Differentiation of Choquet integral for nonnegative measurable function and its application to capital investment decision making

Differentiation of a Choquet integral for a nonnegative measurable function taken with respect to a fuzzy measure over a real fuzzy measure space is proposed. It is applied to the capital investment decision making problem. Kaino and Hirota (1999) proposed the X axis real interval limited Choquet integral for a preparation to define a fuzzy measure shift differentiation of the Choquet integral. Here, the upper differential coefficient, the lower differential coefficient, the differential coefficient, and the derived function of the X axis real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space along the domain (a subset of real numbers) are defined by the limitation process of the X axis real interval limited Choquet integral. Two examples of differentiation are given, where the nonnegative measurable functions are either a simple function or a triangular function. Moreover, they are applied to the automobile factory capital investment decision making problem. Assume that an automobile company has a sales plan of a new car. The current factory line has a capacity to manufacture 3,200 new cars, additional to the current car lines. Then, it is easy to make a decision to invest or not using the differentiation of the X axis real interval limited Choquet integral over X, which give us the increase or decrease rate of the expected sales revenue (also profit) per new car