We apply a primal-dual simplex algorithm for solving the Biobjective min imum cost-time network flow problem such that the total shipping cost and the total shipping fixed time are considered as the first and second objective functions, respectively. To convert the proposed model into a single-objective parametric one, the weighted sum scalarization technique is commonly used. This problem is a mixed-integer programming, which the decision variables are directly dependent together. Generally, the previous works have consid ered the linear Biobjective problem with the traditional network flow con straints, while in this paper, corresponding to each flow variable, a binary variable is defined. These zero-one variables are utilized to describe a fixed shipping time for positive flows. The proposed method is successful in finding all supported efficient solutions of a real numerical example.

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.

In this paper the concept of the Minimum Universal Cost flow (MUCF) for an  infeasible flow network is introduced. A new mathematical model in which the objective function includes the total costs of changing arc capacities and sending flow is built and analyzed. A polynomial time algorithm is presented to find the MUCF.    

In this paper the multicommodity network flow problem with equal flow on some predetermined arcs is considered. The equal flow constraints require that the amounts of arcs flow of some commodities on some subsets of predetermined arcs are equal. For solving this problem, first, using the capacity allocation algorithm, we define a starting solution. Then by lagrangian relaxation technique on general constraints we find a lower bound on the objective function value. An upper bound is also found by using embedded network simplex method.Finally, the optimal or near optimal solution is found when the upper and lowers bounds are made close enough to each other.

In this paper the minimum cost flow problem with additional linear constraints on some arcs' flows has been considered. The additional constraints show that the flow on arcs which are belonged to specific subsets of arcs have to be linearly depended on the flow on a specific arc(called reference arc) in the subset. Since the basis structure in this problem is not a spanning tree, we introduce a basis spanning graph and call it a good (q+1)-forest for the problem. Then by regarding the optimality conditions, we restructure the network simplex algorithm to solve the above problem.

A holding or a multi-business corporate seeks to coordinate its supply for minimum overall costs. A Corporate Supply Optimizer (CSO), as a central entity taking advantage of the notion of flow networks, gathers necessary operational information from members of the corporate supply chain. The CSO then guides supply chain members on ordering decisions for a minimum overall cost for the entire supply chain. Its computational engine models the entire supply chain with multiple members in four stages to satisfy customer demand. The CSO seeks a solution with minimum total costs, unlike non cooperative supply chains where individual members compete to optimize their local costs. The existing literature stays with restrictive assumptions on the number of supply chain stages, disallowing a case of multiple products. Simulation results indicate an approximately 26% reduction in total costs of the supply chain utilizing the CSO.