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Chapter 9: Problem 6

Assume the following probability distribution of daily demand for bushels of strawberries: $$ \begin{array}{lllll} \hline \text { Daily demand } & 0 & 1 & 2 & 3 \\ \text { Probability } & 0.2 & 0.3 & 0.3 & 0.2 \\ \hline \end{array} $$ Further assume that unit cost is \(\$ 3\) per bushel, selling price is \(\$ 5\) per bushel, and salvage value on unsold units is \(\$ 2\). We can stock \(0,1,2\), or 3 units. Assume that units from any single day cannot be sold on the next day. Build a decision tree and determine how many units should be stocked each day to maximize net profit over the long haul.

Short Answer

Expert verified
Stock 3 bushels daily for maximum profit.

Step by step solution

01

Identify Possible Actions

The possible actions are stocking 0, 1, 2, or 3 bushels of strawberries. For each stocking choice, we'll compute the expected profit.
02

Calculate Payoffs for Different Scenarios

For each possible stocking amount, calculate the profit for each level of daily demand.- **Stock 0 units:** Profit = 0 (any demand). - **Stock 1 unit:** - Demand = 0: Profit = 0 + salvage (2) = 2. - Demand = 1: Profit = 5. - Demand = 2 or 3: Profit = 5 (can only sell 1).- **Stock 2 units:** - Demand = 0: Profit = 0 + salvage (2) \(\times\) 2 = 4. - Demand = 1: Profit = 5 + salvage (2) = 7. - Demand = 2: Profit = 10. - Demand = 3: Profit = 10 (can only sell 2).- **Stock 3 units:** - Demand = 0: Profit = 0 + salvage (2) \(\times\) 3 = 6. - Demand = 1: Profit = 5 + salvage (2) \(\times\) 2 = 9. - Demand = 2: Profit = 10 + salvage (2) = 12. - Demand = 3: Profit = 15.
03

Determine Expected Profit for Each Stock Level

Calculate the expected profit for each stocking level using the probabilities given:- **Stock 0 units:** Expected Profit = 0.- **Stock 1 unit:** \[\] \text{Expected Profit} = (0.2 \times 2) + (0.3 \times 5) + (0.3 \times 5) + (0.2 \times 5) = 0.4 + 1.5 + 1.5 + 1 = 4.4 \[\] - **Stock 2 units:** \[\] \text{Expected Profit} = (0.2 \times 4) + (0.3 \times 7) + (0.3 \times 10) + (0.2 \times 10) = 0.8 + 2.1 + 3 + 2 = 7.9 \[\]- **Stock 3 units:** \[\] \text{Expected Profit} = (0.2 \times 6) + (0.3 \times 9) + (0.3 \times 12) + (0.2 \times 15) = 1.2 + 2.7 + 3.6 + 3 = 10.5 \[\]
04

Select Optimal Stocking Level

Compare the expected profit of each stocking level. Stocking 3 units provides the highest expected profit of 10.5. Thus, the optimal decision is to stock 3 bushels of strawberries per day.

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Key Concepts

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

Decision Trees
A decision tree is a visual and analytical tool used to map out potential decisions and their possible outcomes. Imagine it as a branching diagram that helps you identify the best course of action by considering various possible scenarios and their impacts. In our exercise, we used a decision tree to decide how many bushels of strawberries to stock each day. This tool breaks down complex decisions into simpler ones by listing possible actions and their corresponding outcomes. By visualizing the different stocking amounts and their results, we can easily compare to find the most profitable choice. In this case, the decision tree helped us evaluate the outcomes for stocking 0, 1, 2, or 3 bushels. One advantage of decision trees is the ability to clearly see the trade-offs between different decisions. By mapping out potential profits and their associated probabilities, you can determine which decision aligns best with your goals, such as maximizing profits.
Expected Profit
Expected profit is a financial concept that helps forecast the potential earnings of a decision, given various possible outcomes and their probabilities. In our strawberry exercise, expected profit is calculated for each stocking level by multiplying the probability of each demand scenario with its corresponding profit and summing these products. Formally, the expected profit for a decision is given by:\[\text{Expected Profit} = \sum ( \text{Probability of Demand} \times \text{Profit in Demand} )\]This calculation allows you to predict the average profit over time, taking into account different possible levels of demand each day. For example, when stocking two bushels, the expected profit was calculated to be 7.9, considering all demand possibilities from 0 to 3 with their respective probabilities. Understanding expected profit helps decision-makers evaluate the likely profitability of a variety of stock levels and choose the option with the highest potential return. It integrates uncertainty into the decision-making process, making it a crucial tool in probability distribution analysis.
Net Profit Maximization
Net profit maximization focuses on choosing the option that delivers the maximum possible profit after all costs have been considered. In our scenario, we examined different stocking strategies to identify which one yielded the highest net profit. By calculating the expected profit for stocking 0, 1, 2, or 3 bushels, and then comparing the outcomes, we picked the option to stock 3 units daily as it offered the highest expected profit of 10.5. Key aspects of net profit maximization:
  • Considering both revenue and cost, including loss from unsold stock and gains from salvage value.
  • Balancing potential high gains with the risk of overstocking and incurring costs.
  • Focusing on the long-run average outcomes to ensure sustainable profitability.
By strategically selecting choices that maximize expected profits, businesses can ensure efficient resource utilization and financial health over time. This exercise demonstrates that careful analysis of probabilities and outcomes can direct toward optimal decision-making for profit maximization.

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Most popular questions from this chapter

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