The manager of a regional warehouse must decide on the number of loading docks t
ID: 354445 • Letter: T
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? $250 per day and that each dock plus loading crew represents a cost of? $1,138 per day.
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. The manager is considering renting new equipment that would speed up the loading rate to 5.71 trucks per day. The equipment would cost an additional? $100 per day for each crew working at a dock. What is the optimal daily total cost with the new? equipment? What is the optimal daily total cost with the current? (original) equipment? (from your analysis in part a? )? ?(Round your cost amount to 2 decimal places and all other calculations to 4 decimal places. Omit the? "$" sign in your? response.)
The optimal daily total cost with the new equipment would be? $ _______?, and the optimal dailty total cost with the original equipment is? $ ________
Explanation / Answer
This is M/M/s queue model with following parameters
Arrival rate, ? = 4 trucks per day
Service rate, ? = 5 trucks per day
Server cost (dock plus loading crew cost), Cs = $ 1186 per day
Waiting cost (truck-driver combination cost), Cw = $ 259 per day
(a)
With 1 dock, this is M/M/1 queue model.
Average number of trucks-driver in system, L = ?/(?-?) = 4/(5-4) = 4
Total operating cost per day = Cs*M + Cw*Lq = 1138*1 + 250*4 = $ 2138
With 2 docks, Dock plus loading crew cost will be 2*1138 = 2276 . This itself is higher than the cost with 1 dock.
Therefore, optimal number of docks is 1.
Number of docks = 1
(b) With the new equipment, service rate, ? = 5.71 trucks per day
Server cost, Cs = 1138+100 = $ 1238 per day
?/? = 4/5.71 = 0.70
With 1 dock, this is M/M/1 queue model.
Average number of trucks-driver in system, L = ?/(?-?) = 4/(5.71-4) = 2.339
Total operating cost per day = Cs*M + Cw*Lq = 1238*1 + 250*2.339 = $ 1822.75 per day
This cost is lower than the cost determined in part a. Therefore, the manager should invest in the new equipment.
The optimal daily total cost with the new equipment would be? $ __1822.75__, and the optimal daily total cost with the original equipment is $ __2138___
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