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probabilities were determined with a least square estimation method applied to production
data from the Canadian farming company. These 6 growth models are all based on the same
Markov process with different transition probabilities reflecting faster growth with increased
smolt quality. Tables 1 shows the transition probabilities for pen class p = 1. Fig. 3 shows
the development of the average weight of fish for the different smolt quality classes, and
how the fish of different smolt quality becomes marketable at different times during Ihe year.
The Optimization Models
In the following two optimization models of aquaculture production planning are presented,
taking full account of the size distribution of each of the cages operated by the fish farm.
The models can be thought of as in two levels:
1. The lower level is a growth model for each smolt quality, giving the size distribution
in each cage over the whole period of growing and harvesting, taking into account
partial harvesting of the cages. A Markov process approach is proposed for this level.
2. The upper level model is an optimization model built on top of the growth model. The
first such model presented here is a Linear Programming model for strategic product-
ion planning. To assure feasible harvest plans in the harvest-scheduling model, this
will be formulated as a Mixed Integer Programming model.
Notation
The indices used in the niodels are:
t = time periods 1 ..T, in our case of salmon farming we used months 1-27, the last 12 months being
the harvesting period H where t e H (September-August next year).
í = size classes 1..S, in our case 1 to 8, the last 5 are the marketable sizes M where se M.
p= pen group I ..P, which also can be interpreted as smolt quality classes, 1-6.
The data coefficients are:
Ss(t) = Sales Price for ftsh of size ,v in period t, CAD/kg, ,v e M and t e H, see Fig. 1.
Vfj = Weight of fish in size class s, kg/fish.
Cp(t) = Variable cost (mainly feed cost) accumulated per fish of smolt quality class p harvested in
period t e H, CAD/fish, see Fig. 2.
Rps(t) = Number of físh (1000 fish) of smolt quality p, and size ,v, released in period t. In our case
fish is released only as smolt and only in the first period, so all these coefficients are 0 except
Rpi(l) = Rp which represents the smolt quality distribulion.
Pps(t) = Transition probability for fish of size ,v and smolt class p of moving to size class ,v+l in
period t. The largest size class 5=8 is an absorbing state with Pps(t) = 0. See Table 1 for p = I.
Dmm(t) and D"'ax(t) = bounds on demand in tons in period (e H, Dmax(t) can also be the harvest-
ing capacity in tons if this is less than the upper bound on demand.
The variables in the first of our models are:
fps(t) = state variables indicating the number of fish (1000 ftsh) of size ,v left in pen category p at
the end of time period t. For t=\,fps(l) = 0 except for ,s=l where fp/(l) = Rp, i.e. the smolt.
hpS(t) = decision variables, the number of fish (1000 ftsh) of size ,v g M harvested from pen cate-
gory p in time period t e H.