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Decision theory
- 1. Decision Theory 14MBA14 Module3 Dr. Surekha Prabhu Professor& Head Dept of Management Studies, K.V.G. College of Engineering, Sullia
- 2. Contents Fundamentals of decisiontheory Decision environment Decision making under uncertainty Decision making under risk Decision tree Experimental design-basics References Operations Research: Theory and Applications by J K Sharma http://www.r-tutor.com/elementary-statistics
- 3. Decision Theory representsa general approach to decision making which is suitable for a wide range of management decisions, including: product and service design equipment selection location planning Decision Theory Capacity planning Product –mix Credit policies
- 4. 4 Problem Formulation • Adecision problem is characterized by decision alternatives, states of nature, and resulting payoffs. • The decision alternatives are the different possible strategies the decision maker can employ. • The states of nature refer to future events, not under the control of the decision maker, which will ultimately affect decision results. • States of nature should be defined so that they are mutually exclusive and contain all possible future events that could affect the results of all potential decisions.
- 5. 5 Payoff Tables • Theconsequence resulting from a specific combination of a decision alternative and a state of nature is a payoff. • A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table. • Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.
- 6. Fundamentals of decisiontheory Decision alternatives States of nature Payoff Courses of action or strategies An occurrence over which decision maker has no control Quantitative measure of the outcome
- 7. Three-egg omelette Problem •You are preparing a three-egg omelette. Having already broken two good eggs into the pan, you are suddenly assailed by doubts about the quality of the third. As yet unbroken.egg, two things may happen: either the egg is good or it is rotten.
- 8. Egg omelette problem EventsProbability Break 3rd egg into pan Break 3rd egg into saucer & inspect Throw away 3rd egg Third egg is good 0.9 3-egg omlette 3-egg omelette, one saucer to wash 2-egg omelette, one good egg destroyed Third egg is rotten 0.1 No egg omlette 2 good eggs destroyed 2-egg omelette, one saucer to wash 2-egg omelette States of nature Acts (strategies) Pay-offs
- 9. • Certainty -Environment in which relevant parameters have known values • Risk - Environment in which certain future events have probable outcomes • Uncertainty - Environment in which it is impossible to assess the likelihood of various future events Decision Environments
- 10. Risk vs. Uncertainty •Risk – Must make a decision for which the outcome is not known with certainty – Can list all possible outcomes & assign probabilities to the outcomes • Uncertainty – Cannot list all possible outcomes – Cannot assign probabilities to the outcomes • Certainty -is an environment in which future outcomes or state of nature are known. • Eg: Investment in Bank FD, there is CERTAINTY regarding FUTURE PAYMENTS on maturity • Investment in shares is risky • Investment in shares FETCHING returns higher than FD in another 2 years, is uncertain
- 11. Criteria of decisionmaking under uncertainity 0ptimism(Maximax or Minimin) Pessimism(Maximin or Minimax) Equal probabilities(Laplace) Coefficient of optimism(Hurwicz) Regret(Salvage)
- 12. Optimism (Maximax orMinimin criterion) • Choose the alternative with the best possible payoff • Locate the maximum or minimum payoff values corresponding to each alternatives • The maximax is 7000 ,hence the company should adopt strategy S1 Strategies States of nature N1 N2 N3 ROW MAXIMUM S1 7000 3000 1500 7000 S2 5000 4500 0 5000 S3 3000 3000 3000 3000
- 13. Pessimism Maximin orMinimax criterion • Choose the alternative with the best of the worst possible payoffs • Locate the minimum payoff values corresponding to each alternatives • The act/decision with higher minimum value is 3000 ,hence the company should adopt S3 criterion strategy S3 Strategies States of nature N1 N2 N3 Row Minimum S1 7000 3000 1500 3000 S2 5000 4500 0 0 S3 3000 3000 3000 3000
- 14. Laplace criterion(Equal probabilities) •Under this assumption ,all states of nature are equally likely. • decision maker can compute the average payoff for each row (the sum of the possible consequences of each alternative is divided by the number of states of nature) and, then, select the alternative that has the highest row average
- 15. LAPLACE CRITERION Strategies States ofnature N1 N2 N3 ROW MAXIMUM S1 7000 3000 1500 3,833.33 S2 5000 4500 0 3166.66 S3 3000 3000 3000 3000 The largest expected return is from Strategy S1, THE EXECUTIVE MUST SELECT S1
- 16. Coefficient of optimism(Hurwicz) •This criterion represents a compromise between the optimistic and the pessimistic approach to decision making under uncertainty. • For each alternative select the largest &lowest payoff values and multiply these with α and (1- α) values respectively. • Then calculate the weighted average using the formula: H Coefficient of optimism = α (maximum in column)+ (1-α)(minimum in column) • Select the best answer
- 17. Hurwicz Criterion Strategy Maximum pay-off Minimum pay-off H S17000 1500 4800 S2 5000 0 3000 S3 3000 3000 3000 Assuming degree of optimisim α = 0.6 and (1- α )=0.4 H Coefficient of optimism = α (maximum in column) + (1-α)(minimum in column) The maximum value is 4800, adopt S1
- 18. Regret (Salvage rule) •This rule represents a pessimistic approach. • The opportunity loss reflects the difference between each payoff and the best possible payoff in a column (it can be defined as the amount of profit foregone by not choosing the best alternative for each state of nature). • For each course of action identify the maximum regret value, record this no in a row • Select the course of action with Smallest anticipated opportunity loss value
- 19. Strategies States of nature N1N2 N3 Row max S1 7000 – 7000 = 0 4500-3000= 1500 3000-1500=1500 1500 S2 7000- 5000 = 2000 4500-4500=0 3000-0=3000 3000 S3 7000-3000 = 4000 4500-3000= 1500 3000-3000=0 4000 Col max 7000 4500 3000 The company should adopt minimum opportunity loss strategy S1 Strategies States of nature N1 N2 N3 S1 7000 3000 1500 S2 5000 4500 0 S3 3000 3000 3000 Column max 7000 4500 3000
- 20. Decision table andtree 1 Outcome 1 outcome2 2 Outcome 3 outcome 4 States of nature Strategies State 1 State 2 Strategy 1 Outcome 1 Outcome2 Strategy 2 Outcome 3 Outcome 4
- 21. Decision tree • Decisiontree is a network which exhibits graphically the relationship between the different parts of the complex decision process. • It is a graphical model of each combination of various acts and states of nature along with their payoffs, probability distribution • It is extremely useful in multistage situations which involve a number of decisions ,each depending on the preceding one. • A decision tree analysis involves the construction of a diagram that shows , at a glance, when decisions are expected to be made- in what sequence, their possible outcomes, & corresponding payoffs.
- 22. • A DTconsists of nodes, branches, probability estimates and pay-offs • Three types of “nodes” – Decision nodes - represented by squares (□) It represents a point of action where a decision maker must select one alternative course of action among the available – Chance nodes - represented by circles (Ο) It indicates a point of time where the decision maker will discover the response to his decision – Terminal nodes - represented by triangles (optional) • Solving the tree involves pruning all but the best decisions at decision nodes, and finding expected values of all possible states of nature at chance nodes • Create the tree from left to right • Solve the tree from right to left
- 23. Decision tree example Staycomfortable and dry Bear unnecessary trouble of carrying umbrella Get wet and uncomfortable Remain dry and comfortable
- 24. 24 Elements of DecisionTheory • States of nature: The states of nature could be defined as low demand and high demand. • Alternatives: VGK could decide to build a small, medium, or large Flour processing mill . • Payoffs: The profit for each alternative under each potential state of nature is going to be determined. We develop different models for this problem on the following slides.
- 25. 25 VGK Flour mill: Payoff Table Alternatives Low High Small 8 8 Medium 5 15 Large -11 22 States of Nature (Profits in LAKHS of Rs ) THIS IS A PROFIT PAYOFF TABLE
- 26. 1 n i i i E( X) Expected value of X p X Decision making under risk Expected monetary value Where Xi is the ith outcome of a decision, pi is the probability of the ith outcome, and n is the total number of possible outcomes Each possible state of nature has an assumed probability pi States of nature are mutually exclusive Probabilities must sum to 1 Determine the expected monetary value (EMV) for each alternative
- 27. EMV Example 1. EMV(A1)= (.5)($200,000) + (.5)(-$180,000) = $10,000 2. EMV(A2) = (.5)($100,000) + (.5)(-$20,000) = $40,000 3. EMV(A3) = (.5)($0) + (.5)($0) = $0 States of Nature Favorable Unfavorable Alternatives Market Market Construct large plant (A1) $200,000 -$180,000 Construct small plant (A2) $100,000 -$20,000 Do nothing (A3) $0 $0 Probabilities .50 .50 Best Option
- 28. Expected Opportunity Loss(EOL) • It is the opposite of EMV • EOL is defined as the difference between the highest profit or pay-off and the actual profit due to choosing a particular course of action in a particular state of nature • The conditional opportunity loss (EOL) for a particular course of action is determined by taking the difference between the payoff value of the most favourable course of action and some other course of action.
- 30. act Cold weatherWarm weather Sell cold drinks 60-40= 20 90-90 =0 Sell ice cream 60-60 = 0 90-40 =50 Sell cold drinks 0.3 x 20 + 0.7 x 0 = 6 Sell ice cream 0.3 x 0 + 0.7 x 50 = 35 Opportunity loss matrix EOL for each alternative course of action is computed as below Since EOL is minimum in case of selling cold drinks ,this is the best act Sell cold drinks 0.3 x 40 + 0.7 x 90 = 75 Sell ice cream 0.3 x 60 + 0.7 x 40 = 46 Since EMV is more for selling cold drinks, it is recommended
- 31. Expected Value ofPerfect Information Expected value of perfect information: the difference between the expected payoff under certainty and the expected payoff under risk Expected value of perfect information Expected payoff under certainty Expected payoff under risk = - EVPI is defined as the maximum amount one would pay to obtain perfect information about the state of nature that would occur.
- 32. EXPERIMENTAL DESIGN Experiment somemanagement examples Sales Productivity Experiment Will an increase in the average number of sales calls per customer from six to eight per year significantly improve sales? Shelf Space Experiment Will decreasing the shelf space allocated to brand X detergent by 25 percent significantly lower its sales? Direct Mail Experiment Will it be worthwhile to mail last year's donors an attractive (but expensive) brochure describing the company’s activities and soliciting higher contributions for this year? Pricing Experiment Can a company improve the profitability of its fashion clothing line by increasing its price by 10 percent?
- 33. Experiment • An experimentis a procedure in which a company manipulates one (or sometimes more than one) independent or cause variable and collects data on the dependent or effect variable while controlling for other variables that may influence the dependent variable
- 34. Variables are measures thatchange . The independent variable is the variable that is purposely changed. It is the manipulated variable. The dependent variable changes in response to the independent variable. It is the responding variable. Independent variables increase in the average number of sales calls per customer decreasing the shelf space allocated to brand X detergent mail last year's donors an attractive brochure Dependent variables sales sales higher contributions
- 35. Constants &control inan Experiment What are constants in an experiment? Factors that are kept the same and not allowed to change What is a control? The part of the experiment that serves as the standard of comparison. It is the unchanged part of the experiment that detects the effects of hidden variables Levels of the Independent Variable How many different levels of the independent variable should we test?
- 36. Writing A Statementof the Problem for the Experiment It should state: “The Effect of the Independent Variable on the Dependent Variable”. What should it state?
- 37. An Introduction toExperimental Design A factor is a variable that the experimenter has selected for investigation (the independent variable). A treatment is a level of a factor. Experimental units are the objects of interest in the experiment. A completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units.
- 38. Fast food Franchisemust decide which menu item to market. Are the three Menu items equally effective? A Completely Randomized Experimental Design Factor . . . New menu Treatments . . . Item 1, Item 2 , Item 3 Experimental units . . . Franchise restaurants Response variable . . . sales volume A fast food franchise is test marketing 3 new menu items. To find out if they the same popularity, 18 franchisee restaurants are randomly chosen for participation in the study. In accordance with the completely randomized design, 6 of the restaurants are randomly chosen to test market the first new menu item, another 6 for the second menu item, and the remaining 6 for the last menu item.
- 39. Item1 Item2 Item3 2252 16 42 33 24 44 8 19 52 47 18 45 43 34 37 32 39 Suppose the following table represents the sales figures of the 3 new menu items in the 18 restaurants after a week of test marketing. Purpose is to…. test whether the mean sales volume for the 3 new menu items are all equal.
- 40. Randomized Block Design •In a randomized block design, there is only one primary factor under consideration in the experiment. Similar test subjects are grouped into blocks. Each block is tested against all treatment levels of the primary factor at random order. This is intended to eliminate possible influence by other extraneous factors.
- 41. Item1 Item2 Item3 R131 27 24 R2 31 28 31 R3 45 29 46 R4 21 18 48 R5 42 36 46 R6 32 17 40 A fast food franchise is test marketing 3 new menu items. To find out if they have the same popularity, 6 franchisee restaurants are randomly chosen for participation. For the randomized block design, each restaurant will be test marketing all 3 new menu items. A restaurant will test market only one menu item per week, and it takes 3 weeks to test market all menu items. The testing order of the menu items for each restaurant is randomly assigned Purpose is to…. test whether the mean sales volume for the 3 new menu items are all equal.
- 42. Factorial Experiments Insome experiments we want to draw conclusions about more than one variable or factor. Factorial experiments and their corresponding results are valuable designs when simultaneous conclusions about two or more factors are required. For example, for a levels of factor A and b levels of factor B, the experiment will involve collecting data on ab treatment combinations. The term factorial is used because the experimental conditions include all possible combinations of the factors.
- 43. Factorial Design • Afast food franchise is test marketing 3 new menu items in North Indian & South Indian States. To find out if they the same popularity, 12 franchisee restaurants from each part are randomly chosen for participation in the study. In accordance with the factorial design, within the 12 restaurants from north India, 4 are randomly chosen to test market the first new menu item, another 4 for the second menu item, and the remaining 4 for the last menu item. The 12 restaurants from North India are arranged likewise.
- 44. Following tables representthe sales figures of the 3 new menu items after a week of test marketing • Item1 Item2 Item3 N1 25 39 36 N2 36 42 24 N3 31 39 28 N4 26 35 29 Item1 Item2 Item3 S1 51 43 42 S2 47 39 36 S3 47 53 32 S4 52 46 33 Purpose is to………. test whether the mean sales volume for the new menu items are all equal Decide also whether the mean sales volume of the two regions differs.