Bstat11 - Lessons in Business Statistics Prepared By P.K....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lessons in Business Statistics Prepared By P.K. Viswanathan Chapter 11: Decision Analysis Introduction Managers must be capable of making decisions under conditions of uncertainty. They must also be capable of using information, which may be inadequate for decisionmaking. The question that must be asked before making the final decision is, "is it worth gathering additional information?" This Chapter provides a framework for decision making with minimum risk under uncertain environment. 1) Systematic Problem Solving-A Conceptual Framework Many decisions will have to be made very often with a little bit of incomplete data in an environment of uncertainty. It is in this context that the role of systematic problem solving looms large. Schematic Diagram-Problem Solving 2) How to Structure a Decision Problem A decision problem involves a scientific and systematic problem solving approach in which a quantitative analysis using monetary value becomes a basis for decision-making. 2) How to Structure a Decision Problem-Continues A decision problem is characterized by a set of alternatives states of nature, and the consequences expressed in monetary value. The alternatives pave the way for possible strategies the decision maker can implement. The states of nature refer to events that are not under the control of the decision maker. States of nature should be so defined that they become mutually exclusive and collectively exhaustive. For each alternative and state of nature, there is a resulting monetary value. These are displayed in matrix form called a payoff table. Example A small fruit merchant has got a problem on hand. He has to decide how many dozens of a particular type of fruit to stock on a given day. Total demand per day is uncertain. He has analyzed the past data and found the following pattern of demand distribution based on 360 days. Total demand Number of Probability of per day days each demand (based (In dozens) demand level on relative was recorded frequency) 25 30 35 40 72 90 108 90 0.20 0.25 0.30 0.25 Fruits not sold on any day perish and have to be thrown out. Selling price of the fruit per dozen is $ 30. Cost of procurement and other incidentals add to $ 20 per dozen. How many dozens per day should the merchant stock? Pay Off Matrix($ Value) Demand (States of Nature) 25(0.20) 30(0.25) 35(0.30) 40(0.25) If you decide to stock (Decision Alternatives) 25 30 35 40 250 150 50 -50 250 300 200 100 250 300 350 250 250 300 350 400 3) Expected Monetary value If probabilities regarding the states of nature are available, one can use the expected monetary value (EMV) criterion to select the best alternative. EMV for each decision is computed by summing the products of the pay off of each state of nature and the probability of the respective state of nature taking place. The alternative giving the highest expected monetary value (EMV) is selected as the best decision. EMV for the Example Demand (States of Nature) 25(0.20) 30(0.25) 35(0.30) 40(0.25) EMV If you decide to stock (Decision Alternatives) 30 35 40 25 250 150 50 -50 250 300 200 100 250 300 350 250 250 300 350 400 250 270 252.5 190 EMV Optimum Expected Value of Perfect Information(EVPI) Very often in a decision problem, any manager of an enterprise has to answer the question “whether to take action now, or gather additional information and act later. Is the cost of additional information justified in terms of the additional profit you may get? This issue can be resolved using EVPI. EVPI is calculated as the difference between two factors- expected profit with perfect information (EPPI) and EMV optimum. Pay Off Matrix with Perfect Information for the Example Demand (States of Nature) 25(0.20) 30(0.25) 35(0.30) 40(0.25) (Best Decision Alternatives) Stock 30 35 25 40 250 300 350 400 EVPI for the Example Now use probabilities of state of nature as weights and compute the expected value as before. You get EPPI. EPPI = (250)(0.20)+(300)(0.25)+(350)(0.30)+(400)(0.25) =330. Hence EVPI = EPPI-EMV optimum = 330-270 = $60. EVPI acts as an upper bound measuring worth of gathering additional information. Opportunity Loss Table If the pay off matrix is expressed in terms of opportunity loss for every consequence, it is called a loss table. Opportunity loss is based on the concept of opportunity cost that measures the cost of sacrificing one alternative against another. Opportunity Loss Table for the Example(figs in $) Demand (States of Nature) 25(0.20) 30(0.25) 35(0.30) 40(0.25) EOL If you decide to stock (Decision alternatives) 25 30 35 40 0 50 100 150 100 0 50 100 200 100 0 50 300 200 100 0 $80 $60 $77.5 $140 Please note that the optimum decision is to stock 30 dozens because it gives the minimum EOL($60). Also note that Minimum EOL =EVPI! 4) Decision Tree A decision tree is a graphical representation of a decision problem. Each decision tree has two types of nodes; circles represent the states of nature and squares represent the decision alternatives. The branches emanating from each circle denote different states of nature. The branches emanating from squares denote the different decision alternatives. At the end of each extremity of a tree, the payoffs from all the branches are displayed. Decision Tree for the Example Explanation on the Decision Tree for the Example Branches emanating from the square represent decision alternatives of stocking 25, 30, 35, and 40 dozens. For each decision alternative, there are four states of nature (demand pattern) with associated probabilities. The branches that emanate from the circle represent the states of nature of the uncertain demand that could be 25, 30, 35, or 40 dozens. The figures with in brackets for each demand represent the probability of demand occurring. The pay offs for each decision alternative is given at the extremity of the tree for every branch of states of nature. Explanation on the Decision Tree for the Example For every decision option, working backwards, expected monetary value (EMV) is calculated as the sum product of two terms namely state of nature (demand) and probability of state of nature (demand). The pay offs for each decision alternative is given at the extremity of the tree for every branch of states of nature. EMV optimum = $270 because this is the highest you can get. So, stock 30 dozens is the optimal solution to the problem. 5) Value of Sample Information This is calculated using expected value of sample information (EVSI). EVSI intuitively and logically must be = Expected profit with sample information (EPSI)- EMV optimum. In order to get EPSI, we have to revise the probabilities of the states of nature, based on sample information we hope to get. Value of Sample Information- Example The Marketing manager of a company producing consumer durable wants to decide whether to launch a new product or not. The sales quantity of the new product has two states of nature namely high sales or low sales. The pay off table is as follows (Figures in $ Million). States of Probability Launch Nature High Sales 0.4 5 Do not Launch 0 Low Sales 0.6 0 -4 Value of Sample Information- Example A market research agency is willing to do a study that will cost $0.3million. There are two possibilities. The agency may predict high sales for the new product or it may predict low sales for the new product. Past record of the agency in terms of predicting ability is tabulated below: States of Nature Prior Agency Predicts Agency Predicts Probability High Sales given Low Sales given States of Nature States of Nature High Sales 0.4 Low Sales 0.6 0.6 0.3 0.4 0.7 Value of Sample Information- Example Questions: a) Should the product be launched based on EMV criterion? b) Should the marketing manager take the help of the agency or not? Solution to the Example Problem a) The product should not be launched. EMV for launching = (5)(0.4)+(-4)(0.6) = $-0.4ml. Since this is less than $ 0(EMV optimum) that you get if product is not launched, the product should not be launched. Solution to the Example Problem b) In order to solve this problem, we need to work out the revised probabilities of the states of nature from prior probabilities of the states nature. This means we want to obtain posterior probabilities of high sales and low sales given agency prediction. This statisticians call as inverse probability approach based on Bayes’ theorem. You can easily get this by constructing joint probability table given in the next slide Solution to the Example Problem Joint Probability Table States of Agency Nature Predicts High Sales Agency Predicts Low Sales High Sales 0.24 0.16 Marginal Probability of States of Nature 0.40 Low Sales 0.18 0.42 0.60 Marginal Probability of Agency Prediction 0.42 0.58 1.00 Solution to the Example Problem Now the calculation of revised probabilities is very simple. Probability of high sales given agency predicts high sales = 0.24/0.42 = 0.57. Probability of low sales given agency predicts high sales =0.18/0.42 =0.43. Extending this logic leads to the following revised probability table. Revised Posterior Probability Table Posterior Probability of Given Agency Predicts High Sales Low Sales High Sales 0.57 0.28 Low Sales 0.43 0.72 Decision Tree for the Example Explanation on the Tree The manager has two options in the first stage of decision namely “go for agency” or “no agency”. This appears in the first square node. If you don’t go for agency, then you have two decision optionslaunch the new product or do not launch the new product. These two options are shown in the next square node. If you launch, states of nature (demand) may be high sales or low sales with prior probabilities 0.4 and 0.6 respectively. Monetary values corresponding to these two states of nature are given as 5 and –4 at the extreme end of the branch. These are in $million. If you don’t launch, you get $0 million. EMV is calculated in the usual way for these two options. They are –0.4 and 0 respectively. So, EMV optimum = 0. This appears in the square node corresponding to no agency. Explanation on the Tree If you decide to go for agency, there are two states of nature. Agency predicts high sales with a probability of 0.42 and low sales with a probability of 0.58. These probabilities are from the revised posterior probability table. If agency predicts high sales, you have again got two-decision options-launch or no launch. Under each of these options, the states of nature could be high sales or low sales. However, the probabilities of these states of nature are posterior probabilities. These are given in brackets. Please refer again revised posterior probability. Explanation on the Tree EMV values are worked out as before by multiplying probabilities of states of nature with corresponding pay offs and then summing them. EMVs are displayed in circle and square nodes in the diagram. Please also note that while working out EMV values, cost of doing the study by agency ($0.3million) is taken out to represent the correct net monetary value. Under the scenario of taking agency help, if you choose not to launch, you still incur cost of $0.3million(-0.3 EMV) When you finally fold back the tree, you find going for agency produces a positive EMV of $0.17 million. This is better than getting $0 million of no agency. Hence go for agency. Final Conclusions for this Example based on the Decision tree EVSI = EPSI-EMV optimum = 0.17-0 =$0.17 million. This is the gain you have got by using sample information. Take the decision of going to the agency. If agency predicts high sales, then launch the new product (EMV $0.83 million) If agency predicts low sales, then do not launch the new product (EMV $-0.3 million) Expected value of sample information (EVSI) is $0.17 million. ...
View Full Document

Ask a homework question - tutors are online