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An integrated quantitative and qualitative FMCDM model for location choices

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Abstract

International logistics is a very popular and important issue in the present international supply chain system. In order to reduce the international supply chain operation cost, it is very important for enterprises to invest in the international logistics centers. Although a number of research approaches for solving decision-making problems have been proposed, most of these approaches focused on developing quantitative models for dealing with objective data or qualitative models for dealing with subjective ratings. Few researchers proposed approaches for dealing with both objective data and subjective ratings. Thus, this paper attempts to fill this gap in current literature by establishing an integrated quantitative and qualitative fuzzy multiple criteria decision-making model for dealing with both objective crisp data and subjective fuzzy ratings. Finally, the utilization of the proposed model is demonstrated with a case study on location choices of international distribution centers.

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Authors and Affiliations

  1. Department of Shipping Technology, National Kaohsiung Marine University, 482, Chung-Chou 3rd Road, Chi-Chin, Kaohsiung, 805, Taiwan, ROC

    Chien-Chang Chou

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  1. Chien-Chang Chou
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Correspondence to Chien-Chang Chou.

Appendices

Appendix A: The representation of multiplication operation on fuzzy numbers

Definition 1

Let \( P(A_{1} \otimes A_{2} \otimes A_{3} ) \) be the representation of \( A_{ 1} \otimes A_{2} \otimes A_{3} .\;P(A_{ 1} \otimes A_{2} \otimes A_{3} ) \) is defined as follows:

$$ \begin{aligned} & P\left( {A_{1} \otimes A_{2} \otimes A_{3} } \right) \\ & \quad = \intop_{0}^{1} {\intop_{0}^{1} {\intop_{0}^{1} {{\frac{1}{8}}\left[ {\left( {h_{A 1} L_{A 1}^{ - 1} (h) \times h_{A 2} L_{A 2}^{ - 1} (h) \times h_{A 3} L_{A 3}^{ - 1} (h)} \right) + \left( {h_{A 1} R_{A 1}^{ - 1} (h) \times h_{A 2} L_{A 2}^{ - 1} (h) \times h_{A 3} L_{A 3}^{ - 1} (h)} \right)} \right.} } } \\ & \qquad + \left( {h_{A 1} L_{A 1}^{ - 1} (h) \times h_{A 2} R_{A 2}^{ - 1} (h) \times h_{A 3} L_{A 3}^{ - 1} (h)} \right) + \left( {h_{A 1} L_{A 1}^{ - 1} (h) \times h_{A 2} L_{A 2}^{ - 1} (h) \times h_{A 3} R_{A 3}^{ - 1} (h)} \right) \\ & \qquad + \left( {h_{A 1} L_{A 1}^{ - 1} (h) \times h_{A 2} R_{A 2}^{ - 1} (h) \times h_{A 3} R_{A 3}^{ - 1} (h)} \right) + \left( {h_{A 1} R_{A 1}^{ - 1} (h) \times h_{A 2} L_{A 2}^{ - 1} (h) \times h_{A 3} R_{A 3}^{ - 1} (h)} \right) \\ & \qquad + \left. {\left( {h_{A 1} R_{A 1}^{ - 1} (h) \times h_{A 2} R_{A 2}^{ - 1} (h) \times h_{A 3} L_{A 3}^{ - 1} (h)} \right) + \left( {h_{A 1} R_{A 1}^{ - 1} (h) \times h_{A 2} R_{A 2}^{ - 1} (h) \times h_{A 3} R_{A 3}^{ - 1} (h)} \right)} \right] \\ & \qquad \times {{{\text{d}}h_{A 1} {\text{d}}h_{A 2} {\text{d}}h_{A 3} } \mathord{\left/ {\vphantom {{{\text{d}}h_{A 1} {\text{d}}h_{A 2} {\text{d}}h_{A 3} } {\left( {\intop_{0}^{1} {h_{A 1} {\text{d}}h_{A 1} \intop_{0}^{1} {h_{A 2} {\text{d}}h_{A 2} \intop_{0}^{1} {h_{A 3} {\text{d}}h_{A 3} } } } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\intop_{0}^{1} {h_{A 1} {\text{d}}h_{A 1} \intop_{0}^{1} {h_{A 2} {\text{d}}h_{A 2} \intop_{0}^{1} {h_{A 3} {\text{d}}h_{A 3} } } } } \right)}} \\& \quad = \intop_{0}^{1} {\intop_{0}^{1} {\intop_{0}^{1} {{\frac{1}{8}}} } } \left\{ {h_{A1} \left[ {c_{1} + \left( {a_{1} - c_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right]} \right. \\ & \qquad + h_{A1} \left[ {b_{1} + \left( {a_{1} - b_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + h_{A1} \left[ {c_{1} + \left( {a_{1} - c_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + h_{A1} \left[ {c_{1} + \left( {a_{1} - c_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + h_{A1} \left[ {c_{1} + \left( {a_{1} - c_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + h_{A1} \left[ {b_{1} + \left( {a_{1} - b_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + h_{A1} \left[ {b_{1} + \left( {a_{1} - b_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad \left. { + h_{A1} \left[ {b_{1} + \left( {a_{1} - b_{1} } \right)h_{A1} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right]} \right\} \\ & \qquad \times {{{\text{d}}h_{A1} {\text{d}}h_{A2} {\text{d}}h_{A3} } \mathord{\left/ {\vphantom {{{\text{d}}h_{A1} {\text{d}}h_{A2} {\text{d}}h_{A3} } {\left( {\intop_{0}^{1} {h_{A1} {\text{d}}h_{A1} \times \intop_{0}^{1} {h_{A2} {\text{d}}h_{A2} } \times } \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\intop_{0}^{1} {h_{A1} {\text{d}}h_{A1} \times \intop_{0}^{1} {h_{A2} {\text{d}}h_{A2} } \times } \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } \right)}} \\ & \quad = \intop_{0}^{1} {\intop_{0}^{1} {{\frac{1}{8}}} } \left\{ {\left[ {{\frac{1}{2}}c_{1} h_{A1}^{2} + \left( {a_{1} - c_{1} } \right)h_{A1}^{3} } \right]} \right. \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left. {\left. {\left[ {{\frac{1}{2}}b_{1} h_{A1}^{2} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)h_{A1}^{3} } \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right]} \right\}} \right] \\ & \qquad {{\left|{_{0}^{1} }{\vphantom {{\left| {_{0}^{1} } \right.{\text{d}}h_{A2} {\text{d}}h_{A3} } {\left( {{\frac{1}{2}}h_{A1}^{2} \left| {_{0}^{1} } \right. \times \intop_{0}^{1} {h_{A2} {\text{d}}h_{A2} \times \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } } \right)}}} \right.{\text{d}}h_{A2} {\text{d}}h_{A3} } \mathord{\left/ {\vphantom {{\left| {_{0}^{1} } \right.{\text{d}}h_{A2} {\text{d}}h_{A3} } {\left( {{\frac{1}{2}}h_{A1}^{2} \left| {_{0}^{1} } \right. \times \intop_{0}^{1} {h_{A2} {\text{d}}h_{A2} \times \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\frac{1}{2}}h_{A1}^{2} \left| {_{0}^{1} } \right. \times \intop_{0}^{1} {h_{A2} {\text{d}}h_{A2} \times \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } } \right)}} \\ & \quad = \intop_{0}^{1} {\intop_{0}^{1} {{\frac{1}{8}}\left\{ {\left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right]} \right.} } \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)} \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right) h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right) h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right) h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right) h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)} \right] \times h_{A2} \left[ {c_{2} + \left( {a_{2} - c_{2} } \right) h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)} \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right) h_{A2} } \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left. {\left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}\left( {a_{1} - b_{1} } \right)} \right] \times h_{A2} \left[ {b_{2} + \left( {a_{2} - b_{2} } \right)h_{A2} } \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right]} \right\} \\ & \qquad \times {{{\text{d}}h_{A2} {\text{d}}h_{A3 } } \mathord{\left/ {\vphantom {{{\text{d}}h_{A2} {\text{d}}h_{A3 } } {\left( {{\frac{1}{2}}h_{A2}^{2} {\text{d}}h_{A2} \left| {_{0}^{1} } \right. \times \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\frac{1}{2}}h_{A2}^{2} {\text{d}}h_{A2} \left| {_{0}^{1} } \right. \times \intop_{0}^{1} {h_{A3} {\text{d}}h_{A3} } } \right)}} \\ & \quad = \intop_{0}^{1} {{\frac{1}{8}}\left\{ {\left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right]} \right.} \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}(a_{1} - c_{1} )} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}\left( {a_{2} - b_{2} } \right)} \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}(a_{1} - c_{1} )} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}(a_{1} - c_{1} )} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}\left( {a_{2} - b_{2} } \right)} \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}(a_{2} - b_{2} )} \right] \times h_{A3} \left[ {c_{3} + \left( {a_{3} - c_{3} } \right)h_{A3} } \right] \\ & \qquad + \left. {\left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}(a_{2} - b_{2} )} \right] \times h_{A3} \left[ {b_{3} + \left( {a_{3} - b_{3} } \right)h_{A3} } \right]} \right\} \\ & \qquad \times {{{\text{d}}h_{A3 } } \mathord{\left/ {\vphantom {{{\text{d}}h_{A3 } } {\left( {{\frac{1}{2}} \times {\frac{1}{2}} \times \intop_{0}^{1} {h_{A3} } {\text{d}}h_{A3} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\frac{1}{2}} \times {\frac{1}{2}} \times \intop_{0}^{1} {h_{A3} } {\text{d}}h_{A3} } \right)}} \\ & \quad = {\frac{1}{8}}\left\{\left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}c_{3} + {\frac{1}{3}}(a_{3} - c_{3} )} \right] \right.\\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}c_{3} + {\frac{1}{3}}(a_{3} - c_{3} )} \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}\left( {a_{2} - b_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}c_{3} + {\frac{1}{3}}(a_{3} - c_{3} )} \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{3} + {\frac{1}{3}}(a_{3} - b_{3} )} \right] \\ & \qquad + \left[ {{\frac{1}{2}}c_{1} + {\frac{1}{3}}\left( {a_{1} - c_{1} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}\left( {a_{2} - b_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{3} + {\frac{1}{3}}(a_{3} - b_{3} )} \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{3} + {\frac{1}{3}}(a_{3} - b_{3} )} \right] \\ & \qquad + \left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}c_{2} + {\frac{1}{3}}\left( {a_{2} - c_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}c_{3} + {\frac{1}{3}}(a_{3} - c_{3} )} \right] \\ & \qquad + {{\left. {\left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}\left( {a_{2} - b_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{3} + {\frac{1}{3}}(a_{3} - b_{3} )} \right]} \right\}} \mathord{\left/ {\vphantom {{\left. {\left[ {{\frac{1}{2}}b_{1} + {\frac{1}{3}}(a_{1} - b_{1} )} \right] \times \left[ {{\frac{1}{2}}b_{2} + {\frac{1}{3}}\left( {a_{2} - b_{2} } \right)} \right] \times \left[ {{\frac{1}{2}}b_{3} + {\frac{1}{3}}(a_{3} - b_{3} )} \right]} \right\}} {\left( {{\frac{1}{2}} \times {\frac{1}{2}} \times {\frac{1}{2}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\frac{1}{2}} \times {\frac{1}{2}} \times {\frac{1}{2}}} \right)}} \\ &\quad= \left\{ {\left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right)} \right. \\ &\qquad + \left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right) \\ &\qquad + \left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right) \\ &\qquad \left. { + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right) \times \left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right) \times \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right)} \right\} \\ &\quad = \left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right)\left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right)\left[ {\left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right)} \right] + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right)\left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right)\left[ {\left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right)} \right] \\ &\qquad + \left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right)\left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right)\left[ {\left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right)} \right] + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right)\left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right)\left[ {\left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right)} \right] \\ &\quad= \left[ {\left( {{\frac{{c_{1} + 2a_{1} }}{6}}} \right) + \left( {{\frac{{b_{1} + 2a_{1} }}{6}}} \right)} \right] \times \left[ {\left( {{\frac{{c_{2} + 2a_{2} }}{6}}} \right) + \left( {{\frac{{b_{2} + 2a_{2} }}{6}}} \right)} \right] \times \left[ {\left( {{\frac{{c_{3} + 2a_{3} }}{6}}} \right) + \left( {{\frac{{b_{3} + 2a_{3} }}{6}}} \right)} \right] \\ &\quad = {\frac{1}{6}}\left( {c_{1} + 4a_{1} + b_{1} } \right) \times {\frac{1}{6}}\left( {c_{2} + 4a_{2} + b_{2} } \right) \times {\frac{1}{6}}\left( {c_{3} + 4a_{3} + b_{3} } \right) \\ \end{aligned} $$

We have that \( P(A_{ 1} \otimes A_{ 2} \otimes A_{3} ) = {\frac{1}{6}}\left( {c_{1} + 4a_{1} + b_{1} } \right) \times {\frac{1}{6}}\left( {c_{2} + 4a_{2} + b_{2} } \right) \times {\frac{1}{6}}\left( {c_{3} + 4a_{3} + b_{3} } \right) \)

Appendix B: The description of criteria for evaluating distribution center location

 

Subjective criteria

Description

C 13

Trade variables

Trade variables will affect the investment of multinational shipping companies

C 21

Exchange rate

Stable exchange rates will attract the investment of multinational shipping companies

C 22

Labor cost

It affects the operation cost of multinational shipping companies

C 23

Transportation cost (inland transportation cost)

It affects the operation cost of multinational shipping companies

C 25

Land cost

It affects the investment cost of multinational shipping companies

C 31

Efficiency of government departments

It will affect the operation cost and time. High efficiency of government departments will attract more multinational corporations and shipping companies

C 32

Co-operative relationship between enterprises and government

It will affect the investment of multinational shipping companies

C 33

Tax break

Tax breaks will attract more multinational corporations and shipping companies

C 34

Other preferential treatment

It will affect the operation costs and profits of multinational shipping companies

C 35

Law on investment and investment restrictions

Law and investment restriction will decrease the multinational shipping companies’ preferences for investing in international distribution centers

C 36

Political stability

Political stability will affect the investment of multinational corporations and shipping companies

C 37

Social stability

Social stability will affect the investment of multinational corporations and shipping companies

C 41

Availablity of land

It will affect the present investment and future expansions

C 43

Labor quality

It will affect the operation cost and time. High labor quality will attract more multinational corporations and shipping companies

C 44

Trade liberalization

Trade liberalization will attract more cargoes and multinational corporations

C 45

Efficiency of Customs

It will affect the operation cost and time. High efficiency of Customs will attract more multinational shipping companies

C 46

Future development

Future development plans for ports will affect the investment of multinational shipping companies

C 47

Banking sector

It will affect the interest cost and banks making loans to multinational shipping companies

C 48

Private ownership of enterprise

It will affect the investment, the operational strategy and the profit of multinational shipping companies

 

Objective criteria

Description

C 11

Present volume of containers (TEU)

It will affect the short-term profit and the transshipment cost of cargo through the effect of economic scale

C 12

Potential volume of containers in the future (TEU)

It will affect the long-term profit of multinational shipping companies

C 24

Operation cost ($UD/TEU)

Cost of loading/discharging one 20-foot container

C 42

Infrastructure quality (wharf)

Excellent infrastructure is necessary attracting shipping companies

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Chou, CC. An integrated quantitative and qualitative FMCDM model for location choices. Soft Comput 14, 757–771 (2010). https://doi.org/10.1007/s00500-009-0463-8

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  • DOI: https://doi.org/10.1007/s00500-009-0463-8

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