In a factory, the machines breakdown on an average rate of 8 machines per hour.

Question

In a factory, the machines breakdown on an average rate of 8 machines per hour. The idle time cost of a machine is estimated to be 50 AED per hour. The factory works 8 hours a day. The factory management is considering 3 mechanic crews for repairing the machines. The crews are working together on the same machine during repair activities. The first crew A consist of one mechanic and takes about 30 minutes on an average to repair a machine and demands wages of 50AED per hour. The second crew B which consist of three mechanics can service 9 machines per hour and demands wages at the rate of 40 AED per hour per mechanic. The third crew which consist of four mechanics takes about 6 minutes on an average to repair a machine and demands wages of 40AED per hour per mechanic. Assuming that the rate of machine breakdown is Poisson distributed and the repair rate is exponentially distributed. Which one of the crews should be considered?

Explanation / Answer

Solution

Given,

the rate of breakdown, = 8/hr

Work hours = 8 per day

Service rate, µ:

Crew 1: µ1 = 2/hr [30 minutes per repair => service rate = 60/30 = 2]

Crew 2: µ2 = 9/hr

Crew 1: µ3 = 10/hr [6 minutes per repair => service rate = 60/6 = 10]

Service cost per hour:

Crew 1: 50 AED [1 mechanic at 50 AED per mechanic]

Crew 2: 120 AED [3 mechanic at 40 AED per mechanic]

Crew 3: 160 AED [4 mechanic at 40 AED per mechanic]

Machine idle time cost per hour = 50 AED.

Back-up Theory

Approach

With each crew, the average total time spent in the system is computed employing M/M/1 Theory and multiplying it by idle time cost per hour, total cost of idle time is obtained.

Then, the total cost per hour = total cost of idle time + service cost for the crew. Decision criterion would be the minimum total cost.   

Machine idle time = waiting time + repair time [= total time spent in the system, v, in Queueing Theory parlance]

For M/M/1, E(v) = {1/(µ - )}.

Now, to work out the solution,

Crew 1: Since average service rate of 2 per hour is much less than the average arrival (machine breakdown) rate, the queue will explode and hence the total cost would be huge.

Hence, Crew 1 is ruled out.

Crew 2: E(v) = {1/(µ - )} = 1/(9 - 8) = 1 hour and hence total cost = 50 + 120 = 170 AED.

Crew 3: E(v) = {1/(µ - )} = 1/(10 - 8) = ½ hour and hence total cost = 25 + 160 = 185 AED.

Since 170 AED is least, the optimum decision is: Crew 2 should be considered. ANSWER

[Note: Since all parameters are converted to per hour basis, the fact that company works 8 hours a day need not be taken into account.]

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