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MBA SEMESTER II MB0032 –Operation Research- 3 Credits Assignment Set- 1 (60 Marks) Q1. What are the essential characteristics of Operation Research? Mention different phases in an Operation Research study. Point out some limitations of O.R? Ans. Characteristics of Operations Research Operations research, an interdisciplinary division of mathematics and science, uses statistics, algorithms and mathematical modeling techniques to solve complex problems for the best possible solutions. This science is basically concerned with optimizing maxima and minima of the objective functions involved. Examples of maxima could be profit, performance and yield. Minima could be loss and risk. The management of various companies has benefited immensely from operations research. Operations research is also known as OR. It has basic characteristics such as systems orientation, using interdisciplinary groups, applying scientific methodology, providing quantitative answers, revelation of newer problems and the consideration of human factors in relation to the state under which research is being conducted. Systems Orientation o This approach recognizes the fact that the behavior of any part of the system has an effect on the system as a whole. This stresses the idea that the interaction between parts of the system is what determines the functioning of the system. No single part of the system can have a bearing effect on the whole. OR attempts appraise the effect the changes of any single part would have on the performance of the system as a whole. It then searches for the causes of the problem that has arisen either in one part of the system or in the interrelation parts. Interdisciplinary groups o The team performing the operational research is drawn from different disciplines. The disciplines could include mathematics, psychology, statistics, physics, economics and engineering. The knowledge of all the people involved aids the research and preparation of the scientific model. Application of Scientific Methodology o OR extensively uses scientific means and methods to solve problems. Most OR studies cannot be conducted in laboratories, and the findings cannot be applied to natural environments. Therefore, scientific and mathematical models are used for studies. Simulation of these models is carried out, and the findings are then studied with respect to the real environment. New Problems Revealed o Finding a solution to a problem in OR uncovers additional problems. To obtain maximum benefits from the study, ongoing and continuous research is necessary. New problems must be pursued immediately to be resolved. A company looking to reduce costs in manufacturing might discover in the process that it needs to buy one more component to manufacture the end product. Such a scenario would result in unexpected costs and budget overruns. Ensuring flexibility for such contingencies is a key characteristic of OR. Provides Quantitative Answers o The solutions found by using operations research are always quantitative. OR considers two or more options and emphasizes the best one. The company must decide which option is the best alternative for it. Human Factors o In other forms of quantitative research, human factors are not considered, but in OR, human factors are a prime consideration. People involved in the process may become sick, which would affect the company’s output. PHASES OPERATIONS RESEARCH · Formulate the problem: This is the most important process, it is generally lengthy and time consuming. The activities that constitute this step are visits, observations, research, etc. With the help of such activities, the O.R. scientist gets sufficient information and support to proceed and is better prepared to formulate the problem. This process starts with understanding of the organizational climate, its objectives and expectations. Further, the alternative courses of action are discovered in this step. Develop a model: Once a problem is formulated, the next step is to express the problem into a mathematical model that represents systems, processes or environment in the form of equations, relationships or formulas. We have to identify both the static and dynamic structural elements, and device mathematical formulas to represent the interrelationships among elements. The proposed model may be field tested and modified in order to work under stated environmental constraints. A model may also be modified if the management is not satisfied with the answer that it gives. Select appropriate data input: Garbage in and garbage out is a famous saying. No model will work appropriately if data input is not appropriate. The purpose of this step is to have sufficient input to operate and test the model. Solution of the model: After selecting the appropriate data input, the next step is to find a solution. If the model is not behaving properly, then updating and modification is considered at this stage. Validation of the model: A model is said to be valid if it can provide a reliable prediction of the system’s performance. A model must be applicable for a longer time and can be updated from time to time taking into consideration the past, present and future aspects of the problem. Implement the solution: The implementation of the solution involves so many behavioural issues and the implementing authority is responsible for resolving these issues. The gap between one who provides a solution and one who wishes to use it should be eliminated. To achieve this, O.R. scientist as well as management should play a positive role. A properly implemented solution obtained through O.R. techniques results in improved working and wins the management support. Limitations Dependence on an Electronic Computer: O.R. techniques try to find out an optimal solution taking into account all the factors. In the modern society, these factors are enormous and expressing them in quantity and establishing relationships among these require voluminous calculations that can only be handled by computers. Non-Quantifiable Factors: O.R. techniques provide a solution only when all the elements related to a problem can be quantified. All relevant variables do not lend themselves to quantification. Factors that cannot be quantified find no place in O.R. models. Distance between Manager and Operations Researcher: O.R. being specialist’s job requires a mathematician or a statistician, who might not be aware of the business problems. Similarly, a manager fails to understand the complex working of O.R. Thus, there is a gap between the two. Money and Time Costs: When the basic data are subjected to frequent changes, incorporating them into the O.R. models is a costly affair. Moreover, a fairly good solution at present may be more desirable than a perfect O.R. solution available after sometime. Implementation: Implementation of decisions is a delicate task. It must take into account the complexities of human relations and behaviour. Q2. What are the common methods to obtain an initial basic feasible solution for a transportation problem whose cost and requirement table is given? Give a stepwise procedure for one of them? Ans. Transportation Problem & its basic assumption This model studies the minimization of the cost of transporting a commodity from a number of sources to several destinations. The supply at each source and the demand at each destination are known. The transportation problem involves m sources, each of which has available. i (i = 1, 2, …..,m) units of homogeneous product and n destinations, each of which requires bj (j = 1, 2…., n) units of products. Here a i and bj are positive integers. The cost cij of transporting one unit of the product from the ith source to the jth destination is given for each i and j . The objective is to develop an integral transportation schedule that meets all demands from the inventory at a minimum total transportation cost.It is assumed that the total supply and the total demand are equal.i.e. Condition (1)The condition (1) is guaranteed by creating either a fictitious destination with a demand equal to the surplus if total demand is less than the total supply or a (dummy) source with a supply equal to the shortage if total demand exceeds total supply. The cost of transportation from the fictitious destination to all sources and from all destinations to the fictitious sources are assumed to be zero so that total cost of transportation will remain the same. Formulation of Transportation Problem The standard mathematical model for the transportation problem is as follows. Let xij be number of units of the homogenous product to be transported from source i to the destination j Then objective is to Theorem: A necessary and sufficient condition for the existence of a feasible solution to the transportation problem (2) is that Q3. a. What are the properties of a game? Explain the “best strategy” on the basis of minmax criterion of optimality. b. State the assumptions underlying game theory. Discuss its importance to business decisions. Ans. a) Minimax (sometimes minmax) is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss while maximizing the potential gain. Alternatively, it can be thought of as maximizing the minimum gain (maximin). Originally formulated for two-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision making in the presence of uncertainty. Game theory In the theory of simultaneous games, a minimax strategy is a mixed strategy which is part of the solution to a zero-sum game. In zero-sum games, the minimax solution is the same as the Nash equilibrium. Minimax theorem The minimax theorem states: For every two-person, zero-sum game with finitely many strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2′s strategy, the best payoff possible for player 1 is V, and (b) Given player 1′s strategy, the best payoff possible for player 2 is −V. Equivalently, Player 1′s strategy guarantees him a payoff of V regardless of Player 2′s strategy, and similarly Player 2 can guarantee himself a payoff of −V. The name minimax arises because each player minimizes the maximum payoff possible for the other—since the game is zero-sum, he also maximizes his own minimum payoff. This theorem was established by John von Neumann,[1] who is quoted as saying “As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved”.[2] See Sion’s minimax theorem and Parthasarathy’s theorem for generalizations; see also example of a game without a value. Example The following example of a zero-sum B chooses B1 B chooses B2 B chooses B3 game, where A and B make A chooses A1 +3 −2 +2 simultaneous moves, illustrates A chooses A2 −1 0 +4 minimax solutions. Suppose each A chooses A3 −4 −3 +1 player has three choices and consider the payoff matrix for A displayed at right. Assume the payoff matrix for B is the same matrix with the signs reversed (i.e. if the choices are A1 and B1 then B pays 3 to A). Then, the minimax choice for A is A2 since the worst possible result is then having to pay 1, while the simple minimax choice for B is B2 since the worst possible result is then no payment. However, this solution is not stable, since if B believes A will choose A2 then B will choose B1 to gain 1; then if A believes B will choose B1 then A will choose A1 to gain 3; and then B will choose B2; and eventually both players will realize the difficulty of making a choice. So a more stable strategy is needed. Some choices are dominated by others and can be eliminated: A will not choose A3 since either A1 or A2 will produce a better result, no matter what B chooses; B will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what A chooses. A can avoid having to make an expected payment of more than 1/3 by choosing A1 with probability 1/6 and A2 with probability 5/6, no matter what B chooses. B can ensure an expected gain of at least 1/3 by using a randomized strategy of choosing B1 with probability 1/3 and B2 with probability 2/3, no matter what A chooses. These mixed minimax strategies are now stable and cannot be improved. b) Brandenburger and Nalebuff discuss how game theory works and how companies can use the principles to make decisions. The authors state that managers can use the principles to create new strategies for competing where the chances for success are much higher than they would be if they continued to compete under the same rules. A classic example used in the article is the case of General Motors. The automobile industry was facing many expenses due to the incentives that were being used at the retailers. General Motors responded by issuing a new credit card where the cardholders could apply a portion of their charges towards purchasing a GM car. GM even went so far as to allow cardholders to use a smaller portion of their charges towards purchasing a Ford car, allowing both companies to be able to raise their prices and increase long term profits. This action by GM created a new system where both GM and Ford could be better off, unlike the traditional competitive model where one company must profit at the expense of another. The authors state that while the traditional win-lose strategy may sometimes be appropriate, but that the win-win system can be ideal in many circumstances. One advantage to win-win strategies is that since they have not been used much, they can yield many previously unidentified opportunities. Another major advantage is that since other companies have the opportunity to come out ahead as well, they are less likely to show resistance. The last advantage is that when other companies imitate the move the initial company benefits as well, in contrast to the initial company losing ground as they would in a win-lose situation. The authors also state that there are five elements to competition that can be changed to provide a more optimal outcome. These elements are: the players (or companies competing), added values brought by each competitor, the rules under which competition takes place, the tactics used, and the scope or boundaries that are established. By understanding these factors, companies can apply different strategies to increase their own odds of success. The first way that companies can increase their chances of success involves changing who the companies are that are involved in the business. One way that companies can improve their odds of success is by introducing new companies into the business. For example, both Coke and Pepsi wanted to get a contract to have Monsanto as a supplier. Since Monsanto had a monopoly at the time, they encouraged Holland Sweetener Company to compete with Monsanto. Since it seemed Monsanto no longer had a monopoly on the market, they were able to get more favorable contracts with Monsanto. Another way that companies can improve their chances is by helping other companies introduce more or better complimentary products. Companies can also change the added values of themselves or their competitors. Obviously, companies can build a better brand or change their business practices so they operate more efficiently. However, the authors discuss how they can also lower the value of reducing the value of other companies as a viable strategy. Nintendo reduced the added value of retailers by not filling all of their orders, thus leaving a shortage and reducing the bargaining power of the stores buying its products. They also limited the number of licenses available to aspiring programmers, lowering their added value. They even lowered the value held by comic book characters when they developed characters of their own that became widely popular, presumably so that they wouldn’t have to pay as much to license these characters. Changing the rules is another way in which companies can benefit. The authors introduce the idea of judo economics, where a large company may be willing to allow a smaller company to capture a small market share rather than compete by lowering its prices. As long as it does not become too powerful or greedy, a small company can often participate in the same market without having to compete with larger companies on unfavorable terms. Kiwi International Air Lines introduced services on its carriers that were of lower prices to get market share, but made sure that the competitors understood that they had no intention of capturing more than 10% of any market. Companies can also change perceptions to make themselves better off. This can be accomplished either by making things clearer or more uncertain. In 1994, the New York Post attempted to make radical price changes in order to get the Daily News to raise its price to regain subscribers. However, the Daily News misunderstood and both newspapers were headed for a price war. The New York Post had to make its intentions clear, and both papers were able to raise their prices and not lose revenue. The authors also show an example of how investment banks can maintain ambiguity to benefit themselves. If the client is more optimistic than the investment bank, the bank can try to charge a higher commission as long as the client does not develop a more realistic appraisal of the company’s value. Finally, companies can change the boundaries within which they compete. For example, when Sega was unable to gain market share from Nintendo’s 8-bit systems, it changed the game by introducing a new 16-bit system. It took Nintendo 2 years to respond with its own 16-bit system, which gave Sega the opportunity to capture market share and build a strong brand image. This example shows how companies can think outside the box to change the way competition takes place in their industry. Brandenburger and Nalebuff have illustrated how companies that recognize they can change the rules of competition can vastly improve their odds of success, and sometimes respond in a way that benefits both themselves and the competition. If companies are able to develop a system where they can make both themselves and their competitors better off, then they do not have to worry so much about their competitors trying to counter their moves. Also, because companies can easily copy each other’s ideas, it is to a firm’s advantage if they can benefit when their competitors copy their idea, which is not usually possible under the traditional win-lose structure. This article has some parallels with the article “Competing on Analytics” by (). The biggest factor that both of these articles have in common is how crucial it is for managers to understand everything they can about their business and the environment in which they work. In “Competing on Analytics”, the authors say that it is important to be familiar with this information so that managers can change the way they compete to improve their chances of success. At the end of “The Right Game: Use Game Theory to Shape Strategy”, the authors discuss how in order for companies to be able to change the environment or rules under which they compete they need to understand everything they can about the constructs under which they are competing. Whether a manager intends to use analytics or game theory to be successful, he or she must first have all available information and use that information to understand how to make the company better off. However, the work shown in “Competing on Analytics” tends to place an emphasis almost exclusively on the use of quantitative data to improve efficiency or market share of the company. “The Right Game”, however focuses more on using information to find creative ways of changing the constructs or rules applied between companies, often yielding a much broader impact. Q4. a. Compare CPM and PERT explaining similarities and mentioning where they mainly differ. Ans. The Major Differences and Similarities between CPM and PERT CPM (Critical Path Method) & PERT(Program Evaluation and Review Technique) 1)PERT is a probabilistic tool used with three 1)CPM is a deterministic tool, with only single Estimating the duration for completion of estimate of duration. 2)This tool is basically a tool for planning 2)CPM also allows and explicit estimate of and control of time. costs in addition to time, therefore CPM can control both time and cost. 3)PERT is more suitable for R&D related 3)CPM is best suited for routine and those projects where the project is performed for projects where time and cost estimates can the first time and the estimate of duration be accurately calculated are uncertain. 4)The probability factor i major in PERT 4)The deterministic factor is more so values or so outcomes may not be exact. outcomes are generally accurate and realistic. Extensions of both PERT and CPM allow the user to manage other resources in addition to time and money, to trade off resources, to analyze different types of schedules, and to balance the use of resources. Tensions of both PERT and CPM allow the user to manage other resources in addition to time and money, to trade off resources, to analyze different types of schedules, and to balance the use of resources. Graphs _ In mathematics, networks are called graphs, the entities are nodes, and the links are edges _ Graph theory starts in the 18th century, with Leonhard Euler _ The problem of Königsberg bridges _ Since then graphs have been studied extensively. Graph Theory _ Graph G=(V,E) _ V = set of vertices _ E = set of edges 2 _ An edge is defined by the two vertices which it connects _ optionally: 1 3 A direction and/or a weight _ Two vertices are adjacent if they are connected by an edge 4 5 _ A vertex’s degree is the number of its edges Graph G=(V,E) 2 V = set of vertices E = set of edges Each edge is now an 1 3 arrow, not just a line -> direction The indegree of a vertex is the number of 5 incoming edges 4 The outdegree of a vertex is the number of outgoing edges