Question 1. LeanAutoInc produces two models of cars: a sedan, and an SUV. Over the years, the company successfully implemented a production system that effectively achieved "make-to-order": a vehicle is produced only after a customer had placed a order, and the production facility keeps less than one day's inventory.

As a result of the global chip shortage during the pandemic and severe backlogs in the supply chain, LeanAutoInc now has to place orders with their chip supplier 6 months in advance. Each SUV requires chip A, and each sedan needs chip B. The chip supplier charges LeanAutoInc $30 for each unit of chip A, and $25 for each unit of chip B.

LeanAutoInc forecasts that the mean quarterly demand 6 months from now would be 100,000 cars for the SUV, and 80,000 cars for the sedan. The standard deviation of the demand is 15,000 and 12,000 for the SUV and the sedan, respectively. The supply chains for other parts needed by LeanAutoInc are reliable and responsive, such that chips are the only factor that can potentially limit LeanAuto's ability to meet customers' demand. Chips that are ordered but not used by the end of a quarter are worth only a half of their purchasing price due to the re-work needed to keep up with the latest technology.

(1a) Assume that LeanAutoInc makes a profit of $6000 per vehicle from selling an SUV, and $5000 per vehicle from selling a sedan (these profit margins already incorporate the cost of all parts, including the chips).

(1ai) To optimize profit, how many units of chip A and chip B would you order from the supplier, for the quarter starting 6 months from now?

(1aii) In this case, what are the stock-out probabilities for the SUV and the sedan, respectively?

(1b) LeanAutoInc cannot produce an SUV using chip B, but a sedan can be produced using chip A. After analyzing the benefits of risk-pooling, the company decides to use chip A for all SUV and sedan production.

The mean demand for chip A is now 100, 000 + 80, 000 = 180, 000. Assume that the demand for the SUV is independent to the that for the sedan, the standard deviation of the demand for chip A is ((15000)^2 + (12000)^2) = 19209.37.

To achieve the same stock-out probability as in case (a), how many A chips should LeanAutoInc order per quarter? In other words, if the stock-out probability for the SUV computed in part (a) is x, compute the stocking level for A chips such that the demand for the chip exceeds the stocking level with probability x.

(1c) Assume that the plan described in (b) is implemented. When the realized total demand for the SUV and the sedan exceeds the amount of A chips LeanAutoInc has in stock, should the company prioritize the production of the SUV or the Sedan, and why? Do you have any thoughts on further improving the company's chip strategy?

(1d) Compare the size of the chip orders placed 6 months in advance by LeanAutoInc, with the amount they would have ordered if they could order after demand is realized. If all auto makers adopt similar strategies, what would be the impact on the supply chain for chips?

Question 2. A restaurant chain negotiated a new contract with a popular food delivery platform. Instead of a fat commission percentage, the platform will charge higher commissions to restaurants for orders that keep a delivery driver waiting. As an owner of 20 franchises, you would like to better understand whether your restaurants are able to prepare the orders received from the delivery platform, before or shortly after the drivers arrive. You collected data on the most recent 16 orders from each restaurant, and recorded in the Excel sheet Q2.xlsx the time elapsed (in minutes) between receiving each order and when each order is ready to be picked-up.

	Order 1	Order 2	Order 3	Order 4	Order 5	Order 6	Order 7	Order 8	Order 9	Order 10	Order 11	Order 12	Order 13	Order 14	Order 15	Order 16
Restaurant 1	4.18	1.24	1.57	4.09	3.09	2.69	6.35	6.18	6.03	7.78	4.71	7.75	8.51	7.15	6.44	4.64
Restaurant 2	9.09	4.34	5.65	6.84	6.3	3.17	3.66	5.76	3.26	5.25	7.12	6.52	2.47	8.66	5.01	3.49
Restaurant 3	11.07	8.44	4.94	6.8	1.93	5.01	10.04	2.49	3.01	6.34	5.94	7.72	7.1	4.33	7.2	7.53
Restaurant 4	7.73	7.44	5.64	6.29	2.37	3.65	6.95	6.48	6.23	8.03	9.7	6.08	6.17	4.08	6.3	5.57
Restaurant 5	2.81	4.2	3.2	8.89	6.65	8.94	7.02	10.84	5.04	4.04	5.58	1.63	7.33	2.05	3.76	4.27
Restaurant 6	1.88	2.23	2.11	4.34	8.14	4.26	1.27	1.06	2.49	1.53	5.07	2.62	5.56	6.16	1.26	5.3
Restaurant 7	5.73	5.78	2.6	2.55	2.14	3.77	3.55	7.08	7.65	7.07	5.22	7.37	2.53	3.4	5.96	2.75
Restaurant 8	5.77	3.39	9.19	8.49	6.2	4.71	6.45	3.71	7.38	5.01	3.36	4.56	5.14	4.51	3.43	5.27
Restaurant 9	1.2	8.97	4.71	4.02	6.05	5.55	7.68	2.97	5.01	3.58	1.59	5.58	7.75	4.5	5.88	5.45
Restaurant 10	1.68	3.06	2.87	5.71	2.35	5.62	4.94	2.49	3.81	6.92	5.78	6.24	6.22	3.37	3.14	8.52
Restaurant 11	5.79	5.38	8.28	3.7	4.84	5.71	8.83	6.29	2.33	3.93	5.52	2.72	8.13	2.44	2.2	6.2
Restaurant 12	7.32	9.57	7.05	1.49	8.58	7.44	4.71	5.07	5.71	8.33	6.46	5.43	7.72	4.16	1.86	5.07
Restaurant 13	8.03	7.36	3.07	2.88	3.9	4.23	6.61	6.95	10.17	3.59	3.11	4.03	10.24	5.03	5.63	2
Restaurant 14	2.95	4.96	7.4	7.93	1.99	5.79	5.01	4.58	7.34	3.67	3.2	1.7	1.23	3	9.37	1.71
Restaurant 15	12.54	12.9	18.98	7.1	6.21	1.63	10.69	9.96	12.53	5.17	6.54	13.22	8.38	4.49	10.73	9.61
Restaurant 16	6.37	10.37	3.27	4.66	3.75	1.54	4.27	4.56	2.55	5.8	2.75	3.11	4.24	4.68	6.1	3.43
Restaurant 17	4.93	8.13	6.21	5.67	4.2	3.72	4.93	4.84	2.26	7.92	11.02	3.88	1.82	4.42	5.32	4.46
Restaurant 18	2.65	2.03	4.17	7.63	5.92	8.17	1.63	6.37	3.81	5.82	5.83	2.01	3.83	7.04	5.51	3.06
Restaurant 19	2.53	5.41	7.8	5.34	4.88	4.21	5.68	2.29	1.13	6.73	3.54	3.65	8.18	10.16	10.29	5.41
Restaurant 20	6.66	7.38	2.18	4.42	4.15	7.29	5.2	1.88	5.93	7.81	5.95	5.98	5.13	6.9	3.25	5.51

(2a) To understand whether some of your restaurants are faster or slower than the others, you generate the X-bar control chart for the preparation times. Is there any restaurant that seems to be significantly slower than the others?

(2b) What would you do to figure out if some of your restaurants are in fact consistently faster than the others?