Stochastic Scheduling History and Challenges

Transcription

1 Stochastic Scheduling History and Challenges Rolf Möhring Eurandom Workshop on Scheduling under Uncertainty Eindhoven, 3 June 2015 DFG Research Center MATHEON mathematics for key technologies

2 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Risk Analysis for a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as example

3 Shutdown and Turnaround Scheduling Turnaround Scheduling: resource allocation and scheduling of large-scale maintenance activities in chemical manufacturing with Nicole Megow & Jens Schulz Phase 1: plan the schedule length t can hire external workers balance turnaround cost vs. out of service cost for t estimate the risk of meeting t Phase 2: calculate a schedule S for t resource leveling risk analysis of S unforeseen repairs may occur

4 An example: turnaround of a cracker (1) ~2000 jobs, turnaround length 4 8 days very detailed, large variation in processing time must respect workers shifts Sa., So., Mo., Di., Mi., Do., Fr., Sa., So., Ohne Titel AUERLUEFTER DEMONT LUEF.DEM 2.R. 01.STOSS DEM.Boden DEM 7,6 Stunden 2.R. 02.STOSS DEM.Boden DEM 1,79 Tage 2.R. KOL.TEILE ABLASSEN KT. ABL 7,6 Stunden 2.R. KOL.-TEILE TRANSP KT. TRA 1,3Std F KOL.-TEILE SPRITZEN KT. SPR 41 Minuten 1.H. KOL.-TEILE KONT.+REP KT.KONTR 2.R. KOLONNE SPRITZEN KOL. SPR 6,48 Stunden 1.H. IU EIGENUEBERWACHUNG ABNAH.IU S KOL.-TEILE AUFZIEHEN KT. AUF 2 Tage 2.R. 01.STOSS MON.Boden MON 2 Tage 2.R. 02.STOSS MON.Boden MON 2 Tage 2.R. MANNLOCHDECKEL MONT ML. MON 2.R. DRUCKPROBE MIT EDELWASSER DP M.EDELW 0Std 2.R.

5 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Risk Analysis for a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as example

6 Scheduling problems Model a set V of jobs j = 1,,n (with or without preemption) a graph (partial order) G of precedence constraints resource constraints, here given by m identical machines release dates r j G 3 m =

7 Uncertainty and objective jobs have random processing times X j with known distribution Q j we know their joint distribution cost function κ( C 1,,C n ) depending on the (random) completion times C 1,,C n examples C max, w j C j, w j F j ( F j = C j r j ) plan jobs over time and minimize expected cost What can I achieve without knowing the future

8 Planning with policies the dynamic view S(t) Decision at time t (non-anticipative) start set S(t) (possibly empty) t t plan. time fix tentative next decision time t plan. (deliberate idleness) next decision time = min { t plan., next completion time }

9 Stability of policies Data deficiencies, use of approximate methods (simulation) require stability condition: Q approximates Q κ approximates κ OPT(Q,κ ) approximates OPT(Q,κ) No stability for optimal policies in general!

10 Excessive use of information yields instability 1 min E(C max ) 2 4 Q : x =(1 +, 4, 4, 8, 4) y =(1, 4, 4, 4, 8) with prob. ½ with prob. ½ 3 5 x ε E Q (C max )=13 y for ε

11 1 ε 0 Q ε Q with Q :!# x = (1,4, 4,8,4) with probability 1 2 " $ # y = (1,4, 4,4,8) with probability 1 2 No info when 1 completes. So start 2 at t = 0 x y E Q (C max ) = 13 = lim 0 E Q (C max ) 12 16

12 Overview uncertainty in scheduling Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Risk Analysis for a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as example

13 Therefore: Stable classes of policies Stability is more important than optimality use good robust policies instead Investigate classes of policies w.r.t. stability and approximation behavior priority policies (list scheduling) no stability in general stochastic approximation results available preselective policies (delay policies) have stability only few results on approximation

14 Priority policies have anomalies minimize makespan on 2 identical machines use priority list 1 < 2 < 3... x = (4,2,2,5,5,10,10) y = x 1 = (3,1,1,4,4,9,9)

15 Preselective policies F is forbidden set : F cannot be scheduled simultaneously but every proper subset can Solve conflict on every F by selecting a waiting job j F F j must wait until any job from F is completed F i j F j selected i j OR condition representing k k local priority do early start scheduling w.r.t. original precedence constraints and OR conditions resulting from preselected waiting jobs

16 No anomalies for preselective policies identical parallel machines F = {4,5,7} is only forbidden set 7 x = (4,2,2,5,5,10,10) y = x 1 = (3,1,1,4,4,9,9)

17 Three views on policies online planning rule make decisions over time function from R n R n combinatorial object maps processing times to start times used for computations (special policies only)

18 Preselective policies and AND/OR networks G 1 3 F F F F F : {3,4,5}, {1,4} 4 4 this choice of waiting jobs defines policy Π AND/OR network representing Π start of a job in policy Π = min/max of path lengths = min/max of sums of processing times Π is continuous and monotone

19 Tasks related to AND/OR networks may contain cycles F 2 6 F 3 F 1 9 test feasibility of waiting job choice compute earliest start detect forced OR conditions 7 F 4 8 F 5 10 (transitivity) fast algorithms available [M., Skutella & Stork 2004] in NP conp for arbitrary arc weights, no fast algo known

20 A special case: partial order policies i j F j selected i j OR condition representing k k local priority select (i,j) add a precedence constraint on F only AND conditions start of a job in policy Π = earliest start w.r.t. to a partial order of precedence constraints = max of sums of processing times Π is continuous, monotone, and convex

21 Nice policies are preselective function from R n R n combinatorial object [Igelmund & Radermacher 1985] Π is monotone iff Π is preselective (up to domination) Π is continuous iff Π is preselective (up to domination) Π is convex iff Π is a partial order policy

22 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Risk Analysis for a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as running example

23 Some special optimality results Minimize E( w C ) on m identical machines for random j j independent processing times 1 machine WSEPT is optimal for w C [Rothkopf 1966] j j m machines independent exponentially distributed processing times LEPT is optimal for C [Weiss 1982] max SEPT is optimal for C [Weiss & Pinedo 1982] j no optimal policy known for w j C j

24 A more general optimality result Set policy: use only completion, being busy and time t for decisions S(t) no tentative decision times t t planned Additive cost function κ( C 1,,C n ) is the integral over cost rate g(u(t)) of the set U(t) g( ) g( ) g( ) of uncompleted jobs at time t X j independent and exponentially distributed optimal set policy [M., Radermacher, Weiss 1985]

25 The non-idleness problem Set policies contain preselective and priority policies may leave resources idle for obtaining information, even for exponential distributions and m parallel machines Open challenge is there an optimal set policy without idle times for w C on m parallel machines and independent j j exponentially distributed X j

26 The LP-based approach for j w j C j Consider the achievable region [M., Schulz & Uetz 1999] {(E[C Π 1 ],...,E[C Π n ]) R n Π policy} Find a polyhedral relaxation P Solve the linear program (LP) min{ j w j C LP j (C LP 1,...,CLP n ) P} Use the list L: induced by C LP i 1 i 1 < i 2 <... < i n... Ci LP n C LP i 2 for defining a preselective policy (C LP 1,...,C LP n )

27 The LP relaxation E[X j ]C LP j 1 (( 2m j A ( (m 1)( 1) 2m ]) 2 ) E[X j + E[X j ] 2 j A j A ) E[X j ] 2 j A for all A V C LP j E[X j ] for all j V Assume CV[X j ] 2 CV[X j ] 2 := VAR[X j] E[X j ] 2 LP can be solved combinatorially in polynomial time

28 Results for parallel machine scheduling Analysis of WSEPT [M., Schulz, Uetz] (weighted shortest expected processing time first) ( ) 2 1 m approximation, combinatorial algorithm, LP approach is crucial Adding release dates [M., Schulz, Uetz] use job-based priority policy ( ) 4 1 m approximation, LP approach is crucial Adding precedence constraints [Skutella, Uetz] use delayed list scheduling ( ) 3 + 2β + m 1 approximation, LP approach is crucial mβ

29 Extended results Stochastic online scheduling jobs arrive online and must be assigned to machines now next day, jobs are scheduled on the assigned machines number of jobs is not known in advance jobs have random processing times on the machines better or matching bounds as in previous model for w C [Megow, Uetz, Vredeveld 06] j j Preemptive scheduling on parallel machines 2-approximation for w j C j [Megow, Vredeveld 07] involved analysis, uses Gittins index

30 Role of the coefficient of variation Assumed CV[X j ] 2 enters performance guarantee [Skutella, Sviridenko, Uetz 2014] (in the context of unrelated machines) The performance ratio of any fixed-assignment policy can be as large as (1 δ) for any δ > 0, for large enough m [Labonté, M., Megow 2014] for every k, there is an instance I k with k, such that the performance ratio of WSEPT is as large as ( 4p k), for large enough n Is this avoidable? Not for some classes of policies

31 More on the coefficient of variation The approximation ratio of SEPT for C j is $(n 1/4 ) and $(m) [Im, Moseley, Pruhs 2015] A clever priority policy achieves O(log 2 n + m log n) for an arbitrary number of machines large lower bound implies that n, m, and grow simultaneously Challenge: does SEPT achieve a constant approximation ration for a constant number of machines?

32 Open challenges Can one beat? i.e. are there policies whose approximation ratio does not depend on? m w C with independent exponentially distributed j j processing times Is there an optimal policy without idle times? Maybe even a priority policy?

33 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Risk Analysis for a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as running example

34 Detailed analysis of makespan distribution Ideal: distribution function F of C max 1 90% 0 t 90 makespan Modest: percentiles t 90 = inf{t Pr{C max t} 0.90} F(t)

35 Obtaining stochastic information is hard Hagstrom 88: The problems below are #P-complete MEAN Given: Stochastic project network with discrete independent processing times Wanted: Expected makespan DF Given: Stochastic project network with discrete independent processing times, time t Wanted: Pr{ makespan t }

36 Approximate methods and bounds Simulation requires (complete) information about distributions difficult to model stochastic dependencies Bounding the distribution function possible with incomplete information permits stochastic dependencies combinatorial algorithms

37 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Analyzing a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as running example

38 Arc diagrams G D Node diagram jobs are nodes of digraph G Arc diagram jobs are arcs of digraph D dummy arcs may be necessary standard graph algorithms apply makespan = longest path

39 Chain minors Let N, N be project networks. N is a chain minor of N if every job of N is represented by (one or several) copies in N every chain of N is contained in a chain of N by taking an appropriate copy N N

40 Chain minors: an example N N extreme case: parallel composition of all maximal chains

41 Bounds based on chain minors M. & Müller `99 Let N be a chain minor of N Give copies of a job from N in N the same processing time distribution as their original Take all processing time distributions in N as stochastically independent Then the makespan of N is stochastically smaller than the makespan of N Q Cmax (N) st Q Cmax (N )

42 Sandwiching a network with minors Given N, look for special networks N 1, N 2, such that N 1 minor N minor N 2 1 F 1 F F 2 0 special = easier to calculate the makespan distribution

43 Series-parallel networks series reduction indegree = outdegree = 1 parallel reduction A network N is series-parallel if it can be reduced by a finite sequence of series and parallel reductions to one job.

44 Special cases of the chain minor principle Kleindorfer 71 Shogan 77 Spelde 76 Dodin 85 Work for stochastically independent processing times Have same underlying combinatorial principle

45 The bounds of Spelde 76 uses special series-parallel networks (parallel chains) yields upper and lower bounds N disjoint chains lower bound all chains upper bound

46 Distribution-free evaluation of Spelde s bounds Large networks chain length normally distributed % = % j and σ 2 = σ 2 j along a chain per job only % j and σ 2 j required Problem: #chains may be exponential Remedie: Consider only relevant chains

47 Relevant chains µ k-max µ 2-max µ max Pr(Y k Y 1 ) Y i = length of i-longest chain w.r.t. mean processing times % j Apply k-longest path algorithms to determine k Return the product of the k 1 normal distribution functions F Special case: PERT, considers only Y 1 Excellent and fast bounds in practice Y i

48 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Analyzing a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as running example

49 Worst case approach for stochastic dependencies [Meilijson & Nadas 79, Klein-Haneveld 86] Consider expected tardiness E Q [(C max t) + ]=E Q [max{0, C max t}] of makespan C max in the worst case, i.e. (t) =sup Q E Q [(C max t) + ] ranges over all joint distributions with the given job processing time distributions as marginals

50 Properties of expected tardiness E Q [(X t) + ].5 2 E Q [X] slope 1 slope 3/ slope 1/ X t E Q [(X t) + ] is piecewise linear and convex for discrete random variables X

51 Properties of (t) =sup Q E Q [(C max t) + ] ψ(t) slope 1 to left of t 0 convex, decreasing to right of t 0 t 0 t (t) =min (x1,...,x n ) j E[(X j x j ) + ] such that C max (x 1,...,x n ) t special convex separable optimization problem solvable by max flow algorithms for discrete X j

52 Overview Part I: Policies for Stochastic Scheduling The Model Classes of Policies Optimality and Approximation Results for Policies Part II: Analyzing a Policy Analyzing the Makespan Distribution Bounds by Network Modification Dealing with Stochastic Dependencies Turnaround Scheduling as running example

53 Solving the turnaround problem: Phase 1 Phase 1: plan the schedule length t solve a time-cost tradeoff problem, relax shifts and assume continuous workers use the breakpoints on the time-cost tradeoff curve to calculate feasible schedules heuristically (no resource leveling) yields alternative rough schedules cost 155, , ,000 rough schedules time

54 Uncertainty in Phase 1 Phase 1: plan the schedule length t present the risk for the rough schedules let the manager decide manager can change t and see the risk change Computation uses the stochastic bounds on the makespan t

55 Solving the turnaround problem: Phase 2 Phase 2: calculates a schedule S for the chosen time t settles neglected side constraints such as time lags levels resources heuristically uses resource flow for defining a network N calculates the risk based on N In addition have compared our algorithm on small instances with results from a MIP solver for a MIP formulation

56 Result of the leveling algorithm Di, Mi, Do, Fr, Sa, So, Mo, Di, % 52% 144% 63% 33% 66% 229% 185% 125% 93% 58% 25% 12% unleveled Di, Mi, Do, Fr, Sa, So, Mo, Di, % 100% 90% % 300 % 33% % 60% % 9% 20% 98% 0% 100% 100% 98% 100% 0% 66% 8% 4% leveled

57 Summary Uncertainty is imminent in practical scheduling problems there are good tools available to analyze risks and implement policies Turnaround problems are an excellent field for many aspects of scheduling time-cost tradeoff, malleable jobs, multi-mode resource leveling, calendars Paper in INFORMS J. Computing 2011 (with Nicole Megow and Jens Schulz) Turnaround instance available ftp://ftp.math.tu-berlin.de/pub/combi/projects/turnaround/

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