help: basic eoq for perishable items
Posted: Mon Oct 08, 2018 1:35 am
can anyone assist in this?
Consider the following modification of the basic EOQ model. When the inventory is at positive level X(t),
in addition to demand draining inventory at rate gamma, the inventory perishes (due to e.g. items expiring or
becoming defective) at infitesimal rate
X(t)dt. Thus the total rate at which the inventory decreases (at
time t) is lamda+gamma.
X(t). Note that the infitesimal rate at which the inventory decreases depends on the current
inventory level. Suppose that, there are parameters k, h > 0 associated with the problem as well.
For simplicity (and in this problem only), suppose that h = k = lamda =
= 1.
a. Compute the long-run average cost for a given Q (as a function of Q), and let us denote
this quantity by C(Q). You may use any relevant results from the basic theory of differential equations
without proof.
b. Show that the second derivative of C(Q) takes negative values on (0;1), namely the function
is NOT convex on (0;1)
c. Compute the optimal choice of Q, and justify this using some properties of the derivative
of C(Q). Comment on why convexity would have implied the properties you use here (even though convex-
ity itself does not hold).
Consider the following modification of the basic EOQ model. When the inventory is at positive level X(t),
in addition to demand draining inventory at rate gamma, the inventory perishes (due to e.g. items expiring or
becoming defective) at infitesimal rate
X(t)dt. Thus the total rate at which the inventory decreases (at
time t) is lamda+gamma.
X(t). Note that the infitesimal rate at which the inventory decreases depends on the current
inventory level. Suppose that, there are parameters k, h > 0 associated with the problem as well.
For simplicity (and in this problem only), suppose that h = k = lamda =
= 1.
a. Compute the long-run average cost for a given Q (as a function of Q), and let us denote
this quantity by C(Q). You may use any relevant results from the basic theory of differential equations
without proof.
b. Show that the second derivative of C(Q) takes negative values on (0;1), namely the function
is NOT convex on (0;1)
c. Compute the optimal choice of Q, and justify this using some properties of the derivative
of C(Q). Comment on why convexity would have implied the properties you use here (even though convex-
ity itself does not hold).