Journal of Mathematical Extension
Volume:14 Issue: 3, Summer 2020

Data envelopment analysis (DEA) is a nonparametric method aimed to estimate production frontiers. This technique has been empirically used to measure the productive efficiency of decision making units (  comprising a wide range of entities (i.e. banks, car makers, schools) in terms of their inputs conversion into outputs. The decision makers are challenged with the problem of restricted resources and optimal allocation of budget to each DMU. The common budget allocating models (except those having series internal processes) do not seriously address the internal processes of the DMUs. That is, they are overlooked and rather passed up as block boxes. This paper proposed a new model for optimal budgeting for designing dynamic network system using DEA implication. The advantages of this model (which can be regarded as the first accomplishment) are expounded through the research body. Within the framework of the dynamic network systems of optimal system design (OSD) via insights gained by DEA, the optimal budgeting, the budget deficit and the budget congestion along with superior features of the proposed model were systematically investigated. Finally, two cases were presented on which the suggested model was applied. In one application, the OSD network of a business entity with 5 DMUs was addressed; while the other one dealt with a business venture with 24 DMUs.

A new method is proposed for finding a set of efficient solutions to bi-objective fractional transportation problems with fuzzy numbers using ranking function. This method is an important tool for the decision makers to obtain efficient solutions and select the preferred optimal solution from the satisfaction level. The procedure allows the user to identify next efficient solution to the problem from the current efficient solution. This new approach enables the decision makers to evaluate the economic activities and make satisfactory managerial decisions when they are handling a variety of logistic problems involving two objectives. An illustrative example is presented to clarify the idea of the proposed approach.

The main object of this article is to discuss Bayesian methodology for linear regression model according to the class of two-piece scale mixture of normal distribution. This model is appropriate for capturing departure from the usual normal assumption of error such as heavy tails, asymmetric and types of heteroscedasticity. Linear regression model is used to analyze data based on the normality assumption. The robust inference for normality assumption as a way to replace the Gaussian assumption for the residual errors with two-piece scale mixture of normal distribution is a Bayesian framework. An efficient way for applying Bayesian methodology is introduced using Markov chain Monte Carlo (MCMC) algorithm as a way to specify the posterior inference which has been used.

In this paper, a new limited memory BFGS is proposed for solving stochastic optimization problems. Since the cost of storing and manipulating $H_k$ is prohibitive in the large scale setting, the L-BFGS algorithms use the strategy of keeping the most recent correction pairs. Besides, in the stochastic regime, due to some noisy information in both gradient vector and Hessian approximation, the second-order model is not an accurate estimation of the function. To overcome this problem, our L-BFGSemploys memory in an optimal manner by storing the correction pairs that have the least violation in the secant equation. Under some standard assumptions, the convergence property of the new algorithm is established for strongly convex functions. Numerical results on the problems arising in machine learning show that the new method is competitive and effective in practice.

In this paper, we discuss about bounded, isometric and hypercyclic composition operators on Cesaro function spaces.

In this paper the notion of relative probability‎ ‎measure of a set E is considered with respect to a multi-dimensional observer of a set‎ X as a superset of E‎. ‎Relative entropy of a multi-dimensional‎ ‎observer for the partitions is defined and the properties of relative entropy is extended to‎ ‎multi-dimensional observers‎. ‎It is shown that the observer of a set plays a‎ ‎role in uncertainty of a partition of it‎. ‎Relative conditional entropy is also considered and its main properties are proved‎. ‎Moreover‎, ‎the relative entropy off‎ ‎a relative measure preserving map is studied as well‎.

In this paper, we first prove a lemma for twice differentiable functions . Then we establish some inequalities for mapping whose second derivatives in absolute value are convex via Riemann-Liouville fractional integrals. These results generalize the midpoint and trapezoid inequalities involving Riemann-Liouville fractional integrals given in earlier studies.

The concept of topological molecular lattices was introduced by Wang as a generalization of ordinary topological spaces, fuzzy topological spaces and L-fuzzy topological spaces in terms of closed elements, molecules, remote neighbourhoods and generalized order homomorphisms. In our previous work, we introduced the concept of generalized topological molecular lattices in terms of open elements and investigated some properties of them. In this paper, we dene and consider the category TDML whose objects are topological De Morgan molecular lattices and whose morphisms are continuous generalized order homomorphisms such that its right adjoins preserve the pseudocomplement operation. We show that this category is complete and cocomplete. In particular, we characterize products, coproducts, equalizers and coequalizers. Also, we show that the category TOP of all topological spaces is a re ective and core ective subcategory of TDML.

In this paper, we derive some new result on Noetherian and Boolean Artinian BL-algebras. We further obtain some relations between local and semilocal BL-algebras and Boolean Artinian BL- algebras.

Equality algebras as a generalization of EQ-algebras introduced by Jenei [8]. Hyper equal- ity algebras introduced and studied in [4], as a generalization of equality algebras. Now, in this paper, we investigate relations among hyper equality algebras and other hyper algebraic structures such as hyper K(BE, MV)-algebras and hyper hoops. Specially, we prove that any linearly ordered hyper MV-algebra is a strongly commutative symmetric hyper equality algebra and under some conditions, any strongly commutative involutive hyper equality algebra is a hyper MV-algebra.

It is necessary to make use of scientific methods when merging the Decision-Making Units (DMUs) in any organization. Tools such as Data Envelopment Analysis (DEA) and network DEA (NDEA) are quite useful for unit mergers in two-stage network processes. In this paper, a two-stage network inverse DEA (InvDEA) process is proposed for the merger of university and bank branches based on linear programming models. It is generally crucial to prioritize the inputs and outputs and find the intermediate vectors in multi-stage networks. Therefore, a two-stage network inverse DEA model is used for the purposes of this study. Finally, some applications of the proposed model are provided in DMU mergers based on vector prioritization using Shannon’s entropy, namely the mergers of 5 universities, 24 insurance companies, and 20 commercial banks.

The family MX ⊆ P(X) is called an ∩-structure, when it is closed under the arbitrary intersection. This concept has been studied and considered in algebra, specially in lattices. Using this concept, we define a quasi topological structure which is called ∩-structure space. By studying this space, we attempt to explain some algebraic concepts through this structure.