A LINEAR-PROGRAMMING SOLUTION TO SALMON MANAGEMENT' 



Brian J. Rothschild^ and James W. Balsiger" 



ABSTRACT 



A linear-programming model was constructed to allocate the catch of salmon among the days of the 

 salmon run. The objective of the model was to derive a management schedule for catching the salmon 

 which would result in maximizing the value of the landings given certain constraints. These con- 

 straints ensured that cannery capacity was not exceeded, and that escapement of both male and fe- 

 male fish was "adequate." In addition to considering the allocation of the catch in the primal problem, 

 the dual problem considered the shadow prices or marginal value of the various sizes of fish, eggs, 

 and cannery capacity, thus enabling the manager to view his decisions in light of the marginal values 

 of these entities. As an example, the model was applied to a run of sockeye salmon in the Bristol Bay 

 system. In the particular example, which was chosen to replicate the 1960 run, the additional value 

 of the catch owing to optimality amounted to an ex-vessel value of a few hundred thousand dollars. 

 In addition it appeared that the required processing time could be reduced by several days. The op- 

 timum allocation was obtained through conformance to the linear-programming model. The cost of 

 this conformance was not, however, determined. 



The Pacific salmon fisheries have been cited as 

 an example of irrational conservation (Crutch- 

 field and Pontecorvo, 1969). Much of this ir- 

 rationality is reflected in the dissipation of 

 a sizable fraction of the available economic 

 rent, a situation which results from the open- 

 access nature of the fishery and legislated in- 

 efficiency. The remedy for this situation is 

 to alleviate the open-access and inefficiency 

 problem. Such alleviation would require the 

 dissolution of rather formidable institutional 

 problems. In the present paper, we examine 

 the salmon problem from a slightly different 

 vantage point than Crutchfield and Pontecorvo. 

 We examine the salmon problem under the 

 status quo; we do not consider the optimal 

 amount of gear or its efficiency (this should 

 not, however, be construed as reflecting any 

 diminution in the importance of these prob- 

 lems); rather we consider, as an interim ap- 

 proach, whether it is possible, under the strin- 

 gent condition of knowing in advance the 

 structure of the run, to increase the value of 

 the fish on the dock by optimally allocating the 



' Contribution No. 333, College of Fisheries, Uni- 

 versity of Washington. 



" Center for Quantitative Science and Fisheries Re- 

 search Institute, University of Washington, Seattle, 

 Wash. 98105. 



' Fisheries Research Institute, University of Wash- 

 ington, Seattle, Wash. 98105. 



catch among the days of the run. 



The traditional approach to salmon manage- 

 ment might be considered, at the risk of several 

 simplifications, as consisting of (1) forecasting 

 the magnitude of the run ; (2) setting an escape- 

 ment goal and a catch implied by the forecast 

 and the escapement; and (3) daily fishing 

 closures and other devices which allocate the 

 catch in varying quantities to the days of the 

 run. The traditional approach, then, also in- 

 volves an allocation of the catch to the days 

 of the run. In the traditional approach, the 

 allocations are usually based on the experience 

 of management biologists. Although the ob- 

 jectives of their allocations are not always 

 clearly and explicitly stated, there is a tendency 

 for the primary objective of management to be 

 simply the attainment of the escapement goal. 

 Our approach is to use the theory of linear pro- 

 gramming to advise on a non-intuitive optimum 

 allocation of the salmon catch among the days 

 of the run where the objective of management 

 does not explicitly involve escapement. Rather, 

 we develop our allocation strategy to maximize 

 the value of the catch on the dock given a va- 

 riety of constraints which include the necessity 

 for a given number of fish to escape the fishery. 

 The objective of maximizing the value of the 

 fish on the dock and the constraints explicitly 

 define the objectives of the management scheme. 



Manuscript received October 1970. 



FISHERY BULLETIN: VOL. 69. NO. I, 1971. 



117 



