How should we estimate the number of engineers our org needs?

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Director of Engineering in Media, 10,001+ employees
It depends on what journey the org is in. For SME / MSME the answers would be different and in addition to that which business they are in i.e user facing or pure B2B. In an MNC which is B2B the focus is on current product releases and the innovation they want to work on. Depending on this engineering leaders define the roadmap and allocate x number of resources accordingly. Additionally please note in some cases the numbers can also fluctuate on the basis of technology stack the team want to work. 
Director of Engineering in Software, 10,001+ employees
It can be decided during the creation of a team based on the project that the team will work upon. This can also be decided based on the Road Map formed by Product Owner or Manager.
Director of Engineering in Services (non-Government), 2 - 10 employees
I would consider the following:
1. Do you buy or build? Building requires engineers, buying requires contract managers and subject matter experts.  
2. On-premise or cloud? The latter requires fewer engineers to maintain infrastructure. 
3. R&D or business IT? The former requires vastly greater number of engineers. 
4. Use a method like RICE to estimate your development effort. 
5. Do you have a new or an experienced team. If the former, double your estimate. 
Director of Engineering in Media, 10,001+ employees
For a rough estimate of engineers for the organization, one need to definitely look at
1. Book to bill ratio and backlog. The extent of work you estimate to do in the near future, by quarter.
2. Is there a budget provision for buffer resources, may be around 5%, to cater to emergencies and scope creep. Is there a ratio for highly experienced vs low experienced staff? 
3. Rough estimate from historical data to give you a rough assumption of per engineer time effort/hour and the total estimated hours of the project. That will give a rough estimate of engineers needed for that project.
4. Engineers can be deployed on various applications and systems. Taking that also into consideration can help.

From all above, you would be able to forecast the engineers needed for your organization. 
Director of Product/Engineering in Media, 501 - 1,000 employees
There are various factors to take into account here:
1. I would start by looking at the current number of engineers and the amount of projects, to try and get a sense of engineer/project ratio.
2. Define a roadmap - how many projects needs to be completed let's say in the next year.
3. For each project - evaluate the complexity (or just try and calculate the average, with multiple projects this might be close enough to exact evaluation)
4. Projects that could be done externally at lower cost - consider outsource them.
5. The remaining projects, multiply them by their complexity (in terms of MD) and then you'll know how many engineers you would need to meet those projects timelines.
(6. You can always take some buffer, and add 5-15% more engineers to be on the safe side)
Chief Information Technology Officer in Finance (non-banking), 51 - 200 employees
A proper defined roadmap with enough level of detail to assign points and a proper cadence should give you an estimated number, based on your time horizon and then, double that...
Senior Vice President, Engineering in Software, 1,001 - 5,000 employees
This is quite a subjective question, to answer at a high level - Look at the number of projects, divide the workforce into dev+QA+BA+SRE+PM and then allocate a proportionate count according to the size of each project. Typically having someone with account ownership and project management experience can help a lot while this estimation is done, you may initially have an accuracy of 50-60% but slowly the processes mature and lead you towards better results.
CTO in Education, 10,001+ employees
It's different for every organization, size of business, and industry it is in. Also could depend on how much overall technology or network operations are outsourced or maybe even cloud-managed. Overall what is the companies future plans and can the current staffing ratio live up to the desired needs of the organization
AVP Innovations & Product in Banking, 10,001+ employees
It is subjective with respect to what business you operate in and what are your organizational goals.  Different org will have different IT needs, I would approach it by creating an org structure which accounts for areas like enterprise architecture, network administration, tech support, dev ops, cyber security and innovations & product. Create a smaller structure in each area which includes specialist, managers and developers and a team lead, outline specific roles and responsibilities and keep some space for new development and back-up and finalize the requirement. This can solve the problem with 70-75% accuracy.   
Senior Director Engineering in Travel and Hospitality, 10,001+ employees
0 - If there is no work and no customers
10's to 10k if there is a strong roadmap, good customer base to ensure they are supported.

In simple words, engineers are needed to solve a problem, the number of engineers are dependent of the scale and size of the problem

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