Linear Programming
Models: Graphical and
Computer Methods
Linear Programming Part 1
Unit 8
Objectives
1. Understand the basic assumptions and properties
of linear programming (LP).
2. Use graphical procedures to solve LP problems
with only two variables to understand how LP
problems are solved.
3. Understand special situations such as redundancy,
infeasibility, unboundedness, and alternate
optimal solutions in LP problems.
4. Understand how to set up LP problems on a
spreadsheet and solve them using Excel’s Solver.
Introduction
• Management decisions involve the
most effective use of resources
• Most widely used modeling technique
is linear programming (LP)
• Deterministic models
Some Applications
• Manufacturing
• Factory planning
• Refinery planning
• Financial Management
• Portfolio selection
• Human Resources
• Scheduling
• Transportation
• Vehicle routing
Developing a LP Model
• Three distinct steps
1. Formulation
• mathematical expressions
2. Solution
• optimal (or best) solution
3. Interpretation
Key Components
• Maximize or minimize some quantity –
max profit, min cost
• Include restrictions or constraints
• Objective and constraints expressed as
linear equations (=, ≤, ≥)
LP Characteristics
• Feasible Region –
• Set of points that satisfies all constraints
• Corner Point Property –
• Optimal, unique solution must lie at one or more
corner points
• Optimal Solution –
• Corner point with best objective function value
Decision Variables
• What we are solving for
• Decision variables can be in different
units of measurement
The Objective Function
• States goal of a problem
• maximize profit or minimize cost
• Single function
Constraints
• Restrictions or limits on our decisions
• As many as necessary
• Can be independent
Our Action Plan (We will solve the
following problems together)
• State the decision variables
• State the objective function
• State the constraints
• Sketch the objective function and constraints on
graph
• Identify the feasible region (all possible solutions)
• Identify the optimal (best, profit maximizing in this
case) solution.
Guidelines
• Recognizing and defining decision variables
• Different variables, different units
• Use only the decision variables in the model
• Difficulties may point to a need for more
variables or better definitions
Guidelines
• One unit of measurement per expression
• Example – wiring time, testing time, units
demanded
• Constraints are separate
• Translate words into expressions
Example – Electrotech Corporation
• The Electrotech Corporation manufactures two
industrial sized electrical devices: generators
and alternators. Both of these products require
wiring and testing during the assembly
process. Each generator requires 2 hours of
wiring and 1 hour of testing and can be sold for
a $250 profit. Each alternator requires 3 hours
of wiring and 2 hours of testing and can be sold
for a $150 profit. There are 260 hours of wiring
time and 140 hours of testing time available in
the next production period, and Electrotech
wants to maximize profit. Formulate an LP
model for this problem. Solve graphically.
Cost Minimization – American Auto
• American Auto is evaluating its marketing plan for the
sedans, SUVs, and trucks the company produces. A TV
ad featuring it’s SUV has been developed. The company
estimates each showing of this commercial will cost
$500,000 and increase sales of SUVs by 3%, but reduce
sales of trucks by 1%, and have no effect on the sales of
sedans. The company also has a print ad campaign
developed that it can run in various nationally
distributed magazines at a cost of $750,000 per title. It
is estimated that each magazine title the ad runs in will
increase the sales of sedans, SUVs, and trucks by 2%,
1%, and 4%, respectively. The company desires to
increase sales of sedans, SUVs, and trucks by at least 3%,
14%, and 4%, respectively, in the least costly manner.
Formulate and graph the LP model for this problem.
Detailed Solutions to Text
Problems
• The following problems are solved in detail in the
text but I’m including them here in case you don’t
have the second edition.
• There are detailed screen shots of the excel
instructions – VERY USEFUL IF YOU CAN’T
REMEMBER WHAT WE DID IN CLASS OR NEED A
REFERENCE!!!
Example - Flair Furniture Company
• Flair Furniture produces inexpensive tables
and chairs. Each table takes 3 hours of
carpentry work and 2 hours of painting work.
Each chair requires 4 hours of carpentry and 1
hour of painting. As well, 2400 hours of
carpentry time and 1000 hours of painting time
are available. The forecasted demand for
chairs is 450 and we are contractually
obligated to provide at least 100 tables. Each
table sold results in a profit contribution of $7
and each chair sold yields a profit contribution
of $5. How many tables and chairs should the
company produce?
The Objective Function
• For Flair Furniture
Profit = ($7 profit per table)
x (number of tables produced)
+ ($5 profit per chairs)
x (numbers of chairs produced)
• Using decision variables T and C
Maximize $7T + $5C
Constraints
• For carpentry time
(3 hours per table)
x (number of tables produced) +
(3 hours per chair)
x (number of chairs produced)
• There are 2,400 hours of time
available
3T + 4C ≤ 2,400
Constraints
• All four constraints
Carpentry time –
Painting time –
Chairs sold –
Tables provided –
3T + 4C ≤ 2,400
2T + 1C ≤ 1,000
C ≤ 450
T ≥ 100
Nonnegativity and Integers
• Decision variables must be ≥ 0, so
T ≥ 0, and
C≥0
• Decision variables may have to be integers
Guidelines
• Recognizing and defining decision variables
• Different variables, different units
• Use only the decision variables in the model
• Difficulties may point to a need for more
variables or better definitions
Guidelines
• One unit of measurement per expression
• Example – carpentry time, chairs demanded
• Constraints are separate
• Translate words into expressions
Graphical Solution
• Complete model
Maximize profit = $7T + $5C
Subject to
3T + 4C
2T + 1C
C
T
T, C
≤
≤
≤
≥
≥
2,400(carpentry time)
1,000(painting time)
450 (max chairs allowed)
100 (min tables allowed)
0
(nonnegativity)
Graphical Representation
Number of Chairs (C)
1,000 –
–
(T = 0, C = 600)
800 –
–
Carpentry Constraint Line
600 –
–
400 –
(T = 400, C = 300)
–
200 –
(T = 800, C = 0)
–
0 –|
0
|
|
200
|
|
400
|
|
600
|
|
800
Number of Tables(T)
|
|
1,000
|
Graphical Representation
Number of Chairs (C)
1,000 –
–
Region Satisfying
3T + 4C ≤ 2,400
800 –
–
(T = 300, C = 200)
600 –
–
400 –
(T = 600, C = 400)
–
200 –
–
0 –|
0
|
|
200
|
|
400
|
|
600
|
|
800
Number of Tables(T)
|
|
1,000
|
Graphical Representation
(T = 0, C = 1,000)
Number of Chairs (C)
1,000 –
(T = 100, C = 700)
–
(T = 0, C = 600)
800 –
Painting Constraint
–
600 –
(T = 300, C = 200)
Carpentry Constraint
–
400 –
–
(T = 500, C = 200)
200 –
(T = 500, C = 0)
–
0 –|
0
(T = 800, C = 0)
|
|
200
|
|
400
|
|
600
|
|
800
Number of Tables(T)
|
|
1,000
|
Graphical Representation
Painting Constraint
Number of Chairs (C)
1,000 –
Infeasible Solution (T = 50, C = 500)
–
800 –
Maximum Tables Required Constraint
–
600 –
Maximum Chairs Allowed Constraint
–
400 –
–
200 –
–
0 –|
0
|
Feasible
Region
|
|
|
200
400
(T = 300, C = 200)
Carpentry Constraint
Infeasible Solution
(T = 500, C = 200)
|
|
600
|
|
800
Number of Tables(T)
|
|
1,000
|
Using Level Lines
800 –
(T = 0, C = 560)
–
(T = 0, C = 420)
Number of Chairs
(C)
600 –
–
Feasible Region
400 –
–
(T = 300, C = 0)
200 –
–
0–
0
(T = 400, C = 0)
|
|
200
|
|
|
400
Number of
Tables(T)
|
600
|
|
800
|
|
1,000
Using Level Lines
800 –
Optimal Level Profit Line
–
Number of Chairs
(C)
600 –
–
Carpentry Constraint
2
3
Optimal Corner Point Solution
400 –
4
–
Level Profit Line with No
Feasible Points ($7T + $5C =
$4,200)
200 –
Painting Constraint
–
0–
0
|
1
|
200
|
|
|
|
400 5
600
Number of
Tables(T)
|
|
800
|
|
1,000
Using All Corner Points
Point 1
(T = 100, C = 0)
Profit = $7 x 100 + $5 x 0
=
$700
Point 2
(T = 100, C = 450)
Profit = $7 x 100 + $5 x 450 = $2,950
Point 3
(T = 200, C = 450)
Profit = $7 x 200 + $5 x 450 = $3,650
Point 4
(T = 320, C = 360)
Profit = $7 x 320 + $5 x 360 = $4,040
Point 5
(T = 500, C = 0)
Profit = $7 x 500 + $5 x 0
= $3,500
Minimization (Cost) Problem
• The Holiday Meal Turkey Ranch has two different
brands of turkey feed – brand A and brand B. Each
feed contains different quantities of three
nutrients (protein, vitamins, and iron) essential for
fattening turkeys. Brand A feed costs $0.10 per lb
and brand B costs $0.15 per lb. The ranch wants to
determine the quantity of feed to use to meet the
minimum monthly requirements at the lowest cost.
Minimization Problem
• Minimize cost
• Holiday Meal Turkey Ranch
• Two types of feed
Minimize cost = $0.10A + $0.15B
subject to
5A + 10B ≥
4A + 3B ≥
0.5A
≥
A,B
≥
45
24
1.5
0
(protein required)
(vitamin required)
(iron required)
(nonnegativity)
Minimization Problem
10 –
Iron Constraint
Pounds of Brand B (B)
9–
8–
7–
Feasible Region is Unbounded
6–
Vitamin Constraint
5–
4–
1
3–
2
Protein Constraint
2–
1–
0–
0
|
3
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
Pounds of Brand A (A)
Figure 2.8
Calculating a Solution
• Optimal point 2 is the intersection
of two constraints, vitamin and
protein
• Solving simultaneously
4(5A + 10B = 45)
– 5(4A + 3B = 24)
implies
20A + 40B = 180
implies – (20A + 15B = 120)
25B = 60
implies
B = 2.4
and
A = 4.2
Special Situations
• Redundant Constraints
• Do not affect the feasible region
• Changed constraint in Flair
Furniture problem
T ≥ 100 becomes T ≤ 100
Special Situations
C ≤ 450
–
800 –
–
Carpentry Constraint Is Redundant
600 –
–
400 –
–
200 –
–
0 –|
0
Feasible Region
Number of Chairs (C)
1,000 –
Constraint Changed to T ≤ 100
Painting Constraint
Is Redundant
|
|
200
|
|
400
|
|
600
|
|
800
Number of Tables(T)
|
|
|
1,000
Figure 2.10
Special Situations
• Infeasibility
• No one solution satisfies all
the constraints
• Changed constraint in Flair
Furniture problem
T ≥ 100 becomes T ≥ 600
Special Situations
Constraint Changed
to T ≥ 600
Number of Chairs (C)
1,000 –
C ≤ 450
–
800 –
Two Regions
Do Not
Overlap
–
600 –
Region
Satisfying
Fourth
Constraint
3T + 4C ≤ 2,400
–
400 –
–
200 –
–
0 –|
0
2T + C ≤ 1,000
Region
Satisfying
Three
Constraints
|
|
200
|
|
400
|
|
600
|
|
800
Number of Tables(T)
|
|
|
1,000
Figure 2.11
Special Situations
• Alternate Optimal Solutions
• More than one solution satisfies all the
constraints
• Changed objective in Flair Furniture
problem
$7T + $5C becomes $6T + $3C
Special Situations
800 –
Level Profit Line for Maximum Profit
Overlaps Painting Constraint
–
Number of Chairs
(C)
600 –
–
Level Profit Line Is Parallel to Painting Constraint
2
3
$6T + $5C = $2,100
400 –
4
–
Optimal Solution Consists of All
Points Between Corner
Points 4 and 5
200 –
Feasible
Region
–
0–
0
|
1
|
200
|
|
|
|
400 5
600
Number of
Tables(T)
|
|
800
|
|
1,000
Figure 2.12
Special Situations
• Unbounded Solution
• May or may not have a finite
solution
• Usually improper formulation
• Changed objective in Holiday Meal
problem
Minimize = $0.10A + $0.15B
becomes
Maximize = 8A + 12B
Special Situations
10 –
Iron Constraint
Pounds of Brand B (B)
9–
8–
Unbounded
Feasible Region
7–
6–
5–
4–
3–
2–
1–
0 –|
0
Vitamin
Constrain
t Protein Constraint
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
Pounds of Brand A (A)
Figure 2.13
Using Excel’s Solver
• Excel’s built-in LP solution tool for LP
• Commonly available and easy access
• Familiar software
Using Solver
Using Solver
Using Solver
Using Solver
Using Solver
Screenshot 2-1D
Using Solver
Using Solver
Using Solver
Using Solver
Using Solver