The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, we address the primal-dual column generation technique, which relies on well-centred non-optimal solutions of the restricted master problems that are obtained by a primal-dual interior point method. Although good computational results are reported for this technique, it was only applied in a particular class of problems. Moreover, no theoretical analysis to guarantee its convergence is available. Here, we further investigate the primaldual column generation technique and present extensive computational experiments in the context of integer programming, where column generation schemes are widely employed. The results show that the primal-dual technique usually leads to substantial reductions in the number of iterations as well as less running time when compared to the classical and also analytic centre approaches.

Extreme events such as disasters cause partial or total disruption of basic services such as water, energy, communication and transportation. In particular, roads can be damaged or blocked by debris, thereby obstructing access to certain aected areas. Thus, restoration of the damaged roads is necessary to evacuate victims and distribute emergency commodities to relief centers or aected areas. The Crew Scheduling and Routing Problem (CSRP) addresses decisions in postdisaster situations with the aim of minimizing the time that aected areas remain inaccessible. The integration of crew scheduling and routing decisions makes this problem too complicated to be eectively solved for practical instances using mixed integer programming (MIP) formulations recently proposed in the literature. Therefore, we propose a branch-and-Benders-cut (BBC) algorithm that decomposes the integrated problem into a master problem (MP) with scheduling decisions and subproblems with routing decisions. Computational tests based on instances from the literature show that the proposed exact method improves the results of MIP formulations and other exact and metaheuristic methods proposed in literature. The BBC algorithm provides feasible solutions and optimality gaps for instances that thus far have not been possible to solve by exact methods in the literature.

Branch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the Vehicle Routing Problem with Time Windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm.

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The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.

This paper addresses the inventory routing problem (IRP), which consists in defining the customer visit schedule, the delivery quantities, and the vehicle routing plan to meet the demands of a set of customers over a given time horizon. We consider the variant with a single item, a single supplier, multiple vehicles, and a finite multiperiod planning horizon, minimizing the sum of inventory and travel costs. In addition, we address an alternative objective function that minimizes the logistic ratio, defined as the total travel cost divided by the total quantity delivered to customers. This second objective function, while more realistic in some logistics settings, poses a challenge for integer programming models and exact methods because of its nonlinearity. To our knowledge, no heuristic method has been proposed to address this objective in the IRP variant addressed in this paper. To solve this problem with each of these objective functions, we propose effective metaheuristic algorithms based on iterated local search and simulated annealing. Computational experiments show that these algorithms provide reasonably high-quality solutions in relatively short running times for both objective functions when compared to other methods for well-known instances from the literature. Moreover, the algorithms produce new best solutions for some of these instances.

In this paper, we address the constrained two‐dimensional rectangular guillotine single large placement problem (2D_R_CG_SLOPP). This problem involves cutting a rectangular object to produce smaller rectangular items from orthogonal guillotine cuts. In addition, there is an upper limit on the number of copies that can be produced of each item type. To model this problem, we propose a new pseudopolynomial integer nonlinear programming (INLP) formulation and obtain an equivalent integer linear programming (ILP) formulation from it. Additionally, we developed a procedure to reduce the numbers of variables and constraints of the integer linear programming (ILP) formulation, without loss of optimality. From the ILP formulation, we derive two new pseudopolynomial models for particular cases of the 2D_R_CG_SLOPP, which consider only two‐staged or one‐group patterns. Finally, as a specific solution method for the 2D_R_CG_SLOPP, we apply Benders decomposition to the proposed ILP formulation and develop a branch‐and‐Benders‐cut algorithm. All proposed approaches are evaluated through computational experiments using benchmark instances and compared with other formulations available in the literature. The results show that the new formulations are appropriate in scenarios characterized by few item types that are large with respect to the object's dimensions.

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem's objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution-VSS-and the expected value of perfect information-EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-andsee and expected value approaches, even under small variations in the parameters of the problem.In the one-dimensional cutting stock problem, a stock of available objects, such as bars, bobbins and reels must be cut into smaller items in order to exactly meet the demand, while at-