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Linear Programming Models:
Graphical Methods
5/4/1435 (1-3 pm)
noha hussein elkhidir
• Requirements of a Linear Programming
Problem
•
Formulating Linear Programming
Problems.
• Graphical Solution to a Linear
Programming Problem
- Graphical Representation of Constraints
-
Corner-Point Solution Method
5/4/1435 (1-3 pm)
noha hussein elkhidir
Linear Programming
Applications
 Production.
 Diet Problem Example
 Labor Scheduling Example.
5/4/1435 (1-3 pm)
noha hussein elkhidir
When you complete this module
you should be able to:
- Formulate linear programming models,
including an objective function and
constraints
- Graphically solve an LP problem with the
corner-point method.
- Construct and solve a minimization and
maximization problem
5/4/1435 (1-3 pm)
noha hussein elkhidir
- Formulate production-mix, diet, and labor
scheduling problems
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noha hussein elkhidir
What about Linear
Programming?
- A mathematical technique to help plan and
make decisions relative to the trade-offs
necessary to allocate resources
- Will find the minimum or maximum value of
the objective
- Guarantees the optimal solution to the
model formulated
5/4/1435 (1-3 pm)
noha hussein elkhidir
Requirements of an
LP Problem
1. LP problems seek to maximize or
minimize some quantity (usually profit or
cost) expressed as an objective function
2. The presence of restrictions, or
constraints, limits the degree to which we
can pursue our objective
5/4/1435 (1-3 pm)
noha hussein elkhidir
3. There must be alternative courses of
action to choose from
4. The objective and constraints in linear
programming problems must be
expressed in terms of linear equations or
inequalities
5/4/1435 (1-3 pm)
noha hussein elkhidir
Steps in Developing a Linear Programming
(LP) Model
1) Formulation
2) Solution
5/4/1435 (1-3 pm)
noha hussein elkhidir
Properties of LP Models
1) Seek to minimize or maximize
2) Include “constraints” or limitations
3) There must be alternatives available
4) All equations are linear
5/4/1435 (1-3 pm)
noha hussein elkhidir
Example LP Model Formulation:
The Product Mix Problem
1) Decision: How much to make of > 2
products?
2) Objective: Maximize profit
3) Constraints: Limited resources
5/4/1435 (1-3 pm)
noha hussein elkhidir
Example: Flair Furniture Co.
Two products: Chairs and Tables
Decision: How many of each to make this
month?
Objective: Maximize profit
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noha hussein elkhidir
Flair Furniture Co. Data
Tables
Chairs
(per table)
(per chair)
Profit
Contribution
$7
$5
Hours
Available
Carpentry
3 hrs
4 hrs
2400
Painting
2 hrs
1 hr
1000
Other Limitations:
Make no more than 450 chairs
Make at least 100 tables
5/4/1435 (1-3 pm)
noha hussein elkhidir
Decision Variables:
T = Num. of tables to make
C = Num. of chairs to make
Objective Function: Maximize Profit
Maximize $7 T + $5 C
5/4/1435 (1-3 pm)
noha hussein elkhidir
Constraints:
• Have 2400 hours of carpentry time
available
3 T + 4 C < 2400 (hours)
• Have 1000 hours of painting time available
2 T + 1 C < 1000 (hours)
5/4/1435 (1-3 pm)
noha hussein elkhidir
More Constraints:
• Make no more than 450 chairs
C < 450
(num. chairs)
• Make at least 100 tables
T > 100
(num. tables)
Nonnegativity:
Cannot make a negative number of chairs or tables
T>0
C>0
5/4/1435 (1-3 pm)
noha hussein elkhidir
Model Summary
Max 7T + 5C
(profit)
Subject to the constraints:
3T + 4C < 2400
(carpentry hrs)
2T + 1C < 1000
(painting hrs)
T
C < 450
(max # chairs)
> 100
(min # tables)
T, C > 0
5/4/1435 (1-3 pm)
noha hussein elkhidir
(nonnegativity)
Graphical Solution
• Graphing an LP model helps provide
insight into LP models and their solutions.
• While this can only be done in two
dimensions, the same properties apply to
all LP models and solutions.
5/4/1435 (1-3 pm)
noha hussein elkhidir
Carpentry
Constraint Line
C
3T + 4C = 2400
Infeasible
> 2400 hrs
600
Intercepts
(T = 0, C = 600)
(T = 800, C = 0)
Feasible
< 2400 hrs
0
0
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noha hussein elkhidir
800 T
C
1000
Painting
Constraint Line
2T + 1C = 1000
600
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)
0
0
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noha hussein elkhidir
500
800 T
Max Chair Line
C
1000
C = 450
Min Table Line
600
450
T = 100
Feasible
0
Region
0 100
5/4/1435 (1-3 pm)
noha hussein elkhidir
500
800 T
C
Objective
Function Line
7T + 5C = Profit
500
Optimal Point
(T = 320, C = 360)
400
300
200
100
0
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noha hussein elkhidir
0
100
200
300
400
500 T