Final answer:
The reliability function for each of the five individual stations can be determined using the formula Reliability = e^(-t/MTTF). The overall reliability of the assembly line can be calculated as the product of the reliabilities of the individual stations.
Step-by-step explanation:
(a) Reliability function for each station:
For the first three stations with a mean time to failure (MTTF) of 1.5, since the distribution is exponential, the reliability function can be calculated using the formula:
Reliability = e^(-t/MTTF)
Substituting t=0, we get Reliability = e^(-0/1.5) = 1
For the last two stations with a MTTF of 2.4, following the same formula, the reliability function is:
Reliability = e^(-t/MTTF)
Substituting t=0, we get Reliability = e^(-0/2.4) = 1
(b) Reliability function for the assembly line:
Since the assembly line is made up of five independent stations, the overall reliability of the assembly line is the product of the reliabilities of the individual stations:
Reliability of assembly line = Reliability of station 1 * Reliability of station 2 * Reliability of station 3 * Reliability of station 4 * Reliability of station 5
(c) Mean time to failure for the assembly line:
Mean time to failure for the assembly line is the reciprocal of the failure rate, which is the sum of the failure rates of the individual stations:
MTTF of assembly line = 1 / (Failure rate of station 1 + Failure rate of station 2 + Failure rate of station 3 + Failure rate of station 4 + Failure rate of station 5)
(d) Hazard function for the assembly line:
The hazard function for the assembly line is the derivative of the reliability function with respect to time t:
Hazard function = (d/dt) (1 - Reliability function)
The reliability function for each station with exponentially distributed lifetimes is given by R(t) = e-λt, where λ is the inverse of MTTF. The overall reliability of the assembly line as a series system is the product of individual reliability functions. The MTTF and hazard function for the assembly line require advanced mathematical operations.
The question involves concepts from probability and statistics associated with the exponential distribution for modeling equipment lifetimes. Each station on the assembly line has equipment whose lifetimes are exponentially distributed with different mean times to failure (MTTF). Station 1-3 have an MTTF of 1.5 (hundreds of hours), and Stations 4-5 have an MTTF of 2.4.
(a) The reliability function R(t) for exponentially distributed lifetimes is given by R(t) = e-λt where λ is the rate parameter, λ = 1/MTTF. For the first three stations, it is R(t) = e-(2/3)t, and for the last two stations, R(t) = e-(5/12)t.
(b) The assembly line works as a series system, meaning that the overall reliability is the product of the reliabilities of individual components. The reliability function for the assembly line is thus R(t) = (e-(2/3)t)3 × (e-(5/12)t)2.
(c) To find the mean time to failure (MTTF) for the assembly line, one would need to integrate the overall reliability function over all time t from 0 to ∞. This is non-trivial due to the product of exponentials and would typically be calculated using advanced integration techniques, or more practically, using computational methods.
(d) The hazard function h(t), which represents the instantaneous rate of failure at time t, for an exponential distribution is constant and equal to the rate parameter λ. For the assembly line with combined reliabilities, however, calculating the hazard function involves taking the derivative of the overall reliability function and dividing by the reliability function itself, which requires further mathematical operations.