Job Release-Time Design in Stochastic Manufacturing Systems Using Perturbation Analysis By: Dongping Song Supervisors: Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering University of Newcastle upon Tyne March, 2000

Overview 1. Introduction 2. Problem formulation 3. Perturbation analysis (PA) 4. PA algorithm 5. Numerical examples 6. Conclusions

Introduction -- a real example Number of jobs = 113; Number of resources=13.

Introduction -- a simple structure product component WIP

Introduction -- job release times S i -- job release times Result in waiting time if {S i } is not well designed.

Introduction -- backwards scheduling Not good if uncertain processing times or finite resource capacity.

distribution of completion time tardy probability Introduction -- uncertainty problem Processing times follow probability distributions.

Introduction -- resource problem Job 2 and job 3 use the same resource  job 2 is delayed, job 1 is delayed  resulting in waiting times and tardiness.

Problem formulation Find optimal S=(S 1, S 2, …, S n ) to minimise expected total cost: J(S) = E  WIP holding costs + product earliness costs + product tardiness costs)} Key step of stochastic approximation is:  J(S)/  S i = ?

Perturbation analysis -- references Ho,Y.C. and Cao, X.R., 1991, Perturbation Analysis of Discrete Event Dynamic Systems, Kluwer. Glasserman,P., 1991, Gradient Estimation Via Perturbation Analysis, Kluwer. Cassandras,C.G. 1993, Discrete Event Systems: Modeling and Performance Analysis, Aksen.

Perturbation analysis -- general problem Consider to minimise: J(  ) = EL( ,  ) J(.) -- system performance index. L(.) -- sample performance function.  -- a vector of n real parameters.  -- a realization of the set of random sequences. PA aims to find an unbiased estimator of gradient --  J(  )/  i, with as little computation as possible.

Perturbation analysis -- main idea Based on a single sample realization Using theoretical analysis sample function gradient Calculate  L( ,  )/  i, i = 1, 2, …, n Exchange E and  : ? E  L( ,  )/  i  L( ,  )/  i =  J(  )/  i

PA algorithm -- concepts Sample realization for {S i }-- nominal path (NP) Sample realization for {S i +  S j  j  i} -- perturbed path (PP), where  is sufficiently small. All perturbed paths are theoretically constructed from NP rather than from new experiments

PA algorithm -- Perturbation rules Perturbation generation rule -- When PP starts to deviate from NP ? Perturbation propagation rule -- How the perturbation of one job affects the processing of other jobs? -- along the critical paths -- along the critical resources Perturbation disappearance rule -- When PP and NP overlaps again ?

PA algorithm -- Perturbation rules If S 2 is perturbed to be S 2 + . Cost changes due to the perturbation. perturbation generation perturbation disappearance

PA algorithm -- Perturbation rules If S 3 is perturbed to be S 3 + . Cost changes due to the perturbation. perturbation generation perturbation propagation

PA algorithm -- gradient estimate From PP and NP to calculate sample function gradient :  L( S,  )/  S i -- usually can be expressed by indicator functions. Unbiasedness of gradient estimator: E  L( S,  )/  S i =  J( S )/  S i Condition: processing times are independent continuous random variables.

Stochastic approximation Iteration equation:  k+1 =  k+1 +  k  J k step size gradient estimator of  J Robbins-Monro (RM) algorithm: if E  J k =  J. Kiefer-Wolfowitz (KW) algorithm: if  J k is finite difference estimate. RM is faster than KW (Fu and Hu, 1997).

Time comparison for gradient estimate Finite difference estimator of gradient: PA estimator of gradient -- where  1,  2, …,  K is a sequence of sample processes.

Time comparison for gradient estimate Time needed to obtain gradient estimator with K=1000. time (second) number of job simulation method PA method

Example 1 -- two stage uniform distribution Two stage serial system with uniform distributions Compare with theoretical results (Yano, 1987)

Example 1 -- two stage uniform distribution Convergence of planned parameters (S 1, S 2 ) (6.96, 8.44) S1S1 S2S2

Example 2 -- two stage exponential distribution Two stage serial system with exponential distributions Compare with theoretical results (Yano, 1987)

Example 2 -- two stage exponential distribution (7.22, 8.42) Convergence of planned parameters (S 1, S 2 ) S2S2 S1S1

Example 3 -- multi-stage system Assume: Normal distribution for processing times; Infinity capacity model. Product structure:

Convergence of cost in PA+SA J(S) iteration number

The maximum gradient in PA+SA (+/-) max {|  J(S)/  S i |, i=1,…, n} iteration number

Compare with simulated annealing time(second) J(S) Compare the convergence of cost over time (second). simulated annealing PA+SA method Where simulated annealing uses four different settings (initial step sizes and number for check equilibrium)

Example 4 -- complex system Assume: Normal distribution and finite capacity model.

Resource constraints ResourcesJob sequences 1000: 247, 243, 239, 234, 231, 246, 242, 238, 230, 245, 237, 229, : 236:1, 236:2, 236:3, 236:4, 236:5, 236:6, 236:7, 226:1, 236:8, 226:2, 226:3, 226:4, 226:5, 226:6, 236:11, 226:7, 232:1, 226:8, 235:1, 232:2, 236:12, 235:2, 226:9, 232:3, 235:3, 240:1, 235:4, 240:2, 226:10, 232:5, 236:13, 233:2, 235:5, 240:3, 233:3, 235:6, 240:4, 232:7, 226:11, 233:4, 235:7, 240:5, 232:8, 233:5, 235:8, 240:6, 232:9, 233:6, 240:7, 226:12, 232:10, 235:9, 240:8, 233:8, 240:9, 233:9, 226:13, 235:10, 240:10, 236:15, 226:14, 240:11, 236:16, 226: : 236:9, 236:10, 232:4, 232:6, 236:14, 232:11, 232: : 233:1, 233:7, 233:11.

Resource constraints ResourcesJob sequences 1129: 233: : 233: : 244:1, 244:3, 244:5, 241:1, 241:2, 241:3, 248:2, 248:3, 248:5, 248: : 244:2, 241:4, 241:5, 248: : 241:6, 241: : 244: : 244:6, 244: : 244:8, 248:7, 248: : 244:9, 248:1. Total number of jobs: 113; Number of resources: 13.

Convergence of cost in PA+SA iteration number J(S)

The maximum gradient in PA+SA (+/-) max {|  J(S)/  S i |, i=1,…, n} iteration number

Compare with simulated annealing time(minute) J(S) Compare the convergence of cost over time (minute). simulated annealing PA+SA method with four different settings

Conclusions Effective algorithm to design job release times. Can deal with complex systems beyond the ability of analytical methods. Faster to obtain gradient estimator than simulation method Faster than simulated annealing to optimise parameters Not depend on particular distributions and can include other stochastic factors.

Further Work Convexity of the cost function and global optimization problem The effect of different job sequences on job release time design Further compare with other optimisation methods