What Is The Equation To Use When Measuring Equipment Reliability Improvement?

The accepted definition of reliability is: the probability that an equipment item will perform its duty without failure over a designated time.
You also need to be clear as to whether you are measuring reliability of a complete item of equipment, or of its individual components. This was the big mistake made with Reliability Centered Maintenance when Nolan and Heap investigated aircraft failure data in the 1970s to develop the RCM methodology. They mixed-up parts failure data together with systems failure data and came up with RCM. That mistake has now lead to much stupid and wasteful maintenance work being done by those companies around the world that apply RCM.
If you want to use a much better way than RCM that is a lower cost, faster, highly accurate process to identify the correct and proper maintenance to do on your plant and equipment you ought to use our Physics of Failure Maintenance Strategy selection process.
In reliability engineering a component is a replaceable part. If your company practise is to replace a failing rolling bearing in a gearbox with a new one, then the bearing is taken to be the part. If instead your practice is to replaced the whole gearbox with a new one (replacing it with a refurbished gearbox is not an identical situation) then the gearbox is taken to be the part.
As your question is not specific about what is being measured, I will address both scenarios—system reliability first and then component reliability.
If reliability is improving for a particular machine (a machine is a working system of parts) then there will be an increase in the time between outages that require maintenance. That is how  reliability improvement is typically measured—Mean Time Between Failure (MTBF), with failure defined as anytime the machine cannot perform its duty. This does not only mean breakdowns, it means every event where the equipment cannot be used.
You should not only calculate MTBF. The mean time between failure metric used alone is misleading. You also need to know the two extremes of the failure events distribution—both the shortest time between failure (STBF) and the longest time between failure (LTBF).
By using MTBF, STBF and LTBF together you get a more complete understanding of the shape of the probable failure distribution curve and of the equipment operating life expectations.
System reliability improvement can also be measured using Crow-AMSAA reliability growth modelling as described in our simple guidebook Reliability Growth Plotting in Excel that you can buy at our online store.
When measuring component reliability the MTBF (along with STBF and LTBF) can be used so long as you clearly state which part in the machine you are talking about.
For components you can also use the Weibull Analysis life parameters to monitor reliability improvement. Weibull Analysis is only to be undertaken by trained mathematicians and statisticians. Even for degree qualified engineers there are many probability traps to fall into.
You must NEVER use Weibull Analysis for systems; unless the entire system is replaced every time that it fails. Weibull Analysis only works for individual ‘parts’.