A supplier needs to supply 10 retailers (indexed as i = 1,..., 10). In each day,...

 

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A supplier needs to supply 10 retailers (indexed as i = 1,..., 10). In each day, each retailer i has probability pi = 0.7 to place an order (to the supplier), with order quantity D; following a discrete uniform distribution U{1,, 10}. Retailers' ordering decisions are independent of each other. The supplier's unit production cost is $1K per unit. In each day, the supplier first produces x = 50 units of the product at unit cost $1K; then it receives all the orders from the retailers and satisfies them as much as possible. The wholesale price is $10K per unit, and each unit of leftover costs the supplier $1K to dispose. Let V(x) be the supplier's expected daily profit under production quantity x. Assume sample size N = 365. 1. What is the distribution (histogram) of the daily total order quantities received by the supplier? Plot it. 2. What is the 90% confidence interval of the expected daily profit for the supplier? 3. What is the probability that the supplier experiences shortage? What is the probability that the supplier has leftover? 4. What is the quantity * that maximizes the supplier's expected profit? Graph V(x) against x to support your conclusion. 5. When the retailers consolidate their orders, they can get the discounted wholesale price $7K. They coordinate their ordering process as follows: in each day, each supplier i first observes their individual demand Di~ U{1,..., 10}. Then if the consolidated demand Di 35, all orders will be placed; otherwise, no order will be placed. The supplier's operation and other parameters remain the same. In this case, what is the profit-maximizing production x* for the supplier? A supplier needs to supply 10 retailers (indexed as i = 1,..., 10). In each day, each retailer i has probability pi = 0.7 to place an order (to the supplier), with order quantity D; following a discrete uniform distribution U{1,, 10}. Retailers' ordering decisions are independent of each other. The supplier's unit production cost is $1K per unit. In each day, the supplier first produces x = 50 units of the product at unit cost $1K; then it receives all the orders from the retailers and satisfies them as much as possible. The wholesale price is $10K per unit, and each unit of leftover costs the supplier $1K to dispose. Let V(x) be the supplier's expected daily profit under production quantity x. Assume sample size N = 365. 1. What is the distribution (histogram) of the daily total order quantities received by the supplier? Plot it. 2. What is the 90% confidence interval of the expected daily profit for the supplier? 3. What is the probability that the supplier experiences shortage? What is the probability that the supplier has leftover? 4. What is the quantity * that maximizes the supplier's expected profit? Graph V(x) against x to support your conclusion. 5. When the retailers consolidate their orders, they can get the discounted wholesale price $7K. They coordinate their ordering process as follows: in each day, each supplier i first observes their individual demand Di~ U{1,..., 10}. Then if the consolidated demand Di 35, all orders will be placed; otherwise, no order will be placed. The supplier's operation and other parameters remain the same. In this case, what is the profit-maximizing production x* for the supplier?