06_integer_prog

06_integer_prog - Lecture 6 Integer Programming Models...

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8/14/04 J. Bard and J. W. Barnes Operations Research Models and Methods Copyright 2004 - All rights reserved Lecture 6 – Integer Programming Models Topics General model Logic constraint Defining decision variables Continuous vs. integral solution Applications: staff scheduling, fixed charge, TSP Piecewise linear approximations to nonlinear functions
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2 Linear Integer Programming - IP Maximize/Minimize   z  =  c 1 x 1  +  c 2 x 2  +       +  c n x n   {   =   } b i ,    i  = 1,…, m s.t. a i 1 x 1  +  a i 2 x 2  +       +  a in x n   x j     u j ,      j  = 1,…, n x j  integer for some or all  j  =1,…, n
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3 An IP is a  mixed integer program  (MIP)  if some but not all decision variables are integer. If all decision variables are integer we have a  pure  IP. A binary decision variable must be 0 or 1  (a yes-no decision variable). If all decision variables are binary the IP is a binary IP ( BIP ) Decision variables not integer-constrained are continuous decision variables Decision Variables in IP Models
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4 Why study IP? (1) LP divisibility assumption (fractional  solutions are permissible) is  not  always  valid. (2) Binary variables allow powerful new  techniques like  logical  constraints.  
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5 Call Center Employee Scheduling Day is divided into 6 periods, 4 hours each Demand/period = (15, 10, 40, 70, 40, 35) Workforce consists of full-timers and part-timers FT = 8-hour shift, $121.6/ shift PT = 4-hr shift, $51.8/shift One PT = 5/6 FT In any period, at least 2/3 of the staff must be FT employees Problem : Find minimum cost workforce
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6 Decision variables : x t  = # of full-time employees that work shift  t y t  = # of part-time employees that work shift  t   Min    z  =  121.6( x 1  +     + x 6 )  +  51.8( y 1    + y 6 ) Call Center Employee IP Model s.t. x 6  +  x 1 +   5 6   y 1   15 x 1  +  x 2 +   5 6   y 2   10 . . . x 5  +  x 6 +   5 6   y 6       35 x 6  +  x 1    2 3  ( x 6  +  x 1   y 1 ) . . . x 5  +  x 6    2 3  ( x 5  +  x 6   y 6 )   x t    0,    y t    0,     t  = 1,2, …,6 
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Optimal LP solution x  = [ 7.06,  0,  40,  12.94,  27.06,  7.94 ] y   = [ 0,  3.53,   0,  20.47,   0,  0 ] z  = 12,795.2    Not feasible to IP model A correction method: round continuous solution  x  = [  8,  0,  40,  13,  27,  8 ] y  = [  0,  3,  0,  21,  0,  0 ] z  = 12,916.8 Feasible – Yes, Optimal? We do not know!
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06_integer_prog - Lecture 6 Integer Programming Models...

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