Hemnalini Memorial College of Engineering
(APPROVED BY AICTE & AFFILIATED TO MAKAUT)
Gate No. 7 Balindi, haringhata (Near BSF Camp, Jaguli Kalyani More)
Project report on
LINEAR PROGRAMMING PROBLEMS
Paper code: HM-HU 601
Paper Name: HUMANITIES II (OPERATIONS RESEARCH)
Submitted By
Name: SOUVIK DEY
Registration no 213430100720011
Roll no. 34300721018
Year: 3rd year 6th SEM
DEPARTMENT OF MECHANICAL ENGINEERING
MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY
WEST BENGAL
ACKNOWLEDGEMENT
This project report is being prepared on the topic “Linear Programming Problems"
from the subject HUMNAITIES II (Operations Research) which appears in our 6th
Semester Syllabus. This will be submitted for partial fulfillment of requirements for
award of the degree Bachelor of Technology (B. Tech) in MECHANICAL
ENGINEERING of Maulana Abul Kalam Azad University of Technology.
I would like to express my earnest obligation and gratitude to our project coordinator Ms.
Sruti Sen Sarma for her benevolent guidance, suggestions and constructive criticism that
helped me to carry out this dissertation work.
I would also like to thank our departmental course coordinator Mr. Debabrata Roy for his
support and encouragement.
I am also thankful to all the faculty members, staffs and my batch mates of the department
of Mechanical Engineering for free exchange of ideas and discussions which proved
helpful.
All the sources used to prepare this report has been cited appropriately and the same is not
submitted elsewhere for pursuing a degree.
CONTENTS
SL NO.
1.
TOPIC
Page No.


Introduction
Aim
2



Data Collection
Collection
Uses
3



Advantages
Principles
Definition of Terms




Optimal Solution
Integer Programming
Graphical L.P Solution
Formation of Mathematical Model of L.P.P
6




General Form of L.P.P
Canonical Form of L.PP
Standard Form of L.P.P
Application of Linear Programming
7
8
8
9





Basic Requirements of Linear Programming
L.P.P Special Cases
Case of Degeneracy, Cycling and Duality
Uses of Dual Concept
Solution of L.P.P
7.


Limitations
Analysis
14
15
8.



Results and Discussions
Conclusion
References
16
16
17
2.
3.
4.
4-5
5.
6.
10
11
12
12-13
13
1
INTRODUCTION
Mathematics is the queen of science. In our daily life, planning is required on various
occasions, especially when the resources are limited. Any planning is meant for attaining
certain objectives. The best strategy is one that gives a maximum output from a minimum
input. The objective which is in the form of output may be to get the maximum profit,
minimum cost of production or minimum inventory cost with a limited input of raw
material, manpower and machine capacity. Such problems are referred to as the problems
of constrained optimization. Linear programming is a technique for determining an
optimum schedule of interdependent activities in view of the available resources.
Programming is just another word for 'planning' and refers to the process of determining a
particular plan of action from amongst several alternatives.
Linear programming applies to optimization models in which objective and constraint
functions are strictly linear. The technique is used in a wide range of applications, including
agriculture, industry, transportation, economics, health systems, behavioral and social
sciences and the military. It also boasts efficient computational algorithms for problems with
thousands of constraints and variables. Indeed, because of its tremendous computational
efficiency, linear programming forms the backbone of the solution algorithms for other
operative research models, including integer, stochastic and non-linear programming. The
graphical solution provides insight into the development of the general algebraic simplex
method. It also gives concrete ideas for the development and interpretation of sensitivity
analysis in linear programming.
Linear programming is a major innovation since World War II in the field of business
decision making, particularly under conditions of certainty. The word 'linear' means the
relationships handled are those represented by straight lines, i.e. the relationships are of the
form y = a + bx and the word 'programming' means taking decisions systematically. Thus,
linear programming is a decision making technique under given constraints on the
assumption that the relationships amongst the variables representing different phenomena
happen to be linear.
A linear programming problem consists of three parts. First, there objective function which
is to be either maximized or minimized. Second, there is a set of linear constraints which
contains thee technical specifications of the problems in relation to the given resources or
requirements. Third, there is a set of non-negativity constraints - since negative production
has no physical counterpart.
AIM
 To find and know more about the importance and uses of 'linear programming'.
 To formulate a linear programming problem and solve in simplex method and dual problem.
2
DATA COLLECTION
Linear programming is a versatile mathematical technique in operations research and a plan
of action to solve a given problem involving linearly related variables in order to achieve the
laid down objective in the form of minimizing or maximizing the objective function under a
given set of constraints
CHARACTERISTICS






Objectives can be expressed in a standard form viz. maximize/minimize z = f(x) where z is
called the objective function.
Constraints are capable of being expressed in the form of equality or inequality viz. f(x) =
or ≤ or ≥ k, where k = constant and x ≥ 0.
Resources to be optimized are capable of being quantified in numerical terms.
The variables are linearly related to each other.
More than one solution exist, the objectives being to select the optimum solution.
The linear programming technique is based on simultaneous solutions of linear equations.
USES
There are many uses of L.P. It is not possible to list them all here. However L.P is very
useful to find out the following:











Optimum product mix to maximize the profit.
Optimum schedule of orders to minimize the total cost.
Optimum media-mix to get maximum advertisement effect.
Optimum schedule of supplies from warehouses to minimize transportation costs.
Optimum line balancing to have minimum idling time.
Optimum allocation of capital to obtain maximum R.O.I
Optimum allocation of jobs between machines for maximum utilization of machines.
Optimum assignments of jobs between workers to have maximum labor productivity.
Optimum staffing in hotels, police stations and hospitals to maximize efficiency.
Optimum number of crew in buses and trains to have minimum operating costs.
Optimum facilities in telephone exchange to have minimum break downs. The above list
is not an exhaustive one; only an illustration.
3
ADVANTAGES






Provide the best allocation of available resources.
Meet overall objectives of the management.
Assist management to take proper decisions.
Provide clarity of thought and better appreciation of problem.
Improve objectivity of assessment of the situation.
Put across our view points more successfully by logical argument supported by scientific
methods.
PRINCIPLES
Following principles are assumed in L.P.P
 Proportionality: There exist proportional relationships between objectives and constraints.
 Additivity: Total resources are equal to the sum of the resources used in individual
activities.
 Divisibility: Solution need not be a whole number viz decision variable can be in fractional
form.
 Certainty: Coefficients of objective function and constraints are known constants and do
not change viz parameters remain unaltered.
 Finiteness: Activities and constraints are finite in number.
 Optimality: The ultimate objective is to obtain an optimum solution viz 'maximization' or
'minimization'.
DEFINITION OF TERMS
 Basic solution: There are instances where number of unknowns (p) are more than the
number of linear equations (q) available. In such cases we assign zero values to all surplus
unknowns. There will be (p-q) such unknowns. With these values we solve 'q' equations
and get values of 'q' unknowns. Such solutions are called Basic Solutions.
 Basic variables: The variables whose value is obtained from the basic solution is called
basic variables
 Non-basic variables: The variables whose value are assumed as zero in basic solutions are
called non-basic variables.
 Solution: A solution to a L.P.P is the set of values of the variables which satisfies the set
of constraints for the problem.
4
 Feasible solution: A feasible solution to a L.P.P the set of values of variables which
satisfies the set of constraints as well as the non-negative constraints of the problem.
 Basic feasible solution: A feasible solution to a L.P.P in which the vectors associated with
the non-zero variables are linearly independent is called basic feasible solution
 Note: Linearly independent: variables x1, x2, x3......... are said to be linearly independent
if k1x1+k2x2+.........+knxn=0, implying k1=0, k2=0,........
 Optimum (optimal) solution: A feasible solution of a L.P.P is said to be the optimum
solution if it also optimizes the objective function of the problem.
 Slack variables: Linear equations are solved through equality form of equations.
Normally, constraints are given in the "less than or equal" (≤) form. In such cases, we add
appropriate variables to make it an "equality" (=) equation. These variables added to the
constraints to make it an equality equation in L.P.P is called stack variables and is often
denoted by the letter 'S'.
 E.g.: 2x1 + 3x2 ≤ 500
 2x1 + 3x2 + S1 = 500, where S1 is the slack variable
 Surplus variables: Sometime, constraints are given in the "more than or equal" (≥) form.
In such cases we subtract an approximate variable to make it into "equality" (=) form.
Hence variables subtracted from the constraints to make it an equality equation in L.P.P is
called surplus variables and often denoted by the letter 's'.
 E.g.: 3x1 + 4x2 ≥ 100
 3x1 + 4x2 - S2 = 0,
where S2 is the surplus variable.
 Artificial variable: Artificial variables are fictitious variables. These are introduced to help
computation and solution of equations in L.P.P. There are used when constraints are given
in (≥) "greater than equal" form. As discussed, surplus variables are subtracted in such
cases to convert inequality to equality form. In certain cases, even after introducing surplus
variables, the simplex tableau may not contain an 'Identity matrix' or unit vector. Thus, in a
L.P.P artificial variables are introduced in order to get a unit vector in the simplex tableau
to get feasible solution. Normally, artificial variables are represented by the letter 'A'.
 Big-M-method: Problems where artificial variables are introduced can be solved by two
methods viz.
o Big-M-method and
o Two phase method.
 Big-M-method is modified simplex method for solving L.P.P when high penalty cost (or
profit) has been assigned to the artificial variable in the objective function. This method is
applicable for minimizing and maximizing problems.
 Two Phase method: L.P.P where artificial variables are added can be solved by two phase
method. This is a modified simplex method. Here the solution takes place in two phases as
follows:
o Phase I - Basic Feasible solution: Here, simplex method is applied to a specially
constricted L.P.P called Auxiliary L.P.P and obtain basic feasible solution.
o Phase II - Optimum Basic solution: From basic feasible solution, obtain optimum feasible
5
solution.
 Simplex Tableau: This is a table prepared to show and enter the values obtained for basic
variables at each stage of Iteration. This is the derived values at each stage of calculation.
Optimal solution
An optimal solution of a linear programming problem is the set of real values of the
decision variables which satisfy the constraints including the non-negativity conditions, if
any and at the same time optimize the objective function.
A vector (x1, x2... xn) which satisfies the constraints A x ≤ or ≥ b only is called a solution.
And a solution which satisfies all the constraints including the non- negativity condition is
called a feasible solution. The set of all feasible solutions is called feasible space.
Integer Programming
A L.P.P in which solution requires integers is called an integer programming problem. A
mathematical programming in which the objective fn is quadratic but all the constraints are
linear un the decision variable is called a quadratic programming.
e.g.: Max z = x21 + x22
Subject to 2x1 + x2 ≤ 6 7x1 + 8x2 ≤ 28
X1, x2 ≥ 0
Graphical L P solution
The graphical procedure includes two steps:
1. Determination of the solution space that defines all feasible solutions of the model.
2. Determination of the optimum solution from among all the feasible points in the solution
space.
The proper definition of the decision variables is an essential first step in the development of
the model. Once done, the task of constructing the objective function and the constraints is
straighter forward.
FORMATION OF MATHEMATICAL MODEL OF L.P.P
There are three forms:



General form of L.P.P
Canonical form of L.P.P
Standard form of L.P.P
These are written in 'statement form' or in 'matrix' form as explained in subsequent paragraphs.
6
General form of L.P.P
Statement form: This is given as follows "Find the values of x1, x2... xn which optimize z = c1x1 + c2x2 + ... + cnxn subject to :
a11x1 + a12x2 + ... + a1nxn ≤ (or = or ≥) b1 a21x1 + a22x2 + ... + a2nxn ≤ (or = or ≥) b2
am1x1 + am2x2 + ... + amnxn ≤ (or = or ≥) bm
x1, x2,... xn ≥ 0
where all the coefficients (cj, aij, bi) are constants and xj's are variables. (i = 1,2,... m)
(j = 1,2,... n ) "
b)
Matrix form of general L.P.P. This is stated as follows "Find the values of x1, x2, ... xn to maximize: z = c1x1 + c2x2 + ... +cnxn Let z be a linear
function on a Rn defined by
a)
z = c1x1 + c2x2 + ... cnxn
where cj are constants. Let aij be m*n matrix and let { b1, b2, ... bm } be set of constraints such
that
ii.
a11x1 + a12x2 + ... + a1nxn ≤ (or = or ≥) b1 a21x1 + a22x2 + ... + a2nxn ≤
(or = or ≥) b2 am1x1 + am2x2 + ... + amnxn ≤ (or = or ≥) bm And let
iii.
xj ≥ 0 j = 1,2,... n
i.
The problem of determining an n-tuple (x1, x2,... xn) which make z a minimum or a maximum
is called 'General linear programming problem'.
Canonical Form of L.P.P
a) Statement form: This form is given as follows:
"Maximize
z=c1x1 + c2x2 +....... + cnxn
subject to constraints ai1x1 + ai2x2 +........+ ainxn ≤ bi ;
x1x2......xn ≥ 0 "
(i= 1,2,....,m)
Characteristics of canonical form:
1.
Objective function is of the "maximization" type.
Note: minimization of function f(x) is equivalent to maximization of
function {-f(x)}
..
.
Minimize
f(x) = Maximize {-f(x)}
2.
All constraints are of the type "less than or equal to" viz "≤"
except the non-negative restrictions.
Note: An inequality of more than (≥) can be replaced by less than (≤) type
by multiplying both sides by -1 and vice versa.
eg:
3.
b)
2x1 + 3x2 ≥ 100 can be written as -2x1-3x2 ≤ -100
All variables are non-negative viz xj ≥ 0
Canonical form of L.P.P with matrix notations:
7
“Maximize Z = CX , subject to the constraints AX ≤ b
X ≥ 0 Where X= (x1, x2 ,.......,xn); C= (c1
,c2 ,......,cn)
bT = (b1 ,b2,..... ,bm) ;
A= (aij)
j= 1, 2,...., n "
where i= 1, 2,..., m
The Standard Form of L.P.P
a) Statement form
“Maximize Z= c1x1 + c2x2 +....+ cnxn
Subject to the constraints ai1x1 + ai2x2 +.....+ ainxn = bi (i= 1, 2,...., m)
x1x2....xn ≥ 0 "
Characteristics of Standard form
1. Objective function is of maximization type.
2. All constraints are expressed in the function of equality form except the restrictions.
3. All variables are non- negative.
Note: constraints given in the form of "less than or equal" (≤) can be converted to the equality
form by adding "slack" variables. Similarly, those given in "more than or equal" (≥) form can
be converted to the equality form by subtracting "surplus" variables.
b) Standard form of L.P.P in matrix
notations: " Maximize Z = CX
Subject to the
constraints AX =b
b ≥0 and X
≥0
Where X = (x1, x2, ..... xn) ; C = (c1, c2, ....,
cn ) bT = (b1, b2,....., bm)
; A= (aij)
i = 1, 2, ....., m ; j = 1, 2, ....., n "
Note: coefficients of slack and surplus variables in objective function are
always assumed to be zero.
8
APPLICATION OF LINEAR PROGRAMMING
The primary reason for using linear programming methodology is to ensure that limited
resources are utilized to the fullest extent without any waste and that utilization is made in such
a way that the outcomes are expected to be the best possible.
Some of the examples of linear programming are:
 A production manager planning to produce various products with the given resources of raw







materials, man-hours, and machine-time for each product must determine how many
products and quantities of each product to produce so as to maximize the total profit.
An investor has a limited capital to invest in a number of securities such as stocks and
bonds. He can use linear programming approach to establish a portfolio of stocks and bonds
so as to maximize return at a given level of risk.
A marketing manager has at his disposal a budget for advertisement in such media as
newspapers, magazines, radio and television. The manager would like to determine the
extent of media mix which would maximize the advertising effectiveness.
A Farm has inventories of a number of items stored in warehouses located in different parts
of the country that are intended to serve various markets. Within the constraints of the
demand for the products and location of markets, the company would like to determine
which warehouse should ship which product and how much of it to each market so that the
total cost of shipment is minimized.
Linear programming is also used in production smoothing. A manufacturer has to determine
the best production plan and inventory policy for future demands which are subject to
seasonal and cyclical fluctuations. The objective here is to minimize the total production
and inventory cost.
A marketing manager wants to assign territories to be covered by salespersons. The
objective is to determine the shortest route for each salesperson starting from his base,
visiting clients in various places and then returning to the original point of departure. Linear
programming can be used to determine the shortest route.
In the area of personnel management, similar to the travelling salesperson problem, the
problem of assigning a given number of personnel to different jobs can be solved by this
technique. The objective here is to minimize the total time taken to complete all jobs.
Another problem in the area of personnel management is the problem of determining the
equitable salaries and sales-incentive compensation. Linear programming has been used
successfully in such problems.
9
BASIC REQUIREMENTS OF A LINEAR
PROGRAMMING MODEL
 The system under consideration can be described in terms of a series of activities and





outcomes. These activities (variables) must be competing with other variables for limited
resources and the relationships among these variables must be linear and the variables must
be quantifiable.
The outcomes of all activities are known with certainty.
A well-defined objective function exists which can be used to evaluate different outcomes.
The objective function should be expressed as a linear function of the decision variables. The
purpose is to optimize the objective function which may be maximization of profits or
minimization of costs, and so on.
The resources which are to be allocated among various activities must be finite and limited.
These resources may be capital, production capacity, manpower, time etc.
There must not be just a single course of action but a number of feasible courses of action
open to the decision maker, one of which would give the best result.
All variables must assume non-negative values and be continuous so that fractional value of
the variables are permissible for the purpose of obtaining an optimal solution.
L.P.P- Special cases
 We have seen a L.P.P is given in the form Maximize Z= CX
 Subject to constraints AX = b; X ≥ 0 And the solution obtained is X= A-1b
 Thus we see that simplex solution obtained in the Tableau which is declared as optimum basic
feasible solution (O.B.F.S) contains inverse matrix A.
 Case of unbounded solutions: when L.P.P does not give finite values of variable X, viz x1,
x2,∞, such solutions are called UNBOUNDED. Here variable X can take very high values
without violating the conditions of the constraints.
 In such cases the final Tableau shows all Ratios viz R values are negative so that no minimum
ratio condition can be applied.
Establish the UNBOUND condition. Maximize Z= 2x1 + x2
Subject to the constraints x1-x2 ≤ 10
2x1-x2 ≤ 40 x1, x2 ≥ 0
10
Case of more than one optimum solution:
This is the case where L.P.P gives solutions which are optimum but not UNIQUE. This
means, more than one optimum solution is possible.
Case of Degeneracy: A basic feasible solution of a L.P.P is said to degenerate if at least one
of the basic variables is zero.
Types of Degeneracy

First Iteration: Degeneracy can occur right in the first (initial tableau). This normally
happens when the number of constraints are less than number of variables in the
objective function. Problem can be overcome by trial and error method.
 Subsequent Iteration: Degeneracy can also occur in subsequent iteration. This is due to
the fact that minimum Ratio values are the same for two rows. This will make choice
difficult in selecting the 'Replacing Row'.
Case of cycling:
We have seen that when degeneracy exists, replacement of vector in the BASIS does not
improve objective function. Another difficulty encountered in degeneracy is that the
simplex Tableau gets repeated without getting optimum solution. Such occurrences in
L.P.P are called CYCLING. Fortunately, such L.P.Ps are very rare. Also, in such cases,
certain techniques are developed to prevent cycling and reach optimum solution.
Case of duality:
Every L.P.P is associated with another L.P.P which is called the DUAL of the original
L.P.P. The original L.P.P is called the PRIMAL. If the DUAL is stated as a given problem,
PRIMAL becomes its DUAL and vice-versa. The solution of one contains the solution of the
other. In other words, when optimum solution of PRIMAL is known, the solution of DUAL
is also obtained from the very same optimum tableau.
USES OF DUAL CONCEPT
There are many uses of this concept. However, one of the very important use is solve difficult
L.P.P. Many times, a given L.P.P is complicated having large number of constraints. In such
cases, one method is to design its dual which invariably will have less number of constraints.
Dual is now subjected to solution by simplex method.
Solution of a L.P.P
In general we use the following two methods for the solution if a linear programming
problem.
11
I.
Geometrical (or graphical) method: If the objective function z is a function of two variables
only; then the problem can be solved by the graphical method. A problem involving three
variables can also be solved graphically, but with complicated procedure.
Simplex method: This is the most powerful tool of the linear programming as any problem
can be solved by this method. Also, this method is an algebraic procedure which
progressively approaches the optimal solution. The procedure is straight forward and
requires only time and patience to execute manually. Nowadays, computational methods
(using computers), are available for solving such problems.
Geometrical (or Graphical) method for solving a L.P.P
 If the L.P.P is two variable problem, it can be solved graphically. The steps required for
solving a L.P.P by graphic method are :
 Formulate the problem into a L.P.P
 Each inequality in the constraint may be written as equality.
 Draw straight lines corresponding to the equations obtained in step 2. So there will be as
many straight lines as there are equations.
 Identify the feasible region. Feasible region is the area which satisfies all constraints
simultaneously.
 The permissible region or feasible region is a many sided figure (a polygon). The corner
points of the figure are to be located and their co-ordinates (ie. x1 and x2 values) are to be
measured.
 Calculate the value of the objective function Z at each corner point.
 The solution is given by the co-ordinates of that corner point which optimizes the objective
function.
Dual Simplex Method
Any primal iteration zj - cj, the objective equation coefficient of xj, equals the
difference between the left and right sides of the associated dual constraint. When, in
the case of maximization, the primal iteration is not optimal, zj - cj < 0 for t least one
variable. Only at the optimum do we have zj -cj ≥ 0 for all j.
12
LIMITATIONS
 There is no guarantee that linear programming will give integer valued equations. For
instance, solution may result in producing 8.291 cars. In such a situation, the manager will
examine the possibility of producing 8 as well as 9 cars and will take a decision which
ensures higher profits subject to given constraints. Thus, rounding can give reasonably good
solutions in many cases but in some situations we will get only a poor answer even by
rounding. Then, integer programming techniques alone can handle such cases.
 Under linear programming approach, uncertainty is not allowed. The linear programming
model operates only when values for costs, constraints etc. are known but in real life such
factors may be unknown.
 The assumption of linearity is another formidable limitation of linear programming. The
objective functions and the constraint functions in the L.P model are all linear. We are thus
dealing with a system that has constant returns to scale. In many situations, the input-output
rate for an activity varies with the activity level. The constraints in real life concerning
business and industrial problems are not linearly related to the variables, in most economic
situations, sooner or later, the law of diminishing marginal returns begins to operate. In this
context, it can, however be stated that non-linear programming techniques are available for
dealing with such situations.
 Linear programming will fail to give a solution if management have conflicting multiple
goals. In L.P model, there is only one goal which is expressed in the objective function.
o E.g. maximizing the value of the profit function or minimizing he cost function, one should
resort to Goal programming in situations involving multiple goals.
 All these limitations of linear programming indicate only one thing- that linear
programming cannot be made use of in all business problems. Linear programming is not a
panacea for all management and industrial problems. But for those problems where it can be
applied, the linear programming is considered a very useful and powerful tool.
13
ANALYSIS
Linear programming is a resources allocation model that seeks the best allocation of limited
resources to a number of competing activities. L.P has been applied with considerable
success to a multitude of practical problems.
The suitability of the graphical L.P solution is limited to variable problems. However, the
graphical method reveals the important result that for solving L.P problems it is only
necessary to consider the corner (or extreme) points of the solution space. This result is the
key point in the development of the simplex method, which is an algebraic procedure
designed to solve the general L.P problem.
Sensitivity analysis should be regarded as an integral part of solving any optimization
problem. It gives the L.P solutions dynamic characteristics that are absolutely necessary for
making sound decisions in a constantly changing decision- making environment.
According to Ferguson and Sargent:
"Linear programming is a technique for specifying how to use limited resources or capacities
of a business to obtain a particular objective such as the least cost, the highest margin or the
least time, when these resources have alternative uses. It is a technique that systemizes certain
conditions, the process of selecting the most desirable course of action from a number of
available courses of action, thereby giving the management the information for making a more
effective decision about the resources under control."
Since the objective of any organization is to make the best utilization of the given resources,
linear programming provides powerful technique for effective utilization of these given
resources under certain well-defined circumstances. For instance, an industrial process consists
of a number of activities relating to the capital invested and the capital required for operational
activities, products to be produced and marketed, raw materials to be used, machines to be
utilized, products to be stored and consumed or a combination of the above activities. Some or
all of these activities are interdependent and inter-related so that there are many ways where by
these resources can be allocated to various competing demands. Linear programming helps the
decision maker in arranging for such combination of resources which results in optimization of
objectives.
Linear programming method was first formulated by a Russian mathematician Shri. L.V.
Kantorovich. Today, this method is being used in solving a wide range of practical business
problems. The advent of electronic computers has further increased its applications to solve
many other problems in industry. It is being considered as one of the most versatile
management techniques.
14
RESULTS & DISCUSSIONS
The simplex method shows that a corner point is essentially identified by a basic solution of
the standard form of linear programming. The optimality and feasibility conditions of the
simplex method that, starting from a feasible corner point (basic solution), the simplex
method will move to an adjacent corner point which has the potential to improve the value
of the objective function. The maximum number of iterations (basic solutions or corner
points) until the optimum is reached is limited by n!/[(n-m)!m!] in an n-variable m-equation
standard L.P model.
The special case of alternative optima point to the desirable adoption of one optimum
solution over another, even though both may have the same optimum objective value
depending, for example, on the activity "mix" that each solution offers.
An unbounded optimum or a solution space, as well as a non-existing solution, points to the
possibility that some irregularities exist in the original formulation of the model. As a result,
the model must be checked.
The optimum tableau offers more than just the optimum values of the variables.
Additionally, it gives the status and worth (shadow prices) of the different resources. The
sensitivity study shows the resources can be changed within certain limits while maintaining
the same activity mix in the solution. Also, the marginal profits/costs can be changed within
certain ranges without changing the optimum values of the variables.
The method of this project is by collecting data as much as possible and analyzing it. By this
analysis, we can understand that importance and use of Linear Programming Problem in
mathematics. It is decided to observe and collect various books in our library. Net sources,
books and instruments are my teaching aids. We can conclude the project by knowing the
various purposes and brief sketch of linear programming.
CONCLUSION
From this project we came to a conclusion that 'Linear programming' is like a vast ocean where
many methods, advantages, uses, requirements etc. can be seen. Linear programming can be
done in any sectors where there is less waste and more profit. By this, the production of
anything is possible through the new methods of L.P. As we had collected many data about L.P,
we came to know more about this, their uses, advantages and requirements. Also, there are
many different ways to find out the most suitable L.P.
Also, we formulate an example for linear programming problem and done using the two
methods simplex method and dual problem. And came to a conclusion that L.P is not just a
technique but a planning the process of determining a particular plan of action from amongst
several alternatives. Even there are limitations, L.P is a good technique, especially in the
business sectors.
15
REFERENCES

F.S. Hillier, G.J. Lieberman, B. Nag and P. Basu, Introduction to Operation Research,
10th Edition, McGraw Hill, 2017.

C. Mohan and K. Deep, Optimization Techniques, New Age, 2009.

N.D. Vohra, Quantitative Techniques in Management, 5th Edition, McGraw-Hill.

K.V. Mittal and C. Mohan, Optimization Methods in Operations Research and Systems
Analysis, New Age, 2003.

H.A. Taha, Operations Research - An Introduction, 7th Edition, Prentice Hall, 2002.

A Ravindran, D.T. Phillips and J.J. Solberg, Operations Research: Principles and
Practice, 2nd Edition, John Willey and Sons, 2009.

K. Bedi, Production and Operations Management, Oxford University Press, 2004.

S.J. Chandra and A. Mehra, Numerical Optimization with Applications, Narosa, 2009.


J.K. Sharma, Operation Research: Theory and Applications, 5th Edition, Macmillan Pub.,
2013.

L.W. Wayne, Operations Research Applications and Algorithms, 4th Edition,
Brooks/Cole

G¨artner, Bernd. Matou˘sek, Ji˘r´ı. Understanding and Using Linear Programming,

p. 3

G¨artner, Bernd. Matou˘sek, Ji˘r´ı. Understanding and Using Linear Programming,

p. 60-67.

GLPK (GNU Linear Programming Kit), [website], 2012,

https://www.gnu.org/software/glpk/, (accessed 15 October 2018)

A system of linear inequalities defines a polytope as a feasible region. The simplex

algorithm begins at a starting vertex and moves along the edges of the

polytope until it reaches the vertex of the optimal solution, [online photograph],

https://en.wikipedia.org/wiki/Simplex algorithm, (accessed 15 December 2018)
16