Moehl 1 Martin B. Moehl 1/26/10 IE 417 Dr. Parisay’s comments are in red. This is done professionally, though needs some corrections. Assigned Problem by Dr. Parisay The following information is provided for a Newsboy problem similar to the example in the class. Buy a paper for 30 cents, sell a paper for 55 cents, and lost sale (customer) 70c. Probability of having Six to Nine customers is 0.3, 0.2, 0.4, and 0.1, respectively. a) Create the return table (show details of some of your calculations) and use all the possible decision making methods. Indicate what will be your decision based on each method. b) Create the regret table (show details of some of your calculations) and use all the possible decision making methods. Indicate what will be your decision based on each method. c) Create the utility table for a risk indifferent (neutral) person (show details of some of your calculations) and use expected utility method. Indicate what will be your decision. d) Form a summary table listing all different methods used in parts (a), (b), and (c) with your final decision for each one. Comment on the results in this table. How and why your decision varies based on the different methods. e) Perform a sensitivity analysis and draw a graph. The x-axis is different values of lost sale (from 65c to 80c in increments of 5c) and the y-axis is the "expected return" for each possible decision (i.e. buy 7 newspaper). Notice your best decision may be different for different values of the lost sale. It is expected to use Excel (or any other spread sheet) as much as possible. For calculation of expected returns you can formulate the Excel's cells for return table so that it reads the value of lost sale from a specific cell. Then you can simply change the value of lost sale and Excel will update all the calculations. It is required to have a print out from tables prepared in Excel for each one of the values of the lost sale. Later, you can create a table for different values of the lost sale and the selected expected return. The sensitivity graph will be based on this table. Please refer to my notes on web (Notes and Transparencies for IE301 course) for how to draw this graph with Excel if you do not know how to do it. f) Write a comment on the sensitivity graph. g) Assume he has a risk-seeking personality. Draw a utility function of your choice for this personality. Create utility table and find the best decision. Compare this decision with decision of a risk-neutral person and comment on it. 1 Moehl 2 Professional Decision Making by Mr. Moehl Introduction: Dr. Parisay, I recently heard that you are opening up a Newspaper business. From what I gathered you are particularly interested in the current demand at the intersection of A Street and 1 Avenue. I heard that you are trying to determine the optimum number of newspapers to buy in order to make the greatest profit at this intersection. To save you some time, I took the liberty of gathering some historical data and making some calculations so as to aid you in this process. What follows is a summary of my work. I trust you will find this helpful. Note: The conclusions drawn in this packet do not draw a definite decision, but can be used to shed new insight and point in the right direction. Given Information: Newspaper Demand Demand Probability Purchase Cost 30 cents Sales Price 55 cents 6 7 8 9 0.3 0.2 0.4 0.1 Lost Opportunity 70 cents Calculations: By weighing the possible purchase levels with the possible sales levels and the price of a lost sale if it applies, various tables can be created. Return Table Buy 6 7 8 9 Prob. 6 150 120 90 60 0.3 Demand 7 8 80 10 175 105 145 200 115 170 0.2 0.4 9 -60 35 130 225 0.1 E(ret) 59 116.5 149 131.5 Laplace Maximin Maximax 45 -60 150 108.75 35 175 141.25 90 200 142.5 60 225 2 Moehl 3 Based on this first table different alternative can be chosen based off of different ways of calculating the optimum choice. E(ret): The Expected Return method weighs the different profit levels by the probability that the different profit possibilities will happen. The Expected Return method indicates what the return will be over time. Based on this method, Buying 8 Newspapers is best, and will yield an expected return of 149 cents. However in fact, each day you will have a return of either: 90, 145, 200, or 130 cents. This method is used if the personality is risk-neutral (risk-indifferent). Laplace: Laplace is just a fancy word for average. This method is used if we do not have probability values for possible outcomes and the personality is risk-neutral. When the profit for the different purchasing levels is averaged, the best alternative is buying 9 newspapers, and the estimated return is 142.5 cents (do not round up/down). Maximin: The Maximin method is essential the worst case scenario. It chooses the purchasing alternative that yield the largest return between the lowest profit level for each category. Based on this method the best alternative is to buy 8 newspapers and the estimated return is 90 cents. This method is used if the personality is risk-averse (pessimistic). Maximax: The Maximax method picks the best possible situation in each category weighs them against each other. Based on this method alternative 9 is best and yields a profit of 225 cents. This method is used if the personality is risk-seeking (optimistic). Regret Table Buy 6 7 8 9 Prob. 6 0 30 60 90 0.3 Demand 7 8 95 190 0 95 30 0 60 30 0.2 0.4 9 285 190 95 0 0.1 E(reg) 123.5 66 33.5 51 Minimax 285 190 95 90 The Regret is very similar to the Return table except that is focuses on how much profit is potentially lost, instead of how much is potentially gained. E(reg): Expected Regret is the amount of profit that is statistically predicted to be lost in each purchasing category. It is calculated in the same way that expected return is calculated. Based on this method the best alternative is 8 newspapers, and the expected regret is 33.5 cents. This method is used if the personality is pessimistic and you have probability values for possible outcomes. 3 Moehl 4 Minimax: The Minimax method chooses the category with the least loss between purchasing levels with the most loss. Based on this method buying 9 newspapers is the best alternative, and produces a regret of 90 cents. This method is used if the personality is pessimistic and you do not have probability values for possible outcomes. Notice “regret” is not the same as “loss”. Use right term. Utility Table for risk-neutral person Buy 6 7 8 9 Prob. Demand 7 8 0.491 0.246 0.825 0.579 0.719 0.912 0.614 0.807 0.2 0.4 6 0.737 0.632 0.526 0.421 0.3 9 0.000 0.333 0.667 1.000 0.1 E(utility) 0.418 0.619 0.733 0.672 The Utility removes the effect of dollar amounts in decision making. Each outcome is replaced by a value on a scale from 0 -1. Doing this tends to remove bias in decision making. {This sentence is wrong: This method remains very statistically accurate and useful.} The Utility values for a risk-neutral person are determined with the following equation. p (utility(max)) + ((1-p) x (utility(min))) = rij Where Utility(max) = Max Profit Utility (min) = Min Profit rij = The current return of the field to be converted p = The utility of that field (Be careful: p is not probability) The purpose of this equation is to solve for p. E(utility): Expected Utility is essentially the same as Expected Return, it just uses the utility values instead of the various profit levels. Based on this method the best alternative is to buy 8 newspapers. This produces an expected utility of 0.733, where 1 would be the best possible utility. 4 Moehl 5 It is also helpful to plot all utilities on a graph to show the relationship between the original profit points and the new utility values. This graph can also be used chose different price points based on acceptable utilities. This utility is equivalent to expected return of 149 cents. (Always use the graph to find the dollar/cent value related to the expected utility. Do not mention expected utility in report to manager.) The trend line presented on this graph can be curved up or down to accommodate different risk type. The current trend (linear) line caters to a risk neutral personality type. At this point all of the different calculations can be summarized. What follows is a summary table of all the different conclusions drawn from the various measurement methods. Summary Table Method E(ret) Laplace Maximin Maximax E(reg) Minimax E(utility) for riskneutral Suggested Number of Newspapers to Buy 8 8 8 9 8 8 Reason 8 will return the most profit over time at 149 c 8 has the highest average at 142 c 8 has the highest minimum profit at 90 c 9 has the highest maximum profit at 225 c 8 will produce the least regret over time at 34 c 98has the lowest maximum regret at 90 c 8 8 will have the highest utility over time at 0.733 5 Moehl 6 This table indicates that either 8 or 9 newspapers should be purchased. Of these two options we select the one that matches our personality. There is no right/wrong decision, only decision that matches the personality. At this point, however, it is pertinent to do a sensitivity analysis. While collecting historical data, the initial calculation for loss per sale was not concrete. It is of importance to consider how different costs per lost sale may affect the final decision. To do this all table were recalculated with following costs per lost sale. Lost Opportunity 65 c 70 c 75 c 80 c Notice that the values of lost sales for sensitivity analysis (SA) are within a logical/practical range. We do not perform SA for lost sale of 10c or lost sale of 200c. The new Expected Utilities were calculated as follows. Buy 6 7 8 9 65 c 65.5 119.5 149.5 131.5 E(return) 70 c 75 c 59.0 52.5 116.5 113.5 149.0 148.5 131.5 131.5 80 c 46.0 110.5 148.0 131.5 These values where then graphed to see which purchasing alternative seemed best under different loss levels. The following graph demonstrates this. 6 Moehl 7 As you can see this graph clearly shows that no matter what the loss per sale is, buying 8 newspapers still seems to be the wiser choice. For this reason it is safe to draw your conclusion based on the aforementioned summary table. (It just happened that we do not have a turning point of decision.) Now you have all the information to make a wise and educated decision. I trust this document was helpful in aiding your new newspaper business. g) Assume he has a risk-seeking personality. Draw a utility function of your choice for this personality. Create utility table and find the best decision. Compare this decision with decision of a risk-neutral person and comment on it. This part is by Mr. Nuno Summary Table for Utility Values: (this is a good summary table) Utility Utility Values Values Profit (Risk (Risk Values Seeking Neutral cents Personality) Personality) 0.000 0.000 -60 0.010 0.246 10 0.030 0.333 35 0.060 0.421 60 0.100 0.491 80 0.130 0.526 90 0.160 0.579 105 0.190 0.614 115 0.218 0.632 120 0.248 0.667 130 0.288 0.719 145 0.310 0.737 150 0.420 0.807 170 0.450 0.825 175 0.680 0.912 200 1.000 1.000 225 7 Moehl 8 Utility Table: Risk Seeking Personality Decision 6 7 8 9 State of demand 6 7 8 9 0.31 0.1 0.01 0 0.21 0.45 0.16 0.03 0.13 0.28 0.68 0.24 0.06 0.19 0.42 1 E(utility) 0.12 0.22 0.39 0.32 Parisay did not check if the values are obtained correctly from the graphs. I assume it is. Comment: For a risk-Seeking personality the best decision is buying 8 newspapers (highest expected utility is o.39) (It happened to be the same decision as risk-neutral just by chance.) The total profit from it is .39 Cents (this is wrong. 0.39 is utility not cent. You need to use the graph and find the related cent value. In this case it is about 170 cents. You do not need to mention the value of utility in a manager report as it does not make sense to a manager.) . It is important to understand that this personality could select to buy more or less papers during this period since this personality is not afraid of taking any chances. (This previous sentence is not right! You will miss points in exam if your explanation is wrong even if your calculation is right!) 8 Moehl 9 Notice: The decision for risk neutral and risk seeking person are the same, and just by chance, and it is fine to be the same. However, with this decision a risk neural person expects a profit of 149 cents and a risk seeking person expects a profit of 170 cents!!! That is fine, everyone can expect as much as they like! And no one knows what will happen in the future. If the next day there is only 6 customers both these personalities will have 90 cents no matter how much they expected!!! This explanation is very important!! The same discussion holds if we use utility function for decision tree. 9

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