A branch-and-cut method for the bi-objective bi-dimensional knapsack problem

International audienceMulti-objective multi-dimensional knapsack problems (pOmDKP) are widely used to represent practical problems as capital budgeting or allocating processors. It aims to select a subset of n items such that the sum of weight of the selected items does not exceed the capacity on any of the m dimensions, while maximizing p objective functions. Each item has a weight on each dimension and a profit for each objective function. This problem is known for being particularly difficult as soon as the number of dimensions exceeds one, even in its single-objective version.There are many published papers focusing on the exact solution of multi-objective single-dimensional knapsack. The solutions methods are often two-phases methods. The second phase is either a branch-and-bound method (as in [1] for the bi-objective case or in [2] for the three-objective case), either a dynamic programming method [3], or a dedicated ranking method [2].Only a few works have studied exactly the multi-objective multi-dimensional case. Concerning the single-objective multi-dimensional knapsack problem, many works have investigated cutting inequalities to speed-up the computation of solution [4].In this work we are interested in the exact solution of the bi-objective bi-dimensional knapsack problem (2O2DKP), using a branch-and-cut method. A branch-and-cut method is a combination of a cutting plane method and a branch-and-bound method. According to its name, one of the main component of a branch-and-bound method aims at computing bounds of the problem. Convex relaxation has been a key component for successful bi-objective branch-and-bound algorithm (see for example [5]). It defines indeed a tight upper bound set, which can be computed easily if the single-objective version of the problem can be solved in (pseudo-)polynomial time. However, this is not the case for 2O2DKP. On the contrary, the linear relaxation remains relatively easy to compute, but the resulting bound set is less tight, which makes more difficult the exploration of nodes and leads to larger search-trees. To improve the quality of the upper bound set based on linear relaxation, we introduce cover inequalities at each node of the branch-and-bound method, turning it to a branch-and-cut method. Cover inequalities consist of cuts defined for single-objective binary problems [6]. A cover is a set of objects such that the sum of the weights associated to these objects exceeds the capacity. In [6], the authors remark that computing all possible cover inequalities would be time-consuming and even impossible to implement. Instead, they consider the optimal solution of the linear relaxation and solve a smaller binary problem to find a cover inequality that is violated. In the bi-objective context, the linear relaxation is described by a set of extreme points, which are associated to efficient solutions. Moreover, each of these efficient solutions may be fractional and have a different subset of fractional variables. The generation of cover inequalities is therefore more complex, particularly to get a good tradeoff between quality of the improved upper bound set defined and computational time. This leads to numerous strategies to generate cover inequalities. This presentation will describe the mechanisms used in the multi-objective branch-and-cut method that we have developed (separation procedure, bound sets, generation of cover inequalities...). These strategies have been then experimentally validated. [1] Visée, M., Teghem, J., Pirlot, M., Ulungu, E. L., March 1998. Two-phases method and branch and bound procedures to solve the bi–objective knapsack problem. Journal of Global Optimization 12, 139–155. [2] Jorge, J., May 2010. Nouvelles propositions pour la résolution exacte du sac à dos multi-objectif unidimensionnel en variables binaires. Thèse, Université de Nantes.[3] Delort, C., Spanjaard, O., 2010. Using bound sets in multiobjective optimization: Application to the biobjective binary knapsack problem. In: Festa, P. (Ed.), SEA. Vol 6049 of Lecture Notes in Computer Science. Springer, 253-265.[4] Osorio, M. A., Glover, F., Hammer, P., 2002. Cutting and surrogate constraint analysis for improved multidimensional knapsack solutions. Annals of Operations Research 117 (1-4), 71–93.[5] Sourd F. and Spanjaard O., 2008. A multi-objective branch-and bound framework: Application to the biobjective spanning tree problem. INFORMS Journal on Computing, 20:472-484.[6] Crowder, H., Johnson, E. L., Padberg, M. W., 1983. Solving large-scale zero-one linear programming problems. Operations Research 31 (5), 803–834

The bi-objective travelling salesman problem with profits and its connection to computer networks.

This is an interdisciplinary work in Computer Science and Operational Research. As it is well known, these two very important research fields are strictly connected. Among other aspects, one of the main areas where this interplay is strongly evident is Networking. As far as most recent decades have seen a constant growing of every kind of network computer connections, the need for advanced algorithms that help in optimizing the network performances became extremely relevant. Classical Optimization-based approaches have been deeply studied and applied since long time. However, the technology evolution asks for more flexible and advanced algorithmic approaches to model increasingly complex network configurations. In this thesis we study an extension of the well known Traveling Salesman Problem (TSP): the Traveling Salesman Problem with Profits (TSPP). In this generalization, a profit is associated with each vertex and it is not necessary to visit all vertices. The goal is to determine a route through a subset of nodes that simultaneously minimizes the travel cost and maximizes the collected profit. The TSPP models the problem of sending a piece of information through a network where, in addition to the sending costs, it is also important to consider what “profit” this information can get during its routing. Because of its formulation, the right way to tackled the TSPP is by Multiobjective Optimization algorithms. Within this context, the aim of this work is to study new ways to solve the problem in both the exact and the approximated settings, giving all feasible instruments that can help to solve it, and to provide experimental insights into feasible networking instances

Xqx Based Modeling For General Integer Programming Problems

We present a new way to model general integer programming (IP) problems with in- equality and equality constraints using XQX. We begin with the definition of IP problems folloby their practical applications, and then present the existing XQX based models to handle such problems. We then present our XQX model for general IP problems (including binary IP) with equality and inequality constraints, and also show how this model can be applied to problems with just inequality constraints. We then present the local optima based solution procedure for our XQX model. We also present new theorems and their proofs for our XQX model. Next, we present a detailed literature survey on the 0-1 multidimensional knapsack problem (MDKP) and apply our XQX model using our simple heuristic procedure to solve benchmark problems. The 0-1 MDKP is a binary IP problem with inequality con- straints and variables with binary values. We apply our XQX model using a heuristics we developed on 0-1 MDKP problems of various sizes and found that our model can handle any problem sizes and can provide reasonable quality results in reasonable time. Finally, we apply our XQX model developed for general integer programming problems on the general multi-dimensional knapsack problems. The general MDKP is a general IP problem with inequality constraints where the variables are positive integers. We apply our XQX model on GMDKP problems of various sizes and find that it can provide reasonable quality results in reasonable time. We also find that it can handle problems of any size and provide fea- sible and good quality solutions irrespective of the starting solutions. We conclude with a discussion of some issues related with our XQX model

A biobjective method for sample allocation in stratified sampling

The two main and contradicting criteria guiding sampling design are accuracy of estimators and sampling costs. In stratified random sampling, the sample size must be allocated to strata in order to optimize both objectives. In this note we address, following a biobjective methodology, this allocation problem. A two-phase method is proposed to describe the set of Pareto-optimal solutions of this nonlinear integer biobjective problem. In the first phase, all supported Pareto-optimal solutions are described via a closed formula, which enables quick computation. Moreover, for the common case in which sampling costs are independent of the strata, all Pareto-optimal solutions are shown to be supported. For more general cost structures, the non-supported Pareto-optimal solutions are found by solving a parametric knapsack problem. Bounds on the criteria can also be imposed, directing the search towards implementable sampling plans. Our method provides a deeper insight into the problem than simply solving a scalarized version, whereas the computational burden is reasonable.Ministerio de Ciencia y Tecnologí

A multi-level search strategy for the 0–1 Multidimensional Knapsack Problem

AbstractWe propose an exact method based on a multi-level search strategy for solving the 0–1 Multidimensional Knapsack Problem. Our search strategy is primarily based on the reduced costs of the non-basic variables of the LP-relaxation solution. Considering that the variables are sorted in decreasing order of their absolute reduced cost value, the top level branches of the search tree are enumerated following Resolution Search strategy, the middle level branches are enumerated following Branch & Bound strategy and the lower level branches are enumerated according to a simple Depth First Search enumeration strategy. Experimentally, this cooperative scheme is able to solve optimally large-scale strongly correlated 0–1 Multidimensional Knapsack Problem instances. The optimal values of all the 10 constraint, 500 variable instances and some of the 30 constraint, 250 variable instances of the OR-Library were found. These values were previously unknown

Adaptive Improvements of Multi-Objective Branch and Bound

Branch and bound methods which are based on the principle "divide and conquer" are a well established solution approach in single-objective integer programming. In multi-objective optimization branch and bound algorithms are increasingly attracting interest. However, the larger number of objectives raises additional difficulties for implicit enumeration approaches like branch and bound. Since bounding and pruning is considerably weaker in multiple objectives, many branches have to be (partially) searched and may not be pruned directly. The adaptive use of objective space information can guide the search in promising directions to determine a good approximation of the Pareto front already in early stages of the algorithm. In particular we focus in this article on improving the branching and queuing of subproblems and the handling of lower bound sets. In our numerical test we evaluate the impact of the proposed methods in comparison to a standard implementation of multiobjective branch and bound on knapsack problems, generalized assignment problems and (un)capacitated facility location problems

Reoptimization in lagrangian methods for the quadratic knapsack problem

International audienceThe 0-1 quadratic knapsack problem consists in maximizing a quadratic objective function subject to a linear capacity constraint. To solve exactly large instances of this problem with a tree search algorithm (e.g. a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0-1 linear knapsack subproblems. Thus, taking this huge number of solvings into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0-1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach