National Electronics distributes a wide variety of electronic products. National has been experiencing cash flow problems; hence it is concerned about its investment in inventories.

A management consultant, Mr. P. C. Snow, has been hired to suggest ordering decision rules. As a preliminary step, he selects for study an important group of 10 products all bought from the same supplier, Harolds Corp. Based on an analysis of ordering costs, Mr.

Snow estimates that the replenishment cost, A, in dollars, is given by A = 46 + 5.5(V − 1,000)/1,000 (10.18)

where V is the total dollar value of the order and a minimum order size of $1,000 is required by Harolds. The National Electronics Controller, Mr. P. J. Schmeidler, tells Snow that the company uses an r value of 0.24 $/$/year. The sales forecast for each of the next 12 months, as well as the unit values, are shown in Exhibit 1. Snow ascertains that the requirements for any particular month must be on hand at the beginning of the month; hence he can restrict his attention to replenishments covering an integer number of months. In addition, at least initially, he decides to consider a deterministic version of the problem, namely, where lead times are known and the forecasts represent actual demands. Snow observes that because a marked seasonality exists, a simple EOQ strategy is inappropriate.

He considers two possible types of decision rules: P-96 EXHIBIT 1 Forecast (Units)

2016 2017 Unit Value Item vi $/Unit S O N D J F M A M J J A 170.40-.-

4 5 5 5 5 3 2 3 2 2 2 2 2 8.40 30 30 40 40 40 30 30 20 10 10 10 10 3 49.00 15 15 15 15 15 15 11 11 9 9 10 12 4 51.40 1 1 2 2 2 1 1 1 1 1 1 1 5 15.70 24 30 30 30 30 18 12 18 12 12 12 12 6 12.40 30 30 30 30 30 20 20 20 15 15 15 15 7 5.30 31 31 35 35 35 21 21 21 15 15 10 10 8 10.10 36 50 50 50 45 25 25 18 18 15 15 15 9 18.10 33 33 44 44 44 33 33 22 22 11 11 11 10 110.50 10 10 10 10 10 10 8 8 8 6 6 6 1. Order the same fixed time supply (e.g., 2 months coverage) of all items.

2. Use a modified version of the Silver–Meal heuristic (Section 5.6 of Chapter 5), a modification is necessary because of the nature of the setup cost A which can be shared among two or more items involved in the same order.

Snow argues that any reasonable decision rule would require a group replenishment to arrive at the beginning of September, the start of the peak demand period. Hence, he assumes 0 inventories of all the items just prior to the beginning of September 2016.

a. For the fixed time supply (T months) strategy, he evaluates the order quantities, replenishment costs, and carrying costs for each of T = 1, 2, and 3 months. Develop these quantities and select what appears to be the best T value.

b. In an effort to appropriately modify the Silver–Meal heuristic, Snow reasons as follows.
Suppose that a particular order involves n items and has a total value of V . Then, the setup cost A is given by Equation 10.18. Hence, an approximate apportioned cost for each item involved in the order is 46 + 5.5(V − 1,000)/1,000 n The problem is that V depends on the order quantities, which, in turn, should depend upon the setup cost. This suggests the use of an iterative procedure. Historically, the average total value of an order involving n items has been approximately $600n. Thus he starts with the Silver–Meal heuristic for each individual item using an A value of A = 46 + 5.5(60V − 1,000)/1,000 n = 3.30 + 40.5/n (10.19)
The actual total value of the resulting group replenishment is determined from the individual order quantities. This implies a new value of A, the heuristic is reused, a new V is obtained, etc. until convergence takes place. Illustrate in detail these computations for the first replenishment (at the start of September), which must include all 10 items. Once a stable set of order quantities is established for a particular group replenishment, Snow moves on to the next time where at least one item requires another replenishment. The above iterative procedure involving A and V is used for that group of items. Complete, in this fashion, the 12 months of the modified heuristic computations. Determine the total replenishment and carrying costs and compare with your answer in part a.

c. Rather than iterating as above, suppose instead that for each group replenishment the individual item A was established according to Equation