This paper presents a study of inventory replenishment strategy for efficiently managing sales of a deteriorating item in a retail store. The study addresses pertinent effect on sales pattern due to promotional initiatives. The memory effect generated in the consumers’ mind due to various factors like branding and the stock visibility to customers is incorporated in our model by formulating it as a Caputo-Fabrizio fractional differential equation. Even, in practice, consumers’ purchase patterns are noticed to get influenced by the reliability of product, the same is modelled through demand rate formulation. Influence of both these factors is incorporated into the proposed formulation by representing them as model parameters. The study aims at determining the optimal replenishment quantity and its reordering time for the addressed item in terms of said factors estimated as parameters. Results of the study are analyzed through the data set obtained from a retail store. The analysis of model-parameters infers some managerial insights which match the reality of sales patterns. Our study provides a decision support framework for determining replenishment quantities along with an estimate of replenishment time in connection with promotional initiatives and reliability of the product for achieving minimal total cost incurred while keeping the selling price of the product as fixed.
{"title":"Effect of reliability and memory on fractional inventory model incorporating promotional effort on demand","authors":"P. Santra, G. Mahapatra, Akhil Kumar","doi":"10.1051/ro/2023095","DOIUrl":"https://doi.org/10.1051/ro/2023095","url":null,"abstract":"This paper presents a study of inventory replenishment strategy for efficiently managing sales of a deteriorating item in a retail store. The study addresses pertinent effect on sales pattern due to promotional initiatives. The memory effect generated in the consumers’ mind due to various factors like branding and the stock visibility to customers is incorporated in our model by formulating it as a Caputo-Fabrizio fractional differential equation. Even, in practice, consumers’ purchase patterns are noticed to get influenced by the reliability of product, the same is modelled through demand rate formulation. Influence of both these factors is incorporated into the proposed formulation by representing them as model parameters. The study aims at determining the optimal replenishment quantity and its reordering time for the addressed item in terms of said factors estimated as parameters. Results of the study are analyzed through the data set obtained from a retail store. The analysis of model-parameters infers some managerial insights which match the reality of sales patterns. Our study provides a decision support framework for determining replenishment quantities along with an estimate of replenishment time in connection with promotional initiatives and reliability of the product for achieving minimal total cost incurred while keeping the selling price of the product as fixed.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78706303","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}
We address the three-machine flowshop scheduling problem to minimize makespan where processing times are uncertain within some lower and upper bounds. We propose sixteen algorithms based on Johnson's algorithm, which is known to yield the optimal solution for the three-machine flowshop problem under certain cases. The proposed algorithms are computationally evaluated based on randomly generated data. Computational experiments indicate that one of the proposed algorithms, algorithm AL-7, significantly performs better than the rest. Tests of hypotheses were performed to statistically confirm the results. In algorithm AL-7, more weight is given to the processing times of jobs on the first and the third machines compared to those of the second machine. Moreover, both the lower and upper bounds of job processing times on all three machines are utilized. Furthermore, algorithm AL-7 is shown to perform the best regardless of the extreme distributions considered. Hence, it is recommended as the best algorithm.
{"title":"Algorithms for three-machine flowshop scheduling problem to minimize makespan with uncertain processing times","authors":"A. Allahverdi, Muberra Allahverdi","doi":"10.1051/ro/2023091","DOIUrl":"https://doi.org/10.1051/ro/2023091","url":null,"abstract":"We address the three-machine flowshop scheduling problem to minimize makespan where processing times are uncertain within some lower and upper bounds. We propose sixteen algorithms based on Johnson's algorithm, which is known to yield the optimal solution for the three-machine flowshop problem under certain cases. The proposed algorithms are computationally evaluated based on randomly generated data. Computational experiments indicate that one of the proposed algorithms, algorithm AL-7, significantly performs better than the rest. Tests of hypotheses were performed to statistically confirm the results. In algorithm AL-7, more weight is given to the processing times of jobs on the first and the third machines compared to those of the second machine. Moreover, both the lower and upper bounds of job processing times on all three machines are utilized. Furthermore, algorithm AL-7 is shown to perform the best regardless of the extreme distributions considered. Hence, it is recommended as the best algorithm.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76551038","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}
{"title":"Erratum to: Necessary optimality conditions for a fractional multiobjective optimization problem","authors":"N. Gadhi","doi":"10.1051/ro/2023092","DOIUrl":"https://doi.org/10.1051/ro/2023092","url":null,"abstract":"Erratum to: RAIRO-Oper. Res. https://doi.org/10.1051/ro/2020049","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79770622","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}
Large-scale epidemics impose significant burdens globally and cause an imbalance of medical supplies among different regions owing to the dissimilarly and unevenly distributed prevalence of the infection. Along with rebalancing the limited medical supplies to meet the demand and supply requirements, ensuring that the supplies are allocated to support the affected regions is also important. Hence, this study focuses on the collaborative medical supply rebalancing and allocating process to balance the demand and supply. The law of diminishing marginal utility is incorporated in this study to quantify the principle of fairness in rebalancing and allocating medical supplies. Accordingly, under uncertainty, a marginal-utility-oriented optimization model is proposed to formulate the rebalancing and allocation of collaborative medical supplies. Because the proposed model is nonlinear and computationally intractable, a linearization approach is adopted to obtain the global optimum that supports decision-making in response to epidemics. Furthermore, a real case study of the United States is implemented, where the sensitivity analysis of critical parameters is conducted on the coronavirus disease 2019. Computational results indicate that additional medical supplies, stock levels, and scenario constructions significantly influence the supply/demand point identification and outgoing/incoming shipments. Moreover, this study not only validates the effectiveness and feasibility of the method but also highlights the importance of incorporating the law of diminishing marginal utility into the collaborative medical supply rebalancing and allocating problem.
{"title":"Marginal-utility-oriented optimization model for collaborative medical supply rebalancing and allocating in response to epidemics","authors":"Xuehong Gao, Cejun Cao, Zhijin Chen, Guozhong Huang, Huiling Jiang, Liang Zhou","doi":"10.1051/ro/2023089","DOIUrl":"https://doi.org/10.1051/ro/2023089","url":null,"abstract":"Large-scale epidemics impose significant burdens globally and cause an imbalance of medical supplies among different regions owing to the dissimilarly and unevenly distributed prevalence of the infection. Along with rebalancing the limited medical supplies to meet the demand and supply requirements, ensuring that the supplies are allocated to support the affected regions is also important. Hence, this study focuses on the collaborative medical supply rebalancing and allocating process to balance the demand and supply. The law of diminishing marginal utility is incorporated in this study to quantify the principle of fairness in rebalancing and allocating medical supplies. Accordingly, under uncertainty, a marginal-utility-oriented optimization model is proposed to formulate the rebalancing and allocation of collaborative medical supplies. Because the proposed model is nonlinear and computationally intractable, a linearization approach is adopted to obtain the global optimum that supports decision-making in response to epidemics. Furthermore, a real case study of the United States is implemented, where the sensitivity analysis of critical parameters is conducted on the coronavirus disease 2019. Computational results indicate that additional medical supplies, stock levels, and scenario constructions significantly influence the supply/demand point identification and outgoing/incoming shipments. Moreover, this study not only validates the effectiveness and feasibility of the method but also highlights the importance of incorporating the law of diminishing marginal utility into the collaborative medical supply rebalancing and allocating problem.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84513685","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}
Yong Peng, Shue Gao, Dennis Z. Yu, Yun Xiao, Yishan Luo
We study a multi-objective optimization model of a stochastic multimodal transportation network considering key impact factors such as transit cost, time, and transport mode schedule while minimizing total transportation cost and transportation time. In this study, we apply the Monte Carlo simulation to deal with the stochastic transportation time in the network and propose a data-driven approach that combines historical data and the dataset generated by the data mining algorithm to accelerate the search for the nondominated solution in the simulation. To validate the effectiveness of the proposed Data-Driven Multi-Objective Simulation Ant Colony (DD-MSAC) algorithm, we compare the optimum-seeking performance and the running time consumption of the Nondominated Sorting Genetic Algorithm-II (NSGA-II) and the Multi-Objective Simulation Ant Colony (MSAC) algorithm. Then, the MSAC algorithm is adopted as the benchmark for the comparison study on the solving performance of the proposed DD-MSAC algorithm. We conducted 30 times simulation run under different network scales in our numerical examples to show that the DD-MSAC algorithm can be equally effective as the non-data-driven MSAC algorithm in finding a nondominated solution as the average error does not exceed 5%. Meanwhile, we analyze the impact of different data-driven approaches, including data pool and support vector machine, on the solution quality and the running time. Finally, we use an example of China’s Belt Road Initiative to verify the effectiveness of the proposed algorithm.
研究了随机多式联运网络的多目标优化模型，考虑了运输成本、运输时间和运输方式调度等关键影响因素，同时使总运输成本和运输时间最小化。在本研究中，我们应用蒙特卡罗模拟来处理网络中的随机运输时间，并提出了一种数据驱动的方法，该方法将历史数据与数据挖掘算法生成的数据集相结合，以加速模拟中非支配解的搜索。为了验证数据驱动多目标模拟蚁群(DD-MSAC)算法的有效性，我们比较了非支配排序遗传算法- ii (NSGA-II)和多目标模拟蚁群(MSAC)算法的寻优性能和运行时间消耗。然后，以MSAC算法为基准，对本文提出的DD-MSAC算法的求解性能进行比较研究。在我们的数值示例中，我们在不同网络规模下进行了30次模拟运行，表明DD-MSAC算法在寻找非主导解方面与非数据驱动的MSAC算法同样有效，平均误差不超过5%。同时，分析了不同的数据驱动方法(包括数据池和支持向量机)对解决方案质量和运行时间的影响。最后，以中国“一带一路”倡议为例，验证了算法的有效性。
{"title":"Multi-objective optimization for multimodal transportation routing problem with stochastic transportation time based on data-driven approaches","authors":"Yong Peng, Shue Gao, Dennis Z. Yu, Yun Xiao, Yishan Luo","doi":"10.2139/ssrn.4093003","DOIUrl":"https://doi.org/10.2139/ssrn.4093003","url":null,"abstract":"We study a multi-objective optimization model of a stochastic multimodal transportation network considering key impact factors such as transit cost, time, and transport mode schedule while minimizing total transportation cost and transportation time. In this study, we apply the Monte Carlo simulation to deal with the stochastic transportation time in the network and propose a data-driven approach that combines historical data and the dataset generated by the data mining algorithm to accelerate the search for the nondominated solution in the simulation. To validate the effectiveness of the proposed Data-Driven Multi-Objective Simulation Ant Colony (DD-MSAC) algorithm, we compare the optimum-seeking performance and the running time consumption of the Nondominated Sorting Genetic Algorithm-II (NSGA-II) and the Multi-Objective Simulation Ant Colony (MSAC) algorithm. Then, the MSAC algorithm is adopted as the benchmark for the comparison study on the solving performance of the proposed DD-MSAC algorithm. We conducted 30 times simulation run under different network scales in our numerical examples to show that the DD-MSAC algorithm can be equally effective as the non-data-driven MSAC algorithm in finding a nondominated solution as the average error does not exceed 5%. Meanwhile, we analyze the impact of different data-driven approaches, including data pool and support vector machine, on the solution quality and the running time. Finally, we use an example of China’s Belt Road Initiative to verify the effectiveness of the proposed algorithm.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74129593","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}
Facing the strengthening of consumer environmental awareness, we investigate the green investment and green subsidy problem of an incumbent supply chain (ISC) taking into account whether the external manufacturer (EM) encroach. Green investment strategies are explored under three scenarios: no-green investment, ex-ante and ex-post green subsidies by the incumbent manufacturer (IM), and green investment by the supplier. The results show that market size does not influence on supplier channel selection and investment decisions. The green investment strategy is significantly affected by investment cost efficiency. Meanwhile, contrary to expectation, the more the investment is, the more willing the incumbent supply chain is to invest. Moreover, under a single-channel format, the incumbent supply chain can’t always achieve Pareto equilibrium. However, in the presence of the supplier green investment, although each green investment scenario can improve the profit, it cannot realize Pareto equilibrium. In addition, when the dual-channel format is adopted, the local areas can achieve Pareto equilibrium under different scenarios. The supplier plays an important role as it holds a monopoly upstream in the supply chain. As a result, the supplier's green investment generates excellent profit and consumer surplus; however, there will be fluctuations in the optimal strategy of the incumbent manufacturer.
{"title":"Supply chain's green investment strategy to cope with an entrant threat considering differentiated competitiveness","authors":"Chunyu Li, Peng Xing, Yanting Li","doi":"10.1051/ro/2023086","DOIUrl":"https://doi.org/10.1051/ro/2023086","url":null,"abstract":"Facing the strengthening of consumer environmental awareness, we investigate the green investment and green subsidy problem of an incumbent supply chain (ISC) taking into account whether the external manufacturer (EM) encroach. Green investment strategies are explored under three scenarios: no-green investment, ex-ante and ex-post green subsidies by the incumbent manufacturer (IM), and green investment by the supplier. The results show that market size does not influence on supplier channel selection and investment decisions. The green investment strategy is significantly affected by investment cost efficiency. Meanwhile, contrary to expectation, the more the investment is, the more willing the incumbent supply chain is to invest. Moreover, under a single-channel format, the incumbent supply chain can’t always achieve Pareto equilibrium. However, in the presence of the supplier green investment, although each green investment scenario can improve the profit, it cannot realize Pareto equilibrium. In addition, when the dual-channel format is adopted, the local areas can achieve Pareto equilibrium under different scenarios. The supplier plays an important role as it holds a monopoly upstream in the supply chain. As a result, the supplier's green investment generates excellent profit and consumer surplus; however, there will be fluctuations in the optimal strategy of the incumbent manufacturer.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75702798","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}
Despite the fact that there is a large body of literature on the Production Routing Problem (PRP), we were struck by the dearth of research on outsource planning and lateral transshipment. This paper presents a mixed-integer linear programming model for incorporating outsourcing, lateral transshipment, back ordering, lost sales, and time windows into production routing problems. Then a robust optimization model will be introduced to overcome the detrimental effects of demand uncertainty. Considering the scale and complexity of the suggested problem, addressing it in a reasonable time was a challenge. Therefore, three matheuristic algorithms, including Genetic Algorithm, Simulated Annealing, and Modified Simulated Annealing, are developed for solving large-scale problems. Eventually, computational experiments on disparate instances are performed, and the results show the effectiveness and efficiency of the proposed algorithms. In other words, our recommended algorithms outperform the CPLEX solver in terms of the quality and time of obtaining the solutions.
{"title":"A robust optimization approach for the production-routing problem with lateral transshipment and outsourcing","authors":"Pedram Farghadani-Chaharsooghi, Behrooz Karimi","doi":"10.1051/ro/2023083","DOIUrl":"https://doi.org/10.1051/ro/2023083","url":null,"abstract":"Despite the fact that there is a large body of literature on the Production Routing Problem (PRP), we were struck by the dearth of research on outsource planning and lateral transshipment. This paper presents a mixed-integer linear programming model for incorporating outsourcing, lateral transshipment, back ordering, lost sales, and time windows into production routing problems. Then a robust optimization model will be introduced to overcome the detrimental effects of demand uncertainty. Considering the scale and complexity of the suggested problem, addressing it in a reasonable time was a challenge. Therefore, three matheuristic algorithms, including Genetic Algorithm, Simulated Annealing, and Modified Simulated Annealing, are developed for solving large-scale problems. Eventually, computational experiments on disparate instances are performed, and the results show the effectiveness and efficiency of the proposed algorithms. In other words, our recommended algorithms outperform the CPLEX solver in terms of the quality and time of obtaining the solutions.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86238608","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 research is focused on solving the Cutting Stock with Limited Open Stacks Problem (CS-LOSP). The CS-LOSP is an optimization problem which consists of the classical Cutting Stock Problem (CSP) paired with the additional constraint that the maximum number of open stacks from the sequencing of the cutting patterns obtained from the CSP solution is equal or lower than a preset limit. Despite being a problem with great practical importance, the literature lacks models for this problem, and only one-dimensional problems are addressed. In this paper, we propose two integer linear programming formulations for the CS-LOSP that are valid for solving instances of the CSP of any dimension. In order to eliminate symmetrical solutions to the problem, the proposed formulations sequence sets of cutting patterns instead of sequencing the cutting patterns individually, thus, the search space for solutions is reduced. A set of randomly generated instances for the two-dimensional problem is used to perform computational experiments in order to validate the proposed mathematical formulations.
{"title":"Mathematical models for the cutting stock with limited open stacks problem","authors":"Gabriel Gazzinelli Guimarães, K. C. Poldi","doi":"10.1051/ro/2023079","DOIUrl":"https://doi.org/10.1051/ro/2023079","url":null,"abstract":"This research is focused on solving the Cutting Stock with Limited Open Stacks Problem (CS-LOSP). The CS-LOSP is an optimization problem which consists of the classical Cutting Stock Problem (CSP) paired with the additional constraint that the maximum number of open stacks from the sequencing of the cutting patterns obtained from the CSP solution is equal or lower than a preset limit. Despite being a problem with great practical importance, the literature lacks models for this problem, and only one-dimensional problems are addressed. In this paper, we propose two integer linear programming formulations for the CS-LOSP that are valid for solving instances of the CSP of any dimension. In order to eliminate symmetrical solutions to the problem, the proposed formulations sequence sets of cutting patterns instead of sequencing the cutting patterns individually, thus, the search space for solutions is reduced. A set of randomly generated instances for the two-dimensional problem is used to perform computational experiments in order to validate the proposed mathematical formulations.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82516189","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}
Private brands are frequently introduced by online platforms. Motivated by this phenomenon, this study aims to examine the strategic introduction of private brand involving the selling format choice in the online retail platform circumstance. The retail platform determines the private label introduction as well as the selling formats, i.e., reselling or agency selling. We develop a game-theoretic model to examine the private brand introduction problem for the online retail platform. The results indicate that when the potential market for private brand is moderate, the introduction of a private brand will interact with the choice of selling format. Clearly, it makes sense for the platform to exclude or include the private brand strategy based on the size of the potential market when it is extremely small versus relatively large. However, in the case of a medium-sized market for private brands, the platform's private brand introduction and selling format will be determined by both the cost of order fulfillment and the intensity of competition between national brands and private brands. This study contributes to uncovering under what conditions an online retail platform can be beneficial by introducing a private brand.
{"title":"Strategic introduction of private brand in the online retail platform","authors":"Zongsheng Huang, Wei Zhang, Bin Liu","doi":"10.1051/ro/2023080","DOIUrl":"https://doi.org/10.1051/ro/2023080","url":null,"abstract":"Private brands are frequently introduced by online platforms. Motivated by this phenomenon, this study aims to examine the strategic introduction of private brand involving the selling format choice in the online retail platform circumstance. The retail platform determines the private label introduction as well as the selling formats, i.e., reselling or agency selling. We develop a game-theoretic model to examine the private brand introduction problem for the online retail platform. The results indicate that when the potential market for private brand is moderate, the introduction of a private brand will interact with the choice of selling format. Clearly, it makes sense for the platform to exclude or include the private brand strategy based on the size of the potential market when it is extremely small versus relatively large. However, in the case of a medium-sized market for private brands, the platform's private brand introduction and selling format will be determined by both the cost of order fulfillment and the intensity of competition between national brands and private brands. This study contributes to uncovering under what conditions an online retail platform can be beneficial by introducing a private brand.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90227405","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}
A path-factor of a graph G is a spanning subgraph of G whose components are paths. A P≥d-factor of a graph G is a path-factor of G whose components are paths with at least d vertices, where d is an integer with d ≥ 2. A graph G is P≥d-factor covered if for any e ∈ E(G), G admits a P≥d-factor including e. A graph G is (P≥d,n)-factor critical deleted if for any Q ⊆ V (G) with |Q| = n and any e ∈ E(G − Q), G − Q − e has a P≥d-factor. A graph G is (P≥d,n)-factor critical covered if for any Q ⊆ V (G) with |Q| = n, G − Q is a P≥d-factor covered graph. In this paper, we verify that (i) an (n + t + 2)-connected graph G of order p with p ≥ 4t + n + 7 is (P≥3,n)-factor critical deleted if for any independent set {v1,v2,··· ,v2t+1} of G, where
{"title":"Degree conditions for path-factor critical deleted or covered graphs","authors":"Hong-xia Liu","doi":"10.1051/ro/2023078","DOIUrl":"https://doi.org/10.1051/ro/2023078","url":null,"abstract":"A path-factor of a graph G is a spanning subgraph of G whose components are paths. A P≥d-factor of a graph G is a path-factor of G whose components are paths with at least d vertices, where d is an integer with d ≥ 2. A graph G is P≥d-factor covered if for any e ∈ E(G), G admits a P≥d-factor including e. A graph G is (P≥d,n)-factor critical deleted if for any Q ⊆ V (G) with |Q| = n and any e ∈ E(G − Q), G − Q − e has a P≥d-factor. A graph G is (P≥d,n)-factor critical covered if for any Q ⊆ V (G) with |Q| = n, G − Q is a P≥d-factor covered graph. In this paper, we verify that (i) an\u0000(n + t + 2)-connected graph G of order p with p ≥ 4t + n + 7 is (P≥3,n)-factor critical deleted if\u2028 for any independent set {v1,v2,··· ,v2t+1} of G, where","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75723820","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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