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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India DECISION THEORY 14 CHAPTER

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 Learning Objectives Decision Making Environment Decision Making Under Uncertainty Decision Making Under Risk –Decision Matrix –Expected Value of Perfect information –Utility as a decision criterion –Decision Trees –Expected Value of Sample information

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 Decisions A decision is a choice amongst alternatives Steps in Decision Making –List out possible alternatives / options available. These are called strategies. –List out future events which can occur and over which the decision maker has no control. These are called States of Nature. –Construct a payoff table. Evaluate options for the best payoff using various criteria.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Decision Making Environment Decision Making Under Conditions of Certainty. In such situations only one ‘state of nature’ exists. There is complete certainty about the future. Decision Making Under Conditions of Uncertainty. In such an environment, more than one state of nature exists but the decision maker has little knowledge about them and is unable to assign any probability for their occurrence. Decision Making Under risk. In such a situation, more than one state of nature exists, but the decision maker has sufficient knowledge and information to be able to assign probabilities to the occurrence of each state.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 6 Decision Making Under Uncertainty - Example Consider the case of a fashion designer, who is planning her fall collection. She has three options open to her with regard to the length of skirts: –Maxi skirts. –Midi skirts –Mini skirts. Future events related to demand as perceived by the fashion designer are –High demand. The fashion is accepted by the masses. –Medium demand. Only a segment of the customers accept the fashion. –Low demand. The length of the skirt is not acceptable to most people.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 7 The payoff table is given below: Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi SkirtsRs 5,00,000Rs 3,20,000- Rs 1,00,000 Midi SkirtsRs 7,50,000Rs 3,00,000- Rs 2,00,000 Mini SkirtsRs 4,00,000Rs 1,50,000- Rs 1,25,000 Decision is taken according to certain rules. The decision maker adopts a rule based on his personality and risk taking profile.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 The Maximax Criterion /Rule of Optimism / best of the best Choose the maximum payoff for each strategy and choose the maximum of the maximums. Launch Midi skirts (Maximum pay off Rs 7,50,000) - Rs 1,25,000Rs 1,50,000Rs 4,00,000Mini Skirts - Rs 2,00,000Rs 3,00,000Rs 7,50,000Midi Skirts - Rs 1,00,000Rs 3,20,000Rs 5,00,000Maxi Skirts Low DemandModerate Demand High Demand States of NatureAlternative Strategies

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 The Maximin Criterion /Rule of Pessimism / best of the worst Choose the minimum payoff for each strategy and choose the maximum of the minimums. Launch Maxi skirts (Worst pay off Rs – 1,00,000) - Rs 1,25,000Rs 1,50,000Rs 4,00,000Mini Skirts - Rs 2,00,000Rs 3,00,000Rs 7,50,000Midi Skirts - Rs 1,00,000Rs 3,20,000Rs 5,00,000Maxi Skirts Low DemandModerate Demand High Demand States of NatureAlternative Strategies

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 The Minimax Regret Criterion /Savage’s Rule Take the highest value for each state of nature and subtract other values of payoffs for that state of nature from it. Make a new table Take the maximum regret for each alternative and select the minimum of the maximums. Launch Midi skirts. - Rs 1,25,000Rs 1,50,000Rs 4,00,000Mini Skirts - Rs 2,00,000Rs 3,00,000Rs 7,50,000Midi Skirts - Rs 1,00,000Rs 3,20,000 Rs 5,00,000 Maxi Skirts Low DemandModerate Demand High Demand States of NatureAlternative Strategies Rs 2,50,000 Rs 0 Rs 3,50,000 Rs 0 Rs 20,000 Rs 1,70,000 Rs 0 Rs 1,00,000 Rs 25,000

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi SkirtsRs 5,00,000Rs 3,20,000- Rs 1,00,000 Midi SkirtsRs 7,50,000Rs 3,00,000- Rs 2,00,000 Mini SkirtsRs 4,00,000Rs 1,50,000- Rs 1,25,000 The Realism Criterion / Hurwicz’s Rule. If is 0.7 then payoff for maxi skirts = Attempts a balance between optimism and pessimism. Decision maker chooses an index of optimism between 0 and 1. For each alternative Payoff for Midi skirts Payoff for Mini skirts Launch Midi Skirts as it has maximum payoff

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi SkirtsRs 5,00,000Rs 3,20,000- Rs 1,00,000 Midi SkirtsRs 7,50,000Rs 3,00,000- Rs 2,00,000 Mini SkirtsRs 4,00,000Rs 1,50,000- Rs 1,25,000 Criterion of Insufficient Reason / Laplace’s Rule Maxi Skirts = 2,40,000; Midi Skirts = 2,83,333; Mini skirts = 1,41,667 Launch Midi skirts. Assumes that there is equal likelihood of occurrence of all states of nature. The mean value for all alternatives is calculated and the alternative yielding the maximum value is selected.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 13 Decision Rules All decisions are correct as per the rule applied. Generally a decision maker applies a decision rule consistently. A study of his/her past behaviour can help to predict the decision that he / she will take in a particular situation. This is helpful if he/she is a competitor.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 Decision Making Under Risk Assumes that the decision maker can list out the various states of nature and can also assign probabilities to the event of their occurrence. Most commonly used criterion for decision making under risk is the criterion of expected value or the Baye’s criterion. Expected payoff for each alternative is the sum of the weighted payoffs for that alternative where the weights are the probability values assigned to the occurrence of different states of nature by the decision maker.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Decision Making Under Risk Example Assume that the fashion designer has assigned probabilities to the occurrence of various states of nature. Alternative Strategies States of Nature High Demand Moderate Demand Low DemandExpected Value Probability Maxi SkirtsRs 5,00,000Rs 3,20,000- Rs 1,00,000Rs 3,86,000 Midi SkirtsRs 7,50,000Rs 3,00,000- Rs 2,00,000Rs 5,20,000 Mini SkirtsRs 4,00,000Rs 1,50,000- Rs 1,25,000Rs 2,72,500

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 Decision Making Under Risk Different probabilities of occurrence may be assigned to different states of nature for different alternatives. Alternative Strategies States of Nature High Demand Moderate Demand Low DemandExpected Value Probability Maxi SkirtsRs 5,00,000Rs 3,20,000- Rs 1,00,000Rs 3,86,000 Probability Midi SkirtsRs 7,50,000Rs 3,00,000- Rs 2,00,000Rs 3,75,000 Probability Mini SkirtsRs 4,00,000Rs 1,50,000- Rs 1,25,000Rs 1,67,500

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 17 Example Raju is a milk vendor who supplies milk in a small colony. He buys milk from a local dairy farm at Rs 10 per litre and sells it for Rs 12 per litre. Any unsold milk at the end of the day has to be thrown away as Raju does not have any refrigerating facility and the milk curdles. Raju’s demand fluctuates between 31 and 36 litres a day. Past data on demand reveals Demand in litres No of Days Probability of Demand

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 Construct a payoff table (conditional profit table). Stock Policy DemandExpected Value Probability If Raju stocks 32 litres and demand is 31 litres, he will be able to sell 31 litres and make a profit of Rs 62. But he will have to throw away 1 litre and incur a loss of Rs 10. Net profit Rs 52. Use expected value criterion. Stock 32 litres.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 19 Expected Value of Perfect Information If Raju knows the exact demand for any day he will buy only that much milk. Stock Policy DemandExpected Value Probability He can now earn an average profit of Rs Without this information his profit was Rs The value of perfect information is Rs 3.90.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 20 Minimising Expected Loss Obsolescence Loss – Loss due to over stocking by way of having to throw away surplus items, or sell them at a discount. Opportunity Loss – Loss due to understocking and thus losing an opportunity to make more profit.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 21 Loss table for Raju Stock Policy DemandExpected Value Probability If Raju stocks 32 litres and demand is 31 litres, he has to throw away 1 litre of milk and loses Rs 10. If demand is 33 litres, he loses Rs 2 of profit that he would have made if he had stocked 33 litres. Take least expected value of loss. He should stock 32 litres

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 22 Marginal Analysis Let p be the probability of selling an additional unit Then 1 – p is the probability that the unit will not be sold. Let MP be the profit of selling one additional unit (marginal profit) and ML be the loss if the unit is not sold (marginal loss). Expected profit is p(MP) and expected loss is (1 – p)(ML). It would be worthwhile stocking an additional unit as long as there is no loss; or

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 23 Applying this to Raju’s problem As long as the probability of selling an additional unit is or more, an additional unit of milk can be stocked. DemandProbability that demand will be Probability that demand will be equal to or greater than Raju should stock 32 litres as the probability of selling more is less than 0.833

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 24 Utility as a Decision Criterion Money is not entirely an objective measure of value. Money does not mean the same thing to everybody. One may argue that Rs 50 should have the same value for every one, but a pay raise of Rs 50 per month means little to someone earning Rs per month but means a great deal to another person who may be earning a meagre Rs 1000 per month. Each of us assign a particular value to money. The utility that we attach to it, that determines its value for us. The value varies according to the risk taking profile of the decision maker.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 25 Risk Taking Profile RISK AVOIDER RISK TAKER For the risk taker, gains mean much more than losses. If his losses increased from – Rs 8000 to – Rs the utility would change from -9.5 to -10, whereas a gain from Rs 800 to Rs would increase his utility from 6 to 10. To the risk avoider losses are much more important in terms of utility than gains. For instance if his losses increased from – Rs 8000 to – Rs 10000, the utility would reduce from -6 to -10 as can be seen from the above diagram. If his gains increased from Rs 8000 to Rs his utility would only increase from 9.5 to 10.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 26 Decision Trees The decision tree is a means of representing the sequential, multi-stage logic of a decision problem. It uses two symbols – a box to represent a decision node and a circle to represent a chance node. The outcomes emanating from chance nodes are the various states of nature.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 27 Decision Tree - Example A company is introducing a new product. It may set up a commercial plant now or set up a pilot plant at present and set up a commercial plant later depending on the performance of the product. Present cost of setting up the pilot plant will be Rs 2 lakhs and the cost of setting up a commercial plant will be Rs 21 lakhs. If the commercial plant is set up three months later it will cost Rs 25 lakhs. The probability of the product giving a high yield during the pilot stage is 0.9 and that of giving a low yield is 0.1. If the product is introduced commercially without going through a pilot plant stage, it is likely to give a high yield of profits with a probability of 0.7. If the pilot plant shows a high yield, then the probability that the commercial plant will also give a high yield is 0.85; but if the pilot plant gives a low yield the probability that the commercial plant will give a high yield is only 0.1. The estimated profits from high yield at the commercial stage are Rs lakhs and if the yield is low the company will suffer a loss of Rs lakhs.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page Build Commercial Plant Build Com Plant Stop Low 0.3 High 0.85 High 0.7 Low 0.15 High 0.1 Low 0.9 Low 0.1 High Build Pilot Plant Value at Node 2 Cost of plant Rs 21 Lakhs Net Value Rs lakhs Value at Node 6 Cost of plant Rs 25 Lakhs Net Value Rs lakhs Value at Node 7 Cost of plant Rs 25 Lakhs Net Value Rs lakhs Value at Node 4 Higher of the two branch values Value at Node 5 Higher of the two branch values 0 Value at Node 3 Cost of pilot plant Rs 2 Lakhs Net Value Rs lakhs Value at Node 1 Higher of the two branch values Decisions Build a pilot plant If yield is high, build main plant else stop.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 29 Expected Value of Sample Information The building of the pilot plant has provided us with additional information If commercial production of the product is started without going through the pilot plant stage, the expected value would be Rs lakhs. The information provided by the pilot plant has increased this value to Rs lakhs. The cost of the pilot plant is Rs 2 lakhs. Net increase in value is – = 8.99 lakhs. If the cost of the pilot plant is more than Rs 8.99 lakhs then it is not worth while to construct a pilot plant. The expected cost of sample information is Rs 8.99 lakhs.

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