1. Analytic Prediction of the Waiting Time

2. Unavailability Sensitivity calculation by MCS  3. The Hybrid link   3. Marginal Analasys  4. The penalty of the compromise  5. The limitation to spare parts and repair teams  

The starting point of the analytic approach (Waiting Time) suggests a single component that reaches a storage facility at a rate . This rate is assumed to be constant⁽³⁾, representing the flow of units from the field. Upon arrival the failed unit is repaired and returned to storage. The time required for this process is the recycling time Tc. If there are initially q spares in the storage, then a lack of spares will be caused only if upon the arrival of the (q+1)^th unit to the storage the first one has not yet returned from the recycling process, namely,

is the time interval between the arrival of the and the (i+1)^th unit. Since may be assumed to be exponentially distributed, T has a Gamma distribution with q degrees of freedom. Hence, for where and q is an integer.

Ifit means that the system is awaiting a spare and the time it would take is. Thus, the average waiting time in a history would be

represents the cumulative Gamma distributionwhere K is the order of the distribution andis the scale parameter.

If the component is a discarded component (namely is condemned upon failure) it will have a slightly different development, but essentially its turn around time can be regarded as infinity or even by considering it as the length of the service time (further developments of this theme can be found in Dubi, 2003).

2. Unavailability Sensitivity calculation by MCS

1. Analytic Prediction of the Waiting Time  3. The Hybrid link   3. Marginal Analasys  4. The penalty of the compromise  5. The limitation to spare parts and repair teams  

The ability of a Monte Carlo calculation to obtain time dependent statistical quantities enables the calculation of additional important metrics other than the system performance. One of these quantities is the Sensitivity. Here the Sensitivity is an array of fractions, such that every type has its own sensitivity. The Sensitivity can be related to any defined performance target. For example, the Unavailability Sensitivity of type i is defined as the relative portion of the down time (the time at which the system is unavailable), contributed by any component of type i. The following examples may further clarify this definition. Consider a system which comprises five components of three different types as shown in Figure 3. A dark component represents a failed component. The sensitivity will be calculated as follows. Each time the system is failed, for each failed component in the system, repair the component ad hoc, and if the system is repaired as a result of this, then this component is responsible for the failure. If the Sensitivity is a Failure Sensitivity then a unity is added to the type of the component. If the Sensitivity is an Unavailability Sensitivity then when the system will be repaired again, the downtime since the last failure will be added to all those types which were responsible for the failure. The following three cases in Figure 3 are examples for three different Sensitivity outcomes

Figure 3 different system failure configurations and responsible components.

In case A, components 1 and 5 are failed. However, repairing (ad hoc) component 5 will not bring the system back to operation, so this component is not regarded as "responsible" for the system failure. Only component 1 (Radio type) is. In case B, 2 and 5 are failed. Repairing either one of them will repair the system. Hence, both (Antenna and Radar types) are regarded as responsible for the system failure. In case C, 2, 4 and 5 are failed. Note that this can happen only if 2 failed last or else the failure point of the system would have been when 2 failed and only one of 4 or 5 failed too (as in case B). In this case repairing 4 or 5 will not repair the system, so only 2 (Antenna type) will be held the failure cause.

This concept does not have a simple explicit mathematical definition but experience indicates that it is an excellent measure of the relative contribution of each type to systems failures. The average system down time for each type, Tdi, accumulated by the above process is then normalised by dividing it by the sum of all types, to obtain the sensitivity of type i, namely:

(n is the number of different components types in the system).

The total average down time of the system, over the system's life time Tmax is now assumed to be

and the average unavailability is the ratio of the down time to the mission time

In fact, the Sensitivity array is an output related to the system. It summarises the whole scenario, with data, system structure, logical rules, Resources policy and whatever else affects the system performance through its life history. Therefore, the analytic calculation does not "care" about the system structure or any of the problem complexities. The sensitivity and the unavailability are the only quantities, which are relevant for the optimisation process as parameters.

Let us define a new quantity uj that is the unavailability contributed by type i, asuj=U x sj. Approximatelyand by its definition

The former is thus a function of the latter:

3. The Hybrid link

1. Analytic Prediction of the Waiting Time  2. Unavailability Sensitivity calculation by MCS  4. The penalty of the compromise  5. The limitation to spare parts and repair teams  

The assumption is that uj is a linear function of the Waiting Time (the average time interval since demanding a spare, until the spare is available) of type i, , , namely:, where Ai and Bi are constants to be determined in the course of the optimisation.

Thus, it is assumed that the unavailability as function of spare parts is a surface in the spare parts domain, having the form:

Letdenote a specific MCS calculation with a given storage strategy. Letandbe the partial type unavailability in theandcalculations respectively, then the following equations provide two unknowns Ai and Bi.

One extreme,denotes a calculation with unlimited spares (the calculation provides the maximum number of spares used, and this finite number is denoted by ). In this case, since it will result in never needing to wait for spares. The other extreme would be zero spares qi = 0, denoted by

The equations then take the forms

Eq. (2) is the starting point of the process, Ai and Bi are inserted into expression (1) and marginal analysis is performed analytically. Moreover, an additional benefit of andis that they indicate the boundaries of the performance. The process is then to search the cheapest way while progressing towards. . It is worth mentioning that the MC simulation provides the lower bound with the least cost out of the very large number of combinations. Therefore, it suggests that progressing towards that, doing the "best" step every time, will end up in meeting the optimum.

3. Marginal Analasys

1. Analytic Prediction of the Waiting Time  2. Unavailability Sensitivity calculation by MCS  4. The penalty of the compromise  5. The limitation to spare parts and repair teams  

Once the unavailability can be predicted for any given allocation, the optimisation algorithm is carried out by marginal analysis; that is allocating at each step a resource (a spare parts or a repair team) to the type which seems to contribute the highest availability per cost unit. So, the more sensitive and cheaper a certain type is, the more likely that this kind is to be added. Let us denote the unavailability as function of its Resources by U(Q) andqij is the j^th element in the row of the matrix Q, representing the number of Resources of type i in field j. Denote bythe unavailability with a unit added to the ij^thijth element in the matrix Q, then its marginal contribution to the availability will be

, where cij is the cost of the unit added. The optimisation algorithm will select the element with the maximal marginal contribution and will add one unit to it. Then, this process will keep going until a new simulation will provide accurate results with the yielded unavailability and its corresponding set of sensitivities. This will be repeated until the required criteria have been reached.

4. The penalty of the compromise

1. Analytic Prediction of the Waiting Time  2. Unavailability Sensitivity calculation by MCS  3. The Hybrid link   3. Marginal Analasys  5. The limitation to spare parts and repair teams  

Notwithstanding the optimum may never be found, since a level of accuracy must be sacrificed by making the analytical predictions, however, the justification for this sacrifice lays in the argument that the optimum point is lying in a flat-wide valley, and reaching that valley is as effective. This argument can be supported by the idea that a system which does not have such a flat-wide valley but a rather dramatic behaviour around the real optimum would be too sensitive to survive a justifiable period of service time, particularly where the metrics are stochastic.

5. The limitation to spare parts and repair teams

1. Analytic Prediction of the Waiting Time  2. Unavailability Sensitivity calculation by MCS  3. The Hybrid link   3. Marginal Analasys  4. The penalty of the compromise  

The Hybrid Optimizer is very efficient for problems involving spare parts and repair teams optimisation in multi-field and multi-echelon logistic environment. Yet, it is limited to optimise only those resources since it does not consider other supportive resources or parametric factors, such as preventive maintenance or different policy of management etc. This stems from the difficulty in developing analytic approaches for predicting the performance for multi-variable environment. It demands an extensive study of how dependently or independently each resource generically influences the performance whereas the nature of every model can be totally different from another, which makes the development of a general recipe unrealistic.

 Random Samples 

A "Constant failure rate" leads to an exponentially distributed PDF.  Back to Text

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last update:  December 12, 2006