# Developing a mathematical formula and proof

Cancelled

## Job Description

I need to calculate the change in probability resulting from the following scenario

A company buys a piece of equipment A to make widgets There is an insatiable demand for the widgets so the more widgets that the company makes the more they can sells. So the company buys more equipment type A to make more widgets. The equipment once installed is run at design capacity so there is no excess capacity available on any of the equipment when repairs or equipment failure occurs. All type A equipment is the exact same make and model.

The equipment is 98% reliable and available and that extends to 99% when the preventative maintenance program is removed.

The equipment has only three particular failure modes (e.g bearing failure, metal fatigue, gasket failure) which result in varying amounts of downtime (3 days, 6 days and 9 days respectively) resulting from the mean time to repair when spares are held on site.

The downtime can be extended if the spares (bearings, casings and gaskets) are not held on site. So that when the failure occurs, the downtime is increased by the lead time between the point of failure and the delivery of the spare part from the manufacturer. The lead times on delivery are 10 days for the bearings, 20 days for the casings and 30 days for the gaskets
The cost of each spare type is expensive. The bearings are \$1m, casings \$5m and gaskets \$10m each

The question that the company now seeks to answer is how many bearings, casings and gaskets does it need to hold on site as its fleet of equipment type A grows from 1 to 10 to 20 to 30....

Deliverables

Mathematical formula
Example demonstrated in an excel spreadsheet
mathematical logic/ proof
english language explanation of why the formula works

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Skills: design, english