9781848003811-c1-3.pdf - Chapter 2 Multi-objective...

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Chapter 2 Multi-objective Optimization Abstract In this chapter, we introduce multi-objective optimization, and recall some of the most relevant research articles that have appeared in the international litera- ture related to these topics. The presented state-of-the-art does not have the purpose of being exhaustive; it aims to drive the reader to the main problems and the ap- proaches to solve them. 2.1 Multi-objective Management The choice of a route at a planning level can be done taking into account time, length, but also parking or maintenance facilities. As far as advisory or, more in general, automation procedures to support this choice are concerned, the available tools are basically based on the “shortest-path problem”. Indeed, the problem to fi nd the single-objective shortest path from an origin to a destination in a network is one of the most classical optimization problems in transportation and logistic, and has deserved a great deal of attention from researchers worldwide. However, the need to face real applications renders the hypothesis of a single-objective function to be optimized subject to a set of constraints no longer suitable, and the introduction of a multi-objective optimization framework allows one to manage more informa- tion. Indeed, if for instance we consider the problem to route hazardous materials in a road network (see, e.g., Erkut et al., 2007), de fi ning a single-objective function problem will involve, separately, the distance, the risk for the population, and the transportation costs. If we regard the problem from different points of view, i.e., in terms of social needs for a safe transshipment, or in terms of economic issues or pol- 11
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12 2 Multi-objective Optimization lution reduction, it is clear that a model that considers simultaneously two or more such objectives could produce solutions with a higher level of equity. In the follow- ing, we will discuss multi-objective optimization and related solution techniques. 2.2 Multi-objective Optimization and Pareto-optimal Solutions A basic single-objective optimization problem can be formulated as follows: min f ( x ) x S , where f is a scalar function and S is the (implicit) set of constraints that can be de fi ned as S = { x R m : h ( x ) = 0 , g ( x ) 0 } . Multi-objective optimization can be described in mathematical terms as follows: min [ f 1 ( x ) , f 2 ( x ) ,..., f n ( x )] x S , where n > 1 and S is the set of constraints de fi ned above. The space in which the objective vector belongs is called the objective space , and the image of the feasible set under F is called the attained set . Such a set will be denoted in the following with C = { y R n : y = f ( x ) , x S } . The scalar concept of “optimality” does not apply directly in the multi-objective setting. Here the notion of Pareto optimality has to be introduced. Essentially, a vector x S is said to be Pareto optimal for a multi-objective problem if all other vectors x S have a higher value for at least one of the objective functions f i , with i = 1 ,..., n , or have the same value for all the objective functions. Formally speak-
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