Question

A chemical plant stores spare parts for maintenance in a large warehouse. Throughout the working day, maintenance personnel go to the warehouse to pick up supplies needed for their jobs. The warehouse receives a request for supplies, on average, every three minutes. The average request requires 2.75 minutes to fill a request. Maintenance employees are paid $21.50 per hour and warehouse employees are paid $16 per hour. The warehouse operates 8 hours per day. a) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there is only 1 warehouse employees working? b) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there are 2 warehouse employees working? c) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there are 3 warehouse employees working? d) What is the optimal number of warehouse employees to staff the warehouse?

Answer 1

a) The number of maintenance employees is not specified in the problem, so consider that it is an infinite number. So, it is a M/M/1 system with following parameters,

Arrival rate, \lambda = 1/interarrival time = 1/3 per minute = (1/3)*60 = 20 per hour

Service rate, \mu = 1/service time = (1/2.75)*60 = 21.82 per hour

Number of maintenance employees in the system, L = \lambda/(\mu-\lambda) = 20/(21.82-20) = 11

System cost per hour = Cw*L + s*Cs

= 21.5*11+1*16

= $ 252.5

System cost per day = 252.5*8

= $ 2,020

b) With more than 1 warehouse employees (servers) working, it is a multi-channel M/M/s system , where

Number of servers, s = 2

Operating characteristics are computed using spreadsheet as below:

[17:29, 4/13/2019] nag: EXCEL FORMULAS:
[17:29, 4/13/2019] nag: Cell   Formula   Copy to
H3   =($D$4/$D$5)^G2/FACT(G2)+H2   H3:H26
D9   =1/((VLOOKUP(D6+1,$G$2:$H$26,2,0)-VLOOKUP(D6,$G$2:$H$26,2,0))/(1-D4/(D5*D6))+VLOOKUP(D6,$G$2:$H$26,2,0))  
D10   =D4/D5/D6  
D11   =(VLOOKUP(D6+1,$G$2:$H$26,2,0)-VLOOKUP(D6,$G$2:$H$26,2,0))*D9*D10/(1-D10)^2+D4/D5  
D12   =D11-D4/D5  
D13   =D11/$D$4  
D14   =D12/$D$4  
L = 1.1604

Total system cost per hour = 1.1604*21.5+2*16 = 56.95

Total system cost per day = 56.95*8 = $ 455.6

b) With more than 1 warehouse employees (servers) working, it is a multi-channel M/M/s system , where

Number of servers, s = 3

Operating characteristics are computed using spreadsheet as below:

L = 0.9489

Total system cost per hour = 0.9489*21.5+3*16 = 68.40

Total system cost per day = 68.40*8 = $ 547.2

d) We see that total system cost per day of 2 warehouse employees working is the lowest.

Therefore, optimal number of warehouse employees to staff the warehouse = 2