**Answer:**

**a .**The application of Operations research methods helps in making decisions in such complicated situations. Evidently the main

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

Definition of Operations Research: Churchman, Ackoff and Aruoff have defined “Operations research as the application of scientific methods, techniques and tools to operation of a system with optimum solutions to the problems”. Here Optimum implies the one, which is best of all possible alternatives. Another definition is that, “Operations research is the use of scientific methods to provide criteria for decisions regarding man, machine, systems involving repetitive operations”. This definition is more general and comprehensive and seems to be more exhaustive than the previous definition.

In general, whenever there is any problem simple or complicated, the OR techniques may be applied to find the best solution. In this section we shall try to find the scope of OR by seeing its application in various fields of everyday life.

i)

**In Defence Operations:**In modern warfare the defence operations are carried out by a number of independent components namely Air Force, Army and Navy. The activities in each of these components can be further divided in four subcomponents viz.: administration, intelligence, operations and training, and supply. The application of modern warfare techniques in each of the components of military organizations requires expertise knowledge in respective fields. Further more, each component works to drive maximum gains from its operations and there is always a possibility that strategy beneficial to one component may have an adverse effect on the other. Thus in defence operations there is a necessity to coordinate the activities of various components which gives maximum benefit to the organization as a whole, having maximum use of the individual components. The final strategy is formulated by a team of scientists drawn from various disciplines who study the strategies of different components and after appropriate analysis of the various courses of actions, the best course of action, known as optimum strategy, is chosen.ii)

**In Industry:**The system of modern industries are so complex that the optimum point of operation in its various components cannot be intuitively judged by an individual. The business environment is always changing and any decision useful at one time may not be so good some time later. There is always a need to check the validity of decisions continually, against the situations. The industrial revolution with increased division of labour and introduction of management responsibilities has made each component an independent unit having their own goals. For example: Production department minimize cost of production but maximizes output. Marketing department maximizes output but minimizes cost of unit sales.Finance department tries to optimize capital investment and personnel department appoints good people at minimum cost. Thus each department plan their own objectives and all these objectives of various department or components come to conflict with each other and may not conform to the overall objectives of the organization. The application of OR techniques helps

in overcoming this difficulty by integrating the diversified activities of various components so as to serve the interest of the organization as a whole efficiently.

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

iii)

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

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

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

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

**Research and Development:**Control of R and D projects, product introduction planning etc. and many more applications.The basic dominant characteristic feature of operations research is that it employs mathematical representations or model to analyze problems. This distinctive approach represents an adaptation of the scientific methodology used by the physical sciences. The scientific method translates a real given problem into a mathematical representation which is solved and retransformed into the

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

1. Definition of the problem.

2. Construction of the model.

3. Solution of the model.

4. Validation of the model.

5. Implementation of the final result.

**1 Definition of the problem**

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

**2 Construction of the model**

Depending on the definition of the problem, the operations research team should decide on the most suitable model for representing the system. Such a model should specify quantitative expressions for the objective and the constraints of the problem in terms of its decision variables.

A model gives a perspective picture of the whole problem and helps tackling it in a well organized manner. If the resulting model fits into one of the common mathematical models, a convenient solution may be obtained by using mathematical techniques. If the mathematical relationships of the model are too complex to allow analytic solutions, a simulation model may be more appropriate. There are various types of models which can be constructed under different conditions.

**3 Solution of the model**

Once an appropriate model has been formulated, the next stage in the analysis calls for its solution and the interpretation of the solution in the context of the given problem. A solution to a model implies determination of a specific set of decision variables that would yield an Optimum solution. An Optimum solution is one which maximize or minimize the performance of any measure in a model subject to the conditions and constraints imposed on the model.

**4 Validation the model**

A model is a good representative of a system, then the Optimal solution must improve the system’s performance. A common method for testing the validity of a model is to compare its performance with some past data available for the actual system. The model will be valid if under similar conditions of inputs, it can reproduce the past performance of the system. The problem here is that there is no assurance that future performance will continue to duplicate past behaviour. Also, since the model is based on careful examination of past data, the comparison should always reveal favorable results. In some instances this problem may be overcome by using data from trial runs of the system. It must be noted that such a validation method is not appropriate for nonexistent systems, since data will not be available for comparison.

**5 Implementation of the final result**

The optimal solution obtained from a model should be applied practice to improve the performance of the system and the validity of the solution should be verified under changing conditions. It involves the translation of these results into detailed operating instructions issued in an understandable form to the individuals who will administer and operate the recommended system. The interaction between the operations research team and the operating personnel will reach its peak in this phase.

**b. the nature of Operations Research and its limitations:**

Many industrial and business situations are concerned with planning activities. In each case of planning, there are limited sources, such as men, machines, material and capital at the disposal of the planner. One has to make decision regarding these resources in order to either maximize production, or minimize the cost of production or maximize the profit etc. These problems are referred to as the problems of constrained optimization. Linear programming is a technique for

determining an optimal schedule of interdependent activities, for the given resources.

Programming thus means planning and refers to the process of decisionmaking

Regarding particular plan of action amongst several available alternatives.

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

- Decision Variables,

- Objective function,

- Constraints.

**1 Decision variables and parameters**

The decision variables are the unknowns to be determined from the solution of the model. The parameters represent the controlled variables of the system.

**2 Objective functions**

This defines the measure of effectiveness of the system as a mathematical function of its decision variables. The optimal solution to the model is obtained when the corresponding values of the decision variable yield the best value of the objective function while satisfying all constraints.

Thus the objective function acts as an indicator for the achievement of the optimal solution.

While formulating a problem the desire of the decision maker

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

- Maximization of contribution or profit

- Minimization of cost

- Maximization of production rate or minimization of production time

- Minimization of labour turnover

- Minimization of overtime

- Maximization of resource utilization

- Minimization of risk to environment or factory etc.

**3 Constraints**

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

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

The mathematical models in OR may be viewed generally as determining the values of the decision variables x J, J = 1, 2, 3, n, which will optimize Z = f (x 1, x 2, xn).

Subject to the constraints:

g i (x 1, x 2 xn) ~ b i, i = 1, 2, m

and xJ ≥ 0 j = 1, 2, 3 n

where ~ is ≤, ≥ or =.

The function f is called the objective function, where X j ~ b i, represent the i th constraint for i = 1, 2, 3 m

where b i is a known constant. The constraints x j ≥ 0 are called the nonnegativity

condition, which restrict the variables to zero or positive values only.

**4 Diet Problem**

Formulate the mathematical model for the following:

Vitamin – A and Vitamin –

*B*are found in food – 1 and food – 2. One unit of food –1 contains 5 units of vitamin – A and 2 units of vitamin –*B*. One unit of food – 2 contains 6 units of vitamin –A and 3 units of vitamin –*B*. The minimum daily requirement of a person is 60 units of vitamin –A and 80 units of Vitamin –*B*. The cost per one unit of food – 1 is*Rs*. 5/and one unit of food –2 is*Rs*. 6/.Assume that any excess units of vitamins are not harmful. Find the minimum cost of the mixture (of food–1 and food–2) which meets the daily minimum requirements of vitamins.

Mathematical Model of the Diet Problem: Suppose

*x*1 = the number of units of food–1 in the mixture,*x*2 = the number of units of food–2 in the mixture.Now we formulate the constraint related to vitamin A.

Since each unit of food –1 contains 5 units of vitamin – A, we have that

*x*1 units of food – 1 contains 5*x*1 units of vitamin – A. Since each unit of food – 2 contains 6 units of vitamin – A, we have that*x*2 units of food – 2 contains 6*x*2 units of vitamin – A. Therefore the mixture contains 5*x*1 + 6*x*2 units of vitamin A. Since the minimum requirement of vitamin – A is 60 units, we have that 5*x*1 + 6*x*2 ≥ 60.Now we formulate the constraint related to vitamin –

*B*. Since each unit of food – 1 contains 2 units of vitamin –*B*we have that*x*1 units of food – 1 contains 2*x*1 units of vitamin*B*.Since each unit of food – 2 contains 3 units of vitamin –

*B*, we have that*x*2 units of food – 2 contains 3*x*2 units of vitamin –*B*. Therefore the mixture contains 2*x*1 + 3*x*2 units of vitamin –*B*. Since the minimum requirement of vitamin –*B*is 80 units, we have that 2*x*2 + 3*x*2 ≥ 80.Next we formulate the cost function. Given that the cost of one unit of food –1 is

*R's*. 5/and one unit of food – 2 is*R's*. 6/.Therefore

*x*1 units of food–1 costs*Rs*. 5*x*1, and*x*2 units of food – 2 costs*Rs*. 6*x*2. Therefore the cost of the mixture is given*b*y Cost = 5*x*1 + 6*x*2. If we write*z*for the cost function, then we have*z*= 5*x*1 + 6*x*2. Since cost is to be minimized, we write min*z*= 5*x*1 + 6*x*2.Since the number of units (

*x*1 or*x*2) are always nonnegative we have that*x*1 ≥ 0,

*x*2 ≥ 0. Therefore the mathematical model is

5

*x*1 + 6*x*2 ≥ 602

*x*1 + 3*x*2 ≥ 80*x*1 ≥ 0,

*x*2 ≥ 0, min

*z*= 5

*x*1 + 6

*x*2.

**10 limitations of OR**

The limitations are more related to the problems of model building, time and money factors.

i) Magnitude of computation: Modern problem involve large number of variables and hence to find interrelationship, among makes it difficult.

ii) Non – quantitative factors and Human emotional factor cannot be taken into account.

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

iv) Time and Money factors when the basic data is subjected to frequent changes then incorporation of them into OR models is a costly affair.

v) Implementation of decisions involves human relations and behaviour.

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