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# A-III - Optimization Models Draft of III Beyond Linear...

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Optimization Models Draft of August 26, 2005 III. Beyond Linear Optimization Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston, Illinois 60208-3119, U.S.A. (847) 491-3151 [email protected] http://www.iems.northwestern.edu/ ˜ 4er/ Copyright c 1989–2005 Robert Fourer

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A–96 Optimization Models — § 7.0
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

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A–98 Optimization Models — § 7.2 assignment model with the new objective might be written as follows: Maximize h min i = 1 ,...,m m j = 1 c ij x ij i 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 The objective no longer has the proper form for a linear program. Nevertheless, the problem can be reduced to one of linear programming.
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A-III - Optimization Models Draft of III Beyond Linear...

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