# Optimal Quantity for Midwestern Hardware

What is the optimal quantity for Midwestern Hardware to order if the forecast calls for a 40% chance of a harsh winter?

- The optimal quantity to order for Midwestern if the forecast calls for a 40% chance of a harsh winter is 3000.
- The company must order 3000 shovels in anticipation of a harsh winter to maximize the expected payoff.

## Calculating the Optimal Quantity

**Given Data:**

- Cost of each shovel: $15.00
- Selling price of each shovel: $29.95
- Discount selling price of each shovel: $10.00
- Probability of harsh winter: 40%
- Probability of mild winter: 60%

## Probability Distribution of Regular Price Demand:

- Regular price demand: 250, 300, 350, 400, 450, 500, 550
- Probabilities: 0.2, 0.3, 0.1, 0.1, 0.2, 0.05, 0.05

## Calculation of Probability of Demand:

- Various probabilities of demand for different quantities of shovels can be calculated based on the given data.

## Payoff Table Construction:

- Expected payoff is calculated by multiplying the probability with the difference between revenue and cost.
- For Midwestern Hardware, expected payoffs for different order quantities can be calculated using the provided data.
- The order quantity with the highest expected payoff is considered the optimal quantity to order.

## Result:

- The optimal quantity for Midwestern Hardware to order in anticipation of a 40% chance of a harsh winter is 3000 shovels.
- This quantity maximizes the expected payoff and ensures the company is prepared for the potential demand during a harsh winter season.