06. IP (OR Models)

06. IP (OR Models) - Lecture 6 Integer Programming Models...

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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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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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An IP is a  mixed integer program  (MIP) if some but not all  of the 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, then the IP is a binary  IP ( BIP ) Decision variables that are not required to be integer- valued are  continuous variables Decision Variables in IP Models
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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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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 (FT) and part- timers (PT) 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 (this is a headcount constraint) Problem : Find minimum cost workforce
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x t  = # of full-time employees that begin the day at the start of  interval  t  and work for 8 hours y t  = # of part-time employees that are assigned interval  t   Min    z  =  121.6( x 1  +  . . .    + x 6 )  +  51.8( y 1 . . .    + y 6 ) Call Center Employee IP Model s.t. x 1  +  x 6 +   5 6   y 1   15 x 1  +  x 2 +   5 6   y 2   10 . . . x 5  +  x 6 +   5 6   y 6       35 x 1  +  x 6      2 3  ( x 1  +  x 6   y 1 ) . . . x 5  +  x 6    2 3  ( x 5  +  x 6   y 6 )   x t    0,    y t    0,    t  = 1,2,…,6  Decision variables :
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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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This note was uploaded on 12/19/2011 for the course M E 366l taught by Professor Staff during the Spring '08 term at University of Texas.

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06. IP (OR Models) - Lecture 6 Integer Programming Models...

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