To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Supplement S9 Linear Programming To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Model consisting of linear relationships representing a firms objectives & resource constraints Model consisting of linear relationships representing a firms objectives & resource constraints Decision variables are mathematical symbols representing levels of activity of an operation Decision variables are mathematical symbols representing levels of activity of an operation Objective function is a linear relationship reflecting objective 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 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. 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. 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. 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. 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. 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. 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. Computing Optimal Values Z = $50(24) + $50(8) Z = $1,360 x 1 +2x 2 =40 4x 1 +3x 2 =120 4x 1 +8x 2 =160 -4x 1 -3x 2 =-120 5x 2 =40 x 2 =8 x 1 +2(8)=40 x 1 =24 A. 8 B C x 1 + 2x 2 = 40 4x 1 + 3x 2 = 120 | x1x1x1x1 x2x2x2x 0 0 Example S9.2 To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. 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. 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. Farmers Hardware CHEMICAL CONTRIBUTION BrandNitrogen (lb/bag)Phosphate (lb/bag) Super-gro24 Crop-quik43 Minimize Z = $6x 1 + $3x 2 subject to 2x 1 +4x 2 16 lb of nitrogen 4x 1 +3x 2 24 lb of phosphate x 1, x 2 0 Example S9.3 To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Farmers Hardware 8 8 6 6 4 4 2 2 0 0 |22|222 |44|444 |66|666 |88|888 | x1x1x1x1 x2x2x2x2 A B C Z = 6x 1 + 3x 2 x 1 = 0 bags of Super-gro x 2 = 8 bags of Crop-quik Z = $24 Example S9.3 To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. The Simplex Method A mathematical procedure for solving linear programming problems according to a set of steps A mathematical procedure for solving linear programming problems according to a set of steps Based on solving simultaneous equations and matrix algebra Based on solving simultaneous equations and matrix algebra Computers use the simplex method to solve linear programming problems 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. Converting Model Constraints Slack variables added to constraints to represent unused resources Slack variables added to constraints to represent unused resources x 1 + 2x 2 + s 1 = 40 hours of labor x 1 + 2x 2 + s 1 = 40 hours of labor 4x 1 + 3x 2 + s 2 = 120 lb of clay 4x 1 + 3x 2 + s 2 = 120 lb of clay Slack variables have a 0 coefficient in the objective function Slack variables have a 0 coefficient in the objective function Z = $40x 1 + $50x 2 + 0s 1 + 0s 2 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. 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. Converting Model Constraints Surplus variables subtracted from constraints to represent excess above resource requirement 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. Solving LP Problems Exhibit S9.1 Exhibit S9.2 To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. 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. Solving LP Problems Exhibit S9.4 After all parameters and constraints have been input, click on Solve. Objective function Decision variables C6*B10+D6*B1140 C7*B10+D7*B11120 Click on Add to insert contraints. To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Solving LP Problems Exhibit S9.5