Constraint Programming Based Robust Sensor Network Design

Ind. Eng. Chem. Res. 2007, 46, 5985-5999

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Constraint Programming Based Robust Sensor Network Design† Prakash R. Kotecha, Mani Bhushan, and Ravindra D. Gudi* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai Mumbai 400 076, India

The optimal placement of sensors based on different criteria, namely, precision, reliability, cost, fault unobservability has received considerable attention in literature. Most of the sensor location problems proposed in the literature have been solved either using graph theoretic approaches or the geometry based mathematical programming techniques. However, these techniques have not been able to satisfactorily address the issues of determination of global optima and the determination of all multiple globally optimal solutions despite using a large number of additional discrete variables. In this article, we show the suitability of Constraint Programming (CP), an intelligent enumeration based technique, to solve such combinatorial sensor network problems. The power of CP to efficiently model a problem with fewer variables and the ease of determination of all multiple global optima makes it highly suitable for sensor location problems. To demonstrate the expressive modeling power of CP, we have presented an efficient reformulation of the minimum unobservability problem along with the robustness criteria of Bhushan et al. [Comput. Chem. Eng. 2007, DOI:10.1016/ j.compchemeng.2007.06.020]. Also, we exploit the capability of CP to solve feasibility problems for the determination of (i) multiple optima and (ii) the evaluation of tradeoffs between conflicting objectives. All these ideas have been demonstrated on the benchmark Tennessee Eastman problem using the ILOG CP solver. 1. Introduction An optimal placement of sensors in a chemical plant is desirable from the viewpoints of plant safety, fault detection and diagnosis, optimal control, and process economics. Generally, a subset of the thousands of plant variables is measured and the other unmeasured variables can typically be estimated using a process model. The measured variable set should be selected to satisfy certain design criteria, such as observability, reliability, robustness, and estimation accuracy. The problem of choosing appropriate variables to measure is known as the sensor network design or sensor location problem. Vaclavek and Loucka2 gave procedures to select sensor locations that ensured observability for a certain specified set of plant variables. Meyer et al.3 extended this approach to additionally include cost of measurements as an objective to solve the problem of appropriate sensor placement, using a branch and bound method. Ali and Narasimhan4 were the first to introduce the concept of network reliability by considering the probability of sensor failure. This formulation inherently contained the concepts of observability of all the plant variables and redundancy in sensor locations. They also developed a greedy search algorithm to determine the optimal sensor network. Though their initial work was limited to non-redundant linear systems, it was later extended to handle redundant linear5 and non-redundant bilinear6 systems. Bagajewicz7 designed a minimal cost network, subject to constraints on precision, availability, resilience, and error detectability, with the help of a tree type enumeration procedure. Bagajewicz and Sanchez8 have shown that the problems of minimizing variance subject to cost constraints and the problem of minimizing the sensor network cost subject to precision constraints are equivalent. In a subsequent paper,9 this equivalence was shown to hold between reliability and cost as well, thereby enabling the use of their earlier tree based enumerative approach for solving * To whom correspondence should be addressed. Tel.: +91 22 2576 7204. Fax: +91 22 2572 6895. E-mail address: [email protected]. † A shorter version of this manuscript appeared in the 17th ESCAPE Symposium, Romania, 2007.

reliability problems. Bagajewicz and Cabrera10 posed the sensor network design problem as an explicit optimization problem with the minimization of cost as the objective function and requirements of precision, error detectability, resilience, and availability as constraints. Bhushan et al.1 quantified the unobservability of faults and used the minimization of fault unobservability as a criterion to design robust networks. Some novel formulations, which consider the cost advantage obtained in terms of a network’s capability to resolve between faults and its ability to yield desired precision and adequate control, have also been reported.11 From the above literature, it can be seen that different techniques have been used to solve the optimal sensor placement problem. These techniques can be broadly classified as either graph based techniques or explicit optimization techniques. The graph based greedy search algorithms of Ali and Narasimhan4-6 are critically dependent on appropriate initial guesses and therefore do not guarantee global optimality. The tree based enumerative algorithms of Bagajewicz7 are computationally expensive and hence not suitable for large realistic flow sheets. As opposed to these graph based approaches, the explicit optimization techniques result in either MILP,1 MINLP,10 or LMI12 type problems and are solved by the conventional geometry based mathematical programming techniques. The term mathematical programming techniques includes all such techniques that either implicitly or explicitly make use of the gradient information. The drawbacks of these techniques include restrictive modeling power (during formulation), issues of global optimality of solutions (for non-convex MINLPs), and the determination of all multiple global optimal solutions. This latter issue of multiple global optimal solutions is important because most of the proposed criteria for sensor location design are known to yield multiple solutions.1,4,10 From a designer’s perspective, it is desirable to find all these multiple solutions so as to enable a wider choice of decisions. In this article, we propose to address the above deficiencies using Constraint Programming (CP) methods, which are emerging as efficient alternatives to conventional mathematical programming strategies. Specifically, we propose to harness

10.1021/ie061569x CCC: $37.00 © 2007 American Chemical Society Published on Web 08/08/2007

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some of the important merits of CP, namely, (i) rich expressive power of modeling, (ii) generation of global optima, and (iii) ease of generation of multiple solutions, toward solving the complex sensor location problems. To demonstrate these ideas, we consider the fault unobservability problem which has been proposed in the literature.1 In the context of fault unobservability, we show the use of high expressive modeling power of CP by efficiently reformulating the minimum unobservability, lexicographic optimization problems proposed by Bhushan et al.1 This reformulation results in a substantial decrease in the number of binary variables (and potentially the computational burden). Further, we harness the ability of CP to generate multiple solutions, to not only eliminate the lexicographic constants (as these can result in scaling problems) but also to better understand the tradeoffs between various sensor network design objectives used by Bhushan et al.1 The efficacy of the proposed optimization technique has been demonstrated on a realistic case study, namely, the benchmark Tennessee Eastman (TE) challenge problem. The rest of the article is organized as follows: We start with a brief introduction of CP followed by the reformulation of the model for minimizing fault unobservability with uncertainty in the fault occurrence and sensor failure probabilities, as well as the process model. We demonstrate the utility of these formulations on the TE case study to show the superior modeling power of CP, demonstrate the determination of multiple solutions with an explicit knowledge of the precedence in objective functions, and also show the determination of the pareto optimal front to evaluate the tradeoffs between various objectives in the absence of an explicit precedence ordering. 2. Constraint Programming CP13-16 is emerging as a strong candidate for combinatorial optimization in operations research which otherwise was dominated by mathematical programming techniques such as branch and bound and cutting plane methods. The origins of CP can be traced to the Artificial Intelligence and the Computer Science community15 for its use in solving feasibility problems or more appropriately constraint satisfaction problems (CSPs). It has found applications in diverse areas like computer graphics, software engineering, databases, hybrid systems, finance, engineering, circuit design, and combinatorial optimization.13 In combinatorial optimization, CP has been found to be particularly successful in solving feasibility problems such as planning, resource allocation, and scheduling.17-19 Unlike deterministic mathematical programming techniques, CP does not depend on the geometry of the problem but relies more on an intelligent enumerative search, wherein the constraints are used in pruning the search space. Hence CP does not categorize problems as MILP or MINLP. We next briefly present the principles of CP and demonstrate it working on a three-variable problem.15 Principle of CP. CP is a tree based enumerative search technique, which uses constraint propagation as its inference engine to reduce the domain of the variables. A CSP can be defined as the problem of selecting values for the decision variables such that they satisfy all the constraints of a problem. Thus, a CSP can be considered to consist of two interlinked components, namely, the constraints and the decision variables. The constraints are mathematical relations between the decision variables, which in turn need to take values from their domains so as to satisfy the constraints. To mathematically define a CSP, consider a set of n decision variables x1,x2,...,xn and let each of the variables xj be associated with a domain Dj, j ) 1,2,...,n. The domain Dj consists of all possible values for xj and need

not be restricted to real numbers or integers but can also be a set with holes.15 A constraint c(x1,x2,...,xn) is a mathematical expression that defines the relation between the variables x1,x2,...,xn and is satisfied for a subset (S) of the set D1 × D2 × ... × Dn. In other words, a constraint is a mathematical function f: D1 × D2 × ... × Dn f {0,1} such that f(x1,x2,...,xn) ) 1 if and only if c(x1,x2,...,xn) is satisfied. A general procedure for finding a feasible solution using domain reduction via constraint propagation is given below. Let the set V denote the set of all variables in the given problem. Let the set C denote the set of variables that have been assigned a specific value from their domains; that is, set C is the set of variables fixed as choice points. Initialize C ) {L} (a null set). Step 1: Constraint Propagation. Update Dj ∀j ∈ (V\C) by propagating the constraints. Step 2: Checking for Infeasibilities. Is Dj ) {L} for any j ∈ (V\C)? If yes, go to Step 4. Step 3: Checking for Solution. Is Dj ) 1 ∀j ∈ (N\C)? If yes, the problem is feasible and the current set of values corresponds to one feasible solution. If no, go to Step 5. Step 4: Backtracking. Backtrack from the end of C (latest entries first) until different choices for some variable xk ∈ C can be made. If no such xk exists, then the problem is infeasible; otherwise, go to Step 1. Step 5: Branching on a New Variable. Choose any variable xk as a choice point, where xk ∉ C. Update C ) C ∪ xk. Go to Step 1. We now show the working of CP for solving a small CSP with the help of an example.15 Consider the CSP

solve y < z

(1)

x-y)1

(2)

x*z

(3)

x, y, z ∈ {1,2,3}

(4)

Note that the constraints in the above problem are not in the form of inequalities/equalities. The geometry based techniques would have required reformulating the problem in terms of inequalities, thereby introducing several additional variables. From Figure 1, it can be seen that the problem has 27 possible nodes (choices for decision variables), and a blind enumeration would have checked for the satisfiability of the constraints 1-4 at all these 27 possible nodes. But, CP with its domain reduction technique explores a much lesser number of nodes and yet successfully finds the exact solution. The first step in the solution of any CSP is the initial constraint propagation to reduce the domain of the variables, thereby reducing the search space of a problem. Consider the constraint x - y ) 1. This constraint ensures that y * 3 and x * 1. Thus the domains of x and y are restricted to {2,3} and {1,2}, respectively. The constraint y < z restricts the domain of z to {2,3}. Thus we see that at the end of the initial constraint propagation the number of nodes has been reduced from 27 to 8 as shown in Figure 1b. Now, as there is no further possibility of reduction of domains using constraint propagation, we resort to a search technique. We start by making an arbitrary choice (often referred to as choice point in the CP literature) of x ) 2. After generating a choice point, we can again resort to domain reduction by using constraint propagation. Now the constraint x - y ) 1 w y ) 1and the constraint x * z w z ) 3. Thus, all the decision variables have been specified with values from their domains and the problem has been solved with the solution x ) 2; y )

Ind. Eng. Chem. Res., Vol. 46, No. 18, 2007 5987

Figure 1. (a) Nodes to be explored.15 (b) Nodes to be explored after initial domain reduction.15

1; z ) 3 without exploring the entire tree. However, if all the possible solutions are required, we need to explore the other branches also. To realize this, we backtrack to the initial choice point of x ) 2. At this node, we now make a choice of x ) 3 instead of x ) 2. The constraint x - y ) 1 w y ) 2. We can now easily check that the decision variable z cannot take any value from its domain of {2,3} without violating the constraints. Thus, we see that there is no feasible solution for the choice point x ) 3. Such points are referred to as failure points. It can be seen that the entire tree has been searched, and hence there is no chance of any solution being missed. Thus, we can see that CP with a complete tree search enables finding all feasible solutions for CSP more efficiently than blind enumeration because nodes that are not feasible are not explored and a branch is fathomed as soon as the domain of a variable is emptied. For optimization problems (specifically minimization), CSP handles the associated objective function using the following search strategies. Standard Search. This procedure is fairly straightforward and solves the optimization problem as a CSP. In the first step, the objective function is neglected and a feasible solution is searched. The objective function is evaluated at the first feasible point, and an upper bound is obtained. In the next step, a constraint is then added to the original set of constraints requiring the next feasible point to have a better (or at least the same if multiple solutions need to be determined) objective value than the current feasible point. Dichotomic Search. This procedure is particularly useful when a lower bound L for the problem is known a priori. The first step is to evaluate the value of the objective U at a feasible point. The procedure involves the evaluation of a mid-value M given by (U + L)/2. Now, the constraint set of the original problem is appended with a constraint g(x1,x2,...,xn) < M (where g(x1,x2,...,xn) is the objective function) and the CSP is solved ignoring the objective function. A feasible solution will indicate a better solution than the current feasible point; hence, the value of U is updated, and the procedure is continued with a new value of the mid-value M. An infeasible solution will necessitate the updating of the lower bound L (and thereby the value of M), and the search is carried on as mentioned above. 2.1. Motivation for the Use of CP. As discussed earlier, the merits of CP stem from smart enumeration procedures and include the rich expressive power of modeling and ability to generate multiple, globally optimal solutions for a variety of problems. The following limitations of mathematical programming techniques have strengthened the case for the use of CP particularly for combinatorial optimization problems: (1) The conventional mathematical programming techniques have restrictive modeling power. Thus, the requirement for the

constraints to be in the form of a mathematical equation forces the user to introduce additional variables and hence increases the size of the problem, thereby potentially increasing the computational burden. (2) There is a lack of robust solvers and subsequently the guarantee of global solutions for non-convex MINLPs. Many of the solvers have been observed to converge to different optimal solutions based on the initial guesses and are thus not robust. At times, it may be possible to convert MINLPs to MILPs,10 but at the cost of increasing the problem size both in terms of number of variables and constraints. However, for large MINLPs, this conversion can also lead to MILPs that are completely intractable. (3) Many of the engineering problems are associated with multiple optimal solutions, and the importance of determining such solutions have been discussed in a subsequent section. However, the current state of the art solvers (for even MILPs) stop with the determination of the first global optimal solution. The ability of CP in addressing the above issues makes it amenable for formulating and solving optimal sensor network design problems, which are often non-convex and combinatorial in nature, require implication constraints,1 and involve tradeoffs between various objectives. In the literature, a comparison of CP and Integer Programming (IP) has been studied for the modified generalized assignment problem,20 the template design problem,21 the progressive party problem,22 and the change problem.23 A recent development in the field of combinatorial problems is the development of hybrid methods which use the complimentary properties of IP and CP to efficiently handle combinatorial problems. Some of the available CP solvers are ILOG Solver,24 CHIP,25 and ECLiPSe.26 3. Multiple Solutions and Multi-Objective Optimization In engineering problems, very often there are multiple objectives to be satisfied and many times some/all of these objectives are conflicting; that is, the improvement in one of the objective leads to the deterioration of the other objective. In the context of sensor networks, an appropriate example would be the maximization of network reliability and the minimization of the cost of a sensor network. From a designer’s perspective, it is desirable to establish these tradeoffs between the conflicting objectives. In literature, such tradeoff solutions are popularly referred to as pareto optimal solutions.27 The pareto optimal solutions are the set of non-dominated feasible solutions. A solution is said to be non-dominated if it is feasible and there is no other feasible solution which has better values for all the objectives.

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Example 1. Consider two feasible sensor networks A and B for a sensor network design problem. Let the network reliabilities of A and B be 0.89 and 0.9 and their respective costs be 80 and 500 units. If these solutions are known, the designer can make an informed choice by balancing the tradeoff in both the objectives. Hence the designer can very well select solution A (though it has a lower reliability than B) because the cost added to obtain a small advantage in the network reliability is very high. Thus we need a solution procedure, which can list all the tradeoff solutions. The CSP based intelligent search procedure of CP can be used to effectively determine these solutions. 3.1. Lexicographic Optimization. Lexicographic optimization (or preemptive optimization28) is a special form of multiobjective optimization to characterize tradeoffs, in which the various objectives are given a precedence ordering. The primary objective is given the highest priority followed by decreasing priorities on subsequent objectives. The basic philosophy in this approach is that even a marginal improvement in a higher precedence objective is considered more valuable than an arbitrarily large improvement in a lower ranked objective. Example 2. Consider example 1 again but with an additional solution C with network reliability of 0.9 and network cost of 100 units. If network reliability is the primary objective and cost of the sensor network is a secondary objective, the optimum will be solution C. While solutions B and C outperform solution A as a result of their higher reliabilities (irrespective of cost), solution C is better than B because of its lower cost even though B and C have the same reliability. Lexicographic optimization can be performed by solving a series of optimization problems, where the optimization of one objective is considered at each step. In this approach, a new equality constraint gets added in each subsequent step to ensure that the values of the objective functions considered in the previous steps are maintained at their optimal values. Some of the issues involved in lexicographic optimization are discussed next. 3.1.1. Issues with Lexicographic Optimization. (1) The lexicographic approach is appropriate for cases where a clear precedence of objectives is known; however, when this precedence ordering is not known, it may not be suitable because it forces the designer to assign arbitrary relative importance to different objectives. This can at times lead to designs that may not truly reflect the implicit design criteria. Example 3. Again consider example 1 with solutions A and B as candidates for sensor network design. If the network reliability is considered as the primary objective in the lexicographic optimization approach, solution B would be the optimal solution. However, if both the solutions are known the designer might select the solution A because the cost difference between A and B is very high even though the reliabilities differ only marginally. (2) The lexicographic approach is meaningful only when the preceding objective function at an individual step has multiple optima. Example 4. Consider example 2 with only A and C as solutions. It is clear that solution C is the optimal solution as it has a higher primary objective than A. The issue of cost does not enter into consideration as the primary objective (network reliability) does not have multiple optima. Thus, when the preceding problem has single optima, solution of subsequent optimization problems does not lead to better solutions. As discussed earlier, the solution of the lexicographic optimization problem involves the solution of a series of optimization problems. Sherali28 proposed an alternative ap-

proach where the individual objectives were weighted and combined into a single objective function, thereby requiring the solution of a single optimization problem. In this approach, the weights have to be chosen suitably so as to preserve the precedence ordering of the individual objectives. However, this single step lexicographic optimization involves issues related to the choice of weights (these have a bearing on the computational characteristics29); also, as the number of objectives increases, the values of lexicographic constants can vary by several orders of magnitude. This may lead to numerical difficulties in the single step lexicographic optimization. In summary, characterizing the tradeoffs between various objectives in a multi-objective problem is not straightforward. Most of the traditional approaches require a priori specification of some constants (goals, lexicographic weights) to characterize the tradeoffs, which may not reflect the implicit design criteria. On the other hand evolutionary methods such as genetic algorithms, while not requiring this a priori specification of the constants, do not guarantee optimality and also require proper tuning of parameters (related to mutation and crossover operations, etc.) which have a bearing on their performance. For more information on multiobjective optimization techniques, the reader can refer to Deb’s book.27 The CP approaches, through their systematic and intelligent domain reduction strategies, enable the generation of all multiple, globally optimal solutions and hence can be very effective for several design problems. In the context of multi-objective optimization, another important aspect is the possibility of having multiple solutions in the decision space which map onto the same point on the pareto optimal front, that is, have the same tradeoffs in the specified objectives. These multiple solutions if known to the designer can provide additional flexibility in satisfying the various implicit design criteria. For example, consider a case wherein the designer has not accounted for the corrosiveness of various streams in the objective function. If the designer gets to know all the multiple optimal solutions, the network which is more tolerant to corrosion can be selected. On the other hand, the conventional mathematical techniques would have given only one solution and the designer may not have had any additional flexibility. In conventional mathematical programming methods for combinatorial optimization problems, the multiple global optima can be determined by solving a series of optimization problems resulting from the addition of new cuts that avoid the previously visited solutions. A linear cut can be added by the conversion of all integer variables to binary variables, as shown in Tawarmalani and Sahinidis.30 But this method increases the number of binary variables. Alternatively, a nonlinear cut can be added to avoid this conversion. However, this makes the problem nonlinear and hence may lead to problems in achieving global optimality. Nevertheless, this nonlinear cut can be linearized but at the expense of increase in the problem size. This approach of adding integer cuts requires the solution of a number of branch and bound problems, thereby increasing the computational burden. Another flexible way to determine multiple global optimal solutions is to modify the standard rule of branch and bound of fathoming as in BARON.30 In contrast to these methods, when used for solving a feasibility problem, CP by its very enumerative nature visits all the global optima without the addition of any cuts and is therefore more suited for generation of all the multiple solutions. We now present an efficient CP based re-formulation of the problem of design of robust sensor networks with minimum unobservability.


Formulation II:

4. Sensor Location with Fault Diagnosis Perspective Fault detection and diagnosis play an important role in the optimal and safe operation of a chemical plant. Fault detection involves the detection of a fault as soon as it occurs, whereas fault diagnosis as a next step involves identifying the root cause. Any fault detection and diagnosis strategy depends on the appropriate placement of sensors, and this area of sensor placement for fault diagnosis has received some attention in the recent past. Bhushan and Rengaswamy31 were among the first researchers to present strategies for designing sensor networks for reliable fault diagnosis. In their approach, every fault i is associated with its fault occurrence probability fi and every sensor j is associated with a failure probability sj. The ith fault remains undetected if the fault occurs and the associated sensors fail at the same time. Bhushan and Rengaswamy31 have termed this event as the unobservability of fault i which can be calculated as n

Ui ) fi

(sj)b x ∏ j)1

ij j

(5)

Formulation I: min U

(6)

xj

n

bijxj log sj ∑ j)1

∀i ) 1, ..., m

(7)

n

cjxj e C* ∑ j)1 xj ∈ Z+

j ) 1, ..., n

(10)

n

cjxj + xs ) C* ∑ j)1

s.t.

(11)

∀i ∈ I\(If∪Is)

U g log(Ui)

∀i ∈ (If∪Is)

U ) log(Ui) + φi

(8) (9)

In the above formulation, eq 7 ensures that the maximum unobservability among all the m faults is minimized, and eq 8 represents the cost constraint. In addition to the minimization of the unobservability, Bhushan et al.1 also considered other objectives related to cost saving, robustness to the probability data, and the underlying cause-effect model of the process. In particular, they considered the additional objectives: (i) maximization of the minimum slack in the unobservability constraints of the uncertain faults, (ii) maximization of the slack in the cost constraint, and (iii) maximization of the network distribution. These were considered in decreasing order of precedence, leading to a lexicographic optimization problem. Using the ideas of Sherali,28 these objectives were combined to obtain a single step lexicographic optimization problem as shown in Formulation II.1

(12) (13)

φ/i ) φfi/

∀i ∈ (If\Is)

(14)

φ/i ) φsi/

∀i ∈ (Is\If)

(15)

φ/i ) φfi/ + φsi/

∀i ∈ (Is∩If)

(16)

Myi g φi - φ/i

∀i ∈ (Is∪If)

(17)

M(yi - 1) e φi - φ/i φ e φi + φ*yi

∀i ∈ (Is∪If)

∀i ∈ (Is∪If)

φsi/

(18) (19) (20)

φ e φ*

In the above expression, n denotes the number of variables, xj denotes the number of sensors on the jth variable and can be greater than 1 (in the case of hardware redundancy), and bij is the (i,j)th entry of the cause-effect bipartite matrix B. The faults form the rows of this matrix, and the columns form the variables. If the ith fault affects jth variable, then the (i,j)th entry is 1 and is 0 otherwise. As shown in Bhushan and Rengaswamy29 the bipartite matrix B can be generated from the signed digraph of the process. For the purpose of sensor network design Bhushan and Rengaswamy31 defined the network unobservability as the highest unobservability among all faults and presented the following MILP formulation for minimizing the network unobservability in the presence of cost constraint.

s.t. U g log fi +

min[R1U - R2φ - R3N - xs]

Bij(log sj)xj ∑ j∈J

)-

∀i ∈ Is

(21)

s

j ) 1, ..., n

n j e xj

(22)

n

N)

nj ∑ j)1

(23)

yi,nj ∈ {0,1}; U ∈ R-; xj ∈ Z+ (Ui,xs,φ,φ/i ,φi,φsi/ ,N) ∈ R+

(24)

In the above formulation, R1, R2, and R3 are the lexicographic constants, I is the set of all faults, If is the set of faults with uncertain occurrence probabilities, and Is is the set of faults that affect variables which can only be measured by sensors with uncertain failure probabilities. Note that the ith fault is characterized as uncertain if (i) its fault occurrence probability is not known accurately (i ∈ If), (ii) failure probability of any sensor available for measuring the variables affected by that fault (the corresponding bij is 1) is uncertain (i ∈ Is), or (iii) both i and ii hold (i ∈ (Is∩If)). In eq 13, φi is the slack in the unobservability constraint for the ith uncertain fault. A nonzero φi ensures that the system unobservability is robust to changes in the data used to estimate the unobservability of the ith uncertain fault. φ/i is the corresponding value of the slack that is necessary to ensure complete robustness of the system unobservability to the uncertain data used to calculate the unobservability of fault i and is required because the upper values of the probabilities are bounded by 1. Depending on the type of uncertainty in the faults (cases i, ii, and iii), eqs 14-16 can be used to determine the value of φ/i . The secondary objective φ represents the network’s ability to tolerate uncertainty in the probability data without affecting the nominal system unobservability and is defined as the minimum of all those φi which are below their corresponding φ/i values. Equations 17-19 ensure that if φi is greater than φ/i , then the value of φ is not restricted by φi and any further increase in φi will not affect φ. On the other hand, if φi e φ/i then the value of φ is bounded by φi. Further, eq 20 bounds the value of φ by a constant φ* that is calculated on the basis of the given probability and cost data. This constraint is required to ensure

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that additional cost is not spent in increasing φ to arbitrary large values for situations when φi > φ/i , ∀i ∈ (Is∪If). As explained in Bhushan et al.1 the values of φ* and φfi/ can be determined a priori by the following equations

φ* ) max

{

max (φfi/ ), max(i∈If\Is

i∈Is\If

max (φfi/ i∈Is∩If

-

∑ j∈J

Bij(log sj)x/j ), ∑ j∈J s

Bij(log sj)x/j )

s

φ*fi ) -log fi

∀i ∈ If

}

(25)

(26)

In the above equation, x/j is an upper bound on xj and corresponds to the maximum number of sensors that can be installed on the jth variable on the basis of the available cost and is given by (C*/cj) (where x indicates rounding off x to the nearest integer not higher than x). In Formulation II, while φ denotes the robustness to uncertainty in the probability data, N is the network distribution and is a measure of robustness to modeling errors. Network distribution is defined as the total number of measured variables and is calculated as shown in eq 23. In this equation, nj is a binary variable that takes a value of 1 if at least one sensor is placed on variable j and is zero (0) otherwise (eq 22). The variable xs in the objective function represents the cost saving and is equal to the difference in the available and the used cost (eq 11). Bhushan et al.1 showed the utility of the above formulation for the design of robust sensor networks by applying it to the TE problem for various lexicographic orderings and available cost. However, this formulation suffers from the following drawbacks: (1) The optimal sensor network depends on the precedence ordering of the various objectives. As discussed earlier, this precedence ordering may not adequately reflect the implicit design criteria. Even when this precedence ordering is explicitly known, the conventional solvers select only one of the several possible optimal solutions that may satisfy the precedence ordering. These issues are discussed in subsequent sections. (2) The formulation adds n binary variables (nj, j ) 1, ..., n) required to calculate the network distribution N, m1 binary variables (yi, i ) 1, ..., m1) required to calculate φ, and the associated n (eq 22) and 3m1 (eqs 17-19) constraints, respectively. Here, m1 and n correspond to the total number of the uncertain faults and the total number of variables in the process. Although these additions increased the dimensionality of the problem, they were incorporated to cast the optimization problem as an MILP formulation. Now, we present an efficient CP reformulation of the MILP in Formulation II that addresses the above issues. We first address the latter issue by harnessing the expressive modeling power of CP, thereby achieving a substantial reduction in the problem size. Toward this end, we exploit the ability of CP to handle implication constraints and thereby replace eqs 17-19 by the following constraints.

φi < φi g

φ/i φ/i

w φ e φi w φ e φ*

}

∀i ∈ (Is∪If)

(27)

This replacement allows us to eliminate the yi binary variables (m1 in number) and reduce the number of associated constraints from 3m1 (eqs 17-19) to 2m1 (eq 27). As discussed earlier, m1 is the number of uncertain faults and can be quite large for a realistic process where the probability data related to several

Table 1. Comparison of Dimensionality of Formulations II and III binary variables

integer variables

continuous variables

constraints

Formulation II

m1 + n

n

Formulation III

0

n

4 + 2m1 + m3 + m 4 + 2m1 + m3 + m

3 + 4m1 + m3 + n + m 3 + 3m1 + m3 + m

faults and sensors may not be accurately known. Hence, for such processes, this modification can lead to significant computational advantages. Additionally, the set of equations in eq 22 used for calculating the nj binary variables can be excluded, and eq 23 can be accordingly modified to n

N)

min(xj,1) ∑ j)1

(28)

It can be seen that the term min(xj,1) is equivalent to nj because it takes a value of 1 if at least one sensor is placed on the jth variable and is 0 otherwise. This modification enables elimination of n binary variables and n constraints in the process. For a realistic process, the total number of variables is quite large, and this modification can again significantly reduce the computational burden. Thus, the CP reformulation can be summarized as

Formulation III: min[R1U - R2φ - R3N - xs]

(29)

n

s.t.

cjxj + xs ) C* ∑ j)1

(30)

∀i ∈ I\(If∪Is)

U g log(Ui)

∀i ∈ (If∪Is)

U ) log(Ui) + φi

(31) (32)

φ/i ) φfi/

∀i ∈ (If\Is)

(33)

φ/i ) φsi/

∀i ∈ (Is\If)

(34)

φ/i ) φfi/ + φsi/

∀i ∈ (Is∩If)

φi < φ/i w φ e φi φi g φ/i w φ e φ* φsi/ ) -

}

Bij(log sj)xj ∑ j∈J

(35)

∀i∈(Is∪If)

(36)

∀i ∈ Is

(37)

s

φ e φ*

(38)

n

N)

min(xj,1) ∑ j)1

U ∈ R- ; xj ∈ Z+; (Ui,xs,φ,φ/i ,φi,φsi/ ,N) ∈ R+

(39) (40)

A comparison of the number of variables and constraints involved in Formulations II and III is given in Table 1. As can be seen from Table 1, a significant reduction in problem dimensionality is achieved by this CP based reformulation. This reduction in model dimensionality is demonstrated later on the TE problem. It is also important to note that this CP based reformulation does not compromise on the rigor of representation. It can also be seen from above that incorporating logical constraints (such as max-min and implication constraints) is simpler in CP as compared with conventional optimization


techniques where they are posed as inequalities and increase the dimensionality of the problem. As a result, while the number of integer variables remain the same, all the binary variables get eliminated in the CP reformulation (Table 1). Remark. The above formulation can be further simplified by substituting eq 39 in eq 29, and the minimization of the term -xs in eq 29 can be replaced by the minimization of the term n cjxj. These changes will result in the objective function as ∑j)1 shown below n

n

min(xj,1) + ∑ cjxj] ∑ j)1 j)1

min[R1U - R2φ - R3

(41)

Using this objective function will further reduce the number of continuous variables by two (xs and N) and the number of constraints by one (eq 39). However, to be consistent with the existing literature, we will use the objective function as given in Formulations II and III instead of eq 41. We next discuss generation of multiple solutions using CP for the design of robust sensor networks for reliable fault diagnosis. 4.1. Determination of Multiple Solutions. Formulation II (and therefore Formulation III) may have multiple global optima, and the designer may be interested in evaluating possible tradeoffs in these solutions with respect to additional objectives. If these objectives are explicitly known, then Formulation III can potentially be modified by adding these additional criteria in a lexicographic fashion to uncover more promising solutions. However, even if these objectives are not explicitly known, it will still be desirable to present several choices (all of which are optimal to Formulation III) to the designer to choose from. Bhushan et al.1 have considered four objectives in their lexicographic formulation; however, the discovery of multiple solutions for this problem that could satisfy additional criteria was not considered in their work. In this section, we propose the application of CP to Formulation III to generate all multiple optimal solutions. In a typical CP framework, the procedure of generating these multiple solutions involves the following two steps, namely, solving an optimization problem followed by a CSP. The first step involves the solution of the single step lexicographic optimization problem as presented in Formulation III. This formulation will determine the optimal values of the various individual objectives as Uoptimal, φoptimal, xsoptimal, and Noptimal. The second step is the solution of a feasibility problem involving a CSP. This CSP is formulated as follows:

Formulation IV: solve U,Ui,xs,φ,φi,φsi/ ,φ/i ,xj,N ∈ Ω

(42)

U ) Uoptimal

(43)

φ)φ

(44)

optimal

N)N

optimal

xs ) x s

optimal

(45) (46)

In this formulation, Ω represents the set of all the constraints (eqs 30-40) of Formulation III. The last four constraints (eqs 43-46) ensure that all the feasible solutions for Formulation IV are optimal to Formulation III. Hence, solving this CSP will result in generation of all the multiple global optimal solutions of Formulation III, which can be further analyzed by the designer to select more promising sensor networks. It can be easily seen that this idea of solving CSPs to generate all multiple

solutions is generic in nature and can be applied to several other design problems. Remarks. (1) Although the CSP involves additional effort to generate all multiple solutions, the set of constraints (eqs 43-46) facilitate the efficient solution of the CSP by reducing the feasible space as compared to the optimization problem (Formulation III) solved in the first step. (2) It is to be noted that Formulation IV is guaranteed to have at least one feasible solution, namely, the single optimal solution reported for Formulation III. If there are no additional feasible solutions then it confirms that Formulation III has only a single global optimum. This implies that the designer has no additional flexibility for solving the design problem and the incorporation of any additional objective in the lexicographic sense will not change the optimal solution. 4.2. Multi-Objective Optimization. In the above discussion, it was assumed that the lexicographic ordering of the objective functions was explicitly known. However, this may not always be the case. For example, Bhushan et al.1 have considered formulations that have different hierarchical orderings for cost saving and network distribution. In such scenarios, it will be desirable to generate solutions quantifying the tradeoffs between various objectives. These tradeoff solutions are referred to as pareto optimal solutions. In this section, we propose to solve a two-step multi-objective optimization problem to generate the pareto optimal solution set for the sensor network design problem. The first step is to solve an optimization problem (Formulation V) with only the primary criterion (assuming that this is explicitly known) to be optimized, followed by a CSP enhanced with a constraint ensuring the optimality of the primary objective (Formulation VI).

Formulation V: min U

(47)

U,Ui,xs,φ,φi,φsi/ ,φ/i ,xj,N ∈ Ω

(48)

The solution to Formulation V will yield a network with minimum unobservability as Uoptimal.

Formulation VI: solve U,Ui,xs,φ,φi,φsi/ ,φ/i ,xj,N ∈ Ω

(49)

U ) Uoptimal

(50)

The solution to Formulation VI yields all feasible solutions with unobservability Uoptimal but with different values of network distribution N, robustness φ, and cost saving xs. This set of solutions inherently contains the pareto optimal solutions characterizing tradeoffs between these three objectives. These pareto optimal solutions can be extracted from the set of all feasible solutions by straightforward analysis of this set. Alternatively, the CSP procedure can be modified to efficiently generate only the pareto optimal solutions by adding appropriate constraints during the run time. This issue is currently under investigation. Further, it is also possible that different points in the decision space have identical objective function values. For example, two different sensor networks (differing in decision variables xj) can have the same values for each of the objectives xs, φ, and N. Solving the CSP of Formulation VI will generate all multiple solutions that correspond to the same point (in terms of objective function values) on the pareto front. Hence, the proposed multi-objective optimization procedure not only enables the designer to evaluate the tradeoffs between various

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Figure 2. Schematic of the TE problem.1 Table 2. Sensor Cost and Failure Probabilities for Potential Measurements1

1 2 3 4 5 6 7 8 9 10

var

log sj

cost

Pr Ps Pm F6 yA,6 yB,6 yC,6 yD,6 yE,6 yF,6

-3 -3 -3 -3 -3 -3 -3 -3 -3 -3

100 100 100 300 800 800 800 800 800 800

11 12 13 14 15 16 17 18 19 20

var

log sj

cost

yG,6 yH6 F7 yA,7 yB,7 yC,7 yD,7 yE,7 yF,7 yG,7

-3 -3 -3 -3 -3 -3 -3 -3 -3 -3

800 800 300 800 800 800 800 800 800 800

21 22 23 24 25 26 27 28 29 30

var

log sj

cost

yH,7 yA,8 yB,8 yC,8 yD,8 yE,8 yF,8 yG,8 yH,8 xD,10

-3 -3 -3 -3 -3 -3 -3 -3 -3 -3

800 800 800 800 800 800 800 800 800 700

objectives but also identifies the multiple solutions in the decision space for a given tradeoff point and thus empowers the designer with enhanced flexibility. Remark. In the above discussion, it was assumed that the primary objective was explicitly known whereas the hierarchical ordering among the other three objectives was not specified. In situations where the ordering between all the objectives is completely unknown, the solutions quantifying the tradeoff between all objectives can still be generated, by directly solving Formulation VI without the constraint U ) Uoptimal. 5. Case Study We now present application of the formulations discussed in this article on the TE challenge problem. The TE problem was originally proposed by Downs and Vogel32 to serve as a challenge problem in areas such as control, diagnosis, and optimization and has been used by several researchers in the general area of sensor network design. Musulin et al.33 have used this case study for the design of sensor networks in dynamic systems while Bhushan and Rengaswamy29 have used this for reliable sensor network design. Bhushan et al.1 have demonstrated the design of a robust sensor network for fault diagnosis on this flowsheet. In this article, we evaluate the expressive modeling power of CP to simplify the formulation

31 32 33 34 35 36 37 38 39 40

var

log sj

cost

xE,10 xF,10 xG,10 xH,10 xG,11 xH,11 xD,r xE,r xF,r xG,r

-3 -3 -3 -3 -3 -3 -3 -3 -3 -3

700 700 700 700 700 700 700 700 700 700

41 42 43 44 45 46 47 48 49 50

var

log sj

cost

xH,r F10 F11 Ts VLrXm VLre VLsXm VLse VLpXm VLpe

-3 -3 -3 -2 -2 -4 -2 -4 -2 -4

700 200 200 500 150 100 150 100 150 100

Table 3. Types of Faults and Their Occurrence Probability1 fault no.

description

log fj

1, 9 2, 10 3, 11 4, 12 5, 13 6, 14 7, 15 8

F1 high, low F2 high, low F3 high, low F4 high, low F8 high, low F9 high, low Tr high, low Cd low

-2 -2 -2 -2 -2 -2 -1 -2

fault no.

description

log fj

16, 25 17, 26 18, 27 19, 28 20, 29 21, 30 22, 31 23, 32 24, 33

VLrm,bias high, low high, low VLrset m,bias VLrVP,bias high, low VLsm,bias high, low VLssetm,bias high, low VLsVP,bias high, low VLpm,bias high, low VLpsetm,bias high, low VLpVP,bias high, low

-2 -2 -2 -2 -2 -2 -2 -2 -2

of the robust sensor network design problem for the TE process. We further demonstrate the generation of pareto optimal front and multiple solutions and highlight their utility over the solutions existing in literature.1 The results presented in this case study have been generated using the CP solver of ILOG. TE Flowsheet. The TE flowsheet shown in Figure 2 has 50 potential measurements and 33 faults. Table 2 shows the cost of the available sensors along with their failure probability data. The set of 33 faults (16 bidirectional and 1 unidirectional) along with their corresponding fault occurrence probability data are given in Table 3. The cost and probability data have been taken from Bhushan et al.1 whereas the faults and fault sets have been taken from Maurya et al.34 For the sake of brevity, further details

Ind. Eng. Chem. Res., Vol. 46, No. 18, 2007 5993 Table 4. Comparison of Dimensionality for the TE Case Study

Formulation II Formulation III

binary variables

integer variables

continuous variables

constraints

0 58

50 50

20 20

39 97

of the TE problem are not presented here, and the interested reader is referred to Downs and Vogel.32 As has been done in Bhushan et al.,1 we have considered that the sensor failure probabilities of sensors 3 and 4 and occurrence probabilities of faults 1 and 9 are not known accurately. The results have been presented for available costs of 5500 and 1000 units. Under this scenario, we demonstrate the utility of the proposed ideas on the TE case study with respect to the following three issues: (i) dimensionality of the problem, (ii) multiple solutions with explicit knowledge of the precedence, and (iii) pareto front and multiple pareto optimal solutions. Dimensionality of the Problem. In this section, we demonstrate the superiority of CP to model the sensor location problem in an efficient way (i.e., resulting in fewer variables and constraints) which may reduce the computational effort. The results presented in this section are for the single-fault resolution case, and the number of variables and constraints correspond to those obtained after the removal of redundant constraints. The single-fault resolution refers to the ability of a network to distinguish between various faults under the assumption that only one fault can occur at a time. In the methodology followed by Bhushan and Rengaswamy,31 designing sensor networks to ensure single-fault resolution involves the use of pseudo- (or fictitious) faults. For every pair of original faults, a pseudo-fault is appropriately generated. The variables affected by the pseudo-fault are the variables which can distinguish between the two original faults. For a process with k faults, the single-fault resolution strategy leads to the creation of kC2 additional faults. Because in the sensor network design formulation the number of constraints is governed by the number of faults, these additional faults lead to a significant increase in the number of constraints. However, some of these constraints are redundant, and Bhushan et al.1 have discussed the removal of these constraints. This removal of redundant constraints is based on the fact that a fault can be removed if its unobservability is ensured to be less than or equal to that of some other fault irrespective of the chosen sensor network. It can be seen that this procedure does not affect the optimality of the solution because the objective is to minimize the maximum unobservability and the removal of a fault with lower unobservability does not change the optimal solution. Care is taken while applying this procedure to faults that involve uncertain probabilities to ensure that the feasible region is not changed as a result of redundant constraint removal. This procedure was carried out for both the MILP and the CP based formulation, and it leads to equal reduction of redundant constraints in both the formulations. Table 4 gives the comparison between the MILP based formulation (Formulation II) and the CP based formulation (Formulation III) for the TE case study presented above. As discussed earlier, the reduction in dimensionality can be attributed to the expressive modeling power of CP as it can directly incorporate logical constraints (eqs 36) without having to represent them in terms of inequalities. The Ui were not considered as variables in the formulations (both MILP and CP) and were instead replaced by their corresponding expressions in terms of xj (eq 5) for the purpose of variable count as well as during result generation. From Table 4, it can be seen that the dimensionality of the CP based Formulation III is smaller than that of the MILP based

Formulation II. These formulations were solved using the ILOG Solver for CP and the ILOG CPLEX MILP solver. While for this case study the solution times were negligibly small and hence no benefits of the improved formulation were observed, in general, this reduction in dimensionality can be expected to translate into significant computational benefits. Further, as will be shown via Formulation IV, the CP based formulation was able to generate all the multiple optimal solutions as compared to the single solution obtained in the MILP based formulation. Multiple Solutions with Explicit Knowledge of the Precedence. In this section, we utilize the ability of CP to generate all feasible solutions of a CSP to determine all the multiple optima for the sensor network design problem. As the aim is to demonstrate the effective generation of multiple solutions, we have assumed an explicit precedence ordering between various objectives as given in Formulation II (or Formulation III). The multiple solutions are determined by a two-step procedure. The first step is the solution of Formulation III to determine Uoptimal, φoptimal, Noptimal, and xsoptimal followed by the solution of the CSP in Formulation IV to determine all the multiple optimal solutions. Table 5 lists the various multiple solutions under different scenarios of available cost. In this table, the number within brackets in the sensor configuration column denotes the hardware redundancy. For example, 4(3) denotes that 3 sensors are used to measure variable 4. From Table 5, it can be seen that there is only one optimal solution for C* ) 5500. This solution could have been obtained by solving the MILP based formulation in Formulation II, but it would have not given any further insights into the problem or even an idea on the number of multiple solutions. However, with the solution of the CP based Formulation IV (in association with Formulation III), it can be seen that there is only one optimal solution and hence the addition of any further objective in a lexicographic fashion will not result in the determination of any better solution. For the case when C* ) 1000, the CP based formulation was able to identify 10 multiple solutions (listed in Table 5). This enables the designer to include additional criteria that were not considered explicitly in the formulation to select a promising solution. Pareto Optimal Front and Multiple Pareto Optimal Solutions. In this section, we show the use of CP to generate the pareto optimal front when the precedence ordering in some of the objectives is not explicitly specified and the designer is interested in determining the tradeoffs between these objectives. Specifically, the primary objective is considered to be the minimization of the unobservability, and the tradeoffs between φ, xs, and N are evaluated. To determine the pareto front, we first solve Formulation V to obtain the minimum unobservability Uoptimal. This is followed by the solution of the CSP in Formulation VI to obtain all the feasible solutions. This set of feasible solution contains all the points on the pareto front. A straightforward post-optimality analysis can be performed on the set of these solutions to obtain the pareto front. Table 6 shows the number of feasible solutions for two scenarios with available costs of 1000 and 5500 units. A significant number of these points have the same objective function values but differ in the sensor network configuration (multiple feasible solutions). Of the various distinct solutions, some solutions are superior (non-dominated) to the other solutions and form the pareto front. A solution i is said to be non-dominated if there is no other feasible solution with identical or better values than i for all the three objectives. Figures 3 and 4 show all the distinct solutions along with the pareto front for the two different costs

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Table 5. Optimal Solutions for the TE Case Study with Explicit Knowledge of the Precedence C*

Uoptimal

φoptimal

Noptimal

xsoptimal

no. of optimal (multiple) solutions

sensor configuration variable measured (number of sensors)

5500

-5

6

14

0

1

1000

-2

6

4

0

10

{3(3),4(3),5(1),8(1),9(1),13(1),42(1),43(1),45(2), 46(1),47(2),48(1),49(2),50(1)} {3(2),4(2),48(1),50(1)}, {3(2),4(2),46(1),50(1)}, {3(2),4(2),46(1),48(1)}, {2(1),3(2),4(2),50(1)}, {2(1),3(2),4(2),48(1)}, {2(1),3(2),4(2),46(1)}, {1(1),3(2),4(2),50(1)}, {1(1),3(2),4(2),48(1)}, {1(1),3(2),4(2),46(1)}, {1(1),2(1),3(2),4(2)}

Table 6. Details of Feasible Points for Different Costs

C*

Uoptimal

total no. of feasible solutions

5500 1000

-5 -2

19103 74928

no. of distinct feasible solutions

total no. of pareto points

no. of distinct pareto points

101 231

10 102

8 17

respectively. The pareto points are marked distinctly and have been labeled with an alphabetical tag. Table 7 lists the values of each of the objectives for the points on the pareto front. As mentioned earlier, it is quite possible for a pareto point to have multiple realizations (same objective function values but with different sensor network configurations). The number of such realizations for each point on the pareto front has been indicated in the figures and has also been listed in Table 7 for easy reference. However, the realizations themselves have not been reported for the sake of brevity. From Table 7, it can be seen that, for C* ) 5500, pareto points such as D and E have two realizations. This effect is more pronounced for C* ) 1000 as some points have more than 10 realizations. These realizations allow greater flexibility to the designer for the selection of promising sensor networks by exploiting the multiple choices available for the pareto optimal solutions. The identification of such realizations has been possible by an intelligent enumeration of the feasible space based on the powerful domain reduction capabilities of CP. We now discuss the solutions for C* ) 5500 in some detail. Consider the pareto solutions A, C, H, and E presented in Table 7. It can be seen that solution A has the highest cost savings but does not have any robustness to the uncertainties in the probability data. On the other hand, solution C is characterized by a high robustness with respect to the uncertain sensor and

Figure 3. Feasible points along with the pareto front for C* ) 5500.

fault probabilities but does not result in any cost savings. However, both A and C suffer from low network distribution and hence may be vulnerable to the modeling uncertainties. It can also be seen that the solution H has high robustness with respect to the modeling uncertainties but is not robust to the uncertainties in the probability data. However, as seen in Table 7, solution E is characterized by moderate values of all the three objective functions. Depending on the design specifications, an appropriate choice of the sensor network can be made from the list of these pareto solutions. For example, if the designer is not overly concerned by the uncertainty in the probability data but wants to ensure high robustness to modeling uncertainty, then a solution such as H can be selected. We next discuss the selection of a particular network for a pareto point having different realizations. Table 8 shows the different sensor configurations for the pareto points D and E for C* ) 5500. Let us assume that the designer has decided to implement a sensor network configuration corresponding to the point D. Now, there are two choices for the designer, and either of them can be selected on the basis of factors that were not considered explicitly during the optimization formulation. Here, it can be seen that both the sensor configurations for point D differ only in the selection of sensors 1 and 2. These sensors correspond to the pressure variables in the reactor and separator, respectively. If the designer is specifically interested in measuring one of these, then the corresponding sensor network can be chosen. Additional criteria such as sensor precision (noise characteristics) and sensor-response characteristics can be considered to further screen these options. Similar analysis has been done for point E on the pareto front and is reported in Table 8. It is interesting to note that the two solutions corresponding to point E also differ in the location of sensors 1 and 2. Thus, we see that


Figure 4. Feasible points along with the pareto front for C* ) 1000. Table 7. Details of Pareto Front Points for Different Costs pareto points N xs

number of multiple pareto points

C*

tag

φ

5500

A B C D E F G H

0 3 6 0 3 0 3 0

14 14 14 15 15 16 16 17

800 400 0 700 300 600 200 100

1 1 1 2 2 1 1 1

1000

A B C D E F G H I J K L M N O P Q

0 0 3 6 0 3 6 0 3 6 0 3 0 3 0 3 0

1 2 2 2 3 3 3 4 4 4 5 5 6 6 7 7 8

900 800 600 200 700 500 100 600 400 0 500 300 400 200 250 100 100

2 9 1 1 16 5 5 14 10 10 6 10 1 5 3 1 3

Table 8. Sensor Configurations for Pareto Points with Multiple Realizations pareto point tag D

E

sensor configuration {1(1),3(2),4(2),5(1),8(1),9(1),13(1),42(1),43(1),45(2), 46(1),47(2),48(1),49(2),50(1)} {2(1),3(2),4(2),5(1),8(1),9(1),13(1),42(1),43(1),45(2), 46(1),47(2),48(1),49(2),50(1)} {1(1),3(1),4(1),5(1),8(1),9(1),13(1),42(1),43(1),45(2), 46(1),47(2),48(1),49(2),50(1)} {2(1),3(1),4(1),5(1),8(1),9(1),13(1),42(1),43(1),45(2), 46(1),47(2),48(1),49(2),50(1)}

additional flexibility is offered to the designer through the different realizations corresponding to the same point on the pareto front. Though the above analysis was performed on the basis of the results presented in Table 7, such an analysis will become tedious for a large number of pareto points or problems with a

large number of objectives. A pictorial view of Table 7 may be useful in such cases. The three-dimensional (3D) plots presented in Figures 3 and 4 are one such graphical representation but are difficult to interpret. Alternatively, the pareto solutions can be viewed either on a two-dimensional (2D) plot or on a parallel coordinate system as discussed below. 2D Plots. This technique involves the use of 2D plots to view the tradeoffs between any two objectives. Thus to view the tradeoffs between n objectives, an analysis of nC2 plots would be needed, and this may prove cumbersome if a large number of objectives are to be considered. However, in the current case we need to analyze three plots only for each of the available cost values. These plots are relatively easier to interpret as compared to the 3D plots presented earlier. It should be noted that the set of pareto points collectively obtained from such 2D plots may only form a subset of the actual pareto solutions, and it is possible that some pareto points present in the 3D plots are not reflected as pareto points in any of the three 2D plots. We next present the 2D plot based analysis of the pareto points for available cost C* ) 5500. Figure 5 shows the tradeoffs between the network distribution and the slack in the cost constraint and is characterized by four solutions. Figure 6 depicts the tradeoffs between robustness to the uncertainties in probability data and the slack in the cost constraint and is characterized by three solutions. Figure 7 shows the three solutions of the pareto front for the objectives of network distribution and robustness to uncertainties in the probability data. If we combine all the pareto points obtained from the above three plots, we see that solution E has not been represented as a pareto point in any of the 2D plots. A similar analysis of 2D plots for the cost C* ) 1000 shows that several pareto optimal points are not represented as pareto optimal points in any of the three plots. This loss in information is due to the projection of higher dimensional data onto lower dimensional coordinates. Remark. It should be noted that, in the above results, the third objective (say f) which is not represented in a given 2D plot (for example, f ) φ in Figure 5) is not constrained at any specific value. Alternatively the tradeoff between any pair of objectives can also be analyzed after ensuring a minimum performance in the third objective (f g fdesired) as was done in Formulation VI.

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Figure 5. Pareto front for network distribution (N) and slack in the cost constraint (xs) for C* ) 5500.

Figure 6. Pareto front for slack in the cost constraint (xs) and robustness (φ) for C* ) 5500.

While the 3D plots (in Figures 3 and 4) contained all the necessary information but were not amenable to easy interpretation, the 2D plots were easier to interpret and analyze but lead to loss of information in terms of the pareto points. Hence, it would be desirable to use a representation which is easier to interpret without the loss of any information. The parallel coordinate system discussed below is one such system that employs a 2D representation and yet allows the evaluation of tradeoffs in all the three objective functions simultaneously. Parallel Coordinate Systems. The parallel coordinate system was used by Inselberg35 for visualization of multivariate data.

Recently, Bagajewicz and Cabrera36 have applied this technique to the sensor network design and upgrade problem. This technique is advantageous because parallel coordinates in two dimensions are used to represent information present in higher dimensions. For an optimization problem involving n objectives, the parallel coordinates consist of n lines placed parallel and equidistant to each other and perpendicular to the horizontal axis. Each of these n lines corresponds to an objective function whose tradeoff is to be evaluated. Each point of the pareto front is represented by lines connecting these three parallel coordinates. For the case study presented in this article, we represent


Figure 7. Pareto front for network distribution (N) and robustness (φ) for C* ) 5500.

Figure 8. Parallel coordinates representation for C* ) 5500.

the three objectives N, xs, and φ by three parallel coordinates. As shown later, this representation enables simultaneous yet convenient analysis of the tradeoffs between the various objectives. Figures 8 and 9 show the parallel coordinate system for the pareto fronts listed in Table 7 for C* ) 5500 and 1000 units, respectively. The A-H lines in Figure 8 correspond to the eight pareto points presented in Table 7. It may be worth

Figure 9. Parallel coordinates representation for C* ) 1000.

noting that each line corresponding to a pareto point in the parallel coordinates system intersects with all the other lines. This is due to the fact that for each pair of pareto optimal

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solutions (i,j), solution i will be inferior to solution j in at least one objective, and similarly solution j will be inferior to solution i in at least one objective. In general, the parallel coordinate representation is easier to analyze than the earlier methods. For example, if the designer is interested in pareto solutions with objectives more than some specified threshold values, then lines can be drawn corresponding to these threshold values on the parallel coordinates so as to obtain solutions satisfying these thresholds. The solutions which lie on or above (for maximization problems) the line are the pareto optimal solutions for such a scenario. Consider a specific case where the designer is interested in obtaining solutions with a network distribution of at least 15, a cost saving of at least 200, and a robustness of at least 3. On the parallel coordinate representation, these requirements get translated into points X, X′, and X′′ as shown in Figure 8. For the optimization involving the above three objectives, the solutions E and G that lie above the line segments XX′ and X′X′′ satisfy these thresholds. The thresholds can as well be required in only two objectives. The line YY′ shows one such scenario which requires that the network distribution should be at least 14 with a cost savings of at least 400. All the pareto optimal solutions except solutions C, E, G, and H satisfy these thresholds. Similar analysis can be done for C* ) 1000. Thus, it can be seen that the parallel coordinates system can be effectively used for easy analysis of pareto optimal solutions. 6. Conclusion In this article, we have demonstrated the use of CP for the sensor location design problem. We have presented a CP based formulation for the minimum unobservability problem in the presence of uncertainties in the sensor failure and fault occurrence probability data as well as in the underlying cause-effect model. We have also shown that the expressive modeling power of CP results in a superior model formulation (in terms of reduced problem dimensionality) when compared with the MILP based formulation available in literature. We further exploit the capability of CP to determine all the feasible solutions of a problem and have used this feature to determine all the multiple optimal solutions without the addition of integer cuts and also to determine the pareto optimal front between conflicting objectives. The multiple solutions and the pareto front help the designer in selecting promising sensor networks. The above issues were demonstrated on the benchmark TE problem. The future work includes the effective exploitation of the complimentary strengths of MILP and CP in a hybrid framework37-39 for solving various sensor network design problems. The capability of CP to determine the pareto fronts can also be beneficial in other multi-objective problems encountered in the general area of process systems engineering. Nomenclature B ) bipartite matrix between faults and variables bij ) (i,j)th element of bipartite matrix B cj ) cost of the jth sensor C* ) total cost allowed for sensor location Fi ) fault i fi ) fault occurrence probability of the ith fault I ) set of all faults If ) set of faults whose occurrence probability is uncertain Is ) set of faults that affects variables which can only be measured by sensors with uncertain failure probability Js ) set of inaccurate sensors

N ) number of variables measured in the process m ) total number of faults m1 ) cardinality of the set If∪Is, i.e., number of uncertain faults m1′ ) cardinality of the set If∩Is m2 ) cardinality of the set If m3 ) cardinality of the set Is m2,3 ) cardinality of the set (If\Is) m3,2 ) cardinality of the set (Is\If) n ) number of variables in the process M ) a large positive constant sj ) sensor failure probability of the jth sensor U ) network unobservability Ui ) unobservability of the ith fault xj ) number of sensors to measure the jth variable yi ) binary variable corresponding to inaccurate fault i R1, R2, R3 ) lexicographic constants φ ) maximum of all the individual slacks φi ) slack for the ith inaccurate fault Literature Cited (1) Bhushan, M.; Narasimhan, S.; Rengaswamy, R. Robust Sensor Network Design for Fault Diagnosis. Comput. Chem. Eng. 2007, DOI: 10.1016/j.compchemeng.2007.06.020. (2) Vaclavek, V.; Loucka, M. Selection of Measurements Necessary to Achieve Multicomponent Mass Balances in Chemical Plant. Chem. Eng. Sci. 1976, 31, 1199. (3) Meyer, M.; Lann, J. M.; Koehret, B.; Enjalbert, M. Optimal Selection of Sensor Location on a Complex Plant Using a Graph Oriented Approach. Comput. Chem. Eng. 1994, 18, S535. (4) Ali, Y.; Narasimhan, S. Sensor Network Design for Maximizing Reliability of Linear Processes. AIChE J. 1993, 39, 820. (5) Ali, Y.; Narasimhan, S. Redundant Sensor Network Design for Linear Processes. AIChE J. 1995, 41, 2237. (6) Ali, Y.; Narasimhan, S. Sensor Network Design for Maximizing Reliability of Bilinear Processes. AIChE J. 1996, 42, 2563. (7) Bagajewicz, M. Design and Retrofit of Sensor Networks in Process Plants. AIChE J. 1997, 43, 2300. (8) Bagajewicz, M.; Sanchez, M. Duality of Sensor Network Design Models for Parameter Estimation. AIChE J. 1999, 45, 661. (9) Bagajewicz, M.; Sanchez, M. Cost-Optimal Design of Reliable Sensor Networks. Comput. Chem. Eng. 2000, 23, 1757. (10) Bagajewicz, M.; Cabrera, E. New MILP Formulation for Instrumentation Network Design and Upgrade. AIChE J. 2002, 48, 2271. (11) Bagajewicz, M.; Chmielewski, D.; Rengaswamy, R. Integrated Process Sensor Network Design. Presented at the Annual AIChE Meeting, Austin, TX, 2004. (12) Chmielewski, D.; Palmer, T. E.; Manousiouthakis, V. On the Theory of Optimal Sensor Placement. AIChE J. 2002, 48, 1001. (13) Van Hentenryck, P. Constraint and Integer Programming in OPL. INFORMS J. Comput. 2002, 14, 345. (14) Marriott, K.; Stuckey, P. J. Introduction to Constraint Logic Programming; MIT Press: Cambridge, MA, 1999. (15) Lustig, I. J.; Puget, J. F. Program Does Not Equal Program: Constraint Programming and Its Relationship to Mathematical Programming. Interface 2001, 31, 29. (16) Hooker, J. N. Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction; Wiley: New York, 2000. (17) Baptiste, P.; Le Pape, C.; Nuijten, W. Constrained-Based Scheduling: Applying Constraint Programming to Scheduling Problems; Kluwer Academic Publishers: Boston, 2001. (18) Jain, V.; Grossmann, I. E. Algorithms for Hybrid MILP/CP Models for a Class of Optimization Problems. INFORMS J. Comput. 2000, 13, 258. (19) Maravelias, C. T.; Grossmann, I. E. A Hybrid MILP/CP Decomposition Approach for the Continuous Time Scheduling of Multipurpose Batch Plants. Comput. Chem. Eng. 2004, 28, 1921. (20) Darby-Dowman, K.; Little, J.; Mitra, G.; Zaffalon, M. Constraint Logic Programming and Integer Programming Approaches and Their Collaboration in Solving an Assignment Scheduling Problem. Constraints 1997, 1, 245. (21) Proll, L.; Smith, B. Integer Linear Programming and Constraint Logic Programming Approaches to a Template Design Problem. INFORMS J. Comput. 1998, 10, 265.

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ReceiVed for reView December 7, 2006 ReVised manuscript receiVed April 16, 2007 Accepted May 23, 2007 IE061569X

Constraint Programming Based Robust Sensor Network Design

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