ROTHSCHILD: DEFINITION OF FISHING EFFORT 



we eliminate the variables corresponding to the 

 catch of small boats from our optimal solution. 

 The interpretations of the sensitivity of the 

 85,140-ton constraint upon the maximum catch 

 of the 400-ton boats and the 31,840-ton con- 

 straint upon the maximum catch of the 500-ton 

 boats are identical. 



It is perhaps more subtle that the full utiliza- 

 tion of the excess yellowfin tuna capacity is im.- 

 possible because the 85,140 tons of fish that 

 would be caught by the 400-ton boats consists of 

 85,140 X 0.48 = 40,867 tons of yellowfin tuna 

 (the 0.48 is the appropriate technological con- 

 straint). To use up the yellowfin tuna surplus 

 we would need to catch roughly an additional 

 50,000 tons of yellowfin tuna, but if we did this 

 we would need, by virtue of our technological 

 constraint, .to catch a total of 90,000 (0.48) "^ 

 tons of fish which clearly exceeds the fleet ca- 

 pacity. With respect to the technological con- 

 straints, we could in the 300-ton boats, for ex- 

 ample, increase the right-hand side of the equal- 

 ity to 23,460, which would modify the solution by 

 eliminating any catch of skipjack by the small 

 boats (in other words, 7/i2 would be eliminated 

 from the optimal solution). On the other hand, 

 we could reduce the equality to — 30,967, and if 

 we did this, the catch of yellowfin by small boats 

 would be eliminated from the solution [ (30,967) 

 (1.32)-!] = 23,460. The negative right-hand 

 constraint reflects more upon the nature of the 

 solution than reflecting any physical meaning. 



It is clear that since we used all the capacity 

 of our hypothetical fleet that any increase in 

 profits will not induce us to catch more fish. On 

 the other hand, by inducing a negative profit we 

 can show that in these instances some of the 

 boat-species combinations should not be filled to 

 capacity (Table 3) . Thus we would have to lose 



Table 3. — The lower bound of profit and "sensitivity" 

 for yellowfin and skipjack tuna caught by various size 

 classes of fishing boats. The results are reported in 

 dollars. 



Species 

 of tuna 



Boat 

 class 



Profit 



per ton 



in problem 



Lower 



bouncJ 



of profit 



"Sensitivity" 



$5.54 per ton of yellowfin to generate empty ca- 

 pacity space in class 1 vessels. The diflference be- 

 tween the lower bound profit and the profit used 

 in the problem is a measure of sensitivity. We 

 note for example that the behavior of the fleet is 

 most sensitive in class 3 boats where a $12.00 de- 

 cline in profits would generate excess fleet ca- 

 pacity, or a $16.82 decline in yellowfin profit 

 would also generate excess fleet capacity. 



Now let us make an apparently slight but im- 

 portant modification in our problem. We will 

 keep everything the same, but we will increase 

 the capacity of the small boats from 23,460 tons 

 to 65,000 tons. In the first example we were in- 

 terested, primarily, in the sensitivities of our 

 model to changes in the constraints. Now, how- 

 ever, it is of interest to compare the optimal so- 

 lutions in the two examples (Table 4) . Thus by 

 adding an extra 42,000 tons of capacity to the 

 small boats, we increase the skipjack catch by 

 only 16,000 tons and the yellowfin catch by 

 22,000 tons. We have not, owing to the con- 

 straints, caught an additional 42,000 tons of fish. 

 We have caught proportionately more yellowfin 

 than skipjack, increasing the optimal solution 

 from $1,248,835 to $1,562,133. In the second 

 example, in contrast to the first, we have used 



Table 4. — A comparison of optimal solutions in Example I where the capacity of the small boats is 23,460 tons and 

 in Example II where the capacity of the small boats is 65,000 tons. The comparison shows the optimal allocation 

 in tons of fish for each example. 



675 



