Linear Programming: Algebraic Formulation and Optimization

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Linear Algebra
Student Name
Institution Name

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Introduction
Linear programming by definition is the section of arithmetic which is designed to offer
solutions for optimization issues where all the limiting factors as well as the objective function
can be expressed as linear functions. The concept was developed by George B. Denting in 1947.
The idea of linear programming was initially applied in the second world war from where it
gained popularity and is currently being applied in several other fields (Arsham, 2013). The
idea under linear programming is meant to assist decision makers under situations of certainty.
That is when all the courses of options that are available to a firm can be identified and the firm’s
objective together with its constraints can be assigned numerical values. The course of action is
selected at the optimal point out of all the possible valid alternatives (Gerard & Yori, 2015).
Linear programming can also be applied to conduct verifications and to check mechanisms to
ascertain their accuracy. The reliability of decisions taken can also be vetted by applying linear
programming. This way the managers experience can be applicable hand in hand with linear
programming and hence yield an optimal decision. In this report the main objective is to assist
the production manager to make an effective planning for the four months period so as to achieve
an optimal profitability from the production and sales of items X, Y and Z.
Formulation of algebraic linear programming
Linear programming model is composed of 4 parts that is the decision variables,
objective function, constraints as well as the variable value restriction. Each of these components
are discussed in details below.
Decision variables
The decision variables also termed as activity variable is the activities which are
competing with other variables for the limited production and sales resources. For instance, in
the business case study in question the variables are the units of products X, Y and Z that the
firm needs to produce (Gerard & Diptesh, 2010). The variables are dependent when it comes
to utilization of the scarce resources and hence there is need to provide a simultaneous solution
to them. The relationship between the variables are assumed to be linear. The table below
represents the decision variables for the excel model.

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Decision Variables
Month Month 1 Month 2 Month 3 Month 4
Production (units)
x 1350 1000 1400 1500
y 4000 4000 8800 9000
z 4000 2100 5400 4800
Objective function
In a linear programming model, we ought to have a clearly defines and unambiguous
objective function which will be optimized by the solution. The function is expressed as linear
function of the decision variables (Williams, 2013). The single objective optimization is one of
the most important features when developing a linear programming model. In this model the
objective is to maximum the profits from the production and sales of items X, Y and Z. this is
represented by;
Objective function
Total profits
£17,897,500.6
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The total profit is given by the function =SUM (B76:E78) which makes it a linear function of the
decision variables.
Constraints
Constraints are the different kinds of limitations on the available resources. For all the
organisations that are in the manufacturing sector important factors such as raw materials,
machine hours as well as labour are always limited in supply (Wang, 2014). For this model,
there are two categories of resources that are limited that is the raw material and machine hours
necessary for the production of the three items. The table presents the constraints of the model.
Constraint
Total raw material
Steel 814600 <= 1000000
Protection 183.3 <= 200000