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 three 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, once 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. 9780357432907, Quantitative Methods for Business, Thirteenth Edition, David R. Anderson, - © Cengage Learning. All Rights Reserved. No distribution allowed without express authorization. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-247. Distributed by Grand Canyon University. 718 Chapter 15 Waiting Line Models OEI is planning to expand its business. Within one year, OEI projects that it will have 20 customers, and within two 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 three-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 three-hour waiting time guarantee at the lowest possible total cost. Managerial Report Develop a managerial report summarizing your analysis of the OEI 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? 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. 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? 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 ● Average number of customers in the waiting line ● Average number of customers in the system ● Average time a customer waits until the service technician arrives ● Average time a customer waits until the machine is back in operation ● Probability that a customer will have to wait more than one hour for the service technician to arrive ● The total cost per hour for the service operation Do you agree with OEI management that one technician can meet the average threehour service call guarantee? Explain. 5. 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 part (4) to justify your answer. 6. 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 part (4) to justify your answer. 7. What are the annual savings of your recommendation in part (6) compared to the planning committee’s proposal that 30 customers will require three service technicians

Answer 1

Technician arrival time at customer site: average 3 hours from time of notifying OEI

No. of customer with contract with OEI: 10

No. of technician to handle all service calls (n):1

Average no. of customer call / 50 hours: 1

Travel time to customer by technician: 1 hour

Time to complete repair: 1.5 hour

Order of completing calls: current service call > waiting call > new call

Cost of service technician: $80 /hour

Downtime and service time cost for customer: $100/hour

OEI plan to expand:

1 year – 20 customer- 2 technician

2 years - 30 customer – 3 technician

1. Arrival time is the average number of customers per hour

Average no. of customer arrival / 50 hours: 1

1^st year:

No. of customers: 20

In 50 hours - 1 customer arrives

In 50 hours – 20 customers arrive if all are serviced without wait time

In hour – 20/50 customer arrive

Thus Arrival rate (a) = 0.4 customers per hour

2^nd year:

No. of customers: 30

In 50 hours - 1 customer arrives

In 50 hours – 30 customers arrive if all are serviced without wait time

In hour – 30/50 customer arrive

Thus Arrival rate (a) = 0.6 customers per hour

1. Service time is average number of customers that can be served during the hour4

Travel time to customer by technician: 1 hour

Time to complete repair: 1.5 hour

Total service time for one customer = 2.5 hours

So,

In 2.5 hours, he services 1 customer

Service rate (s) = no. of customer served per hour = 1 / 2.5 = 0.4 customers/hour

1. 1^st year:

Average utilization (u) = Arrival time / (no. of resources * Service time) = 0.4/ (2*0.4) = 0.5 or 50%

Average waiting time for customer= average no. of customer waiting in line/ arrival rate = 0.22 /0.4 = 0.54 hour

1. a) Probability of no customers in system (p0) = summation of all probability of arrival rate/service rate = (((0.4/0.4) ^0)/ 0! + ((0.4/0/4)/1!) + (((0.4/0.4)^2)/ 2!) * (1/0.5)) ) ^-1 = 1/3 = 0.33

b) Average no. of customers in waiting line (W) = ((p0 (a/s) ^n) u)/ (n!(1 – u)^2) = (0.33 ^2)/ (2*0.25) = 0.22

c) Average no. of customers in system: 20

d) Average time customer waits until service technical arrives= = average no. of customer waiting in line/ arrival rate = 0.22 /0.4 = 0.54 hour

e) Average time customer waits until machine is back in operation = time for fixing the machine + arrival time for technician + waiting time = 1.5 +1+0.22 = 2.72 hours

f) Total cost per hour for the service operation = technician cost + downtime cost

= 80*2.5 + 2.72 *100 = $292