Many-objective de Novo water supply portfolio planning under deep uncertainty

https://doi.org/10.1016/j.envsoft.2011.04.003Get rights and content

Abstract

This paper proposes and demonstrates a new interactive framework for sensitivity-informed de Novo planning to confront the deep uncertainty within water management problems. The framework couples global sensitivity analysis using Sobol’ variance decomposition with multiobjective evolutionary algorithms (MOEAs) to generate planning alternatives and test their robustness to new modeling assumptions and scenarios. We explore these issues within the context of a risk-based water supply management problem, where a city seeks the most efficient use of a water market. The case study examines a single city’s water supply in the Lower Rio Grande Valley (LRGV) in Texas, using a suite of 6-objective problem formulations that have increasing decision complexity for both a 10-year planning horizon and an extreme single-year drought scenario. The de Novo planning framework demonstrated illustrates how to adaptively improve the value and robustness of our problem formulations by evolving our definition of optimality while discovering key tradeoffs.

Introduction

Climate change, population growth, and increased urbanization pose serious challenges to urban water supply management (Frederick and Schwarz, 1999, Lane et al., 1999, Vorosmarty et al., 2000, Milly et al., 2008, Brekke et al., 2009). These changes lead to increased water demands and amplified hydrologic variability, subsequently leading to higher risks for water supply failures (Kundzewicz et al., 2007). The severe costs associated with structural adaptations such as building new reservoirs motivate the need for innovative nonstructural adaptation techniques such as water marketing (Anderson and Hill, 1997). A water market seeks to allocate water to its “highest value use” through transfers between regions (Israel and Lund, 1995) or different user sectors (Hadjigeorgalis, 2008). The presence of a water market, though, does not imply that the city knows how to develop the most efficient water supply portfolio that fulfills all of its planning goals Kasprzyk et al. (2009). This difficulty can be attributed to severe uncertainties within water supply portfolio planning, including future supply and demand (Ng and Kuczera, 1993), the form of economic instruments within the water market itself (Williamson et al., 2008), and the quantitative model that relates decision making within the market to decision maker’s utility and preferences.

Many of these uncertainties can be characterized as deep (Lempert et al., 2006) or Knightian (Knight, 1921) in nature, since it may not be straightforward (or even possible) to construct a probability model of likely vulnerabilities for these systems. Since the seminal work by Knight, there has been much discussion on the proper definition of deep uncertainty (Ellsberg, 1961, LeRoy and Singell, 1987, Langlois and Cosgel, 1993, Runde, 1998). According to Langlois and Cosgel (1993), the most prevalent interpretation is exemplified by Friedman (1976), who states that Knight’s risk describes events that can be characterized by a “knowable” probability distribution while deep uncertainty describes events where it is not possible to characterize a probability distribution. Our interpretation follows Langlois and Cosgel (1993), where deep uncertainty refers to decision makers’ limitations in conceptualizing the full range of possible types of risk in addition to quantitative probabilities of those risks. This paper explores the deep uncertainty in water portfolio planning by rigorously testing alternative formulations of a city’s decision strategies and carefully exploring the effect of modeling assumptions by constructing challenging planning scenarios. The goal of this analysis is to develop robust solutions that have good performance under many different modeling conditions (Lempert, 2002) and aid decision makers in understanding the effects of their use of estimated probabilities (Savage, 1972) on the planning process.

To explore the deep uncertainties of water portfolio planning, we present an adaptive decision making framework termed the sensitivity-informed de Novo planning framework. The framework continually updates planning objectives, constraints, and decision variables with the goal of aiding the decision maker in rigorously attaining a more complete understanding of their planning problem (Zeleny, 1989). Our work is an extension of the de Novo programming paradigm developed by Zeleny, in which the formulation of resource allocation problems is modified to avoid squandered resources (Zeleny, 1981, Zeleny, 2005). While recent studies have acknowledged the “statistical nonstationarity” of historical data (Milly et al., 2008), we posit that there is a more general mathematical nonstationarity in how water management problems are actually defined. This nonstationarity lies in the definition of “optimality” itself and reflects that knowledge gained in solving a problem will lead to iterative changes in the definitions of planning objectives, decisions, and constraints (i.e., a nonstationary problem topology because the definition of optimality is changing). Our work builds on the growing body of work in the area of constructive decision aiding (Roy, 1990, Lund and Palmer, 1997, Roy, 1999, Castelletti and Soncini-Sessa, 2006), which emphasizes the role of quantitative environmental policy models as a means of promoting and improving problem understanding rather than strictly focusing on providing a single definitive answer (Liebman, 1976).

This paper uses a case study of a single city’s use of market-based transfers to augment its water supply in the Lower Rio Grande Valley (LRGV) of Texas, USA (Characklis et al., 2006, Kirsch et al., 2009, Kasprzyk et al., 2009). The case study is used to demonstrate the planning framework with the goal of promoting improved problem understanding of water marketing within the LRGV. Each step in the framework (presented in Fig. 1) uses the concept of many-objective (Reed and Minsker, 2004, Fleming et al., 2005) tradeoffs of three or more objectives. Solutions in the tradeoffs are found using the concept of nondomination or Pareto optimality; solutions in the tradeoff set are better than all other solutions in at least one objective. Step 1 begins with an a priori problem formulation that represents planners’ initial conception of the problem through a quantitative model, decision variables that control strategies within the model, and objectives and constraints that measure strategies’ performance. In step 2, the framework diagnoses the effect of decision variables and model parameters using Sobol’ variance decomposition (Sobol’, 1993). The illustration in Fig. 1 shows that different variables can have a wide range of sensitivity performance across different evaluative metrics. Step 3 uses insights from the sensitivity analysis to construct a new many-objective planning problem. Objectives and constraints can be removed or added depending on their sensitivity structure or insights learned from previous iterations of the framework. Additionally, a suite of decision variable formulations of increasing complexity is used to explore the implications of the sensitivity analysis results. This framework seeks a balance between the complexity and effectiveness of a planning formulation.

Step 4 solves the de Novo formulations using a multiobjective evolutionary algorithm (MOEA) (Coello Coello et al., 2007, Nicklow et al., 2010). After a quantitative tradeoff comparing performance across decision variable formulations is developed, step 5 uses interactive visual analytics (Keim et al., 2006, Thomas and Cook, 2006, Thomas and Kielman, 2009, Andrienko et al., 2010) to view the tradeoffs interactively when evaluating the competing decision variable formulations. Exploration of decision variables’ impact on many-objective tradeoffs has been successfully demonstrated in prior work (van Werkhoven et al., 2009). Use of interactive visual analytics represents a posteori decision making, where decision makers explore our approximate Pareto optimal sets to negotiate a choice of alternative as a final decision (Stump et al., 2003, Kollat and Reed, 2007b, Lotov, 2007, Castelletti et al., 2010). A major benefit to this approach is that it allows the decision makers to modify their preferences and perform experiments through setting thresholds on objective function values and adding unmodeled objectives (Loughlin et al., 2001) to their analysis. Within step 5, the planners can choose the decision variable formulation that provides preferred performance compared to the other formulations. In this manner, formulations themselves can be considered non-dominated with respect to each other if they provide non-dominated solutions of interest to the decision maker and/or increase the diversity of alternatives that can be considered (Brill et al., 1990). This focus on finding the non-dominated problem formulation (as compared to the classical focus on non-dominated solutions within a single static formulation) is a unique contribution of this work. Selected solutions within this preferred formulation are also used to further interrogate the effect of deeply uncertain model assumptions on the solutions’ performance. Step 6 shows how deviations in model assumptions can change the performance of the selected solutions. For our case study, we use modifications of model assumptions within a drought scenario. Note that this process is iterative, and further improvements can be made to the problem formulation after this scenario analysis (i.e., the formulation from step 6 becomes a new a priori formulation for the next investigation). Overall, the de novo planning framework seeks to facilitate learning and innovation in decision making problems solved under deep uncertainty.

Section snippets

Lower Rio Grande case study

The case study used to demonstrate our de Novo planning framework focuses on water marketing (Anderson and Hill, 1997) in the Lower Rio Grande Valley (LRGV) of Texas, USA. The LRGV’s water market is described in Schoolmaster, 1991, Characklis et al., 1999 and Levine (2007). Due to limited regional groundwater storage, the primary sources of water in the LRGV are the Falcon and Amistad reservoirs, in which the water supply is shared between the United States and Mexico (Schoolmaster, 1991). The

Performance metrics

The three categories of performance metrics considered in this work are presented in Table 1. Efficiency metrics evaluate costs and the volumes of water carried over or not used for supply, risk indicators focus on water portfolios’ modes of failure and recovery, and market use metrics quantify the extent to which portfolios rely on the water market to provide supply. Full descriptions of the equations used to calculate these metrics are given in the Appendix A.

A priori problem formulation

In Kasprzyk et al. (2009),

Sobol’ sensitivity indices

This section presents the results from our Sobol’ sensitivity analysis of the a priori problem formulation. The goal of the sensitivity analysis is to identify the relative importance of decision variables controlling the city’s water supply portfolio, the demand growth parameter, and the city’s initial water supply on groups of output performance metrics. Fig. 3 summarizes the total sensitivity for the 10-year scenario (Fig. 3a) and the drought (Fig. 3b). Each row represents a different

Conclusion

This work supports the view that decision variable and objective formulations are constantly changing and being improved by new learning or decision maker preferences (Zeleny, 1989). Typical environmental planning problems are solved within a static formulation using a quantitative model with a fixed set of decision variables that determine the planning strategy. The sensitivity analysis using Sobol’ variance decomposition, however, showed that several variables were insensitive across many

Acknowledgements

This research was made possible by the US EPA STAR Graduate Fellowship program. Its contents are solely the responsibility of the grantee and do not necessarily represent the official views of the USEPA. Further, USEPA does not endorse the purchase of any commercial products or services mentioned in the publication. Also, this work was supported in part through instrumentation funded by the National Science Foundation through grant OCI-0821527. We would like to thank the helpful comments of

References (79)

  • A. Saltelli et al.

    How to avoid a perfunctory sensitivity analysis

    Environmental Modelling and Software

    (2010)
  • I.M. Sobol’

    On the distribution of points in a cube and the approximate evaluation of integrals

    USSR Computational Mathematics and Mathematical Physics

    (1967)
  • K. van Werkhoven et al.

    Sensitivity-guided reduction of parametric dimensionality for multi-objective calibration of watershed models

    Advances in Water Resources

    (2009)
  • G. Andrienko et al.

    Space, time, and visual analytics

    International Journal of Geographical Information Science

    (2010)
  • G. Archer et al.

    Sensitivity measures, anova-like techniques and the use of bootstrap

    Journal of Statistical Computation and Simulation

    (1997)
  • L.D. Brekke et al.

    Assessing reservoir operations risk under climate change

    Water Resources Research

    (2009)
  • E.D. Brill et al.

    MGA: a decision support system for complex, incompletely defined problems

    IEEE Transactions on Systems, Man, Cybernetics

    (1990)
  • A. Castelletti et al.

    Visualization-based multi-objective improvement of environmental decision-making using linearization of response surfaces

    Environmental Modelling and Software

    (2010)
  • G. Characklis et al.

    Developing portfolios of water supply transfers

    Water Resources Research

    (2006)
  • G.W. Characklis et al.

    Improving the ability of a water market to efficiently manage drought

    Water Resources Research

    (1999)
  • K. Deb et al.

    Introducing robustness in multi-objective optimization

    Evolutionary Computation

    (2006)
  • K. Deb et al.

    A fast and elitist multiobjective genetic algorithm: NSGA-II

    IEEE Transactions on Evolutionary Computation

    (2002)
  • D. Ellsberg

    Risk, ambiguity, and the savage axioms

    The Quarterly Journal of Economics

    (1961)
  • P.J. Fleming et al.

    Many-objective optimization: an engineering design perspective

  • H.J. Fowler et al.

    Modeling the impacts of climatic change and variability on the reliability, resilience, and vulnerability of a water resource system

    Water Resources Research

    (2003)
  • K.D. Frederick et al.

    Socioeconomic impacts of climate change on U.S. water supplies

    Journal of the American Water Resources Association

    (1999)
  • M. Friedman

    Price Theory

    (1976)
  • E. Hadjigeorgalis

    Managing drought through water markets: Farmer preferences in the rio grande basin

    Journal of the American Water Resources Association

    (2008)
  • Harik, G., Cantu-Paz, E., Goldberg, D., Miller, B., 1997. The gambler’s ruin problem, genetic algorithms, and the...
  • T. Hashimoto et al.

    Reliability, resiliency and vulnerability criteria for water resource system performance evaluation

    Water Resources Research

    (1982)
  • R.M. Hogarth

    Beyond discrete biases: Functional and dysfunctional aspects of judgemental heuristics

    Psychological Bulletin

    (1981)
  • A. Inselberg

    The plane with parallel coordinates

    The Visual Computer

    (1985)
  • M. Israel et al.

    Recent California water transfers: Implications for water management

    Natural Resources Journal

    (1995)
  • J.R. Kasprzyk et al.

    Managing population and drought risks using many-objective water portfolio planning under uncertainty

    Water Resources Research

    (2009)
  • D. Keim et al.

    Challenges in visual data analysis

  • Kenny, J., Barber, N., Hutson, S., Linsey, K., Lovelace, J., Maupin, M., 2009. Estimated use of water in the United...
  • B.R. Kirsch et al.

    More efficient optimization of long-term water supply portfolios

    Water Resources Research

    (2009)
  • Cited by (130)

    • Sustainable aquifer management for food security

      2023, Agricultural Water Management
    • Integrated risk assessment and decision support for water-related disasters

      2023, Hydro-Meteorological Hazards, Risks, and Disasters
    • Addressing climate uncertainty and incomplete information in transboundary river treaties: A scenario-neutral dimensionality reduction approach

      2022, Journal of Hydrology
      Citation Excerpt :

      In a similar manner, we use hierarchical clustering here to refine an initial vector of candidate objectives to update the dimensionality of the problem before identifying Pareto-optimal solutions. Kasprzyk et al. (2012) continues with a scenario analysis to test the robustness of solutions to model assumptions – a step that mirrors our proposed scenario-neutral analysis. We demonstrate our combined approach using the Ganges Water Sharing Agreement (GWSA) between India and Bangladesh as an illustrative example.

    View all citing articles on Scopus
    View full text