In this paper, a step fixed charge transportation problem is developed where the products are sent from the sources to the destinations in existence of both unit and step fixed-charges. The proposed model determines the amount of products in the existing routes with the aim of minimizing the total cost (sum of unit and step fixedcharges) to satisfy the demand of each customer. As the problem is NP-hard, a moderate sized instance of this problem becomes intractable for general-purpose solvers. In order to overcome this difficulty, a Lagrangian relaxation approach is proposed. The computational experiments show that the Lagrangian relaxation algorithm is able to solve large sized problems with optimality gap compared to general-purpose solvers.

In the literature hardly any attention is paid to solving a fuzzy fixed charge transportation problem. In this paper, we consider the fully fixed-charge transportation problem and try to find both the lower and upper bounds on the fuzzy optimal value of such a problem in which all of the parameters are triangular fuzzy numbers. To illustrate the proposed method, a numerical example is presented.

Fixed charge solid transportation problem (FCSTP) is one of the main and most important problems in transportation and network research areas. To tackle such an NP-hard problem, An Electromagnetism-like algorithm (EM) is employed. To the best of our knowledge, EM has been considered for any kind of transportation problems. Due to the significant role of parameters on the algorithm’s performance, a calibration in EM is carried out with the aid of a set of experimental design. The efficiency of employed parameters is measured by the experimental design method. To evaluate the performance of the proposed EM, a computational study has been conducted and the associated results obtained by the EM are compared with simulated annealing algorithm (SA).

The fixed charge transportation problem (FCTP) is a deployment of the classical transportation problem in which a fixed cost is incurred, independent of the amount transported, along with a variable cost that is proportional to the amount shipped. Since the problem is considered as an NP-hard, the computational time grows exponentially as the size of the problem increases. In this paper, we propose a new heuristic along with well-known metaheuristic like Genetic algorithm (GA), simulated annealing (SA) and recently developed one, Keshtel algorithm (KA) to solve the FCTP. Contrary to previous works, we develop a simple and strong heuristic according to the nature of the problem and compare the result with metaheuristics. In addition, since the researchers recently used the priority-based representation to encode the transportation graphs and achieved very good results, we consider this representation in metaheuristics and compare the results with the proposed heuristic. Furthermore, we apply the Taguchi experimental design method to set the proper values of algorithms in order to improve their performances. Finally, computational results of heuristic and metaheuristics with different encoding approaches, both in terms of the solution quality and computation time, are studied in different problem sizes.

Nowadays، one of the most significant aims of supply chain، and also one of the most substantial evaluation criteria of any organization’ s performance is making customers' satisfaction. Therefore، delivering products to the right place، at the right time، and with the lowest cost are considered to be important goals in supply chain management. The transportation cost is one of the most important and effective factor for pricing goods and the final price for customers. Also، fixed-charge transportation problem (FCTP) is a primary and important problem which attracts researchers in the last decade. Therefore، focusing on transportation costs in order to reducing the final price of goods is necessary to increase the place of industry among the competitors and the satisfaction of customers. In the problem، when a route in a solution is used، both fixed and variable costs are calculated for opening the rout and also for the amount of the transferred goods in the route. Contrary to the general TP، the FCTP is more difficult to solve because of the fixed costs that result in discontinuities in the objective function and makes it indissoluble by the straight application of the transportation algorithms. In this research، the fixed cost transportation problem with considering the discount limitation is modeled and solved. Due to NP-hardness of the problem، three metaheuristics Simulated Annealing (SA)، Genetic Algorithm (GA) and Whale Optimization Algorithm (WOA) are developed. Prufer number encoding is utilized to represent the solution in all algorithms. Because of importance of the parameters calibration، Taguchi method is used for tuning the parameters in algorithm designing. Besides، 28 test problems with different sizes are solved and compared with the results of GAMS.

Lithium and oxygen interaction plays a cornerstone role in lithium ion and lithium air batteries and lithium based technologies. In this way, oxidation and reduction of neutral and charged lithium is the key process in its application as power source and sustainable energies. Since both oxidation and reduction are based on charge transfer in molecular scale, they can be analyzed via electronic structure changes. Two types of fragmentations for charged complexes were suggested, in which the positive charge was located on either lithium or oxygen fragment. Deformation density analysis is a recently developed technique for identification of different intermolecular interactions in the context of quantum chemical language. In this analysis, the molecular orbitals of isolated fragments were employed to build non-interacting and anti-symmetrized fragments and the corresponding density matrices to find deformation density matrix of each one. In the present study, two types of deformation density including kinetic energy pressure and relaxation analyses were accomplished for lithium and oxygen interaction at B3LYP/6-311+G* theoretical level. Electronic deformation orbitals responsible for charge transfer were identified with respect to their eigenvalues. The results showed how these two competed with each other in neutral and charged complexes with different fragmentations.