IntroductionIntroduction Optimization is the act of obtaining the best result under Optimization is the act of obtaining the best result under

any given circumstance. In various practical problems we any given circumstance. In various practical problems we may have to take many technical or managerial decisions may have to take many technical or managerial decisions at several stages.at several stages.

The ultimate goal of all such decisions is to either The ultimate goal of all such decisions is to either maximize the desired benefit or minimize the effort maximize the desired benefit or minimize the effort required.required.

We make decisions in our every day life without even We make decisions in our every day life without even noticing them.noticing them.

Decision making is one of the main activity of a manager Decision making is one of the main activity of a manager or executive.or executive.

In simple situations decisions are taken simply by common In simple situations decisions are taken simply by common sense, sound judgment and expertise without using any sense, sound judgment and expertise without using any mathematics.mathematics.

IntroductionIntroduction But here the decisions we are concerned with are rather complex But here the decisions we are concerned with are rather complex

and heavily loaded with responsibility.and heavily loaded with responsibility.• Examples of such decision are finding the appropriate Examples of such decision are finding the appropriate

product mix when there are large numbers of products with product mix when there are large numbers of products with different profit contributions and productional requirement or different profit contributions and productional requirement or planning public transportation network in a town having its own planning public transportation network in a town having its own layout of factories, apartments, blocks etc.layout of factories, apartments, blocks etc.

The search of a decision may also be done by trial and error but The search of a decision may also be done by trial and error but such a search may be cumbersome and costly.such a search may be cumbersome and costly.

Preparative calculations may avoid long and costly research.Preparative calculations may avoid long and costly research. Doing preparative calculations is the purpose of Operations Doing preparative calculations is the purpose of Operations

research.research. Operations research does mathematical scoring of consequences Operations research does mathematical scoring of consequences

of a decision with the aim of optimizing the use of time, efforts of a decision with the aim of optimizing the use of time, efforts and resources and avoiding blunders.and resources and avoiding blunders.

IntroductionIntroduction Learning objectives:Learning objectives: After studying this unit, you should be able to understand After studying this unit, you should be able to understand

the following.the following.

1. Know the significant features in O.R.1. Know the significant features in O.R.

2. Understand the Methodology of O.R.2. Understand the Methodology of O.R.

3. Define the structure of a mathematical model in O.R.3. Define the structure of a mathematical model in O.R.

4. Know the significance of the Objective function4. Know the significance of the Objective function

Historical BackgroundHistorical Background During Second World War in United Kingdom a team of scientists During Second World War in United Kingdom a team of scientists

from different disciplines studied the strategic and tactical from different disciplines studied the strategic and tactical problems associated with air and land defence of the country.problems associated with air and land defence of the country.

Their objective was to determine the most effective utilization of Their objective was to determine the most effective utilization of limited military resources to win the battle and the technique limited military resources to win the battle and the technique they developed was named as Operations research.they developed was named as Operations research.

After the war, Operations research techniques rapidly developed After the war, Operations research techniques rapidly developed in the fields of industrial, academic and government in the fields of industrial, academic and government organizations.organizations.

The main The main objective of Operations research is to provide a objective of Operations research is to provide a scientific basis to the decision makers for solving the scientific basis to the decision makers for solving the problems involving the interaction of various components problems involving the interaction of various components of organization, by employing a team of scientists from of organization, by employing a team of scientists from different disciplines, all working together for finding a different disciplines, all working together for finding a solution which is the best in the interest of the solution which is the best in the interest of the organization as a whole.organization as a whole.

Historical BackgroundHistorical Background Definition of Operations Research: Churchman, Definition of Operations Research: Churchman,

Ackoff and Aruoff have defined “Operations research Ackoff and Aruoff have defined “Operations research as the application of scientific methods, techniques as the application of scientific methods, techniques and tools to operation of a system with optimum and tools to operation of a system with optimum solutions to the problems”.solutions to the problems”.

Optimum implies the one, which is best of all possible Optimum implies the one, which is best of all possible alternatives.alternatives.

Another definition is that, Another definition is that, “Operations research is the “Operations research is the use of scientific methods to provide criteria for use of scientific methods to provide criteria for decisions regarding man, machine, systems involving decisions regarding man, machine, systems involving repetitive operations”.repetitive operations”.

Scope of Operations Scope of Operations Research (OR)Research (OR)

In general, whenever there is any problem simple or complicated, In general, whenever there is any problem simple or complicated, the OR techniques may be applied to find the best solution.the OR techniques may be applied to find the best solution.

In Defence Operations: In modern warfare the defence In Defence Operations: In modern warfare the defence operations are carried out by a operations are carried out by a number of independent number of independent components namely Air Force, Army and Navy. The activities in each components namely Air Force, Army and Navy. The activities in each of these components can be further divided in four subcomponents of these components can be further divided in four subcomponents viz.: administration, intelligence, operations and training, and viz.: administration, intelligence, operations and training, and supply. The application of modern warfare techniques in each of the supply. The application of modern warfare techniques in each of the components of military organizations requires expertise knowledge components of military organizations requires expertise knowledge in respective fields. Further more, each component works to drive in respective fields. Further more, each component works to drive maximum gains from its operations and there is always a possibility maximum gains from its operations and there is always a possibility that strategy beneficial to one component may have an adverse that strategy beneficial to one component may have an adverse effect on the other. Thus in defence operations there is a necessity effect on the other. Thus in defence operations there is a necessity to coordinate the activities of various components which gives to coordinate the activities of various components which gives maximum benefit to the organization as a whole, having maximum maximum benefit to the organization as a whole, having maximum use of the individual components. use of the individual components.

Scope of Operations Scope of Operations Research (OR)Research (OR)

The final strategy is formulated by a team of scientists drawn The final strategy is formulated by a team of scientists drawn from various disciplines who study the strategies of different from various disciplines who study the strategies of different components and after appropriate analysis of the various components and after appropriate analysis of the various courses of actions, the best course of action, known as courses of actions, the best course of action, known as optimum strategy, is chosen.optimum strategy, is chosen.

In Industry: The system of modern industries are so In Industry: The system of modern industries are so complex that the optimum point of complex that the optimum point of operation in its various operation in its various components cannot be intuitively judged by an individual. The components cannot be intuitively judged by an individual. The business environment is always changing and any decision business environment is always changing and any decision useful at one time may not be so good some time later. There useful at one time may not be so good some time later. There is always a need to check the validity of decisions continually, is always a need to check the validity of decisions continually, against the situations. The industrial revolution with against the situations. The industrial revolution with increased division of labour and introduction of management increased division of labour and introduction of management responsibilities has made each component an independent responsibilities has made each component an independent unit having their own goals.unit having their own goals.

Scope of Operations Scope of Operations Research (OR)Research (OR)

For example: Production department minimize cost of For example: Production department minimize cost of production but maximizes output. Marketing department production but maximizes output. Marketing department maximizes output but minimizes cost of unit sales. Finance maximizes output but minimizes cost of unit sales. Finance department tries to optimize capital investment and personnel department tries to optimize capital investment and personnel department appoints good people at minimum cost. Thus each department appoints good people at minimum cost. Thus each department plan their own objectives and all these objectives department plan their own objectives and all these objectives of various department or components come to conflict with of various department or components come to conflict with each other and may not conform to the overall objectives of each other and may not conform to the overall objectives of the organization. The application of OR techniques helps in the organization. The application of OR techniques helps in overcoming this difficulty by integrating the diversified overcoming this difficulty by integrating the diversified activities of various components so as to serve the interest of activities of various components so as to serve the interest of the organization as a whole efficiently.the organization as a whole efficiently.

OR methods in industry can be applied in the fields of OR methods in industry can be applied in the fields of production, inventory controls and marketing, purchasing, production, inventory controls and marketing, purchasing, transportation and competitive strategies etc.transportation and competitive strategies etc.

Scope of Operations Scope of Operations Research (OR)Research (OR)

Planning: In modern times it has become necessary Planning: In modern times it has become necessary for every government to have careful for every government to have careful planning, for planning, for economic development of the country. OR techniques can economic development of the country. OR techniques can be fruitfully applied to maximize the per capita income, be fruitfully applied to maximize the per capita income, with minimum sacrifice and time. A government can thus with minimum sacrifice and time. A government can thus use OR for framing future economic and social policies.use OR for framing future economic and social policies.

Agriculture: With increase in population there is a Agriculture: With increase in population there is a need to increase agriculture output. But need to increase agriculture output. But this cannot be this cannot be done arbitrarily. There are a number of restrictions under done arbitrarily. There are a number of restrictions under which agricultural production is to be studied. Therefore which agricultural production is to be studied. Therefore there is a need to determine a course of action, which there is a need to determine a course of action, which serves the best under the given restrictions. The problem serves the best under the given restrictions. The problem can be solved by the application of OR techniques.can be solved by the application of OR techniques.

Scope of Operations Scope of Operations Research (OR)Research (OR)

In Hospitals: The OR methods can be used to solve In Hospitals: The OR methods can be used to solve waiting problems in outpatient waiting problems in outpatient department of big department of big hospitals. The administrative problems of hospital hospitals. The administrative problems of hospital organization can also be solved by OR techniques.organization can also be solved by OR techniques.

In Transport: Different OR methods can be applied to In Transport: Different OR methods can be applied to regulate the arrival of trains and regulate the arrival of trains and processing times, processing times, minimize the passengers waiting time and reduce minimize the passengers waiting time and reduce congestion, formulate suitable transportation policy, congestion, formulate suitable transportation policy, reducing the costs and time of transshipment.reducing the costs and time of transshipment.

Research and Development: Control of R and D Research and Development: Control of R and D projects, product introduction planning etc. projects, product introduction planning etc. and many and many more applications.more applications.

Features of ORFeatures of OR Important features of OR are:Important features of OR are:

i.i. It is System oriented: OR studies the problem from over all It is System oriented: OR studies the problem from over all point of view of organizations or situations since optimum point of view of organizations or situations since optimum result of one part of the system may not be optimum for some result of one part of the system may not be optimum for some other part.other part.

ii.ii. It imbibes Inter – disciplinary team approach. Since no single It imbibes Inter – disciplinary team approach. Since no single individual can have a thorough knowledge of all fast developing individual can have a thorough knowledge of all fast developing scientific knowhow, personalities from different scientific and scientific knowhow, personalities from different scientific and managerial cadre form a team to solve the problem.managerial cadre form a team to solve the problem.

iii.iii. It makes use of Scientific methods to solve problems.It makes use of Scientific methods to solve problems.

iv.iv. OR increases the effectiveness of a management Decision OR increases the effectiveness of a management Decision making ability.making ability.

v.v. It makes use of computer to solve large and complex problems.It makes use of computer to solve large and complex problems.

vi.vi. It gives Quantitative solution.It gives Quantitative solution.

vii.vii. It considers the human factors also.It considers the human factors also.

Phases of ORPhases of OR The scientific method in OR study generally involves the following The scientific method in OR study generally involves the following

three phases:three phases:

1. Judgment Phase: This phase consists of1. Judgment Phase: This phase consists of

a)a) Determination of the operation.Determination of the operation.

b)b) Establishment of the objectives and values related to the Establishment of the objectives and values related to the operation.operation.

c)c) Determination of the suitable measures of effectiveness andDetermination of the suitable measures of effectiveness and

d)d) Formulation of the problems relative to the objectivesFormulation of the problems relative to the objectives

2. Research Phase: This phase utilizes2. Research Phase: This phase utilizes

a)a) Operations and data collection for a better understanding of the Operations and data collection for a better understanding of the problems.problems.

b)b) Formulation of hypothesis and model.Formulation of hypothesis and model.

c)c) Observation and experimentation to test the hypothesis on the Observation and experimentation to test the hypothesis on the basis of additional data.basis of additional data.

Phases of ORPhases of OR

d)d) Analysis of the available information and verification of Analysis of the available information and verification of the hypothesis using reestablished measure of the hypothesis using reestablished measure of effectiveness.effectiveness.

e)e) Prediction of various results and consideration of Prediction of various results and consideration of alternative methods.alternative methods.

3. 3. Action Phase: It consists of making Action Phase: It consists of making recommendations for therecommendations for the

decision process by those decision process by those who first posed the who first posed the problem for considerationproblem for consideration

or by anyone in a position to make a decision, or by anyone in a position to make a decision, influencing the operation ininfluencing the operation in

which the problem is occurred.which the problem is occurred.

Types of OR ModelsTypes of OR Models A model is a representation of the reality.A model is a representation of the reality. It is an idealized representation or abstraction of a real life It is an idealized representation or abstraction of a real life

system.system. The objective of the model is to identify significant factors and The objective of the model is to identify significant factors and

their interrelationship.their interrelationship. A model is helpful in decision making as it provides a simplified A model is helpful in decision making as it provides a simplified

description of complexities and uncertainties of a problem in description of complexities and uncertainties of a problem in logical structure.logical structure.

Major advantage of models is it does not interfere with real Major advantage of models is it does not interfere with real system.system.

1.1. A broad classification of OR modelsA broad classification of OR models

1. 1. Physical ModelPhysical Model

2. Mathematical or symbolic model2. Mathematical or symbolic model

3. Models by nature of Environment3. Models by nature of Environment

4. Models by the extent of generality4. Models by the extent of generality

Types of OR ModelsTypes of OR Models

1. Physical Modes include all form of diagrams, graphs 1. Physical Modes include all form of diagrams, graphs and charts.and charts.

- - They are designed to deal with specific problems. They They are designed to deal with specific problems. They bring out significant factors and interrelationship in pictorial bring out significant factors and interrelationship in pictorial firm so as to facilitate analysis. There are two types i) Iconic firm so as to facilitate analysis. There are two types i) Iconic models and ii) Analog models. Iconic model is an image of models and ii) Analog models. Iconic model is an image of an object or system, represented on a small scale. These an object or system, represented on a small scale. These models can simulate the actual performance of a product.models can simulate the actual performance of a product.

2. Mathematical Model or symbolic models employ a set 2. Mathematical Model or symbolic models employ a set of mathematicalof mathematical

symbols to representsymbols to represent

- The decision variable of the system. The variables are - The decision variable of the system. The variables are related by mathematical system Eg: related by mathematical system Eg: Allocation, sequencing, Allocation, sequencing, replacement models etc.replacement models etc.

Methodology of OR Methodology of OR ModelsModels

The OR approach to problem solving consists of the following steps:The OR approach to problem solving consists of the following steps:

1. Definition of the problem.1. Definition of the problem.

2. Construction of the model.2. Construction of the model.

3. Solution of the model.3. Solution of the model.

4. Validation of the model.4. Validation of the model.

5. Implementation of the final result.5. Implementation of the final result.

1. Definition of the problem1. Definition of the problem

-The first and the most important requirement is that the root -The first and the most important requirement is that the root problem shouldproblem should

be identified and understood.be identified and understood.

-The problem should be identified properly, this indicates three -The problem should be identified properly, this indicates three major aspects: (1) a description of the goal or the objective of the major aspects: (1) a description of the goal or the objective of the study, (2) an identification of the decision alternative to the system, study, (2) an identification of the decision alternative to the system, and (3) a recognition of the limitations, restrictions and and (3) a recognition of the limitations, restrictions and requirements of the system.requirements of the system.

Methodology of OR Methodology of OR ModelsModels

2. Construction of the model2. Construction of the model

- Depending on the definition of the problem, the operations research - Depending on the definition of the problem, the operations research teamteam

should decide on the most suitable model for representing the should decide on the most suitable model for representing the system. system.

- Such a model should specify quantitative expressions for the - Such a model should specify quantitative expressions for the objective and the constraints of the problem in terms of its decision objective and the constraints of the problem in terms of its decision variables.variables.

- A model gives a perspective picture of the whole problem and helps - A model gives a perspective picture of the whole problem and helps tacklingtackling

it in a well organized manner.it in a well organized manner.

- If the resulting model fits into one of the common mathematical - If the resulting model fits into one of the common mathematical models, amodels, a

convenient solution may be obtained by using mathematical convenient solution may be obtained by using mathematical techniques.techniques.

- If the mathematical relationships of the model are too complex to - If the mathematical relationships of the model are too complex to allow analytic solutions, a simulation model may be more appropriate.allow analytic solutions, a simulation model may be more appropriate.

Methodology of OR Methodology of OR ModelsModels

3. 3. Solution of the modelSolution of the model

- - Once an appropriate model has been formulated, the next stage in Once an appropriate model has been formulated, the next stage in the analysis calls for its solution and the interpretation of the the analysis calls for its solution and the interpretation of the solution in the context of the given problem.solution in the context of the given problem.

- A solution to a model implies determination of a specific set of - A solution to a model implies determination of a specific set of decision variables that would yield an Optimum solution.decision variables that would yield an Optimum solution.

- An Optimum solution is one which maximize or minimize the - An Optimum solution is one which maximize or minimize the performance of any measure in a model subject to the conditions performance of any measure in a model subject to the conditions and constraints imposed on the model.and constraints imposed on the model.

4. 4. Validation the modelValidation the model

--A model is a good representative of a system, then the Optimal A model is a good representative of a system, then the Optimal solution must improve the system’s performance.solution must improve the system’s performance.

- A common method for testing the validity of a model is to compare- A common method for testing the validity of a model is to compare

itsperformance with some past data available for the actual itsperformance with some past data available for the actual system.system.

Methodology of OR Methodology of OR ModelsModels

-The model will be valid if under similar conditions of inputs, it -The model will be valid if under similar conditions of inputs, it can reproduce the past performance of the system.can reproduce the past performance of the system.

- The problem here is that there is no assurance that future - The problem here is that there is no assurance that future performance will continue to duplicate past behavior.performance will continue to duplicate past behavior.

-Also, since the model is based on careful examination of past -Also, since the model is based on careful examination of past data, the comparison should always reveal favorable results. In data, the comparison should always reveal favorable results. In some instances this problem may be overcome by using data from some instances this problem may be overcome by using data from trial runs of the systemtrial runs of the system

- It must be noted that such a validation method is not appropriate - It must be noted that such a validation method is not appropriate for non for non

existent systems, since data will not be available for comparison.existent systems, since data will not be available for comparison.

5. 5. Implementation of the final resultImplementation of the final result

--The optimal solution obtained from a model should be applied The optimal solution obtained from a model should be applied practice to improve the performance of the system and the practice to improve the performance of the system and the validity of the solution should be verified under changing validity of the solution should be verified under changing conditions.conditions.

Methodology of OR Methodology of OR ModelsModels

-It involves the translation of these results into detailed -It involves the translation of these results into detailed operating instructionsoperating instructions

issued in an understandable form to the individuals who issued in an understandable form to the individuals who will administer andwill administer and

operate the recommended system.operate the recommended system.

-The interaction between the operations research team and -The interaction between the operations research team and the operatingthe operating

personnel will reach its peak in this phase.personnel will reach its peak in this phase.

Techniques of OR Techniques of OR 1. Linear Programming:1. Linear Programming: It is used to find a solution for optimizing a given objective. It is used to find a solution for optimizing a given objective.

Objectives may be to maximize profit or minimize cost. Objectives may be to maximize profit or minimize cost. Both objective function and constraints must be capable of Both objective function and constraints must be capable of being expressed as linear expression of decision variables. being expressed as linear expression of decision variables. Its various use will be seen in Chapter2.Its various use will be seen in Chapter2.

2. Inventory control Methods:2. Inventory control Methods: The production, purchasing and material managers are The production, purchasing and material managers are

always confronted with question of when to buy, how much always confronted with question of when to buy, how much to buy and how much to keep in stock. Inventory models to buy and how much to keep in stock. Inventory models aims at optimizing inventory levels.aims at optimizing inventory levels.

Techniques of OR Techniques of OR 3. Goal Programming:3. Goal Programming: Single objective function is taken in the linear Single objective function is taken in the linear

programming and all other factors are considered as programming and all other factors are considered as constraints, but in actual practice there may be number of constraints, but in actual practice there may be number of important objective functions. Goal programming has important objective functions. Goal programming has several objective functions, each having a target value and several objective functions, each having a target value and programme models are developed to minimize deviation programme models are developed to minimize deviation from these targets.from these targets.

4. The tools, namely, queuing model, sequence model, 4. The tools, namely, queuing model, sequence model, transportation and assignment model, network analysis transportation and assignment model, network analysis are discussed in detail in later chapters.are discussed in detail in later chapters.

Structure of Structure of Mathematical Model Mathematical Model

Many industrial and business situations are concerned with Many industrial and business situations are concerned with planning activities. In each case of planning, there are limited planning activities. In each case of planning, there are limited sources, such as men, machines, material and capital at the disposal sources, such as men, machines, material and capital at the disposal of the planner. One has to make decision regarding these resources of the planner. One has to make decision regarding these resources in order to either maximize production, or minimize the cost of in order to either maximize production, or minimize the cost of production or maximize the profit etc. These problems are referred production or maximize the profit etc. These problems are referred to as the problems of constrained optimization. Linear to as the problems of constrained optimization. Linear programming is a technique for determining an optimal schedule of programming is a technique for determining an optimal schedule of interdependent activities, for the given resources. Programming interdependent activities, for the given resources. Programming thus means planning and refers to the process of decision making thus means planning and refers to the process of decision making regarding particular plan of action amongst several available regarding particular plan of action amongst several available alternatives.alternatives.

Any business activity of production activity to be formulated as a Any business activity of production activity to be formulated as a mathematical model can best be discussed through its constituents; mathematical model can best be discussed through its constituents; they are:they are:

1. Decision Variables and parameters, 1. Decision Variables and parameters,

2. Objective functions,2. Objective functions,

3. Constraints.3. Constraints.

Structure of Structure of Mathematical Model Mathematical Model

1. 1. Decision variables and parametersDecision variables and parameters The decision variables are the unknowns to be determined The decision variables are the unknowns to be determined

from the solution of the model. The parameters represent from the solution of the model. The parameters represent the controlled variables of the system.the controlled variables of the system.

2. 2. Objective functionsObjective functions This defines the measure of effectiveness of the system as This defines the measure of effectiveness of the system as

a mathematical function of its decision variables. The a mathematical function of its decision variables. The optimal solution to the model is obtained when the optimal solution to the model is obtained when the corresponding values of the decision variable yield the corresponding values of the decision variable yield the best value of the objective function while satisfying all best value of the objective function while satisfying all constraints. Thus the objective function acts as an constraints. Thus the objective function acts as an indicator for the achievement of the optimal solution.indicator for the achievement of the optimal solution.

Structure of Structure of Mathematical Model Mathematical Model While formulating a problem the desire of the decision While formulating a problem the desire of the decision

maker is expressed as a function of ‘n’ decision variables. maker is expressed as a function of ‘n’ decision variables. This function is essentially a linear programming problem This function is essentially a linear programming problem (i.e., each of its item will have only one variable raise to (i.e., each of its item will have only one variable raise to power one). Some of the Objective functions in practice power one). Some of the Objective functions in practice are:are:

- Maximization of contribution or profit- Maximization of contribution or profit

- Minimization of cost- Minimization of cost

- Maximization of production rate or minimization of - Maximization of production rate or minimization of production timeproduction time

- Minimization of labour turnover- Minimization of labour turnover

- Minimization of overtime- Minimization of overtime

- Maximization of resource utilization- Maximization of resource utilization

- Minimization of risk to environment or factory etc.- Minimization of risk to environment or factory etc.

Structure of Structure of Mathematical Model Mathematical Model 3. Constraints3. Constraints To account for the physical limitations of the system, the To account for the physical limitations of the system, the

model must include constraints, which limit the decision model must include constraints, which limit the decision variables to their feasible range or permissible values. variables to their feasible range or permissible values. These are expressed in the form of constraining These are expressed in the form of constraining mathematical functions.mathematical functions.

For example, in chemical industries, restrictions come For example, in chemical industries, restrictions come from the government about throwing gases in the from the government about throwing gases in the environment. Restrictions from sales department about the environment. Restrictions from sales department about the marketability of some products are also treated as marketability of some products are also treated as constraints. A linear programming problem then has a set constraints. A linear programming problem then has a set of constraints in practice.of constraints in practice.

Structure of Structure of Mathematical Model Mathematical Model The mathematical models in OR may be viewed generally The mathematical models in OR may be viewed generally

as determining the values of the decision variables x J, J = as determining the values of the decision variables x J, J = 1, 2, 3, n, which will optimize1, 2, 3, n, which will optimize

Z = f (x_ 1, x_ 2, x_ n).Z = f (x_ 1, x_ 2, x_ n).

Subject to the constraints:Subject to the constraints:

g i (x 1, x 2 x g i (x 1, x 2 x n) ~ b i, i = 1, 2, m n) ~ b i, i = 1, 2, m and xJ ³ 0 j = 1, 2, 3 n and xJ ³ 0 j = 1, 2, 3 n where ~ is £, ³ or =.where ~ is £, ³ or =.

The function f is called the objective function, where X j ~ The function f is called the objective function, where X j ~ b i, represent the b i, represent the

i th constraint for i =1, 2, 3 m where b i is a known i th constraint for i =1, 2, 3 m where b i is a known constant. The constraints x j ³ 0 are called the nonnegative constant. The constraints x j ³ 0 are called the nonnegative condition, which restrict the variables to zero or positive condition, which restrict the variables to zero or positive values only.values only.

Limitations of OR Limitations of OR The limitations are more related to the problems of model building, The limitations are more related to the problems of model building,

time and money factors.time and money factors.

i) Magnitude of computation: Modern problem involve large i) Magnitude of computation: Modern problem involve large number ofnumber of

variables and hence to find interrelationship, among makes it variables and hence to find interrelationship, among makes it difficult.difficult.

ii) Non – quantitative factors and Human emotional factor cannot ii) Non – quantitative factors and Human emotional factor cannot be takenbe taken

into account.into account.

iii) There is a wide gap between the managers and the operation iii) There is a wide gap between the managers and the operation researches.researches.

iv) Time and Money factors when the basic data is subjected to iv) Time and Money factors when the basic data is subjected to frequentfrequent

changes then incorporation of them into OR models is a costly changes then incorporation of them into OR models is a costly affair.affair.

v) Implementation of decisions involves human relations and v) Implementation of decisions involves human relations and behavior.behavior.

Summary Summary The OR approach needs to be equally developed in various The OR approach needs to be equally developed in various

agricultural problems on a regional or international basis. agricultural problems on a regional or international basis. With the explosion of population and consequent shortage With the explosion of population and consequent shortage of food, every country faces the problem of optimum of food, every country faces the problem of optimum allocation of land in various crops in accordance with allocation of land in various crops in accordance with climate conditions and available facilities. climate conditions and available facilities.

The problem of optimal distribution of water from a The problem of optimal distribution of water from a resource like a reservoir for irrigation purposes is faced by resource like a reservoir for irrigation purposes is faced by each developing country, and a good amount of scientific each developing country, and a good amount of scientific work can be done in this direction.work can be done in this direction.

OR JournalsOR Journals Operations Research Operations Research Management Science Management Science MS/OR Today (Management Science/Operations Res.)MS/OR Today (Management Science/Operations Res.) European Journal of Operational Research European Journal of Operational Research Journal of the Operational Research Society Journal of the Operational Research Society Mathematical Programming Mathematical Programming Journal of Optimization Theory and Applications Journal of Optimization Theory and Applications InterfacesInterfaces OR - SpektrumOR - Spektrum International Transactions in Operational ResearchInternational Transactions in Operational Research Annals of Operations ResearchAnnals of Operations Research Central European Journal of Operations ResearchCentral European Journal of Operations Research