ROTHSCHILD and BALSIGER: LINEAR-PROGRAMMING SOLUTION 



tion for the first 3 days of the season. This 

 is not true for entity 3 or 4 since the egg-catch 

 constraint is binding, indicating that no more 

 eggs can be "caught" without breaking the con- 

 straint. Hence, when an entity 4 female is 

 caught on a day 1 to 3 (catching 4,384 eggs), 

 4,384 eggs must be released on some other day 

 of the season. This can be done with the smallest 

 loss by releasing 4,384/3,700 entity 3 females 

 (entity 3 females have 3,700 eggs). To further 

 decrease the loss, the above fractional parts of 

 an entity 3 should be released on a day on which 

 there are entity 1 or 2 available to be caught, 

 since this will keep the cannery constraint in- 

 tact and not violate the male catch constraint 

 since it is not binding. Remembering that the 

 price differential between entity 1 and entity 3 

 increases over the season, it is desirable to catch 

 the additional entity 1 fish as early as possible, 

 which is on day 7. Then (note that all of the 

 larger males are already in the catch scheme) : 



RSP4J = c-4.,- ^Q,7+ ^ ci,7 

 4,384 



= $1,941 



3,700 



$1,419) 



+ AMI ($1,324) 

 = $1,828, 



which can be seen on the graph as the shadow 

 price for entity 4 day 1. To calculate the value 

 of the run shadow price for entity 3 in days 1 

 to 3, we must realize that catching an entity 3 

 female requires the release of a female some 

 other time during the season in order to avoid 

 breaking the egg catch constraint. Since the 

 entity 3 females are of lesser value than the 

 fractional part of an entity 4 female which must 

 be released to account for the 3,700 extra eggs 

 caught, the release of this fish on a day on which 

 there are entity 1 fish available will lessen the 

 loss. As before, the price differential between 

 entities 1 and 3 is smaller early in the season, 

 and the earliest date on which an entity 1 male 

 is available is day 7. For example, 



RSPy,i = C3.1 - r3.7 + ci.7 



= $1,453 - $1,419 + $1,324 

 = $1,358, 



which is the value shown in Figure 8. 



The run shadow price for entity 1 day 4 re- 

 sults from the catch of entity 1 on day 4 requir- 

 ing the release of entity 3 (to avoid breaking 

 the cannery constraint) which, in turn, allows 

 the catch of an entity 3 and the release of an 

 entity 1 on day 11 (the last day on which there 

 are entity 3's not in the catch scheme). Essen- 

 tially, what we have done is to exchange the 

 catch of entity 1 and entity 3 for a time when 

 the price differential is more favorable. 



RSP\4 =C],4 - C3,4 + C3,7 - <^1.7 



= $1,353 - $1,445 + $1,419 - $1,324 

 = $0,003, 



which is the value shown in Figure 8. Run 

 shadow prices for entity 1 days 5 and 6 can be 

 figured similarly. For entity 1 days 7 to 15, 

 the daily run constraints are not binding, and 

 hence, the run shadow prices are zero. 



Run shadow prices for entity 2 days 4 to 15 

 (still referring to Figure 8) can be calculated 

 as the difference between the value of an entity 

 2 and entity 1 on those days. The entity 1 must 

 be released in order to satisfy the cannery con- 

 straints for those days. In addition, for days 

 4 to 6, calculating the value of this new avail- 

 ability of an entity 1 released in order to catch 

 an entity 2 is essentially the same situation as 

 calculating the shadow price of entity 1 on those 

 days, and hence, the value of the scheme and 

 the run shadow price is increased by that addi- 

 tional amount. 



Run shadow prices for entity 3 days 4 to 7 

 in Figure 8 are zero since the run constraints 

 of entity 3 for those days are not binding. For 

 entity 3 days 8 to 15, they can be calculated as 

 follows: If another entity 3 is caught on day 8, 

 an entity 1 must be released on day 8 to main- 

 tain the cannery constraints. The catch requires 

 the escape of an entity 3 fish on another day to 

 maintain the egg catch constraint. In turn, if 



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