A three-phase algorithm for the cell suppression problem in two-dimensional statistical tables

To obtain full cooperation from respondents, statistical offices must guarantee that confidential data will not be disclosed when their reports are published. For tabular data, cell suppression is one of the preferred techniques to control statistical disclosure. When suppressing only confidential values does not guarantee the desired data protection, it is also necessary to suppress the values in some non-confidential cells. The problem of finding an optimal set of complementary suppressions—the cell suppression problem (CSP)—is NP-hard. We present a three-phase algorithm for the CSP based on a binary relaxation derived from row and column protection conditions. To enforce violated single cell conditions, integer cuts are added to the CSP relaxation. The numerical results obtained in 1410 instances with up to more than 250?000 cells, which were generated to reproduce two classes of real-world data, indicate that the algorithm is quite effective for both classes of instances and that it outperforms state-of-the-art algorithms for one of them.     

Stochastic nuclear outages semidefinite relaxations

This paper deals with stochastic scheduling of nuclear power plant outages. Focusing on the main constraints of the problem, we propose a stochastic formulation with a discrete distribution for random variables, that leads to a mixed 0/1 quadratically constrained quadratic program. Then we investigate semidefinite relaxations for solving this hard problem. Numerical results on several instances of the problem show the efficiency of this approach, i.e., the gap between the optimal solution and the continuous relaxation is on average equal to 53.35 % whereas the semidefinite relaxation yields an average gap of 2.76 %. A feasible solution is then obtained with a randomized rounding procedure.     

Tighter Bounds for the Gap and Non-IRUP Constructions in the One-dimensional Cutting Stock Problem

The one-dimensional cutting stock problem is investigated with respect to the difference between the optimal function values of the integer programming problem and its continuous relaxation. A tighter bound for this gap is presented, followed by some non-IRUP constructions. Finally, instances with gap 7/6 are constructed, the largest gap known so far.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

Integrality gaps for colorful matchings

We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali–Adams “lift-and-project” technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Füredi (1981). We further leverage this to show upper bounds on the performance of the Sherali–Adams hierarchy when applied to the natural LP relaxation of the BCM problem.     

Solving the CLSP by a Tabu Search Heuristic

The multi-item, single-level, capacitated, dynamic lot-sizing problem, commonly abbreviated as CLSP, is considered. The problem is cast in a tight mixed-integer programming model (MIP); tight in the sense that the gap between the optimal value of MIP and that of its linear programming relaxation (LP) is small. The LP relaxation of MIP is then solved by column generation. The resulting feasible solution is further improved by adopting the corresponding set-up schedule and re-optimizing variable costs by solving a minimum-cost network flow (trans-shipment) problem. Subsequently, the improved solution is used as a starting solution for a tabu search procedure, with the worth of moves assessed using the same trans-shipment problem. Results of computational testing of benchmark problem instances are presented. They show that the heuristic solutions obtained are effective, in that they are extremely close to the best known solutions. The computational efficiency makes it possible to solve realistically large problem instances routinely on a personal computer; in particular, the solution procedure is most effective, in terms of solution quality, for larger problem instances.     

A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem

The multi-activity shift scheduling problem involves assigning a sequence of activities to a set of employees. In this paper, we consider the variant where the employees have different qualifications and each activity must be performed in a specified time window; i.e., we specify the earliest start period and the latest finish period. We propose a matheuristic in which Lagrangian relaxation is used to identify a subset of promising shifts, and a restricted set covering problem is solved to find a feasible solution. Each shift is represented by a context-free grammar. Computational tests are carried out on two sets of instances from the literature. For the first set, the matheuristic finds a solution with an optimality gap less than 0.01% for 70% of the instances and improves the best-known solution for 16% of them; for the second set, the matheuristic reaches the best-known solutions for 55% of the instances and finds better solutions for 37.5% of them.     

A Beam Search approach for the optimization version of the Car Sequencing Problem

The Car Sequencing Problem (CSP) is a feasibility problem that has attracted the attention of the Constraint Programming community for a number of years now. In this paper, a new version (opt-CSP) that extends the original problem is defined, converting this into an optimization problem in which the goal is to satisfy the typical hard constraints. This paper presents a solution procedure for opt-CSP using Beam Search. Computational results are presented using public instances that verify the goodness of the procedure and demonstrate its excellent performance in obtaining feasible solutions for the majority of instances while satisfying the new constraints.     

Traditional heuristic versus Hopfield neural network approaches to a car sequencing problem

This paper considers the problem of optimally sequencing different car models along an assembly line according to some contiguity constraints, while ensuring that the demands for each of the models are satisfied. This car sequencing problem (CSP) is a practical NP-hard combinatorial optimisation problem. The CSP is formulated as a nonlinear integer programming problem and it is shown that exact solutions to the problem are difficult to obtain due to the indefinite quadratic form of the CSP objective function. Two traditional heuristics (steepest descent and simulated annealing) are employed to solve the CSP approximately. Several Hopfield neural network (HNN) approaches are also presented. The process of mapping an optimisation problem onto a HNN is demonstrated explicitly, and modifications to the existing neural approaches are presented which guarantee feasibility of solutions. Further modifications are proposed to improve the solution quality by permitting escape from local minima in an attempt to locate the global optimum. Results from all solutions techniques are compared on a set of instances of the CSP, and conclusions drawn.     

A note on the integrality gap of cutting and skiving stock instances

4OR - In this paper, we consider the (additive integrality) gap of the cutting stock problem (CSP) and the skiving stock problem (SSP). Formally, the gap is defined as the difference between the...     

Solving one-dimensional cutting stock problems exactly with a cutting plane algorithm

When solving the one-dimensional cutting stock problem (1D CSP) as an integer linear programming problem one has to overcome computational difficulties arising from the integrality condition and a huge number of variables. In the Gilmore–Gomory approach the corresponding continuous relaxation is solved via column generation techniques followed by an appropriate rounding of the in general non-integer solution. Obviously, there is no guarantee of obtaining an optimal solution in this way but it is extremely effective in practice. However, in two- and three-dimensional cutting stock problems the heuristics are not so good which necessitates the research of effective exact methods. In this paper we present an exact solution approach for the 1D CSP which is based on a combination of the cutting plane method and the column generation technique. Results of extensive computational experiments are reported.     

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

This paper considers a pricing problem on a network with connected toll arcs and proposes a Dantzig-Wolfe reformulation for it. The model is solved with column generation and the gap between the optimal integer value and the linear relaxation optimal value is shown to be at least as good as the one from the mixed-integer formulation proposed in the literature. Numerical results on different sets of instances are reported, showing that in many cases the proposed model performs strictly better.     

Upper and lower bounds for the sales force deployment problem with explicit contiguity constraints

The sales force deployment problem arises in many selling organizations. This complex planning problem involves the concurrent resolution of four interrelated subproblems: sizing of the sales force, sales representatives locations, sales territory alignment, and sales resource allocation. The objective is to maximize the total profit. For this, a well-known and accepted concave sales response function is used. Unfortunately, literature is lacking approaches that provide valid upper bounds. Therefore, we propose a model formulation with an infinite number of binary variables. The linear relaxation is solved by column generation where the variables with maximum reduced costs are obtained analytically. For the optimal objective function value of the linear relaxation an upper bound is provided. To obtain a very tight gap for the objective function value of the optimal integer solution we introduce a Branch-and-Price approach. Moreover, we propose explicit contiguity constraints based on flow variables. In a series of computational studies we consider instances which may occur in the pharmaceutical industry. The largest instance comprises 50 potential locations and more than 500 sales coverage units. We are able to solve this instance in 1273 seconds with a gap of less than 0.01%. A comparison with Drexl and Haase (1999) shows that we are able to halve the solution gap due to tight upper bounds provided by the column generation procedure.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.     

New geometry-inspired relaxations and algorithms for the metric Steiner tree problem

Determining the integrality gap of the bidirected cut relaxation for the metric Steiner tree problem, and exploiting it algorithmically, is a long-standing open problem. We use geometry to define an LP whose dual is equivalent to this relaxation. This opens up the possibility of using the primal-dual schema in a geometric setting for designing an algorithm for this problem. Using this approach, we obtain a 4/3 factor algorithm and integrality gap bound for the case of quasi-bipartite graphs; the previous best integrality gap upper bound being 3/2 (Rajagopalan and Vazirani in On the bidirected cut relaxation for the metric Steiner tree problem, 1999). We also obtain a factor \({\sqrt{2}}\) strongly polynomial algorithm for this class of graphs. A key difficulty experienced by researchers in working with the bidirected cut relaxation was that any reasonable dual growth procedure produces extremely unwieldy dual solutions. A new algorithmic idea helps finesse this difficulty—that of reducing the cost of certain edges and constructing the dual in this altered instance—and this idea can be extracted into a new technique for running the primal-dual schema in the setting of approximation algorithms.     

A fast solution method for the time-dependent orienteering problem

This paper introduces a fast solution procedure to solve 100-node instances of the time-dependent orienteering problem (TD-OP) within a few seconds of computation time. Orienteering problems occur in logistic situations were an optimal combination of locations needs to be selected and the routing between the selected locations needs to be optimized. In the time-dependent variant, the travel time between two locations depends on the departure time at the first location. Next to a mathematical formulation of the TD-OP, the main contribution of this paper is the design of a fast and effective algorithm to tackle this problem. This algorithm combines the principles of an ant colony system (ACS) with a time-dependent local search procedure equipped with a local evaluation metric. Additionally, realistic benchmark instances with varying size and properties are constructed. The average score gap with the known optimal solution on these test instances is only 1.4% with an average computation time of 0.5 seconds. An extensive sensitivity analysis shows that the performance of the algorithm is insensitive to small changes in its parameter settings.     

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.      

The capacitated minimum spanning tree problem: On improved multistar constraints

In this paper, we propose a new set of valid inequalities for the Capacitated Minimum Spanning Tree (CMST) problem with general node demands. These inequalities are stronger versions of the well-known multistar constraints (see Hall [Informs J. Comput. 8 (1996) 219]). We also propose and test one lagrangean relaxation scheme that dualizes the new constraints. Our computational results involving instances with up to 80 nodes plus the root node, show that the new inequalities are worth using for random cost and sparse instances. These instances appear to be more difficult to solve than Euclidean ones. We also propose a new arc elimination test for the CMST problem. Besides helping to reduce the size of the instances being solved, the new test improves the performance of the lagrangean relaxation proposed in this paper for the random cost instances.     

Maximizing the Net Present Value of a Project Under Resource Constraints Using a Lagrangian Relaxation Based Heuristic with Tight Upper Bounds

Resource-constrained project scheduling under a net present value objective attracts growing interest. Because this is an NP-hard problem, it is unlikely that optimum solutions can be computed for large instances within reasonable computation time. Thus, heuristics have become a popular research field. Up to now, however, upper bounds are not well researched. Therefore, most researchers evaluate their heuristics on the basis of a best known lower bound, but it is unclear how good the performance really is. With this contribution we close this gap and derive tight upper bounds on the basis of a Lagrangian relaxation of the resource constraints. We also use this approach as a basis for a heuristic and show that our heuristic as well as the cash flow weight heuristic proposed by Baroum and Patterson yield solutions very close to the optimum result. Furthermore, we discuss the proper choice of a test-bed and emphasize that discount rates must be carefully chosen to give realistic instances.     

Approximations and Randomization to Boost CSP Techniques

In recent years we have seen an increasing interest in combining constraint satisfaction problem (CSP) formulations and linear programming (LP) based techniques for solving hard computational problems. While considerable progress has been made in the integration of these techniques for solving problems that exhibit a mixture of linear and combinatorial constraints, it has been surprisingly difficult to successfully integrate LP-based and CSP-based methods in a purely combinatorial setting. Our approach draws on recent results on approximation algorithms based on LP relaxations and randomized rounding techniques, with theoretical guarantees, as well on results that provide evidence that the runtime distributions of combinatorial search methods are often heavy-tailed. We propose a complete randomized backtrack search method for combinatorial problems that tightly couples CSP propagation techniques with randomized LP-based approximations. We present experimental results that show that our hybrid CSP/LP backtrack search method outperforms the pure CSP and pure LP strategies on instances of a hard combinatorial problem.