Balancing bilinearly interfering elements

Many decisions in various fields of application have to take into account the joint effects of two elements that can interfere with each other. This happens, for example, in Medicine (synergic or antagonistic drugs), Agriculture (anti-cryptogamics), Public Economics (interfering economic policies), Industrial Economics (where the demand of an asset can be influenced by the supply of another asset), Zootechnics, and so on. When it is necessary to decide about the dosage of such elements, there is sometimes a primary interest for one effect rather than another; more precisely, it may be of interest that the effects of an element are in a certain proportion with respect to the effects of the other. It may also be necessary to take into account minimum quantities that must be assigned. In Carfì, Gambarelli and Uristani (2013), a mathematical model was proposed to solve the above problem in its exact form. In this paper, we present a solution in closed form for the case in which the function of the effects is bilinear.