02. LP1 (OR Models) - LinearProgramming Topics...

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Linear Programming Topics • General optimization model • LP model and assumptions • Manufacturing example • Characteristics of solutions  • Sensitivity analysis • Excel add-ins
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Most of the deterministic OR models can be formulated as mathematical  programs. "Program" in this context, has to do with a “plan” and not a computer  program.  Mathematical Program Maximize / Minimize z  =  f   ( x 1 , x 2 ,…, x n Subject to {   =   } b ,   i  =1,…, m x j  ≥ 0,    j  = 1,…, n   Deterministic OR Models g i ( x 1 , x 2 ,…, x n )
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    x j  are called  decision variables .  These are things that you  control   { =   }   b i are called  structural   (or functional or technological) constraints  x j  ≥ 0 are  nonnegativity  constraints  Model Components ( x 1 , x 2 ,…, x n ) is the  objective function g i ( x 1 , x 2 ,…, x n )
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(   x 1 . . n x A feasible solution    x  =   satisfies all the constraints (both structural and nonnegativity)  The objective function ranks the feasible solutions; call them  x 1 x 2 , . . . ,  x . The optimal solution is the best among these. For a  minimization objective, we have  z * = min{   f   ( x 1 ),  f   ( x 2 ), . . . ,  f   ( x )   }. . )   Feasibility and Optimality
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Linear Programming A linear program is a special case of a mathematical program where  f ( x ) and  g 1 ( x )   ,…,  g m ( x )   are  linear  functions Linear Program : Maximize/Minimize    z  =  c 1 x 1  +  c 2 x 2  +  • • •  +  c n x n   Subject to   a i 1 x 1  +  a i 2 x 2  +  • • •  +  a in x n   { = }   b i  ,     i  = 1,…, m x j     u j ,    j  = 1,…, n x j    0,    j  = 1,…, n  
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   x j     u j  are called simple bound constraints     x  = decision vector = "activity levels" a ij  c ,   b u j   are all known data  goal is to find  x  =  ( x 1 , x 2 ,…, x n ) (the symbol “   T   ” means) LP Model Components
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Linear Programming Assumptions (i) proportionality (ii) additivity   linearity (iii) divisibility (iv) certainty
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j ’s contribution to objective function is  c j x j   and usage in constraint  i  is  a ij x j   both are  proportional  to the level of activity  j (volume discounts, set-up charges, and nonlinear efficiencies are potential sources of violation) (ii) 1 2 no “cross terms” such as   
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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 at Austin.

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02. LP1 (OR Models) - LinearProgramming Topics...

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