Scheduling under Uncertainty: Solution Approaches Frank Werner Faculty of Mathematics Outline of the talk 1. 2. 3. 4. 5. 6. Introduction Stochastic approach Fuzzy approach Robust approach Stability approach Selection of a suitable approach St. Etienne / France | November 23, 2012 2 1. Introduction Notations J J1 ,...,J n - setof jobs M M1 ,...,M m - setof machines Q Oij | Ji J , j 1,...,ni 1,...,q - setof operations pij - processingtimeof Oij wi , ri , di ,..., - furtherdatafor J i J St. Etienne / France | November 23, 2012 3 • 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: pijL pij pijU St. Etienne / France | November 23, 2012 for allOij Q 4 T p Rq | pijL pij pijU , Oij Q - setof scenarios problem | , pijL pij pijU | pijL pijU for all Oij Q deterministic problem | | Relationship between stochastic and uncertain problems: L Distribution function 0 if t p Density function Fij (t ) P( pij t ) 1 0 f ij (t ) F 'ij (t ) ? 0 St. Etienne / France | November 23, 2012 ij U ij if t p if t pijL if pijL pij pijU if t pijU 5 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. St. Etienne / France | November 23, 2012 6 Two-phase decision-making procedure 1) Off-line (proactive) phase construction of a set S * 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 S * 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) St. Etienne / France | November 23, 2012 7 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.. 2004 • Herroelen and Leus, EJOR, 2005 • Demeulemeester and Herroelen, Special Issue, J. Scheduling, 2007 St. Etienne / France | November 23, 2012 8 2. 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) St. Etienne / France | November 23, 2012 9 Some results for single-stage problems (see Pinedo 1995) Single machine problems (a) Problem 1 || E wiCi pi ~ arbitrarily distributed WSEPT rule: order the jobs according to non-increasing ratios wi E pi Theorem 1: The WSEPT rule determines an optimal solution in the class of NSL as well as ND policies. (b) Problem 1 || ELmax pi ~ arbitrarily distributed, di fixed Theorem 2: The EDD rule determines an optimal solution in the class of NSL, ND and PD policies. St. Etienne / France | November 23, 2012 10 (c) Problem1 | di d | E wiUi pi ~ exponentiallydistributed, d fixed 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 pi . Parallel machine problems pi ~ exponentiallydistributed (a) Problem P2 || ECmax Theorem 4: The LEPT rule determines an optimal solution in the class of NSL policies. St. Etienne / France | November 23, 2012 11 (b) Problem P | pmtn| ECmax Theorem 5: The non-preemptive LEPT policy determines an optimal solution in the class of PD policies. (c) Problem P | pmtn| ECi Theorem 6: The non-preemptive SEPT policy determines an optimal solution in the class of PD policies. St. Etienne / France | November 23, 2012 12 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., 1990 St. Etienne / France | November 23, 2012 13 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, 2011 St. Etienne / France | November 23, 2012 14 3. Fuzzy approach • Fuzzy scheduling techniques either fuzzify existing scheduling rules or solve mathematical programming problems ~ ~ • Often: fuzzy processing times pi , fuzzy due dates di • Examples triangular fuzzy processing times ~p " pi is aroundpiM " i 1 .0 trapezoidal fuzzy processing times ~p i ~ pi 1 .0 ~ pi 0 piL piM piU St. Etienne / France | November 23, 2012 0 pL i p p piU 15 Often: possibilistic approach (Dubois and Prade 1988) PosV x ~p ( x), x R PosV a, b sup ~p ( x) xa ,b Nec V a, b inf 1 ~p ( x) xa ,b Chanas and Kasperski (2001) ~ ~ Problem 1 | prec, pi , di | f max ~ fi Ci ( ) min! Objective: max i ~ Assumption: fi F - monotonicw.r.t. Ci → adaption of Lawler‘s algorithm for problem 1 | prec | f max St. Etienne / France | November 23, 2012 16 Special cases: a) b) c) d) ~ ~ minPosC ( ) d max! ~ maxE L ( ) min! ~ ~ max Pos Ci ( ) d i min! i ~ ~ max Nec Ci ( ) d i min! i i i i i i Alternative goal approach ~ ~ ~ ~ max! Pos max w L ( ) G G - fuzzy goal, Objective: i i i Chanas and Kasperski (2003) ~ ~ ~ E Ti ( ) min! Problem 1| pi , di | maxETi Objective: max i → adaption of Lawler‘s algorithm for problem 1 | prec | f max St. Etienne / France | November 23, 2012 17 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, 2000 St. Etienne / France | November 23, 2012 18 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., 2008 St. Etienne / France | November 23, 2012 19 4. Robust approach Objective: Find a solution, which minimizes the „worst-case“ performance over all scenarios. Notations (single machine problems) Fp ( ) - functionvalueof sequence J k1 ,...,J kn for p T Fp* - optimalfunctionvaluefor p T S - setof feasiblejobsequences maximal regret of S Z ( ) max Fp ( ) Fp* pT Minmax regret problem (MRP): Find a sequence * such that Z * minZ ( ) S St. Etienne / France | November 23, 2012 20 Some polynomially solvable MRP 1 | prec, piL pi piU , diL di diU | Lmax (Kasperski 2005) 1 | prec, piL pi piU , diL di diU , wiL wi wiU | maxwiTi (Volgenant and Duin 2010) (Averbakh 2006) Fm | n 2, p L p pU | C ij ij ij max 1| pi 1, wiL wi wiU | Ui (Kasperski 2008) Some NP-hard MRP 1| piL pi piU | Ci is NP - hard (Lebedev and Averbakh 2006) (for a 2-approximation algorithm, see Kasperski and Zielinski 2008) F 2 | pijL pij pijU | Cmax is stronglyNP - hard (Kasperski, Kurpisz and Zielinski 2012) St. Etienne / France | November 23, 2012 21 Kasperski and Zielinski (2011) Consideration of MRP‘s using fuzzy intervals Z ' ( ) min Fp ( ) Fp* Deviation interval I Z ' ( ), Z ( ) Known: deviation z ( ) I Application of possibility theory (Dubois and Prade 1988) possibly optimal if Z ' ( ) 0 necessarily optimal if Z ( ) 0 St. Etienne / France | November 23, 2012 22 Fuzzy problem ~ Nec z( ) G max! or equivalently ~C Pos z( ) G min! ~ ~C ~ where G is a fuzzy interval and G is the complement of G with membership function 1 G~ ( x). The fuzzy problem can be efficiently solved if a polynomial algorithm for the corresponding MRP exists. St. Etienne / France | November 23, 2012 23 Solution approaches a) Binary search method - repeated exact solution of the MRP - applications: ~ 1 | prec, ~ pi , di | Lmax : O(n4 log 1 ) algorithm ~ | maxw T : O(n3 log 1 ) algorithm 1 | prec, w i i i ~ ~ 1| prec, wi , di | maxwiTi : O(n4 log 1 ) algorithm ~ | wU : O(n mind , n dlog 1 ) algorithm 1 | pi 1, di d , w i i i F2 | ~ pij | Cmax: binary search subroutine in B&B algorithm St. Etienne / France | November 23, 2012 24 b) Mixed integer programming formulation - use of a MIP solver - application: 1| ~ p| C i i c) Parametric approach - solution of a parametric version of a MRP (often time-consuming) - application: ~ 1 | prec, di | Lmax St. Etienne / France | November 23, 2012 25 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, 2007 St. Etienne / France | November 23, 2012 26 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, 2012 St. Etienne / France | November 23, 2012 27 5. Stability approach 5.1. Foundations 5.2. General shop problem 5.3. Two-machine flow and job shop problems 5.4. Problem 1 | p pi p | wiCi L i U i 28 5.1. Foundations Mixed Graph G (V , A, E) Example: p 75 p 11 12 11 50 12 p13 40 13 00 ** p00 0 p** 0 21 22 p21 60 p22 55 23 p23 30 (G) Gs | Gs (V , A Es , ) G1, G2 ,...,G - set of digraphs Gs (G) semiactiveschedules c1 (s), c2 (s),...,cq (s) St. Etienne / France | November 23, 2012 29 Example (continued) c11 75 G1 c12 125 11 12 c13 165 13 ** 00 21 22 c21 60 c22 130 Cmax G1 165 23 c23 160 C G 325 i 1 G G1, G2 ,...,G5 St. Etienne / France | November 23, 2012 30 Stability analysis of an optimal digraph Definition 1 The closed ball O ( p) with R1 and p Rq is called a stability ball of Gs (G) if for any p' O ( p) R q , Gs ( p' ) remains optimal. The maximal value s ( p) max R1 | Gs optimalfor any p'O ( p) Rq is called the stability radius of digraph Gs . Known: • Characterization of the extreme values of s ( p) • Formulas for calculating s ( p) for Cmax , Ci • Computational results for job shop problems with n 10 and m 8 (see Sotskov, Sotskova and Werner, Omega, 1997) St. Etienne / France | November 23, 2012 31 L U G | p p p 5.2. General shop problem ij ij ij | Definition 2 * (G) (G) is called a G-solution for problem G | pijL pij pijU | if for any fixed p T , * (G) contains an optimal digraph. * * If any (G) (G) is not a G-solution, (G) is called a minimal G-solution denoted as T (G). Introduction of the relative stability radius: 0 pij pijL pij pijU polytope T (G) B (G) St. Etienne / France | November 23, 2012 32 Cmax lsp - criticalweightinGs for p T Definition 3 Let Gs B (G) be such that for any p' O ( p) T lsp' min lkp' | Gk B . The maximal value of of such a stability ball O ( p) is B called the relative stability radius ˆ s p T . Known: • Dominance relations among paths and sets of paths • Characterization of the extreme values of ˆ sB p T St. Etienne / France | November 23, 2012 33 L U Characterization of a G-solution for problem J | pij pij pij | Cmax Definition 4 Gs (strongly) dominates Gk in D R q if lsp lkp lsp lkp for any p D. → dominance relation Gs DGk Gs D Gk Theorem 7: (G) is a G-solution. There exists a finite covering q T D R of polytope by closed convex sets j with d T j 1 D j , d , such that for any Gk (G) and any D j , j 1,...,d , there exists a Gs for which Gs D Gk . Corollary: T (G) Gs Gs T Gk for any Gk (G). j St. Etienne / France | November 23, 2012 34 Theorem 8: * * Let (G) be a G-solution with (G ) 2. Then: * * (G) is a minimal G-solution. For any Gs (G) there exists a vector p( s ) T such that Gs p ( s ) Gk for any Gk * (G ) \ Gs . L U J | p p p Algorithms for problem ij ij ij | - regular criterion,e.g. Cmax , Ci ,... St. Etienne / France | November 23, 2012 35 Several 3-phase schemes: • B&B: implicit (or explicit) enumeration scheme for generating a G-solution B B ' • • SOL: reduction of B by generating a sequence ˆ1 ˆ 2 ... ˆ I of Oˆi ( p) with the same p T and different B (G) * MINSOL: generation of a minimal G-solution T (G) by a repeated application of algorithm SOL T (G ) T St. Etienne / France | November 23, 2012 36 Some computational results: Cmax (n, m) Degree of uncertainty (4,4) 1, 3, 5, 7 Exact solution * T ' 41.8 6.4 2.4 2, 6, 8, 10 79.0 14.7 5, 10, 15, 20 434.9 43.5 ' Heuristic solution * T 19.9 3.8 2.4 9.5 27.3 6.9 4.4 34.8 112.8 25.7 20.0 Ci (n, m) Degree of uncertainty (4,4) 1, 3, 5, 7 ' Exact solution * T ' 34.2 7.5 6.3 2, 6, 8, 10 88.3 16.1 5, 10, 15, 20 477.7 30.8 Heuristic solution * T 24.1 6.5 5.5 14.5 52.9 13.5 12.0 30.1 132.0 24.8 24.0 Exact sol.: n m 24 , Heuristic sol.: n m 50 (n 10, m 8) St. Etienne / France | November 23, 2012 37 5.3. Two-machine problems with interval processing times a) Problem F 2 | pijL pij pijU | Cmax pijL pijU for all (i, j) Q O(n logn) algorithmby Johnson(1954) Johnson permutation: Ji , Ji ,...,Ji 1 2 n with min pik ,1, pil ,2 min pil ,1, pik ,2 for 1 k l n is optimal * Partition of the job set J J 0 J1 J 2 J with J J J \ J | p p J J J \ J | p p J J J | p p , p p J 0 J i J | piL1 piU1 piL2 piU2 1 i 2 i * i 0 U i1 L i2 0 U i2 L i1 U i1 L i2 U i2 St. Etienne / France | November 23, 2012 L i1 38 J - solutionS (T ) : minimalset containinga Johnsonpermutation for any p T Theorem 9: S (T ) 1 (1) for any Ji , J j J1 ( J 2 , respectively) U L U L p p or p p either i1 j1 j1 i1 U L U L p p or p p (either i 2 j2 j2 i 2) and * * * J 1 J , J J (2) and if satisfies i* L U p max p * – i ,1 i1 | J i J1 maxp J U – piL , 2 j2 | J j 2 L L pk for any J k J 0 max p , p – i ,1 i ,2 * * * St. Etienne / France | November 23, 2012 39 Theorem 10: If maxpijL | Ji J ,1 j 2 minpijU | Ji J ,1 j 2 then S (T ) n! Percentage of instances with S (T ) 1 , where pijU pijL L n L 5 10 15 20 25 30 1 99.2 95.2 91.2 86.1 79.2 72.8 2 97.2 89.8 77.6 63.5 51.0 39.6 3 95.0 80.9 66.4 47.6 32.8 20.6 4 91.8 78.6 56.0 39.2 20.3 10.7 5 91.0 69.4 44.9 28.9 14.6 6.0 St. Etienne / France | November 23, 2012 40 General case of problem F 2 | pij pij pij | Cmax Theorem 11: There exists an S (T ) with J v J w inany S (T ) L p U v1 U pwL1 and pvU1 pvL2 or pUw2 pwL1 and pUw2 pvL2 . J v , J w A , J v J w constructthe dominancegraph G J, A in O(n²) time Theorem 12: If J 0 , then A transitive. St. Etienne / France | November 23, 2012 41 Example: n 6 Ji J1 J2 J3 J4 J5 J6 piL1 1 9 10 5 5 10 piU1 piL2 2 12 11 8 6 11 8 14 13 6 4 4 piU2 9 15 17 7 4 4 A : J1 J 2 , J1 J 3 , J1 J 4 , J1 J 5 , J1 J 6 , J 2 J5 , J 2 J 6 , J3 J5 , J3 J6 , J5 J6 , J6 J5 J,A without transitive arcs: J2 J1 J3 J5 J6 J4 St. Etienne / France | November 23, 2012 42 Properties of J, A in the case of J 0 : 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 n 100 if J 0 and for n 1000 if J 0 b) Problem J 2 | ni 2, pijL pij pijU | Cmax → Reduction to two F 2 | pijL pij pijU | Cmax problems: see Sotskov, Sotskova, Lai and Werner, Scheduling under uncertainty, 2010 (Section 3.6) St. Etienne / France | November 23, 2012 43 5.4. Problem 1| p pi p | wiCi L i U i Notations: J J1,...,J n - setof n jobs wi - weightfor J i J T p R | p pi piL , piU - processingtimeof J i J , 0 piL piU n L i p p1 ,..., pn k J k ,...,J k 1 n pi piU , i 1,...,n - setof scenarios - jobsequence S 1,..., n! - setof jobsequences St. Etienne / France | November 23, 2012 44 1|| wiCi : On logn algorithmby Smith (1956) k J k ,...,J k S optimal 1 n wk1 pk1 ... wkn pk n Definition of the stability box: J (ki ) J k1 ,...,J ki1 J ki J ki1 ,...,J kn Ski - setof permutations J ki , J ki , J ki S J ' - permutation of the jobs J ' J Nk N 1,...,n St. Etienne / France | November 23, 2012 45 Definition 5 The maximal closed rectangular box SB k , T ki Nk [lki , uki ] T is a stability box of permutation k J k ,...,J k S , if permutation e J e ,...,J e Sk being optimal for instance 1| p | wiCi with a scenario p p1,..., pn T remains optimal for the i 1 n instance 1| p'| wiCi with a scenario p' pk , pk lk , uk pk , pk j i 1 j 1 for each ki Nk . If there does not exist a scenario p T such that permutation k is optimal for instance 1| p | wiCi , then SB k , T . 1 1 n n i j j i i j j Remark: The stability box is a subset of the stability region. However, the stability box is used since it can easily be computed. St. Etienne / France | November 23, 2012 46 Theorem 13: For the problem 1 | piL pi piU | wiCi , job J u dominates J v if and only if the following inequality holds: wu wv L U pu pv wk i Lower (upper) bound on the range of preserving the pk optimality of k S : i wki wk j d max U , max L , i 1,...,n 1 pki i j n pk j ki wki wk j d min L , min U , i 2,...,n pki 1 j i pk j ki St. Etienne / France | November 23, 2012 kn d k1 d wk n pkUn wk1 pkL1 47 Theorem 14: If there is no job J ki , i 1,...,n 1 , in permutation k ( J k ,...,J k ) S such that inequality wki wk j U L pki pk j 1 n holds for at least one job J k j , j i 1,...,n, then the stability box SB( k , T ) is calculated as follows: wki wki SB( k , T ) d d , ki ki d ki d ki otherwise SB( k , T ) . St. Etienne / France | November 23, 2012 48 Example: Data for calculating SB1, T , 1 ( J1,...,J8 ) St. Etienne / France | November 23, 2012 49 Stability box for SB1 , T w2 w2 w4 w4 , , d2 d2 d4 d4 w6 w6 w8 w8 , , d 6 d 6 d8 d8 3,6 9,10 12,15 19,20 Relative volume of a stability box Maximal ranges li , ui of possible variations of the processing times pi , i 2,4,6,8 , within the stability box SB1, T are dashed. St. Etienne / France | November 23, 2012 wi wi U : pi piL di di 3 1 3 1 1 8 4 9 5 160 50 Sotskov, Egorova, Lai and Werner (2011) Derivation of properties of a stability box that allow to derive an O(n log n) algorithm MAX-STABOX for finding a permutation t with • the largest dimension | Nt | and • the largest volume of a stability box SB t ,T . St. Etienne / France | November 23, 2012 51 Computational results Randomly generated instances with L,U 1,100, wi 1, 50 St. Etienne / France | November 23, 2012 52 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, 2012 St. Etienne / France | November 23, 2012 53 6. Selection of a suitable approach Problem 1 | pij pij pij | wiCi S (T ) - minimaldominantset L U Cardinality of S (T ) Theorem 15: S (T ) k J k1 , J k2 ,...,J kn wi a min U J i J pi wkn1 wkn wk2 wk2 wk3 L , U L ,..., U L U pk1 pk2 pk2 pk3 pkn1 pkn wk1 wi b max L J i J pi wi wi J r J i J r U L , r a, b, r R pi pi St. Etienne / France | November 23, 2012 54 Theorem 16: Assume that there is no r a, b with J r 2. Then: S (T ) n! wi wi max U J i J min L J i J . pi pi Theorem 17: S (T ) uniquelydetermined there is no r a, b with J r 2. S (T ) not uniquely determined Construct an equivalent instance with less jobs for which S (T ) is uniquely determined Assumption: S (T ) uniquely determined. z - instance with the set T of scenarios St. Etienne / France | November 23, 2012 55 Uncertainty measures ( z) 1 n! S (T ) n!1 1 S (T ) n! 0 ( z) 1 Dominance graph G J, A ( z) 1 2A n(n 1) 0 A n(n 1) 2 0 ( z) 1 Recommendations: ( z ), ( z) small ( z ), ( z ) large ( z), ( z) around0.5 use a stability approach use a robust approach use a fuzzy or stochastic approach St. Etienne / France | November 23, 2012 56 Example: n 6 L i U i p wi wi piL wi piU i p 1 5 6 300 60 50 2 4 6 240 60 40 3 6 14 420 70 30 4 2 7 140 70 20 5 10 35 700 70 20 6 5 10 250 50 25 Dominance conditions: w6 w1 50 50 L p1U p6 ( z) 1 n! S (T ) n!1 S (T ) 1 6! 360 2 6!320 359 0.5 6!1 719 apply a stochastic or a fuzzy approach St. Etienne / France | November 23, 2012 57 Example (continued): L i U i wi wi piL wi piU E pi wi E pi i p p 1 5 6 300 60 50 5.5 54 6/11 2 4 6 240 60 40 5 48 3 6 14 420 70 30 10 42 4 2 7 140 70 20 4.5 31 1/9 5 10 35 700 70 20 22.5 31 1/9 6 5 10 250 50 25 7.5 33 1/9 pi ~ U piL , piU for all J i J apply WSEPT rule J1 , J 2 , J 3 , J 6 , J 5 , J 4 ( z) 1 2A n(n 1) 1 2 14 1 6 5 15 n(n 1) A 1, 15 2 (apply a robust approach) Remark: (z ) easier computable than (z ) St. Etienne / France | November 23, 2012 58 Announcement of a book Sequencing and Scheduling with Inaccurate Data Editors: To appear at: Completion: Yuri N. Sotskov and Frank Werner Nova Science Publishers Summer 2013 4 parts: Each part contains a survey and 2-4 further chapters. Part 1: Part 2: Part 3: Part 4: Stochastic approach Fuzzy approach Robust approach Stability approach Contact address: survey: Cai et al. survey: Sakawa et al. survey: Kasperski and Zielinski survey: Sotskov and Werner frank.werner@ovgu.de St. Etienne / France | November 23, 2012 59