Discussion

Parallel Machine Models (Deterministic)

Algorithm 5.2.3 Minimizing Makespan with Preemptions) Step 1

5.7 Discussion

142 5Parallel Machine Models (Deterministic) and

n j=1

Cj(RR) =

n/m

=1

m(2,−1)q.

Once again, the ratio is maximized when all theqare equal, implying that the worst case ratio is exactly 2−2m/(n+m).

Since in online scheduling a competitive ratio is usually not expressed as a function ofn(since the number of jobs is typically not known in advance), the competitive ratio has to hold for any value ofn. It follows that the competitive

ratio for RR is equal to 2.

Actually, there are several other variants of the online scheduling paradigm.

The variant considered in this section assumes that the decision-maker does not know the processing time of a job when it is released. The decision-maker only finds out what the processing time is when the job is completed. This form of online scheduling is at times referred to asnon-clairvoyant online scheduling.

In another variant of online scheduling, the processing time of a job becomes known to the decision-maker immediately upon the job’s release. This variant is often referred to as clairvoyant online scheduling. However, in clairvoyant online scheduling the decision-maker still does not know how many jobs are going to be released and when the releases will occur.

An entirely different class of online algorithms are the so-called randomized online algorithms. A randomized algorithm allows the decision-maker to make random choices (for example, instead of assigning a job to the machine with the smallest load, the decision-maker may assign a job to a machine at random). If randomization is allowed, then it is of interest to know the expected objective value, where the expectation is taken over the randomchoices of the algorithm.

A randomized algorithm isσ-competitive if for each instance this expectation is within a factor ofσof the optimal objective value.

An example of such a very hard problem isQm|sijk|

wjTj. This problem is extremely hard to solve to optimality. It is already hard to find an optimal solution for instances with, say, 5 machines and 30 jobs. However, this problem is of considerable interest to industry and many heuristics have been developed and experimented with. Part III of this book describes several heuristic methods that have been applied to this problem.

Online scheduling in a parallel machine environment has received a signif- icant amount of attention during the last couple of years. Online scheduling is important for several reasons. In practice, it is often the case that a very limited amount of information is available when a decision must be made, see Example 1.1.4. From a theoretical point of view, online scheduling is of interest because it establishes a bridge between deterministic and stochastic scheduling.

In stochastic scheduling decisions also have to be made with only a limited amount of information available. However, the stochastic scheduling paradigm is still quite different fromthe online paradigm. Nevertheless, the bounds ob- tained in online scheduling often give rise to bounds in stochastic scheduling.

Exercises (Computational)

5.1. ConsiderP6||C_max with 13 jobs.

jobs 1 2 3 4 5 6 7 8 9 10 11 12 13 pj 6 6 6 7 7 8 8 9 9 10 10 11 11

(a) Compute the makespan under LPT.

(b) Find the optimal schedule.

5.2. ConsiderP4|prec|C_max with 12 jobs.

jobs 1 2 3 4 5 6 7 8 9 10 11 12 p_j 10 10 10 12 11 10 12 12 10 10 10 10

The jobs are subject to the precedence constraints depicted in Figure 5.10.

(a) Apply the generalized version of the CP rule: every time a machine is freed select the job at the head of the string with the largest total amount of processing.

(b) Apply the generalized version of the LNS rule: every time a machine is freed select the job that precedes the largest total amount of processing.

(c) Is either one of these two schedules optimal?

5.3. ConsiderP3|brkdwn, Mj|C_max with 8 jobs.

144 5Parallel Machine Models (Deterministic)

1 2 3

4 6

5

7 10

8 11

9 12

Fig. 5.10Precedence constraints graph (Exercise 5.2)

jobs 1 2 3 4 5 6 7 8 pj 10 10 7 7 7 7 7 7

Machines 1 and 2 are available continuously. Machine 3 is not available during the interval [0,1]; after time 1 it is available throughout. TheMjsets are defined as follows:

M₁={1,3} M₂={2,3} M₃=M₄=M₅={1} M₆=M₇=M₈={2}

(a) Apply the LPT rule, i.e., give always priority to the longest job that can be processed on the machine freed.

(b) Apply the LFJ rule, i.e., give always priority to the least flexible job while disregarding processing times.

(c) Compute the ratioC_max(LP T)/C_max(LF J).

5.4. ConsiderP3|prmp|

Cj with the additional constraint that the com- pletion of jobj has to be less than or equal to a given fixed deadlinedj. Pre- emptions may occur only at integer times 1,2,3, . . .

1 2

7 3

4 5

8

10 6

9

11 12

13

Fig. 5.11Precedence constraints graph (P∞ |prec|Cmax) for Exercise 5.5

jobs 1 2 3 4 5 6 7 8 9 10 11 p_j 2 3 3 5 8 8 8 9 12 14 16 d_j ∞ ∞ ∞ ∞ ∞ ∞ 11 12 13 28 29 Find the optimal schedule and compute the total completion time.

5.5. ConsiderP∞ |prec|C_max

jobs 1 2 3 4 5 6 7 8 9 10 11 12 13 p_j 5 11 9 8 7 3 8 6 9 2 5 2 9

The precedence constraints are depicted in Figure 5.11. Determine the optimal makespan and which jobs are critical and which jobs are slack.

5.6. ConsiderP5||

h(Cj) with 11 jobs.

jobs 1 2 3 4 5 6 7 8 9 10 11 pj 5 5 5 6 6 7 7 8 8 9 9 The functionh(Cj) is defined as follows.

h(Cj) =

0 ifCj ≤15 Cj−15 if Cj >15

146 5Parallel Machine Models (Deterministic) (a) Compute the value of the objective under SPT.

(b) Compute the value of the objective under the optimal schedule.

5.7. Consider again P5 ||

h(Cj) with the 11 jobs of the previous exercise.

The functionh(Cj) is now defined as follows:

h(C_j) =

C_j ifC_j ≤15 15 ifC_j >15.

(a) Compute the value of the objective function under SPT.

(b) Compute the value of the objective under the optimal schedule.

5.8. ConsiderQ2|prmp|C_max with the jobs jobs 1 2 3 4 p_j 36 24 16 12 and machine speedsv₁= 2 and v₂= 1.

(a) Find the makespan under LRPT when preemptions can only be made at the time points 0, 4, 8, 12, and so on.

(b) Find the makespan under LRPT when preemptions can only be made at the time points 0, 2, 4, 6, 8, 10, 12, and so on.

(c) Find the makespan under LRPT when preemptions can be made atany time.

(d) Compare the makespans under (a), (b) and (c).

5.9. Consider the following example of P3 | prmp, brkdwn |

Cj with 6 jobs. Three jobs have a processing time of 1, while the remaining three have a processing time of 2. There are three machines, but two machines are not available from time 2 onwards. Determine the optimal schedule. Show that SRPT is not optimal.

5.10. Consider the following instance ofP2 | prmp | L_max with 4 jobs. Pre- emptions are allowed at integer points in time. Find an optimal schedule.

jobs 1 2 3 4 dj 5 6 9 10 pj 4 5 7 9

Exercises (Theory)

5.11. ConsiderP m||C_max.

(a) Give an example showing that LPT is not necessarily optimal when the number of jobs is less than or equal to twice the number of machines (n≤2m).

(b) Show that if an optimal schedule results in at most two jobs on any machine, then LPT is optimal.

5.12. ConsiderP m||C_max. Describe the processing times of the instance that attains the worst case bound in Theorem5.1.1 (as a function ofm). (Hint:see Exercise 5.1.)

5.13. Show that the CP rule is optimal forP m|outtree, p_j= 1|C_max. 5.14. Complete the proof of Theorem 5.1.5. That is, show that the CP rule applied toP m | intree, pj = 1 | C_max results in a makespan that is equal to l_max+c.

5.15. ConsiderP m|rj, prmp|C_max. Formulate the optimal policy and prove its optimality.

5.16. ConsiderP m|prmp, brkdwn|C_maxwith the number of machines avail- able a function of time, i.e., m(t). Show that for any function m(t) LRPT minimizes the makespan.

5.17. ConsiderP m|brkdwn|

Cj with the number of machines available a function of time, i.e.,m(t). Show that ifm(t) is increasing the nonpreemptive SPT rule is optimal.

5.18. ConsiderP m|prmp, brkdwn|

C_jwith the number of machines avail- able a function of time, i.e., m(t). Show that the preemptive SRPT rule is optimal ifm(t)≥m(s)−1 for alls < t.

5.19. ConsiderP m|prmp|

Cjwith the added restriction that all jobsmust be finished by some fixed deadline ¯d, where

d¯≥max pj

m , p₁, . . . , pn

.

Find the rule that minimizes the total completion time and prove its optimality.

5.20. Consider P m ||

wjCj. Show that in the worst case example of the WSPT rulew_j has to be approximately equal top_j, for eachj.

5.21. Give a characterization of the class of all schedules that are optimal for P m||

C_j. Determine the number of schedules in this class as a function ofn andm.

148 5Parallel Machine Models (Deterministic) 5.22. ConsiderP2 ||

Cj. Develop a heuristic for minimizing the makespan subject to total completion time optimality. (Hint: Say a job is of Rank j if j−1 jobs follow the job on its machine. With two machines in parallel there are two jobs in each rank. Consider the difference in the processing times of the two jobs in the same rank. Base the heuristic on these differences.)

5.23. ConsiderP m|Mj |γ. The setsMjare given. LetJidenote the set of jobs that are allowed to be processed on machinei. Show, through a counterexample, that the setsMj being nested does not necessarily imply that setsJiare nested.

Give sufficiency conditions on the set structures under which theMjsets as well as theJi sets are nested.

5.24. Show, through a counterexample, that the CP rule is not necessarily optimal forP m|intree, p_j = 1|

C_j.

5.25. Consider P m | r_j, prmp | L_max. Show through a counterexample that the preemptive EDD rule doesnot necessarily yield an optimal schedule.

5.26. ConsiderP m|intree, prmp|C_max with the processing time of each job at level k equal topk. Show that a preemptive version of the generalized CP rule minimizes the makespan.

5.27. Consider Q∞ | prec, prmp | C_max. There are an unlimited number of machines that operate at the same speed. There isone machine that is faster.

Give an algorithm that minimizes the makespan and prove its optimality.

5.28. Consider an online version ofP m|rj, prec|C_max. An online algorithm for this problemcan be described as follows. The jobs are again presented in a list; whenever a machine is freed, the job that ranks highest among the remaining jobs which are ready for processing is assigned to that machine (i.e., it must be a job that already has been released and of which all predecessors already have been completed). Show that the bound presented in Theorem 5.6.1 applies to this more general problem as well.

Comments and References

The worst case analysis of the LPT rule for _{P m} || Cmax is fromthe classic paper by Graham(1969). This paper gives one of the first examples of worst case analyses of heuristics (see also Graham(1966)). It also provides a worst case analysis of an arbitrary list schedule for_{P m}||Cmax. A more sophisticated heuristic for_{P m}||Cmax, with a tighter worst case bound, is the so-called MUL- TIFIT heuristic, see Coffman, Garey and Johnson (1978) and Friesen (1984a).

Lee and Massey (1988) analyze a heuristic that is based on LPT as well as on MULTIFIT. Hwang, Lee and Chang (2005) performa worst case analysis of the LPT rule for _{P m|brkdwn |Cmax}. For results on heuristics for the more gen- eral_Qm||Cmax, see Friesen and Langston (1983), Friesen (1984b), and Dobson

(1984). Davis and Jaffe (1981) present an algorithmfor_Rm||Cmax. The CPM and PERT procedures have been covered in many papers and textbooks, see for example, French (1982). The CP result in Theorem 5.1.5 is due to Hu (1961).

See Lenstra and Rinnooy Kan (1978) with regard to_{P m|pj} = 1, prec |Cmax, Du and Leung (1989) with regard to_P2|tree|Cmax and Du, Leung and Young (1991) with regard to_P2|chains|Cmax. Chen and Liu (1975) and Kunde (1976) analyze the worst case behavior of the CP rule for_{P m}|pj= 1, prec|Cmax. Lin and Li (2004) and Li (2006) do a complexity analysis of_{P m}|pj= 1, Mj|Cmax

and_Qm|pj=p, Mj |Cmax. Apparently, no worst case analysis has been done for the LFJ rule.

For_{P m} |prmp| Cmax, see McNaughton (1959). For _Qm |prmp|Cmax, see Horvath, Lamand Sethi (1977), Gonzalez and Sahni (1978a) and McCormick and Pinedo (1995).

Conway, Maxwell and Miller (1967) discuss the SPT rule for_{P m||Cj}; they also give a characterization of the class of optimal schedules. For a discussion of _Qm||Cj, see Horowitz and Sahni (1976). The worst case bound for the WSPT rule for_{P m||wjCj}is fromKawaguchi and Kyan (1986). Elmaghraby and Park (1974) and Sarin, Ahn and Bishop (1988) present branch-and-bound algorithms for this problem. Eck and Pinedo (1993) present a heuristic for minimizing the makespan and the total completion time simultaneously. The optimality of the CP rule for_{P m|pj} = 1, outtree|

Cj is due to Hu (1961).

For complexity results with regard to_{P m} |prec|

Cj, see Sethi (1977) and Du, Leung and Young (1990).

For an analysis of the_Qm|prmp|

Cj problem, see Lawler and Labetoulle (1978), Gonzalez and Sahni (1978a), McCormick and Pinedo (1995), Leung and Pinedo (2003) and Gonzalez, Leung and Pinedo (2006).

A significant amount of work has been done on_Qm|rj, pj=p, prmp|γ; see Garey, Johnson, Simons and Tarjan (1981), Federgruen and Groenevelt (1986), Lawler and Martel (1989), Martel (1982) and Simons (1983).

For results with regard to_Qm|prmp|Lmax, see Bruno and Gonzalez (1976) and Labetoulle, Lawler, Lenstra and Rinnooy Kan (1984). For other due date related results, see Sahni and Cho (1979b).

Chen and Powell (1999) and Van den Akker, Hoogeveen and Van de Velde (1999) applied branch-and-bound methods (including column generation) to Rm||

wjCj and_Rm||wjUj.

The worst case analysis of an arbitrary list schedule for _{P m} ||Cmax is re- garded as one of the basic results in online scheduling. Theorem5.6.1 is due to Graham(1966). The analysis of the Round Robin rule and the total completion time objective is due to Motwani, Phillips and Torng (1994). Research in online scheduling has focused on other parallel machine scheduling problems as well;

see, for example, Shmoys, Wein and Williamson (1995). For an overview of on- line scheduling on parallel machines with the makespan objective, see Fleischer and Wahl (2000). For comprehensive overviews of online scheduling, see Sgall (1998) and Pruhs, Sgall and Torng (2004).

Chapter 6

Flow Shops and Flexible Flow