Linear Programming Calculator is a free online tool that displays the best optimal solution for the given constraints. BYJU’S online linear programming calculator tool makes the calculations faster, and it displays the best optimal solution for the given objective functions with the system of linear constraints in a fraction of seconds. Linear programming deals with only single objective, whereas in real-life situations a decision problem may have conflicting and multiple objectives. 2.5 APPLICATION AREAS OF LINEAR PROGRAMMING Linear programming is the most widely used technique of decision-making in business and industry and in various other fields. The treatment of applications covers the transportation problem and general linear programming applications, and a final part examines nonlinear programming. Numerical examples and exercises with selected answers appear in every chapter. ADVERTISEMENTS: This article throws light upon the top three examples on the application of linear programming. Production Allocation Problem: A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three. One application of linear programming in marketing is media selection. LP can be used to help marketing managers allocate a fixed budget to various advertising media. The objective is to maximize reach, frequency, and quality of exposure. Restrictions on the allowable allocation usually arise during consideration of company policy, contract.

This article throws light upon the top three examples on the application of linear programming.

Example # 1. Production Allocation Problem:

A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three machines are given in the table below.

It is required to determine the daily no. of units to be manufactured for each product. The profit per unit for product 1, 2 and 3 is Rs. 4, Rs. 3 & Rs. 6 respectively. It is assumed that all the amounts produced are consumed in the market.

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Formulation of Linear Programming Model:

Step 1:

From the study of the situation find the key-decisions to be made. This connection, looking for variables helps considerably. In the given situation key decision is to decide the extent of products 1, 2 and 3, as the extents are permitted to vary.

Step 2:

Assume symbol for variable qualities noticed in step 1. Let the extents of product. 1, 2, and 3 manufactured daily be, x₁, x₂ and x₃ respectively.

Step 3:

Express the feasible alternatives mathematically in terms of variables. Feasible alternatives are those which are physically, economically and financially possible. In the given situation feasible alternatives are sets of values of x₁ x₂ and x₃.

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Where x₁, x₂, x₃ > 0 …(1)

Since negative production has no meaning and is not feasible.

Step 4:

Mention the objective quantitatively and express it as a linear function of variables. In the present situation: objective is to maximize the profit.

i.e., which maximize Z = 4x₁ + 3x₂ + 6x₃ … (2)

Step 5:

Put into words the influencing factors or constraints. These occur generally because of constraints on availability or requirements. Express these constraints also as linear equalities / inequalities in terms of variable.

Here, constraints are on the capacities and can be mathematically expressed as

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2x₁ + 3X₂ + 2X₃ ≤ 440

4x₁ + 01X₂ + 3x₃ ≤ 470 …(3)

2x₁ + 5x₃ + 0x₃ ≤ 430

Example # 2. Production Planning Problem:

A factory manufactures a product each unit of which consists of 5 units of part A and 4 units of part B. The two parts A & B require different raw materials of which 120 units & 240 units respectively are available. Three parts can be manufactured by three different methods. Raw material requirements per production run and the number of units for each part produced are given below.

Determines the number of production runs for each method so as to maximize the total no. of complete units of the final product.

Formulation of Linear Programming Model:

Step 1:

The key decision to be made is to determine the number of production runs for each method.

Step 2:

Let x₁, x₂, x₃ represents the number of production runs for method 1, 2 and 3 respectively

Step 3:

Feasible alternative are the sets of values of x₁, x₂, and x₃ where x_v x₂, x₃ ≥ 0 …(1)

Since negative no. of production runs has no meaning and is not feasible.

Step 4:

The objective is to maximize the total no. of units of the final product. Now the total no. of units of part A produced by different methods is (6x₁ + 5x₂ + 7x₃) and for part B is (4x₁ + 8x₂ + 3x₃). Since each unit of the final product requires 5 units of part A and 4 units of part B, it is evident that the maximum no of units of the final product cannot exceed the smaller value of;

Step 5:

Constraints are on the availability of raw material they are for raw material 1,7x₁ + 4x₂ + 2x₃ ≤ 120 …(3)

& raw material 2, 5x₁ + 7x₂ + 9x₃ ≤ 240

.^.. The LPP of this problem

Example # 3. Product Mix Problem:

A chemical company produces two products x and y, each unit of product x requires 3 hours on operation 1 & 4 hours on operation II. While each unit of product;’ requires 4 hours on operation 1 and 5 hours on operation II. Total available time for operation 1 and II is 20 hours and 26 hours respectively. The production of each unit of product, y also result in two units of a by-product z at no extra cost.

Product x sells at a profit of Rs. 10/ unit, Whiles sells at a profit of 20/unit. By product z brings a unit profit of Rs. 6 if sold; in case of it cannot be sold the destruction cost is Rs. 4 unit. Forecasts indicate that not more than 5 units of z can be sold. Determine the quantities of x and y to be produced keeping z in mind, so that the profit earned is maximum.

Formulation of L. P. Model:

Step 1:

The key decision to be made is to determine the no. of units of products x, y and z to be produced.

Step 2:

Let the no. of units of products x, y, z produced by x₁, x₂, x₃ where

x₃ = no. of units of z produced

= no. of units of z sold + ——- z destroyed

= x₃ + x₄ (say)

Step 3:

Feasible alternatives are sets of values of x₁, x_?, x₃ & x₄, where x₁, x₂, x₃, x₄ ≥ 0

Step 4:

Objective is to maximize the profit, objective function (profit function) for products x and y is a linear because the profits are constant irrespective of the no. of units produced.

Thus the objective function is maximize z = 10x₁ + 20x₂ + 6x₃ – 4x₄

Step 5:

Constraints are on the time available on operation I: 3x₁ + 4x₂ ≤ 20

——II: 4x₁ + 5x₂ ≤ 26

On the number of units of product z sold: x₃ ≤ 5 produced

2x₂ = x₃ + x₄

or -2x₂ + x₃ + x₄ = 0

Example 4: [Diet Problem]:

A person wants to decide the constituents of a diet which will fulfill his daily requirement of proteins, fats and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. The yield per unit of those foods are given below.

Formulate linear programming model for the problem.

Formulation of L. P. Model

Step 1:

Key decision is to determine the number of units of food type 1, 2, 3, & 4 to be used.

Step 2:

Let three units be x₁, x₂, x₃ & x₄ respectively

Step 3:

Feasible alternatives are sets of value of x_j

Where x_j > 0, J = 1,2,3,4 …(1)

Step 4:

Objective is to minimize the cost i.e., minimize z = (54 x₁ +49 x₂ + 89 x₃ + 75 x₄) …(2)

Step 5:

Linear Programming Application In Business Pdf

Constraints are on the fulfillment of the duty requirements of the various constituents.

i. e., for proteins 5x, + 6x₂ + 9x_} + 3x₄ ≥ 900

For fats x₁ + 4x₂ + 4x₃ + 5x₄ ≥ 300 … (3)

For carbohydrates 3x₁+ 2x₂ + 6x₃ + 2x₄ ≥ 800

Thus the L.P. Model is to determine the no. of units of x₁,x₂, x₃ & x₄ that minimize eq. (2) Subject to constraints eq. (3) and non-negatively equations (1).

Example 5:

A ship was there cargo loads – forward after and centre, the capacity limits are:

The following cargoes are offered, the sheep owner accept all or any part of each commodity:

In order to preserve the trim of the ship, the weight in each load must be proportional to the capacity in tonnes. The cargo is to be distributed so as to maximize the profit. Formulate the problem as LP model.

Solution:

Consider the decision variable as:

X_iA, x_iB and x_ic – Weight (in kg.) of commodities A, B, & C to be accommodated in the direction i.e., (1, 2, 3 – forward, centre and after) respectively.

P. N. Ezra¹, A. V. Oladugba¹, F. O. Ohanuba¹, N. O. Igwe², C. A. Okonta²

¹Department of Statistics, University of Nigeria, Nsukka

²Department of Maths / Statistics, Akanu Ibiam Federal Polytechnic, Unwana

Correspondence to: P. N. Ezra, Department of Statistics, University of Nigeria, Nsukka.

Email:

Copyright © 2020 The Author(s). Published by Scientific & Academic Publishing.

Linear Programming Application Problems

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

Nigerian Bottling Company is into production and selling of such products as Orijin Bitter Large bottle, Orijin Bitter Sachet, Harp, Dubic Malt, Small Stout, Medium Stout, and Mac-dowels Large Spirit. Production has to do with transforming raw materials into products in order to maximize a Company’s revenue. It has been the quest of Nigerian Bottling Company to have a structured process with documented steps and measurable results so as to establish a trend for the profit making of the business. In view of this, in this study, linear programming model was used to allocate raw materials (resources) to the production of the above mentioned Company’s products so as to know the right product mix that actually maximizes the Company’s profit and in what capacities. Data collected on these known products were analysed using simplex algorithm of linear programming with the help of Tora Software and the result showed that 1,732 crates of Dubic Malt with the objective coefficient (490), 144 cartons of Orijin bitter sachets with the objective coefficient (370) and 8,227 crates of Mac-dowel Spirit with the objective coefficient (1000) sold in a day yielded a profit of ₦9,129,117.24. Also, sensitivity analysis was carried out to know the maximum and minimum unit profits (i.e, the objective function cost coefficients) of the concerned products and the resource quantities within which the maximum profit of the Company should be maintained. The result of the sensitivity analysis also revealed the shadow price (dual price) which is the cost of acquiring an additional resource in case the Company has any of such needs. Based on the results obtained from this research, it is recommended that Nigerian Bottling Company should invest more in the production of those products that maximize their profit.

Keywords: Linear Programming, Optimization, Objective function, Inequality Constraints and Optimal Solution

Cite this paper: P. N. Ezra, A. V. Oladugba, F. O. Ohanuba, N. O. Igwe, C. A. Okonta, Application of Linear Programming on the Profit Maximization of Nigerian Bottling Company, American Journal of Operational Research, Vol. 10 No. 2, 2020, pp. 39-43. doi: 10.5923/j.ajor.20201002.03.

1. Introduction

Every business organization seeks the best way to go about their business in order to maximize profit with their available limited resources. Optimal production planning and care are therefore required in order to sustain the optimal profit making and existence of such organization: See [9] and [11]. Production planning in this regard has to do with putting in place various activities and measures in order to ensure optimal production that satisfies customers’ demands, considering the fact that the real world resources such as material, money, manpower, space and time are limited: See [7], [8] [12] and [14]. Every company has a number of product lines which consist of the product mix that such a company deals on. Product-mix determination is very important as it helps a given company to focus its production on relevant products that enhance profit making. Nevertheless, a company may be tempted to add more product lines in their quest to have more customers; thereby entering into the risk of including such products that appeal only to a fraction of its customer base. Considering the fact that different products require different amounts of production resources having different costs and revenues at different stages of production, there is need for a mathematical technique that can determine the product mix that will maximize the total profit: See [5] and [14]. This mathematical technique is called “Linear Programming (LP) model.”Linear Programming also called linear optimization is a technique for the optimization of a linear objective function, subject to linear inequality and sometimes equality constraints: See [2], [3] and [4]. It is an operational research technique used to allocate limited production resources for a firm’s best practices: See [1], [3] and [12]. This calls for model formulation which is very important as it represents a real life situation or system for competent decision analysis: See [2], [3] and [7]. Most times, production mix problems reach complexity when for instance about five (5) or more products of the same department are involved and this makes algebraic solution difficult and cumbersome. A procedure for solving any large problem of such nature is the simplex method (tableau) which uses iterative procedures in solving linear programming problems: See [4]. The manual solution of a linear programming model using the simplex method can be lengthy and tedious but can be quicker and accurate when computer (TORA software or any other software that can handle LP) is used. [11] used LP model to schedule drivers in a transport co-operation so as to determine the minimum number of drivers needed for each shift in a day in order to reduce the amount spent for the reserved drivers. [15] considered linear programming model as a quantitative decision making tool needed for proper optimization of product mix in an apparel industry. [8] used big M and dual simplex methods as alternative methods of finding solutions to linear programming that can reduce the number of iteration and also save valuable time. [1] used the simplex algorithm of linear programming for the optimal allocation of raw materials for the production of different sizes of bread in a bakery industry. [14] discovered that intercept values could be used to find out the equality constraints among the inequality constraints in order to achieve optimal solutions while solving linear programming problems. [10] used simplex method of linear programming to determine which bread size contributed the highest to the profit maximization of the company. Sensitivity analysis was also carried out to determine the minimum and maximum cost coefficients of the bread sizes within which the company’s optimal profit could be maintained.The purpose of this research is to determine how the limited resources (raw materials) of Nigeria Bottling Company could be allocated in order to maximize profit. It is aimed at determining the product(s) that contribute(s) maximally to such profit. The data on which this is based are quantities of raw materials available in stock, cost and selling prices and also the profits from crates of each product. The profit constitutes the objective function while raw materials available in stock are used as constraints. In every optimization problem, due to the limited nature of resources, it is always of interest not only to obtain an optimal solution of the problem but also to find how stable that optimal solution would be, assuming there changes either the objective function’s cost coefficient or the right hand side resource constraint. In view of this, we carried out sensitivity analysis so as to determine the quantities of the variables (the company’s products) that would give the optimal solution, the range of values of such variables within which the optimal solution should not deteriorate and also the shadow price (also known as the dual price) which is the price for acquiring an additional unit of each resource (in case the company runs short of any) for optimal production.

2. Material and Methodology

2.1. Material

Table 1. Quantity of raw materials available

Table 2. Quantity of raw materials needed to produce a crate of each product

Table 3. Products and their prices (in naira) per crate

2.2. Methodology

2.2.1. Linear Programming Model

Linear Programming Applications Doc

In order to grant the quest of the Nigerian Bottling Company on establishing a structured process with documented steps and measurable results in allocating their raw materials for optimal production that will enhance the profit making of the business, the maximization linear programming model given below has been considered the most efficient.

(1)

The function, Z = C^Tx which represents the profit to be maximized is called the objective function. The inequalities Ax ≤ b and x ≥ 0 are the functional and non-negative constraints respectively, which specify the feasible region within which the objective function is optimized.x = (x₁, x_2,…,x_n) represents the vector of variables (to be determined), C^T = (C₁, C₂, …, C

Linear Programming Applications Examples

_n) is the vector of the cost coefficients of the objective function, b = (b_1, b₂, …, b_m) is the vector of the right hand side (RHS) constraints (resources), A = (a_ij) is a (known) matrix of the constraint coefficients. The above model in a standard form is as given below:

(2)

For this study, x_i, i

Linear Programming Applications Pdf

= 1,2,…,nrepresents the number of crates needed of each of the company’s products; C_i is the per unit (crate) profit of Product _i; a_ij is the amount of resource j needed for the production of one crate of product i

Linear Programming Applications In Business

and b

Linear Programming Applications

_j, j =1,2,…,m is the total quantity of resource _j available. The decision variables, x_i, i = 1,2,…,n are as defined below:

3. Data Analysis

In order to achieve the purpose of this study which is to obtain the best product mix that will lead to maximization of the Company’s profit, simplex method procedures outlined in Section 2 were applied to the data presented in Tables 1, 2, and 3. The models consist of the objective function and the constraints as shown by (2). For the profit maximization of the concerned company, the LP model was formulated using the data values in Tables 1, 2, and 3 as follows: The problem in standard form is as shown below:Where x₈, x₉, x₁₀ and x₁₁ are the added slack variables.

Table 4. Results of the Primal analysis

3.1. Sensitivity Analysis

In this section, sensitivity analysis was carried out to know the extent to which any of the input data (that is, the per unit profit of any of the products (variables) or the available resources) of the company can change without affecting the optimal solution. The result of the sensitivity analysis also showed the dual prices also known as the shadow prices to enable the company know the cost of acquiring an additional unit of any of their resources in case there is need for that. In carrying out sensitivity analysis using the dual variables, the dual problem was formulated from the original (primal) problem above as follows:where y_is ≥ 0, for i=1,2,…,4 are the dual variables.The problem in standard form is as shown below:where y₅, y₆, y₇…y₁₁ are the added surplus variables. The analyses (accomplished in 10 iterations) were done using Tora Software.

Table 5. Results of the dual analysis

Table 6. Result of Sensitivity Analysis showing the mini. and max. limits for Objective coefficients and RHS

4. Result Discussion and Conclusions

In this research, the major aims were to decide how the limited resources of Nigeria Bottling Company would be allocated in order to maximize profit and also determine the right product mix that would maximize such profit. From the result of the analysis, it was discovered that Dubic Malt, Orijin bitter sachet and Mac-dowel Spirit contributed maximally to the profit of the company. Their production quantities on daily basis should be 1,732, 144 and 8,227 crates respectively yielding the total profit of ₦9,129,117.24. Also, the dual problem formulated from the original (primal) problem was used to conduct sensitivity analysis in order to determine the feasibility of the already obtained optimal solution above. The result of the sensitivity analysis presented in Table 6, showed the various reduced costs (that is, the ranges of increments) of the objective cost coefficients of the involved variables and the shadow prices of the RHS resource constraints within which the solution remains optimal. The result of the sensitivity analysis also revealed the shadow price (dual price) which is the cost of acquiring an additional resource in case the Company has any of such needs. Finally, based on the results obtained from this research, it is recommended that Nigerian Bottling Company should focus more on producing those products that maximize their profit.

References

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