Appendix A
Equivalency
Proof of the equivalency of minimizing a difference and maximizing its ratio (Haie
and Keller 2012):
Let Z1 be the minimum of the difference (F − H) as follows:
• Z1 = min (F − H), with F and H positive
The solution (i.e., real values of F and H) is the minimum value of F and the
maximum value of H. These two values of F and H yield the minimum for (F − H),
or Z1 = min (F) − max (H).
Now let Z2 be the maximum of the ratio (H/F) as follows:
• Z2 = max (H/F)
The solution is the minimum value of F and the maximum value of H. Therefore,
although Z1 and Z2 are different, the two solutions (i.e., the values for F and H) are
the same.
Reference
Haie N, Keller A (2012) Macro, meso, and micro-efficiencies in water resources
management: a new framework using water balance. J Am Water Resour Assoc
(JAWRA) 48(2):235–243
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
N. Haie, Transparent Water Management Theory,
Water Resources Development and Management,
https://doi.org/10.1007/978-981-15-6284-6
107
Equivalency
Proof of the equivalency of minimizing a difference and maximizing its ratio (Haie
and Keller 2012):
Let Z1 be the minimum of the difference (F − H) as follows:
• Z1 = min (F − H), with F and H positive
The solution (i.e., real values of F and H) is the minimum value of F and the
maximum value of H. These two values of F and H yield the minimum for (F − H),
or Z1 = min (F) − max (H).
Now let Z2 be the maximum of the ratio (H/F) as follows:
• Z2 = max (H/F)
The solution is the minimum value of F and the maximum value of H. Therefore,
although Z1 and Z2 are different, the two solutions (i.e., the values for F and H) are
the same.
Reference
Haie N, Keller A (2012) Macro, meso, and micro-efficiencies in water resources
management: a new framework using water balance. J Am Water Resour Assoc
(JAWRA) 48(2):235–243
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
N. Haie, Transparent Water Management Theory,
Water Resources Development and Management,
https://doi.org/10.1007/978-981-15-6284-6
107