## 数学代写|凸优化作业代写Convex Optimization代考|The Analysis of Exploratory Guess

The visualization applied to the available data gives a reason to guess the variable $x_2$ as the most significant variable defining a point on the Pareto front. However, the visual exploratory approach does not provide reliable conclusions, and a further analysis is necessary either to prove or reject this guess.

If the guess above is correct, a clearly expressed dependency between the value of $x_2$ and the position of the corresponding point on the Pareto front should exist. To indicate such a dependency in Figure 9.4a, the values of $x_2$ are shown depending on the index of a Pareto optimal solution where the latter are sorted according to the increase of $f_1\left(\mathbf{x}_i\right)$.

The linear dependency between $x_2$ and the index of a Pareto optimal decision is clearly seen in Figure 9.4a for the indices which belong to the interval $(20,170)$, which is much longer than the interval of interest indicated above. Since the points of the discrete representation are distributed over the Pareto front quite densely and uniformly, all characteristics of interest can be presented as functions of the index as an independent variable. However, such a parametric description of the problem data has a disadvantage: the independent variable has no interpretation in the engineering terms of problem formulation.

A variable $t$ varying along the line in Figure 9.4a seems well suited for use as an independent variable for parametric description of the data of interest. The value of $t$ can be interpreted as a value of $x_2$ smoothed along the Pareto front. The values of $t$ in the interval $0.15 \leq t \leq 0.4$ correspond to the kink of the Pareto front. The relationship between $f_1\left(\mathbf{x}_i\right), f_2\left(\mathbf{x}_i\right)$ and the corresponding value of $t$ is presented by Figure 9.4b. The graphs of $x_2(t)$ and $x_4(t) 0.15 \leq t \leq 0.4$ are presented in Figure 9.5a.

To highlight the location of the most interesting part of the Pareto optimal decisions corresponding to the kink of the Pareto front, the part of image of Figure $9.2$ is presented in Figure 9.5b. The images of the closest vertices are marked by 0 and 1 .

Before the visualization-based analysis, the engineers expected a different impact of the design variables to the trade-off between the Pareto optimal solutions. We refer to [263] where a priori opinion and the final conclusions are discussed in engineering terms.

## 数学代写|凸优化作业代写Convex Optimization代考|Multi-Objective Optimization Aided

Graphs are very popular models of many research subjects. Graphical presentation of a problem is advantageous for heuristic perception and understanding of relations between the objects considered. On the other hand, many efficient algorithmic techniques are available to attack mathematically stated graph problems. Therefore, graph models are especially useful where heuristic abilities of a human user in the formulation of a problem are combined with its algorithmic solution in the interactive mode. In the present chapter we consider the graph models of business processes which, in the literature on the management of business processes, are called business process diagrams (BPDs). To be more precise, a problem of drawing aesthetically pleasing layouts of BPDs is considered. The research of this problem was motivated by the fact that the aesthetic layouts are not only well readable but also most informative and practical [12].

The considered business process management methodology is oriented to managers and consultants either designing a new Small/Medium Enterprise (SME) or searching for the possibilities to improve an existing one. The Business Process Modeling Notation (BPMN) is accepted as a standard for drawing the business process diagrams [146]. The graph drawing aesthetics is comprehensively discussed, e.g., in $[15,171]$. Although the problem of graph drawing attracts many researchers, and plenty of publications are available, special cases of that problem frequently cannot be solved by straightforward application of the known methods and algorithms. We cite [172]: “Few algorithms are designed for a specific domain, and there is no guarantee that the aesthetics used for generic layout algorithms will be useful for the visualization of domain-specific diagrams.”

The original problem is reduced to a problem of the combinatorial multi-objective optimization. We start with the analysis of the attitude of the potential users of the supposed algorithms towards the relative importance of the widely acceptable criteria of aesthetics. The latter are used as the objectives of the multi-objective optimization problem which formalizes the problem of BPD drawing as an optimization problem.

General rules for drawing BPDs are defined by the standards of BPMN. These standards, however, leave sufficient freedom for choosing the techniques of BPD drawing that are most relévant to the supposed conditions of business processs management. We consider in the present chapter the methods for drawing BPDs aimed at the application by the business process management consultants who advise managers of small and medium enterprises planned to be established or undergoing the re-organization.

# 凸优化代考

.凸优化 .凸优化

## 数学代写|凸优化作业代写凸优化代考|多目标优化辅助

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