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‎In 2021, a novel degree-based topological index was introduced by Gutman, called the  Sombor index. Recently Kulli and Gutman introduced a vertex-edge variant of the Sombor index, is caled KG-Sombor index. In this paper, we establish lower bound on the KG-Sombor index and determine the extremal trees achieve this bound.

Remarkable efforts are made to develop the job shop scheduling problem up to now. As a novel generalization, the stage shop can be defined as an environment, in which each job is composed of some stages and each stage may include one operation or more. A stage can be defined a subset of operations of a job, such that these operations can be done in any arbitrary relative order while the stages should be processed in a predetermined order. In other words, the operations of a stage cannot be initiated until all operations of the prior stage are completed. In this paper, an innovative lower bound based on solving the preemptive open shop (using a linear programming model in polynomial time) is devised for the makespan in a stage shop problem. In addition, three metaheuristics, including firefly, harmony search and water wave optimization algorithms are applied to the problem. The results of the algorithms are compared with each other, the proposed lower bound, and a commercial solver.

One of the most popular graphs in computational geometry is Yao graphs, denoted by $Y_k$, For every point set $S$ in the plane and an integer $k\geq 2$, the Yao graph $Y_k$ is constructed as follows. Around each point $p\in S$, the plane is partitioned into $k$ regular cones with the apex at $p$. The set of all these cones is denoted by ${\cal C}_p$. Then, for each cone $C\in {\cal C}_p$, an edge $(p,r)$ is added to $Y_k$, where $r$ is a closest point in $C$ to $p$. In this paper, we provide a lower bound of 3.8285 for the stretch factor of $Y_4$. This partially solves an open problem posed by Barba et al. (L.~Barba et al., New and improved spanning ratios for Yao graphs. Journal of computational geometry}, 6(2):19--53, 2015).

Stability of foundations near slopes is one of the important and complicated problems in geotechnical engineering, which has been investigated by various methods such as limit equilibrium, limit analysis, slip-line, finite element and discrete element. The complexity of this problem is resulted from the combination of two probable failures: foundation failure and overall slope failure. The current paper describes a lower bound solution for estimation of bearing capacity of strip footings near slopes. The solution is based on the finite element formulation and linear programming technique, which lead to a collapse load throughout a statically admissible stress field. Three-nodded triangular stress elements are used for meshing the domain of the problem, and stress discontinuities occur at common edges of adjacent elements. The Mohr-Coulomb yield function and an associated flow rule are adopted for the soil behavior. In this paper, the average limit pressure of strip footings, which are adjacent to slopes, is considered as a function of dimensionless parameters affecting the stability of the footing-on-slope system. These parameters, particularly the friction angle of the soil, are investigated separately and relevant charts are presented consequently. The results are compared to some other solutions that are available in the literature in order to verify the suitability of the methodology used in this research.

In this paper we establish the best lower bound for the weighted Jensen's discrete inequality with ordered variables applied to a convex function ¦, in the case when the bound depends on ¦, weights and three fixed variables. Some applications for particular cases of interest are provided.

Flexible flow shop scheduling problem (FFS) with unrelated parallel machines contains sequencing in flow shop where, at any stage, there exists one or more processors. The objective consists of minimizing the maximum completion time. Because of NP-completeness of FFS problem, it is necessary to use heuristics method to address problems of moderate to large scale problem. Therefore, for assessment the quality of this heuristic, this paper develop a global lower bound on FFS makespan problems with unrelated parallel machines.