## Abstract

Shortest path problem in real life applications has to deal with multiple criteria. This article intends to solve a proposed multi-criteria shortest path problem of a weighted connected directed network whose associated edge weights are represented as rough variables in order to tackle the imprecision. We have exhibited two different approaches to determine the optimum path(s) of the proposed problem. The first approach is the proposed modified rough Dijkstra’s labelling algorithm. The second approach considers the rough chance constrained programming technique to formulate the proposed multi-criteria shortest path problem which is eventually solved by two different methods: the goal attainment method and the nondominated sorting genetic algorithm II. These methodologies are numerically illustrated for a multi-criteria weighted connected directed network. Moreover, the simulated results on similar networks of large order and size are analyzed to show the efficiency of the algorithms.

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## Acknowledgements

The authors are very much grateful to the Editor and referees for their constructive and valuable suggestion to enhance the quality of the manuscript. Saibal Majumder, an INSPIRE fellow \(\left( {{\text{IF}}150410} \right)\) would like to thank DST, Govt. of India, for providing him financial support for the work.

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## Appendix

### Appendix

We consider a WCDN with a source and a terminal vertex. Besides, the associated edge weights of the WCDN are considered as rough variables. A suitable example in support of such a WCDN is presented below.

###
*Example A1*

Consider a WCDN of four cities. A city is represented by a vertex and an edge connecting two cities is associated with the cost incurred while travelling from city \({v_i}\) to \({v_j}.\) The cost of travelling from \({v_i}\) to \({v_j}\) depends on several factors like fuel price, toll charges, labour costs, etc., and as each of these factors fluctuate from time to time, it is not always possible to determine exact travelling cost between a pair of cities. Suppose that four experts are chosen to determine the possible cost\(~{c_{i,j}}\) while traversing from \({v_i}\) to \({v_j}\) during a certain time period. The experts feedback are obtained as intervals which are, \([100,~150],\) \([108,146.90],\) \([110,155.50]\) and \([105.50,149.50]\) respectively.

Considering all these intervals, the lower approximation of \(~{c_{i,j}}\) becomes \(\left[ {110,~146.90} \right]\), i.e., every members of \([110,146.90]\) are certain values of \(~{c_{i,j}}\). In other words, the interval selected as lower approximation of \(~{c_{i,j}}\), satisfies the consent opinion of all the experts. The interval, \(\left[ {100,155.50} \right]~\) becomes the upper approximation of \({c_{i,j}},\) which includes all the feedback received from the experts such that the elements of \([100,155.50]\) become the possible values of \(~{c_{i,j}}\). Hence \({c_{i,j}}\) is represented as a rough variable \(~\left( {\left[ {110,146.90} \right],\left[ {100,155.50} \right]} \right).\) Figure 8 depicts a four city weighted directed rough graph (or a WCDN) and the corresponding costs as determined by the experts are listed in Table 12. The rough costs in Table 12 which are associated with every edge of \(S\) in Fig. 8 are determined as rough variables and are estimated from the opinion of experts.

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Majumder, S., Kar, S. Multi-criteria shortest path for rough graph.
*J Ambient Intell Human Comput* **9**, 1835–1859 (2018). https://doi.org/10.1007/s12652-017-0601-6

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DOI: https://doi.org/10.1007/s12652-017-0601-6