Primal and dual valuation of our natural resources using O.R.

There are many ways in which one can devise the importance of O.R. to protect our environment, many of which dealing with optimization problems related to directly reducing the costs to prevent its destruction and so on. However, what about the environmental impacts from our patterns of consumption? Shall we change our way of living dramatically or rather find a balance between what we want and what we can use from our environment? Maybe O.R. can help us on that.

Roughly speaking, Operations Research (O.R.) deals mostly with finding the best way of doing something subject to a lot of different kinds of restrictions. Thus, one can indirectly consider the protection of the environment whilst solving a wide range of different optimization problems related to the daily needs of our society. For that sake, I figure two possibilities to consider the protection of the environment:

  • pricing natural resources appropriately as a subtraction to the profit of the operation;
  • limiting their use so as to avoid that we steal the share that belongs to the future generations.

I’ve already written about the first possibility in my post about optimizing public policies for urban planning. My fiancee and I devised a model to consider the environmental costs of subsidizing low income housing units at different parts of the city in what comes to daily displacement to work. However, finding the right data to run the model turned out to be our biggest problem. When one defines a penalty to the environmental impact related to the profit of an operation – for instance, using the value of carbon credits – it represents a cost to the problem. However, sometimes we might not have data to price it. But if the consumption of a given natural resource is limited by a constraint instead of penalized in the profit, it is still possible to figure the economic importance of such resource through duality. Therefore, let’s take a look at the second possibility as an alternative to finding the price of natural resources – as well as avoiding an excessive use of them.

The concept of duality in linear programming allows us to associate costs to our constraints. Suppose that our optimization problem is about finding the amount of goods of each kind to produce in order to maximize the profit that they generate subject to the limited resources available. The dual of this problem consists of finding the price of each unity of our limited resources in order to define the minimum price at which it is worthier to sell them instead of processing subject to how much profit each finished good would give us. The relationship between those two problems is quite strong: if a resource is not used up to its limit, its dual cost is zero – meaning that it does not have an economic importance according to the model. Therefore, duality can help us devising how much a limited resource is worth (if it is worth something) and thus provide a way of valuating resources according to their limitation and importance.

As a matter of fact, the more you understand the relation between primal and dual problems, the easier it becomes to talk face-to-face to economists. Indeed, this topic has a lot to do with the 1975 Nobel Prize in Economics. If you want to know more about that, the prize lectures from Kantorovich and Koopmans are a good starting point.

This post was written to the September’s INFORMS blog challenge: O.R. and the Environment.

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