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: