Question

SCENARIO The Forest Stewardship Council (FSC) was formed in 1993 to promote sustainable management of the world’s forests. The FSC quickly began to certify lumber based on whether the forest that it was taken from was managed according to its guidelines. Soon thereafter, several builders in California began to specialize in the construction of "Green" buildings that only used FSC-certified lumber. This was seen as a viable business because some customers were willing to pay a premium to have their projects completed with FSC lumber. These builders have an opportunity to order this lumber once every 3 months because the forests involved must be harvested in accordance with certain restrictions. Consequently, builders who focused on this market were forced to hold large inventories. On the other hand, builders who only used "traditional" wood which was not FSC-certified could order on a just-in-time basis, meaning they did not have to hold any lumber in their own lumberyards. Consider the following 3 scenarios and related questions.

1. A green builder must decide how much FSC lumber to purchase to meet demand for the next 3 months. Demand is normally distributed with a mean of 40,000 board-feet and a standard deviation of 15,000 board feet. (A board-foot is a standard unit for lumber.) The purchase price for the builder is $4.00 per board-foot. At the end of a 3-month period the wood will dry and may warp, reducing its value. Of any lumber remaining in the builder’s lumber yard at the end of the 3-month period, approximately half will be worthless. The builder will use any wood that is not warped in the next period. However, buying the wood now, rather than in the next period incurs a holding cost of 4% of the purchase cost. If the builder has too little FSC certified wood to meet demand, he will be forced to substitute traditional lumber which he can buy for $3.35 per board foot. In addition, the green builder assigns a shortage cost of $2.00 per board foot for the loss of good will and damage to his reputation. How many board feet of FSC certified lumber should the builder purchase?
2. Suppose a lumber-yard (Nice Lumber) agrees to serve as a distributor for a builder. This means Nice Lumber will stock the FSC-certified lumber for one green builder. Nice Lumber will pay $4.00 per board foot for FSC-certified wood and sell it to the builder for $4.20 per board foot. If demand exceeds the inventory, the green builder will buy traditional wood from a different lumber yard to meet the demand at price of $3.20 per board-foot. In addition to the lost sale, Nice assigns a cost of $2.00 per board foot of shortage of FSC lumber. If the inventory of FSC-certified lumber exceeds demand, Nice will immediately substitute the excess FSC certified lumber to meet demand from other customers and reduce its purchases of traditional lumber accordingly. Nice pays $3.20 per board foot for traditional lumber. How many board feet of FSC certified lumber should Nice Lumber purchase?
3. Suppose Nice Lumber will stock the FSC certified lumber for 10 green builders. For each of these builders, demand is normally distributed with a mean of 40,000 board feet and a standard deviation of 15,000 board feet, and each builder’s demand is independent of other builders’ demand. How many board feet of FSC certified lumber should Nice lumber purchase per builder?

Answer 1

Answer is explained in the explanation section below.

Step-by-step explanation:

Solution:

Board-Feet = Mean + Z*SD

Where SD = Standard Deviation

Mean = 40,000;

SD= 15,000;

Z = NORMSINV(SL); SL (Service Level) = Cu/(Cu + Co)

1.

Solution:

Cu = $(4 - 3.35 + 2)

Since he would gain $4 - $3.35 by substituting lumber for conventional wood, he would lose $2 in goodwill and credibility loss.

Cu = 2.65

And,

Co = (4 - 2 + 4% x 4)

Since half of the lumber becomes scrap after three months and he pays 4% as a holding cost for keeping $4/unit as inventory, he incurs a holding cost of 4%.

Co = 2.16

Service Level SL = Cu/(Cu+Co)

Service Level SL = 2.65/(2.65+2.16)

Service Level SL = 0.550936

Z = NORMSINV(0.550936)

Z = 0.12

Hence,

Board-Feet = Mean + Z*SD

Board-Feet = 41,920.38

2.

Solution:

Cu = $(2 - 4.2 + 3.2)

Since he would gain $4.2-$3.2 by substituting lumber for conventional wood, he would lose $2 in goodwill and credibility loss.

Cu = 1

In the event of overstocking, he does not specify the price at which he will replace FSC lumber with conventional lumber. Only his price, which is $3.2/board-foot for traditional lumber, is issued.

Co = 4 - 3.4

since he'll have to market the excess FSC lumber inventory at conventional wood's price; ASSUMING traditional lumber rate at the rate $3.4/board-feet

Co = 0.6

So,

Service Level SL = Cu/(Cu+Co) = 1/(1+0.6)

SL = 0.625

Z = NORMSINV(0.625)

Z = 0.318639

Board-Feet = 44,779.59

3.

Solution:

Here, everything will be same except the formula for calculate the Board -Feet. New formula is:

Board-Feet = Mean +
`(Z * SD)/(√(n) )`

Here, n = 10

Just plugging in the values, we get:

Board-Feet = 40,000 +
`(0.318639 * 15000)/(√(10) )`

Board-Feet = 41,511.44