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Decision sciences
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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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