In this paper, we provide a straightforward proof of a conjecture proposed in cite{Duxbury2021} regarding the optimal solutions of a non-convex mathematical programming model of the Golomb ruler problem. Subsequently, we investigate the computational efficiency of four new binary mixed-integer linear programming models to compute optimal Golomb rulers. These models are derived from a well-known nonlinear integer model proposed in cite{Kocuk2019}, utilizing the reformulation-linearization technique. modif{Finally, we provide the correct outputs of the greedy heuristic proposed in cite{Duxbury2021} and correct some false conclusions stated or implied therein.}
{"title":"Partial reformulation-linearization based optimization models for the Golomb ruler problem","authors":"H. Ouzia","doi":"10.1051/ro/2024121","DOIUrl":"https://doi.org/10.1051/ro/2024121","url":null,"abstract":"In this paper, we provide a straightforward proof of a conjecture proposed in cite{Duxbury2021} regarding the optimal solutions of a non-convex mathematical programming model of the Golomb ruler problem. Subsequently, we investigate the computational efficiency of four new binary mixed-integer linear programming models to compute optimal Golomb rulers. These models are derived from a well-known nonlinear integer model proposed in cite{Kocuk2019}, utilizing the reformulation-linearization technique. modif{Finally, we provide the correct outputs of the greedy heuristic proposed in cite{Duxbury2021} and correct some false conclusions stated or implied therein.}","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"6 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141267010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alireza Amirteimoori, T. Allahviranloo, Asunur Cezar
Clean production of electricity is not only cost-effective but also effective in reducing pollutants. Toward this end, the use of clean fuels is strongly recommended by environmentalists. Benchmarking techniques, especially data envelopment analysis, are an appropriate tool for measuring the relative efficiency of firms with environmental pollutants. In classic data envelopment analysis models, decision-makers are faced with production processes in which reducible inputs are used to produce expandable outputs. In this contribution, we consider production processes when the input and output data are given in stochastic form and some throughputs are reducible and some others are expandable. A stochastic directional distance function model is proposed to calculate the relative technical efficiency of firms. In order to evaluate firm-specific technical efficiency, we apply bootstrap DEA. We first calculate the technical efficiency scores of firms by classic DEA model. Then, the double bootstrap DEA model is applied to determine the impact of explanatory variables on firm efficiency. To demonstrate the applicability of the procedure, we present an empirical application wherein we employ power plants.
清洁发电不仅具有成本效益，还能有效减少污染物。为此，环保人士强烈建议使用清洁燃料。基准技术，特别是数据包络分析法，是衡量企业环境污染物相对效率的合适工具。在经典的数据包络分析模型中，决策者面对的是用可减少的投入生产可扩大的产出的生产过程。在本文中，我们考虑的是以随机形式给出输入和输出数据的生产流程，其中一些吞吐量是可还原的，而另一些则是可扩展的。我们提出了一个随机定向距离函数模型来计算企业的相对技术效率。为了评估特定企业的技术效率，我们采用了自引导 DEA 方法。我们首先通过经典的 DEA 模型计算企业的技术效率得分。然后，运用双重自举 DEA 模型确定解释变量对企业效率的影响。为了证明该程序的适用性，我们以发电厂为例进行了实证应用。
{"title":"A double bootstrap data envelopment analysis model with stochastic reducible outputs and expandable inputs: An application to power plants","authors":"Alireza Amirteimoori, T. Allahviranloo, Asunur Cezar","doi":"10.1051/ro/2024119","DOIUrl":"https://doi.org/10.1051/ro/2024119","url":null,"abstract":"Clean production of electricity is not only cost-effective but also effective in reducing pollutants. Toward this end, the use of clean fuels is strongly recommended by environmentalists. Benchmarking techniques, especially data envelopment analysis, are an appropriate tool for measuring the relative efficiency of firms with environmental pollutants. In classic data envelopment analysis models, decision-makers are faced with production processes in which reducible inputs are used to produce expandable outputs. In this contribution, we consider production processes when the input and output data are given in stochastic form and some throughputs are reducible and some others are expandable. A stochastic directional distance function model is proposed to calculate the relative technical efficiency of firms. In order to evaluate firm-specific technical efficiency, we apply bootstrap DEA. We first calculate the technical efficiency scores of firms by classic DEA model. Then, the double bootstrap DEA model is applied to determine the impact of explanatory variables on firm efficiency. To demonstrate the applicability of the procedure, we present an empirical application wherein we employ power plants.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"10 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141267337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is devoted to the study of duality results for optimization problems with vanishing constraints in nonsmooth case. We formulate Wolfe type dual and establish weak, strong, converse, restricted converse and strict converse duality results for mathematical programs with vanishing constraints involving quasidifferentiable functions. Under the assumption of invex and strictly invex functions with respect to a convex compact set
{"title":"Wolfe type duality on quasidifferentiable mathematical programs with vanishing constraints","authors":"Shashi Kant Mishra, Vandana Singh","doi":"10.1051/ro/2024115","DOIUrl":"https://doi.org/10.1051/ro/2024115","url":null,"abstract":"This article is devoted to the study of duality results for optimization problems with vanishing constraints in nonsmooth case. We formulate Wolfe type dual and establish weak, strong, converse, restricted converse and strict converse duality results for mathematical programs with vanishing constraints involving quasidifferentiable functions. Under the assumption of invex and strictly invex functions with respect to a convex compact set","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"2 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141099225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a novel network design for the Hub Location Problem, inspired by the round-trip structure commonly used by transport service providers. Our design integrates spoke nodes assigned to a central hub node, creating round-trips where the hub node serves as the starting point, visits all assigned spoke nodes, and returns to the hub. To enhance transportation services and provide additional redundancy, we introduce a new type of nodes called runaway nodes to the network. The motivation for this research arises from two real-life cases encountered during consultancy projects, underscoring the necessity for an optimized network design in transportation services. To address the proposed problem, we introduce a mixed-integer linear programming (MIP) mathematical model. However, due to the problem's complexity, the feasibility of the MIP model is limited to small-scale instances. To tackle medium and large-scale instances, we introduce two hyper-heuristic approaches based on reinforcement learning. These hyper-heuristic approaches harness the power of reinforcement learning to guide the selection of low-level heuristics and improve solution quality. We conduct extensive computational experiments to evaluate the efficiency and effectiveness of the proposed approaches. The results of our experiments affirm the efficiency of the proposed hyper-heuristic approaches, showcasing their ability to discover high-quality solutions for the Hub Location Problem.
{"title":"Round-trip hub location problem","authors":"Omar Kemmar, karim bouamrane, S. Gelareh","doi":"10.1051/ro/2024116","DOIUrl":"https://doi.org/10.1051/ro/2024116","url":null,"abstract":"In this paper, we introduce a novel network design for the Hub Location Problem, inspired by the round-trip structure commonly used by transport service providers. Our design integrates spoke nodes assigned to a central hub node, creating round-trips where the hub node serves as the starting point, visits all assigned spoke nodes, and returns to the hub. To enhance transportation services and provide additional redundancy, we introduce a new type of nodes called runaway nodes to the network. The motivation for this research arises from two real-life cases encountered during consultancy projects, underscoring the necessity for an optimized network design in transportation services.\u0000To address the proposed problem, we introduce a mixed-integer linear programming (MIP) mathematical model. However, due to the problem's complexity, the feasibility of the MIP model is limited to small-scale instances. To tackle medium and large-scale instances, we introduce two hyper-heuristic approaches based on reinforcement learning. These hyper-heuristic approaches harness the power of reinforcement learning to guide the selection of low-level heuristics and improve solution quality. We conduct extensive computational experiments to evaluate the efficiency and effectiveness of the proposed approaches.\u0000The results of our experiments affirm the efficiency of the proposed hyper-heuristic approaches, showcasing their ability to discover high-quality solutions for the Hub Location Problem.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"1 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141101182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Derya Deliktaş, O. Ustun, Ezgi Aktar Demirtaş, Rifat Aykut Arapoglu
In multi-stage processes, classical Data Envelopment Analysis (DEA) acts like a black box and measures the efficiency of decision-making units (DMUs) without considering the internal structure of the system. Unlike classical DEA, recent studies have shown that the overall system efficiency scores are more meaningful if researched using the Network DEA (NDEA) methodology. NDEA performs simultaneous efficiency evaluations of sub-processes and the entire system. Recently, the composition method integrated with multi-objective programming (MOP) has been preferred by many authors to alleviate the drawbacks of earlier methods such as decomposition, slack-based measure (SBM) and the system-centric approach. This study proposes a novel approach incorporating Multi-Choice Conic Goal Programming into the NDEA (MCCGP-NDEA). It provides a more accurate representation of the Pareto front by revealing potential Pareto optimal solutions which are overlooked by the composition methods. Due to its ability to modify stage weights based on the decision makers' (DMs) preferences, it is likely to gather more samples from the Pareto surface. Computational results on available benchmark problems confirm that the proposed MCCGP-NDEA is a viable alternative to existing methods.
{"title":"Multi-choice conic goal programming model-based network data envelopment analysis","authors":"Derya Deliktaş, O. Ustun, Ezgi Aktar Demirtaş, Rifat Aykut Arapoglu","doi":"10.1051/ro/2024117","DOIUrl":"https://doi.org/10.1051/ro/2024117","url":null,"abstract":"In multi-stage processes, classical Data Envelopment Analysis (DEA) acts like a black box and measures the efficiency of decision-making units (DMUs) without considering the internal structure of the system. Unlike classical DEA, recent studies have shown that the overall system efficiency scores are more meaningful if researched using the Network DEA (NDEA) methodology. NDEA performs simultaneous efficiency evaluations of sub-processes and the entire system. Recently, the composition method integrated with multi-objective programming (MOP) has been preferred by many authors to alleviate the drawbacks of earlier methods such as decomposition, slack-based measure (SBM) and the system-centric approach. This study proposes a novel approach incorporating Multi-Choice Conic Goal Programming into the NDEA (MCCGP-NDEA). It provides a more accurate representation of the Pareto front by revealing potential Pareto optimal solutions which are overlooked by the composition methods. Due to its ability to modify stage weights based on the decision makers' (DMs) preferences, it is likely to gather more samples from the Pareto surface. Computational results on available benchmark problems confirm that the proposed MCCGP-NDEA is a viable alternative to existing methods.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"3 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141099568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples.
{"title":"Higher-order optimality conditions with separated derivatives and sensitivity analysis for set-valued optimization","authors":"Tian Tang, Guolin Yu","doi":"10.1051/ro/2024118","DOIUrl":"https://doi.org/10.1051/ro/2024118","url":null,"abstract":"In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"3 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141100681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a graph. If $G$ has exactly $r$ distinct sizes of maximal independent sets, $G$ belongs to a collection called $mathcal{M}_r$. If $G in mathcal{M}_{r}$ and the distinct values of its maximal independent sets are consecutive, then $G$ belongs to $mathcal{I}_{r}$. The independence gap of $G$ is the difference between the maximum and the minimum sizes of a maximal independent set in $G$. In this paper, we show that recognizing graphs in $mathcal{I}_r$ is $mathcal{NP}$-complete, for every integer $r geq 3$. On the other hand, we show that recognizing trees in $mathcal{M}_r$ can be done in polynomial time, for every $r geq 1$. Also, we present characterizations of some graphs with girth at least 6 with independence gap at least 1, including graphs with independence gap $r-1$, for $rgeq 2$, belonging to $mathcal{I}_r$. Moreover, we present a characterization of some trees in $mathcal{I}_3$.
{"title":"Distinct sizes of maximal independent sets on graphs with restricted girth","authors":"Márcia Cappelle, Julliano Nascimento, Vinícius Santos","doi":"10.1051/ro/2024110","DOIUrl":"https://doi.org/10.1051/ro/2024110","url":null,"abstract":"Let $G$ be a graph. If $G$ has exactly $r$ distinct sizes of maximal independent sets, $G$ belongs to a collection called $mathcal{M}_r$. If $G in mathcal{M}_{r}$ and the distinct values of its maximal independent sets are consecutive, then $G$ belongs to $mathcal{I}_{r}$. The independence gap of $G$ is the difference between the maximum and the minimum sizes of a maximal independent set in $G$. In this paper, we show that recognizing graphs in $mathcal{I}_r$ is $mathcal{NP}$-complete, for every integer $r geq 3$. On the other hand, we show that recognizing trees in $mathcal{M}_r$ can be done in polynomial time, for every $r geq 1$.\u0000Also, we present characterizations of some graphs with girth at least 6 with independence gap at least 1, including graphs with independence gap $r-1$, for $rgeq 2$, belonging to $mathcal{I}_r$. Moreover, we present a characterization of some trees in $mathcal{I}_3$.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"66 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140972277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Economic growth and excessive fossil energy consumption have direct effects on environmental destruction and greenhouse gas increments. The existing appropriate pattern for economic performance increase as well as pollution emissions abatement is a basic issue in industry activities. In this paper, a data envelopment analysis (DEA) model is introduced for estimating the directional marginal profit maximization of supply chain divisions based on wasted energy and power losses. The purpose of this study is to estimate the directional marginal productivity in the supply chain, which enables us to find the optimal direction of efficient divisions on the frontier. This makes the allocation of resources create a marginal profit increase and the pollution emissions be abated simultaneously. Indeed, the proposed model considers the synergistic effects of each input on MP estimation in predetermined directions. The model is able to estimate the marginal profit maximization of desirable output and undesirable output decrease for each input simultaneously. The results suggested that the gas field division of one of the supply chains had fundamental capacities for energy production increments and flare gas decrements. Furthermore, the gas field division of this supply chain also had a considerable structure for the marginal profit maximization of outputs based on flare gas decreases. Additionally, the distribution lines of 0.80% supply chains provided wasted energy reduction by adding one extra unit to the line's capacity in the determined direction. Especially, there were supply chains that had investment opportunities for an acceptable abatement of power losses. This not only enables divisions to respond to fluctuations in demand as they produce more energy in critical situations like climate change but also decreases harmful emissions as wasted energy in supply chain divisions.
{"title":"Marginal profit maximization estimation of supply chains by waste energy decrement: a case study of the power industry","authors":"M. Pouralizadeh","doi":"10.1051/ro/2024103","DOIUrl":"https://doi.org/10.1051/ro/2024103","url":null,"abstract":"Economic growth and excessive fossil energy consumption have direct effects on environmental destruction and greenhouse gas increments. The existing appropriate pattern for economic performance increase as well as pollution emissions abatement is a basic issue in industry activities. In this paper, a data envelopment analysis (DEA) model is introduced for estimating the directional marginal profit maximization of supply chain divisions based on wasted energy and power losses. The purpose of this study is to estimate the directional marginal productivity in the supply chain, which enables us to find the optimal direction of efficient divisions on the frontier. This makes the allocation of resources create a marginal profit increase and the pollution emissions be abated simultaneously. Indeed, the proposed model considers the synergistic effects of each input on MP estimation in predetermined directions. The model is able to estimate the marginal profit maximization of desirable output and undesirable output decrease for each input simultaneously. The results suggested that the gas field division of one of the supply chains had fundamental capacities for energy production increments and flare gas decrements. Furthermore, the gas field division of this supply chain also had a considerable structure for the marginal profit maximization of outputs based on flare gas decreases. Additionally, the distribution lines of 0.80% supply chains provided wasted energy reduction by adding one extra unit to the line's capacity in the determined direction. Especially, there were supply chains that had investment opportunities for an acceptable abatement of power losses. This not only enables divisions to respond to fluctuations in demand as they produce more energy in critical situations like climate change but also decreases harmful emissions as wasted energy in supply chain divisions.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"32 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140974205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M, e)$ be the set of pairs $(x, y)$ with a vertex $x$ of $M$ and a vertex $y$ of $V(G)$ such that $d_{G}(x, y)neq d_{G-e}(x, y)$. %For a vertex $x$, let $EM(x)$ be the set of edges $e$ such that there exists a vertex $v$ in $G$ with $(x, v)in P({x}, e)$. If $e in EM(x)$, then we say that $e$ can be monitored by the vertex $x$. Given a vertex $x$, an edge $e$ is said to be monitored by $x$ if there exists a vertex $v$ in $G$ such that $(x, v)in P({x}, e)$, and the collection of such edges is $EM(x)$. A set $M$ of vertices of a graph $G$ is distance-edge-monitoring (DEM for short) set if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The DEM number $operatorname{dem}(G)$ of a graph $G$ is defined as the smallest size of such a set in $G$. The vertices of $M$ represent distance probes in a network modeled by $G$; when the edge $e$ fails, the distance from $x$ to $y$ increases, and thus we are able to detect the failure. In this paper, we first give some bounds or exact values of line graphs of trees, grids, complete bipartite graphs, and obtain the exact values of DEM numbers for some graphs and their line graphs, including the friendship and wheel graphs. Next, for each $n, m>1$, we obtain that there exists a graph $G_{n,m}$ such that $operatorname{dem}(G_{n,m})=n$ and $operatorname{dem}(L(G_{n,m}))=4 or 2n+t$, for each integer $tgeq 0$. In the end, the DEM number for the line graph of a small world network (DURT) is given.
{"title":"Realizability problem of distance-edge-monitoring numbers","authors":"Zhen Ji, Yaping Mao, Eddie Cheng, Xiaoyan Zhang","doi":"10.1051/ro/2024106","DOIUrl":"https://doi.org/10.1051/ro/2024106","url":null,"abstract":"Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M, e)$ be the set of pairs $(x, y)$ with a vertex $x$ of $M$ and a vertex\u0000$y$ of $V(G)$ such that $d_{G}(x, y)neq d_{G-e}(x, y)$.\u0000%For a vertex $x$, let $EM(x)$ be the set of edges $e$ such that there exists a vertex $v$ in $G$ with $(x, v)in P({x}, e)$. If $e in EM(x)$, then we say that $e$ can be monitored by the vertex $x$.\u0000Given a vertex $x$, an edge $e$ is said to be monitored by $x$ if there exists a vertex $v$ in $G$ such that $(x, v)in P({x}, e)$, and the collection of such edges is $EM(x)$.\u0000A set $M$ of vertices of a graph $G$ is distance-edge-monitoring (DEM for short) set if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The\u0000DEM number $operatorname{dem}(G)$ of a graph $G$ is defined as the smallest size of such a set in $G$.\u0000The vertices\u0000of $M$ represent distance probes in a network modeled by $G$; when the edge $e$ fails, the\u0000distance from $x$ to $y$ increases, and thus we are able to detect the failure.\u0000In this paper, we first give some bounds or exact values of line graphs of trees, grids, complete bipartite graphs, and\u0000obtain the exact values of DEM numbers for some graphs and their line graphs, including the friendship and wheel graphs.\u0000Next, for each $n, m>1$, we obtain that there exists a graph $G_{n,m}$ such that $operatorname{dem}(G_{n,m})=n$ and $operatorname{dem}(L(G_{n,m}))=4 or 2n+t$, for each integer $tgeq 0$. In the end, the DEM number for the line graph of a small world network (DURT) is given.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"53 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140972542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bike-sharing system has become an indispensable element of sustainable urban transportation, effectively resolving the "last mile" transportation challenge for city dwellers. A major daily operational task in these systems is planning a fleet to rebalance the bikes over time, ensuring the optimal availability of bikes and docks to users. Recycling is also a daily work with the an increase in the number of broken bikes. However, rebalancing or recycling operation is always regarded as an independent tasks. They are separately studied in existing papers. Thus, this paper develops an operational strategy for recycling broken bikes during the rebalancing process, and studies the combination of the station inventory and vehicle routing problems. First, an inventory routing model is constructed with the aim of minimizing the total costs including procurement, expected user loss, inventory and transportation costs. Then, a two-stage iterative algorithm is developed with both exact and heuristic algorithms. We use real-world data from Capital Bikeshare to test our proposed model and approach, which shows the two-stage iterative algorithm is efficient and outperforms existing solutions in reducing total costs. Finally, the sensitivity analysis is performed on key parameters such as the vehicle’s capacity, unit penalty costs for customer dissatisfaction events, unit inventory holding costs and the observation period of rebalancing. It shows that enterprises can reduce the total cost by altering vehicle’s capacity, reducing the unit inventory holding costs or changing the observation period of rebalancing.
共享单车系统已成为可持续城市交通不可或缺的要素，有效解决了城市居民 "最后一英里 "的交通难题。这些系统的一项主要日常运营任务是规划车队，随着时间的推移重新平衡自行车，确保向用户提供最佳的自行车和停靠点。随着破损自行车数量的增加，回收也是一项日常工作。然而，重新平衡或回收操作一直被视为独立的任务。在现有的论文中，它们是分开研究的。因此，本文开发了一种在重新平衡过程中回收破损自行车的运营策略，并研究了站点库存和车辆路由问题的结合。首先，本文构建了一个库存路由模型，目的是使包括采购、预期用户损失、库存和运输成本在内的总成本最小化。然后，利用精确算法和启发式算法开发了一种两阶段迭代算法。我们使用来自 Capital Bikeshare 的真实数据来测试我们提出的模型和方法，结果表明两阶段迭代算法是高效的，在降低总成本方面优于现有的解决方案。最后，对车辆容量、客户不满意事件的单位惩罚成本、单位库存持有成本和再平衡观察期等关键参数进行了敏感性分析。结果表明，企业可以通过改变车辆容量、降低单位库存持有成本或改变再平衡的观察期来降低总成本。
{"title":"Joint optimization of the inventory routing problem considering the recycling of broken bikes in the bike-sharing system","authors":"Aigui Wu","doi":"10.1051/ro/2024109","DOIUrl":"https://doi.org/10.1051/ro/2024109","url":null,"abstract":"Bike-sharing system has become an indispensable element of sustainable urban transportation, effectively resolving the \"last mile\" transportation challenge for city dwellers. A major daily operational task in these systems is planning a fleet to rebalance the bikes over time, ensuring the optimal availability of bikes and docks to users. Recycling is also a daily work with the an increase in the number of broken bikes. However, rebalancing or recycling operation is always regarded as an independent tasks. They are separately studied in existing papers. Thus, this paper develops an operational strategy for recycling broken bikes during the rebalancing process, and studies the combination of the station inventory and vehicle routing problems. First, an inventory routing model is constructed with the aim of minimizing the total costs including procurement, expected user loss, inventory and transportation costs. Then, a two-stage iterative algorithm is developed with both exact and heuristic algorithms. We use real-world data from Capital Bikeshare to test our proposed model and approach, which shows the two-stage iterative algorithm is efficient and outperforms existing solutions in reducing total costs. Finally, the sensitivity analysis is performed on key parameters such as the vehicle’s capacity, unit penalty costs for customer dissatisfaction events, unit inventory holding costs and the observation period of rebalancing. It shows that enterprises can reduce the total cost by altering vehicle’s capacity, reducing the unit inventory holding costs or changing the observation period of rebalancing.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"74 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140973863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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