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Newsboy Problem IE417 Operations Research II Cal Poly Pomona Elias Angulo Hans Carlo Domingo Ismael Reyes Jr.

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Problem Statement » Determine how many newspapers the boy should buy to maximize profit. » The possible decisions are to buy: six, seven, eight or nine newspapers.

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Given Information » Prices ˃Buy newspaper 30 cents ˃Sell newspaper 55 cents ˃Lost sale70 cents » Customer Probabilities ˃6 customers P(6)=0.30 ˃7 customers P(7)=0.20 ˃8 customers P(8)=0.40 ˃9 customers P(9)=0.10

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Return Table The first step is to find the return values for all of the given demands. Example: (-30cents*6newspapers) = -180 (55cents*6newspapers) = = 150 cents Demand P(x)= Decision6789E(return)LaplaceMAXIminMAXImax Answer

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E(Return): P(x)= Decision6789E(return) Answer Risk-neutral personality Best decision : To buy 8 Newspapers since it gives us the highest Expected Return value, 149 cents This value is obtained by:.3(90) +.2(145) +.4(200) +.1(130) =149 Note: expected value

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Laplace: P(x)= Decision6789Laplace Answer Risk-neutral personality and Probability values for possible outcomes are not provided Best decision: would be to buy 9 Newspapers since this gives us the highest return average value, cents This number is obtained by: ( )/4 = 142.5

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Maximin: P(x)= Decision6789MAXImin Answer Risk-averse personality Best decisions: would be to buy 8 Newspapers since it yields the highest value, 90 cents This value is obtained by choosing the lowest number from each row.

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Maximax: P(x)= Decision6789MAXImax Answer Risk-seeking personality Best decision: would be to buy 9 Newspapers, since it yields the highest return value, 225 cents This value is obtained by choosing the highest number from each row.

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Return Table Demand P(x)= Decision6789E(return)LaplaceMAXIminMAXImax Answer Summary of all methods

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Regret Method » Based on regret » Focus is more on what will be LOST than what will be gained. Demand P(x)= Decision6789E(regret)MINImax Answer33.590

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Regret Table Demand Buy6789E(ret) Laplac e Maximi n Maxima x Prob Highest = – 150 = – 120 = – 90 = – 60 = 90

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Regret Method » Expected Regret – E (regret) ˃Calculated the same as expected return ˃Average Value of Regret ˃This method yields 8 (buy 8 newspapers) as the best decision and the expected regret would be 33.5 cents. Demand P(x)= Decision6789E(regret) Answer33.5

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Regret Method » Minimax ˃the highest regret for each decision, and then choosing a decision based on the lowest of those high regrets. ˃the “best” among the worst case scenario ˃the best decision is to buy 9 newspapers with a maximum regret of 90 cents. Demand P(x)= Decision6789MINImax Answer90

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Utility Function » The utility function is a method used to exclude emotion and personality from decision making. » Each return value is replaced by utility function value that is within 0-1. » The lowest return value is automatically changed to 0. » The highest return value is changed to 1.

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Utility Function » Use equation to calculate Utility Function value from Return value » P(LR) + (1-P)(SR) = R ij ˃R ij = the return value on row i, column j ˃P = Utility function value of R ij ˃LR = Largest return value ˃SR = Smallest return value ***P IS NOT PROBABILITY!!!!

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Utility Function » R 32 = 145 P(x)= Decision Answer P(225)+(1-P)(-60) = 145**Solve for P 225P + 60P – 60 = P – 60 = P = 205 P =205/285 =.719 **P = Utility function value

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Utility Function Demand P(x)= Decision6789E(utility) Answer0.733 very similar to Expected return. calculated exactly the same as expected return and expected regret. the best decision is to buy 8 newspapers, and expected utility value of

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Risk Neutral Utility

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Risk-Seeking Utility

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Demand P(x)= Decision6789E(utility) Max E =0.464 » The decision outcome is the same as Risk Neutral Utility. » Expected values are all smaller.

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Decision Summary » E(return) – Risk Neutral with historical information » Laplace – Risk Neutral without historical information » MAXImin – Risk Averse » MAXImax – Risk-Seeking MethodResult Decision (Newspapers) E(return)1498 Laplace MAXImin908 MAXImax2259 E(regret)33.58 MINImax909 E(utility) risk neutral0.733 = 1498 E(utility) risk seeking0.464 = 1578

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Decision Summary » E(regret) – Make decisions based on regret » MINImax – Chose by minimizing regret MethodResultDecision (Newspapers) E(return)1498 Laplace MAXImin908 MAXImax2259 E(regret)33.58 MINImax909 E(utility) risk neutral0.733 = 1498 E(utility) risk seeking0.464 = 1578

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Decision Summary » E(utility) – Risk Neutral, removes emotion from the equation MethodResult Decision (Newspapers) E(return)1498 Laplace MAXImin908 MAXImax2259 E(regret)33.58 MINImax909 E(utility) risk neutral0.733 = 1498 E(utility) risk seeking0.464 = 1578

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Sensitivity Analysis

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Conclusion » We believe the best decision would be to buy 8 newspapers per day. » No RIGHT or WRONG answer. Decisions correspond to the decision maker’s personality.

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Questions? ¿Preguntas? سوالات ?

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