ROTHSCHILD and BALSIGER: LINEAR-PRCXSRAMMING SOLUTION 



The Pi's and the L/s are constants appropriate 

 to a particular problem. The details of the LP 

 (linear-programming) procedure can be found 

 in the many treatises on the subject (e.g., Gass, 

 1964) or in most texts on operations research 

 (e.g., Hillier and Lieberman, 1967). 



In our application of the LP model, we max- 

 imize the following objective function 



A/ N 



Z = I Z CijXij, (1) 



where M refers to the total number of age-sex 

 categories and N refers to the days of the run. 

 The variable A',,- is the number of fish caught 

 in the ith entity on the jth day of the run and 

 cij corresponds to the value of the fish caught 

 in the ith entity on the yth day (Table 1). The 

 age-sex category classification results from the 

 fact that salmon runs are comprised of a va- 

 riety of age-groups. Because each age-group 

 is usually of a different average size, the indi- 



Table 1. — Linear program model notation. 



Af — The total number of age-sex categories. 



N — The total number of days in the run. 



Xij — The number of fish in the t'th oge-sex category which are 

 caught on day ; of the run. 



C-- — The value of a fish caught in the tth oge-sex category on 



day y of the run. 



R — The number of fish in the ith age-sex category which run past 



the fishery on day ; of the run. 



Kj — The copocity, in numbers of fish, of the canneries on day ;' 



of the run, 



K — The total seosonol copocity of the canneries in numbers of fish, 



J¥^j — The number of fish of the ith oge-sex category in the escape- 

 ment on day ;' of the run. 



a — The overage number of eggs in each fish of the ith oge-sex 



category. 



T — The total number of eggs contained in the escopement and 



catch. 



£ — The minimum number of eggs required in the escapement, 



^ — The totol number of moles in the escapement and catch. 



F — The average fecundity of the female oge-sex cotegories, ex- 



pressed in number of eggs, 



f{ — The sex ratio desired in the escapement, expressed as the 



number of females per mole. 



/.,- — The number of fish of the ith oge-sex category desired in 



the season's escapement. 



S — The number of fish in the totol season run of the ifh oge-sex 



category, 



P(j) — The proportion of the run thot arrives by doy ; of the run. 



P' (j) — The proportion of the run that orrives on day > of the run. 



viduals in each age-group also have a different 

 average value which we denote by cij. It should 

 be mentioned that size is not the only criterion 

 which can be used for classification. For ex- 

 ample, in the Naknek-Kvichak run of Bristol 

 Bay, the sex of the fish can also be used be- 

 cause within an age-group the male fish tend to 

 be larger than the female fish and thus more 

 valuable in terms of weight of fish-flesh; but, 

 on the other hand, the eggs of the females are 

 a valuable commodity and thus the per-pound 

 value of females may be greater than the per- 

 pound price of males. If the value of the fish 

 were constant during the course of the run, we 

 could replace the C;j with Cj and the allocation 

 problem would become rather uninteresting. 

 But the value, however, does tend to vary dur- 

 ing the course of the run. One reason for this 

 is a deterioration of the quality of fish, as in- 

 dicated by declining oil content and reduction 

 in color intensity with the progression of the 

 run. Another way in which Cy could vary is 

 that the average value of the fish on a par- 

 ticular day would tend to vary during the course 

 of the run because of a within-entity trend in 

 the average size of the fish during the course 

 of the run; this, however, is not considered in 

 the present paper. It is obvious that, if we 

 had sufficient information, we could establish 

 a large number of different c,/s. 



As indicated previously, equation (1) is max- 

 imized subject to a variety of constraints. For 

 the salmon problem, the first set of constraints is 

 rather obvious and constrains the catch, of any 

 entity, on any day, to be less than, or equal to, 

 the number of fish in that entity in the run. 

 These constraints are of the form 



<Y„ 



(2) 



Xij is always ^ and Rij is the number of fish 

 of the ith entity which run past the fishery 

 on the yth day. There can be as many asM x N 

 constraints of this form, but in some applica- 

 tions, either the number of entities or the num- 

 ber of days will be collapsed owing to either the 

 nature of the problem or a lack of information. 

 Note also we can easily "close" the fishery for 



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