384 



Fishery Bulletin 91(2), 1993 



cja-u,) 

 c 9 ,(b+v-m) . 



(9) 



For a survey design subject to a given total cost C, 

 r,* is the optimal subsampling ratio required to reach 

 the minimum loss function value (min. £). Thus, sub- 

 stitute Eq. 9 into Eq. 6 and solve for the optimal set of 

 N* and n,*: 



AT: 



c x +%c 2l r\ 



n, = r\ AT, 



l /a-u, \ b+v-m 

 min. £ = X — 1 + 



'=' n, 



AT 



(10) 



For a survey design subject to a desired precision level 

 of £, r,' is the optimal subsampling ratio to reach the 

 minimum total cost (min. C). The optimal set of N* and 

 n,* can be obtained by substituting Eq. 9 into Eq. 5: 



(11) 



Similarly, the above derivation can be extended to the 

 traditional fixed- and proportional-age subsampling 

 schemes. For these age subsampling schemes, the per- 

 unit cost for ageing a fish is not length-specified (i.e., 

 c 2i =c 2 for all i's). The loss and cost functions in Eq. 5 

 and 6 are modified according to the definition of the 

 two age subsampling schemes: (1) n=n/L for fixed- 

 age subsampling, and (2) n, =n 1, for proportional-age 

 subsampling, where n = Sn,. The loss function for fixed- 

 age subsampling is 



X L (a ,-«,) b+v-m 



£ - H + 



n N 



and that for a proportional-age subsampling is 



(12) 



£= t 



X L (a r u,)l l t b+v-m 



(13) 



n 



N 



C = Cl AT+ c,n 



(14) 



Using the Cauchy-Schwarz inequality, the optimal 

 subsampling ratio (r*) for either minimizing J^'at a fixed 

 total cost C or minimizing total cost C at a desired 

 precision level of £ for a fixed-age subsampling is: 



r = 



AT 



c,IL (a r u,) 

 c, (b+v-m) 



(15) 



The optimal set of N* and n* and min. £ subject to 

 fixed cost is: 



AT = c l + c 2 r , 



n =r'AT, (16) 



L 



X L (a,-u,) b+v-m 



min. £ = nl + . 



n N 



and the optimal set of N* and n' and min. C subject to 

 a desired precision level of ^'is 



AT 



Z(a— u t ) 



M + b+v-m 



n'= r'N", 

 min. = 0,^+ c 2 n' 



(17) 



For proportional-age subsampling, the optimal 

 subsampling ratio (r) for either minimizing £ at a 

 given total cost C, or minimizing total cost C at a 

 desired precision level of £, is: 



c x X (a-u,) I l t 

 Co (b+v-m) 



(18) 



The optimal set of N' and n' and min. £ subject to a 

 given total cost C is 



(19) 



The cost function for both age subsampling schemes 

 is: 



and the optimal set of N* and n' and min. C subject to 

 a desired precision level of ^'is: 



