question archive A product manufacturing plant at a city distributes its products by trucks, loaded at the factory warehouse
A product manufacturing plant at a city distributes its products by trucks, loaded at the factory warehouse
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A product manufacturing plant at a city distributes its products by trucks, loaded at the factory warehouse. It has its own fleet of trucks plus trucks of a private transport company. This transport company has complained that sometimes its trucks have to wait in line and thus the company loses money paid for a truck and driver of waiting truck. The company has asked the plant manager either to go in for a second warehouse or discount prices equivalent to the waiting time. The data available is:
- Average arrival rate of all trucks = 3 per hour.
- Average service rate is = 4 per hour.
The transport company has provided 40% of the total number of trucks. Assuming that these rates are random according to Poisson distribution, determine:
(a) The probability that a truck has to wait?
(b) The waiting time of a truck that has to wait,
(c) Assume that the cost of hiring another warehouse will be Rs. 3,000 per day but it will increase the service rate to 6 per hour. The trucks are having opportunity cost of Rs. 200 per hour of the company’s own trucks and Rs. 400 of those of the private transport company. If we assume that there are 10 hours working in a day, will you like to suggest the company to have another warehouse?
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Answer:
Arrivak rate lambda = 3 per hour
Service rate mu = 4 per hour
Utilisation rho = lambda /mu = 0.75
(i) Probability that the truck has to wait =0.75
(ii) Waiting time Wq = Lambda /mu*(mu-lambda) = 3/4(4-3) = 0.75 hour = 45 minutes
Waiting time for all trucks = lambda *Wq = 0.75*3 =2,25 hours
Total cost of waiting per day = 10 *2.25 ( 0.6*200+0.4*400) = 6300
If a warehouse is hired, the new waiting time = 3/6*(6-3) = 1/6 hours =10 minutes
Total waiting time per hour = 1/6*3 = 0.5 hours
Total cost of waiting per day = 0.5*10 ( 0.6*200+0.4*400) =1400
Total cost = 3000+1400 =4400
Which is less than the cost in option I.
Since hiring of the new warehouse will save money for the company, company should go for it.
