Problem 1. Assume that demand is normally distributed with a mean μ and variance σ2. Find the form of the optimal policy for the newsvendor model in this case in terms of the standard normal cumulative distribution function. For given values of r, c and v, explain how you would find the optimal order quantity, y∗. Find the optimal order quantity if r = 15,c = 1,v = 0.2,μ = 10 and σ = 2.
Problem 2. To consider the impact of lost sales on the expected cost, consider the following, alternative cost structure. Let c and r be as defined above. In addition define two additional costs. Let h+ be the holding cost per unit of inventory remaining in stock at the end of the period. Let h− be a penalty cost per unit of lost sales. The holding cost could represent the cost of disposing of any excess inventory. The penalty cost could represent the loss of goodwill the newsvendor experiences as a result of lost sales. The newsvendor incurs a holding cost of h+(y−D) if D < y, and a penalty cost of h−(D − y) if D > y. Use this fact to construct a new cost function for the newsvendor problem, and repeat the analysis in Section 2 to find an expression for the optimal order quantity. Interpret the critical fractile in your solution in terms of overage and underage costs.