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.

Survivable Capacitated NETWORKs Design PROBLEM (SCNDP for short) is one of the most essential issues in telecommunication and transportation NETWORKs. A survivable NETWORK is a NETWORK that is designed to remain operational in the event of a component(s) failure(s) (cable cuts, electronic failures on switching centers and so forth). In SCNDP, researchers usually consider survivability with two different ways named diversification and reservation. Diversification consists of dividing the demand of source-sink node pairs into two or more node-disjoint (or arc-disjoint) paths, and in reservation after the failure of a component(s) of the NETWORK, part of the demand for node pairs can still be satisfied by rerouting. To implement reservation, the links on the NETWORK ought to have enough capacity to support the new FLOWs after the rerouting has been made. ‎Therefore, in this paper, we first present one model for the PROBLEM of survivable capacitated NETWORK design based on the reservation strategy. In many real-world applications, the observed values are often uncertain, for example random, fuzzy, etc. Therefore, we consider the demand vector (the right hand side values) randomly and then by using joint chance-constrained and probability concepts, obtain the corresponding deterministic model. Then, we propose an approximation optimization approach and use piecewise tangent approximation and piecewise linear methods to obtain the lower and upper bounds for it, respectively. Finally, computational results show the efficiency of the proposed model and approximation methods for relatively large-scale NETWORKs.

In this work, we consider a ‎multi-objective‎ ‎minimal ‎cost ‎FLOW (MMCF) ‎PROBLEM where there are several commodities to transport from‎ sources ‎to ‎destinations and there is more than one conveyance for those transporting. We also assume that in each conveyance, there are distinct capacities for each commodity. The obtained model is not necessary balanced, and we introduced a method to solve this model without converting it to a balanced model. Some advantages of the proposed method is discussed.

In this paper, we state the PROBLEM of the NETWORK FLOW interdiction in a set of initial and destination nodes so that each initial is capable of only delivering products to certain pre-determined destinations. The NETWORK user’s purpose is to deliver the highest value of FLOW from the sources to the sinks and the NETWORK interdictor’s purpose is to reduce the highest value of FLOW being used. In this paper, the NETWORKs FLOW interdiction in multi-source and multi-sink conditions are addressed in a way that the parameters of arc capacity are trapezoidal fuzzy sets.

We consider the maximum FLOW NETWORK interdiction PROBLEM. We provide a new interpretation of the PROBLEM and define a concept called ”optimalcut”. We propose a heuristic algorithm to obtain an approximated cut, and we also obtain its error bound. Finally, we show that our heuristic is an α-approximation algorithm for a class of NETWORKs. By implementing it on three NETWORK types, we show the advantage of it over solving the model by CPLEX.

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.

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.

The NETWORK Design PROBLEM (NDP) is one of the important PROBLEMs in combinatorial optimization. Among the NETWORK design PROBLEMs, the Multicommodity Capacitated NETWORK Design (MCND) PROBLEM has numerous applications in transportation, logistics, telecommunication, and production systems. The MCND PROBLEMs with splittable FLOW variables are NP-hard, which means they require exponential time to be solved in optimality. With binary FLOW variables or unsplittable MCND, the complexity of the PROBLEM is increased significantly. With growing complexity and scale of real world capacitated NETWORK design applications, metaheuristics must be developed to solve these PROBLEMs. This paper presents a simulated annealing approach with innovative representation and neighborhood structure for unsplittable MCND PROBLEM. The parameters of the proposed algorithms are tuned using Design of Experiments (DOE) method and the Design-Expert statistical software. The performance of the proposed algorithm is evaluated by solving instances with different dimensions from OR-Library. The results of the proposed algorithm are compared with the solutions of CPLEX solver. The results show that the proposed SA can find near optimal solution in much less time than exact algorithm.

In this paper, we report a new mathematical programming model for the tactical planning of freight railways, which addresses the train formation PROBLEM. The PROBLEM results in a MULTI-COMMODITY NETWORK FLOW formulation with both integer and continuous variables. The modular structure of the model allows considering special conditions of the NETWORK under study and testing various operating policies.