VARI-METRIC

Like it is stated in assumption 6 in section 4.1.1, the METRIC method minimizes the expected backorders, because the assumption is that item importance among items is equally. In practice there is a structure of items. This structure is recognized by Muckstadt. (Muckstadt, 1973) Muckstadt solely uses the end-items in a multi-echelon supply chain. This method is called MOD- METRIC.

Sherbrooke responses that the MOD-METRIC is underestimating the total backorders. Muckstadt used a Poisson distribution to calculate the number of items in the pipeline. This is true for the number of items in the de depot pipeline. The pipelines of the bases however could have a different distribution. Sherbrooke showed in (Sherbrooke, 2004) that the pipelines of the depot are indeed Poisson distributed, but the base pipeline is not always Poisson distributed. The distribution of the base pipeline is depending on the backorders of the depot. When the Variance-to-mean ration is exactly one then the pipeline is Poisson distributed, but most of the times the variance-to-mean is larger than one. Therefore Sherbrooke responded with VARI-METRIC. This method checks what kind of distribution the pipeline needs to calculate the backorders. The following distributions are used in Inventri for the pipelines. Whereby the E[X] is the mean of random variable X and Vx is the variance- to-mean.

Table 16: Four types of distributions related to Vx and E[X] (Rustenburg, 2000, p. 58)

Combinations of E[X] and Vx Type of distribution

1- E[X] ≤ Vx < 1 Mixture of two binominal distribution

Vx = 1 Poisson distribution

1 < Vx≤ 1 + E[X] Negative binominal distribution

J.L. Schmal The transition from operational availability to mission availability Case Study 31 The VARI-METRIC model uses the multi-echelon and multi-indenture structure. A review is made of a single-site, multi-indenture model to keep the theory understandable. Most of the derivations can be found in appendix C, the complete derivations can be found (Rustenburg, 2000) and (Sherbrooke, 2004).

Single-site, multi-indenture model

When an LRU fails then this is caused by an SRU i with a probability qi. The LRU could only be defect by one of the SRUs so the sum of all the probabilities qi must be equal to one. An LRU could be in three states, namely: Operational, in repair or on stock. Figure 14 shows the steady state model. The assumption in this model is that the repair success rate is 100%.

Figure 14: Steady state single-site, multi-indenture model

The single LRU demand is the sum of the demand of the SRUs. The expected pipeline of the LRU is the demand of the LRU multiplied by the repair lead-time of the LRU plus the summation of the expected backorders of the SRUs. This is also true for the variance of the LRU’s pipeline, but the summation is over the variance in backorders of the SRUs. The formulas are as followed:

 

_

 

 

_

 

Multi-echelon, multi-indenture model

Sherbrooke uses multiple demand calculations for the multi-echelon, multi-indenture model. Each demand calculation depends on the location and the indenture. Sherbrooke also takes into account that the base could repair the system, otherwise the depot repairs the LRUs. The probability whether the base could repair the LRU is rij. Figure 15 shows how the formula is adjusted to fit each location and indenture.

32 Case Study The transition from operational availability to mission availability J.L. Schmal Figure 15: Demand formulas per location per indenture (Sherbrooke, 2004)

The demands of the LRU at the depot and the SRU at the depot can be calculated from the demand of the LRU at base. From the demand of the SRU at base and the demand of the LRU at depot the demand of the SRU at the depot can be calculated.

Now the demands are explained the pipelines could be explained. The calculations of the pipelines are made in the opposite direction of Figure 15. The pipeline variance is among other dependent on the variance of backorders (VBO). The VBO can be calculated with the following formula:

 







 



The pipeline for the LRU can be calculated as followed: Start with the calculations of the SRUs in the depot repair. The SRUs in the depot repair have no influences from other items and locations. The numbers of the SRUs in the pipeline have a Poisson distribution with a mean mi0Ti0. The calculations of the mean and the variance of the SRU i at the depot could be done by EBOi0(si0), VBOi0(si0). The following step is the calculations of the depot pipeline of the LRUs. This pipeline is dependent on the SRU pipeline. The third step is the SRU pipeline at the base. The SRU pipeline at the base is dependent on the SRU backorders of the depot. The final step is to calculate the pipeline of the LRUs at the base. The numbers of the LRUs in the base pipeline are dependent on the LRU backorders at the depot and the SRU backorders at the base. All the SRU backorders at base j arise from the LRU demand at base j.