Finding the Optimal Service Provider in a Market

The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can be hired at a rate of $C per hour. The manager estimates that, on the average, each customer’s time is worth $1 per hour and should be accounted for in the model.

Assume customers arrive at a Poisson rate of 10 per hour

(a) What is the average cost per hour if Mary is hired? If Alice is hired?

(b) Find C if the average cost per hour is the same for Mary and Alice.

In a market, hiring the right service provider can have a significant impact on the bottom line. The manager of a market is considering hiring either Mary or Alice, each of whom gives service at different rates and costs. In this blog post, we will explore the average cost per hour if Mary or Alice is hired and find the value of C if the average cost per hour is the same for both.

Calculating the Average Cost per Hour:

To calculate the average cost per hour if Mary or Alice is hired, we need to take into account their service rates and costs, as well as the value of each customer's time. In this case, Mary gives service at an exponential rate of 20 customers per hour and can be hired at a rate of $3 per hour. Alice gives service at an exponential rate of 30 customers per hour and can be hired at a rate of $C per hour. The manager estimates that each customer's time is worth $1 per hour.

The average cost per hour if Mary is hired can be calculated as follows:

Average cost per hour (Mary) = ($3 per hour) + (10 customers per hour) * ($1 per hour per customer) = $13 per hour

The average cost per hour if Alice is hired can be calculated as follows:

Average cost per hour (Alice) = ($C per hour) + (10 customers per hour) * ($1 per hour per customer) = $C + $10 per hour

Finding C if the Average Cost per Hour is the Same for Mary and Alice:

To find the value of C if the average cost per hour is the same for Mary and Alice, we need to set the two costs equal to each other and solve for C.

$13 per hour = $C + $10 per hour

$C = $3 per hour

The manager of a market must consider the service rates and costs of different service providers to determine the optimal choice. In this example, the average cost per hour was found to be the same for Mary and Alice when C = $3 per hour. By understanding these calculations, the manager can make an informed decision about which service provider to hire and minimize costs for the market.