unit i notes.pdf

OPERATION RESEARCH UNITED COLLEGE OF ENGG & RESEARCH, ALLAHABAD

GAURAV SONKAR DEPARTMENT OF BUSINESS ADMINISTRATION 1



(Q1) Define OR and discuss its scope. (Scope is areas of application) (03-04) (ANS) O.R. simply defined as the research of operations.

By E.L. ARNOFF & M.J.NETZORG O.R. is the systematic, method oriented study of the basic structure, characteristics, functions

and relationship of an organization to provide the executive with sound, scientific and

quantitative basis for decision making.

By C. KITTEL O.R. is an aid for the executive in making his decisions by providing him with the needed

quantitative information based on the scientific method of analysis. By J.O.R. SOCIETY, U.K O.R. is the application of modern methods of mathematical science to complex problems

involving management of large systems of men, machine, materials and money in industry,

business, government and defence. This distinctive approach is to develop a scientific model of

the system incorporating measurement of factors such as chance and risk to predict and

compare the outcomes of alternative decisions, strategies or controls.

By FABRYCKY & TORGERSEN O.R. is the application of scientific methods to the problem arising from operations involving

integrated systems of men, machine and materials. It normally utilizes the knowledge and skills

of interdisciplinary research team to provide the managers of such systems with optimum

operating solutions. By CHURCHMANN, ACOFF, ANNOFF O.R. is the application of scientific methods, techniques and tools to problems involving the

operations of system so as to provide those in control of the operations with optimum solutions

to the problem.

SCOPE OF O.R.

(I) Allocation and Distribution: Optimal allocation of limited resources such as men, machine, and material.

Location and size of warehouses, distribution centre, retail depot etc.

Distribution policy.

(II) Production and Facility Planning: Selection, location and design of production plant.

Project scheduling & allocation of resources.



Forecasting.

Maintenance policy.

Scheduling & sequencing.

(III) Procurement:

What, when and how to purchase at minimum procurement cost

Bidding and replacement policies.

(IV) Marketing:

Product selection, timing & competitive action

Selection of advertising media.

Demand forecast and stock level.

Customer’s preference for size, colour & packaging of various products.

(V) Finance:

Capital requirement, cash flow analysis.

Credit policies, credit risks etc.

Profit plan of the company.

Determination of optimum replacement policies.

(VI) Personnel:

Selection of personnel, determination of retirement age and skills

Recruitment of policies & assignments of jobs.

(VII) Research and Development: Determination of areas of Research and Development

Reliability & control of development of projects.

Selection of projects & preparation of their budgets.



(Q2) What is Operations Research? What is the methodology of OR? (06-07) (ANS)

The effective use of OR techniques, requires to follow the systematic sequence of steps. In general there are following six steps

METHODOLOGY OF OR

ESTABLISHING CONTROL OVER THE

TEST THE MODEL

SOLVE THE MODEL

CONSTRUCT A MATHEMATICAL MODEL

FORMULATE THE PROBLEM

IMPLEMENTATION OF SELECTED STRATEGY



STEP I: FORMULATE THE PROBLEM

The problem formulation phase is generally lengthy, requiring considerable time and efforts. It involves the process to identify, understand and describe the problem or problems being faced by an organization. In formulation a problem for OR study, analysis must be made on the four major components.

Environment Decision Maker Objective Alternative course of action and constraints.

STEP II: CONSTRUCT A MATHEMATICAL MODEL

After the problem is clearly defined and understood, the next step is to collect required data and construct a mathematical model. A mathematical model consists of a set of equations which describes the system or problem. These equations represent effectiveness function or objective function and constraints.

STEP III: SOLVE THE MODEL

Once a mathematical model of the problem has been constructed, the next step is to solve it, that is, to obtain the numerical value of decision variables. A solution may be extracted from a model either by conducting experiments on it, that is, by simulation or by mathematical analysis. Here the aim is to find out the optimal solution of the problem.

STEP IV: TEST THE MODEL

After solving a mathematical model, it is important to review the solution carefully to see that the values make sense and that the resulting decisions can be implemented. When the model is complete, it should be tested again as a whole for obvious or oversight errors. This may be done by re-examining the formulation of the problem and comparing it with the model that may help to reveal any mistakes.

STEP V: ESTABLISHING CONTROL OVER THE SOLUTION

A solution derived from a model remains a solution only so long as the uncontrolled variables retain their values and the relationship between the variables does not changes. The solution itself goes ‘out of control’ if the values of one or more controlled variables vary or relationship between variables undergoes a change. Therefore, controls must be established to indicate the limit within which the model and its solution can be considered as reliable.

STEP VI: IMPLEMENTATION OF SELECTED STRATEGY The decision maker has not only to identify good decision alternatives but also to select alternatives that are capable of being implemented. It is important to ensure that any solution implemented is continuously reviewed and updated in the light of changing environment. In any case, the decision-maker who is in the best position to implement the results must be aware of the objective, assumption and limitations of the model.



(Q3) Describe the necessity of operation research in industry. (03-04) (Ans) Operation research is the science of managing. As is known, management is most of the time making decisions. It is thus a decision science which helps management to make better decisions. As one of the basic function of the management of an organization is decision making regarding integrated system of men, machine, material and money so as to minimize cost, losses, risk and uncertainty and to maximize production capacity, profits, skills of employees etc which must be based on some scientific methods, rules, techniques and models and OR is that kit of scientific and programmable rules providing the management a quantitative basis for decisions regarding the operations under its control.

As operation research comes into existence in connection with war operations, but its need has been equally felt by the industry because of following reasons:

Complexity: In a big industry, the numbers of factor influencing a decision have been increased. Situation has become big & complex because these factors interact with each other in complicated fashion. There is, thus, great uncertainty about the outcomes of interaction of factors like technological, environmental, competitive, etc. For instance, consider a factory production schedule which has to take into account.

a) Customer demand b) Requirement of raw materials c) Equipment capacity and possibility of equipment failure, and d) Restriction on manufacturing process.

Evidently, it is not easy to prepare a schedule which is both economical and realistic where as operation research provides mathematical models to analyze the problem and gives the executive a quantitative basis for decision making.

Scattered responsibility and authority: In a big industry, responsibility and authority of decision- making is scattered throughout the organization and thus the organization, if it is not conscious, may be following inconsistent goals. Mathematical quantification of OR overcomes this difficulty to a great extent.

Uncertainty: There is a great uncertainty about the economic and general environment. With economic growth, uncertainty is also growing. This makes the decision costlier and time-consuming. OR is, thus, quite essential from reliability point of view.

Knowledge explosion:

Knowledge is increasing at a very fast rate. Majority of the industries are not up-to-date with the latest knowledge and are, therefore, at a disadvantage. OR teams collect the latest information for analysis purposes which are quite useful for the industries.



In short OR is research of operations and is a problem solving and decision making science.

(Q4) Describe the rules for drawing the network diagram (decision tree) with suitable illustration. (07-08) (Ans) A decision tree is a graphical representation of the decision process indicating decision alternative,

states of nature, probabilities attached to the states of nature and conditional benefits and losses.

It consists of a network of nodes and branches. Two types of nodes are used- decision node

represented by a square and state of nature (chance or event) node represented by a circle. Alternative

course of action (strategies) originate from the decision node as main branches (decision branches). At

the end of each decision branch, there is a state of nature node from which emanates chance events in

the form of sub-branches (chance branches). The respective payoffs & the probabilities associated with

alternative courses and the chance events are shown alongside these branches.

At the terminal of the chance branches are shown the expected values of the outcomes.

The general approach used in decision tree analysis is to work backward through the tree from left to

right, computing the expected value of each chance node. We then choose the particular branch

leaving a decision node which leads to chance node with the highest expected value. This is known as

roll back or fold back process. Steps in Decision Tree Analysis (i) Identify the decision points and the alternative courses of action at each decision point

systematically.

(ii) At each decision point determine the probability and the payoff associated with each course of

action.

(iii) Starting from the extreme right end, compute the expected payoff (EMV) for each course of

action.

(iv) Choose the course of action that yields the best payoff for each of the decisions.

(v) Proceed backwards to the next stage of decision points.

(vi) Repeat above steps till the first decision points point is reached.

(vii) Finally, select the course of action which yields maximum possible EMV.

DECESION TREE



Illustration: Suppose we have the decision-making problem represented by the following table:

States of Nature

Probability Alternative Actions Produce 25 units

(A1) Produce 75 units (A2)

S1 (High Demand) 0.6 4000 10,000 S2 (Low Demand) 0.4 4000 - 5,000

The decision tree for the above illustration is shown below:

For a decision alternative (strategy) the EMV is calculated by summing the products of payoff of each

state and its probability. EMV on node 2= Rs. (4,000X0.6 + 2,000X0.4) = Rs.3, 200 EMV on node 3= Rs. (10,000X0.6 - 5000X0.4) = Rs.3, 000 EMV on node 1= Max {3200, 3000} = Rs.3200. Thus the optimal alternative action is A1 i.e. produce 25 units.

A2

A2

S1 p=0.6

p=0.4

S2

p=0.6

S2

CHANCE NODE

DECISION NODE

S1 DECISION NODE

CHANCE NODE

DECISION NODE

Rs.4, 000

Rs.10, 000

- Rs.5000

Rs.4.000

Rs.4, 000

S1 p=0.6

p=0.4

S2

p=0.6

S2

p=0.4

S1

CHANCE NODE 1

2

3

A1

A2

DECISION NODE



Maximax Criterion or Criterion of optimism

Maximin Criterion or Criterion of pessimism

Minimax Criterion or Regret Criterion

Laplace Criterion or Criterion of Rationality

Hurwicz Criterion or Criterion of Realism

(Q5) What techniques are used to solve a decision making problems under uncertainty? Illustrate any one technique with an example. (07-08)

(Ans) Under condition of uncertainty, the decision maker has knowledge about states of nature that

happens but lacks the knowledge about the probabilities of their occurrence. Situations like launching a product fall under this category.

Under conditions of uncertainty, a few decision criterions are available which could be of help

to the decision maker.

i) Maximax Criterion:

This criterion provides the decision maker with optimistic criterion. The working method is summarizing as follow.

(a) Locate the maximum payoff values corresponding to each alternative (or course of action or strategy), then

(b) Select an alternative with maximum payoff value.

(ii) Maximin Criterion:

This criterion provides the decision maker with pessimistic criterion. The working method is summarizing as follow.

(a) Locate the minimum payoff values corresponding to each alternative (or course of action or strategy), then




(iii) Minimax Criterion:

This criterion is also known as opportunity loss decision criterion or minimax regret criterion because decision maker feels regret after adopting wrong course of action resulting in an opportunity loss of payoff. Thus we have intends to minimize this regret. The working method is summarizing as follow.

(a) Determine the amount of regret corresponding to each alternative for each state of nature. The regret for jth event corresponding to ith alternative is given by

(b) Determine the maximum regret amount for each alternative. (c) Choose the alternative which corresponds to the minimum of the maximum regrets.

(iv) Hurwicz Criterion:

Also called weighted average criterion, it is a compromise between the maximax (optimistic) and minimax (pessimistic) decision criterion. This concept allows the decision maker to take into account both maximum and minimum for each alternative and assign them weights according to his degree of optimism (or pessimism). The working method is summarizing as follow:

(a) Choose an appropriate degree of optimism, α so that (1-α) represents degree of pessimism.

(b) Determine the maximum as well as minimum of each alternative and obtain

for each alternative. (c) Choose the alternative that yields the maximum value of P.

(v) Laplace Criterion:

Also known as equal probabilities criterion or criterion of rationality. Since the probability of states of nature are not known, it is assumed that all states of nature will occur with equal probability, i.e. assign an equal probability. The working method is summarizing as follow:

(a) Determine expected value for each alternative; if n denotes the number of events and P’s denote the payoffs, then expected value is given by 1\n[P1+P2+….+Pn]

(b) Choose the alternative that yields the maximum value of P.

ith regret = (maximum payoff – ith payoff) for the jth event

P = α. Maximum + (1-α). Minimum



Illustration: Considering a manufacturing company that is thinking of various alternatives to increase its production to meet the increasing market demand.

Alternative State of Nature

High (Rs.)

Moderate (Rs.)

Low (Rs.)

Nil (Rs.)

Expand 50,000 25,000 - 25,000 - 45,000 Construct 70,000 30,000 - 40,000 - 80,000

Subcontract 30,000 15,000 - 1,000 - 10,000

Here we are using Maximax Criterion; Steps involved are

(a) Locate the maximum payoff values corresponding to each alternative (or course of action or strategy), then


Alternative State of Nature Max of Row

High (Rs.)

Moderate (Rs.)

Low (Rs.)

Nil (Rs.)

Expand 50,000 25,000 - 25,000 - 45,000 50,000 Construct 70,000 30,000 - 40,000 - 80,000 70,000

Subcontract 30,000 15,000 - 1,000 - 10,000 30,000

Thus, according to maximax criterion the executive will choose alternative –“Construct”



(Q6) What techniques are used to solve a decision making problems under risk? Illustrate anyone technique with an example.

(Ans) Here more than one state of nature exists and the decision maker has sufficient information to

assign probabilities to each of these states. These probabilities could be obtained from the past

records or simply the subjective judgment of the decision maker. Under conditions of risk, a few

decision criterions are available which could be of help to the decision maker.

Expected Value Criterion:

The expected monetary value for a given course of action is the weighted sum of possible payoffs for each alternative. It is obtained by summing the payoffs for each course of action multiplied by the probabilities associated with state of nature. It consists of following steps:

(a) Construct a payoff table listing the alternative decisions and the various state of nature. Enter the conditional profit for each decision event combination along with the associated probabilities. (Construct Conditional profit table).

(c) Calculate the EMV for each decision alternative by multiplying the conditional profits by assigned probabilities and adding the resulting conditional values. (Construct expected profit table).

(d) Select the alternative that yields the highest EMV.

Expected Value Criterion or Expected Monetary Value Criterion

Expected Opportunity Loss Criterion or Expected Value of Regret

Expected Value for Perfect Information

Conditional Profit Table Expected Profit Table

Conditional Profit Table Conditional Loss table Expected Loss Table

Conditional Profit Table with P.I.

Expected Profit Table with P.I.



Expected Opportunity Loss Criterion: EOL represents the amount by which maximum possible profit will be reduced under various possible stock actions. The course of action that minimizes these losses or reductions is the optimal decision alternative. The procedure to calculate expected opportunity losses is as follows: 1. Prepare the conditional profit table for each decision-event combination and write associated

probabilities. (Construct Conditional profit table). 2. For each event, determine the conditional opportunity loss (COL) by subtracting the payoff from the

maximum payoff for that event. (Construct Conditional loss table).

3. Calculate the expected opportunity loss for each decision alternative by multiplying the COL’s by the associated probabilities and then adding the values. (Construct Expected loss table).

4. Select the alternative that yields the lowest EOL.

Expected Value of Perfect Information:

EVPI represents the maximum amount of money the decision maker has to pay to get this additional information about the occurrence of various state of nature before a decision has to be made. The procedure to calculate expected value of perfect information is as follows: 1. Construct conditional profit table with perfect information. 2. Construct expected profit table with perfect information.

3. Determine EVPI from relation; EVPI = EPPI – max EMV Illustration: A newspaper boy has the following probabilities of selling a magazine:

No. of copies sold Probability 10 0.10 11 0.15 12 0.20 13 0.25 14 0.30

Cost of the copy is 30 paisa and sale price is 50 paisa. He cannot return the unsold copies. How many should he order?

Solution from EMV approach:

Cost Price = 30 paisa.

Selling Price = 50 paisa.

Profit = Selling price – Cost price = 20 paisa.



Step 1: Construct Conditional Profit Table:

Possible Demand (No. of Copies)

Probability Possible Stock Action 10

Copies 11 Copies 12

Copies 13

Copies 14

Copies

10 11 12 13 14

0.10 0.15 0.20 0.25 0.30

200 200 200 200 200

170 220 220 220 220

140 190 240 240 240

110 160 210 260 260

80 130 180 230 280

Step 2: Construct Expected Profit Table:

Possible Demand (No. of Copies)

Probability Possible Stock Action 10

Copies 11 Copies 12

Copies 13

Copies 14

Copies

10 11 12 13 14

0.10 0.15 0.20 0.25 0.30

20 30 40 50 60

17 33 44 55 66

14 28.5 48 60 72

11 24 42 65 78

8 19.5 36

57.5 84

Total Expected Profit (paise)

200 215 222.5 220 205

Thus, therefore, the newspaper boy must order 12 copies to earn the highest possible average daily profit of 222.5 paisa.

unit i notes.pdf

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