ROTHSCHILD and BALSIGER: LINKAR-PROGRAMMING SOLUTION 



This constraint can also be formulated in a va- 

 riety of ways. 



In addition to escapement goals established 

 in terms of eggs and a sufficient number of males 

 to fertilize the eggs, we desired to establish, for 

 some sample problems, escapement goals in 

 terms of total numbers of fish of each entity 

 in the escapement; this would greatly simplify 

 the simulation of the actual escapements for any 

 historic salmon season. Hence, we developed 

 the constraint 



m 



^W„ 



maximize 2- r,vV/ 



(11) 



( = / 



m 



subject to Z, fl//'V, ^ /), 

 / = l 



j=\ n (14) 



where we have used slightly different notation 

 than in the salmon problem, but the analogue 

 between (14) and the salmon problem should, 

 nevertheless, be quite clear. If (14) is the 

 primal, then the dual of this primal is 



where L; is an escapement goal set for entity 

 i which would ensure escapements identical with 

 any historic year. To make (11) conform to 

 our constraint set, we note that 



inimize L ^jYj 



Zn-'y +Y.Xij = Si 



(12) 



where Wij and Xij are the escapement and catch 

 (respectively) of entity / on day j and Si is the 

 total season's run of entity (', we can substitute 

 (11) into (12) and get the constraint in the 

 proper form: 





Li. 



(13) 



Thus a seasonal limit can be placed upon the 

 catch of any entity /. 



As indicated earlier, once the objective func- 

 tion and the constraints have been formulated, 

 then optimization of the objective function, given 

 the constraints, is a standard procedure out- 

 lined in some detail in the literature, and, with 

 a computer facility, is a relatively simple task. 

 There are. however, interpretations which are 

 of further interest than the solution of the ob- 

 jective function. These interpretations rest in 

 the primal-dual relationship of the LP problem. 



In order to demonstrate this relation, we will 

 denote a general form of the primal problem as 



./ = l 



subject to L aijYj ^ c, 



/ = l m (15) 



In the primal we are allocating scarce resources, 

 the hj's (number of fish in the run. cannery 

 capacity, egg complement, and male fish) , among 

 the / activities (which consist in our problem 

 of catching a particular entity of fish on a par- 

 ticular day). The intensity of the activity is 

 Xi, the catch of the ith entity on the yth day 

 and the "profit" per unit of the activity is, of 

 course, cj. It might be mentioned at this point 

 that most LP computer codes provide as out- 

 put the value of slack variables which in (14) 

 is the difference between the right-hand side 

 and the left-hand side of the constraint equa- 

 tions. The slack variables have a rather im- 

 portant interpretation for the salmon problem 

 in that the slack variable in the run constraint 

 (2) is the escapement; in the cannery con- 

 straint (3), it is the unused capacity of the 

 cannery which we might want, in viewing the 

 problem from a different context, to minimize; 

 in the egg constraint (7), it is the number of 

 eggs which are not caught that could be caught 

 — another quantity which we might wish to 

 minimize — and finally we have the slack var- 

 iable associated with constraint (10) denoting 

 the number of males which are not caught, but 



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