Representing uncertain time on space-time cube for bus movement

The movement of a city bus is represented visually on a space-time cube by a space-time path connecting mathematical points associating the particular locations of bus stops with the times the bus visits them. In reality, most buses do not visit stops to drop down and pick up passengers on regulation time, thus the points indicating their visiting times on the time axis of a space-time cube may be fuzzified according to the fuzzy set theory. In this article, a fuzzy-time cube which is a space-time cube having the fuzzified time axis is studied to represent the movement of buses. On a fuzzy-time cube, a regulation visiting-time point t_i is fuzzified to a pair (T_i, m_i). A process of estimating the fuzzy set (T_i, m_i) is proposed to indicate times on the fuzzified time axis of a fuzzy-time cube. With the method dividing T_i into time units T_i^j, the membership function is estimated as a step function. The membership grade of each point on the one-day time axis at a bus stop is estimated by the statistical method based on the data recorded from a 30-day surveying. The representation of the bus movement on a fuzzy-time cube enables passengers to take buses more effectively and bus managers rank drivers as well as calculate the benefit of each route.