1 IOE/MFG 543 Chapter 10: Single machine stochastic models Sections 10.1 and 10.4 You may skip Sections

2 Expected weighted completion time 1||E(  w j C j ) The expected completion times are the sum of the expected processing times The expected completion times are the sum of the expected processing times WSEPT rule WSEPT rule –Sequence the jobs in decreasing order of the ratio w j /E(X j )= j w j –Also called the w rule Theorem Theorem –The WSEPT rule minimizes E(  w j C j ) (i) in the class of nonpreemptive static policies and (ii) in the class of nonpreemptive dynamic policies Breakdowns at an exponential rate can also be easily handled Breakdowns at an exponential rate can also be easily handled

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

4 Deterministic due dates Minimizing the maximum lateness L max when the processing times are stochastic Minimizing the maximum lateness L max when the processing times are stochastic Theorem Theorem –The EDD rule minimizes the (expected) maximum lateness L max for arbitrarily distributed processing times and deterministic due dates

5 Minimizing h max Suppose we want to minimize max{E(h 1 (C 1 )),…, E(h n (C n ))} Suppose we want to minimize max{E(h 1 (C 1 )),…, E(h n (C n ))} The backward algorithm for minimizing h max can be modified to include stochastic processing times The backward algorithm for minimizing h max 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 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 h j (C j ) is linear in C j since E(h j (C j ))=h j (E(C j )) in that case The task becomes easier if h j (C j ) is linear in C j since E(h j (C j ))=h j (E(C j )) in that case

6 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 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 The problem is in some cases even easier than when the processing times are deterministic! The problem is in some cases even easier than when the processing times are deterministic!

7 Expected total weighted number of tardy jobs 1|d=d j |  w j U j ) The deterministic version of 1|d=d j |  w j U j is NP-hard The deterministic version of 1|d=d j |  w j U j is NP-hard Theorem Theorem –The WSEPT rule minimizes the expected weighted number of tardy jobs in the classes of (i) nonpreemptive static list policies (ii) nonpreemptive dynamic policies (iii) preemptive dynamic policies A similar result holds for geometric (discrete) processing times (see Theorem ) A similar result holds for geometric (discrete) processing times (see Theorem )

8 Breakdowns and release dates The WSEPT rule also applies in the case of breakdowns 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 If the jobs have stochastic release dates the preemptive version of WSEPT rule is optimal 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

9 Expected total weighted tardiness 1|d=d j |  w j T j ) Theorem Theorem –The WSEPT rule minimizes the expected total weighted tardiness in the classes of (i) nonpreemptive static list policies (ii) nonpreemptive dynamic policies (iii) preemptive dynamic policies