"Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semidry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision about the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine, it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & 0.05 & 0.50 \\ \text { Strong } & 0.25 & 0.20 \end{array} \end{aligned}$$

Revenue projections show an annual contribution to profit of $20,000 if Seneca Hill plants only Chardonnay grapes and demand is weak for Chardonnay wine, and$70,000 if Seneca plants only Chardonnay grapes and demand is strong for Chardonnay wine. If Seneca plants only Riesling grapes, the annual profit projection is $25,000 if demand is weak for Riesling grapes and$45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & \$ 22,000 & \$ 40,000 \\ \text { Strong } & \$ 26,000 & \$ 60,000 \end{array} \end{aligned}$$

Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand to also be strong in this case. Suppose the probability of strong demand for Chardonnay and weak demand for Riesling is 0.05 and that the probability of strong demand for Chardonnay and strong demand for Riesling is 0.40. How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still 0.05 and 0.50."

"Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semidry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision about the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine, it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & 0.05 & 0.50 \\ \text { Strong } & 0.25 & 0.20 \end{array} \end{aligned}$$

Revenue projections show an annual contribution to profit of $20,000 if Seneca Hill plants only Chardonnay grapes and demand is weak for Chardonnay wine, and$70,000 if Seneca plants only Chardonnay grapes and demand is strong for Chardonnay wine. If Seneca plants only Riesling grapes, the annual profit projection is $25,000 if demand is weak for Riesling grapes and$45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & \$ 22,000 & \$ 40,000 \\ \text { Strong } & \$ 26,000 & \$ 60,000 \end{array} \end{aligned}$$

What is the decision to be made, what is the chance event, and what is the consequence? Identify the alternatives for the decisions and the possible outcomes for the chance events"

A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately 5% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1.

Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default.

What was the purpose of the Seneca Falls Convention?

Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semidry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision about the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine, it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments:

Revenue projections show an annual contribution to profit of $\$ 20,000$ if Seneca Hill plants only Chardonnay grapes and demand is weak for Chardonnay wine, and $\$ 70,000$ if Seneca plants only Chardonnay grapes and demand is strong for Chardonnay wine. If Seneca plants only Riesling grapes, the annual profit projection is $\$ 25,000$ if demand is weak for Riesling grapes and $\$ 45,000$ if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are as shown in the following table:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & \$ 22,000 & \$ 40,000 \\ \text { Strong } & \$ 26,000 & \$ 60,000 \end{array} \end{aligned}$$

What is the decision to be made, what is the chance event, and what is the consequence? Identify the alternatives for the decisions and the possible outcomes for the chance events.

Demand function. Given the demand function

$$p=D(x)=\frac{1,000}{x}$$

find the average price (in dollars) over the demand interval $[400, 600]$.

A stock's returns have the following distribution:

$$\small{ \begin{array}{lcc} \begin{array}{l} \textbf{ Demand for the } \\ \textbf{ Company's Products } \end{array} & \begin{array}{c} \textbf{ Probability of This } \\ \textbf{ Demand Occurring } \end{array} & \begin{array}{c} \textbf{ Rate of Retum if This } \\ \textbf{ Demand Occurs } \end{array} \\ \hline \text { Weak } & 0.1 & (50 \%) \\ \text { Below average } & 0.2 & (5) \\ \text { Average } & 0.4 & 16 \\ \text { Above average } & 0.2 & 25 \\ \text { Strong } & \underline{0.1} & 60\\ &\underline{\underline{0.1}}\\\ \end{array}}$$

Calculate the stock's expected return, standard deviation, and coefficient of variation.

Other members of the management team expect the Chardonnay market to become saturated at some point in the future, causing a fall in prices. Suppose that the annual profit projections fall to $\$ 50,000$ when demand for Chardonnay is strong and only Chardonnay grapes are planted. Using the original probability assessments, determine how this change would affect the optimal decision.

"Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semidry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision about the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine, it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & 0.05 & 0.50 \\ \text { Strong } & 0.25 & 0.20 \end{array} \end{aligned}$$

Revenue projections show an annual contribution to profit of $20,000 if Seneca Hill plants only Chardonnay grapes and demand is weak for Chardonnay wine, and$70,000 if Seneca plants only Chardonnay grapes and demand is strong for Chardonnay wine. If Seneca plants only Riesling grapes, the annual profit projection is $25,000 if demand is weak for Riesling grapes and$45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table:

$$\begin{aligned} &\text { Riesling Demand }\\ &\begin{array}{lcc} \text { Chardonnay Demand } & \text { Weak } & \text { Strong } \\ \text { Weak } & \$ 22,000 & \$ 40,000 \\ \text { Strong } & \$ 26,000 & \$ 60,000 \end{array} \end{aligned}$$

Use the expected value approach to recommend which alternative Seneca Hill Winery should follow in order to maximize expected annual profit."

Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand to also be strong in this case. Suppose that the probability of strong demand for Chardonnay and weak demand for Riesling is 0.05 and that the probability of strong demand for Chardonnay and strong demand for Riesling is 0.40 . How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still 0.05 and 0.50 .

The distance from Potsdam to larger markets and limited air service have hindered the town in attracting new industry. Air Express, a major overnight delivery service, is considering establishing a regional distribution center in Potsdam. But Air Express will not establish the center unless the length of the runway at the local airport is increased. Another candidate for new development is Diagnostic Research, Inc. (DRI), a leading producer of medical testing equipment. DRI is considering building a new manufacturing plant. Increasing the length of the runway is not a requirement for DRI, but the planning commission feels that doing so will help convince DRI to locate their new plant in Potsdam. Assuming that the town lengthens the runway, the Potsdam planning commission believes that the probabilities shown in the following table are applicable.

$$\begin{matrix} \text{ } & \text{DRI Plant} & \text{No DRI Plant}\\ \text{Air Express Center} & \text{.30} & \text{.10}\\ \text{No Air Express Center} & \text{.40} & \text{.20}\\ \end{matrix}$$

For instance, the probability that Air Express will establish a distribution center and DRI will build a plant is .30. The estimated annual revenue to the town, after deducting the cost of lengthening the runway, is as follows:

$$\begin{matrix} \text{ } & \text{DRI Plant} & \text{No DRI Plant}\\ \text{Air Express Center} & \text{\$600,000} & \text{\$150,000}\\ \text{No Air Express Center} & \text{\$250,000} & \text{-\$200,000}\\ \end{matrix}$$

If the runway expansion project is not conducted, the planning commission assesses the probability DRI will locate their new plant in Potsdam at .6; in this case, the estimated annual revenue to the town will be $450,000. If the runway expansion project is not conducted and DRI does not locate in Potsdam, the annual revenue will be$0 since no cost will have been incurred and no revenues will be forthcoming. a. What is the decision to be made, what is the chance event, and what is the consequence? b. Compute the expected annual revenue associated with the decision alternative to lengthen the runway. c. Compute the expected annual revenue associated with the decision alternative to not lengthen the runway. d. Should the town elect to lengthen the runway? Explain. e. Suppose that the probabilities associated with lengthening the runway were as follows:

$$\begin{matrix} \text{ } & \text{DRI Plant} & \text{No DRI Plant}\\ \text{Air Express Center} & \text{.40} & \text{.10}\\ \text{No Air Express Center} & \text{.30} & \text{.20}\\ \end{matrix}$$

What effect, if any, would this change in the probabilities have on the recommended decision?

Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full price service using the company’s new fleet of jet aircraft and a discount service using smaller capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based upon two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars).

$$\begin{matrix} \text{ } & \text{Demand} & \text{for Service}\\ \hline \text{Service} & \text{Strong} & \text{Weak}\\ \text{Full price} & \text{\$960} & \text{-\$490}\\ \text{Discount} & \text{\$670} & \text{\$320}\\ \end{matrix}$$

a. What is the decision to be made, what is the chance event, and what is the consequence for this problem? How many decision alternatives are there? How many outcomes are there for the chance event? b. Suppose that management of Myrtle Air Express believes that the probability of strong demand is .7 and the probability of weak demand is .3. Use the expected value approach to determine an optimal decision. c. Suppose that the probability of strong demand is .8 and the probability of weak demand is .2. What is the optimal decision using the expected value approach?

"Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full-price service using the company’s new fleet of jet aircraft and a discount service using smaller-capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based on two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars):

$$\begin{aligned} &\text { Demand for Service }\\ &\begin{array}{lcr} \text { Service } & \text { Strong } & \text { Weak } \\ \text { Full Price } & \$ 960 & -\$ 490 \\ \text { Discount } & \$ 670 & \$ 320 \end{array} \end{aligned}$$

What is the decision to be made, what is the chance event, and what is the consequence for this problem? How many decision alternatives are there? How many outcomes are there for the chance event?"

Consider the following network representation of a transportation problem:

The supplies, demands, and transportation costs per unit are shown on the network. What is the optimal (cost minimizing) distribution plan?