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Lecture Notes 1

MSc in Opera t iona l Research

Opera t iona l Techn iques 1

Lecture 1OPERATI ONAL RESEARCH

I n t r o d u c t i o n

The term, operations research was first coined in 1940 by McClosky and Trefthen in a small town

Bowdsey, of the United Kingdom. This new science came into existence in a military context. During

World War II, military management called on scientists from various disciplines and organized them

into teams to assist in solving strategic and tactical problems, relating to air and land defence of the

country. Their mission was to formulate specific proposals and plans for aiding the Military commands

to arrive at decisions on optimal utilization of scarce military resources and efforts and also to

implement the decisions effectively. This new approach to the systematic and scientific study of the

operations of the system was called Operations Research or operational research. Hence OR can be

associated with "an art of winning the war without actually fighting it."

Scope o f Opera t ion s Research

There is a great scope for economists, statisticians, administrators and the technicians working as a

team to solve problems of defence by using the OR approach. Besides this, OR is useful in the various

other important fields like:

1. Agriculture.

2. Finance.

3. Industry.

4. Marketing.

5. Personnel Management.

6. Production Management.

7. Research and Development.

Phases o f Opera t iona l Research

The procedure to be followed in the study of OR, generally involves the following major phases.

1. Formulating the problem.

2. Constructing a mathematical model.3. Deriving the solution from the model.

4. Testing the model and its solution (updating the model).

5. Controlling the solution.

6. Implementation.


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Lecture Notes 2

Models In Operations Research

A model in OR is a simplified representation of an operation, or is a process in which only the basic

aspects or the most important features of a typical problem under investigation are considered. The

objective of a model is to identify significant factors and interrelationships. The reliability of the solution

obtained from a model, depends on the validity of the model representing the real system.

A good model must possess the following characteristics:

(i) It should be capable of taking into account, new formulation with out having any changes in its

frame.

(ii) Assumptions made in the model should be as small as possible. (iii) Variables used in the model

must be less in number ensuring that it is simple and coherent.

(iv) It should be open to parametric type of treatment.

(v) It should not take much time in its construction for any problem.

Advant ages o f A Mode lThere are certain significant advantages gained when using a model these are:

(i) Problems under consideration become controllable through a model.(ii)It provides a logical and systematic approach to the problem.(iii)It provides the limitations and scope of an activity.

(iv) It helps in finding useful tools that eliminate duplication of methods applied to solve problems.

(v) It helps in finding solutions for research and improvements in a system.

(vi) It provides an economic description and explanation of either the operation, or the systems they

represent.

M ode ls by St r uc t u r e

M at hem at i ca l m ode ls Are most abstract in nature. They employ a set of mathematical symbols to

represent the components of the real system. These variables are related together by means of math-

ematical equations to describe the behavior of the system. The solution of the problem is then obtained

by applying well-developed mathematical techniques to the model.

I n t r o d u c t i o n o f L i n ea r P r o g r am m i n g

Linear programming deals with the optimisation (maximisation or minimisation) of a function of

variables known as objective functions. It is subject to a set of linear equalities and /or inequalities

known as constraints. Linear programming is a mathematical technique, which involves the allocation

of limited resources in an optimal manner, on the basis of a given criterion of optimality.

In this section properties of Linear Programming Problems (LPP) are discussed. The graphical

method of solving a LPP is applicable where two variables are involved. The most widely used method


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Lecture Notes 3

for solving LP problems consisting of any number of variables is called s im p l e x m e t h o d .

For m u la t i on o f LP P r ob lem s

The procedure for mathematical formulation of a LPP consists of the following steps:

Step 1 To write down the decision variables of the problem.

Step 2 To formulate the objective function to be optimised (Maximised or Minimised) as a linear

function of the decision variables.

Step 3 To formulate the other conditions of the problem such as resource limitation, market

constraints, interrelations between variables etc, as linear in equations or equations in terms of the

decision variables.

Step 4 To add the non-negativity constraint from the considerations so that the negative values of the

decision variables do not have any valid physical interpretation.

The objective function, the set of constraint and the non-negative constraint together form a Linearprogramming problem.

Genera l Form ula t io n o f LPP

The general formulation of the LPP can, be stated as follows:

In order to find the values of n decision variables n321 x,.....,x,x,x to maximize or minimize the

objective function.

nn332211 xc.....xcxcxcZ ++++= (1)

and also satisfy m-constraints

=++++

=++++

=++++

=++++

mnmn33m22m11m

inin33i22i11i

2nn2323222121

1nn1313212111

bxa.....xaxaxa

..............................................................

bxa.....xaxaxa

..................................................................

bxa.....xaxaxa

bxa.....xaxaxa

(2)

where constraints may be in the form of inequality < or > or even in the form an equation (=) and

finally satisfy the non negative restrictions

,0x.....,x,x,x n321 (3)

E x a m p l e 1 A manufacturer produces two types of models M1 and M2.,. Each model of the type M1

requires 4 hrs of grinding and 2 hours of polishing; where as each model of the type M2 requires 2 hours

of grinding and 5 hours of polishing. The manufacturers have 2 grinders and 3 polishers. Each grinder

works 40 hours a week and each polisher works for 60 hours a week. Profit on M1 model is Rs.3.00 and


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Lecture Notes 4

on model M2, is Rs.4.00. Whatever is produced in a week is sold in the market. How should the

manufacturer allocate his production capacity to the two types of models, so that he may make

the maximum profit in a week?

So lu t i on

Decision Var iab lesLet 1x and 2x be the number of units of 1M and 2M model.

Ob ject i ve f u nc t i on Since the profit on both the models are given, we have to maximize the profit

21 x4x3)Z(Max +=

Const ra in ts There are two constraints one for grinding and the other for polishing.

No. of hrs. available on each grinder for one week is 40 hrs. There are 2 grinders. Hence the

manufacturer does not have more than 2 x 40 = 80 hrs of grinding. M1 requires 4 firs of grinding and

M2, requires 2 hours of grinding.

The grinding constraint is given by

80x2x4 21 + .

Since there are 3 polishers, the available time for polishing in a week is given by 3 x 60= 180. M1

requires 2 hrs of polishing and M2,requires 5 hrs of polishing. Hence we have 180x5x2 21 +

Finally we have

21 x4x3)Z(Max +=

Subject to 80x2x4 21 +

180x5x2 21 +

Gr aph ica l So lu t i on


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Lecture Notes 5

Example 2 A company manufactures two products A and B. These products are processed in the same

machine. It takes 10 minutes to process one unit of product A and 2 minutes for each unit of product

B and the machine operates for a maximum of 35 hrs in a week. Product A requires 1 kg. and B 0.5 kg.

of raw material per unit the supply of which is 600 kg. per week. Market constraint on product B is

known to be 800 unit every week. Product A costs Rs.5 per unit and sold at Rs.10. Product B costs Rs.6

per unit and can be sold in the market at a unit price of Rs.8. Determine the number of units of A and

B per week to maximize the profit.

S o l u t i o n

Dec ision Var iab les Let 1x and 2x be the number of products A and B.

Ob ject i ve f u nc t i on Costs of product A per unit is Rs.5 and sold at Rs. 10 per unit. Profit on one unit

of product A. = 10-5=5

1x units of product A contributes a profit of Rs.5 x1 profit contribution from one unit of product

B=8-6=2

2x units of product B contribute a profit of Rs.2 2x . The objective function is given by

,x2x5)Z(Max 21 +=

Const ra in ts Time requirement constraint is given by)6035(x2x10 21 +

2100x2x10 21 +

Raw material constraint is given by

600x5.0x 21 +

Market demand on product Bis 800 units every week

800x 2

The complete LPP is

,x2x5)Z(Max 21 +=

Subject to 2100x2x10 21 +

600x5.0x 21 +

800x 2

,0x,0x 21


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Lecture Notes 6


Ex a m p l e 3 A person requires 10,12, and 12 units chemicals A, B respectively for his garden. One unit

of liquid product contains 5, 2 and 1 units of A, B and C respectively. One unit of dry product contains 1,2

and 4 units of A, B, C. If the liquid product sells for Rs.3 and the dry product sells for Rs.2 , how many

of each should be purchased, in order to minimize the cost and meet the requirements?

Solu t ion

Decision var iableLet 1x and 2x be the number of units of liquid and dry products.

Ob ject i ve f u nc t i on Since the cost for the products are given we have to minimize the cost

,x2x3)Z(Min 21 +=

Const ra in ts As there are 3 chemicals and its requirement are given. We have three constraints for

these three chemicals.

10xx5 21 +


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Lecture Notes 7

12x2x2 21 +

12x4x 21 +

,0x,0x 21

Finally the complete L.P.P is

,x2x3)Z(Min 21 +=

Subject to 10xx5 21 +

12x2x2 21 +

12x4x 21 +

,0x,0x 21



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Lecture Notes 8

Example 4 A paper mill produces two grades of paper namely X and Y. Because of raw material

restrictions, it cannot produce more than 400 tons of grade X and 300 tons of grade Y in a week. There

are 160 production hours in a week. It requires 0.2 and 0.4 hours to produce a ton of products X and Y

respectively with corresponding profits of Rs.200 and Rs.500 per ton. Formulate the above as a LPP to

maximize profit and find the optimum product mix.

Solu t ion

Dec is ion var iab les Let 1x and 2x be the number of units of two grades of paper of X and Y

Ob jec t i ve f unc t i on Since the profit for the two grades of paper X and Yare given, the objective

function is to maximize the profit ,x500x200)Z(Max 21 +=

Const ra in ts There are 2 constraints one w.r t. to raw material, and the other w.r t .to production

hours.

The complete LPP is

,x500x200)Z(Max 21 +=

Subject to 400x1

300x 2

160x4.0x2.0 21 +

,0x,0x 21



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Lecture Notes 9

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