The models for uncoordinated service and coordination with a common headway are formulated as
nonlinear programming problems (NLP). Since constraints in the proposed models are not convex
functions, standard heuristic algorithms for solving these NLPs can guarantee convergence only to a local
minimum. The model of integer-ratio coordination including both integer and linear variables (i.e. integer
ratio multipliers) with nonlinear cargo time values is known as a mixed-integer nonlinear program
(MINLP). The optimization of such models is typically difficult due to their combinatorial nature and
potential existence of multiple local minima.
Many previous studies apply genetic algorithms (GAs) to solve scheduling and schedule coordination
problems. Shrivastava et al. (2002) formulate scheduling and schedule coordination problems as
conflicting objectives with user's costs and operator's costs. Torabi et al. (2006) investigate the delivery
schedule that would minimize the average of holding, setup, and transportation costs per unit time for the
supply chain. Cao (2008) presents a vehicle routing problem with time windows constraints and
simultaneous delivery and pick-up operations. A hybrid optimization algorithm is proposed based on the
combination of differential evolution techniques and GAs.