200 K. E. F. WATT 



Now it so happens that many problems in modern society are essentially 

 analogous to this problem of biological yield, and therefore, happily, the 

 mathematicians have given such problems a great deal of thought. Un- 

 fortunately, the fruits of this thought have almost entirely been pubhshed in 

 books and journals that the average biologist would not come across in his 

 entire career. A somewhat more complete introduction to this hterature is 

 given elsewhere (Watt, 19606), but I shall attempt a succinct resume here. 



When the biologist is confronted with the problem of fmding values of 

 independent variables which yield a maximum or minimum for a complex 

 function, there are three general areas of mathematics to which he can look 

 for help. (The problem under discussion, incidentally, is referred to by 

 modern mathematicians as the 'extremum' problem.) 



Briefly, these areas are: 

 (i) Classical analysis (e.g. Lagrange multipliers). 



(2) Mathematical programming (more specifically, non-linear and dynamic 

 programming). 



(3) Electronic computer technology (systematic hypersurface-exploring 

 techniques as proposed by Box and associates). 



Any good text on advanced calculus will elucidate the theory and apphca- 

 tion of Lagrange multipliers. Parke (1958) has pubhshed an excellent anno- 

 tated bibliography, complete to about 1955, which will serve the biologist 

 as an excellent guide to the hterature in these unfamiliar areas. 



Mathematical programming is a very recent branch of applied mathe- 

 matics created to cope with complex extremum problems arising in the 

 utihzation of transportation networks, petroleum refming, and various 

 manufacturing processes, etc. 



Specifically, it was invented because classical analysis cannot handle yield, 

 production or allocation extremum problems where there are inequality 

 constraints. That is, suppose we can increase productivity by increasing some 

 variable up to, but not beyond a certain value. Our constraint is, in such cases, 

 not of the form 



Vi = 2,700, 



but rather, is of the form 



V^ < 2,700. 



George Dantzig first published on such problems in the volume edited by 

 Koopmans (195 1), and there is now a vast literature on the subject. The 

 biologist would be advised to begin his reading in one of the simple intro- 

 ductory texts such as those by Kemeny et ah (1957) or Vajda (1956)- More 

 complete expositions are by Dorfman et ah (1958), Gass (1958) or Vazsonyi 

 (1958). Finally, the biologist (or his mathematical and statistical consultant) 



