Shortest path problem is one of the most fundamental and well-known optimization problems in graph theory due to its various real-world applications. Fuzzy set can manage the uncertainty, associated with the information of a problem, where conventional mathematical models may fail to reveal satisfactory result. In most cases, shortest path problem in fuzzy graph, called fuzzy shortest path problem, uses type-1 fuzzy set as arc length. The uncertainty associated with the linguistic description of information is not represented properly by type-1 fuzzy set due to inexactness of human perception in the evaluation of membership degrees having crisp values. An interval type-2 fuzzy set is able to tackle this type of uncertainty. In this paper, we have proposed an algorithmic approach based on genetic algorithm for finding shortest path from a source node to a destination node in a fuzzy graph with interval type-2 fuzzy arc lengths. We have designed a new crossover operator which does not need mutation operation. The purpose of mutation operation has been taken care by the proposed crossover operation. We have compared our algorithm with two other existing genetic algorithms for the fuzzy shortest path problem, where superiority of the proposed algorithm is shown. To the best of our knowledge, no algorithm based on genetic algorithm exists in the literature for fuzzy shortest path problem with interval type-2 fuzzy arc lengths. A numerical example is used to illustrate the effectiveness of the proposed approach.

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Abstract. Fuzzy graph model can represent a complex, imprecise and uncertain problem, where classical graph model may fail. In this paper, we propose a fuzzy graph model to represent the examination scheduling problem of a university and introduce a genetic algorithm based method to find the robust solution of the scheduling problem that remains feasible and optimal or close to optimal for all scenarios of the input data. Fuzzy graph coloring method is used to compute the minimum number of days to schedule the examination. But problem arises if after the examination schedule is published, some students choose new courses in such a way that it makes the schedule invalid. We call this problem as fuzzy robust coloring problem (FRCP). We find the expression for robustness and based on its value, robust solution of the examination schedule is obtained. The concept of fuzzy probability of fuzzy event is used in the expression of robustness, which in turn, is used for fitness evaluation in genetic algorithm. Each chromosome in the genetic algorithm, used for FRCP, represents a coloring function. The validity of the coloring function is checked keeping the number of colors fixed. Fuzzy graphs with different number of nodes are used to show the effectiveness of the proposed method.

The latent heat phase change materials (PCMs) have emerged as an important and effective thermal storage material. The massive change in volume and the requirement for specialized containment of PCMs limits the thermal storage application of the common PCM like paraffin wax. Herein, we report the encapsulation of paraffin wax with the crosslinked poly(styrene-divinylbenzene-acrylic acid) shell via suspension polymerization. The optical and SEM images of the prepared encapsulated phase change materials (EPCMs) were in the size range of 200-500 µm. The thermal chargingdischarging rate of the EPCM was studied using a facile water bath technique. It was found that the optimized EPCM showed about 1.7 times faster-charging and 3.5 times faster discharging rate compared to reference PCM (without encapsulation). The latent heat for the EPCM was found to be about 3.3 times higher than the reference PCM. The thermal charging-discharging ability and latent heat values are higher for the encapsulated paraffin wax when compared with reference paraffin wax. The encapsulation of paraffin wax by polymeric materials via facial suspension polymerization using polymerizable vinyl monomers could find its application as heat storage materials.

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