By Hans Kaper
Our planet is being stressed. For some 150 years, we have run an uncontrolled experiment, assuming (wrongly) that our resources are infinite and that nature will adjust to our needs and desires. As a result, our climate is changing, ocean levels are rising, and the impact of natural disasters has been increasing. As mathematical and computational scientists, we should be concerned: We have a responsibility. The only tools available for studying future scenarios are mathematical models and computational experiments; large-scale controlled experiments are essentially impossible, and there is no planet B. The newly-formed SIAM Activity Group on Mathematics of Planet Earth (see announcement here) is SIAM’s forum for discussions of mathematical and computational issues of climate, sustainability, ecology, socioeconomic systems, and the environment.
In this article I give an example of MPE in action. The example comes from my native country, the Netherlands, and makes an excellent case for the importance of MPE. (I have not been involved in the research; the topic was suggested by an article in the June 2014 issue of the Nieuw Archief voor Wiskunde, published by the Netherlands Mathematical Society [1]. I thank the authors for permission to use their results.)
Flood Control in the Netherlands
Flood control is an important issue for the Netherlands—about two thirds of its area is vulnerable to flooding, and it is among the most densely populated countries in the world. Natural sand dunes and human-made dikes, dams, and floodgates provide defense against storm surges from the sea. River dikes prevent flooding from the major Rhine and Meuse Rivers, while a complicated system of drainage ditches, canals, and pumping stations (historically, windmills) keep the low-lying parts dry for habitation and agriculture.
The current sea defenses are stronger than ever, but in 2008 a government committee reported that an expected sea level rise of 65 to 130 cm by the year 2100 might make further upgrades to the flood control and water management infrastructure necessary.
The first of the committee’s 12 recommendations was that the present flood-protection levels of all diked areas be improved by a factor of 10. The national government requested a cost–benefits analysis, which would find an optimal balance between investment costs and the benefits of reduced flood damages. The researchers who took on the task developed an optimization model, generalizing a model produced in the 1950s by Van Dantzig after a catastrophic flood in 1953 [3].
The project, which is estimated to yield savings of €7.8 billion, received the prestigious international Franz Edelman Award in 2013. What follows is a brief overview of the research; details can be found in technical papers by project participants [2, 4].
Model Formulation
Figure 1. Dike-ring areas in the Netherlands, color-coded according to flood protection standards. From Klijn, van Buuren, and van Rooij (2004) / Rijkswaterstaat [7].
Flood-protection levels have been defined for each of 95 flood-prone areas in the Netherlands. These areas, which are referred to as “dike-ring areas,” are protected from flooding by a ring of contiguous segments. A ring can have up to ten segments, which can be a mix of dunes, dikes, and human-made structures; each segment has different characteristics with respect to investment costs, flood probabilities, water level rise, etc. Figure 1 shows some of the main dike-ring areas with their flood-protection standards. Not all segments need to be raised by the same amount, or at the same time; the decision problem concerns the question of when and how much to invest in each individual segment. The objective is to find an investment plan that minimizes the total expected costs for a given finite planning horizon \([0, T)\).
The number of segments in a ring is denoted by \(L\). The height of a segment can be changed only at discrete times \(t_k\) and must remain unchanged during the subinterval \((t_k, t_k + 1)\), where \(0 = t_0 < t_1 < \cdots < t_K < T\); it also does not change beyond the planning horizon. An investment plan is an ordered pair \((\mathbf{U}, \mathbf{t})\), where \(\mathbf{U}\) is a matrix whose (positive) entries \(u_{lk}\) correspond to the increase in height of segment \(l\) at time \(t_k\), and \(\mathbf{t}\) is a vector \((t_0, . . . t_K)^T\). The objective is to find an investment plan that minimizes the sum of the investment costs and the expected damage costs,
\[\begin{equation}\tag{1}
\mathrm{minimize}~ \mathcal{I}(\mathbf{U}, \mathbf{t}) + \mathcal{E}(\mathbf{U}, \mathbf{t}) + \mathcal{R}(\mathbf{U}, \mathbf{t}).
\end{equation}\]
Here, \(\mathcal{I}\) represents the total discounted investment cost within the planning horizon \([0, T)\); \(\mathcal{E}\), the total expected damage in the same planning horizon; and \(\mathcal{R}\), the total damage after the planning horizon.
Figure 2. Cumulative height increases of dike segments in a six-segment ring. From Figure 1 in [1].
Numerical Results
The solutions of the nonlinear optimization problem (1) can be found numerically (after some approximations) for dike rings with modest numbers of segments. Figure 2 shows the cumulative height increases for a six-segment ring during a 300-year planning horizon; Figure 3 shows the resulting segment flood probabilities. These numerical results were obtained with realistic parameter values from Deltares, an independent institute in the Netherlands for applied research on water, subsurface, and infrastructure. Notice that at \(t = 20\) the height of segments 2, 3, 4, and 6 is increased, while segments 1 and 5 remain unchanged. Figure 3 shows why it is not necessary to increase the height of these two segments: Their flood probabilities are still very low compared to the other segments. Figure 3 also shows that simultaneously increasing the height of all segments is not necessary but generally leads to good or even optimal results.
Figure 3. Flood probabilities for the six segments of the ring from Figure 2. From Figure 2 in [1].
Practical Impact
The solution to the optimization problem (1) has been used to construct flood- protection standards [5, 6]. Increasing the legal protection standards of all dike-ring areas tenfold, as recommended in 2008, was found to be unnecessary. The current protection standards are (more than) adequate, except for three specific regions that the optimizers identified. For these three regions, new standards have been developed and endorsed by parliament.
References
[1] R.C.M. Brekelmans, C.J.J. Eijgenraam, D. den Hertog, and C. Roos, A mixed integer non-linear optimization approach to optimize dike heights in the Netherlands, Nieuw Archief voor Wiskunde, Series V, 15 (2014), 113–118.
[2] R.C.M. Brekelmans, D. den Hertog, C. Roos, and C.J.J. Eijgenraam, Safe dike heights at minimal costs: The non-homogeneous case, Oper. Res., 60 (2012), 1342–1355.
[3] D. van Dantzig, Economic decision problems for flood prevention, Econometrica, 24 (1956), 276–287.
[4] C.J.J. Eijgenraam, R.C.M. Brekelmans, D. den Hertog, and C. Roos, Flood prevention by optimal dike heightening, submitted for publication, 2014.
[5] C.J.J. Eijgenraam, J. Kind, C. Bak, R.C.M. Brekelmans, D. den Hertog, M. Duits, C. Roos, P.J. Vermeer, and W. Kuijken, Economically efficient standards to protect the Netherlands against flooding, Interfaces, 44 (2014), 7–21; http://pubsonline.informs.org/doi/abs/10.1287/inte.2013.0721.
[6] J.M. Kind, Economically efficient flood protection standards for the Netherlands, J. Flood Risk Management, 7 (2014), 103–117; doi:10.1111/jfr3.12026.
[7] F. Klijn, M. van Buuren, and S.A.M. van Rooij, Flood-risk management strategies for an uncertain future: Living with Rhine River floods in The Netherlands? AMBIO, 33:3 (2004), 141–147.
Hans Kaper, founding chair of SIAG/MPE and editor-in-chief of SIAM News, is an adjunct professor of mathematics at Georgetown University.