Line planning with minimal travel and transit times

In this paper we study passenger's oriented line planning problem of public transport with main goal to create a comfortable line concept (limited set of lines and vehicles). Our approach is non-traditional by using an iterative generation of small potential lines on a given transportation network. We suppose that OD-matrix was measured on a given set of lines, where each entry gives the number of passengers that want to travel from one stop in the network to another stop within a fixed time horizon. The real vehicle fleet (consisting of buses, trams or trolleybuses) is limited, so some passengers have to pass from one line to another. Our algorithm is based on solving multi-commodity circulation problem for line planning problem (MCCLPP) via a mixed linear problem. Object function is the total travel time of passengers that use a path with or without line change. In the first iteration we solve MCCLPP via existing set of links and its OD matrix. The solution is the line concept with these lines, vehicles and total travel time. In the second iteration we solve MCCLPP where lines from the first iteration are extended by lines created from OD pair (s, t) so that the shortest s-t path does not lie on any chosen lines. In the (k+1)-th iteration we solve MCCLPP from chosen links of k-th iteration and extended links as in the second iteration. Algorithm ends if two consequent iterations have same line concept. This approach is demonstrated via illustrative example.