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Chapter 3: Problem 219

A large bakery can produce rolls in lots of either \(0,\) \(1000,2000,\) or 3000 per day. The production cost per item is \(\$ 0.10 .\) The demand varies randomly according to the following distribution: $$ \begin{array}{lcccc} \text { Demand for rolls } & 0 & 1000 & 2000 & 3000 \\ \text { Probability of demand } & 0.3 & 0.2 & 0.3 & 0.2 \end{array} $$ Every roll for which there is a demand is sold for \(\$ 0.30 .\) Every roll for which there is no demand is sold in a secondary market for \(\$ 0.05 .\) How many rolls should the bakery produce each day to maximize the mean profit?

Short Answer

Expert verified
Produce 2000 rolls per day to maximize profit.

Step by step solution

01

Understand the Cost and Revenue Structure

For every roll produced, the bakery incurs a cost of $0.10. If there is demand, the roll is sold for $0.30, providing a profit of $0.20 per roll. If there is no demand, the roll is sold for $0.05, resulting in a loss of $0.05 per roll compared to the production cost.
02

Calculate Expected Profit for Each Production Level

For each production level (0, 1000, 2000, 3000), calculate expected profit by considering all possible demand scenarios, their probabilities, and profit outcomes. **Example for 1000 rolls:** - **Demand = 0:** All 1000 rolls go unsold, profit = 1000 * (-0.05) = -$50. - **Demand = 1000:** All are sold, profit = 1000 * 0.20 = $200. - **Demand = 2000 or 3000:** Only 1000 rolls are produced and sold, profit = $200 (same as demand=1000). Expected Profit for 1000 rolls = 0.3 * (-$50) + 0.2 * $200 + 0.3 * $200 + 0.2 * $200 = -$15 + $40 + $60 + $40 = $125.
03

Repeat Profit Calculation for Other Production Levels

Calculate the expected profit for producing 0, 2000, and 3000 rolls using similar approach: **Production = 0 rolls:** - Profit = 0 for any demand as nothing is produced. Expected profit = $0. **Production = 2000 rolls:** - **Demand = 0:** 2000 unsold rolls, profit = -$100. - **Demand = 1000:** 1000 sold at $0.20, 1000 unsold at $0.05, profit = 1000 * 0.20 + 1000 * (-0.05) = $150. - **Demand = 2000:** All sold, profit = 2000 * 0.20 = $400. - **Demand = 3000:** 2000 sold, profit same as demand = 2000, $400. Expected Profit for producing 2000 rolls = 0.3 * (-$100) + 0.2 * $150 + 0.3 * $400 + 0.2 * $400 = -$30 + $30 + $120 + $80 = $200. **Production = 3000 rolls:** - **Demand = 0:** 3000 unsold, profit = -$150. - **Demand = 1000:** 1000 sold, 2000 unsold, profit = $100. - **Demand = 2000:** 2000 sold, 1000 unsold, profit = $250. - **Demand = 3000:** All sold, profit = 3000 * 0.20 = $600. Expected Profit for producing 3000 rolls = 0.3 * (-$150) + 0.2 * $100 + 0.3 * $250 + 0.2 * $600 = -$45 + $20 + $75 + $120 = $170.
04

Compare Expected Profits and Choose Optimal Decision

Compare calculated expected profits: - 0 rolls: $0 - 1000 rolls: $125 - 2000 rolls: $200 - 3000 rolls: $170 The bakery should produce 2000 rolls each day to maximize the mean profit, as this yields the highest expected profit ($200).

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Key Concepts

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

Probability Distribution
Understanding probability distribution is crucial when dealing with varying outcomes and their respective likelihoods. In this context, it refers to knowing how likely different demands will occur for the bakery's rolls. The exercise gives a probability distribution, where demand for 0, 1000, 2000, and 3000 rolls occurs with probabilities of 0.3, 0.2, 0.3, and 0.2 respectively.
This means:
  • There's a 30% chance no demand exists.
  • A 20% chance the demand equals 1000 rolls.
  • Another 30% chance sees demand rising to 2000 rolls.
  • Finally, a 20% chance that all 3000 rolls are in demand.

The bakery uses this distribution to estimate overall demand and decide on the number of rolls to produce. By assessing each possible scenario and its probability, the bakery can predict expected profits for every production level, guiding them towards an optimal decision on how many rolls to bake each day.
Profit Optimization
Profit optimization is about making the best decision to achieve the highest profit possible, given constraints like production capacity and demand variability. The bakery's task is to optimize profits under uncertainty of demand.
In the exercise, profit is calculated for each scenario:
  • If demand matches production, the profit is maximized as every roll produced is sold at full price.
  • If demand is less than production, excess rolls are sold at a loss on the secondary market, reducing overall profitability.
  • If demand exceeds production, the bakery loses potential profit, as it cannot meet the full demand.

The calculations showed varying profits for each production scenario: - 0 Rolls: Expected profit = $0 - 1000 Rolls: Expected profit = $125 - 2000 Rolls: Expected profit = $200 - 3000 Rolls: Expected profit = $170
The optimum choice in this scenario is to produce 2000 rolls, as this results in the highest expected profit of $200.
Decision Making in Uncertainty
Decision making in uncertainty involves choosing actions in situations where outcomes aren't guaranteed. It requires balancing potential rewards with risks. Here, the bakery must decide how many rolls to produce each day without knowing beforehand the actual demand.
Several factors are considered:
  • Cost of production per roll versus potential revenue from sales at different demand levels.
  • Probability distribution of demand helps estimate expected outcomes for each production scenario.
  • Understanding the risk of producing too many or too few rolls.

The goal is to make a decision that not just maximizes profit but also accounts for variability in customer demand. By methodically calculating expected profit for each production scenario and comparing them, the bakery identifies the most financially favorable option, which is producing 2000 rolls daily. This approach exemplifies strategic decision-making under uncertainty, relying heavily on probabilistic reasoning and financial calculations.

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