Thanks to all discussants for the positive reaction to our paper. We do appreciate the comments—most of which are very complimentary—and the stimulating suggestions for further research. Due to the strict time constraints for finishing this rejoinder we will not be able to give detailed answers to the questions raised by the discussants, but will formulate some thoughts on these questions, including conjectures on results and additional ideas, which may be followed up by the authors of this paper, the discussants or readers of TOP, or—even better—any combination of members of these groups.

The Steiner multi-ring network design problem with revenues consists of designing node-disjoint multiple rings connected by a specific node (hub) and passing through all the nodes with high priority of service and some of the nodes with low priority of service. The number of nodes in each ring has an upper bound to assure a certain level of service. Besides the usual arc link costs, we also consider revenues between each pair of nodes in the same ring, even when they are not connected by a direct link. The objective is to minimize the difference between the total connection cost and total revenue. The problem is a generalization of the problem studied in Gouveia and Pires (Eur J Oper Res 133:21–31, 2001a) and it can also be seen as a combination of variants of two NP-Hard problems, the vehicle routing problem and the maximum edge-weighted clique problem. We introduce and discuss two types of integer linear programming formulations and propose some valid inequalities to strengthen the linear programming relaxation. Computational results are presented to evaluate the quality of the linear programming relaxation bounds associated with these formulations as well as efficiency of the models to obtain the optimal integer solutions.

Pulido et al. (Annals Oper Res 158:133–141, 2008) present an extension of the classical bankruptcy problem (O’Neill in Math Social Sci 2:345–371, 1982) where the involved agents have, apart from the claims vector, an additional reference vector. To analyze this extended problem, they propose the extreme and the diagonal approaches, both of them restricted to the case in which the reference vector is lower than the claims vector. We note that if the claims and the reference vectors are interchanged, the allocation proposed by the extreme approach varies. Therefore, by introducing the idea of impartiality, in the current approach, we propose an extension of their model in which no relation is assumed between the claims and reference vectors.

A parallel computing implementation of a Serial Stochastic Dynamic Programming approach referred to as the S-SDP algorithm is introduced to solve large-scale multiperiod mixed 0–1 optimization problems under uncertainty. The paper presents Inner and Outer Parallelization versions of the S-SDP algorithm, referred to as Inner P-SDP and Outer P-SDP, respectively, so that the problem solving elapsed time and gap reduction is analyzed. The basic idea of Inner P-SDP consists of parallelizing the optimization of variations of the MIP subproblems attached to the sets of scenario clusters created by the modeler-defined stages to decompose the original problem. The Outer P-SDP performs simultaneous interconnected executions of the serial algorithm, so that a wider feasibility area is explored using iterative communication to redefine search directions. Strategies are presented to analyze the performance of parallel computation based on Message-Passing Interface threads to solve stage-related subproblems versus the serial version of SDP methodology. The results of using the parallelization are remarkable, as not only faster but also better solutions than the serial version are obtained. In particular, we report up to 10 times speedup for 12 threads on the Inner P-SDP algorithm. The new approach allows problems to be solved using less computing time than a state-of-the-art MIP solver. It can thus solve very large-scale problems that could not otherwise be achieved by plain use of the solver or by the S-SDP algorithm in acceptable elapsed time, if any.

The support vector machine (SVM) is one of the most popular classification algorithms. In the SVM, the minimum distance from the separating hyperplane to training samples of one class is called margin and the SVM is trained so that the margin is maximized under the constraint that the margin of one class is the same as that of the other. (The resulting separating hyperplane is called optimal separating hyperplane.)

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discrete-time models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.

In this paper, a transportation problem comprising stochastic demands, fixed handling costs at the origins, and fixed costs associated with the links is addressed. It is assumed that uncertainty is adequately captured via a finite set of scenarios. The problem is formulated as a two-stage stochastic program. The goal is to minimize the total cost associated with the selected links plus the expected transportation and fixed handling costs. A prototype problem is initially presented which is then progressively extended to accommodate capacities at the origins and multiple commodities. The results of an extensive set of computational tests are reported and discussed.

Trade credit financing plays a vital role in current business operations. Vendors extend payment dates to encourage sales, and buyers are not required to pay immediately after receiving products. This is equivalent to receiving a price reduction. Besides, buyers reduce their need for capital from bank loans. In addition, a number of defective products are produced during the production process. The number of defective items influences the on-hand inventory levels of buyers, service levels, and frequency of orders. To ensure that the analysis incorporates a realistic production environment, we developed an integrated inventory model with a two-part trade credit and considered an imperfect production process that can be improved by capital investment. The objective was to determine the optimal ordering, shipping, and quality improvement policies to maximize joint total profit. An iterative algorithm was established to determine the optimal strategy. Furthermore, a sensitivity analysis was conducted to examine the effects of changing parameter values on the optimal solution.

In these comments on the excellent survey by Dinh and Jeyakumar, we briefly discuss some recently developed topics and results on applications of extended Farkas’ lemma(s) and related qualification conditions to problems of variational analysis and optimization, which are not fully reflected in the survey. They mainly concern: Lipschitzian stability of feasible solution maps for parameterized semi-infinite and infinite programs with linear and convex inequality constraints indexed by arbitrary sets; optimality conditions for nonsmooth problems involving such constraints; evaluating various subdifferentials of optimal value functions in DC and bilevel infinite programs with applications to Lipschitz continuity of value functions and optimality conditions; calculating and estimating normal cones to feasible solution sets for nonlinear smooth as well as nonsmooth semi-infinite, infinite, and conic programs with deriving necessary optimality conditions for them; calculating coderivatives of normal cone mappings for convex polyhedra in finite and infinite dimensions with applications to robust stability of parameterized variational inequalities. We also give some historical comments on the original Farkas’ papers.

In this paper, we propose a novel autonomous intelligent tool for the optimum design of a wireless relayed communication network deployed over disaster areas. The so-called dynamic relay deployment problem consists of finding the optimum number of deployed relays and their location aimed at simultaneously maximizing the overall number of mobile nodes covered and minimizing the cost of the deployment. In this paper, we extend the problem by considering diverse relay models characterized by different coverage radii and associated costs. To efficiently tackle this problem we derive a novel hybrid scheme comprising: (1) a Harmony Search (HS)-based global search procedure and (2) a modified version of the well-known K-means clustering algorithm as a local search technique. Single- and bi-objective formulations of the algorithm are proposed for emergency and strategic operational planning, respectively. Monte Carlo simulations are run over a emulated scenario based on real statistical data from the Castilla La Mancha region (center of Spain) to show that, in comparison with a standard implementation of the K-means algorithm followed by a exhaustive search procedure over all relay-model combinations, the proposed scheme renders on average better coverage levels and reduced costs providing, at the same time, an intelligent tool capable of simultaneously determining the number and models of the relays to be deployed.