Here are the key steps to evaluate the cost-efficiency and energy-efficiency of a boiler or heating system:

1. Calculate the boiler's annual fuel utilization efficiency or AFUE. This is the percentage of fuel energy that is converted to heat. A higher AFUE means more efficient. Boilers with AFUE of 80% or less are considered not very efficient. Newer boilers will typically be in the 90%+ range.

2. Calculate the boiler's thermal efficiency. This is the percentage of the fuel's energy that is converted to usable heat. Again, a higher percentage is more efficient. For natural gas boilers, a thermal efficiency of 85% or more is considered efficient.

3. Determine the heating requirements for your home in terms of BTUs or kilowatts needed per hour. This depends on factors like square footage, insulation, climate, etc. You can use an online heating load calculator to estimate your requirements.

4. Calculate the operating costs of your boiler or heating system based on the efficiency numbers and your usage requirements. Also account for the cost of fuel to get the total annual operating cost. Compare the costs of different systems to evaluate relative cost-efficiency.

5. Also consider other factors like upfront costs, lifetime, and environmental impact to evaluate the overall efficiency and cost-effectiveness of different heating options for your home.

In general, higher efficiency boilers, heat pumps, and other systems will be more energy-efficient and cost-effective in the long run compared to less efficient boilers, even though they may have higher upfront costs. But you have to run the numbers for your own home and requirements to determine the best choice. Let me know if you have any other questions!
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