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Unformatted text preview: Lecture 19 Reading: 8.3 of text book. Computational Methods for Management and Economics Carla Gomes Outline Assignment Problem Minimum Cost Flow Problem The assignment problem Special type of linear programming problem where assignees are assigned to tasks . Assignees and tasks people, machines, plants, vehicles, locations, etc Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task. The assignment problem To fit the model for an assignment problem, the following assumptions need to be satisfied: 1. The number of assignees and the number of tasks are the same. 2. Each assignee is to be assigned to exactly one task. 3. Each task is to be performed by exactly one assignee. 4. There is a cost associated with each combination of an assignee performing a task. 5. The objective is to determine how all the assignments should be made to minimize the total cost. Note: These assumptions appear quite restrictive. However by using e.g., dummy assignees and/or tasks we can apply it to several problems. Sellmore Company Assignment Problem The marketing manager of Sellmore Company will be holding the companys annual sales conference soon. He is hiring four temporary employees: Ann; Ian; Joan; Sean Each will handle one of the following four tasks: Word processing of written presentations Computer graphics for both oral and written presentations Preparation of conference packets, including copying and organizing materials Handling of advance and onsite registration for the conference Question: Which person should be assigned to which task? Data for the Sellmore Problem Required Time per Task (Hours) Temporary Employee Word Processing Graphics Packets Registrations Hourly Wage Ann 35 41 27 40 $14 Ian 47 45 32 51 12 Joan 39 56 36 43 13 Sean 32 51 25 46 15 The Network Representation A 2 A 1 T 4 A 4 T 3 A 3 T 2 T 1 Assignees Tasks 490 540 468 690 (Ann) (Ian) (Joan) (Sean) (Word processing) (Graphics) (Packets) (Registrations) 574 378 560 564 384 612 507 728 559 480 765 375 Algebraic Formulation Let x ij = 1 if employee i performs task j ; 0 otherwise c ij = cost of required by i to perform task j Minimize Cost = i j c ij x ij subject to j x ij = 1 for each employee I; (each employee performs a single task ) i x ij = 1 for each task j (each task assigned to a single employee) and x ij 0 for all i and j . The Assignment Problem Model Decision variables: 1 if assignee i performs task j x ij otherwise i,j = 1,2 , n Each x ij is a binary variables it takes only two values: 0 or 1. The Assignment Problem Model j and i all for ij x or ij x n i for n j ij x n j for n i ij x t s n i n j ij x ij c Z Min 1 ,..., 2 , 1 1 1 ,..., 2 , 1 1 1 . ....
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This note was uploaded on 02/25/2011 for the course AEM 4120 taught by Professor Gomes,c. during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 GOMES,C.

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