# Question

Fancy Paints is a small paint store. Fancy Paints stocks 200 different SKUs (stock-keeping units) and places replenishment orders weekly. The order arrives one month (let's say four weeks) later. For the sake of simplicity, let's assume weekly demand for each SKU is Poisson distributed with mean 1.25. Fancy Paints maintains a 95 percent in-stock probability.

a. What is the average inventory at the store at the end of the week?

b. Now suppose Fancy Paints purchases a color-mixing machine. This machine is expensive, but instead of stocking 200 different SKU colors, it allows Fancy Paints to stock only five basic SKUs and to obtain all the other SKUs by mixing. Weekly demand for each SKU is normally distributed with mean 50 and standard deviation 8. Suppose Fancy Paints maintains a 95 percent in-stock probability for each of the five colors. How much inventory on average is at the store at the end of the week?

c. After testing the color-mixing machine for a while, the manager realizes that a 95 percent in-stock probability for each of the basic colors is not sufficient: Since mixing requires the presence of multiple mixing components, a higher in-stock probability for components is needed to maintain a 95 percent in-stock probability for the individual SKUs. The manager decides that a 98 percent in-stock probability for each of the five basic SKUs should be adequate. Suppose that each can costs $14 and 20 percent per year is charged for holding inventory (assume 50 weeks per year). What is the change in the store's holding cost relative to the original situation in which all paints are stocked individually?

a. What is the average inventory at the store at the end of the week?

b. Now suppose Fancy Paints purchases a color-mixing machine. This machine is expensive, but instead of stocking 200 different SKU colors, it allows Fancy Paints to stock only five basic SKUs and to obtain all the other SKUs by mixing. Weekly demand for each SKU is normally distributed with mean 50 and standard deviation 8. Suppose Fancy Paints maintains a 95 percent in-stock probability for each of the five colors. How much inventory on average is at the store at the end of the week?

c. After testing the color-mixing machine for a while, the manager realizes that a 95 percent in-stock probability for each of the basic colors is not sufficient: Since mixing requires the presence of multiple mixing components, a higher in-stock probability for components is needed to maintain a 95 percent in-stock probability for the individual SKUs. The manager decides that a 98 percent in-stock probability for each of the five basic SKUs should be adequate. Suppose that each can costs $14 and 20 percent per year is charged for holding inventory (assume 50 weeks per year). What is the change in the store's holding cost relative to the original situation in which all paints are stocked individually?

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