Question

Consider a firm (retailer) that faces a constant annual demand D. Each time the firm places an order to its supplier it incurs a fixed cost  per order. The firm also incurs a holding cost h $per unit kept in inventory per unit time. Along with cost, the firm is sensitive to the amount of carbon emitted during its operations. For instance, the firm will emit  carbon unit[1]per order and   carbon unit per unit hold in stock per time unit.

We assume that orders are delivered with zero lead-time. The firm must satisfy all the demand without shortage and backorders.

[1]Carbon unit could be in Tons of CO₂, kilogram of CO₂, cubic meter of CO₂etc. What is important is that  are given and representing one of these measures.

In this question we assume that the government is imposing an Emission Tax Policy: that is every company has to pay T $per unit of carbon emitted. Describe the firm optimal strategy; this includes (1) order quantity, (2) total cost, (3) associated emission level.

Answer 1

Let order quantity = Q

Number of orders per year = D/Q

Average inventory = Q/2

Annual cost of carbon emitted due to number of orders per year = (D/Q)*T

Annual cost of carbon emitted due to average inventory held in stock = (Q/2)*T

Total cost of carbon emitted = (D/Q)*T + (Q/2)*T = T(D/Q + Q/2)

For minimum total cost, first oder derivative of total cost function wrt Q should be equal to 0.

d(DT/Q + QT/2)/dQ = 0

-DT/Q² + QT/2 = 0

Q² = 2DT/T

(1) Order quantity, Q = 2D

(2) Total cost = T(D/Q + Q/2)T = T(D/(2D) + (2D) /2) = T2D

(3) Associated emission level = 2D