Transcript LINEAR PROGRAMMING

LINEAR PROGRAMMING
Linear Programming
Linear programming is a mathematical
technique. This technique is applied for
choosing the best alternative from a set
of feasible alternatives. This technique
is designed to help managers in
planning, decision making and to
allocate the resources.
Definitions
“The analysis of problems in which a
linear function of a number of variables
is to be maximised (or minimised) when
these variables are subject to number
of restraints in the form of linear
inequalities.”
- Samuelson & Slow
In the words of LOOMBA :
“LP is only one aspect of what has been called a
system approach to management where in all
programmes are designed and evaluated
in the terms of their ultimate affects in the
realization of business objectives.”
Terminology & Requirements
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Objective Function
Presence of constraints or restrictions
Alternative course of action
Non-negativity Constraints
Linearity
Finite number of variables
Certainty
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Additivity
Divisibility
Decision Variables
Basic Variables
Slack Variables
Surplus Variables
Artificial Variables
Optimum Solution
DISTINCTION BETWEEN VARIABLES
CONTINUED...
General Form of LPP
(Max./ Min.) Z = a1x1+a2x2+...+anxn
(objective function)
Subject to constraints :
a11x1+a12x2<,=,> b1 ;
a21x1+a22x2<,=,> b2 ;
& x1, x2 are not equal to 0 (non-negativity
constraint)
Applications
Linear programming is a technique for
optimizing or making most effective use of
given conditions. During the last 40 years
applications of the technique in a number of
areas have been developed.
Some important situations where it is very
popular are discussed further.
1. Product Mix : Most popular application of linear
programming is product mix decisions i.e. On deciding
how much quantity of each product should be made
to maximise profit or minimise cost which making
best use of the given production facility or considering
the limited availability of men, material, machines,
money, markets etc.
2. Diet Problem: The diet problem arises whenever the
manager is to determine that optimal combination of
diet of feed mix which will satisfy specified nutritional
requirements and minimize and maximize the total
cost of purchasing the diet.
3. The portfolio selection problem : A portfolio selection
problem arises whenever a given amount of money is to be
allocated among served investment opportunities the
portfolio selection problem is quite general in the sense that
is can exist for individual fund managers or opportunities.
The components of the portfolio can be bonds savings
certificates.
4. Media Selections : Lpp has also been used in
advertisements field as a decision aid in selecting the
effective media. Media helps
the marketing managers in allocations a fixed budget across
various advertising media like newspaper, magazines, radio
and televisions etc. in most of these cases the objective is
taken to be the maximization of audience of exposure.
5. Blending Problems : The manufacturing process involves mixing
of three various types of materials in specific quantity. Supply of raw
materials and these specification are the constraints in obtaining the
minimum raw materials cost. Our solution indicates the number of
units of each raw materials which are to be blended in one unit of
product.
6. Transportation problems : Physical movement of goods from plants
to warehouse warehouses to wholesalers, wholesalers to retailers, and at
last retailers to customers. solutions of the transportation models
requires the determinations of how many units should be transported
from each supply origin to each demand destinations in order to satisfy
all the destinations in order to satisfy all the destinations demands
while minimizing the total associated cost of transportations
Administrative Applications
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Linear programming can be used for
administrative applications.
Administrative Applications of LPP are
concerned with optimal usage of resources
like men, machine and material.
Linear programming helps an administrative
incharge in the best possible manner to utilize
his resources.
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Linear programming can be used to
make decisions for departmental staff
requirements for given period of time.
It can be used for work distribution
among staff members according to their
efficiency, so as to obtain optimum
result.
Solution Procedure of
LPP
For the solution of LPP mathematical model is
developed. The mathematical model serves
as a basis for initial solution and is improved
step by step, till an optimal solution is
obtained.
There are three stages of solution:
 Feasible solution
 Basic Feasible solution
 Optimal solution
FORMULATION OF
LINEAR
PROGRAMMING
PROBLEM
Problem 1.
Problem: A diet for a sick person must contain at least 4000 units
of vitamins, 50 units of minerals and 1400 calories. Two foods A
and B are available at a cost of Rs. 4 and Rs.3 per unit
respectively. If one unit of food B contains 100 units of vitamins,
2 units of minerals and 40 calories, find by graphical method,
what combination of food be used to have least cost?
Solution
The mathematical formulation of the problem is as under:
Min: Z= 4x1 + 3x2
(objective function)
Subject to:
200x1 + 100x2 > 4000
x1 + 2x2 > 50
40x1 + 40x2 > 1400
}
} constraints
}
Where x1,x2> 0
(non-negativity constraint)
The above constraints can be simplified as under:
2x1+x2> 40
x1+x2 > 35
Now converting the inequalities of the above constraints to
equalities in order to plot the constraints on the graph, we
get
2w1+x2=40…….(i)
x1+2x2=50……..(ii)
x1+x2=35……..(iii)
Problem 2.
The Manager of an oil refinery must decide on the optimal mix of 2
possible blending processes of which the inputs and outputs per
production run as follows:
The maximum availability crude A & B are 250 units and 200 units
respectively. The market requirement shows that at least 150 units of
gasoline X and 130 units of gasoline Y must be produced. The profit per
production run from process 1&2 are Rs. 40 and Rs.50 respectively.
Formulate the problem for maximising the profit.
Solution
• Step-1 Let x ,y designate the number of production
of the two processes respectively.
Step-2 The constraints of the problem are:
6x + 5y ≤ 250
3x+ 6y ≤ 200
6x+ 5y ≤ 150
9x+ 5y ≤ 130
Step-3 The objective is to maximise the total profit.
Z = 40x + 50 y
Step-4 The feasible alternatives are:
xᶟ 0 , yᶟ 0
So, the formulated problem is
Max. Z = 40x + 50 y
Subject to
6x
3x
6x
9x
+
+
+
+
5y
6y
5y
5y
≤
≤
≤
≤
250
200
150
130
Where as x, y ᶟ 0
Merits of LPP
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1.The linear programming technique helps to make the best
possible use of available productive resources (such as time,
labour, machines etc.)
2. In a production process, bottle necks may occur. For
example, in a factory some machines may be in great demand
while others may lie idle for some time. A significant advantage
of linear programming is highlighting of such bottle necks.
3.Linear programming helps in studying the information of an
organization in such a way that it depicts clear picture of a problem.
This scientific approach is very valuable.
4. Linear programming helps the manager to plan and execute the
policies of top management in such a way that cost involved are
minimum. It helps to make use of limited resources.
 5. Linear programming highlights the various bottlenecks in
problem process and it helps to remove such bottlenecks.
6. Linear programming is one of the best technique to be used
under changing circumstances. It has flexibility to adjust in changing
conditions.
Limitations of LPP
• Linear Relationship:- One of the basic assumptions of L.P.P is
the relationship between variables. But in actual practice many
objective functions can be expressed linearly.
• Time Effect:- L.P Models do not consider the effect of time. The
solution to the problems are good in case of static situation not in
case of dynamic situation.
• Conflicting multiple goals:- LPP also fails to give a solution if the
management has conflicting multiple goals. In LP models there is
always one goal which is expressed in the objective function.
• Divisibility of resources :- We have assumed that L.P models
are based on perfect divisibility of resources. It means all
solution variables should have any value, but in certain situation
like in a product mix problem, it can be fractional units.
• Complexity:- If there are large number of variables and
constraints, then the formulated mathematical model become
complex.
• Single objective:- L.P deals with the single objective where in
real life situation we may come across more than one objective.