1. Operations Research

2.  Operations Research (OR) aims to having the optimization solution for some administrative problems, such as transportation, decision-making, inventory Copyright 2006 John Wiley & Sons, Inc.Supplement 13-2

3. Operations Research Models  Linear Programming  Markov Chains  Network Optimization  Decision Analysis  Transportation  Inventory Copyright 2006 John Wiley & Sons, Inc.Supplement 13-3

4. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-4 A model consisting of linear relationships representing a firm’s objective and resource constraints Linear Programming (LP) LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

5. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-5 LP Model Formulation (cont.) Max/min z = c 1 x 1 + c 2 x 2 +... + c n x n subject to: a 11 x 1 + a 12 x 2 +... + a 1n x n (≤, =, ≥) b 1 a 21 x 1 + a 22 x 2 +... + a 2n x n (≤, =, ≥) b 2 : a m1 x1 + a m2 x 2 +... + a mn x n (≤, =, ≥) b m a m1 x1 + a m2 x 2 +... + a mn x n (≤, =, ≥) b m x j = decision variables b i = constraint levels c j = objective function coefficients a ij = constraint coefficients

6. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-6 LP Formulation: Example Maximize Z = 40 x 1 + 50 x 2 Subject to x 1 +2x 2  40 4x 1 +3x 2  120 x 1, x 2  0

7. x 1 +2x 2  40 400 x1 020x2 Copyright 2006 John Wiley & Sons, Inc.Supplement 13-7 4x 1 +3x 2  120 300 x1 040x2

8. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-8 Graphical Solution: Example x 1 + 2 x 2  40 50 50 – 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – |10 60 50 20 30 40 x1x1x1x1 x2x2x2x2

9. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-9 Graphical Solution: Example 4 x 1 + 3 x 2  120 x 1 + 2 x 2  40 50 50 – 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – |10 60 50 20 30 40 x1x1x1x1 x2x2x2x2

10. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-10 Graphical Solution: Example 4 x 1 + 3 x 2  120 x 1 + 2 x 2  40 Area common to both constraints 50 50 – 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – |10 60 50 20 30 40 x1x1x1x1 x2x2x2x2

11. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-11 Computing Optimal Values 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 4 x 1 + 3 x 2  120 lb x 1 + 2 x 2  40 hr 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – |10 20 30 40 x1x1x1x1 x2x2x2x2 A B c D

12. Z = 40 x 1 + 50 x 2 (X1, X2) 0 + 0 = 0 (0, 0)A 1200 + 0 = 1200(30, 0)B 960 + 400 = 1360(24,8)C 0 + 1000 = 1000 (0, 20)D Copyright 2006 John Wiley & Sons, Inc.Supplement 13-12

13. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-13 Minimization Problem Minimize Z = 6x 1 + 3x 2 subject to 2x 1 +4x 2  16 4x 1 +3x 2  24 x 1, x 2  0

14. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-14 14 14 – 12 12 – 10 10 – 8 8 – 6 6 – 4 4 – 2 2 – 0 0 – |22|222 |44|444 |66|666 |88|888 |10 12 14 x1x1x1x1 x2x2x2x2 A B C Graphical Solution