4.1 Proof of Sefficiency Indicators
41
(Macro, Meso, and Micro-Efficiencies in Water Resources Management: A New
Framework Using Water Balance 2012) and Haie (Sefficiency (Sustainable efficiency): a Systemic Framework for Advancing Water Security 2013). To start, let us
be comprehensive and write the water balance in terms of these two perspectives.
The generic WUS depicted in Fig. 2.1 is composed of Inflow and Outflow with
negligible change in storage (FIW1a), which according to the principle of water
balance (Inflow = Outflow) can be written as Eq. (4.1):
(V 1 + O S + P P) − (E T + N R + V 2 + R P) = 0
(4.1)
For variable definitions refer to Chap. 2 or the Abbreviations and Symbols in the
beginning of this book. Water balance of a WUS can also be presented in terms of
consumptive and non-consumptive flows (Table 2.2) as in Eq. (4.2):
(V 1 + O S + P P − V 2 − R P) − (E T + N R) = 0
( 4 . 2 )
In order to write an alternative arrangement of water balance that embodies the
above two equations and keeps their forms (inflow and consumptive), the binary
index ic is introduced, which gives Eq. (4.3):
[(V 1 + O S + P P) − (1 − ic)(V 2 + R P)]
− [(E T + N R) + ic(V 2 + R P)] = 0, ic = {0, 1}
(4.3)
Note that if ic = 0 (i.e., consumptive type), Eq. (4.3) becomes Eq. (4.2), and if
ic = 1 (i.e., inflow type), we get Eq. (4.1). However, water balance equations are
descriptive and do not give any information regarding the performance of a WUS.
At this point, we need to introduce the other two Pillars of water management,
i.e., quality and benefit attributes into Eq. (4.3). To do so, we apply the Usefulness
Criterion given in Sect. 2.2 to Eq. (4.3) as shown in Eq. (4.4):
[(V 1 + O S + P P) − (1 − ic)(V 2 + R P)]−
−[(E T + N R) + ic(V 2 + R P)]
s
= Λ, ic = {0, 1}
(4.4)
Subscript ‘s’ stands for the useful part of all the WPTs and their corresponding
WPIs within the curly brackets. This means that X S = W sX * X, with X being a WPI,
X S its useful part, and W sX its Useful Criterion, which is presented in Eq. 2.1. For
example, if X = ET, then its useful part is X S = ET S = W sET * ET.
Because X is greater than or equal to X S and the usefulness of the inflow is more
than or equal to the outflow, a non-negative undesirable (Sect. 2.5) factor called
Lambda, , is inserted in the right hand side of the equation to maintain the equality.
The undesirables of a WPI are non-beneficial and pollution and consequently a
fundamental aim of water management is to minimize them. It should be noted that
Eq. (4.3) is a special case of the Eq. (4.4), with W bX = W qX = 1 (unitary Usefulness
Criterion) leading to = 0, which indicates that there are no undesirables. In a
41
(Macro, Meso, and Micro-Efficiencies in Water Resources Management: A New
Framework Using Water Balance 2012) and Haie (Sefficiency (Sustainable efficiency): a Systemic Framework for Advancing Water Security 2013). To start, let us
be comprehensive and write the water balance in terms of these two perspectives.
The generic WUS depicted in Fig. 2.1 is composed of Inflow and Outflow with
negligible change in storage (FIW1a), which according to the principle of water
balance (Inflow = Outflow) can be written as Eq. (4.1):
(V 1 + O S + P P) − (E T + N R + V 2 + R P) = 0
(4.1)
For variable definitions refer to Chap. 2 or the Abbreviations and Symbols in the
beginning of this book. Water balance of a WUS can also be presented in terms of
consumptive and non-consumptive flows (Table 2.2) as in Eq. (4.2):
(V 1 + O S + P P − V 2 − R P) − (E T + N R) = 0
( 4 . 2 )
In order to write an alternative arrangement of water balance that embodies the
above two equations and keeps their forms (inflow and consumptive), the binary
index ic is introduced, which gives Eq. (4.3):
[(V 1 + O S + P P) − (1 − ic)(V 2 + R P)]
− [(E T + N R) + ic(V 2 + R P)] = 0, ic = {0, 1}
(4.3)
Note that if ic = 0 (i.e., consumptive type), Eq. (4.3) becomes Eq. (4.2), and if
ic = 1 (i.e., inflow type), we get Eq. (4.1). However, water balance equations are
descriptive and do not give any information regarding the performance of a WUS.
At this point, we need to introduce the other two Pillars of water management,
i.e., quality and benefit attributes into Eq. (4.3). To do so, we apply the Usefulness
Criterion given in Sect. 2.2 to Eq. (4.3) as shown in Eq. (4.4):
[(V 1 + O S + P P) − (1 − ic)(V 2 + R P)]−
−[(E T + N R) + ic(V 2 + R P)]
s
= Λ, ic = {0, 1}
(4.4)
Subscript ‘s’ stands for the useful part of all the WPTs and their corresponding
WPIs within the curly brackets. This means that X S = W sX * X, with X being a WPI,
X S its useful part, and W sX its Useful Criterion, which is presented in Eq. 2.1. For
example, if X = ET, then its useful part is X S = ET S = W sET * ET.
Because X is greater than or equal to X S and the usefulness of the inflow is more
than or equal to the outflow, a non-negative undesirable (Sect. 2.5) factor called
Lambda, , is inserted in the right hand side of the equation to maintain the equality.
The undesirables of a WPI are non-beneficial and pollution and consequently a
fundamental aim of water management is to minimize them. It should be noted that
Eq. (4.3) is a special case of the Eq. (4.4), with W bX = W qX = 1 (unitary Usefulness
Criterion) leading to = 0, which indicates that there are no undesirables. In a