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A new approach of Romero’s extended lexicographic goal programming: fuzzy extended lexicographic goal programming

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Abstract

Goal programming (GP) is perhaps one of the most widely used approaches in the field of multicriteria decision making. The major advantage of the GP model is its great flexibility which enables the decision maker to easily incorporate numerous variations on constraints and goals. Romero provides a general structure, extended lexicographic goal programming (ELGP) for GP and some multiobjective programming approaches. In this work, we propose the extension of this unifying framework to fuzzy multiobjective programming. Our extension is carried out by several methodologies developed by the authors in the fuzzy GP approach. An interval GP model has been constructed where the feasible set has been defined by means of a relationship between fuzzy numbers. We will apply this model to our fuzzy extended lexicographic goal programming (FELGP). The FELGP is a general primary structure with the same advantages as Romero’s ELGP and moreover it has the capacity of working with imprecise information. An example is given in order to illustrate the proposed method.

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Notes

  1. A fuzzy set is normal if the supreme of its membership function is equal to 1.

  2. Heilpern (1992) defines the expected interval of a fuzzy number \( \tilde{N} \), which will be noted \( \hbox{EI}\left( {\tilde{N}} \right) \). In terms of α-cuts the expected interval is: \( \hbox{EI}\left( {\tilde{N}} \right) = \left[ {\hbox{EI}\left( {\tilde{N}} \right)^{L} - \hbox{EI}\left( {\tilde{N}} \right)^{R} } \right] = \left[ {\int_{0}^{1} {n_{\alpha }^{L} {\text{d}}\alpha ,\int_{0}^{1} {n_{\alpha }^{R} {\text{d}}\alpha } } } \right] \).

    The expected interval of a fuzzy vector \( \tilde{a}_{i} = \left( {\tilde{a}_{i1} ,\tilde{a}_{i2} , \ldots ,\tilde{a}_{in} } \right) \), as a vector composed of the expected intervals of each fuzzy number of the vector \( \tilde{a}_{i} \), that is, \( \hbox{EI}\left( {\tilde{a}_{i} } \right) = \left( {\hbox{EI}\left( {\tilde{a}_{i1} } \right),\hbox{EI}\left( {\tilde{a}_{i2} } \right), \ldots ,\hbox{EI}\left( {\tilde{a}_{in} } \right)} \right) \).

  3. In practice, input fuzzy data are often assumed to be triangular fuzzy numbers. A triangular fuzzy number can be denoted as \( \tilde{N} = \left( {n_{1} ,n_{2} ,n_{3} } \right) \):

    For a triangular fuzzy number \( \tilde{N} = \left( {n_{1} ,n_{2} ,n_{3} } \right) \) the expected interval is obtained as:

    \( \hbox{EI}\left( {\tilde{N}} \right) = \left[ {\hbox{EI}\left( {\tilde{N}} \right)^{L} - \hbox{EI}\left( {\tilde{N}} \right)^{R} } \right] = \left[ {{\frac{{n_{1} + n_{2} }}{2}},{\frac{{n_{2} + n_{3} }}{2}}} \right]. \)

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Acknowledgments

We would like to thank two anonymous referees for their valuable suggestions and comments which have contributed to the improvement of the paper. A previous version of this research was presented and published in the proceedings of the Second International Congress in Soft Methods in Probability and Statistics, Oviedo 2004. The authors wish to gratefully acknowledge financial support from the Spanish Ministry of Education, project MTM2007-67634.

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Authors and Affiliations

  1. Departamento de Economía Cuantitativa, Facultad de CC Económicas y Empresariales, University of Oviedo, Avenida del Cristo s/n, Oviedo, Asturias, Spain

    Mar Arenas-Parra, Amelia Bilbao-Terol, Blanca Pérez-Gladish & María Victoria Rodríguez-Uría

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  1. Mar Arenas-Parra
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  2. Amelia Bilbao-Terol
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  3. Blanca Pérez-Gladish
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  4. María Victoria Rodríguez-Uría
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Correspondence to Blanca Pérez-Gladish.

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Arenas-Parra, M., Bilbao-Terol, A., Pérez-Gladish, B. et al. A new approach of Romero’s extended lexicographic goal programming: fuzzy extended lexicographic goal programming. Soft Comput 14, 1217–1226 (2010). https://doi.org/10.1007/s00500-009-0533-y

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