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Nov 11, 2021Open Access
In this paper, we address the problem of solving discrete alternative multiple criteria decision making (MCDM) problems where some or all of the criteria might have indifference regions. We utilize the classical cone-dominance approach and significantly extend the associated theory. We then develop a convergent solution method based on cone-dominance for achieving the most preferred choice. The convergent method utilizes pairwise comparisons of the alternatives by the decision maker (DM) to elim...
Oct 20, 2021Open Access
The core of Shape of numbers is formal calculation, which has three forms. This paper proves the equivalence of these forms and extends the formula to the general case. Some properties of the coefficients are summarized and some new conclusions are drawn. The coefficient matrix is studied and the corresponding results are obtained. Using the formal method, the calculation formula of ∑ n-0N-1Π i-1M (K...
Mar 26, 2021Open Access
This paper redefines the Shape of numbers, makes it more natural and concise, and the domain of definition is extended to ring. The inconvenient PCHG() and PH() are removed. The concept of subsets is also removed. The new definition can be used to calculate ∑ n-0N-1Π i-1M (K i n×D i)
∑ ni,j-0j-N-1Π i-1...
Jan 26, 2021Open Access
This article extends the concept of the shape of numbers. Originally, a shape was defined as [1, K1, K2, ···], 1<K1<K2< ···, Ki∈N. In this paper, the domain of a shape is extended from N to Z, the low bound is extended from 1 to Z, and Ki<Ki 1, Ki=Ki 1, K...
Dec 31, 2020Open Access
This article is based on the concept of Shape of numbers, introduces subset of the Shape and obtains its calculation formula. This article also makes some analysis and draws new conclusions, especially the calculation method of 1 M 2 M 3 M ··· N M. The Shape’s concept becomes clearer and richness.
Sep 22, 2020Open Access
This article is based on the concept of Shape of numbers, introduce more shapes, obtain the calculation formulas and find an explanation of the formulas. By observing and associating, show a law about the symmetry of Ring.
May 21, 2020Open Access
In this paper, we find the polynomials, indices and average distance for Schultz and modified Schultz of vertex identification chain for 4-cycle and 4- cycle complete.
Aug 27, 2019Open Access
For a connected graph G, the Schultz and modified Schultz polynomials are defined as, respectively, where the summations are taken over all unordered pairs of distinct vertices in V(G), is the degree of vertex u, is the distance between u and v and V(G) is the vertex set of G. In this paper, we find Schultz and modified Schultz polynomials of the Cog-special graphs such as a complete graph, ...
Jul 11, 2017Open Access
In this paper, multiplicative version of degree
distance of a graph is defined and tight upper bounds of the graph operations
have been found.
Dec 09, 2016Open Access
The objective of this paper is to prove by simple
construction, generalized by induction, that the bounded areas on any map, such
as found on the surface of a sheet of paper or a spherical globe, can be
colored completely with just 4 distinct colors. Rather than following the
tradition of examining each of tens of thousands of designs that can be
produced on a planar surface, the approach here is to all the ways that any given
pla ...
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