Linear Programming: Building the Model
The Problem Scenario:
EZ Co. produces two types of running shoes, Aeros and Ballistics. These shoes are
produced in two distinct departments--basic design and finishing. Aeros require 2 hours in the
basic department, 1 hour in finishing, and produce a profit per unit of $6. Ballistics require 2
hours in basic, 2 hours in finishing, and produce a profit of $8 per unit. Currently, the basic and
finishing departments have 20 and 12 hours available, respectively, for production each day.
In addition, marketing research has revealed that no more than 9 Aeros and 4 Ballistic
can be sold in a day. EZ Co. would like to determine the production mix of Aeros and Ballistics
that will maximize profits.
(1) Problem (or Opportunity) Statement: “The undesirable (or unexplored) situation which
should be addressed is…”
EZ Co may not be using their resources in the most efficient manner. (How is
efficiency defined here?) By changing their resource allocation, EZ may be able to
increase their profits.
(2) Formulate a model to represent the problem
Objective Statement: “The objective or purpose of any action the organization takes is
to…”
Maximize Profits
Decision Alternatives: “There are several possible actions that this organization can
take in response to the problem (or opportunity). These include …”
Every possible production mix of Aeros and Ballistics that is feasible with respect to
the constraints placed on the production process represents a decision alternative.
Criterion Measures: “The quantitative meaure(s) that will be used to judge the
“goodness” of any of the decision alternatives is (are)…”
Profit = 6 x (# of Aeros made) + 8 x (# of Ballistics made)
Data: “The available data sets describing important exogenous variables (to be used with
the model) and parameters (to be used as fixed inputs), along with an assessment of the
data validity for this decision situation, are …”
•
•
•
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The profit per unit for Aeros and Ballistics.
The number of hours in the two departments required to make Aeros and
Ballistics.
The number of hours available in the basic and finishing shops.
The market capacities.
Our Model:
What are the decision variables in this problem?
x A  Number of Aeros to produce
x B  Number of Ballistics to produce
How can we write our objective in terms of these variables?
max z  6 xA  8xB
What are the constraints? Can we express them in terms of our decision variables?
2 x A  2 xB  20
[1]
x A  2 xB  12
[ 2]
xA
xA
9
[3]
xB  4
[ 4]
0
[5]
xB  0
[ 6]
A Graphical Representation (Note: this is not a necessary step in the decision making
approach.)
In summary, here are the steps to solve a two-variable problem graphically:
(1) Develop an algebraic representation.
(2) Graph each constraint as an equality. This should be a straight line.
(3) Determine which side of each constraint is feasible. If the original constraint was an
(4)
(5)
(6)
(7)
(8)
(9)
equality, then only the line represents feasible points.
Determine set of points in the intersection of all constraints. This is the feasible region.
Select an (semi) arbitrary value for the objective, and graph the points which yield this value.
This should also be a straight line.
Perform step 5 again for another value of the objective. This should be a straight line parallel
to the one in step 5.
Determine the direction which improves the objective function (called the gradient). Notice
that the direction of improvement is perpendicular to the two parallel lines in steps 5 and 6.
Determine which point in the feasible region yields the best objective function value. This
will occur at the intersection of two constraints which hold as equalities.
Solve the two equalities for the two unknowns (optimal values of our decision variables).
Substitute these values into the objective function to determine the optimal value of the
objective function.
(3) Develop a computer-based approach to develop problem solutions from the model.
Next class!
(4) Results: “Based upon the model, the following numerical results were found and sensitivity
analyses of key input estimates showed the following...”
EZ Co should produce 8 Aeros and 2 Ballistics for a total profit of $64.
(5) Recommendations: “Based on the above analyses, the recommended or most attractive
alternative(s) is (are)...and, additional analysis of the following issues may be useful…”
Taking into account all aspects of the situation, the recommended production is….
(6) Implement the adopted recommendations.
Homework:
(1) Kelli Kennels provides overnight lodging for a variety of pets. A particular feature at Kelli’s
is the quality of care the pets receive, including excellent nutrition. The kennel’s dog food is
made by mixing two special dog food products to obtain what the kennel calls the “wellbalanced doggie diet.” Data of importance for the two dog foods are as follows:
Dog Food Type
Cost/lb.
Protein (oz/lb)
Fat (oz/lb)
Bark bits
.80
3
2
Canine Chow
.60
2
4
If Kelli wants to be sure that the dogs receive at least 11 ounces of protein and at least 10
ounces of fat per day, what is the minimum cost mix of the two dog food products?
(a) Formulate the problem algebraically.
(b) Solve the problem graphically. State in a sentence what the optimal solution means, and
determine the minimum cost mix.
(2) Solve the following problem graphically.
max 300𝐷 + 500𝑊
subject to:
𝐷
≤4
2𝑊 ≤ 12
3𝐷 + 2𝑊 ≤ 18
𝐷 ≥ 0, 𝑊 ≥ 0