We consider the problem of scheduling non preemptively a set of jobs on parallel identical machines with prior setup operations on a single shared server, where the objective is to minimise the makespan. We develop an arc-flow formulation to the problem with two multigraphs, one for the machines and one for the server, with a same set of nodes representing points in time, and arcs associated with job execution, and with machines or server idleness. The resulting formulation, called Flow–Flow formulation (FFF), and its tuned version (FFT) are compared with the best existing model in the literature, a time-indexed variable formulation (), on benchmark instances with up to 200 jobs and 10 machines. Computational results showed that our Flow–Flow models outperformed especially for instances with more than 50 jobs and optimally solved a majority of problems with 150 and 200 jobs for which found only very few optimal solutions.
{"title":"Efficient arc-flow formulations for makespan minimisation on parallel machines with a common server","authors":"Alessandro Druetto , Andrea Grosso , Jully Jeunet , Fabio Salassa","doi":"10.1016/j.cor.2024.106911","DOIUrl":"10.1016/j.cor.2024.106911","url":null,"abstract":"<div><div>We consider the problem of scheduling non preemptively a set of jobs on parallel identical machines with prior setup operations on a single shared server, where the objective is to minimise the makespan. We develop an arc-flow formulation to the problem with two multigraphs, one for the machines and one for the server, with a same set of nodes representing points in time, and arcs associated with job execution, and with machines or server idleness. The resulting formulation, called Flow–Flow formulation (FFF), and its tuned version (FFT) are compared with the best existing model in the literature, a time-indexed variable formulation (<span><math><mrow><mi>F</mi><mn>2</mn></mrow></math></span>), on benchmark instances with up to 200 jobs and 10 machines. Computational results showed that our Flow–Flow models outperformed <span><math><mrow><mi>F</mi><mn>2</mn></mrow></math></span> especially for instances with more than 50 jobs and optimally solved a majority of problems with 150 and 200 jobs for which <span><math><mrow><mi>F</mi><mn>2</mn></mrow></math></span> found only very few optimal solutions.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"174 ","pages":"Article 106911"},"PeriodicalIF":4.1,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1016/j.cor.2024.106910
Li Guan , Gilbert Laporte , José M. Merigó , Stefan Nickel , Iman Rahimi , Francisco Saldanha-da-Gama
Computers & Operations Research (COR) is a leading international journal in the field of Operations Research, established with a vision to provide a platform for emphasising and promoting the application of computers and operations research techniques to problems of world concern and general interest. The journal published its first issue in 1974 and in 2024 celebrated its 50th anniversary. Motivated by this special event, this paper aims to present a complete bibliometric overview of the most significant development patterns and trends of the journal during its first half-century of publishing history from 1974 to 2023. The study uses the Web of Science Core Collection database to collect bibliographic information and analyse the data, complemented by the Scopus database and the journal’s webpage. Based on a wide range of bibliometric indicators, the results of the bibliometric analysis highlight the publication and citation structure of COR, the most cited documents, the leading authors, institutions, countries/territories, and supranational regions, and the most popular keywords and research topics in the journal. Additionally, the work also graphically maps the bibliographic material with techniques of co-citation, bibliographic coupling, and co-occurrence of author keywords by using the Visualization of Similarities (VOS) viewer software. The findings of the study provide strong evidence of the significant growth of COR through its lifetime development and its international diversity having publications from all over the world. The study is also useful for understanding the substantial contributions of the journal it has made to the scientific community.
{"title":"50 years of Computers & Operations Research: A bibliometric analysis","authors":"Li Guan , Gilbert Laporte , José M. Merigó , Stefan Nickel , Iman Rahimi , Francisco Saldanha-da-Gama","doi":"10.1016/j.cor.2024.106910","DOIUrl":"10.1016/j.cor.2024.106910","url":null,"abstract":"<div><div><em>Computers & Operations Research</em> (COR) is a leading international journal in the field of Operations Research, established with a vision to provide a platform for emphasising and promoting the application of computers and operations research techniques to problems of world concern and general interest. The journal published its first issue in 1974 and in 2024 celebrated its 50th anniversary. Motivated by this special event, this paper aims to present a complete bibliometric overview of the most significant development patterns and trends of the journal during its first half-century of publishing history from 1974 to 2023. The study uses the Web of Science Core Collection database to collect bibliographic information and analyse the data, complemented by the Scopus database and the journal’s webpage. Based on a wide range of bibliometric indicators, the results of the bibliometric analysis highlight the publication and citation structure of COR, the most cited documents, the leading authors, institutions, countries/territories, and supranational regions, and the most popular keywords and research topics in the journal. Additionally, the work also graphically maps the bibliographic material with techniques of co-citation, bibliographic coupling, and co-occurrence of author keywords by using the Visualization of Similarities (VOS) viewer software. The findings of the study provide strong evidence of the significant growth of COR through its lifetime development and its international diversity having publications from all over the world. The study is also useful for understanding the substantial contributions of the journal it has made to the scientific community.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"175 ","pages":"Article 106910"},"PeriodicalIF":4.1,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.cor.2024.106897
Tanner Nixon , Robert M. Curry , Phanuel Allaissem B.
Various applications in contested logistics and infrastructure restoration require dynamic flow solutions characterized by a schedule of network flows consecutively transmitted over a sequence of successive periods. For these schedules, we assume flows transmit via arcs during periods while flows reside at nodes from one period to the next. Within this context, we introduce the Maximum Value Dynamic Network Flow Problem (MVDFP) in which we seek to maximize the cumulative value of a non-simultaneous network flow schedule that accumulates node value whenever some minimum amount of flow resides at a node between periods. For solving the MVDFP, we first introduce a large mixed-integer program (MIP). As this MIP can become computationally-expensive for large networks, we present a trio of computationally-effective, easy to implement heuristic approaches that solve a series of smaller, more manageable MIPs. These heuristic approaches typically determine high-quality solutions significantly faster than the MIP obtains an optimal solution by dividing the full network flow schedule into a sequence of consecutive shorter network flow subschedules. In many cases, at least one of our heuristic approaches produces an optimal solution in a fraction of the MIP’s computational time. We present extensive computational results to highlight our heuristics’ efficacy, discuss for what instances each approach may be most applicable, and detail future research avenues.
{"title":"Mixed-integer programming models and heuristic algorithms for the maximum value dynamic network flow scheduling problem","authors":"Tanner Nixon , Robert M. Curry , Phanuel Allaissem B.","doi":"10.1016/j.cor.2024.106897","DOIUrl":"10.1016/j.cor.2024.106897","url":null,"abstract":"<div><div>Various applications in contested logistics and infrastructure restoration require dynamic flow solutions characterized by a schedule of network flows consecutively transmitted over a sequence of successive periods. For these schedules, we assume flows transmit via arcs <em>during</em> periods while flows <em>reside</em> at nodes from one period to the next. Within this context, we introduce the Maximum Value Dynamic Network Flow Problem (MVDFP) in which we seek to maximize the cumulative <em>value</em> of a non-simultaneous network flow schedule that accumulates node <em>value</em> whenever some minimum amount of flow resides at a node between periods. For solving the MVDFP, we first introduce a large mixed-integer program (MIP). As this MIP can become computationally-expensive for large networks, we present a trio of computationally-effective, easy to implement heuristic approaches that solve a series of smaller, more manageable MIPs. These heuristic approaches typically determine high-quality solutions significantly faster than the MIP obtains an optimal solution by dividing the full network flow schedule into a sequence of consecutive shorter network flow subschedules. In many cases, at least one of our heuristic approaches produces an optimal solution in a fraction of the MIP’s computational time. We present extensive computational results to highlight our heuristics’ efficacy, discuss for what instances each approach may be most applicable, and detail future research avenues.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"175 ","pages":"Article 106897"},"PeriodicalIF":4.1,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.cor.2024.106892
Haimin Lu, Jiayan Huang, Chenxu Lou, Zhi Pei
Single machine scheduling aims at determining the job sequence with the best desired performance, and provides the basic building block for more advanced scheduling problems. In the present study, a single machine scheduling model with uncertain processing time is considered by incorporating the job release time and due date. The job processing time follows unknown probability distribution, and can be estimated via the historical data. To model the uncertainty, the processing time distribution is defined over a Wasserstein ball ambiguity set, which covers all feasible probability distributions within the confidence radius of the empirical distribution, known as the center of the ball. Then a data-driven distributionally robust scheduling model is constructed with individual chance constraints. In particular, two equivalent reformulations are derived with respect to the -norm and -norm metrics of the Wasserstein ball, namely, a mixed-integer linear programming and a mixed-integer second order cone programming model, respectively. To accelerate the solving of large-scale instances, a tailored constraint generation algorithm is introduced. In the numerical analysis, the proposed distributionally robust scheduling approach is compared with the state-of-the-art methods in terms of out-of-sample performance.
{"title":"Distributionally robust single machine scheduling with release and due dates over Wasserstein balls","authors":"Haimin Lu, Jiayan Huang, Chenxu Lou, Zhi Pei","doi":"10.1016/j.cor.2024.106892","DOIUrl":"10.1016/j.cor.2024.106892","url":null,"abstract":"<div><div>Single machine scheduling aims at determining the job sequence with the best desired performance, and provides the basic building block for more advanced scheduling problems. In the present study, a single machine scheduling model with uncertain processing time is considered by incorporating the job release time and due date. The job processing time follows unknown probability distribution, and can be estimated via the historical data. To model the uncertainty, the processing time distribution is defined over a Wasserstein ball ambiguity set, which covers all feasible probability distributions within the confidence radius of the empirical distribution, known as the center of the ball. Then a data-driven distributionally robust scheduling model is constructed with individual chance constraints. In particular, two equivalent reformulations are derived with respect to the <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-norm and <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-norm metrics of the Wasserstein ball, namely, a mixed-integer linear programming and a mixed-integer second order cone programming model, respectively. To accelerate the solving of large-scale instances, a tailored constraint generation algorithm is introduced. In the numerical analysis, the proposed distributionally robust scheduling approach is compared with the state-of-the-art methods in terms of out-of-sample performance.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"174 ","pages":"Article 106892"},"PeriodicalIF":4.1,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-10DOI: 10.1016/j.cor.2024.106883
Akang Wang , Xiandong Li , Jeffrey E. Arbogast , Zachary Wilson , Chrysanthos E. Gounaris
Inventory management, vehicle routing, and delivery scheduling decisions are simultaneously considered in the context of the inventory routing problem. This paper focuses on the continuous-time version of this problem where, unlike its more traditional discrete-time counterpart, the distributor is required to guarantee that inventory levels are maintained within the desired intervals at any moment of the planning horizon. In this work, we develop a compact mixed-integer linear programming formulation to model the continuous-time inventory routing problem. We further discuss means to expedite its solution process, including the adaptation of well-known rounded capacity inequalities to tighten the formulation in the context of a branch-and-cut algorithm. Through extensive computational studies on a suite of 90 benchmark instances from the literature, we show that our branch-and-cut algorithm outperforms the state-of-the-art approach. We also consider a new set of 63 instances adapted from a real-life dataset and show our algorithm’s practical value in solving instances with up to 20 customers to guaranteed optimality.
{"title":"A novel mixed-integer linear programming formulation for continuous-time inventory routing","authors":"Akang Wang , Xiandong Li , Jeffrey E. Arbogast , Zachary Wilson , Chrysanthos E. Gounaris","doi":"10.1016/j.cor.2024.106883","DOIUrl":"10.1016/j.cor.2024.106883","url":null,"abstract":"<div><div>Inventory management, vehicle routing, and delivery scheduling decisions are simultaneously considered in the context of the inventory routing problem. This paper focuses on the continuous-time version of this problem where, unlike its more traditional discrete-time counterpart, the distributor is required to guarantee that inventory levels are maintained within the desired intervals at any moment of the planning horizon. In this work, we develop a compact mixed-integer linear programming formulation to model the continuous-time inventory routing problem. We further discuss means to expedite its solution process, including the adaptation of well-known rounded capacity inequalities to tighten the formulation in the context of a branch-and-cut algorithm. Through extensive computational studies on a suite of 90 benchmark instances from the literature, we show that our branch-and-cut algorithm outperforms the state-of-the-art approach. We also consider a new set of 63 instances adapted from a real-life dataset and show our algorithm’s practical value in solving instances with up to 20 customers to guaranteed optimality.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"174 ","pages":"Article 106883"},"PeriodicalIF":4.1,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-09DOI: 10.1016/j.cor.2024.106894
Teresa Corberán , Isaac Plana , José María Sanchis
This paper studies the Min Max Multi-Trip drone Location Arc Routing Problem (MM-MT-dLARP), an arc routing problem that combines trucks and drones. We have a set of lines (usually curved) that have to be flown over by drones to perform a service (inspection, for example). There is a depot from which the trucks leave, each one carrying a drone, and a set of potential launching points where the truck can launch and pick up the drone. Drones have limited autonomy, but they can make several flights. We consider a min–max objective, in which the makespan, or time necessary to complete the service, must be minimized. Using aerial drones instead of ground vehicles allows to travel off the network: drones can enter a line through any of its points, service only a portion of that line and then exit through another of its points, without following the lines of the network. This allows for finding better solutions but also increases the difficulty of the problem. This issue can be addressed by digitizing the MM-MT-dLARP instances, approximating each line by a polygonal chain with a finite number of intermediate points, and requiring that drones can only enter and exit a line through those intermediate points. Thus, an instance of a discrete Min Max Multi-Trip Location Arc Routing Problem (MM-MT-LARP) is obtained. Here, an integer formulation for the MM-MT-LARP is proposed, some families of valid inequalities are proved to be facet-inducing of a relaxed polyhedron, and a branch-and-cut algorithm based on the strengthened formulation is developed. This algorithm has only been applied to small instances without intermediate points on the lines. In addition, we have developed a matheuristic algorithm for the MM-MT-dLARP that combines a construction phase, four local search procedures integrated into a Variable Neighborhood Descent (VND) algorithm, and a set of rules for selecting intermediate points to improve the solutions. We present the results obtained on a set of randomly generated instances involving up to 6 launching points and 88 original lines.
{"title":"The min max multi-trip drone location arc routing problem","authors":"Teresa Corberán , Isaac Plana , José María Sanchis","doi":"10.1016/j.cor.2024.106894","DOIUrl":"10.1016/j.cor.2024.106894","url":null,"abstract":"<div><div>This paper studies the Min Max Multi-Trip drone Location Arc Routing Problem (MM-MT-dLARP), an arc routing problem that combines trucks and drones. We have a set of lines (usually curved) that have to be flown over by drones to perform a service (inspection, for example). There is a depot from which the trucks leave, each one carrying a drone, and a set of potential launching points where the truck can launch and pick up the drone. Drones have limited autonomy, but they can make several flights. We consider a min–max objective, in which the makespan, or time necessary to complete the service, must be minimized. Using aerial drones instead of ground vehicles allows to travel off the network: drones can enter a line through any of its points, service only a portion of that line and then exit through another of its points, without following the lines of the network. This allows for finding better solutions but also increases the difficulty of the problem. This issue can be addressed by digitizing the MM-MT-dLARP instances, approximating each line by a polygonal chain with a finite number of intermediate points, and requiring that drones can only enter and exit a line through those intermediate points. Thus, an instance of a discrete Min Max Multi-Trip Location Arc Routing Problem (MM-MT-LARP) is obtained. Here, an integer formulation for the MM-MT-LARP is proposed, some families of valid inequalities are proved to be facet-inducing of a relaxed polyhedron, and a branch-and-cut algorithm based on the strengthened formulation is developed. This algorithm has only been applied to small instances without intermediate points on the lines. In addition, we have developed a matheuristic algorithm for the MM-MT-dLARP that combines a construction phase, four local search procedures integrated into a Variable Neighborhood Descent (VND) algorithm, and a set of rules for selecting intermediate points to improve the solutions. We present the results obtained on a set of randomly generated instances involving up to 6 launching points and 88 original lines.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"174 ","pages":"Article 106894"},"PeriodicalIF":4.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-09DOI: 10.1016/j.cor.2024.106896
Suzan Iloglu , Laura A. Albert , Carla Michini
Effective recovery of interdependent infrastructure systems after natural disasters requires coordination between multiple infrastructure owners, such as power and telecommunications utilities. If infrastructure owners make restoration decisions in isolation from one another, then recovery may be piecemeal. A fundamental understanding of these interdependencies can provide insights to incentivize shared restoration that benefit all infrastructure users, with the goal to maximize the social welfare even in a non-cooperative setting. We introduce a non-cooperative facility location and restoration game on a layered network, where each layer belongs to a player, to model the recovery of interdependent infrastructure systems after disasters. The goal of the model is to plan short term post-disaster recovery. The players want to minimize the cost to satisfy their own demand by restoring network components, and each player can serve the other players’ demands if they are paid a fee to do so. We propose exact and approximate algorithms to set incentives (fees) so that the players’ actions at equilibrium are aligned with a social optimum of the system, which minimizes the total cost. We present a case study in which we consider the recovery efforts of telecommunication infrastructure companies and provide results for the facility location and restoration games. The models and proposed algorithms can be used to set policy, inform the structure of inter-agency mutual aid partnerships to support disaster recovery, and negotiate inter-agency usage fees prior to a disaster to ease shared recovery efforts.
{"title":"Facility location and restoration games","authors":"Suzan Iloglu , Laura A. Albert , Carla Michini","doi":"10.1016/j.cor.2024.106896","DOIUrl":"10.1016/j.cor.2024.106896","url":null,"abstract":"<div><div>Effective recovery of interdependent infrastructure systems after natural disasters requires coordination between multiple infrastructure owners, such as power and telecommunications utilities. If infrastructure owners make restoration decisions in isolation from one another, then recovery may be piecemeal. A fundamental understanding of these interdependencies can provide insights to incentivize shared restoration that benefit all infrastructure users, with the goal to maximize the social welfare even in a non-cooperative setting. We introduce a non-cooperative facility location and restoration game on a layered network, where each layer belongs to a player, to model the recovery of interdependent infrastructure systems after disasters. The goal of the model is to plan short term post-disaster recovery. The players want to minimize the cost to satisfy their own demand by restoring network components, and each player can serve the other players’ demands if they are paid a fee to do so. We propose exact and approximate algorithms to set incentives (fees) so that the players’ actions at equilibrium are aligned with a social optimum of the system, which minimizes the total cost. We present a case study in which we consider the recovery efforts of telecommunication infrastructure companies and provide results for the facility location and restoration games. The models and proposed algorithms can be used to set policy, inform the structure of inter-agency mutual aid partnerships to support disaster recovery, and negotiate inter-agency usage fees prior to a disaster to ease shared recovery efforts.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"174 ","pages":"Article 106896"},"PeriodicalIF":4.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-08DOI: 10.1016/j.cor.2024.106893
Mohammad H. Shekarriz , Dhananjay Thiruvady , Asef Nazari , Rhyd Lewis
For , a -happy vertex in a coloured graph has at least same-colour neighbours, and a -happy colouring (aka soft happy colouring) of is a vertex colouring that makes all the vertices -happy. A community is a subgraph whose vertices are more adjacent to themselves than the rest of the vertices. Graphs with community structures can be modelled by random graph models such as the stochastic block model (SBM). In this paper, we present several theorems showing that both of these notions are related, with numerous real-world applications. We show that, with high probability, communities of graphs in the stochastic block model induce -happy colouring on all vertices if certain conditions on the model parameters are satisfied. Moreover, a probabilistic threshold on is derived so that communities of a graph in the SBM induce a -happy colouring. Furthermore, the asymptotic behaviour of -happy colouring induced by the graph’s communities is discussed when is less than a threshold. We develop heuristic polynomial-time algorithms for soft happy colouring that often correlate with the graphs’ community structure. Finally, we present an experimental evaluation to compare the performance of the proposed algorithms thereby demonstrating the validity of the theoretical results.
对于 0<ρ≤1，彩色图 G 中的ρ-快乐顶点 v 至少有ρ⋅deg(v) 个同色相邻顶点，G 的ρ-快乐着色（又称软快乐着色）是使所有顶点都ρ-快乐的顶点着色。社群是一个子图，其顶点之间的相邻关系多于其他顶点之间的相邻关系。具有群落结构的图可以用随机图模型来建模，如随机块模型（SBM）。在本文中，我们提出了几个定理，说明这两个概念是相关的，并在现实世界中有着大量应用。我们证明，如果模型参数的某些条件得到满足，随机块模型中的图群落很有可能会在所有顶点上诱发 ρ 快乐着色。此外，还推导出了ρ的概率阈值，从而使随机块模型中的图群落诱发ρ-快乐着色。此外，我们还讨论了当 ρ 小于阈值时，图的群落诱导的 ρ 快乐着色的渐近行为。我们为软快乐着色开发了启发式多项式时间算法，这种算法通常与图的群落结构相关。最后，我们通过实验评估来比较所提算法的性能，从而证明理论结果的正确性。
{"title":"Soft happy colourings and community structure of networks","authors":"Mohammad H. Shekarriz , Dhananjay Thiruvady , Asef Nazari , Rhyd Lewis","doi":"10.1016/j.cor.2024.106893","DOIUrl":"10.1016/j.cor.2024.106893","url":null,"abstract":"<div><div>For <span><math><mrow><mn>0</mn><mo><</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn></mrow></math></span>, a <span><math><mi>ρ</mi></math></span>-happy vertex <span><math><mi>v</mi></math></span> in a coloured graph <span><math><mi>G</mi></math></span> has at least <span><math><mrow><mi>ρ</mi><mi>⋅</mi><mo>deg</mo><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> same-colour neighbours, and a <span><math><mi>ρ</mi></math></span>-happy colouring (aka soft happy colouring) of <span><math><mi>G</mi></math></span> is a vertex colouring that makes all the vertices <span><math><mi>ρ</mi></math></span>-happy. A community is a subgraph whose vertices are more adjacent to themselves than the rest of the vertices. Graphs with community structures can be modelled by random graph models such as the stochastic block model (SBM). In this paper, we present several theorems showing that both of these notions are related, with numerous real-world applications. We show that, with high probability, communities of graphs in the stochastic block model induce <span><math><mi>ρ</mi></math></span>-happy colouring on all vertices if certain conditions on the model parameters are satisfied. Moreover, a probabilistic threshold on <span><math><mi>ρ</mi></math></span> is derived so that communities of a graph in the SBM induce a <span><math><mi>ρ</mi></math></span>-happy colouring. Furthermore, the asymptotic behaviour of <span><math><mi>ρ</mi></math></span>-happy colouring induced by the graph’s communities is discussed when <span><math><mi>ρ</mi></math></span> is less than a threshold. We develop heuristic polynomial-time algorithms for soft happy colouring that often correlate with the graphs’ community structure. Finally, we present an experimental evaluation to compare the performance of the proposed algorithms thereby demonstrating the validity of the theoretical results.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"174 ","pages":"Article 106893"},"PeriodicalIF":4.1,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-07DOI: 10.1016/j.cor.2024.106898
Banu Soylu , Betül Yıldırım
Relocation involves the repositioning of idle Emergency Service (ES) vehicles among stations in order to reduce the response time. It is well-known in the literature that relocating idle vehicles provides better coverage in the network, which in turn reduces the response time to the next call. In classical emergency service networks, idle vehicles can be relocated between any two stations. This can cause long delays and increase the response times. In this study, we proposed for the first time a hub-and-spoke network to efficiently realize the relocation of idle vehicles. The proposed hub-and-spoke structure consolidates relocations among hubs, while hub-spoke relocations are implemented as needed. Such a structure helps to better organize the simultaneous movements of ES vehicles for relocation. We have developed a mathematical model to maximize the expected safely covered population. The model provides both the hub-spoke topology and the relocation plan (a compliance table), which shows the desired stations of idle vehicles depending on the system state. In the literature, the relocation plan does not show the relocation paths (movements) of the vehicles. We have presented an exact algorithm that computes the relocation paths for all possible call cases and system levels in advance. This helps the dispatcher to manage the system effectively. We performed a detailed simulation study for ES vehicles of a natural gas distributor to demonstrate the real-life suitability of the proposed system. Compared to the classical relocation network structure, the proposed system has improved the response time, relocation time, and travel time especially when the system is busy.
重新定位涉及在站点之间重新定位闲置的紧急服务（ES）车辆，以缩短响应时间。众所周知，重新定位闲置车辆可提高网络的覆盖率，从而缩短对下一个呼叫的响应时间。在传统的紧急服务网络中，闲置车辆可以在任意两个站点之间重新定位。这会造成长时间的延迟，增加响应时间。在本研究中，我们首次提出了一种集线器-辐条网络，以有效实现闲置车辆的重新定位。所提出的 "枢纽-辐条 "结构在枢纽之间整合搬迁，而 "枢纽-辐条 "搬迁则根据需要实施。这种结构有助于更好地组织 ES 车辆同时移动，进行重新安置。我们建立了一个数学模型，以最大限度地提高预期安全覆盖人口。该模型提供了轮辐式拓扑结构和搬迁计划（合规表），根据系统状态显示了闲置车辆的理想站点。在文献中，搬迁计划并不显示车辆的搬迁路径（移动）。我们提出了一种精确算法，可以提前计算所有可能的呼叫情况和系统级别的重新安置路径。这有助于调度员有效地管理系统。我们对一家天然气分销商的 ES 车辆进行了详细的模拟研究，以证明所提系统在现实生活中的适用性。与传统的搬迁网络结构相比，建议的系统改善了响应时间、搬迁时间和旅行时间，尤其是在系统繁忙时。
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Pub Date : 2024-11-06DOI: 10.1016/j.cor.2024.106886
Marc Goerigk , Jannis Kurtz
We study iterative constraint and variable generation methods for (two-stage) robust combinatorial optimization problems with discrete uncertainty. The goal of this work is to find a set of starting scenarios that provides strong lower bounds early in the process. To this end we define the Relevant Scenario Recognition Problem (RSRP) which finds the optimal choice of scenarios which maximizes the corresponding objective value. We show for classical and two-stage robust optimization that this problem can be solved in polynomial time if the number of selected scenarios is constant and NP-hard if it is part of the input. Furthermore, we derive a linear mixed-integer programming formulation for the problem in both cases.
Since solving the RSRP is not possible in reasonable time, we propose a machine-learning-based heuristic to determine a good set of starting scenarios. To this end, we design a set of dimension-independent features, and train a Random Forest Classifier on already solved small-dimensional instances of the problem. Our experiments show that our method is able to improve the solution process even for larger instances than contained in the training set, and that predicting even a small number of good starting scenarios can considerably reduce the optimality gap. Additionally, our method provides a feature importance score which can give new insights into the role of scenario properties in robust optimization.
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