To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming A. A. Elimam

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, 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, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Applications  Disentangling Bankruptcies  Production Planning  Gasoline Blending  Diet Design  Product Distribution  Bus Scheduling  Resource Allocation

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. SAKARA Pipe Company

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Type 1 Type 2 How much of each type do I make?

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Two Plastic Pipes Two Plastic Pipes Type 1 Type 2 Type 1 Type 2 Profit, $/ 100 ft 34 40Profit, $/ 100 ft Extrusion, hrs/100 ft 4 6Extrusion, hrs/100 ft 4 6 Packaging, hrs/100 ft 2 2Packaging, hrs/100 ft 2 2 Additive Mix, lbs/100 ft 2 1Additive Mix, lbs/100 ft 2 1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, 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

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Step 2 - Define the objective function Max Z = Objective

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Step 2 - Define the objective function Max Z = x 1 + x 2 Decision variables

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Step 2 - Define the objective function Max Z = $34 x 1 + $40 x 2 Coefficients

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Extrusion Time, 48 hrs Packaging Time, 18 hrs Additive Mix, 16 lbs

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, 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

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, 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

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, 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=

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, 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

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Step 3 - Formulate the constraints Max Z = $34 x 1 + $40 x 2 4 x x 2  48 (extrusion) Coefficients

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Step 3 - Formulate the constraints Max Z = $34 x 1 + $40 x 2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) x 1, x 2  0 x 1, x 2  0

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Wershon Suit Company Jackets Jackets Slacks Available Profit,$/unit Profit,$/unit Material, Material, Square yards Square yards Person Hours Person Hours How many jackets and slacks should be produced ?

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Wershon Suit Company Type of Objective Function Type of Objective Function Maximize Profit Variable Definition Variable Definition JK = number of jackets produced / week SK = number of slacks produced / week

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Wershon Suit Company Max Z = $10 JK + $15 SK ST 2 JK + 5 SK  50 (Material) 4 JK + 2 SK  36 4 JK + 2 SK  36 (sewing)  0 JK, SK  0 (Nonnegativity)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Farmer Spanky Pigs Alfalfa Buckwheat Min. Units/lb units/lb Daily Units/lb units/lb Daily Cost, $/lb 7 9 Cost, $/lb 7 9 Vitamin, Vitamin, A A B B C C Find the minimum cost mix that satisfies the daily vitamin requirements.

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Farmer Spanky Pigs Type of Objective Function Type of Objective Function Minimize Cost Variable Definition Variable Definition Alfa Alfa = Pounds of Alfalfa used in mix Bukw Bukw = Pounds of Buckwheat used in mix

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Farmer Spanky Pigs Min Z = $ 7 Alfa + $ 9 Bukw ST 3 Alfa + 5 Bukw  15 (Vitamin A) 8 Alfa + 8 Bukw  32 (Vitamin B) 7 Alfa + 2 Bukw  14 (Vitamin C)  0 ( Alfa, Bukw  0 ( Nonnegativity)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Gasoline Blending Prem. Reg. CostStock gal. gal.$/gal. gal. gal. gal.$/gal. gal. Price, $/gal Price, $/gal Light Oil Light Oil Heavy Oil Heavy Oil At least 60% of all gasoline must be Regular Find the maximum profit gasoline mix.

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Gasoline Blending Type of Objective Function ? Type of Objective Function ? Maximize Net Revenue Variable Definition Variable Definition P = Number of premium gallons produced R = Number of regular gallons produced H = Number of heavy crude gallons used L = Number of light crude gallons used

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Gasoline Blending Max Z = 1.4 P R L H Max Z = 1.4 P R L HST H= 0.75 P R H= 0.75 P R L= 0.50 P R H  6000 L  4000 R  0.6 (R+P) R, P, H, L  0

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Gasoline Blending: Two Variables Max Z = 1.4 P R - Max Z = 1.4 P R (0.50 P R) 1.0 (0.75 P R) ST (0.75 P R)  6000 (0.75 P R)  6000 (0.50 P R)  4000 R  0.6 (R+P) R, P, H, L  0

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Gasoline Blending: Two Variables Max Z = 1.4 P R - Max Z = 1.4 P R (0.50 P R) 1.0 (0.75 P R) ST (0.75 P R)  6000 (0.75 P R)  6000 (0.50 P R)  4000 R (R+P)  0 R (R+P)  0 R, P  0 R, P  0

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Gasoline Blending: Two Variables Max Z = 1.4 P R - Max Z = 1.4 P R (0.50 P R) 1.0 (0.75 P R) ST (0.75 P R)  6000 (0.75 P R)  6000 (0.50 P R)  R - 0.6P  R - 0.6P  0 R, P  0 R, P  0

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x2x2 ||||||||| x1x1 4 x x 2  48 (extrusion)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) (12, 0) (0, 8)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 x1x1 x2x2 4 x x 2  48 (extrusion) (12, 0) (0, 8) ||||||||

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) (9, 0) (0, 9) 2 x x 2  18 (packaging)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) (9, 0) (0, 9) 2 x x 2  18 (packaging)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) (8, 0) (0, 16) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) (8, 0) (0, 16) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| x 1 + x 2  10 x2x2 x1x1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| x 1 + x 2  10 2 x x 2  18 x2x2 x1x1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| x 1 + x 2  10 2 x x 2  18 x 1  7 x2x2 x1x1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| x 1 + x 2  10 2 x x 2  18 x 1  7 x 2  5 x2x2 x1x1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| x x 2  5 2 x 1 + x 2  10 2 x x 2  18 x 1  7 x 2  5 x2x2 x1x1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| Feasible region - 6 x x 2  5 2 x 1 + x 2  10 2 x x 2  18 x 1  7 x 2  5 x2x2 x1x1

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Feasible region 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| Feasible region - 6 x x 2  5 2 x 1 + x 2  10 2 x x 2  18 x 1  7 x 2  5 x2x2 x1x1 Test point

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) Feasible region A B C D E

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8) 34 x x 2 = $272

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8) 34 x x 2 = $320

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8) 34 x x 2 = $342

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Graphical solution 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E (8,0) (0,6.8) 34 x x 2 = $342 Optimal solution (3,6)

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Sensitivity Analysis How sensitive the Optimal solution to changes in problem parameters Expect Approximate model data “ What if ? “ Questions Changes in profit / cost data Changes in RHS Changes in Constraint Coefficients

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. SA on Objective Function Coefficients Changes in objective function coefficients (OFC) Solution stays optimal if value of OFC within a given range If outside this range, need to find a new optimal solution

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. SA of RHS Values Use SA software output SHADOW PRICE of a constraint: Amount of change in optimum objective function value caused by adding one unit to the RHS.

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) (Slope = - 2 / 3 ) 2 x x 2  18 (packaging) (Slope = -1) 2 x 1 + x 2  16 (additive mix) A B C D E

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Optimal solution before rotation 20 x x 2 = 160

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E Optimal solution before rotation 20 x x 2 = 160 Optimal solution after rotation

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 2 x x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C E 4 x x 2  48 (extrusion) D

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (original packaging constraint) 2 x 1 + x 2  16 (additive mix) A B C D E 2 x x 2  19 (relaxed packaging constraint) Increase in feasible region C

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x 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, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x x 2  18 (packaging) 2 x 1 + x 2  16 (additive mix) A B C D E CC Packaging constraint for upper bound

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Sensitivity analysis 18 — 16 — 14 — 12 — 10 — 8 — 6 — 4 — 2 — 0 ||||||||| x1x1 x2x2 4 x x 2  48 (extrusion) 2 x 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

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Infeasible Solution 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| Y X Max 2 X + 3 Y ST 4 X + 3 Y  12 X + Y  8

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Unbounded Solution 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| Y X Max 2 X + 3 Y ST X - Y  1 -X + Y  1 Objective Function Line

To accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Fourth Edition  1996 Addison-Wesley Publishing Company, Inc. All rights reserved. Linear Programming Alternative Optima 12 — 11 — 10 — 9 — 8 — 7 — 6 — 5 — 4 — 3 — 2 — 1 — 0 |||||||||||| Y X Min 2 X + 3 Y ST X Y  3 X + Y  6 Objective Function Line