1. Department of Management Information systems
An-Najah N. University
Faculty of Engineering and Information Technology
Department of Management Information systems
Operations Research and Applications MIS:
Prepared by Mr.Maher Abubaker
Fall 2015/2016
Resources
Daniel J. Epstein Department of Industrial and Systems Engineering University of Southern California Andrew and Erna Viterbi School of Engineering
Operations Research: An Introduction, 9/EHamdy A. Taha, University of ArkansasISBN-10: X ISBN-13:  ©2011 • Prentice Hall • Cloth, 832 ppPublished 08/29/2010
™ INFORMS – ™
ORMS - ™
Science of Better -

2. Chapter 3 Sensitivity Analysis

3. Shadow Prices Unit Worth ( Shadow Price ) of a resource:
is measured in terms of the units of the objective and it measures the improvement of the objective, as a result of increasing the resource by 1 unit.
( It is the shadow price of the resource or the dual price ).
IF a slack variable take a positive value, this mean that there is a resource that not used totally.
Any resource that is not fully consumed is called Abundant resource and its shadow price =0.
Any resource that is fully consumed by the activity is called Scarce resource.
Unit worth or shadow price is determined using the coefficient of the slack variables in the final Tableau Z-Row.

4. Example Subj to: Resource # 1 x1 + x2 + x3  120
Production Planning Problem A company produced 3 products , using 3 resources
Max z = 2 x1 + x2 + 3 x3
  Subj to:
Resource # x x x3  120
Resource # x1 + 3 x2 + 3 x3  150
Resource # x x3  90
x1 , x2 , x3  0

5. Optimal! The solution Shadow Prices:
Basic
Eq Var x x x s s s RHS
0 z 8/ /
1 s1 1/ /
2 x2 -4/ / /
3 x /
Solution:
x1=0,
x2=5,
x3=45,
s1=70,
s2=0,
s3= 0,
z=140
Shadow Prices:
Resource # 1 ( S1 ) is an abundant resource and its shadow price = 0
( look at the coff. Of s1 in Z-Row ) . Initially S1= 120 , currently S1= 70 this means
( = 50 ) 50 units of resource # 1 is consumed and it is not consumed totally .
Therefore, Resource #1 (s1) is an abundant resource , if we add one unit of this resource
the objective function will be z = ( Shadow Price ) = * 0 = 140 , no any effect
Resource # 2 ( S2 ) is an Scarce resource and its shadow price = 1/3
( look at the coff. Of s1 in Z-Row ). Initially S2 = 150, currently S2 = 0 this means
( 150-0=150) 150 units of resource #2 is consumed, it is consumed totally.
Therefore, Resource # 2 is a scarce resource, if we add one unit of this resource the objective
function will be Z = ( shadow price ) = * 1/3 = z increased by 1/3
What about Resource # 3 ?????

6. Sensitivity Analysis
It is a tool used for evaluating different scenarios by analyzing the effect of changing some of the model parameters on solution feasibility and optimality.
Changing the RHS will change the feasibility
Changing the objective function coefficient will change the Optimality

7. Bicycles R’ US, a small midwest manufacturer of bicycles,
makes four different models: a racing bike called the Racer,
a standard 10 speed, called the standard, a mountain bike, and
a dirt bike, called the ultimate.
All four bikes use the same seat assembly. There are only 180
available per day.
There is also a limitation of 100 reinforced steel alloy frames
used on the mountain and dirt bike.
The company desires to maximize profit. Other relevant data are:
Racer Standard Mountain Ultimate Available
Assembly Time 8 hrs
Inspection Time
Unit Profit

8. The Formulation Max z = 40 x1 + 60x2 + 80 x3 + 100 x4 subj to:

9. The Initial and Final Tableau
Bas z x x x x4 x x x x8 RHS z x x x x
final
Bas z x x x x4 s s s s RHS
z ,320 x x x s
The dual solution

10. Look here! I want to know what will happen if
Final
Bas z x x x x4 s s S S4 RHS
z ,320 x x x s
Look here! I want to know what will happen if
I hire more workers. Should I hire assemblers or
inspectors? How many? Why can’t I produce the
Racer? How much do I have to increase my profit
on that bike? I want to produce a new model called
the Traveler. Will it be part of this production plan?
Come on! I want
President
Bicycles R’Us

11. Let’s add some assembly hours
initial
Bas z x x x x4 x x x x8 RHS z x x x x
Delta b1 1
final
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b1 2 .1 .2
-.3

12. z = 12,320 + 2 delta b1 If delta b1 = 10, then x2 = 116 + .1 delta b1
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b1 2 .1 .2
-.3
z = 12, delta b1
x2 = delta b1
x4 = delta b1
x3 = delta b1
x8 = delta b1
If delta b1 = 10, then
z = 12, (10) = 12,340
x2 = (10) = 117
x4 = (10) = 14
x3 = (10) = 49
x8 = (10) = 37
>= 0
delta b1 >= -116/.1 = -1160
delta b1 >= -12 / .2 = -60
delta b1 <= -52/-.3 =
delta b1 >= -36/.1 = -360
-60 <= delta b1 <=
1940 <= b1 <=

13. Gosh this is great! Do another one.
Will ya? Will Ya?
“The Standard”

14. Let’s add some test hours
initial
Bas z x x x x4 x x x x8 RHS z x x x x
Delta b2 1
final
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b2 8
-.6
-.2 .8

15. z = 12,320 + 8 delta b2 If delta b2 = 10, then x2 = 116 - .6 delta b2
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b2 8
-.6
-.2 .8
z = 12, delta b2
x2 = delta b2
x4 = delta b2
x3 = delta b2
x8 = delta b2
If delta b2 = 10, then
z = 12, (10) = 12,400
x2 = (10) = 110
x4 = (10) = 10
x3 = (10) = 60
x8 = (10) = 30
>= 0
delta b2 <= -116/-.6 =
delta b2 <= -12 / -.2 = 60
delta b2 >= -52/.8 = -65
delta b2 <= -36/-.6 = 60
-65 <= delta b2 <= 60
435 <= b2 <= 560

16. What about more bike seats?
initial
Bas z x x x x4 x x x x8 RHS z x x x x
Delta b3 1
final
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b3 24
1.2
-1.6
1.4 .2

17. z = 12,320 + 24 delta b3 If delta b3 = 10, then
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b3 24
1.2
-1.6
1.4 .2
z = 12, delta b3
x2 = delta b3
x4 = delta b3
x3 = delta b3
x8 = delta b3
If delta b3 = 10, then
z = 12, (10) = 12,560
x2 = (10) = 118
x4 = (10) = -4 infeasible
x3 = (10)
x8 = (10)
>= 0
delta b3 >= -116/1.2 =
delta b3 <= -12 / -1.6 = 7.5
delta b3 >= -52/1.4 =
delta b3 >= -36/.2 = -180
<= delta b3 <= 7.5
<= b3 <= 187.5

18. Let us do the steel alloy frames.
initial
Bas z x x x x4 x x x x8 RHS z x x x x
Delta b4 1
final
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b4 1

19. z = 12,320 + 0 delta b4 If delta b4 = 10, then x2 = 116 + 0 delta b4
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
Delta b4 1
z = 12, delta b4
x2 = delta b4
x4 = delta b4
x3 = delta b4
x8 = delta b4
If delta b4 = 10, then
z = 12,320
x2 = 116
x4 = 12
x3 = 52
x8 = (10) = 46
>= 0
delta b4 >= -36/1 = -36
-36 <= delta b4
64 <= b4

20. What about my Racer? Optimal if 16 > 0 or < 16 Or c1 <= 56
Bas z x x x x4 x x x x8 RHS z x x x x
initial
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x
final
Optimal if > 0 or < 16
Or c1 <= 56

21. The Standard profit coefficient
Bas z x x x x4 x x x x8 RHS z x x x x
initial
final
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x

22. 16 + .8 >= 0 >= -16/.8 = -20 2 + .1 >=0 >= -2/.1 = -20
Bas z x x x x4 x x x x8 RHS z x x x x
>= 0
>= -16/.8 = -20
>=0
>= -2/.1 = -20
>=0
<= -8/-.6 = 13.33
>=0
>= -24/1.2 = -20
-20 <= <=
40 <= c2 <= 73.33

23. The Ultimate profit coefficient
Bas z x x x x4 x x x x8 RHS z x x x x
initial
final
Bas z x x x x4 x x x x8 RHS
z ,320 x x x x

24. 16 -.4 >= 0 <= 16/.4 = 40 2 + .2 >=0 >= -2/.2 = -10
Bas z x x x x4 x x x x8 RHS z x x x x
>= 0
<= 16/.4 = 40
>=0
>= -2/.2 = -10
>=0
<= -8/-.2 = 40
>=0
<= -24/-1.6 = 15
-10 <= <= 15
90 <= c4 <= 115

25. We can use the dual variable values to analyze this:
What about the boss’ new Traveler bike? I understand
that it will take 10 assembly hours and 3 hours to test.
Profit will be $70 per unit. It uses the standard seat
assembly but not the steel alloy frame.
We can use the dual variable values to analyze this:
y1 = value of one assembly hour = $2
y2 = value of one test hour = $8
y3 = value of one seat assembly = $24
y4 = value of one steel alloy frame = 0
Therefore, the objective function coefficient of the traveler in
the final tableau is:
-70 + ($2 x 10 + $8 x 3 + $24 x 1 + 0) = = -2
We would no longer be optimal! Relative to current
basic solution, the Traveler would now entering the basis.

26. The Initial and “Final” Tableau with the Traveler
Bas z x x x x x x x x8 RHS z x x x x
Traveler
-70 10 3 1
Bas z x x x x x x x x8 RHS
z ,320 x x x x
Traveler -2 x
The Red X=(10*0.1) + (3 * -0.6) + (1*1.2)+(0*0) = 0.4 and so on for the rest