Problem $1.$ Average inventory. An assumption we will be making on the sales of our

hypothetical product is that they occur at a constant rate, and that we will not reorder until

the quantity in inventory goes to zero. Suppose, for instance, that we order a quantity of

$500$ units four times in a given year $($once every three months$).$ The following figure is what

an inventory graph for a three$-$month period $($an order interval$)$ would look like.

$123$

$100$

$200$

$300$

$400$

$500$

Time $($in months$)$

Quantity $($inventory$)$

$($a$)$ Why is it reasonable to regard the dashed line in the figure as representing the average

inventory level for the three$-$month period? $($Many answers are possible, i$.$e$.,$ there

is no one correct answer. The point is that defining the average of a continuously

varying $($even linear$)$ quantity is not such an easy thing to do$.)$

$($b$)$ If the quantity per order is Q$,$ then the average inventory level is given by $.$

$($c$)$ Does this average pertain only to one order interval or for the entire year? Explain.

Problem $2.$ Carrying cost. If C is the carrying cost per unit per year, use your answer

from Problem $1$ to determine an expression estimating the total carrying cost per year in

terms of C and Q:

total carrying cost $=.$

$2$Problem $1.$ Average inventory. An assumption we will be making on the sales of our

hypothetical product is that they occur at a constant rate, and that we will not reorder until

the quantity in inventory goes to zero. Suppose, for instance, that we order a quantity of

$500$ units four times in a given year $($once every three months$).$ The following figure is what

an inventory graph for a three$-$month period $($an order interval$)$ would look like.

$123$

$100$

$200$

$300$

$400$

$500$

Time $($in months$)$

Quantity $($inventory$)$

$($a$)$ Why is it reasonable to regard the dashed line in the figure as representing the average

inventory level for the three$-$month period? $($Many answers are possible, i$.$e$.,$ there

is no one correct answer. The point is that defining the average of a continuously

varying $($even linear$)$ quantity is not such an easy thing to do$.)$

$($b$)$ If the quantity per order is Q$,$ then the average inventory level is given by $.$

$($c$)$ Does this average pertain only to one order interval or for the entire year? Explain.

Problem $2.$ Carrying cost. If C is the carrying cost per unit per year, use your answer

from Problem $1$ to determine an expression estimating the total carrying cost per year in

terms of C and Q:

total carrying cost $=.$

$2$Problem $1.$ Average inventory. An assumption we will be making on the sales of our

hypothetical product is that they occur at a constant rate, and that we will not reorder until

the quantity in inventory goes to zero. Suppose, for instance, that we order a quantity of

$500$ units four times in a given year $($once every three months$).$ The following figure is what

an inventory graph for a three$-$month period $($an order interval$)$ would look like.

$123$

$100$

$200$

$300$

$400$

$500$

Time $($in months$)$

Quantity $($inventory$)$

$($a$)$ Why is it reasonable to regard the dashed line in the figure as representing the average

inventory level for the three$-$month period? $($Many answers are possible, i$.$e$.,$ there

is no one correct answer. The point is that defining the average of a continuously

varying $($even linear$)$ quantity is not such an easy thing to do$.)$

$($b$)$ If the quantity per order is Q$,$ then the average inventory level is given by $.$

$($c$)$ Does this average pertain only to one order interval or for the entire year? Explain.

Problem $2.$ Carrying cost. If C is the carrying cost per unit per year, use your answer

from Problem $1$ to determine an expression estimating the total carrying cost per year in

terms of C and Q:

total carrying cost $=.$

$2$