Palmer Cheese Company is a small manufacturer of several different cheese products. One product is a cheese spread sold to retail outlets. Susan Palmer must decide how many cases of cheese spread to manufacture each month. The probability that demand will be 6 cases is .1, for 7 cases is .3, for 8 cases is .5 and for 9 cases is .1. The cost of every case is $45, and the price Susan gets for each case is $95. Unfortunately, any cases not sold by the end of the month are of no value as a result of spoilage. How many cases should Susan manufacture each month?
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To determine how many cases Susan should manufacture each month, we need to calculate the expected profit for each possible number of cases produced.
Let's break down the problem step by step:
1. Identify the possible demand levels and their associated probabilities. According to the problem, the probabilities of demand are as follows:
- Demand for 6 cases: 0.1
- Demand for 7 cases: 0.3
- Demand for 8 cases: 0.5
- Demand for 9 cases: 0.1
2. Define the selling price and the cost per case:
- Selling price per case: $95
- Cost per case: $45
3. Calculate the profit for each demand level:
- Profit = (Selling price - Cost per case) * Demand
Let's calculate the expected profit for each possible demand level:
- For 6 cases:
Profit = (95 - 45) * 6 = $300
- For 7 cases:
Profit = (95 - 45) * 7 = $350
- For 8 cases:
Profit = (95 - 45) * 8 = $400
- For 9 cases:
Profit = (95 - 45) * 9 = $450
4. Calculate the overall expected profit:
- Expected profit = (Profit for 6 cases * Probability for 6 cases) + (Profit for 7 cases * Probability for 7 cases) + (Profit for 8 cases * Probability for 8 cases) + (Profit for 9 cases * Probability for 9 cases)
Expected profit = ($300 * 0.1) + ($350 * 0.3) + ($400 * 0.5) + ($450 * 0.1)
Expected profit = $30 + $105 + $200 + $45 = $380
The expected profit from manufacturing each month is $380.
Therefore, to maximize profit, Susan should manufacture the number of cases that yields the highest expected profit. In this case, Susan should manufacture the number of cases that corresponds to the highest probability, which is the demand for 8 cases (with a probability of 0.5).
Hence, Susan should manufacture 8 cases of cheese spread each month to maximize her expected profit.