A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract-The Vehicle Routing Problem with Time Windows (VRPTW) is an important logistics problem which in the realworld appears to be multi-objective. Most research in this area has been carried out using classic datasets designed for the single-objective case, like the well-known Solomon's problem instances. Some unrealistic assumptions are usually made when using these datasets in the multi-objective case (e.g. assuming that one unit of travel time corresponds to one unit of travel distance). Additionally, there is no common VRPTW multiobjective oriented framework to compare the performance of algorithms because different implementations in the literature tackle different sets of objectives. In this work, we investigate the conflicting (or not) nature of various objectives in the VRPTW and show that some of the classic test instances are not suitable for conducting a proper multi-objective study. The insights of this study have led us to generate some problem instances using data from a real-world distribution company. Experiments in these new dataset using a standard evolutionary algorithm (NSGA-II) show stronger evidence of multi-objective features. Our contribution focuses on achieving a better understanding about the multi-objective nature of the VRPTW, in particular the conflicting relationships between 5 objectives: number of vehicles, total travel distance, makespan, total waiting time, and total delay time.

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Four selfreciprocal integral transformations of Hankel type are defined through(ℋi,μf)(y)=Fi(y)=∫0∞αi(x)ℊi,μ(xy)f(x)dx,   ℋi,μ−1=ℋi,μ,where i=1,2,3,4; μ≥0; α1(x)=x1+2μ, ℊ1,μ(x)=x−μJμ(x), Jμ(x) being the Bessel function of the first kind of order μ; α2(x)=x1−2μ, ℊ2,μ(x)=(−1)μx2μℊ1,μ(x); α3(x)=x−1−2μ, ℊ3,μ(x)=x1+2μℊ1,μ(x), and α4(x)=x−1+2μ, ℊ4,μ(x)=(−1)μxℊ1,μ(x). The simultaneous use of transformations ℋ1,μ, and ℋ2,μ, (which are denoted by ℋμ) allows us to solve many problems of Mathematical Physics involving the differential operator Δμ=D2+(1+2μ)x−1D, whereas the pair of transformations ℋ3,μ and ℋ4,μ, (which we express by ℋμ*) permits us to tackle those problems containing its adjoint operator Δμ*=D2−(1+2μ)x−1D+(1+2μ)x−2, no matter what the real value of μ be. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation∫0∞f(x)g(x)dx=∫0∞(ℋμf)(y)(ℋμ*g)(y)dy,which is now valid for all real μ

The simultaneous consideration of two integral transf ormations verifying a mixed Parseval relation suggests a new method of defining the Schwartz's Hankel transformation on certain spaces of generalized functions of slow growth. The results obtained extend prior analysis of the generalized Hankel transformation .