A hot dog vendor must decide on Monday how many hot dogs to have available for the coming Saturday's football game. Each hot dog costs the vendor $3.00 and is sold for $5.00. After the game any unsold hot dogs are discounted and sold to the university cafeteria for $1.75. The vendor believes that the demand for hot dogs follows the probability distribution shown below: The optimal number of hot dogs the vendor should order for next Saturday's game is
A) 1000
B) 1500
C) 2000
D) 3000

Question

A hot dog vendor must decide on Monday how many hot dogs to have available for the coming Saturday's football game. Each hot dog costs the vendor $3.00 and is sold for $5.00. After the game any unsold hot dogs are discounted and sold to the university cafeteria for $1.75. The vendor believes that the demand for hot dogs follows the probability distribution shown below:  The optimal number of hot dogs the vendor should order for next Saturday's game is
A) 1000 
B) 1500 
C) 2000 
D) 3000

Quizplus · Accepted Answer

The vendor's goal is to maximize their profit, which is revenue minus cost. The revenue from selling a hot dog is $5.00, and the cost is $3.00, so the profit is $2.00 per hot dog sold.

To determine the optimal number of hot dogs to order, the vendor needs to consider the expected demand and the potential for unsold hot dogs to be sold to the university cafeteria.

Using the probability distribution provided, we can calculate the expected demand as:

Expected demand = (0.1 x 1000) + (0.3 x 1500) + (0.4 x 2000) + (0.2 x 2500) + (0.1 x 3000) = 1950

This means that on average, the vendor can expect to sell 1950 hot dogs during the game.

However, the vendor also needs to consider the potential for unsold hot dogs. To do this, they should calculate the expected number of unsold hot dogs as:

Expected unsold hot dogs = 2500 - 1950 = 550

This means that on average, there will be 550 unsold hot dogs after the game.

The vendor can sell these unsold hot dogs to the university cafeteria for $1.75 each, so the revenue from these sales would be:

Revenue from unsold hot dogs = 550 x $1.75 = $962.50

Now we can calculate the total profit for each possible number of hot dogs ordered:

Profit for 1000 hot dogs = (1000 x $2) - $3000 = -$1000
Profit for 1500 hot dogs = (1500 x $2) - $4500 = $1500
Profit for 2000 hot dogs = (1950 x $2) + ($962.50) - $6000 = $725
Profit for 2500 hot dogs = (1900 x $2) + ($962.50) - $7500 = $362.50
Profit for 3000 hot dogs = (1850 x $2) + ($962.50) - $9000 = -$187.50

Based on these calculations, the optimal number of hot dogs for the vendor to order is 2000, which would result in a profit of $725. Ordering any more hot dogs would result in a decrease in profit due to the increased cost of extra inventory, and ordering fewer hot dogs would result in lost sales and potential lost profit.

A Hot Dog Vendor Must Decide on Monday How Many