Question

The answers were given for this problem, but I am not sure how to derive them. Would need any kind of help

Suppose that you sell Christmas trees each holiday season for $30 a tree. Peak selling time is the 2 weeks leading up to Christmas, but since harvesting real pine trees takes time, your supplier requires a 1 month lead time.

Your purchase cost per tree is $20. Anytime a customer comes to your store requesting a tree and it is unavailable, you give them a $5 credit (per tree) to spend on other products. (You should assume they always use this credit.) Any trees not sold before Christmas are sold to a local lumber yard at $10/tree.

Demand for this Christmas is forecasted in the following table.

Demand	Probability
600	0.2
700	0.3
800	0.3
900	0.2

a) (3 points) To the nearest tree, what is the expected demand for this Christmas? (Remember, as with all questions, you should show work for full credit.) 750 trees

b) (3 points) Suppose you order exactly 800 trees, how many trees would you expect to sell? (Remember, as with all questions, you should show work for full credit.) 730 trees

c) (3 points) Suppose you order exactly 800 trees while the actual demand turns out to be 700 trees, what is your expected profit (including goodwill costs if there are any)? $6000

d) (3 points) What is the underage cost? What is the overage cost? Just by comparing the two costs, should we order more or less than the average demand? Briefly explain.

Underage: $15

Overage: $10

We should order more than average, but why? I don't understand the explanation

e) (3 points) What is the optimal order quantity given the demand in part a)? As with all questions, please show work for full credit. 800 trees

f) (1 point) What is the effective service level for the quantity you suggested in part e)? 80%

Answer 1

a) We get the expected number of trees by multiplying demands and the probability a ssociated with them.

b) Suppose we order 800 trees, the probability and number of trees sold are as following,

(Please note that we have only 800 trees. So even if the demand is 900, we will only be able to sell 800 trees)

c)

Number of trees bought = 800

Cost of buying trees = 800*20 = $16,000

Number of trees sold = 700

Revenue from sales = 700* 30 = 21,000

Number of trees sent to lumber yard= 100

Revenue from lumber yard = 100*10 = $1,000

Therefore, total revenue = 21,000+1,000 = $22,000

Expected profit = 22,000 - 16,000 = $6,000

d)

Underage cost is the cost incurred when the material is out of stock. Here, we have to forego a profit of $10 and a $5 credit when the trees are stock out. Hence, underage cost =10+5 = $15

Overage cost the loss we incurr if there is excess quantity available. We spend $20 for a tree and we get $10 when we give the excess trees to the lumber yard. Hence, Overage cost = 20-10 =$10.

Here, average demand is 750.

To calculate the optimal demand, we use the formula,

Criticle Fractile = Underage cost/(Underage cost + overage cost) = 15/25 = 60%.

Whenever the uderage cost is more than overage cost, we will need to order more than average.

e)

Criticle Fractile = Underage cost/(Underage cost + overage cost) = 15/25 = 60% (0.6).

We shoud order trees such that the quantity is atleast above the criticle fractile. And we always round up. Here, if we go for 600, only 0.2 is covered. With 700, we cover 0.5 and with 800 we cover 0.8.

Hence, the order quantity should be 800.

f) When we order 800, there is 80% chance that everyones requirement is met. Hence the service level is 80%