Table of Content

30 December 2021, Volume 9 Issue 4

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Competitive Equilibria and Benefit Distributions of Population Production Economies with External Increasing Returns

Zhe Yang, Xian Zhang

2021, 9(4):  723-740.  doi:10.1007/s40305-021-00340-1

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Inspired by the work of population games, we establish the model of population production economies with external increasing returns and introduce the notion of competitive equilibria. We first prove the existence of competitive equilibria under some regular assumptions. Furthermore, we assume that there exists the cooperative behavior of different populations. By proving the existence of transferable utility (TU) core, we analyze the benefit distributions of population production economies with external increasing returns.

Coopetition Between TTA and OTA Based on Multinomial Logit Choice Model

Hui-Li Yan, Hao Xiong

2021, 9(4):  741-756.  doi:10.1007/s40305-021-00343-y

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This paper proposes a framework to analyse the impact of online travel agency (OTA) when it steps into an original market of a traditional travel agency (TTA). Based on the multinomial logit choice model, the demand model and the profit model are presented. Then, the demand squeeze, the total demand increase and the cooperation range of wholesale price are analysed. From the analysis, the results indicate that: (1) OTA can increase the demand of the whole market while it squeezes the demand of TTA; (2) The demand squeeze, total demand increase and the range of cooperation wholesale price are all positive with the perceived value from OTA and negative with the perceived value from TTA. (3) The more immature the market is the more necessary for TTA to cooperate with OTA. In addition, numerical example and sensitivity analysis of perceived value and price are presented to illustrate the demand squeeze, demand increase and cooperation range of wholesale price.

Modelling Economic Order Quantities, Considering Buy and Repair Options for Defective Items, and Allowing for Shortages and Inspection Errors

Harun Öztürk

2021, 9(4):  757-795.  doi:10.1007/s40305-021-00339-8

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Nowadays, it is common for suppliers to be separated from their buyers by great distances. This means that, if some items in the lot turn out to be defective, the distance makes it uneconomical to order replacements from the original supplier. Moreover, both Type I and Type Ⅱ errors may occur in the screening process for eliminating defective items and the combination of defective items and inspection errors may lead to shortages. Working with these assumptions, this paper develops two distinct models. Under the first model, defective items are repaired by a local repair shop subject to a repair charge and a mark-up margin. Under the second model, a local supplier replaces the defective with good ones, but at a higher cost. The expected total profit per cycle is developed, together with the expected cycle time, and, employing the renewal reward theorem, the objective function is derived, from which the optimum values are obtained for the order and shortage quantities. The paper presents numerical results and a discussion for both models. The study finds that repairing defective items generally leads to greater total profit than purchasing local replacements.

A Levenberg–Marquardt Method for Solving the Tensor Split Feasibility Problem

Yu-Xuan Jin, Jin-Ling Zhao

2021, 9(4):  797-817.  doi:10.1007/s40305-020-00337-2

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This paper considers the tensor split feasibility problem. Let C and Q be non-empty closed convex set and $\mathcal{A}$ be a semi-symmetric tensor. The tensor split feasibility problem is to find x ∈ C such that $\mathcal{A} x^{m-1} \in Q$. If we simply take this problem as a special case of the nonlinear split feasibility problem, then we can directly get a projection method to solve it. However, applying this kind of projection method to solve the tensor split feasibility problem is not so efficient. So we propose a Levenberg– Marquardt method to achieve higher efficiency. Theoretical analyses are conducted, and some preliminary numerical results show that the Levenberg–Marquardt method has advantage over the common projection method.

Disruption Recovery at Airports: Ground Holding, Curfew Restrictions and an Approximation Algorithm

Prabhu Manyem

2021, 9(4):  819-852.  doi:10.1007/s40305-020-00338-1

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We study disruptions at a major airport. Disruptions could be caused by bad weather, for example. Our study is from the perspective of the airport, the air services provider (such as air traffic control) and the travelling public, rather than from the perspective of a single airline. Disruptions cause flights to be subjected to ground holding, or they cause the flights to violate airport curfew hours. We consider curfew and arrival capacities applicable at a single airport. After proving that the problem is NP-hard, we present a polynomial time approximation algorithm based on the primal–dual schema and show that if the problem is feasible, the algorithm finds a feasible solution that is both within a certain additive bound and within a certain multiplicative factor of the optimal solution. The algorithm returns a solution mix of which flights suffer no delay, which ones to be ground-held and which ones may violate the curfew (and hence pay a curfew penalty). Computational results are positive; our heuristic outperforms the integer programming solver by a wide margin.

Optimized Filling of a Given Cuboid with Spherical Powders for Additive Manufacturing

Zoya Duriagina, Igor Lemishka, Igor Litvinchev, Jose Antonio Marmolejo, Alexander Pankratov, Tatiana Romanova, Georgy Yaskov

2021, 9(4):  853-868.  doi:10.1007/s40305-020-00314-9

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In additive manufacturing (also known as 3D printing), a layer-by-layer buildup process is used for manufacturing parts. Modern laser 3D printers can work with various materials including metal powders. In particular, mixing various-sized spherical powders of titanium alloys is considered most promising for the aerospace industry. To achieve desired mechanical properties of the final product, it is necessary to maintain a certain proportional ratio between different powder fractions. In this paper, a modeling approach for filling up a rectangular 3D volume by unequal spheres in a layer-by-layer manner is proposed. A relative number of spheres of a given radius (relative frequency) are known and have to be fulfilled in the final packing. A fast heuristic has been developed to solve this special packing problem. Numerical results are compared with experimental findings for titanium alloy spherical powders. The relative frequencies obtained by using the imposed algorithm are very close to those obtained by the experiment. This provides an opportunity for using a cheap numerical modeling instead of expensive experimental study.

A General Jury Theorem on Group Decision Making

Yu-Da Hu

2021, 9(4):  869-881.  doi:10.1007/s40305-020-00330-9

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This paper established a general jury theorem on group decision making where the probabilities of the individuals in making correct choice between two alternatives can be different. And we proved that the higher the probability of any decision maker in the group correctly choosing between two alternatives, the higher the probability of the group correctly choosing the same two alternatives. The general jury theorem also indicates that given two groups of individuals with the same average probability of making the correct choice, the one with a more varied or diverse distribution of probabilities will have a higher probability of making the correct choice. In particular, we proved that as the number of decision makers in the group increases to infinity, this probability tends to the limit 1. The general jury theorem presented in this paper substantially generalizes the well-known Condorcet jury theorem in the group decision making theory, which has not been generalized for 200 years until now.

Approximation Algorithms on k-Cycle Transversal and k-Clique Transversal

Zhong-Zheng Tang, Zhuo Diao

2021, 9(4):  883-892.  doi:10.1007/s40305-020-00335-4

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Given a weighted graph $G=(V, E)$ with weight $w: E \rightarrow \mathbb{Z}^{+}$, a $k$-cycle transversal is an edge subset $A$ of $E$ such that $G-A$ has no $k$-cycle. The minimum weight of $k$ cycle transversal is the weighted transversal number on $k$-cycle, denoted by $\tau_{k}\left(G_{w}\right)$. In this paper, we design a $(k-1 / 2)$-approximation algorithm for the weighted transversal number on $k$-cycle when $k$ is odd. Given a weighted graph $G=(V, E)$ with weight $w: E \rightarrow \mathbb{Z}^{+}$, a $k$-clique transversal is an edge subset $A$ of $E$ such that $G-A$ has no $k$-clique. The minimum weight of $k$-clique transversal is the weighted transversal number on $k$-clique, denoted by $\widetilde{\tau_{k}}\left(G_{w}\right)$. In this paper, we design a $\left(k^{2}-k-1\right) / 2-$ approximation algorithm for the weighted transversal number on $k$-clique. Last, we discuss the relationship between $k$-clique covering and $k$-clique packing in complete graph $K_{n}$.

Nonuniqueness of Solutions of a Class of ℓ₀-minimization Problems

Jia-Liang Xu

2021, 9(4):  893-908.  doi:10.1007/s40305-020-00336-3

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Recently, finding the sparsest solution of an underdetermined linear system has become an important request in many areas such as compressed sensing, image processing, statistical learning, and data sparse approximation. In this paper, we study some theoretical properties of the solutions to a general class of $\ell_{0}$-minimization problems, which can be used to deal with many practical applications. We establish some necessary conditions for a point being the sparsest solution to this class of problems, and we also characterize the conditions for the multiplicity of the sparsest solutions to the problem. Finally, we discuss certain conditions for the boundedness of the solution set of this class of problems.

Unbounded Serial-Batching Scheduling on Hierarchical Optimization

Cheng He, Hao Lin, Li Li

2021, 9(4):  909-914.  doi:10.1007/s40305-020-00334-5

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The paper considers a serial-batching scheduling problem on hierarchical optimization with two regular maximum costs, where hierarchical optimization means the primary objective function is minimized, and keeping the minimum value of the primary objective function, the secondary objective function is also minimized. In serial-batching machine environment, the machine processes the jobs in batch, and the jobs in the identical batch are processed by entering into the machine together and leaving the machine together. The time taken to process a batch amounts to the total processing time of the jobs in the batch. Moreover, a fixed switching time s is inserted when a machine begins to process a new batch. We only study the unbounded model, i.e., the batch capacity is unbounded. We give an algorithm that can solve the hierarchical optimization problem in $O\left(n^{4}\right)$ time, where n denotes the number of jobs.