Question

A farmer raises chickens industrially. Chicken coops have a certain degree of automation. In particular, the feeding system is automatic, to prevent food illnesses, a nutritional additive is added. The farmer has an inventory of this additive, which he reviews periodically. Monthly demand has a normal distribution with a mean of 50 lb and a standard deviation of 25 lb. The delivery time is 4 weeks and the farmer estimates that the cost of maintaining the inventory is 20% per year. The order cost is $ 170 per order and the Administration established a service policy in which the probability of a shortage must not be greater than 0.005

Assuming the inventory is reviewed every 6 weeks, find the target inventory.

What is the expected annual cost of this policy that the farmer follows?
What is the safety inventory and the target inventory?
How often should an order be placed and how many orders should be placed?
You have arrived to place an order, when there are 300 units in inventory you have an order of 300 to arrive, and there is a shortage of 100 units to deliver, how much should you buy?

Answer 1

Average demand, d = 50 lb per month
Stdev of monthly demand, σ = 25 lb
Lead time, L = 4 weeks = 1 month
In-stock probability, F(z) = 1 - 0.005 = 0.995, so, z = normsinv(0.995) = 2.58

(a)

Review period, P = 6 weeks = 1.5 months

Target inventory level, T = d.(L+P) + z.σ.√(L+P) = 50*(1+1.5) + 2.58*25*√(1+1.5) = 227 lbs.

(b)

Average inventory level, AIL = d.P/2 = 50*1.5/2 = 37.5 lbs.

Number of orders placed in a year (n) = 12 / 1.5 = 8 orders

Ordering cost, K = $170
Inventory holding cost, h = 20% of 'Purchase cost per lbs=C' = 0.2C [But 'C' is not given anywhere]

So, the total annual cost = AIL * 0.2C + n * K = 37.5*0.2*C + 8*170 = 7.5*C + 1360

The numeric value of total cost can be calculated from above only after knowing the value of 'C' which is not given anywhere in the question.

(c)

Safety stock = z.σ.√(L+P) = 2.58*25*√(1+1.5) = 102 lbs.

The target inventory level has been already computed in part-a.

(d)

An order is placed every 1.5 months i.e. 6 weeks (as per the periodic review policy).

Total number of orders placed in a year (n) = 12 / 1.5 = 8 orders

(e)

You are dealing with 'lbs' as the unit of measurement from the beginning and suddenly you cannot change the unit to 'number of units'.

T = 227
On-order = 300
On-hand = 300
Backorder = 100

So, order size would be =T - On-hand - On-order + Backorders = 227 - 300 - 300 + 100 = -273 i.e. we don't have to order anything.