INTRODUCTION /NATURE / DEFINITIONS / FEATURES / METHODOLOGY/APPROACHES /PRINCIPAL /APPLICATIONS OF OPERATIONS RESEARCH

 INTRODUCTION TO OPERATIONS RESEARCH

CONTENTS:- 

 INTRODUCTION TO OPERATIONS RESEARCH

OPERATIONS RESEARCH IN INDIA

NATURE OF OPERATIONS RESEARCH

DEFINITIONS OF OPERATIONS RESEARCH

FEATURES OF OR

METHODOLOGY/APPROACHES OF OR

TYPE OF OPERATIONS RESEARCH

PRINCIPAL OF OPERATIONS RESEARCH

TYPICAL APPLICATIONS OF OR/ SCOPE OF OR

TECHNIQUES OR TOOLS OF OR 

I. INTRODUCTION

Operation Research may simply be viewed as a systematic and analytical approach to decision-making and problem-solving. O.R. as termed in the USA, Canada, Africa, and Australia, and Operations Research as termed in Europe is an interdisciplinary branch of applied mathematics that uses techniques and statistics to arrive at Optimal or near Optimal solutions to complex problems. It is typically concerned with determining the maximum profit, sale, output, crop yield and efficiency, and minimum losses, risk, cost, and time of some objective function. Although it is a distinct discipline in its own sight, it has also become an integral part of the Industrial Engineering profession. Industrial Engineers typically consider O.R. techniques to be a major part of their tool set. Some of the primary tools used by operations researchers are statistics, probability theory, game theory, graph theory, decision theory, and optimization techniques with computers. Sometimes O.R. is referred to as Management Science in order to better reflect its role as a scientific method in solving management problems but it appears that the terminology is more popular with business professionals.

II. HISTORY OF O.R.

While there is no clear date that marks the birth of O.R., it is generally accepted that the field originated in England during World War II. Some say that Charles Babbage (17911871) is the father of Operations Research because his research into the cost of transportation and sorting of mail led to England's Universal Penny Post in 1840. Modern Operations Research originated at the Bowdsey Research Station in the U.K. in 1937 to analyze and improve the working of the UK's early warning radar system. During the Second World War about 1000 men and women were engaged to work for British Army. After World War II, military operational research in the U.K. become known as Operational Analysis (OA) within the UK Ministry of Defence with expanded techniques and graving awareness, military O.R. or OA was no longer limited to only operations but was extended to encompass equipment procurement, training, and infrastructure.

III. OPERATIONS RESEARCH IN INDIA

In India OR society was founded in 1959, and also became a member of the International Federation of OR Societies in 1959. The Journal OPSEARCH was published for the first time in 1963. Now OR techniques are used in almost all walks of our life and OR is emerging as an interdisciplinary area of knowledge that can make a contribution to the solution of the problems in diversified areas of interest. There is too much impact of OR in economics, management, engineering, and other social and behavioral sciences.

Industries in India are also becoming conscious of the role of operations research and a good number of them have well-trained OR teams. Some of the Indian organizations using OR techniques are Indian Airlines, Railways, Defence Organisations, TATA, TELCO, DCM, STC, etc. Assignment Models have been used by Kirloskar company for allocation of their salesmen to different areas so as to maximize the profits. DCM, Calico, and Binnys' have been using linear programming techniques for cotton blending. Many organizations are making use of PERT/CPM techniques for effective management and control of their construction projects.

IV. NATURE OF OPERATIONS RESEARCH

As stated earlier, Operations Research involves research on (military) operations. This indicates the approach as well as the area of applications in the field. Thus it is an approach to problems of how to coordinate and control the operations or activities within an organization. Following is an example that will elaborate on the nature of operations research.

In order to run an organization effectively as a whole, the problem that arises frequently is coordination among the conflicting goal of its various functional departments. For example, consider the problem of stocks of finished goods. The various departments of the organization may like to handle this problem differently. To the marketing department, stock of a large variety of products is a means of supplying the company's customers with what they want, and where they want it. Clearly, the fully stocked warehouse is of prime importance to the company. The production department argues for long production runs preferably on a smaller product range, particularly if there is a significant time loss when production is switched from one variety to another. On the other hand, the finance department sees stocks kept as capital tied up unproductively and argues strongly for their reduction. Finally, the personnel department sees a great advantage in labor relations if there is a steady level of production leading to steady employment. To optimize the whole system, the decision-maker must decide on the best policy keeping in view the relative importance of objectives and validity of conflicting claims of various departments from the perspective of the whole organization.

Operations research thus helps to seek the optimal solution to a problem and not merely one which gives better solutions than the one currently in use. The decision taken by the decision-maker may not be acceptable to every department but it should be optimal for the organization as a whole or at least for a large portion of the total organization. In order to obtain such types of solutions, the decision-maker must follow the effects and interactions of a particular decision.

V. DEFINITION OF OPERATIONS RESEARCH

It is not possible to give a uniformly acceptable definition of Operations Research. However, a few definitions which are commonly used and widely accepted are given below.

(a) Operations Research is the application of scientific methods, techniques, and tools to problems involving the operation of a system so as to provide those in control of the system with optimum solution to the problem".C.W. Churchman

(b) "Operations research is the art of giving bad answers to problems which otherwise have worse answers". T.L.Saaty

(c) "Operations research is a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control." P.H. Morse and G.E. Kimball

(d) "Operation management is a scientific approach to problem-solving for H.M. Wagner

(e) "Operation research is concerned with scientifically deciding how best to design and operate man-machine systems usually under conditions requiring the allocation of scarce resources". OR Society of America

O.R. is an aid for the executive in making his decision by providing him with the needed quantitative information based on the scientific method of analysis." C. Kittel

(g) "O.R. is the application of scientific methods to problems arising from operations involving an integrated system of men, machines, and material. It normally utilizes the knowledge and skill of an interdisciplinary research team to provide the managers of such systems with optimum operation solutions." Febrycky and Torgersen

(h) "O.R. is the application of the scientific method by interdisciplinary teams to problems involving the control of organized (men-machines) systems so as to provide a solution which best serves the purpose of the organization as a whole." Ackoff & Sasieni

VI. FEATURES OF OR

From the various definitions of OR the important features or characteristics of OR can be listed below:

1.     System orientation. OR study the situation or problem as a whole. This means that any action or activity has the same effect on the other part of the organization. The optimum result of one part of a system may not be the optimum for some other part. Therefore to evaluate any decision, one must identify all possible interactions and determine their impact on the organization as a whole.

2.     Inter-disciplinary team approach. OR is inter-disciplinary in nature, and is performed by a team of scientists whose individual members have been drawn from different scientific and engineering disciplines. No single individual can have a thorough knowledge of all aspects of the undertaking. Managerial problems have economic, physical, psychological, biological, sociological, and engineering aspects. This requires a number of people which expertise in the areas of mathematics, statistics, engineering, economics, management, computer science, and so on. Another reason for the existence of OR teams is that knowledge is increasing at a very fast rate. No single person can collect all the useful scientific information from all managerial areas. So, OR team brings the latest scientific know-how to analyze the problem and helps in providing better results.

3.      Scientific approach. OR uses scientific methods to solve the problems. Most science studies such as Chemistry, Physics, Biology, etc. can be carried out in laboratories, without much interference from the outside world.) But same is not true in the systems under study by OR teams., For example, no company can risk its failure in order to conduct a successful experiment.,

So, OR is a formalized process of reasoning. Under OR the problem is to be analyzed and defined clearly. Observations are made under different conditions to study the behavior of the system. On the basis of these observations, a hypothesis describing how the various factors involved are believed to interact and the best solution to the problem is formulated. To test the hypothesis experiment is designed and executed. Observations are made and measurements are recorded. Finally, the results of the experiments are studied and the hypothesis is accepted or rejected. So, OR is the use of the scientific method to solve the problem under study.

4. Decision-making. OR increases the effectiveness of management decisions. Management is most of the time making decisions. It is thus a decision science that helps management to make better decisions. So, the major premise of OR is the decision-making, irrespective of the situation involved. So decision-making is a systematic process and consists of the following steps.

(a) Diagnose the problem, and establish the criterion. The criterion may be the maximization of profits, utility, minimization of cost, etc. (b) Select the alternative course of action for consideration. values of the parameters.

(c) Determine the model to be used and the value of the parameters.

(d) Evaluation of various alternatives.

(e) Selecting the best and the optimum alternative.

5.      Use of computer. Operations research (OR) often requires a computer to solve a complex mathematical model or to perform a large number of computations that are involved. The use of a digital computer has become an integral part of the operations research approach to decision making.

6.      Objectives. Operations research always attempts to find the best and optimal solution to the problem. For these purposes, the objectives of the organization are defined and analyzed. These objectives are then used as the basis to compare the alternative courses of action.

7.     Quantitative solution. Operations research provides managers with a quantitative basis for decision-making. OR attempts to provide a systematic and rational approach for quantitative solutions to the various managerial problems.

8.      Human factors. In deriving quantitative solutions, we do not consider human factors, which doubtlessly play a great role in the problems. So the study of the OR is incomplete without a study of human factors.

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VII. METHODOLOGY/APPROACHES OF O.R.

Given that O.R. represents an integrated framework to help make decisions, it is important to have a clear understanding of this framework so that it can be applied to generic problems. The approach or methodology of O.R. comprises the following seven consequential steps.

•   (a)   Orientation

(b) Problem Definition

      (c) Data collection

      (d) Model Formulation

      (e) Solution/results

      (f) Analysis and interpretation of results

     (g) Implementation & Monitoring.

(a) Orientation. The first step in the O.R. approach is referred to as problem orientation. The primary objective of this step is to have a clear picture of the relevant issues. This phase also involves a study of documents and literature relevant to the problems. The aim of the orientation phase is to obtain a clear understanding of the problem and its relationship to different operational aspects of the system, so that problem can be properly defined.

(b) Problem Definition. This is the second and most important and difficult step of the O.R. process. The objective here is to have a clear definition of the problem in terms of its scope and the results desired. A clear definition of the problem has three broad components viz.

(1) Statement of an unambiguous objective.

(2) Specification of factors that will affect the objective.

(3) Specification of the constraints on the causes of action. The proper definition will help in data correlation in the right direction.

(c) Data Collection. In the third phase of the O.R. process data is collected with the objective of translating the problem defined in the second phase into a model that can then be objectively analyzed. Data typically comes from two sources - and standards i.e. primary and secondary sources. Data collection can have an important effect on the previous steps mentioned. Now we have ready access to - observation previously very hard to obtain. Even though the data is all present data that was somewhere and in some form, extracting useful information from these sources is often very difficult.

(d) Model Formulation. In this fourth phase of the O.R. process, modeling is the process of defining the characteristics of all operations. A model may be defined formally as a selective abstraction of reality that helps in the working of the original system. Models may be classified as :

 (i) Physical Models

(ii) Analogic Models

(iii) Mathematical Models

(e) Solution Results. After formulating an appropriate model, and collecting data, the next stage is to find the solution to the model and then interpret the solutions this stage we arrive at the answers to the variables and the objective function solution can be classified as :

(i) Feasible solution

(ii) Infeasible solution

(iii) Optimal solution

(iv) Non- optimal solution

(v) Unique solution

(vi) Multiple solution

(f) Analysis and interpretation of results. This stage requires determining whether the model can adequately and reliably predict the behavior of the real system. It involves testing the structural assumptions of the model. Analysis and interpretation of the performance of the model give no assurance that the future performance of the system will continue to be in the same manner as in the past.

(g) Implementation and Monitoring. It is the last process of implementing the solution in the organization, implementation of the solution is often more difficult as it does not assures that the solution obtained would be automatically implemented. The impact of the decision may affect various segments of the organization which needs proper monitoring of the various changes and taking suitable measures. The methodology of O.R. can be depicted in the following diagram.

 

VIII. TYPES OF OPERATIONS RESEARCH MODELS

A Model is a representation of reality. Most of our thinking of operations research in business takes place in the context of models. A model is a general term denoting any idealized representation or abstraction of a real-life system or situation. The objective of the model is not to identify all aspects of the situation but to identify significant factors and their interrelationship. A model can be helpful in decision making as it provides a simplified description of the complexities and uncertainties of a problem in hand in a logical structure. A major advantage of modeling is that it permits the decision-maker to examine the behavior of a system without interfering with going operations.

A broad classification of operations research models is given below:

(A) Physical Models. These models include all forms of diagrams, drawings graphs, and charts. Most of which are designed to deal with specific types of problems. By presenting significant factors and inter-relationships in pictorial terms, physical models are able to indicate problems in a manner that facilitates analysis. For example, a BAR chart can be used effectively as a summary presentation of a company's monthly production forecast. These models are easy to observe, build and describe, but cannot be manipulated and used for prediction. There are two types of physical models - Iconic models and Analog models.

(i) Iconic Models. An icon is an image or likeness of an object or system it represents. An iconic model the least abstract physical replica of a system, is usually based on a smaller scale than the original. The range of management problem areas where these models can be used effectively is extremely narrow. However, it consists largely of these fields that are oriented toward engineering and science. For instance, Indian Airlines as well as Air India use flight simulators to train their pilots and members of the crew. These flight simulators are Iconic models of different types of aircraft and the trainee really feels as if he is piloting the actual aircraft. Thus an iconic model has all the operating features of the real system. These models can simulate the actual performance of a product thereby availing the tremendous expense of designing full-scale experimental models.

(ii) Analog models. Analog models are closely associated with iconic models. However, they are not replicas of problem situations. Rather, they are small physical systems that have similar characteristics and work like an object or system it represents e.g. children's toys model of rails, roads, etc., whereas the actual objects are complex and might not allow direct handling or manipulation. The objective of constructing these models is to understand by analogy.

(B) Mathematical Model Or Symbolic model. The symbolic model employs a set of mathematical symbols to represent the decision variable of the system under study. These variables are related together by mathematical equations. In other words, equations are mathematical models commonly used in operations research Simple demand curve in economics is a symbolic model predicting buyer's behavior at different price levels. Similarly, the profit and loss statements and budget for next; year are both symbolic models. The profit and loss account is just reproduced on one sheet of paper, and it summarises the result of a year but does not recreate every action, which took place during the year. Following are the examples of mathematical models which have been applied to business and industry.

INTRODUCTION TO OPERATIONS RESEARCH

- Allocation model

- Routing model

-Queuing model

-Simulation

- Replacement model

- Markov analysis.

- Sequencing model

-Competitive model

- Dynamic Programming model

- Decision theory

-Goal programming

(C) By Nature of Environment.

(i) Deterministic Models. In this model, everything is defined and the results are certain. Certainty is the state of nature assumed in these models. For any given input variable there shall be the same output variable, e.g. in EOQ models, one can easily determine economic lot size, and one can apply sensitivity analysis, where a change of one variable shall cause a certain change in the outcome.

(ii) Probabalistic Model. In cases of risk and uncertainty, the input and output variables take the form of a probability distribution. In fact, such models are semiclosed models and reveal the probability of occurrence of an event. In fact, they reveal the complexity of the real world and uncertainty prevailing in it, e.g. in a game theory where saddle points or equilibrium points of the player do not exist, we apply the probabilistic model. A similar kind of model can be applied to inventory control.

(D) By the Extent of Generality.

 

(i)         General Models. The general model is one that does not apply to one situation only rather it has got general applications. For example, the linear programming model is known as a general model since it can be used for product mix, production scheduling, marketing, transportation, and assignment problems.

(ii)         Specific Model. The specific model is applicable under specific conditions only, e.g. sales response curve, or equation as a function of advertising is applicable in

IX. PRINCIPLES OF OPERATION RESEARCH MODELLING

The following principles must be kept in mind while formulating models.

1. Principle of Simplicity: Mathematicians are in the habit of making complex models whereas one should go in for a simple model if it is sufficient. It means the model must be kept simple & understandable.

2. Understand the problem, only then apply the appropriate technique: If we do not understand the problem properly, we can not apply the appropriate technique of Operation Research. For example, in the case of allocation of scarce resources, the technique of LPP may be applied.

For the queuing problem, we must apply queuing theory. So, 50% of the problem is solved, if it is properly understood and the required technique is recommended.

3. Model must be validated before implementation, otherwise, it can be implemented in phases for validation e.g. Two-Phase Simplex Method.

4. A model can analyze the data but it can not be better than the information that goes into it. It means data can be interpreted but can not be generated.

5. Models are for the decision-makers: OR models are to aid the decision-maker but not to replace them. The decision is to be taken by the management e.g.,

X₂ = 0                    X₂ = 9

X₂ = 6                    X₂ = 3

Say, in the above multiple optimal solutions are provided by LPP giving profits. It is for the management to decide whether to discontinue the product X₁ or continue to produce X₁ because the total profits are the same.

6. A model should never be taken too literally: It means it does not provide a unique answer. In sensitivity analyses, or under simulation, we can find still better results. 7. Flexible Model: The model should be flexible enough to incorporate changes. It should give a range where one solution is valid as in the case of sensitivity analyses.

8. Seek Cooperation from Operation Research Experts: As we know, the tested results of the model are implemented. It is to be done by Operation Research Experts. Hence it must be according to the expectation of those who are to execute so that cooperation of Operation Research Experts may be sought to implement the models.

9. Use of Computers: Use of computers is made wherever possible in the case of implementation of models. OR techniques are usually complex and only computers can solve them. Hence, steps should be clearly stated to enable the OR researcher to develop computer software for future implementation of the techniques. Without computers, perhaps the OR experts will be like illiterate people in the years to come.

X, TYPICAL APPLICATIONS OF O.R./SCOPE OF O.R.

In the field of industrial management, there is a chain of problems starting from the purchase of raw materials to the dispatch of finished goods. The management is interested in having an overall view of the method of optimizing profits. In order to make the decision on a scientific basis, the operation research team will have to consider various alternative methods of producing the goods and the return in each case. An operation research study should also point out the possible changes in the overall structure like the installation of a new machine, the introduction of more automation, etc. OR has been successfully applied in industry, in the fields of production, blending, product mix, inventory control, demand forecast, sale and purchase, transportation, repair and maintenance, scheduling and sequencing, planning, scheduling and control of projects, and scores of other associated areas.

The operation research approach is equally applicable to big and small organizations. For example, whenever a departmental store faces a problem like employing additional sales girls, purchasing an additional van, etc., techniques of operation research can be applied to minimize cost and maximize the benefit for each such decision.

 Some of the areas of management where techniques of Operations Research are applied are listed below:

1.     Finance, Budgeting, and Investments

(a) Cash flow analysis, long-range capital requirements, investment portfolios, dividend policies, etc.

2.     Purchasing, Procurement, and Exploration

• (a)   Determining the quantity and timing of the purchase of raw materials, machinery, etc.
• (b)  Rules for buying and supplies under varying prices.
• (c)   Bidding policies.
• (d)  Equipment replacement policies.
• (e)   Determination of quantities and timings of purchases.

3.      Production Management

(i) Project Planning:

(a) Location and size of warehouses, distribution centers, retail outlets, etc.            

(b)  Distribution policy.

(ii) Manufacturing and Facility Planning:

(a) Production scheduling and sequencing.

(b) Project scheduling and allocation of resources.

(c) Selection and location of factories, warehouses, and their sizes.

(d) Determining the optimal production mix.

(e) Maintenance policies and preventive maintenance.

(f) Maintenance crew sizes.

(g) Scheduling arid sequencing the production run by proper allocation of machines.

4. Marketing Management

(a) Product selection, timing, competitive actions.

(b)/Advertising strategy and choice of different media of advertising.

(c) Number of salesmen, frequency of calling of accounts, etc.

(d) Effectiveness of market research.

(e) Size of the stock to meet the future demand.

5. Personnel Management

(a) Recruitment policies and assignment of jobs.

(b) Selection of suitable personnel with due consideration for age and skills, etc.

(c) Establishing equitable bonus systems.

6. Research and Development

(a) Determination of areas of concentration of research and development.

(b)/Reliability and evaluation of alternative designs.

(c) Control of development projects.

(d) Co-ordination of multiple research projects.

(e) Determination of time and cost requirements.

XI. TECHNIQUES OR TOOLS OF OPERATIONS RESEARCH

An operation researcher may use the following techniques. This is not exhaustive, because still more and more techniques can be added to the said list.

1. Linear Programming. This technique is used to find a solution for optimizing a given objective. Objectives may be maximizing profit or minimizing cost. The objective function and boundary conditions are linear in nature. Linear programming techniques solve product mix and distribution problems of enterprises. LPP techniques are used to allocate scarce resources in an optimum manner in problems of scheduling, product mix, etc. It consists of an objective function which is the same measure of effectiveness as profit, loss, or return investment, and several boundary conditions putting restrictions on the use of resources.

2. Queuing Theory. This theory deals with the situations in which There are different types of situations in which queue is formed, e.g. customers waiting for service, machines waiting for repairmen and air crafts waiting for landing strips, etc. The waiting line theory aims at minimizing the overall cost due to servicing and waiting. On the other hand, if the queue becomes long there will be a cost due to waiting for units in the queue. This technique is used to analyze the feasibility of adding facilities and to assess the amount and cost of waiting time. These calculations can be used to determine the desired number of service facilities.

3. Inventory Control Models. When to buy, how much to buy, and how much to keep in stores are some of the questions that production managers, purchase managers, and material managers address themselves. Inventory control aims at optimizing inventory levels. Inventory may be defined as a useful idle resource that has economic value in raw materials, spare parts, finished products, etc. Inventory control models help the above-said managers to decide on reordering time, reordering level and optimal ordering quantity, etc. Inventory planning answers two questions viz., how much to buy and when to buy? The approach is to prepare a model of the situation that expresses total inventory costs in terms of demand, size of the order, possible over or under stocking, and other relevant factors and then to determine optimal order size, reorder level, etc. using inventory control models.

4. Network analysis. Net work models are a very popular and widely used quantitative technique. This model helps the managers to plan, schedule, monitor, and control large projects, such as the construction of a building, making a ship, or planning for a space flight. The Network analysis helps the managers to determine total project completion time, the probability that a project will be completed by a certain date, the least-cost way of shortening total project completion time, etc. Program Evaluation and Review Technique (PERT), Critical Path Method (CPM), and other network techniques such as Gantt Chart come under network analysis.

5. Replacement Problems. The theory of replacement is concerned with situations that arise when some items such as machines, men, electric light bulbs, or any other equipment require replacement due to their decreasing efficiency, failure, or breakdown. Sooner or later all types of equipment require to be replaced. The replacement may be necessary because of obsolescence due to new discoveries and better design of the equipment. In a replacement decision the cost of the equipment to be installed and the one being replaced, scrap value, useful life, return and other relevant aspects have to be considered. The replacement theory helps to solve all replacement problems.

6. Sequencing. Models have been developed to find a sequence for processing jobs so that the total elapsed time for all the jobs will be a minimum. The models also help to resolve the conflict between the objectives of maximizing machine utilization and complying with predetermined delivery dates.

7. Integer Programming. Integer means complete or whole number. One important limitation of linear programming models is the assumption that all the variables can take any value, maybe decimals or fractions. Sometimes some of the variables are restricted and can take only integer values e.g. the number of taxis in a flea, the number of power plugs in a factory, and so on. With the techniques of linear programming sometimes it is difficult to get the best solution when the figures are rounded off. Rounding off figures to the nearest integer may lead to a poor choice of solution. Integer programming has been developed to meet the increments of such situations. By using the integer programming algorithm a series of continuous linear programming problems are solved in such a way that the solution containing un-acceptable non-integer value is ruled out and the best higher programming solution is obtained. These are certain techniques that are applied for the purpose of getting the best solution to the given problem.

8. Assignment Problems. An assignment problem is a special type of linear programming problem. It deals with allocating the various resources or items to various activities on a one-to-one basis in such a way that the time or cost involved in minimizes one; sale or profit is maximized e.g. Manager may like to know which job should be assigned to which person so that all jobs can be completed in the shortest possible time. Similarly in marketing set up by making an estimate of sales performance for different territories one can assign a particular salesman to a particular territory with a view to maximizing overall sales.

9. Transportation Problems. Transportation problem deals with the transportation of a product from a number of sources, with limited supplies, to a number of destinations with specified demands, at the minimum total transportation cost. In other words, it is a special procedure for finding the minimum cost for distributing homogeneous units of a product from several points of sources to a number of points of demand destinations. The main objective of transportation is to schedule shipments from sources to destinations in such a way as to minimize the total transportation cost.

10. Decision Theory and Games Theory. Decision theory is primarily concerned with decision-making under the conditions of risk and uncertainty. However, the theory of games is concerned with decision-making under conflict. Hence, both decision theory and game theory assist the decision-maker in analyzing problems with numerous alternative courses of action and consequences. The decision-maker is to identify the best course of action on the basis of the above techniques. The game theory assists the decision-maker to have the knowledge of course action available to his opponent. In decision theory, we use decision tables and decision trees, which can be used to graphically represent and solve decision-making problems.

11. Simulation. In many cases, the goal has been to determine the optimal solution. All real-life problems cannot be stated in mathematical form because of stochastic relationships, complexity, etc. such situations can be tackled by simulation. Hence, simulation is a general technique that allows us to develop a dynamic model that acts like a real process. Developing a good simulation model can be difficult but simulation allows us to solve problems that are difficult or impossible to solve otherwise.