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# Farm Grown, Inc., produces cases of perishable food products. Each case contains an assortment of vegetables and other farm products. Each case costs $5 and sells for$15. If there are any not sold by the end of the day, they are sold to a large food processing company for $3 a case. The probability that daily demand will be 100 cases is 0.30, the probability that daily demand will be 200 cases is 0.40, and the probability that daily demand will be 300 cases is 0.30. Farm Grown has a policy of always satisfying customer demands. If its own supply of cases is less than the demand, it buys the necessary vegetables from a competitor. The estimated cost of doing this is$16 per case. $a$ Draw a decision table for this problem $b$ What do you recommend?

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## Answer to a math question Farm Grown, Inc., produces cases of perishable food products. Each case contains an assortment of vegetables and other farm products. Each case costs $5 and sells for$15. If there are any not sold by the end of the day, they are sold to a large food processing company for $3 a case. The probability that daily demand will be 100 cases is 0.30, the probability that daily demand will be 200 cases is 0.40, and the probability that daily demand will be 300 cases is 0.30. Farm Grown has a policy of always satisfying customer demands. If its own supply of cases is less than the demand, it buys the necessary vegetables from a competitor. The estimated cost of doing this is$16 per case. $a$ Draw a decision table for this problem $b$ What do you recommend?

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$a$ Decision Table for Farm Grown, Inc. Demand Produce Sell Loss/Gain 100 100 100 $300 100 150 100$150 100 200 100 -$50 100 250 100 -$150 200 100 100 -$200 200 150 150 -$50 200 200 200 $0 200 250 200 -$50 300 100 100 -$400 300 150 150 -$250 300 200 200 -$100 300 250 200 -$50 300 300 300 \$0 Notes: The "Produce" column denotes the number of cases Farm Grown should produce. The "Sell" column denotes the number of cases Farm Grown actually sells. The "Loss/Gain" column shows the profit/loss for each scenario. A positive value indicates a gain, while a negative value indicates a loss. The price of buying vegetables from a competitor is not explicitly shown in the table, but it is factored into the "Loss/Gain" calculation. $b$ Recommendation Based on the decision table, it is difficult to make a definitive recommendation without considering some additional factors: Risk tolerance: If Farm Grown is risk-averse, they may want to produce closer to the expected demand $around 200 cases$ to minimize the potential for losses. However, this could also mean missing out on potential profits if demand is higher than expected. Inventory holding costs: Storing unsold cases incurs additional costs for Farm Grown. This factor might sway the decision towards producing closer to the expected demand. Competitor's supply: If the competitor from whom Farm Grown buys vegetables has limited supply, it might be risky to rely on them to always be able to fulfill the additional demand. Here are some potential recommendations based on different priorities: Maximizing expected profit: Based on the expected demand probabilities, producing 200 cases seems to be the best option in terms of maximizing expected profit. However, this strategy comes with the risk of missing out on profits if demand is higher than expected. Minimizing risk: Producing 150 cases would ensure that Farm Grown avoids losses in all scenarios except when demand is 300. However, this strategy also sacrifices potential profits from higher demand. Considering inventory holding costs: If inventory holding costs are significant, the optimal production level might be even lower than 150 cases. Ultimately, the best decision for Farm Grown will depend on their specific preferences and constraints. They should carefully consider all the relevant factors before making a choice.
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