Marcus Poggi

Improved branch-cut-and-price for capacitated vehicle routing

The best performing exact algorithms for the capacitated vehicle routing problem developed in the last 10 years are based in the combination of cut and column generation. Some authors only used cuts expressed over the variables of the original formulation, in order to keep the pricing subproblem relatively easy. Other authors could reduce the duality gaps by also using a restricted number of cuts over the master LP variables, stopping when the pricing becomes prohibitively hard. A particularly effective family of such cuts are the subset row cuts. This work introduces a technique for greatly reducing the impact on the pricing of these cuts, thus allowing much more cuts to be added. The newly proposed branch-cut-and-price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.

Improved bounds for large scale capacitated arc routing problem

The Capacitated Arc Routing Problem (CARP) stands among the hardest combinatorial problems to solve or to find high quality solutions. This becomes even more true when dealing with large instances. This paper investigates methods to improve on lower and upper bounds of instances on graphs with over 200 vertices and 300 edges, dimensions that, today, can be considered of large scale. On the lower bound side, we propose to explore the speed of a dual ascent heuristic to generate capacity cuts. These cuts are next improved with a new exact separation enchained to the linear program resolution that follows the dual heuristic. On the upper bound, we implement a modified Iterated Local Search procedure to Capacitated Vehicle Routing Problem (CVRP) instances obtained by applying a transformation from the CARP original instances. Computational experiments were carried out on the set of large instances generated by Brandao and Eglese and also on the regular size sets. The experiments on the latter allow for evaluating the quality of the proposed solution approaches, while those on the former present improved lower and upper bounds for all instances of the corresponding set.

Harvest planning in the Brazilian sugar cane industry via mixed integer programming

This work addresses harvest planning problems that arise in the production of sugar and alcohol from sugar cane in Brazil. The planning is performed for two planning horizons, tactical and operational planning, such that the total sugar content in the harvested cane is maximized. The tactical planning comprises the entire harvest season that averages seven months. The operational planning considers a horizon from seven to thirty days. Both problems are solved by mixed integer programming. The tactical planning is well handled. The model for the operational planning extends the one for the tactical planning and is presented in detail. Valid inequalities are introduced and three techniques are proposed to speed up finding quality solutions. These include pre-processing by grouping and filtering the distance matrix between fields, hot starting with construction heuristic solutions, and dividing and sequentially solving the resulting MIP program. Experiments are run over a set of real world and artificial instances. A case study illustrates the benefits of the proposed planning.

New benchmark instances for the Capacitated Vehicle Routing Problem

The recent research on the CVRP is being slowed down by the lack of a good set of benchmark instances. The existing sets suffer from at least one of the following drawbacks: (i) became too easy for current algorithms; (ii) are too artificial; (iii) are too homogeneous, not covering the wide range of characteristics found in real applications. We propose a new set of 100 instances ranging from 100 to 1000 customers, designed in order to provide a more comprehensive and balanced experimental setting. Moreover, the same generating scheme was also used to provide an extended benchmark of 600 instances. In addition to having a greater discriminating ability to identify “which algorithm is better”, these new benchmarks should also allow for a deeper statistical analysis of the performance of an algorithm. In particular, they will enable one to investigate how the characteristics of an instance affect its performance. We report such an analysis on state-of-the-art exact and heuristic methods.

Efficient elementary and restricted non-elementary route pricing

Column generation is involved in the current most efficient approaches to routing problems. Set partitioning formulations model routing problems by considering all possible routes and selecting a subset that visits all customers. These formulations often produce tight lower bounds and require column generation for their pricing step. The bounds in the resulting branch-and-price are tighter when elementary routes are considered, but this approach leads to a more difficult pricing problem. Balancing the pricing with route relaxations has become crucial for the efficiency of the branch-and-price for routing problems. Recently, the ng-routes relaxation was proposed as a compromise between elementary and non-elementary routes. The ng-routes are non-elementary routes with the restriction that when following a customer, the route is not allowed to visit another customer that was visited before if they belong to a dynamically computed set. The larger the size of these sets, the closer the ng-route is to an elementary route. This work presents an efficient pricing algorithm for ng-routes and extends this algorithm for elementary routes. Therefore, we address the Shortest Path Problem with Resource Constraint (SPPRC) and the Elementary Shortest Path Problem with Resource Constraint (ESPPRC). The proposed algorithm combines the Decremental State-Space Relaxation technique (DSSR) with completion bounds. We apply this algorithm for the Generalized Vehicle Routing Problem (GVRP) and for the Capacitated Vehicle Routing Problem (CVRP), demonstrating that it is able to price elementary routes for instances up to 200 customers, a result that doubles the size of the ESPPRC instances solved to date.

Improved Branch-Cut-and-Price for Capacitated Vehicle Routing

The best performing exact algorithms for the Capacitated Vehicle Routing Problem are based on the combination of cut and column generation. Some authors could obtain reduced duality gaps by also using a restricted number of cuts over the Master LP variables, stopping separation before the pricing becomes prohibitively hard. This work introduces a technique for greatly decreasing the impact on the pricing of the Subset Row Cuts, thus allowing much more such cuts to be added. The newly proposed Branch-Cut-and-Price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.

A branch-cut-and-price algorithm for the capacitated arc routing problem

Arc routing problems are among the most challenging combinatorial optimization problems. We tackle the Capacitated Arc Routing Problem where demands are spread over a subset of the edges of a given graph, called the required edge set. Costs for traversing edges, demands on the required ones and the capacity of the available identical vehicles at a vertex depot are given. Routes that collect all the demands at minimum cost are sought. In this work, we devise a Branch-Cut-and-Price algorithm for the Capacitated Arc Routing problem using a column generation which generates non-elementary routes (usually called q-routes) and exact separation of odd edge cutset and capacity cuts. Computational experiments report one new optimal and twelve new lower bounds.

Limited memory Rank-1 Cuts for Vehicle Routing Problems

Pecin etal. (2016) introduced a limited memory technique that allows an efficient use of Rank-1 cuts in the Set Partitioning Formulation of Vehicle Routing Problems, motivating a deeper investigation of those cuts. This work presents a computational polyhedral study that determines the best possible sets of multipliers for cuts with up to 5 rows. Experiments with CVRP instances show that the new multipliers lead to significantly improved dual bounds and contributes decisively for solving an open instance with 420 customers.

A more human-like portfolio optimization approach

Black and Litterman proposed an improvement to the Markowitz portfolio optimization model. They suggested the construction of views to represent investor’s opinion about the future of stocks’ returns. However, conceiving these views can be quite confusing. It requires the investor to quantify several subjective parameters. In this article, we propose a new way of creating these views using Verbal Decision Analysis. Questionnaires were designed with the intent of making it easier for investors to express their vision about stocks. Following the ZAPROS methodology, the investor answers sets of questions allowing to determine a Formal Index of Quality (FIQ). The views are then derived from the resulting FIQ. Our approach was implemented and tested on data from the Brazilian Stocks. It allows investors to create a personal risk-return balanced portfolio without the help of an expert. The experiments show that the proposed method mitigates the impact of poor view estimation. Also, one must notice that the method is qualitative and its aim is to create a more efficient portfolio considering the investor’s vision.

A Computational Study of Conflict Graphs and Aggressive Cut Separation in Integer Programming

Abstract This work explores the fast creation of densely populated conflict graphs at the root node of the search tree for integer programs. We show that not only the Generalized Upper Bound (GUB) constraints are useful for the fast detection of cliques: these can also be quickly detected in less structured constraints in O ( n log ⁡ n ) . Routines for the aggressive separation and lifting of cliques and odd-holes are proposed. Improved bounds and a faster convergence to strong bounds were observed when comparing to the default separation routines found in the current version of the COmputation INfrastructure for Operations Research (COIN-OR) Branch and Cut solver.