Question

A construction company distributes its products by trucks loaded at its loading station. A backacter in conjunction with trucks are used for this purpose. If it was found out that on an average of 12 trucks per hour arrived and the average loading time was 3 minutes for each truck. A truck must queue until it is loaded. The backacter’s daily all-in rate is GH¢ 1000 and that of the truck is GH¢ 400.

a) Compute the operating characteristics: L, Lq, W, Wq, and P.

b) The company is considering replacing the backacter with a bigger one which will have an average service rate of 1.5 minutes to serve trucks waiting to have their schedules improved. As a manager, would you recommend the new backacter if the daily all-in rate is GH¢ 1300.

c) The site management is considering whether to deploy an extra backwater to assist the existing one. The daily all-in-rate and efficiency of the new backwater is assumed to be the same as that of the existing backwater. Should the additional backwater be deployed?

Answer 1

a)
`L = 1.5`

`L_q = 0.9`

`W = (1 )/(8 ) \, hour`

`W_q = (3)/(40 ) \, hour`

`P = (3)/(5 )`

b) The new backacter should be recommended

c) The additional backacter should not be deployed

Step-by-step explanation:

a) The required parameters are;

L = The number of customers available

`L = (\lambda )/(\mu -\lambda )`

μ = Service rate

`L_q` = The number of customers waiting in line

`L_q = p* L`

W = The time spent waiting including being served

`W = (1 )/(\mu -\lambda )`

`W_q` = The time spent waiting in line

`W_q = P * W`

P = The system utilization

`P = (\lambda )/(\mu )`

From the information given;

λ = 12 trucks/hour

μ = 3 min/truck = 60/3 truck/hour = 20 truck/hour

Plugging in the above values, we have;

`L = (12 )/(20 -12 ) = (12 )/(8 ) = 1.5`

`P = (12 )/(20 ) = (3)/(5 )`

`L_q = (3)/(5 ) * (3)/(2 ) = (9)/(10 ) = 0.9`

`W = (1 )/(20 -12 ) = (1 )/(8 ) \ hour`

`W_q = (3)/(5 ) * (1)/(8 ) = (3)/(40 ) \, hour`

(b) The service rate with the new backacter = 1.5 minutes/truck which is thus;

μ = 60/1.5 trucks/hour = 40 trucks/hour

`P = (12 )/(40 ) = (3)/(10)`

`W = (1 )/(40 -12 ) = (1 )/(38 ) \, hour`

`W_q = (3)/(10 ) * (1)/(38 ) = (3)/(380 ) \, hour`

λ = 12 trucks/hour

Total cost =
`mC_s + \lambda WC_w`

m = 1

`C_s` = GH¢ = 1300

`C_w` = 400

Total cost with the old backacter is given as follows;

`1 * 1000 + 12 * (1)/(8) * 400 = \$ 1,600.00`

Total cost with the new backacter is given as follows;

`1 * 1300 + 12 * (1)/(38) * 400 = \$ 1,426.32`

The new backacter will reduce the total costs, therefore, the new backacter is recommended.

c)

Here μ = 3 min/ 2 trucks = 2×60/3 truck/hour = 40 truck/hour

`\therefore W = (1 )/(40 -12 ) = (1 )/(38 ) \, hour`

Total cost with the one backacter is given as follows;

`1 * 1000 + 12 * (1)/(8) * 400 = \$ 1,600.00`

Total cost with two backacters is given as follows;

`2 * 1000 + 12 * (1)/(38) * 400 = \$ 2,126.32`

The additional backacter will increase the total costs, therefore, it should not be deployed.