Robust Scheduling: A General View

1. Robust Scheduling: A General View Heng-Soon GAN and Andrew WIRTH When scheduling information is moderately incomplete and will deviate during the execution phase, a proactive-reactive scheduling method, such as robust scheduling is preferred. In this seminar, I will define and discuss (analytically and empirically) five different robust scheduling performance measures, namely schedule effectiveness, schedule predictability, heuristic efficiency, heuristic robustness and schedule nervousness. If time permits, I will make a comparison between stochastic and robust scheduling techniques and empirically justify the use of deterministic, robust and online scheduling techniques via the entropy measure.

2. Outline • The scheduling environment • Terminologies • Schedule execution costs • Heuristic robustness (or stability) • Schedule robustness (effectiveness and predictability) • Schedule nervousness (frequent rescheduling) • Integer program formulation • A more practical robust scheduling approach • Some empirical results • Stochastic scheduling • Scheduling and uncertainty (the entropy concept) • Conclusions and future directions

3. Scheduling Environment Schedule Planning Phase Schedule Execution Phase Schedule Deployment Time Local disruption or information update from other dependent sources. Information sent to other dependent sources.

4. Terminologies • Initial schedule • the schedule generated in the planning phase (off-line) • referred to as initial off-line schedule • OR the schedule prior to a perturbation event • Perturbed schedule • the schedule produced after a decision is made and executed in reaction to a perturbation event • Perturbation event • may occur during the planning and execution phases • described by the event time and disruption magnitude • e.g. machine breakdowns, change in operation processing times, arrival and removal of new operations

5. Terminologies (cont’d) • Perturbation scenario • a set of perturbation events • In-process operations, completed operations and operations that have not started current time completed operations in-process operations not-started operations

6. Terminologies (cont’d) • Shift-rescheduling (Sh) • regarded as the simplest possible repair procedure a b c d Processing time of operation a is updated, replaced with a’. a’ b c d Shift operation b to the left. a’ b c d 0 current time

7. Terminologies (cont’d) • Heuristic-rescheduling (H) • repair/regeneration of schedule using algorithms or heuristics or local search methods. a b c d Processing time of operation a updated, replaced with a’. a’ b c d Use LPT to reschedule. a’ d c b 0 current time

8. Schedule Execution Costs • Schedule effectiveness • the degree of optimality of a perturbed schedule, e.g. makespan, flowtime, earliness, tardiness etc. • this is the main cost to be optimised if no disruption occurs • Schedule predictability • the closeness of the perturbed schedule performance relative to the initial off-line schedule performance • reduces costs of under-utilisation or overtime • Heuristic efficiency • the computational complexity of the schedule generation/repair method • timeliness of response to perturbation events • Degree of re-arrangement (heuristic robustness or stability) • the degree of alteration to the operations’ arrangement • reduces costs of replanning and rerouting • Schedule nervousness • the frequency of H-rescheduling • reduces the number of plan revisions in other parts of the supply chain

9. Schedule Execution Costs (cont’d) • If no rescheduling is allowed (only perform shift), we want to minimise • If rescheduling is allowed, we want to minimise

10. Heuristic Robustness • A heuristic is said to berobust if the sequences of the operations do not change drastically when this heuristic is used for rescheduling after a disruption. • If local search method is used to generate/repair schedules, this measure can be embedded in the objective function. • Possible measures • Sum of the absolute changes in start-time and completion times of operations • Minimal Perturbation (El Sakkout et al.-2000) • Neighbourhood-based Robustness (Jensen-1999,2000,2001,2003; Jensen and Hansen-1999) • Predictable Scheduling (O’Donovan et al.-1999) • Rescheduling with effectiveness and stability as criteria (Wu et al.-1993) • Sum of the absolute changes in the precedence of operation • Neighbourhood-based Robustness (Jensen-1999,2000,2001,2003; Jensen and Hansen-1999) • Rescheduling under random disruptions (Abumaizar and Svestka-1997)

11. Heuristic Robustness (cont’d) • Sum of operations reassigned • Matchup Scheduling (Bean et al.-1991) • Sum of the absolute changes in sequence/positions of operations • Spearman’s footrule as measure of disarray (Diaconis and Graham-1977) • Artificial Immune System (Hart et al.-1997) • Most of the work provide definitions, but lack of analyses of the measure provided.

12. Heuristic Robustness (cont’d) Increasing start time Before perturbation 1 2 3 4 5 6 2 4 1 1 4 2 After perturbation and application of some heuristic 3 6 2 5 1 4 Increasing start time

13. Heuristic Robustness (cont’d) ……. ……. (Sh) (H) (H) (H) (Sh) (H) (H) (Sh) time ……. ……. ωj ωj+1 ωj+2 ωj+3 ωj+4 ωj+5 ωj+6 ωj+7

14. Heuristic Robustness (cont’d) • General bounds Diaconis and Graham (1977)

15. Heuristic Robustness (cont’d) • Longest Processing Time heuristic (P| |Cmax) • change in processing time of one operation (unbounded) or removal of one operation or addition of one operation • change in processing time of k operations (unbounded)

16. Heuristic Robustness (cont’d) • change in processing time of one operation (bounded)

17. Heuristic Robustness (cont’d) • mean analysis (all scenarios are equally likely) • a change in processing time • addition of one operation • removal of one operation • addition or removal or change in processing time of one operation • Diaconis and Graham (1977)

18. s1 s2 sk-1 sk sk+1 s2k-1 s2k s1 s2 sk sk+1 sk+2 s2k s2k+1 Heuristic Robustness (cont’d) • Abdekhodaee and Wirth equal length algorithm (P2|si + pi = a|Cmax) • the algorithm S2k-1 S2k-3 S3 S1 S2 S2k-2 S2k Even number of jobs S2k S2k-2 S2 S1 S3 S2k-1 S2k+1 Odd number of jobs

19. Heuristic Robustness (cont’d) • change in processing time of one operation (unbounded) or removal of one operation or addition of one operation • mean analysis

20. Heuristic Robustness (cont’d) • Johnson’s algorithm (F2| |Cmax) • Nt refers to jobs (equivalent to nt = 2Ntoperations)

21. Heuristic Robustness (cont’d) • Multifit heuristic (P| |Cmax) RLPT,A = 18 n = 10, m = 4 RMF7,A = 42

22. Heuristic Robustness (cont’d)

23. Heuristic Robustness (cont’d)

24. Schedule Robustness • In a weaker sense, an initial off-line schedule is said to be robust if • the perturbed schedule is effective • low cost • the absolute deviation of the perturbed schedule performance relative to that of the initial off-line schedule is small • predictability Z 0 time DZ 0 time

25. Schedule Robustness (cont’d) time ……. ……. ωj ωj+1 ωj+2 ωj+3 ωj+4 ωj+5 ωj+6 ωj+7

26. Schedule Robustness (cont’d) • Suppose that and are quantities to be minimised, the schedule produced by heuristic A is more robust than that of heuristic B (in a weaker sense) if,

27. Schedule Nervousness ……. ……. (Sh) (H) (H) (H) (Sh) (H) (H) (Sh) time ……. ……. ωj ωj+1 ωj+2 ωj+3 ωj+4 ωj+5 ωj+6 ωj+7

28. Integer Program Formulation • A robust initial off-line schedule (in a stronger sense) is a schedule which • minimises the total schedule execution cost and do not require any H-rescheduling when disruption occurs; • costs consist of effectiveness, predictability and stability (shift robustness) • More formally, a robust initial off-line schedule is a schedule S* which minimises and only Sh is performed when disruption occurs.

29. Integer Program Formulation (cont’d) • Previous integer program formulation attempts • Daniels and Kouvelis (1995) • single machine problem (SPT is optimal for the deterministic case) • minimising maximum absolute deviation of perturbed schedule total flowtime from the optimal schedule • suggested solution procedures (for processing time intervals): B&B algorithm and 2 heuristics (endpoint sum and endpoint product + pairwise interchange) • Book: Kouvelis and Yu (1997) • described robust formulation for various problems such as scheduling (single machine and flowshop), facility layout etc. • presented 3 variations of objective function formulations: • minimise the maximum perturbed schedule performance over all perturbation scenarios

30. Integer Program Formulation (cont’d) • minimise the maximum absolute deviation of perturbed schedule performance from the optimal schedule over all perturbation scenarios • minimise the maximum relative deviation of perturbed schedule performance w.r.t the optimal schedule over all perturbation scenarios • Kouvelis, Daniels and Vairaktarakis (2000) • two-machine flowshop problem (Johnson’s algorithm provide optimal schedule for the deterministic case) • absolute deviation robust schedule (makespan) • suggested solution procedures: B&B algorithm and a heuristic approach • Kuo and Lin (2002) • single machine problem, an extension of Daniels and Kouvelis (1995) • relative deviation robust schedule • solution procedure: B&B algorithm • Yang and Yu (2002) • single machine problem, also an extension of Daniels and Kouvelis (1995)

31. Integer Program Formulation (cont’d) • revealed that three types of robust formulation (absolute, absolute deviation and relative deviation) can be solved using a common solution procedure - generalisation of Daniels and Kouvelis(1995) and Kuo and Lin(2002) • suggested solution procedures (for discrete processing times): dynamic programming, surrogate relaxation procedure and greedy heuristic • Conclusions from the literatures and some open questions • the problem (single machine and two machine flowshop) is NP-hard. • most solution procedures use extreme processing time information (lower and upper bounds) and this has been proven to be sufficient. • is this sufficient for more complex scheduling problems?

32. Integer Program Formulation (cont’d) • objective function assumes the existence of optimal solution to the problem • challenge: the optimal solution to most (more complex) scheduling problems is unknown, even for identical parallel machines. • can lower bounds be used? • only effectiveness is considered • need to include other measures such as predictability and stability • perturbation scenarios are not time dependent • if only effectiveness and predictability is considered, perturbation scenarios need not be time dependent • but if stability is to be included, perturbation scenarios has to be time dependent.

33. Practical Robust Scheduling • Since finding a robust schedule is NP-hard (even for the simplest scheduling problem), we propose the following scheduling procedure: • create a initial off-line schedule using heuristic or local search (in consideration of an objective) • create a rescheduling policy, i.e. decide to use either H or Sh when disruptions occur • decide the robust scheduling scheme (which initial off-line schedule and rescheduling policy) to be used, i.e. the scheme which minimises the average or maximum cost

34. Practical Robust Scheduling (cont’d) • cost to be minimised • in real time, to decide whether to shift or to regenerate the schedule • map the current state of disruptions (magnitude, time etc.) to the database of the robust scheduling scheme chosen OR • use the best “heuristic + 0-look-ahead procedure” and apply it myopically at each disruption OR • game-theoretic control approach (Leon, Wu and Storer-1994)

35. Practical Robust Scheduling (cont’d) • Other practical scheduling approaches • contingency schedules • Artificial Immune System (Hart et al.-1997) • Proactive rescheduling analysis (Guo and Nonaka-1999) • least commitment scheduling • Preprocess-First-Schedule-Later (Byeon et al.-1998; Kutanoglu and Wu-1998; Wu et al.-1999) • Generating initial off-line schedule • choice of deterministic-(near-)optimal OR robust-(near-) optimal initial off-line schedule • attempts (mostly for machine breakdowns): • ARS, ADRS and RRS (Daniels and Kouvelis etc.) – as discussed earlier • capacity hedging method (Yellig and Mackulak-1997) • schedule sensitivity analysis (Morikawa et al.-1993)

36. Practical Robust Scheduling (cont’d) • neighbourhood-based robustness (Jensen-1999,2000,2001,2003; Jensen and Hansen-1999) • slack-based techniques (Chiang and Fox-1990; Gao et al.-1995; Davenport et al.-2001) • fuzzy evaluation of expected delay (Dorn et al.-1995; Chen and Muraki-1997) – for uncertain processing times • Assuming the perturbation scenario is known, rescheduling policies can be constructed via methods such as • IP formulation (no attempts yet) • the problem is likely to be intractable • B&B algorithm: computationally exhaustive • Genetic Algorithm • easy coding of chromosomes: …1010…  …H,Sh,H,Sh • The a-look-ahead heuristic • 2(a + 1) possibilities • for a = 0, i.e. 0-look-ahead heuristic can be used in real-time scheduling

37. Practical Robust Scheduling (cont’d) . . . . . . . . . . . .

38. Practical Robust Scheduling (cont’d) • Off-line procedures to create a robust scheduling scheme: • When heuristic (e.g. LPT, MFk etc.) is used, • use heuristic to generate initial off-line schedule and repair schedule when disruptions occur • the initial off-line schedule created is myopic. • the rescheduling policy can be constructed via methods described earlier. • this procedure is myopic if 0-look-ahead is used (but suitable in real-time). • When local search method (e.g. GA, SA etc.) is used, • if stability (heuristic robustness) is important, embed this measure into the objective function. • initial off-line schedule created can be “long-sighted” • application of LSM similar to that of a heuristic • It is possible to combine both heuristic and local search methods into the robust scheduling scheme.

39. Some Empirical Results • Heuristic A is better than heuristic B if C(A,B)  1, where

40. Some Empirical Results (cont’d) • Use a-look-ahead heuristic, where a = 0. • Perform Sh if

41. Some Empirical Results (cont’d) • Compare the use of LPT, MF7 and SPT on identical parallel machines • minimising makespan • subjected to changes in processing times and machine breakdowns. • 10 sets of n = 30 operations, where processing times are randomly generated from U(1,100). • m = 6 identical parallel machines. • 10 sets of perturbations with 20 events each, • change in processing time • probability of 0.5 that an operation will change its processing time • range: U(0.1pi, 2pi) • occurrence time ~ U(0, 200) • machine breakdown • Time between failure ~ neg-exp(0.005) • Downtime ~ neg-exp(0.08)

42. Some Empirical Results (cont’d) • Cost coefficients used:

43. Some Empirical Results (cont’d) • Results

44. Stochastic Scheduling • Stochastic dominance • almost surely larger • P(X1 X2) = 1 • larger in likelihood ratio sense • P(X1 = t)/P(X2 = t) is nondecreasing in t, t  0 and f1(t) and f2(t) are p.d.f.’s. • stochastically larger • P(X1 > t) P(X2 > t) for all t • larger in expectation (often used in stochastic scheduling) • E(X1)  E(X2) • Types of policies • static list policy • puts all operations in a list at time 0 and this list does not change during schedule execution (perform Sh whenever disruptions occur) • dynamic list policy • no fixed list; the decision maker allowed to make decisions during schedule execution (perform H whenever disruptions occur) • could be preemptive or non-preemptive

45. Stochastic Scheduling (cont’d) • Most results for stochastic scheduling depends on the following • optimality in expectation (the crudest form of stochastic optimality) • “simple” distribution • some nice results (extracted from Pinedo’s “Scheduling: Theory, Algorithms and Systems”-1995) • 1|pi ~ general|SwiCi (nonpreemptive static/dynamic list policies) • WSEPT is optimal in expectation (also optimal for general machine breakdowns on single machines) • 1|pi ~ general|Lmax (dynamic & nonpreemptive static list policies) • EDD is optimal almost surely • P2| pi ~ exp(lj)|Cmax (nonpreemptive static list policies) • LEPT is optimal in expectation • P|preempt|Cmax (preemptive dynamic list policies) • LEPT is optimal in expectation • P|pi ~ general|SCi (preemptive dynamic list policies) • SEPT is optimal stochastically

46. Stochastic Scheduling (cont’d) • Stochastic vs Robust scheduling • robust: hedge against uncertainty in expectation and/or worst case & optimisation of other criteria such as efficiency, stability, predictability and nervousness • stochastic: hedge against uncertainty in expectation (usually) • some comments: • both stochastic and robust scheduling are addressing the same problem, i.e. uncertainty in scheduling • which is preferable? depends on what is to be optimised and the availability of optimal solutions • the stochastic analysis only provide optimal solutions for simple shop-floor configurations and restricted uncertainty distributions • but at least this formulation gives more optimistic results than the robust formulation • both stochastic and robust formulations are difficult to solve • need a more practical approach, e.g. the more practical robust scheduling, contingency schedules etc. • need more flexibility in deciding whether to reschedule or not (from the stochastic scheduling point of view, this is in fact switching between static and dynamic list policies)

47. Scheduling and Uncertainty • Certain event • Event happens with no variability (I AM ABSOLUTELY SURE……) • Uncertain event • Some information on the event available, but with variability (MAYBE…..) • Unexpected event • Information on the event revealed at the time it occurs (I DON’T KNOW…..)

48. Scheduling and Uncertainty (cont’d) Low Uncertainty Medium Uncertainty High Uncertainty Unexpected Deterministic Scheduling Robust Scheduling On-line Scheduling Reactive Proactive

49. Scheduling and Uncertainty (cont’d) • Heuristic applied in a deterministic sense (static list policy) • all operations committed to the initial off-line schedule • perform shift when disruption occurs • Heuristic applied in a robust sense • all (or partial) operations are committed to the initial off-line schedule • perform shift or H-rescheduling when disruption occurs • perform shift when (0-look-ahead heuristic) • Heuristic applied in a online sense (dynamic list policy) • operations not committed to the initial off-line schedule • operation assigned over time according to a specified rule

50. Scheduling and Uncertainty (cont’d) • We measure the uncertainty associated with scheduling information using the entropy concept. • schedule stability radius: Sotskov et al. (1997,1998), Lai et al. (1997) • empirical testing on static and dynamic applications of optimal and heuristic solution to job shop problem: Lawrence and Sewell (1997) • Recalling the entropy concept: • finite scheme with mutually exclusive events, A1, A2, …, An with probabilities p1, p2, …,pn respectively, where Sipi = 1 • the amount of uncertainty associated with the finite scheme is given by and if pk = 0, pk log pk = 0