Question

Trucks arrive at a warehouse at a rate of 12 per hour during business hours. Crews can unload the trucks at a rate of 2 per hour. Recent changes in wage rates have caused the warehouse   manager to reexamine the question of how many crews to use as that will affect their profitability and productivity. The new rates are: Crew and dock cost is $90 per hour; truck and driver cost is $140 per hour. The crew and dock cost represents the server cost and the truck and driver cost represents the waiting cost.

How many crews does the warehouse need to minimize the total cost per hour?

Set up a model in Excel and find the optimal number of crew members needed to perform the operation with a minimum cost. A formatting given below may be helpful as a starting point. You do not need to use the exact formatting.

Answer 1

Average arrival rate, λ = 12 per hour
Average service rate, μ = 2 per hour

Use the following two formulae for computing the idle server probability (P₀) and the average number of trucks and drivers waiting (L_q) for different server levels (s) and compute the total cost.

s	P0	Lq	Cost of waiting = 140*Lq	Cost of servers = 90*s	Total cost per hour
7	0.0016	3.683	$515.6	$630.0	$1,145.6
8	0.0021	1.071	$149.9	$720.0	$869.9
9	0.0024	0.392	$54.9	$810.0	$864.9
10	0.0024	0.152	$21.3	$900.0	$921.3

So, note that s = 9 is the ideal arrangement because the total cost per hour is minimum.