ROTHSCHILD and BALSIGER: LINEAR-PROGRAMMING SOLUTION 



the static nature of the LP approach provides 

 a challenge to application in the sense that the 

 unit values in the objective function, the con- 

 straints, and the right-hand sides of the con- 

 straint equations must not only be known in 

 advance, but also must not change as a result 

 of any of the allocations in the model. 



The above assumptions can be handled in a 

 variety of ways such as those indicated to handle 

 problems of the deterministic nature. For ex- 

 ample, we might, in some instances, use qua- 

 dratic programming to handle the problem of 

 non-linearity or d>-namic programming or apply 

 the outlined procedure in real time to handle the 

 static nature of the programming problem, but 

 unfortunately these approaches will present 

 what can be quite complicated computational 

 difficulties which may, in some instances, be in- 

 surmountable. It is thus clear that we have 

 made certain approximations, trading off real- 

 ism for an easily computable solution which 

 certainly provides management guidance. 



As we implied previously, we do not consider 

 our departures from realism to seriously affect 

 the utility of the model to provide guidance for 

 decision making. Thus we believe that, for ex- 

 ample, fixing the cannery capacity independent 

 of the entities involved (or we could consider 

 the cannery capacity to be fixed at a level which 

 would accept a reasonable mixture of the en- 

 tities) or using a simple average fecundity of 

 the female entities to represent the average 

 fecundity of the spawning females materially 

 affects our conclusions. These, however, can 

 be evaluated in direct applications by a sensi- 

 tivity analysis. 



Having outlined some cautions with respect 

 to assumptions, we can now examine some of 

 the indications provided by the various trials 

 of the procedure. These involve the value of the 

 fish on the dock, a reduction in processing-season 

 length, changing value of entities during the run, 

 and finally future data needs. 



First with respect to the value of the total 

 catch on the dock, we experimented with three 

 value functions which set the daily value of each 

 entity. Using the value functions to determine 

 the value for each entity and day, and the actual 

 distribution of the catch over the 1960 season, 



a total value of the catch was calculated which 

 corresponds to the use of each of the three value 

 functions. These values of the actual allocation 

 of the catch were compared with the value of the 

 optimal allocation as determined by the linear 

 program as an indication of the value of op- 

 timally allocating the catch over the season. The 

 increased value of the optimally allocated catch 

 ranged from approximately $350,000 to $420,000 

 dependent on which value-function curve was 

 considered. Table 3 shows these results. In 

 the table, a fourth value function is indicated, 

 which is simply a straight-line function such 

 that the value of each entity remained constant 

 through the season. Each of the other value 

 functions was determined such that the average 

 value of each curve was equal to the constant 

 value for that entity for the season. 



Table 3. — Comparison of the value of the optional al- 

 location with the value of the actual allocation of the 

 catch for the 1960 season. 



^ After doy 6, the price dropped 3(f per pound. 



2 The price was reduced by subtracting a logistic curve that reduced 



the price of eoch entity by 3tf per pound over the season. 



^ The price was reduced by subtracting a quodratic curve that reduced 



the price of each entity by 3^ per pound over the season. 



* The price for each entity remained constant through the season (actual 

 situation.) 



^ Difference due to rounding in the linear programming algorithm. 



All three value functions had the effect of 

 placing emphasis, in the optimal solution, on 

 catching fish on the early days of the season. 

 For tw'o of the functions the value for any entity 

 of fish on a given day is less than the value for 

 that entity on the previous day. This is not true 

 in the step function and thus we do not have 

 a unique allocation, but rather a set of alloca- 

 tions under the high values and a set of allo- 

 cations under the low values. But results are 

 exactly the same; optimal allocations of fish are 

 identical under the three value functions, al- 

 though the total value of the catch changes some- 

 what, according to the exact shape of the value- 

 function curve. Again, we emphasize that these 

 gains from allocation can only be obtained by 



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