Consider the problem of Inventory Management for a new retail refrigerator store. The store manager needs to consider at least two things: the maximum size of inventory that can be stored on site, say I, and the number of shipments of new refrigerators per year, say N, as required to keep inventory on hand. One usually assumes that each shipment of refrigerators will arrive just as the existing inventory runs out, so each shipment would have exactly 'I' refrigerators. There are costs associated to how the inventory is managed, apart from the costs of the actual refrigerators. Let's suppose that the shipping company charges a flat fee of $ 2000 per shipment (remember, we're assuming all the shipments contain 'I' refrigerators.) Another cost is storage: let's assume storage costs are $ 20 per square foot per year and that each refrigerator stored takes 5 square feet of space (remember, there are never more than 'I' refrigerators on site). Finally, let's assume that the store expects to sell 500 refrigerators each year. (This tells you that the two variable 'I' and 'N' are RELATED to one another.) The sum of all the costs associated to managing inventory is often called the Inventory Control Cost . It is usually more complicated than this example.

Question

Consider the problem of Inventory Management for a new retail refrigerator store. The store manager needs to consider at least two things: the maximum size of inventory that can be stored on site, say I, and the number of shipments of new refrigerators per year, say N, as required to keep inventory on hand. One usually assumes that each shipment of refrigerators will arrive just as the existing inventory runs out, so each shipment would have exactly 'I' refrigerators. There are costs associated to how the inventory is managed, apart from the costs of the actual refrigerators. Let's suppose that the shipping company charges a flat fee of $ 2000 per shipment (remember, we're assuming all the shipments contain 'I' refrigerators.) Another cost is storage: let's assume storage costs are $ 20 per square foot per year and that each refrigerator stored takes 5 square feet of space (remember, there are never more than 'I' refrigerators on site). Finally, let's assume that the store expects to sell 500 refrigerators each year. (This tells you that the two variable 'I' and 'N' are RELATED to one another.) The sum of all the costs associated to managing inventory is often called the Inventory Control Cost . It is usually more complicated than this example.

a) Give a formula for the annual inventory control costs, C, as a function of the maximum storage size, I

b) Give a formula for the annual inventory control costs, C, as a function of the number of shipments, N

c) Suppose the store manager chooses to have at most 25 refrigerators on site at any one time. Calculate the annual inventory control costs

d) If you look at a graph, you should be able to figure out the OPTIMUM number of shipments, the OPTIMUM storage capacity for the refrigerator store and the minimum possible annual inventory control costs.

d-a) Optimum number of shipments
d-b) Optimum storage capacity
d-c) Minimum possible inventory control costs

Answer 1

I'll use x for the size of inventory, so the upper-case I does not get confusing to read. Then we have

C(x,n) = 2000n + 20*5x
But, since n=500/x, that makes
C(x) = 1,000,000/x + 100x

Now suppose you try the other parts. Show what you tried if you get stuck.

Answer 2

a) The formula for the annual inventory control costs, C, as a function of the maximum storage size, I, can be calculated as follows:

C = (N * 2000) + (I * 5 * 20)

Explanation: The first component of the formula calculates the cost of shipments, which is the number of shipments (N) multiplied by the flat fee per shipment ($2000). The second component calculates the cost of storage, which is the maximum storage size (I) multiplied by the number of refrigerators per square foot (5) multiplied by the storage cost per square foot per year ($20).

b) The formula for the annual inventory control costs, C, as a function of the number of shipments, N, can be calculated as follows:

C = (N * 2000) + (I * 5 * 20)

Explanation: The formula remains the same as in part a) because the number of shipments doesn't directly affect the storage cost per se. The only change is the variable N, which represents the number of shipments.

c) If the store manager chooses to have at most 25 refrigerators on site at any one time (I = 25), we can calculate the annual inventory control costs using the formula from part a):

C = (N * 2000) + (25 * 5 * 20)
C = 2000N + 2500

d) To find the optimum number of shipments, optimum storage capacity, and minimum possible inventory control costs, we can examine the graph of the inventory control cost function.

d-a) Optimum number of shipments: The optimum number of shipments can be found by identifying the point on the graph that minimizes the cost. This can be determined by looking for the lowest point on the graph.

d-b) Optimum storage capacity: The optimum storage capacity can also be found by identifying the point on the graph that minimizes the cost. This can be determined by looking for the lowest point on the graph.

d-c) Minimum possible inventory control costs: The minimum possible inventory control costs can be determined by finding the lowest point on the graph, which represents the minimum cost.