Question

Office Equipment, Inc. (OEI) leases automatic mailing machines to business customers in Fort Wayne, Indiana. The company built its success on a reputation of providing timely maintenance and repair service. Each OEI service contract states that a service technician will arrive at a customer’s business site within an average of 3 hours from the time that the customer notifies OEI of an equipment problem.

Currently, OEI has 10 customers with service contracts. One service technician is responsible for handling all service calls. A statistical analysis of historical service records indicates that a customer requests a service call at an average rate of one call per 50 hours of operation. If the service technician is available when a customer calls for service, it takes the technician an average of 1 hour of travel time to reach the customer’s office and an average of 1.5 hours to complete the repair service. However, if the service technician is busy with another customer when a new customer calls for service, the technician completes the current service call and any other waiting service calls before responding to the new service call. In such cases, after the technician is free from all existing service commitments, the technician takes an average of 1 hour of travel time to reach the new customer’s office and an average of 1.5 hours to complete the repair service. The cost of the service technician is $80 per hour. The downtime cost (wait time and service time) for customers is $100 per hour.

OEI is planning to expand its business. Within 1 year, OEI projects that it will have 20 customers, and within 2 years, OEI projects that it will have 30 customers. Although OEI is satisfied that one service technician can handle the 10 existing customers, management is concerned about the ability of one technician to meet the average 3-hour service call guarantee when the OEI customer base expands. In a recent planning meeting, the marketing manager made a proposal to add a second service technician when OEI reaches 20 customers and to add a third service technician when OEI reaches 30 customers. Before making a final decision, management would like an analysis of OEI service capabilities. OEI is particularly interested in meeting the average 3-hour waiting time guarantee at the lowest possible total cost.

Managerial Report

Develop a managerial report (1,000-1,250 words) summarizing your analysis of the OEI service capabilities. Make recommendations regarding the number of technicians to be used when OEI reaches 20 and then 30 customers, and justify your response. Include a discussion of the following issues in your report:

1. What is the arrival rate for each customer?
2. What is the service rate in terms of the number of customers per hour? (Remember that the average travel time of 1 hour is counted as service time because the time that the service technician is busy handling a service call includes the travel time in addition to the time required to complete the repair.)
3. Waiting line models generally assume that the arriving customers are in the same location as the service facility. Consider how OEI is different in this regard, given that a service technician travels an average of 1 hour to reach each customer. How should the travel time and the waiting time predicted by the waiting line model be combined to determine the total customer waiting time? Explain.
4. OEI is satisfied that one service technician can handle the 10 existing customers. Use a waiting line model to determine the following information: (a) probability that no customers are in the system, (b) average number of customers in the waiting line, (c) average number of customers in the system, (d) average time a customer waits until the service technician arrives, (e) average time a customer waits until the machine is back in operation, (f) probability that a customer will have to wait more than one hour for the service technician to arrive, and (g) the total cost per hour for the service operation.
5. Do you agree with OEI management that one technician can meet the average 3-hour service call guarantee? Why or why not?
6. What is your recommendation for the number of service technicians to hire when OEI expands to 20 customers? Use the information that you developed in Question 4 (above) to justify your answer.
7. What is your recommendation for the number of service technicians to hire when OEI expands to 30 customers? Use the information that you developed in Question 4 (above) to justify your answer.
8. What are the annual savings of your recommendation in Question 6 (above) compared to the planning committee's proposal that 30 customers will require three service technicians? (Assume 250 days of operation per year.) How was this determination reached?

Answer 1

Develop a managerial report summarizing your analysis of the OEl service capabilities. Make recommendations regarding the number of technicians to be used when OEI reaches 20 customers and when OEI reaches 30 customers. Include a discussion of the following issues in your report.

1. What is the arrival rate for each customer per hour?

Answer: call/50 hours = 0.02 calls per hours

2. What is the service rate in terms of the number of customers per hour? Note that the average travel time of 1 hour becomes part of the service time because the time that the service technician is busy handling a service call includes the travel time plus the time required to complete the repair?
Answer: Average Service time = travel time + repair time = 1+ 1.5= 2.5 hours

customers per hour

3. Waiting line models generally assume that the arriving customers are in the same location as the service facility. Discuss the OEI situation in light of the fact that a service technician travels an average of 1 hour to reach each customer. How should the travel time and the waiting time predicted by the waiting line model be combined to determine the total customer waiting time?
Answer: We know that the travel time is 1 hour. Since this is considered as a part of service time, it means that the customer will have to wait during the first hour of the service time. Therefore, travel time must be included in the time spent in line as a predicted model so as to determine the total customer's waiting time.

4.OEI is satisfied that one service technician can handle the 10 existing customers. Use a waiting line model to determine the following information:

• Probability that no customers are in the system

Answer: The probability that no customers are in the system is 0.5380

• Average number of customers waiting in line

Answer: Average number of customers waiting in line is 0.2972

• Average number of customers in the system

Answer: Average Number of customers in the system is 1- 0.2972 = 0.7593

• Average time a customer waits until the service technician arrives

Answer: Average time a customer waits until the service technician arrives is 1.6082 hours

• Average time a customer waits until the machine is back in operation

Answer: Average time a customer waits until the machine is back in operation is 4.1082 hours

• Probability that a customer will have to wait more than one hour for the service technician to arrive

Answer: Probability that customers will have to wait is 0.4620

5. What is your recommendation for the number of service technicians to hire when OEl expands to 20 customers? Use the information that you developed in part (4) to justify your answer.
Answer: If the company continues to use one technician when the customer base expands to 20 customers, the average time in the waiting line will increase to 6.9454 hours. With an average travel time of 1hour, the average total waiting time will be 6.9454 +1 7.9454 hours The total cost will be$397.78 per hour. However, this average total waiting time is too long and a second technician is definitely necessary.

If The company hires two service technicians then the output is as follows:
Probability that no customers are in the system is 0.3525
Average number of customers in the waiting line is 0.2104
Average number of customers in the system is 1.1527
Average time a customer spends in the waiting line is 0.5581 hours*
Average time until the machine is back in operation is 3.0581 hours
Probability of a wait more than one hour is 0.2949
Hours a week the technicians are not on service calls
PO = 0.3525(0.3525) x 2 technicians x 40 hours = 28.2 hours
P1 = 0.3525(0.3525) x 1 technician x 40 hours = 14.1 hours
Total = 42.3 hours
Total Cost per hour of service operation $275.27

Form the above output; we see that the average waiting time one customer spends is 0.5581 hours. This is the time that the service technician takes to complete his all previous commitments and is ready to move to the new customer. As we know that the average travel time for the service technician is 1-hour to reach to his new destination, the customer's total
waiting time is 0.5581 +1 -1.5581 hours. Therefore, company requires two service technicians to meet the company's 3-hour service guideline when the company expands its base to 20 customers. Hence, the total cost is $275.27 per hour.

6. What is your recommendation for the number of service technicians to hire when OEl expands to 30 customers? Use the information that you developed in part (4) to justify your answer
Answer: When the customer base expands to 30 customers, we see that the average total waiting time with two technicians will be 2.6895 hours and the average total waiting time with three technicians will be 1.2626 hours. The cost per hour with two technicians is $391.94 and the cost per hour with three technicians is $397.08. While three technicians provide a smaller waiting time, two technicians are able to meet the 3-hour service guideline for a total
lower cost. Thus, the company should continue to use two technicians when the customer base expands to 30 customers.
If The company hires two service technicians then the output is as follows:
Probability that no customers are in the system is 0.1760
Average number of customers in the waiting line is 0.9353
Average number of customers in the system is 2.3194
Average time a customer spends in the waiting line is 1.6895 hours
Average time until the machine is back in operation is 4.1895 hours
Probability of a wait more than one hour is 0.5600
Hours a week the technicians are not on service calls
PO = 0.1760(0.1760) x 2 technicians x 40 hours = 14.08 hours
P1 = 0.2640(0.2640) x 1 technician x 40 hours = 10.56 hours
Total = 24.64 hours
Total Cost per hour of service operation $391.94

Form the above output; the average time a customer spends in the waiting line is 1.6895
hours. While the average travel time is 1-hour for the service technician to reach the new
customer's office, the average total customer waiting time is 1.6895 +1= 2.6895 hours.

7.What are the annual savings of your recommendation in part (6) compared to the planning committee's proposal that 30 customers will require 3 service technicians? Assume 250 days of operation per year.
Answer: The OEl planning committee's proposal concluded that hiring three technicians will have a total cost of $397.08 per hour. Therefore, it is recommended to stay with two technicians which will have an annual savings of (397.08 - 391.94) x 8 hours/day x 250 days/year = $10,280