LP - Linear Programming LP is a general method to solve...

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Subhash Suri UC Santa Barbara Linear Programming LP is a general method to solve optimization problem with linear objective function and linear constraints. Diet Problem Example: Feeding an army. 4 menu choices: Fish, Pizza, Hamburger, Burrito. Each choice has benefits (nutrients) and a cost (calories). A C D Calories F 200 90 100 600 P 75 80 250 800 H 275 80 510 550 B 500 95 200 450 RDA 2000 300 475 Determine least-caloric diet that satisfies RDA of nutrients.
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Subhash Suri UC Santa Barbara LP Formulation Introduce variables x F , x P , x H , x B for the amount (units) consumed of each food. Obj. Function is total calories. minimize 600 x F + 800 x P + 550 x H + 450 x B Each of the nutrient constraint can be expressed as a linear function: 200 x F + 75 x P + 275 x H + 500 x B 2000 90 x F + 80 x P + 80 x H + 95 x B 300 100 x F + 250 x P + 510 x H + 200 x B 475 Finally, all food quantities consumed should be non-negative: x F 0; x P 0; x H 0; x B 0 This is an example of Linear Prog.
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Subhash Suri UC Santa Barbara Manufacturing Example A plant has 3 types of machines, A, B, C . Choice of 4 products: 1, 2, 3, 4. Resource consumption (how much machine time a product uses) and profit per unit. A B C Profit 1 2 0.5 1.5 3.5 2 2 2 1 4.2 3 0.5 1 3 6.5 4 1.5 2 1.5 3.8 Avail. 20/wk 30/wk 15/wk Machine shop operates 60 hrs/week. Determine most profitable product mix.
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Subhash Suri UC Santa Barbara LP Formulation Introduce variables x 1 , x 2 , x 3 , x 4 for the units of different products manufactured. Obj. Function is total profit. maximize 3 . 5 x 1 + 4 . 2 x 2 + 6 . 5 x 3 + 3 . 8 x 4 Each of the machines provides a constraint, expressed as a linear function: 2 x 1 + 2 x 2 + 0 . 5 x 3 + 1 . 5 x 4 1200 0 . 5 x 1 + 2 x 2 + x 3 + 2 x 4 1800 1 . 5 x 1 + x 2 + 3 x 3 + 1 . 5 x 4 900 Finally, all product quantities should be non-negative: x 1 , x 2 , x 3 , x 4 0
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Subhash Suri UC Santa Barbara Transportation Example m supply sites, and n demand sites . Supply site i produces a i units; demand site j requires b j units. c ij cost of shipping one unit from i to j . Demand Nodes Supply Nodes b1 a1 a2 am bn b2 . . . . Determine optimal shipping cost.
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Subhash Suri UC Santa Barbara LP Formulation Let x ij be the amount shipped from i to j . Then, the LP is min m X i =1 n X j =1 c ij x ij subject to n X j =1 x ij = a i , i = 1 , 2 , . . . , m m X i =1 x ij = b j , j = 1 , 2 , . . . , n m X i =1 a i = n X j =1 b j x ij 0
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Subhash Suri UC Santa Barbara General Form of LP n unknowns x 1 , . . . , x n , and m constraints . max c 1 x 1 + c 2 x 2 + · · · + c n x n subject to a 11 x 1 + a 12 x 2 + · · · + a 1 n x n b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n b 2 . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n b m x 1 , x 2 , · · · , x n 0 A company making n products , each requiring some amount each of m resources. Item i has profit c i . b j is total resource of type j . a ij is the amount of resource j needed by item i .
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Subhash Suri UC Santa Barbara Matrix Form maximize cx subject to Ax b x 0 The input data are: – the m × n coefficient matrix A , – the m -vector of right hand sides b , and – the n -vector of objective costs c .
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