Flow Shops and Flexible Flow Shops (Deterministic)
6.4 Discussion
This chapter has an emphasis on the makespan objective. In most machine environments the makespan is usually the easiest objective. In the flow shop environment, the makespan is already hard when there are three or more ma- chines in series. Other objectives tend to be even more difficult.
In any case, some research has been done on flow shops with the total comple- tion time objective. Minimizing the total completion time in a two machine flow shop, i.e., F2 ||
Cj, is already strongly NP-Hard. Several integer program- ming formulations have been proposed for this problem and various branch- and-bound approaches have been developed. Still, only instances with 50 jobs can be solved in a reasonable time. Minimizing the total completion time in a proportionate flow shop is, of course, very easy and can be solved via the SPT rule. Even the minimization of the total weighted completion time in a proportionate flow shop can be solved in polynomial time.
Exercises 173 Flow shops with due date related objective functions have received very little attention in the literature. On the other hand, more complicated flow shops, e.g., robotic cells, have received a considerable amount of attention.
Exercises (Computational)
6.1. Consider F4 | prmu | Cmax with the following 5 jobs under the given sequencej1, . . . , j5.
jobs j1 j2 j3 j4 j5 p1,jk 5 3 6 4 9 p2,jk 4 8 2 9 13 p3,jk 7 8 7 6 5 p4,jk 8 4 2 9 1
Find the critical path and compute the makespan under the given sequence.
6.2. Write the integer programming formulation ofF4|prmu|Cmaxwith the set of jobs in Exercise 6.1.
6.3. Apply the Slope heuristic to the set of jobs in Exercise 6.1. Is (Are) the sequence(s) generated actually optimal?
6.4. Consider F4 | block | Cmax with 5 jobs and the same set of processing times as in Exercise 6.1. Assume there is no buffer in between any two successive machines. Apply the Profile Fitting heuristic to determine a sequence for this problem. Take jobj1as the first job. If there are ties consider all the possibilities.
Is (any one of) the sequence(s) generated optimal?
6.5. ConsiderF4|prmu|Cmaxwith the following jobs jobs 1 2 3 4 5 p1,j 18 16 21 16 22 p2,j 6 5 6 6 5 p3,j 5 4 5 5 4 p4,j 4 2 1 3 4
(a) Can this problem be reduced to a similar problem with a smaller num- ber of machines and the same optimal sequence?
(b) Determine whether Theorem 6.1.4 can be applied to the reduced prob- lem.
(c) Find the optimal sequence.
6.6. Apply Algorithm3.3.1 to find an optimal schedule for the proportionate flow shopF3|pij=pj|
Uj with the following jobs.
jobs 1 2 3 4 5 6
pj 5 3 4 4 9 3
dj 17 19 21 22 24 24
6.7. Find the optimal schedule for the instance of the proportionate flow shop F2|pij=pj|hmax with the following jobs.
jobs 1 2 3 4 5
pj 5 3 6 4 9
hj(Cj) 12√
C1 72 2C3 54 +.5C4 66 +√ C5
6.8. Apply a variant of Algorithm3.4.4 to find an optimal schedule for the instance of the proportionate flow shopF2|pij =pj|
Tj with the following 5 jobs.
jobs 1 2 3 4 5 pj 5 3 6 4 9 dj 4 11 2 9 13
6.9. ConsiderF2|block|Cmax with zero intermediate storage and 4 jobs.
jobs 1 2 3 4 p1,j 2 5 5 11 p2,j 10 6 6 4
(a) Apply Algorithm4.4.5 to find the optimal sequence.
(b) Find the optimal sequence when there is an unlimited intermediate storage.
6.10. Find the optimal schedule for a proportionate flexible flow shopF F2| pij =pj |
Cj with three machines at the first stage and one machine at the second stage. There are 5 jobs. Determine whether SPT is optimal.
jobs 1 2 3 4 5 pj 2 2 2 2 5
Exercises 175
Exercises (Theory)
6.11. Consider the problemF m||Cmax. Assume that the schedule does allow one job to pass another while they are waiting for processing on a machine.
(a) Show that there always exists an optimal schedule that does not require sequence changes between machines 1 and 2 and between machinesm−1 and m. (Hint: By contradiction. Suppose the optimal schedule requires a sequence change between machines 1 and 2. Modify the schedule in such a way that there is no sequence change and the makespan remains the same.) (b) Find an instance ofF4||Cmax where a sequence change between ma- chines 2 and 3 results in a smaller makespan than in the case where sequence changes are not allowed.
6.12. ConsiderF m|prmu|Cmax. Let
pi1=pi2=· · ·=pin=pi
fori= 2, . . . , m−1. Furthermore, let
p11≤p12≤ · · · ≤p1n and
pm1≥pm2≥ · · · ≥pmn.
Show that sequence 1,2, . . . , n, i.e., SPT(1)-LPT(m), is optimal.
6.13. Consider F m | prmu | Cmax where pij = ai+bj, i.e., the processing time of job j on machine i consists of a component that is job dependent and a component that is machine dependent. Find the optimal sequence when a1≤a2≤ · · · ≤amand prove your result.
6.14. ConsiderF m|prmu|Cmax. Letpij =aj+ibj withbj>−aj/m.
(a) Find the optimal sequence.
(b) Does the Slope heuristic lead to an optimal schedule?
6.15. ConsiderF2||Cmax.
(a) Show that the Slope heuristic for two machines reduces to sequencing the jobs in decreasing order ofp2j−p1j.
(b) Show that the Slope heuristic is not necessarily optimal for two ma- chines.
(c) Show that sequencing the jobs in decreasing order of p2j/p1j is not necessarily optimal either.
6.16. ConsiderF3||Cmax. Assume max
j∈{1,...,n}p2j≤ min
j∈{1,...,n}p1j
and
max
j∈{1,...,n}p2j≤ min
j∈{1,...,n}p3j.
Show that the optimal sequence is the same as the optimal sequence forF2||
Cmax with processing timespij wherep1j=p1j+p2j andp2j =p2j+p3j. 6.17. Show that in the proportionate flow shop problemF m|pij =pj |Cmaxa permutation sequence is optimal in the class of schedules that do allow sequence changes midstream.
6.18. Show that if in a sequence forF2||Cmax any two adjacent jobsjandk satisfy the condition
min(p1j, p2k)≤min(p1k, p2j)
then the sequence minimizes the makespan. (Note that this is a sufficiency condition and not a necessary condition for optimality.)
6.19. Show that for F m | prmu | Cmax the makespan under an arbitrary permutation sequence cannot be longer thanm times the makespan under the optimal sequence. Show how this worst case bound actually can be attained.
6.20. Consider a proportionate flow shop with two objectives, namely the total completion time and the maximum lateness, i.e.,F m|pij =pj|
Cj+Lmax. Develop a polynomial time algorithm for this problem. (Hint: Parametrize on the maximum lateness. Assume the maximum lateness to be z; then consider new due dates dj +z which basically are hard deadlines. Start out with the SPT rule and modify when necessary.)
6.21. Consider a proportionate flow shop withn jobs. Assume that there are no two jobs with equal processing times. Determine the number of different SPT-LPT schedules.
6.22. ConsiderF m |prmu, pij =pj |
wjCj. Show that if wj/pj > wk/pk
and pj < pk, then there exists an optimal sequence in which job j precedes jobk.
6.23. Consider the following hybrid between F m | prmu | Cmax and F m | block | Cmax. Between some machines there is no intermediate storage and between other machines there is an infinite intermediate storage. Suppose a job sequence is given. Give a description of the graph through which the length of the makespan can be computed.
Comments and References
The solution for theF2||Cmaxproblemis presented in the famous paper by S.M.
Johnson (1954). The integer programming formulation ofF m||Cmax is due to
Comments and References 177 Wagner (1959) and the NP-Hardness proof forF3||Cmaxis fromGarey, Johnson and Sethi (1976). A definition of SPT-LPT schedules appears in Pinedo (1982).
Theorem6.1.9 is fromEck and Pinedo (1988). For results regarding propor- tionate flow shops see Ow (1985), Pinedo (1985), and Shakhlevich, Hoogeveen and Pinedo (1998). For an overview of F m||Cmax models with special struc- tures that can be solved easily, see Monma and Rinnooy Kan (1983); their framework includes the results obtained earlier by Smith, Panwalkar and Dudek (1975, 1976) and Szwarc (1971, 1973, 1978). The slope heuristic for permutation flow shops is fromPalmer (1965). Many other heuristics have been developed forF m ||Cmax; see, for example, Campbell, Dudek and Smith (1970), Gupta (1972), Baker (1975), Dannenbring (1977), Widmer and Hertz (1989) and Tail- lard (1990). For complexity results with regard to various objective functions, see Gonzalez and Sahni (1978b) and Du and Leung (1993a, 1993b).
The flow shop with limited intermediate storageF m|block|Cmax is studied in detail by Levner (1969), Reddy and Ramamoorthy (1972) and Pinedo (1982).
The reversibility result in Lemma 6.2.1 is due to Muth (1979). The Profile Fit- ting heuristic is from McCormick, Pinedo, Shenker and Wolf (1989). Wismer (1972) establishes the link betweenF m|nwt|Cmax and the Travelling Sales- man Problem. Sahni and Cho (1979a), Papadimitriou and Kannelakis (1980) and R¨ock (1984) obtain complexity results for F m | nwt | Cmax. Goyal and Sriskandrajah (1988) present a review of complexity results and approximation algorithms forF m|nwt|γ. For an overview of models in the classesF m||γ, F m|block|γ and F m|nwt|γ, see Hall and Sriskandarajah (1996).
Theorem6.3.2 is fromEck and Pinedo (1988). For makespan results with regard to the flexible flow shops see Sriskandarajah and Sethi (1989). Yang, Kreipl and Pinedo (2000) present heuristics for the flexible flow shop with the total weighted tardiness as objective. For more applied issues concerning flexible flow shops, see Hodgson and McDonald (1981a, 1981b, 1981c).
For research concerning the two machine flow shop with the total completion time objective, see van de Velde (1990), Della Croce, Narayan and Tadei (1996), Shakhlevich, Hoogeveen and Pinedo (1998), Della Croce, Ghirardi and Tadei (2002), Akkan and Karabati (2004), and Hoogeveen, van Norden and van de Velde (2006).
Dawande, Geismar, Sethi and Sriskandarajah (2007) present an extensive overview of one of the more important application areas of flow shops, namely robotic cells.