mitted coverage areas of the individual pivots
to overlap and the second did not allow an
overlap. The engineering techniques devel-
oped by Anderson et al. to simulate field
coverage served as a framework for address-
ing both the economic incentive to irrigate
and the optimal irrigation investment deci-
sion questions evaluated in this paper.
3
- ∑ Yk + US = O;
k=ι
N 3 M
(4) ( Σ Σ BjVij - Σ Wh)Z
i=l j=l h=l
+ S + A≤F;
(5) Y1 ≤ Q; and
MODEL
A mixed integer linear programming model
was constructed to determine the optimal
number, size, and location of center pivots
using profit maximization as the objective.
Data for peanut production in southeast Ala-
bama were used to illustrate the technique.
The field was represented by a series of grid
points and mixed integer programming was
used to optimize field coverage. A grid point
refers to a potential pivot location and a grid
area is the portion of the field associated with
a particular grid point.
Several decision variables were included
in the model. They were: selection of pivot
size from among three alternative sizes (96
acres, 138 acres, and 188 acres)1; selection
of the location for each pivot; whether to
produce peanuts without irrigation; whether
to rent out available land for other uses; and
the selling of peanuts at quota, contract, and
world prices.
The mixed integer programming model
used to solve the questions of how many,
what size, and what locations for center pivot
irrigation systems may be expressed mathe-
matically as follows:
3
(1) Maximize Σ Pk Yk+TA
k=l
N 3 Cj V4-XS
- Σ Σ
i=l j=l
Subject to:
3
(2) Σ Vlj≤ 1 for all i ;
j=l
N 3 M
(3) ∑ ∑ Rj Vlj - ∑ DW1
i=l j=l h=l
(6) Y2 ≤ G;
where:
P = price received per pound of pea-
nuts;
B = area irrigated by a pivot;
Z = conversion factor from grid area to
acreage;
Y = total pounds of peanuts sold;
A = number of acres rented out;
T = rent received per acre;
N = number of potential pivot loca-
tions;
C = annual cost of a center pivot of
size j ɑ/pivot) plus the cost of
producing irrigated peanuts for the
area covered;
X = cost per acre of producing non-
irrigated peanuts;
S = number of acres of non-irrigated
peanuts;
V = zero-one integer variable indicat-
ing whether a center pivot system
of size j is placed at point i;
M = number of grid areas in the field;
R = peanut yield expected for a center
pivot of size j;
D = peanut yield expected for each grid
area if it is irrigated;
W = Overwatervariablewhichprohibits
multiple yields if a single grid area
is watered more than once;
U = yield per acre for non-irrigated
peanuts;
F = total land available;
Q = peanut quota;
G = peanut volume that may be sold at
the contract price;
i = pivot locations (i = 1, 2,..., N);
j = pivot sizes (j = 1, 2, 3);
к = peanut price levels (quota = 1,
contract = 2, world = 3); and
h = grid areas (h = 1, 2,..., m).
1 Center pivot irrigation technology permits the selection of numerous system sizes. Only three were used in
this analysis since the authors felt that enough alternatives would be provided to illustrate the procedure. Any
number of alternative sizes could be easily included for an actual analysis.
164