Question

The manager of a regional warehouse must decide on the number of loading docks to request for a new facility in order to minimize the sum of dock costs and driver-truck costs. The manager has learned that each driver-truck combination represents a cost of $270 per day and that each dock plus loading crew represents a cost of $1,116 per day. Use Table 1 and Table 2.

a. How many docks should be requested if trucks arrive at the rate of four per day, each dock can handle five trucks per day, and both rates are Poisson?

Number of dock(s)            

b. An employee has proposed adding new equipment that would speed up the loading rate to 5.71 trucks per day. The equipment would cost an additional $100 per day for each dock. Should the manager invest in the new equipment? (Round your cost amount to 2 decimal places and all other calculations to 3 decimal places. Omit the "$" sign in your response.)

(Click to select)  Yes  No  , because the daily total cost with the new equipment is $
which is lower than without the new equipment.

Answer 1

Part a:

Given:

	Arrival rate (trucks/day)	λ = 4 trucks per day
	Service rate (trucks/day)	µ = 5 trucks per day
	Individual dock Utilization ρ = λ/µ	ρ = λ/µ = 4/5 = 0.80 ρ = 0.80

From table, for λ/µ = 0.80, three dock options are available – 1, 2, and 3 docks

Number Docks	No. of trucks waiting	No. of trucks waiting and served	Total Trucks-driver combination cost per day	Total Dock cost per day	Total cost of system per day
(M)	Lq (from table)	Ls = Lq + λ/µ	Cw = Ls x $270/day	Cs = M x $1116)	TC = Cw + Cs
1	3.2	4	$1,080	$1,116	$2,196
2	0.152	0.952	$257	$2,232	$2,489
3	0.019	0.819	$221	$3,348	$3,569

Lowest system cost is $2,196 per day for single dock.

Thus, to minimize the cost single dock is required.

Part b.

Given

	Arrival rate (trucks/day)	λ = 4 trucks per day
	Service rate (trucks/day)	µ = 5.71 trucks per day
	Individual dock Utilization ρ = λ/µ	ρ = λ/µ = 4/5.71 = 0.70 ρ = 0.70

                                   

From table, for λ/µ = 0.70, three dock options are available – 1, 2, and 3 docks

Dock cost per day per dock = $1116 + $100 = $1216 per day per dock

Number Docks	No. of trucks waiting	No. of trucks waiting and served	Total Trucks-driver combination cost per day	Total Dock cost per day	Total cost of system per day
(M)	Lq (from table)	Ls = Lq + λ/µ	Cw = Ls x $270/day	Cs = M x $1216)	TC = Cw + Cs
1	1.633	2.334	$630	$1,216	$1,846
2	0.098	0.799	$216	$2,432	$2,648
3	0.011	0.712	$192	$3,648	$3,840

Lowest system cost is $1,846 per day for single dock, which is less than above option (a).

Total cost of system with new equipment = $1,846

Thus, to minimize the cost new equipment should be used

ANS:

Yes  No  , because the daily total cost with the new equipment is $1,846 per day, which is lower than without the new equipment.