| | | Obtaining the Density of the Resources Space by Random Samples
 |
|
Our starting point for an improved approach is to sample points in the Resources space to create Unavailability density in that space. By doing so, we should be able to perceive what the density of the average unavailability in the Resources space is, and to comprehend what the order of magnitude of the computational cost is of building up such scaffoldings for optimisation or search process (a deeper discussion can be found in Keane and Nair, 2005). |
|
Here the Resources space is transformed into cost with a simple Cost function. Therefore, the density diagram will show Unavailability against Cost. It is worth mentioning that the graphic illustration of the density as a function of the Cost is not as informative as that of the Resources, because some similar Costs may result from completely different Resources allocations (distant in the Resources space). However, it is straight forward to break down the Costs to their corresponding Resources allocations when required. Also, apparently it is the only way for viewing that density in a two-dimensional diagram. |
|
The number of allocated Resources strongly affects system performance and for every single allocation, a complete Monte Carlo simulation which comprises a significant number of Histories must be carried out. In the current process we randomly sample the number of each Resource and execute a simulation. Of course, if we sample each Resource independently and uniformly in its corresponding region then the resulting Cost will not be uniformly distributed since the Cost is a summation over 25 independent variables and hence has the properties of the Sample Average where according to the Central Limit Theorem its probability density function approaches a Normal distribution around the mean value. Then, certainly this would not be a uniform distribution. In order to gain a more even coverage we carry out conditional sampling, starting with the Cost (Uniformly) as illustrated in figure 4 and walking backwards. The range of Costs lays between zero and the maximum Cost. The finite maximum Cost is determined by the Cost function applied for the 98.5% Sufficiency point. Each type is then allocated with maximum needed spares (with 98.5% probability of never having a lack when required), and the same principle is applied for the repair teams. |
|
Figure 4: Sampling the Cost uniformly between zero to Max. |
|
Having sampled the Cost we then investigate what the distribution among the types would need to be. This is done by sampling a random number for each type and normalising it with the sampled cost. The Pie chart in figure 5 illustrates an example for the cost distribution.
|
|
Figure 5: Pie chart of the Cost distribution among the different Line Replaceable Units types in the problem |
|
The number of spares of each type will be the size of its corresponding segment divided by the cost of that type and rounded. The same principle is then applied to obtain the distribution among the four different storages of spares of a given type.
|
|
Having done that, we have a spares allocation for all storages and we obtain the Unavailability as a function of the Cost using Monte Carlo simulation. Repeating this process many times, each time with new sampled allocation, we then create the topology of the Unavailability in the Resources space. This is clearly seen in figure 6.
|
|
Figure 6: Unavailability in the Resources space when sampled uniformly, from which the resulting Pareto Front is compared with the Hybrid Curve,  T is added to illustrate a common allocation which is far away with respect to the optimum |
|
These results show the Unavailability density in the Cost space, plus the optimisation curve obtained by the Hybrid Optimizer and a Pareto front fitted on the sampled space for comparison. One can conclude that most points in the space are far away from the optimal region and indeed it is not trivial to obtain the optimum and that indeed the Hybrid optimisation curve meets this pattern quite well. Still, the Hybrid curve is not the best curve and some better allocations can be proposed. |
|
last update: December 12, 2006 |
|
|
| | |
