66 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS describe the effects of cosmetic products even by a rough mathematical model, and the optimization procedure can only be performed on the given final characteristics. In this case, a more global optimization approach of the "black box" type would be better, and the real goal is not to find the exact optimum but to narrow down the area for each parameter in the system around the expected optimum. METHODS Among the investigation procedures, the ANTICOMPLEX method (3), derived from the COMPLEX method (4,5), was chosen. Initially developed for technological pro- cesses, this algorithm has been used without modification for cosmetic formulation studies for the following reasons: ß A global analysis of the hyper space of the parameters. To a certain extent, the random exploration of the whole parameter space is similar to a screening technique that has the advantage over the other optimization methods. One important conse- quence is the capability to locate and discard possible false optima regions. ß A rapid attainment of optimum conditions acceptable by the user. This is essentially due to the serial procedure and the strategy oriented to a fast convergence toward the optimum region. ß A versatile algorithm specially written for technological processes with convenient options such as the discretization of parameter values. ß A very easy way to take into account implicit and explicit constraints for each parameter or for specific subsets of parameters. In the ANTICOMPLEX terminology, the implicit constraints refer to the boundary limits of the parameters, while the explicit ones refer to any constraints imposed by the technological process or deter- mined by the user. For example, the formulation constraint is a typical explicit constraint. This strategy is illustrated here by the microemulsion study of a high concentration of lipid material in an aqueous phase using a ternary system with specific surfactants. The criterion to be optimized is a combination of somewhat contrary characteristics. Due to the complexity of the problem, non-standard options of the algorithm were used and the strategy was adapted at some important steps according to the results already obtained and the accuracy desired. METHODOLOGICAL ASPECTS In this approach, the optimum operating conditions of a technological process are accessed by a "black box" strategy. The global description of the algorithm is mentioned elsewhere in detail (3), and we shall briefly summarize its main features. This program is based on an iterative procedure with at each step a random sampling of the space of the parameters. After each series of measurements (step), a statistical analysis of the result leads, under normal conditions, to successive restrictions of the boundary limits. This methodology is very flexible since the investigation can be stopped after any series of experiments according to the desired accuracy for the optimum region. Preliminary trials of formulation were performed to determine the internal variable

FORMULA OPTIMIZATION 67 values for the ANTICOMPLEX, acceptable from a practical point of view. The follow- ing two main parameters values were chosen: p (number of parameters) = 6 n (number of trials at each series) = 20 Furthermore, the definition of the boundary limits and the explicit constraints, plus an estimating criterion, have to be defined before using the program. DEFINITION OF FORMULATION PARAMETERS In regard to microemulsion, a physical system with six parameters was considered as a good model for this formulation, easily transposable on an industrial scale. The selected parameters are the following: x•: lipophilic active compound x2: oil x 3: water x4: surfactant 1 xs: surfactant 2 x6: surfactant 3 For this formulation study, the corresponding values will be expressed in weighted percentage. CONSTRAINTS ON PARAMETER VALUES For each parameter, the boundary limits are implicit constraints that must be defined, prior to any calculation in the algorithm. We have chosen the widest boundary limits for the parameters that will be kept (or reduced) for a greater number of series. Another strategy would consist in first choosing rather narrower limits that would be kept for a limited number of series and that would then be reexamined for subsequent iterative procedure. This way is similar to experimental design techniques, but it presupposes some knowledge of the formulation. In our study, the former way was chosen since we wanted to use the ANTICOMPLEX procedure as a searching algorithm as well as a screening method. Furthermore, little information was known about the studied for- mulation and it was decided that the algorithm would not be biased so that its capa- bilities could be tested. The boundary limits used and the corresponding steps for each parameter will be described in detail in the Results section. Due to these large boundary limits, the explicit constraints on specific sets of parameters were carefully defined. The first constraint involves the two most important compounds and thus corresponds to economic consideration: x• + x 2 • 50 On the contrary, the second constraint involves the last three parameters (surfactants) and corresponds to safety consideration: X 4 -1 t- X 5 -1 t- X 6 • 50 It must be mentioned that these constraints are large enough to encompass any physical formulation, even those that are not acceptable for further manufacturing developments.