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
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.
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 :
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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