Data envelopment analysis is a nonparametric method for measuring of the performance of decision-making units—which do not need to have or compute a firm’s production function, which is often difficult to calculate. For any manager, the progress or setback of the thing they manage is important because it makes planning and adoption of future policies for the organization or decision-making unit more rational and scientific. Different methods have been used to calculate the improvements and regressions using Malmquist Index. In this article, we evaluate the units under review in terms of economic efficiency, and the units in terms of spending, production, revenue and profit over several periods, and the rate of improvement or regression of each of these units. Considering the minimal use of resources and consuming less money, generating more revenue, and maximizing profits, the improvement or retreat of the recipient’s decision unit in terms of cost, revenue, and profit was examined by presenting a method based on solving linear programming models using the productivity index is Malmquist and Malmquist Global. Finally, by designing and solving a numerical example, we emphasize and test the applicability of the material presented in this article.
In this paper, the notions of rank−k numerical range and k−spectrum of rectangular complex matrices are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for -higher rank numerical ranges of rectangular matrices is also introduced and studied. The proposed definitions yield a natural generalization of standard higher rank numerical ranges.Λ k (A) = {λ ∈ C : P AP = λP, f or some rank − k orthogonal projection P on C n }.
Let $$M_{n}({\mathbb {R}}_{+})$$
M
n
(
R
+
)
be the set of all $$n \times n$$
n
×
n
nonnegative matrices. Recently, in Tavakolipour and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let $$1 \le k \le n$$
1
≤
k
≤
n
be a positive integer and $$C \in M_{ n}({\mathbb {R}}_{+}).$$
C
∈
M
n
(
R
+
)
.
We introduce the notions of max $$k-$$
k
-
numerical range and max $$C-$$
C
-
numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range $$W_\text {max}(\varSigma )$$
W
max
(
Σ
)
of a bounded set $$\varSigma$$
Σ
of $$n \times n$$
n
×
n
nonnegative matrices is introduced and some of its properties are also investigated.
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