Question

Jill, the office manager of a desktop publishing outfit, stocks replacement toner cartridges for laser printers. Demand for cartridges is approximated 30 per year and follows Poisson distribution. Cartridges cost $100 each and require 3 weeks to obtain from the vendor. Jill uses a (Q,r) approach to control stock levels. (Hint: you find the values on page 26 of LN 3 helpful in solving this problem) a. If Jill wants to restrict replenishment orders to tice per year on average, what batch size Q should she use? Using this batch size, what reorder point r should she use to ensure a service level (i.e., probability of having the cartridge in stock when needed) of at least 98 percent? b. If Jill is willing to increase the number of replenishment orders per year to six, how do Q and r change? Explain the difference in r. c. If the supplier of toner cartridge offers a quantity discount of $10 per cartridge for orders of 15 or more, how does this affect the relative attractiveness of ordering twice per year versus six

Answer 1

To determine the optimal batch size (Q) and reorder point (r) for controlling stock levels of toner cartridges, Jill can use the (Q,r) approach. In this approach, the batch size is equal to the average demand during the lead time for replenishment orders.

Step-by-step explanation:

To determine the optimal batch size (Q) and reorder point (r) for controlling stock levels of toner cartridges, Jill can use the (Q,r) approach. In this approach, the batch size is equal to the average demand during the lead time for replenishment orders. Since Jill wants to restrict replenishment orders to twice per year on average, she needs to calculate the average demand during the lead time and set Q accordingly. The average demand during the lead time can be calculated by multiplying the average demand per year (30 cartridges) by the time it takes to obtain a replenishment order (3 weeks). So, the average demand during the lead time would be 30 cartridges/year * (3 weeks/52 weeks) = 1.73 cartridges.

To ensure a service level of at least 98 percent, Jill needs to determine the reorder point (r). The reorder point is the number of cartridges at which Jill should place a replenishment order. To calculate the reorder point, Jill can use the table on page 26 of LN 3. According to the table, for a service level of 98 percent, the number of standard deviations (Z value) is 2.06. The formula to calculate the reorder point is r = d * L + Z * √(d * L), where d is the average demand during the lead time and L is the lead time. Plugging in the values, r = 1.73 * 1 + 2.06 * √(1.73 * 1) = 3.57 cartridges. Rounded to the nearest whole number, Jill should set the reorder point to 4 cartridges.

If Jill is willing to increase the number of replenishment orders per year to six, the batch size (Q) and reorder point (r) would change. The new batch size would be the average demand per year (30 cartridges) divided by the number of replenishment orders (6), which is 5 cartridges. The new reorder point would be calculated using the same formula as before, but with the new average demand during the lead time and lead time. The new average demand during the lead time would be 30 cartridges/year * (3 weeks/52 weeks) = 1.73 cartridges. Substituting the values, r = 1.73 * 1 + 2.06 * √(1.73 * 1) = 3.57 cartridges. Rounded to the nearest whole number, Jill should still set the reorder point to 4 cartridges, as it remains the same.

If the supplier offers a quantity discount of $10 per cartridge for orders of 15 or more, it may affect the relative attractiveness of ordering twice per year versus six. With the quantity discount, the cost per cartridge would be reduced by $10, making it more cost-effective to order larger batches. This means that ordering six times per year would become more attractive, as Jill would be able to take advantage of the quantity discount more frequently. However, it is important to consider other factors such as storage space and potential obsolescence of cartridges when deciding on the optimal ordering strategy.