Pareto solution

Figure 5 OF mean value (left) and standard deviation (right) and Pareto solutions plotted in the design vector space. Green points are for < I and red ones are for > I...

Figure 1 shows the Pareto curve for the deterministic case. This curve was obtained maximizing the NPV and constraining the consumer satisfaction. The curve shows that only above a 66% consumer satisfaction level some trade off between the objectives exists. Below 66% of requested consumer satisfaction the solution is the same as that of the model without constraining consumer satisfaction and therefore all the pareto solutions accumulate at the end point on the left. Figure 2 shows the same curve for the stochastic model. Figures 3 and 4 show the corresponding consumer satisfaction and financial risk curves of the pareto solutions of the multiobjective stochastic problem. Unsupported solutions are suspected to exist, but this could also be the effect of the small number of scenarios (100) used. In future work this matter will be resolved. 

The solutions of the problem are situated on section BDC of the boundary of the feasible region. These are called non-dominated solutions, Pareto ideal solutions or the compromise set that is, no single objective function can be increased without causing the simultaneous decrease of at least one other function. In effect, the difficulty is that there are a great number of Pareto solutions. It is therefore necessary to apply other procedures for the selection of one solution, called the preferable solution. ... 

Another method is based on the selection of a point on the curve of Pareto solutions, which is closest to the ideal solution in the space of normalized functions. In that second method the definition of the closest point needs to be justified, again using some new arguments, for example, from the conditions of execution in practice. 

Effective searching process used in obtaining Pareto solutions of two-objective network design problem... 

Making a decision reasonably needs to consider plural measures when we evaluate these network systems. In our study, we consider allterminal reliability and construction cost for network as important measures for evaluation. However, there exists a trade-off between reliability and construction cost, so it is a rare case that a solution (network system) makes all-terminal reliability and construction cost optimum simultaneously. Therefore we consider an algorithm for obtaining Pareto solutions of a two-objective network with all-terminal reliability and construction cost. The reliability and cost problems for the network systems have been studied in a long... 

In order to construct an efficient algorithm, we researched differences between the optimum networks, which become Pareto solutions, and the other networks. Based on these obtained properties, we reduce the number of networks whose reh-ability and cost must be calculated, and propose a new dgorithm for obtaining the subset of Pareto solutions. And we evaluate the efficiency and accuracy of a proposed algorithm by numerical experiments. 

It is a rare case that a network makes allterminal reliability and construction cost optimal simultaneously. For solving this problem, we search for Pareto solutions. Let G be the set of feasible solutions. For g, g G, g becomes Pareto solutions if there are no other feasible solution g which satisfies either 3 conditions as follows ... 

That is, our considered optimal network design problem means obtaining Pareto solutions from all the sub-networks g, where g = iV, E) is all the sub-networks of g (or g = iK E), E q E. 

We consider network design problem with allterminal reliability and construction cost. For this problem, proposed algorithm by Akiba et al. (2007) can reduce computing time for all-terminal reliability. However, their algorithm doesn t consider the relation between reliability and cost, and must calculate reliabilities and costs for all sub-networks. This process expands search space for Pareto solutions and makes computing time increase. 

Property hg +i constructed fromg (G j fGPf ) is likely to be Pareto solutions GPf ... 

Property 1 means networks close to each Pareto solutions GPf are likely to be candidate networks for construction of Gf R is lower boundary of considered rank as candidate. In an experiment with networks of 5 nodes, all Pareto solutions GPf could be constructed by gpj g GPf GPf ... 

Next, we take notice of the shape of Pareto front and propose restricting the range by the slope of Pareto solutions. This restriction adopts the straight line connecting origin point and one of Pareto solutions. Two properties are assumed below. 

For obtained Pareto solutions GPf, we sum up included times for each edge and this measure is applied to select edge. Using binary variable x which takes 1 if edge e. exists or else takes 0, we define the following index. 

Property 3 Pareto solutions include networks as the degree of node satisfies following equation. 

Next, we consider adding edge to selected networks. Pareto solutions depend on two objective functions, reliability and cost of each edge. Therefore this section mentions how two values of edge give influence on Pareto solutions. For e., we define following index. 

In the following explanation, we describe ALs v of Table 1. f and GP f signify considered networks by the slope of Pareto solutions, and G f and GP f by the degree of nodes. Among G f, networks which satisfy the range for considered networks are set in GP f, where paretoiG f) is set in... 

Moreover, changing combination of network properties constructs three other algorithms which are AL y, ALrov and AL y of Table 1. ALjy (Takahashi et al. 2013) excludes STEPs on thie node degree from ALsjjy. AL py and AL y substitute rank of Pareto solution for STEPs on slope of Pareto solution of AL py and ALj y, respectively. Varying parameters R and ec as shown in Table 2, numerical experiments are conducted. For example, ALjjpy takes values of 3 patterns on R and 10 patterns on ec, so 30 experiment patterns are conducted. Table 1 also indicates where the result... 

In ALj s y, slope of Pareto solutions is used as the range for considered networks when k <... 

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