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03_linear_prog2 - Topics Feedmixproblem Cuttingstockproblem...

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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
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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
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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 $12.95. Employees may  start work only at the beginning of one of 6 shifts.  • At least two-thirds of the employees working at any one time  must be full-time employees. • Part-time employees can only answer 5 calls in the time a full- time employee can answer 6 calls.  (i.e., a part-time employee is  only 5/6 of a full-time employee.)  Formulate an LP to determine how to staff the  hotline at minimum cost.   Constraints for Employee Scheduling
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4 Decision Variables x t  = # of full-time employees that begin work in shift  t y t  = # of part-time employees that work shift  t min    121.6 ( x 1  +  • • •  +  x 6 )  + 51.8 ( y 1  +  • • •  + y 6 ) s.t. 5 6   y 1   15 5 6   y 2   10 5 6   y 3   40 5 6   y 4   70 5 6   y 5   40 5 6   y 6   35 (8  ×  15.20) (4  ×  12.95) All shifts must be covered PT employee is 5/6 FT employee x 6     +     x 1     +   x 1     +     x 2     +   x 2     +     x 3     +   x 3     +     x 4     +   x 4     +     x 5     +   x 5     +     x 6     +  
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5 ( x 6  +  x 1 ) 2 3   2 3   . . . 2 3 x t   ≥ 0, y t   ≥ 0 t  =1,2, …,6 At least 2/3 workers must be full time More constraints: Nonnegativity ( x 6  +  x 1  + y 1 ) ( x 1  +  x 2 ) ( x 1  +  x 2  + y 2 ) ( x 5  +  x 6  + y 6 ) ( x 5  +  x 6 )
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6 2.  Energy Generation Problem (with piecewise linear objective)   Austin Municipal Power and Light (AMPL) would like to determine optimal operating levels for  their electric generators and associated distribution patterns that will satisfy customer demand.   Consider the following prototype system Plants The two plants (generators) have the following (nonlinear) efficiencies: Plant 1 [ 0, 6 MW] [ 6MW, 10MW] Unit cost ($/MW) $10 $25 Plant 2 [ 0, 5 MW] [5MW, 11MW] Unit cost ($/MW) $8 $28 The table is to be read as follows.  For plant #1 if you generate at a rate of 8MW 
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