Let A and B be two associative rings, let I be an ideal of B and let f∈Hom(A,B){finmathrm{Hom}(A,B)}. In this paper, we give a complete description of generalized derivations over A⋈fI{Abowtie^{f}I}. Furthermore, when A is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of A⋈fI{Abowtie^{f}I}.
让 A 和 B 是两个关联环，让 I 是 B 的一个理想，让 f ∈ Hom ( A , B ) {finmathrm{Hom}(A,B)} 。在本文中，我们将完整地描述 A ⋈ f I {Abowtie^{f}I} 上的广义推导。此外，当 A 是质数或半质数时，我们给出了关于广义推导的几个同素异形，这些同素异形提供了 A ⋈ f I {Abowtie^{f}I} 的交换性。
{"title":"Generalized derivations over amalgamated algebras along an ideal","authors":"Brahim Boudine, Mohammed Zerra","doi":"10.1515/gmj-2023-2108","DOIUrl":"https://doi.org/10.1515/gmj-2023-2108","url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two associative rings, let <jats:italic>I</jats:italic> be an ideal of <jats:italic>B</jats:italic> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>Hom</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0167.png\" /> <jats:tex-math>{finmathrm{Hom}(A,B)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we give a complete description of generalized derivations over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{Abowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, when <jats:italic>A</jats:italic> is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{Abowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"6 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexander Domoshnitsky, Elnatan Berenson, Shai Levi, Elena Litsyn
A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions.
{"title":"Floquet theory and stability for a class of first order differential equations with delays","authors":"Alexander Domoshnitsky, Elnatan Berenson, Shai Levi, Elena Litsyn","doi":"10.1515/gmj-2023-2119","DOIUrl":"https://doi.org/10.1515/gmj-2023-2119","url":null,"abstract":"A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ω be a doubling weight and 0<p≤q<∞{0<pleq q<infty}. The essential norm of Riemann–Stieltjes operator Tg{T_{g}} from the weighted Bergman space Aωp{A^{p}_{omega}} to Aωq{A^{q}_{omega}} was investigated in the unit ball of ℂn{mathbb{C}^{n}}.
设 ω 为加倍权重，且 0 < p ≤ q < ∞ {0<pleq q<infty} 。在 ℂ n {mathbb{C}^{n} 的单位球中研究了从加权伯格曼空间 A ω p {A^{p}_{omega}} 到 A ω q {A^{q}_{omega}} 的黎曼-斯蒂尔杰斯算子 T g {T_{g}} 的基本规范。｝ .
{"title":"Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights","authors":"Lian Hu, Songxiao Li, Rong Yang","doi":"10.1515/gmj-2023-2110","DOIUrl":"https://doi.org/10.1515/gmj-2023-2110","url":null,"abstract":"Let ω be a doubling weight and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0161.png\" /> <jats:tex-math>{0<pleq q<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The essential norm of Riemann–Stieltjes operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0210.png\" /> <jats:tex-math>{T_{g}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from the weighted Bergman space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>A</m:mi> <m:mi>ω</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0172.png\" /> <jats:tex-math>{A^{p}_{omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>A</m:mi> <m:mi>ω</m:mi> <m:mi>q</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0173.png\" /> <jats:tex-math>{A^{q}_{omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> was investigated in the unit ball of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℂ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0250.png\" /> <jats:tex-math>{mathbb{C}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cristian Conde, Fuad Kittaneh, Hamid Reza Moradi, Mohammad Sababheh
Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.
{"title":"Numerical radii of operator matrices in terms of certain complex combinations of operators","authors":"Cristian Conde, Fuad Kittaneh, Hamid Reza Moradi, Mohammad Sababheh","doi":"10.1515/gmj-2023-2112","DOIUrl":"https://doi.org/10.1515/gmj-2023-2112","url":null,"abstract":"Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that if B and B′{B^{prime}} are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then B and B′{B^{prime}} differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of B and B′{B^{prime}} in view of their positive equivalence.
众所周知，如果 B 和 B ′ {B^{/prime}} 是平移不变的凸密度微分基，并且与它们相关的最大算子在局部上相互大化，那么 B 和 B ′ {B^{/prime} 就微分同一类非负函数的积分。我们证明，在同样的条件下，鉴于 B 和 B ′ {B^{prime} 的正等价性，不可能断言它们的微分性质有更多的相似性。
{"title":"On the comparison of translation invariant convex differentiation bases","authors":"Irakli Japaridze","doi":"10.1515/gmj-2023-2070","DOIUrl":"https://doi.org/10.1515/gmj-2023-2070","url":null,"abstract":"It is known that if <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in view of their positive equivalence.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"81 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
我们研究局部最大振荡积分算子 T α , β ∗ ( f ) ( x ) = sup 0 < t <；1 | ∫ ℝ n e i | t ξ | α | t ξ | β Ψ ( | t ξ | ) f ^ ( ξ ) e 2 π i 〈 x 、ξ 〉𝑑 ξ | , displaystyle T_{alpha,beta}^{ast}(f)(x)=sup_{0<；t<；1}Bigg{|}int_{mathbb{% R}^{n}}frac{e^{i|txi|^{alpha}}}{|txi|^{beta}}Psi(|txi|)widehat{f}(xi)% e^{2pi ilangle x,其中 α ∈ ( 0 , 1 ) {alphain(0,1)}, β >；0 {beta>0} Ψ 是在原点附近消失的截止函数。首先，在 0 < p < 1 {0<p<1} 的情况下，我们可以得到 H p ( Ψ) 。 我们得到 H p ( ℝ n ) → L p ( ℝ n ) {{{H^{p}}({{mathbb{R}^{n}})}rightarrow{{L^{p}({{mathbb{R}^{n}})}} T α 的有界性、β∗ {T_{alpha,beta}^{ast}} 与 α , β {alpha,beta} 和 p 之间的尖锐关系。然后，利用插值法，当 p > 1 {p>1} 时，我们得到 L p ( ℝ n ) {{{L^{p}({{mathbb{R}^{n}})}}} 对 T α , β∗ {T_{alpha,beta}^{ast}} 的约束性。｝ 这是对凯尼格和斯陶巴赫最新结果的改进。在临界情况 p = 1 {p=1} 和 β = n α 2 {beta=frac{nalpha}{2}} 下，我们证明了 T α , β = n α 2 {beta=frac{nalpha}{2}} 和 β = n α 3 {beta=frac{nalpha}{2}} 我们证明 T α , β ∗ : B q ( ℝ n ) → L 1 , ∞ ( ℝ n ) {T_{alpha,beta}^{ast}:B_{q}({mathbb{R}^{n}})rightarrow L^{1,infty}({% mathbb{R}^{n}}} 其中 B q ( ℝ n ) {B_{q}({mathbb{R}^{n}})} 是 Lu、Taibleson 和 Weiss 为研究 Bochner-Riesz 均值在临界指数处的几乎每次收敛而引入的块空间。作为进一步的应用，我们得到了分数薛定谔算子 { e i t k | △ | α } 组合的收敛速度。 {{e^{itk|triangle|^{alpha}}}} .
The present study introduces the notions of statistical convergence of order α and strong p-Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.
{"title":"On statistical convergence of order α in partial metric spaces","authors":"Erdal Bayram, Çiğdem A. Bektaş, Yavuz Altın","doi":"10.1515/gmj-2023-2116","DOIUrl":"https://doi.org/10.1515/gmj-2023-2116","url":null,"abstract":"The present study introduces the notions of statistical convergence of order α and strong <jats:italic>p</jats:italic>-Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α z t t - Δ z t t - ( ξ 1 + ξ 2 ∥ ∇ z ∥ 2 + σ ( ∇ z , ∇ z t ) ) Δ z - Δ z t + β ( x ) f ( z t ) + g ( z ) = 0 |z_{t}|^{alpha}z_{tt}-Delta z_{tt}-bigl{(}xi_{1}+xi_{2}|nabla z|^{2}+% sigma(nabla z,nabla z_{t})bigr{)}Delta z-Delta z_{t}+beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.
摘要 本文研究了 Balakrishnan-Taylor 准线性波方程 | z t | α z t t - Δ z t t - ( ξ 1 + ξ 2 ∥ ∇ z ∥ 2 + σ ( ∇ z , ∇ z t ) )Δ z - Δ z t + β ( x ) f ( z t ) + g ( z ) = 0 |z_{t}|^{alpha}z_{tt}-Delta z_{tt}-bigl{(}xi_{1}+xi_{2}|nabla z|^{2}+%sigma(nabla z、Delta z-Delta z_{t}+beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {mathbb{R}^{n}} with Dirichlet boundary conditions.通过使用 Faedo-Galerkin 方法，我们证明了全局弱解的存在性。借助扰动能量法，我们还建立了解的指数稳定性。
{"title":"Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping","authors":"Zayd Hajjej","doi":"10.1515/gmj-2023-2105","DOIUrl":"https://doi.org/10.1515/gmj-2023-2105","url":null,"abstract":"Abstract In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α z t t - Δ z t t - ( ξ 1 + ξ 2 ∥ ∇ z ∥ 2 + σ ( ∇ z , ∇ z t ) ) Δ z - Δ z t + β ( x ) f ( z t ) + g ( z ) = 0 |z_{t}|^{alpha}z_{tt}-Delta z_{tt}-bigl{(}xi_{1}+xi_{2}|nabla z|^{2}+% sigma(nabla z,nabla z_{t})bigr{)}Delta z-Delta z_{t}+beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"9 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138943947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gia Sirbiladze, Bezhan Ghvaberidze, Bidzina Midodashvili, Bidzina Matsaberidze, Irina Khutsishvili
Route planning problems are among the activities that have the highest impact in emergency logistical planning, goods transportation and facility location-distribution because of their effects on efficiency in resource management, service levels and client satisfaction. In the extreme conditions, such as disaster-stricken zones, the difficulty of vehicle movement between nearest different affected areas (demand points) on planning routes cause the imprecision of time of movement and the uncertainty of feasibility of movement. In this paper, the imprecision is presented by triangular fuzzy numbers and the uncertainty is presented by a possibility measure. A new two-stage, fuzzy bi-criterion optimization approach for the vehicle routing problem (VRP) is considered. On the first stage, the sample of so-called “promising” closed routes are selected based on a “constructive” approach. On the second stage, triangular fuzzy valued Choquet aggregation (TFCA) operator is constructed for the selected closed routes. The evaluation of constructed routes, levels of failure and possibility of vehicle movement on the roads are aggregated by the TFCA operator by the new criterion – minimization of infeasibility of movement. The new criterion together with the classic criterion – minimization of the total distance traveled – creates a bi-criteria fuzzy VRP. The constructed VRP is reduced to the bi-criteria fuzzy partitioning problem, and an 𝜀-constraint approach is developed for solving it. For numerical experiments, a parallel algorithm is created on the basis of D. Knuth’s algorithm of Dancing Links (DLX). An example is presented with the results of our approach for the VRP, where all Pareto-optimal solutions are found from the set of promising routes. The optimal solutions tend to avoid roads that are problematic because of extreme situations.
{"title":"A new fuzzy approach of vehicle routing problem for disaster-stricken zones","authors":"Gia Sirbiladze, Bezhan Ghvaberidze, Bidzina Midodashvili, Bidzina Matsaberidze, Irina Khutsishvili","doi":"10.1515/gmj-2023-2097","DOIUrl":"https://doi.org/10.1515/gmj-2023-2097","url":null,"abstract":"Route planning problems are among the activities that have the highest impact in emergency logistical planning, goods transportation and facility location-distribution because of their effects on efficiency in resource management, service levels and client satisfaction. In the extreme conditions, such as disaster-stricken zones, the difficulty of vehicle movement between nearest different affected areas (demand points) on planning routes cause the imprecision of time of movement and the uncertainty of feasibility of movement. In this paper, the imprecision is presented by triangular fuzzy numbers and the uncertainty is presented by a possibility measure. A new two-stage, fuzzy bi-criterion optimization approach for the vehicle routing problem (VRP) is considered. On the first stage, the sample of so-called “promising” closed routes are selected based on a “constructive” approach. On the second stage, triangular fuzzy valued Choquet aggregation (TFCA) operator is constructed for the selected closed routes. The evaluation of constructed routes, levels of failure and possibility of vehicle movement on the roads are aggregated by the TFCA operator by the new criterion – minimization of infeasibility of movement. The new criterion together with the classic criterion – minimization of the total distance traveled – creates a bi-criteria fuzzy VRP. The constructed VRP is reduced to the bi-criteria fuzzy partitioning problem, and an 𝜀-constraint approach is developed for solving it. For numerical experiments, a parallel algorithm is created on the basis of D. Knuth’s algorithm of Dancing Links (DLX). An example is presented with the results of our approach for the VRP, where all Pareto-optimal solutions are found from the set of promising routes. The optimal solutions tend to avoid roads that are problematic because of extreme situations.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"103 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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