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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) = 330 -180 + 330 = 150 cents Demand P(x)=0.30.20.40.1 Decision6789E(return)LaplaceMAXIminMAXImax 61508010-605945-60150 712017510535116.5108.7535175 890145200130149141.2590200 960115170225131.5142.560225 Answer150175200225149142.590225

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E(Return): P(x)=0.30.20.40.1 Decision6789E(return) 61508010-6059 712017510535116.5 890145200130149 960115170225131.5 Answer150175200225149 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)=0.30.20.40.1 Decision6789Laplace 61508010-6045 712017510535108.75 890145200130141.25 960115170225142.5 Answer150175200225142.5 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, 142.5 cents This number is obtained by: (60+115+170+225)/4 = 142.5

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Maximin: P(x)=0.30.20.40.1 Decision6789MAXImin 61508010-60 712017510535 89014520013090 96011517022560 Answer15017520022590 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)=0.30.20.40.1 Decision6789MAXImax 61508010-60150 712017510535175 890145200130200 960115170225 Answer150175200225 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)=0.30.20.40.1 Decision6789E(return)LaplaceMAXIminMAXImax 61508010-605945-60150 712017510535116.5108.7535175 890145200130149141.2590200 960115170225131.5142.560225 Answer150175200225149142.590225 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)=0.30.20.40.1 Decision6789E(regret)MINImax 6095190285123.5285 73009519066190 8603009533.595 990603005190 Answer33.590

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Regret Table Demand Buy6789E(ret) Laplac e Maximi n Maxima x 61508010-605945-60150 712017510535116.5108.7535175 890145200130149141.2590200 960115170225131.5142.560225 Prob.0.30.20.40.1 Highest = 150 150 – 150 = 0 150 – 120 = 30 150 – 90 = 60 150 – 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)=0.30.20.40.1 Decision6789E(regret) 6095190285123.5 73009519066 8603009533.5 9906030051 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)=0.30.20.40.1 Decision6789MINImax 6095190285 730095190 86030095 9906030090 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)=0.30.20.40.1 Decision6789 61508010-60 712017510535 890145200130 960115170225 Answer150175200225 P(225)+(1-P)(-60) = 145**Solve for P 225P + 60P – 60 = 145 285P – 60 = 145 285P = 205 P =205/285 =.719 **P = Utility function value

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Utility Function Demand P(x)=0.30.20.40.1 Decision6789E(utility) 60.7370.4910.2460.0000.418 70.6320.8250.5790.3330.619 80.5260.7190.9120.6670.733 90.4210.6140.8071.0000.672 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 0.733.

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Risk Neutral Utility

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Risk-Seeking Utility

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Demand P(x)=0.30.20.40.1 Decision6789E(utility) 60.4300.2050.0600.0000.194 70.3200.5600.2800.1000.330 80.2300.4000.7000.3500.464 90.1600.3000.5301.0130.421 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 Laplace142.59 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 Laplace142.59 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 Laplace142.59 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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