42
2. Formulating
the
Problem
where
= capital needed for building reservoir j in year t
Kjt
= capital needed for building canal j in year t
Μ
= maximum number of dams that can be built
Ν
= number of dams that exist at the beginning of the planning
horizon
= length of the planning period
Xijt
= return from reservoir j in month i of year t
a
= discount factor
β»
= a Heavyside function; 1 designates that a return is available
from project j in year t, while 0 indicates that no return is
available
\j t
= a Heavyside function; 1 designates capital must be provided to
build dam j in year t, while 0 indicates capital does not have
to be provided
= a Heavyside function; 1 designates that capital must be provided
to build a canal to supply imported water to reservoir j in
year t, while 0 indicates that capital does not have to be
provided
The subscript i refers to the month of operation, the subscript j to a particular dam, and the subscript t to the year of operation. The units for each
of the symbols will be found in the list of notation in Appendix C. Other
benefits and costs can be added to the objective function by analogy with
the given terms. A specific example of an objective function will be found
in Section 4.3.
2.3. Constraints
Constraints exist that limit the range of variation of each of the variables,
prescribe their relationships to each other, and delineate the external influences on the planning. Constraints generally are of two basic types:
equality or inequality constraints. Another type of constraint, as we shall
see, is the restriction of a variable to being either 0 or 1. An exact mathematical representation of a water resources development project, even if
possible, would lead to hopeless mathematical complexity. Therefore, in
writing down the constraints it is necessary to attain a reasonable balance
between accurate representation and mathematical manageability. Some
variables have been deliberately omitted from the constraints because their
2. Formulating
the
Problem
where
= capital needed for building reservoir j in year t
Kjt
= capital needed for building canal j in year t
Μ
= maximum number of dams that can be built
Ν
= number of dams that exist at the beginning of the planning
horizon
= length of the planning period
Xijt
= return from reservoir j in month i of year t
a
= discount factor
β»
= a Heavyside function; 1 designates that a return is available
from project j in year t, while 0 indicates that no return is
available
\j t
= a Heavyside function; 1 designates capital must be provided to
build dam j in year t, while 0 indicates capital does not have
to be provided
= a Heavyside function; 1 designates that capital must be provided
to build a canal to supply imported water to reservoir j in
year t, while 0 indicates that capital does not have to be
provided
The subscript i refers to the month of operation, the subscript j to a particular dam, and the subscript t to the year of operation. The units for each
of the symbols will be found in the list of notation in Appendix C. Other
benefits and costs can be added to the objective function by analogy with
the given terms. A specific example of an objective function will be found
in Section 4.3.
2.3. Constraints
Constraints exist that limit the range of variation of each of the variables,
prescribe their relationships to each other, and delineate the external influences on the planning. Constraints generally are of two basic types:
equality or inequality constraints. Another type of constraint, as we shall
see, is the restriction of a variable to being either 0 or 1. An exact mathematical representation of a water resources development project, even if
possible, would lead to hopeless mathematical complexity. Therefore, in
writing down the constraints it is necessary to attain a reasonable balance
between accurate representation and mathematical manageability. Some
variables have been deliberately omitted from the constraints because their