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Table of Content

    30 March 2021, Volume 9 Issue 1
    Two-Stage Stochastic Variational Inequalities: Theory, Algorithms and Applications
    Hai-Lin Sun, Xiao-Jun Chen
    2021, 9(1):  1-32.  doi:10.1007/s40305-019-00267-8
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    The stochastic variational inequality (SVI) provides a unified form of optimality conditions of stochastic optimization and stochastic games which have wide applications in science, engineering, economics and finance. In the recent two decades, one-stage SVI has been studied extensively and widely used in modeling equilibrium problems under uncertainty. Moreover, the recently proposed two-stage SVI and multistage SVI can be applied to the case when the decision makers want to make decisions at different stages in a stochastic environment. The two-stage SVI is a foundation of multistage SVI, which is to find a pair of “here-and-now” solution and “wait-and-see” solution. This paper provides a survey of recent developments in analysis, algorithms and applications of the two-stage SVI.
    Multi-Objective Vendor Selection Problem of Supply Chain Management Under Fuzzy Environment
    Srikant Gupta, Irfan Ali, Aquil Ahmed
    2021, 9(1):  33-62.  doi:10.1007/s40305-018-0226-2
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    Survival of a company in today’s competitive business environment depends mainly on its supply chain. An adequate supply chain gives a competitive edge to a company. Sourcing, which is the initial stage of a supply chain, can be made efficient by making an appropriate selection of vendors. Appropriate vendor selection results not only in reduced purchasing costs, decreased production lead time, increased customer satisfaction but also in improved corporate competitiveness. In general, the vendor selection problem is a multi-objective decision-making problem that involves some quantitative and qualitative factors. So, we have considered a multi-objective vendor selection problem (MOVSP) with three multiple objective goals: minimization of net ordering price, minimization of rejected units and minimization of late delivered units. In most of the cases, information about the price of a unit, percentage of rejected units, percentage of late delivered units, vendor rating value and vendor quota flexibility may not be known precisely due to some reasons. In this paper, imprecision in input information is handled by the concept of a simulation technique, where the parameter follows the uniform distribution. Deterministic, stochastic, α-cut and ranking function approaches are used to get the crisp value of the simulated data sets. The four different algorithms, namely—fuzzy programming, goal programming, lexicographic goal programming and D1-distance algorithm, have been used for solving the MOVSP. In last, three different types of simulated data sets have been used to illustrate the work.
    Scheduling Jobs with Release and Delivery Times Subject to Nested Eligibility Constraints
    Yu-Zhong Zhang, Shu-Guang Li
    2021, 9(1):  63-77.  doi:10.1007/s40305-019-00268-7
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    The problem of scheduling jobs with release and delivery time subject to machine eligibility constraints is considered. The eligible sets of the jobs are nested, and preemptions are not allowed. The goal is to minimize the maximum delivery completion time, i.e., the time by which all jobs are delivered. For the special case of equal release time, a 2-approximation algorithm is presented whose running time depends linearly on the number of jobs. For the general case of unequal release time, a polynomial time approximation scheme is derived.
    Continuity of Solutions for Parametric Set Optimization Problems via Scalarization Methods
    Pei-Pei Liu, Hong-Zhi Wei, Chun-Rong Chen, Sheng-Jie Li
    2021, 9(1):  79-97.  doi:10.1007/s40305-018-0230-6
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    The aim of this paper is to investigate the continuity of solution mappings for parametric set optimization problems with upper and lower set less order relations by scalarization methods. First, we recall some linear and nonlinear scalarization properties used to characterize the set order relations. Subsequently, we introduce new monotonicity concepts of the set-valued mapping by linear and nonlinear scalarization methods. Finally, we obtain the semicontinuity and closedness of solution mappings for parametric set optimization problems (both convex and nonconvex cases) under the monotonicity assumption and other suitable conditions. The results achieved do not impose the continuity of the set-valued objective mapping, which are obviously different from the related ones in the literature.
    Optimal Algorithms for Integer Inverse Undesirable p-Median Location Problems on Weighted Extended Star Networks
    Esmaeil Afrashteh, Behrooz Alizadeh, Fahimeh Baroughi
    2021, 9(1):  99-117.  doi:10.1007/s40305-018-0229-z
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    This paper is concerned with the problem of modifying the edge lengths of a weighted extended star network with n vertices by integer amounts at the minimum total cost subject to be given modification bounds so that a set of p prespecified vertices becomes an undesirable p-median location on the perturbed network. We call this problem as the integer inverse undesirable p-median location model. Exact combinatorial algorithms with $\mathcal{O}({p^2}\;n\;\log \;n)$ and $\mathcal{O}({p^2}(n\;\log \;n + n\;\log \;\eta {}_{\max }))$ running times are proposed for solving the problem under the weighted rectilinear and weighted Chebyshev norms, respectively. Furthermore, it is shown that the problem under the weighted sum-type Hamming distance with uniform modification bounds can be solved in $\mathcal{O}({p^2}\;n\;\log \;n)$ time.
    Sufficient Conditions for Maximally Edge-Connected Hypergraphs
    Lin-Ken Tong, Er-Fang Shan
    2021, 9(1):  119-129.  doi:10.1007/s40305-018-0224-4
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    The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected. A graph or hypergraph is called maximally edge-connected if the edge-connectivity equals its minimum degree. In this paper, we show that some classical sufficient conditions for graphs to be maximally edge-connected can be generalized to hypergraphs.
    Bi-level Programming for Stackelberg Game with Intuitionistic Fuzzy Number: a Ranking Approach
    Sumit Kumar Maiti, Sankar Kumar Roy
    2021, 9(1):  131-149.  doi:10.1007/s40305-018-0234-2
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    This paper introduces a ranking function procedure on a bi-level programming for Stackelberg game involving intuitionistic fuzzy parameters. Intuitionistic fuzzy number is considered in many real-life situations, so it makes perfect sense to address decision-making problem by using some specified intuitionistic fuzzy numbers. In this paper, intuitionistic fuzziness is characterized by a normal generalized triangular intuitionistic fuzzy number. A defuzzification method is introduced based on the proportional probability density function associated with the corresponding membership function, as well as the complement of non-membership function. Using the proposed ranking technique, a methodology is presented for solving bi-level programming for Stackelberg game. An application example is provided to demonstrate the applicability of the proposed methodology, and the achieved results are compared with the existing methods.
    Well-Posedness and Structural Stability on a System of Simultaneous Generalized Vector Quasi-Equilibrium Problems
    Xi-Cai Deng, Wen-Sheng Jia, Yan-Long Yang
    2021, 9(1):  151-161.  doi:10.1007/s40305-018-0228-0
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    In this paper, we establish the stable results for a system of simultaneous generalized vector quasi-equilibrium problems (SSGVQEP) by using its bounded rationality model. Under the abstract frame, a unified well-posedness on Hadamard types and Tikhonov types well-posedness for SSGVQEP is introduced. Moreover, sufficient condition for the well-posedness of SSGVQEP is given. Finally, we prove that the majority (in Baire category sense) of SSGVQEP is structural stability.
    Optimal Stopping Time of a Portfolio Selection Problem with Multi-assets
    Xian-Ping Wu, Seakweng Vong, Wen-Xin Zhou
    2021, 9(1):  163-179.  doi:10.1007/s40305-018-0223-5
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    In this work, we study a right time for an investor to stop the investment among multiassets over a given investment horizon so as to obtain maximum profit. We formulate it to a two-stage problem. The main problem is not a standard optimal stopping problem due to the non-adapted term in the objective function, and we turn it to a standard one by stochastic analysis. The subproblem with control variable in the drift and volatility terms is solved first via stochastic control method. A numerical example is presented to illustrate the efficiency of the theoretical results.
    Linear Arboricity of Outer-1-Planar Graphs
    Xin Zhang, Bi Li
    2021, 9(1):  181-193.  doi:10.1007/s40305-019-00243-2
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    A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. Zhang et al. (Edge covering pseudoouterplanar graphs with forests, Discrete Math 312:2788-2799, 2012; MR2945171) proved that the linear arboricity of every outer-1-planar graph with maximum degree △ is exactly 「△/2」 provided that △ = 3 or △ ≥ 5 and claimed that there are outer-1-planar graphs with maximum degree △ = 4 and linear arboricity 「(△ + 1)/2」 = 3. It is shown in this paper that the linear arboricity of every outer-1-planar graph with maximum degree 4 is exactly 2 provided that it admits an outer-1-planar drawing with crossing distance at least 1 and crossing width at least 2, and moreover, none of the above constraints on the crossing distance and crossing width can be removed. Besides, a polynomial-time algorithm for constructing a path-2-coloring (i.e., an edge 2-coloring such that each color class induces a linear forest, a disjoint union of paths) of such an outer-1-planar drawing is given.
    Generalized Krasnoselskii–Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces
    You-Cai Zhang, Ke Guo, Tao Wang
    2021, 9(1):  195-206.  doi:10.1007/s40305-018-0235-1
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    The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings, and it is well known that the classic Krasnoselskii-Mann iteration is weakly convergent in Hilbert spaces. The weak convergence is also known even in Banach spaces. Recently, Kanzow and Shehu proposed a generalized Krasnoselskii-Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces. In this paper, we show that the generalized Krasnoselskii-Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces. As applications, we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.
    Binary Operations for the Lattice Structure in a Many-to-Many Matching Model
    Paola Belén Manasero
    2021, 9(1):  207-228.  doi:10.1007/s40305-019-00246-z
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    The lattice structure of the set of stable matchings in many-to-many matching model is well known in literature. If preferences of the agents are substitutable, this result can be obtained by fixed-point methods, for that purpose an algorithm for finding a fixed-point matching is defined. Since the fixed-point set equals the set of stable matchings, the latter has a lattice structure too. In this paper, we consider a many-to-many matching market where the preferences of firms satisfy substitutability and the law of aggregate demand, and workers have responsive preferences. In this many-to-many matching market, we explicitly compute for any pair of stable matchings the least upper bound and the greatest lower bound, without using fixed-point methods.