In: Statistics and Probability
Problem 11-21 (Algorithmic)
Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.7 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. Agan's management would like to evaluate two alternatives:
If the consultants are paid $13 per hour and the customer waiting time is valued at $21 per hour for waiting time prior to service, should Agan expand to the two-consultant system?
What is the total cost for each scenario? Round your answers to the nearest cent.
The total cost for the first scenario where there is one consultant with an average service time of 8 minutes per customer is $ . The total cost for the second scenario where there are two consultants with an average service time of 10 minutes per customer is $ .
Note: Use P0 values from Table 11.4 to answer the questions below.
Given:
If the consultants are paid $13 per hour and the customer waiting time is valued at $21 per hour for waiting time prior to service, should Agan expand to the two-consultant system NO
Now
Average customer arrival = λ =2.7 customers/hour
Average consultant time1 = μ = 8 minutes/customer
Average consultant time2 = μ = 10 minutes/customer
Cost of consultants service = $13
Cost of customer waiting time = $21
Average consultant time1 = μ = 60/8 = 7.5 hours/customer
Average consultant time2 = μ = 60/10 = 6 hours/customer
Calculations for one consultant
Average no. of customers waiting for service = Lq = λ²/μ(μ-λ)
Average no. of customers waiting for service = Lq = (2.7)²/7.5(7.5 - 2.5)
Average no. of customers waiting for service = Lq = 0.1944
Average no. of customers in the system = L = Lq + λ/μ
==> L = 0.1944 + 2.7/7.5
==> L = 0.5544
Total cost = (customer waiting time cost)*L + consultant service cost
Total cost = $21*0.5544 + $13
Total cost = $26.6424
Calculations for two consultants
Average no. of customers waiting for service = Lq = ((λ/μ)^k(λμ)/(k -1)(kμ - λ))*P₀
Where k = 2 is the number of consultants and P₀ is the probability that all of the k consultants are idle. The value of P₀ can be found in the tables with λ/μ = 2.7/6 = 0.45 and k = 2, P₀ ≅ 0.6327
Average no. of customers waiting for service = Lq = (2.7/6)2(2.7*6)/(2-1)(2*6 - 2.7)*0.6327
Average no. of customers waiting for service = Lq = 0.2231
Average no. of customers in the system = L = Lq + λ/μ
==> L = 0.2231 + 2.7/6
==> L = 0.6731
Total cost = (customer waiting time cost)*L + consultant service cost
Total cost = $21*0.6731 + ($13)*2 (since there are 2 consultants now)
= 14.1351 + 26
Total cost = $40.1351
It is recommended to use one consultant with an average service time of 8 minutes per customer since the total cost is less than the other option.
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