Linear_Programming.pdf

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Components of Linear
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Steps for Mathematical
Formulation of LPP’s
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Examples on Graphical
method for solving LPP
Linear Programming
Dr. Tushar Bhatt
Department of Science and Humanities,
Faculty of Engineering and Technology,
Atmiya University, Rajkot
March 2, 2023
Dr. Tushar Bhatt (Atmiya University, Rajkot) Linear Programming 1 / 49

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Table of Contents
1 Introduction to LPP
Components of Linear Programming Problem
Basic Assumption in LPP
Examples of LPP
2 Formulation of LPP
Steps for Mathematical Formulation of LPP’s
Examples on Formulation of LPP
3 Basic Definitions
4 Graphical Method for solving LPP
Examples on Graphical method for solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Table of Contents
Examples of LPP
3 Basic Definitions

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
1. Introduction
Operation Research (O.R.) is a scientific approach to problem solving for
executive management. – H.M. Wagner.
O.R.= Operation + Research, in which the operation means identify
problems and in scientific way to represent it and research means inventing
some new ideas techniques, models for solving desired problems.
O.R. is the application of the methods of science to solve complex problems
arising in the management of large systems of man, machine, and materials
in industry.
It measures the factors such as risk, chances, comparisons, controlling and
decision-making extra. It is experimental.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
1. Introduction to LPP
In O.R. our main focus to obtain an optimal solution of any problem, the
techniques for obtaining an optimal solution will be known as optimization.
Optimization is the act of obtaining the best result under given
circumstances.
Optimization can be defined as the process of finding the conditions that give
maximum or minimum value of a function.
Any problem that requires a positive decision to be made can be classified as
an operation research (OR) type problem and solutions can be made
according to them as known as Optimization Techniques.
The linear programming is one of the best tool or technique of OR.
George B. Dantzig is generally recognized as the pioneer of linear
programming.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
1.1 Components of Linear Programming Problem
A linear programming problem (LPP) consists of three components namely,
1 Decision Variables or Activities: It refers to the activities that are
competing one another for sharing the resources available. These variables
are usually inter-related in terms of utilization of resources and need
simultaneous solutions. All the decision variables are considered as
continuous, controllable and non-negative.
2 The Objective or Goal: A linear programming problem must have an
objective which should be clearly identifiable and measurable in quantitative
terms. It could be of profit (sales) maximization, cost (time) minimization
and so on. The relationship among the variables representing objective must
be linear.
3 The Constraints or Restrictions: There are always certain limitations or
constraints on the use of resources, such as labor, space, raw material, money
and many more that limit the degree to which an objective can be achieved.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
1.2 Basic Assumption in LPP
The following four basic assumptions are necessary for all linear programming
problems:
1 Certainty: In all LPP’s, it is assumed that all the parameters such as
availability of resources, profit (or cost) contribution of a unit of decision
variable and consumption of resources by a unit decision variable must be
known and fixed.
2 Divisibility / Continuity: This implies that solution values of the decision
variables can take only non-negative values, including fraction values.
3 Proportionality: This requires the contribution of each decision variable in
both the objective function and the constraints to be directly proportional to
the value of the variable.
4 Additivity: The value of the objective function for the given values of
decision variables and the total sum of resources used, must be equal to the
sum of the contributions (profit or cost) earned from each decision variable
and the sum of the resources used by each decision variable respectively.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
1.3 Examples of LPP
Manufacturing Problems: Determine the number of units that should be
produced and sold in order to maximize profit, when each product requires a
fixed manpower, machine hours and raw materials.
Diet Problems: Determine the amount of different kinds of to be included
in the diet, minimizing the cost and subject to the availability of food and
their prices.
Transportation Problems: Determine a transportation schedule to find the
cheapest way of transporting a product from plants or factories situated at
different location to different markets.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Table of Contents
Examples of LPP
3 Basic Definitions

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2. Steps for Mathematical Formulation of LPP’s
Step - 1: To identify the decision variables from the problem.
Step - 2: To construct the objective function as a linear combination of the
decision variables.
Step - 3: To identify the constraints of the problem such as resources
limitations, Inter-relationship between variables etc. Formulae of these
constraints as linear equations or inequations in terms of non-negative
decision variables.
Thus, LPP is a collection of objective function, the set of constraints and the set
of the non – negative constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2.1 Examples on Formulation of LPP
Ex - 1: A manufacturer produce two types of models M and N each M
model requires 4 hours grinding and 2 hours for polishing whereas each N
model requires 2 hours of grinding and 5 hours for polishing. The
manufacturer has 2 grinders and 3 polishers. Each grinder works for 40 hrs
a week and each polisher’s works for 60 hrs a week. Profit on an M model
is 3 RS. and on an N model is 4 Rs. whatever is produced in a week sold
in the market. How should the manufacturers allocate this production
capacity to the two types of models? So that he may make the maximum
profit in a week?

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2.1 Examples on Formulation of LPP - Solution - 1 ...
Solution:
Step – 1: Here first we identify the decision variables

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x1 = No.of units produced by model M.
x2 = No.of units produced by model N.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Step – 2: Construct the Objective Function

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Maximize Z = 3x1 + 4x2.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Step – 3: Identify the Constraints

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Subject to constraints
4x1 + 2x2 ≤ 80
2x1 + 5x2 ≤ 180
And x1, x2 ⩾ 0.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Ex - 2: A firm manufacturers 3 products A, B and C. The profits are 3 Rs,
2Rs, and 4 Rs, respectively. The firm has two machines M1 and M2 below
is the required time in minutes for each machine on each product.
Products
A B C
M1 4 3 5
M2 2 2 4
Machines M1 and M2 have 2000 and 2500 machine minutes respectively.
The firm must manufacture 100A’s, 200B’s and 50C’s but not more than
150A’s. Set up an LPP to maximize profit.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2.1 Examples on Formulation of LPP - Example - 2 solution...
Solution:
Step - 1: Here first we identify the decision variables

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x1 = No.of units of product A.
x2 = No.of units of product B.
x3 = No.of units of product C.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Maximize Z = 3x1 + 2x2 + 4x3.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
4x1 + 3x2 + 5x3 ≤ 2000,

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
4x1 + 3x2 + 5x3 ≤ 2000, 2x1 + 2x2 + 4x3 ≤ 2500,

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
4x1 + 3x2 + 5x3 ≤ 2000, 2x1 + 2x2 + 4x3 ≤ 2500, And 100 ≤ x1 ≤ 150,

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
4x1 + 3x2 + 5x3 ≤ 2000, 2x1 + 2x2 + 4x3 ≤ 2500, And 100 ≤ x1 ≤ 150, x2 ⩾ 200,

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
4x1 + 3x2 + 5x3 ≤ 2000, 2x1 + 2x2 + 4x3 ≤ 2500, And 100 ≤ x1 ≤ 150, x2 ⩾ 200,
x3 ⩾ 50.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Ex-3: A mine company has two different mines X and Y that produce an
ore which after being crushed, is graded into three classes: high, medium
and low grade. The company has contracted to provide a smelting plant
with 12 tons of high grade, 8-tons of medium grade and 24-tons of low
grade ore per week. The two mines have different operating characteristic
as detailed below:
Production(Tons/day)
Mine Cost/day (Rs’ 000) High Medium Low
X 180 6 3 4
Y 160 1 1 6
How many days per week should each mine be operated to fulfils the
smelting plant contract? For minimizing the cost of product.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2.1 Examples on Formulation of LPP - Example - 3 Solution...
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x = No.of days per week mine X is operated.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
y = No.of days per week mine Y is operated.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Minimize Z = 180x + 160y.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Subject to constraints 6x + y ⩾ 12, 3x + y ⩾ 8, 4x + 6y ⩾ 24
Days per week constraints we cannot work more than a certain maximum number
of days a week consider 5 working days in a week therefore x ≤ 5 and y ≤ 5. It is
obvious x, y ⩾ 0.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Ex-4: A calculator company produces a handheld calculator and a scientific
calculator. Long-term projections indicate an expected demand of at least
150 scientific and 100 handheld calculators each day. Because of
limitations on production capacity, no more than 250 scientific and 200
handheld calculators can be made daily.
To satisfy a shipping contract, a minimum of 250 calculators must be
shipped each day. If each scientific calculator sold, results in a 20 rupees
loss, but each handheld calculator produces a 50 rupees profit; then how
many of each type should be manufactured daily to maximize the net
profit?

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x = number of scientific calculators produced.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
y = number of handheld calculators produced.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Minimize Z = −20x + 50y.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x + y ⩾ 250, 150 ≤ x ≤ 250, 100 ≤ y ≤ 200, x ⩾ 0, y ⩾ 0.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Ex-5: Universal Corporation manufactures two products- P1 and P2. The
profit per unit of the two products is Rs. 50 and Rs. 60 respectively. Both
the products require processing in three machines. The following table
indicates the available machine hours per week and the time required on
each machine for one unit of P1 and P2. Formulate this product mix
problem in the linear programming form.
Machine Product Product Available Time
P1 P2 (in machine hours per week)
1 2 1 300
2 3 4 509
3 4 7 812
Profit Rs. 50 Rs. 60

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x1 = amounts manufactured of products P1.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
Minimize Z = 50x1 + 60x2.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
2x1 + x2 ≤ 300, 3x1 + 4x2 ≤ 509, 4x1 + 7x2 ≤ 812, x1 ⩾ 0, x2 ⩾ 0.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Table of Contents
Examples of LPP
3 Basic Definitions

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
3. Basic Definitions
Definition: Solution
A set of variables {x1, x2, ...xn+m} is called a solution of LPP if it satisfies its
constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
constraints.
Definition: Feasible solution
A set of variable {x1, x2, ...xn+m} is called a feasible solution of LPP if it satisfies
its constraints as well as non-negativity restrictions.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
constraints.
Definition: Optimal/optimum feasible solution
The basic feasible solution is said to be optimal feasible solution if it optimizes the
objective function.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
constraints.
Definition: Optimal/optimum feasible solution
The basic feasible solution is said to be optimal feasible solution if it optimizes the
objective function.
Definition: Unbounded solution
If the objective function can be increased or decreased indefinitely, the solution is
called an unbounded solution.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Table of Contents
Examples of LPP
3 Basic Definitions

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
4. Graphical Method for solving LPP
Linear programming problems involving two decision variables can be easily solved
by graphical method. If the problem has more than two variables, graphical
method is either impossible or impractical. Graphical solution to a two variables
LP problem can be obtained using following steps:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
1 To convert all constraints into equations.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2 To plot the graph of equations and apply the conditions given in the
constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
constraints.
3 To identify feasible region (that satisfies all the conditions) and find it’s
corners (extreme points).

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
constraints.
3 To identify feasible region (that satisfies all the conditions) and find it’s
corners (extreme points).
4 To test the maximum / minimum value of objective function for these
corners.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
4.1 Examples on Graphical method for solving LPP
Ex - 1: Solve the following LP problem by Graphical method.
Maximize Z = 5x1 + 3x2
Subject to
3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
And x1, x2 ⩾ 0.
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
And x1, x2 ⩾ 0.
Solution:
Step – 1: Convert all constraints into equations

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
And x1, x2 ⩾ 0.
Solution:
3x1 + 5x2 = 15 (1)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
And x1, x2 ⩾ 0.
Solution:
3x1 + 5x2 = 15 (1)
In equation (1), if we put x1 = 0 ⇒ x2 = 3 and for x2 = 0 ⇒ x1 = 5.
x1 0 5
x2 3 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
And x1, x2 ⩾ 0.
Solution:
3x1 + 5x2 = 15 (1)
x1 0 5
x2 3 0
Hence, we get two points A(0, 3) and B(5, 0).

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
4.1 Examples on Graphical method for solving LPP; Ex - 1: Solu...
5x1 + 2x2 = 10 (2)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
5x1 + 2x2 = 10 (2)
x1 0 2
x2 5 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
5x1 + 2x2 = 10 (2)
x1 0 2
x2 5 0
Hence, we get two points C(0, 5) and D(2, 0).
Step - 2: To plot the graph of equations and apply the conditions given in
the constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Consider equation − − − − − − (1) × 2 – equation − − − − − − (2) × 5 we get
(1) =⇒ 3x1 + 5x2 = 15 =⇒ x1 + 10x2 = 30
(2) =⇒ 5x1 + 2x2 = 10 =⇒ 25x1 + 10x2 = 50
−19x1 = −20
=⇒ x1 =
20
19
Put x1 = 20
19 in equation (1), we get
(2) =⇒ 3x1 + 5x2 = 15
=⇒ 3(20/19) + 5x2 = 15
=⇒ 60 + 95x2 = 285
=⇒ x2 =
45
19

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Now the feasible region is: OACBO
Corner points Co-ordinates Z = 5x1 + 3x2
A (0, 3) 9
B (2, 0) 10
C 20
19, 45
19

12.36= Max. Value
O (0, 0) 0
Therefore the point C 10
19, 45
19

follows Maximize Z,and hence x1 = 10
19 and
x2 = 45
19, are the required solutions.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Maximize Z = 8x + y
Subject to
x + y ≤ 40
2x + y ≤ 60
And x, y ⩾ 0.
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Maximize Z = 8x + y
Subject to
x + y ≤ 40
2x + y ≤ 60
And x, y ⩾ 0.
Solution:
x + y = 40 (1)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Maximize Z = 8x + y
Subject to
x + y ≤ 40
2x + y ≤ 60
And x, y ⩾ 0.
Solution:
x + y = 40 (1)
In equation (1), if we put x = 0 ⇒ y = 40 and for y = 0 ⇒ x = 40.
x 0 40
y 40 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Maximize Z = 8x + y
Subject to
x + y ≤ 40
2x + y ≤ 60
And x, y ⩾ 0.
Solution:
x + y = 40 (1)
x 0 40
y 40 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2x + y = 60 (2)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2x + y = 60 (2)
x 0 30
y 60 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2x + y = 60 (2)
x 0 30
y 60 0
the constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Consider equation − − − − − − (1) – equation − − − − − − (2) we get
(1) =⇒ x + y = 40
(2) =⇒ 2x + y = 60
−x = −20
=⇒ x = 20
Put x = 20 in equation (1), we get
(1) =⇒ x + y = 40
=⇒ 20 + y = 40
=⇒ y = 40 − 20
=⇒ y = 20

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Feasible Region points Co-ordinates Z = 8x + y
A (0, 40) 40
B (30, 0) 240= Max. Value
C (20, 20) 180
O (0, 0) 0
Therefore the point B (30, 0) follows Maximize Z,and hence x = 30 and y = 0,
are the required solutions.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Minimize Z = 5x1 + 4x2
Subject to
4x1 + x2 ⩾ 40
2x1 + 3x2 ⩾ 90
And x1, x2 ⩾ 0.
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
4x1 + x2 ⩾ 40
2x1 + 3x2 ⩾ 90
And x1, x2 ⩾ 0.
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
4x1 + x2 ⩾ 40
2x1 + 3x2 ⩾ 90
And x1, x2 ⩾ 0.
Solution:
4x1 + x2 = 40 (1)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Subject to
4x1 + x2 ⩾ 40
2x1 + 3x2 ⩾ 90
And x1, x2 ⩾ 0.
Solution:
4x1 + x2 = 40 (1)
In equation (1), if we put x1 = 0 ⇒ x2 = 40 and for x2 = 0 ⇒ x = 10.
x 0 10
y 40 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2x1 + 3x2 = 90 (2)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2x1 + 3x2 = 90 (2)
x1 0 45
x2 30 0

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
2x1 + 3x2 = 90 (2)
x1 0 45
x2 30 0
the constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Consider equation − − − − − −(1)–2 × equation − − − − − −(2) we get
(1) =⇒ 4x1 + x2 = 40
(2) =⇒ 4x1 + 6x2 = 180
−5x2 = −140
=⇒ x2 = 28
Put x2 = 28 in equation (1), we get
(1) =⇒ 4x1 + x2 = 40
=⇒ 4x1 + 28 = 40
=⇒ 4x1 = 40 − 28
=⇒ x1 =
12
4
=⇒ x1 = 3

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Feasible Region points Co-ordinates Z = 5x1 + 4x2
A (45, 0) 225
B (3, 28) 127= Min. Value
C (0, 40) 160
Therefore the point B (30, 0) follows Minimize Z,and hence x1 = 3 and x2 = 28,
are the required solutions.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Ex-4: At a university, Professor Symons wishes to employ two people, John
and Mary, to grade papers for his classes. John is a graduate student and
can grade 20 papers per hour; John earns $ 15 per hour for grading
papers. Mary is an post-doctoral associate and can grade 30 papers per
hour; Mary earns $ 25 per hour for grading papers. Each must be
employed at least one hour a week to justify their employment.
If Prof. Symons has at least 110 papers to be graded each week, how
many hours per week should he employ each person to minimize the cost?

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
x = The number of hours per week John is employed.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
y = The number of hours per week Mary is employed.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Solution:
20x + 30y ⩾ 110, x ⩾ 1, y ⩾ 1, x ⩾ 0, y ⩾ 0.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
4.1 Examples on Graphical method for solving LPP; Ex- 4 Solu...
Solution:
The problem has been formulated as follows.
20x + 30y ⩾ 110,
x ⩾ 1,
y ⩾ 1,
x ⩾ 0,
y ⩾ 0.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
4.1 Examples on Graphical method for solving LPP; Ex-4: Solu...

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
x = 1 (1)
y = 1 (2)
20x + 30y = 110 (3)

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
x = 1 (1)
y = 1 (2)
20x + 30y = 110 (3)
In equation (3), if we put x = 1 ⇒ y = 110−20
30 = 90
30 = 3 and
for y = 1 ⇒ x = 110−30
20 = 80
20 = 4.
x 1 4
y 3 1

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
the constraints.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Now the feasible region is: AB
Feasible Region points Co-ordinates Z = 15x + 25y
A (1, 3) $ 90
B (4, 1) $ 85= Min. Value
Therefore the point B (4, 1) follows Minimize Z,and hence x = 4 and y = 1, are
the required solutions.

Linear
Programming
Dr. Tushar
Bhatt
Introduction to
LPP
Programming Problem
Basic Assumption in
LPP
Examples of LPP
Formulation of
LPP
Examples on
Formulation of LPP
Basic
Definitions
Graphical
Method for
solving LPP
Thank you!

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