OPERATION RESEARCHHemal Rajyaguru

IntroductionOne has to take decisions for himself and

for others.(E.g.-students which specialization they

should take.)

Everyone has to develop their talents in such a way so that they can take correct decision at a proper time.

(E.g. selection of proper engg. branch,Investments in proper property,etc.)

Decision DependencyAn effective decision depends on

following factors:Economic,Social,Political

Starting of a new factory

Economic factors would be construction cost, labour cost, raw material cost, transportation cost, taxes, energy, etc.

Social factors would be number of persons available for labour, engineers, experts, etc.

Political factors would be government policies, rules & regulations (E.g. Gujarat is the best)

Decision maker has to faced with…..

(i)large number of interacting variables as discussed in factors

(ii)actions of the other competitors, over which we don’t have control

People make decisions…..Based on their past experience

and intuitions(understanding) but they are not always enough.

Some formal system is needed to determine an effective course of actions.

Operation ResearchQuantitative methods,Management science,Decision science

provides scientific approach to the executives for making better decisions for operation under their control.

Operation research is not decision making

It hopefully improve the decisions made by the management in rational, logical, systematic and scientific way.

Operations Research Operations research is “a scientific approach to decision

making which seeks to determine how best to design and

operate a system, under conditions requiring the allocation of

scarce resources.”

Provides a set of algorithms that act as tools for effective

problem solving and decision making

Extensive applications in engineering, business and public

systems.

Used extensively by manufacturing and service industries in

decision making that’s why also useful in IT industry.

Operations Research

Origin during World War II when the British military

asked scientists to analyze military problems

The application of mathematics and scientific method to

military applications was called Operations Research.

Operational analysis

System analysis

Cost benefit analysis

Management science

Decision Science

PurposeHelps in determining the best(optimum)

solution(course of action) to problems where decision has to be taken under the restriction of limited resources.

It is possible to convert any real life problem into a mathematical model.

Basic feature of OR is to formulate a real world problem as a mathematical model.

Any industry want to lower their labour or production costs to achieve higher profits and OR can implied that.

OR=problem solving technique =science+art where, science is using mathematical techniques for solving decision problems Art is the ability to develop good report with good supplying information

Quantitative approach to decision-making: or/ms

Decision under certainty or uncertainty

Static decision: Decisions for one time period

only come under this.

E.g.: (1)Toss result in match

(II)Match result

Dynamic decision: Decisions over several time

periods comes under this.

E.g.: Bowling, Batting, Fielding strategies at

different stage of match.

Methodology of Operations Research

Methodology of Operations Research Problem Formulation

Model Building

◦ Physical Models

◦ Mathematical Models

Obtaining input Data

Solution of Model

◦ Feasible and Infeasible Solutions

◦ Optimal and Non-optimal Solutions

◦ Unique and Multiple Solutions

Fundamentals of Operations ResearchLinear Programming –

FormulationsLinear Programming – SolutionDuality and Sensitivity AnalysisTransportation ProblemAssignment ProblemDynamic ProgrammingDeterministic Inventory Models

Linear ProgrammingFirst conceived by George B Dantzig

around 1947.The work of Kantorovich (1939) was

published in 1959Dantzig’s first paper was titled

“Programming in Linear Structure”Koopmans coined the term “Linear

Programming” in 1948Simplex method was published in

1949 by Dantzig

LINEAR PROGRAMMING FORMULATIONS

Example 1 – Maximisation Problem(Product Mix Problem)

Consider a small manufacturer making two products A and B.

Two resources R1 and R2 are require to make these products.

Each unit of product A requires 1 unit of R1 and 3 units of R2.

Each unit of product B requires 1 unit of R1 and 2 units of R2.

The manufacturer has 5 units of R1 and 12 units of R2 available,

The manufacturer also makes a profit of◦ Rs. 6 per unit of product A sold and◦ Rs. 5 per unit of product B sold.

Product Resource 1 Resource 2 Profit

A 1 3 Rs. 6

B 1 2 Rs. 5

TOTAL 5 12

Solution The manufacturer has to decide on the number of units of products A

and B to produce.

It is acceptable that the manufacturer would like to make as much

profit as possible and would decide on the production quantities

accordingly.

The manufacturer has to ensure that the resources needed to make

the products are available.

Before we attempt to find out the decisions of the manufacturer, let

us redefine the problem in an algebraic form.

The manufacturer has to decide on the production quantities. Let us

call them x and y and define

Let x be the number of units of product A made

Let y be the number of units of product B made.

The profit associated with x units of product A and y units of product B is 6x + 5y.

The manufacturer would determine x and y such that this function has a

maximum value.

The requirements of two resources are x + y for R1 and 3x + 2y for R2 and the

manufacturer has to ensure that these are available.

The problem is to find x and y such that 6x + 5y is maximized and

x + y ≤ 5 and 3x + 2y ≤ 12 are satisfied. Also it is necessary that x, y ≥ 0 so

that only nonnegative quantities are produced.

If we redefine the production quantities as x1 and x2 (for reasons of uniformity

and consistency) then problem is to

Maximize Z = 6x1 + 5x2

Subject to

x1 + x2 ≤ 5

3x1 + 2x2 ≤ 12

x1, x2 ≥ 0

Terminology The problem variables x1 and x2 are called decision variables

and they represent the solution or the output decision from the problem.

The profit function that the manufacturer wishes to increase, represents the objective of making the decisions on the production quantities and is called the objective function.

The conditions matching the resource availability and resource requirement are called constraints.◦ These usually limit (or restrict) the values the decision variables can

take.

We have also explicitly stated that the decision variable should take non negative values.◦ This is true for all Linear Programming problems.◦ This is called non negativity restriction

The problem that we have written down in algebraic form represents the mathematical model of the given system and is called the Problem Formulation.

The problem formulation has the following steps.◦ Identifying the decision variables.◦ Writing the objective function◦ Writing the constraints.◦ Writing the non negativity restrictions.

In the above formulation, the objective function and the constraints are linear.◦ Therefore the model that we formulated is a linear

programming problem.

A Linear Programming problem has◦ A linear objective function,◦ linear constraints and◦ the non negativity constraints on all the decision

variables.

Example – 2 Minimization Problem The Agricultural Research institute suggested to a farmer to

spread out at least 4800 kg of a special phosphate fertilizer and not less than 7200 kg of a special nitrogen fertilizer to raise productivity of crops in his fields.

There are two sources two obtaining these-mixtures A and B. Both of these are available in bag weighing 100 kg each and they cost Rs 40 and Rs 24 respectively.

Mixture A contains phosphate and nitrogen equivalent of 20 kg and 80 kg respectively.

While mixture B contains these ingredients equivalent of 50 kg each.

Write this as a linear programming problem to determine how many bags of each type the farmer should buy in order to obtain the required fertilizer at minimum cost.

Bag Type (100 Kg)

Price per Bag

Phosphate/Bag

Nitrogen / Bag

A 40 20 80

B 24 50 50

Min Requirement

4800 7200

SolutionThe objective functionThe constraintsNon-negativity condition

Minimise Z = 40x1 + 24x2 Cost

Subject to 20x1 + 50x2 ≥ 4800 Phosphate requirement

80x1 + 50x2 ≥ 7200 Nitrogen requirement

x1, x2 ≥ 0 Non-negativity restriction

Examples A firm is engaged in producing two products, A and B.

Each unit of product A requires 2 kg of raw material and 4 labour hours for processing, whereas each unit of product B requires 3 kg of raw material and 3 hours of labour, of the same type. Every week, the firm has an availability of 60 kg of raw material and 96 labour hours. One unit of product A sold yields Rs 40 and one unit of product B sold gives Rs 35 as profit.

Formulate this problem as a linear programming problem to determine as to how many units of each of the products should be produced per week so that the firm can earn the maximum profit. Assume that there is no marketing constraint so that all that is produced can be sold.

Product Raw Material Labour Profit

A 2 4 40

B 3 3 35

Total 60 96

A company is manufacturing two different types of products, A and B. Each product has to be processed in 3 different departments – casting, machining and finally inspection. The capacity of 3 departments is limited to 35 hrs., 32 hrs and 24 hrs. per week respectively. Product A requires 7 hrs. in casting department, 8 hrs. in machining shop and 4 hrs. in inspection, whereas product B requires 5 hrs., 4 hrs. and 6 hrs. respectively in respective shop. The profit contribution for a unit product of A and B is Rs. 30 and Rs. 40 respectively.

Formulate the problem.

Department Product CapacityPer weekA B

Casting 7 5 35

Machining 8 4 32

Inspection 4 6 24

Profit contribution

Per unit

Rs. 30 /- Rs. 40 /-

A24-hour supermarket has the following minimal requirements for cashiers:

Period 1 follows immediately after period 6. A cashier works eight consecutive hours, starting at the beginning of one of the six time periods. Determine a daily employee Worksheet which satisfies the requirements with the least number of personnel. Formulate the problem as a linear programming problem.

Period: 1 2 3 4 5 6

Time of day (24-hour clock):

3-7 7-11 11-15

15-19 19-23 23-3

Minimum number required:

7 20 14 20 10 5