The optimal scheduling is as old as transportation systems, established in some European and American cities during the 19th century using horse-drawn coaches.

From the very first moment, the managers of those businesses faced the problem of scheduling their services, because a better planning of coaches, drivers and horses could bring great profit.

Usually, the scheduling problems in transportation are considered as problems of multi-objective combinatorial optimization:

TRANSPORT OPTIMITATION– Multi-objective: the quality of the solutions depends on many different objectives or criteria.

– Combinatorial: the quantity of possible solutions is normally considered finite (although very large).

– Optimization: the best possible solution is sought.

The theoretical basis of the multi-objective combinatorial optimization is the theory of the economic equilibrium, commonly attributed to the famous work “The Wealth of Nations” by Adam Smith, written in 1776.

The works of Edgeworth and Pareto at the end of the 19th century about the theory of utility, welfare and equilibrium are regarded as the origin of the meaning given to optimization nowadays. The Pareto optimality is described as a situation in which consumers and producers cannot increase their satisfaction without decreasing the satisfaction of the other part.

Another essential milestone in the origins of the multi-objective optimization is the creation of the game theory and the notion of strategy. Already in the 20th century, John von Neumann and Oskar Morgernstern mentioned in their work “Theory of Games and Economic Behavior” that they had faced an optimization problem in economics which was “a curious and puzzling mix of various problems in conflict with each other” and which could not be solved using the classical mathematical methods for optimization.

Last, the concept of vector maximum problem introduced by Kuhn and Tucker (1951) meant the beginning of the multi-objective optimization as a new mathematical discipline.

New challenges

Nowadays, multi-objective optimization is applied to a great deal of domains, such as biology, for many analysis of DNA sequences; inorganic chemistry, in the synthesis of compounds which must have specific properties; in engineering, to analyze and enhance systems like water pipes and electrical networks, and of course, in the optimal scheduling of transportation systems: in this field, GOAL SYSTEMS is a world leader, always researching and innovating in order to obtain the most satisfaction of users and employees with the minimal costs for the clients.

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