Operations Research

Operations Research 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.

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

Linear Programming (LP) A model consisting of linear relationships representing a firm’s objective and resource constraints 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 Copyright 2006 John Wiley & Sons, Inc.

LP Model Formulation (cont.) Max/min z = c1x1 + c2x2 + ... + cnxn subject to: a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1 a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2 : am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bm xj = decision variables bi = constraint levels cj = objective function coefficients aij = constraint coefficients Copyright 2006 John Wiley & Sons, Inc.

LP Formulation: Example Maximize Z = 40 x1 + 50 x2 Subject to x1 + 2x2 40 4x1 + 3x2 120 x1 , x2 0 Copyright 2006 John Wiley & Sons, Inc.

x1 + 2x2 40 40 x1 20 x2 4x1 + 3x2 120 30 x1 40 x2 Copyright 2006 John Wiley & Sons, Inc.

Graphical Solution: Example x1 + 2 x2 40 50 – 40 – 30 – 20 – 10 – 0 – | 10 60 50 20 30 40 x1 x2 Copyright 2006 John Wiley & Sons, Inc.

Graphical Solution: Example 4 x1 + 3 x2 120 x1 + 2 x2 40 50 – 40 – 30 – 20 – 10 – 0 – | 10 60 50 20 30 40 x1 x2 Copyright 2006 John Wiley & Sons, Inc.

Graphical Solution: Example 4 x1 + 3 x2 120 x1 + 2 x2 40 Area common to both constraints 50 – 40 – 30 – 20 – 10 – 0 – | 10 60 50 20 30 40 x1 x2 Copyright 2006 John Wiley & Sons, Inc.

Computing Optimal Values x1 + 2x2 = 40 4x1 + 3x2 = 120 4x1 + 8x2 = 160 -4x1 - 3x2 = -120 5x2 = 40 x2 = 8 x1 + 2(8) = 40 x1 = 24 4 x1 + 3 x2 120 lb x1 + 2 x2 40 hr 40 – 30 – 20 – 10 – 0 – | 10 20 30 40 x1 x2 D c A B Copyright 2006 John Wiley & Sons, Inc.

Z = 40 x1 + 50 x2 (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.

Minimization Problem Minimize Z = 6x1 + 3x2 subject to 2x1 + 4x2  16 Copyright 2006 John Wiley & Sons, Inc.

Graphical Solution x2 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – A B C | 2 | 4 Copyright 2006 John Wiley & Sons, Inc.