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.

Understanding the Cost of Machine Failures in a Factory

Machines in a factory break down at an exponential rate of six per hour. There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost production when machines are out of service is $10 per hour per machine. What is the average cost rate incurred due to failed machines?

In a factory, machine breakdowns are a common and costly issue. The cost of machine failures can have a significant impact on a factory's operations and bottom line. In this blog post, we will explore the average cost rate incurred due to failed machines in a factory where machines break down at an exponential rate of six per hour and there is a single repairman who fixes machines at an exponential rate of eight per hour.

Calculating the Average Cost Rate:

To calculate the average cost rate incurred due to failed machines, we need to take into account the cost of lost production when machines are out of service. In this case, the cost incurred in lost production is $10 per hour per machine. To calculate the average cost rate, we need to know the average number of machines that are out of service at any given time.

The average number of machines that are out of service can be calculated as follows:

Average number of machines out of service = (Breakdown rate) / (Repair rate – Breakdown rate) = (6) / (8 – 6) = 6

The average cost rate incurred due to failed machines can then be calculated as follows:

Average cost rate = (Average number of machines out of service) * ($10 per hour per machine) = (6) * ($10) = $60 per hour

The cost of machine failures can have a significant impact on a factory's operations and bottom line. By understanding the average cost rate incurred due to failed machines, factories can take steps to minimize these costs and improve their operations. In this example, the average cost rate was found to be $60 per hour, which highlights the importance of having an efficient repair process and minimizing machine downtime.