1. Optimization of production planning in fish farming Páll Jensson University of Iceland Presented at IFORS 2002 in Edinbourgh

2. Background Aquaculture is fast growing New measurement technology (Vaki) => more detailed and better data Market price fluctuations, by fish size More market driven operations Need for software and DSS

3. Goal Decision system for salmon farming (applies also to other farming) Release, growth and harvesting Smolt quality, feeding Size grading, different harvest schemes Processing, distribution to markets Software prototype

4. Literature Review of modeling and IT: Cacho (1997), El-Gayar (1997) MIP (single size): Shaftel and Wilson (1990), Clayton, E.R. (1995) Markov approach, LP: Sparre (1976), Leung et al. (1993), Forsberg (1996)

5. Price fluctuation example (East Canada Salmon)

6. Growth (average weight)

7. Harvesting Methods BH: Batch Harvesting Partial BH: Same size distribution as stock. Full BH: Whole cage harvested once. Special case of GH, SH and Partial BH. SH: Selective Harvesting. Any sizes can be selected from the stock in a cage. GH: Graded Harvesting. Usually “thinning from above”.

8. Data S s (t) = Sales Price size s time per. T W s = Weight of fish in size class s C p (t) = Variable Cost (mainly feed) R ps (t) = No of fish of smolt class p P ps (t) = Transition Prob. s -> s+1 D min (t), D max (t) = Harvest bounds, (Sales Aggrements, Capacities,...)

9. Variables f ps (t) = state variables, no of fish of smolt class p and size s left at t. h ps (t) = decision variables, no of fish harvested y p (t) = 1 if class p is harvested at time t, 0 else.  s  M h ps (t)  R p y p (t) z ps (t) = 1 if size s is smallest size harvested, 0 else.  s  M z ps (t) = y p (t)

10. The Markov model h ps (t) R ps-1 (t-1) s s-1 P ps-1 (t-1) 1-P ps-1 (t-1) f ps (t) t-1 t

11. Two level approach for size graded modeling Lower level: Markov type growth model with size variables for GH and SH (constant distribution for BH): f ps (t) = [1-P ps (t-1)] f ps (t-1) + P ps-1 (t-1) f ps-1 (t-1) Upper level: Objective function and harvesting constraints

12. LP for Aggregate Planning

13. MIP for GH (size graded harvesting)

14. Graded Harvesting Z ps (t) : 0 0 0 0 1 0 0 0  Z ps (t): 0 0 0 0 1 1 1 1 s F ps (t) h ps (t)  F ps (t)  s r=1 z pr (t) h ps (t) = 0 h ps (t) = F ps (t) 0<h ps (t)<F ps (t)

15. No of fish harvested (000)

16. Harvest from a single pen

17. Extensions Various harvesting methods Feeding limits (Norway) Smolt release as decision variables Multi-site operation, transport costs, processing capacities Product mix in processing Other fish species than salmon