21
Scheduling problems
In this section we consider a very important class of problems involving
integer variables, arising from scheduling. We suppose we have
n
jobs, where
job
j
takes time
p
j
≥
0 to process.
We cannot process two or more jobs
simultaneously. We must choose an order in which to schedule the jobs, and
our goal is minimize the average completion time for the jobs.
For example, consider five jobs with processing times shown below.
j
1
2
3
4
5
p
j
13
4
19
20
3
If we chose the scheduled order 5
,
2
,
1
,
3
,
4, then the completion times would
be 3
,
7
,
20
,
39
,
59, giving an average completion time of
128
5
.
Suppose we now add the following “precedence” constraints:
(1
,
3)
,
(1
,
4)
,
(2
,
4)
,
(2
,
5)
,
(4
,
5)
.
The pair (1
,
3), for example, means that job 1 must be finished before job 3
starts. We will try to model such problems as
mixed integer programs
: some
variables must be integers.
As a first try, for each job
j
we introduce a variable
c
j
that specifies the
completion time for that job. Our objective is then clear:
minimize
1
n
n
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 Fall '07
 BLAND
 Optimization, Book of Job, Book of Proverbs, cj yjk yjk

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