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Structural analysis of Conditional Models
V. Rico-Ramírez and A. W. Westerberg^1
Institute for Complex Engineered Systems
Carnegie Mellon University, Pittsburgh, PA 15213
Abstract: Structural analysis is applied to exploit sparsity in the solving of a system of equations (Duff et al. 1989). Zaher (1995) studied the issues involved in the structural analysis of conditional models and presented a methodology to ensure consistency in a conditional model, the complexity of such an analysis being combinatorial. In that work, Zaher considered only cases in which the number of variables and equations of all the alternatives in a conditional model are the same. In this chapter, an extension to Zaher’s consistency analysis is presented. This extension allows the consistency analysis to be applied to conditional models in which the number of variables and equations for each of the alternatives may not be the same. Also, we show how, by taking advantage of the structure of the problem, it is sometimes possible to reduce the effort required by such an analysis. In particular, the cases of the existence of repeated structures and common incidence pattern among alternatives are discussed.
Keywords: Structural Analysis, variable partitioning, conditional models.
- Author to whom all correspondence should be addressed
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INTRODUCTION /MOTIVATION
1 INTRODUCTION /MOTIVATION
Before attempting to solve a model, a structural analysis has to be performed to determine if the model formulation is well posed. Techniques are available which can be used to detect if any structural inconsistency exists among the equations of the model (Duff et al .,1989; Zaher, 1995). However, consistency is generally assumed and structural analysis is rarely discussed in the literature.
Still, in practice it is very difficult to create large models of equations without introducing structural inconsistencies. We believe that structural analysis is an indispensable tool in an equation-based modeling environment, and, as we show in section 3, conditional models make the need for this tool an even stronger requirement.
2 T ERMINOLOGY IN STRUCTURAL ANALYSIS
We provide here a brief and general description of the terminology employed throughout this paper. Also, we use the system of equations given in Example 1 to illustrate each of the definitions.
Variables appearing in an equation are said to be incident in that equation and incident in the problem containing that equation. Assume that m is the number of equations in a model and n is the number of variables incident in those equations. For most engineering models,. In order to solve a problem, the problem has to be square ; that is, the number of equations and the number of variables to be calculated in the problem has to be the same. Accordingly, in order to solve a problem containing n variables and m equations, it is necessary to provide the values of n- m variables, so that we can calculate the rest. Thus, the difference between the number of variables and the number of equations gives us the number of degrees of freedom , DOF=n-m, of the problem. The m variables to be calculated in the problem are called dependent variables , while the n-m variables whose values are provided by the modeler are called independent or decision variables. Because of structural considerations (as we explain below), not every variable can be designated an independent variable. The set of variables whose values can be provided by the modeler (that is, the set of candidates to become an independent variable) is called the eligible set.
n > m
TERMINOLOGY IN STRUCTURAL ANALYSIS EXAMPLE 1 A System of Equations to Illustrate the Terminology in Structural Analysis.
For the system of equations in Example 1:
Set of incident variables in first equation
Set of incident variables in the problem
Number of variables in the problem =n = 4
Number of equations in the problem =m = 3
Number of degrees of freedom = DOF = 4 - 3 = 1
The equations of a model are expected to have different sets of variables incident in them. Furthermore, they are expected to involve only a few of the variables in the problem. This observation supports the idea that models are sparse. An effective representation of a sparsity pattern of a system of equations is given by an incidence matrix. The rows of an incidence matrix correspond to the equations of the problem. Similarly, the columns of an incidence matrix correspond the variables incident in the equations. An element in row i and column j of an incidence matrix is nonzero if and only if the variable of column j is incident in the equation of row i. The incidence matrix of the system of equations given by Example 1 is shown in Figure 1.
FIGURE 1 Incidence matrix of Example 1.
x 1 = 1 x 2 + x 4 = 5 x 3 – x 4 + x 2 = 3
f (^) 1 = x 1 – 1 = 0 f (^) 2 = x 2 + x 4 – 5 = 0 f (^) 3 = x 3 – x 4 + x 2 – 3 = 0
ê
= { x 1 }
= I ={ x 1 , x 2 , x 3 , x 4 }
x 1 x 2 x 3 x (^4) f (^1) f (^2) f (^3)
n n n n n n
TERMINOLOGY IN STRUCTURAL ANALYSIS FIGURE 3 A Steward path based on the output assignment of Figure 2.
The eligible set of variables for the problem of Example 1 is:
Eligible set =Set of variables eligible to be chosen as decisions
For the system of equations given in Example 1, selecting one of the variables in the eligible set E as being a independent variable results in a square structurally consistent system of equations.
2.1 N OTATION
The following notation is used in the remainder of this paper. For an alternative set of equations i where , and s is the number of alternatives in a conditional model:
Ei Eligible set Set of variables eligible to be chosen as independent variables in the
alternative i.
I (^) i Incidence set Set of variables incidents in the equations constituting the alternative i.
M Maximal set Union of the incidence sets of all of the alternatives.
DOFi Number of degrees of freedom left to be assigned.
e Intersection of the eligible sets of all the alternatives.
While assigning degrees of freedom in a structural analysis, every time that a variable is chosen to be an independent variable, the elements change in the eligible set for the selection of the remaining degrees of freedom. The number of elements in the new eligible set is at least one less than the previous set. Moreover, the new eligible set is always a subset of the eligible set previous to the selection of the independent variable. For that reason, we use the index k to indicate a k-th step in the selection of the independent variables while performing structural analysis. Note that
x 1 x 2 x 3 x (^4) f (^1) f (^2) f (^3)
n n n n ên n
ê ê
ê
= E ={ x 2 , x 3 , x 4 }
i ∈{ 1 … s }
STRUCTURAL C ONSISTENCY
the sets I (^) i and M are independent of this index k , while the sets Ei and their intersection e change at each step k
k k-th step in the assignment of the degrees of freedom.
2.1.1 Set Operators ∪ Union is all elements either in A , B , or both.
∩ Intersection is all elements in both A and B.
\ Minus A\B is all elements from A not in B.
3 S TRUCTURAL C ONSISTENCY
Duff et al. (1989) and Zaher (1995) describe algorithms for the systematic structural consistency analysis in conventional models. They give a step by step procedure to:
- Generate an incidence matrix.
- Perform an output assignment in order to test structural consistency.
- Collect all the eligible variables by following all the Stewards path in a problem.
The interested reader may refer to those works for a detailed description of the procedures. Our attention in the rest of this paper is focused in the structural consistency of conditional models.
3.1 C ONDITIONAL M ODELS
In conditional models, the sparsity pattern is expected to change from one alternative set of equations to another. This implies that a consistent set of independent variables for one alternative set of equations may not be valid for another one.
Zaher (1993, 1995) also addressed the structural analysis of conditional models. A necessary condition for structural consistency in conditional models is that each of the alternative sets of equations must be structurally consistent. Hence, consistency of a conditional model is assessed by finding at least one consistent partitioning of the variables (independent-dependent) such that output assignment of all of the equations in each alternative can be performed. This requirement makes the problem combinatorial, since we have to perform the analysis for all of the alternative
A ∪ B
A ∩ B
EXTENSION OF THE CONSISTENCY ANALYSIS FOR CONDI-
equations ( e k ) can be regarded as eligible to become independent in the context of the overall conditional model.
In a general situation, however, the number of equations in each alternative of a conditional model may change and so may the incidence set of variables.
4 E XTENSION OF THE C ONSISTENCY ANALYSIS FOR
C ONDITIONAL MODELS
For the case developed by Zaher in which the incident variables of all the alternatives are the same, the result of applying equation (1) is the elimination of all those variables which are eligible to be chosen as independent variables in some alternatives but non eligible to be chosen as independent variables in some other alternatives. That is readily accomplished by using the intersection of the individual eligible sets since all the variables are incident in all the alternatives.
For the general case, however, since we expect
we cannot use the intersection of the eligible set of each alternative to generate the eligible set for the overall conditional model. If we would do that, we would immediately remove variables which are not incident in some of the alternatives, since they would not be eligible for an alternative in which they are not incident.
A detailed derivation of an equivalent to (1) when the alternatives of a conditional model have different incident variables is presented in Appendix A. In Appendix A, we show that, in general, for any alternative , the set of “truly” eligible variables (ineligible variables are eliminated) for each alternative in the context of the overall conditional problem is given by:
(2)
and that the union of these individual sets gives the set of eligible variables from which we can safely select the independent variables of a conditional model:
I 1 ≠ I 2 ≠ I 3 … ≠ I (^) s
i ∈{ 1 … s }
E (^) ik ′ E (^) ik \ E (^) ik^ ( I (^) j \ E kj ) j , j ≠ i
s
EXTENSION OF THE CONSISTENCY ANALYSIS FOR CONDI- (3)
Hence, by using (3) instead of (1), we can apply the structural consistency algorithm described in 3.1 to a general conditional model having alternatives with different incident variables.
Furthermore, in Appendix A we also show that we do not have to perform the analysis for the general case as defined in (3). A simpler analysis can be used instead. We demonstrate that, if we augmented the eligible set of each alternative E (^) ik^ with the non incident variables of that alternative:
(4)
and find the intersection of the augmented sets ,
then the resulting set is equivalent to the set given by (3). Recall that M is the maximal set of variables,
(6)
so that represents the set of non incidences in alternative i.
Therefore, the use of (5) instead of (1) also allows us to apply the structural consistency algorithm described in 3.1 to a general conditional model having alternatives with different incident variables. As a final result in Appendix A, we also demonstrate that both (5) and (3) reduce to (1) when the alternatives of a conditional model have the same incident variables.
e k ′ E (^) ik ′ i
s
E (^) ik ′′ = E (^) ik^ ∪( M \ I (^) i )
E (^) ik ′′
e k ′′ E (^) ik ′′ i
s
∩ i [^ E^ ik^ ∪(^ M \ I^ i )]
s
e k ′′ e k ′
M I 1 I 2 … I (^) s I (^) j j
s
( M \ I (^) i )
EXTENSION OF THE CONSISTENCY ANALYSIS FOR CONDI-
EXAMPLE 2 A simple disjunctive set of equations.
- 1 Introduction/Motivation........................................................................... Table of Contents
- 2 Terminology in Structural Analysis.........................................................
- 2.1 Notation........................................................................................
- 2.1.1 Set Operators.................................................................
- 3 Structural Consistency.............................................................................
- 3.1 Conditional Models......................................................................
- 3.1.1 Limitations of Zaher’s Consistency Analysis...............
- 4 Extension of the Consistency Analysis for Conditional Models.............
- 4.1 The Main Result...........................................................................
- 4.2 An Implementation......................................................................
- 4.3 An Illustrative Example...............................................................
- Necessary ?.................................................................................. 4.4 Is the Combinatorial Consistency Analysis Really
- 5 Simplifying the Consistency Analysis of Conditional Models................
- 5.1 Common Incidence Pattern..........................................................
- 5.1.1 An Implementation.......................................................
- 5.2 Repeated Structures......................................................................
- 5.2.1 An Approach for Conventional Models........................
- of Repeated Structures................................................... 5.2.2 Equation-Based Modeling and the Modulo
- Equations....................................................................... 5.2.3 Repeated Structures Containing Conditional
- 5.2.4 Illustrative Examples.....................................................
- 6 Summary..................................................................................................
- 7 Acnowledgments......................................................................................
- 7 References................................................................................................ - with Different Incident Variables................................................. Appendix A Eligible Set for a Conditional Model Having Alternatives - x 18 = FIGURE 4 4 Alternatives in the Example 2. - x 1 = 0.8 ⋅ x - x 3 = 10 – x - x 5 = x 1 + x - x 6 = x 5 – x - x 8 = x 7 + x - x 7 =0.9 ⋅ x - x 9 = x 8 + x 10 + x - x 10 = 40 – x - x 13 = x - x 12 = 3 ⋅ x 21 ⋅ x ∨ - x 14 = x 7 + x - x 16 = x 24 + x - x 15 = x 16 + x - x 23 = x - x 17 = x - x 19 = x - x 20 = x - x 18 = ∨ - x 1 =0.8 ⋅ x - x 3 = 10 – x - x 5 = x 1 + x - x 6 = x 5 – x - x 8 = x 7 + x
- x 12 = 3 ⋅ x 21 ⋅ x - x 14 = x 7 + x
- x 16 = x 24 + x
- x 15 = x 16 + x - x 23 = x - x 18 = - x 1 =0.8 ⋅ x - x 3 = 10 – x - x 5 = x 1 + x - x 6 = x 5 – x - x 8 = x 7 + x - x 17 = x - x 19 = x - x 20 = x - x 18 = - x 1 =0.8 ⋅ x - x 3 = 10 – x - x 7 =0.9 ⋅ x - x 9 = x 8 + x 10 + x - x 10 = 40 – x - x 13 = x - x 12 = 3 ⋅ x 21 ⋅ x - x 14 = x 7 + x - x 16 = x 24 + x - x 15 = x 16 + x - x 23 = x - x 18 = - x 1 =0.8 ⋅ x - x 3 = 10 – x - x 7 =0.9 ⋅ x - x 9 = x 8 + x 10 + x - x 10 = 40 – x - x 13 = x - x 17 = x - x 19 = x - x 20 = x
- Alternative 1 Alternative 2 Alternative 3 Alternative
EXTENSION OF THE CONSISTENCY ANALYSIS FOR CONDI-
Table 1 shows an analysis of the degrees of freedom left to be assigned for each of the alternatives. Note that the number of equations, the number of incident variables, and the number of degrees of freedom left to be assigned are different for each alternative.
The eligible set for each of the alternatives, obtained from the eligibility analysis we described, is shown in Table 2.
If we would try to apply Zaher’s structural analysis in order to obtain a consistent set of independent variables for the overall problem, we would conclude structural inconsistency, since the intersection of the individual eligible sets is empty:
Table 1: Degrees of freedom left to be assigned in Example 2. Alternati ve
Number of equations
Number of incidences
Number of DOF
DOF left to be assigned 1 11 17 6 3 2 9 13 4 1 3 12 18 6 3 4 10 15 5 2
Table 2: Incidence and eligible sets in Example 2. Alternative Incidences,^ Ii Eligible set,^ Ei 1 x 1 , ... x8, x11, x12, x14,...x16, x18, x 21 , x23, x
x11, x12, x15, x16, x21, x
2 x 1 , ... x8, x17, ..., x20, x22 x19, x20, x 3 x 1 , ... x4, x7, ..., x16, x18, x21, x23, x (^24)
x11, x12, x13, x15, x16, x21, x
4 x 1 , ... x4, x7, ..., x10, x13, x17,...x22 x13, x19, x20, x21, x
e k^ E kj j
s
SIMPLIFYING THE CONSISTENCY ANALYSIS OF C ONDITION-
Hence, we believe that it is a valid methodology to consider the structural analysis of only those alternatives encountered during the solution of a conditional model if such procedure reaches a satisfactory result. However, such methodology cannot guarantee that a consistent set of independent variables selected for an alternative will also be consistent with respect to any another alternative visited later during the iterative solution technique.
In general, in order to ensure the structural consistency of a conditional model, the combinatorial consistency analysis must be performed.
5 S IMPLIFYING THE CONSISTENCY A NALYSIS OF
C ONDITIONAL MODELS
The analysis required for partitioning the variables in a conditional model was outlined in section 3.1. We see the most serious disadvantage of this analysis to be the combinatorial nature of the search consistency algorithm, which requires the analysis of all of the alternatives every time that a selection of an independent variable is made.
However, we have observed special features of some problems which can contribute to simplifying the analysis. Even though they represent very particular cases, if found alone or in any combination, we can take advantage of them in order to reduce the computational effort needed to perform the analysis.
5.1 C OMMON INCIDENCE PATTERN
It happens sometimes that the equations of some alternatives in a conditional model are different only because of the difference in value of some parameters, such as cost factors, mass balance coefficients, power of a correlation, etc. That is, the incidence pattern of those alternatives is the same. In such a case, the combinatorial structural analysis can be greatly simplified. Consider the conditional model defined in Example 3. In that example, there are 6 disjunctions in the problem. Hence, the number of alternative set of equations is:
Number of alternatives = 2 6 = 64
SIMPLIFYING THE CONSISTENCY ANALYSIS OF C ONDITION- EXAMPLE 3 Taking advantage of a common incidence pattern.
However, it is trivial to observe that, in all but one of the disjunctions, the incidence pattern is the same. As a consequence, for the purposes of structural analysis, only two different alternatives have to be considered.
Number of alternatives for structural analysis
Hence, for instance, the structural analysis for the problem of Example 3, could be simplified to one of the system of equations shown in Figure 5.
5.1.1 An Implementation In parallel with the implementation of our extension to the consistency analysis of a conditional model, we have also implemented a computer tool whose goal is identifying all those conditional structures with the same incidence pattern. This tool has been incorporated within the ASCEND
x 1 = x 6 + x 12 x 9 = x 10 + x 11 x 6 =1.15 ⋅ x 7 x 10 =0.1 ⋅ x 7 x 7 ≤ 8
x 6 =1.2 ⋅ x 7 x 10 =0.2 ⋅ x 7 x 7 ≥ 8
∨
x 2 =0.47 ⋅ x 8 x 7 =0.75 ⋅ x 8 x 8 ≤ 10
x 2 =0.45 ⋅ x 8 x 7 =0.7 ⋅ x 8 x 8 ≥ 10
∨
x 8 =1.8 ⋅ x 4 x 9 =0.7 ⋅ x 2 x 4 ≤ 11
x 8 =1.87 ⋅ x 4 x 9 = x 1 x 4 ≥ 11
∨
x 3 =1.15 ⋅ x 13 x 12 =0.25 ⋅ x 13 x 13 ≤ 9
x 3 =1.10 ⋅ x 13 x 12 =0.3 ⋅ x 13 x 13 ≥ 9
∨
x 11 =0.35 ⋅ x 14 x 13 =1.25 ⋅ x 14 x 14 ≤ 8
x 11 =0.3 ⋅ x 14 x 13 =1.3 ⋅ x 14 x 14 ≥ 8
∨
x 14 =1.10 ⋅ x 5 x 5 ≤ 4
x 14 =1.02 ⋅ x 5 ∨ x 5 ≥ 4
SIMPLIFYING THE CONSISTENCY ANALYSIS OF C ONDITION-
5.2.1 An Approach for Conventional Models In their work, Allen and Westerberg used a representative incidence matrix to perform a systematic analysis of a conventional model containing repeated structures. They provide a criterion to decide whether or not, after a consistent output assignment has been found for the representative matrix, the result obtained for the representative matrix can be expanded to a system containing any number of the repeated structures. Consider the simple case illustrated in Figure 6. In this example, the output assignment of a system containing two equations has been performed, resulting in the selection of a structurally consistent set of 3 independent variables (marked as I in Figure 6). The problem consists in finding if we can expand this partitioning to a system containing n blocks of the same two equations. First, it is necessary to introduce the definition of the modulo of the expansion. Modulo is the number of positions that each new block added to the structure is going to move from the left-most entry (or right-most entry if the expansion is upward) of the previous block. In other words, modulo is the number of columns that each new block is going to be displaced with respect to the previous one. In the case of Figure 6, the modulo of the expansion is equal to 3.
Once the modulo of the expansion is known, Allen and Westerberg propose to enumerate the columns of the representative block successively from 0 to modulo-1 until reaching the last column of the block. Then, the necessary condition for expanding the result to n blocks is that no two columns representing a dependent variable in the representative matrix can have the same column number. In Figure 6, this criterion is satisfied since the two columns representing the dependent variables has been enumerated as 1 and 0 , and, therefore, the expansion can take place. Practically speaking, what this condition means is that we cannot allow, after an expansion to n blocks, the overlapping of any two columns chosen as dependent variables. If we allow that, we would have a variable assigned to more than one equation, a situation that violates the structural consistency requirement explained in section 2 (definition of the output assignment).
SIMPLIFYING THE CONSISTENCY ANALYSIS OF C ONDITION- FIGURE 6 Expanding the result of a representative matrix.
5.2.2 Equation-Based Modeling and the Modulo of Repeated Structures In the context of an equation-based modeling, the set of overlapping variables and, therefore, the modulo of an array of repeated structures, are given by the connections among the repeated structures. In most of the existing equation-based environments currently available, there exist language constructs which allow the representation of connections defining the flow of information. So, for instance the IS operator of gPROMS (Barton, 1992) and the ARE_THE_SAME operator of ASCEND (Piela, 1989) serve this purpose.
5.2.3 Repeated Structures Containing Conditional Equations As stated earlier, the combinatorial complexity of the consistency analysis of systems containing repeated structures with conditional equations would be practically unmanageable.
What we propose here is to perform the consistency analysis of this type of conditional model by using a representative structure of the problem. The difference with respect to the work presented by Allen and Westerberg is that, in our problem, the basic structure to be considered in the analysis contains conditional equations, and, therefore, a consistency analysis over all the possible
n n n n n
n n
I I I
n n n n n
n n
n n n n n
n n
n n n n n
n n
0 1 2 0 1
MODULO = 3
. ..
i) Output assignment in representative matrix
ii) Expanding the result to n blocks
