位置12的问题 (Average Inventory Part B)* A warehouse orders 400 boxes of papers at a time to a manufacturer. The unit time is one week and the manufacturer takes 4 weeks to deliver. (Hint: The order quantity is given. You do not need to calculate it.)The warehouse manager finds that the weekly demand is random, and normally distributed with a standard deviation of 100. She targets a service level of 97.73%. (Hint: Demand is random in Part B.)The warehouse holds an average inventory of ___. (*Show calculations.)600625650675700单项选择题
A
600
B
625
C
650
D
675
E
700

Question

位置12的问题 (Average Inventory Part B)* A warehouse orders 400 boxes of papers at a time to a manufacturer. The unit time is one week and the manufacturer takes 4 weeks to deliver. (Hint: The order quantity is given. You do not need to calculate it.)The warehouse manager finds that the weekly demand is random, and normally distributed with a standard deviation of 100. She targets a service level of 97.73%. (Hint: Demand is random in Part B.)The warehouse holds an average inventory of ___. (*Show calculations.)600625650675700单项选择题

A

600

B

625

C

650

D

675

E

700

BlackTom AI · Accepted Answer

Average inventory consists of cycle stock (Q/2) plus safety stock (z × σ_leadtime).

Relevant knowledge:
- Cycle stock = Q/2 where Q is the fixed order quantity.
- Safety stock for random demand during lead time = z × σ_L, where σ_L is the standard deviation of demand during lead time and z is the standard normal value for the desired service level.
- For independent weekly demand with standard deviation σ_week and lead time L weeks, σ_L = σ_week × sqrt(L).

Step-by-step reasoning and calculations:
1. Given Q = 400 boxes, so cycle stock = Q/2 = 400/2 = 200 boxes.
2. Weekly demand standard deviation σ_week = 100 boxes; lead time L = 4 weeks, so
   σ_L = 100 × sqrt(4) = 100 × 2 = 200 boxes.
3. Target service level = 97.73%. For the standard normal distribution, the z-value corresponding to cumulative probability 0.9773 is z ≈ 2.0 (Φ(2.0) ≈ 0.97725).
4. Safety stock = z × σ_L = 2.0 × 200 = 400 boxes.
5. Average inventory = cycle stock + safety stock = 200 + 400 = 600 boxes.

Why the correct answer is correct:
- The safety stock calculation directly uses the service-level z-value and the demand variability over the 4-week lead time. Adding cycle stock Q/2 yields the long-run average on-hand inventory, which computes to 600 boxes.

Why the other options are incorrect:
- 625, 650, 675, 700 would require different z-values (e.g., safety stocks of 425, 450, 475, 500 respectively), none of which correspond to the given 97.73% service level (z ≈ 2.0). The only option consistent with z = 2 and the given parameters is 600.