FISHERY BULLETIN: VOL. 85. NO. 2 



sampling, in which the size of age subsample in all 

 length strata is constant (i.e., rij = nIL, where n = 

 Zn^ is total age subsample size), and 2) random age 

 subsampling, in which the size of the age subsam- 

 ple in each length stratum is proportional to the 

 length sample size for all length strata (i.e., n^ - 

 nlj). Thus applying Equations (2) and (3), Vartot 

 for a fixed age subsample (Appendix A) is 



«! 0-2 



Vartot = — + -- 



n N 



(4) 



and Vartot of a random age subsample is 



where a^ 



ttp 



b, 



(5) 



,)] 



p,r] 



It should be noted that a^, a2, 6i, and 69 are all 

 positive values. 



The total cost of a survey can be expressed as 

 some function of the costs of the two sampling 

 stages. The commonly used form of the cost func- 

 tion is a simple linear equation relating the unit cost 

 of observing the length and the age of a fish to N 

 and n (Tanaka 1953; Kutkuhn 1963; Southward 

 1963): 



C = CiN + C2W 



(6) 



where C is the total cost, Cj is the unit cost of 

 observing the length of a fish, and Cg is the unit 

 cost of determining the age of a fish. For optimum 

 allocation, the problem becomes one of determin- 

 ing the values of A'^ and n, which will provide an 

 estimated age composition with a minimum Vartot 

 subject to a given total cost C. 



Cauchy-Schwarz inequality, which is frequently 

 used in sampling theory (Cochran 1977; Kendall and 

 Stuart 1977; Schweigert and Sibert 1983), is applied 

 to find optimum set ofN* and n* for an ALK. The 

 set of N* and n* is obtained when they minimize 

 the product of Z)- (= Vartot) and C. 



For a fixed age subsample, using Equations (4) 

 and (6) and the Cauchy-Schwarz inequality (Appen- 

 dix B), we obtain 



(n/N)* = \fa^cja^2- 



(7) 



This quantity is the optimum subsampling ratio (r*) 

 required to reach the minimum (min.) Vartot sub- 

 ject to the cost function given in Equation (6). There- 

 fore, N* and n* are dependent on Equation (6): 



(8) 

 (9) 



(10) 



Similarly, the optimum allocation of a random age 

 subsample can be obtained using Equations (5) and 

 (6) and the Cauchy-Schwarz inequality. The solutions 



of r*, AT* 



n 



and min. Vartot are 



r* = {nlNf = \/b^cJb^2 

 N* = C/(Ci + c.r*) 



n* = r*N* 



bi &2 

 min. Vartot = ^ -1- 



n 



* ' N*' 



(11) 

 (12) 



(13) 

 (14) 



Generally, survey designs are based on two con- 

 straints (Schweigert and Sibert 1983). The first, as 

 derived previously, relates to obtaining the preci- 

 sion of p,, viz., to minimize Vartot at a fixed total 

 cost. The second determines the total cost required 

 to achieve a given precision, that is, to minimize total 

 cost (min. C) at a desired level of Vartot. In the 

 latter problem, r * in Equations (7) and (11) will also 

 minimize the product of D^C irrespective of the 

 value of C and D'~, i.e., it will minimize D'^ for fixed 

 C or C for fixed D" (Kendall and Stuart 1977, Sec- 

 tion 39.20). The solutions of n* and N* are now 

 dependent on the desired level of Vartot. 



For a fixed age subsample, substituting r* of 

 Equation (7) into Equation (4), we obtain 



A^* = (tti/r* + a.)//)- 



n 



= r*N* 



min. C = c^N* + Cj 



n 



(15) 

 (16) 



(17) 



180 



