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(SwjCj)


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
Theorem 10.1.1
–
The WSEPT rule minimizes E(SwjCj)
(i)
(ii)

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

If all processing time distribution are ICR
– lw increases as the job is being processed
– WSEPT is optimal

If some processing time distribution are DCR
– lw decreases as the job is being processed
– WSEPT may not be optimal

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
Theorem 10.1.5
– The EDD rule minimizes the (expected)
maximum lateness Lmax for arbitrarily
distributed processing times and
deterministic due dates
4
Minimizing hmax




Suppose we want to minimize
max{E(h1(C1)),…, 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
The problem is in some cases even
easier than when the processing times
are deterministic!
6
Expected total weighted number of
tardy jobs 1|d=dj|E(SwjUj)


The deterministic version of 1|d=dj|SwjUj is
NP-hard
Theorem 10.4.1
–
The WSEPT rule minimizes the expected
weighted number of tardy jobs in the classes of
(i)
(ii)
(iii)

nonpreemptive static list policies
nonpreemptive dynamic policies
preemptive dynamic policies
A similar result holds for geometric
(discrete) processing times (see Theorem
10.4.2)
7
Breakdowns and release
dates

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
– The WSEPT rule is not necessarily optimal when
preemptions are not allowed
8
Expected total weighted
tardiness 1|d=dj|E(SwjTj)
Theorem 10.4.5

–
The WSEPT rule minimizes the
expected total weighted tardiness in the
classes of
(i)
(ii)
(iii)
nonpreemptive static list policies
nonpreemptive dynamic policies
preemptive dynamic policies
9