Theoretical studies on the necessary number of components in mixtures

Theoretical studies on the optimal numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to phenotypic yield stability (measured by the parameter ‘variance’). For each component i, i = 1, 2,..., n, a parameter u_i with 0 ≦ u_i ≦ 1 has been introduced reflecting the different survival and yielding ability of the components. For the stochastic analysis the mean of each u_i is denoted by u ₁ and its variance by σ ²_i For the character ‘total yield’ the phenotypic variance V can be explicitly expressed dependent on 1) the number n of components in the mixture, 2) the mean \(\overline {\sigma ^2 } \) of the σ ²_i 3) the variance of the σ ²_i 4) the ratio \({{\overline {\sigma ^2 } } \mathord{\left/ {\vphantom {{\overline {\sigma ^2 } } {\lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\lambda ^2 }}\) and 5) the ratio σ ²_i /χ² where χ denotes the mean of the u _i and σ ²_u is the variance of the u _j. According to the dependence of the phenotypic stability on these factors some conclusions can be easily derived from this V-formula. Furthermore, two different approaches for a calculation of necessary or optimal numbers of components using the phenotypic variance V are discussed: A. Determination of ‘optimal’ numbers in the sense that a continued increase of the number of components brings about no further significant effect according to stability. B. A reduction of b % of the number of components but nevertheless an unchanged stability can be realized by an increase of the mean χ of the u _i by 1% (with \(\overline {\sigma ^2 } \) and σ ²_u assumed to be unchanged). Numerical results on n (from A) and 1 (from B) are given. Computing the coefficient of variation v for the character ‘total yield’ and solving for the number n of components one obtains an explicit expression for n dependent on v and the factors 2.-5. mentioned above. In the special case of equal variances, σ ²_i = σ ²_o for each i, the number n depends on v, x = (σ₀/χ)² and y = (σ_u/χ)². Detailed numerical results for n = n (v, x, y) are given. For x ≦ 1 and y ≦ 1 one obtains n = 9, 20 and 79 for v = 0.30, 0.20 and 0.10, respectively while for x ≦ 1 and arbitrary y-values the results are n = 11, 24 and 95.