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3. A Procedure for Solving the Optimal Expansion
Problem
(a)
BRANCHING PROCEDURE
At time t = 0 there are Ν facilities in use. At time ί = 1 it may be decided to introduce no new facility (node 0), facility Ν + 1, or any one of
the other possible facilities (N + 2, Ν + 3, . .. , Μ). At time t = 2, for
each one of the alternatives in period 1 (except 0), there are Μ — Ν alternatives corresponding to each of the nodes of (t = 1). The alternatives
are (1) to introduce no new facility (node 0), or (2) to introduce a new
facility Ν + 1, Ν + 2,. .., M 9 except that no new facility may be repeated.
The structure continues until all possibilities are exhausted or until period
is reached.
(b)
CALCULATION OF THE BOUNDS
A bound Bj t is associated with each node j of year t. It is the upper limit
on the economic return from the use of the dam(s) associated with the
node added to (1) the cumulative net return CR t -i from the system to the
end of year t — 1 plus (2) the return TR t from operating the system as it
exists at the beginning of year t for the years t through T ma x . CR t -\ and
TR t are defined as follows:
CR^ = Σ [* *—i
<=-i
i-J\r+i
^m&x
TR t = X t Σ [1/(1+ r)«D
(3.2)
t~t
The bound B Jt is calculated by assuming that:
1. The dam j corresponding to the given node is introduced and operated
at maximum efficiency for every time period, from the time of its introduction into the system through all the remaining time periods. This assumption leads to the net operational return for reservoir j if introduced at the
beginning of year t and operated at maximum efficiency for the years t to
Τ m *x y
OR jt = (MB,) Σ) [1/(1 + r)«] - λ,Α.[1/(1 + Ό«]
(3.3)
where MR; is the maximum annual return from operating reservoir j.
2. In the following time period the best remaining dam (i.e., the dam
with the highest bound) is introduced into the system and is operated at
maximum efficiency until the end of the time horizon T ma x , to give ORj> tt +i,
and so on for all the remaining dams and time periods.
3. A Procedure for Solving the Optimal Expansion
Problem
(a)
BRANCHING PROCEDURE
At time t = 0 there are Ν facilities in use. At time ί = 1 it may be decided to introduce no new facility (node 0), facility Ν + 1, or any one of
the other possible facilities (N + 2, Ν + 3, . .. , Μ). At time t = 2, for
each one of the alternatives in period 1 (except 0), there are Μ — Ν alternatives corresponding to each of the nodes of (t = 1). The alternatives
are (1) to introduce no new facility (node 0), or (2) to introduce a new
facility Ν + 1, Ν + 2,. .., M 9 except that no new facility may be repeated.
The structure continues until all possibilities are exhausted or until period
is reached.
(b)
CALCULATION OF THE BOUNDS
A bound Bj t is associated with each node j of year t. It is the upper limit
on the economic return from the use of the dam(s) associated with the
node added to (1) the cumulative net return CR t -i from the system to the
end of year t — 1 plus (2) the return TR t from operating the system as it
exists at the beginning of year t for the years t through T ma x . CR t -\ and
TR t are defined as follows:
CR^ = Σ [* *—i
<=-i
i-J\r+i
^m&x
TR t = X t Σ [1/(1+ r)«D
(3.2)
t~t
The bound B Jt is calculated by assuming that:
1. The dam j corresponding to the given node is introduced and operated
at maximum efficiency for every time period, from the time of its introduction into the system through all the remaining time periods. This assumption leads to the net operational return for reservoir j if introduced at the
beginning of year t and operated at maximum efficiency for the years t to
Τ m *x y
OR jt = (MB,) Σ) [1/(1 + r)«] - λ,Α.[1/(1 + Ό«]
(3.3)
where MR; is the maximum annual return from operating reservoir j.
2. In the following time period the best remaining dam (i.e., the dam
with the highest bound) is introduced into the system and is operated at
maximum efficiency until the end of the time horizon T ma x , to give ORj> tt +i,
and so on for all the remaining dams and time periods.