OR CHAPTER TWO II.PPT

2.3. Sensitivity Analysis
 Sensitivity analysis is defined as a
study of how the uncertainty in the
output of a mathematical model can
be apportioned, qualitatively or
quantitatively, to different sources of
variation in the input of a model.
 A main purpose of sensitivity
analysis is to identify the sensitive
parameters.
1

Cont…
 When linear programming was formulated, it
was implicitly assumed that the parameters of
the model were known with certainty.
 However, rarely does a manager know all of
these parameters exactly.
 In reality the model parameters are simply
estimates that are subject to change.
 For this reason it is of interest to the manager
to see what effects a change in a parameter
will have on the solution to the model.
2

Cont…
Change may be either reactions
to anticipated uncertainties in
the parameters or reactions to
information.
The analysis of parameter
changes and their effects on the
model solution is known as
sensitivity.
3

Cont…
To ascertain the effect of a
change in the parameter of a
model is:
I. to make the change in the
original model
II. resolve the model and
III. compare the solution results
with original. 4

Cont…
 Some parameters of change:
I. Objective function coefficient
II. Constraints quantity value ( RHS)
III. Coefficient of the constraints
IV. Deletion of constraints
V. Addition of constraints
VI. Addition and deletion of variables etc
5

Cont…
Example 1:
Max Z = $40x1 +50x2
S T: x1 + 2x2 ≤ 40 hours of labour
4x1 + 3x2 ≤ 120 pound of clay
x1, x2, ≥ 0
6

Cont…
I. Objective Function Coefficient Ranges
 The first model parameter change
we will analyse is a change in an
objective function coefficient. We
will use the above given model to
illustrate this change.
 Graphical solution for this problem is
shown below
7

Cont…
 The optimal solution point is shown to be
at corner point B (x1 = 24, x2 =8), which is
the last point the objective function line
touches as it leaves the feasible solution
area.
 However, what will happen if we change
the profit of a bowl, x1, from $40 to $100?
How would that affect the solution
identified previous?
9

Cont…
 Increasing profit for a bowl , i.e., the x1
coefficient , from $40 to $100 made the
objective function line steeper , so much
so that the optimal solution point
changed from corner point B to corner
point C.
 If we had increased the profit for a mug ,
the x2 coefficient , from $50 to $100 , the
objective function line would have
become flatter to the extent that corner
point A would become optimal with x1=0,
x2 = 20 and Z = $2000. 11

Cont…
The objective of sensitivity
analysis in this case is to
determine the range of values
for a specific objective
function coefficient over which
the optimal solution point, x1
and x2, will remain optimal.
12

Cont…
 E.g. the coefficient of x1 in the
objective function is originally $40,
but at some value greater than $40,
point C will become optimal, and at
some value less than $40, point A
will become optimal. The focus of
sensitivity analysis is to determine
those two values, referred to as the
sensitivity range for x1 coefficient,
which we will designate as C1. 13

Cont…
 Look at the graph above and determine
the sensitivity range for x1 coefficient.
The slope of the objective function is
currently -4/5, determined as follows.
 Z = 40x1 +50x2, So, X2 = Z/50 – 4x1/5
 The objective function is now in the form
of the equation of a straight line,
Y =a+Bx, where the intercept, a, equal
1/50 and the slope, b, is -4/5.
14

Cont…
 If the slope of the objective function
increases into -4/3, the objective function
line becomes exactly parallel to the
constraint line.
4x1 + 3x2 = 120
And point C becomes optimal (along with B).
The slope of this constraint line is -4/3, so we
ask ourselves what objective function
coefficient for x1 will make the objective
function slope equal -4/3. The answer is
determined as follows, where C1 is the
objective function coefficient for x1,
15

Cont…
 -C1/50 = -4/3,
 -3C1=-200 , C1 = $66.67,
 X2 Slope = -4/3
16

Cont…
 If the coefficient of x1 is 66.67 then the objective
function will have a slope of -66.67 or -4/3
 We have determined that the upper limit of the
sensitivity range for c1, the x1 coefficient, is 66.67.
 If profit for a bowl increase to exactly $66.67, the
solution points will be both B and C.
 If the profit for a bowl is more than $66.67, point C
will be the optimal solution point.
 The lower limit for the sensitivity range can be
determined by observing the above graph (b).
19

Cont…
 In this case if the objective function line slope
decreases (becomes flatter) from -4/5 to the
same slope as the constraint line, x1 + 2x2 = 40
 Then point A becomes optimal (along with B).
The slope of this constraint line is -1/2 (i.e. x2 =
20-1x1/2) in order to have an objective function
slope of -1/2, the profit for a bowl would have to
decrease to $25 as follows:
-C1/50 = -1/2, -2C1 = -50, C1= $25
 This is the lower limit of the sensitivity range for
the x1 coefficient.
 The complete sensitivity range for the x1
coefficient can be expressed as 25 ≤ c1≤ 66.67 20

Cont…
This means that the profit for a
bowl can vary anywhere between
$25 and $66.67 and the optimal
solution point, x1 = 24 and X2 = 8,
will not change. Of course, the
total profit, or Z value, will
change depending on whatever
value c1 actually is.
21

Cont…
 Performing the same type of graphical
analysis will provide the sensitivity range
for the x2 objective function coefficient,
c2. This range is 30 ≤ c2 ≤ 80. This means
that the profit for a mug can vary
between $30 and $80 and the optimal
solution Point, B, will not change.
 when we say that profit for a mug can
vary between $30 and $80; this is true
only if C1 remains constant.
22

Cont…
 Simultaneous changes can be made in the
objective coefficients as long as change
taken together does not exceed 100% of
the total allowable changes for all the
ranges combined
 E.g. The maximum allowable increase in
the profit for a bowl, according to the
upper limit of the sensitivity range, $66.67
- $40 = $26.67.the maximum allowable
increase for a mug is $80-$50 = $30.
23

Cont…
 What if the profit for a bowl is
increased by $10 and the profit for a
mug is increased by $15? Ten dollar
is $10/26.67 x 100 = 37.5% of the
maximum allowable increases for a
bowl’s profit, and $15 is 15/30 = 50%
of the maximum allowable increase
in the profit for a mug.
24

Cont…
 The total percentage increase is 37.5 +50 =
87.5%, which is less than 100%. Therefore,
these simultaneous changes are feasible and
acceptable within the sensitivity ranges.
 If the increase in profit for a bowl had been
$20, then the percentage increase would be
20/26.67 = 75%. When taken with a
simultaneous increase of 50% in the profit for a
mug, the total increase would then be 125%.
Since this exceeds 100%, it would not be
feasible.
25

Simplex methods to determine the
range for the coefficients of the OF
The range of optimality is the range
over which a basic decision variable
coefficient in the objective function
can change without changing the
optimal solution mix. However, this
change will change only the optimal
value of the objective function.
26

Cont…
Eg. For the foregoing example:
Max z = 40x1 + 50x2
st x1 + 2x2 ≤ 40 hours of labor
x1, x2 ≥ 0,
the optimal simplex tableau is
presented in the next side.
27

Cont…
Cj BV QTY 40 50 0 0
X1 X2 S1 S2
50 X2 8 0 1 4/5 -1/5
40 X1 24 1 0 -3/5 2/5
Zi $1360 40 50 16 6
Cj-Zi 0 0 -16 -6
28

Cont…
Que : Determine the range of optimality for
the coefficients of the basic decision
variables
I. Analysis of X1 coefficient (C1)
Steps
a. Take the Cj-Zj row of the optimal solution
of the non-basic variables
b. Take the X1 row of the non-basic variables
c. Cj-Zj row
X1 row
29

Cont…
Non-basic variables
S1 S2
Cj-Zj row -16 -6
X1 row -3/5 2/5
Cj-Zj row /X1 row 26.67 -15
30

Cont…
 Upper limit
The smallest positive number in the Cj-
Zj row tells how much the profit of x1
can be increased before the solution is
changed
 Upper limit= CV + the smallest positive
value of Cj-Zj/x1 row , cv = 40
therefore, UL = 40 + 26.67
= 66.67
31

Cont…
 Lower limit: The largest negative number
closest to zero.
 LL = CV + the largest negative value of
Cj-Zj row of Cj-Zj/x1 row
 LL = 40 + (-15)= 25
Therefore, the range of optimality for the
coefficient of X1 is 25 ≤ C1≥ 66.67
[25,66.67]
32

Cont…
II. Analysis for X2
 Upper limit:
 UL = CV + smallest positive number (SP)
 UL = 50 + 30 = 80
 LL= CV + largest negative (RMN)
 50+(-20) = 30; 30 ≤ C2 ≥ 80 = [30,80]
Non-basic variables
s1 S2
Cj-Zj row -16 -6
X2 row 4/5 -1/5
Cj-Zj/x2 row -20 30
33

II. Changes in Constraint Quantity Values (RHS)
Using graphical solution method
Shadow prices: how much should a firm be
willing to pay to make additional resources
available?
Shadow price signify the changes in the optimal
value of the objective function for 1 unit
increases in the value of the RHS of the
constraint that represent the availability of
scarce resources.
The negative number of Cj –Zj row is its slack
variable columns provide as with shadow prices
RHS ranging is the range over which shadow
prices remain valid
34

Cont…
Max z = 40x1 + 50x2
st x1 + 2x2 ≤ 40 hours of labor
x1, x2 ≥ 0,
35

Cont…
 Consider a change in which the manager of
the company can increase the labour hours
from 40 to 60.
 The effect of this change in the model is
graphically displayed in the preceding
section.
 By increasing the available labour hours
from 40 to 60, the feasibility solution space
changed.
 It was originally OABC and now it is OA’B’C.
B’ is new optimal solution instead of B.
36

Cont…
 However, the important consideration in this
type of sensitivity analysis is that the solution
mix (or variables that do not have zero values)
including slack variables, did not change even
though the values of x1 and x2 did change (from
x1 = 24, x2 = 8 to x1 = 12, x2 = 24).
 The focus of sensitivity analysis for certain
quantity values is to determine the range over
which the constraint quantity values can
change without changing the solution variable
mix, especially including the slack variables.
37

Cont…
 If the quantity value for the labour
constraint is increased from 40 to 80
hours, the new solution space is
OA’C, and a new solution variable
mix occurs at A’ , as shown in graph
(a), whereas at the original optimal
point, B, both X1 and X2, are in the
solution, at the new optimal point,
A’ , only X2 is produced (i.e., X1 = 0,
X2 = 40, S = 0, and S2 = 0). 39

Cont…
 The upper limit of the sensitivity range
for the quantity value of the first
constraint, which we will refer to as q1
is 8 hours. At this value the solution
mix changes such that bowls are no
longer produced. Further, as q1
increases past 80 hours, s1 increases,
i.e., slack hours are created. Similarly,
if the value for q1 is decreased to 30
hours, the new feasible solution space
42

Cont…
is OA’C, as shown in graph (b) above. The
new optimal is at C, where no mugs (x2)
are produced. The new solution is x1 =
30, x2 = 0, S1 = 0, S2 = 0, and Z = $1200.
Again, the variable mix is changed.
Summarising, the sensitivity range for
the constraint quantity value for labour
hours is
30 ≤ q1≤ 80 hours
43

Cont…
 The sensitivity range for clay can be
determined graphically in a similar manner. If
the quantity value for the clay constraint, 4x1
+ 3x2 ≤ 120, is increased from 120 to 160,
shown in graph (a) below, then the new
solution space ,OAC’ , results , with a new
optimal point, C’. Alternatively, if the quantity
value is decreased from 120 to 60, as shown
in graph (b) below, the new solution space is
OAC’ and the new optimal point A ( x1, 20, x2
=0, S1=0, S2= 0, Z= 800)
44

Cont…
 Summarizing, the sensitivity range for q1 and q2
are
30 ≤ q1 ≤ 80 hr
60 ≤ q2 ≤ 160 Ib
 As was the case with the sensitivity ranges for the
objective function coefficient, these sensitive
ranges are valid for only one q1 value and assume
all the qi values are held constant. However,
simultaneous changes can occur as long as they
do not exceed 100% of the total allowable
increase or decrease in the right hand side
changes, as we demonstrated with the objective
function coefficient ranges.
47

Simplex Solution Method for RHS range
values
 As we have discussed above in this chapter,
sensitivity analysis can be also determined using
simplex method.
e.g. Max Z = 3x1+ 5x2+ 4x3
Subject to:
2x1 + 3x2 ≤ 8
2x2+ 5x3 ≤10
3x1+ 2x2 + 4x3≤ 15
x1, x2, x3 ≥0
48

Cont…
Cj BV QTY
3 5 4 0 0 0
x1 x2 x3 s1 s2 s3
5 X2 50/41 0 1 0 15/41 8/41 -10/41
4 X3 62/41 0 0 1 -6/41 5/41 4/41
3 X1 89/41 1 0 0 -2/41 -22/41 -15/41
Zj 765/41 3 5 4 45/41 24/41 11/41
Cj-jz 0 0 0 -45/41 -24/41 -34/41
49

Cont…
 Required
1. Determine the shadow prices
2. Determine the RHS ranges over which
the shadow prices are valid
Ans for que #1
The shadow prices of the 1
st
, 2
nd
and 3
rd
constrained resources are equals to the
Cj-Zj row values regardless of their sign
or values of Zj i.e. 45/41, 24/41 and
11/41 respectively. 50

Cont…
Ans for que #2
I. Analysis of the first constraint (S1)
 Lower limit =bj-the smallest positive number in
the /Sj column
 Upper limit = bj-the largest negative number in
the Q/Sj column
Qty s1 Qty/s1
50/41 15/41 3.33
62/41 -6/41 -10.33
89/41 -2/41 -44.5
51

Cont…
 LL= Current RHS – SP
LL = 8-3.33 = 4.67
 UL= Current RHS – RMN
UL= 8-(-10.33)= 18.33
Therefore, 4.67 ≤ Q1 ≤ 18.33
= [4.67, 18.33]
Interpretation, the range over which the
shadow prices of 45/41 per unit is valid.
52

Cont…
II. Analysis of the 2nd constrained
resources ( s2)
Qty S2 Qty/s2
50/41 8/41 6.25
62/41 5/41 12.4
89/41 -22/41 -4
53

Cont…
 LL = CV RHS –SP, LL = 10-6.25
LL = 3.75
 UL = CV RHS – RMN, UL = 10- (-4)
UL = 14
Hence, 3.75 ≤ q2 ≤ 2 14
[3.75, 14]
 Interpretation, the range over which
the shadow prices of 24/41 per unit is
valid.
54

Cont…
III. Analysis of the 3rd constrained
resources ( s3)
Qty s3 Qty/s3
50/41 -10/41 -5
62/41 4/41 15.5
89/41 -15/41 -5.9
55

Cont…
 LL = CV RHS –SP, LL = 15-15.5
LL = 0
 UL = CV RHS – RMN, UL = 15- (-5.9)
UL = 20.9
Hence, 0 ≤ q3 ≤ 20.9
[0 , 20.9]
 Interpretation, the range over which
the shadow prices of 11/41 per unit is
valid. 56

Characteristics of OFC and RHS Change
Characteristics of Objective Function Coefficient
Changes
 There is no effect on the feasible region
 The slope of the level profit line changes
 If the slope changes enough, a different corner
point will become optimal
 There is a range for each OFC where the current
optimal corner point remains optimal.
 If the OFC changes beyond that range a new
corner point becomes optimal.
57

Cont…
Characteristics of RHS Changes
 The constraint line shifts, which
could change the feasible region
 Slope of constraint line does not
change
 Corner point locations can change
 The optimal solution can change
58

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