Methods of decision support and optimization

2.4.1 Qualitative methods

There are several qualitative decision support systems for different kinds of particular problems. Existing methods have often to be adapted for the application in a specific context.

Participatory methods

A multi-objective project needs the participation of a maximal number of the concerned parties. Nowadays participatory methods are even implemented in legal processes. The project, the involved parties and the links between project and parties have to be defined first (Luyet, 2005). A specific method for management of hydraulic projects has been developed (Leach and Pelkey, 2001). The actors have to be chosen by a method based on seven criteria (Mason and Mitroff, 1981; Banville et al., 1998). Only by applying these identification rules the integrity can be guaranteed. This method has been adapted and completed by several authors (Kapoor, 2001; Luyet, 2005). Participation levels with different competences are defined as well as a number of general rules to support a successful decision making process.

Expert judgment

An expert judgment (e.g., Delphi method) is similar to a participatory method. Decisions are taken by specialists, whose performances are usually higher than those of non-experts. Good decisions are only possible by a formal communication method between the experts (Rowe and Wright, 1996). Anonymous communication structures as well as iterative procedures guided by a group coordinator leading to a consensus are the main pillars of successful objective decision making. The precision and the reliability of the results cannot be easily defined. Thus, the consensus of experts is an approach that comes close to the objective optimum (Landeta, 2006).

System dynamics

System dynamics take into account temporal aspects for decision making. The approach solves a problem in a complex and non-linear system analytically (Park et al., 2004). The definition of the impact of every factor on others allows the identification of retroaction and loops (Maani and Maharaj, 2004). Modeling by system dynamics is divided into four steps: conceptualization, formulation, test and implementation (Luna- Reyes and Andersen, 2003). Qualitative models are often explained by diagrams, where the links show the causality of the relations. The use of system dynamics asks for a well-developed data set. Human variables are complicated to model and the interactions not easily defined.

Heller (2007) qualifies the applicability of the different methods in various contexts. System dynamics is generally the most appropriate. If social aspects have to be taken into account, a participatory method has to be implemented in a second step of the analysis for guaranteeing the acceptance by the concerned parties. The presented methods are limited on comparison of alternatives and neglect the aspect of optimization. They just attempt to achieve a compromise and a consensus. Heller et al. (2010) applied the method of Gomez and Probst (1995) to analyze the system of a multipurpose run-of-river hydropower plant. The holistic method, initially developed

Chapter 2

for economic contexts, allows a global overview on a complex system for adequate decision making (Figure 2.6).

Figure 2.6. Main steps of decision making by the method of Gomez and Probst (1995).

2.4.2 Mixed methods

Mixed methods link qualitative aspects to quantitative ones. In case of multi-criteria

criteria decision making (MCDM), decision makers often face problems with incomplete and vague information. Fuzzy set approaches, recognized as an important problem modeling and solution techniques, are suitable when modeling of human knowledge is necessary and when human evaluations are needed. Fuzzy set theory in

MCDM has been successfully applied to many problems in recent years (Kahraman,

2008).

Multi-criteria analysis

Multi-criteria decision making is a modeling and methodological tool is well suited for dealing with complex problems. It neglects aspects of system modeling and is confined to the objective of the optimal solution. The technique consists in four potentially cyclic steps (Mena, 2000): list of potential solutions, list of considered criteria, table of performances and aggregation of the performances. Most applications have the first three steps in common, showing mainly an objective character. For the last step, various methods exist to define the aggregation function, because of the bias between the performances and the consequent subjectivity (Schärlig, 1985).

Fuzzy sets

Fuzzy sets are systems where the links between the variables are not known or not sophisticatedly quantifiable. Fuzzy logic was invented by Zadeh (1965). Since then, the theory has been used for various applications. The goal of fuzzy logic is to quantify imprecise responses, which allows implementation in software tools. Fundamentals of the theory are the expressions of interference rule and attribution functions.

Identification of the problem

Participants, relations, limits

Comprehension of the problem

Factors, integration, drive

Generation of solutions

Key factors, scenarios

Evaluation of the solutions

Qualitative and quantitative

Realisation of the solutions

State of the art

2.4.3 Quantitative methods

After having defined the problem, its components and the relation between them, the output of the system has to be quantified. A quantitative method allows for optimization, which defines the set of model variables leading to the highest general performance. The comparison of different alternatives is done by an objective function.

Complex problems, such as optimization of dynamic processes described by partial differential equations, cannot be analytically solved. The problems can be continuous or discrete, depending on the type of equations to be solved. The simulation and the optimization can be linked by two different means (Bock et al., 2007):

− Black box approaches treat the two tools separately. An outer optimization loop calculates the objective function by iterating over the decision variables only, whereas an inner simulation loop iteratively determines the state variables describing the dynamic system behavior (Figure 2.7a).

− Simultaneous approaches closely couple the optimization aspect of the overall algorithm with the computation of the state variables of the dynamic system in the simulation. With an adequate resolution algorithm, this method is rapidly executable (Figure 2.7b).

a) b)

Figure 2.7. Interaction between optimization tool and model for black box (a) and simultaneous approach

(b) (Heller 2007), with objective function (Fobjective), external constraints (G), input variable at time step i

Xi and input variable at time step (i+1) Xi+1.

An optimization of a problem is characterized by the mathematical equations used in the model. The objective function and its conditions are defined by linearity or non- linearity. The variables can be continuous or integer. Thus, the optimization has to be done by an adequate solver:

Mathematical optimization methods

Mathematical optimization methods can be used in strictly and well defined environments, where the solving algorithm is known as either polynomial or exponential. The following rules are generally applicable (Heller, 2007):

− Linear models can be easily solved and the optimum is fast defined.

− Non-linear models with continuous variables can be solved by mathematical methods. Limits of validity of the results and of problem formulation exist.

Model Xi+1= f(Xi) Fobjective= F(Xi+1) Optimization min Fobjective(Xi) Fobjective Xi Optimization min Fobjective(Xi) by G(Xi) = 0 Model Fobjective= F(Xi) G(Xi) Fobjective , G(Xi) Xi

Chapter 2

− Non-linear models with discrete variables cannot generally be solved by mathematical methods. Heuristic approaches have to be applied.

Heuristic optimization methods

Heuristic methods (artificial intelligence, expert systems) are flexible and close to programming tools. They generate solutions which have been tested by simulations. The different types of problem settings lead to various algorithms, as e.g., genetic algorithms, neuronal networks, expert systems, fuzzy methods, simulated annealing, dynamic and goal programming. In hydrology and hydraulics, several of these methods have been applied (Chang and Moore, 1997; Cheng, 1999; Faber and Stedinger, 2001; Sharma et al., 2004; Chandramouli and Deka, 2005; Chang, 2008).

Between mathematical and heuristic optimization techniques, a large number of hybrid models have been developed.

2.4.4 Multi-objective optimization

For multi-objective optimization of complex systems, a problem description by a qualitative method is recommended (Heller, 2007), as shown in Figure 2.8. The alternative to be tested is defined by internal variables and depends on external variables. For every set of internal variables, a simulation with the corresponding models (global model) is undertaken. The performance or efficiency is expressed by objective functions. Economic, ecological, social or other indicators allow a rating and therefore an optimization of different alternatives. When faced with multiple objectives, Pareto curves (Pareto, 1896) define the boundary along which multiple different solutions lie and help users to compare and decide.

Figure 2.8. Flowchart of a decision making procedure: The effect of external and internal variables of the

alternative is defined by model simulations (A...N) and their output rated by indicators (1...n). An optimization procedure allows the definition of the optimal alternative(s) for comparison and decision making (Heller, 2007).

Model A

Objective functions

Optimization Comparison Decision

Model B Model C Model N

Global model

Indicator 1 Indicator 2 Indicator 3 Indicator n

Model simulations Variables

Internal variables External variables

3