ROTHSCHILD and BALSIGER: LINEAR-PROGRAMMING SOLUTION 



"UV 





— 



.45-1 



1.40- 



1.35- 



1.30- 



.25- 



1.20- 



Cannery constraints 

 binding ttiese days 



1 1 1 1 



5 10 15 18 



Day of season 



Figure 7. — Shadow prices for daily cannery constraints 

 — an example in which seasonal entity limits are not 

 imposed. 



As an example of this situation where we have 

 an escapement of 5 billion eggs, consider Figure 

 7, where the curve representing the cannery 

 shadow prices is exactly the value function 

 curve for entity 1 males. If, however, we con- 

 trast the 5 billion egg constraint situation with 

 the actual 1960 run where the seasonal limit 

 constraints are binding for all entities, then the 

 improvement in the objective function attributed 

 to the one unit of increased cannery capacity will 

 only be equal to 



CSPj 



<^ii 



since to catch a fish of entity / on day j another 

 fish of entity i must be released on day j* (i Vj) 

 to avoid breaking the seasonal limit constraint. 

 Figure 6 gives an example of this situation. 

 The particular example is taken from the LP 

 model using the logistic value function and 

 actual escapements of 1960 by entity (thus the 

 seasonal limit constraints). In this case, the 



total number of fish processed remains the same 

 (since season limits are binding for all entities 

 already) , but processing a fish earlier in the 

 season can result in an increased profit because 

 of the shape of the value-function curve. 



The implications of Figure 6 are quite subtle. 

 In Figure 6, until day 4, the cannery is not at 

 capacity, and hence there is obviously no value 

 associated with an increased unit of capacity. 

 All fish are included in the catch allocation in 

 the early-season, high-value situation. On day 4 

 the cannery is at capacity and some fish must 

 be included in the escapement (excluded from 

 the catch). Intuitively, we would expect the 

 highest value fish to be caught. This is partly 

 reflected by the continued catch of all entities 



2 and 4 fish (the large males and females), but, 

 keeping in mind the fact that catches in all 

 entities are being limited by the seasonal limit 

 constraints and the idea of the decreasing pi'ice 

 diff'erential between entity 3 and entity 1 fish, 

 entity 3 fish become part of the escapement. 

 This says that on days 4 to 6, an increase in the 

 capacity of the cannery will result in an entity 



3 fish being caught on that day and a less valu- 

 able entity 3 fish released later in the season 

 (to avoid breaking the season's limit constraint) . 

 We can see in Figure 5 that the last day on 

 which an entity 3 fish can be released is day 11. 

 The value of an entity 3 fish on day 4 is $1,445, 

 the value of an entity 3 fish on day 11 is $1,356. 

 Hence we would expect to gain exactly $0,089 

 by increasing the cannery capacity by one unit 

 on day 4. Checking Figure 6, we see this is 

 exactly what the graph shows for day 4. Days 

 5 and 6 can be calculated similarly. 



On days 7 to 10, entity 1 daily run constraints 

 are no longer binding, and hence an additional 

 unit of cannery capacity could result in an entity 

 1 fish being caught on day 7. From Figure 5 

 the least valuable entity 1 fish in the catch scheme 

 is on day 10, and one of these would go into 

 the escapement. The increased value is 



or 



C|.7 - f'l.io 

 $1,324 - $1,270 



or a profit increase of $0,054. In Figure 6, the 



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