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Solving Linear
Programming Models
Topics

Computer Solution

Sensitivity Analysis
Product mix problem - Beaver Creek Pottery
Example (1 of 2)
Resource Requirements
Labor
(Hr./Unit)
Clay
(Lb./Unit)
Profit
($/Unit)
Bowl
1
4
40
Mug
2
3
50
Product

Product mix problem - Beaver Creek Pottery Company

How many bowls and mugs should be produced to maximize profits given
labor and materials constraints?
Resource Availability:
40 hrs of labor per day (labor constraint)
120 lbs of clay (material constraint)

Product mix problem - Beaver Creek Pottery
Example (2 of 2)
Complete Linear Programming Model:
x1 = number of bowls to produce per day
x2 = number of mugs to produce per day
Maximize
Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Beaver Creek Pottery Example
Excel Spreadsheet – Data Screen (1 of 5)
Click on “Data” tab
to invoke “Solver.”
=C6*B10+D6*B11
Decision variable—
bowls (x1)=B10;
mugs (x2)=B11
Objective function
=C4*B10+D4*B11
=G6-E6
=G7-E7
=C7*B10+D7*B11
Beaver Creek Pottery Example
“Solver” Parameter Screen (2 of 5)
Objective function
Decision variables
=C7*B10+D7*B11<20
Click on “Add” to add
model contraints.
=C6*B10+D6*B11<40
x1, x2 >0
Select “Simplex LP”
method
Solver parameters
Beaver Creek Pottery Example
Adding Model Constraints (3 of 5)
=40
=C6*B10+D6*B11
Labor constraint
Beaver Creek Pottery Example
“Solver” Settings (4 of 5)
Slack—S1=0
and S2=0
Slack S1 = 0
and S2 = 0
Solution screen
Beaver Creek Pottery Example
Solution Screen (5 of 5)
Answer report
Beaver Creek Pottery Example
Graphical Solution
Maximize Z = $40x1 + $50x2
subject to:
x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Optimal solution point
Sensitivity Analysis

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.
Objective Function Coefficient
Sensitivity Range

The sensitivity range for an objective function coefficient is the
range of values over which the current optimal solution point will
remain optimal.
objective function Z = $40x1 + $50x2
sensitivity range for:
x1: 25  c1  66.67
x2: 30  c2  80
Objective Function Coefficient Ranges
Excel “Solver” Results Screen
Solver results screen
Objective Function Coefficient Ranges
Beaver Creek Example Sensitivity Report
Sensitivity ranges for
objective function
coefficients
sensitivity range for:
x1: 25  c1  66.67
x2: 30  c2  80
Changes in Constraint Quantity Values
Sensitivity Range

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.

Recall the Beaver Creek Pottery example.
Maximize Z = $40x1 + $50x2 subject to:
x1 + 2x2  40 hr of labor
4x1 + 3x2  120 lb of clay
x1, x2  0
Constraint Quantity Value Ranges by Computer
Excel Sensitivity Range for Constraints
Sensitivity ranges
for constraint
quantity values
the sensitivity range for the labor hours q1 is
30 ≤q1 ≤80 hr.
the sensitivity range for clay
quantity q2 is 60≤ q2 ≤160 lb.
Excel Sensitivity Report for Beaver Creek Pottery
Shadow Prices Example
Maximize Z = $40x1 + $50x2 subject to:
x1 + 2x2  40 hr of labor
4x1 + 3x2  120 lb of clay
x1, x2  0
Shadow prices
(dual values)
The shadow price (or marginal value) for labor is $16 per hour, and the shadow
price for clay is $6 per pound. This means that for every additional hour of labor
that can be obtained, profit will increase by $16 and for every additional lb of clay
the profit increases by $6. The upper limit of the sensitivity range for the labor &
clay are 80 hours & 160 lb and the lower limits are 30 hours & 60 lb, before the
optimal solution mix changes.
Duality (Shadow Prices)
 With every linear programming problem, there is associated
another linear programming problem which is called the dual of
the original (or the primal) problem.

Shadow price is also called 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.
The Primal-Dual Relationship
Primal and Dual problems for
Beaver Creek Pottery Example
Primal Problem
Maximize Z = $40x1 + $50x2
subject to:
x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Dual Problem
Minimize P = 40y1 + 120y2
subject to:
y1 + 4y2 ≥ 40
2y1 + 3y2 ≥ 50
y1, y2  0
Flair Furniture Company
The Flair Furniture Company produces tables and chairs. The
production process for each is similar in that both require a
certain number of hours of carpentry work and a certain
number of labor hours in the painting and varnishing
department. Each table takes 4 hours of carpentry and 2 hours
in the painting and varnishing shop. Each chair requires 3
hours in carpentry and 1 hour in painting and varnishing.
During the current production period, 240 hours of carpentry
time are available and 100 hours in painting and varnishing
time are available. Each table sold yields a profit of $70; each
chair produced is sold for a $50 profit.
Formulate the LP Model.
Solve the model graphically.
Solve this model by using Excel.
Flair Furniture Company
T = number of tables to be produced per week
C = number of chairs to be produced per week
Maximize profit Z= $70T + $50C
subject to the constraints
4T + 3C ≤ 240 (carpentry constraint)
2T + 1C ≤ 100 (painting and varnishing constraint)
T, C ≥0 (non-negativity constraints)
Flair Furniture CompanySolver Solution
Target Cell (Max)
Cell
Name
$B$12 Profit=
Original Value Final Value
0
4100
Adjustable Cells
Cell
Name
$B$10 Tables=
$B$11 Chairs=
Original Value Final Value
0
30
0
40
Constraints
Cell
Name
$E$6 Carpentry Usage
$E$7 Painting Usage
Cell Value
Formula
Status Slack
240$E$6<=$G$6 Binding
0
100$E$7<=$G$7 Binding
0
Sensitivity Report
Adjustable Cells
Cell
$B$10
$B$11
Name
Tables=
Chairs=
Final Reduced
Objective
Allowable
Value
Cost
Coefficient
Increase
30
0
70
30
40
0
50
2.5
Allowable
Decrease
3.333333333
15
Name
Carpentry Usage
Painting Usage
Final
Value
240
100
Allowable
Decrease
Constraints
Cell
$E$6
$E$7
Shadow
Price
Constraint
Allowable
R.H. Side
Increase
15
240
60
5
100
20
sensitivity range for T: 66.7 c1  100 C: 35 c2  52.5
The sensitivity range for the carpentry hours q1 is 200 ≤q1 ≤ 300
The sensitivity range for painting hours q2 is 80≤ q2 ≤120
The shadow price (or marginal value) for carpentry is $15 per hour, and the
shadow price for painting is $5 per hour. This means that for every additional hour
of carpentry that can be obtained, profit will increase by $15 and for every
additional hour of painting the profit increases by $5.
40
20
Transportation Problem – Example
The Zephyr Television Company ships
televisions from three warehouses to three
retail stores on a monthly basis. Each
warehouse has a fixed supply per month, and
each store has a fixed demand per month. The
manufacturer wants to know the number of
television sets to ship from each warehouse to
each store in order to minimize the total cost
of transportation.
Demand & Supply
Each warehouse has the following supply of televisions available
for shipment each month:
Warehouse
Supply (sets)
1. Cincinnati
2. Atlanta
3. Pittsburgh
300
200
200
700
Each retail store has the following monthly demand for television
sets:
Store
Demand (sets)
A. New York
B. Dallas
C. Detroit
150
250
200
600
Cost Matrix
Costs of transporting television sets from the warehouses to
the retail stores vary as a result of differences in modes of
transportation and distances. The shipping cost per television
set for each route is as follows:
To Store
From
Warehouse
1
2
3
A
$16
14
13
B
$18
12
15
C
$11
13
17
Model Summary
minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 13x3A +
15x3B + 17x3C
subject to
The transportation model can also be optimally
solved by Linear Programming
Computer Solution with Excel
Computer Solution with Excel
Computer Solution with Excel
The solution is
x1C = 200 TVs shipped from Cincinnati to Detroit
x2B = 200 TVs shipped from Atlanta to Dallas
x3A = 150 TVs shipped from Pittsburgh to New York
x3B = 50 TVs shipped from Pittsburgh to Dallas
Z = $7,300 shipping cost