FISHERY BULLETIN; VOL, 69. NO. I 



We consider these problems in three additional 

 sections. In the first, we describe the linear- 

 programming allocation model, which we be- 

 lieve to be applicable, with simple modifications, 

 to a variety of salmon management situations. 

 In the second, we consider how the model might 

 be applied to a run of salmon in the Naknek- 

 Kvichak system of Bristol Bay, Alaska. As an 

 example, we choose data from the 1960 run 

 to that system and obtain an optimum allocation 

 of large and small, male and female fish, on each 

 day of the run to the daily catch. This optimum 

 allocation served to maximize the value of the 

 fish on the dock subject to constraints which 

 ensured that the catch did not exceed the daily 

 run, that the catch would be less than the can- 

 nery capacity, and that an "adequate" escape- 

 ment, both in terms of the number of eggs and 

 sex ratio, passed the fishery. Thus, in addition 

 to managing the run by a non-intuitive optimum 

 allocation and satisfying an escapement goal, 

 we also considered the quality of the run in terms 

 of its sex and age composition. In order, how- 

 ever, to achieve this optimum allocation we 

 needed certain data on the structure of the run 

 in advance and we also needed a mechanism 

 by which we could select large and small male 

 and female fish. It would most likely be im- 

 practical to have either a precise prediction of 

 the daily run or an ability to select, with high 

 precision, large or small, male or female fish. 

 We show that even if we had the necessary data, 

 a technique for precise selection of the various 

 entities of fish, and maintained the 1960 escape- 

 ment and sex-ratio conditions, optimum alloca- 

 tion would yield us a catch having a value of 

 several hundred thousand dollars more than the 

 actual catch. Thus given the cost of obtaining 

 the necessary information to perform the op- 

 timum allocation and the constraints extant in 

 1980, it is questionable whether biological man- 

 agement could yield a better allocation than that 

 which was obtained. This serves to re-empha- 

 size the approach of Crutchfield and Pontecorvo, 

 indicating that the system is most sensitive to 

 variables which lie outside the objective and 

 constraint equations specified in the present 

 paper. On the other hand, our results show 

 that it is possible, at least in terms of the model. 



to reduce the number of days during which the 

 cannery operates and yet process the same num- 

 ber of fish. Furthermore as previously indicated, 

 we constrained our example to fit the statistics 

 of the 1960 run and thus we had, in our ex- 

 ample, a nearly 1:1 sex ratio; but as we indi- 

 cate later, we could have caught a considerably 

 larger number of male fish and still would have 

 had sufficient male fish in the escapement to en- 

 sure the efficient production of fertilized eggs. 

 And finally the model was quite sensitive to de- 

 creasing the escapement but unfortunately there 

 is little guidance in the literature which would 

 indicate the optimum escapement for the Nak- 

 nek-Kvichak system and furthermore there ap- 

 pears to be little hope of learning the magnitude, 

 in the reasonably near future, of the optimum 

 escapement for the Naknek-Kvichak system. 

 Thus evaluation of the cannery processing time, 

 catch problem, and relaxation of sex ratio and 

 escapement constraints might result in an ad- 

 ded value to the catch which would make some 

 attempts at allocation practical. We also, in 

 the second section, place some stress on in- 

 terpretation of the shadow prices of the var- 

 ious variables in the problem. This is of in- 

 terest to operations researchers because it 

 provides an example, in addition to those con- 

 ventionally used, of an application of the inter- 

 pretation of the linear-programming primal- 

 dual relation. The shadow prices are of interest 

 to the fishery manager because from them it is 

 possible to impute values to the various resources 

 under the manager's control, and, in making a 

 decision, the manager can thus consider these 

 values which, as we show, are not always intui- 

 tively obvious. In the third and final section 

 we conclude the paper with a general discussion 

 of salmon management in a linear-programming 

 setting. 



MODEL 



Most linear-programming models generally 

 involve finding values A'j which maximize (or 

 minimize) an objective function i;r,A',-, subject 

 to a set of constraints each of which has the 

 form :^PiXi « Lj, where the inequality can be 

 in either direction or can, in fact, be an equality. 



118 



