Linear Programming for Production Planning Analysis Report

This report delves into the application of linear programming to optimize production planning and maximize profitability. It begins with an introduction to linear programming, defining its core components: decision variables, objective function, and constraints. The report uses a case study involving the production of items X, Y, and Z over a four-month period to illustrate the model's practical application. It formulates the algebraic linear programming model, detailing decision variables, the objective function aimed at maximizing total profits, and various constraints related to raw materials, machine hours, and product demand. The report also addresses variable value restrictions, emphasizing the non-negativity and integer constraints. The report concludes by highlighting the importance of linear programming as a decision-making tool, particularly in optimizing profitability using tools like Excel Solver, and suggests that managers can improve their decision-making by using linear programming. The report also includes references to relevant academic sources.

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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

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Total machine activity
A 13550 <= 20000
B
20520.8333
3 <= 35000
C
34508.3333
3 <= 40000
In addition to the items mentioned in the table above, the productions and sales of X, Y
and Z are also affected by the polices and demand of the products. For instance, in every month
the business’s policy is to meet at least 50% of the product’s demand (Arsham, 2009). The firm
on the other hand have no storage facilities hence production is limited to the quantity of goods
that are projected to be demanded in the four months period.
Variable value restriction
In a production process, the quantity of items being produced can never be a negative
number, for this reason, all the decision variables ought to resume a non-negative value.
Moreover, products are produced in whole units hence the quantity being produced need to be an
integer (Lin, 2010). This is because an item cannot be sold in a fraction form. These two value
restrictions will form part of the constraints when solving the linear programming model using
excel solver.
Conclusion
Linear programming is a vital tool that can be very effective in assisting the managers
make critical decisions. The linear model is executable in simple business software like Excel
and therefore should be adopted by firms operating in various industries. From the model that
was developed in the case study, we have noted how vital the solver tool is when it comes to
optimising the profitability of a firm. The model output can also be applied in identifying the
fatal constraints that are causing the firm a fortune. For constraints that are within the managers
say such as business policies, they need to be revised so as to ensure the firm is earning more
profit with every unit being produced.

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References
Arsham, H., 2009. A simplified algebraic method for system of linear inequalities with LP applications.
International Journal of Management Science, Volume 37, p. 876–882.
Arsham, H., 2013. An interior boundary pivotal solution algorithm for linear programs with the optimal
solution-based sensitivity region. International Journal of Mathematics in Operational
Research.
Gerard, S. & Diptesh, G., 2010. Networks in Action; Text and Computer Exercises in Network
Optimization, s.l.: Springer.
Gerard, S. & Yori, Z., 2015. Linear and Integer Optimization: Theory and Practice, s.l.: CRC Press.
Lin, C., 2010. Computing shadow proces/costs of dengernerate LP problems with reduced simplex
tables. Expert Systems with Applicationsplex tables, Volume 37, p. 5848–5855..
Wang, J., 2014. Management Science, Logistics, and Operations Research. 2014 ed. s.l.:IGI-Global
Publisher,.
Williams, H. P., 2013. Model Building in Mathematical Programming. Fifth ed. s.l.:s.n.