To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc1  Model consisting of linear relationships representing a firm’s objectives & resource constraints  Decision variables are mathematical symbols representing levels of activity of an operation  Objective function is a linear relationship reflecting objective of an operation  Constraint is a linear relationship representing a restriction on decision making Linear Programming

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc2 General Structure of a Linear Programming (LP) Model Max/min z = c 1 x 1 + c 2 x c n x n subject to:a 11 x 1 + a 12 x a 1n x n  b 1 (or , =) a 21 x 1 + a 22 x a 2n x n  b 2 : a m1 x 1 + a m2 x a mn x n  b m a m1 x 1 + a m2 x a mn x n  b m x j = decision variables b i = constraint levels c j = objective function coefficients a ij = constraint coefficients

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc3 Linear Programming Model Formulation LaborClayRevenue PRODUCT(hr/unit)(lb/unit)($/unit) Bowl1440 Mug2350 There are 40 hours of labor and 120 pounds of clay available each day Decision variables x 1 = number of bowls to produce x 2 = number of mugs to produce RESOURCE REQUIREMENTS Example S9.1

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc4 Objective Function and Constraints Maximize Z = $40 x x 2 Subject to x 1 +2x 2  40 hr(labor constraint) 4x 1 +3x 2  120 lb(clay constraint) x 1, x 2  0 Solution is x 1 = 24 bowls x 2 = 8 mugs Revenue = $1,360 Example S9.1

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc5 Graphical Solution Method 1.Plot model constraint on a set of coordinates in a plane 2.Identify the feasible solution space on the graph where all constraints are satisfied simultaneously 3.Plot objective function to find the point on boundary of this space that maximizes (or minimizes) value of objective function

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc6 Graph of Pottery Problem 4 x x 2  120 lb x x 2  40 hr Area common to both constraints – – – – – 0 0 – | x1x1x1x1 x2x2x2x2 Example S9.2

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc7 Plot Objective Function – – – – 0 0 – $800 = 40x x 2 Optimal point B | x1x1x1x1 x2x2x2x2 Example S9.2

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc8 Extreme Corner Points x 1 = 224 bowls x 2 =  8 mugs Z = $1,360 x 1 = 30 bowls x 2 =  0 mugs Z = $1,200 x 1 = 0 bowls x 2 =  20 mugs Z = $1,000 A B C | x1x1x1x1 x2x2x2x – – – – 0 0 – Example S9.2

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc9 Objective Function Determines Optimal Solution 4x 1 + 3x 2  120 lb x 1 + 2x 2  40 hr – – – – 0 0 –B | x1x1x1x1 x2x2x2x2 C A Z = 70x x 2 Optimal point: x 1 = 30 bowls x 2 =  0 mugs Z = $2,100 Example S9.2

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc10 The Simplex Method  A mathematical procedure for solving linear programming problems according to a set of steps  Based on solving simultaneous equations and matrix algebra  Computers use the simplex method to solve linear programming problems

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc11 Converting Model Constraints  Slack variables added to  constraints to represent unused resources  x 1 + 2x 2 + s 1 =  40 hours of labor  4x 1 + 3x 2 + s 2 =  120 lb of clay  Slack variables have a 0 coefficient in the objective function  Z = $40x 1 + $50x 2 + 0s 1 + 0s 2

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc12 Solutions at Extreme Points – – – – 0 0 –B | x1x1x1x1 x2x2x2x2 C A x 1 = 0 x 2 =  20 s 1 = 0 s 2 =  60 x 1 = 24 x 2 =  8 s 1 = 0 s 2 =  0 x 1 = 30 x 2 =  0 s 1 = 10 s 2 =  0 4x 1 + 3x 2 + s 2 =  120 Figure S9.1 x 1 + 2x 2 + s 1 = 40

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc13 Converting Model Constraints  Surplus variables subtracted from  constraints to represent excess above resource requirement 2x 1 +4x 2  16 2x 1 +4x 2 -s 1 =  16 But s 1 is negative 2(0) + 4(0) - s 1 =  16 s 1 = -16

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc14 Solving LP Problems Exhibit S9.3 Click on “Tools” to invoke “Solver.” Objective function Decision variables – bowls (x 1 )=B10; mugs (x 2 )=B11

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc15 Solving LP Problems Exhibit S9.4 After all parameters and constraints have been input, click on “Solve.” Objective function Decision variables C6*B10+D6*B11≤40 C7*B10+D7*B11≤120 Click on “Add” to insert contraints.

To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved by Prentice-Hall, Inc16 Solving LP Problems Exhibit S9.5