Reliability Allocation and Cost/Penalty Functions
Traditional reliability allocation requires that the reliabilities of the components that make up the system be allocated such that the overall system reliability reaches the desired goal. As an example, consider a very simple system with two components arranged reliability-wise in series. The system reliability equation is the product of the reliability equations of the two components. In other words, at a fixed point in time, the reliability of the system is the product of the reliabilities of the components at that point in time, or R_S=R₁xR₂. There are multiple solutions that would allow you to allocate the reliability of R₁ and R₂ such that R_S=98%. To arrive at the best solution to the problem, however, additional information must be added to the formulation. This information would be some measure that tells you how difficult (or how expensive, etc.) it is to increase the reliability of each component in the system. Based on this, you can then allocate the reliability such that this difficulty (or cost) is minimized. Obviously, the allocation scheme that takes this cost into account would result in the best solution.

At this point, we introduce a concept called the Cost Function (or Penalty Function), which describes cost (or difficulty) as a function of the reliability for each component. As shown in Figure 1, BlockSim supports any user-defined function to describe this relationship and the software also has a set of predetermined functions. To simplify the discussion of the analysis, we utilize these predetermined functions in the example that follows. BlockSim's standard cost functions are exponential in nature and are bounded by a minimum and maximum reliability. The following cost function will be used: C(R)=exp{1- f [(R-Rmin)/(Rmax -R)]}. In this case, we will vary f to define the degree of difficulty/cost to increase the component's reliability. The value f=0.9 will be defined as "easy" (least costly) and f=0.1 will be defined as "hard" (most costly), with the numbers in between representing a cost/difficulty somewhere between easy and hard. With this basic understanding of the use of cost functions to quantify the cost to improve a component's reliability, we will proceed with a simple reliability allocation example.

Simple Reliability Allocation Example
Consider a simple system composed of two subsystems: Subsystem A and Subsystem B. Each subsystem is composed of assemblies (Assembly A through Assembly D). Each assembly is composed of components (Component 0 through Component 9). The reliability of the system is based on the reliability characteristics of the subsystems, the reliabilities of the subsystems are based on the reliability characteristics of the assemblies and so forth. At the lowest level (i.e. the component level) we will assume that a life distribution can be
obtained or assumed. Figure 2 displays the reliability block diagram for this example, created using a sub-diagram approach, along with the assumed life distribution and parameters for each component.

The objective is to determine the reliability of the system and then if the system reliability does not meet the goal for the system, to allocate reliability for the subsystems, assemblies and components to meet the system goal. Let us assume that our time measure is in hours and that our system reliability goal is 98% at 400 hours.

Determining System Reliability and Subsystem Allocations
The first step in the analysis is to compute the current system reliability. Issues 2 and 3 of ReliaSoft's Reliability Hotwire eMagazine (http://www.weibull.com/hotwire) discuss ways of doing this. The analysis yields a system reliability of R(t=400)=95.39%, which is below the stated requirement of 98%. Since the requirement is not met, our next step is to determine what reliability should be allocated to each subsystem so that the system reliability target can be achieved. In this allocation, we want to use the cost function concept discussed previously, so we will assign difficulty factors to each subsystem (based on business, design and other constraints). We determine that increasing the reliability of Subsystem A is easier than increasing the reliability of Subsystem B and we assign an "easy" classification (f=0.9) to A and a "hard" classification (f=0.1) to B.

Based on the system reliability block diagram and the definition of the cost function for increasing the reliability of each subsystem, BlockSim's optimization engine is utilized to determine the optimum allocation of reliabilities at the subsystem level. The allocation results (minimum cost based on the C(R) cost function described previously and the given reliability requirement) are found to be:

• Increase Subsystem A's reliability from 96.71% to 99.25%.
• Increase Subsystem B's reliability from 98.64% to 98.74%.

Determining Assembly and Component Allocations
The next step is to repeat this analysis for each one of the subsystems to allocate the reliability for each of the assemblies. We start by looking at Subsystem A. Subsystem A is composed of two assemblies, Assembly A and Assembly B. There is one Assembly A unit and two of the three Assembly B units in the subsystem must operate in order for the subsystem to operate (a 2-out-of-3 redundancy). We will assume that the cost function is the same for both Assembly A and Assembly B ("easy," f=0.9). Based on the previous analysis, the reliability target for Subsystem A is 99.25% at 400 hours. Using the same procedure, we determine that the best allocation scheme for Subsystem A is to increase the reliability of Assembly A from 0.9676 to 0.9931 and make no change to Assembly B. This result was expected because Assembly B is redundant and has a much lower reliability importance metric than Assembly A. Note that for more complex systems, reliability importance graphs can also be utilized to determine which components to focus on for improvement. An article in Volume 1, Issue 1 of the Reliability Edge (http://www.ReliaSoft.com/newsletter) discusses this technique in greater detail.

Once the reliability target for Assembly A has been set, the next step is to allocate this reliability among the components that comprise the assembly. Repeating the same allocation procedure, we see that Assembly A is made up of Component 0 and Component 1. We assume a higher cost for increasing the reliability of Component 0 (f=0.5) and a lower cost for Component 1 (f=0.9). The optimum allocation of reliabilities is found to be:

• Increase Component 0's reliability from 99.36% to 99.66%. 
• Increase Component 1's reliability from 97.38% to 99.65%.

We have now set reliability specifications for the components and assemblies to achieve the desired reliability for Subsystem A. The same procedure can be repeated for Subsystem B. When the Subsystem B analysis is complete, realistic goals and targets will be have been set for each component in order to achieve the desired system reliability.