To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming.

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Basic Concepts  Objective function  Decision variables  Constraints  Feasible region  Parameters  Linearity  Nonnegativity

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. x 1 =amount of type 1 pipe produced and sold next week, 100-foot increments x 2 =amount of type 2 pipe produced and sold next week, 100-foot increments Linear Programming Step 1–Define the decision variables Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Max Z = Objective Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Max Z = Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Max Z = x 1 + x 2 Decision variables Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Max Z = x 1 + x 2 Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Max Z = $34 x 1 + $40 x 2 Coefficients Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 2—Define the objective function Max Z = $34 x 1 + $40 x 2 Example D.1

Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 48 (extrusion) 48 (extrusion) Right-hand side value Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 48 (extrusion) 48 (extrusion) RHS value Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 48 (extrusion) 48 (extrusion) Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2  48 (extrusion)  48 (extrusion) Type of limit Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2  48 (extrusion)  48 (extrusion) Type of limit Upper limit  Lower limit  Equality= Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2  48 (extrusion)  48 (extrusion) Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 x 1 x 2  48 (extrusion) x 1 x 2  48 (extrusion) Decision variables Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 x 1 x 2  48 (extrusion) x 1 x 2  48 (extrusion) Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 4 x 1 + 6 x 2  48 (extrusion) Coefficients Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 4 x 1 + 6 x 2  48 (extrusion) Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Step 3—Formulate the constraints Max Z = $34 x 1 + $40 x 2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) Example D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) Figure D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) (12, 0) (0, 8) Figure D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) (12, 0) (0, 8) ||||||||| 24681012141618 Figure D.1

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) (9, 0) (0, 9) 2 x 1 + 2 x 2  18 (packaging) Example D.2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) (8, 0) (0, 16) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) Example D.2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) Figure D.2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 x2x2 x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 2 x 1 + x 2  10 x2x2 x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 2 x 1 + x 2  10 2 x 1 + 3 x 2  18 x2x2 x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 2 x 1 + x 2  10 2 x 1 + 3 x 2  18 x1  7x1  7x1  7x1  7 x2x2 x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 2 x 1 + x 2  10 2 x 1 + 3 x 2  18 x1  7x1  7x1  7x1  7 x2  5x2  5x2  5x2  5 x2x2 x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 - 6 x 1 + 5 x 2  5 2 x 1 + x 2  10 2 x 1 + 3 x 2  18 x1  7x1  7x1  7x1  7 x2  5x2  5x2  5x2  5 x2x2 x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 Feasible region – 6 x 1 + 5 x 2  5 2 x 1 + x 2  10 2 x 1 + 3 x 2  18 x1  7x1  7x1  7x1  7 x2  5x2  5x2  5x2  5 x2x2 x1x1x1x1 Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| 123456789101112 Feasible region – 6 x 1 + 5 x 2  5 2 x 1 + x 2  10 2 x 1 + 3 x 2  18 x1  7x1  7x1  7x1  7 x2  5x2  5x2  5x2  5 x2x2 x1x1 Test point Figure D.3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) Feasible region A B C D E Figure D.4

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8) 34 x 1 + 40 x 2 = $272 Figure D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8)

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8) Optimal solution (3,6) Figure D.6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Optimal corner point 4 x 1 +6 x 2 =48 2 x 1 +2 x 2 =18

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Optimal corner point 4 x 1 +6 x 2 =48 –(4 x 1 +4 x 2 =36)

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Optimal corner point 4 x 1 +6 x 2 =48 –(4 x 1 +4 x 2 =36) 2 x 2 =12

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Optimal corner point 4 x 1 +6 x 2 =48 –(4 x 1 +4 x 2 =36) 2 x 2 =12 x 2 =6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Optimal corner point 4 x 1 +6(6)=48 –(4 x 1 +4 x 2 =36) 2 x 2 =12 x 2 =6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Graphical solution ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Optimal corner point 4 x 1 +6(6)=48 4 x 1 =12 x 1 =3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Slack and Surplus Variables ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Slack variables

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Slack and Surplus Variables ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Slack variables 2 x 1 + x 2 =16

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Slack and Surplus Variables ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Slack variables 2 x 1 + x 2 =16 2 x 1 + x 2 + s 1 =16

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Slack and Surplus Variables ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Slack variables 2 x 1 + x 2 =16 2(3)+6+ s 1 =16

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Slack and Surplus Variables ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) D E (8,0) 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2A B C (0,6.8) Optimal solution (3,6) Example D.4 Slack variables 2 x 1 + x 2 =16 s 1 =4

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z 40 x 2 =– 34 x 1 + Z

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z 40 x 2 =– 34 x 1 + Z x 2 = –+ 34 x 1 Z40

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z 40 x 2 =– 34 x 1 + Z x 2 = –+ c1x1Zc2c2c1x1Zc2c2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z 40 x 2 =– 34 x 1 + Z x 2 = –+ c1x1Zc2c2c1x1Zc2c2 If c 1 increases

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z 40 x 2 =– 34 x 1 + Z x 2 = –+ c1x1Zc2c2c1x1Zc2c2 If c 1 decreases

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Objective function coefficients 34 x 1 +40 x 2 = Z 40 x 2 =– 34 x 1 + Z x 2 = –+ c1x1Zc2c2c1x1Zc2c2 If c 1 decreases New Optimal Point

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Range of optimality (Slope = – 1) (Slope = – 2/3) Figure D.7

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  342 c 2 3 Example D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  – – 1  – 342 c 2 3 34 c 2 Example D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  – c 2  34 342 c 2 3 Example D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  – –  – 342 c 2 3 342 c 2 3 Example D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E 2c232c23 Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  – 34  342 c 2 3 Example D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  – 51  c 2 342 c 2 3 Example D.5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Example D.5 Range of optimality (Slope = – 1) (Slope = – 2/3) – 1  –  – 34  c 2  51 342 c 2 3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E x2x2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Figure D.8 (Slope = – 1) (Slope = – 2/3) Optimal solution after rotation Optimal solution before rotation x2x2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (Slope = – 1) (Slope = – 2/3) Optimal solution after rotation Optimal solution before rotation Coefficient sensitivity Example D.6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (Slope = –1) (Slope = – 2/3) Optimal solution after rotation Optimal solution before rotation Coefficient Sensitivity Example D.6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Example D.6 (Slope = – 1) (Slope = – 2/3) Optimal solution after rotation Optimal solution before rotation Coefficient Sensitivity

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C E 4 x 1 + 6 x 2  48 (extrusion) D

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9 Optimal corner point

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9 Optimal corner point 4 x 1 +6 x 2 =48 2 x 1 +2 x 2 =19

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9 Optimal corner point 4 x 1 +6 x 2 =48 2 x 1 +2 x 2 =19 x 1 =4.5 x 2 =5

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9 Optimal corner point 4 x 1 +6 x 2 =48 2 x 1 +2 x 2 =19 x 1 =4.5 x 2 =5 Z = $34(4.5) + $40(5) = $353

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9 Optimal corner point 4 x 1 +6 x 2 =48 2 x 1 +2 x 2 =19 x 1 =4.5 x 2 =5 Z = $34(4.5) + $40(5) = $353 $353 - $342 = $11

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x 1 + 2 x 2  19 (relaxed packaging constraint) Increase in feasible region C Figure D.9 Optimal corner point 4 x 1 +6 x 2 =48 2 x 1 +2 x 2 =19 x 1 =4.5 x 2 =5 Z = $34(4.5) + $40(5) = $353 $353 - $342 = $11 Shadow price for packaging constraint

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C E D

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 x2x2 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E C˝ Packaging constraint for upper bound Packaging constraint for lower bound Figure D.10

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E CC Packaging constraint for upper bound Packaging constraint for lower bound Lower limit of shadow price Example D.7

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E CC Packaging constraint for upper bound Packaging constraint for lower bound Lower limit of shadow price B is the lowest feasible limit Example D.7

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E CC Packaging constraint for upper bound Packaging constraint for lower bound Lower limit of shadow price B is the lowest feasible limit For packaging, B, x 1 = 0 and x 2 = 8 2(0) + 2(8) = 16 hours Example D.7

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| 24681012141618 x1x1 4 x 1 + 6 x 2  48 (extrusion) 2 x 1 + 2 x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E CC Packaging constraint for upper bound Packaging constraint for lower bound Example D.7 Lower limit on $11 shadow price for packaging Lower limit of shadow price B is the lowest feasible limit For packaging, B, x 1 = 0 and x 2 = 8 2(0) + 2(8) = 16 hours

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Example D.8 Should Stratton increase the capacity of extrusion or packaging or buy more additive? Cost of Areaincrease Extrusion$8 Packaging$6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Example D.8 Should Stratton increase the capacity of extrusion or packaging or buy more additive? Cost of Areaincrease Extrusion$8 Packaging$6 No, too expensive. $8 > $3

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Example D.8 Should Stratton increase the capacity of extrusion or packaging or buy more additive? Cost of Areaincrease Extrusion$8 Packaging$6

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Example D.8 Should Stratton increase the capacity of extrusion or packaging or buy more additive? Cost of Areaincrease Extrusion$8 Packaging$6 Yes, increased revenue. $6 < $11

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming Example D.8 Should Stratton increase the capacity of extrusion or packaging or buy more additive? More additive will not help, already more than required.

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Applications Linear Programming Aggregate planning Production, Staffing, Blends Distribution Shipping Inventory Stock control, Supplier selection Location Plants or warehouses Process management Stock cutting Scheduling Shifts, Vehicles, Routing

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Solved Problem 1 Figure D.13 15 15 — 10 10 — 5 5 — 0 0 — ||| A51015 x 1 + x 2 ≤ 12 (gate) x 1 ≤ 9 (market) B C D 15 x 1 + 10 x 2 ≤ 150 (labor) 2,500 x 1 +2,000 x 2 = $20,000 (iso-profit line) E x1x1x1x1 x2x2x2x2

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Solved Problem 1 Figure D.14 ||| A51015 x 1 + x 2 ≤ 12 (gate) x 1 ≤ 9 (market) B C D´ 15 x 1 + 10 x 2 ≤ 150 (labor) E´ x1x1 x2x2 15 — 10 — 5 — 0 —

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Solved Problem 1 Figure D.15 ||| A51015 x 1 + x 2 ≤ 12 (gate) x 1 ≤ 9 (market) B F 15 x 1 + 10 x 2 ≤ 165 (labor) 2,500 x 1 +2,000 x 2 = $20,000 (iso-profit line) E x1x1 x2x2 15 — 10 — 5 — 0 —

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Solved Problem 2 Figure D.16 750 — 500 — 250 — 0 — ||| 51015 0.02 x 1 + 0.04 x 2 ≥ 20 (minimum order) B x 1 + x 2 ≥ 750 (minimum number) $0.30 x 1 + $0.20 x 2 = $150.00 (isocost line) A x1x1 x2x2 C Feasible region has no upper bound.

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming.

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To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming.

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Presentation on theme: "To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Seventh Edition © 2004 Prentice Hall, Inc. All rights reserved. Linear Programming."— Presentation transcript:

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