Question

A pharmacist wants to establish an optimal inventory policy for a new antibiotic that requires refrigeration in storage. The pharmacist expects to sell 4800 packages of this antibiotic at a steady rate during the next year. She plans to place several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is $ 12, and carrying​ costs, based on the average number of packages in​ inventory, amount to $ 8 per year for one package. Use this information to answer the following questions.
a) let x be the order quantity and r the number of orders placed during the year. Find the inventory cost (ordering cost plus carrying cost) in terms of x and r

C= ?

b) Find the constraint function

c)Determine the economic order quantity that minimizes the inventory cost and then find the minimum inventory cost.

x=? C=$?

Answer 1

Answer:

a) Inventory cost = 12r + 4x

b) Constraint

r = (D/x) = (4800/x)

c) Optimal order quantity = x = 120

Minimum Inventory Cost = C = $960

Optimal number of orders in a year = r = 40

Step-by-step explanation:

x be the order quantity and r the number of orders placed during the year.

We will use the following variables:

x = Quantity being ordered per run

D = annual Demand for the item, over the year = 4800

O = ordering cost, regardless of the number of units in the order (fixed cost per order) = $12

H = annual cost to Hold/carry one unit = $8

r = the number of orders placed during the year

It is important to note which variables are annualized, which are per-order and which are per-unit.

Inventory cost = Ordering Cost + Holding or Carrying Cost

Ordering cost = (ordering cost per order) × (number of orders) = 12r

Holding Cost = (annual unit Holding cost × order Quantity)/2 (because throughout the year, on average the warehouse is half full).

Holding cost = (8 × x)/2 = 4x

Inventory cost = 12r + 4x

b) The constraint equation

Total demand = (order quantity per order) × (number of orders)

D = rx

r = (D/x)

r = (4800/x)

c) Inventory Cost = C = 4x + 12r = 8x + 12(4800)/x

C = 4x + 57600/x

At minimum inventory cost, (dC/dx) = 0

(dC/dx) = 4 - 57600/x² = 0

57600 = 4x²

x² = (57600/4)

x² = 14400

x = √14400

x = 120

C = 4x + 57600/x

C = 4(120) + (57600/120)

C = 480 + 480 = $960

Hope this Helps!!!