2026 Vol.30

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A survey on research advances in consensus-based optimization algorithm WEI Jiazhen, BIAN Wei Operations Research Transactions    2026, 30 (1): 1-23.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.001 Abstract （250）      PDF（pc） （811KB）（144）       Knowledge map    Save Global optimization problems have widespread applications across various fields such as scientific research, engineering, economics, and artificial intelligence. Consensus-based optimization algorithm is a class of multi-agent, meta-heuristic and derivative-free algorithms. It is designed to solve global nonsmooth and nonconvex optimization problems, while also being conducive to theoretical analysis and algorithm implementation. In this paper, we first introduce the fundamental principles and analytical results of the original algorithm. Subsequently, the latest development of the consensus-based optimization algorithms and their variants are discussed in detail. And the applications in fields such as machine learning and image processing are briefly described. Finally, we explore future research directions across three key areas: theoretical innovation, algorithm design, and application expansion. Reference | Related Articles | Metrics | Comments（0）

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Profit allocation model of service-oriented manufacturing hybrid supply chain under uncertain demand YU Xiaohui, ZHOU Weiqing, WU You Operations Research Transactions    2026, 30 (1): 24-40.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.002 Abstract （122）      PDF（pc） （1027KB）（39）       Knowledge map    Save New models such as product and service bundling bring more uncertainty to the customer demand of service-oriented manufacturing, while the service-oriented manufacturing hybrid supply chain is a coalition structure cooperation game with uncertain payoff. In order to solve this uncertain coalition structure cooperative game, a new profit distribution mechanism is proposed based on the internal distribution proportion. It allows each supply chain of the hybrid supply chain to distribute the total profit according to the coalition contribution, while within a single supply chain, the coalition profit is dynamically allocated according to a certain distribution coefficient. This solution reflects the uncertainty of demand in the service-oriented manufacturing hybrid supply chain during the establishment period, and accurately describes the impact of market demand fluctuation on cooperation profit. At the same time, it decreases the impact of incomplete information of coalition profit on the division of total profit. Reference | Related Articles | Metrics | Comments（0）

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Differential game models for a new energy vehicle closed-loop supply chain under the applications of blockchain XU Jianteng, MA Keke, BAI Qingguo, ZHANG Yuzhong Operations Research Transactions    2026, 30 (1): 41-60.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.003 Abstract （158）      PDF（pc） （818KB）（42）       Knowledge map    Save The application of blockchain technology in the new energy vehicle supply chain can to a certain extent solve the problems of low utilization rate and difficult recycling of waste power batteries. In this context, this paper considers the dynamic strategy for a new energy vehicle closed-loop supply chain consisting of one power battery supplier, one general component supplier and one manufacturer. In this system, blockchain technology is adopted to trace the lifecycle process management of the power battery. The stochastic evolution process of the traceability level is described to show the dynamic changes of the power battery. For the three cases that the power battery supplier and the manufacturer are respectively responsible for recycling products and the vertical integration of both parties is implemented, we construct the corresponding leader-follower stochastic differential game models. By solving the feedback equilibrium, we compare the state variables, decision variables and feedback profits of the three models when the system reaches the steady situation. Numerical examples are provided to conduct the performance of the supply chain in the unsteady-state and dynamic environments. These findings provide some guides for the new energy vehicle enterprises to adopt the blockchain technology, recycle the waste power batteries and integrate the supply chain. Reference | Related Articles | Metrics | Comments（0）

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Blockchain empowering the digital transformation: Study on the coordination strategies of supply chain ZHOU Yunxu, YAO Fanjun, GAO Hongwei Operations Research Transactions    2026, 30 (1): 61-74.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.004 Abstract （185）      PDF（pc） （724KB）（64）       Knowledge map    Save We used differential game to study the coordination problem of supply chain, taking the digital transformation of the supply chain as the starting point, in which the knowledge accumulation of blockchain is the state variable. We solved and compared the quality improvement, blockchain technology investment, the trajectory of knowledge accumulation, market demand and supply chain profit in non-cooperative and cost-sharing scenarios. We also explored the blockchain technology in the supply chain from the perspective of game theory. Combined with numerical simulation, the sensitivity analysis of relevant parameters was carried out. The study found that the cooperation between players can improve the retailer's blockchain technology investment level without affecting the supplier's quality improvement strategy, effectively alleviate the “double marginal effect” of the non-cooperative case, achieve Pareto improvement of supply chain performance, and then improve the social welfare. Reference | Related Articles | Metrics | Comments（0）

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Research on diversification portfolio optimization model and method ZHAO Hongxin, KONG Lingchen Operations Research Transactions    2026, 30 (1): 75-92.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.005 Abstract （255）      PDF（pc） （706KB）（89）       Knowledge map    Save Portfolio selection is an important topic in the financial field. Under a series of basic assumptions, economist Markowitz established the mean-variance model in 1952. Then, the study of modern portfolio theory began. Effective diversification is the key to reduce risks and increase returns. This paper starts from mean-variance model and reviews the diversified portfolio strategies. We focus on the portfolio optimization model and solution method under regularization. Finally, we briefly introduce some of our recent work and put forward prospects and ideas based on current research hotspots. Reference | Related Articles | Metrics | Comments（0）

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The economics of waiting-area entertainment SUN Ke, WANG Jinting, WANG Zhongbin Operations Research Transactions    2026, 30 (1): 93-107.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.006 Abstract （225）      PDF（pc） （861KB）（71）       Knowledge map    Save In order to relieve customers' waiting anxiety when queueing, many service providers provide waiting-area entertainment (WAE) by charging a service fee. This measure can effectively reduce customers' waiting cost and the queueing anxiety of them has been also greatly alleviated. By providing this option, it can not only attract more customers to join the system, but also add additional revenue to service providers. So it is widely favored. This paper establishes a queueing game model by focusing on the popular operation mode of “waiting-area entertainment” in the current service industry, and analyzes the impact of this option on the equilibrium behavior of customers and the revenue of service providers theoretically. The following results are derived in this paper. First, the Nash equilibrium joining strategy of customers is explored under observable and unobservable information cases. Second, the revenue-maximizing information disclosure strategy is derived, and it is optimal to reveal (conceal) queue length information to customers when market size is large (small). The revenue-maximizing service fee increases with the market size. Third, although WAE is provided an additional service option for releasing customers' anxiety for waiting, we find that when the market size is large, this option may hurt consumer surplus. Reference | Related Articles | Metrics | Comments（0）

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A faster SDP relaxation for two-period financial derivatives liquidation problem HUANG Yin, LUO Hezhi Operations Research Transactions    2026, 30 (1): 108-120.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.007 Abstract （98）      PDF（pc） （613KB）（34）       Knowledge map    Save In this paper, we consider a two-period financial derivative liquidation problem without the convexity assumption. Its optimization model is a non-convex quadratic programming problem with a single quadratic constraint and linear constraints, which is NP-hard. We propose a faster new semi-definite programming (SDP) relaxation for this model by making use of its special structure, and estimate the gap between it and the original problem. We also show that it provides a tighter lower bound than the existing SDP relaxation in the literature. Numerical experiments show that this SDP relaxation can fast provide a very tight lower bound for the original problem, thus can provide effective lower bounds in branch-and-bound algorithm to find the global optimal solution of the problem. Reference | Related Articles | Metrics | Comments（0）

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A coordinated multi-agent production and transportation scheduling on parallel machines based on auction algorithm XU Ke, JI Lanping, GONG Hua, LIU Peng, SUN Wenjuan Operations Research Transactions    2026, 30 (1): 121-136.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.008 Abstract （112）      PDF（pc） （828KB）（41）       Knowledge map    Save Under the background of resource sharing, in order to solve the competition between different customers' jobs for machine resources, the coordinated production and transportation scheduling problem on parallel machines based on multi-agent is studied. Multiple manufacturers put idle, similar machines on a shared platform, forming a production environment of parallel machines. Jobs from multiple customers need to be processed on machines on the shared platform. Multiple customers with their own optimization objectives are regarded as multiple agents. After the job is processed, the distribution of finished products needs to be considered because the location of machine and customers is dispersed. Reference | Related Articles | Metrics | Comments（0）

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Strategic analysis in repairable retrial queueing systems with Bernoulli vacations HAN Yunna, TIAN Ruiling Operations Research Transactions    2026, 30 (1): 137-155.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.009 Abstract （109）      PDF（pc） （790KB）（32）       Knowledge map    Save This paper studies the M/M/1 constant retrial queueing model with Bernoulli vacations and server unreliability, where the server breaks down at different rates in normal and idle states. The system has no waiting space and service starts immediately if the arriving customer finds the server is idle. Otherwise, if the system is busy, on vacation, and in a breakdown state, the customer decides whether or not to join the orbit based on the different levels of information provided by the system. After each completed service, the system starts to go on vacation or remains available. The system rejects new customers from entering the system in the event of server breakdown. Based on the different levels of information provided by the system, we study the steady-state indicators in the almost unobservable case and fully unobservable case, as well as the equilibrium strategies of customers in both cases based on the reward-cost structure. Finally, we use numerical examples to show that revealing server status information does not increase the social benefit. Reference | Related Articles | Metrics | Comments（0）

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Equilibrium analysis of the fluid model with two types of parallel customers and delayed repair WANG Jing, XU Xiuli Operations Research Transactions    2026, 30 (1): 156-170.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.010 Abstract （101）      PDF（pc） （1493KB）（30）       Knowledge map    Save In this paper, the fluid model with two types of parallel customers and fault delay repair is economically analyzed. The normal working state, fault delay repair state and fault repair state are alternately carried out. When the fluid reaches the system, the net benefit based on the obtained information is calculated, and then whether to enter the system is determined. In the case of complete feasibility and almost visibility, the fluid equilibrium balking strategies and the social benefit optimal strategy per unit time are discussed respectively. Numerical examples are given to analyze the influence of the arrival rate and service rate on average social benefits per unit time. Reference | Related Articles | Metrics | Comments（0）

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Dynamic programming algorithms for single machine supply chain scheduling CHEN Rongjun, LIU Yongcai, HUANG He, TANG Guochun Operations Research Transactions    2026, 30 (1): 171-178.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.011 Abstract （101）      PDF（pc） （586KB）（37）       Knowledge map    Save In this paper, an integrated scheduling model of production and distribution operations is studied. In this model, a set of jobs (i.e., customer orders) are first processed on single machine and then delivered in batches to the downstream customers in different regions. The problem is to find a joint schedule of production and distribution such that an objective function that takes into account both production cost and distribution cost is optimized. Production cost is measured by a function of delivery times, namely, the times when the jobs are delivered to the customers. The distribution cost of a delivery shipment consists of a fixed charge and a variable cost proportional to the total distance of the route taken by the shipment. This paper considers different production costs. For the model with the sum of weighted delivery times as production costs, the strong NP-Hardness has been proven and a dynamic programming algorithm is developed for consistency constraints on jobs' processing time and weight. For the model with production costs related to the due date, the NP-Hardness is analyzed and two dynamic programming algorithms are designed. Reference | Related Articles | Metrics | Comments（0）

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Single-machine online scheduling with NDP constraint and deterioration MENG Lanmeng, MA Ran, ZHANG Yuzhong Operations Research Transactions    2026, 30 (1): 179-187.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.012 Abstract （92）      PDF（pc） （720KB）（48）       Knowledge map    Save This paper studies the single machine online production scheduling with non-delayed processing (NDP) constraint as well as deterioration. Jobs arriving over time on-line are processed non-preemptive on machine. The model of job processing time is $p_{j}=a+b_{j}t (a>0$). Its objective is to minimize the maximum weighted completion time. Firstly, we derive the lower bound of the considered problem is $1+b_{\max}$ by means of adversary method. Then we design a online algorithm with the competitive ratio of $2+b_{\max}$. Moreover, these online models are simulated to verify the effectiveness and correctness of online algorithm. Reference | Related Articles | Metrics | Comments（0）

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Approximation algorithm for mixed batch parallel machine scheduling with nested processing set restrictions WU Hongyi, WANG Dong, WAN Long, LUO Wenchang Operations Research Transactions    2026, 30 (1): 188-196.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.013 Abstract （106）      PDF（pc） （591KB）（36）       Knowledge map    Save In this paper, we investigate the mixed batch parallel machine scheduling problem in which a set of jobs should be processed on one of the parallel batch machines with nested processing set restrictions. Each job has its processing time and its machine set for processing with these machine sets satisfying the nested processing set restrictions. Each machine can process a group of jobs as a batch simultaneously, as long as the total number of jobs in this batch does not exceed the capacity of the machine. For a given batch, its processing time is equal to the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective function is to minimize the makespan. The problem includes the classic parallel machine scheduling problem as a special case, which is strongly NP-hard. For the studied problem, we derive an approximation algorithm with a performance ratio of $\left({2 + \alpha} \right)$, where $\alpha$ is a given parameter for weight with $0\leq\alpha\leq 1$. Reference | Related Articles | Metrics | Comments（0）

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A variance reduced gradient descent ascent algorithm for a class of nonconvex-nonconcave minimax problems WANG Ziqi, WANG Junlin, XU Zi Operations Research Transactions    2026, 30 (1): 197-206.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.014 Abstract （104）      PDF（pc） （543KB）（66）       Knowledge map    Save In this paper, we consider a class of stochastic nonconvex-nonconcave minimax problems, i.e., NC-PL minimax problems, for which we assume that the objective function satisfies the Polyak-Łojasiewicz (PL) condition with respect to the inner variable. We propose a variance reduced gradient descent ascent (VRGDA) algorithm for solving NC-PL minimax problems under the stochastic setting. The number of iterations to obtain an $\varepsilon$-stationary point of the VRGDA algorithm for solving NC-PL minimax problems is upper bounded by $\mathcal{O}(\varepsilon^{-3})$. The VRGDA algorithm owns the best iteration complexity in first-order algorithms for solving stochastic NC-PL problems. Reference | Related Articles | Metrics | Comments（0）

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A modified conjugate gradient algorithm with its applications in image recovery problems LIU Cong, JIAN Ailun, YUAN Gonglin Operations Research Transactions    2026, 30 (1): 207-216.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.015 Abstract （115）      PDF（pc） （14406KB）（49）       Knowledge map    Save It is well-known that, under the WWP (weak Wolfe-Powell) line search technique, the global convergence of the PRP conjugate gradient method for non-convex functions is still open. In this paper, a hybrid conjugate gradient method is proposed for large-scale unconstrained optimization problems. In this method, the modified BFGS method is mixed with the modified PRP conjugate gradient method, and the weak Wolfe-Powell line search technique is used to find the step size, and the search direction has the property of sufficient descent. Theoretically, the global convergence of nonconvex functions is ensured by assuming reasonable conditions. In numerical experiments, the parameter estimation of the Muskingum model reduces the amount of computation and storage, which illustrates the effectiveness of MPRP. In different noise situations, the MPRP is proved to be highly competitive by comparing the recovery of multiple images. In addition, the restoration of the image is more significant under the image of low impulse noise. Reference | Related Articles | Metrics | Comments（0）

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Convergence analysis of an adaptive proximal gradient-subgradient algorithm for square-root-loss regression problems YANG Jinji, SHEN Chungen, YU Zhensheng Operations Research Transactions    2026, 30 (1): 217-234.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.016 Abstract （88）      PDF（pc） （793KB）（32）       Knowledge map    Save Square-root-loss regression problems have attracted great attention since the choice of its regularization parameter does not rely on the prior knowledge of the deviation of the noise. However, the square-root loss function has a non-differentiable point, which brings difficulties to numerical algorithms. In this paper, we improve proofs of local smoothness and locally restricted strong convexity of the square-root loss function, which is based on the work of Li et al. (2020). To overcome numerical difficulties caused by nonsmoothness of the loss function, we develop an adaptive proximal gradient-subgradient algorithm (APGSA). Under some assumptions, the global convergence of the proposed algorithm is guaranteed with high probability. In addition, we also prove that the algorithm can accurately identify the active manifold in finite iterations, and then the linear rate of local convergence with high probability is established. Finally, simulation experiments were conducted to verify both the effectiveness and the fast local linear convergence of the algorithm APGSA. Reference | Related Articles | Metrics | Comments（0）

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The study of the stability of KKMS points CUI Ruiqi, ZHANG Shu, SONG Qiqing Operations Research Transactions    2026, 30 (1): 235-246.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.017 Abstract （96）      PDF（pc） （567KB）（25）       Knowledge map    Save KKMS Theorem is the generalization of the famous Sperner Lemma. In the study of existence of many kinds of cores in cooperative game theory, and in the equilibrium analysis of mathematical economics, KKMS Theorem plays an important and fundamental role. Based on the importance of KKMS Theorem in the application of game theory and economics, this paper introduces the conception of KKMS points, constructs the space of KKMS mappings, and studies the stability of the KKMS points, and obtains the semi-continuous and continuous results of KKMS mappings. The results show that KKMS mappings have upper semi-continuity. Further, by constructing a specific counterexample in 2-simplices, this shows that KKMS mappings do not have lower semi-continuity generally. This also gives a sufficient and necessary condition for the continuity of KKMS mappings. Furthermore, this obtains that the KKMS points have generic stability and essential stability, which includes the existing stability results on KKM points as special cases. Reference | Related Articles | Metrics | Comments（0）

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Equal division value, equal surplus division value, and differential marginality YU Zhiqiang, CUI Zeguang, SHAN Erfang Operations Research Transactions    2026, 30 (1): 247-255.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.018 Abstract （72）      PDF（pc） （540KB）（26）       Knowledge map    Save For cooperative games with transferable utility, equal division value and equal surplus division value are two eminent solutions, and both of them satisfy two standard axioms, additivity and symmetry. To eliminate the controversial additivity, Casajus (2011) proposes differential marginality axiom and explores the relationship between this proposed axiom and both additivity and symmetry. Inspired by Casajus (2011), we employ differential marginality to characterize the equal division value, the equal surplus division value, and their convex combinations. Reference | Related Articles | Metrics | Comments（0）

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Isolated toughness variant and the existence of fractional [a,b]-factor GAO Wei, WANG Weifan Operations Research Transactions    2026, 30 (1): 256-266.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.01.019 Abstract （75）      PDF（pc） （550KB）（26）       Knowledge map    Save The existence of fractional factors in specific settings is an important topic of graph factor theory, and isolated toughness is an important parameter to measure the vulnerability of networks. As the unique variant of isolation toughness, $I'(G)$ is defined as the minimum ratio of $|S|$ and $i(G-S)-1$, where $S$ is the subset of vertices that satisfies $i(G-S)\ge2$. This parameter measures the robustness of the network from the perspective of topology, and recent research reveals that it is closely related to the fractional factor. In this paper, we give an $I'(G)$ condition for the existence of fractional $[a,b]$-factors in a graph, and show that the condition is sharp by counterexample. This result extends the original $I'(G)$ tight bound on the existence of the fractional $k$-factor. Reference | Related Articles | Metrics | Comments（0）

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Proximal-based methods can guarantee blunt local minimizer for nonconvex nonsmooth optimization problem WANG Xiangfeng, ZENG Shangzhi, ZHANG Jin, ZHOU Jinchuan Operations Research Transactions    2026, 30 (2): 1-23.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.001 Abstract （90）      PDF（pc） （907KB）（73）       Knowledge map    Save We propose a general Flexible proxImal-based block-wise First-order Algorithm framework called FIFA for a stochastic composite minimization problem with two nonconvex function components in the objective while only one of them is assumed to be differentiable. Under some per-block Lipschitz-like conditions based on Bregman distance, but without the global Lipschitz continuity of the gradient of the differentiable function, we prove that any accumulation point of the sequence is a stationary point of the model. We further show that the stationarity is the "best" one if the global Lipschitz continuity is additionally assumed, and that it is even the local minimizer for some special cases. Convergence analysis without the global Lipschitz continuity and the enhanced stationarity analysis make our results different from existing results in both the convex and nonconvex contexts. Reference | Related Articles | Metrics | Comments（0）

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Fast algorithm for OWL1 norm constrained regression model MEN Yanchao, LI Xudong Operations Research Transactions    2026, 30 (2): 24-44.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.002 Abstract （54）      PDF（pc） （748KB）（34）       Knowledge map    Save As machine learning algorithms continue to evolve and incorporate more features, effective variable selection has become a critical issue. To control the false discovery rate of regression coefficients in the variable selection, Bogdan et al. proposed the SLOPE model in 2015, which considers a linear regression model regularized by the OWL1 norm. In this paper, we consider a new regression model that uses an OWL1 norm constraint, enabling more flexible control of the false discovery rate through two parameters, $\lambda$ and $\tau$. We designed a fast algorithm that efficiently solves this model by using the dual semismooth Newton based proximal point algorithm (PPDNA). The algorithm's outer layer uses the proximal point algorithm, while the inner layer employs the semismooth Newton method to solve the subproblems efficiently. Meanwhile, the special structure of the generalized Jacobian induced by the projector onto the OWL1 norm ball is exploited to efficiently handle the corresponding Newton linear systems in the inner algorithm. Finally, we demonstrate the effectiveness and robustness of PPDNA by comparing it with two popular algorithms using both simulated data and large-scale real datasets. Reference | Related Articles | Metrics | Comments（0）

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Solution to strong partition of 2-balanced regular multipartite tournaments AI Jiangdong, HE Fankang, LIU Yihang Operations Research Transactions    2026, 30 (2): 45-57.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.003 Abstract （47）      PDF（pc） （535KB）（18）       Knowledge map    Save We call a partition of a $c$-partite tournament into tournaments of order $c$ strong if each tournament is strongly connected. The strong partition number, denoted as $ST(r)$, represents the minimum integer $c'$ such that every regular $r$-balanced $c$-partite tournament has a strong partition for all $c \geq c'$. Figueroa, Montellano-Ballesteros, and Olsen showed the existence of $ST(r)$ for all $r\geq 2$ and proved that $5\leq ST(2)\leq 7$. In this note, we establish that $ST(2)=6$ and we also show the unique $2$-balanced $5$-partite tournament which has no strong partition. Reference | Related Articles | Metrics | Comments（0）

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The well-posedness of traffic equilibrium problems ZENG Jing, XIANG Yao, ZHANG Wenyan Operations Research Transactions    2026, 30 (2): 58-68.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.004 Abstract （52）      PDF（pc） （563KB）（24）       Knowledge map    Save In the study of traffic assignment theory, the development of equilibrium model theory has attracted significant attention. The traffic equilibrium problem aims to determine the flow distribution and various performance metrics in a traffic network by using given O/D pair demand and cost functions, based on established path selection criteria. The core of this methodology lies in the fact that, under a traffic equilibrium state, no participant in the system can improve their travel time or cost by unilaterally changing their path choice, thereby maximizing the efficiency and performance of the traffic system. However, in reality, due to the diversity of traffic participants' behavior and the complexity of traffic networks, achieving a true equilibrium state is often challenging. This paper defines an approximate form of traffic equilibrium ${\varepsilon}$-equilibrium flow and verifies its existence. Additionally, it introduces a type of well-posedness for ${\varepsilon}$-equilibrium flow and establishes sufficient conditions for the validity of this well-posedness, providing new theoretical tools and methods for understanding and solving traffic equilibrium problems. Reference | Related Articles | Metrics | Comments（0）

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Research of touring route planning based on spatio-temporal awareness WANG Feng, HANG Bo, HUANG Jinzhou, XU Degang, ZHANG Zeyu, LIU Jiamou Operations Research Transactions    2026, 30 (2): 69-78.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.005 Abstract （48）      PDF（pc） （1348KB）（15）       Knowledge map    Save Tourist route planning is a challenging task that requires considering both temporal and spatial dimensions of tourism data. On one hand, it involves obtaining the spatial distribution of tourist attractions within the scenic area, and on the other hand, it takes into account the tourists' visiting behaviors during their exploration of the area. Therefore, in addition to collecting attribute information of various attractions in the scenic area, tourism route planning also requires a significant amount of data on tourists' visiting behaviors. This paper collects the aforementioned data and extracts the travel behavior information of tourists between attractions, proposing important indicators for measuring tourist travel behavior. Based on a comprehensive consideration of these indicators, the Travel Route Planning algorithm (TRP) is introduced with the aim of significantly reducing travel time between attractions. The experiments yield distinct results and corresponding travel times between scenarios without travel route planning and scenarios with travel route planning. Furthermore, by further optimizing the route planning, a comparison of travel times before and after optimization is obtained. The results indicate that the Travel Route Planning algorithm not only greatly reduces travel time but also provides valuable insights for determining optimal locations for tourist buses to make stops. Compared to three representative path planning algorithms currently available, the algorithm proposed in this paper exhibits significant advantages in response time and average solution quality. Reference | Related Articles | Metrics | Comments（0）

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Two RMIL-type conjugate gradient methods with sufficient descent property and applications in image restoration WU Xiaoyu, SHAO Hu, LIU Pengjie, ZHOU Jincheng Operations Research Transactions    2026, 30 (2): 79-92.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.006 Abstract （48）      PDF（pc） （4846KB）（21）       Knowledge map    Save The conjugate gradient method possesses the advantages of lower storage requirement and simplicity to iterate, therefore it has been widely used for solving the large-scale optimization problems. Based on the Rivaie-Mustafa-Ismail-Leong (RMIL) conjugate coefficient, we propose two extended RMIL-type coefficients and establish the corresponding conjugate gradient algorithms for solving unconstrained optimization problems. Under the strong Wolfe line search, we prove that the search direction sequence generated by the first algorithm satisfies the descent property. The global convergence property is established under the normal assumptions. The descending property of the second algorithm is independent of any line search condition. By using the standard Wolfe line search, the global convergence of the second algorithm is obtained. To test the numerical effects of two proposed algorithms, we apply them to solve unconstrained optimization problems and restore the blurred images affected by impulse noise. Compared with some existing conjugate gradient methods, experimental results show that the two proposed algorithms are promising. Reference | Related Articles | Metrics | Comments（0）

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The full-Newton step feasible interior-point method for general Fisher market equilibrium problems based on a kernel function CHI Xiaoni, YANG Yuping, LIU Sanyang, YANG Qili Operations Research Transactions    2026, 30 (2): 93-108.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.007 Abstract （40）      PDF（pc） （612KB）（8）       Knowledge map    Save In this paper, we design and analyze a full-Newton step interior-point method (IPM) for solving the weighted linear complementarity problem (WLCP), which is general optimization of the Fisher market equilibrium problem. As a non-trivial generalization of the complementarity problem (CP), weight complementarity problem (WCP) can be used to model a wide range of equilibrium problems in economics, science and engineering. Since there are nonnegative weight vectors in WCP, the theory and algorithms of WCP are more complicated than CP. In this paper, the IPM for CP is extended to solve WCP. Based on a kernel function, search directions are obtained by applying Newton's method to the equivalent system, which defines the central path. Thus, a full-Newton step feasible IPM for general Fisher market equilibrium problem is proposed. At each iteration we only use full-Newton steps, which avoids the calculation of the step size. Under suitable assumptions, the algorithm is shown to have global convergence and polynomial complexity. Some numerical results are provided to support the practical efficiency of the proposed algorithm. Reference | Related Articles | Metrics | Comments（0）

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Research on altruistic equilibria in altruistic strategy-form games WANG Nengfa, YANG Zhe Operations Research Transactions    2026, 30 (2): 109-124.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.008 Abstract （49）      PDF（pc） （594KB）（20）       Knowledge map    Save In this paper, we assume that the players are altruistic in strategy-form games. Following the idea, we introduce the altruistic games and altruistic generalized games with preference correspondences, and prove the existence theorems of altruistic equilibria. As applications, we obtain the existence theorems of altruistic equilibria for normal-form games and generalized games. Reference | Related Articles | Metrics | Comments（0）

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The proper jury theorem of voting theory HU Yuda Operations Research Transactions    2026, 30 (2): 125-136.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.009 Abstract （41）      PDF（pc） （499KB）（19）       Knowledge map    Save The famous Condorcet jury theorem provides the theoretical foundation for voting theory. Based on this theorem, it is only limited that all voters must have the same probability in their preference for the alternatives, which is impossible to happen in real voting, so in fact it only gives a special case of the general situation. This paper substantively extends the Condorcet jury theorem, and establishes a proper jury theorem for the relationship between the probability of the voting group using the majority preference rule to make a strict preference choice for the alternatives when each voter has its own different preference probabilities for the alternatives. At the same time, some important properties of the group strict preference probability determined by the established theorem are given. Finally, it is also proved that when the number of voters increases infinitely, the group strict preference probability determined by this theorem will tend to its maximum limit value of 1. Reference | Related Articles | Metrics | Comments（0）

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The inspection and preventive maintenance policy for a δ-shock model which has two types of failures GAO Qiaoqiao, ZHANG Junnan Operations Research Transactions    2026, 30 (2): 137-148.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.010 Abstract （37）      PDF（pc） （652KB）（11）       Knowledge map    Save The preventive maintenance strategy of a single component repairable system is studied in this paper. The system is subject to random external shocks during operation. The shock has two types: extreme shock and $\delta$-shock. When the working time reaches to a certain value $T$, the preventive maintenance will be carried out, preventive maintenance restores the state of the system to the state it was in after the last repair. There is a certain probability of delayed repair after the system failure, and the system will be replaced by a completely new one after the $N$th failure.According to the renewal reward theory, an expression for the expected cost per unit time of long-term operation is derived by using the replacement strategy $(T,N)$. Finally the feasibility of the model is verified by numerical examples, and the sensitivity analysis of some parameters is made, which can guide enterprises to carry out preventive maintenance of different systems. Reference | Related Articles | Metrics | Comments（0）

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Solving the optimal order quantity for an inventory system with unknown parameter values——Based on stock-dependent demand rate GUO Zhanbing, ZHANG Yejie Operations Research Transactions    2026, 30 (2): 149-158.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.011 Abstract （35）      PDF（pc） （2023KB）（23）       Knowledge map    Save Considering the difficulty in obtaining the accurate parameter values in the inventory models, this paper provides a two-stage ordering strategy for one kind of inventory system, where the demand rate depends on the inventory level. By constructing a dynamic order quantity and analyzing its related properties, this two-stage order strategy realizes the estimation of critical value for control parameter, and finally obtains the optimal order quantity. Moreover, this two-stage ordering strategy not only draws on the simple mathematical form of the classic EOQ model, but also does not require the retailer to know the exact model parameter values in advance. Theoretical analysis and numerical simulation demonstrate the feasibility of this strategy. Sensitivity analysis further provides the impact of the estimation error of the control parameter critical value on the effectiveness of this strategy, and the results show that this two-stage ordering strategy is robust to the misestimation of critical value for control parameter. Reference | Related Articles | Metrics | Comments（0）

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Pseudo-E-convex functions without differentiability condition and optimality of pseudo-E-convex programming HUANG Yingquan, JIANG Xuyu, SONG Guihua, YANG Han Operations Research Transactions    2026, 30 (2): 159-168.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.012 Abstract （33）      PDF（pc） （594KB）（17）       Knowledge map    Save Firstly, a class of pseudo-E-convex functions without differentiability condition is defined in this paper, which is a proper generalization of E-convex functions and pseudoconvex functions without differentiability condition. The existence of such functions is verified with an example. Furthermore, the relationships between pseudo-E- convex functions and other functions are discussed. These indicate that pseudo-E-convex functions without differentiability condition are more general and have a wider range of applications. Secondly, some properties of pseudo-E-convex functions are obtained. Finally, in terms of optimality, the relationship between the fixed points of the mapping E and their global optimal solutions is discussed for the pseudo-E-convex programming problems ($\mathrm{P}$) and ($\mathrm{P}_{E}$), the local-global property of the optimal solutions and the uniqueness of the global optimal solution for programming problem ($\mathrm{P}$) are obtained, and the examples are provided to verify the results respectively. Reference | Related Articles | Metrics | Comments（0）

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New characterizations of the two-step Shapley-solidarity value and its application YUAN Meng, LIU Tao, SHAN Erfang Operations Research Transactions    2026, 30 (2): 169-178.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.013 Abstract （38）      PDF（pc） （615KB）（11）       Knowledge map    Save In 2022, Zou et al. proposed a two-step Shapley-solidarity value for cooperative games with coalition structure, which distributes the total worth in two steps. Firstly, players within one union obtain the solidarity value in the subgame restricted to the corresponding union. Then, the surplus of the difference of between the Shapley value of the union obtained in the quotient game, and the worth of the union, will also be allocated to the players in the same union equally. This research proposes a new axiom called the grand coalition solidarity property, and proves that the two-step Shapley-solidarity value can be uniquely characterized by four axioms: efficiency, coalitional balanced contributions, population solidarity within unions and grand coalition solidarity property. Besides, this paper shows that the two-step Shapley-solidarity value is the only value that satisfies efficiency, coalitional symmetry, coalitional marginality, population solidarity within unions and grand union solidarity property. Finally, comparing the two-step Shapley-solidarity value with other values by a numerical example, this paper shows that the two-step Shapley-solidarity value can take care of the weak players better, while ensuring the fair allocation within unions. It turns out that the two-step Shapley-solidarity value reflects a higher degree of solidarity than those values. Reference | Related Articles | Metrics | Comments（0）

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An adaptive algorithm for solving quasimonotone variational inequality problems YU Sijie, LONG Xianjun Operations Research Transactions    2026, 30 (2): 179-193.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.014 Abstract （37）      PDF（pc） （655KB）（16）       Knowledge map    Save In this paper, we propose a new adaptive forward-backward algorithm to solve the quasimonotone variational inequality problem in real Hilbert spaces. Under reasonable assumptions, we prove that the iterative sequence generated by the algorithm converges strongly to an element of the solution set of the variational inequality problem. Finally, numerical experiments show the effectiveness and superiority of the new algorithm. Reference | Related Articles | Metrics | Comments（0）

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Inexact proximal point algorithms and projection methods for monotone variational inequalities CUI Hengxin, JIANG Fan Operations Research Transactions    2026, 30 (2): 194-208.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.015 Abstract （41）      PDF（pc） （600KB）（14）       Knowledge map    Save n this paper, we propose a class of inexact proximal point algorithms with relative error criterion for solving monotone variational inequalities. The next iterate in the proposed methods can be obtained in two ways. Under general hypothetical conditions, the global convergence of the new algorithms is established. By choosing a special form for the error, the proposed inexact proximal point algorithms reduce to a class of projection and contraction methods with linesearch, which reveals the connection between inexact proximal point algorithms and a class of projection methods. Numerical experiments demonstrate the efficiency of the new methods. Reference | Related Articles | Metrics | Comments（0）

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Improved q-trust region algorithm for unconstrained optimization problems QIU Yingming, PENG Jianwen Operations Research Transactions    2026, 30 (2): 209-224.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.016 Abstract （47）      PDF（pc） （744KB）（19）       Knowledge map    Save In this paper, we propose an improved $q$-trust region algorithm for solving unconstrained optimization problems. The algorithm has a new rule for updating the radius of the trust region. We establish the convergence of the improved $q$-trust region algorithm for solving unconstrained optimization problems under the conditions that the function is continuously $q$-differentiable and so on. Finally, numerical experiments show that our algorithm is effective. Compared with the improved $q$-trust region algorithm proposed by Zhou, our proposed improved $q$-trust region algorithm not only iterates to the optimal point faster, but also solves optimization problems with multiple optimal solutions. The results obtained in this paper extend and improve some existing results in the literature. Reference | Related Articles | Metrics | Comments（0）

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A sufficient condition for hamiltonicity in t-tough graphs CHEN Tao Operations Research Transactions    2026, 30 (2): 225-231.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.017 Abstract （34）      PDF（pc） （524KB）（13）       Knowledge map    Save Let $t $ be a nonnegative real number, $S$ be a subset of $V(G)$ and $c(G-S)$ denote the number of components of $G-S$. The graph $G$ is said to be $t$-tough if $|S|\geq t\cdot c(G-S)$ with $c(G-S)\geq 2$ for each vertex set $S$. The toughness is the largest real number $t$ satisfying the above condition. The following result will be proved in this paper. Let $G$ be a $t$-tough graph on $n\geq 3$ vertices with $t\geq 1$. If it holds that $\max \{d(u),d(v)\}>\frac{n}{1+t}+2t-2$ for any two nonadjacent vertices, then $G$ is Hamiltonian. Reference | Related Articles | Metrics | Comments（0）

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Edge colorings of planar graphs without 5-fans XUE Ling, WU Jianliang Operations Research Transactions    2026, 30 (2): 232-236.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.018 Abstract （45）      PDF（pc） （518KB）（16）       Knowledge map    Save A $k$-edge-coloring of a graph is an assignment of colors from a set of $k$ colors to the edges of $G$ such that adjacent edges receive distinct colors. $\chi'(G)$ denotes the smallest $k$ for which $G$ admits such a coloring. It is proved here that if a planar graph $G$ contains no $5$-fan $F_5$ as a subgraph, where $F_5$ is a graph of order $5$ with a vertex $v\in V(F_5)$ such that $d(v)=4$ and $F_5-v$ is a path, then $\chi'(G) \leq \max\{6, \Delta(G)\}$. Reference | Related Articles | Metrics | Comments（0）

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A review of intelligent branch and bound algorithms for mixed integer linear programming problems ZHANG Xuefeng, PENG Xiao, CHEN Liangyu, YANG Zhengfeng, ZENG Zhenbing Operations Research Transactions    2026, 30 (2): 237-270.   DOI: 10.15960/j.cnki.issn.1007-6093.2026.02.019 Abstract （73）      PDF（pc） （1033KB）（36）       Knowledge map    Save Mixed integer linear programming problems are spread across various fields in the real world. Solving mixed integer linear programming problems is an NP-hard problem. Current advanced solvers generally use the branch and bound method as the core framework for solving mixed integer linear programming problems. But the inherent exponential nature of branch and bound means that one wrong decision during its execution can double the size of the search tree and fail to improve the search process. Such a complex and data-rich environment, combined with a lack of formal understanding, makes it possible to leverage machine learning techniques to improve branch and bound algorithms. Therefore, combining data-driven machine learning methods with branch and bound algorithms to improve their decision-making processes has received increasing attention. In this article, we first introduce the branch and bound algorithm and analyze the possible decision-making process therein. Afterwards, the research work on integrating machine learning methods into branch and bound algorithms in recent years is mainly analyzed from two aspects: deep learning methods based on behavioral cloning that imitate existing expert strategies and reinforcement learning methods based on the idea of discovering new strategies. Finally, we discuss possible future directions and challenges in combining machine learning with branch and bound algorithms. Reference | Related Articles | Metrics | Comments（0）