We consider BIOBJECTIVE SHORTEST PATH problems in networks with fuzzy arc lengths. Considering the available studies for single objective SHORTEST PATH problems in fuzzy networks, using a distance function for comparison of fuzzy numbers, we propose three approaches for solving the BIOBJECTIVE problems. The rst and second approaches are extensions of the labeling method to solve the single objective problem and the third approach is based on dynamic programming. The labeling methods usually producing several nondominated PATHs, we propose a fuzzy number ranking method to determine a fuzzy SHORTEST PATH. Illustrative examples are worked out to show the e ectiveness of our algorithms.

This paper develops an algorithm for finding the dynamic SHORTEST PATH from the source node to the sink node in stochastic dynamic networks, in which the arc lengths are independent random variables with exponential distributions. In each node there is an environmental variable, which evolves in accordance with a continuous time Markov process. The parameter of the exponential distribution of the transition time of each arc is also a function of the state of the environmental variable of its initiating node. It is also assumed that upon arriving at each node, we know the state of its environmental variable and also the states of the environmental variables of its adjacent nodes. Upon arriving at each node, we can move toward the sink node through the best outgoing arc or wait for encountering the better state of its environmental variable, which reduces the expected transition times of the outgoing arcs. In this paper, we apply the stochastic dynamic programming for finding the dynamic SHORTEST PATH from the source node to the sink node by obtaining the optimal strategy of movement in each node of the network.

In this paper, we use a classical theorem of Hardy, Littlewood and Polya on rearrangements of functions for solving well known SHORTEST PATH problem. We first introduce a special .class of variational problems. To solye this kind of problems, a classical theorem of Hardy, Littlewood and  Polya on rearrangements of functions is used to prove.the equivalence of this class of variational problems. As a consequence, solutions of this class of variational, problems are solution of SHORTEST PATH problem that can be computed numerically via quadratic.programming In this paper we suggest an algorithm' for the SHORTEST PATH problem when this PATH, called (t,F(t)) , starting from given Initial point (0,F(0)) and reaching destination (1,F(1)) while remaining between boundaries defined by (t,A(t)) and (t,B(t)). One extension of the problem can be considering SHORTEST PATH surrounded by the given boundaries while a circle lies in the phase plane between two given boundaries and we should consider this circle. In the solution of Problem A useful application of this problem is optimal, battery interchange policy for solar cars.

For a network with cycle, where at least one cycle exists, the Floyd-War shall algorithm is probably the most used algorithm to determine the least cost PATH between every pair of nodes on this network, i.e. the solution for the SHORTEST PATH problem with cycle.In this paper, a new algorithm for this problem which requires less computational effort than the Floyd- War shall algorithm has been developed Furthermore, it can be shown that the basis of our algorithm is much easier to be learnt and understood which might be an advantage for educational purposes. A small example validates our algorithm and shows its implementation.

We propose a dynamic program to find the SHORTEST PATH in a network having gamma probability distributions as arc lengths. Two operators of sum and comparison need to be adapted for the proposed dynamic program. Convolution approach is used to sum two gamma probability distributions being employed in the dynamic program.