IOEMFG 543 Chapter 10 Single machine stochastic models

IOE/MFG 543 Chapter 10: Single machine stochastic models Sections 10. 1 and 10. 4 You may skip Sections 10. 2 -10. 3 1

Expected weighted completion time 1||E(Swj. Cj) n n n The expected completion times are the sum of the expected processing times WSEPT rule – – Sequence the jobs in decreasing order of the ratio wj/E(Xj)=ljwj Also called the lw rule – The WSEPT rule minimizes E(Swj. Cj) Theorem 10. 1. 1 (i) (ii) n in the class of nonpreemptive static policies and in the class of nonpreemptive dynamic policies Breakdowns at an exponential rate can also be easily handled 2

Preemptions and nonpreemptive WSEPT n If all processing time distribution are ICR – lw increases as the job is being processed – WSEPT is optimal n If some processing time distribution are DCR – lw decreases as the job is being processed – WSEPT may not be optimal n Example 10. 1. 2 Xj=0 Xj=exp(lj) w. p. pj w. p. (1 -pj) 3

Deterministic due dates Minimizing the maximum lateness Lmax when the processing times are stochastic n Theorem 10. 1. 5 n – The EDD rule minimizes the (expected) maximum lateness Lmax for arbitrarily distributed processing times and deterministic due dates 4

Minimizing hmax n n Suppose we want to minimize max{E(h 1(C 1)), …, E(hn(Cn))} The backward algorithm for minimizing hmax can be modified to include stochastic processing times The implementation can be difficult because the completion time of the job being scheduled last is unknown and depends on the distribution of the processing times of all the earlier jobs The task becomes easier if hj(Cj) is linear in Cj since E(hj(Cj))=hj(E(Cj)) in that case 5

Section 10. 4 Exponential processing times Some problems that cannot be solved to optimality for arbitrary processing time distributions can be solved in the special case of exponential processing time distributions n The problem is in some cases even easier than when the processing times are deterministic! n 6

Expected total weighted number of tardy jobs 1|d=dj|E(Swj. Uj) n n The deterministic version of 1|d=dj|Swj. Uj is NP-hard Theorem 10. 4. 1 – The WSEPT rule minimizes the expected weighted number of tardy jobs in the classes of (i) (iii) n nonpreemptive static list policies nonpreemptive dynamic policies A similar result holds for geometric (discrete) processing times (see Theorem 10. 4. 2) 7

Breakdowns and release dates n The WSEPT rule also applies in the case of breakdowns – The effect of the breakdown is to shorten the time until the due date – The WSEPT rule is independent of this time n If the jobs have stochastic release dates the preemptive version of WSEPT rule is optimal – The WSEPT rule is not necessarily optimal when preemptions are not allowed 8

Expected total weighted tardiness 1|d=dj|E(Swj. Tj) n Theorem 10. 4. 5 – The WSEPT rule minimizes the expected total weighted tardiness in the classes of (i) (iii) nonpreemptive static list policies nonpreemptive dynamic policies 9