Module 1. Introduction to operations research

Lesson 2

APPLICATIONS OF OPERATIONS RESEARCH IN DECISION MAKING

2.1 Introduction

The Operations Research may be regarded as a tool which is utilized to increase the effectiveness of management decisions. Scientific method of OR is used to understand and describe the phenomena of operating system. Mathematical and logical means of Operations Research provides the executive with quantitative basis for decision making and enhance ability to make long range plans and to solve everyday problems of industry with greater efficiency and competence.

2.2 Features (Characteristics) of OR

OR is a tool employed to increase the effectiveness of managerial decisions as an objective supplement to the subjective feeling of the decision makers. There are five salient features of OR.

2.2.1 Decision making

Primarily OR is addressed to managerial decision making or problem solving.

2.2.2 Scientific approach

OR employs scientific methods for the purpose of solving problems. It is a formalized process of reasoning.

2.2.3 Objective

OR attempts to locate the best or optimal solution to the problem under consideration. For this purpose, it is necessary that a measures of effectiveness is defined which is based on the goals of the organization. This measure is then used as the basis to compare the alternative courses of action. The examples are profit, net returns, cost of production etc.

2.2.4 Inter-disciplinary team approach

OR is inter-disciplinary in nature and requires a team approach to a solution of the problem. Managerial problems have economic, physical, psychological, biological, sociological and engineering aspects. This requires a blend of people with expertise in the areas of Mathematics, Statistics, Engineering, Economics, Management, Computer Science and so on.

2.2.5 Digital computer

Use of a digital computer has become an integral part of the OR approach to decision making. The computer may be required due to the complexity of the model, volume of data required and the computations to be made.

2.3 Modeling in Operations Research

A model as used in OR is defined as idealized representation of a real-life system. It shows the relationships (direct or indirect) and inter-relationships of action and reaction in terms of cause and effect. A model in OR is 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. The construction of a model helps in putting the complexities and possible uncertainties in a decision making problem, into a logical framework amenable to comprehensive analysis. With such a model we can have several decision alternatives, their anticipated effects indicating the need for relevant data for analyzing the alternatives that leads to informative conclusions.Models can be broadly classified according to following characteristics:

2.3.1 Classification by structure

2.3.1.1 Iconic models

These models are pictorial representation of real systems and have the appearance of the real thing. They represent the system as it is by scaling it up or down (i.e., by enlarging or reducing the size). In other words it is an image. Examples of such models are child’s toy, photograph, a schedule of operations, histogram, pictogram, cartogram, a physical model such as small scale model of a dairy plant, an engine etc. These kinds of models are called ‘Iconic’ because they are look alike items to understand and interpret the real things. An iconic model is said to be ‘scaled down’ or ‘scaled up’ according to the dimensions of the model and are smaller or greater than those of the real item. Iconic models are easy to observe, build and describe, but are difficult to manipulate and not very useful for the purpose of prediction. Commonly these models represent a static event. These models can be constructed up to three dimensions (e.g., atom, globe, small aeroplane, cube, etc.) while it is not possible to construct them physically for higher dimensions. Further these models do not include those aspects of the real system that are irrelevant for the analysis.

2.3.1.2 Analogue models

These models are more abstract than the iconic ones as there is no ‘look alike’ correspondence between these models and real items. These models are the one in which one set of properties is used to represent another set of properties. After the problem is solved, the solution is re-interpreted in terms of the original system. For example, graphs and maps in various colours are analogue models in which different colours correspond to different characteristics e.g., blue representing water, brown representing land, yellow representing production etc. Further, graphs are very simple analogues because distance is used to represent such properties as time, number, per cent, age, weight, and many other properties. Contour lines on the map represent the rise and fall of the heights. Demand curves, flow charts in production control and frequency curves in statistics are analogue models of the behaviour of events. Graphs of time series, bar diagrams, stock market changes are other examples of analogue models. Analogue models are less specific, less concrete but easier to manipulate and can represent dynamic situations. These are, generally, more useful than the iconic ones because of their vast capacity to represent the characteristics of the real system under investigation.

2.3.1.3 Mathematical models (Symbolic Model)

The Symbolic or mathematical model is one which employs a set of mathematical symbols (i.e. letters, numbers etc.) to represent the decision variables of the system. These variables are related together by means of a mathematical equation or set of equations to describe the behavior (or properties) of the system. These models are most general and precise. The solution to these models is then obtained by applying well developed mathematical techniques to the model. The symbolic model is usually the easiest to manipulate experimentally and the most general and abstract. Its function is more often explanatory rather than descriptive. Explanatory model is that which contains controlled variables, while the descriptive model does not contain controlled variables.

2.3.2 Classification by purpose

2.3.2.1 Descriptive model

It describes the some aspects of a situation based on observation, survey, questionnaire results or other available data.

2.3.2.2 Predictive models

Such models can answer ‘what if’ type of questions i.e. they can make predictions regarding certain events.

2.3.2.3 Prescriptive models

When a predictive model has been repeatedly successful, then it can be used to prescribe a source of action. For example, linear programming is a prescriptive model because it prescribes what the managers ought to do.

2.3.3 Classification by nature of environment

2.3.3.1 Deterministic model

Such models assume conditions of complete certainty and perfect knowledge for example; linear programming, transportation and assignment models are deterministic models.

2.3.3.2 Probabilistic (or Stochastic) model

These models handle those situations in which consequences or payoff of managerial actions cannot be predicted with certainty.

2.3.4 Classification by Behavior

2.3.4.1 Static models

These models do not consider the impact of changes that take place during the planning horizon i.e. they are independent of time.

2.3.4.2 Dynamic models

These models consider time as one of the important variables and depict the impact of changes generated over time. In this instead of one decision, a series of independent decisions are required during the planning horizons.

2.4 Characteristics of a Good Model

· A good model should be capable of taking into account new formulations without having any significant change in its frame.

· Assumptions made in the model should be as few as possible.

· It should be simple and coherent having less number of variables.

· It should be open to parametric type treatments.

2.5 Advantages of a Model

· Through the model, the problem under consideration becomes controllable.

· It provides some logical and systematic approach to the problem.

· It indicates the limitations and scope of an activity

· Models help in incorporating useful tools that eliminate duplication of methods applied to solve any specific problem.

· Models help in finding avenues for new research and improvements in a system.

2.6 Methodology of Operations Research

The major phases of an O.R. study are as follows:

Phase I: Formulation of the Problem

In this phase, the problem is formulated in an appropriate form. This phase should give a statement of the problem’s elements that include the controllable (decision) variables, the uncontrollable parameters, the restrictions or constraints on the variables and the objectives for defining a good or improved solution.

Phase II: Constructing a mathematical model

The second phase of the investigation is concerned with the choice of proper data inputs and the design of the appropriate information output. In this phase, both static and dynamic structural elements and the representation of inter-relationship among the elements in terms of mathematical formulae need to be specified. A mathematical model should include mainly the following three basic sets of elements:

(i) Decision Variables and Parameters

(ii) Constraints or Restrictions

(iii) Objective Function

Phase III: Deriving the solution from the model

This phase of the study deals with the mathematical calculations for obtaining the solution to the model. A solution of the model means those values of the decision variables that optimize one of the objectives and give permissible levels of performance on any other of the objectives.

Phase IV: Testing the model and its solution

This phase of the study involves checking the validity of the model used. A model may be said to be valid if it can give a reliable prediction of the system’s performance.

Phase V: Controlling the solution

This phase of the study establishes control over the solution by proper feedback of the information on variables which deviated significantly. As soon as one or more of the controlled variables change significantly, the solution goes out of control. In such a situation the model may accordingly be modified.

Phase VI: Implementing the solution

This phase of the study deals with the implementation of the tested results of the model. This would basically involve a careful explanation of the solution to be adopted and its relationship with the operating realities.

2.7 General Methods of Deriving the Solution

There are three methods to derive the solution to an OR model.

2.7.1 Analytical method

Analytical methods involve expressions of the model by graphic solutions or by mathematical computations. For example, area indicated by mathematical function may be evaluated through the use of integral calculus. Solutions of various inventory models are obtained by using the analytical procedure. The analytical methods involve all the tools of classical mathematics, such as calculus, finite differences, etc. The kind of mathematics required for a particular OR study depends upon the nature of the model.

2.7.2 Numerical or iterative method

Numerical methods are concerned with the iterative or trial and error procedures, through the use of numerical computations at each step. These numerical methods are used when some analytical methods fail to derive the solution. The algorithm is started with a trial (initial) solution and continued with the set procedure to improve the solution towards optimality. The trial solution is then replaced by the improved one and the process is repeated until the solution converges or no further improvement is possible. Thus the numerical methods are hit and trial methods that end at a certain step after which no further improvement to the solution is possible.

2.7.3 Monte Carlo method

This method is essentially a simulation technique, in which statistical distribution functions are created by generating a series of random numbers. Various steps associated with a Monte Carlo method are:

(a) For appropriate model of the system, sample observations are made and then the probability distributions for the variables of interest are determined.

(b) Convert the probability distribution to a cumulative distribution.

(d) Determine the sequence of values of variables of interest with the sequence of random numbers obtained in the previous step

(e) Fit an appropriate mathematical function to the values obtained in step (d)

2.8 Advantages/ Merits of OR Techniques

2.8.1 Optimum use of production factors

Linear programming techniques indicate how a manager can utilize most effectively his inputs/ factors and by more efficiently selecting and distributing these elements.

2.8.2 Improved quality of decision

The effect on the profitability due to changes in the production pattern will be clearly indicated in the simplex table. These tables give a clear picture of the happenings within the basic restrictions and possibilities of behaviour of compound elements involved in the problem.

2.8.3 Preparation of future managers

These methods substitute a means for improving the knowledge and skill of young managers.

2.8.4 Modification of mathematical solution

OR presents a possible practical solution when one exists, but it is always a responsibility of the manager to accept or modify the solution before its use. The effect of these modifications may be evaluated from the computation steps and tables.

2.8.5 Alternative solutions

OR techniques suggest all the alternative solutions available for the same profit so that the management may decide on the basis of its strategies.

2.9 Limitations of OR

2.9.1 Practical application

Formulation of an industrial problem to an OR set programme is a difficult task.

2.9.2 Reliability of the proposed solution

A non-linear relationship is changed to linear for fitting the problem to linear programming. This may disturb the solution.

2.9.3 Money and time cost

When the basic data is subject to frequent changes, the cost of changing programmes manually is a costly affair.

2.9.4 Combining two or more objective functions

Very frequently maximum profit does not come from manufacturing the maximum quantity of the most profitable product at the most convenient machine and at the minimum cost, since this may lead to underutilization of certain lines of production.

The aim is not to optimize individual objective function. It is, therefore, necessary to have a single objective function which can cover several objective functions at the same time. Despite all the above limitations, OR is a powerful tool and an analytical process that offers the presentation of an optimal solution.