Asia-Pacific Journal of Operational Research

Based on the redistributed technique of bundle methods and the auxiliary problem principle, we present a redistributed bundle method for solving a generalized variational inequality problem which consists of finding a zero point of the sum of two multivalued operators. The considered problem involves a nonsmooth nonconvex function which is difficult to approximate by workable functions. By imitating the properties of lower-$C^2$ functions, we consider approximating the local convexification of the nonconvex function, and the local convexification parameter is modified dynamically in order to make the augmented function produce nonnegative linearization errors. The convergence of the proposed algorithm is discussed when the sequence of stepsizes converges to zero, any weak limit point of the sequence of serious steps $x^k$ is a solution of problem (P) under some conditions. The presented method is the generalization of the convex bundle method [Salmon, G, JJ Strodiot and VH Nguyen (2004). A bundle method for solving variational inequalities. SIAM Journal on Optimization, 14(3), 869–893].

In a framework where both power structure and consumer heterogeneity (brand loyalty and different willingness-to-pay for perceived quality) count, we study six noncooperative games (two Stackerberg and one Nash games with and without a store brand) between a national-brand manufacturer and a store-brand retailer. Our results contribute to the theoretical literature on store brand in four aspects: (a) revealing that the mechanism whereby store brands alleviate double-marginalization problem varies with supply-chain power structures; (b) finding that it is the power manufacturer with a low-loyalty national brand that suffers most from the store brand entry; (c) finding that a category follower retailer is more inclined to introduce its store brand than a power one, and should optimally position its store brand close to the low-loyalty national brand supplied by a power manufacturer; (d) showing that the store brand may lead the manufacturer and the retailer to be trapped in the prisoners’ dilemma.

The multi-vehicle covering tour problem ($m$-CTP) is defined on a graph $G=(V\cup W,E)$, where $V$ is a set of vertices that can be visited and $W$ is a set of vertices that must be covered but cannot be visited. The objective of the $m$-CTP is to obtain a set of total minimum cost tours on subset of $V$, while covering all $v\in W$ by up to $m$ vehicles. In this paper, we first generalize the original $m$-CTP by adding a realistic constraint, and then propose an algorithm for the generalized $m$-CTP using a column generation approach. Computational experiments show that our algorithm performs well and outperforms the existing algorithms.

We investigate a setting in which two retailers sell substitutable products to a market. With the rise of group-buying (GB) consumers, each retailer has the option to open a GB channel to cater to their demand and complement its regular sales. We find that the retailers will forfeit this option to only sell in regular channels when they have scarce capacities, but both open GB channels when they have ample capacities. The situation is more intricate when their capacity levels are intermediate, in which case the equilibrium channel structure is influenced by the substitution effect and the consumers’ channel preference. A higher capacity level does not necessarily incentivize a retailer to open group-buying channel. When given the authority, the retailers under strong competition pressure prefer to commit to regular prices over tailoring regular prices to fit the channel structure. Circumstances exist in which both retailers strategically open GB channels but keep them inactive by inducing all consumers to only buy from regular channels. Moreover, the sequence whereby the retailers follow to set regular prices and decide on GB channels is inconsequential on market equilibrium. Our main findings are robust with respect to model assumptions.

In this paper, the focus is on the role of the municipality, as an enabler of a collaboration between freight forwarders and the municipality in which the consolidation of goods is considered as a means for goods flow improvement in urban freight transportation. We present a cost allocation model that is based on solution concepts from cooperative game theory, for allocating the operational costs associated with the collaboration. It is assumed that the municipality is willing to carry some cost to ensure a stable collaboration for the potential benefits received, e.g., reduced traffic congestion in the city. The model is applied to some illustrative examples, and the cost allocation results are discussed. It is shown that the role of the municipality may be decisive in achieving a stable collaboration between the freight forwarders, and further that the municipality does not necessarily need to contribute to covering the costs.

This paper studies online scheduling on $m$ identical parallel machines under the KRT environment, where jobs arrive over time and “KRT” means that in the online setting no jobs can be released when all of the machines are busy. The goal is to determine a feasible schedule to minimize the total of weighted completion times. When $m=1$, we prove that WSPT is an optimal online algorithm. When $m\ge 2$, we first present a lower bound $\frac{m^2-2+\sqrt{m^4-4m+4}}{2m(m-1)}$, and then show that WSPT is a 2-competitive online algorithm for the case $m=2$. For the case in which $m=2$ and all jobs have equal processing times, we provide a best possible online algorithm with a competitive ratio of $\frac{1+\sqrt{3}}{2}$.

The present paper develops a weighted least-square dissimilarity approach to address the Multiple Criteria ABC inventory classification problem, when the different criteria ranking makes impossible to achieve a group consensus. The proposed approach effectively eliminates drawbacks regarding subjective judgements of multiple decision makers on the criteria importance, and comprehensively aggregates all rankings of the criteria importance to provide a more reasonable and effective classification mechanism. The common weights associated with all rankings are determined. An illustrative example is presented to compare the performance of our approach with the existing studies.

In this paper, we consider the online scheduling of incompatible family jobs with equal length on an unbounded parallel-batch machine with job delivery. The jobs arrive online over time and belong to $f$ incompatible job families, where $f$ is known in advance. The jobs are first processed in batches on an unbounded parallel-batch machine and then the completed jobs are delivered in batches by a vehicle with infinite capacity to their customers. The jobs from distinct families cannot be processed and delivered in the same batch. The objective is to minimize the maximum delivery completion time of the jobs. For this problem, we present an online algorithm with the best competitive ratio of $1+\frac{\sqrt{4f^2+1}-1}{2f}$.

In this paper, we propose a new proof for smoothing homotopy method based on the Fischer–Burmeister function to solve the nonlinear complementarity problem under a nonmonotone solution condition. Under this assumption condition imposed on the defined mapping $F$, global convergence of a smooth curve determined by the referred homotopy equation is established for almost all initial points in $R_{+}^n$ and it is actually regarded as an interior point method. Besides, if the initial point is expanded to $R^n$, the global convergence of the homotopy method is ensured under a similar condition. The numerical results are reported and illustrate that the method is efficient for some nonlinear complementarity problems.

In the present paper, we introduce a pair of multiobjective second-order symmetric variational control programs over cone constraints and derive weak, strong and converse duality theorems under second-order $F$-convexity assumption. Moreover, self-duality theorem is also discussed. Our results extend some of the known results in literature.