Optimization with multiple responses

Lack of fit of the obtained model has to be statistically checked, so that, if needed, the polynomial degree may be augmented. Knowing the mathematical model of the research subject for several responses is a prerequisite in solving optimization with multiple responses. The computation of this is solved geometrically or by use of computers and the method of linear algebra. 

Fig. 2.1 Lead discovery requires an integrated approach to address a multiple response surface optimization problem properties related to pharmacokinetics, toxicology and pharmacodynamics require optimization for novel chemotypes. The increased difficulty of finding such optima is suggested by darker backgrounds. Modified from [4] with permission.

In the introduction, we have already classified the optimization problems as deterministic and stochastic. It is evident that deterministic problems are based on functional models or models that disregard experiment error. Problems where one cannot neglect experiment error are stochastic ones and, as established, they are the subject of this book. Besides, optimization problems are by the number of factors divided into one-dimensional and more-dimensional. The Optimization problem grows with dimension. The problem becomes even more complicated if optimization is not done by one but by more responses simultaneously-multiple response processes. 

When integrating multiple legal entities into an SCM organization, ownership of inventory should be strongly consolidated to simplify geographical mobility and avoid local optimization moves. An SCM organization with full responsibility for operational performance should also develop, standardize, and implement company-wide practices that go beyond the responsibility of single functions. 

Chowdhury and Fard (2001) presented a method for estimating dispersion effects from robust design experiments with right censored data. Kim and Lin (2002) proposed a method to determine optimal design factor settings that take account of both location and dispersion effects when there are multiple responses. They based their approach on response surface models for location and dispersion of each response variable. 

The desirability function is a guide in optimizing a process or a formulation using multiple response data from a statistically planned experiment. It should be used with caution. Always try a number of different individual desirability functions for the different responses, and when an apparently optimum point or region has been found, go back to the plots of the original responses to check that it is really what is required. 

For instance, Yadav and Ahuja prepared nanoparticles using gum cordia as the polymer and to evaluate them for ophthalmic delivery of fluconazole. A w/o/w emulsion containing fluconazole and gum cordia in aqueous phase, methylene chloride as the oily phase, and dioctyl sodium sulfosuccinate and polyvinyl alcohol as the primary and secondary emulsifiers, respectively, were cross-linked by the ionic gelation technique to produce a fluconazole-loaded nanoreservoir system. The formulation of nanoparticles was optimized using response surface methodology. Multiple response simultaneous optimizations using the desirability approach were used to find optimal experimental conditions. The optimal conditions were found to be concentrations of gum cordia (0.85%, w/v), di-octyl sodium sulfosuccinate (9.07%, w/v), and fluconazole (6.06%, w/v). On comparison of the optimized nanosuspension formulation with commercial formulation, it was found to provide comparable in vitro corneal permeability of... 

The PDS capability permits performing Monte Carlo simulations with multiple input random variables. Geometric, material, and load parameters can be varied by assigning statistical distribution functions to them. Response variables are defined so that the device can be optimized by minimizing or maximizing them. The PDS analysis can then be used to perform sensitivity analysis to determine which random input variables should be altered and how, in order to optimize the selected response variables. 

In general, inventory can be positioned at each tier of the supply chain. While it increases responsiveness, it causes decision making to become complex. Such a system with multiple tradeoffs is modeled as multi echelon inventory optimization. 

The model can be extended by considering other objectives such as supply chain risk, responsiveness in forward and reverse supply chains, and environmental impact. Interactive optimization methodologies with multiple objectives can be developed. We have not considered any uncertainties in the return parameters. The model can also be extended by considering uncertainties in demand and returned products. 

---