Characterizing Heuristic Performance in Large Scale Dual Resource Constrained Job Shops

Transcription

1 Characterizing Heuristic Performance in Large Scale Dual Resource Constrained Job Shops Benjamin J. Lobo a, James R. Wilson b, Kristin A. Thoney b, Thom J. Hodgson b, and Russell E. King b May 20 th, 2013 a U.S. Army Research Laboratory, Aberdeen Proving Ground, MD b North Carolina State University, Raleigh, NC 1 / 18

2 Introduction Problem Definition Job Shop Scheduling & DRC Systems Determine an optimal schedule given a performance objective Active area of research for more than 60 years [Potts & Strusevich, 2009] Most job shop scheduling problems are NP-Hard [Lenstra et al., 1977] Dual Resource Constrained (DRC) system One in which shop capacity may be constrained by machine and labor capacity or both [Treleven & Elvers, 1985] 2 / 18

3 DRC Job Shop Solution Approach [Lobo et al., 2013a,b] Solution Approach Objective & Solution Approach Objective: Partition job shop into machine groups, assign workers to machine groups Determine the worker allocation ϑ that minimizes L max, the maximum lateness Solution approach [Lobo et al., 2013a,b]: Lower bound on L max given allocation ϑ, LB ϑ Lower Bound Search Algorithm (LBSA) finds ϑ ϑ = arg min{lb ϑ : ϑ is feasible} Virtual Factory [Hodgson et al., 1998]: generates a feasible schedule aimed at minimizing L max ; VF ϑ is the smallest L max value found using allocation ϑ 3 / 18

4 DRC Job Shop Solution Approach [Lobo et al., 2013a,b] Solution Approach Allocation ϑ HSP Allocation ϑ does not always satisfy optimality criteria Heuristic Search Procedure (HSP): searches for promising (hopefully optimal) allocations, returns ϑ HSP If VF ϑ HSP LB ϑ > 0, how good is allocation ϑ HSP? Question: What is the quality of an arbitrary allocation ϑ? VF ϑ VFB forms a benchmark where allocation ϑ VFB is achieved through enumeration 4 / 18

5 DRC Job Shop Solution Approach [Lobo et al., 2013a,b] Solution Approach A Probabilistic Approach Probabilistic characterization of the data set VFB composed of differences PLB(ϑ VFB ) = VF ϑ VFB LB ϑ Allows estimation of the probability that a given allocation is in fact a VF-best allocation Example: 90% of observations in data set VFB are larger than PLB(ϑ) 90% confident that allocation ϑ is a VF-best allocation Generally used continuous and mixed distributions to characterize data set VFB Sometimes a discrete probability mass function was used 5 / 18

6 DRC Job Shop Solution Approach [Lobo et al., 2013a,b] Solution Approach Fitting Data Set VFB Data set VFB : PLB(ϑ VFB ) values, Q 500 Fitting criteria: Continuous: p 0 < 0.05, greater than 50 non zero values Mixed: p , greater than 50 non zero values Discrete: p , less than 50 non zero values Continuous: Beta, Gamma, Johnson-SB, Johnson-SU, Lognormal, Weibull Mixed: F (x) = p 0 F 0 (x) + (1 p 0 )F c (x) F 0 (x) = 0 for x < 0 and F 0 (x) = 1 for x 0 for < x < F c : Continuous distribution fit to non zero values 6 / 18

7 DRC Job Shop Solution Approach [Lobo et al., 2013a,b] Data set VFB Issue with data set VFB Requires enumeration Feasible allocation search space grows exponentially with an increase in: 1 The number of machine groups 2 The number of machines Impractical or infeasible to obtain data set VFB for large scale DRC job shop scheduling problems Need: another data set with the same characteristics as data set VFB 7 / 18

8 An Alternate Data Set Data Set HSP An Alternate Data Set Experimental results point to data set HSP composed of differences PLB(ϑ HSP ) = VF ϑ HSP LB ϑ Compare data set VFB and HSP do they result in the same probabilistic characterization? Because VF ϑ VFB VF ϑ HSP, data set VFB might have more zero-valued data points Continuous vs. mixed distribution criteria: value of p 0 8 / 18

9 An Alternate Data Set Data Set HSP Comparing Data Sets Three comparison cases: VFB HSP Case 1 Continuous Continuous Case 2 Mixed Continuous Case 3 Mixed Mixed Mixed distribution: Composed of probability mass at 0 and the continuous distribution fit to the non zero data points Notation: δ VFB or δ HSP used to refer to reduced data set of non zero data points 9 / 18

10 An Alternate Data Set Data Set HSP Demonstrating the similarity of data sets VFB and HSP Empirical cumulative distribution functions (c.d.f.s) Degree of overlap (subjective judgement) is an indicator of the likelihood that both data sets are drawn from the same underlying distribution Mann-Whitney one-sided test [Conover, 1999] Null hypothesis: Both data sets come from the same underlying distribution Alternate hypothesis: The underlying distribution of data set HSP or δ HSP is stochastically greater than the underlying distribution of data set VFB or δ VFB 10 / 18

11 Experimental Results Results Experimental design 192 designated DRC job shop scheduling problems: 1 3 size job shops, increasingly larger 2 2 loading of machine groups (balanced and unbalanced) 3 4 staffing levels (ratio of workers to machines) 4 8 due-date ranges for job due-dates Q = 500 randomly generated simulation replications based on each designated problem Data sets formed from the Q Q simulated replications where the optimality criteria were not satisfied HSP2: modified version of HSP to ensure fair comparison of performance in different size job shops 11 / 18

12 Experimental Results Results Example empirical c.d.f. fits and Mann-Whitney p-values PLB(# VFB ) PLB(# HSP2 ) PLB(# VFB ) PLB(# HSP2 ) 0 0 VFB HSP2 ıvfb HSP2 PLB(#) PLB(#) PLB(# VFB ) PLB(# HSP2 ) 0 ıvfb ıhsp2 PLB(#) 12 / 18

13 Experimental Results Results General Results for the Continuous Fits Empirical c.d.f.s and Mann-Whitney p-values indicate that, in general, both data sets being compared were drawn from the same underlying distribution Specifically: Case 1: Data set VFB and HSP2 are equivalent (99) Case 2: Data set δ VFB and HSP2 are equivalent (7) Case 3: Data set δ VFB and δ HSP2 are equivalent (44) Case 1: can use data set HSP2 in place of data set VFB Case 2 & Case 3: more investigation needed must compare overall fit 13 / 18

14 Experimental Results Results Bias Introduced in Case 2 & Case 3 Fits In general p 0,VFB > p 0,HSP2 Assumption: F c (x) is the same given continuous fit results Case 2: F VFB (x) = p 0,VFB F 0 (x) + (1 p 0,VFB )F c (x) F HSP2 (x) = F c (x) Case 3: F VFB (x) = p 0,VFB F 0 (x) + (1 p 0,VFB )F c (x) F HSP2 (x) = p 0,HSP2 F 0 (x) + (1 p 0,HSP2 )F c (x) In both cases F VFB (x) F HSP2 (x) for a given value of x F HSP2 (x) will overestimate the quality of the allocation F HSP2 (x) forms an upper bound on the quality 14 / 18

15 Experimental Results Example Large Asymmetric Job Shop, 70% Staff., 200 DDR PLB(# HSP2 ) PLB(# HSP2 ) 0 ıhsp2 0 ıhsp2 PLB(# HSP2 ) PLB(# HSP2 ) F HSP2(x) = 0.301F 0(x) F c(x) where F c( ) Weibull(1.47, 51.86, 1.00) 15 / 18

16 Experimental Results Example Large Asymmetric Job Shop, 70% Staff., 200 DDR LB ϑ = 3,702, VF ϑ = 3,825, and PLB(ϑ ) = 123: PLB(ϑ VFB ) 0 { } f HSP2(x) dx = Pr PLB(ϑ VFB ) 123 [ { = = exp VF ϑ HSP2 = 3,712 and PLB(ϑ HSP2 ) = 10: ( ) }] { } { } Pr PLB(ϑ VFB ) PLB(ϑ HSP2 ) = Pr PLB(ϑ VFB ) 10 = From at most 2% to at most 64% confident that ϑ HSP2 is a VF-best allocation! 16 / 18

17 Conclusions Conclusions Data set HSP2 is a viable substitute for data set VFB Data set HSP2 can be obtained even for large scale DRC job shop scheduling problems Consistent bias introduced by using data set HSP2 Other results indicate this bias generally on the order of 10% In general you won t know what fitting case is being satisfied must assume a bias 17 / 18

18 Questions Any questions? 18 / 18