Newsboy Problem

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

2. 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.

3. Given Information • Prices • Buy newspaper 30 cents • Sell newspaper 55 cents • Lost sale 70 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

4. 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

5. E(Return): • 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

6. Laplace: • 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

7. Maximin: • 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.

8. Maximax: • 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.

9. Return Table • Summary of all methods

10. Regret Method • Based on regret • Focus is more on what will be LOST than what will be gained.

11. Regret Table • Highest = 150 • 150 – 150 = 0 • 150 – 120 = 30 • 150 – 90 = 60 • 150 – 60 = 90

12. 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.

13. 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.

14. 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.

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

16. Utility Function • R32 = 145 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

17. Utility Function • 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.

18. Risk Neutral Utility

19. Risk-Seeking Utility

20. Risk-Seeking Utility • The decision outcome is the same as Risk Neutral Utility. • Expected values are all smaller.

21. Decision Summary • E(return) – Risk Neutral with historical information • Laplace – Risk Neutral without historical information • MAXImin – Risk Averse • MAXImax – Risk-Seeking

22. Decision Summary • E(regret) – Make decisions based on regret • MINImax – Chose by minimizing regret

23. Decision Summary • E(utility) – Risk Neutral, removes emotion from the equation

24. Sensitivity Analysis

25. 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.

26. Questions? ¿Preguntas? سوالات ?