Problem 1: Consider a perishable product whose daily demand can range from a low of 31 to a high of 50. Demand is uniformly distributed over this range, so that sales of the product are equally likely to be anywhere from 31 to 50. Unsold units at the end of the day must be discarded. a) The retailer selling the product wishes to maintain an 85% service level (i.e., probability of fulfilling demand should be at least 0.85). How many units should the retailer order each day? b) What is the minimum number of additional units that the retailer should order each day if the desired service level is increased to 95%? c) If the service level is 85%, what is the average numbers of units that are unsold each day? d) If the retailer sells each unit for $10, what is the average daily revenue if the service level is 85%? e) The retailer purchases each unit at a cost of $7. If the retailer is interested in maximizing expected profit, what is the optimal order quantity? f) If the retailer were able to lower the unit purchase cost by 10%, what would be the corresponding percentage increase in profit? g) The retailer sells a second product with a similar price and purchase cost, but whose demand is uniformly distributed over the range of 36 to 45 (i.e., the mean of the two products are the same but their variances are different). If $10 is the selling price and $7 is the purchase cost, which product is more profitable (has a higher expected profit)? Compare the optimal order quantities for the two products. h) Generate the following plots using software of your choice for the two products discussed in g): i) Expected profit versus order quantity (i.e., vary from 31 to 50 and plot the corresponding expected cost) for both products on the same figure, ii) Expected revenue versus order quantity (i.e., vary Q from 31 to 50 and plot the corresponding expected cost) for both products on the same figure, iii) Optimal order quantity versus the purchase cost (i.e., vary the purchase costs from 0 to 10 and plot the corresponding optimal order quantity). Discuss the results observed