1/24: Linear Programming &
Sensitivity Analysis
• Review:
– LP Requirements
– Graphical solutions
• Using MS Excel for Linear Programming
• Sensitivity Analysis
Requirements of an LP Problem
• Objective Function
– LP problems seek to maximize or minimize some
quantity .
• Constraints
– restrictions that limit the degree to which we can
pursue our objective.
• Alternative courses of action.
• Linear equations or inequalities.
Graphical Solutions for LP
•
•
•
•
Must be no more than 2 decision variables.
Plot the limits of the constraints.
Find the feasible region.
Determine the optimized solution (using corner
point method).
Graphical Solutions for LP:
Corner Point Method
• Theory behind LP states that the optimal
solution to any problem will lie at a corner
point, or extreme point of the feasible region.
• Find result for each corner of the feasible
region, pick the one that best satisfies the goal.
EX: The Brewery
• The microbrewery produces beer and ale. Beer
sells for $5 per barrel, and ale for $2/barrel.
Producing a barrel of beer requires 5 pounds of
corn and 2 pounds of hops. Producing a barrel
of ale requires 2 pounds of corn and 1 pound of
hops. Every day, the microbrewery has 60
pounds of corn and 25 pounds of hops.
• Maximize the daily profit for the microbrewery.
EX: The Brewery
Beer
Revenue 5
Corn
5
Hops
2
Ale
2
2
1
available
60
25
EX: The Brewery
Max Z ( revenue ) = 5 x1 + 2 x2
S. T. 5 x1 + 2 x2 <= 60
2 x1 + 1 x2 <= 25
x1, x2 >= 0
30
EX: The Brewery
25
Max Z ( revenue ) = 5 x1 + 2 x2
0, 25 = $50
S. T. 5 x1 + 2 x2 <= 60
2 x1 + 1 x2 <= 25
10, 5 = $60
3
12, 0 = $60
3
12,12.5
x1, x2 >= 0
Spreadsheet Solutions for LP:
MS Excel
• Formulate the optimization equation and the
constraints.
• Build spreadsheet with formulas showing the
relationships between the various elements.
• Using MS Excel Solver, define the cell to
optimize, and the constraints on it.
• In “Options”, use “Assume Linear Model”
Using MS Excel for Linear
Programming: the Answer Report
• Binding & non-binding constraints
– Binding: limits the present solution
– Non-binding: does not limit the present solution
• Slack:
– How much extra capacity there is for a constraint
Using MS Excel for LP:
Sensitivity Analysis: Adjustable Cells
• Reduced Cost
– how much a coefficient would need to change to
affect the optimal solution (distrust sign)
• Allowable increase & decrease
– how much a coefficient can change without
changing the optimal combination for the solution
(but it can change the actual optimized value)
Using MS Excel for LP:
Sensitivity Analysis: Constraints
• Shadow Price (also called Dual Price)
– How much, per unit, the constraint affects the
optimized value
• Allowable increase & decrease
– In what range the shadow price holds true.
Using MS Excel for LP:
Sensitivity Analysis
• All data in the sensitivity analysis are based on
changing ONE THING AT A TIME.
Using MS Excel for LP:
Troubleshooting
• Infeasibility: two possibilities:
– An answer cannot be found that satisfies all
constraints.
– Part of the LP was input incorrectly. (More likely)
• Unboundedness:
– An answer has no limit to its goal.
– Part of the LP was input incorrectly. (See above)
Using MS Excel for LP: Example
• A laboratory orders pig heart valves for use in mechanical
heart valves for humans. The 3 sizes ordered come from 3
suppliers in differing ratios. The lab needs 500 large, 300
medium, and 300 small valves per month. Each supplier can
send no more than 500 valves total per month. The product
mixes & cost/valve are below. Minimize the cost.
$ / valve
Supplier 1
$5
Supplier 2
$4
Supplier 3
$3
% large
20
35
% med
40
35
% small
40
30
60
20
20
Using MS Excel for LP:
Group Exercise
• A telemarketing company needs to contact 150 wives,
120 husbands, 100 single males, and 110 single females.
A day call costs $2.00, and an evening call costs $5.00.
No more than half the calls can be made at night. See
the table for percentage of answers. No more than half
the calls can be made at night. Minimize the cost.
Who
answers
Day
Wife
Husband
Single
female
10%
No one
10%
Single
male
10%
30%
Night
30%
30%
15%
20%
5%
40%
Using MS Excel for LP:
Group Exercise
• Go to lab to solve the problem using Excel.
• Print out main sheet, formulas, answer report, and
sensitivity report.
• Discuss the meanings of the answer report & sensitivity
report items.
Using MS Excel for LP:
Homework
• Ch. 3 #25, 26, 27, 34
Along with working up an Excel Solver answer to
the problems, answer all questions in the problems –
type them on the spreadsheets, or add a worksheet to
answer each problem’s questions.
• Turn in printouts of solved spreadsheets, formulas,
and answers, and email the Excel files by
BEGINNING OF CLASS Thursday, January 31