Table 2 Problem 18-7 The manager of a regional warehouse must decide on the numb
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Table 2
Problem 18-7 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 $259 per day and that each dock plus loading crew represents a cost of $1,186 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) , because the daily total cost with the new equipment is $ which is lower than without the new equipment.Explanation / Answer
This is M/M/s queue model with following parameters
Arrival rate, = 4 trucks per day
Service rate, = 5 trucks per day
/ = 4/5 = 0.8
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, refer table for / = 0.8 and M = 1, Lq = 3.2
Total cost per day = Cs*M + Cw*Lq = 1186*1 + 259*3.2 = $ 2014.8
With 2 docks, refer table for / = 0.8 and M = 2, Lq = 0.152
Total cost per day = Cs*M + Cw*Lq = 1186*2 + 259*0.152 = $ 2411.4
Total cost of 1 dock is lesser. 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 = 1186+100 = $ 1286 per day
/ = 4/5.71 = 0.70
With 1 dock, refer table for / = 0.7 and M = 1, Lq = 1.633
Total cost per day = Cs*M + Cw*Lq = 1286*1 + 259*1.633 = $ 1709 per day
This cost is lower than the cost determined in part a. Therefore, the manager should invest in the new equipment.
Total cost with the new equipment is $ 1709 , which is lower than without the new equipment.
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