1. a. The simulation indicates that 584 is the optimum stocking quantity. Daily profit at this stocking quantity is $331.4346. b. Using the newsvendor model, Cu = 1 – 0.2 = 0.8 and Co = .2. Cu /(Cu + Co) = .8. Using the spreadsheet, we found Q* = NORM.INV(.8,500,100) = 584.16. The simulation and newsvendor model give the same optimal stocking quantity.
2. a. According to the simulation spreadsheet, 4 hours of investment in creation maximizes daily profit at $371.33. b. Sheen would choose an effort level where the marginal benefit gained by the effort is equal to her marginal cost of expending the effort. To calculate the effort level, h, we equalize marginal cost and marginal benefit. Here (.8 * 50) / (2√h) = 10. Solving gives h = 4, or the same as the simulation. c. The optimal profit derived in this scenario is $371.33 per day, which is a $40 increase from the profit derived in problem #1, of $331.43.
3. a. Using the spreadsheet, Ralph’s optimal stocking quantity to maximize his profit is 516. b. The optimal stocking quantity differs from problem #2 because Ralph is incurring the cost of overstocking, which changes the critical ratio from .8 in problem #2 to .2. Because of the critical ratio change, Anna’s profit decreases as Ralph’s increases. This is consistent with the Newsvendor Model, which gives Cu=.2, Co=.8, for a critical ratio of .2. Using the formula in the spreadsheet, Q*=NORM.INV(.2,600,100)=515.837, gives the optimal stocking quantity of 516.
c. Assuming that we only use whole numbers for her amount of time, Anna’s optimal effort is 2 hours with a profit of $261.93, a decrease from problem #2 of 4 hours. This is because Anna is now sharing her profit. d. If you decrease the transfer price, Anna’s effort level also decreases, and Ralph will increase his stocking quantity, adding to his profit. Anna’s effort level decreases because her profit decreases when Ralph buys the newspapers for less than $0.80. When the transfer price increases, the opposite occurs; Anna’s effort level increases and there is a decrease in Ralph’s stocking quantity and profit.
4. a. The optimal stocking quantity is 409 according to the spreadsheet in the simulation, which is a decrease from 516 in problem #3 because in the event that the Express stocks out, Ralph still makes a profit from 40% of customers who will buy the Private. Therefore, because he makes more profit off of the Private, his risk decreases because of cost of understocking of the Express. b. For problems #1 and #2 there were no profitable alternatives to understocking, whereas in problem #3, Ralph has a profitable alternative for understocking since 40% of customers will buy the Private.
The different critical ratios from each problem produce a different optimal stocking quantity. c. This decreases his optimal stocking quantity because Ralph is allocating $0.03 to the cost of each newspaper, making his cost of understocking now 1-.83-40%*.4=.01. Co=.83 Critical ratio 0.01/.83= 0.012 According to the data, the optimal stocking quantity is Q*=NORMINV(.012,500,100).
5. a. A lower buy-back price means a lower stocking quantity, because it affects the cost of overstocking. Ralph wants to stock a lower quantity in order to lower his risk of overstocking. The optimal buy-back price is $0.75, which gives a stocking quantity of 659 and channel profits of $369.80. b. The optimal transfer price is $0.99, giving a buy-back price of $0.988, and channel profits of $372.62. However, this is an unrealistic scenario because Ralph’s profits are negative at -$24 and Anna is making almost the full $1 price on each sale.
The channel profit is very close to the $371.33 profit from problem #2. This is because the transfer price is almost the same as the selling price to customers of $1, eliminating Anna’s cost of under or overstocking. c. If Ralph had to pay a franchise fee, he would no longer have an incentive to understock. Anna’s effort would remain the same because the marginal benefit of her effort would not change given the additional fixed profit from Ralph’s fee.