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In real life, multiple attribute decision problems (MADM) can be applied in different areas and numerous related extensions and methodologies have been proposed by researchers. Combining three-way TOPSIS decision ideas with MADM is a feasible and meaningful research direction. In light of this, this paper generalizes the classical TOPSIS method with the help of mean and standard deviation and proposes the so-called modified three-way TOPSIS. First, using a pair of thresholds which is derived by mean and standard deviation, we divide decision alternatives into three segments, and then a preliminary rank results of decision alternatives can be obtained. Furthermore, in each decision region, we use two ranking regulations (one-way TOPSIS or modified two-way TOPSIS method) to rank decision alternatives. A practical example of urban expressway route selection illustrates the feasibility of the proposed method. Finally, we test the feasibility and validity of the modified three-way TOPSIS method by comparing with some existing method.

We consider families of life distributions with the first three moments belonging to small neighborhoods of respective powers of a positive number. For various shapes of the neighborhoods, we determine exact convergence rates of their Prokhorov radii to zero. This provides a refined evaluation of the effect of specific moment convergence on the weak one.

A color cast correction algorithm based on improved Frankle-McCann Retinex is proposed to correct images which are influenced by illumination. To improve on the original algorithm, the distance-weighting factor with Gauss function is introduced, and a linear stretch with the mean and the standard deviation is carried out. Experimental results demonstrate that the algorithm in this paper has improved correction effect on the color cast image.

In this and next two chapters, we will use advanced calculus to formalize and extend the probability theory introduced in Chapter 2. The use of calculus enables us to investigate probability more deeply. A number of quantitative-oriented probability concepts will be introduced. In this chapter, we first introduce the concept of a random variable and characterize the probability distributions of a random variable and functions of a random variable by the cumulative distribution function, the probability mass function or probability density function, the moment generating function and the characteristic function, respectively. We also introduce a class of moments and discuss their relationships with a probability distribution. This chapter focuses on univariate distributions.