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Table of Content

    15 September 2023, Volume 27 Issue 3
    Optimal reinsurance investment strategies of two competing insurance companies under VaR constraints
    Xinya HE, Ailing GU
    2023, 27(3):  1-20.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.001
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    This paper investigates the optimal reinsurance-investment strategies of two competing insurers under VaR constraints. We assume that the insurers' dynamic surplus processes are described by the classic Cramer-Lundberg (C-L) risk model, in which the premiums are determined by the loss-dependent premium principle. Moreover, the insurers can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset, where the price process of the risky asset is described by the geometric Brownian motion. Firstly, we aim to maximize the expected utility of the insurers' relative terminal wealth and then establish optimization problems with the VaR constraints. In the next, we solve the corresponding constrained optimization problems by using the optimal control theory and the dynamic programming principle. Specially, we get three different Nash equilibrium strategies under exponential utility. Finally, we illustrate the effects of some parameters on the optimal reinsurance strategy and the optimal investment strategy through specific numerical analysis, and find some interesting results.

    A modified inventory pricing model with uncertainty demand
    Ke SU, Xiaohui REN
    2023, 27(3):  21-36.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.002
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    The economic order quantity (EOQ) model seeks the optimal order quantity to minimize the total inventory cost, so as to solve the inventory management problem with independent demand. In this work, a modified inventory pricing model based on the EOQ model is proposed. In the presented model, the original certain demand is replaced by a perturbation demand. Different from the classical EOQ model, the goal of modified model is to find the optimal price maximum the total profit. Because the modified model has perturbation set $\zeta$, it is hard to obtain the optimal solution. We get the optimal solution by solving its robust counterpart model. In the context of low carbon, in the meantime, the government and enterprises must take positive measures to reduce carbon emissions. The carbon tax is added to the modified inventory pricing model in order to reduce carbon emission. Finally, numerical examples is provided to support that enterprises should appropriately reduce carbon emissions in order to obtain more profits under the carbon tax policy.

    A class of inertial symmetric regularization alternating direction method of multipliers for nonconvex two-block optimization
    Jianwen PENG, Hongwang LEI
    2023, 27(3):  37-52.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.003
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    The alternating direction method of multipliers(ADMM) is an valid method for solving separable convex optimization problems, nevertheless, when the objective function has a nonconvex function, ADMM may not converge. This paper proposes an inertial symmetric regularization alternating direction method of multipliers for nonconvex two-block optimization problem with linear equality constraints. Under the appropriate hypothesis conditions, the global convergence of the algorithm is established. Secondly, When the benefit function satisfies the Kurdyka-Łojasiewicz(KL) property, the strong convergence of the algorithm is established. Finally, numerical experiments are performed on the algorithm, and the results show that the algorithm is an effective method.

    Research on city unmanned logistics distribution with simultaneous delivery and pickup
    Yunwei ZHANG, Shuguang HAN
    2023, 27(3):  53-67.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.004
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    With the requirement for the protection of energy saving and emission reduction and also the development of the artificial intelligence, the city logistics distribution that uses unmanned aerial vehicle as distribution device has gradually turned into reality. Considering the battery capacity constraints of device, the charging decision and the pickup and delivery simultaneous demands, a mathematical programming model is constructed with the objective of minimizing the distribution cost (E-VRPSDP). A new branch-cut-and-price algorithm is provided to solve the exact solution based on the column generation method and bi-directional dynamic programming. To solve the large size E-VRPSDP, an improved simulated annealing algorithm is designed. A new operator is introduced as the constructing feasible solution operator which evolves the original solution into a feasible solution, thereby improving the searching ability of the algorithm. Finally, the suitable data examples are generated based on the standard test set to verify the proposed algorithms. The theoretical guidance and algorithm support for city logistics enterprises is provided to develop the driver-less logistics distribution.

    Transit vehicle scheduling model and 3M evolutionary algorithm based on super spatiotemporal network
    Shengxue HE
    2023, 27(3):  68-82.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.005
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    In order to reduce the number of deadheading trips and realize the equality of working hours with the given bus timetable, this paper formulated a super spatiotemporal based transit vehicle scheduling model and designed an evolutionary algorithm including mixed creation, mutation and mature operators. Firstly, this study used the super-network conception to combine pulling out arcs, pulling in arcs, connectors, actual trips and deadheading trips into a super spatiotemporal network. Based on the flow conservation conception in the super-network, this study built a vehicle scheduling model and transformed the constraint of working time equality into an addable item of objective. In view of the topological feature of the set of trip coverings, this paper designed a mixed creation operator to generate new solution with several known feasible solutions. By searching the connectors with loop feature, this paper realized the mutation operator to the elements of feasible solution. The mature operator consists of formulating an assignment network, computing the cost of links in the assignment network, and obtaining the optimal match by Hungarian algorithm. Based on the above three operators, this paper proposed a new "3M" evolutionary algorithm. The numerical analysis verified the rationality of the new model and the effectiveness of the new algorithm. This research discovers that there is a mutual restriction relationship between reducing deadheading trips and equalizing the actual trip times among trip chains, but the both have no obvious connection with the fleet size.

    A two-echelon facility location problem with choice of facility size
    Tingying WU, Yao WANG, Zhili ZHOU, Yating REN
    2023, 27(3):  83-95.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.006
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    The facility location and size are important factors that affect operation cost and service quality of supply chain, and also two decisive factors for enterprises to gain competitive advantage. In order to optimize facility location and size simultaneously, a mixed integer programming model is formulated to minimize the total costs, and to decide the location of plants and depots, select sizes for the plants, determine the product flows from the plants to the depots and assign the customers to the depots. According to characteristics of the problem model, a Lagrangian relaxation algorithm is designed to solve the problem, and a hybrid simulated annealing tabu search algorithm is developed to further improve the solution quality. To test the validity of the proposed algorithm, a large number of randomly generated instances of different sizes and parameters are provided. The numerical results indicate that the proposed algorithm is effective and efficient for the two-echelon facility location problem with choice of facility size.

    Two new accelerated proximal gradient algorithms for Toeplitz matrix completion
    Chuanlong WANG, Jianhua NIU, Qianying SHEN
    2023, 27(3):  96-108.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.007
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    In this paper, we propose two modified accelerated proximal gradient algorithms for Toeplitz matrix completion in which the iterative matrices keep the Toeplitz structure in each step to decrease SVD times. Furthermore, we prove the convergence of the new algorithms under some reasonal conditions. Finally, numerical experiments show that the new algorithms are much more effective than the accelerated proximal gradient (APG) algorithm for Toeplitz matrix completion in CPU times.

    Analysis of waiting time of customers in an M/M/c/m + c queueing system with customer interjections
    Wenqing WU, Qilin KE, Yinghui TANG, Lin CHEN
    2023, 27(3):  109-120.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.008
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    This paper studies waiting time distributions of customers in a multi-server queueing system with customer interjections and finite buffer. The customers who enter the system are divided into regular customers and interjection customers according to whether they interject the queue or not. After entering the system, the regular customers queue up at the end of the waiting line and wait for service, while the interjection customers queue up as close as possible to the head of the waiting line to receive service. By using the properties of negative exponential distribution and phase type distribution, the matrix expressions of waiting time distribution of customers in waiting queue position $n$, regular customers and interjection customers are derived. Further, the plots of waiting time distributions with time $t$ are given.

    Stability of solutions for a class of non-convex vector optimization problems with mapping differences
    Jing ZENG, Ruowen DING
    2023, 27(3):  121-128.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.009
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    The data of problem are often perturbed in real life. We often calculate the solution of a perturbed problem to approximate the original problem solution. Therefore, the stability of the solution set of the original problem is an important issue. In this paper, we consider a class of non-convex vector optimization problems with two mapping differences. By taking advantage of appropriate convergence and convexity of the two mappings, the stability results of the nonconvex vector optimization problem is obtained, when the approximate problem data converge to the original problem data in the sense of Painlevé-Kuratowski's convergence.

    The method for a class of Nash equilibrium game
    Jian HOU, Mengmeng LI, Zhu WEN
    2023, 27(3):  129-136.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.010
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    The Nash equilibrium game has been applied to many fields, the algorithm for the problem has attracted much attention in recent years. But since the Nash equilibrium game is a complex system composed by a series of optimization problems, the standard optimization method cannot be directly used for the problem, this leads to the difficulty of solving it. For a class of Nash equilibrium games with strongly convex payoff functions, using the Nikaido-Isoda function to transform the Nash equilibrium game into a smooth constraint optimization problem is an effective method to solve the Nash equilibrium game. Based on the gradient strongly monotonity of the Nash equilibrium payoff function, we present a Nikaido-Isoda algorithm for the game and the global convergence of the algorithm is proved. Finally, by solving two kinds of standard Nash equilibrium problems, the feasibility and effectiveness of the Nikaido-Isoda algorithm are verified.

    Unrelated parallel-machine scheduling with deteriorating maintenance activities and job rejection
    Jie GAO, Juan ZOU, Yukang SUI, Yuzhong ZHANG
    2023, 27(3):  137-149.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.011
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    We study unrelated parallel-machine scheduling problems with deteriorating maintenance activities and job rejection. Each machine has at most one deteriorating maintenance activity throughout the scheduling horizon. The duration of the maintenance activity increases linearly with its starting time. Each job is either processed and it incurs a production cost, or is rejected with a penalty cost. The objective is to find the position of the maintenance activity of each machine and the sequence of the accepted jobs such that the scheduling cost of all accepted jobs plus the total cost of rejecting and processing jobs is minimized. When the scheduling cost is the makespan, we provide an approximation algorithm with worst-case ratio of 2. When the scheduling costs are the total completion time, the total machine load and the total absolute deviation of completion times, we show that the three versions of the scheduling problem can be optimally solved in polynomial time.

    On the largest root of the matching polynomials of (n, m)-graphs without even cycles
    Ling YUAN, Wenhuan WANG
    2023, 27(3):  150-158.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.012
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    Let $G$ be a simple connected graph with $n$ vertices. The matching polynomial of $G$ is given by $\sum_{k=0}^{[n/2]}(-1)^km(G, k)x^{n-2k}$, where $m(G, k)$ is the number of $k$-matchings in $G$ with $0\leq k\leq [n/2]$. Let $\Phi_{n, m}$ be the set of graphs with $n$ vertices and $m$ edges having no even cycles, where $n\leq m\leq \frac{3(n-1)}{2}$. In this paper, four new transformations for comparing the largest roots of matching polynomials are introduced and the graph with the largest root of matching polynomial is characterized among graphs in $\Phi_{n, m}$.

    The Banzhaf value for hypergraph communication situations
    Wenrong LYU, Erfang SHAN
    2023, 27(3):  159-168.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.013
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    Alonso-Meijide and Fiestras-Janeiro(2006) introduced TU games with restricted cooperative structure represented by an undirected graph, or simple graph games, and present the Banzhaf value of the graph game, that extend the Banzhaf value. In this paper, we first generalize the Banzhaf value to the hypergraph game, define the Banzhaf value of the hypergraph game. Secondly, we prove that the Banzhaf value of the hypergraph game satisfies the property of component decomposability, component total contribution, fairness, balanced contribution, and isolation, and propose two characterizations of this value. Finally, we give an example to illustrate the properties satisfied by the Banzhaf value of the hypergraph game.

    Sufficient conditions on Wiener index and Harary index for pancyclic graphs
    Huicai JIA, Hongye SONG
    2023, 27(3):  169-177.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.014
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    Let G be a simple and connected graph. A graph G is pancyclic if it contains all possible cycles with length from three to the number of vertices. In this paper, we provide sufficient conditions for a graph to be pancyclic in terms of the Wiener index, the Harary index, the distance spectral radius and the Harary spectral radius of a graph, respectively. These results establish the relationship between algebraic properties and structural properties of graphs.

    Extremal problems for Steiner Wiener index of unicyclic graphs
    Jie ZHANG, Yan JI
    2023, 27(3):  178-184.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.015
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    Wiener index is an important chemical index in chemical graph theory, defined as the sum of distances between all pairs of vertices. A generalization of the Wiener index, called the Steiner Wiener index, takes the sum of the Steiner distances over all sets S of cardinality k. The Steiner distance of vertices in a set S is the minimum size of a connected subgraph that contain these vertices. We consider the extremal problems with respect to the Steiner Wiener index among all unicyclic graphs.

    Some remarks about dominating number of network
    Jianxiu HAO
    2023, 27(3):  185-190.  doi:10.15960/j.cnki.issn.1007-6093.2023.03.016
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    (d, w) -dominating number is an important measuring parameter for the reliability of sharing common source in a network. (1, 1) -dominating number is also known as dominating number which is a classical parameter in graph theory. (d, w) -dominating number is a simple generalization of (1, 1) -dominating number. In this paper we present a lower bound and an upper bound for the calculating of (1, w) -dominating number. Using these two bounds, we find the (1, n - 1)-dominating number and (1, n)-dominating number for hypercube, we find the (1, 2n - 1) -dominating number and (1, 2n)-dominating number for 4-ary n-cube, and the (1, n)-dominating number for n dimensional folded hypercube.