INTRODUCTION

Since time immemorial, man has struggled for survival

and in this; he produces his wants/needs. The resources for

the production of these wants are not so abundant, so he

tries to maximize the limited resources available and

minimizes cost of producing his needs. To do this, he must

be conversant with the technology available as for its

products, services and method of operation concerned.

Production can be defined as the creation and

distribution of goods and services. While production planning

is made in order to utilize the limited amount of resources

available for use. As a result of this, there is need therefore

to produce those goods which is most pressing i.e making a

scale of preference. Opportunity cost is not left out since we

have limited resources thus we forego the production of the

less pressing wants.

To help assist the manager’s ability to take a wise

decision constitutes what is known as Operations Research.

The knowledge of Operations Research helps companies to

maximize their profit by managing their resources effectively.

Nature of Operations Research

The first formal activities of Operations Research were

initiated in England during World War II when a team of

British scientists set out to make decisions regarding the

most effective allocation of limited military resources to the

various military operations and to the activities within each

operation. Gupta (1979).

A cornerstone of Operations Research is mathematical

modeling. Though the solution of the mathematical model

provides a basis for making a decision, unquantifiable

factors such as human behavior must be accounted for

before a final decision can be reached. Mathematical models

are used in their functional forms as linear equations or

inequalities such that several of them are combined to form

a system. Constraint equation retaining the variable in a set

of objective function, which is to be optimized, depends on

the goals of the firm and assuming some other necessary

properties of the variables (e.g non-negative assumption).

The most important contribution Operations Research

makes is in decision of lower, middle and top management

level, based on the application of its output. The

mathematical model and the linear programming which is the

main focus of this study is an important Operation Research

technique that produce output, which form the best

combination of planning, organizing, directing and controlling

of the firm’s activities which are essential for the

management.

Brief History of Phinomar Farms

Phinomar farms are a private-owned farm established

in the 1990’s. It is located in Ngwo, Enugu to help provide

the basic proteinous needs of the residents of the area and

the state in entirety. Their range of products include Broiler

breeder, Layer breeder, Commercial layer, Turkey, Day Old

Chicks, Frozen chicken and parts. The majority of the staff

was recruited from the local community as a way of creating

employment opportunities for them. They often meet with

representatives of the various community groups to foster

ways of moving the estate forward as regarding

developments. 

At Phinomar they continually engage in the upgrading

and redesigning of the operations and facilities to meet the

latest standards. They embrace the use of technology not

only to improve the efficiency of the operations but in

safeguarding the health of the customers making sure only

the best gets to their doorstep. They have been engaging

the intensive information technology to further improve the

efficiency of the operations.

Scope of the study

This project has the intention of limiting its scope to cover

the production set up (i.e factory activities of transforming

the raw materials into finished goods and not the

administration welfare of the company. After all these, it will

cover the best product mixture that will maximize the firm’s

profit given the necessary constraints and most effective

decision variables to be employed and discharged.

Statement of the problem

Most problems faced by the companies are the problem

concerning their inability to apply the right tool (total input in

production and its yield). This had led to profit loss in most

organizations or not being able to maximize their profit.

This gives rise to Linear Programming which has proved

to be the right tool for solving these problems, if the following

conditions are satisfied;

1. There must be a well defined objective function (profit,

cost of quantities produced) which is to be either

maximized or minimized and which can be expressed

as a linear function of decision variables.

2. There must be constraints on the amount or extent of

attainment of the objective and these constraints must

be capable of being expressed as linear equations or

inequalities in terms of variables.

3. The decision variables should be inter-correlated and

non-negative. The non-negative condition shows that

Linear Programming deals with real-life situations for

which negative quantities are generally illogical.

Objective of the Study

The main objective is to develop a Linear Programming

model that will enable Phinomar Farms to maximize her total

profit so that the company will allocate more resources to the

production of such product amongst the brands of broiler

feed; pre-starter, starter and finisher.

Secondly, to carry out a sensitivity analysis to ascertain the

stability of the firm’s optimal solution or profit.

Relevance of this study

This work will enlighten most industries who are

traditional in their methods of decision making that the

contemporary World has at its disposal modern techniques

of decision making in optimizing profit and cost (minimization

and maximization). It will also motivate them to use such

technique which eventually boosts the country’s economy,

Nigeria inclusive.

LITERATURE REVIEW

It is imperative for broiler producers to source for cheap

alternative feedstuffs without affecting the quality of the feed,

productive performance of the birds and the economics of

production. One of the major problems facing broiler

producers is high prices and non-availability of feed

ingredients. The feed cost incurred about 60-65% of the total

cost of broiler production. Availability of quality feed at a

reasonable cost is a key to successful poultry operation

(Hodge and Rowland, 1978).Linear Programming is one of

the most important techniques to allocate available

feedstuffs in a least-cost broiler ration formulation (Dantzig,

1951 a,b; Alector, 1986; Ali and Lesson, 1995)

Linear Programming (LP) is a technique for optimization

of a linear objective function, subject to linear equality and

linear inequality constraints (Keuster and Mize, 1973).

Informally, Linear Programming determines the way to

achieve the best outcome (such as maximum profit or lower

cost) in a given mathematical model and given some list of

requirements represented as linear equations. Patrick and

Schaible (1980) stated that Linear Programming is

technically a mathematical procedure for obtaining a value-

weighting solution to a set of simultaneous equations. Linear

Programming was first to put into significant use during the

World War II when it was used to determine the most

effective way of deploying troops, ammunitions, machineries

which were all scarce resources (Chv’atal, 1983). There are

hundreds of applications of Linear Programming in

agriculture (Taha, 1987). Olurunfemi et al (2001) also

applied Linear Programming into duckweed utilization in

least-cost feed formulation for broiler starter.

Gonzalez-Alcorta et al (1994) developed a profit

maximization model that uses non-linear and separable

programming to determine the precise energy and protein

levels in the feed that maximizes profit. Their model is

distinguished by the assumption that body weight is not fixed

at a pre-determined level. Feed cost is not determined by

least cost feed formulation. Rather feed cost is determined

as a variable of the profit maximization model in a similar to

that described in Pesti et al. (1986). Costa et al. (2001)

developed a 2-step profit maximization model that minimizes

feed cost and maximizes profit in broiler production. Their

model indicates the optimal average feed consumed, feed

cost, live and processed body weight of chickens, as well as

the optimal length of time that the broilers must stay in the

house and other factors, for given temperature, size of the

house, costs of inputs and outputs and for certain pre-

determined protein level, source and processing decisions.

Njideka (2006) in her project “Establishing a production

quota for Hardis and Dromedas Nig Ltd used Linear

Programming to maximize profit.

Osigwe (2010) in his project titled “Profit Maximization

of Bread production using Linear Programming technique”

used the Simplex method to determine the optimal quantities

of bread to be produced at UAC Foods, Nigeria, PLC, Ikeja.

Also, John (2010) in her project “Optimization in Soap

Production” advised PZ Cussons, Aba Soap factory on the

need for the company to use the technique of Linear

Programming and also employ the aid of Operations

Researchers to keep the company afloat.

Meaning and Concept of Linear Programming

Linear programming is that branch of mathematical

programming which is designed to solve optimization

problems where all the constraints as well as the objectives

are expressed as linear function. It was developed by

George B. Dantzig in 1947.

Linear programming is a technique for making

decisions under certainty i.e; when all the courses of options

available to an organization are known and the objective of

the firm along with its constraints are quantified. That course

of action is chosen out of all possible alternatives which yield

the optimal results. Linear programming can also be used as

a verification and checking mechanism to ascertain the

accuracy and the reliability of the decisions which are taken

solely on the basis of manager’s experience without the aid

of a mathematical model.

Thus, it can be defined as a method of planning and

operation involved in the construction of a model of a real-

life situation having the following elements;

a) Variables which denote the available

b) The related mathematical expressions which relate the

variables to the controlling conditions, which reflect

clearly the criteria to be employed for measuring the

benefits flowing out of each course of action and

providing an accurate measurement of the

organization’s objective.

Definitions of Some Basic Terminologies

Simplex Method; This is a mathematical procedure that

uses addition, subtraction, multiplication and division in

a particular sequence to solve a linear programming

problem. It requires the use of iterative method to

reaching the optimal solution.

Objective Function; This is the quantity to be

maximized or minimizes. It is, in general, the function

which represents the goal of the economic agent (firm).

Constraint; This allows the unknowns activities to take

on certain values (raw materials). It does not make

sense to spend a negative amount of capital, labour on

any activity, so we constraint all the production

resource to be non-negative. They are unknown values

or activities to be determined.

Slack Variables; Also known as disposable variables,

they are variables included in the mathematical

procedure of simplex method which are non-negative

and which transformed the inequalities in the constraint

equation to equalities.

Basic Variable; A variable is said to be a basic variable

in a given equation if it appears with a unit coefficient in

that equation and zeros in all other, otherwise it is non-

basic.

Basic Solution; The solution obtained from a canonical

system by setting the non-basic to zero and solving for

the basic variable.

Feasible solution; A solution is said to be feasible when

it satisfies all the constraint equations.

Basic feasible equation; This is a basic solution in

which the values of the basic variables are non-

negative.

Basis; The levels of constraints and unutilized

constraints in any one solution form a basis.

Entering variable; This is also called incoming activity,

that must be introduced in the basis.

Leaving variable; This is the element or variable at the

intersection of the incoming activity and outgoing

variable or activity.

Pivot row; This is the row that will be occupied by the

incoming activity, that the place of the outgoing activity.

ASSUMPTIONS OF LINEAR PROGRAMMING

In linear programming, the following assumptions are

made.

PROPORTIONALITY ASSUMPTION;

This assumption indicates that the level of each activity

is directly proportional to the quantity of the material

resources in that equation. In view of this, if one wants to

increase the effect of that activity by one unit, he/she just

increases the level by one unit.

ADDITIVITY ASSUMPTION;

It is assumed that the total profitability and the total

amount of each resource utilized would be exactly equal to

the sum of the respective individual amounts. Thus, the

function or the activities must be additive and the interaction

among the activities of the resources does not exist.

DIVISIBILITY ASSUMPTION;

Variables may be assigned fractional values i.e they

need not always be integers. If a fraction of a product cannot

be produced, an integer programming problem exists.

CERTAINTY ASSUMPTION;

It is assumed that conditions of certainty exists i.e all

the relevant parameters or coefficients in linear

programming model are fully and completely known and that

they do not change during the period. However, such an

assumption may not hold good at all times.

FINITENESS;

Linear programming assumes the presence of a finite

number of activities and constraints without which it is not

possible to obtain the optimal solution.

THE STANDARD-LINEAR PROGRAMMING

The standard form of a linear programming problem

with m-constraint equations and n-decision variables can be

represented as follows;

CHAPTER TWO

METHOD OF DATA COLLECTION

Data collection is an activity aimed at getting

information to satisfy some decision objectives. The data

collection can be done through experiment, questionnaire,

personal interview, survey e.t.c.

However, the data used in this study was collected from

the Deputy Manager, Feed Mills Department of the case

study (Phinomar Farms, Ltd, Ngwo, Enugu) on the raw

materials used and the amount of feed produced via

recorded data and means of interview.

Problems Encountered

The problems encountered in this study was mainly the

sourcing of the materials the researcher actually needed,

like the production quota and total raw materials used for

feed production.

Also the time of going from school amidst lectures to

the farm in Enugu was a major challenge.

Finally, some of the data gotten had to be processed as

they were in its very raw state. The researcher was able to

do this through some very important conversions in order to

get the desired and accurate results.

Data Presentation

Let;

1 unit = 10bags of broiler feed (for all three activities

Input (kg)

Raw materials Activity(Products) Availability

of raw

materials

Pre-starter

feed

Starter

feed

Finisher

feed

Maize 114.5 113.5 83.8 401.9

G/Corn 25 25 50 261

Fish meal 12 5.3 - 90

Soya meal 62.5 62.5 35 237

Full Fat Soya 12.5 25 33.8 102

Soya Oil 3 6.3 8.5 28

DCP 3 2.9 1 36

Limestone 3 3.5 2.3 19

Wheat Offal - 8 10 30

Methionine liquid

(litres)

0.5 0.5 0.5 12

Acidomix Acid 1 1 0.5 15

INPUT PRICES AND THE PROFIT MADE

METHODOLOGY

THE SIMPLEX METHOD;

The Simplex method is an iterative technique starting

with known basic feasible solution to a new decision variable

Activity 1 Activity 2 Activity 3

Cost of Production ₦18854.12 ₦18054.98 ₦13343.06

Selling price ₦22500 ₦22000 ₦19000

Profit ₦3645.88 ₦ ₦3945.02 ₦5656.94

called the entering variable and the selection of another

variable called the leaving variable to leave the basis and

finally calculate the solution that optimized the objective

function. Because each successive solution improves upon

the current one, it is not possible to consider the same

solution twice and the procedure terminates in a finite

number of iteration, since it embodies a sequence of specific

instruction.

Outline of the Simplex method

Initialization steps;

Identify an initial basic feasible solution. Firstly, we

introduce slack variables Si’s (i= 1,2,3,…,n), then select the

original variable Xi’s (i=1,2,3,…,n) to be the initial non-basic

variables, then set them equal zero and the slack to initial

basic variables. When solving, it is convenient to use the

following procedure;

Iterative Step; This involves moving to the better adjacent

basic feasible solution. The iterative steps are in part.

Part 1; Determine the entering variable by selecting the

variable (automatically the non-basic variable) with the

largest negative coefficient. That is the non-basic variable

that will increase the objective function Z at its fastest rate.

We indicate the variable with a pivotal point and indicate the

column (pivotal column).

Part 2; Determine the leaving basic variable by;

I. Picking out each coefficient in the basic column that is

strictly positive greater than zero.

II. Divide each of these coefficients of the entering

variable with the coefficient of the right-hand side (b-

value) for the same row.

III. Identify the equation (row) that has the smallest ratio of

the quotient.

IV. Select the basic variable for the equation (this is the

basic variable that reaches zero first as the entering

basic variable is increased). Put a box around this row

in the tableau to the right of Z column and call the

boxed row the pivot row.

Part 3; Determine the new basic feasible solution by

repeating the same procedure in part 1 and part 2 in the

simplex tableau below the current one. The first other

columns are unchanged except that the leaving basic

variable in the first column is replaced by the entering basic

variable.

The new pivot numbers =

Part 4; Stopping rule: Stop when an adjacent feasible Z

solution is better. The current basic feasible solution is

optimal if and only if every coefficient in the basic variable of

the Z equation is non-negative.

Fundamental Condition of the Simplex Method

The basis for simplex method which guarantees the

generation of such a sequence of basic feasible solution is

based on

1. Feasibility Condition;

This guarantees that starting with basic feasible solutions;

only basic feasible solutions are encountered during

computation and iteration.

2. Optimality Condition;

This ensures that no inferior solutions relative to the

current solution are encountered during computation

and iteration.

Sensitivity Analysis

Sensitivity analysis in Linear Programming refers

to changes in the parameters (input data) within limits

without causing the optimal solution to change. The

parameters of LP models are usually not exact. With

sensitivity analysis, we can ascertain the impact of this

uncertainty on the quality of the optimal solution.

The changes in the LPP can be considered in four

types;

1. Changes in the objective function

2. Addition of a new variable

3. Changes in the constant column vector

4. Addition of a new variable to the problem

Duality Theory

The dual problem is an LP defined directly and

systematically from the primal (or original) LP model.

The main focus of a dual problem is to find for each

resource its best marginal value. This value (also

called the shadow price) reflects the scarcity of the

resources. If a resource is not completely used i.e

there is no slack, then its marginal profit is zero.

The format of the Simplex method is such that

solving one type of problem is equivalent to solving the

others simultaneously as they both provide optimal

solutions to each other. (Sharma 2009).

Thus the primal in matrix notation is

The dual problem is constructed as

Relationship between the primal and dual problem

1. The objective function coefficients of the primal

problem have become the right-hand side

constraint value of the dual. Furthermore, the

right-hand side constraint of the primal has

become the cost coefficient of the dual.

2. The inequalities have been reversed in the

constraint

3. The objective function is changed from

maximization to minimization (and vice versa).

4. Each column in the primal corresponds to a

constraint (row) in the dual. Thus, the number of

dual constraints is equal to the number of primal

variables.

5. Each constraint (row) in the primal corresponds to

a column in the dual.

6. The dual of a dual is the primal problem.

(Advanced Statistics for Higher Education. A.I

Arua et al).

CHAPTER THREE

DATA ANALYSIS

The study uses Linear Programming (LP) technique to

determine the optimum level of profit for the three brands of

broiler feeds. Sensitivity analysis is applied to the objective

function coefficients and resource vector in order to

determine how robust the optimal solution is.

Linear programming model for Phinomar Farms, Nigeria

Limited

The farm produces 3 brands of broiler feeds which have

some limitations (raw materials and demand). All these

amount to 11 constraints. Hence, the model for Phinomar

Farm is n = 3 decision variables and m = 11 constraints.

Thus;

The Model

The model for the three brands of broiler feeds in

Phinomar Farms, Nigeria Limited is given below;

For all

Then introducing the slack variables

are the positive slack variables commonly referred to

as ‘Surplus’ (of the LHS over the RHS)

CHAPTER FOUR

ANALYSIS OF RESULT, CONCLUSION AND

RECOMMENDATION

The Tora output of the linear programming model for

Phinomar Farms is divided into two major sections, namely

1. The Optimum Solution Summary

2. Sensitivity analysis

Optimum Solution Summary

This part is sub-divided into three parts; the first

displays the number of iterations (three in this maximization

problem) that gave the maximum objective function of

₦19843.85.

The second part gives the maximum values of the

maximized variables; namely 1.78 Units of Pre-Starter broiler

feed and 2.36 Units of finisher broiler feed. However, the

second variable; Starter broiler feed did not contribute

anything at all. Also shown are the relative profit coefficients

of the activities (variables) in the objective function and their

respective contributions. Pre-Starter broiler feed made a

profit of ₦6505.67 and the Finisher broiler feed made a profit

of ₦13338.18, while the Starter broiler feed did not

contribute any profit at all. Thus, Phinomar farms will make a

profit of ₦19843.85 if they produce at the specified quantities

excluding the Starter broiler feed for every 1unit.

The last section of this Optimum Solution Summary

accounts for the current right-hand-side values of the

variables and their slacks(-). There are no surpluses since all

the constraints are of the ≤ type. These slacks are the

amounts by which the constraints are over-satisfied.

Sensitivity Analysis

The Sensitivity analysis section of the output deals with

individual changes in the coefficient of the objective function

and the right-hand-side of the constraints. The specified

limits define the boundaries which the variable must not

exceed for the solution to remain optimal. For instance, the

profit of Starter broiler feed can be reduced to ₦2092.06 but

must not exceed ₦7729.35 when it’s been increased with

other products still being produced. Also, the current optimal

solution will remain unchanged so long as the right-hand-

side of Maize and Full Fat Soya lies between

(256.90,474.28) and(43.88,112.19) respectively.

In this Sensitivity section, there are the reduced costs

and the dual price which are of significance to any profit-

maximizing firm. The reduced cost per unit activity can be

defined as cost of consumed resources per unit less revenue

per unit. If the activity reduced per unit is positive, then its

unit cost of consuming a resource is higher than its unit profit

and such activity should be given less attention. It implies

that the variables with zero-reduced cost can be produced

without making loss. Thus, the Pre-Starter and the Finisher

broiler feeds can be produced without making loss at this

one unit.

The dual price, also known as the shadow price,

measures the unit worth of the resources i.e. the contribution

of the objective function of a unit increase or decrease in

availability of the resources. Specifically, the zero dual prices

associated with constraints at (2, 3, 4, 6… 11) implies that

they are in excess. Hence, an increase in any of them brings

no economic profit. Also, the dual price corresponding to

constraints (1, 5) is an indication that a unit increase or

decrease in the availability of Maize and Full Fat Soya will

add ₦18.61 and ₦121.23 to the total profit respectively and

reduces the total profit by the same amount.

CONCLUSION

From the analysis we see there is need for the farm to

employ the use of Linear Programming technique and the

aid of Operations researchers to keep the farm up and

doing. For the farm to maximize profit, they should produce

more of X1 and X 3 (Pre-Starter and finisher broiler feed). This

solution indicates that the farm should produce Pre-Starter,

Starter and Finisher broiler feed at the rate of 1.78, 0.00 and

2.36 respectively for every 1unit with a resulting profit of

₦19843.85.

RECOMMENDATION

Based on the findings of this research, the following

recommendations are made;

In order to maximize profit, the resources of broiler feed

should be combined in the best optimum way. That is,

the decision variables (X1,X3) should be combined in

their best optimum way. If the farm still wants to go into

the production of variable X2 (Starter broiler feed), it

can ensure the product becomes profitable by;

A. Increasing the revenue by increasing the price of the

product

B. Decreasing the cost of consumed resources (raw

materials).

Some raw materials are in excess and it is advisable

for the farm to make maximum use of these resources

in such a way that waste will be minimized.

The Sensitivity parameters of the model for the

research work, that is, the unit profit of the decision

variables and the resource input should be ensured

that they do not exceed their boundaries. However, if it

becomes impossible to control them within these limits,

then the solution should be changed.