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Simplex Method Initial Set Up. Introduction to Operations Research. Wyndor Glass Co. Example. Corner Point solutions represent all points of intersection Maximize Z = 3x 1 + 5x 2 Subject to: x 1 ≤ 4 2x 2 ≤ 12 3x 1 + 2x 2 ≤ 18 x 1 ≥ 0, x 2 ≥ 0. Wyndor Glass Co.
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Simplex Method Initial Set Up Introduction to Operations Research
Corner Point solutions represent all points of intersection Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0 Wyndor Glass Co.
Wyndor Glass Co. Corner Point Feasible solutions represent all points of intersection in the feasible region Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0
Choose (0,0) as initial CPF solution Check adjacent CPF solutions - (0,0) NOT optimal Consider two edges from (0,0). Move in direction with largest improvement of Z (x2) Solve for the intersection of the new set of boundaries Choose (0,6) as new CPF solution Check adjacent CPF solutions - (0,6) NOT optimal Consider two edges from (0,6). Move in direction with largest improvement of Z Solve for the intersection of the new set of boundaries Choose (2,6) as new CPF solution Check adjacent CPF solutions - (2,6) IS OPTIMAL Wyndor Glass Co.
The Simplex method: • Focuses solely on CPF solutions • Is an iterative algorithm (repetition of fixed series of steps) • When possible, chooses the origin as the initial CPF solution • Chooses adjacent CPF solutions to move to • Computes rate of improvement of Z along an edge • Selects an optimal CPF solution if no edges give a positive rate of improvement of Z Key Solution Concepts
Original Form Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 ≥ 0, x2 ≥ 0 Setting up the Simplex Method Augmented Form Maximize Z = 3x1 + 5x2 Subject to: • Z - 3x1 - 5x2 = 0 • x1 + x3= 4 • 2x2 + x4 = 12 • 3x1 + 2x2 + x5 = 18 • x1 , x2 , x3 , x4 , x5, ≥ 0
Slack Variables – Added to convert functional inequality constraints to equivalent equality constraints • Augmented Solution – Solution for the original variables that is augmented by the slack variables • (3,2) → (3,2,1,8,5) x3 = 1, x4 = 8, x5 = 5 • Basic Solution – Augmented corner-point solution • Basic Feasible (BF) Solution – Augmented CPF solution • Degrees of Freedom – Number of variables – number of equations • 5 – 3 = 2 • Number of variables that are set equal to zero • Nonbasic Variables – Variables that are set equal to zero • Basic Variables – Variables that are not set equal to zero Simplex Method Definitions
Each variable is either a basic or nonbasic variable • Number of basic variables equals the number of functional constraints • Nonbasic variables are set equal to zero • Values of the basic variables are obtained as the simultaneous solution of the system of equations • If the basic variables satisfy the non-negativity constraints, the basic solution is a BF solution. Example • Choose x1 and x4 as nonbasic variables and set equal to 0 • Solve system of equations for values of basic variables: x3 = 4, x2 = 6, and x5 = 6 • Since all three basic variables are non-negative, then this basic solution is a BF solution. Basic Solution Properties
Two BF solutions are adjacent if all but one of their nonbasic variables are the same Example • Consider (0,0) and (0,6) • Augmented solutions are (0,0,4,12,18) and (0,6,4,0,6) • Nonbasic variables are (x1,x2) and (x1,x4) • x1 is same, therefore they are adjacent corner points Adjacent BF Solutions
