Figure 3. The values after one step in the gradient search process.

used to generate the accuracy necessary to make the results useful. This approach can be used for more than two variables, but the number of formulations for which data must be gathered also grows.

This sketch of a gradient search method leaves out many details of how the estimates are generated and used to determine the next formulation. More information may be found in Hanrez-LaGrange (8) and Hanrez-LaGrange and Norback (9). This approach has some similarity to response surface methods (RSM). The main difference is that in RSM, a formula for the objective measure must be generated, and this estimation of the objective measure then may be searched for a maximizing value. Because this function is maximized over the entire range of feasible formulations, it usually requires more expensive data collection than direct estimates of the gradient for a specific formulation and is of little value beyond its use in finding a maximum. Besides finding the maximum, the gradient search approach shows the impact of different components on the objective measure.

The weaknesses in this method are in the determination of the step size at each iteration, in the noise intrinsic to sampling from a population of consumers, and the fact that it may be suboptimal if the objective measure is multimodal. The latter difficulty can be avoided by careful pre-screening of the product and its potential formulations followed by restriction of the feasible formulations so that only one maximum may occur over the range of ingredient values being considered. In every test case, the procedure has yielded good solutions to formulation problems in two or three iterations. The gradient search procedure may be used for any functional objective measure, not just for the acceptor set size measure.

Making and adjusting formulations will have important economic consequences in high-volume situations. In this case, the food technology constrained the formulation, whereas the business objective of making the acceptor set size largest adjusted the formulation among the feasible alternatives. The situation could also be constrained by the cost of the inputs to the product, by company policy, or by government regulation. Decision support methods (gradient search) were applied to assist in determining the right formulation for the product. Such methods could be applied again if the product were to be reformulated for a substitute ingredient. Small variations in a formulation can have important consequences to the consumer's response to the product and to its profitability. With infinitely many formulations to choose from, these methods provide an organized means of choosing a good one.

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