Question

1. Crestview sells a particularly popular Christmas card once a year and distributes the cards to gift shops. It costs Crestview $1 per card to order from a printing company, and Crestview receives $2 for each card sold. Each card that is not sold is discarded, and Crestview receives $0.1 for each. Crestview has estimated that the demand for the coming Christmas season follows a normal distribution with a mean of 100,000 and standard deviation of 30,000.

a) Determine the optimal number of cards Crestview should order for the coming Christmas season.

b) Suppose Crestview has purchased a printing machine and will print cards themselves. The cost to print each card is $0.75. Determine the optimal number of cards Crestview should print for the coming Christmas season.

c) Interpret the difference between the results in a) and b).

2. Consider the following information on an inventory management system:

​​Item Cost:​​​$10

​​Order Cost:​​​$300

​​Annual Holding Cost:​​30% of item cost

​​Annual Demand:​​15,000 units based on 300 working days

​​Average Demand:​​50 units

​​Std. Dev. of Demand:​​12 units per day

​​Leadtime:​​​16 days​

a) Ignoring the uncertainty in the demand (i.e. looking only at average values), find the optimal order quantity and the reorder point. What is the annual inventory holding and ordering cost for this policy?

b) Consider now the uncertainty. The order quantity remains the same. If the target is to have a 99% fill rate, what should be the reorder point? What is the safety stock? How much additional inventory cost is incurred due to the safety stock? What should be the reorder point?

Answer 1

1.

(a)

Cost of underage, Cu = Selling price - Purchase price = 2 - 1 = 1
Cost of overage, Co = Purchase price - Salvage value = 1 - 0.1 = 0.90

So, newsvendor critical ratio = Cu / (Cu + Co) = 1 / (1 + 0.90) = 0.5263

For optimal order size, F(z) should be equal to 0.5263
or, z = normsinv(0.5263) = 0.066

So,

Optimum order quantity, Q = mean demand + z * stdev = 100000 + 0.066*30000 = 1,01,980

(b)

Cost of underage, Cu = Selling price - Manufacturing cost = 2 - 0.75 = 1.25
Cost of overage, Co = Manufacturing cost - Salvage value = 0.75 - 0.1 = 0.65

So, newsvendor critical ratio = Cu / (Cu + Co) = 1.25 / (1.25 + 0.65) = 0.6579

For optimal order size, F(z) should be equal to 0.6579
or, z = normsinv(0.6579) = 0.407

So,

Optimum order quantity, Q = mean demand + z * stdev = 100000 + 0.407*30000 = 1,12,210

(c)

The order quantity is increasing in (b) becasue the manufacturing cost os less than the purchase price. Due to this, the risk of loss for left-over quantity is reducing and at th same time the profit per unit sold is also inreasing. So, the company can now take more risk and order more.

2.

Item Cost, C = ​​​$10
​​Order Cost, K = ​​​$300
Unit Holding Cost, h = ​​30% of C = $3
​​Annual Demand, D = ​​15,000 units
Daily demand, d = 15,000 / 300 = 50 units
​​Stdev of daily demand, σ = ​​12 units
Average lead time:​​, L = 16 days​

(a)

Optimal order quantity, Q = (2.D.K / h)^1/2 = sqrt(2*15000*300 / 3) = 1,732 units

Reorder point = d.L = 50*16 = 800 units

Total annual cost of inventory = (D/Q).K + (Q/2).h = (15000 / 1732)*300 + (1732/2)*3 = $5,196 (round off)

(b)

Stdev of lead time demand, σ_LTD = σ√L = 12*√16 = 48

Fill rate = 99%

Fill rate = 1 - (Expected shortage per cycle, ESC / Q)
or, 0.99 = 1 - L(z)*σ_LTD / Q
or, 099 = 1 - L(z)*48 / 1732
or, L(z) = 1732*(1 - 0.99) / 48 = 0.36

Look up a standard normal loss function table to find the corresponding 'z' wll be close to 0.08

So,

Safety stock = z.σ_LTD = 0.08*48 = 3.84 or 4 units

Additional carrying cost for the safety stock = 4.h = $12

Reorder point = d.L + safety stock = 50*16 + 4 = 804 units