emelie project work
TRANSCRIPT
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.