Question

A farmer raises chickens on an industrial basis. The chicken coops have a certain degree of automation. Specifically, the food distribution system is automated. To prevent chicken disease, the farmer adds nutritional additive to the chicken food. . ". He holds inventory for this additive and manages it using a periodic review system. Monthly demand is normally distributed with mean of 50 lb and standard deviation of 75 lb. Delivery time is four weeks, and the farmer estimates his inventory holding cost to be 20 percent annually. Ordering cost is $170 per order and the cost of the additive is $60 per pound. A fill rate of 98 percent is required. 1. Assume that inventory is reviewed every six weeks, and find the appropriate inventory target. 2. What is the expected number of "pounds short" per year?

Answer 1

Average demand, d = 50 lbs per month
Stdev of monthly demand, σ = 75 lb.
Lead time, L = 4 weeks = 1 month
Review period, P = 6 weeks = 1.5 months
Unit cost, C = $60 per lb
Unit carrying cost, h = 20% of C = $12 per lb. per annum
Ordering cost, K = $170
Required fill rate, FR = 0.98

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1.

Stdev in the (P+L) period, σ_L+P = σ.√(L+P) = 75*√(1+1.5) = 118.59 lbs

FR = 1 - L(z)*σ_L+P / d.(L+P)

or, 0.98 = 1 - L(z)*118.59/(50*(1+1.5))

or, L(z) = (1 - 0.98)*(50*(1+1.5))/118.59

or, L(z) = 0.021

Use the normal loss function table, to note the appropriate value for the 'z'. In this case, this will be z = 1.64 as shown.

So,

Appropriate target inventory level, T = d.(P+L) + z.σ.√(L+P) = 50*(1+1.5) + 1.64*75*√(1+1.5) = 319.48 lbs.

2.

The Expected shortages per cycle = L(z)*σ.√(L+P) = 0.021*75*√(1+1.5) = 2.49 lbs.

Number of cycles per year = 12 months / (L+P) = 12/2.5 = 4.8

So, the expected shorategs per year = Expected shortages per cycle * Number of cycles per year = 2.49*4.8 = 11.95 lbs.