FISHERY BLLLETIN: VOL. 69, NO. 1 



any entity or all entities on any day simply by 

 setting the appropriate Rij = 0. 



The second set of constraints constrains the 

 catch on any day to be less than the daily ca- 

 pacity of the canneries. Thus, we have the set 

 of constraints, 



M 

 i = \ 



K: 



(3) 



where Kj is the capacity of the cannery or can- 

 neries on the yth day. Another form of this 

 constraint might be incorporated in situations 

 such as in the Bristol Bay fishery, which has a 

 short season, is remote from supply points, and 

 thus has a finite seasonal capacity; these con- 

 straints can be expressed by 



I L A', 



(4) 



/ = i 



= 1 



The constraint is redundant if K ^ L Kj. 



J = I 

 and hence is only used when the season's ca- 

 pacity is less than the sum of the daily capa- 

 cities, i.e., 



K' 



Ki. 



y=i 



The next set of constraints results from the 

 need to ensure that an adequate number of fish 

 escape the fishery and are thus permitted to 

 spawn. We formulate this constraint in terms 

 of the egg complement of the number of females 

 escaping the fishery rather than the number of 

 females escaping per se or the total number 

 of fish (males and females) escaping. 



In order to formulate this constraint set, we 

 define T as 



XL ajWij + LXa//Yy = T 



(5) 



where a; is the average number of eggs in each 

 fish in the ith entity, Wij is the escapement of 

 fish in the /th entity on the jth day and thus 



T is the egg complement of the escapement and 

 the catch. Now if we need a minimum number 

 of eggs to represent the escapement, a quantity 

 which we denote by E, we must have 



ZZa,H',/ & E. 



(6) 



So substitution of (6) into (5) yields the con- 

 straint set conformable to the Xi/s of our other 

 constraints, viz. 



ZI a,Xii ^ T-E. 



(7) 



We can see that by using the same reasoning we 

 could construct a consti'aint set which would 

 constrain the egg complement to be less than 

 some maximum egg complement. 



Our next constraint involves the sex ratio of 

 the spawning fish. The utility of allowing a 

 particular egg complement to escape the fishery 

 could be negated by not allowing a sufficient 

 number of males to escape for the purpose of 

 fertilizing the eggs. In order to ensure that 

 an adequate number of males escape the fishery, 

 we formulate the necessary constraint by noting 

 that 



Z M'/ + Z Xj = .^i 



(8) 



where in this particular equation the TF's refer 

 to the number of males in the escapement and 

 the A'''s to the number of males in the catch and 

 M to the total number of males in the run. Now 

 to satisfy our constraint, we must have 



ZlV, 



£ X /y 



(9) 



where the sum extends over all male entities 

 and days of the run; F is the average fecundity 

 of the female entities in the run, and H is the 

 desired male to female sex ratio. Substitution 

 of (9) into (8) then yields the desired con- 

 straint 



Z,V,y 



J( 



£,■ X W 



(10) 



120 



