ROTHSCHILD and BALSIGER: LINEAR-PROGRAMMING SOLUTION 



Table 2.— Computations used to determine the value 

 of each entity classification for the model. 



Entity 



Sex 



Ago 



Average 

 weight' 



Average 

 no. of eggs'* 



Average 

 value 



Male 

 Male 

 Female 

 Female 



2-ccean 

 3-ocean 

 2-ocean 

 3-ocean 



Lb. 

 5.1 

 7A 

 4.5 

 6.2 



Number 



3,700 

 4,384 



S 



),23 

 1.85 

 1.38 

 1.84 



1 Willi the price of salmon at $0.25/pound, the value of each entity 

 can bo calculated as follows: 



entity 1 - 5,1 X $0.25 = $1.28; 



entity 2 - 7.4 X $0,25 = $1.85; 



entity 3 - 4.5 X $0.25 = $1.13; 



entity 4 - 6.2 X $0.25 = $1.55. 



2 With the price of salmon eggs at $0.50/pound, the additional value 

 of each female entity con be calculated as follows (it should be noted 

 that eggs were not processed in I960): 



entity 3 — 3,700 eggs X ,061 g/egg -^ 

 453,6 g/lb X $0,50 = $0.25 



entity 4 — 4,384 eggs X .061 g/egg -1- 

 453.6 g/Ib X $0.50 = $0.29. 



valuable than others. Thus, we deduced that 

 this fixed price must reflect an average value 

 and we note, parenthetically, the important 

 point that the bias and precision (we take the 

 liberty of using these terms in the statistical 

 sense even though the estimation procedure may 

 not be statistical in nature) with which this 

 average is estimated is a subject of significance 

 to the management of the salmon stocks. 



In addition to the value of salmon differing 

 among entities, the value of salmon usually de- 

 teriorates within an entity during a season. 

 Thus, even though a fixed price is paid for 

 salmon during a season, the value decreases 

 owing to a reduction in quality. For example, 

 the value of pink salmon may be 25 7c less near 

 the end of the run than near the beginning of 

 the run. The decline in value of red salmon 

 is not so severe, amounting to a range of about 

 $0.03/pound from the beginning to the end of 

 the season. (While a few cents decline in value 

 during the course of the season may seem to 

 be a negligible quantity, we must remember that 

 this factor must be multiplied by the several 

 pounds in weight of each fish and the several 

 million fish that are involved in the value re- 

 duction.) So just as we deduced $0.25/pound 

 to be an average price among entities, we must 

 likewise deduce that the values tabulated in 

 Table 2 are average values for each entity for 

 the season. In the Naknek-Kvichak run, which 



usually begins around June 27 and ends around 

 July 15, the decline in value appears to be 

 centered on July 4. 



In order to arrive at a unique allocation, we 

 must deduce how the c,y's for each of the ;/ days 

 of the run differ from the average of the cj/s 

 listed in Table 2. The ideal way of doing this 

 would be to develop a model which is descriptive 

 of the value change during the run. Unfortu- 

 nately, we have no information upon which to 

 base such a model, so we used three arbitrarily 

 chosen functions to de.scribe the day-to-day value 

 change of the salmon. An example of these 

 is shown in Figure 1. 



2.0i 



1.9- 



=^0= 





1.8 



1.7- 



Average value 

 of all functions-i 



Function I ( step)' 



Function II ( logistic )'" 



Function III (quadratic)- 



1.64 







5 

 Day 



10 

 of season 



15 



18 



Figure 1. — Value functions as objective function co- 

 efficients in the linear-programming model for entity 2 

 (large male fishes). 



Next we needed to determine the quantity, in 

 the objective function, of N, the number of 

 days of the run. We utilized an empirical equa- 

 tion presented by Royce (1965) to obtain both 

 N and the daily run for each entity Rij. The 

 function Royce used for each of several years 

 to describe the temporal change in the Naknek- 

 Kvichak catch is 



PkU) 



1 



1 +e 



-(ak + biJ) 



(16) 



123 



