### Article

## Modeling of multi depot vehicle routing problem for petroleum products

The paper is devoted to modeling multi depot vehicle routing problem (VRP) with capacity constraints for petroleum products delivery. Applying efficient metaheuristics algorithms combined with local search procedures, we present how to get suboptimal solutions for this NP-hard problem in an acceptable time. Some parallel computing techniques are also used to reduce the execution time. Experimental results are performed by the case of VRP for petroleum products.

Motor fuel distribution problem is considered. Accepting some assumptions it can be reduced to a well-known vechicle routing problem with capacity constraints. Ant colony optimization approach is suggested for solving CVRP. Modified ant algorithms are performed. Computational results for some benchmarks are given in compare with classical ant algorithms.

The companies that are IT-industry leaders perform from several tens to several hundreds of projects simultaneously. The main problem is to decide whether the project is acceptable to the current strategic goals and resource limits of a company or not. This leads firms to an issue of a project portfolio formation; therefore, the challenge is to choose the subset of all projects which satisfy the strategic objectives of a company in the best way. In this present article we propose the multi-objective mathematical model of the project portfolio formation problem, defined on the fuzzy trapezoidal numbers. We provide an overview of methods for solving this problem, which are a branch and bound approach, an adaptive parameter variation scheme based on the epsilon-constraint method, ant colony optimization method and genetic algorithm. After analysis, we choose ant colony optimization method and SPEA II method, which is a modification of a genetic algorithm. We describe the implementation of these methods applied to the project portfolio formation problem. The ant colony optimization is based on the max min ant system with one pheromone structure and one ant colony. Three modification of our SPEA II implementation were considered. The first adaptation uses the binary tournament selection, while the second requires the rank selection method. The last one is based on another variant of generating initial population. The part of the population is generated by a non-random manner on the basis of solving a one-criterion optimization problem. This fact makes the population more strongly than an initial population, which is generated completely by random. Comparing of ant colony optimization algorithm and three modifications of a genetic algorithm was performed. We use the following parameters: speed of execution and the C-metric between each pair of algorithms. Genetic algorithm with non-random initial population show better results than other methods. Thus, we propose using this algorithm for solving project portfolio formation problem.

In this paper we consider application of ant colony optimization techniques for capacitated vehicle routing problem. Modified ant colony optimization algorithm is proposed, computational results are reported.

The companies that are IT-industry leaders perform from several tens to several hundreds of projects simultaneously. The main problem is to decide whether the project is acceptable to the current strategic goals and resource limits of a company or not. This leads firms to an issue of a project portfolio selection; therefore, the challenge is to choose the subset of all projects which satisfy the strategic objectives of a company in the best way. In this present article we propose the multi-objective mathematical model of the project portfolio selection problem, defined on the fuzzy trapezoidal numbers. We provide an overview of methods for solving this problem, which are a branch and bound approach, an adaptive parameter variation scheme based on the epsilon-constraint method, ant colony optimization method and genetic algorithm. After analysis, we choose ant colony optimization method and SPEA II method, which is a modification of a genetic algorithm. We describe the implementation of these methods applied to the project portfolio selection problem. The ant colony optimization is based on the max min ant system with one pheromone structure and one ant colony. Three modification of our SPEA II implementation were considered. The first adaptation uses the binary tournament selection, while the second requires the rank selection method. The last one is based on another variant of generating initial population. The part of the population is generated by a non-random manner on the basis of solving a one-criterion optimization problem. This fact makes the population more strongly than an initial population, which is generated completely by random.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.