Question

Consider the following newsvendor environment with sales season in August (school year start). We are now in late-April and the best forecast for demand in August is that it is normally distributed with a mean of 4000 units and a standard deviation of 1000. We can buy now from a Chinese supplier at 6 $ per unit. Lead time for this order is 3 months, so an order placed now will be delivered before August. The item sells for 12 $ per unit. Inventory left over at end of August has to be discounted with a salvage value 2 $ per unit.

1.How many units do you buy now from China (only one order is placed)? If the Chinese supplier’s variable cost per unit is 50 cents, calculate the Chinese supplier’s profit for your order quantity. Do you expect to make more or less profit than the Chinese supplier? Explain.

2. Suppose in late June we will get to know the demand for August perfectly (our major customers place early orders); this is the demand forecast update. In late June, we can buy from a quicker but more expensive local supplier. The unit cost is 8 $ per unit and the lead time for orders is one month (so delivery is by late July, before the August selling season). In this case, how many units do you buy now from the Chinese supplier (knowing you can buy again later from the local supplier)? Explain your logic.

Answer 1

Average demand, d = 4000 units

Standard deviation of demand, s = 1000 units

Unit cost, c = $ 6

Selling price, p = $ 12

Salvage value, v = $ 2

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1)

Under-stocking cost, Cu = p - c

= 12 - 6

= $ 6

Overstocking cost, Co = c - v

= 6 - 2

= $ 4

Critical ratio = Cu/(Cu+Co)

= 6/(6+4)

= 0.6

z value = NORMSINV(0.6)

= 0.2533

Optimal quantity you buy now from Chinese supplier = d + z*s

= 4000+0.2533*1000

= 4,253 units

Chinese supplier's profit = 4253*(6-0.5)

= $ 23,391.5

You expect to make lesser profit than the chinese supplier, because Value of I(z) corresponding to z value of 0.2533 is 0.538 (from standard normal table), which means expected leftover inventory = 0.538*1000 = 538

Expected sales = order quantity - expected leftover inventory = 4253 - 538 = 3715

So, expected profit you would make = Expected sales * Cu - Expected leftover inventory * Co

= 3715*6 - 538*4

= $ 20,138

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2)

Under-stocking cost, Cu = Difference between the cost of quicker supplier and chinese supplier

= 8 - 6

= $ 2

Overstocking cost remains same, i.e. Co = $ 4

Critical ratio = Cu/(Cu+Co)

= 2/(2+4)

= 0.333

z value = NORMSINV(0.333)

= -0.4308

Quantity to buy from chinese supplier = 4000+(-0.4308)*1000

= 3569

We see that this order quantity is lesser than that determined in part 1. This is because, given the opportunity to buy from a quicker supplier, gives us freedom to take a lesser risk of having leftover inventory, because the opportunity cost of under-stocking is now lesser.