Question

This comes from the book Decision Technology: Modeling, Software and Applications by Matthew J. Liberatore & Robert L. Nydick ISBN: 0471-41712-2

The Subject is actually Systems Engineering

TOOLCRIB is a production shop of a large industrial corporation that services skilled workers. Attendants dispense tools on a first-come-first serve basis to the skilled workers who need tools during an eight-hour workday. Recently, the vice president of production has been receiving complaints of excessively long waiting lines for the skilled workers. A subsequent study indicates that approximately 30 workers per hour arrive at the tool crib, yet an attendant can service only approximately 20 workers per hour. Assume that the arrival time distribution is exponential and the service time distribution is Erlang with a k value of 3. The company has found that it costs $30.00 per hour for each attendant. They also have found that it costs $45.00 per hour to have their skilled workers wait in the que. (In assessing this cost, they have decided that the service time should not be included in the waiting cost.)

a. What is the minimum number of attendants that must be hired

b. Perform an analysis to determine the number of attendants that are needed to minimize average total costs. You can use either trial and error or optimization?

Answer 1

arrival rate lambda = 30 per hour

service rate mu = 20 per hour

utilisation rho = lambda / mu = 30/20 =1.5

SInce the rho value is 1.5, minimum two servers are needed for a sustainable line.

Scenario 1 - 2 servers

Lq for rho = 1.5 and 2 servers =1.929

avg waiting time per customer = 1.929 / lambda = 1.929 /30 =

Total waiting time = 1.929/30 x30 = 1.929 hours

cost of waiting = 1.929 x 45 = 86.8

Total cost = 30x2 +86.8 = 146.8

Scenaio 2 - 3 servers

Lq for rho = 1.5 and 3 servers = 0.237

Total waiting time = 0.237 hours

cost of waiting = 0.237x45 =10.665

Total cost = 30x3+10.665 = 100.665

Calculation with 4 servers is irrelevant as the cost would be more than the cost of employing 4 servers which is 120.

The min cost of 100.66 is with 3 servers.