an optimal solution (i.e., a generalization of Algorithm4.1.3).

4.14. Consider the same problem as described in the previous exercise. How- ever, now the due date is not far out and the machine does have to start pro- cessing a job immediately at time zero. Describe a heuristic that would yield a good solution (i.e., a generalization of Algorithm4.1.4).

4.15. Consider an instance where each job is subject to earliness and tardiness penalties andw_j =w_j=wj for all j. However,wj is not necessarily equal to wk. The jobs have different due dates. Prove or disprove that EDD minimizes the sumof the earliness and tardiness penalties.

4.16. Describe the optimal schedule for 1 ||

wjC_j⁽¹⁾, L⁽²⁾_max and prove its optimality.

4.17. Describe the optimal schedule for 1 ||

w_jC_j⁽¹⁾,

U_j⁽²⁾ and prove its optimality.

4.18. Describe the algorithmfor 1 || L⁽¹⁾_max,

wjC_j⁽²⁾. That is, generalize Lemma 4.2.1 and Algorithm 4.2.2.

4.19. Show that the maximum number of Pareto-optimal solutions for 1 ||

θ₁

Cj+θ₂L_max isn(n−1)/2.

4.20. Describe the optimal schedule for 1||θ₁

Uj+θ₂L_maxunder the agree- ability conditions

d₁≤ · · · ≤dn, and

p₁≤ · · · ≤p_n. 4.21. Prove Lemma 4.5.4.

4.22. Prove Lemma 4.6.1.

4.23. Prove Lemma 4.6.7.

4.24. Prove Lemma 4.6.9.

110 4 Advanced Single Machine Models (Deterministic)

Comments and References

The survey paper by Baker and Scudder (1990) focuses on problems with ear- liness and tardiness penalties. The text by Baker (1995) has one chapter ded- icated to problems with earliness and tardiness penalties. There are various papers on timing algorithms when the optimal order of the jobs is a given. The optimal timing Algorithm 4.1.8 is based on the paper by Szwarc and Mukhopad- hyay (1995). An algorithmto find the optimal order of the jobs as well as their optimal start times and completion times, assuming_wj =wj= 1 for all _j, is presented by Kim and Yano (1994). For more results on models with earliness and tardiness penalties, see Sidney (1977), Hall and Posner (1991), Hall, Kubiak and Sethi (1991), and Wan and Yen (2002).

A fair amount of research has been done on single machine scheduling with multiple objectives. Some single machine problems with two objectives allow for polynomial time solutions; see, for example, Emmons (1975), Van Wassen- hove and Gelders (1980), Nelson, Sarin and Daniels (1986), Chen and Bulfin (1994), and Hoogeveen and Van de Velde (1995). Potts and Van Wassenhove (1983) as well as Posner (1985) consider the problem of minimizing the total weighted completion time with the jobs being subject to deadlines (this prob- lemis strongly NP-hard). Chen and Bulfin (1993) present a detailed overview of the state of the art in multi-objective single machine scheduling. The book by T’kindt and Billaut (2002, 2006) is entirely focused on multi-objective schedul- ing.

The material in Section 4.4 dealing with the Travelling Salesman Problem is entirely based on the famous paper by Gilmore and Gomory (1964). For more results on scheduling with sequence dependent setup times see Bianco, Ricciardelli, Rinaldi and Sassano (1988), Tang (1990) and Wittrock (1990).

Scheduling with the jobs belonging to a given (fixed) number of families has received a fair amount of attention in the literature. At times, these types of models have also been referred to as batch scheduling models (since the consecutive processing of a set of jobs from the same family may be regarded as a batch). Monma and Potts (1989) discuss the complexity of these scheduling problems. An excellent overview of the literature on this topic is presented in the paper by Potts and Kovalyov (2000). Brucker (2004) in his book also considers this class of models and refers to it as_s-batching (batching with jobs processed in series).

When the machine is capable of processing multiple jobs in parallel, the machine is often referred to as a batching machine. An important paper con- cerning batch processing and batching machines is the one by Brucker, Gladky, Hoogeveen, Kovalyov, Potts, Tautenhahn, and van de Velde (1998). Potts and Kovalyov (2000) provides for this class of models also an excellent survey.

Brucker (2004) considers this class of models as well and refers to them as p-batching (batching with jobs processed in parallel).

Parallel Machine Models