Consider the problem of inventory Management for a store that sells frozen foods

Question

Consider the problem of inventory Management for a store that sells frozen foods. The store expects to sell 50,000 pounds of frozen foods in a year and they need to decide what size refrigeration unit to purchase. The cost of the refrigeration unit is 8000+0.4s dollars, where s is the maximum number of pounds of frozen food that the refrigeration unit will hold. It costs the same amount to run the refrigeration unit no matter how much frozen food is in it, but that cost is 6s dollars per year. We assume that the store will arrange to have new shipments of frozen food arrive just as their existing inventory runs out( this is called just in time inventory control). This means each shipment of new inventory should contain s pounds of frozen food. Let N be the number of deliveries required in a given year. The delivery costs are 300+0.06s dollars per delivery. Finally it is likely that the refrigeration unit has a useful life of only five years, so lets assume that the cost of the refrigeration unit is spread out evenly over a five-year period.

A) Give a formula for the annual inventory control costs, C, as a function of the maximum storage size, s. (This would only be valid for the first five years.)

C=______ dollars

B) To the nearest pound, what size refrigeration unit should the store purchase ?

_________ pounds.

Suppose now that the store owner is worried about power outages spoiling all his inventory. He decides to insure his inventory against loss. His insurance agent agrees to insure the total value of the inventory at any given time for 50 cents per pound per year. Calculate on the maximum number of pounds of inventory that are ever in the refrigeration unit.

C) Taking into account the insurance costs, to the nearest pound, what size refrigeration unit should the store purchase?

_______ pounds.

Remark It is unlikely that one could actually purchase a refrigeration unit to hold the exact number fo pounds of frozen foods we are calculating in these problems. But is should be clear from the analysis which refrigeration unit should choose if your choices are 100 pounds, 1500 pounds or 200 pounds .

Explanation / Answer

A>

We are given that the store intends to sell = 50000 pounds for frozen food in a yeas.

And we are given that 's' is the maximum amount of frozen food in pounds that the refrigration unit could store.

=> The total number of deliveries required to deliver 50000 bounds of frozen food per year is, N = 50000/s

We are gieven that the delivery costs are 300+0.06s dollars per delivery

=> Total delivery cost is = N*(delivery cost per delivery) =50000/s*(300+0.06s) = 15000000/s + 3000   ----> (1)

The cost of the refrigeration unit is = 8000+0.4s dollars

The life of the refrigration unit is = 5 years

and the refrigration unit cost is spread out evenly in the five years

=> The refrigration unit cost per year is = (8000+0.4s)/5 = 1600 + 0.08s -------> (2)

Cost to run the refrigration unit is = 2s   -----------> (3)

Hence the annual inventory control cost . C is = (1) + (2) + (3)

=> C(s) = 15000000/s + 3000 + 1600 + 0.08s + 2s

C(s) = 15000000/s +4600 + 2.08s dollars

B>

Here we need to find s that is the maximum amount of frozen food in pounds that the refrigration unit could store in one go.

So we would differentiate the function C(s) in order to minimize the cost C(s)

=> C '(s) = -15000000/s^2 + 2.08

find the critical number(s) by solving C '(s) = 0

=> -15000000/s^2 + 2.08 = 0

2.08s^2 = 15000000

s^2 = 15000000/2.08

s = +- 2685.43

as s cannot be negative as s represents that weight of the frozen food

=> s = 2685.43 = 2686 pounds

=> The refrigration unit should be big enough to hold 2686 pounds for frozen food in one go.

Hence the size of unit is = 2686 pounds

C>

First we need to find the total cost and to do that we just need to add 0.5s dollars to the C(s) from part (a)

We know that the maximum amount of frozen food or inventory in the unit refrigration at any time is = s pounds

and the insurance agent agrees to insure the total value of the inventory at any ginve time for = 50 cents = 0.5 dollars per pound per year

=> Total insurance cost per year = 0.5 dollars

=> The total cost is , C = (15000000/s +4600 + 2.08s) + 0.5s

or C = 15000000/s +4600 + 2.58s dollars

=> Next we find the maximum value of 's'

and to do so we would try to optimize C

=> C ' = -15000000/s^2 + 2.58

next find the critical number(s) by solving C ' = 0

=> -15000000/s^2 + 2.58 = 0

s^2 = 15000000/2.58

or s = +- 2411.214

as s cannot be negative as s represents that weight of the frozen food

=> s = 2411.214 = 2412 pounds

Hence the size of the refrigration unit should be big enough to hold 2412 pounds

Hence the size of unit is = 2412 pounds

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