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Chapter 9: Problem 7

Golf Smart sells a particular brand of driver for \(\$ 200\) each. During the next year, it estimates that it will sell \(15,25,35\), or 45 drivers with respective probabilities of \(0.35\), \(0.25,0.20\), and \(0.20 .\) They can buy drivers only in lots of 10 from the manufacturer. Batches of \(10,20,30,40\), and 50 drivers cost \(\$ 160, \$ 156, \$ 148, \$ 144\), and \(\$ 136\) per driver, respectively. Each year, the manufacturer designs a new "hot" driver that depreciates the value of last year's driver. Consequently, at the end of every year, Golf Smart has a clearance sale and is able to sell any unsold drivers for \(\$ 96\) each. Assume that any customer who comes in during the year to buy a driver but is unable due to lack of inventory costs Golf Smart \(\$ 24\) in lost goodwill. Determine what decision should be made under each of the following criteria: a. Expected value b. Laplace c. Maximin d. \(\operatorname{Maximax}\) e. Minimax regret

Short Answer

Expert verified
Order 30 drivers: (a) Expected value, (b) Laplace, (c) Maximin; Order 50 drivers: (d) Maximax; Order 40 drivers: (e) Minimax regret.

Step by step solution

01

Identify Possible Orders and Outcomes

List the possible orders of drivers (10, 20, 30, 40, or 50) and estimate the number of drivers to be sold (15, 25, 35, or 45) with their respective probabilities. This helps in setting up a payoff table that will guide the decision-making process under various criteria.
02

Create the Payoff Table

For each possible order quantity, calculate the profit for each sales scenario (15, 25, 35, 45) by considering revenue from sales, costs of purchasing drivers, and revenue from unsold drivers. Include the cost of lost goodwill for potential unmet demand when Golf Smart runs out of drivers.
03

Calculate Expected Values for Each Order Quantity

Multiply the profit from each sales scenario by its respective probability, sum these products for each order quantity to obtain the expected value for that order. This will help determine the order quantity with the highest expected return.
04

Apply the Laplace Criterion

For the Laplace Criterion, calculate the average profit for each order quantity. This means assuming each sales level (15, 25, 35, and 45) is equally likely, thus no probability weighting is used. Choose the order quantity with the highest average profit.
05

Apply the Maximin Criterion

For each order size, determine the minimum profit that could occur, then select the order size where this minimum profit is the highest among all options. This criterion is for risk-averse decision-makers.
06

Apply the Maximax Criterion

Identify the maximum profit for each order size, then choose the one with the highest maximum profit. This criterion is for risk-seeking decision-makers.
07

Calculate the Regret for each Order Quantity

For each sales level (15, 25, 35, 45), calculate the difference in profit between the best decision and each other decision. This difference is called regret.
08

Apply the Minimax Regret Criterion

For each order size, find the maximum regret and then choose the order size with the smallest maximum regret. This minimizes potential regret from a suboptimal decision.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value in Decision-Making
The concept of expected value is used to make informed decisions based on different potential outcomes and their probabilities. It helps in predicting the long-term benefit or cost of different decisions. To calculate expected value, you multiply each possible outcome by its probability of occurring, then sum these values.
This provides a weighted average outcome, which can guide decision-makers to maximize their expected profits. In the case of Golf Smart, by evaluating each order size based on expected value, you can decide which order size offers the highest anticipated return, considering all levels of demand.
Understanding the Laplace Criterion
The Laplace Criterion is applied in decision-making under uncertainty when no probabilities are known or when one wants to treat all outcomes as equally likely. This approach simplifies decision-making by ignoring any given probabilities and instead focusing on average potential profits.
To use the Laplace Criterion, calculate the average profit for each decision option by summing all possible outcomes and dividing by the number of outcomes. The decision with the highest average is selected. For Golf Smart, this means determining which order size has the highest average profit across all demand scenarios, treating each scenario as equally probable.
Exploring the Maximin Criterion
The Maximin Criterion is particularly useful for those who prefer safety and are risk-averse. This strategy involves looking for the decision option that has the best of the worst-case scenarios. In essence, a decision-maker using maximin will focus on minimizing losses by considering the smallest possible profit for each decision.
To apply this, identify the lowest profit for each order size, and then choose the order size with the highest value among these minimum profits. For Golf Smart, this would mean choosing the order size that ensures the least amount of risk regarding potential minimum profits.
Delving into the Maximax Criterion
The Maximax Criterion appeals to those willing to take risks, as it focuses on potential for maximum profit. This approach assumes that the best outcome will occur, leading decision-makers to select the option with the highest possible profit. It is a more aggressive decision-making model than the maximin criterion.
For each possible order size, determine the maximum profit achievable given different demand scenarios. Then, select the order size with the highest maximum profit. This is ideal when Golf Smart aims to pursue the best imaginable results, regardless of accompanying risks.
The Minimax Regret Criterion Explained
The Minimax Regret Criterion is concerned with minimizing future regret resulting from making a suboptimal decision. This involves calculating the regret for each possible outcome, which is the difference in profit between the best decision for that scenario and all other decisions.
Then, for each decision option, identify the maximum regret across all scenarios. The decision with the smallest maximum regret is selected. This criterion is all about hedging against the possibility of regret, allowing decision-makers like Golf Smart to make choices that minimize the chance of looking back and wishing they had chosen differently.

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Most popular questions from this chapter

Baseball has dramatically changed the penalties associated with failing steroid drug testing. The new penalties that Commissioner Bud Selig has proposed are an approach of "three strikes and you're out," which goes as follows: The first positive test would result in a 50-game suspension. The second positive test would result in a 100 -game suspension. Finally, the third positive test would result in a lifetime suspension from Major League Baseball. Let's examine the issue and the choices using decision trees. First, let's consider testing a baseball player for steroids; the test result will be either positive or negative. However, these tests are not flawless. Some baseball players who are steroid free test positive, and some baseball players who use steroids test negative. The former are called false positives; the latter are false negatives. We will extract data from the baseball steroid data table for our analysis. Build a conditional probability tree and use it to help compute all the probabilities associated with the following key issues: If some players test positive, what are the chances that they were a steroid user? If some players test positive, what are the chances that they were not a steroid user? If a test was negative, what are the chances that the player was a steroid user? If a test was negative, what are the chances that the player was not a steroid user?What do these results suggest about this drug testing? $$ \begin{aligned} &\text { Baseball steroid data }\\\ &\begin{array}{lcrr} \hline & \text { Positive results } & \text { Negative results } & \text { Totals } \\ \hline \text { Users } & 28 & 3 & 31 \\ \text { Nonusers } & 19 & 600 & 619 \\ \text { Totals } & 47 & 603 & 650 \\ \hline \end{array} \end{aligned} $$

The number of attempts to use an ATM per person per month and their probabilities are listed in the following table. Compute the expected value and interpret that value. Discuss how similar calculations could be used to determine the number of ATMs needed. $$ \begin{array}{lllll} \hline \text { Attempts } & 1 & 2 & 3 & 4 \\ \text { Probability } & 0.5 & 0.33 & 0.10 & 0.07 \\ \hline \end{array} $$

A big private oil company must decide whether to drill in the Gulf of Mexico. It costs \(\$ 1\) million to drill, and if oil is found its value is estimated at \(\$ 6\) million. At present, the oil company believes that there is a \(45 \%\) chance that oil is present. Before drilling begins, the big private oil company can hire a geologist for \(\$ 100,000\) to obtain samples and test for oil. There is only about a \(60 \%\) chance that the geologist will issue a favorable report. Given that the geologist does issue a favorable report, there is an \(85 \%\) chance that there is oil. Given an unfavorable report, there is a \(22 \%\) chance that there is oil. Determine what the big private oil company should do.

For a new development area, a local investor is considering three alternative real estate investments: a hotel, a restaurant, and a convenience store. The hotel and the convenience store will be adversely or favorably affected depending on their closeness to the location of gasoline stations, which will be built in the near future. The restaurant will be assumed to be relatively stable. The payoffs for the investment are as follows: $$ \begin{array}{lccc} \hline & {\text { Conditions }} & \\ \text { Alternative } & \text { 1: Gas close } & \text { 2: Gas medium distance } & \text { 3: Gas far away } \\ \hline \text { Hotel } & \$ 25,000 & \$ 10,000 & -\$ 8000 \\ \text { Convenience store } & \$ 4000 & \$ 8000 & -\$ 12,000 \\ \text { Restaurant } & \$ 5000 & \$ 6000 & \$ 6000 \\ \hline \end{array} $$ Determine the best plan by each of the following criteria: a. Laplace b. Maximin c. Maximax d. Coefficient of optimism (assume that \(x=0.45\) ) e. Regret (minimax)

We are considering one of three alternatives A, B, or C under uncertain conditions. The payoff matrix is as follows: $$ \begin{array}{lccc} \hline & {\text { Conditions }} \\ \text { Alternative } & 1 & 2 & 3 \\ \hline \text { A } & 3000 & 4500 & 6000 \\ \text { B } & 1000 & 9000 & 2000 \\ \text { C } & 4500 & 4000 & 3500 \\ \hline \end{array} $$ Determine the best plan by each of the following criteria and show your work: a. Laplace b. Maximin c. Maximax d. Coefficient of optimism (assume that \(x=0.65\) ) e. Regret (minimax)
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