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Scheduling under Uncertainty: Solution Approaches Frank Werner Faculty of Mathematics.

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1 Scheduling under Uncertainty: Solution Approaches Frank Werner Faculty of Mathematics

2 St. Etienne / France | November 23, 2012 Outline of the talk 1.Introduction 2.Stochastic approach 3.Fuzzy approach 4.Robust approach 5.Stability approach 6.Selection of a suitable approach 2

3 St. Etienne / France | November 23, Introduction Notations 3

4 St. Etienne / France | November 23, 2012 Deterministic models: all data are deterministically given in advance Stochastic models: data include random variables In real-life scheduling: many types of uncertainty (e.g. processing times not exactly known, machine breakdowns, additionally ariving jobs with high priorities, rounding errors, etc.) Uncertain (interval) processing times: 4

5 St. Etienne / France | November 23, Relationship between stochastic and uncertain problems: Distribution function Density function

6 St. Etienne / France | November 23, 2012 Approaches for problems with inaccurate data: Stochastic approach: use of random variables with known probability distributions Fuzzy approach: fuzzy numbers as data Robust approach: determination of a schedule hedging against the worst-case scenario Stability approach: combination of a stability analysis, a multi-stage decision framework and the concept of a minimal dominant set of semi-active schedules → There is no unique method for all types of uncertainties. 6

7 St. Etienne / France | November 23, 2012 Two-phase decision-making procedure 1)Off-line (proactive) phase construction of a set of potentially optimal solutions before the realization of the activities (static scheduling environment, schedule planning phase) 2)On-line (reactive) phase selection of a solution from when more information is available and/or a part of the schedule has already been realized → use of fast algorithms (dynamic scheduling environment, schedule execution phase) 7

8 St. Etienne / France | November 23, 2012 General literature (surveys) Pinedo: Scheduling, Theory, Algorithms and Systems, Prentice Hall, 1995, 2002, 2008, 2012 Slowinski and Hapke: Scheduling under Fuzziness, Physica, 1999 Kasperski: Discrete Optimization with Interval Data, Springer, 2008 Sotskov, Sotskova, Lai and Werner: Scheduling under Uncertainty; Theory and Algorithms, Belarusian Science, 2010 For the RCPSP under uncertainty, see e.g. Herroelen and Leus, Int. J. Prod. Res Herroelen and Leus, EJOR, 2005 Demeulemeester and Herroelen, Special Issue, J. Scheduling,

9 St. Etienne / France | November 23, Stochastic approach Distribution of random variables (e.g. processing times, release dates, due dates) known in advance Often: minimization of expectation values (of makespan, total completion time, etc.) Classes of policies (see Pinedo 1995) Non-preemptive static list policy (NSL) Preemptive static list policy (PSL) Non-preemptive dynamic policy (ND) Preemptive dynamic policy (PD) 9

10 St. Etienne / France | November 23, 2012 Some results for single-stage problems (see Pinedo 1995) Single machine problems (a)Problem WSEPT rule: order the jobs according to non-increasing ratios Theorem 1: The WSEPT rule determines an optimal solution in the class of NSL as well as ND policies. (b)Problem Theorem 2: The EDD rule determines an optimal solution in the class of NSL, ND and PD policies. 10

11 St. Etienne / France | November 23, 2012 (c)Problem Theorem 3: The WSEPT rule determines an optimal solution in the class of NSL, ND and PD policies. Remark: The same result holds for geometrically distributed Parallel machine problems (a)Problem Theorem 4: The LEPT rule determines an optimal solution in the class of NSL policies. 11

12 St. Etienne / France | November 23, 2012 (b)Problem Theorem 5: The non-preemptive LEPT policy determines an optimal solution in the class of PD policies. (c)Problem Theorem 6: The non-preemptive SEPT policy determines an optimal solution in the class of PD policies. 12

13 St. Etienne / France | November 23, 2012 Selected references (1) Pinedo and Weiss, Nav. Res. Log. Quart., 1979 Glazebrook, J. Appl. Prob., 1979 Weiss and Pinedo, J. Appl. Prob., 1980 Weber, J. Appl. Prob., 1982 Pinedo, Oper. Res., 1982; 1983 Pinedo, EJOR, 1984 Pinedo and Weiss, Oper. Res., 1984 Möhring, Radermacher and Weiss, ZOR, 1984; 1985 Pinedo, Management Sci., 1985 Wie and Pinedo, Math. Oper. Res., 1986 Weber, Varaiya and Walrand, J. Appl. Prob., 1986 Righter, System and Control Letters, 1988 Weiss, Ann. Oper. Res.,

14 St. Etienne / France | November 23, 2012 Selected references (2) Weiss, Math. Oper. Res., 1992 Righter, Stochastic Orders, 1994 Cai and Tu, Nav. Res. Log., 1996 Cai and Zhou, Oper. Res., 1999 Möhring, Schulz and Uetz, J. ACM, 1999 Nino-Mora, Encyclop. Optimiz., 2001 Cai, Sun and Zhou, Prob. Eng. Inform. Sci., 2003 Ebben, Hans and Olde Weghuis, OR Spectrum, 2005 Ivanescu, Fransoo and Bertrand, OR Spectrum, 2005 Cai, Wu and Zhou, IEEE Transactions Autom. Sci. Eng., 2007 Cai, Wu and Zhou, J. Scheduling, 2007; 2011 Cai, Wu and Zhou, Oper. Res., 2009 Tam, Ehrgott, Ryan and Zakeri, OR Spectrum,

15 St. Etienne / France | November 23, Fuzzy approach Fuzzy scheduling techniques either fuzzify existing scheduling rules or solve mathematical programming problems Often: fuzzy processing times, fuzzy due dates Examples triangular fuzzy processing times trapezoidal fuzzy processing times 15

16 St. Etienne / France | November 23, 2012 Often: possibilistic approach (Dubois and Prade 1988) Chanas and Kasperski (2001) Problem Objective: Assumption: → adaption of Lawler‘s algorithm for problem 16

17 St. Etienne / France | November 23, 2012 Special cases: a) b) c) d) Alternative goal approach - fuzzy goal, Objective: Chanas and Kasperski (2003) Problem Objective: → adaption of Lawler‘s algorithm for problem 17

18 St. Etienne / France | November 23, 2012 Selected references (1) Dumitru and Luban, Fuzzy Sets and Systems, 1982 Tada, Ishii and Nishida, APORS, 1990 Ishii, Tada and Masuda, Fuzzy Sets and Systems, 1992 Grabot and Geneste, Int. J. Prod. Res., 1994 Han, Ishii and Fuji, EJOR, 1994 Ishii and Tada, EJOR, 1995 Stanfield, King and Joines, EJOR, 1996 Kuroda and Wang, Int. J. Prod. Econ., 1996 Özelkan and Duckstein, EJOR, 1999 Sakawa and Kubota, EJOR,

19 St. Etienne / France | November 23, 2012 Selected references (2) Chanas and Kasperski, Eng. Appl. Artif. Intell., 2001 Chanas and Kasperski, EJOR, 2003 Chanas and Kasperski, Fuzzy Sets and Systems, 2004 Itoh and Ishii, Fuzzy Optim. and Dec. Mak., 2005 Kasperski, Fuzzy Sets and Systems, 2005 Inuiguchi, LNCS, 2007 Petrovic, Fayad, Petrovic, Burke and Kendall, Ann. Oper. Res.,

20 St. Etienne / France | November 23, Robust approach Objective: Find a solution, which minimizes the „worst-case“ performance over all scenarios. Notations (single machine problems) maximal regret of Minmax regret problem (MRP): Find a sequence such that 20

21 St. Etienne / France | November 23, 2012 Some polynomially solvable MRP (Kasperski 2005) (Volgenant and Duin 2010) (Averbakh 2006) (Kasperski 2008) Some NP-hard MRP (Lebedev and Averbakh 2006) (for a 2-approximation algorithm, see Kasperski and Zielinski 2008) (Kasperski, Kurpisz and Zielinski 2012) 21

22 St. Etienne / France | November 23, 2012 Kasperski and Zielinski (2011) Consideration of MRP‘s using fuzzy intervals Deviation interval Known: deviation Application of possibility theory (Dubois and Prade 1988) possibly optimal if necessarily optimal if 22

23 St. Etienne / France | November 23, 2012 Fuzzy problem or equivalently where is a fuzzy interval and is the complement of with membership function The fuzzy problem can be efficiently solved if a polynomial algorithm for the corresponding MRP exists. 23

24 St. Etienne / France | November 23, 2012 Solution approaches a)Binary search method - repeated exact solution of the MRP - applications: : binary search subroutine in B&B algorithm 24

25 St. Etienne / France | November 23, 2012 b)Mixed integer programming formulation - use of a MIP solver - application: c)Parametric approach - solution of a parametric version of a MRP (often time-consuming) - application: 25

26 St. Etienne / France | November 23, 2012 Selected references (1) Daniels and Kouvelis, Management Sci., 1995 Kouvelis and Yu, Kluwer, 1997 Kouvelis, Daniels and Vairaktarakis, IEEE Transactions, 2000 Averbakh, OR Letters, 2001 Yang and Yu, J. Comb. Optimiz., 2002 Kasperski, OR Letters, 2005 Kasperski and Zielinski, Inf. Proc. Letters, 2006 Lebedev and Averbakh, DAM, 2006 Averbakh, EJOR, 2006 Montemanni, JMMA,

27 St. Etienne / France | November 23, 2012 Selected references (2) Kasperski and Zielinski, OR Letters, 2008 Sabuncuoglu and Goren, Int. J. Comp. Integr. Manufact., 2009 Aissi, Bazgan and Vanderpooten, EJOR, 2009 Volgenant and Duin, COR, 2010 Kasperski and Zielinski, FUZZ-IEEE, 2011 Kasperski, Kurpisz and Zielinski, EJOR,

28 28 5. Stability approach 5.1. Foundations 5.2. General shop problem 5.3. Two-machine flow and job shop problems 5.4. Problem

29 St. Etienne / France | November 23, Foundations Mixed Graph Example: **

30 St. Etienne / France | November 23, 2012 Example (continued) **

31 St. Etienne / France | November 23, 2012 Stability analysis of an optimal digraph Definition 1 The closed ball is called a stability ball of if for any remains optimal. The maximal value is called the stability radius of digraph Known: Characterization of the extreme values of Formulas for calculating Computational results for job shop problems with (see Sotskov, Sotskova and Werner, Omega, 1997) 31

32 St. Etienne / France | November 23, General shop problem Definition 2 is called a G-solution for problem if for any fixed contains an optimal digraph. If any is not a G-solution, is called a minimal G-solution denoted as Introduction of the relative stability radius: 32

33 St. Etienne / France | November 23, 2012 Definition 3 Let be such that for any The maximal value of of such a stability ball is called the relative stability radius Known: Dominance relations among paths and sets of paths Characterization of the extreme values of 33

34 St. Etienne / France | November 23, 2012 Characterization of a G-solution for problem Definition 4 (strongly) dominates in → dominance relation Theorem 7: is a G-solution. There exists a finite covering of polytope by closed convex sets with such that for any and any there exists a for which Corollary: 34

35 St. Etienne / France | November 23, 2012 Theorem 8: Let be a G-solution with Then: is a minimal G-solution. For any there exists a vector such that Algorithms for problem 35

36 St. Etienne / France | November 23, 2012 Several 3-phase schemes: B&B: implicit (or explicit) enumeration scheme for generating a G-solution SOL: reduction of by generating a sequence with the same and different MINSOL: generation of a minimal G-solution by a repeated application of algorithm SOL 36

37 St. Etienne / France | November 23, 2012 Some computational results: Exact sol.:, Heuristic sol.: 37 Degree of uncertainty Exact solutionHeuristic solution (4,4)1, 3, 5, , 6, 8, , 10, 15, Degree of uncertainty Exact solutionHeuristic solution (4,4)1, 3, 5, , 6, 8, , 10, 15,

38 St. Etienne / France | November 23, Two-machine problems with interval processing times a)Problem Johnson permutation: Partition of the job set 38

39 St. Etienne / France | November 23, 2012 Theorem 9: (1) for any either (either ) and (2) and if satisfies – 39

40 St. Etienne / France | November 23, 2012 Theorem 10: If then Percentage of instances with, where

41 St. Etienne / France | November 23, 2012 General case of problem Theorem 11: There exists an Theorem 12: 41

42 St. Etienne / France | November 23, 2012 Example: without transitive arcs: J1J1 J1J1 J4J4 J4J4 J3J3 J3J3 J2J2 J2J2 J5J5 J5J5 J6J6 J6J6

43 St. Etienne / France | November 23, 2012 Properties of in the case of see Matsveichuk, Sotskov and Werner, Optimization, 2011 Schedule execution phase: see Sotskov, Sotskova, Lai and Werner, Scheduling under uncertainty, 2010 (Section 3.5) Computational results for and for b)Problem → Reduction to two problems: see Sotskov, Sotskova, Lai and Werner, Scheduling under uncertainty, 2010 (Section 3.6) 43

44 St. Etienne / France | November 23, Problem Notations: 44

45 St. Etienne / France | November 23, 2012 Definition of the stability box: 45

46 St. Etienne / France | November 23, 2012 Definition 5 The maximal closed rectangular box is a stability box of permutation, if permu- tation being optimal for instance with a scenario remains optimal for the instance with a scenario for each If there does not exist a scenario such that permutation is optimal for instance, then Remark: The stability box is a subset of the stability region. However, the stability box is used since it can easily be computed. 46

47 St. Etienne / France | November 23, 2012 Theorem 13: For the problem, job dominates if and only if the following inequality holds: Lower (upper) bound on the range of preserving the optimality of : 47

48 St. Etienne / France | November 23, 2012 Theorem 14: If there is no job, in permutation such that inequality holds for at least one job, then the stability box is calculated as follows: otherwise 48

49 St. Etienne / France | November 23, 2012 Example: Data for calculating 49

50 St. Etienne / France | November 23, 2012 Stability box for Relative volume of a stability box 50 Maximal ranges of possible variations of the processing times, within the stability box are dashed.

51 St. Etienne / France | November 23, 2012 Sotskov, Egorova, Lai and Werner (2011) Derivation of properties of a stability box that allow to derive an algorithm MAX-STABOX for finding a permutation with the largest dimension and the largest volume of a stability box 51

52 St. Etienne / France | November 23, 2012 Computational results Randomly generated instances with 52

53 St. Etienne / France | November 23, 2012 Selected references Lai, Sotskov, Sotskova and Werner, Math. Comp. Model., vol. 26, 1997 Sotskov, Wagelmans and Werner, Ann. Oper. Res., vol. 38, 1998 Lai, Sotskov, Sotskova and Werner, Eur. J. Oper. Res., vol. 159, 2004 Sotskov, Egorova and Lai, Math. Comp. Model., vol. 50, 2009 Sotskov, Egorova and Werner, Aut. Rem. Control, vol. 71, 2010 Sotskov, Egorova, Lai and Werner, Proceedings SIMULTECH, 2011 Sotskov and Lai, Comp. Oper. Res., vol. 39, 2012 Sotskov, Lai and Werner, Manuscript,

54 St. Etienne / France | November 23, Selection of a suitable approach Problem Cardinality of Theorem 15: 54

55 St. Etienne / France | November 23, 2012 Theorem 16: Assume that there is no Then: Theorem 17: not uniquely determined Construct an equivalent instance with less jobs for which is uniquely determined Assumption: uniquely determined. - instance with the set of scenarios 55

56 St. Etienne / France | November 23, 2012 Uncertainty measures Dominance graph Recommendations: use a stability approach use a robust approach use a fuzzy or stochastic approach 56

57 St. Etienne / France | November 23, 2012 Example: Dominance conditions: apply a stochastic or a fuzzy approach

58 St. Etienne / France | November 23, 2012 Example (continued): (apply a robust approach) Remark: easier computable than / / / /9

59 St. Etienne / France | November 23, 2012 Announcement of a book Sequencing and Scheduling with Inaccurate Data Editors: Yuri N. Sotskov and Frank Werner To appear at: Nova Science Publishers Completion:Summer parts: Each part contains a survey and 2-4 further chapters. Part 1:Stochastic approach survey: Cai et al. Part 2:Fuzzy approach survey: Sakawa et al. Part 3: Robust approach survey: Kasperski and Zielinski Part 4: Stability approach survey: Sotskov and Werner Contact address: 59


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