03_linear_prog2

03_linear_prog2 - 8/14/04 J. Bard and J. W. Barnes...

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Unformatted text preview: 8/14/04 J. Bard and J. W. Barnes Operations Research Models and Methods Copyright 2004 - All rights reserved Lecture 3 Classic LP Examples Topics Employee scheduling problem Energy distribution problem Feed mix problem Cutting stock problem Regression analysis Model Transformations 2 ( More) Examples of LP Formulations 1. Employee Scheduling Macrosoft has a 24-hour-a-day, 7-days-a-week toll free hotline that is being set up to answer questions regarding a new product. The following table summarizes the number of full- time equivalent employees (FTEs) that must be on duty in each time block. Shift Time FTEs 1 0-4 15 2 4-8 10 3 8-12 40 4 12-16 70 5 16-20 40 6 20-0 35 3 Macrosoft may hire both full-time and part-time employees. The former work 8-hour shifts and the latter work 4-hour shifts; their respective hourly wages are $15.20 and $12.95. Employees may start work only at the beginning of one of 6 shifts. At least two-thirds of the employees working at any one time must be full-time employees. Part-time employees can only answer 5 calls in the time a full- time employee can answer 6 calls. (i.e., a part-time employee is only 5/6 of a full-time employee.) Formulate an LP to determine how to staff the hotline at minimum cost. Constraints for Employee Scheduling 4 Decision Variables x t = # of full-time employees that begin work in shift t y t = # of part-time employees that work shift t min 121.6 ( x 1 + + x 6 ) + 51.8 ( y 1 + + y 6 ) s.t. 5 6 y 1 15 5 6 y 2 10 5 6 y 3 40 5 6 y 4 70 5 6 y 5 40 5 6 y 6 35 (8 15.20) (4 12.95) All shifts must be covered PT employee is 5/6 FT employee x 6 + x 1 + x 1 + x 2 + x 2 + x 3 + x 3 + x 4 + x 4 + x 5 + x 5 + x 6 + 5 ( x 6 + x 1 ) 2 3 2 3 . . . 2 3 x t 0 , y t 0 t =1,2, ,6 At least 2/3 workers must be full time More constraints: Nonnegativity ( x 6 + x 1 + y 1 ) ( x 1 + x 2 ) ( x 1 + x 2 + y 2 ) ( x 5 + x 6 + y 6 ) ( x 5 + x 6 ) 6 2. Energy Generation Problem (with piecewise linear objective) Austin Municipal Power and Light (AMPL) would like to determine optimal operating levels for their electric generators and associated distribution patterns that will satisfy customer demand. Consider the following prototype system Plants The two plants (generators) have the following (nonlinear) efficiencies: Plant 1 [ 0, 6 MW] [ 6MW, 10MW] Unit cost ($/MW) $10 $25 Plant 2...
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This note was uploaded on 01/26/2011 for the course IEOR 4004 taught by Professor Sethuraman during the Fall '10 term at Columbia.

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03_linear_prog2 - 8/14/04 J. Bard and J. W. Barnes...

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