Árbók VFÍ/TFÍ - 01.06.1998, Blaðsíða 320
318 Ritrýndar greinar
Additional 0/1 variables:
yp(t) = 1 if pen p is harvested in time period /, 0 else.
zps(t) = 1 if size class s is the smallest size class harvested from pen p in period t in a “largest ani-
mal” harvesting, 0 else.
The HS-model is as follows, noting that the model is run only for the last periods where ftsh
is harvested, with a known starting size distribution fps(0) and all Rps(t) = 0.
(1 a) max z = X/;X,X.V [5,(f) Ws - Cp(t)] hps(t) - XPX, Kp yp(t)
(2) fps(t) = Rps(t) + [1 -Pps(t-\)]fps(t-\) + Pps-,(t-\)fps.,(t-\) - hps(t), \/p,s,t
(3) Dmin(t) < X;,X, Ws hps(t) < Dmax(t), Vr e H
(4) Lp yp(t) < X.veAí hpÁO < Rp yp(t), Vp,t e H
(5) Mp<l,yp(t)<Np,VP
(6) X.veA/ ZpÁt) ~ yP0) , V/?,/eH
(7) hps(t) < Fps(t) Xsr=/ zpÁt), Vp,se M,íe H
(8) hps(t) > [\-Pps(t-\)]fps(t-\) + Pps-i(t-\)fps-,(t-\) - Fps(t) [l-F-'r=/ ZpÁDl Vp,íeM,/eH
fps(t), hpseM(teH) > 0, yp(teH), zpseM(teH) = 0/1
Restrictions (4) ensure that the number of fish in each harvest from pen p is within bounds,
and that nothing is harvested in periods where yp(t) is 0, and (5) restrict the number of har-
vests from each pen. With (6) the zps(t) variables are all 0 if yJt) is 0 (no harvest), otherwise
one of them, for each p and t, is equal to 1, indicating the grading or size selection limit of
the “largest animal” harvesting. The purpose of (7) is to give as tight as possible upper bounds
on the hps(t) variables, so that harvesting is 0 for fish smaller than or equal to size .v-1, and
not greater than Fps(t), the number of fish of size s in the pen without any harvesting, for
size class s and above.
Again the reader is reminded that in (2) and (8) the term [ 1 -Pps(t-1)] fps(t-1 )+Pps_j(t-1) fps
(/-1) is the number of harvestable fish of size s in pen p in time period t. This helps to under-
stand restrictions (8), which give tight lower bounds on the hps(t) variables, i.e. less than 0
(and thereby hps(t) > 0) for size classes up to and including ,v, and equal to the number of
harvestable fish for size ,v+1 and above. This ensures full harvesting of sizes greater than s
but size class s can be harvested partially, fully or not at all.
The HS-model is a mixed integer programming model with P*H + M*P*H 0/1 variables,
P*S*H + P*M*H continuous variables and P*S*H + 2*H + 2*P*H + P + 2*P*M*H restrict-
ions. The model can be applied to individual pens or pen groups, and to all 12 harvesting
months or the next n months. If we use the HS-model for the 6 pen groups and the whole
harvesting period we have 432 0/1 variables, 936 continuous variables and 1470 restrictions.
The number of integer variables can be reduced by eliminating the yp(t) variables, see
restrictions (6), and instead adding the restrictions Xs zps(t) < 1.