Transcript Beaver Creek Pottery Example Sensitivity Analysis

Department of Business Administration
FALL 2007-08
Management Science
by
Asst. Prof. Sami Fethi
© 2007 Pearson Education
Ch 3: Computer Solution and Sensitivity analysis
Chapter Topics

Standard form

Sensitivity Analysis

Dual Problem

Example Problems
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Ch 3: Computer Solution and Sensitivity analysis
Linear Programming Problem: Standard Form
Standard form requires all variables in the constraint
equations to appear on the left of the inequality (or
equality) and all numeric values to be on the right-hand
side.
Examples: (Equation)
x3  x1 + x2 must be converted to x3 - x1 - x2  0
x1/(x2 + x3)  2 becomes x1  2 (x2 + x3)
and then x1 - 2x2 - 2x3  0
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Ch 3: Computer Solution and Sensitivity analysis
Linear Programming Problem: Standard Form
Models are also transformed into Standard form.
Examples: (model)
Having defined the profit or cost function as well as constraints functions within
the system eqn, standard form can be formulated as follows:
Z= 12x1 + 16 x2
Subject to: 3x1 + 2x2  500
4x1 + 5x2  800
x1 x2  0
Standard form:
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Ch 3: Computer Solution and Sensitivity analysis
Beaver Creek Pottery Example
Sensitivity Analysis (1 of 4)

Sensitivity analysis determines the effect on the optimal
solution of changes in parameter values of the objective
function and constraint equations.

Changes may be reactions to anticipated uncertainties in
the parameters or to new or changed information
concerning the model.
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Beaver Creek Pottery Example
Sensitivity Analysis (2 of 4)
Ch 3: Computer Solution and Sensitivity analysis
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Figure 3.1 Optimal Solution Point
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Beaver Creek Pottery Example Ch 3: Computer Solution and Sensitivity analysis
Change x1 Objective Function Coefficient (3 of 4)
Maximize Z = $100x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Figure 3.2 Changing the x1 Objective Function Coefficient
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Beaver Creek Pottery Example Ch 3: Computer Solution and Sensitivity analysis
Change x2 Objective Function Coefficient (4 of 4)
Maximize Z = $40x1 + $100x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Figure 3.3 Changing the x2 Objective Function Coefficient
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Ch 3: Computer Solution and Sensitivity analysis
Objective Function Coefficient
Sensitivity Range (1 of 3)

The sensitivity range for an objective function coefficient is
the range of values over which the current optimal solution
point will remain optimal.

The sensitivity range for the xi coefficient is designated
as ci.
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Objective Function Coefficient Ch 3: Computer
Sensitivity Range for c1 and c2 (2 of 3)
Solution and Sensitivity analysis
objective function Z = $40x1 + $50x2
sensitivity range for:
x1: 25  c1  66.67
x2: 30  c2  80
Figure 3.4 Determining the Sensitivity Range for c1
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Objective Function Coefficient Ch 3: Computer Solution and Sensitivity analysis
Fertilizer Cost Minimization Example (3 of 3)
Minimize Z = $6x1 + $3x2
subject to:
2x1 + 4x2  16
4x1 + 3x2  24
x1, x2  0
sensitivity ranges:
4  c1  
0  c2  4.5
Figure 3.5 Fertilizer Cost Minimization Example
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Ch 3: Computer Solution and Sensitivity analysis
Changes in Constraint Quantity Values
Sensitivity Range (1 of 4)

The sensitivity range for a right-hand-side value is the
range of values over which the quantity’s value can
change without changing the solution variable mix,
including the slack variables.
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Changes in Constraint Quantity ValuesCh 3: Computer
Increasing the Labor Constraint (2 of 4)
Solution and Sensitivity analysis
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Figure 3.6
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Increasing the Labor Constraint Quantity
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3: Computer Solution and Sensitivity analysis
Changes in Constraint Quantity Ch
Values
Sensitivity Range for Labor Constraint (3 of 4)
Sensitivity range for:
30  q1  80 hr
Figure 3.7
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Determining the Sensitivity Range for Labor Quantity
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3: Computer Solution and Sensitivity analysis
Changes in Constraint Quantity Ch
Values
Sensitivity Range for Clay Constraint (4 of 4)
Sensitivity range for:
60  q2  160 lb
Figure 3.8
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Determining the Sensitivity Range for Clay Quantity
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Ch 3: Computer Solution and Sensitivity analysis
Other Forms of Sensitivity Analysis
Topics (1 of 4)

Changing individual constraint parameters

Adding new constraints

Adding new variables
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Ch 3: Computer Solution and Sensitivity analysis
Other Forms of Sensitivity Analysis
Changing a Constraint Parameter (2 of 4)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Figure 3.9
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Changing the x1 Coefficient in the Labor Constraint
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Ch 3: Computer
Other Forms of Sensitivity Analysis
Adding a New Constraint (3 of 4)
Solution and Sensitivity analysis
Adding a new constraint to Beaver Creek Model:
0.20x1+ 0.10x2  5 hours for packaging
Original solution: 24 bowls, 8 mugs, $1,360 profit
To find out Optimal solution
coordinates, we use the both
constraints.
x1 = 40 -2x2
4(40 -2x2) + 3x2 =120
160-8x2 + 3x2 =120
-5x2 = -40
X2 = 8
x1 = 24
Z = $40 (24) + $50 (8)
Z = $ 1360 max daily profit possible
Exhibit 3.17
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Ch 3: Computer
Other Forms of Sensitivity Analysis
Adding a New Variable (4 of 4)
Solution and Sensitivity analysis
Adding a new variable to the Beaver Creek model, x3, a third
product, cups
Maximize Z = $40x1 + 50x2 + 30x3
subject to:
x1 + 2x2 + 1.2x3  40 hr of labor
4x1 + 3x2 + 2x3  120 lb of clay
x1, x2, x3  0
Solving model shows that change has no effect on the original
solution (i.e., the model is not sensitive to this change).
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Ch 3: Computer Solution and Sensitivity analysis
Shadow Prices (Dual Variable Values)

Defined as the marginal value of one additional unit of
resource.

The sensitivity range for a constraint quantity value is
also the range over which the shadow price is valid.
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Ch 3: Computer Solution and Sensitivity analysis
The Dual Problem and Shadow prices
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Every linear programming problem, called the primal
problem, has a corresponding or symmetrical problem
called the dual problem.
A profit-maximization primal problem has a costminimization dual problem, and vice versa. The solution
of a dual problem yields the shadow prices.
They give the change in the value of the objective
function per unit change in each constraint in the primal
problem.
According to the duality theorem, the optimal value of
the objective function is the same in the primal and in
the corresponding dual problems.
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Ch 3: Computer Solution and Sensitivity analysis
Dual of the Profit Maximization Problem
Maximize
Subject to
 = $30QX + $40QY
(objective function)
1QX + 1QY  7
(input A constraint)
0.5QX + 1QY  5
(input B constraint)
0.5QY  2
(input C constraint)
QX, QY  0
Minimize
Subject to
(nonnegativity constraint)
C = 7VA + 5VB + 2VC
1VA + 0.5VB  $30
1VA + 1VB + 0.5VC  $40
VA, VB, VC  0
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Ch 3: Computer Solution and Sensitivity analysis
Dual of the Profit Maximization Problem
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In the dual problem we seek to minimize the shadow prices
of inputs A, B, and C used by the firm. Defining VA, VB, as
the shadow prices of inputs and C as the total imputed
values of the fixed quantities of inputs available to the firm,
we can write the dual objective function as minimize
C=7 VA+5 VB+2 VC
Thus the constraints of the dual problem can be written as follow:
1VA + 0.5VB  $30
1VA + 1VB + 0.5VC  $40 so VC=0 due to slack variable.
1VA + 0.5VB = $30
1VA + 1VB
= $40 therefore 0.5VB=$10, VB=$20 and
VA=$20
C=7 ($20)+5 ($20) +2 (0)=$240 this is the minimum cost.
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Ch 3: Computer Solution and Sensitivity analysis
Dual of the Cost Minimization Problem
Minimize
Subject to
C = $2QX + $3QY
(objective function)
1QX + 2QY  14
(protein constraint)
1QX + 1QY  10
(minerals constraint)
1QX + 0.5QY  6
(vitamins constraint)
QX, QY  0
Maximize
Subject to
(nonnegativity constraint)
 = 14VP + 10VM + 6VV
1VP + 1VM + 1VV  $2
2VP + 1VM + 0.5VV  $3
VP, VM, VV  0
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Ch 3: Computer Solution and Sensitivity analysis
Dual of the Cost Minimization Problem

Since we know that from the solution of the primal
problem that vitamin constraint is a slack variable, so
that VV=0, subtracting the first from the second
constraint, we can get the solution of the dual problem
as follow:
2VP + 1VM =3
1VP + 1VM = 2 , therefore VP =$1 and VM =$1
The profit as follows:
 = 14($1)+ 10($1) + 6($0)
= $24.
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Example Problem 1
Problem Statement (1 of 2)




Ch 3: Computer Solution and Sensitivity analysis
Two airplane parts: no.1 and no. 2.
Three manufacturing stages: stamping, drilling, milling.
Decision variables: x1 (number of part no.1 to produce)
x2 (number of part no.2 to produce)
Model: Maximize Z = $650x1 + 910x2
subject to:
4x1 + 7.5x2  105 (stamping,hr)
6.2x1 + 4.9x2  90 (drilling, hr)
9.1x1 + 4.1x2  110 (finishing, hr)
x1, x2  0
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Example Problem 1
Graphical Solution (2 of 2)
Ch 3: Computer Solution and Sensitivity analysis
Maximize Z =
$650x1 + $910x2
subject to:
4x1 + 7.5x2  105
6.2x1 + 4.9x2  90
9.1x1 + 4.1x2  110
x1, x2  0
s1 = 0, s2 = 0, s3 = 11.35 hr
485.33  c1  1,151.43
137.76  q1  89.10
Figure 3.10 Graphical Solution
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 2
Problem Statement


Southern
Sporting
Goods
Company makes basketballs and
footballs.
Each
product
is
produced from two resources—
rubber and leather. The resource
requirements for each product and
the total resources available are as
follows:
Each basketball produced results in
a profit of $12, and each football
earns $16 in profit.
Operations Research
Product
Resource Requirements per
Unit
Rubber(lb)
Leather (ft.2)
Basketball
3
4
Football
2
5
Total resources
available
500 lb.
800 ft.2
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 2
Problem Statement
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
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





a. Formulate a linear programming model to determine the number of basketballs and
footballs to produce in order to maximize profit.
b. Transform this model into standard form.
c. Solve the model formulated in the Problem for Southern Sporting Goods Company
graphically.
d. Identify the amount of unused resources (Le., slack) at each of the graphical
extreme points.
e. What would be the effect on the optimal solution if the proflt for a basketbali
changed from $12 to $13? What would be the effect if the profit for a footbali
changed from $16 to$15?
f. What would be the effect on the optirnal solution if 500 additional pounds of
rubber could be obtained? What would be the effect if 500 additional square feet of
leather could be obtained?
For the linear programming model for Southern Sporting Goods Company,
formulated in section a and solved graphically in section b:
g. Determine the sensitivity ranges for the objective function coeffıcients and
constraint quantity values, using graphical analysis.
h. Determine the shadow prices for the resources and cxplain their meaning.
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 2
Problem Solution
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 2
Problem Solution
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 2
Problem Solution
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 2
Problem Solution
(h)
Maximize Z = $12x1 + $16x2
Min C= $500v1 + $800v2
subject to:
3v1 + 4v2  12
2v1 + 5v2  16
v1, v2  0
subject to:
3x1 + 2x2  500 (rubber)
4x1 + 5x2  800 (leather)
x1, x2  0
A profit-max primal problem has a
cost-min dual problem and vice-versa.
The soln. Of a dual problem yields the
shadow prices. They give the change
in the value of the obj. Function per
unit change in each constraint in the
primal problem.
Operations Research
3v1 + 4v2  12
v2 = 3-(3/4) v1
2v1 + 5(3-(3/4) v1=16
v1=-4/7, v1  0 so v1=0
v2 = 16/5=3.20
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 3
Problem Statement


An Aluminum Company produces
three grades (high, medium, and low)
of aluminum at two milis. Each miii
has a different production capacity (in
tons per day) for each grade, as follows:
The company has contracted with a
manufacturing flrm to supply at least
12 tons of high-grade aluminum, 8
tons of meclium-grade aluminum, and
5 tons of low-grade aluminum. It costs
United $6,000 per day to operate miii 1
and $7,000 per day to operate miii 2.
The company wants to know the
number of days to operate each miii in
order to meet the contract at the
minimum cost.
Operations Research
Mill
Aluminum Grade
High
Medium
Low
1
2
6
2
2
2
4
10
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 3
Problem Statement





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a. Formulate a linear programming model for this problem.
b. Solve the linear programming model formulated in section a
for United Aluminum Company graphically.
c. How much extra (i.e., surplus) high-, medium-, and low-grade
aluminum does the company produce at the optimal solution?
d. What would be the effect on the optimal solution if the cost
of operating mill 1 increased frorn $6,000 to $7,500 per day?
e. What would be the effect on the optimal solution if the
company could supply only 10 tons of high-grade aluminum?
f. Identify and explain the shadow prices for each of aluminum
grade contract requirements
g. Determine the sensitivity ranges for the objcctive function
coefficients and for the constraint quantity values.
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 3
Problem Solution
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 3
Problem Solution
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Ch 3: Computer Solution and Sensitivity analysis
Example Problem 3
Problem Solution
(f)
Min C= $6000x1 + $7000x2
Min Z= $12v1 + $8v2+ $5v3
subject to:
6x1 + 2x2  12 (high)
2x1 + 2x2  8 (medium)
4x1 + 10x2  5 (low)
x1, x2  0
subject to:
6v1 + 2v2 + 4v3  6000
2v1 + 2v2 + 10v3  7000
v1, v2 ,, v3 0
Since v3 is a slack variable,
we set v3=0
A cost-min primal problem has a profitmax dual problem and vice-versa.
The soln. Of a dual problem yields the
shadow prices. They give the change
in the value of the obj. Function per
unit change in each constraint in the
primal problem.
Operations Research
6v1 + 2v2 =6000
v2 = 3000-3 v1
2v1 +6000-(3- 6 v1=16
v1=- 250, v1  0 so v1=0
v2 = 3000
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Ch 3: Computer Solution and Sensitivity analysis
End of Chapter
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