ROTHSCHILD and BALSIGER: LINEAR-PROGRAMMING SOLUTION 



ber of fish of any entity is fixed by the seasonal 

 limit, it is more profitable to catch the small, 

 less valuable, males earlier in the season, which 

 may seem contrary to the intuition. The impli- 

 cations of the optimal allocation are considered 

 in the discussion and conclusions section. 



=vk 





125- 



.100- 



,075- 



§ 



■S .050- 



I 025- 



Cannery conslroints 

 binding these days 



o-^ 



Entity 1 daily run 

 constroints not binding 

 day 7 or after 



10 

 Day of season 



15 



18 



Figure G. — Shadow prices for daily cannery constraints 

 — an e.xample in which seasonal entity limits are imposed. 



As indicated previously, the shadow prices 

 are useful in considering various management 

 implications. We consider, as examples, the egg 

 shadow prices, the cannery shadow prices, and 

 the run shadow prices. It might be mentioned 

 somewhat parenthetically that although the 

 shadow prices can be explained and interpreted 

 as in the following paragraphs, in the LP calcu- 

 lation, they are not found in this manner. The 

 shadow prices are calculated as simultaneous 

 results of an iterative solution procedure and 

 include the results of pi-evious iterations. In 

 fact, the shadow prices associated with each 

 constraint at the end of each iteration are used 

 to determine how to manipulate the matrix to 

 improve the objective function in the subsequent 



iteration. With very involved problems, it might 

 not be possible to examine the shadow prices 

 as below, and in any case, only a good deal of 

 insight into the problem permits their delinea- 

 tion in this manner. One other caution is that 

 while applying the following equations to deter- 

 mine a total increase in profit, care must be taken 

 to see that the same constraints remain binding 

 or nonbinding. Once a constraint changes from 

 binding to nonbinding, the solution basis 

 changes, also changing the relationships between 

 variables and constraints. 



The increase in value of the objective func- 

 tion corresponding to a relaxed egg constraint 

 (allowing one more egg in the catch) is the shad- 

 ow price associated with the egg constraint. The 

 shadow price is dependent upon which, if any, 

 of the constraints in the LP model are binding. 

 If the egg catch constraint' is not binding, indi- 

 cating that the value of this catch scheme is not 

 being limited by this constraint, then the shadow 

 price associated with the egg constraint is zero. 

 If the egg catch constraint is binding, shadow 

 prices associated with the egg constraint, de- 

 pending upon which of the other constraints 

 are effective, can be calculated. 



The imputed value of an egg, its shadow price, 

 thus depends on whether the cannery constraint 

 is binding. Now, if the cannery constraint is 

 binding on day j (indicating that the max- 

 imum number of fish are being processed and 

 the addition of a single or marginal fish to the 

 processed catch requires that a fish already ex- 

 isting in the catch must, to maintain the con- 

 straint, be replaced by the marginal fish), and 

 the entity 4 run constraints are binding through- 

 out the season (indicating that all large females 

 are caught), then the shadow price associated 

 with the egg catch (ESP) constraint is 



ESP = (C7.J -f|,,)/3,700 

 where day / is a day on which the entity 3 run 



' Although the egg constraint arose from a minimum 

 egg escapement requirement, it was necessary to con- 

 vert it to a maximum egg catch requirement constraint 

 for use with the LP model, as was described earlier. 

 It is convenient to think of the constraints in terms of 

 "egg catch" for purposes of discussing the shadow 

 prices. 



127 



