Make sure to Calculate Co and Cu, then the critical ratio of Cu/(Cu+Co).

Scenario 1: Trash Pick-up in Columbia Residents of Columbia, MD place their trash at the curb each Wednesday morning to be picked up by municipal crews. Experience shows that the total amount of trash put out has a normal distribution with a mean of 40 tons and a standard deviation of 9 tons. Crews of full-time city employees collect the trash. Each crew can collect 6 tons of trash per day. The city has plenty of trucks of the kind used for trash collection. The marginal cost of operating one trash crew for one day including both personnel-related and truck-related costs is $800. Whatever trash remains at the end of the work day must be collected that evening by an outside contractor who charges $650 per ton.

1. How many crews should the city assign to trash collection to minimize expected total cost? In other words what number of crews (1, 2, 3, etc.) produces the lowest expected cost?

Scenario 2: Forest Stewardship Council (FSC) The Forest Stewardship Council (FSC) was formed in 1993 to promote sustainable management of the worlds 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 larger 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.

Scenario 2A: Builders Order (FSC) 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 35,000 board-feet and a standard deviation of 17,000 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 builders 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 3% of the purchase cost. (Only apply this to the wood that is held over for the following period, and not to the wood that is thrown away.) 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 $3.40 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.

2. How many board feet of FSC certified lumber should the builder purchase?

Scenario 2B: Middle-mans Order (1 customer) 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 $4.25 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 a price of $3.40 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 $3.40 per board foot for traditional lumber.

3. How many board-feet of FSC certified lumber should Nice Lumber purchase?

Scenario 2C: Middle-mans Order (12 customers) Suppose Nice Lumber will stock the FSC certified lumber for 12 green builders. For each of these builders, demand is normally distributed with a mean of 35,000 board feet and a standard deviation of 17,000 board feet, and each builders demand is independent of other builders demand.

4. How many board-feet of FSC certified lumber should Nice Lumber purchase per builder?