December 1998
QT211: QUANTITATIVE ANALYSIS FOR MANAGEMENT

QUESTION 4

Total Marks: 20 Marks

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(a) The personnel manager of the Acme Machine shop needs to hire a machine to fix certain type of copier when it is in need of repair. Breakdowns on the copiers are Poisson-distributed and occur at an average of 3 per hour. Any copier out of service, either being repaired or waiting to be repaired, costs the company $80 per hour.

The personnel manager can consider hiring either one of two applicants. The first applicant is a very speedy worker who can repair this type of copier at an average rate of 5 per hour (exponentially distributed) and expected to be paid $18 per hour. The second applicant is somewhat slower, performing an average 4 repairs per hour (also exponentially distributed service times) and he expects a wage of $10 per hour.

Should the company hire either or both workers in order to minimise total cost? Assume that if both are hired, they would work as a team with mean service rate equal to the sum of two individual service rates.

 

 [13]
For the fast worker, we compute the mean 'out of service' time as:
Ws = 1/(m-1) (1 mark) = 1/(5-3) = 1/2 hr per copier (1 mark)
Expenses = (3 lathes/hr)(1/2hr/lathe)($80/hr) + $18/hr (1 mark) = $138/hr (1 mark)

 

 [4]
With the slower worker, we compute the mean 'out of service' time as:
Ws = 1/(m-1) (1 mark) = 1/(4-3) = 1 hr per copier (1 mark)
Expenses = (3 lathes/hr)(1hr/lathe)($80/hr) + $10/hr (1 mark) = $250/hr (1 mark)

 

 [4]
With both hired assumed to be a team of one: m = 5 + 4 = 9 repairs per hour (1 mark)
Ws = 1/(9-3) = 1/6 hr per copier (1 mark)
Expenses  = (3 lathes/hr)(1/6/lathe)($80/hr) + $18 + $10/hr (1 mark) = $68/hr (1 mark)
Therefore both workers should be hired (1 mark).

 

 [5]
(b) State seven characteristics of a simple queue. [7]
Characteristics:
  • The queue discipline is First come First served.
  • The queue has infinite capacity
  • Arrivals are random and follow a Poisson distribution
  • No simultaneous arrivals
  • Service times are random and follow a negative exponential distribution
  • Discrete customers from an infinite population of potential customers
  • Single, follow-on service discipline
  • The system must have been in operation long enough to settle down and the traffic intensity must be less than 1

Any seven, 1 mark each