Lecture 5

Asthisexampleindicatesaslackvariablecanbeusedasabasicv

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Unformatted text preview: nishing and carpentry. The amount of each resource needed to make each type of furniture is given in the table below. At present, 48 board feet of lumber, 20 finishing hours, 8 carpentry hours are available. A desk sells for $60, a table for $30, and a chair for $ 20. Dakota believes that demand for desks and chairs is unlimited, but at most 5 tables can be sold. Since the available resources have already been purchased, Dakota wants to maximize total revenue. 4.3 – The Simplex Algorithm (max LPs) Define: x1 = number of desks produced x2 = number of tables produced x3 = number of chairs produced. The LP is: max z = 60x1 + 30x2 + 20x3 s.t. 8x1 + 6x2 + x3 ≤ 48 (lumber constraint) 4x1 + 2x2 + 1.5x3 ≤ 20 (finishing constraint) 2x1 + 1.5x2 + 0.5x3 ≤ 8 (carpentry constraint) x2 ≤ 5 (table demand constraint) x1, x2, x3 ≥ 0 4.3 – The Simplex Algorithm (max LPs) Step 1 ‐ Convert the LP to Standard Form Basic Variable Canonical Form 0 Row 0 z – 60x1 – 30x2 – 20x3 Row 1 8x1 + 6x2 + Row 2 4x1 + 2x2 + 1.5x3 Row 3 2x1 + 1.5x2 + 0.5x3 Row 4 x2 =0 = 48 + s2 + s3 + s4 s1 = 48 = 20 x3 + s1 z=0 s2 = 20 =8 s3 = 6 =5 s4 = 5 If we set x1 = x2 = x3 = 0, we can solve for the values s1, s2, s3, s4. Thus, BV = {s1, s2, s3, s4} and NBV = {x1, x2, x3 }. Since each constraint is then in canonical form (BVs have a coefficient = 1 in one row and zeros in all other rows) with a nonnegaWve rhs, a bfs can be obtained by inspecWon. 4.3 – The Simplex Algorithm (max LPs) Step 2 – Obtain a Basic Feasible SoluWon To perform the simplex algorithm, we need a basic (although not necessarily nonnegaW...
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This note was uploaded on 09/20/2011 for the course ENG 300 taught by Professor Albin during the Fall '11 term at Rutgers.

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