In conventional DEA methods, all inputs and outputs are assumed to be continuous. However, the implied presumption of continuous data may not maintain an acceptable level of precision in practical data. Many practical situations such as number of teachers in a school, number of students in a college, or research papers only take Integer values. Presented work presumes subsets of input and output variables in traditional , ve-pronged axioms in DEA model to be Integer values for the , rst time in the , eld. An additional e, ort in this work expands axioms and a new model is presented. This paper presents a novel two phase model that in the , rst phase produces more precise e, ciency values and the best benchmark in its second phase, i. e. the nearest Integer point to the e, ciency boundary is selected based on constant returns to scale. A case study is presented that demonstrates a comparison of e, ciency values obtained with this model compared to prior models that is show in table 1. And in Table 2, the values of the slack are compared with each other.

In traditional Data Envelopment Analysis (DEA) approaches, inputs and outputs are usually considered as exact and real values. The relative efficiency of the Decision Making Units (DMUs) is evaluated and it is known that the factors are inputs and/or outputs. However, there are some conditions under which the efficiency of DMUs should be calculated while the data are Integer and ambiguous. Therefore, various Integer DEA models have been proposed to determine the performance of DMUs when Integer data and fuzzy factors are available. In addition, there are cases where the efficiency of DMUs should be determined when Integer data and flexible factors are available. Therefore, some Integer DEA methods have been proposed to calculate the performance of DMUs and specify the role of flexible measures when some of the data are Integer and flexible factors are available. However, there are some situations where there are Integer data, fuzzy Integer measures and flexible factors. Therefore, this paper sheds light on the nature of the model to determine the efficiency of DMUs when there are Integer inputs and/or outputs, flexible factors and fuzzy Integer measures, and determines the role of factors with uncertain inputs or outputs. In fact, slacks are addressed and a slack-based efficiency measure (SBM) is defined to compute the performance of DMUs in the presence of flexible factors, Integer data and fuzzy Integer measures. The proposed approaches are demonstrated and illustrated using an example.

‎Sombor index of a graph $G$ is defined by $SO(G) = \sum_{uv \in E(G)} \sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the degree of the vertex $v$ of $G$. An $r$-degree graph is a graph whose degree sequence includes exactly $r$ distinctive numbers. In this article, we study $r$-degree connected graphs with Integer Sombor index for $r \in \{5, 6, 7\}$. We show that: if $G$ is a 5-degree connected graph of order $n$ with Integer Sombor index then $n \geq 50$ and the equality occurs if only if $G$ is a bipartite graph of size 420 with $SO(G) = 14830$; if $G$ is a 6-degree connected graph of order $n$ with Integer Sombor index then $n \geq 75$ and the equality is established only for the bipartite graph of size $1080$; and if $G$ is a 7-degree connected graph of order $n$ with Integer Sombor index then $n \geq 101$ and the equality is established only for the bipartite graph of size $1680$.

We study the problem of reconstructing binary images from four projections data in a fuzzy environment. Given the uncertainly projections, we want to find a binary image that respects as best as possible these projections. We provide an iterative algorithm based on fuzzy Integer programming and linear membership functions.