Linear Programming

LINEAR PROGRAMMING
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LINEAR PROGRAMMING
Name of student:
Name of Institution:
Date:

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Question one
a. Explain why a linear programming model would be suitable for this case study
Solution:
A linear programming model is a mathematical technique that is used to solve problems that
require the optimization of resources. The optimization can be either minimization or
maximization of the resources (Carmana, et al., 2018). Minimization is done to reduce the
cost of materials or a production process. On the other hand, maximization is done to increase
the net profits by ensuring a proper utilization of the resources that are at hand. The case study in
our scenario is a linear programming model because the aim of the factory is to minimize the
cost producing beverages. The case study has decision variables, the constraint variables and the
objective function. The decision variables are the amounts of mangoes, limes and oranges that
are used in making the beverage. The constraints are the available amounts or units or mangoes,
limes and oranges. The objective function is the cost function (Tamas & Viola, 2011).
b. Formulate a linear programming model
Solution:
A linear programming is formed by stating the objective function, the constraints and the
decision variables. Let beverage A be denoted by A and beverage B denoted by B. Further, let
oranges be denoted by or. Let mangoes be denoted by ma. Let lime be li (Ozdaglar, et al.,
2011).
The decision variables:
Lime
Oranges
Mangoes

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Constraints:
or<=4.5
ma<=5
li<=6
4.5or+5ma+6li<=100
The Objective function:
8A+7B
c. Finding the optimum solution by graphical method
Solution:
Lime Orange Mango Cost
A 3 6 4 8
B 8 4 6 7
Constraints A B
Lime 5 6 70
Orange 5 4 70
Mango 5 5 70

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1 2 3
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
5 6
0
70
0
5 4
0
70
0
5 5
0
70
0
Graphical Solution
Lime Orange Mango
a. Is there a range for the cost ($) of A that can be changed without affecting the
optimum solution obtained above?
Solution:
The range of cost of A cannot be changed since all the constraint variables have a similar value.
Question Two
a. Choose appropriate decision variables, formulate a linear programming model to
determine the optimal product mix that maximizes the profit.
Solution:
The objective of the factory is to maximize the profit, while satisfying the cotton ad wool
proportion constraints. To formulate a linear programming model, the objective function, the
constraints and the decision variables must be determined (Fouad & Hotem, 2010). Let spring