1. Prove that the normal distribution and the exponential distribution have increasing...

1. Prove that the normal distribution and the exponential distribution have increasing generalized failure rate (IFGR).

2. As we have shown in Sect. 12.2, wholesale contracts do not coordinate the supply chain in general. Now assume that the supplier is willing to provide an all-unit quantity discount. Design an all-unit quantity discount contract coordinating the supply chain; that is, find a per-unit wholesale price w(q) as a decreasing function of the order quantity q such that the optimal ordering quantity of the retailer and the optimal production quantity of the supplier equal the optimal production quantity of the whole system.