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MBA-H2040 Quantitative Techniques for Managers UNIT I 1 INTRODUCTION TO OPERATIONS RESEARCH LESSON STRUCTURE 1.1 Introduction 1.2 History of Operations Research 1.3 Stages of Development of Operations Research 1.4 Relationship Between Manager and OR Specialist 1.5 OR Tools and Techniques 1.6 Applications of Operations Research 1.7 Limitations of Operations Research 1.8 Summary Objectives 1.9 Key Terms After Studying this lesson, you should be able 1.10 Self Assessment Questions to: 1.11 Further References  Understand the meaning, purpose, and tools of Operations Research  Describe the history of Operations Research  Describe the Stages of O.R  Explain the Applications of Operations Research  Describe the Limitations of Operation Research  Understand the OR specialist and Manager relationship LESSON MBA-H2040 Quantitative Techniques for Managers 1.1 Introduction The British/Europeans refer to "operational research", the Americans to "operations research" - but both are often shortened to just "OR" - which is the term we will use. Another term which is used for this field is "management science" ("MS"). The Americans sometimes combine the terms OR and MS together and say "OR/MS" or "ORMS". Yet other terms sometimes used are "industrial engineering" ("IE") and "decision science" ("DS"). In recent years there has been a move towards a standardization upon a single term for the field, namely the term "OR". Operation Research is a relatively new discipline. The contents and the boundaries of the OR are not yet fixed. Therefore, to give a formal definition of the term Operations Research is a difficult task. The OR starts when mathematical and quantitative techniques are used to substantiate the decision being taken. The main activity of a manager is the decision making. In our daily life we make the decisions even without noticing them. The decisions are taken simply by common sense, judgment and expertise without using any mathematical or any other model in simple situations. But the decision we are concerned here with are complex and heavily responsible. Examples are public transportation network planning in a city having its own layout of factories, residential blocks or finding the appropriate product mix when there exists a large number of products with different profit contributions and production requirement etc. Operations Research tools are not from any one discipline. Operations Research takes tools from different discipline such as mathematics, statistics, economics, psychology, engineering etc. and combines these tools to make a new set of knowledge for decision making. Today, O.R. became a professional discipline which deals with the application of scientific methods for making decision, and especially to the allocation of scarce resources. The main purpose of O.R. is to provide a rational basis for decisions making in the absence of complete information, because the systems composed of human, machine, and procedures may do not have complete information. Operations Research can also be treated as science in the sense it describing, understanding and predicting the systems behaviour, especially man-machine system. Thus O.R. specialists are involved in three classical aspect of science, they are as follows: i) Determining the systems behaviour ii) Analyzing the systems behaviour by developing appropriate models iii) Predict the future behaviour using these models The emphasis on analysis of operations as a whole distinguishes the O.R. from other research and engineering. O.R. is an interdisciplinary discipline which provided solutions to problems of military operations during World War II, and also successful in other operations. Today business applications are MBA-H2040 Quantitative Techniques for Managers primarily concerned with O.R. analysis for the possible alternative actions. The business and industry befitted from O.R. in the areas of inventory, reorder policies, optimum location and size of warehouses, advertising policies, etc. As stated earlier defining O.R. is a difficult task. The definitions stressed by various experts and Societies on the subject together enable us to know what O.R. is, and what it does. They are as follows: 1. According to the Operational Research Society of Great Britain (OPERATIONAL RESEARCH QUARTERLY, l3(3):282, l962), Operational Research is the attack of modern science on complex problems arising in the direction and management of large systems of men, machines, materials and money in industry, business, government and defense. Its distinctive approach is to develop a scientific model of the system, incorporating measurements of factors such as change and risk, with which to predict and compare the outcomes of alternative decisions, strategies or controls. The purpose is to help management determine its policy and actions scientifically. 2. Randy Robinson stresses that Operations Research is the application of scientific methods to improve the effectiveness of operations, decisions and management. By means such as analyzing data, creating mathematical models and proposing innovative approaches, Operations Research professionals develop scientifically based information that gives insight and guides decision- making. They also develop related software, systems, services and products. 3. Morse and Kimball have stressed O.R. is a quantitative approach and described it as “ a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control”. 4. Saaty considers O.R. as tool of improving quality of answers. He says, “O.R. is the art of giving bad answers to problems which otherwise have worse answers”. 5. Miller and Starr state, “O.R. is applied decision theory, which uses any scientific, mathematical or logical means to attempt to cope with the problems that confront the executive, when he tries to achieve a thorough-going rationality in dealing with his decision problem”. 6. Pocock stresses that O.R. is an applied Science. He states “O.R. is scientific methodology (analytical, mathematical, and quantitative) which by assessing the overall implication of various alternative courses of action in a management system provides an improved basis for management decisions”. 1.2 History of Operations Research Operation Research is a relatively new discipline. Whereas 70 years ago it would have been possible to study mathematics, physics or engineering (for example) at university it would not have been possible to study Operation Research, indeed the term O.R. did not exist then. It was really only in the late 1930's that operational research began in a systematic fashion, and it started in the UK. As such it would be interesting to give a short history of O.R. 1936 MBA-H2040 Quantitative Techniques for Managers Early in 1936 the British Air Ministry established Bawdsey Research Station, on the east coast, near Felixstowe, Suffolk, as the centre where all pre-war radar experiments for both the Air Force and the Army would be carried out. Experimental radar equipment was brought up to a high state of reliability and ranges of over 100 miles on aircraft were obtained. It was also in 1936 that Royal Air Force (RAF) Fighter Command, charged specifically with the air defense of Britain, was first created. It lacked however any effective fighter aircraft - no Hurricanes or Spitfires had come into service - and no radar data was yet fed into its very elementary warning and control system. It had become clear that radar would create a whole new series of problems in fighter direction and control so in late 1936 some experiments started at Biggin Hill in Kent into the effective use of such data. This early work, attempting to integrate radar data with ground based observer data for fighter interception, was the start of OR. 1937 The first of three major pre-war air-defence exercises was carried out in the summer of 1937. The experimental radar station at Bawdsey Research Station was brought into operation and the information derived from it was fed into the general air-defense warning and control system. From the early warning point of view this exercise was encouraging, but the tracking information obtained from radar, after filtering and transmission through the control and display network, was not very satisfactory. 1938 In July 1938 a second major air-defense exercise was carried out. Four additional radar stations had been installed along the coast and it was hoped that Britain now had an aircraft location and control system greatly improved both in coverage and effectiveness. Not so The exercise revealed, rather, that a new and serious problem had arisen. This was the need to coordinate and correlate the additional, and often conflicting, information received from the additional radar stations. With the outbreak of war apparently imminent, it was obvious that something new - drastic if necessary - had to be attempted. Some new approach was needed. Accordingly, on the termination of the exercise, the Superintendent of Bawdsey Research Station, A.P. Rowe, announced that although the exercise had again demonstrated the technical feasibility of the radar system for detecting aircraft, its operational achievements still fell far short of requirements. He therefore proposed that a crash program of research into the operational - as opposed to the technical - aspects of the system should begin immediately. The term "operational research" RESEARCH into MBA-H2040 Quantitative Techniques for Managers (military) OPERATIONS was coined as a suitable description of this new branch of applied science. The first team was selected from amongst the scientists of the radar research group the same day. 1939 In the summer of 1939 Britain held what was to be its last pre-war air defence exercise. It involved some 33,000 men, 1,300 aircraft, 110 antiaircraft guns, 700 searchlights, and 100 barrage balloons. This exercise showed a great improvement in the operation of the air defence warning and control system. The contribution made by the OR team was so apparent that the Air Officer Commander-in-Chief RAF Fighter Command (Air Chief Marshal Sir Hugh Dowding) requested that, on the outbreak of war, they should be attached to his headquarters at Stanmore in north London. Initially, they were designated the "Stanmore Research Section". In 1941 they were redesignated the "Operational Research Section" when the term was formalised and officially accepted, and similar sections set up at other RAF commands. 1940 On May 15th 1940, with German forces advancing rapidly in France, Stanmore Research Section was asked to analyses a French request for ten additional fighter squadrons (12 aircraft a squadron - so 120 aircraft in all) when losses were running at some three squadrons every two days (i.e. 36 aircraft every 2 days). They prepared graphs for Winston Churchill (the British Prime Minister of the time), based upon a study of current daily losses and replacement rates, indicating how rapidly such a move would deplete fighter strength. No aircraft were sent and most of those currently in France were recalled. This is held by some to be the most strategic contribution to the course of the war made by OR (as the aircraft and pilots saved were consequently available for the successful air defense of Britain, the Battle of Britain). 1941 onward In 1941, an Operational Research Section (ORS) was established in Coastal Command which was to carry out some of the most well-known OR work in World War II. The responsibility of Coastal Command was, to a large extent, the flying of long-range sorties by single aircraft with the object of sighting and attacking surfaced U-boats (German submarines). The technology of the time meant that (unlike modern day submarines) surfacing was necessary to recharge batteries, vent the boat of fumes and recharge air tanks. Moreover U-boats were much faster on the surface than underwater as well as being less easily detected by sonar. MBA-H2040 Quantitative Techniques for Managers Thus the Operation Research started just before World War II in Britain with the establishment of teams of scientists to study the strategic and tactical problems involved in military operations. The objective was to find the most effective utilization of limited military resources by the use of quantitative techniques. Following the end of the war OR spread, although it spread in different ways in the UK and USA. In 1951 a committee on Operations Research formed by the National Research Council of USA, and the first book on “Methods of Operations Research”, by Morse and Kimball, was published. In 1952 the Operations Research Society of America came into being. Success of Operations Research in army attracted the attention of the industrial mangers who were seeking solutions to their complex business problems. Now a days, almost every organization in all countries has staff applying operations research, and the use of operations research in government has spread from military to wide variety of departments at all levels. The growth of operations research has not limited to the U.S.A. and U.K., it has reached many countries of the world. India was one the few first countries who started using operations research. In India, Regional Research Laboratory located at Hyderabad was the first Operations Research unit established during 1949. At the same time another unit was set up in Defense Science Laboratory to solve the Stores, Purchase and Planning Problems. In 1953, Operations Research unit was established in Indian Statistical Institute, Calcutta, with the objective of using Operations Research methods in National Planning and Survey. In 1955, Operations Research Society of India was formed, which is one of the first members of International Federation of Operations Research societies. Today Operations Research is a popular subject in management institutes and schools of mathematics. 1.3 Stages of Development of Operations Research The stages of development of O.R. are also known as phases and process of O.R, which has six important steps. These six steps are arranged in the following order: Step I: Observe the problem environment Step II: Analyze and define the problem Step III: Develop a model Step IV: Select appropriate data input Step V: Provide a solution and test its reasonableness Step VI: Implement the solution MBA-H2040 Quantitative Techniques for Managers Step I: Observe the problem environment The first step in the process of O.R. development is the problem environment observation. This step includes different activities; they are conferences, site visit, research, observations etc. These activities provide sufficient information to the O.R. specialists to formulate the problem. Step II: Analyze and define the problem This step is analyzing and defining the problem. In this step in addition to the problem definition the objectives, uses and limitations of O.R. study of the problem also defined. The outputs of this step are clear grasp of need for a solution and its nature understanding. Step III: Develop a model This step develops a model; a model is a representation of some abstract or real situation. The models are basically mathematical models, which describes systems, processes in the form of equations, formula/relationships. The different activities in this step are variables definition, formulating equations etc. The model is tested in the field under different environmental constraints and modified in order to work. Some times the model is modified to satisfy the management with the results. Step IV: Select appropriate data input A model works appropriately when there is appropriate data input. Hence, selecting appropriate input data is important step in the O.R. development stage or process. The activities in this step include internal/external data analysis, fact analysis, and collection of opinions and use of computer data banks. The objective of this step is to provide sufficient data input to operate and test the model developed in Step_III. Step V: Provide a solution and test its reasonableness This step is to get a solution with the help of model and input data. This solution is not implemented immediately, instead the solution is used to test the model and to find there is any limitations. Suppose if the solution is not reasonable or the behaviour of the model is not proper, the model is updated and modified at this stage. The output of this stage is the solution(s) that supports the current organizational objectives. Step VI: Implement the solution MBA-H2040 Quantitative Techniques for Managers At this step the solution obtained from the previous step is implemented. The implementation of the solution involves mo many behavioural issues. Therefore, before implementation the implementation authority has to resolve the issues. A properly implemented solution results in quality of work and gains the support from the management. The process, process activities, and process output are summarized in the following Table 1-1. Process Activities Process Process Output Site visits, Conferences, Sufficient information and Step 1: Observations, Research support to proceed Observe the problem environment Define: Use, Objectives, Clear grasp of need for and limitations Step 2: nature of solution requested Analyze and define the problem Define interrelationships, Models that works under stated Formulate equations, environmental constraints Step 3: Develop a Model Use known O.R. Model , Search alternate Model Analyze: internal-external data, Sufficient inputs to operate and Step 4: facts test model Select appropriate data input Collect options, Use computer data banks Test the model Solution(s) that support current Step 5: organizational goals Provide a solution and find limitations test its reasonableness update the model Step 6: Implement the solution MBA-H2040 Quantitative Techniques for Managers Resolve behavioural issues Improved working and Management support for longer Sell the idea run operation of model Give explanations Management involvement Table 1-1: Process, Process activities, Process output of O.R. development stages 1.4 Relationship between the Manager and O.R. Specialist The key responsibility of manager is decision making. The role of the O.R. specialist is to help the manager make better decisions. Figure 1-1 explains the relationship between the O.R. specialist and the manager/decision maker. STEPS IN PROBLEM RECOGNITION, INVOLVEMENT: O.R. SPECIALIST or FORMULATION AND SOLUTION MANAGER Recognize from organizational Manager symptoms that a problem exists. Decide what variables are involved; state the problem in quantitative relationships Manager and O.R. Specialist among the variables. Investigate methods for solving the O.R. Specialist problems as stated above; determine appropriate quantitative tools to be used. Attempt solutions to the problems; find various solutions; state assumptions O.R. Specialist underlying these solutions; test alternative solutions. Determine which solution is most effective because of practical constraints Manager and O.R. Specialist within the organization; decide what the solution means for the organization. MBA-H2040 Quantitative Techniques for Managers Choose the solution to be used. Manager ‘Sell’ the decision to operating managers; Manager and O.R. Specialist get their understanding and cooperation. Figure 1-1 Relationship between Manager/Decision Maker and O.R. Specialists 1.5 O.R. Tools and Techniques Operations Research uses any suitable tools or techniques available. The common frequently used tools/techniques are mathematical procedures, cost analysis, electronic computation. However, operations researchers given special importance to the development and the use of techniques like linear programming, game theory, decision theory, queuing theory, inventory models and simulation. In addition to the above techniques, some other common tools are non-linear programming, integer programming, dynamic programming, sequencing theory, Markov process, network scheduling (PERT/CPM), symbolic Model, information theory, and value theory. There is many other Operations Research tools/techniques also exists. The brief explanations of some of the above techniques/tools are as follows: Linear Programming: This is a constrained optimization technique, which optimize some criterion within some constraints. In Linear programming the objective function (profit, loss or return on investment) and constraints are linear. There are different methods available to solve linear programming. Game Theory: This is used for making decisions under conflicting situations where there are one or more players/opponents. In this the motive of the players are dichotomized. The success of one player tends to be at the cost of other players and hence they are in conflict. Decision Theory: Decision theory is concerned with making decisions under conditions of complete certainty about the future outcomes and under conditions such that we can make some probability about what will happen in future. Queuing Theory: MBA-H2040 Quantitative Techniques for Managers This is used in situations where the queue is formed (for example customers waiting for service, aircrafts waiting for landing, jobs waiting for processing in the computer system, etc). The objective here is minimizing the cost of waiting without increasing the cost of servicing. Inventory Models: Inventory model make a decisions that minimize total inventory cost. This model successfully reduces the total cost of purchasing, carrying, and out of stock inventory. Simulation: Simulation is a procedure that studies a problem by creating a model of the process involved in the problem and then through a series of organized trials and error solutions attempt to determine the best solution. Some times this is a difficult/time consuming procedure. Simulation is used when actual experimentation is not feasible or solution of model is not possible. Non-linear Programming: This is used when the objective function and the constraints are not linear in nature. Linear relationships may be applied to approximate non-linear constraints but limited to some range, because approximation becomes poorer as the range is extended. Thus, the non-linear programming is used to determine the approximation in which a solution lies and then the solution is obtained using linear methods. Dynamic Programming: Dynamic programming is a method of analyzing multistage decision processes. In this each elementary decision depends on those preceding decisions and as well as external factors. Integer Programming: If one or more variables of the problem take integral values only then dynamic programming method is used. For example number or motor in an organization, number of passenger in an aircraft, number of generators in a power generating plant, etc. Markov Process: Markov process permits to predict changes over time information about the behavior of a system is known. This is used in decision making in situations where the various states are defined. The probability from one state to another state is known and depends on the current state and is independent of how we have arrived at that particular state. MBA-H2040 Quantitative Techniques for Managers Network Scheduling: This technique is used extensively to plan, schedule, and monitor large projects (for example computer system installation, R & D design, construction, maintenance, etc.). The aim of this technique is minimize trouble spots (such as delays, interruption, production bottlenecks, etc.) by identifying the critical factors. The different activities and their relationships of the entire project are represented diagrammatically with the help of networks and arrows, which is used for identifying critical activities and path. There are two main types of technique in network scheduling, they are: Program Evaluation and Review Technique (PERT) – is used when activities time is not known accurately/ only probabilistic estimate of time is available. Critical Path Method (CPM) – is used when activities time is know accurately. Information Theory: This analytical process is transferred from the electrical communication field to O.R. field. The objective of this theory is to evaluate the effectiveness of flow of information with a given system. This is used mainly in communication networks but also has indirect influence in simulating the examination of business organizational structure with a view of enhancing flow of information. 1.6 Applications of Operations Research Today, almost all fields of business and government utilizing the benefits of Operations Research. There are voluminous of applications of Operations Research. Although it is not feasible to cover all applications of O.R. in brief. The following are the abbreviated set of typical operations research applications to show how widely these techniques are used today: Accounting: Assigning audit teams effectively Credit policy analysis Cash flow planning Developing standard costs Establishing costs for byproducts Planning of delinquent account strategy Construction: Project scheduling, monitoring and control Determination of proper work force Deployment of work force Allocation of resources to projects Facilities Planning: Factory location and size decision Estimation of number of facilities required Hospital planning MBA-H2040 Quantitative Techniques for Managers International logistic system design Transportation loading and unloading Warehouse location decision Finance: Building cash management models Allocating capital among various alternatives Building financial planning models Investment analysis Portfolio analysis Dividend policy making Manufacturing: Inventory control Marketing balance projection Production scheduling Production smoothing Marketing: Advertising budget allocation Product introduction timing Selection of Product mix Deciding most effective packaging alternative Organizational Behavior / Human Resources: Personnel planning Recruitment of employees Skill balancing Training program scheduling Designing organizational structure more effectively Purchasing: Optimal buying Optimal reordering Materials transfer Research and Development: R & D Projects control R & D Budget allocation Planning of Product introduction 1.7 Limitations of Operations Research Operations Research has number of applications; similarly it also has certain limitations. These limitations are mostly related to the model building and money and time factors problems involved in its application. Some of them are as given below: i) Distance between O.R. specialist and Manager Operations Researchers job needs a mathematician or statistician, who might not be aware of the business problems. Similarly, a manager is unable to understand the complex nature of Operations Research. Thus there is a big gap between the two personnel. ii) Magnitude of Calculations MBA-H2040 Quantitative Techniques for Managers The aim of the O.R. is to find out optimal solution taking into consideration all the factors. In this modern world these factors are enormous and expressing them in quantitative model and establishing relationships among these require voluminous calculations, which can be handled only by machines. iii) Money and Time Costs The basic data are subjected to frequent changes, incorporating these changes into the operations research models is very expensive. However, a fairly good solution at present may be more desirable than a perfect operations research solution available in future or after some time. iv) Non-quantifiable Factors When all the factors related to a problem can be quantifiable only then operations research provides solution otherwise not. The non-quantifiable factors are not incorporated in O.R. models. Importantly O.R. models do not take into account emotional factors or qualitative factors. v) Implementation Once the decision has been taken it should be implemented. The implementation of decisions is a delicate task. This task must take into account the complexities of human relations and behavior and in some times only the psychological factors. 1.8 Summary Operations Research is relatively a new discipline, which originated in World War II, and became very popular throughout the world. India is one of the few first countries in the world who started using operations research. Operations Research is used successfully not only in military/army operations but also in business, government and industry. Now a day’s operations research is almost used in all the fields. Proposing a definition to the operations research is a difficult one, because its boundary and content are not fixed. The tools for operations search is provided from the subject’s viz. economics, engineering, mathematics, statistics, psychology, etc., which helps to choose possible alternative courses of action. The operations research tool/techniques include linear programming, non-linear programming, dynamic programming, integer programming, Markov process, queuing theory, etc. Operations Research has a number of applications. Similarly it has a number of limitations, which is basically related to the time, money, and the problem involves in the model building. Day-by- day operations research gaining acceptance because it improve decision making effectiveness of the managers. Almost all the areas of business use the operations research for decision making. 1.9 Key Terms MBA-H2040 Quantitative Techniques for Managers OR: Operations Research. MS: Management Science. Symbolic Model: An abstract model, generally using mathematical symbols. Criterion: is measurement, which is used to evaluation of the results. Integer Programming: is a technique, which ensures only integral values of variables in the problem. Dynamic Programming: is a technique, which is used to analyze multistage decision process. Linear Programming: is a technique, which optimizes linear objective function under limited constraints. Inventory Model: these are the models used to minimize total inventory costs. Optimization: Means maximization or minimization. 1.10 Self Assessment Questions Q1. Define Operations Research. Q2. Describe the relationship between the manager and O.R. specialist. Q3. Explain the various steps in the O.R. development process. Q4. Discuss the applications of O.R. Q5. Discuss the limitation of O.R. Q6. Describe different techniques of O.R. Q7. Discuss few areas of O.R. applications in your organization or organization you are familiar with. 1.11 Further References Hamdy A Taha, 1999. Introduction to Operations Research, PHI Limited, New Delhi. Sharma, J.K., 1989. Mathematical Models in Operations Research, Tata McGraw Hill Publishing Company Ltd., New Delhi. Beer, Stafford, 1966. Decision and Control, John Wiley & Sons, Inc., New York. Levin, Rubin, Stinson, Gardner, 1992. Quantitative Approaches to Management, Tata McGraw Hill Publishing Company Ltd. New Delhi. Wagner, Harvery M., 1975. Principles of Operations Research, PHI, Egnlewood Cliffs, N.J. MBA-H2040 Quantitative Techniques for Managers UNIT I 2 LINEAR PROGRAMMING –GRAPHICAL METHOD LESSON STRUCTURE 2.1 Introduction to Linear Programming 2.2 Linear Programming Problem Formulation 2.3 Formulation with Different Types of Constraints 2.4 Graphical Analysis of Linear Programming 2.5 Graphical Linear Programming Solution 2.6 Multiple Optimal Solutions 2.7 Unbounded Solution 2.8 Infeasible Solution 2.9 Summary 2.10 Key Terms 2.11 Self Assessment Questions 2.12 Key Solutions 2.13 Further References LESSON MBA-H2040 Quantitative Techniques for Managers Objectives After studying this lesson, you should be able to:  Formulate Linear Programming Problem  Identify the characteristics of linear programming problem  Make a graphical analysis of the linear programming problem  Solve the problem graphically  Identify the various types of solutions MBA-H2040 Quantitative Techniques for Managers 2.1 Introduction to Linear Programming Linear Programming is a special and versatile technique which can be applied to a variety of management problems viz. Advertising, Distribution, Investment, Production, Refinery Operations, and Transportation analysis. The linear programming is useful not only in industry and business but also in non-profit sectors such as Education, Government, Hospital, and Libraries. The linear programming method is applicable in problems characterized by the presence of decision variables. The objective function and the constraints can be expressed as linear functions of the decision variables. The decision variables represent quantities that are, in some sense, controllable inputs to the system being modeled. An objective function represents some principal objective criterion or goal that measures the effectiveness of the system such as maximizing profits or productivity, or minimizing cost or consumption. There is always some practical limitation on the availability of resources viz. man, material, machine, or time for the system. These constraints are expressed as linear equations involving the decision variables. Solving a linear programming problem means determining actual values of the decision variables that optimize the objective function subject to the limitation imposed by the constraints. The main important feature of linear programming model is the presence of linearity in the problem. The use of linear programming model arises in a wide variety of applications. Some model may not be strictly linear, but can be made linear by applying appropriate mathematical transformations. Still some applications are not at all linear, but can be effectively approximated by linear models. The ease with which linear programming models can usually be solved makes an attractive means of dealing with otherwise intractable nonlinear models. 2.2 Linear Programming Problem Formulation The linear programming problem formulation is illustrated through a product mix problem. The product mix problem occurs in an industry where it is possible to manufacture a variety of products. A product has a certain margin of profit per unit, and uses a common pool of limited resources. In this case the linear programming technique identifies the products combination which will maximize the profit subject to the availability of limited resource constraints. Example 2.1: Suppose an industry is manufacturing tow types of products P1 and P2. The profits per Kg of the two products are Rs.30 and Rs.40 respectively. These two products require processing in three types of machines. The following table shows the available machine hours per day and the time required on each 18MBA-H2040 Quantitative Techniques for Managers machine to produce one Kg of P1 and P2. Formulate the problem in the form of linear programming model. Profit/Kg P1 P2 Total available Machine Rs.30 Rs.40 hours/day Machine 1 3 2 600 Machine 2 3 5 800 Machine 3 5 6 1100 Solution: The procedure for linear programming problem formulation is as follows: Introduce the decision variable as follows: Let x = amount of P1 1 x = amount of P2 2 In order to maximize profits, we establish the objective function as 30x + 40x 1 2 Since one Kg of P1 requires 3 hours of processing time in machine 1 while the corresponding requirement of P2 is 2 hours. So, the first constraint can be expressed as 3x + 2x ≤ 600 1 2 Similarly, corresponding to machine 2 and 3 the constraints are 3x + 5x ≤ 800 1 2 5x + 6x ≤ 1100 1 2 In addition to the above there is no negative production, which may be represented algebraically as x ≥ 0 ; x ≥ 0 1 2 Thus, the product mix problem in the linear programming model is as follows: Maximize 30x + 40x 1 2 Subject to: 3x + 2x ≤ 600 1 2 3x + 5x ≤ 800 1 2 5x + 6x ≤ 1100 1 2 x ≥ 0, x ≥ 0 1 2 2.3 Formulation with Different Types of Constraints 19MBA-H2040 Quantitative Techniques for Managers The constraints in the previous example 2.1 are of “less than or equal to” type. In this section we are going to discuss the linear programming problem with different constraints, which is illustrated in the following Example 2.2. Example 2.2: A company owns two flour mills viz. A and B, which have different production capacities for high, medium and low quality flour. The company has entered a contract to supply flour to a firm every month with at least 8, 12 and 24 quintals of high, medium and low quality respectively. It costs the company Rs.2000 and Rs.1500 per day to run mill A and B respectively. On a day, Mill A produces 6, 2 and 4 quintals of high, medium and low quality flour, Mill B produces 2, 4 and 12 quintals of high, medium and low quality flour respectively. How many days per month should each mill be operated in order to meet the contract order most economically. Solution: Let us define x1 and x2 are the mills A and B. Here the objective is to minimize the cost of the machine runs and to satisfy the contract order. The linear programming problem is given by Minimize 2000x + 1500x 1 2 Subject to: 6x + 2x ≥ 8 1 2 2x + 4x ≥ 12 1 2 4x + 12x ≥ 24 1 2 x ≥ 0, x ≥ 0 1 2 2.4 Graphical Analysis of Linear Programming This section shows how a two-variable linear programming problem is solved graphically, which is illustrated as follows: Example 2.3: Consider the product mix problem discussed in section 2.2 Maximize 30x + 40x 1 2 Subject to: 3x + 2x ≤ 600 1 2 3x + 5x ≤ 800 1 2 5x + 6x ≤ 1100 1 2 x ≥ 0, x ≥ 0 1 2 20

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