DECISION SCIENCES II

Faculty: Sanjay Sinha J K Nanda,

Neena Nanda Bhavika Thakker

INSTITUTE FOR TECHNOLOGY & MANAGEMENTKharghar, NAVI MUMBAI

Institute for Technology and Management,

Kharghar, Navi Mumbai

TITLE : Decision sciences II

Course code : 162

DURATION : 30 Hrs (24 Sessions)

PGDM Batch : 2013 - 2015 (Oct- Jan) Sem I part IICourse faculty : Sanjay Sinha ,Neena Nanda,

J K Nanda, Bhavika Thakker

No. of Credits : 3

COURSE OBJECTIVES

The course has been designed to enable the student to study and apply various techniques of operations research in business world.

Emphasis have been given to conceptual understanding of the topic so that the students can model business situations into mathematical models and reach to a decision.

This course will highlight the benefits of quantitative approach to decision making or how to reach to an optimal decision in the light of uncertain or risky environments.

Analysis, Interpretation and Conclusion Drawing.

COURSE CONTENTS

1. Introduction – Role of Quantitative Analysis in Decision Making and Operations Research Techniques their Nature and Characteristics feature.

2. Theory Of Games

3. Linear Programming

4. Transportation Models

5. Assignment Models

6. Simulation

7. Decision Theory

8.Queuing Theory

9.Index numbers

10. Excel applications using solver

11.PPT is attached for all chapter.

12. Additional Practice questions.

Text Book: Quantitative Techniques- N D Vohra

Reference Books:

1) Operation Research - Hamdy A. Taha, 8th

Edition, 2008, Pearson Education.2) Quantitative Analysis for Management - Barry Render, Ralph M. Stair, Jr., 3)Michael E. Hanna, T N

Badri, 10th

Edition, 2009, Pearson, Education.

Quantitative Methods for Business - Anderson, Sweeney & Williams, Cengage Learning, 9th

Edition, 2008.

4) Essentials of Operations Research & Quantitative Techniques – K Shridhara Bhat, 1st

Edition, 2007, Himalaya Publishing House.

5) Operations Research- Algorithms and Applications – Rathindra P Sen, 1st

Edition, 2010, PHI Learnings.

6) Quantitative Techniques for Managerial Decisions – R B Khanna, 1st

Edition, 2007, PHI Learnings.

7) Operations Research - S D Sharma, 15th

Edition, 2007, Kedar Nath Ram Nath, Meerut.

8) Applied statistics-Gupta and Kapoor,2011 , Sultan Chand and co, New Delhi

COURSE OUTLINE

Institute for Technology and Management,

Kharghar, Navi Mumbai

TITLE : Decision sciences I

Course code : 162

DURATION : 30 Hrs (24 Sessions)

PGDM Batch : 2013 - 2015 (June- Sept) Sem I part ICourse faculty : Sanjay Sinha ,Neena Nanda,

J K Nanda, Bhavika Thakker

No. of Credits : 3

PGDM/GLC 2013-14

FACULTY : Sanjay Sinha, J K Nanda, Neena Nanda, Bhavika Thakker

1. COURSE OBJECTIVES

The course has been designed to enable the student to study and apply various techniques of operations research in business world.

Emphasis have been given to conceptual understanding of the topic so that the students can model business situations into mathematical models and reach to a decision.

This course will highlight the benefits of quantitative approach to decision making or how to reach to an optimal decision in the light of uncertain or risky environments.

2) TEACHING METHOD AND MATERIAL

Teaching Materials will include the prescribed Text Book, Problem Situations. The Course will use the following teaching methods :(a) Interactive Discussion and Understanding of the theoretical design.(b) Understanding the problem situation or the various environments under which decisions are to be made. (c) Quantitative results (d) Analysis, Interpretation and Conclusion Drawing.

3)Course Format and Discussions

Classes will follow inductive and deductive method of learning. Entire pedagogy is envisaged to be learner- centric. Case discussion, news analysis, class tests. Quiz etc, will take place during the class.

Case study evaluation criteria:

Analysis should include these sequential steps:

Presentation of the facts pertaining to techniques used in the case. Identification of the key issues. Listing alternative courses of action that could be taken. Evaluation of alternative courses of action. Recommendation of the best course of action.

Assignment evaluation criteria : There are 2 types of assignments that can be given:list of different types of questions to give them an idea of practical applications of

various techniques in various industries. And/ Or

a.)Students to clip an example of a presentation from a recent magazine or newspaper (if it is from library make a photocopy instead).b.) They have to cite the exact source where they found the display.c.)They have to think and write what is its presumed purpose.?d.Then they write a short,critical evaluation of its strengths and weaknesses.and they have to make sure to attach the original clipping to their analysis.

Quiz evauation criteria: Students prepare a quiz and conduct in the class.It mainly consists of MCQs.

Class Participation evaluation criteria: Students have to select a topic of their choice and speak on that topic for 2 minutes.This helps them to browse and look for variety of topics through books/magazines/internet which they can share with their classmates.Every ttalk delivered is followed by a question/answer session and student is evaluated on the basis of reasoning given for queries along with examples given in support of topic.This very often leads to a good discussion and indepth learning of the related topic. EVALUATION

S No Criterion of Evaluation Weightage ( in %)

1 Mid Term Examination 20

2 Assignment (*) 5

4 Quiz 5

5 Class Participation 5

6 Class Attendance 5

Term End Examination 60

Total 100

(*) Assignment will be based upon Business situation understanding and quantification of the results. Hard Copy has to be submitted. (Will be used for Internal Marking)

9) COURSE CONTENTS

1. Introduction – Role of Quantitative Analysis in Decision Making and Operations Research Techniques their Nature and Characteristics feature. Different types of OR problems.

2. Theory Of Games – Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2. games, Games of Pure and Mixed Strategy, Principle of Dominance.

3. Linear Programming – Formulation of LPP, Graphical Solutions, Simples Methods, Post Optimality Analysis, Duality.

Case Study: Planning the product mix at Panchtantra Corporation.

4. Transportation Models – Methods of Feasible Solution : NW Corner Method, LCM, VAM, Row / Column Minima Methods, Optimal Solutions, Balanced and Unbalanced Types, Maximization and Minimization Types. Case study: Red Brand Canners

5. Assignment Models – Maximization and Minimization Types, HAM, Traveling Salesman Problems

6. Simulation – Process of Simulation, Monte Carlo Simulation techniques and Simulation queuing models, Inventory models, planning etc. : stat wide Development corporation.

7. Decision Theory – Payoff table, Opportunity loss or regret table, decision rules, decision making under the conditions of certainty, uncertainty, risk and competition or conflict.

Case study: Starting Right Corporation.

Case study: Blake Electronics.

8.Queuing Theory – General Structure of Queuing Models, Operations Characteristics of Queuing Models. Examples and their solutions.

Case Study: Wilkins,A Zurn Company: Aggregate production planning.

9.Index numbers- basic concepts , calculation of Indices..., Major indices used in business world WIP, CPI,Sensex.

6)Text Book: Quantitative Techniques- N D Vohra

Reference Books:

1) Operation Research - Hamdy A. Taha, 8th

Edition, 2008, Pearson Education.2) Quantitative AnalLinear Programming – Formulation of LPP, Graphical Solutions, Simples Methods, Post Optimality Analysis, Duality. ysis for Management - Barry Render, Ralph M. Stair, Jr.,

Michael E. Hanna, T N Badri, 10th

Edition, 2009, Pearson, Education.

3) Quantitative Methods for Business - Anderson, Sweeney & Williams, Cengage Learning, 9th

Edition, 2008.

4) Essentials of Operations Research & Quantitative Techniques – K Shridhara Bhat, 1st

Edition, 2007, Himalaya Publishing House.5) Operations Research- Algorithms and Applications – Linear Programming – Formulation of LPP,

Graphical Solutions, Simples Methods, Post Optimality Analysis, Duality. Rathindra P Sen, 1st

Edition, 2010, PHI Learnings.

6) Quantitative Techniques for Managerial Decisions – R B Khanna, 1st

Edition, 2007, PHI Learnings.7) Operations ReLinear Programming – Formulation of LPP, Graphical Solutions, Simples

Methods, Post Optimality Analysis, Duality. search - S D Sharma, 15th

Edition, 2007, Kedar Nath Ram Nath, Meerut.

8) Applied statistics-Gupta and Kapoor,2011 , Sultan Chand and co, New Delhi

7) SESSION WISE PLAN

Session Topic Suggested Reading (SR); Numerical for Practice (NP) ; Classroom Session (CS)

1 Introduction to Operations Research Subject matter and Importance of quantification for achieving better results or decisions.Sensitivity Analysis

CSNPSR

2,3,4 Game Theory CSNPSR

5,6,7 Decision theory CSNPSR

8,9 Simulation 10,11,12 Linear Programming CS

NPSR

13,14 Transportation Problems Theory Of Games – Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2. games, Games of Pure and Mixed Strategy, Principle of Dominance.

CSNPSR

15,16 Assignment Models CSNPSR

17,18 Index Numbers19,20 Queuing Theory21,22 Case Studies CS

NPSR

23,24 Excel applications using solver

CH.1 Introduction – Role of Quantitative Analysis in Decision Making and Operations Research Techniques their Nature and Characteristics feature. Different types of OR problems.

INTRODUCTION

Operations Research (OR) is a science which deals with problem, formulation, solutions and finally

appropriate decision making. This subject is new and started after World War II, when the failures of

missions were very high. Scientists and technocrates formed team to study the problem arising out of

difficult situations and at the later stage solutions to these problems. It is research designed to determine

most efficient way tLinear Programming – Formulation of LPP, Graphical Solutions, Simples Methods, Post Optimality Analysis, Duality. o do something new. OR is the use of mathematical models, statistics and algorithm to

aid in decision-making. It is most often used to analyze complex real life problems typically with the goal

of improving or optimizing performance. Decision making is the main activity of an engineer/manager.

Some decisions can be taken by common sense, sound judgement and experience without using math-

ematics, and some cases this may not be possible and use of other techniques is inevitable.

With the growth of technology, the World has seen a remarkable changes in the size and complexity

of organisations. An integral part of this had been the division of labour and segmentation of manage-

ment responsibilities in these organisations. The results have been remarkable but with this, increasing

specialisation has created a new problem to meet out organisational challenges. The allocation of

limited resources to various activities has gained significant importance in the competitive market.

These types of problems need immediate attention which is made possible by the application of OR

techniques.

The tools of operations research are not from any one discipline, rather Mathematics, Statistics,

Economics, Engineering, Psychology, etc. have contributed to this newer discipline of knowledge.

Today, it has become a professional discipline that deals with the application of scientific methods for

decision-making, and especially to the allocation of scare resources.

In India first unit of OR started in the year 1957 with its base at RRL Hyderabad. The other group

was set up in Defence Science Laboratory which was followed by similar units at different parts of the

country. The popular journal of OPSEARCH was established in 1963, to promote research in this field.

Keeping in view the critical economic situation which required drastic increase in production

efficiency, OR activities were directed, in all areas of business activities. In the late 50’s OR was

introduced at university level. With the development of PC’s the use of OR techniques became promi-

nent and effective tool as large amount of computation is required to handle complex problems. In

recent years application of OR techniques have achieved significance in all walk of life, may it be

industry or office work for making strategical decisions more scientifically.

2. BACKGROUND OF OPERATIONS RESEARCH

The effectiveness of operations research in military spread interest in it to other governmental depart-

ments and industry. In the U.S.A. the National Research Council formed a committee on operations

research in 1951, and the first book on the subject “Methods of Operations Research”, by Morse and

Kimball, was published. In 1952 the Operations Research Society of America came into being.

Today, almost all organisations make use of OR techniques for decision-making at all levels. This

general acceptance to OR has come as managers have learned the advantage of the scientific approach

to all industrial problems. Some of the Indian organisations using operations research techniques to

solve their varied complex problems are: Railways, Defence, Indian Airlines, Fertilizer Corporation of

India, Delhi Cloth Mills, Tata Iron and Steel Co. etc.

A purpose of OR is to provide a rational basis for making decisions in the absence of complete

information. OR can also be treated as science devoted to describing, understanding and predicting the

behaviour of systems, particularly man-machine systems.

3. MEANING OF OR

Defining OR is difficult task as its boundaries and content are not yet fixed. It can be regarded as use

of mathematical and quantitative techniques to substantiate the decision being taken. Further, it is

multidisciplinary which takes tools from subjects like mathematics, statistics, engineering, economics,

psychology etc. and uses them to score the consequences of possible alternative actions. Today it has

become professional discipline that deals with the application of scientific methods to decision-making.

Salient aspects related to definition stressed by various experts on the subject are as follows:

(a) Pocock stresses that OR is an applied science; he states “OR is scientific methodology-analyti-

cal, experimental, quantitative—which by assessing the overall implication of various alterna-

tive courses of action in a management system, provides an improved basis for management

decisions’’.

(b) Morse and Kimball have stressed the quantitative approach of OR and have described it as “a

scientific method of providing executive departments with a quantitative basis for decisions

regarding the operations under their control”.

(c) Miller and Starr see OR as applied decision theory. They state “OR is applied decision theory.

It uses any scientific, mathematical or logical means to attempt to cope with the problems that

confront the executive, when he tries to achieve a thorough—going rationality in dealing with

his decision problem”.

(d) Saaty considers O Theory Of Games – Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2. games, Games of Pure and Mixed Strategy, Principle of Dominance. R as tool of improving the quality of answers to problems. He say, “OR is

the art of giving bad answers to problems which otherwise have worse answers”.

Few other definitions of OR are as follows:

• “OR is concerned with scientifically deciding how to best design and operate man-machine

system usually requiring the allocation of scare resources.”

– Operations Research Society, America

• “OR is essentially a collection of mathematical techniques and tools which in conjunction with

system approach, are applied to solve practical decision problems of an economic or engineer-

ing nature’’.

– Daellenbach and George

• “OR utilizes the planned approach (updated scientific method) and an interdisciplinary team in

order to represent complex functional relationships as mathematical models for the purpose of

providing a quantitative analysis’’.

– Thieraub and Klekamp

• “OR is a scientific knowledge through interdisciplinary team effort for the purpose of deter-mining the best utilization of limited resources.”– H.A. Taha

• “OR is a scientific approach to problem solving for executive management”.– H.M. Wagner

4. FEATURES OF OR

The significant features of operations research include the followings:

(i) Decision-making. Every industrial organisation faces multifacet problems to identify best

possible solution to their problems. OR aims to help the executives to obtain optimal solution

with the use of OR techniques. It also helps the decision maker to improve his creative and

judicious capabilities, analyse and understand the problem situation leading to better control,

better co-ordination, better systems and finally better decisions.

(ii) Scientific Approach. OR applies scientific methods, techniques and tools for the purpose of

analysis and solution of the complex problems. In this approach there is no place for guess

work and the person bias of the decision maker.

(iii) Inter-disciplinary Team Approach. Basically the industrial problems are of complex nature

and therefore require a team effort to handle it. This team comprises of scientist/mathematician

and technocrates. Who jointly use the OR tools to obtain a optimal solution of the problem. The

tries to analyse the cause and effect relationship between various parameters of the problem

and evaluates the outcome of various alternative strategies.

(iv) System Approach. The main aim of the system approach is to trace for each proposal all

significant and indirect effects on all sub-system on a system and to evaluate each action in

terms of effects for the system as a whole.

The interrelationship and interaction of each sub-system can be handled with the help of

mathematical/analytic Theory Of Games – Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2. games, Games of Pure and Mixed Strategy, Principle of Dominance. al models of OR to obtain acceptable solution.

(v) Use of Computers. The models of OR need lot of computation and therefore, the use of

computers becomes necessary. With the use of computers it is possible to handle complex

problems requiring large amount of calculations.

The objective of the operations research models is to attempt and to locate best or optimal solution

under the specified conditions. For the above purpose, it is necessary that a measure of effectiveness has

to be defined which m Theory Of Games – Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2. games, Games of Pure and Mixed Strategy, Principle of Dominance. ust be based on the goals of the organisation. These measures can be used to

compare the alternative courses of action taken during the analysis.

5. PHASES OF OR STUDY

OR is a logical and systematic approach to provide a rational basis for decision-making. The phases of

4 PRINCIPLES OF OPERATIONS RESEARCH

OR must be logical and systematic. The various steps required for the analysis of a problem under OR

are as follows:

Step I. Observe the Problem Environment

The first step of OR study is the observation of the environment in which the problem exists. The

activities that constitute this step are visits, conferences, observations, research etc. with the help of

such activities, the OR analyst gets sufficient information and support to proceed and is better prepared

to formulate the problem.

Step II. Analyse and Define the Problem

In this step not only the problem is defined but also uses, objectives and limitations of the study that are

stressed in the light of the problem. The end results of this step are clear grasp of need for a solution

and understanding of its nature.

Step III. Develop a Model

The next step is to develop model, which is representation of same real or abstract situation. OR models

are basically mathematical models representing systems, process or environment in form of equations,

relationships or formulae. The activities in this step is to defining interrelationships among variables,

formulating equations, using known OR models or searching suitable alternate models. The proposed

model may be field tested and modified in order to work under stated environmental constraints. A

model may also be modified if the management is not satisfied with the answer that it gives.

Step IV. Selection of Data Input

It is a established fact that without authentic and appropriate data the results of the OR models cannot

be trusted. Hence, taping right kind of data is a vital step in OR process. Important activities in this step

are analysing internal-external data and facts, collecting opinions and using computer data banks. The

purpose of this step is to have sufficient input to operate and test the model.

Step V. Solution and Testing

In this step the solution of the problems is obtained with the help of model and data input. Such a

solution is not implemented immediately and this solution is used to test the model and to find its

limitations if any. If the solution is not reasonable or if the model is not behaving properly, updating and

modification of the model is considered at this stage. The end result of this step is solution that is

desirable and supports current organisational objectives.

Step VI. Implementation of the Solution

This is the last phase of the OR study. In OR the decision-making is scientific but implementation of

decision involves many behavioural issues. Therefore, implementation authority has to resolve the

behavioural issues, involving the workers and supervisors to avoid further conflicts. The gap between

management and OR scientist may offer some resistance but must be eliminated before solution is

accepted in totality. Both the parties should play positive role, since the implementation will help the

organisation as a whole. A properly implemented solution obtained through OR techniques results in

improved working conditions and wins management support.

6. OUTLINES OF OR MODELS

In OR the problem is expressed in the form of a model. Where, a model is a theoretical abstraction

(approximation) of a real-life problem. It can be defined as a simplified representation of an operation

or a process in which only the basic aspects or the most important features of a typical problem under

investigation are considered. OR analysts have given special impetus to the development and use of

techniques like, linear programming, waiting line theory, game theory, inventory controls and simulation.

In addition, some other common tools are non-linear programming, integer programming, dynamic

programming, sequencing theory, Markov process, network scheduling—PERT and CPM, symbolic

logic, information theory and utility/value theory. The list, of course, is not exhaustive. The detailed

discussion on above will be presented in appropriate chapters, however, brief explanation of these is

given below:

(i) Linear Programming (L.P.)

Linear programming is basically a constrained optimisation technique which tries to optimise some

criterion within some constraints. It consists of an objective function which is some measure of effec-

tiveness like profit, loss or return on investment and several boundary conditions putting restriction on

the use of resources. Objective function and boundary conditions are linear in nature. There are

meth-

ods available to solve a linear programming problem.

(ii) Waiting Line or Queuing Theory

This deals with the situation in which queue is formed or the customers have to wait for service or

machines wait for repairmen and therefore concept of a queue is involved. If we assume that there are

costs associated with waiting in line, and if there are costs of adding more service facilities, we want

to minimize the sum of costs of waiting and the costs of providing service facilities. Waiting line theory

helps to make calculations like number of expected member of people in queue, expected waiting time

in the queue, expected idle time for the server, etc. These calculations then can be used to determine the

desirable number of service facilities or number of servers.

(iii) Game Theory

It is used for decision-making under conflicting situations where there are one or more opponents. The

opponents, in game theory, are called players. The motives of the players are dictomized. The success

of one player tends to be at the cost of others and hence they are in conflict. Game theory models, a

conflict situation arises and helps to improve the decision process by formulating appropriate strategy.

(v) Simulation

It is basically data generating technique, where sometimes it is risky, cumbersome, or time consuming

to conduct real study or experiment to know more about situation or problem. The available analytical

methods cannot be used in all situations due to large number of variables or large number of inter-

relationships among the variables and the complexity of relationship, it is not possible to develop an

analytical model representing the real situation. Some times, even building of model is possible but its

solution may not be possible. Under such situations simulation is used. It should be noted that simula-

tion does not solve the problem by itself, but it only generates the required information or data needed

for decision problem or decision-making.

(vi) Non-Linear Programming

These models may be used when either the objective function or some of the constraints are not linear

in nature. Non-linearity may be introduced by such factors as discount on price of purchase of large

quantities and graduated income tax etc. Linear programming may be employed to approximate the

non-linear conditions, but the approximation becomes poorer as the range is extended. Non-linear

methods may be used to determine the approximate area in which a solution lies and linear methods

may be used to obtain a more exact solution.

(vii) Integer Programming

This method can be used when one or more of the variables can only take integer values. Examples are

the number of trucks in a fleet, the number of generators in a power house and so on. Approximate

solutions can be obtained without using integer programming methods, but the approximation

generally becomes poorer as the number becomes smaller. There are techniques to obtain solution of

integer programming problems.

((ix) Sequencing Theory

This is related to waiting line theory and is applicable when the facilities are fixed, but the order of

servicing may be controlled. The scheduling of service or the sequencing of jobs is done to minimize

the relevant costs and time.

(x) Markov Process

It is used for decision-making in situations where various states are defined. The probability of going

from one state to another is known and depends on the present state and is independent of how we have

arrived at that state. Theory of Markov process helps us to calculate long run probability of being in a

particular state (steady state probability), which is used for decision-making.

7. SCOPE OF OPERATIONS RESEARCH

As presented in the earlier paragraphs, the scope of OR is not only confined to any specific agency like

defence services but today it is widely used in all industrial organisations. It can be used to find the best

solution to any problem be it simple or complex. It is useful in every field of human activities, where

optimisation of resources is required in the best way. Thus, it attempts to resolve the conflicts of interest

among the components of organization in a way that is best for the organisation as a whole. The main

fields where OR is extensively used are given below, however, this list is not exhaustive but only

illustrative.

(i) National Planning and BudgetingOR is used for the preparation of Five Year Plans, annual budgets, forecasting of income and expenditure,

scheduling of major projects of national importance, estimation of GNP, GDP, population, employment

and generation of agriculture yields etc.

(ii) Defence Services

Basically formulation of OR started from USA army, so it has wide application in the areas such as:

development of new technology, optimization of cost and time, tender evaluation, setting and layouts

of defence projects, assessment of “Threat analysis”, strategy of battle, effective maintenance and

replacement of equipment, inventory control, transportation and supply depots etc.

(iii) Industrial Establishment and Private Sector Units

OR can be effectively used in plant location and setting finance planning, product and process planning,

facility planning and construction, production planning and control, purchasing, maintenance manage-

ment and personnel management etc. to name a few.

(iv) R & D and Engineering

Research and development being the heart of technological growth, OR has wide scope for and can be

applied in technology forecasting and evaluation, technology and project management, preparation of

tender and negotiation, value engineering, work/method study and so on.

8. DEVELOPMENT OF OR IN INDIA

OR being a new discipline started a bit late in India with its inception at Regional Research Laboratory

at Hyderabad and at the same time a group was established in Defence Science Laboratory to solve the

problems of stores, purchase and planning. OR society was formed in 1953. Today OR subject is very

popular and is being taught at graduation and post graduation level in all the university of the country.

It is also being used in industrial establishment extensively to improve decision-making process.

9. COMPUTERS IN OR

As has been presented earlier that OR tries to find optimal solutions with multiple variables. In most of

the cases a large number of iterations are required to reach optimal solution. Manually this task be-

comes time consuming and single mistake at any point can generate erroneous results. With the devel-

opment of computers and P.C’s this has reduced manual efforts considerably and solutions can be

obtained in a short period of time and possibility of errors is also minimised considerably.

Storage of information/data is easy and faster with the use of computers because of its memory. The

computational time requirements are also less and no paper work is required. Transfer of data from one

place to another is also possible through net/computers. The reliability of solutions is also high. For the

large size problems, where simulation was to be used, it was not possible to carry it out manually,

which

is now possible with the use of computers. To handle linear programming problem with multiple

variables use to be cumbersome and time taking, can be done at wink of moment without any manual

efforts.

10.LIMITATIONS OF OPERATIONS RESEARCH

OR has some limitations however, these are related to the problem of model building and the time and

money factors involved in application rather then its practical utility. Some of them are as follows:

(i) Magnitude of Computation. Operations research models try to find out optimal solution

taking into account all the factors. These factors are enormous and expressing them in quantity

and establishing relationships among these require voluminous calculations which can be

handled by computers.

(ii) Non-Quantifiable Factors. OR provides solution only when all elements related to a problem

can be quantified. All relevant variables do not lend themselves to quantification. Factors

which cannot be quantified, find no place in OR study. Models in OR do not take into account

qualititative factors or emotional factors which may be quite important.

(iii) Distance between User and Analyst. OR being specialist’s job requires a mathematician or

statistician, who might not be aware of the business problems. Similarly, a manager fails to

understand the complex working of OR. Thus there is a gap between the two. Management

itself may offer a lot of resistance due to conventional thinking.

(iv) Time and Money Costs. When basic data are subjected to frequent changes, incorporating

them into the OR models is a costly proposition. Moreover, a fairly good solution at present

may be more desirable than a perfect OR solution available after sometime. The computational

time increases depending upon the size of the problem and accuracy of results desired.

(v) Implementation. Implementation of any decision is a delicate task. It must take into account

the complexities of human relations and behaviour. Sometimes, resistance is offered due

to psychological factors which may not have any bearing on the problem as well as its

solution.

Ch-2 Game theory Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2. games, Games of Pure and Mixed Strategy, Principle of Dominance.

Game theory is a branch of mathematics with direct applications in economics, sociology, and psychology. The theory was first devised by John Von Neumann . Later contributions were made by John Nash, A. W. Tucker, and others.

Game-theory research involves studies of the interactions among people or groups of people. Because people make use of an ever-increasing number and variety of technologies to achieve desired ends, game theory can be indirectly applied in practical pursuits such as engineering, information technology, and computer science.

So-called games can range from simple personal or small group encounters or problems to major confrontations between corporations or superpowers. One of the principal aims of game theory is to determine the optimum strategy for dealing with a given situation or confrontation. This can involve such goals as maximizing one's gains, maximizing the probability that a specific goal can be reached, minimizing one's risks or losses, or inflicting the greatest possible damage on adversaries.

Ch-3 Linear Programming – Formulation of LPP, Graphical Solutions, Simples Methods, Post Optimality Analysis, Duality.

What is a Linear Programming Problem?A linear program (LP) is a minimization problem where we are asked to minimize a given linearfunction subject to one or more linear inequality constraints. The linear function is also calledthe objective function.Formulation:n MinimizeCi Xi(where Ci and are constants and Xi and are variables)∈ ∈i=1Subject to constraints:a11 x1 + a12 x2 + ........ + a1n xn ≥ b1a21 x1 + a22 x2 + ........ + a2n xn ≥ b2a31 x1 + a32 x2 + ........ + a3n xn ≥ b3...an1 x1 + an2 x2 + ........ + ann xn ≥ bnAlternately, we can rewrite the above formulation as:Minimize C T X(where C, X and are column vectors)∈Subject to constraints:AX ≥ b(where b m,A m×n)∈ ∈Given C, A and b the above LP can be solved in time poly (inputlength).

PPT is Attached

Case study : Planning the product Mix at Panchtantra corporation

CH -4 Transportation Problem:Methods of Feasible Solution : NW Corner Method, LCM, VAM, Row / Column Minima Methods, Optimal Solutions, Balanced and Unbalanced Types, Maximization and Minimization Types.

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Cost Table

Dist Center D

Dist Center E

Plant A 80 215Plant B 100 108Plant C 102 68Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

Let be the amount of cars to be shipped from source i to destination j. Then our objective is to minimize the total cost which is . The constraints are the ones imposed by the amount of cars to be transported from each plant and the amount each center can absorb.

The whole model is:

Minimize z =

subject to,

;

;

;

;

;

and integer, i = 1,2,3, j = 1,2.

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

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Case study: Red Brand Canners

CH:5. Assignment Models – Maximization and Minimization Types, HAM, Traveling Salesman Problems

The Assignment Problem: Suppose we have n resources to which we want to assign to n tasks on a one-to-one basis. Suppose also that we know the cost of assigning a given resource to a given task. We wish to find an optimal assignment–one which minimizes total cost. The Mathematical Model: Let ci,j be the cost of assigning the ith resource to the jth task. We define the cost matrix to be the n × n matrix An assignment is a set of n entry positions in the cost matrix, no two of which lie in the same row or column. The sum of the n entries of an assignment is its cost. An assignment with the smallest possible cost is called an optimal assignment.

The Hungarian Method: The Hungarian method is an algorithm which finds an optimal assignment for a given cost matrix. The Hungarian Method: The following algorithm applies the above theorem to a given n × n cost matrix to find an optimal assignment. Step 1. Subtract the smallest entry in each row from all the entries of its row. Step 2. Subtract the smallest entry in each column from all the entries of its column. Step 3. Draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used. Step 4. Test for Optimality: (i) If the minimum number of covering lines is n, an optimal assignment of zeros is possible and we are finished. (ii) If the minimum number of covering lines is less than n, an optimal assignment of zeros is not yet possible. In that case, proceed to Step 5. Step 5. Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to Step 3.

CH :6Simulation – Process of Simulation, Monte Carlo Simulation techniques and Simulation queuing models, Inventory models, planning etc. :

Case study :stat wide Development corporation.

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CH : 7 Decision Theory – Payoff table, Opportunity loss or regret table, decision rules, decision making under the conditions of certainty, uncertainty, risk and competition or conflict. Case study: Starting Right Corporation.

Case study: Blake Electronics.

Decision theory is the study of principles and algorithms for making correct decisions—that is, decisions that allow an agent to achieve better outcomes with respect to its goals. Every action at least implicitly represents a decision under uncertainty: in a state of partial knowledge, something has to be done, even if that something turns out to be nothing (call it "the null action"). Even if you don't know how you make decisions, decisions do get made, and so there has to be some underlying mechanism. What is it? And how can it be done better? Decision theory has the answers.

A core idea in decision theory is that of expected utility maximization, usually intractable to directly calculate in practice, but an invaluable theoretical concept. An agent assigns utility to every possible outcome: a real number representing the goodness or desirability of that outcome. The mapping of outcomes to utilities is called the agent's utility function. (The utility function is said to be invariant under affine transformations: that is, the utilities can be scaled or translated by a constant while resulting in all the same decisions.) For every action that the agent could take, sum over the utilities of the various possible outcomes weighted by their probability: this is the expected utility of the action, and the action with the highest expected utility is to be chosen.

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CH:8.Queuing Theory – General Structure of Queuing Models, Operations Characteristics of Queuing Models. Examples and their solutions.

Case Study: Wilkins,A Zurn Company: Aggregate production planning. is attached.

Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be: banks/supermarkets - waiting for service computers - waiting for a response failure situations - waiting for a failure to occur e.g. in a piece of machinery

public transport - waiting for a train or a bus

As we know queues are a common every-day experience. Queues form because resources are limited. In fact it makes economic sense to have queues. For example how many supermarket tills you would need to avoid queuing? How many buses or trains would be needed if queues were to be avoided/eliminated?

In designing queueing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers).

In essence all queuing systems can be broken down into individual sub-systems consisting of entities queuing for some activity (as shown below).

Typically we can talk of this individual sub-system as dealing with customers queuing for service. To analyse this sub-system we need information relating to:

arrival process: how customers arrive e.g. singly or in groups (batch or bulk arrivals) how the arrivals are distributed in time (e.g. what is the probability distribution of time

between successive arrivals (the interarrival time distribution)) whether there is a finite population of customers or (effectively) an infinite number

The simplest arrival process is one where we have completelCase Study: Wilkins,A Zurn Company: Aggregate production planning. is attached.y regular arrivals (i.e. the same constant time interval between successive arrivals). A Poisson stream of arrivals corresponds to arrivals at random. In a Poisson stream successive customers arrive after intervals which independently are exponentially distributed. The Poisson stream is important as it is a convenient mathematical model of many real life queuing systems and is described by a single parameter - the average arrival rate. Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).

service mechanism: a description of the resources needed for service to begin how long the service will take (the service time distribution) the number of servers available whether the servers are in series (each server has a separate queue) or in parallel (one

queue for all servers) whether preemption is allowed (a server can stop processing a customer to deal with

another "emergency" customer)

Assuming that the service times for customers are independent and do not depend upon the arrival process is common. Another common assumption about service times is that they are exponentially distributed.

queue characteristics: how, from the set of customers waiting for service, do we choose the one to be served

next (e.g. FIFO (first-in first-out) - also known as FCFS (first-come first served); LIFO (last-in first-out); randomly) (this is often called the queue discipline)

do we have: balking (customers deciding not to join the queue if it is too long) reneging (customers leave the queue if they have waited too long for service)

jockeying (customers switch between queues if they think they will get served faster by so doing)

a queue of finite capacity or (effectively) of infinite capacity

Changing the queue discipline (the rule by which we select the next customer to be served) can often reduce congestion. Often the queue discipline "choose the customer with the lowest service time" results in the smallest value for the time (on average) a customer spends queuing.

Note here that integral to queuing situations is the idea of uncertainty in, for example, interarrival times and service times. This means that probability and statistics are needed to analyse queuing situations.

In terms of the analysis of queuing situations the types of questions in which we are interested are typically concerned with measures of system performance and might include:

How long does a customer expect to wait in the queue before they are served, and how long will they have to wait before the service is complete?

What is the probability of a customer having to wait longer than a given time interval before they are served?

What is the average length of the queue? What is the probability that the queue will exceed a certain length? What is the expected utilisation of the server and the expected time period during which he will

be fully occupied (remember servers cost us money so we need to keep them busy). In fact if we can assign costs to factors such as customer waiting time and server idle time then we can investigate how to design a system at minimum total cost.

These are questions that need to be answered so that management can evaluate alternatives in an attempt to control/improve the situation. Some of the problems that are often investigated in practice are:

Is it worthwhile to invest effort in reducing the service time? How many servers should be employed? Should priorities for certain types of customers be introduced? Is the waiting area for customers adequate?

In order to get answers to the above questions there are two basic approaches:

analytic methods or queuing theory (formula based); and simulation (computer based).

The reason for there being two approaches (instead of just one) is that analytic methods are only available for relatively simple queuing systems. Complex queuing systems are almost always analysed using simulation (more technically known as discrete-event simulation).

The simple queueing systems that can be tackled via queueing theory essentially:

consist of just a single queue; linked systems where customers pass from one queue to another cannot be tackled via queueing theory

have distributions for the arrival and service processes that are well defined (e.g. standard statistical distributions such as Poisson or Normal); systems where these distributions are derived from observed data, or are time dependent, are difficult to analyse via queueing theory

The first queueing theory problem was considered by Erlang in 1908 who looked at how large a telephone exchange needed to be in order to keep to a reasonable value the number of telephone calls not connected because the exchange was busy (lost calls). Within ten years he had developed a (complex) formula to solve the problem.

Additional queueing theory information can be found here and here

Case Study: Wilkins,A Zurn Company: Aggregate production planning. is attached.

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CH9.Index numbers- basic concepts , calculation of Indices..., Major indices used in business world WIP, CPI,Sensex.

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