Comprehensive Design of a Sensor Network for Chemical PlantsBased on Various Diagnosability and Reliability Criteria. 1.Framework

Mani Bhushan and Raghunathan Rengaswamy*

Department of Chemical Engineering, Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India

Fault diagnosis is an important area in the chemical process industry and has attractedconsiderable attention from researchers in the recent past. All approaches for fault diagnosisdepend critically on the sensors measuring the important process variables in the system. Inthis paper, a reliability maximization based optimization framework for sensor location from afault diagnosis perspective is presented. The formulation is aimed toward maximizing thereliability of the fault monitoring system while satisfying the constraints imposed on the system.A minimum-cost model which minimizes the cost of the fault monitoring system while ensuringthat the solution provides a minimum threshold reliability is also presented. A one-stepoptimization formulation which maximizes reliability and, among the various solutions withthe same reliability, chooses the one with minimum cost is discussed in this paper. A methodologyfor obtaining the “best” sensor location irrespective of the single/multiple fault assumption isalso presented. In the first part of this two-part series of papers, the sensor location frameworkis discussed. In the second part, the sensor location procedure is applied to a large flowsheet,the Tennessee-Eastman flowsheet. Various issues involved in the application of the reliabilitymaximization based optimization procedure are explained using this case study.

1. Introduction and Literature Survey

For safe and optimal operation of a chemical plant, itis essential to quickly detect and identify faults whenthey occur. Hence, an efficient fault diagnosis methodol-ogy is very useful for modern day complex chemicalplants. The increasing importance of fault diagnosis inthe chemical process industry has led several research-ers to work in this area. Whenever a process encountersa fault, the effect of the fault is propagated to all or someof the process variables. The main objective of the faultdiagnosis step is to observe these fault symptoms anddetermine the root cause for the observed behavior. Thefault detection step involves a comparison of the ob-served behavior of the process to a reference model. Thisobserved fault-symptom pattern forms the basis for thefault identification step. Thus, the efficiency of thediagnostic system depends critically on the location ofthe sensors monitoring important process variables.With hundreds of process variables available for mea-surement in any chemical plant, selection of crucial andoptimum sensor positions poses a unique problem.Hence, there is a need for an automated procedure todesign a cost-optimum, fool-proof, and highly reliablefault monitoring system for the safe operation of thechemical processes.

In our previous work, we have developed proceduresto locate sensors for fault diagnostic observability basedon digraph (DG)1 and signed digraph (SDG)2 represen-tations of the process. However, these approaches werelargely qualitative and hence did not use the quantita-tive information that might be available about thesystem. Further, the notion of reliability as discussed

in this paper was not discussed. Hence, the aim of thefirst part of this two-part series of papers is thedevelopment of a comprehensive design strategy for thedesign of sensor locations that takes into considerationthe available quantitative information such as faultoccurrence and sensor failure probabilities while han-dling the various constraints that might be imposed onthe design problem.

There have been a few researchers who have workedon the problem of sensor location. Lambert3 usedprobabilistic importance of events in fault trees to decideoptimal sensor locations. Ali and Narasimhan4 intro-duced the concept of reliability of a variable. Theydescribed a graph-theoretic procedure for maximizingthe reliability of linear processes in the presence ofsensor failures. The reliability of the process was definedas the smallest reliability among all of the variables.They also extended this procedure for the optimal designof a redundant sensor network for linear processes.5 Adesign procedure for a nonredundant sensor networkfor bilinear processes was also discussed.6 Sen et al.7presented a genetic algorithm based approach that canbe applied for the design of nonredundant sensornetworks using different objective functions.

Bagajewicz8 proposed an optimization formulation toobtain cost optimal sensor networks for linear systemssubject to constraints on precision, residual precision,and error detectability. Bagajewicz and Sanchez9 mergedthe concepts of the degree of redundancy for measure-ments and the degree of observability for unmeasuredvariables into a single concept, the degree of estimat-ibility of a variable. They presented optimization for-mulations for the design of sensor networks to achievedifferent degrees of estimatibility of key variables. Aminimum-cost model and a generalized reliability modelfor the design of a reliable sensor network was alsopresented by them.10 The connection of the minimum-

* To whom correspondence should be addressed. Currentaddress: Department of Chemical Engineering, Clarkson Uni-versity, Potsdam, NY 13699-5705. E-mail: raghu@clarkson.edu.

1826 Ind. Eng. Chem. Res. 2002, 41, 1826-1839

10.1021/ie0104363 CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 03/03/2002

cost model to the maximum-reliability model,10 and tothe maximum-precision model,11 was also established.Alheritiere et al.12 dealt with the optimization of re-sources allocated to various sensors for improving theprecision of a parameter. Recently, Bagajewicz andSanchez13 presented a framework to perform realloca-tion and upgradation of existing instrumentation toachieve maximum precision of selected parameters.

The optimization approaches summarized here are forlocating sensors to maximize objectives such as preci-sion, estimatibility of variables, and so on. In this paper,in contrast, the sensor location strategy is presentedfrom a fault diagnosis perspective. The sensor locationproblem is formulated as an integer programmingoptimization problem which maximizes the systemreliability from a fault diagnosis perspective whilesatisfying the constraints imposed on the system. Thereliability is defined in terms of the probability of a faultoccurring and remaining undetected. After presentingthe reliability formulation, a cost minimization modelis also discussed. A one-step optimization procedurewhich generates the most reliable sensor network and,among the multiple solutions, chooses the one withminimum cost is then presented. Use of these formula-tions by incorporating various process-specific con-straints is also discussed. Various other related formu-lations are also presented.

2. General Solution Philosophy

Most of the previous work in the literature pose thesensor location problem as an optimization problem.One of the key ideas in our approach to solve the sensorlocation problem from the fault diagnosis perspectiveis the decoupling of the cause-effect modeling from theoptimization formulation. This is done to facilitate theuse of various techniques for cause-effect modeling inconjunction with various optimization formulations tosolve a particular sensor network design problem ofinterest. This two-level strategy is illustrated in Figure1.

The cause-effect modeling is integrated with theoptimization formulation through the use of the idea offault sets. These are sets of sensors that are generatedbased on cause-effect modeling which then form thebasis for sensor location. For example, if an observabilityproblem is solved, these sets are simply sets of variablesthat are affected by the faults; if a resolution problemis being solved, then the sets consist of variables thatcan discriminate between the various faults. The gen-eration of these fault sets based on a DG model wasdiscussed by us in work by Raghuraj et al.,1 andgeneration of these sets based on a SDG model wasdiscussed by us in work by Bhushan and Rengaswamy.2In general, in DG and SDG methods, given a process

with its faults and measurable variables (where sensorsmay be placed), the cause-effect information is repre-sented in a matrix, which will be referred to as thebipartite matrix, D. The rows of this matrix correspondto faults, and the columns correspond to measurablenodes (or sensor nodes). The (i,j)th entry (dij) of thismatrix is 1 if fault i affects node j and is zero otherwise.This bipartite matrix forms the basis for the generationof the fault sets.

However, these are not the only methods for thegeneration of fault sets. The fault sets may also begenerated by, say, querying an experienced processengineer or operator. Also, modifications to process SDGmay be performed to remove spurious effects. The orderof magnitude approach14 is one way of achieving this.For the case study presented in part 2 of this series, wehave used a modified SDG representation of the processto generate this information. The process SDG wasmodified by not only considering the signs of arcs butalso associating a gain with the arc. This enables useof order of magnitude arguments to achieve betterresolution of the faults. This will be discussed in part 2of this series. These fault sets could also be generatedbased on a completely quantitative model of the process.

The next important advantage of this approach is thatthe generation of the fault sets can be tailored to theproblem that is being solved. Hence, various diagnosticcriteria such as observability, single-fault resolution,and multiple-fault resolution can be used in the opti-mization formulation. Optimization formulations canalso be based on the reliability of the sensor networkdesigned, cost of the fault monitoring system, and/orcombinations thereof. Further, various information suchas fault probability and sensor failure probability canbe incorporated in our framework. Hence, the proposedapproach provides a transparent framework that canbe used to solve various sensor location problems whilebeing generally applicable for a wide variety of cause-effect models.

In this paper, we will focus on the optimizationformulations for the sensor network design problem. Asmentioned before, cause-effect modeling for fault di-agnosis has been discussed in detail in our previouswork and hence will not be dealt with in this part ofthe series of papers. However, in part 2 of this series,where the application of the proposed approach isdemonstrated on the Tennessee-Eastman (TE) casestudy, we will discuss the cause-effect modeling andthe generation of fault sets for the case study in detail.Hence, in this part, we will discuss various optimizationformulations and the use of the fault sets in theseformulations for a comprehensive solution to the sensornetwork design problem.

2.1. Optimization Formulation for ReliabilityMaximization. The aim of any sensor network designis to maximize the system reliability. A sensor networkis highly reliable if the probability of any fault occurringwithout being detected is low. The formulation that wepropose is based on maximizing the minimum reliabilityamong all of the faults. The system reliability is definedas the lowest reliability among all faults. This is basedon the philosophy that a chain can be no stronger thanits weakest link.4

For a given process, the faults of that process havecertain occurrence probabilities. The various availablesensors also have certain failure probabilities, whichdepend on the type of the sensor and the variable being

Figure 1. Two-level strategy.

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1827

measured. The only way in which a fault can occurwithout being detected is that the fault occurs andsimultaneously the sensors covering that fault fail. Theprobability of such an event taking place is the productof the fault occurrence and corresponding sensor failureprobabilities. This product Ui, is referred to as theunobservability value of that fault:

The concept of unobservability is illustrated in thefollowing example.

Example 1. Consider Figure 2, which is a cause-effect (bipartite) representation of a process with threefaults and two measurable nodes. In this process, ifsensors are placed at both nodes S1 and S2, thenunobservability values of faults F1, F2, and F3 are0.0001 (0.01 × 0.01), 0.000 02 (0.02 × 0.01 × 0.1), and0.001 (0.01 × 0.1), respectively. On the other hand, ifonly node S1 is measured, the unobservability valuesof faults F1, F2, and F3 are 0.0001 (0.01 × 0.01), 0.0002(0.02 × 0.01), and 0.01, respectively.

The reliability of detecting a fault is inversely pro-portional to the unobservability value of that fault.Maximizing the reliability of the system is then equiva-lent to minimizing the unobservability of the system.Because we want to maximize the minimum reliabilityof the system, this is equivalent to minimizing themaximum unobservability of the system. With this aim,the optimization formulation for sensor network designto maximize the system reliability is as follows:

Problem Ia.

subject to

where

In the above formulation, Ui is as given by eq 1.Constraint (3) ensures that the cost of the fault moni-toring system is not more than the available resource,C*, where cj is the cost of placing a sensor at node j.For the problems considered in this work, cj’s will beconsidered to be positive constants. The decision vari-ables (xj) are allowed to take nonnegative integer valueswhich may be greater than 1. In other words, hardwareredundancy is allowed. For example, xk ) 2 means thattwo sensors have to be placed on node k. This feature

makes the approach practical because it makes senseto use more than one sensor to measure a variable ifthat particular sensor has a high failure probability orif the covered fault has a high occurrence probability.Another important point to note is that in the formula-tions presented in this paper (including the one pre-sented above), we are considering only one type ofmeasurement for a given variable. Cases where avariable may be measured using different types ofsensors (with possibly different costs and failure prob-abilities) can be easily incorporated in the formulationspresented here.

The optimal solution of problem Ia will give sensorlocations which maximize the system reliability. Asformulated above, problem Ia is a nonlinear (objectivefunction is nonlinear) integer programming (decisionvariables are nonnegative integers) problem which isnot easy to solve exactly. It turns out that, by a suitabletransformation, the problem can be converted to a linearinteger programming problem. This is discussed below.

Equivalent Linear Objective Function. The ob-jective function (2) can be replaced by a linear objectivefunction:

where

ln(Ui) is linear in the decision variables xj and isobtained by taking the natural log on both sides of eq1.

Claim I. Objective function (5) is equivalent toobjective function (2).

Proof of Claim I. The equivalence of the two objec-tive functions is based on the fact that the natural log,ln(x), is a monotonically increasing function of x for x >0. This follows from the fact that the derivative of ln(x)is a positive quantity for positive x.

Hence,

The unobservability of a fault i, Ui ) fi(∏j)1n sj

dijxj), isalways nonnegative, i ) 1, ..., m. Given a set of selectedsensors, the fault i for which Ui is maximum will alsogive the maximum value of ln(Ui). Hence, minimizingthe maximum Ui is the same as minimizing the maxi-mum ln(Ui), i ) 1, ..., m. Therefore, objective function(5) is equivalent to objective function (2).

The objective as given by eq 5 is still not in thestandard integer linear programming (ILP) form be-cause it involves minimization of the maximum value.By a simple modification, the problem is converted tothe standard ILP form.

Problem I.

Figure 2. Example to explain unobservability.

Ui ) fi∏j)1

n

sjdijxj (1)

minxj

[max∀ i

Ui] (2)

∑j)1

n

cjxj e C* (3)

xj ∈ Z+, j ) 1, ..., n (4)

minxj

[max∀ i

ln(Ui)] (5)

ln(Ui) ) ln(fi) + ∑j)1

n

dijxj ln(sj) (6)

ddx

ln(x) ) 1x, which is >0, ∀ x > 0 (7)

x > y w ln(x) > ln(y), ∀ x, y > 0 (8)

minxj

U (9)

1828 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

subject to

Claim II. Objective function (5) is equivalent toobjective function (9) with the constraint (10).

Proof of Claim II. The proof is obvious becauseconstraint (10) ensures that the quantity U beingminimized in objective function (9) is g{max (ln(Ui), i) 1, ..., m)}. Because U is being minimized, it will beequal to {max (ln(Ui), i ) 1, ..., m)}. This is equivalentto using objective function (5).

Based on claims I and II, the equivalence of problemsI and Ia is proved next.

Claim III. Problem I is equivalent to problem Ia.Proof of Claim III. By claim II, the objective

function (9) together with constraint (10) of problem Iis equivalent to objective function (5). Claim I estab-lishes the equivalence of objective function (5) to objec-tive function (2). Hence, objective function (9) along withconstraint (10) is equivalent to objective function (2).The cost constraint (3) and the integer requirements (xj∈Z+) are present in both problems I and Ia. Therefore,problem I is equivalent to problem Ia.

Problem I is now in standard ILP form. The solutionto the sensor location design problem, as posed here,gives the optimal set of sensors for maximizing thesystem reliability (minimizing the system unobservabil-ity), subject to the cost constraint. It is also importantto note that the cost constraint can be written in termsof the total number of available sensors. Also, dependingon the problem at hand, different constraints for dif-ferent sensors can be imposed. A minimum-cost modelis presented next.

3. Minimum-Cost Model

The minimum-cost model is posed as the followingoptimization problem:

Problem II.

subject to

In the above formulation, U is the system unobserv-ability (maximum unobservability among all faults), Uiis the unobservability value of fault i as defined in eq1, and U* is the threshold value for the system unob-servability. This formulation selects the minimum-costsensor network which achieves the required systemreliability. Note that, instead of having a bound on thesystem unobservability as above, the designer maychoose to have bounds on the individual fault unobserv-

abilities. In the next section, some issues related to theapplication of the proposed formulations for a givenprocess are discussed.

4. Application of the Proposed Formulation

Two main issues are discussed in this section: (i) useof the optimization framework to come up with the“best” design and (ii) reduction in the number ofconstraints for the optimization problem.

4.1. Obtaining the “Best” Sensor Location. Animportant aspect of the sensor location formulationspresented above is that their application is not restrictedto just the observability (of faults) case. As shown byRaghuraj et al.,1 any other problem (single-fault resolu-tion, double-fault resolution, etc.) can be converted to asuitable observability problem. This is briefly discussedbelow:

(i) Single-fault resolution: For each pair (i, j) of faults,the fault set

is generated. In the above expression, Ai and Aj are thesets of measurable nodes affected by faults i and j,respectively. Bij represents the set of nodes which canbe used for differentiating between faults i and j. EachBij is treated as a pseudofault and added to the bipartitematrix D, discussed earlier in this paper. Now, theobservability of Bij refers to the ability of the faultmonitoring system to distinguish between faults i andj.

The probability fij of occurrence of the pseudofault Bijis defined as the minimum of the probabilities of faultsi and j:

The above definition of fij is based on an intuitiveunderstanding of the concept of the pseudofault Bij. Thisfault affects only those sensors which may be used forresolving faults i and j and, hence, represents theprobability with which the two original faults wouldneed to be resolved. The use of the “min” function inthis definition conveys the idea that the probability ofrequiring resolution between faults fi and fj is no morethan the minimum probability among the two faults,and hence its “weightage” (as given by its probability)in the sensor location framework is less than equals thatof the two individual faults themselves. This makessense because the sensor network designer’s aim wouldbe to first observe the individual faults and then worryabout resolving them from each other. This probabilitycould also be defined strictly from probabilistic argu-ments and similar analysis could be carried out usingthe same formulations as proposed in this paper.

(ii) Multiple-fault resolution: For illustration, con-sider the specific case when a maximum of two faultscan occur at a time. For each pair (i, j) of faults, the setAij ) Ai ∪ Aj is formed. Aij represents the set of nodeswhich are affected when both of the faults i and j occurtogether. The set Aij is treated as a pseudofault andadded to the original set of faults. The probability ofoccurrence of this fault ) fi × fj, the product of occur-rence probabilities of faults i and j. To generate the setsfor resolution, the single-fault resolution algorithm asdiscussed above is applied to this extended set of faults.

U g ln(Ui), i ) 1, ..., m (10)

∑j)1

n

cjxj e C* (11)

xj ∈ Z+, j ) 1, ..., n (12)

minxj

∑j)1

n

cjxj (13)

U e U* (14)

U g ln(Ui), i ) 1, ..., m (15)

xj ∈ Z+, j ) 1, ..., n (16)

Bij ) Ai ∪ Aj - Ai ∩ Aj (17)

fij ) min (fi, fj) (18)

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1829

The same methodology can be applied to cases whenmore than two faults can occur simultaneously. Ra-ghuraj et al.1 have given the above procedures in detailfor the DG representation of the process. Bhushan andRengaswamy2 have given the procedures when theprocess SDG is used.

Depending on the scenario being considered (singlefault, double fault, etc.), the bipartite matrix D isappropriately generated as discussed above. The reli-ability maximization and cost minimization formula-tions can then be applied to obtain optimal sensorlocation for the case considered. For a given availablecost, for different scenarios, one may obtain differentreliabilities and different sensor networks. The questionthat naturally arises then is, given a process andavailable resource, is it possible to design a “best” sensornetwork irrespective of the assumed scenario (singlefault, double fault, etc.)? It turns out that, by a system-atic application of the maximum-reliability formulation,it is possible to achieve such a design.

The procedure to obtain the best design for a givenprocess is explained through the following example.

Example 2. Consider the bipartite graph shown inFigure 3. The process represented by the graph consistsof three faults (F1, F2, F3), and three measurable nodes(S1, S2, S3). The probabilities of occurrence of faults aref1 ) 0.01, f2 ) 0.02, and f3 ) 0.03. The probabilities offailures of sensors available to measure the threemeasurable nodes are s1 ) 0.1, s2 ) 0.2, and s3 ) 0.3.The set of nodes affected by the three faults are A1 )[S1, S2], A2 ) [S2], and A3 ) [S2, S3]. For the purposeof illustration, we will assume all of the sensors to havethe same cost. The resource available can then be statedin terms of the number of available sensors. Considerthe specific case of two available sensors. Depending onthe scenario considered, different sensor locations maybe obtained:

(i) Only observability of faults is considered: For thiscase, the optimal sensor network is [S2, S2], that is,placing two sensors on node S2. The unobservabilitiesof faults then are U1 ) 4 × 10-4, U2 ) 8 × 10-4, and U3) 1.2 × 10-3. The system unobservability U is themaximum unobservability. Hence, U ) max (U1, U2, U3)) 1.2 × 10-3.

(ii) Resolution for the single-fault case is also consid-ered: For this case, three pseudofaults are constructed.They are

The probabilities of occurrence of these pseudofaults are

These three pseudofaults are added to the original setof faults. Performing sensor location on this extendedsystem gives [S2, S3] as the optimal sensor network.With these sensors, the unobservabilities of faults areU1 ) 2 × 10-3, U2 ) 4 × 10-3, U3 ) 1.8 × 10-3, U12 )0.01, U13 ) 3 × 10-3, and U23 ) 6 × 10-3. The systemunobservability then is U ) max (U1, U2, U3, U12, U13,U23) ) 0.01.

(iii) Simultaneous occurrence of two faults is alsoconsidered: The probabilities of the simultaneous oc-currence of faults are (F1, F2) ) 2 × 10-4, (F1, F3) ) 3× 10-4, and (F2, F3) ) 6 × 10-4. All of these prob-abilities are less than the system unobservability ob-tained for the single-fault resolution case ii above.Hence, even if double faults are considered, the optimalsensor location obtained in the previous case will notchange. It is also clear that consideration of any otherscenario where more than two faults can occur simul-taneously will also not change the optimal sensorlocation obtained by solving the reliability maximizationproblem for the single-fault resolution case. Hence, forthe given cost (the number of sensors available), thesensor location obtained by solving for the single-faultresolution case is the “best” sensor location irrespectiveof the scenario being considered. For a different (higher)given cost, the double-fault case (or other scenarios) mayresult in a different sensor location.

The above example illustrates the methodology to befollowed to obtain the best design for a given processand a given total resource. This approach is summarizedbelow:

1. Solve the problem for the first level (single-faultassumption). Calculate the unobservability value of thesystem.

2. Generate the pseudofaults for the next level (sayup to simultaneous occurrence of r faults at a time). Ifthe probability of occurrence of all of the new faultsgenerated at this level is less than the system unob-servability obtained by solving the reliability maximiza-tion problem at the previous level, then stop. Otherwise,generate sets corresponding to resolution of faults forthe current level, and solve the maximum-reliabilityproblem. Then go to the next level (simultaneousoccurrence of up to r + 1 faults at a time), and continuethis procedure.

It is possible that while generating new faults, someof the faults (corresponding to resolution of faults) maynot affect any sensor node1 (the A set corresponding tothese faults will be empty). These faults will not beconsidered for unobservability calculation. In the nextsection, techniques to reduce the number of constraintsin the optimization problems to be solved are discussed.

4.2. Reduction in the Number of Constraints. Adrawback of the approach discussed above is that thesize of the problem to be solved increases rapidly assimultaneous occurrence of faults is considered. Toillustrate this, consider a system with 10 faults. If onlyone fault at a time is assumed, then corresponding tothese 10 original faults, 10C2 ) 45 Bij sets are generated.

Figure 3. Bipartite graph for example 2.

f12 ) min (f1, f2) ) 0.01

f13 ) min (f1, f3) ) 0.01

f23 ) min (f2, f3) ) 0.02

B12 ) A1 ∪ A2 - A1 ∩ A2 ) [S1]

B13 ) A1 ∪ A3 - A1 ∩ A3 ) [S1, S3]

B23 ) A2 ∪ A3 - A2 ∩ A3 ) [S3]

1830 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

The system for which the reliability maximizationproblem has to be solved therefore now consists of 10 +45 ) 55 faults. If double-fault scenario is assumed, then10C2 ) 45 new faults corresponding to the simultaneousoccurrence of any two original faults are constructed.The system, hence, now consists of 10 + 45 ) 55 faults.Applying a single-fault resolution algorithm to thissystem generates 55C2 ) 1485 new faults. The problem,therefore, now consists of 1485 + 55 ) 1540 faults onwhich the reliability maximization algorithm has to beapplied. Because each fault corresponds to a constraintin the reliability maximization problem (problem I), thisinvolves solving a problem with 1541 constraints (onecost constraint). For the triple-fault case, the numberof constraints would be 15401. It may not be computa-tionally feasible to solve a problem with so manyconstraints using commonly available ILP packages. Itturns out that the number of constraints can be sub-stantially reduced by removing redundant constraints.A systematic procedure to achieve this is given below.This reduction is possible at two levels.

(i) At the first level, some faults in the original processmay be redundant. If two faults, say i and j, affect thesame sensors, that is, Ai ) Aj, then the one with thelower probability is not considered.

(ii) Once redundant faults in the original process havebeen removed, then for the scenario considered (doublefault, triple fault, etc.), appropriate sets are generated(for example, for the double-fault case, Aij sets areconstructed) and added to the original system. This nowbecomes the system on which the resolution algorithmis applied (generation of Bij sets). The second level ofreduction is performed now. Some of the Bij sets maybe empty. These are not considered. Also some otherredundant faults are removed. If any Bij ⊇ Bkl and theprobability of Bij e the probability of Bkl, then fault Bijcan be removed from the problem being solved withoutaffecting the optimal solution. The reason for this is thatwhenever a sensor is selected to reduce the unobserv-ability of fault Bkl, the unobservability of fault Bij alsodecreases because that sensor is affected by fault Bij aswell. Also, to start with (when no sensors have beenselected), the unobservability of fault Bij e the unob-servability of fault Bkl. Hence, fault Bij can be removedfrom the problem without affecting the optimal solution.For example, while locating sensors for the observabilityof all faults for bipartite graph of Figure 3, fault F1 neednot be considered because A1 ⊃ A2 and f1 < f2.

After the above reductions are performed, the numberof constraints in the problem being solved decreasesconsiderably. The reliability maximization algorithm isthen applied to this reduced system.

5. Drawback of Maximum-Reliability andMinimum-Cost Models

The maximum-reliability model as presented in theprevious sections will give the most reliable sensornetwork, but it may not give a cost optimal result. Theresult may not be cost optimal in the sense that theremay be some other network with a lower cost whichmight yield the same reliability. This is explained bythe following example.

Example 3. Consider the bipartite graph of Figure4. Both of the sensors have the same probability offailure (0.1) but have different costs (75 and 100 units,respectively). The fault occurrence probability of thegiven fault is 0.01. For a given available amount of 100

units, one sensor can be placed at either S1 or S2. Theunobservability value in both of the cases is 0.001, andthe cost constraint is also satisfied for both of the cases.Hence, both of the solutions are optimal solutions of thereliability maximization problem. In other words, theproblem has multiple solutions, all of which yieldoptimal results from the point of view of the achievedreliability but have different costs.

Similar cases can occur in the cost minimizationmodel, where one may obtain a solution which is leastcostly and also satisfies the constraint on reliabilitybeing greater than the threshold value but does notresult in the most reliable network.

In such cases, where multiple objectives are present,one would like to obtain a solution which optimizes theobjectives in an ordered way. As a first step, the firstobjective is optimized. Given the optimal value for thefirst objective, the second objective is optimized, subjectto the first objective being equal to its optimal value.Given the optimal values for the first and secondobjectives, the third objective is optimized, and so on.This is popularly known as lexicographic optimization.15

For the maximum-reliability model presented in thispaper, maximizing the reliability is the first objectiveand minimizing the cost is the next objective. On theother hand, for the minimum-cost model, minimizingthe cost is the first objective and maximizing thereliability is the second objective. One way for achievingthe desired solution will be to solve the problemssequentially. For the maximum-reliability model, thiswould involve (i) solving the maximum-reliability prob-lem and then (ii) with the reliability fixed at its optimalvalue solving the minimum-cost problem.

This would imply solving two ILPs which may not bedesirable for large problems. In the next section wepresent a modified maximum-reliability model, whichwill directly yield a solution which will be optimal inthe lexicographic sense.

6. One-Step OptimizationConsider the following modified, maximum-reliability

optimization problem.Problem III.

subject to

where

Figure 4. Bipartite graph for example 3.

minxj

[max∀ i

{ln(Ui)} - Rxs] (19)

∑j)1

n

cjxj + xs ) C* (20)

xj ∈ Z+, j ) 1, ..., n (21)

xs ∈ R+ (22)

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1831

In the above formulation, xs is the slack in the costconstraint, which takes nonnegative real values. R is apositive constant which has to be chosen such that theprimary objective (maximizing reliability) still attainsits earlier optimal value. In such a case, negativecontribution of the second term (Rxs) in the objectivefunction will ensure that the solution is also cost optimalin the lexicographic sense. Among all solutions whichyield maximum reliability, the one which has thehighest xs will be chosen. Thus, if the constant R isappropriately chosen, the solution to problem III willgive a sensor network, which will have the least costamong all of the networks which yield the maximumreliability. The proof of existence of such an R for themaximum-reliability model is presented next.

The basic idea behind the selection of the constant Ris as follows:

Consider the original maximum-reliability problem(problem I). Let the solutions which yield the maximumreliability be elements of a set S. Let all of the solutionswhich yield the optimal value of the modified problem(problem III) be elements of a set S′. The idea is tochoose R such that (i) S′ ⊆ S, that is, no extra optimalsolution is introduced as a result of the modification,and (ii) only the least cost elements of S are elementsof S′.

Note that once condition i is ensured, negativecontribution of the term Rxs in the objective functionensures condition ii. To ensure condition i, it has to beensured that, for any two feasible solutions y and z ofproblem I, if the objective function value with y isgreater (lesser) than the objective function value withz, then the same relationship should hold between theirobjective function values for the modified problem. Thatis,

where ln(Uy) and ln(Uz) are the objective function valuesfor problem I for solutions y and z, respectively, and xsyand xsz are the slacks in cost constraint for y and z,respectively. From eq 24 we get

Now, consider the two cases:(i) xsy - xsz < 0. Then,

where the right-hand side is a negative quantity (fromeq 23). This condition is trivially satisfied because weare considering R to be a positive constant.

(ii) xsy - xsz > 0. Then,

This inequality should hold for all possible pairs (y, z),satisfying eq 23. Each such inequality will give an upperbound on R. To get an R which will satisfy all such

inequalities, we can choose R such that it satisfies thefollowing:

An upper bound on xsy - xsz is C*, the available resource:

To get a lower bound on ln(Uy) - ln(Uz), consider ln-(Uy):

It is reasonable to assume that the constants ln(fi) andln(sj) are rational numbers and, hence, can be expressedas

where pi, qi, hj, and ej are all integers. The aboveequation for ln(Uy) can then be rewritten as

Let

Then, a ln(Uy) is an integer. Similarly, a ln(Uz) will alsobe an integer. Therefore,

However,

From eqs 35 and 36, we get a lower bound on ln(Uy) -ln(Uz) as

From eqs 28, 29, and 37, an R which satisfies thefollowing relation will satisfy inequality (27):

Also, because we considered only positive R, the rangefor R then is

Any R which lies in this range will ensure that thesolution of the modified maximum-reliability model isoptimal in the lexicographic sense. An important pointto be noted is that the range for R as given here is aconservative range, and there may be cases where evenif R is chosen out of the range it may still yield thedesired result.

The key idea behind modifying the maximum-reli-ability formulation was that reliability was the primary

if ln(Uy) > ln(Uz) (23)

then ln(Uy) - Rxsy > ln(Uz) - Rxsz (24)

R(xsy - xsz) < [ln(Uy) - ln(Uz)] (25)

R >ln(Uz) - ln(Uy)

xsz - xsy(26)

R <ln(Uy) - ln(Uz)

xsy - xsz(27)

R <min [ln(Uy) - ln(Uz)]

max (xsy - xsz)(28)

max (xsy - xsz) e C* (29)

ln(Uy) ) ln(fi) + ∑j)1

n

dijxj ln(sj), for some fault i (30)

ln(fi) ) pi/qi (31)

ln(sj) ) hj/ej (32)

ln(Uy) )pi

qi

+ ∑j)1

n

dijxj

hj

ej

, for some fault i (33)

a ) LCM (q1, q2, ..., qm, e1, e2, ..., en) (34)

min {a ln(Uy) - a ln(Uz)} g 1 (35)

min {a ln(Uy) - a ln(Uz)} ) a min {ln(Uy) - ln(Uz)}(36)

min {ln(Uy) - ln(Uz)} g 1/a (37)

R < 1/(aC*) (38)

0 < R < 1/(aC*) (39)

1832 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

objective and cost was the secondary objective. For thecase when cost is the primary objective and reliabilityis the secondary objective, the minimum-cost model canalso be similarly modified by the addition of an R termin the minimum-cost objective function. A valid rangefor R can also be similarly derived.

The idea of converting two (or more) sequential,lexicographic optimization problems into a single modi-fied optimization problem as presented here is quitegeneral and may be applied to a variety of problems.The statement and proof of this for a general nonlinearproblem is presented in the appendix. The applicationof the modified maximum-reliability model will bepresented by applying it to the TE case study in part 2of this series. To illustrate the idea behind the approach,its application to the maximum-precision model pre-sented by Bagajewicz11 is presented next.

6.1. Application of the r Formulation to a Maxi-mum-Precision Model. The maximum-precision modelas presented by Bagajewicz11 is

subject towhere Mp is the set of unmeasured parameters

whose estimate is desired, σi is the standard deviationof the estimate of parameter i, σi* is the threshold valueof the standard deviation of the estimate of parameteri, ai is the weight given to the ith estimate, M1 is theset of variables which can be measured, cj is the(nonnegative) cost of measuring variable j, and x is thevector of decision variables such that xj ) 1 if variablej is measured.

The corresponding minimum-cost model is11

subject to

solution of the maximum precision model will give asolution which gives the maximum precision but maynot give the least cost. To obtain the least cost model,one would be required to solve the minimum-cost modelwith the weighted sum of precisions fixed at the valueobtained by the maximum-precision model. This pointis illustrated by case 6 in Table 3 of Bagajewicz’s11

paper, where the optimal solution of the maximum-precision model gives the objective function value as

0.004982 with a total cost of 2950. Case 5 of Table 2, onthe other hand, has as a solution of the minimum-costmodel, a different sensor network, which gives the samestandard deviation (0.004982) but at a lesser cost of2900.

To obtain this minimum-cost model by solving a singleoptimization problem, the maximum-precision modelcan be modified as in the previous section:

such that

where, as before, R is a positive constant and xs is theslack in the cost constraint.

For this case also, an R can be chosen which willensure that, among all solutions which yield the maxi-mum precision, the least cost solution is obtained. Toobtain a range for R, it is sufficient to note that thevariance of the estimated parameter depends on thevariances of the measurements. Hence, the accuracy inthe parameter’s variance calculation cannot be betterthan the accuracy in the variances of the measurements.In Bagajewicz’s11 paper, standard deviations of themeasurements have been reported to the third decimal-place. So, it is reasonable to assume that the accuracyof the variances of the measurements will also be up tothe third decimal place. The objective function of themaximum-precision model is the weighted sum of thestandard deviations of the estimated parameters. Hence,the accuracy of the objective function value will also beup to the third decimal place. Now, when the sameprocedure as that given in the previous section forfinding a range for R is followed, it is easily seen thatthe following range for R is obtained for the modifiedmaximum-precision model:

Any R which lies in the above range will yield a solutionwhich is optimal in the lexicographic sense. Once again,it is important to note that the range for R as givenabove is a conservative one, and there may be caseswhen an R which lies outside this range may also yieldthe desired results. For case 5, Table 6 of Bagajewicz’spaper, the total available resource is C* ) 3000. So, wemay choose any positive R < 10-3/3000 ) 3.333 × 10-7.We choose R ) 1 × 10-8. For this case, solving themaximum-precision model, with and without R, gives

min ∑i∈Mp

aiσi2(x) (40)

∑j∈M1

cjxj e C* (41)

σi(x) e σi* ∀ i ∈ Mp (42)

xj ) {0, 1} ∀ j ∈ M1 (43)

min ∑j∈M1

cjxj (44)

∑i∈Mp

aiσi2 e ∑

i∈Mp

ai(σi*)2 (45)

σi(x) e σi* ∀ i ∈ Mp (46)

xj ) {0, 1} ∀ j ∈ M1 (47)

min ∑i∈Mp

aiσi2(x) - Rxs (48)

∑j∈M1

cjxj + xs ) C* (49)

σi(x) e σi* ∀ i ∈ Mp (50)

xj ) {0, 1} ∀ j ∈ M1 (51)

xs g 0 (52)

0 < R < 10-3/C* (53)

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1833

the results shown in Table 1. It can be seen that thesolution of the R formulation gives the same precisionbut at a lower cost. (The results do not exactly matchwith those given in Bagajewicz.11 This may be due tonumerical errors.)

In the next section, heuristics for solving the formu-lated problems are presented.

7. Heuristics for Solving the FormulatedProblems

The problems presented in this paper have beenformulated in an ILP format. These can be solved bystandard solvers for moderate size problems. Also, theconstraint reduction techniques presented in this paperhelp in reducing the size of the problem. However, forcases where the size of the problem is large even afterconstraint reduction, it may not be computationallyfeasible to solve the problem exactly. Also, because ofthe availability of only approximate values of the dataused (fault and sensor probabilities), one may only beinterested in a “good enough” solution rather than anexact solution. For these reasons, it is essential to haveheuristics which give a quick and reasonably ap-proximate solution to the posed problems. A greedysearch heuristic has been developed for solving the one-step reliability maximization problem (problem III). Thisis discussed next.

7.1. Solution Strategy for the Reliability Maxi-mization Problem. The greedy search heuristic forsolving problem III is given below. At each step the bestavailable sensor is chosen. The flowchart of the algo-rithm is given in Figure 5. Some of the terms used inthe algorithm are defined below.

Unobservability Value. This term was defined ineq 1.

Redundancy Value. This value provides an ap-proximate measure of the total unobservability coveredby a sensor. It is defined as

With these simple definitions, the steps in the algorithmare discussed below.

1. Initialization step. The following variables areinitialized: current solution, CS ) φ; optimal solution,OS ) φ; total available cost ) C* (problem specification);cost currently utilized, C ) 0; optimal cost utilized, OC) 0; currently available cost ) C* - C.

2. As a first step, the fault with the highest unob-servability value is selected. Label this fault as Fc(critical fault). The unobservability value being thehighest means that the chances of Fc occurring andremaining undetected are maximum. In the case of morethan one fault having the highest unobservability, anyone among these faults may be selected.

3. The aim now is to lower the unobservability valueof fault Fc. Therefore, a sensor which observes Fc is tobe selected. However, there may be more than one

sensor observing Fc. Among these sensors, only thosesensors can be chosen whose cost is less than thecurrently available cost. Let Q be the set of sensorswhich are connected to fault Fc, such that cost of eachmember of Q is less than (or equal to) the currentlyavailable cost. From Q, a sensor with the minimumfailure probability is chosen because this causes amaximum decrease in the unobservability value of thefault under consideration. If more than one element ofQ satisfies this criteria (that is more than one elementof Q has the lowest failure probability), then from those,the one with the highest redundancy value is chosen.In case more than one sensor still satisfies this criteria,then among these, the one with the lowest cost ischosen. If still there is more than one candidate, thenamong those any sensor may be chosen. A crucial pointis that no restriction is placed on the number of sensorsthat can be placed to observe the same node (variable).Hence, it is possible that at some node more than onesensor is placed (xj > 1), while some other node is notselected as a sensor node at all. Let the selected sensorbe the jth node. Then carry out the following updates:current solution, CS ) CS ∪ xj (note that xj may alreadybe present in CS; in that case, increase the count of xjin CS by 1); cost utilized, C ) C + cj; currently availablecost ) C* - C. After this sensor selection, recalculatethe unobservabilities of all faults and the systemunobservability (maximum unobservability). If thissystem unobservability is less than the previous value,then update the optimal solution to be the currentsolution: optimal solution, OS ) CS; optimal cost, OC) C. If the system unobservability does not decrease,the optimal solution and optimal cost are not updated.

4. This procedure (from step 2 onward) is continueduntil the set Q turns out to be empty. This indicatesthat no more sensors can be selected to decrease the

Table 1. Results for the Maximum-Precision Model

formulation sensors selected

standard deviationof the estimated

parametercost

utilized

without R [F1, y22, F3, y13, y23, P] 0.005 2950with R [F1, F2, y22, y13, y23, P] 0.005 2900

Rj ) ∑i)1

m

dijUi (54)

Figure 5. Algorithm for solution to problem III.

1834 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

system unobservability. The set of sensors in OS is theselected sensor location.

Steps 2 and 3 ensure minimization of the systemunobservability. The procedure of updating the optimalsolution only if there is a decrease in the systemunobservability ensures that the solution is optimal inthe lexicographic sense.

The above algorithm is for the one-step lexicographicoptimization problem (problem III). The solution ofproblem III is an optimal solution to problem I also.Hence, no separate heuristics are needed for solvingproblem I. For the minimum cost problem (problem II),heuristics similar to the ones given above may bedeveloped.

8. Additional Constraints for Sensor Location

In the previous sections, a reliability maximizationproblem with cost constraints and a cost minimizationproblem with reliability constraints were presented. Inthis section various other types of problem specificconstraints that may be incorporated while performingsensor location are briefly discussed.

(i) Different availability of different types of sensors:Constraints such as the availability of only four tem-perature sensors, five pressure sensors, etc., may beincorporated in place of/along with the cost constraint.For example, consider a specific case where, besides thecost constraint, the one-step sensor location optimizationis restricted by the availability of only up to fourtemperature sensors. The formulation for this case thenis

subject to

where

In the above formulation, constraint (57) ensures thatthe total number of sensors selected to measure thetemperature variables in the process is not more than4. These specific constraints would be especially relevantwhile performing a sensor network reallocation studyon an existing process.

(ii) Constraints requiring that a particular (critical)fault be observed by more than a fixed number ofsensors, or two faults should be resolvable by more thana given number of sensors, may also be considered whileperforming sensor location. These constraints would beuseful when the underlying causal information (basedon which the fault sensor matrix is constructed) hasuncertainties associated with it. Hence, one would likesome redundancy (more than one sensor) while selectingsensors to resolve various faults. This aspect is high-lighted in the case study in part 2 of the series, where

SDG with gains is used to perform fault modeling forthe TE case study. A related constraint might be toensure that the unobservability of a critical fault is lessthan some threshold value. Another important set ofconstraints may be to ensure that all pairs of faults canbe resolved by the selected sensor location. This par-ticular case may be formulated as

subject to

where

In the above formulation, constraint (62) ensures thatthe selected sensors are able to resolve all resolvablepairs of faults. An important point to note here is thatconstraint (62) is not written for pairs of faults whichare topologically unresolvable (the Bkl set is empty).

9. Lexicographic Optimization with OtherCriteria

In this paper, sensor location with reliability maxi-mization and cost minimization as optimization criteriahas been presented. A lexicographic, one-step optimiza-tion procedure to combine these two objectives was alsopresented. Cost minimization was used as an objectivetoselect the minimum cost network, from the varioussolutions which resulted in maximum system reliability.When this procedure is extended, other criteria may alsobe used, in a lexicographic manner, to select an optimalsolution if there are still multiple sensor networks whichminimize the total cost. One such criterion may be toobtain a “distributed sensor network”. A sensor networkmay be said to be more “distributed” than some othernetwork if the total number of variables being measuredis more in the former case. A more distributed networkwould be preferable for cases where the underlyingcause-effect modeling approach has uncertainties as-sociated with it. Use of a more distributed networkwould impart robustness to the selected sensors evenin the presence of uncertainties in the cause-effectmodeling approach.

Another criterion to select from among the multiplesolutions to the one-step optimization problem may bebased on the use of sensitivity of the selected sensorlocation with respect to data used in the problem. Thedata on fault occurrence and sensor failure probabilitiesmay not be precisely available. Different sensor loca-tions, while being optimal with respect to one set of data,may result in different system unobservabilities when

minxj

[max∀ i

{ln(Ui)} - Rxs] (55)

∑j)1

n

cjxj + xs ) C* (56)

∑t∈Temp.Variables

xt e 4 (57)

xj ∈ Z+, j ) 1, ..., n (58)

xs ∈ R+ (59)

minxj

[max∀ i

{ln(Ui)} - Rxs] (60)

∑j)1

n

cjxj + xs ) C* (61)

∑xkl∈Bkl

xkl g 1, k ) 1, ..., m-1; l ) k + 1, ..., m (62)

Bkl ) Ak ∪ Al - Ak ∩ Al )set of all sensors which can be used

to resolve between fault k and fault l (63)

xj ∈ Z+, j ) 1, ..., n (64)

xs ∈ R+ (65)

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1835

a different data set is used. This performance in thepresence of a different data set may then be used toscreen from among the various multiple solutionsobtained earlier. This idea is illustrated in the casestudy in part 2 of the series.

Choosing a more distributed sensor network andusing sensitivity analysis are just a few possible criteriawhich can be integrated in the lexicographic optimiza-tion framework. Depending on the specific require-ments, the designer may come up with other criteriawhich may be used in place of/along with the criteriapresented in this paper. We are currently investigatingsome of these issues.

10. Optimization Formulations forObservability and Resolution of Faults

Raghuraj et al.1 (for DG-based modeling) and Bhus-han and Rengaswamy2 (for SDG-based modeling) dis-cussed sensor location design methodologies based oncriteria such as fault observability and resolution. It wasmentioned in those papers that the formulated problemswere set cover problems. In this section, those set coverproblem formulations are presented, and different cri-teria for solving those problems are also discussed.

Problem IV: Set Cover Problem with a Mini-mum Number of Selected Sensors.

such that

where xj’s are the binary decision variables. xj ) 1indicates that a sensor has to be placed at node j, anda value of 0 indicates that the corresponding variableneed not be measured. dij is the (i, j)th element of thebipartite matrix D; it is 1 if fault i affects node j and is0 otherwise. This bipartite matrix D has faults as rowsand nodes as columns and is generated from the processDG/SDG during the fault modeling step as explainedearlier in this paper.

The constraint (67) ensures that all of the faults arecovered. Minimization of the objective function thenensures that a minimum number of sensors are selectedto cover all faults.

The observability problem as presented above is a setcover problem. Also, any other resolution problem (evenfor multiple-fault cases) is converted to an appropriateobservability problem. So, the sensor location problemfinally involves solving an appropriate set cover problemof type problem IV. Raghuraj et al.1 gave a greedysearch-based heuristic to solve such set cover problems.

The formulation presented in problem IV involvesminimizing the total number of sensors. It gives equalweightage to all sensors. Instead of this objectivefunction, another objective function which minimizes thetotal cost of the sensors can also be chosen as shown inproblem V:

Problem V: Set Cover Problem with MinimumCost.

such that

where cj is the cost of sensor j.A greedy search heuristic to solve the above problem

is given by Bhushan and Rengaswamy.16 They have alsopresented a methodology for reducing the number ofconstraints in the above problems by systematicallyremoving the redundant constraints. An importantdifference in the formulations presented in this sectionas compared to the reliability maximization basedformulations presented earlier in this paper is that thedecision variables (xj) are allowed to take only binaryvalues in the set cover formulations, and also no faultoccurrence or sensor failure probability data are usedin these formulations.

Some results for the formulations presented in thissection are discussed in part 2 of this series.

11. Use of Fault Measurement in SensorLocation

In the various formulations presented so far, faultmeasurements were not considered. So, even if thereare faults (such as flow rates, temperatures, etc.) whichcan themselves be measured, these were not consideredin the sensor location formulations. In general, a bettersensor location may be obtained by use of these faultmeasurements. In this section, the one-step optimizationproblem (problem III) is reformulated to include faultmeasurements:

Problem III with Fault Measurements.

Subject to

where

where, as before, there are n original measurablevariables and the index j runs over these measurablevariables. The measurable faults are represented by theindex k, the variable yk represents the number of sensorsused to measure fault k, ck is the cost of measuring faultk, and p faults are measurable faults. The fault-sensorbipartite matrix D is appropriately modified to take the

min ∑j)1

n

xj (66)

∑j)1

n

dijxj g 1, i ) 1, ..., m (67)

xj ) {0, 1} j ) 1, ..., n (68)

min ∑j)1

n

cjxj (69)

∑j)1

n

dijxj g 1, i ) 1, ..., m (70)

xj ∈ {0, 1} j ) 1, ..., n (71)

minxj,yk

[max∀ i

{ln(Ui)} - Rxs] (72)

∑j)1

n

cjxj + ∑k)1

p

ckyk + xs ) C* (73)

xj ∈ Z+, j ) 1, ..., n (74)

yk ∈ Z+, k ) 1, ..., p (75)

xs ∈ R+ (76)

1836 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

fault measurements into consideration. This matrix isgenerated from the process SDG which represents thecausal information among process variables. In the SDGframework, this corresponds to the addition of newnodes corresponding to fault measurements. For eachmeasurable fault, a new node is created, and it has anarc from the corresponding fault. The sign of the arcrepresents the relationship between the fault and thecorresponding measurement. For example, if the faultis “increase in flow rate”, then a new variable corre-sponding to this flow rate measurement is added witha positive arc from the fault node to this new node. Onthe other hand, if the fault is “decrease in flow rate”,then a negative arc to this new node is added from thefault. For cases where both positive and negativedeviations in the same variable are represented asseparate faults, the new node will have an arc each fromboth of these faults, with the signs of the arcs being +and -, respectively. As with other measurable nodes,information about the cost and failure probability isrequired for these new nodes corresponding to faultmeasurements. The fault-sensor bipartite matrix is thengenerated from this modified SDG. The ith row of thismatrix corresponds to Ai, the set of sensors affected byfault i. This matrix is then used for calculation of ln Uivalues in the sensor location formulation. Also, asbefore, this matrix is then used to generate the ap-propriate bipartite matrix for the scenario under con-sideration (single-fault assumption, double-fault as-sumption, etc.).

Similar to the above formulation, other formulations(problems I, II, IV, and V) can also be modified toincorporate fault measurements. Some comparisons ofsensor locations with and without consideration of faultmeasurements, for problems IV and V, will be presentedin part 2 of this series.

12. Importance of the Proposed Formulations

The formulations presented in this paper are impor-tant mainly because of their ability to generate optimalsensor locations while handling various constraints offault diagnosis and variable measurements.

(i) The cost constraint was explicitly mentioned in theproposed formulations. Other constraints which may beincorporated are the following:

(1) Different availability of different types of sensors.Thus, constraints such as the availability of only fourtemperature sensors, five pressure sensors, etc., maybe incorporated in place of/along with the cost con-straint.

(2) Constraints requiring that a particular (critical)fault be observed by more than a fixed number ofsensors, or two faults should be resolvable by more thana given number of sensors, may also be considered whileperforming sensor location. A related constraint mightbe to ensure that the unobservability of a critical faultis less than some threshold value. Another importantset of constraints may be to ensure that all pairs offaults can be resolved by the selected sensor location.

(ii) The optimization framework gives flexibility to thedesigner to perform sensor location for various sce-narios.

(iii) The framework presented here may be modifiedto take into account different desirable characteristicsof the fault monitoring system, such as quick detectionof faults. This may be achieved, for example, by modify-

ing the sensor failure probabilities to penalize sensorswhich involve offline analysis or have low samplingrates.

(iv) The concept of system unobservability gives aquantitative measure to compare different sensor loca-tions.

(v) Depending on the process requirements, variousoptimization criteria may be incorporated in the sensorlocation framework.

13. Conclusions

An attempt has been made to come up with acomprehensive design strategy for the problem of find-ing optimum sensor locations. Depending on the re-quirements of the fault monitoring system, appropriateoptimization problems have been posed. Models formaximum-reliability and minimum-cost sensor net-works have been posed. A one-step optimization formu-lation which performs lexicographic optimization withreliability as the primary objective and cost as thesecondary objective has also been presented. Methodol-ogy to use the proposed formulations to obtain the bestdesign for a given process, independent of the single/multiple-fault assumption, has also been discussed.Ways of reducing the size of the problem being solved,by reducing the number of constraints, have beendiscussed. Working heuristics have been provided tosolve the formulated reliability maximization problems.These heuristics provide quick, near-optimal solutionseven for large problems. The use of other optimizationcriteria, such as distributed networks, objectives fromprocess controllability requirements, and formulationsfor performing sensor network retrofit studies on exist-ing plants based on fault diagnostic criteria, is currentlybeing investigated. The aim of this research is toseamlessly incorporate various criteria for sensor loca-tion, so that the resulting sensor location is optimal ina much broader sense and not just from a fault diagnosisperspective.

In the second part of this series of papers, theapplication of the proposed formulations to a largeflowsheet, the TE flowsheet, is presented. Various issuesdiscussed in this paper are demonstrated in this ap-plication, and results for different cases are presentedand compared.

Appendix: Lexicographic Optimization for theGeneral Case

Consider the following optimization problem:Problem P1.

(F takes integral values) such that

where the cost coefficients cj are nonnegative constantsand C* is the total available cost.

The problem P1 may have several optimal solutions,each with possibly different costs. To obtain an optimalsolution in the lexicographic sense, one needs to solve

max F(x1,x2,...,xn) (77)

∑j)1

n

cjxj e C* (78)

xj ∈ Z+, j ) 1, ..., n (79)

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1837

two problems in sequence: the first being problem P1and the second involving minimization of the costsubject to the primary objective F being at its optimalvalue. The second problem P2 is posed below.

Problem P2.

such that

where F* is the optimal value of problem P1.Consider another problem which combines the objec-

tive functions of problems P1 and P2 in a weightedsense.

Problem P3.

such that

where R is a nonnegative constant.The following theorem states sufficient conditions on

R for which solving problems P1 and P2 in sequence isequivalent to solving problem P3.

Theorem 1. Solving problem P3 is equivalent toperforming lexicographic optimization on problem P1(solving problems P1 and P2 in sequence), if

Proof of Theorem 1. The proof is similar to the onepresented for finding a valid range for R for thereliability maximization problem in the paper. For thesake of completeness, it is presented here.

Let the solutions which yield optimal values ofproblem P1 be elements of a set S. Let all of thesolutions which yield the optimal value of problem P3be elements of a set S′. The idea for choosing a validrange for R is that (i) S′ ⊆ S, that is, no extra optimalsolutions are introduced as a result of the modificationof problem P1 to problem P3, and (ii) only the least costelements of S are elements of S′.

Note that once condition i is ensured, positive contri-bution of the term Rxs in the objective function ofproblem P3 ensures condition ii. To ensure condition i,it has to be ensured that, for any two feasible solutionsy and z of Problem P1, if the objective function valuewith y is greater than the objective function value withz, then the same relationship should hold between theirobjective function values for the modified problem:

where F(y) and F(z) are the objective function valuesfor problem P1 for solutions y and z, respectively, andxsy and xsz are the slacks in the cost constraint for y andz, respectively. From eq 89 we get

Now, consider the two cases:(i) xsz - xsy < 0. Then,

where the right-hand side is a negative quantity (fromeq 88). This condition is satisfied if

(ii) xsz - xsy > 0. Then,

R has to be so chosen such that this inequality holdsfor all possible pairs (y, z) which satisfy eq 88. Each suchinequality will give an upper bound on R. To get an Rwhich will satisfy all such inequalities, we can chooseR such that it satisfies the following:

Because F is an integer function (it takes only integralvalues), a lower bound on F(y) - F(z) is 1. An upperbound on xsz - xsy is C*, the total available cost.Therefore, an R which satisfies

will satisfy inequality (94). From inequalities (92) and(95), we get the following range for R:

Hence, an R which satisfies the above inequality willensure that solving problem P3 is equivalent to solvingproblems P1 and P2 in sequence. An important observa-tion is that this is just a sufficient condition on R andnot a necessary one. Depending on the given problem,there may be cases where an R lying outside the aboverange may yield the desired result. Also, the R formula-tion is applicable to even those problems where F isnot an integer function. The idea is to use knowledgeabout the problem to rewrite the objective function interms of a new objective function F′ where F′ takesintegral values. Two such applications (maximum-reliability and maximum-precision) were presented inthis paper.

Nomenclature

Ai ) set of nodes affected by fault iAij ) Ai ∪ AjBij ) Ai ∪ Aj - Ai ∩ AjC* ) available resource for sensor locationcj ) cost of the sensor used to measure node jck ) cost of the sensor used to measure fault kD ) bipartite matrix between fault nodes and sensor nodes

min ∑j)1

n

cjxj (80)

F(x1,x2,...,xn) g F* (81)

xj ∈ Z+, j ) 1, ..., n (82)

max F(x1,x2,...,xn) + Rxs (83)

∑j)1

n

cjxj + xs ) C* (84)

xj ∈ Z+, j ) 1, ..., n (85)

xs ∈ R+ (86)

0 < R < 1/C* (87)

if F(y) > F(z) (88)

then F(y) + Rxsy > F(z) + Rxsz (89)

R(xsz - xsy) < F(y) - F(z) (90)

R >F(z) - F(y)

xsy - xsz(91)

R > 0 (92)

R <F(y) - F(z)

xsz - xsy(93)

R <min (F(y) - F(z))

max (xsz - xsy)(94)

R < 1/C* (95)

0 < R < 1/C* (96)

1838 Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

dij ) (i, j)th entry of D (is 1 if fault i affects sensor j and is0 otherwise)

fi ) occurrence probability of fault im ) number of faults in the optimization formulationsn ) number of measurable nodesp ) number of measurable faultsR+ ) set of nonnegative real numbersRj ) redundancy value of the sensor at node jsj ) failure probability of sensor jxj ) decision variable for the optimization problems; value

indicates the number of sensors to be placed at node jxs ) slack variable for the cost constraintyk ) number of sensors measuring fault kUi ) unobservability value of fault iU* ) threshold value for system unobservabilityZ+ ) set of nonnegative integers

Greek Symbols

R ) constant used for one-step optimization

Literature Cited

(1) Raghuraj, R.; Bhushan, M.; Rengaswamy, R. AIChE J.1999, 45 (2), 310.

(2) Bhushan, M.; Rengaswamy, R. Ind. Eng. Chem. Res. 2000,39 (4), 999.

(3) Lambert, H. E. Chem. Eng. Prog. 1977, Aug, 81.

(4) Ali, Y.; Narasimhan, S. AIChE J. 1993, 39 (5), 820.(5) Ali, Y.; Narasimhan, S. AIChE J. 1995, 41 (10), 820.(6) Ali, Y.; Narasimhan, S. AIChE J. 1996, 42 (9), 2563.(7) Sen, S.; Narasimhan, S.; Deb, K. Comput. Chem. Eng. 1998,

22 (3), 385.(8) Bagajewicz, M. J. AIChE J. 1997, 43 (9), 2300.(9) Bagajewicz, M. J.; Sanchez, M. C. AIChE J. 1999, 45 (9).(10) Bagajewicz, M.; Sanchez, M. Comput. Chem. Eng. 2000,

23, 1757.(11) Bagajewicz, M. J.; Sanchez, M. C. AIChE J. 1999, 45 (3),

661.(12) Alheritiere, C.; Thornhill, N. F.; Fraser, S.; Knight, M. J.

Comput. Chem. Eng. 1998, 22 (Suppl.), S1031.(13) Bagajewicz, M.; Sanchez, M. Comput. Chem. Eng. 2000,

24, 1945.(14) Mavrovouniotis, M. L.; Stephanopoulos, G. Comput. Chem.

Eng. 1988, 12 (9/10), 867.(15) Help Manual, Hyper Lindo/PC; Lindo Systems Inc.: Chi-

cago, IL, 1997.(16) Bhushan, M.; Rengaswamy, R. A framework for sensor

network design for efficient and reliable fault diagnosis. 4th IFACworkshop on On-line Fault Detection and Supervision in theChemical Process Industries, Korea, June 7-8, 2001; p 33.

Received for review May 14, 2001Revised manuscript received October 30, 2001

Accepted November 29, 2001

IE0104363

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1839