When using the stochastic continuous-review model presented in Sec. 19.5, a difficult managerial judgment decision needs to be made on the level of service to provide to customers. The purpose of this problem is to enable you to explore the trade-off involved in making this decision. Assume that the measure of service level being used is L = probability that a stockout will not occur during the lead time. Since management generally places a high priority on providing excellent service to customers, the temptation is to assign a very high value to L. However, this would result in providing a very large amount of safety stock, which runs counter to management's desire to eliminate unnecessary inventory. (Remember the just-in time philosophy discussed in Sec. 19.3 that is heavily influencing managerial thinking today) What is the best trade-off between providing good service and eliminating unnecessary inventory? Assume that the probability distribution of demand during the lead time is a normal distribution with mean μ and standard deviation σ Then the reorder point R is R = μ + k_1-L σ, where K_1L is obtained from Appendix 5. The amount of safety stock provided by this reorder point is K_1L σ Thus, if h denotes the holding cost for each unit held in inventory per year, the average annual holding cost for safety stock (denoted by C) is C = hK_1-L σ

(a) Construct a table with five columns. The first column is the service level L, with values 0.5, 0.75, 0.9, 0.95, 0.99, and 0.999. The next four columns give C for four cases. Case 1 is h = $1 and σ - 1. Case 2 is h - $100 and σ - 1. Case 3 is h - $1.and σ = 100. Case 4 is h = $100 and σ = 100.

(b) Construct a second table that is based on the table obtained in part (a). The new table has five rows and the same five columns as the first table. Each entry in the new table is obtained by subtracting the corresponding entry in the first table from the entry in the next row of the first table. For example, the entries in the first column of the new table are 0.75 0.5 = 0.25, 0.9  0.75 = 0.15, 0.95  0.9 = 0.05, 0.99  0.95 = 0.04, and 0.999  0.99 = 0.009. Since these entries represent increases in the service level L, each entry in the next four columns represents the increase in C that would result from increasing L by the amount shown in the first column.

(c) Based on these two tables, what advice would you give a manager who needs to make a decision on the value of L to use?