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The purpose of this paper is to obtain the reasonable dimensioning for each part and a full-dimension model of assembly dimensions.

The relational path graph of assembly dimension, the shortest-path spanning tree of functional dimension and a revised spanning tree are established in this paper.

The proposed method can obtain reasonable dimensioning of parts and establishment of dimension model in an assembly.

The proposed method can easily realise by computer and be more suitable to automatic dimensioning and establishment of dimension model of parts.

We address the problem of selecting a topological design for a network having a single traffic source and uncertain demand at the remaining nodes. Solving the associated fixed charge network flow (FCF) problem requires finding a network design that limits both the fixed costs of establishing links and the variable costs of sending flow to the destinations. In this paper, we discuss how to obtain a sequence of optimal solutions that arise as the demand intensity varies from low levels to high. One of the network design alternatives associated with these solutions will be chosen based upon the dominant selection criteria of the decision maker. We consider both probabilistic and non-probabilistic criteria and compare the network designs associated with each. We show that the entire sequence of optimal solutions can be identified with little more effort than solving a single FCF problem instance. We also provide solution approaches that are relatively efficient and suggest good design alternatives based upon approximations to the optimal sequence.

The purpose of this chapter is to solve multi-objective formulation for traveling salesman and transportation problems. Computations are based on real data for the road freight transportation of a Spanish company. The company was selected because of its importance in Spanish economy and market. This company is important in the whole country; however, it has its higher importance in the northern part of Spain. The requirements for these models are the minimization of total distance and the CO₂ emissions. To achieve this, it is required to know and carry out the minimization of the total distance traveled by the trucks during the deliveries. The deliveries are going to be executed between the different locations, nodes, in the region, and Elorrio, where the depot is situated. The data have been used to decide the best route in order to obtain a minimization of cost for the company. As it was mentioned earlier, the problems are focused on the reduction of the amount of CO₂ emissions and minimization of total distance; by studying different parameters, the best solutions of route transportation have been obtained. The software used to solve these models is CPLEX solver with AMPL programming language.

In this chapter, four bi-objective vehicle routing problems are considered. Weighted-sum approach optimization models are formulated with the use of mixed-integer programming. In presented optimization models, maximization of capacity of truck versus minimization of utilization of fuel, carbon emission, and production of noise are taken into account. The problems deal with real data for green logistics for routes crossing the Western Pyrenees in Navarre, Basque Country, and La Rioja, Spain.

Heterogeneous fleet of trucks is considered. Different types of trucks have not only different capacities, but also require different amounts of fuel for operations. Consequently, the amount of carbon emission and noise vary as well. Modern logistic companies planning delivery routes must consider the trade-off between the financial and environmental aspects of transportation. Efficiency of delivery routes is impacted by truck size and the possibility of dividing long delivery routes into smaller ones. The results of computational experiments modeled after real data from a Spanish food distribution company are reported. Computational results based on formulated optimization models show some balance between fleet size, truck types, and utilization of fuel, carbon emission, and production of noise. As a result, the company could consider a mixture of trucks sizes and divided routes for smaller trucks. Analyses of obtained results could help logistics managers lead the initiative in environmental conservation by saving fuel and consequently minimizing pollution. The computational experiments were performed using the AMPL programming language and the CPLEX solver.

The purpose of this chapter is to optimize multi-criteria formulation for green vehicle routing problems by mixed integer programming. This research is about the road freight transportation of a Spanish company of groceries. This company has more power in the north of Spain and hence it was founded there. The data used for the computational experiments are focused in the northern region of Spain. The data have been used to decide the best route in order to obtain a minimization of costs for the company. The problem focused on the distance traveled and the altitude difference; by studying these parameters, the best solution of route transportation has been made. The software used to solve this model is CPLEX solver with AMPL programming language. This has been helpful to obtain the results for the research and some conclusions have been obtained from them.

The purpose of this paper is to address the novel issues of executing graph optimization tasks on distributed simple growing biological systems.

The author utilizes biological and physical processes to implement non‐classical, and in principle more powerful, computing devices. The author experimentally verifies his previously discovered techniques on approximating spanning trees during single cell ontogeny. Plasmodium, a vegetative stage of slime mold Physarum polycephalum, is used as experimental computing substrate to approximate spanning trees. Points of given data set are represented by positions of nutrient sources, then a plasmodium is placed on one of the data points. Plasmodium develops and span all sources of nutrients, connecting them by protoplasmic strands. The protoplasmic strands represent edges of the computed spanning tree.

Offers experimental implementation of plasmodium devices for approximation of spanning tree.

The techniques, discussed in the paper, can be used in design and development of soft bodied robotic devices, including gel‐based robots, reconfigurable massively robots, and hybrid wet‐hardware robots.

Discusses original ideas on growing spanning trees, and provide innovative experimental implementation.

A variety of phenomena such as world wide web, social or business networks, interactions are modelled by various kinds of networks (such as the scale free or preferential attachment networks). However, due to the model‐specific requirements one may want to rewire the network to optimize the communication among the various nodes while not overloading the number of channels (i.e. preserving the number of edges). The purpose of this paper is to present a formal framework for this problem and to examine a family of local search strategies to cope with it.

This is mostly theoretical work. The authors use rigorous mathematical framework to set‐up the model and then we prove some interesting theorems about it which pertain to various local search algorithms that work by rerouting the network.

This paper proves that in cases when every pair of nodes is sampled with non‐zero probability then the algorithm is ergodic in the sense that it samples every possible network on the specified set of nodes and having a specified number of edges with nonzero probability. Incidentally, the ergodicity result led to the construction of a class of algorithms for sampling graphs with a specified number of edges over a specified set of nodes uniformly at random and opened some other challenging and important questions for future considerations.

The measure‐theoretic framework presented in the current paper is original and rather general. It allows one to obtain new points of view on the problem.

Discusses the building of a prototype decision‐support system for a container transport company in the light of constraining factors which affect scheduling of trucks and vehicle routeing problems of various kinds. Considers the prototype’s data requirements and describes the information flow analysis, drawing conclusions and offering suggestions, before proceeding to model the problem as a vehicle routeing problem. Describes the development of an heuristic procedure and how it was tested and validated. Summarizes the benefits of the analysis and the prototype’s construction.

The purpose of this paper is to propose a multi-objective differential evolution algorithm named as MOMR-DE to resolve multicast routing problem. In mobile ad hoc network (MANET), multicast routing is a non-deterministic polynomial -complete problem that deals with the various objectives and constraints. Quality of service (QoS) in the multicast routing problem mainly depends on cost, delay, jitter and bandwidth. So the cost, delay, jitter and bandwidth are always considered as multi-objective for designing multicast routing protocols. However, mobile node battery energy is finite and the network lifetime depends on node battery energy. If the battery power consumption is high in any one of the nodes, the chances of network’s life reduction due to path breaks are also more. On the other hand, node’s battery energy had to be consumed to guarantee high-level QoS in multicast routing to transmit correct data anywhere and at any time. Hence, the network lifetime should be considered as one objective of the multi-objective in the multicast routing problem.

Recently, many metaheuristic algorithms formulate the multicast routing problem as a single-objective problem, although it obviously is a multi-objective optimization problem. In the MOMR-DE, the network lifetime, cost, delay, jitter and bandwidth are considered as five objectives. Furthermore, three QoS constraints which are maximum allowed delay, maximum allowed jitter and minimum requested bandwidth are included. In addition, we modify the crossover and mutation operators to build the shortest-path multicast tree to maximize network lifetime and bandwidth, minimize cost, delay and jitter.

Two sets of experiments are conducted and compared with other algorithms for these problems. The simulation results show that our proposed method is capable of achieving faster convergence and is more preferable for multicast routing in MANET.

In MANET, most metaheuristic algorithms formulate the multicast routing problem as a single-objective problem. However, this paper proposes a multi-objective differential evolution algorithm to resolve multicast routing problem, and the proposed algorithm is capable of achieving faster convergence and more preferable for multicast routing.