ENVE 335: Decision Making for Environmental Engineers
Estimated study time: 7 minutes
Table of contents
Sources and References
Primary texts — Revelle, C. S., Whitlatch, E. E., and Wright, J. R., Civil and Environmental Systems Engineering; Hillier, F. S. and Lieberman, G. J., Introduction to Operations Research.
Supplementary texts — Ang, A. H.-S. and Tang, W. H., Probability Concepts in Engineering; Loucks, D. P. and van Beek, E., Water Resource Systems Planning and Management; Kroese, D. P., Taimre, T., and Botev, Z. I., Handbook of Monte Carlo Methods.
Online resources — MIT OpenCourseWare 1.204 Computer Algorithms in Systems Engineering and 15.053 Optimization Methods in Management Science; Stanford EE364A Convex Optimization open course; COIN-OR open-source optimization projects; scikit-learn, SciPy, and PuLP documentation; NIST/SEMATECH Engineering Statistics Handbook.
Chapter 1: Decisions Under Uncertainty
Environmental engineering decisions — sizing a reservoir, siting a landfill, managing a contaminated site — are made with incomplete information about the future and about the system itself. Formal decision methods do not eliminate uncertainty; they make it explicit, enable comparison, and support defensible choices.
1.1 The Decision Process
A decision-analysis cycle identifies objectives, defines alternatives, characterizes uncertainties, models consequences, evaluates trade-offs, and monitors outcomes. Each phase demands both technical analysis and engagement with stakeholders whose values shape the objectives.
1.2 Objectives and Constraints
Objectives are things we want more (or less) of — cost minimization, emissions reduction, reliability. Constraints are limits we must respect — budget, regulations, technology. A well-posed decision problem states both cleanly. Value trees decompose objectives hierarchically; screening analyses rule out dominated alternatives early.
Chapter 2: Multi-Criteria Decision Making
2.1 Additive Value Functions
A common multi-attribute model scores each alternative \(j\) on criteria \(i\) and aggregates
\[ V_j = \sum_i w_i\, v_i(x_{ij}) \]with weights \(w_i\) summing to one and value functions \(v_i\) mapping attribute levels to a common scale. Trade-off weights should be elicited against ranges of attribute levels, not in the abstract — a common mistake that biases results.
2.2 Structured Methods
The Analytic Hierarchy Process (AHP) builds weights from pairwise comparisons with a consistency check via the largest eigenvalue of the comparison matrix. PROMETHEE and ELECTRE handle qualitative preferences and thresholds. TOPSIS ranks by proximity to an ideal solution and distance from a negative-ideal solution.
2.3 Concept and Embodiment Design
In early concept design, decision methods screen diverse options with rough estimates. In later embodiment design, alternatives are fewer and detailed; criteria broaden to include constructability, maintenance, and life-cycle cost. Decision methods evolve accordingly — rough weighted matrices give way to cost–benefit analyses and LCA.
Chapter 3: Probability and Uncertainty
3.1 Random Variables
An environmental variable \(X\) is modelled by a probability density function \(f_X(x)\) with moments
\[ \mu = \int x\,f_X(x)\,dx,\qquad \sigma^2 = \int (x-\mu)^2 f_X(x)\,dx \]Common distributions include normal (many natural variables, CLT), lognormal (concentrations), exponential (times between rare events), Poisson (counts), and extreme-value distributions (annual maxima).
3.2 First-Order Uncertainty Analysis
For \(Y = g(X_1, \dots, X_n)\), linearization around the means gives
\[ \mathrm{Var}[Y] \approx \sum_i \left(\frac{\partial g}{\partial X_i}\right)^2 \mathrm{Var}[X_i] + 2\sum_{i3.3 Monte Carlo Simulation
When nonlinearity matters, Monte Carlo draws samples from joint input distributions, evaluates the model, and empirically approximates the output distribution. The standard error of a mean scales as \(1/\sqrt{N}\). Latin hypercube sampling improves convergence; variance reduction techniques (importance sampling, control variates) can be powerful.
Chapter 4: Risk-Based Performance
4.1 Reliability, Availability, Resilience
Reliability is the probability that a system performs its function over a stated interval. Availability is the long-run fraction of time the system is up. Resilience adds the system’s ability to recover from disruption, often captured in graceful-degradation metrics. Series, parallel, and \(k\)-out-of-\(n\) configurations are analysed by standard reliability block diagrams.
4.2 Performance Metrics
Risk-based metrics include probability of failure \(p_f\), expected annual damage, and conditional value at risk. The reliability index
\[ \beta = \frac{\mu_R - \mu_S}{\sqrt{\sigma_R^2 + \sigma_S^2}} \](with resistance \(R\) and load \(S\) assumed normal and independent) is directly usable in design.
4.3 Trend Tests
Detecting trends in monitored data is central to environmental surveillance. The Mann–Kendall non-parametric test assesses monotonic trend without assuming a distribution; its statistic is
\[ S = \sum_{iChapter 5: Optimization
5.1 Linear Programming
A linear program has the canonical form
\[ \min\ \mathbf{c}^\top\mathbf{x}\quad\text{subject to}\quad A\mathbf{x} \le \mathbf{b},\ \mathbf{x}\ge \mathbf{0} \]Simplex and interior-point methods solve large LPs reliably. Classic applications include blending, transportation, and resource allocation. The dual yields shadow prices that quantify how much the optimum changes per unit relaxation of a constraint — invaluable information for the decision maker.
5.2 Integer and Mixed-Integer Programming
Many environmental problems require integer decisions (a facility is built or not). Mixed-integer programs use branch-and-bound and cutting-plane methods. Siting problems — landfills, pumping stations, monitoring networks — are typically MILPs.
5.3 Nonlinear Programming
Nonlinear objectives or constraints arise with reaction kinetics, diffusion, and economies of scale. Interior-point methods and sequential quadratic programming are standard. Convex problems are particularly tractable: any local optimum is global, strong duality holds, and efficient algorithms scale well. Disciplined convex programming frameworks (CVXPY) lower the barrier.
5.4 Heuristic and Metaheuristic Methods
For intractable combinatorial or highly nonlinear problems, heuristics — greedy, simulated annealing, genetic algorithms, particle swarm, tabu search — seek good solutions within a compute budget. They provide no optimality guarantees and require careful tuning.
Chapter 6: Multi-Objective Optimization and Sensitivity
6.1 Pareto Frontiers
Multi-objective problems rarely have a single optimum. The Pareto frontier collects non-dominated alternatives; a decision maker selects along the frontier by articulating trade-offs. The weighted-sum and \(\varepsilon\)-constraint methods trace the frontier, but the weighted sum can miss non-convex regions.
6.2 Evolutionary Multi-Objective Algorithms
NSGA-II, SPEA2, and related evolutionary algorithms search the decision space in parallel, returning an approximate Pareto set. They accept black-box models and handle nonconvexity, but require evaluation budgets that can become large.
6.3 Sensitivity Analysis
Local sensitivity uses partial derivatives at the nominal point. Global sensitivity — variance-based methods such as Sobol indices — decomposes output variance into first-order and total effects of inputs:
\[ V(Y) = \sum_i V_i + \sum_{i6.4 Descriptive Models and Decision Support
Decision analysis sits at the interface between descriptive models — hydrologic, hydraulic, atmospheric, biogeochemical — and normative choice. Coupling the two in decision-support systems enables scenario exploration, stakeholder deliberation, and adaptive management. The engineer’s competence lies not only in solving the optimization but in structuring a problem that, when solved, gives a community defensible answers to the questions that actually matter.
