FISHERY BULLETIN: VOL. 69. NO. 1 



It is interesting to note that the run shadow 

 prices of entity 4 throughout the season are high- 

 er than those of entity 2 (see Figure 9), even 

 though the value for an entity 4 is less, for any 

 given day, than the value of entity 2. This is 

 due to the optimal catch scheme which has 

 catches of entity 4 until day 13, while entity 2 

 catches are only made until day 11. The effect 

 is that when an additional fish of an entity is 

 caught early in the season (requiring the re- 

 lease of a fish of the same entity later in the 

 season), the entity 2 fish released is from day 

 11 of value $1,812, the entity 4 fish released 

 is from day 13 of value $1,780. The value of 

 the entity scheme then, decreases least (propor- 

 tionately) for releasing the entity 4 fish. 



DISCUSSION AND CONCLUSIONS 



It can be seen that the linear-programming 

 (LP) approach to salmon management, as with 

 all other modelling approaches, involves a va- 

 riety of assumptions which are either intrinsic 

 to the procedure or to the way the procedure 

 is applied to real-world problems. We have gone 

 into some detail to show the richness of inter- 

 pretations that the salmon model affords, and 

 we believe that the application of this procedure 

 to salmon management will provide increased 

 guidance to and widen the spectrum of possible 

 management decisions. The procedure we have 

 used for the Naknek-Kvichak run is widely ap- 

 plicable to a variety of situations both within 

 the Naknek-Kvichak sockeye salmon setting and 

 to other salmon runs as well. The setting, the 

 necessary data, and the formulation of the 

 problem really depend on the problem situation. 

 Our purpose was to demonstrate a conceptual 

 method and we have chosen our data and ex- 

 amples accordingly. 



There will naturally be difl'erences of opinion 

 in the formulation of the model (that is, dif- 

 ferent ways of expressing the constraint equa- 

 tions, some of which are indicated) or the ap- 

 propriate data which should be used for actual 

 management situations. These differences can 

 at times be easily resolved by examining the 

 sensitivity of the model to various data or 

 formulation modifications. 



Nevertheless, we should not ignore the as- 

 sumptions which are implicit in the LP pro- 

 cedure. These are outlined by, for example, Hil- 

 lier and Lieberman (1967), and constitute three 

 concepts that should be recognized. These in- 

 volve the linearity property of the model, the 

 problem of divisibility, and the deterministic 

 nature of the LP approach. In addition, it is 

 important to consider that the LP approach 

 models only static situations. First, the linear- 

 ity property asserts, for example, that the value 

 of any term in the constraint or objective func- 

 tion must be directly proportional to the level 

 of activity involved. Expressed in another way, 

 the relation between the level of an entity and 

 its contribution toward filling the constraint or 

 modifying the objective function must be a 

 straight line passing through the origin, a con- 

 dition not often met in practice but frequently 

 approximated. Furthermore, there should be 

 no synergism among the terms of the objective 

 or constraint functions. For example, the unit 

 catch value on day j for the tth entity, ca, can- 

 not be affected, a posteriori, by the unit catch 

 value on day /-l for the (th entity c;, j.i- The 

 problem of divisibility refers to the fact that 

 the LP approach that we have used gives solu- 

 tion values which are not necessarily integers. 

 A usual practice is to round solutions to the near- 

 est integer value, thus avoiding the embarrass- 

 ment of having, e.g., 7,012,342.631 salmon. In 

 other applications, such as allocating 10 fishing 

 boats, say, to perhaps three fishing grounds, the 

 possibility of having non-integer answers may 

 lead to erroneous conclusions and one of the 

 integer programming techniques would then be 

 most appropriate. Next, the deterministic na- 

 ture of the LP approach is, of course, a deficiency 

 in the probabilistic real world. The manager 

 must realize that a full stochastic treatment of 

 the salmon allocation as an optimization problem 

 would most likely be a very difficult task. As 

 alternates, an error structure could be applied 

 to various elements in the problem, thus enabling 

 one to explore a variety of probabilistic phe- 

 nomena, or chance-constrained programming 

 might be employed. Monte Carlo and simula- 

 tion approaches might also be utilized but these 

 are not per se optimization procedures. Finally, 



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