Fuzzy multi-objective stochastic programming model for disaster relief logistics considering telecommunication infrastructures: a case study

Abstract

In humanitarian relief logistics, providing a safe place for evacuees, supplying relief commodities and designing a proper telecommunication infrastructure, for fast communications during disaster, are important issues. Therefore, in this paper, we develop a fuzzy scenario-based optimization model concerning location of shelters, relief distribution centers and telecommunication towers. Towards effective management and reliable servicing, telecommunication towers and shelters are considered to constitute integrated facilities (shelter-TTs). Moreover, to enhance efficiency of emergency services during disaster, backup relief distribution centers, and to approach the model to the real world, failure probabilities in the routes and the relief distribution centers are considered. The problem is formulated in a nonlinear and multi-objective model. Nonlinearity is treated by applying heuristic arguments in conjunction with Lp-metrics method. Finally, the developed model for the case study of flood disaster in an urban district in Iran is implemented. The results demonstrate that the proposed model can help make decisions on both the preparation and response phases in humanitarian relief logistics.

Appendix

There exist several methods in the literature to solve fuzzy problems (Pishvaee et al. 2012). One of the common methods is proposed by Jiménez et al. (2007). Generally, this method relies on strong mathematical concepts and is computationally efficient to solve fuzzy linear problems since it (1) can be applied to different membership functions; (2) preserves the linearity of the model; (3) does not increase the number of objective functions or inequality constraints; (Pishvaee and Torabi 2010; Shiraz et al. 2015).

Consider the following linear programming problem with fuzzy parameters:

$$Min\,\,\tilde{h}X$$

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$$st.\,\,\,\tilde{a}_{i} X \ge \tilde{b}_{i} ,\quad \forall i = 1, \ldots ,l$$

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$$\tilde{a}_{i} X = \tilde{b}_{i} ,\quad \forall i = l + 1, \ldots ,m$$

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$$X \ge 0$$

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Now, let $\tilde{h} = (h^{p} ,h^{m} ,h^{o} )$ be a triangular fuzzy number. Two following equations are respectively the expected value and the expected interval of triangular fuzzy number $\tilde{h}$.

$$EV(\tilde{h}) = \frac{{E_{1}^{h} + E_{2}^{h} }}{2} = \frac{{h^{p} + 2h^{m} + h^{o} }}{4}$$

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$$EI(\tilde{h}) = \left[ {E_{1}^{h} ,E_{2}^{h} } \right] = \left[ {\frac{1}{2}\left( {h^{p} + h^{m} } \right),\frac{1}{2}\left( {h^{m} + h^{o} } \right)} \right]$$

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According to Jiménez method, and the study of Pishvaee and Torabi (2010), the above model can be reformulated as follows:

$$Min\,EV(\tilde{h})X$$

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$$st.\,\,\,\,\left[ {(1 - \alpha )E_{2}^{{a_{i} }} + \alpha E_{1}^{{a_{i} }} } \right]X \ge \alpha E_{2}^{{b_{i} }} + (1 - \alpha )E_{1}^{{b_{i} }}$$

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$$\left[ {\left( {\frac{\alpha }{2}} \right)E_{2}^{{a_{i} }} + \left( {1 - \frac{\alpha }{2}} \right)E_{1}^{{a_{i} }} } \right]X \le \left( {1 - \frac{\alpha }{2}} \right)E_{2}^{{b_{i} }} + \left( {\frac{\alpha }{2}} \right)E_{1}^{{b_{i} }}$$

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$$\left[ {\left( {1 - \frac{\alpha }{2}} \right)E_{2}^{{a_{i} }} + \frac{\alpha }{2}E_{1}^{{a_{i} }} } \right]X \ge \frac{\alpha }{2}E_{2}^{{b_{i} }} + \left( {1 - \frac{\alpha }{2}} \right)E_{1}^{{b_{i} }}$$

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