Phase II: Product realization
This part of the methodology starts by identifying a list of available ingredients. For instance, cosmetic product designers are constrained to use approved ingredients, or they are limited to choose among available ingredients in the company’s inventory. Here, a binary variable is associated with each potential ingredient, indicating whether it is selected or not as part of the formulation. Then, available heuristics regarding the choice of ingredients and their amounts are listed and modeled as algebraic restrictions. Some heuristics are first stated as logical conditions and then translated into algebraic constraints involving binary variables. These heuristic-related restrictions, together with other known limits (technical and/or legal), help to reduce the design space. The search in this reduced space is then guided by available property models relating product composition to key physicochemical properties or even to sensorial attributes.
Let y be the vector of binary variables associated with the choice of ingredients and x the vector of corresponding their concentration in the final product (i.e. mass fractions). Let pbe the vector of product performance metrics, including well-defined physicochemical properties and metrics related to more subjective sensorial attributes. Product quality is often evaluated in terms of the deviation of p from target values p*(product performance specifications). Then, property models, also known as the property function, are any relationship between metrics pand product composition, here represented by the set of equationsh(x,y,p)=0 .42 The heuristic rules are incorporated in the problem formulation as the set of constraints g(x,y,p) ≤ 0 . Finally, let fbe a global objective function to be minimized, in this case accounting for both product quality deviation and costs. The problem of optimal product formulation may then be stated as follows: