To Violate or Not To Violate Water Quality Standards: A Decision Analytic Approach

In the United States, the manager of an industrial wastewater treatment plant is considering internal plant operation policies in response to a recently-issued NPDES (National Pollutant Discharge Elimination System) permit. The NPDES permit stipulates that total phosphorus concentration should not exceed 1 mg/l in the discharge from the plant. How should the manager operate the plant, or in other words, what should be the design total phosphorus concentration in the wastewater discharge?

First, consider the situation in which engineering design and operation are perfect (without error). Can the plant manager specify the best operating policy under those conditions? The answer is no. To see this, consider first the situation where: (1) treatment cost decreases as total phosphorus concentration removal efficiency decreases, and (2) it is generally understood that there is no enforcement of the permit discharge limits as long as total phosphorus concentration concentrations are not greater than 1 mg/l above the permitted level. Then, on economic arguments alone, the plant manager may consider plant operation to achieve 2 mg/l total phosphorus concentration (1 mg/l + 1 mg/l) in the effluent.

Alternatively, suppose: (1) treatment cost decreases (increases) as total phosphorus concentration removal efficiency decreases (increases) as before, and (2) one, even slight, violation of the permit limits results in the enforcement penalty of plant closure. In this case, the plant manager is likely to consider plant operation to achieve exactly 1 mg/l total phosphorus concentration in the effluent (i.e., come as close as possible to the NPDES limit, but never exceed the limit).

It is apparent that in both situations, some notion of costs or losses was needed for the plant manager to justify a particular operating policy. Once stated, it probably seems obvious that the plant manager would request a cost analysis (including permit violation penalty costs) before deciding. Then, once the cost analysis is complete, the plant manager could implement the least-cost error-free operation. In summary, cost information was needed to supplement the water quality impact assessment, before the decision could be made.

What should the decision be if there is uncertainty in the engineering design and operation? To see how the interplay between scientific uncertainty and net cost should influence decision making, consider the figure below. The bottom graph in the figure presents the scientific and engineering assessments of total phosphorus concentration in the treatment plant discharge. These assessments are presented as probability distributions, using probability as the expression of scientific uncertainty. The two bell-shaped curves in the bottom graph convey uncertainty through their dispersion (i.e., through how "spread-out" they are). A third object in the bottom graph is a line with an upward-pointing arrow. This line/arrow represents the scenario described above - certain science; it is a probability density function with no width (no uncertainty) and infinite height.

In the upper graph on the figure, the lines represent costs (only costs above those required to exactly meet the NPDES permit requirements) in dollars, based on an NPDES effluent limit of 1 mg/l. For simplicity, while not changing the central message presented here, it is assumed that there are only two primary sources of costs: (1) penalty costs or fines associated with NPDES permit violations, and (2) costs that result from excess wastewater treatment, beyond that required to meet the permit limits. The height of the lines indicates the magnitude of the cost at a particular total phosphorus concentration discharge concentration. Thus, all lines drop to zero cost at exactly 1 mg/l total phosphorus concentration, the point at which no fines are levied and treatment is not excessive. On either side of 1 mg/l, the lines rise linearly, indicating a linear increase in cost associated with either: (1) fines for NPDES permit violations (above 1 mg/l), or (2) excessive wastewater treatment (below 1 mg/l). The vertical line just beyond 1 mg/l represents an "infinite" fine for permit violation, which for the example above characterizes plant closure.

Two basic scenarios, A and B, are presented in the figure. Scenario A (solid lines on the figure) is the situation described above. In the lower graph, the scenario A arrow with no width indicates perfect scientific knowledge. In the upper graph, the scenario A cost function has infinite slope above 1 mg/l, indicating that the discharger goes out of business. Below 1 mg/l, the scenario A cost function linearly increases with decreasing total phosphorus concentration, as a consequence of excessive treatment.

The two intertwined graphs in the figure can be used to illustrate decision making under uncertainty in the following way. First, the cost function is determined (graphically and mathematically) and placed on the upper graph in the figure. Next, scientific and engineering knowledge concerning total phosphorus concentration discharge concentration should be characterized probabilistically and the probability distribution placed as a "sliding overlay" on the figure. The sliding overlay is meant as a graphical exercise to move horizontally and ultimately place the distribution of total phosphorus concentration discharge concentration such that expected cost is minimized.

Consider scenario A as an example. Scientific knowledge is certain, indicated by the arrow with zero width. The key management question is: given perfect knowledge, what should be the total phosphorus concentration discharge concentration (i.e., where along the total phosphorus concentration scale should the arrow be placed) to minimize cost? Obviously, it should not be above 1 mg/l, since the cost (penalty) is infinite. In addition, the further below 1 mg/l the discharge is, the greater the cost of overtreatment. Thus, cost is minimized when the discharge concentration (and the arrow) is at precisely 1 mg/l. In other words, with perfect knowledge and the asymmetric cost function (i.e., a different cost rate, in cost per mg/L total phosphorus concentration, associated with overtreatment from the 1 mg/l NPDES limit versus that associated with undertreatment for scenario A) we should treat to just achieve the standard.

Given this understanding, we can now use the linked graphs to gain insight on decision making under scientific uncertainty. In the upper cost graph, again consider scenario A. Now, however, we acknowledge and quantify the scientific uncertainty inherent in the prediction of discharge concentration from a wastewater treatment plant. This uncertainty for the total phosphorus concentration discharge concentration is expressed in the probability distribution in the lower graph. As before, the probability distribution should be thought of as a horizontal sliding overlay that can be placed at any point along the horizontal axis. The optimal location for this sliding distribution is that which minimizes expected cost.

Where should this probability distribution of total phosphorus discharge concentration be placed? (i.e., What should be the expected total phosphorus concentration in the discharge to minimize expected cost?) Well, since cost is infinite if 1 mg/l is exceeded, there must be no chance (zero probability) that total phosphorus concentration discharge concentration will be above 1 mg/l. Thus, the distribution must be to the left of 1 mg/l, and must be far enough to the left so that no portion of the right tail exceeds 1 mg/l (it is assumed that the probability distributions are symmetric and similar to normal density functions, except that the tails go to zero as displayed in the figure). The further to the left the total phosphorus concentration distribution is placed, the higher the cost of overtreatment. Thus, we do not want to place the distribution any further to the left than is necessary to avoid the infinite cost above 1 mg/l. The solution is therefore clear - locate the total phosphorus concentration distribution so that the right tail intersects the horizontal axis (reaches zero probability) at exactly 1 mg/l. This is identified as scenario A' on the lower graph.

What can we learn from this example? First, note that in the absence of scientific uncertainty, we can discharge at the concentration limit (if costs justify this choice). However, once scientific uncertainty is considered (scenario A' versus scenario A), discharging at the concentration limit may no longer be the optimal (least cost) choice (it is still optimal under certain specific conditions as noted below). Mathematically, the solution to this cost minimization problem involves the integration of the cost function with the probability model. This, in effect, weights the cost at each concentration increment by the probability that that concentration increment will occur in the discharge. Knowing that, we can see that the graphical representation of the solution must have zero probability above 1 mg/l to negate the impact of infinite cost.

The final decision results from the fact that, in effect, we hedge decisions away from large (in this case, infinite) losses. This is a general strategy in decision making under uncertainty, whether it is based on formal scientific decision analysis or informal, everyday decision making. As a consequence of hedging from large losses, note that the probability distribution is centered at total phosphorus concentration ~ 0.5 mg/l; thus, the expected total phosphorus concentration in the discharge is less than 1 mg/l. The greater the dispersion (uncertainty) in the total phosphorus concentration probability distribution, the greater will be the difference between the NPDES limit and the expected discharge concentration. As this difference between expected discharge concentration and the permit limit increases, operating costs increase.

In general, reduction in scientific uncertainty should be expected to reduce the dispersion in the total phosphorus concentration distribution, which should result in management strategies that reduce operating cost. Of course, uncertainty reduction comes at a cost. In addition, the new knowledge associated with reduction in uncertainty may actually imply greater cost due to previously unforeseen consequences.

Once we understand the interplay between cost and uncertainty, the decision in scenario A' is relatively simple because of the infinite cost of permit violation. The analysis becomes complicated with a more realistic cost function like that for scenario B in the figure. Yet, while we may not be able to precisely define the optimal solution to scenario B in a graphical sense, we can still gain some insight into the role of uncertainty by considering the general features of the solution.

Since the cost of NPDES permit violation is not infinite in scenario B, the optimal solution will now have a nonzero probability of standard violation, as long as the cost of overtreatment is greater than zero. The exact location of the sliding horizontal overlay distribution for scenario B can best be determined mathematically (by integration). However, it is clear that lower overall cost is achieved when the distribution is positioned allowing a small probability of permit violation cost (upper right tail), as opposed to a correspondingly small probability of higher unit cost of overtreatment (lower left tail). This is apparent when we note that the slope of the cost lines indicates the change in cost as total phosphorus concentration discharge concentration changes, and the height of the line at any point indicates the cost. Thus, the cost of extreme overtreatment (e.g., total phosphorus concentration discharge concentration of 0.5 mg/L) is greater than the cost of a slight permit violation (e.g., total phosphorus concentration of 1.1 mg/L).

The proportion of the symmetric total phosphorus concentration distribution that exceeds 1 mg/l is dependent on the relative slopes of the overtreatment and undertreatment cost functions. Some general conclusions are:

(1) If the probability distribution is symmetric, and the cost function is also symmetric (the slopes on the costs of overtreatment and undertreatment are identical), then the optimal solution will have an expected total phosphorus concentration discharge concentration of exactly 1 mg/l (i.e., the distribution will be centered on 1 mg/l).

(2) If the probability distribution is symmetric, but the cost function is asymmetric (the slopes on the costs of overtreatment and undertreatment are different), then the optimal solution will have an expected total phosphorus concentration discharge concentration on the side of 1 mg/l with the cost function of lesser slope.

(3) If the probability distribution is asymmetric, the solution is more difficult to characterize and describe, since the expected value is no longer the "center of symmetry" for the distribution. However, if the cost function is symmetric, then the asymmetric probability distribution will have its peak (mode) on one side of 1 mg/l and its "stretched-out" tail on the other side, in order to hedge away from the large loss associated with the elongated distribution tail.