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In this article we show how to convert the Probability of Availability to Stock-Out-Risk or Mean-Time-Between-Stock-Out for Spare Parts inventory.

## Problem

1. Spares Calculator can forecast [Stock-Out-Risk],  [Mean Time Between Stock-Out] and [Probability of Availability]
2. Analysts need to understand the relationship between these three parameters

## Background

There is no accepted best practice for setting goals for spare parts inventory. Some contracts specify [Probability of Availability], others prefer [Stock-Out-Risk] and others prefer the [Mean Time Between Stock-Out]. Thankfully, there is a mathematical relationship between each of the goals and in this article we will show you how you can manually convert between each value?

## Proof 1

### [Stock-Out-Risk] and [Probability of Availability]

We now prove the relationship between [Stock-Out-Risk] and [Probability of Availability]

We start with the fundamental statistical axiom that the probability of a certainty is 1.

[Probability of a Certainty] = 1 (eq1)

We also know that it is certain that a part will either be Availability or Unavailability.

Therefore:

[Probability of Availability] + [Probability of Unavailability] = 1 (eq2)

By transposing we get:

[Probability of Availability] = 1 – [Probability of Unavailability] (eq3)

And:

[Probability of Unavailability] = 1 – [Probability of Availability] (eq4)

Now:

Stock-Out-Risk is defined as:

[Stock-Out-Risk] = [Probability of Unavailability] x 100% (eq5)

Or:

[Stock-Out-Risk] = (1 – [Probability of Availability]) x 100% (eq6)

By transposing we get:

[Probability of Availability] = 1 – ([Stock-Out-Risk] / 100%) (eq7)

Therefore, we have proven the relationship between the [Probability of Availability] and [Stock-Out-Risk].

## Proof 2

### [Mean Time Between Stock-Out] and [Probability of Availability]

We now prove the relationship between [Mean Time Between Stock-Out] and [Probability of Availability]

[Mean Time Between Stock-Out] = [Replenishment Delay] / [Probability of Unavailability] (eq8)

But:

[Probability of Unavailability] = 1 – [Probability of Availability] (eq9)

Therefore:

[Mean Time Between Stock-Out] = [Replenishment Delay] / 1 – [Probability of Availability] (eq10)

Or:

[Probability of Availability] = 1 – ( [Replenishment Delay] / [Mean Time Between Stock-Out] ) (eq11)

## Example 1

### Convert [Probability of Availability] to  [Stock-Out-Risk]

A procurement authority states:

All spare parts shall have a [Probability of Availability] of greater than 0.95 (or 95%) measured in a 30-day period.

Convert this into a corresponding [Stock-Out-Risk] goal:

Equation 5a states:

[Stock-Out-Risk] = (1 – [Probability of Availability]) x 100%

[Stock-Out-Risk] = (1 – 0.95) x 100%

[Stock-Out-Risk] = 5%

Therefore, we can convert the statement to read:

All spare parts shall have a [Stock-Out-Risk] of less than 5% measured in a 30-day period.

## Example 2

### Convert [Probability of Availability] to  [Mean Time Between Stock-Out]

A procurement authority states:

All spare parts shall have a [Probability of Availability] of greater than 0.95 measured in a 30-day period.

Convert this into a corresponding [Mean Time Between Stock-Out] goal:

Equation 9 states:

[Mean Time Between Stock-Out] = [Replenishment Delay] / 1 – [Probability of Availability]

[Mean Time Between Stock-Out] = 30/(1-0.95)

[Mean Time Between Stock-Out] = 600 days

Therefore, we can convert the statement to read:

All spare parts shall have a [Mean Time Between Stock-Out] of greater than 600 days.

Notice that the [Mean Time Between Stock-Out] encompasses both the availability figure and the 30-day period in one single parameter.