Operations Research Operations Research w Operations Research OR

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Operations Research

Operations Research

Operations Research w Operations Research (OR) aims to having the optimization solution for some

Operations Research w 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

Operations Research Models w Linear Programming w Markov Chains w Network Optimization w Decision

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

Linear Programming (LP) A model consisting of linear relationships representing a firm’s objective and

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. Supplement 13 -4

LP Model Formulation (cont. ) Max/min z = c 1 x 1 + c

LP Model Formulation (cont. ) Max/min z = c 1 x 1 + c 2 x 2 +. . . + cnxn subject to: a 11 x 1 + a 12 x 2 +. . . + a 1 nxn (≤, =, ≥) b 1 a 21 x 1 + a 22 x 2 +. . . + a 2 nxn (≤, =, ≥) b 2 : am 1 x 1 + am 2 x 2 +. . . + amnxn (≤, =, ≥) bm xj = decision variables bi = constraint levels cj = objective function coefficients aij = constraint coefficients Copyright 2006 John Wiley & Sons, Inc. Supplement 13 -5

LP Formulation: Example Maximize Z = 40 x 1 + 50 x 2 Subject

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

x 1 + 2 x 2 40 x 1 0 40 x 2 20

x 1 + 2 x 2 40 x 1 0 40 x 2 20 0 4 x 1 + 3 x 2 120 x 1 0 30 x 2 40 0 Copyright 2006 John Wiley & Sons, Inc. Supplement 13 -7

Graphical Solution: Example x 2 50 – 40 – 30 – 20 – 10

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

Graphical Solution: Example x 2 50 – 4 x 1 + 3 x 2

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

Graphical Solution: Example x 2 50 – 4 x 1 + 3 x 2

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

Computing Optimal Values x 1 + 2 x 2 = 40 4 x 1

Computing Optimal Values x 1 + 2 x 2 = 40 4 x 1 + 3 x 2 = 120 x 2 40 – 4 x 1 + 3 x 2 120 lb 4 x 1 + 8 x 2 = 160 -4 x 1 - 3 x 2 = -120 30 – 20 – D x 1 + 2 x 2 40 hr 40 8 x 1 + 2(8) = x 1 = 40 24 c 10 – 0– A 5 x 2 = | 10 | | 20 B 30 Copyright 2006 John Wiley & Sons, Inc. | x 1 40 Supplement 13 -11

(X 1, X 2) Z = 40 x 1 + 50 x 2 A

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

Minimization Problem Minimize Z = 6 x 1 + 3 x 2 subject to

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

Graphical Solution x 2 14 – 12 – 10 – 8–A 6– 4– B

Graphical Solution x 2 14 – 12 – 10 – 8–A 6– 4– B 2– 0– | 2 | 4 Copyright 2006 John Wiley & Sons, Inc. | 6 | 8 C | 10 | 12 | 14 x 1 Supplement 13 -14

OPERATIONS RESEARCH INTRODUCTION TO OPERATIONS RESEARCH Management ProcessOPERATIONS RESEARCH INTRODUCTION TO OPERATIONS RESEARCH Management Process
Operations Research Amity School Of Business OPERATIONS RESEARCHOperations Research Amity School Of Business OPERATIONS RESEARCH
OPERATIONS RESEARCH UNITI Introduction to OR Operations researchOPERATIONS RESEARCH UNITI Introduction to OR Operations research
Advanced Operations Research Reference Operations Research Models andAdvanced Operations Research Reference Operations Research Models and
Operations Research and Implementation Science Introduction Operations ResearchOperations Research and Implementation Science Introduction Operations Research
LIMITATIONS OF OPERATIONS RESEARCH LIMITATIONS OF OPERATIONS RESEARCHLIMITATIONS OF OPERATIONS RESEARCH LIMITATIONS OF OPERATIONS RESEARCH
Operations Research I Phases of operations research IngOperations Research I Phases of operations research Ing
IT Operations Management in Banks Operations IT OperationsIT Operations Management in Banks Operations IT Operations
STANDARD OPERATIONS 2 Standard Operations DEFINISI STANDARD OPERATIONSSTANDARD OPERATIONS 2 Standard Operations DEFINISI STANDARD OPERATIONS
Tending Operations Shrinivas N Sabale Tending Operations OperationsTending Operations Shrinivas N Sabale Tending Operations Operations
Overlay Operations Overlay Operations Overlay Operations involve combiningOverlay Operations Overlay Operations Overlay Operations involve combining
Introduction to Operations Management What is Operations OperationsIntroduction to Operations Management What is Operations Operations
Order of Operations PEMDAS Operations Operations mean thingsOrder of Operations PEMDAS Operations Operations mean things