Decision variables

Given a set of existing (x, y) data records, where x is a vector of operating or decision variables, which are believed to influence the values taken on by y, y is a performance metric, usually assumed to be a quality characteristic of the product or process under analysis ... 

An examination of previous classical learning procedures reveals that they differ from each other only with respect to the choices of /, and S. All of them share the same basic format for and the corresponding solution space, S. Let s assume that each (x, y) pair in the problem statement (2) contains a total of M decision variables ... 

By considering such a language to express solutions, one ignores the fact that in the type of problems we want to study decision variables behave as random variables, and there is always some variability associated with them. No matter how good control systems happen to be, in... 

A classification decision tree allows one to predict in a sequential way the y value (or corresponding conditional probabilities) that is associated with a particular x vector of values. At the top node of the tree (A in Fig. 3), a first test is performed, based on the value assumed by a particular decision variable (jCj). Depending on the outcome of this test, vector X is sent to one of the branches emanating from node A. A second test follows, being carried out at another node (B), and over the values of the same or a different decision variable (e.g., JCg). This procedure is... 

The construction starts at the root node of the tree, where all the available (x, y) pairs are initially placed. One identifies the particular split or test, s, that maximizes a given measure of information gain (Shannon and Weaver, 1964), 0(.s). The definition of a split, s, involves both the choice of the decision variable and the threshold to be used. Then, the (x, y) root node pairs are divided according to the best split found, and assigned to one of the children nodes emanating fi-om it. The information gain measure, 0is), for a particular parent node t, is... 

The preceding strategy for the construction of decision trees provides an efficient way for inducing compact classification decision trees from a set of (x, y) pairs (Moret, 1982 Utgoff, 1988 Goodman and Smyth, 1990). Furthermore, tests based on the values of irrelevant variables are not likely to be present in the final decision tree. Thus, the problem dimensionality is automatically reduced to a subset of decision variables that convey critical information and influence decisively the system performance. 

There are four decision variables three different measures of the feed composition ( Ci, Zj, x ) and the value of an unspecified operating condition (.Z4). 

In Fig. 4 we present the final induced decision tree, as well as the partition of the (xi,X4) plane defined by its leaves, together with a projection of all the available (x, y) pairs on the same plane. These two decision variables are clearly influencing the current performance of the refinery unit, and the decision tree leaves perform a reasonable partition of the plane. To achieve better performance, we must look for operating zones that will result in obtaining mostly y = 3 values. Terminal nodes 2... 

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... 

To illustrate how different m(X ) and x may happen to be, let s consider as a specific example (others can be found in Saraiva and Stephanopoulos, 1992c) a Kraft pulp digester. The performance metric y, that one wishes to minimize, is determined by the kappa index of the pulp produced and the cooking yield. Two decision variables are considered H-factor (xj), and alkali charge (X2). Furthermore, we will assume as perfect an available deterministic empirical model (Saraiva and Stephanopoulos, 1992c), /, which expresses y as function of x, i.e., that y =/(xi, X2) is perfectly known. 

Before starting the search for solutions, it is necessary to select among the M decision variables a subset of H variables, Xf, h- 1,..., W, which influence significantly the system performance, and thus will be used by S and included in the definition of the final set of hyperrectangles, X. For this preliminary choice of critical decision variables, other than his or her own specific process knowledge, the decisionmaker can count on a number of auxiliary techniques enumerated in Saraiva and Stephanopoulos (1992c). 

Besides the identification of the decision variables, x, m = and the performance variables, y, the user is asked to... 

The three decision variables, and the corresponding ranges of values in... 

Xpp a generic pointwise decision policy, i.e., a vector whose components are values of decision variables,... 

Xqp a generic interval vector decision policy, whose components are ranges of decision variables... 

As final solution formats, interval vector decision policies, X p, replace their pointwise counterparts, x p. Thus, a decision policy, Xpp, in the context of this section is an interval vector whose components are intervals of decision variables associated with one or more of the infimal decision units. No connection variables or disturbance factors are involved in their definition ... 

If X involves only ranges of decision variables attached to unit it defines a decision policy, and thus one can move directly to the validation and refinement phase. 

However, X, besides decision variables, may also include ranges of connection variables,, that link units k A 1 and k. That being the... 

Let s assume that the inputs to the supremal decision unit are a subset of all the decision variables attached to infimal decision units, consisting of those variables that are believed to be particularly influential with respect to the operation of the overall system. Then, an application of basic to DUj, results directly in the identification of a decision policy, X p. This decision policy is then passed down to the lower level in the hierarchy, where it is submitted to a process of validation and refinement by all infimal decision units that is identical to the one that takes place in the bottom-up approach. 

List of All 23 Decision Variables and Corresponding Windows under Current Operating Conditions... 

Furthermore, we will consider the 23 different decision variables, u, as the DUo inputs. 

The learning process was initiated at the top-digester infimal decision unit, leading to a solution, Xjj, that involves local decision variables and a range of white liquor sulfidity (fraction of active reactants in the white... 

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