Formulating a Cheese Topping

When a new product is fielded, there is the opportunity to establish its formulation. This decision affects the economics associated with the product and the consumer's perception of the product. The challenge is to devise a formulation that is technologically feasible and commercially viable. Because there are many alternatives to any formulation, some strategy has to be adopted to determine the desired formulation. An optimization strategy provides a good way to sift through alternatives and come up with an attractive formulation.

Cheese toppings may be manufactured by making processed cheese in such a way that the substance has many potential consistencies and textures. Such material may be mixed with other ingredients such as onions, chives, bacon bits, red pepper, and many more alternatives. The resulting product may be used by consumers as a topping for vegetables, potatoes, and salads. Properly formulated, it may be used hot or cold in a variety of ways. Details about the formulation of the plain cheese topping may be found in Hanrez-LaGrange (8). For this illustration we will consider the problem of determining what proportions of the noncheese ingredients added to the plain cheese toppings will make the best product. Preliminary studies have guided the decision makers to the point of determining what combination of red pepper pieces and nacho flavor will make the best cheese topping.

With a new product in the laboratory or pilot stage, determination of an acceptable objective measure for optimization is difficult. This measure will be used to compare alternative formulations, but it should indicate the performance of the product when it is ultimately marketed. Because this amounts to a forecast, the measure is beset with intrinsic variability problems of all forecasts. A procedure is to ask a sample of consumers to taste and then rate the product. Presuming this sample represents the intended market, the formulator may use the rating as a guide for adjusting the formulation. The connection between the formulator and the consumer panelist is a ballot requiring the panelist's responses. The questions used to solicit these responses can be used to fashion an objective measure. Although there are many ways to phrase such questions, care should be taken that the responses somehow relate to the potential market for the product. One possibility is ask the consumer to rate the product as acceptable or not acceptable. If the sample of tasters represents the target market for the product, the set of purchasers will be a subset of the set of acceptors. The bigger the set of acceptors, the larger the potential market. Actual sales will depend on many factors not controllable by manipulating the cheese topping formulation, such as advertising, distribution, price, and the competition. This acceptor set size measure can be taken for any potential formulation and becomes a way to compare formulations in order to discard the less desirable ones. Other measures are possible, but for this example, the acceptor set size will be used as the objective measure.

The strategy, then, is to ask a sample of consumers whether various formulations of this product with varying amounts of red peppers and nacho flavoring are acceptable. The objective is to find a formulation that will maximize the acceptor set size when the product is marketed. The acceptor set size is presumed to be a function of the formulation. Because the intention is that the product is to be sold in large volume, this function may be presumed to be continuous and differentiable over a reasonable range of values for input amounts of red pepper and nacho flavor. For such a function the mathematical construct called the gradient exists and may be used as a guide in seeking a better (bigger acceptor set size) formulation. The gradient of a function is a vector of partial derivatives of the function with respect to the variables. It has the property that it always points in the direction in which the function is increasing most rapidly. The advantage here is that we may estimate the partial derivatives of this function without knowing an explicit formula for the acceptor set size function. Thus, the gradient may be estimated and improvements to the acceptor set size function may be made by determining the gradient, taking a step in that direction, and retesting to determine how to adjust the formulation farther. In symbols:

A = fix, y) is the acceptor set size function grad(A)

where x is the amount of nacho flavor, and y is the amount of red pepper

If (xfh yg) is the initial formulation, then yi) = ^o) + k x grad(A)0x;o, y0)

where k is the step size in the gradient direction. This process of estimating the gradient and determining the next test formulation ends when there is no more improvement possible (the gradient is at or near (0,0)) or when the noise in the data overpowers the information in the estimate of the gradient.

Partial derivatives are measures of rates of change in the direction parallel to the axes of the chosen variable. To estimate the value of the partial derivative at a given formulation requires that functional estimates be generated at higher and lower values of the chosen variable while the other variables are held constant. In this example, the initial formulation was x0 = 3% and y0 = 13.75%. To obtain estimates of the partial derivative required functional estimates for 2%, 3%, and 4% nacho flavor while holding the amount of red pepper in the mix at a constant 13.75%. To get the other partial derivative required holding nacho flavor at a constant 3% and getting functional estimates of the acceptor set size with red pepper at 8.75%, 13.75%, and 18.75%. The results of the initial test are shown in Figure 2.

The gradient vector for this test was estimated as:

which implies that a better formulation is possible with more nacho flavor and less red pepper. Choosing the step size depends on the product and ingredient circumstances and the judgment of the investigator. The next step in the optimization process was to test with the formulation set at

The functional estimates for the acceptor set size are shown in Figure 3.

Further adjustments in this formulation are possible providing that big enough samples of consumers can be a) Q. Q.

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