*Episode 9 in this series on principles to follow when ranking environmental projects. It is about how to account for account for multiple types of benefits from the same project.*

Previous posts in this series have assumed that there is only one type of benefit being generated, or that multiple benefits generated by a project have already been converted into a common currency, such as dollars, and added up. What if we have multiple benefits and we want to account for them individually?

Suppose that a single project will generate three types of benefits, due to improvements to three different, but connected, environmental assets (e.g. a threatened species, a river that is suffering from reduced water quality, and an area of riparian vegetation that is attractive and provides habitat). The three assets have independent values *V*_{1}, *V*_{2} and *V*_{3}.

All of the variables we’ve talked about in this series (the effectiveness of works, adoption, time lags and the various risks) could potentially have different values for each of the environmental benefits generated by this one project. For example, if the project is implemented, the risk of failing to improve water quality might be higher than the risk of failing to improve the condition of the riparian vegetation.

If the differences were significant enough, we might think it would be worth estimating three different values for each of the variables: *W*, *A*, *L* and the various *R*s.

If the *V*s were each measured in dollars, the expected benefits for the project would simply be the sum of the formulas for each benefit, as follows. [Three lots of the formula from PD242.]

*Expected benefit* = *V*_{1}(*P*’) × *W*_{1} × *A*_{1} × (1 – *R*_{1}) / (1+*r*)^{L1} +

*V*_{2}(*P*’) × *W*_{2} × *A*_{2} × (1 – *R*_{2}) / (1+*r*)^{L2} +

*V*_{3}(*P*’) × *W*_{3} × *A*_{3} × (1 – *R*_{3}) / (1+*r*)^{L3}

To keep the formula simple, I’ve assumed that the relationship between adoption and benefits is linear, and I’ve combined the various risks into one overall probability of failure. Also, from now on I’m mostly going to use the *V*(*P*’) × *W* version of the formula, rather than the [*V*(*P*_{1}) – *V*(*P*_{0})] version, which is equivalent. This has advantages in shaping the thinking, as outlined in PD239, but also advantages for simplifying the formula where there are multiple benefits, as you’ll see later.

If the values are not measured in money terms, you’ll need to provide weights (*z*_{1}, *z*_{2} and *z*_{3}) to indicate the relative importance of the different benefits. The formula becomes:

*Expected benefit* = *z*_{1} × *V*_{1}(*P*’) × *W*_{1} × *A*_{1} × (1 – *R*_{1}) / (1+*r*)^{L1} +

*z*_{2} × *V*_{2}(*P*’) × *W*_{2} × *A*_{2} × (1 – *R*_{2}) / (1+*r*)^{L2} +

*z*_{3} × *V*_{3}(P’) × *W*_{3} × *A*_{3} × (1 – *R*_{3}) / (1+*r*)^{L3}

This formula is getting pretty big and ugly. It also implies the need for a lot of information: the full equation for each type of benefit. Based on my experience, I’d say that most managers of real-world programs would not be prepared to go to this much detail. In reality, what commonly happens is that some of the variables are assumed to be the same for the different types of benefits. Often, I think that’s a reasonable approximation of reality, or at least one that’s not so bad that it’s worth fighting against. If it seems reasonable to assume that *W*, *A*, *R* and *L* are the same for all three benefit types, then we can simplify the equation for expected benefits, as follows.

*Expected benefit* = [*z*_{1} × *V*_{1}(*P*’) + *z*_{2} × *V*_{2}(*P*’) + *z*_{3} × *V*_{3}(*P*’)] × *W* × *A* × (1 – *R*) / (1+*r*)^{L}

This is just the same as the formula in PD242, but with *V*(*P*’) replaced by the large term in square brackets. Rather than being a single value, it’s now the weighted sum of several values.

In previous posts in this series, I’ve been critical of the common practice of weighting and adding up variables in certain cases. However, this formula shows that it is not always a mistake. If we don’t have dollar values, it’s reasonable to weight and add the separate values to get an indicator of the total value at stake, prior to adjusting it down for *W*, *A*, *R* and *L*, as shown. The big mistake that is commonly made is to also weight and add *W*, *A*, *R* and *L* into the equation, rather than including them in the way shown above. Weighting and adding can be appropriate, but needs to be applied in a way that makes logical sense, rather than indiscriminately to all variables.

If we were weighting the [*V*(*P*_{1}) – *V*(*P*_{0})] version of the formula, it would look like this:

*Expected benefit* = {*z*_{1} × [*V*_{1}(*P*_{1}) – *V*_{1}(*P*_{0})] + *z*_{2} × [*V*_{2}(*P*_{1}) – *V*_{2}(*P*_{0})] + *z*_{3} × [*V*_{3}(*P*_{1}) – *V*_{3}(*P*_{0})]} × *A* × (1 – *R*) / (1+*r*)^{L}

You can see that the other version is more compact.

Choices about the weights need to consider the way that the different benefits are scored. If the values are in dollars, all the weights become 1.0, so you end up just adding up the values.

If the values are not in money terms, the weights reflect the relative importance of the different benefits (a very subjective judgement), but they also need to account for the ranges over which the values are scored. For example, if value scores range from zero to 1.0 for one benefit but zero to 100 for another, the second one should probably have a much smaller weight to avoid it dominating the rankings. If the two benefits were equally important, the weight for the first one would need to be 100 times larger than the weight for the second one.

### Further reading

Pannell, D.J., Roberts, A.M., Park, G. and Alexander, J. (2013). Designing a practical and rigorous framework for comprehensive evaluation and prioritisation of environmental projects, *Wildlife Research* 40(2), 126-133. Journal web page ♦ Pre-publication version at IDEAS

## One comment

“If the values are not in money terms, the weights reflect the relative importance of the different benefits (a very subjective judgement), but they also need to account for the ranges over which the values are scored. For example, if value scores range from zero to 1.0 for one benefit but zero to 100 for another, the second one should probably have a much smaller weight to avoid it dominating the rankings. If the two benefits were equally important, the weight for the first one would need to be 100 times larger than the weight for the second one”

I think it helps to see these “weights” as “exchange rates” for converting between the different “benefits”, so they represent in effect an implict price of one benefit in terms of another.