Life-Cycle Costs and Reliability Allocation in Automated Transit
J. Edward Anderson, Ph.D., P. E.
1. Introduction
This paper addresses specification of the Mean Time between Failures (MTBF) of the subsystems and components of a Personal Rapid Transit (PRT) system in such a way that the life-cycle cost (LCC) of the system is minimized. The method is general and can be applied to any type of system.
Consider the following equation for the Dependability[1] of an automated transit system:
(1)
in which U was referred to as the Undependability. Undependability is the ratio of person-hours of delay due to failures of any kind to the person-hours of operation. U can be expressed as the ratio of vehicle-hours of delay per year due to failures to vehicle-hours of operation per year. In terms of the notation given in Section 7, U can be expressed in the form
(2)
in which
VHyr = NvPk/Day Days/Yr (3)
While precise specification of the MTBFs is meaningless, it is of considerable value in the design process to have a formula for the optimum MTBFs to
- Determine design feasibility,
- Determine the degree to which redundancy is required,
- Clarify factors affecting the optimum MTBFs,
- Determine the reasonableness of the dependability standard, and
- Provide specific objectives for component design.
If some of the MTBFs in equation (2) are known, those terms can be placed on the left side and the method developed can be applied to the remaining unknown terms.
2. Life-Cycle Cost vs. Reliability
The Acquisition Cost of a component or subsystem generally increases with increased reliability, measured here by MTBF, and the Support Cost over the life of the same unit decreases as MTBF increases. Therefore the sum of Acquisition Cost and Support Cost as a function of MTBF is a U-shaped curve with a single minimum point. If the MTBF of each subsystem could be specified close to this minimum point, the LCC of the system would be minimized; and, if the corresponding U were acceptably low, the problem addressed does not arise. Generally, however, the corresponding U in this case is not acceptably low, so it is necessary to design or select components and subsystems with higher MTBFs. At these higher MTBFs, the rate of change of LCC with MTBF is positive.
The mathematical problem is to find the minimum system LCC given a fixed Undependability U smaller than the value associated with the absolute minimum LCC. To shorten the notation, let
(4)
Then, LCC for subsystem i is a function of xi, a fact that is expressed mathematically in the form LCCi(xi). Thus, the system life-cycle cost is
(5)
in which signifies the sum over all of the subsystems, there are ri subsystems of the type designated by i, and there are E separate types of subsystems in the system under consideration. Equation (5) implies that the reliability of one subsystem does not directly affect the LCC of any other subsystem. If the reliability of one type of subsystem is poor, the other types of subsystems will have to be more reliable to attain the required service dependability. How much more is what we wish to determine.
3. Minimization of Life-Cycle Cost
The problem of minimization of LCC reduces to minimization of the function
(6)
given by equation (5), subject to constraint equation (2),
a given constant. (7)
This is called a “constrained minimization problem” and was first solved by the famous French mathematician Joseph Louis Lagrange (1736-1813). To solve it, imagine that you pick arbitrarily one of the xi in the constraint equation (7), say xj, which is, from equation (7), a function of all of the other xi. Imagine that you substitute this solution into expression (6), which would become
Then, the minimum value of LCC is obtained by setting equal to zero the partial derivatives of LCC with respect to each of the xiexcept xj, which is no longer present because it has been removed by substitution. Thus
(8)
Similarly, substitute xj back into equation (7). Partial differentiation of equation (7) then gives
(9)
which is zero because U is a constant. Now, if the right-hand terms of each of equations (8) and (9) are placed on their right sides and equation (8) is divided by equation (9), the result is
(10)
Since i and j were picked arbitrarily, equation (10) holds for any pair of these indices, and thus the quantity is a universal constant of the system, called a Lagrangian constant. In words, we have found that the condition for minimum life-cycle cost of the system is that the rate of change of system life-cycle cost with respect to any subsystem MTBF divided by the rate of change of the constraint with respect to the same system MTBF is the same for all subsystems.
From equation (5) we see that
(11)
and, from equation (2), using equation (4),
(12)
Substitution of equation (11) and equation (12) into equation (10) gives
(13)
in which
(14)
While is in general a function of xi, we solve equation (13) for the value of xi shown explicitly. Thus
(15)
in which, as mentioned earlier, Now, using the definition of xi given by equation (4), substitute the family of E equations (15) into constraint equation (2), and solve for The result is an equation for the Lagrangian constant in terms of the given Undependability and other system parameters:
(16)
Now, having found as a sum including all subsystems, it can be substituted from equation (16) into each of the family of equations (15) to give a family of equations for all subsystem MTBFs. These equations now satisfy Lagrangian condition (10) for minimum life-cycle cost and they also satisfy constraint equation (2). This is the desired solution. In writing down the solution, change the dummy subscript i in equations (15) to j to avoid confusion, use the definition (4), and note from equation (3) that
(17)
where the subscript v corresponds to vehicle subsystems. Further, define
(18)
in which for vehicle subsystems; and for wayside subsystems. Then, the family of equations (15) for the optimum MTBFs of each of the E subsystems becomes
(19)
which is the desired result.
Note, as mentioned, that is in general a function of so that the solution of equations (19) is iterative. However, since the appear under square-root signs and is generally nearly constant in the region of interest, it has been found[2] that the solutions converge quickly.
4. Analysis of the Equation for Optimum MTBF
Equations (19) form a family of equations for each of the MTBFs of all subsystems required to meet the Dependability standard at minimum LCC. These equations include terms related to normal system characteristics, delays due to failures in each subsystem, and the rate of change of LCC of each subsystem with respect to its MTBF. A major value of equations (19) is that it organizes the reliability and maintenance analysis program of any system to which it is applied, and at any stage from early conceptualization through operation. With equations (19) in hand, the utility of specific reliability data is clear and specific.
Consider how varies with changes in the terms equations (19) contains. High means that LCC of subsystem j increases rapidly with . Therefore, the optimum will be lower than if were smaller, which is predicted by equations (19). On the other hand, high, where , means that, if LCC of any subsystem other than subsystem j increases rapidly with MTBF, should be increased because it is relatively cheaper to do so. Data on the are not plentiful but they have been collected for many components at the Reliability Analysis Center at Griffiss Air Force Base, Rome, New York, and can be found on several websites on the Internet.
As where , increases, the optimum increases by a small amount, suppressed because appears under a square-root sign and because it appears in only one of many terms. On the other hand, if increases, the optimum increases in direct proportion to, and also through the j-term of the summation.
5. An Example
To further clarify equations (19), consider a simple example. Let i = v correspond to vehicle subsystems, i = s correspond to station subsystems, i = z correspond to zone controllers, and i = c correspond to the central controller. Assume redundancy in all critical subsystems. Let, , and let be the delay of an average vehicle as a result of a central-computer failure because we can expect that will increase in direct proportion to . Then the summation term of equations (19) becomes
(20)
in which the factor of 2 is shown explicitly to remind the reader that redundancy has been assumed, and it should be noted that the control-facility term decreases as the number of vehicles in the system increases. Now, the four separate subsystem MTBFs are
(21)
and
(22)
If, lacking information on the , we were to assume, for example, that they were all the same, we would have a non-optimum distribution of MTBFs, but one that satisfied constraint equation (2). So, to obtain a feeling for magnitudes, since all of the LCC appear as ratios, set them all to unity. Assume all of the = 0.1 hr, let = 120, and let U = 0.003. Values of in the assumed range imply redundancy; but, with redundancy, these terms will likely be smaller. Then = 0.918, = 97 hrs, and = 150 hrs. Practical solutions of equation (19) will involve many more terms, but the calculation illustrates the procedure.
6. Conclusions
Equations (19) give the mean time between failures of each subsystem or component of a PRT system needed to minimize system life-cycle cost. Section 5 provides a simplified example of how the method is used in a sensitivity analysis that shows how the required MTBFs depend on all relevant parameters including the duration of service interruption. The method presented determines the hardware characteristics required to provide a practical, dependable, and economical PRT system, and provides a readily measurable standard of performance.
7. Notation
ratio of yearly to weekday travel
Enumber of types of subsystems, failure of which causes passenger delay
LCClife-cycle cost
mean time between failures of the i-th type subsystem
number of vehicles in the system
ratio of daily to peak-hour travel
number of units of i-type subsystem in the system
Uratio of person-hours of delay to person-hours of operation
vehicle-hours of operation per year
mean vehicle-hours of delay due to a single failure of i-type subsystem
hours per year of operation of i-type subsystem
1
[1] See the paper “Dependability as a Measure of On-Time Performance in PRT Systems.”
[2] J. E. Anderson, W. L. Garrard, and P. J. Starr “A Technique for Reliability Allocation for Minimum Life-Cycle Cost in Transit Systems,” Contract No. DOT-UT-9012, July 1980.