Árbók VFÍ/TFÍ - 01.06.1998, Síða 319
Optimization Models of Aquaculture Operations 317
The Aggregate Planning model: The first model is an aggregate production-planning
model, called the AP-model. This is a Linear Programming model for the whole growing
period of 27 months, 15 months of grow-out only plus 12 months of growing and harvest-
ing, where te H.
(1) max z = 'LpLiLs [5v(/)W., - Cpm hps{t)
(2) fps(t) = Rps(t) + \\-Pps(t-l)]fps(t-\) + Pps.,(t-l)fps.,(t-l) - hps(t), Vp,s,t
(3) D"ún(t) < LPLS Ws hps(t) < Dmax(t), V/ e H
fps(t). hpse m(JsM) > 0
Restrictions (2) are the Markov type growth model with smolt release R (t) and harvesting
lips(t), and (3) are market and capacity restrictions. We note that during the harvesting time
íeH, the term 11 -Pps(t-1)] fps(t-1) + Pps.,(t-\) fps.,(t-\) in (2) is the number of harvestable
fish of size s in pen category p in time period t, and fps(t) is the nurnber of fish left at the
end of period t after harvesting. Natural mortality can be included in (2) in at least two ways,
either implicitly in the transition probabilities or explicitly like in Forsberg (1996).
This LP-ntodel has P:|:S*T + P*M*H variables and P*S*T + 2*H restrictions, in our case
6*8*27 + 6*5*12 = 1656 variables and 6*8*27 + 24 = 1320 restrictions. Of course, further
constraints on the demand can be added to the model, for example for certain size claSses if
the market conditions require so.
The AP-model is thought for long term strategic planning purpose to analyze how the
different smolt quality categories shoukl be used to meet market requirements in different
periods. In addition to the values of the harvesting variables il gives the following valuable
information:
1. The shadow prices on the smolt categories provide the hatchery part of the company
with information regarding the selling and buying of smolt of different sizes.
2. The shadow prices on the demand (and capacity) restrictions show if buying fish from
other sources should be used as means to satisfy the market, or if investment in increased
market or capacity is justified.
The AP-model can not be used for short-term harvest scheduling, as the size selection
proposed by the model can be infeasible. The model might for example suggest harvesting
size classes 4, 5 and 8 but not 6 and 7. This is not acceptable for the farming companies with
present harvesting technology and practice.
The Harvest Scheduling model: To deal with this an exlended model for harvest schedul-
ing, the HS-model, is proposed, where integer variables have been added to assure feasibility
of size selection. The following is added to the list of data coefficients:
Fps(t) = the number of fish (1000 fish) of each size s that would be in pen p at any time t if no har-
vesting took place. This forecast can be calculated by running the model with hps(t) = 0.
Mp and Np = lower and upper bound on number of harvests from pen p.
Lp and Rp = lower and upper bound on nuniber of fish (1000 fish) in each harvest from pen p.
Kp = Cost of each harvest from pen p.