1. Operations Research

2. 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.

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

4. 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.

5. LP Model Formulation (cont.)
Max/min z = c1x1 + c2x cnxn
subject to:
a11x1 + a12x a1nxn (≤, =, ≥) b1
a21x1 + a22x a2nxn (≤, =, ≥) b2 :
am1x1 + am2x amnxn (≤, =, ≥) bm
xj = decision variables
bi = constraint levels
cj = objective function coefficients
aij = constraint coefficients
Copyright 2006 John Wiley & Sons, Inc.

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

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

8. 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.

9. 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.

10. 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.

11. 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.

12. Z = 40 x x2
(X1, X2)
= 0
(0, 0) A
= 1200
(30, 0) B
= 1360
(24,8) C
= 1000
(0, 20) D
Copyright 2006 John Wiley & Sons, Inc.

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

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