Decision space

The chapter consists of six sections. Section 11.2 describes the general structure of the methodology proposed. Section 11.3 discusses the approach of the consequence assessment of the problem. Section 11.4 presents the MOEA of emergency planning and response optimization around chemical plants. Section 11.5 includes some applications, and finally, section 11.6 offers a summary and the conclusion of this work. 

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Traditional approaches adopt a solution space S that coincides with decision) d thus any final solution or x ) has the same format as x, consisting of a real vector that defines a single point in the decision space. 

Thus, a critical departure from previous approaches, common to all our learning methodologies, is the adoption of a solution format that consists of hyperrectangles (not points) defined in the decision space. 

Solution format, The solution space consists of hyperrectangles ( == X G), instead of points ( = x g, ), defined in the decision space. 

Performance criterion, ft. The quality of any potential solution, X, is determined by the average system performance achieved within the zone of the decision space identified by X, ftiX), not by the individual performance obtained at any particular point x, ftin). 

A solution space, a, consisting of hyperrectangles defined in the decision space, X, is a basic characteristic common to all the learning methodologies that will be described in subsequent sections. The same does not happen with the specific performance criteria tfi, mapping models /, and search procedures 5, which obviously depend on the particular nature of the systems under analysis, and the type of the corresponding performance metric, y. 

Thus, they share exactly the same solution (H) and performance criteria (y ) spaces. Furthermore, since their role is simply to estimate y for a given X, no search procedures S are attached to classical pattern recognition techniques. Consequently, the only element that dilfers from one classification procedure to another is the particular mapping procedure / that is used to estimate y(x) and/ or ply = j x). The available set of (x, y) data records is used to build /, either through the construction of approximations to the decision boundaries that separate zones in the decision space leading to different y values (Fig. 2a), or through the construction of approximations to the conditional probability functions, piy =j ). 

What are hyperrectangles in the decision space, X e inside which one gets only a desired y value, or at least a large fraction of that y (8) valued... 

The solution space thus consists of hyperrectangles in the decision space, X Gand the corresponding performanee criteria are the conditional probabilities of getting any given y value inside X,p(y= X), = 1,..., or the single most likely y value within X. 

The search procedure, S, used to uncover promising hyperrectangles in the decision space, X, associated with a desired y value (e.g., y = good ), is based on symbolic inductive learning algorithms, and leads to the identification of a final number of promising solutions, X, such as the ones in Fig. 2b. It is described in the following subsection. 

Although decision trees contain a number of attractive features, including competitive accuracy, when considered strictly as classification devices (Saraiva and Stephanopoulos, 1992a), the most important point for our purposes is that each of the tree s terminal nodes identifies a particular hyperrectangle in the decision space, X, associated with a given y value. For example, node M defines a y = excellent rectangle that corresponds to the following rule ... 

In this particular problem, one wishes to achieve values of z as high as possible, and thus to identify zones in the decision space where one gets mostly y = 3. 

It can be noticed from the conditional probability estimates that one should expect to get almost only good y values while operating inside these zones of the decision space, as opposed to the current operating conditions, which lead to just 40% of good y values. 

For the reasons already discussed in Section III, our solution space consists of hyperrectangles in the decision space, X g I, not single points, X. The corresponding performance criterion used to evaluate solutions, i/i, is the expected y value within X ... 

Recognizing that, due to unavoidable variability in the decision variables, one has to operate within a zone of the decision space, and not at a single point, we might still believe that finding the optimal pointwise solution, X, as usual, would be enough. The assumption behind such a... 

On the other hand, when the unavoidable variability in the decision space is considered explicitly, the goal of the search becomes the identification of the optimal hyperrectangle, X , which solves the following problem ... 

This solution lies in a zone of the decision space that is quite distant from X = (200 17.9) and targeting the operation around x results in clearly suboptimal performance. 

Finally, it should be added that the conventional problem statement and pointwise solution format can be interpreted as a particular degenerate case of our more general formulations. As the minimum acceptable size for zones in the decision space decreases, the different performance criteria converge to each other and X gets closer and closer to x. Both approaches become exactly identical in the extreme limiting case where Ax = 0, m = 1,..., M, which is the particular degenerate case adopted in traditional formulations. 

For a zone X in the decision space to lead to a small conditional expected loss, E[L(z)lX], it must achieve both precise (reduced cr ) and accurate (fi =z ) performance with respect to z. Finding such robust zones and operating on them results in inoculating the process against the transmission of variation from disturbances and the decision space to the performance space (Taylor, 1991). 

Consequently, the goal of our learning methodology is the identification of hyperrectangles in the decision space, X, that minimize expected total manufacturing cost, (y X), a performance measure that combines in a consistent form and a quantitative basis both operating and quality costs. 

Rather than finding the exact location of the single feasible hyperrectangle that optimizes i/f (X), our primary goal is to conduct an exploratory analysis of the decision space, leading to the definition of a set of particularly promising solutions, X, to be presented to the decisionmaker. 

To identify this set of final feasible solutions, X e 1, with low scores, we developed a greedy search procedure, S (Saraiva and Stephanopoulos, 1992c), that has resulted, within an acceptable computation time, in almost-optimal solutions for all the cases studied so far, while avoiding the combinatorial explosion with the number of (x, y) pairs of an exhaustive enumeration/evaluation of all feasible alternatives. The algorithm starts by partitioning the decision space into a number of isovolu-... 

This solution leads to a considerably worse performance, (y X), than X, and it is also much more distant from the zone of the decision space where the true optimum, X p, is located. 

Before beginning the search for feasible zones of the decision space where the preceding tentative aspiration levels can be achieved, a preliminary check for the possibility of existence of such a zone is conducted. If the perceived ideal, y, does not pass this preliminary check, i.e., there is no commensurable solution to the multiobjective problem, the decisionmaker is asked to relax y, in order to transform the problem into one with commensurable solutions. For instance, if the initial tentative perceived... 

However, conflicts between the fulfillment of different objectives and aspiration levels may prevent any feasible zone of the decision space from leading to satisfactory joint performances. If the search procedure fails to uncover at least one feasible final solution, X, consistent with y, a number of options are available to the decisionmaker to try to overcome this impasse. Namely, the decisionmaker can revise the initial problem definition, by either... 

Additional studies documented in Saraiva and Stephanopoulos (1992b, c) also illustrate how the introduction of changes in preference structures is translated into displacements of the final uncovered solutions in the decision space. 

Green s functions and optimal systems The gradient direction in decision space (with M.M. Denn). Ind. Eng. Chem. Fund. 4, 213-222 (1965). 

The methodology proposed in this chapter, is based on the determination of the Pareto Optimal set of solutions for multi-objective emergency response decision-making around chemical plants. Given the size of the decision space, the efficient set of solutions cannot be... 

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