Draft of August 26, 2005
A–97
7. Max-Min and Min-Max Formulations
There are a few kinds of models whose objectives are not quite linear, but
that can be can be solved by converting them to linear programs. This chap-
ter looks in particular at problems of maximizing the minimum — or similarly
minimizing the maximum — among several linear functions.
7.1 A max-min assignment model
We begin with two examples that illustrate the usefulness of “max-min” and
“min-max” objectives and that demonstrate the principles involved. Then we
summarize the transformations in a more general setting.
A problem of assigning
m
people to
m
jobs was modeled as a linear program
in Section 3(c). The data values are the individual preferences:
c
ij
preference of person
i
for job
j
,
on a scale of 1 (lowest) to 10 (highest),
for
i
=
1
, . . . , m
and
j
=
1
, . . . , m
The variables are the decisions as to who is assigned what job, represented as
follows:
x
ij
=
1
⇒
person
i
is assigned to job
j
x
ij
=
0
⇒
person
i
is not assigned to job
j
We argued that the preference of person
i
for the job assigned to
i
can be
represented as
∑
m
j
=
1
c
ij
x
ij
.
This suggested the following linear program for
maximizing the total preference of all individuals for the jobs to which they are
assigned:
Maximize
∑
m
i
=
1
∑
m
j
=
1
c
ij
x
ij
Subject to
∑
m
j
=
1
x
ij
=
1
,
i
=
1
, . . . , m
∑
m
i
=
1
x
ij
=
1
,
j
=
1
, . . . , m
x
ij
≥
0
,
i
=
1
, . . . , m
;
j
=
1
, . . . , m
We explained how the constraints, together with the integrality property of
transportation models, insure that each person is assigned exactly one job, and
each job is given to exactly one person.
The solution to this model could leave some people quite dissatisfied. Al-
though the overall sum of the preferences is maximized, certain individuals
may be assigned jobs for which their preference is very low. A direct way to
avoid this possibility is to change the objective so that it maximizes the least-
preferred assignment. More precisely, we seek values
x
ij
of the variables so
that the minimum preference of person
i
for the job assigned to
i
, over all
i
=
1
, . . . , m
, is maximized.
Since the preference of person
i
for the job assigned to
i
is
∑
m
j
=
1
c
ij
x
ij
, the