A heuristic for BILP problems: The Single Source Capacitated Facility Location Problem

In the Single Source Capacitated Facility Location Problem (SSCFLP) each customer has to be assigned to one facility that supplies its whole demand. The total demand of customers assigned to each facility cannot exceed its capacity. An opening cost is associated with each facility, and is paid if at least one customer is assigned to it. The objective is to minimize the total cost of opening the facilities and supply all the customers. In this paper we extend the Kernel Search heuristic framework to general Binary Integer Linear Programming (BILP) problems, and apply it to the SSCFLP. The heuristic is based on the solution to optimality of a sequence of subproblems, where each subproblem is restricted to a subset of the decision variables. The subsets of decision variables are constructed starting from the optimal values of the linear relaxation. Variants based on variable fixing are proposed to improve the efficiency of the Kernel Search framework. The algorithms are tested on benchmark instances and new very large-scale test problems. Computational results demonstrate the effectiveness of the approach. The Kernel Search algorithm outperforms the best heuristics for the SSCFLP available in the literature. It found the optimal solution for 165 out of the 170 instances with a proven optimum. The error achieved in the remaining instances is negligible. Moreover, it achieved, on 100 new very large-scale instances, an average gap equal to 0.64% computed with respect to a lower bound or the optimum, when available. The variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality.     

Dynamic programming algorithms for the bi-objective integer knapsack problem

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.     

k-Edge Failure Resilient Network Design

We design a network that supports a feasible multicommodity flow even after the failures of any k edges. We present a mixed-integer linear program (MILP), a cutting plane algorithm, and a column-and-cut algorithm. The algorithms add constraints to repair vulnerabilities in partial network designs. Empirical studies on previously unsolved instances of SNDlib demonstrate their effectiveness.     

Lexicography and degeneracy: can a pure cutting plane algorithm work?

We discuss an implementation of the lexicographic version of Gomory’s fractional cutting plane method for ILP problems and of two heuristics mimicking the latter. In computational testing on a battery of MIPLIB problems we compare the performance of these variants with that of the standard Gomory algorithm, both in the single-cut and in the multi-cut (rounds of cuts) version, and show that they provide a radical improvement over the standard procedure. In particular, we report the exact solution of ILP instances from MIPLIB such as stein15, stein27, and bm23, for which the standard Gomory cutting plane algorithm is not able to close more than a tiny fraction of the integrality gap. We also offer an explanation for this surprising phenomenon.     

A computational study of the cutting plane tree algorithm for general mixed-integer linear programs

The cutting plane tree algorithm provides a finite procedure for solving general mixed-integer linear programs with bounded integer variables. The computational evidence provided in this work illustrates that this algorithm is powerful enough to close a significant fraction of the integrality gap for moderately sized MIPLIB instances.     

A Cut and Branch Approach for the Capacitated <Emphasis Type="Italic">p</Emphasis>-Median Problem Based on Fenchel Cutting Planes

The capacitated p-median problem (CPMP) consists of finding p nodes (the median nodes) minimizing the total distance to the other nodes of the graph, with the constraint that the total demand of the nodes assigned to each median does not exceed its given capacity. In this paper we propose a cutting plane algorithm, based on Fenchel cuts, which allows us to considerably reduce the integrality gap of hard CPMP instances. The formulation strengthened with Fenchel cuts is solved by a commercial MIP solver. Computational results show that this approach is effective in solving hard instances or considerably reducing their integrality gap.      

Models and heuristic algorithms for a weighted vertex coloring problem

We consider a weighted version of the well-known Vertex Coloring Problem (VCP) in which each vertex i of a graph G has associated a positive weight w _i . Like in VCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used. While in VCP the cost of each color is equal to one, in the Weighted Vertex Coloring Problem (WVCP) the cost of each color depends on the weights of the vertices assigned to that color, and it equals the maximum of these weights. WVCP is known to be NP-hard and arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose three alternative Integer Linear Programming (ILP) formulations for WVCP: one is used to derive, dropping integrality requirement for the variables, a tight lower bound on the solution value, while a second one is used to derive a 2-phase heuristic algorithm, also embedding fast refinement procedures aimed at improving the quality of the solutions found. Computational results on a large set of instances from the literature are reported.     

A linear programming approach for linear programs with probabilistic constraints

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.     

A cutting plane algorithm for the General Routing Problem

The General Routing Problem (GRP) is the problem of finding a minimum cost route for a single vehicle, subject to the condition that the vehicle visits certain vertices and edges of a network. It contains the Rural Postman Problem, Chinese Postman Problem and Graphical Travelling Salesman Problem as special cases. We describe a cutting plane algorithm for the GRP based on facet-inducing inequalities and show that it is capable of providing very strong lower bounds and, in most cases, optimal solutions. Received: November 1998 / Accepted: September 2000?Published online March 22, 2001     

Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts

In this paper, we propose to enhance Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations for polynomial programming problems by developing cutting plane strategies using concepts derived from semidefinite programming. Given an RLT relaxation, we impose positive semidefiniteness on suitable dyadic variable-product matrices, and correspondingly derive implied semidefinite cuts. In the case of polynomial programs, there are several possible variants for selecting such particular variable-product matrices on which positive semidefiniteness restrictions can be imposed in order to derive implied valid inequalities. This leads to a new class of cutting planes that we call v-semidefinite cuts. We explore various strategies for generating such cuts, and exhibit their relative effectiveness towards tightening the RLT relaxations and solving the underlying polynomial programming problems in conjunction with an RLT-based branch-and-cut scheme, using a test-bed of problems from the literature as well as randomly generated instances. Our results demonstrate that these cutting planes achieve a significant tightening of the lower bound in contrast with using RLT as a stand-alone approach, thereby enabling a more robust algorithm with an appreciable reduction in the overall computational effort, even in comparison with the commercial software BARON and the polynomial programming problem solver GloptiPoly.     

Resource assignment with preference conditions

This paper deals with a modification of the standard assignment problem, where subsets of resources express preferences in being, or not being, assigned together to the same activity. The problem arises in several real settings, among which the job assignment of the crew personnel of an airline company. We provide an integer programming formulation for both the Split Preference Problem, where couples of assignees do not want to work together, and for the Join Preference Problem, where, oppositely, couples of assignees want to work together. The mathematical nature of the two problems is indeed different, as for the first one it is possible to determine a minimum cost flow formulation on a suitable graph, and thus a polynomial time algorithm, while for the second one we face a NP-hard problem and device some heuristic solution approaches. Experimental tests conducted on instances of variable size confirm the effectiveness of the models and of the algorithms proposed.     

Approximate and exact algorithms for the double-constrained two-dimensional guillotine cutting stock problem

In this paper, we propose approximate and exact algorithms for the double constrained two-dimensional guillotine cutting stock problem (DCTDC). The approximate algorithm is a two-stage procedure. The first stage attempts to produce a starting feasible solution to DCTDC by solving a single constrained two dimensional cutting problem, CDTC. If the solution to CTDC is not feasible to DCTDC, the second stage is used to eliminate non-feasibility. The exact algorithm is a branch-and-bound that uses efficient lower and upper bounding schemes. It starts with a lower bound reached by the approximate two-stage algorithm. At each internal node of the branching tree, a tailored upper bound is obtained by solving (relaxed) knapsack problems. To speed up the branch and bound, we implement, in addition to ordered data structures of lists, symmetry, duplicate, and non-feasibility detection strategies which fathom some unnecessary branches. We evaluate the performance of the algorithm on different problem instances which can become benchmark problems for the cutting and packing literature.     

Parametric integer linear programming: A synthesis of branch and bound with cutting planes

A branch and bound algorithm is designed to solve the general integer linear programming problem with parametric right-hand sides. The right-hand sides have the form b + θd where b and d are comformable vectors, d consists of nonnegative constants, and θ varies from zero to one.The method consists of first determining all possible right-hand side integer constants and appending this set of integer constants to the initial tableau to form an expanded problem with a finite number of family members. The implicit enumeration method gives a lower bound on the integer solutions. The branch and bound method is used with fathoming tests which allow one family member possibly to fathom other family members. A cutting plane option applies a finite number of cuts to each node before branching. In addition, the cutting plane method is invoked whenever some members are feasible at a node and others are infeasible. The branching and cutting process is repeated until the entire family of problem has been solved.     

A branch and cut algorithm for the hierarchical network design problem

The Hierarchical Network Design Problem consists of locating a minimum cost bi-level network on a graph. The higher level sub-network is a path visiting two or more nodes. The lower level sub-network is a forest connecting the remaining nodes to the path. We optimally solve the problem using an ad hoc branch and cut procedure. Relaxed versions of a base model are solved using an optimization package and, if binary variables have fractional values or if some of the relaxed constraints are violated in the solution, cutting planes are added. Once no more cuts can be added, branch and bound is used. The method for finding valid cutting planes is presented. Finally, we use different available test instances to compare the procedure with the best known published optimal procedure, with good results. In none of the instances we needed to apply branch and bound, but only the cutting planes.     

An inexact algorithm for the sequential ordering problem

Given the directed G = (N, A) and the penalty matrix C, the Sequential Ordering Problem (hereafter, SOP) consists of finding the permutation of the nodes from the set N, such that it minimizes a C-based function and does not violate the precedence relationships given by the set A. Strong sufficient conditions for the infeasibility of a SOP's instance are embedded in a procedure for the SOP's pre-processing. Note that it is one of the key steps in any algorithm that attempts to solve SOP. By dropping the constraints related to the precedence relationships, SOP can be converted in the classical Asymmetric Traveling Salesman Problem (hereafter, ASTP). The algorithm obtains (hopefully) satisfactory solutions by modifying the optimal solution to the related Assignment Problem (hereafter, AP) if it is not a Feasible Sequential Ordering (hereafter, FSO). The new solution ‘patches’ the subtours (if any) giving preference to the patches with zero reduced cost in the linking arcs. The AP-based lower bound on the optimal solution to ATSP is tightened by using some of the procedures given in [1]. In any case, a local search for improving the initial FSO is performed; it uses 3- and 4-changed based procedures. Computational results on a broad set of cases are reported.     

A distributed exact algorithm for the multiple resource constrained sequencing problem

Sequencing problems arise in the context of process scheduling both in isolation and as subproblems for general scenarios. Such sequencing problems can often be posed as an extension of the Traveling Salesman Problem. We present an exact parallel branch and bound algorithm for solving the Multiple Resource Constrained Traveling Salesman Problem (MRCTSP), which provides a platform for addressing a variety of process sequencing problems. The algorithm is based on a linear programming relaxation that incorporates two families of inequalities via cutting plane techniques. Computational results show that the lower bounds provided by this method are strong for the types of problem generators that we considered as well as for some industrially derived sequencing instances. The branch and bound algorithm is parallelized using the processor workshop model on a network of workstations connected via Ethernet. Results are presented for instances with up to 75 cities, 3 resource constraints, and 8 workstations.     

A hierarchical algorithm for making sparse matrices sparser

IfA is the (sparse) coefficient matrix of linear equality constraints, for what nonsingularT is ≡TA as sparse as possible, and how can it be efficiently computed? An efficient algorithm for thisSparsity Problem (SP) would be a valuable pre-processor for linearly constrained optimization problems. In this paper we develop a two-pass approach to solve SP. Pass 1 builds a combinatorial structure on the rows ofA which hierarchically decomposes them into blocks. This determines the structure of the optimal transformation matrixT. In Pass 2, we use the information aboutT as a road map to do block-wise partial Gauss-Jordan elimination onA. Two block-aggregation strategies are also suggested that could further reduce the time spend in Pass 2. Computational results indicate that this approach to increasing sparsity produces significant net reductions in simplex solution time.     

The DH/KD algorithm: a hybrid approach for unconstrained two-dimensional cutting problems

We study both weighted and unweighted unconstrained two-dimensional guillotine cutting problems. We develop a hybrid approach which combines two heuristics from the literature. The first one (DH) uses a tree-search procedure introducing two strategies: Depth-first search and Hill-climbing. The second one (KD) is based on a series of one-dimensional Knapsack problems using Dynamic programming techniques. The DH /KD algorithm starts with a good initial lower bound obtained by using the KD algorithm. At each level of the tree-search, the proposed algorithm uses also the KD algorithm for constructing new lower bounds and uses another one-dimensional knapsack for constructing refinement upper bounds. The resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time. Our algorithm is compared to Morabito et al.'s algorithm (the unweighted case), and to Beasley's [2] approach (the weighted case) on some examples taken from the literature as well as randomly generated instances.     

Modeling and solving a Crew Assignment Problem in air transportation

A typical problem arising in airline crew management consists in optimally assigning the required crew members to each flight segment of a given time period, while complying with a variety of work regulations and collective agreements. This problem called the Crew Assignment Problem (CAP) is currently decomposed into two independent sub-problems which are modeled and solved sequentially: (a) the well-known Crew Pairing Problem followed by (b) the Working Schedules Construction Problem. In the first sub-problem, a set of legal minimum-cost pairings is constructed, covering all the planned flight segments. In the second sub-problem, pairings, rest periods, training periods, annual leaves, etc. are combined to form working schedules which are then assigned to crew members.In this paper, we present a new approach to the Crew Assignment Problem arising in the context of airline companies operating short and medium haul flights. Contrary to most previously published work on the subject, our approach is not based on the concept of crew-pairings, though it is capable of handling many of the constraints present in crew-pairing-based models. Moreover, contrary to crew-pairing-based approaches, one of its distinctive features is that it formulates and solves the two sub-problems (a) and (b) simultaneously for the technical crew members (pilots and officers) with specific constraints. We show how this problem can be formulated as a large scale integer linear program with a general structure combining different types of constraints and not exclusively partitioning or covering constraints as usually suggested in previous papers. We introduce then, a formulation enhancement phase where we replace a large number of binary exclusion constraints by stronger and less numerous ones: the clique constraints. Using data provided by the Tunisian airline company TunisAir, we demonstrate that thanks to this new formulation, the Crew Assignment Problem can be solved by currently available integer linear programming technology. Finally, we propose an efficient heuristic method based on a rounding strategy embedded in a partial tree search procedure.The implementation of these methods (both exact and heuristic ones) provides good solutions in reasonable computation times using CPLEX 6.0.2: guaranteed exact solutions are obtained for 60% of the test instances and solutions within 5% of the lower bound for the others.