Class_4a_IntProgram

Class_4a_IntProgram - Integer Programming Models EMSE...

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EMSE 154-254: Applications of Linear and Nonlinear Optimization Theory Integer Programming Models
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EMSE 154 - 254: Applied Optimization Modeling Agenda What are integer programs? Modeling with integer variables Investment Problems If-then / Either-or logic. Set-covering Problems Fixed charge problems gp Using Mosel Personnel scheduling model 2
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EMSE 154 - 254: Applied Optimization Modeling What are integer programs? An integer program is a linear program that requires some decision variables to take integer values. Example: assignment problems seen earlier, ractional values have no useful meaning). (fractional values have no useful meaning). s illustrated in the staff scheduling example As illustrated in the staff scheduling example (see next slide), rounding a fractional solution may result in suboptimal solutions or infeasible olutions 3 solutions
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EMSE 154 - 254: Applied Optimization Modeling Motivation: Personnel Scheduling Revisited Model: mi nx M + x T + x W + x Th + x F + x Sa + x S subject to : x M + x Th + x F + x Sa + x S 17 (Monday) 3 uesday x M + x T + x F + x Sa + x S 13 (Tuesday) x M + x T + x W + x Sa + x S 15 (Wednesday) x M + x T + x W + x Th + x S 19 (Thursday) 4 riday x M + x T + x W + x Th + x F 14 (Friday) x T + x W + x Th + x F + x Sa 16 (Saturday) + x W + x Th + x F + x Sa + x S 11 (Sunday) h a x i 0, i = {M,T,W,Th,F,Sa,S}. x M = 6, x T = 5.33 3 , x W = 0, x Th = 7.33 3 x F = 0, x Sa = 3.33 3 x S = 0.33 3 Optimal solution: 4 Rounding results in an infeasible solution! It is necessary to compute the integer solutions .
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EMSE 154 - 254: Applied Optimization Modeling More Nomenclature IP : A pure integer program is one where all decisions variables are required to take integer values. MIP : A mixed integer program is one where some, ut not all decision variables are required to take but not all, decision variables are required to take integer values. IP A inary teger program or 0 integer program BIP : A binary integer program or 0-1 integer program requires decision variables to take values in the set {0,1}. The LP relaxation of an IP is the LP that results from removing integer or 0-1 constraints. 5
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EMSE 154 - 254: Applied Optimization Modeling A pure integer program (IP): Examples max z = 3 x 1 + 2 x 2 s . t . x 1 + x 2 6, x 1 , x 2 0, x 1 , x 2 integer max z = 3 x 1 + 2 x 2 A mixed integer program (MIP): s . t x 1 + x 2 x 1 , x 2 0, x 1 integer 0- Integer program (BIP): max z = x 1 x 2 s . t x 1 + 2 x 2 2, A 0 1 Integer program (BIP): 6 2 x 1 x 2 1, x 1 , x 2 = 0 or 1
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EMSE 154 - 254: Applied Optimization Modeling Geometrical Interpretation of an IP A pure integer program (IP): 12 max 21 11 . . 7 4 13 , 0, , integer zxx st x x xx =+ +≤ x 2 2.5 3 3.5 = point in feasible region (IP) max 21 11 741 3 t x x Corresponding LP relaxation: 1 1.5 2 ,0 x 1 .5 1.0 1.5 2.0 2.5 3.0 .5 7 Note: The optimal z-value for the LP relaxation is an upper bound for the optimal z-value of the original IP. (since LP region contains all integer feasible solutions, but also contains the non-integer feasible solutions)
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EMSE 154 - 254: Applied Optimization Modeling Geometrical Interpretation of an IP
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This note was uploaded on 01/02/2010 for the course EMSE 254 taught by Professor Hernanabeledo during the Fall '08 term at GWU.

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Class_4a_IntProgram - Integer Programming Models EMSE...

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