This study aims to minimize training costs for staff at Patuakhali Science and Technology University using linear programming. It identifies two decision variables (permanent and non-permanent staff to be trained) and develops constraints based on time available and staff in different departments. The linear programming model is solved to find the optimal solution: 1 permanent staff should be sent for 5 days of training among departments to minimize costs. The research suggests this approach can help determine optimal staffing levels for future training programs.

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Md. Enamul Islam Shemul
Student of Patuakhali Science and Technology University
Faculty of Business Administration and Management
Session 2013-14

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Chapter- 01
Abstract:
The success and failure that an individual or organization experiences, depends to a large extent
on the ability of making appropriate decision. Making of a decision requires an enumeration of
feasible and viable alternatives. It is proposed to apply linear programming techniques in the area
of minimizing the cost of staff training. This model gives an optimal solution to all the methods
formulated. Data collected may not yield a feasible solution, when this occurs the model needs to
be reformed to give an optimum solution. However, this study recommends to the management
of the Patuakhali Science and Technology University, the number of staff to be sent for training
program when there is need for such in the permanent and non-permanent sections of the
institution.
Objective of Study:
This study aims at using Simplex Method, one of the Linear Programming Techniques to create a
Mathematical Model that will minimize the cost of Stuff training for Register sector in
Patuakhali Science and Technology University.
Methodology:
Linear programming model for simplex method has been used for obtaining optimum solution.
Linear model has been used for allocate the stuff properly.
Findings of the study:
Our research paper suggests that 1 permanent stuff should be sent for training among
departments for 5 days.
Future scope of the study:
This research paper will help to determine the number of stuffs should be sent for training
program while there is need for such program in the permanent and non-permanent section of the
institution.

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Chapter- 02
Introduction:
Linear programming is a mathematical technique used to find the best possible solution in
allocating limited resources (constraints) to achieve maximum profit or minimum cost by
modeling linear relationships. Linear programming is a widely used mathematical modeling
technique to determine the optimum allocation of scarce resources among competing demands.
Resources typically include raw materials, manpower, machinery, time and money. The
technique is very powerful and found especially useful because of its application to many
different types of real business problems in areas like finance, production, sales and distribution,
personnel, Stuffs training, marketing and many more areas of management. As its name implies,
the linear programming model consists of linear objectives and linear constraints, which means
that the variables in a model have a proportionate relationship. Training is the process of helping
employee’s develops maximum effectiveness in their present and future jobs. This implies that
the training process is student centered; it is tailored to the individual needs and abilities of the
learners. It also means that training is a continuous process, starting with the introduction of
employees to their first jobs and continuing throughout their careers. This research paper is
conducted on minimizing cost by making optimum combination of permanent and non-
permanent stuffs before training program is in progress in the register sector, Patuakhali Science
and Technology University.
Chapter- 03
Theoretical framework of the study:
Historical origin of linear programming techniques: The 1940s was a time of innovation and
reformation of how products were made, both to make things more efficient and to make a
better-quality product. The Second World War was going on at the time and the army needed a
way to plan expenditures and returns in order to reduce costs and increase losses for the enemy.
George B. Dantzig is the founder of the simplex method of linear programming, another person
who played a key role in the development of linear programming is John von Neumann, who
developed the theory of the duality and Leonid Kantorovich, a Russian mathematician who used
similar techniques in economics before Dantzig and won the Nobel Prize in 1975 in economics.
Actually, several researchers developed the idea in the past. Fourier in 1823 and the well-known
Belgian mathematician de la Vallée Poussin in 1911 each wrote a paper describing today's linear
programming methods, but it never made its way into mainstream use. A paper by Hitchcock in
1941 on a transportation problem was also overlooked until the late 1940s and early 1950s. It
seems the reason linear programming failed to catch on in the past was lack of interest in
optimizing.

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Dantzig's original example of finding the best assignment of 70 people to 70 jobs emphasizes the
practicality of linear programming. The computing power required to test all possible
combinations to select the best assignment is quite large. However, it takes only a moment to
find the optimum solution by modeling problem as a linear program and applying the simplex
algorithm. The theory behind linear programming is to drastically reduce the number of possible
optimal solutions that must be checked. LP has become popular in academic circles, for decision
making.
"Linear programming can be viewed as part of a great revolutionary development which has
given mankind the ability to state general goals and to lay out a path of detailed decisions to take
in order to 'best' achieve its goals when faced with practical situations of great complexity. Our
tools for doing this are ways to formulate real-world problems in detailed mathematical terms
(models), techniques for solving the models (algorithms), and engines for executing the steps of
algorithms.
Basic requirements for the use of linear programming techniques:
 LP problems seek to maximize or minimize some quantity (usually profit or cost)
expressed as an objective function.
 The presence of restrictions, or constraints, limits the degree to which we can pursue our
objective.
 There must be alternative courses of action to choose from.
 The objective and constraints in linear programming problems must be expressed in
terms of linear equations or inequalities.
Assumption in linear programing model:
Before we get too focused on solving linear programs, it is important to review some theory. For
instance, several assumptions are implicit in linear programing problems. These assumptions are:
1. Proportionality: The contribution of any variable to the objective function or constraints
is proportional to that variable. This implies no discounts or economies to scale. For
example, the value of 8x1 is twice the value of 4x1, no more or less.
2. Additivity: The contribution of any variable to the objective function or constraints is
independent of the values of the other variables.
3. Divisibility: Decision variables can be fractions. However, by using a special technique
called integer programming, we can bypass this condition.
4. Certainty: This assumption is also called the deterministic assumption. This means that
all parameters (all coefficients in the objective function and the constraints) are known
with certainty. Realistically, however, coefficients and parameters are often the result of
guess-work and approximation.
Continuity assumption: Variables can take any value within a given feasible range.

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Formation of programing model:
Steps In Formulating A Linear Programming Model
Linear programming is one of the most useful techniques for effective decision making. It is an
optimization approach with an emphasis on providing the optimal solution for resource
allocation. How best to allocate the scarce organizational or national resources among different
competing and conflicting needs forms the core of its working. The effective use and application
of linear programming requires the formulation of a realistic model which represents accurately
the objectives of the decision making subject to the constraints in which it is required to be made.
The basic steps in formulating a linear programming model are as follows:
Step I. Identification of the decision variables. The decision variables (parameters) having a
bearing on the decision at hand shall first be identified, and then expressed or determined in the
form of linear algebraic functions or in equations.
Step II. Identification of the constraints. All the constraints in the given problem which restrict
the operation of a firm at a given point of time must be identified in this stage. Further these
constraints should be broken down as linear functions in terms of the pre-defined decision
variables.
Step III. Identification of the objective. In the last stage, the objective which is required to be
optimized (i.e., maximized or memorized) must be dearly identified and expressed in terms of
the pre-defined decision variables.
Method of solving linear model: There are three methods of linear programming but we learned
only two methods such as simplex method and Graphical method that discussed below.
 Simplex Method: The first class of algorithms to solve LPs attempt to find the optimal
solution by searching the boundary of the constraint poly tope. These methods use” pivot
rules” that indicates pivot Colum and pivot row to determine the next direction of travel
once a basic point is reached. If no direction of travel yields an improvement to the
objective function, then the basic point is the optimal solution. The first algorithm to use
this idea was the Simplex Method from George Dantzig in 1947. Though these methods
perform well in practice, it is not known if any” pivot rule” algorithm can run in
polynomial time.
 Graphical Method: The main objectives of graphical method to determine the feasible
region. This algorithm gives way of finding a feasible solution to an optimization
problem. The idea is to enclose feasible solutions in an ellipse. Then, check to see center
of the ellipse is a feasible solution. If not, find a violating constraint using a separation
oracle. Then, enclose the half of the ellipse where the feasible solutions must lie in
another ellipse. Though this result was a breakthrough in the theory, the algorithm

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usually takes longer than the Simplex Method in practice. Following are the steps in
graphical solution of linear programming problem (LPP):
Formulate LPP by writing the objective function and the constraints.
 Constraints are changed into equalities.
 Plot the constraints on the graph.
 Identify the feasible region and ascertain their coordinates.
 Test which point is most profitable.
Interior Point Method: The next class of algorithms for solving LPs are called” interior point”
methods. As the name suggests, these algorithms start by finding an interior point of the
constraint polytope and then proceeds to the optimal solution by moving inside the polytope. The
first interior point method was given by Karmarkar in 1984. His method is not only polynomial
time like the Ellipsoid Method, but it also gave good running times in practice like the Simplex
Method.
Chapter- 04
Some Basic Application of Linear Programming Technique:
Military: Paradoxically the most appropriate example of an organization is the military and
worldwide, Second World War is considered to be one of the best managed or organized events
in the history of the mankind. Linear Programming is extensively used in military operations.
Such applications include the problem of selecting an air weapon system against the enemy so as
to keep them pinned down and at the same time minimizes the amount of fuel used.
Agriculture: Agriculture applications fall into two broad categories, farm economics and farm
management. The former deals with the agricultural economy of a nation or a region, while the
latter is concerned with the problems of the individual form. Linear Programming can be
gainfully utilized for agricultural planning e:g. allocating scarce limited resources such as capital,
factors of production like labor, raw material etc. in such a way 'so as to maximize the net
revenue.
Environmental Protection: Linear programming is used to evaluate the various possible
a1temative for handling wastes and hazardous materials so as to satisfy the stringent provisions
laid down by the countries for environmental protection. This technique also finds applications in
the analysis of alternative sources of energy, paper recycling and air cleaner designs.
Facilities Location: Facilities location refers to the location nonpublic health care facilities
(hospitals, primary health centers) and’ public recreational facilities (parks, community hal1s)

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and other important facilities pertaining to infrastructure such as telecommunication booths etc.
Apart from these applications, LP can also be used to plan for public expenditure and drug
control. '
Product-Mix: The product-mix of a company is the existence of various products that the
company can produce and sell. However, each product in the mix requires finite amount of
limited resources. Hence it is vital to determine accurately the quantity of each product to be
produced knowing their profit margins and the inputs required for producing them. The primary
objective is to maximize the profits of the firm subject to the limiting factors within which it has
to operate.
Production: A manufacturing company is quite often faced with the situation where it can
manufacture several products (in different quantities) with the use of several different machines.
The problem in such a situation is to decide which course of action will maximize output and
minimize the costs.
Mixing or Blending: Such problems arise when the same product can be produced with the help
of a different variety of available raw-materials each having a fixed composition and cost. Here
the objective is to determine the minimum cost blend or mix (Le.; the cost minimizations) and
the various constraints that operate are the availability of raw materials and restrictions on some
of the product constituents.
Portfolio Selection: Selection of desired and specific investments out of a large number of
investment' options available10 the managers (in the form of financial institutions such as banks,
non-financial institutions such as mutual funds, insurance companies and investment services
etc.) The objective of Linear Programming, in such cases, is to find out the allocation which
maximizes the total expected return or minimizes the total risk under different situations.
Profit Planning & Contract: Linear Programming is also quite useful in profit planning and
control. The objective is to maximize the profit margin from investment in the plant facilities and
machinery, cash on hand and stocking-hand.
Media Selection/Evaluation: Media selection means the selection of the optimal media-mix so as
to maximize the effective exposure. The various constraints in this case are: Budget limitation,
different rates for different media (i.e.; print media, electronic media like radio and T.V. etc.) and
the minimum number of repeated.

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Chapter- 05
Significance of the study:
Scientific Approach to Problem Solving: Linear Programming is the application of scientific
approach to problem solving. Hence it results in a better and true picture of the problems-which
can then be minutely analyzed and solutions ascertained.
Evaluation of All Possible Alternatives: Most of the problems faced by the present organizations
are highly complicated - which cannot be solved by the traditional approach to decision making.
The technique of Linear Programming ensures that’ll possible solutions are generated - out of
which the optimal solution can be selected.
Helps in Re-Evaluation: Linear Programming can also be used in re-evaluation of a basic plan
for changing conditions. Should the conditions change while the plan is carried out only
partially, these conditions can be accurately determined with the help of Linear Programming so
as to adjust the remainder of the plan for best results?
Quality of Decision: Linear Programming provides practical and better quality of decisions’ that
reflect very precisely the limitations of the system i.e.; the various restrictions under which the
system must operate for the solution to be optimal. If it becomes necessary to deviate from the
optimal path, Linear Programming can quite easily evaluate the associated costs or penalty.
Creation of Information Base: By evaluating the various possible alternatives in the light of the
prevailing constraints, Linear Programming models provide an important database from which
the allocation of precious resources can be don rationally and judiciously.
Maximum optimal Utilization of Factors of Production: Linear Programming helps in optimal
utilization of various existing factors of production such as installed capacity, labor and raw
materials etc.
Chapter- 06
Methodology of the study:
This Research paper is completed with help of linear programming simplex model. The area of
this research paper is register sector, Patuakhali Science and Technology University. The
permanent stuffs, non-permanent register sector employees are chosen as sample size of this
research paper.
Decision variables:
These are permanent and non-permanent staff from the institution. It is represented by X1 and
X2 respectively. These variables are used in the two models formulated as
X1 = Permanent Staff, X2 = Non-Permanent Staff

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Objective Function:
In any organization set up the main aim is to minimize cost and in this case; it is a minimization
problem because the cost of training of staff to the establishment has to be minimized. Therefore,
the objective function is given by:
Minimize: Z= X1+ X2
Constraints:
The constraint for this study is the time available for training as the programme is
Department Name Permanent Staff Non-Permanent Staff
Register Office:
Register office
Foundation office
Council office
7
21
1
1
1
0
Assumption of the model:
 The variables values are known with certainty.
 The objective function and constraints exhibit constant cost to scale.
 Τhe additivity assumption: There are no interactions between the decision variables.
Chapter- 07
Data presentation and Analysis:
The data used for this study has been assumed.
List of staff in various departments of the institution
Register sector Research Model
Minimize, Z = X1+X2
Subject to constraints,
7X1 + 1X2 ≥ 5 Register office
21X1 + 1X2 ≥ 5 Foundation office

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6X1 + 0X2 ≥ 5 Council office
X1, X2 ≥ 0
Solution:
Minimize, Z = X1 + X2 + 0.S1 + 0.S2 + 0.S3 + MA1 + MA2 + MA3
Subject to constraints,
7X1 + 1X2 - S1 + A1 = 5
21X1 + 1X2 - S2 + A2 = 5
6X1 + 0X2 - S3 + A3 = 5
Table: 01
CJ-
>
CB
B.V. Cj 1 1 0 0 0 M M M Ratio
S.V X1 X2 S1 S2 S3 A1 A2 A3
M A1 5 7 1 -1 0 0 1 0 0 .71
M A2 5 21 1 0 -1 0 0 1 0 .24
Pivot
row
M A3 5 6 0 0 0 -1 0 0 1 .83
Zj 34M 2M -M -M -M M M M
Cj - Zj 1-34M
Pivot Column
1-
2M
M M M 0 0 0
Table: 02
CJ-
>
CB
B.V. Cj 1 1 0 0 0 M M M Ratio
S.V X1 X2 S1 S2 S3 A1 A2 A3
M A1 10/3 0 3/2 -1 1/3 0 1 0 10
1 X1 5/21 1 1/21 0 -1/21 0 0 0 -5
Pivot
row
M A3 25/7 0 -2/7 0 2/7 0 0 0 25/2
Zj 1 8M/21 + 1/21 -M 13M/21 –
1/21
0 M 0
Cj -
Zj
0 -8M/21+22/21 M -13M/21+1/21
Pivot
Column
0 0 M

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Table: 03
CJ-
>
CB
B.V. Cj 1 1 0 0 0 M M M Ratio
S.V X1 X2 S1 S2 S3 A1 A2 A3
M A1 5 7 1 -1 0 0 1 0 .71
0 S2 -5 -21 -1 0 1 0 0 0 .24
Pivot
row
M A3 5 6 0 0 0 0 0 0 .83
Zj 13M M -M 0 0 M 0
Cj - Zj 1-13M
Pivot
Column
1-M M 0 0 0 M
Table: 04
CJ-
>
CB
B.V. Cj 1 1 0 0 0 M M M Ratio
S.V X1 X2 S1 S2 S3 A1 A2 A3
M A1 10/3 0 -6 -4/3 0 1 0
1 X1 5/21 1 1/21 0 0 0 0
M A3 25/7 0 -2/7 0 0 0 0
Zj 1 -
44M/7
-4M/3 0 M 0
Cj - Zj 0 1+
44M/7
4M/3 0 0 0
Since all the entries in the Cj – Zj row are either positive or zero, the optimum solution to the
problem is obtained which is : X1 = 5/21 = .24 ≈ 1and X2 = 0 with minimum Z = 1

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Chapter- 08
Findings of the study:
From the solution to the model using simplex method the minimize objective function for
register office is given as Z=1, x1(permanent staff) is 1 and x2(non-permanent staff) is 0 which
implies that 1 of the permanent staff should be sent for training, where no non-permanent staff is
eligible for training program.
Recommendations of the study:
As register sector has 34 permanent staff and the simplex method shows 1 permanent staff
should be send for training program so 1 member should be sent for training.
Conclusion:
The objective of the study is to apply the linear programing techniques in the effective use of
resources for staff training in Register office. The study uses the permanent and non-permanent
staff from the units (register office, foundation office and council office) as the decision
variables. It gives the optimum solution, which shows the combination of number of permanent
and non-permanent staff from each units that should be send for training program to reduce cost
and achieve optimum goal.

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References:
1. Kashyap M. Gupta, Application of Linear Programming Techniques for Staff Training,
(IJEIT), Volume 3, (12, June 2014)
2. Fagoyinbo., I. S and Ajibode., & I.A, Application of Linear Programming Techniques in
the Effective Use of Resources for Staff Training, (JETEAS) 1 (2)(2010) 127-132.
3. Raj Kishore Singh., S.P.Varma., & Arvind Kumar, Application of Linear Programming
techniques in Personnel Management , IOSR-JM, Volume 8, (Sep. - Oct. 2013), PP 45-48