I am working on a project. So, i come across one problem is that,
if we have n No. of machines with their utilization as U1, U2 ,.....,Un . And Now if i have the overall plant utilization as, Up. So, can anyone please tell how should i relate Up, with the utilization of the machines?
The typical way to do it (assuming all machine are in parallel) is to multiply each machine's utilization times its capacity sum the result to get total utilization and divide by the total capacity (the sum of the individual capacities),
U overall = (Sum (Ci*Ui))/Sum( Ci) ) or actual total production/Max Production if everything worked 100% of the time.
Often it is a little more complicated by this because some machines may be in series instead of in parallel and some machines may actually be spares used to improve utilization of one or more of the other machine. Overall utilization is still actual production/Max Production.
Everything Rick wrote is correct. I'll add one special case. If all of your machines are used in series, then the actual production/Max Production for the entire plant is also the actual production/Max Production for the one machine that is the bottleneck. Even if not all machines are used in series, there is nearly always one or two that are bottlenecks for the plant. This leads us to one of Eliyahu Goldratt's conclusions in "The Goal": Capacity improvements in a non-bottleneck step are worthless. Capacity improvements in a bottleneck are worth the profit on the increased production.
Ronald is correct if the machines are in series they must be treated as a single unit and both must operate for the unit to count as operating. If there are 3 units (2 operating/1spare) each with a probability of 90% of operating the expected utilization of the unit will be the probability that all 3 are available (.9*.9*.9 = .729) + the probability that 2 of the three are available (.9 *.9*.1*3)= .243 so the overall expected reliability of the unit will be .729+.243 =.972
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