A duality based framework for integrating reliability and precision
for sensor network design
P.R. Kotecha a, Mani Bhushan a, R.D. Gudi
a
a,*
, M.K. Keshari
b
Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
b
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
Abstract
In sensor network design literature, requirements such as maximization of the network reliability [Y. Ali, S. Narasimhan, Sensor network design for maximizing reliability of linear processes, AIChE J. 39 (1993) 820–828; Y. Ali, S. Narasimhan, Redundant sensor network design for linear processes, AIChE J. 41 (1995) 2237–2249] and minimization of cost subject to precision constraints [M.
Bagajewicz, Design and retrofit of sensor networks in process plants, AIChE J. 43 (1997) 2300–2306; M. Bagajewicz, E. Cabrera,
New MILP formulation for instrumentation network design and upgrade, AIChE J. 48 (2002) 2271–2282] have been proposed as a criteria for optimally locating sensors. In this article, we show that the problems of maximizing reliability and maximizing precision (or
minimizing variance) for linear processes are dual of each other. To achieve this duality, we propose transformations which can be used
to convert sensor failure probabilities into equivalent sensor variances and vice versa. Thus, the duality enables working in a single framework with specified criteria on reliability as well as precision. As an application of this duality, we propose two formulations for the
sensor network design problem viz., maximization of the network reliability subject to precision constraints and minimization of the network variance subject to reliability constraints. We also show the utility of these formulations to determine the pareto-front for the combinatorial sensor network design problem. Hydrodealkylation and steam-metering case studies are used to illustrate the proposed ideas.
Keywords: Duality; Reliability; Precision; Optimization; Sensor network design
1. Introduction
The operation of a typical chemical process is characterized by several thousand variables. All these process variables however cannot be measured due to cost and other
constraints. Generally, only a subset of the overall process
variables are measured, and the unmeasured variables can
typically be estimated using their relationships with the
measured variables via the use of a process model. For
choosing the optimal placement of sensors, some of the criteria considered in literature are observability [5], cost [6,7]
and reliability [1]. A greedy-search based graph theoretic
approach was used in [1] to obtain the most reliable sensor
network. The initial reliability criterion was proposed for
non-redundant linear systems and was later extended to
redundant linear [2] and non-redundant bilinear systems
[8]. Bagajewicz [3] used a tree type enumeration procedure
to design a minimal cost network subject to constraints on
precision, availability, resilience and error detectability.
Bagajewicz and Sanchez [9] have shown that the problem
of minimizing variance subject to cost constraints can be
converted into an equivalent problem of minimizing the
sensor network cost subject to precision constraints. Later
they have extended [10] this idea to reliability, i.e. maximization of network reliability subject to cost constraints was
converted into an equivalent problem of minimizing cost
subject to reliability constraints. This enabled them to use
their earlier [3] tree based enumerative approach for solv-
190
Nomenclature
akj
coefficient of the relation between the jth measurement and the kth variable
A
reduced incidence matrix (independent mass
balances) for flow process
L
variance of the fake sensor
n,m
number of variables, number of linearly independent mass balances written across each of
the m units
N,M
set of all and measured variables
P
symmetric matrix of dimension m · m as defined
in the formulation P3
R,R*
network reliability, threshold on network reliability
Ri,Re,i reliability, equivalent reliability of the ith sensor
_ _
R i ; R e;i ; Ri ; Re;i reliability, equivalent reliability, threshold reliability, equivalent threshold reliability
of the ith variable
sj
failure probability of the jth sensor
s
equal failure probability of all sensors
ing reliability problems. Bagajewicz and Cabrera [4] posed
the sensor network problem as an explicit optimization
problem with minimization of cost as the objective function
and requirements of precision, error detectability, resilience
and availability as constraints. Sen et al. [11] integrated
genetic algorithms (GA) with graph–theoretic concepts to
solve the problem of optimal design of sensor network
for linear processes. By the use of GA, they could solve various problems such as optimizing cost, estimation accuracy
and network reliability. Recently, formulations [12] which
take into account the cost advantage obtained in terms of
the ability of the network to resolve between faults, its ability to yield desired precision and adequate control, have
also been reported.
From the above literature survey, it can be seen that in
general, the criteria of reliability and precision have been
handled by different techniques. It must be mentioned that
the problem of maximizing reliability has no reported
explicit optimization formulation and has been solved
using graph theoretic or tree based enumeration
approaches. Although the graph theory based greedysearch algorithms [1] are robust, they do not guarantee global optimality. Moreover, it is not clear how multiple
objectives or sensor network requirements (e.g. maximizing
the reliability with the least possible cost or maximizing the
reliability subject to precision constraints) can be efficiently
accommodated in such a framework. While the tree based
enumeration approach [3] guarantees global optimality,
due to its enumerative nature, it is not suitable for large
flowsheets. On the other hand, for the precision case, Bagajewicz and Cabrera [4] have reported an explicit optimization formulation for the minimization of cost subject to
constraints on variances for some of the variables. These
V
S
set of variables with threshold reliability or variance constraints
sensor variance matrix defined as Sði; jÞ ¼
8
< 0 i 6¼ j 8i; j 2 N
L if i ¼ j and lj;2 ¼ 1
: 2
rj if i ¼ j and lj;2 ¼ 0
_
S
reconciled variance matrix
r2 ; r2
network variance, threshold on equivalent nete
work variance
2
r
equal sensor variance
r2i ; r2e;i variance, equivalent variance of the ith sensor
_2 _2
2
r i ; r e;i ; r2
i ; re;i variance, equivalent variance, threshold
variance, equivalent threshold variance of the
ith variable
lj,1,lj,2 binary variables indicating whether the jth variable is measured by a real (lj,1 = 1, lj,2 = 0) or
fake sensor (lj,1 = 0, lj,2 = 1)
r2scale
scaling factor as used in Eq. (34)
formulations result in mixed integer non-linear programming (MINLP) problems. As global optimality for MINLPs is not guaranteed in general, these problems are
further converted into mixed integer linear programming
(MILP) [4] problems. Such simplifying transformations
increase the size of the problem thereby increasing the computational requirements. Hence, although there has been
much work done to accommodate various requirements
for the sensor networks, there is still no single satisfactory
framework which can accommodate both reliability and
precision. This is essentially due to the lack of equivalence
between reliability and precision.
Our work contributes in establishing the duality between
reliability and precision related formulations, thereby
enabling the user to choose an appropriate formulation
on the basis of the merits of individual solution methods.
We show that a problem of reliability can be posed and
solved as an equivalent variance problem and vice versa.
The solution space of maximizing reliability is in principle
equivalent to the solution space of minimizing equivalent
variance. The variances used in this equivalence are not
the inherent variances of the sensors but are to be determined by an appropriate transformation of the sensor reliabilities. We restrict the analysis to the non-redundant [1]
sensor network design problems for steady-state, linear
processes. We demonstrate the proposed strategies of
equivalence on a hydrodealkylation (HDA) flowsheet and
also demonstrate the use of the duality principle to design
sensor networks that have constraints on reliability as well
as variances. We further show the use of duality theory in
designing sensor networks for a steam-metering network
by explicit optimization. It is thus shown that the proposed
duality can be used to (i) solve reliability and precision
191
problems efficiently, and (ii) evaluate tradeoffs and select
sensor network designs which result in highly reliable as
well as high precision networks. The remainder of the article is structured as follows: To begin with, we present a
brief definition of the terminologies used in the article. This
is followed by the proposed theory of duality between precision and reliability. Subsequently, representative formulations involving both reliability and precision are
proposed. We then show the use of the proposed duality
to transform these formulations into either of these (precision or reliability) frameworks. Finally, case studies on
HDA and steam-metering network are presented to demonstrate the duality concept and the utility of the proposed
formulations.
Remark 1. The relation between variance and precision
can be qualitatively expressed as
2. Preliminary definitions
Network variance: Similar to the approach taken by
Ali and Narasimhan [1] for the reliability case, we
define the network variance to be the maximum variance
amongst all variables. Hence, if r2 is the network variance,
then
We now briefly list the definitions of some of the terms
that have been used in this article [13,14].
Non-redundant sensor network design: A sensor network
that employs minimum number of sensors (number of variables – number of equations) is referred to as non-redundant sensor network design. The following discussion is
for the non-redundant case.
Observability: A sensor network is said to be observable
if all the process variables can be estimated, either based on
their direct measurement or their relationships with other
measured variables.
Reliability: Reliability of a variable is the probability of
estimating that variable for a given sensor network and the
failure probabilities of the sensors [1]. For a measured variable, the reliability is equal to the reliability of the sensor
while for an unmeasured variable, the reliability is equal to
the product of the reliabilities of the sensors used for estimating the unmeasured variable. Hence,
we can write the
_
following expression for reliability Ri for any variable i
as
Y
_
ai
Ri ¼
ð1 sj Þ j 8i 2 N
ð1Þ
j2M
where aij ¼ 1 if measurement on variable j is required to
estimate variable i and is 0 otherwise. For a measured variable i, aij ¼ 1 only for i = j and "i 5 j the value of aij ¼ 0.
The value of aij can be obtained from the reduced incidence
matrix A.
Variance: Variance is the uncertainty in the estimate of a
variable for a given sensor network and noise variances of
the sensors. For a measured variable, the variance is equal
to the variance of the sensor used for measuring the variable while for an unmeasured variable, the variance is
equal to the sum of the variances of the sensors used for
estimating the unmeasured variable. Similar to the reliability case, we can then write the following expression for variance for any variable i
X
_2
ri ¼
aij r2j 8 i 2 N
ð2Þ
j2M
Variance /
1
Precision
i.e., lower the variance of an estimate, the more precise is
the estimate and vice versa. Therefore, in this article, we
have used variance and precision interchangeably.
Network reliability: Ali and Narasimhan [1] have defined
the network reliability to be the minimum reliability
amongst all variables. In other words, if R is the network
reliability, then
_
R ¼ min Rj
8 j2N
_
r2 ¼ max r 2j
8 j2N
ð3Þ
ð4Þ
The above definitions are consistent with the idea of a
chain being as strong as its weakest link [1].
3. Theory of duality
In this section, we present transformations that convert
problems formulated for precision to those for reliability
and vice versa. We first present the theory for transforming
a problem defined in reliability space to precision space.
Subsequently, we also show that the proposed theory of
duality holds for the reverse problem, i.e., transformation
of precision to reliability problems.
3.1. Formulation of reliability problem as an equivalent
precision problem
First, we show that a sensor network design problem for
reliability maximization can be converted into an equivalent sensor network design problem for variance minimization with appropriately chosen sensor variances. In other
words, given the sensor failure probabilities, we want to
select the variables to be measured, so as to maximize network reliability. We pose this problem as one of choosing
sensors so as to minimize the network variance of an
appropriately transformed variance problem. The important issue here is an appropriate selection of these sensor
variances so as to ensure the equivalence of the two
approaches. Theorem 1 presents the necessary transformation to achieve this equivalence.
Theorem 1. Given a mass flow network (in terms of its
reduced incidence matrix A) along with the failure probabilities of the sensors (sj "j 2 N) which can be used to measure
variables in the network, solving the reliability maximization
problem is equivalent to solving an appropriate variance
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minimization problem, where the equivalent variances of the
sensors r2e;j are chosen as
r2e;j ¼ lnð1 sj Þ
ð5Þ
or
r2e;j ¼ lnðRj Þ
ð6Þ
Proof. We present the proof in two steps: In Step 1, we
show that the proposed equivalence holds within a chosen
sensor network M1. In Step 2, the equivalence is shown to
hold across two sensor networks M1 and M2.
Step 1: For a given observable sensor network M1,
consider a situation when the reliability of the ith variable
is the least amongst all the variables i.e.,
_
RM1
i
_
6 RM1
k
8k 2 N
ð7Þ
Thus for M1 the network reliability R is
R
M1
¼
_
min RM1
j
8j2N
¼
_
RiM1
ð8Þ
or in other words, variable i defines the network reliability
for M1. We now show that if variance minimization problem is posed using the variances as specified by the proposed transformation (Eq. (5)), then variable i also
determines the network variance for the sensor network
M1, i.e.,
ðr2e Þ
M1
_
¼ maxðr 2e;j Þ
M1
8j2N
_
¼ ðr 2e;i Þ
M1
ð9Þ
In other words
_
ðr 2e;i Þ
M1
_
P ðr 2e;k Þ
M1
8k 2 N
ð10Þ
Using Eq. (2) the above condition is converted into
X
X
aij r2e;j P
akj r2e;j
j2M1
We now ensure the equivalence between Eqs. (7) and (11)
as follows:
Since the reliability of a variable is a positive number,
Eq. (7) can be written as
_
ln RM1
P ln RM1
i
k
ð12Þ
By making use of Eqs. (1), (12) can be written in terms of
the failure probabilities of the selected sensors as,
"
#
"
#
Y
Y
aij
akj
ln
ð1 sj Þ P ln
ð1 sj Þ
ð13Þ
j2M1
j2M1
Using Eq. (5), we have
!
!
X
X
i 2
k 2
aj re;j P
aj re;j
j2M1
RM1 6 RM2
ð15Þ
Consider that the reliabilities of M1 and M2 are respectively given by the reliabilities of variables p and q
(p,q 2 N) as
_
_
_
_
RM1 ¼ min RM1
¼ RM1
j
p
ð16Þ
8j2N
RM2 ¼ min RM2
¼ RM2
j
q
ð17Þ
8j2N
Then to prove the equivalence it is enough to show that
using the suggested transform, the following also holds
ðr2e Þ
M1
P ðr2e Þ
M2
ð18Þ
where
ðr2e Þ
M1
ðr2e Þ
M2
_
M1
¼ ðr 2e;p Þ
_
M1
ð19Þ
_
M2
¼ ðr 2e;q Þ
_
M2
ð20Þ
¼ maxðr 2e;j Þ
8j2N
¼ maxðr 2e;j Þ
8j2N
are the network variances for networks M1 and M2
respectively.
The steps involved in showing this equivalence are
analogous to those presented in Step 1 and are omitted for
the sake of brevity.
Steps 1 and 2 together complete the proof of Theorem
1. h
ð11Þ
j2M1
_
variance space. In Step 2, we additionally show that the
ordering relationship holds across sensor networks, i.e., the
ordering in reliability space is preserved (as before, in an
inverse sense) in the variance space for different sensor
networks.
Step 2: For two different observable sensor networks M1
and M2, consider a situation when the reliability of M1 is
lower than reliability of M2
ð14Þ
j2M1
which is the desired relation (Eq. (11)).
This step ensures that for a given sensor network M1,
the ordering relation for the variables in reliability space is
preserved (but in an inverse sense as required) in the
Corollary 1. If all the sensor failure probabilities are equal
ðsj ¼ s 8j 2 N Þ, then the equivalent sensor variances can be
chosen as r2e;j ¼ b 8j 2 N where b is any positive number.
3.2. Formulation of precision problem as an equivalent
reliability problem
In this section, we show that a sensor network design
problem for variance minimization can be converted into
an equivalent sensor network design problem for reliability
maximization, with appropriately chosen sensor failure
probabilities.
Theorem 2. Given a mass flow network (in terms of its
reduced incidence matrix A) along with the variances
ðr2j 8j 2 N Þ of the sensors which can be used to measure
variables in the network, solving the variance minimization
problem is equivalent to solving an appropriate reliability
maximization problem, where the equivalent failure probabilities of the sensors se,j are chosen as
se;j ¼ 1 expðr2j Þ
ð21Þ
193
or
4.1. Maximization of network reliability
Re;j ¼ expðr2j Þ
ð22Þ
Proof. The proof of Theorem 2 is analogous to the proof
of Theorem 1 and is not presented here. h
Corollary
2. If all the sensor variances are equal
2
2
8j 2 N , then the equivalent sensor failure probarj ¼ r
bilities can be chosen as se,j = c "j 2 N where c is any number
between 0 and 1.
Remark 2. It is interesting to note that the transformation
suggested in Eq. (5) also makes sense for the limiting cases:
sj ! 0 ) r2e;j ! 0
i.e. a highly reliable sensor transforms to a high equivalent
precision sensor, and a sensor with low reliability is transformed to a sensor with low equivalent precision. Similar
observations also hold for Eqs. (21) and (22), since
r2j
! 1 ) Re;j ! 0
P1
ðor se;j ! 0Þ
ðor se;j ! 1Þ
ð24Þ
which can be interpreted as: when the sensor variance tends
to zero, i.e., the sensor has high precision, the equivalent
reliability tends to one signifying that the sensor is highly
reliable. If the sensor variance increases, the equivalent reliability starts decreasing.
Remark 3. It should be mentioned that while we have used
negative of natural log (or log1/e) and 1/exponential (e1)
transformations in Theorems 1 and 2 respectively, we could
have used loga and power of a, for any a 2 (0, 1). The proof
regarding use of power of a is given in Appendix 1. In this
proof, we prove a much stronger result which states that
any function f which transforms variance to equivalent
2
reliability has to be of the form f ðr2 Þ ¼ ar for any a such
that 0 < a < 1. Similarly it can also be proved that any
function g which transforms reliability to variance problem
has to be of the form g(R) = lnaR for any a such that
0 < a < 1. For the sake of brevity, this proof is not
provided in the article.
4. Applications of the duality theory for sensor network
design
In this section, we describe some of the applications of
the proposed duality. The first two applications have been
addressed in literature by different techniques. Here, we
show that any of the two techniques can be used to solve
these formulations. The formulations presented are only
representative and several other problems can be formulated using the duality proposed in this article.
Maxlj
R
s:t:
R 6 Rj
_
8j 2 N
lj 2 f0; 1g
ð25Þ
8j 2 N
where lj is 1 if the jth variable is measured and zero otherwise. Ali and Narasimhan [1] have addressed this problem
by using greedy-search algorithms in a graph theoretic
framework. This problem can be converted into an equivalent problem in the precision space as follows
ð23Þ
sj ! 1 ) r2e;j ! 1
r2j ! 0 ) Re;j ! 1
This problem involves choosing n m sensors to maximize network reliability for a given set of sensor failure
probabilities. If the network reliability is R, then the design
problem is posed as follows:
P10
Minlj
r2e
s:t:
r2e P r 2e;j
_
lj 2 f0; 1g
8j 2 N
ð26Þ
8j2N
The values of r2e;j are obtained from Rj using the proposed
transformation (Eq. (6)). It can be seen that Problem P1 0 is
then the dual of Problem P1 and can be solved as an explicit MINLP (formulation P3 presented later).
4.2. Minimization of network variance
This problem requires the optimal placement of n m
sensors to minimize network variance, for given sensor
variances. If r2 represents the network variance, then the
problem at hand is
P2
Minlj
r2
s:t:
r2 P r 2j 8j 2 N
lj 2 f0; 1g 8j 2 N
_
ð27Þ
This problem can be converted into its reliability counterpart as
0
P2
Maxlj
Re
s:t:
Re 6 Re;j
_
lj 2 f0; 1g
8j 2 N
8j 2 N
Due to the proposed transformation, problem P2 0 is dual
of P2 and can be solved using the graph theoretic based
greedy-search algorithm proposed by Ali and Narasimhan
[1]. Thus, using the proposed duality, graph theoretic approaches can be used to solve precision problems and similarly explicit optimization techniques can be used to
handle reliability problems.
We now present some problems, which have not been
satisfactorily attempted in literature [15] due to the existing
disconnect between reliability and precision problems. The
equivalence between the reliability and precision problems
developed in this article enables us to easily pose all these
194
problems either as appropriate graph theory problems or
as appropriate explicit optimization formulations.
4.4. Minimization of network variance with constraints on
the reliabilities
4.3. Maximization of network reliability with constraints on
the variances
A robust network may require that the network variance
be minimized subject to reliabilities of some or all of the
variables being above a threshold value. Mathematically,
the problem can be represented as
There are formulations in the literature for maximizing
reliability [1] and minimizing cost [4]. However, a robust
network may require that the network reliability be maximized while ensuring that some or all of the variances are
below threshold values. This problem can be posed as
Maxlj
R
s:t:
R 6 Rj
_
F1
ð28Þ
6 r2
8i 2 V ; V N
i
lj 2 f0; 1g 8j 2 N
This problem can be posed in any of the following two
ways:
(a) Maximize network reliability subject to constraints
that equivalent reliability be greater than or equal
to the equivalent reliability threshold values as
Maxlj
R
s:t:
R 6 Rj
_
8j 2 N
_
Re;i P Re;i
8i 2 V ;
lj 2 f0; 1g
8j 2 N
ð29Þ
V N
It can be seen that the original problem in reliability
and precision space has been converted into a problem in reliability space alone. The values of the equivalent threshold reliabilities and equivalent sensor
reliabilities can be determined by Eq. (22).
(b) Minimize network equivalent variance subject to constraints that the variances be less than or equal to
their threshold values.
F3
Minlj
r2e
s:t:
r2e P r 2e;j
_
_2
ri
r2
s:t:
r2 P r 2j
_
_
Ri P Ri
8j 2 N
r2
i
8j 2 N
ð30Þ
6
8i 2 V ; V N
lj 2 f0; 1g 8j 2 N
This consists of transforming the original problem into its
equivalent problem in precision framework.
Remark 4. Comparing between formulations F1 and F2, it
is seen that the variance threshold constraints in F1 have
been replaced by equivalent threshold constraints on
reliability in F2. Without these constraints the two formulations are the same. Addition of these constraints still
preserves the equivalence between F1 and F2 since any
observable sensor network satisfying (not satisfying) the
constraints on threshold variances in F1, will also satisfy
(not satisfy) the constraints on threshold reliabilities in F2.
Similar arguments hold for equivalence of F1 and F3.
ð31Þ
8i 2 V ; V N
lj 2 f0; 1g
8j 2 N
_2
ri
F2
F4
Minlj
8j 2 N
This problem can equivalently be posed either in the reliability framework or in the precision framework as
(a) Maximize network equivalent reliability subject to
constraints that the reliabilities are greater than or
equal to their threshold values. Let this formulation
which is written using only reliability constraints be
labeled F5.
(b) Minimize network variance subject to constraints
that equivalent variances be less than or equal to their
equivalent threshold values. Let this formulation be
labeled F6. Formulations F5 and F6 can be easily
modified for the case where certain variables are subject to original precision constraints as well.
In all the above formulations, graph theoretic
approaches [1] can be explored for solving the problems
P1, P2 0 , F2 and F5, which are in the reliability framework.
The formulations P1 0 , P2, F3 and F6 are in the precision
framework. These can be posed as MINLPs [4] as shown
later (formulation P3). Any available MINLP solution tool
can be used for solving these. Also, Bagajewicz and Cabrera [4] have shown the conversion of these MINLPs into
equivalent MILPs (at the cost of increasing problem size).
Chmielewski et al. [15] have converted these MINLPs into
equivalent convex-MINLPs using linear matrix inequalities
(LMIs). These approaches can also be tried for solving the
precision problems.
Remark 5. In literature, Bagajewicz and Sanchez [10] have
incorporated simultaneous reliability and precision constraints in a cost minimization approach. This approach
however involves tree based enumeration [3] and not an
explicit optimization strategy, since evaluation of reliability
and variance require constructive constraints (involves a
procedure for reliability and variance calculation for a
given sensor network). Such enumeration based schemes
are not efficient for large systems. Later, even though
Bagajewicz and Cabrera [4] incorporated variance constraints in an explicit optimization formulation, no similar
approach has been presented for the reliability constraints.
The duality proposed in the current work addresses this
shortcoming by providing an equivalence between reliability and precision. This equivalence obviates the need to
195
pose reliability requirements as constructive constraints,
thereby facilitating the use of explicit optimization to solve
the proposed formulations.
4.5. Pareto-optimal solutions
An ideal sensor network should have a high network
reliability as well as low network variance. However, very
often there is a trade-off between these objectives since
maximization of network reliability may lead to an increase
in the network variance and vice versa. In multi-objective
optimization literature, the set of solutions characterizing
trade-offs between competing objectives are typically represented as pareto-optimal solutions. The pareto-optimal
solutions are the set of non-dominated solutions. A solution is said to be non-dominated if it is feasible and there
is no other feasible solution which has better values of both
objectives [16]. In this section, we discuss a special case of
formulation F4 which helps us to determine these trade-offs
typically characterized by a pareto-optimal front. In formulation F4, let the required threshold reliabilities for all
the variables Ri ; 8 i 2 N be R*. The resulting formulation
is labeled F7 for the discussion below. It is easy to see that
R* corresponds to a threshold on network reliability. Formulation F7 can now be solved for different values of
thresholds on network reliability (i.e. different R* values)
to obtain corresponding lowest network variances and thus
a trade-off can be generated. An important point to be
noted is that the optimal solution of formulation F7 need
not necessarily be pareto-optimal since multiple solutions
may exist due to combinatorial nature of the problem.
To understand this, consider a scenario when the solution
of F7 is r2optimal for a threshold on the network reliability
equal to R*. While r2optimal is the best possible network variance under the constraint that the network reliability is
greater than or equal to R*, it may however be possible
to design a network with a higher network reliability without compromising on the network variance. In order to
find such pareto-optimal points, we need to solve the following additional problem in sequence with that in formulation F7.
Maxlj
R
s:t:
r2 P r 2j
_
_
F8
8i 2 N
R 6 Ri
RPR
8j 2 N
ð32Þ
r2 ¼ r2optimal
lj 2 f0; 1g 8j 2 N
The above formulation ensures that the highest network
reliability is obtained for the optimal network variance
r2optimal . It is obvious that if ðR ; r2optimal Þ is a pareto-point,
the solution to F8 will be R = R*. The above formulations
F7 and F8 can be converted into either reliability or variance framework according to the solution technique to be
used. In the steam-metering case study presented later,
the variance framework is used. For easy referencing, let
the equivalent variance-framework formulations for formulations F7 and F8 be labeled F9 and F10 respectively.
Both the formulations F9 and F10 can be solved as explicit
optimization problems and hence it can be seen that the
proposed duality helps in the determination of pareto-front
using an explicit optimization approach.
Remark 6. Instead of formulation F4, formulation F1 can
also be suitably modified to obtain the trade-offs between
network variance and network reliability. These trade-offs
will be exactly similar to those obtained from F7 and F8.
5. Case studies
In this section, we discuss two case studies to substantiate the theory of duality and demonstrate its utility by solving the above formulations. The HDA process is an
example of mid-size flowsheet and the steam-metering network is a large-size flowsheet. The results reported in this
work for the HDA case study are obtained by explicit enumeration. The results for steam-metering network have
been obtained by explicit optimization in GAMS [17] and
have also been verified for global optimality using explicit
enumeration.
5.1. Hydrodealkylation (HDA) network
The Hydrodealkylation (HDA) [2] of toluene is a midsize flowsheet (14 variables and seven equations). This
flowsheet has a total of 14C7 sensor network combinations.
Of these, only 992 combinations form an observable network. Using this process we demonstrate the concept of
equivalence and its application to the formulations of previous section. The necessary data including dimensionless
variances for this case study have been presented in Tables
1 and 2.
(a) Given sensor failure probabilities, design the most reliable network (formulation P1): The cases for equal
and unequal sensor reliabilities in Table 1 are from
Ali and Narasimhan [2]. The results are shown in
Table 1. For the case of equal sensor reliabilities,
there are 55 globally optimal solutions with a network reliability of 0.729. We were able to obtain all
these 55 sensor configurations by solving the problem
of minimization of the equivalent network variance
(formulation P1 0 ). For the case of unequal sensor reliabilities, there is a single globally optimal solution
with network reliability of 0.4336 and this solution
is also obtained by minimizing the equivalent network variance (formulation P1 0 ).
(b) Given sensor variances, design a network with maximum precision (formulation P2): Again consider the
HDA process with the randomly generated variances
as reported in Table 1. The results have been presented in Table 1. For the equal variances case, there
196
Table 1
Data and results for HDA process
Equal and unequal sensor reliabilities
Data
Case 1 Equal Sensor reliabilities
Case 2 Unequal Sensor reliabilities
Results
Objective
Case 1
Case 2
Maximization of reliability
Minimization of equivalent variance
Maximization of reliability
Minimization of equivalent variance
Equal and unequal sensor variances
Data
Case 1 Equal sensor variances
Case 2 Unequal sensor variances
Results
Objective
Case 1
Case 2
a
Minimization of variance
Maximization of equivalent Reliability
Minimization of variance
Maximization of equivalent reliability
0.9 for each sensor
{0.6800, 0.8400, 0.7500, 0.6300, 0.9100, 0.7700, 0.8300, 0.9000, 0.8600, 0.7300, 0.6600, 0.7600, 0.7200,
0.8800}
Objective value
Number feasible solutions
Number optimal solutions
Optimal measurements
0.7290
0.3161
0.4336
0.8356
992
992
992
992
55
55
1
1
{1, 4, 6, 10, 11, 13, 14}a
{1, 4, 6, 10, 11, 13, 14}a
{2,5,6,8,9,10,11}
{2, 5, 6, 8, 9, 10, 11}
0.32 for each sensor
{0.8462, 0.5252, 0.2026, 0.6721, 0.8381, 0.0196, 0.6813, 0.3795, 0.8318, 0.5028, 0.7095, 0.4289, 0.3046,
0.1897}
Objective value
Number feasible solutions
Number optimal solutions
Optimal measurements
0.9600
0.3829
1.3999
0.2466
992
992
992
992
55
55
2
2
{1, 4, 6, 10, 11, 13, 14}a
{1, 4, 6, 10, 11, 13, 14}a
{2, 3, 4, 7, 8, 10, 14}a
{2, 3, 4, 7, 8, 10, 14}a
Sample optimal solution.
Table 2
Results for HDA to maximize reliability subject to precision constraints and minimize variance subject to reliability constraints
Data
Threshold reliabilities
Threshold variances
{0.4570, 0.7881, 0.2811, 0.2248, 0.9089, 0.0073, 0.5887 0.5421, 0.6535, 0.3134, 0.2312, 0.4161, 0.2988, 0.6724}
{6.1068, 3.3057, 3.0011, 1.4202, 7.3270, 1.5161, 2.7363 3.4982, 6.0345, 3.2106, 3.2247, 9.5335, 6.3163, 7.7431}
Results
Formulation
Mixed framework
Reliability framework
Precision framework
Maximize reliability subject to precision constraints
Objective
0.4336
No. of feasible solutions
518
No. of optimal solutions
1
Optimal measurements
{2, 5, 6, 8, 9, 10, 11}
0.4336
518
1
{2, 5, 6, 8, 9, 10, 11}
0.8356
518
1
{2,5,6,8,9,10,11}
Minimize variance subject to reliability constraints
Objective
2.0354
No. of feasible solutions
19
No. of optimal solutions
2
Optimal measurements
{2, 4, 5, 7, 8, 12, 14}a
0.1306
19
2
{2, 4, 5, 7, 8, 12, 14}a
2.0354
19
2
{2, 4, 5, 7, 8, 12, 14}a
a
Sample optimal solution.
are 55 optimal sensor configurations which are identical to those obtained for equal reliabilities situation
in case a. This result is as expected based on Corollary 1. For the unequal case, there are however only
two optimal configurations. The results for the maximization of the equivalent network reliability (formulation P2 0 ) also confirm these findings.
(c) Design the most reliable network with constraints on
variances (formulation F1): This problem involves
the optimal placement of sensors such that the network reliability is maximized while satisfying the precision constraints. The threshold variances are given
in Table 2. We have enumerated the solution space
for all the three frameworks: combined (formulation
F1), exclusively in reliability framework (formulation
F2) and exclusively in precision framework (formulation F3). The results have been given in Table 2. This
problem has one single global optimum and this is
achieved in all the three frameworks thereby validating the proposed equivalence.
(d) Design the most precise network with constraints on
reliability (formulation F4): This problem is complimentary to the above problem. Here the objective is
to design a sensor network with least variance subject
197
to constraints on the reliabilities of variables. In this
example, we have reliability constraints on all the
variables (given in Table 2). The results are listed in
Table 2. It can be seen that there are multiple (2)
optima. These two solutions are obtained in all the
three frameworks (formulations F4, F5 and F6).
12
4
10
11
16
4
3
9
5
2
2
15
1
5
6
22
17
1
3
25
9
24
8
18
13
23
14
6
10
27
5.2. Steam-metering network
The steam-metering system shown in Fig. 1 is of a methanol synthesis unit and has been used in [1,2,18,19]. The
steam-metering network is an example of large-size flowsheet (28 variables, 11 units). Using this flowsheet, we demonstrate the utility of equivalence in solving existing and
novel formulations by means of explicit optimization. This
flowsheet has a total of 28C11 sensor combinations with
1,243,845 observable networks. We solve all the proposed
formulations using explicit optimization approach (i.e.,
consider precision framework for all the proposed formulations). The sensor failure probabilities, sensor variances,
threshold reliabilities, and threshold variances are given
in Table 3. Before discussing the results, the explicit optimization problem for the minimization of network variance is
presented next in some detail.
19
7
26
7
8
5.2.1. Optimization formulation
While formulation P2 states the problem of minimization of network variance, it does not explicitly specify the
equations/constraints required for calculation of
_2
r j 8j 2 N irrespective of the chosen sensor network. Such
constraints are required for this formulation to be posed as
an explicit optimization problem. For this purpose, formulation P2 is modified [4] as shown next:
21
20
11
28
Fig. 1. Steam-metering network [1].
Table 3
Data for steam-metering network
Vars.
Sen. Rel.
Eq. var
Th. Rel.
Eq. Th. var
Sen. Var.
Th. var
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
0.7340
0.8570
0.8890
0.9220
0.6560
0.8750
0.7770
0.8620
0.7070
0.6540
0.6850
0.8630
0.8000
0.8230
0.8550
0.9430
0.7690
0.8560
0.7610
0.6780
0.8770
0.8980
0.6750
0.7990
0.9430
0.7440
0.7250
0.8750
0.3092
0.1543
0.1177
0.0812
0.4216
0.1335
0.2523
0.1485
0.3467
0.4246
0.3783
0.1473
0.2231
0.1948
0.1567
0.0587
0.2627
0.1555
0.2731
0.3886
0.1312
0.1076
0.393
0.2244
0.0587
0.2957
0.3216
0.1335
0.6340
0.5016
0.7890
0.8220
0.3800
0.3813
0.6770
0.7620
0.1872
0.5540
0.5850
0.3480
0.7000
0.7230
0.7550
0.2794
0.1772
0.7560
0.6610
0.5780
0.1824
0.7980
0.5750
0.3436
0.8430
0.6440
0.2822
0.3348
0.4557
0.6900
0.2370
0.1960
0.9676
0.9642
0.3901
0.2718
1.6756
0.5906
0.5361
1.0556
0.3567
0.3243
0.2810
1.2751
1.7305
0.2797
0.4140
0.5482
1.7016
0.2256
0.5534
1.0683
0.1708
0.4401
1.2651
1.0942
6.8611
2.8471
16.7043
7.6741
10.4099
12.7337
1.2750
21.0758
6.3127
25.2903
8.0394
10.9326
14.0090
2.6323
3.6552
3.7080
10.5511
2.6194
4.2917
1.5561
10.7706
6.2972
13.0402
12.9512
9.3099
7.9517
13.5968
12.5028
16.8611
22.8471
37.3823
17.6741
20.4099
22.7337
11.2750
31.0758
55.4944
27.2710
38.2036
20.9326
35.5808
12.6323
13.6552
36.5397
62.7123
12.6194
14.2917
11.5561
22.3745
16.2972
23.0402
35.6120
19.3099
17.9517
35.2836
40.5836
Vars. – Variables; Sen. Rel. – Sensor reliabilities; Eq. var – Equivalent sensor variances; Th. Rel. – Threshold reliabilities; Eq. Th. var. – Equivalent
Threshold variances; Sen. var. – Sensor variances; Th. var. – Threshold variances.
198
Minlj;k ;P
s:t:
P3
r2
_
r P r 2j
_
_2
r j ¼ Sðj;jÞ
_
2
8j 2 N
S ¼ S S:AT :P:A:S
PðA:S:AT Þ ¼ Imm
N
P
lj;1 ¼ n m
ð33Þ
j¼1
N
P
lj;2 ¼ m
j¼1
2
P
lj;k ¼ 1;
8j 2 N
k¼1
lj;k 2 f0; 1g
8j 2 N ; k 2 f1; 2g
b is the covariance matrix of reconciled estimates, P
where S
is the inverse of A.S.AT and has been introduced to avoid
the symbolic inversion of A.S.AT, Im·m is the identity matrix of size m · m, and S is the given sensor variance matrix.
The off-diagonal elements of S are zero (under the assumption that noise in different sensors is uncorrelated) and the
jth diagonal element corresponds to the variance of the sensor selected for measuring jth variable. If variable j is not
measured by its sensor (lj,1 = 0), then it is assumed to be
measured by a fake sensor (lj,2 = 1) with large variance L
(theoretically infinite). The choice of L has been a subject
of discussion in the literature in terms of the scaling issues
it introduces in the formulation. Similar to Bagajewicz and
Cabrera’s work [4] the value of L is taken to be 1000.
The explicit optimization formulation given by Eq. (33)
is an MINLP. This can be reformulated into an MILP as
shown by Bagajewicz and Cabrera [4]. Here, we restrict
ourselves to MINLP because this conversion may have
time issues due to large number of variables. Also, it can
be seen that Eq. (33) depends only on the reduced incidence
matrix A. Moreover, it is also valid for redundant sensor
network design cases. It is to be noted that no such explicit
optimization procedure is available in literature for the reliability framework. Hence, in this case study all formulations (reliability or precision or combined) have been
solved using the framework defined by P3. The resulting
optimization problems presented in this section have been
solved on GAMS [17] platform with SBB as the MINLP
solver. The results for various formulations are presented
below:
(a) Given sensor failure probabilities, design the most reliable network (formulation P1): We transform this
problem from reliability framework to precision
framework. The resulting problem was of type P3
formulated using equivalent variances. The results
are given in Table 4. The minimum equivalent variance of the network was found to be 1.262. This optimal equivalent network variance was converted back
to network reliability as 0.2831 and agreed with the
Table 4
Results for steam-metering network
Results for steam-metering network to maximize reliability (explicit optimization Eq. (25))
Initial guess
Real 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1
Fake 1 0 0 1 0 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 1
Binary variables
56 Continuous variables 121
Optimal measurements
{3,4,5,6,7,8,12,13,14,15,16,18,19,20,23,24,26}
Objective value
Equivalent network variance = 1.262, Corresponding network reliability = 0.2831
No. of optimal solutions
92
Results for steam-metering network to minimize variance (explicit optimization Eq. (27))
Initial guess
Real 1 1 0 1 1 1 1 1 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1
Fake 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1
Binary variables
56 Continuous variables 121
Optimal measurements
{1 4 5 6 7 9 11 13 14 15 16 18 19 20 22 24 26}
Objective value
41.732
No. of optimal solutions
18
Results for steam-metering network to maximize reliability subject to precision constraints (explicit optimization Eq. (28))
Initial guess
Real 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1
Fake 0 0 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1
Binary variables
56 Continuous variables 242
Optimal measurements
{1 2 4 5 6 7 8 14 15 16 18 19 20 22 23 25 26}
Objective value
Equivalent network variance = 1.262, Corresponding network reliability = 0.2831
No. of optimal solutions
2
Results for steam-metering network to minimize variance subject to reliability constraints (explicit optimization Eq. (31))
Initial guess
Real 1 1 1 1 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1
Fake 1 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1
Binary variables
56 Continuous variables 242
Optimal measurements
{1 2 4 5 6 7 9 11 14 15 16 18 20 22 23 26 27}
Objective value
42.172
No. of optimal solutions
1
199
literature value [1]. The optimality of the solution was
also verified by explicit enumeration. This enumeration further revealed the existence of 91 additional
optimal solutions which is once-again in agreement
with the value reported in literature [1].
(b) Given sensor variances, design the maximum precision
network (formulation P2): This problem was posed
as in P3 and the results are reported in Table 4.
The network variance was found to be 41.732. This
was verified by enumeration which also revealed the
existence of 17 additional optimal solutions. The
objective function value (network variance) for these
18 optimal solutions was found to be 45.3038 by enumeration. No reported results are available in literature for this case.
Remark 7. As seen above, there is a difference between the
optimal network variance as calculated by enumeration
and explicit optimization. This difference is due to the
approximation of the variance (L) of fake sensor by 1000
instead of infinity which affects the results in the optimization approach.
(c) Given sensor failure probabilities and variances, design
the most reliable network with constraints on variances
(formulation F1): This problem was posed as in P3
with additional constraints on threshold variances
(listed in column 7, Table 3). On solving, the network
variance was found to be 1.262 and the corresponding network reliability is 0.2831. The results have
been reported in Table 4. It can be seen that the network reliability remains unaffected (case a) despite
the addition of the constraints on variances. However, the number of optimal solutions satisfying the
variance requirement is reduced to two. Hence the
designer should choose one of these two sensor
networks.
(d) Given sensor variances and failure probabilities, design
the most precise network with constraints on reliability
(formulation F4): The results for this scenario with
threshold reliabilities listed in Table 3 are shown in
Table 4. The network variance was found to be
42.172 by optimization but the actual value by enumeration was found to be 45.3038. It can be seen that
while the network variance has remained unaffected
(case b) despite the additional constraints on reliabilities, the number of optimal solutions satisfying the
reliability requirements has been reduced to one.
The designer should choose this sensor network to
satisfy reliability requirements while maximizing precision. The other 17 optimal solutions for case b will
not meet the reliability requirements. Also, other criteria (such as cost) can be easily included in the optimization formulation to select even more promising
sensor networks.
The above four cases (a–d) demonstrate the utility of the
proposed equivalence in formulating and solving various
sensor network design problems. Even though the formulations have been solved in the explicit optimization based
variance framework, the duality acts as a link which
enables formulation of these problems in a graph theory
based reliability framework as well.
5.2.2. Pareto-optimal solutions
To determine the pareto-front, we solve the explicit optimization formulation as presented in formulations F9 and
F10, i.e., we will minimize the network variance with constraints on the network reliability. The thresholds on network reliability (R*) used to determine the pareto
solutions are given in Table 5a. The equivalent threshold
network variance ðr2
e Þ can be obtained by the use of Eq.
(6). Thus, we can now solve formulation F9 to obtain the
minimum network variance r2optimal . Further, to uncover
the pareto-optimal solutions, we solve formulation F10.
The equivalent network variances obtained by solving formulation F10 are given in Table 5a and the corresponding
network reliability is calculated using the sensor configuration. The equivalent network variances could have been
converted to their corresponding network reliability using
Eq. (22) but this may propagate the error due to the
approximation involved in using a finite value for L. Table
5a also lists all the pareto-optimal solutions constituted by
the four points A, B, C and D. A complete inspection of the
solution space also confirmed the existence of these four
pareto-optimal points. Additionally, this inspection
revealed that these pareto-optimal points were characterized by multiple realizations i.e., sensor networks with different configurations but the same objective function
values. The points A, B, C and D are characterized by
19, 9, 2 and 16 realizations respectively. From Table 5a,
it can be seen that the points D and A represent the best
solutions if only the objective of network reliability or
Table 5a
Results of pareto-front
Tag
A
B
C
D
Threshold reliabilities
R*
r2
e
0.1000
0.2000
0.2700
0.2800
2.3026
1.6094
1.309
1.273
r2optimal
42.172
42.547
46.203
47.861
Maximum reliability
Pareto solution (R, r2)
r2e
R
Explicit optimization
Complete inspection
1.823
1.309
1.262
1.262
0.1591
0.2692
0.2793
0.2824
(0.1591,
(0.2692,
(0.2793,
(0.2824,
(0.1591,
(0.2692,
(0.2793,
(0.2824,
42.172)
42.547)
46.203)
47.861)
No. of realizations
45.3038)
45.4944)
50.3081)
51.7458)
19
9
2
16
Note: To minimize the error propagation of L used in explicit optimization, the network reliability in the above table has been calculated from the optimal
network configuration (Table 5b) instead of r2e .
200
Table 5b
Network configuration of pareto-front
Tag
Optimal measurements
A
B
C
D
1
1
1
1
2
2
3
2
4
4
5
4
5
5
6
5
6
6
7
6
7
7
8
7
9
8
11
8
11
11
13
12
Network Variance
D
47
C
46
45
44
43
B
A
42
0.7
0.72
15
15
15
15
16
16
16
16
18
18
18
18
19
19
19
19
20
20
20
20
22
22
22
22
23
23
23
23
26
26
26
26
While the duality does not guarantee optimality for the
redundant case, it can still be used to generate optimal or
close to optimal solutions for such problems. In a simulation case study on an ammonia example [1], the proposed
transformation reached the global optimal network in
73% of the randomly generated test cases for solving reliability maximization problem using variance framework.
Issues related to the formal extension of this duality to
the redundant case are currently under investigation.
49
48
14
14
14
14
0.74
0.76
0.78
0.8
0.82
0.84
1-Network Reliability
Fig. 2. Pareto-front for steam-metering network.
network variance is to be optimized. If there are no rigid
requirements on variance, point B can be chosen in favor
of point A as the increase in reliability is very large compared to the increase in variance. Table 5b shows one realization of the network configuration for each of the pareto
points and Fig. 2 shows a pictorial representation of the
pareto points. In this figure, to facilitate easy interpretation, (1 R) has been plotted versus variance, since the
maximization of network reliability (R) is equivalent
to the minimization of (1 R). This pareto-front can help
the designer in making an informed choice as it shows the
tradeoffs between the network reliability and the network
variance.
Remark 8. The current limitation of working in precision
framework is the lack of availability of robust solvers for
solving MINLP problems. In Table 4, initial guesses are
also listed since the optimal solutions were found to be
dependent on the initial guesses. In general, convergence to
a globally optimal solution is not guaranteed from
arbitrary initial guesses. The transformation of MINLP
to MILP (which are not sensitive to initial guesses)
increases the computational burden [4].
Remark 9. Redundant sensor network design: Though the
duality proposed in this article is applicable for non-redundant sensor networks, there may be situations where additional variables need to be measured (redundant case).
Remark 10. It can be seen from Eq. (22) that the transformation from variance to reliability is essentially mapping
the interval [0,1] to the interval [0, 1]. This may lead to a
situation where as the variance gets larger, the reliability
becomes smaller, leading to numerical difficulties while
solving the problem in the reliability framework. To overcome this problem, Eq. (22) can be modified as
!
r2j
Re;j ¼ exp
ð34Þ
r2scale
where r2scale is a scaling factor (with same dimensions as variance) and can be chosen to be the maximum variance from
the list of sensor variances though it can be any positive
number in general.
6. Conclusions
In this work, we have proposed a duality between the
precision and reliability problems for non-redundant sensor network design problems for linear processes. This
duality provides an elegant theoretical link between the
precision and reliability frameworks. This link enables formulation of reliability problems as equivalent precision
problems and vice versa. It also enables formulation of
the optimal sensor location problem taking into consideration both reliability and precision in a single framework.
This framework can be either in terms of precision or reliability and this flexibility helps the designer in efficiently
evaluating trade-offs and identifying the best sensor network for the problem at hand. At an application level,
the proposed equivalence can be used to solve several other
sensor network design problems such as the design of a
minimal cost network subject to reliability and precision
constraints. Any other combination of cost, reliability
and precision can also be considered, with or without constraints on these objectives. Extension of the proposed
201
duality for redundant and non-linear processes is currently
under investigation.
ðA:1Þ
ðA:6Þ
a c0 < a a
Also, since a > c0 we get (on taking powers of a),
a a < a c0
ðA:7Þ
since 0 < a < 1. Comparing Eqs. (A.6) and (A.7) we get a
contradiction. Hence, the statement f(c) 5 ac is incorrect,
and indeed f(c) = ac.
Now, since f(r) = ar for r rational and irrational, the
claim that f(r) = ar for r 2 [0, 1) is proved. h
ðA:2Þ
References
Appendix 1
For every non-negative real number a, we want to associate another real number f(a) 2 [0, 1] such that the following property holds:
If
a 1 þ a 2 þ þ a n 6 b 1 þ b 2 þ þ bm
The relation c > a ) f(c) < f(a). Since a is rational
f(a) = aa. Then
then
f ða1 Þf ða2 Þ f ðan Þ P f ðb1 Þf ðb2 Þ f ðbm Þ
where ai, bj 2 [0, 1) and incase of strict inequality in Eq.
(A.1), strict inequality in Eq. (A.2) also holds.
Claim. The map f : [0, 1) ! [0, 1] satisfying the above
property is of the form f(a) = aa for some a 2 (0, 1).
Proof. Consider equality in Eq. (A.1), then the equality
should also be valid in Eq. (A.2). Without loss of generality, let f(1) = a where a 2 (0, 1). We claim that f(a) = aa for
every a 2 [0, 1).
Assume that f(0) 5 0. Since, 0 + 0 = 0, we get
f(0)f(0) = f(0). Hence, f(0) = 1. Now, let n 2 N be a positive
integer. Then 1 + 1 + + 1(n times) = n. Hence, we get
n
f ð1Þf ð1Þ f ð1Þ ¼ f ðnÞ ) f ðnÞ ¼ ðf ð1ÞÞ ¼ an
Also,
1
1
1
1
1¼
þ
þ þ
ðn timesÞ ) f
n
n
n
n
1
1
¼ ðf ð1ÞÞn ¼ an
Now let m 2 N be another positive integer. Then
m
m
1
1
1
¼
þ
þ þ
ðm timesÞ ) f
n
n
n
n
n
m
1
m
¼ f
¼ an
n
ðA:3Þ
ðA:4Þ
In other words, for every non-negative rational number q,
f(q) = aq.
We now additionally show that for any non-negative
irrational number c, f(c) = ac. To show this (by contradiction), let f(c) = b 5 ac. Since ax is a continuous function,
9c0 2 ½0; 1Þ s.t. b ¼ ac0 where c 5 c0. Without loss of
generality, we can assume c > c0. Now, since the set of
rational numbers is dense in the real line, there exists a
rational number a, such that
c > a > c0
ðA:5Þ
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