The balanced academic curriculum problem revisited

The Balanced Academic Curriculum Problem (BACP) consists in assigning courses to teaching terms satisfying prerequisites and balancing the credit course load within each term. The BACP is part of the CSPLib with three benchmark instances, but its formulation is simpler than the problem solved in practice by universities. In this article, we introduce a generalized version of the problem that takes different curricula and professor preferences into account, and we provide a set of real-life problem instances arisen at University of Udine. Since the existing formulation based on a min–max objective function does not balance effectively the credit load for the new instances, we also propose alternative objective functions. Whereas all the CSPLib instances are efficiently solved with Integer Linear Programming (ILP) state-of-the-art solvers, our new set of real-life instances turns out to be much more challenging and still intractable for ILP solvers. Therefore, we have designed, implemented, and analyzed heuristics based on local search. We have collected computational results on all the new instances with the proposed approaches and assessed the quality of solutions with respect to the lower bounds found by ILP on a relaxed and decomposed problem. Results show that a selected heuristic finds solutions of quality at 9%–60% distance from the lower bound. We make all data publicly available, in order to stimulate further research on this problem.     

The mixed general routing polyhedron

 In Arc Routing Problems, ARPs, the aim is to find on a graph a minimum cost traversal satisfying some conditions related to the links of the graph. Due to restrictions to traverse some streets in a specified way, most applications of ARPs must be modeled with a mixed graph. Although several exact algorithms have been proposed, no polyhedral investigations have been done for ARPs on a mixed graph. In this paper we deal with the Mixed General Routing Problem which consists of finding a minimum cost traversal of a given link subset and a given vertex subset of a mixed graph. A formulation is given that uses only one variable for each link (edge or arc) of the graph. Some properties of the associated polyhedron and some large families of facet-inducing inequalities are described. A preliminary cutting-plane algorithm has produced very good lower bounds over a set of 100 randomly generated instances of the Mixed Rural Postman Problem. Finally, applications of this study to other known routing problems are described. Received: June 30, 1999 / Accepted: March 2002 Published online: March 21, 2003 Key Words. polyhedral combinatorics – facets – routing – arc routing – rural postman problem – general routing problem – mixed chinese postman problem     

Optimizing departure times in vehicle routes

Most solution methods for the vehicle routing problem with time windows (VRPTW) develop routes from the earliest feasible departure time. In practice, however, temporary traffic congestion make such solutions non-optimal with respect to minimizing the total duty time. Furthermore, the VRPTW does not account for driving hours regulations, which restrict the available travel time for truck drivers. To deal with these problems, we consider the vehicle departure time optimization (VDO) problem as a post-processing of a VRPTW. We propose an ILP formulation that minimizes the total duty time. The results of a case study indicate that duty time reductions of 15% can be achieved. Furthermore, computational experiments on VRPTW benchmarks indicate that ignoring traffic congestion or driving hours regulations leads to practically infeasible solutions. Therefore, new vehicle routing methods should be developed that account for these common restrictions. We propose an integrated approach based on classical insertion heuristics.     

The static bicycle relocation problem with demand intervals

This study introduces the Static Bicycle Relocation Problem with Demand Intervals (SBRP-DI), a variant of the One Commodity Pickup and Delivery Traveling Salesman Problem (1-PDTSP). In the SBRP-DI, the stations are required to have an inventory of bicycles lying between given lower and upper bounds and initially have an inventory which does not necessarily lie between these bounds. The problem consists of redistributing the bicycles among the stations, using a single capacitated vehicle, so that the bounding constraints are satisfied and the repositioning cost is minimized. The real-world application of this problem arises in rebalancing operations for shared bicycle systems. The repositioning subproblem associated with a fixed route is shown to be a minimum cost network problem, even in the presence of handling costs. An integer programming formulation for the SBRP-DI are presented, together with valid inequalities adapted from constraints derived in the context of other routing problems and a Benders decomposition scheme. Computational results for instances adapted from the 1-PDTSP are provided for two branch-and-cut algorithms, the first one for the full formulation, and the second one with the Benders decomposition.     

The Team Orienteering Problem with Time Windows: An LP-based Granular Variable Neighborhood Search

The Team Orienteering Problem (TOP) is a known NP-hard problem that typically arises in vehicle routing and production scheduling contexts. In this paper we introduce a new solution method to solve the TOP with hard Time Window constraints (TOPTW). We propose a Variable Neighborhood Search (VNS) procedure based on the idea of exploring, most of the time, granular instead of complete neighborhoods in order to improve the algorithm’s efficiency without loosing effectiveness. The method provides a general way to deal with granularity for those routing problems based on profits and complicated by time constraints. Extensive computational results are reported on standard benchmark instances. Performance of the proposed algorithm is compared to optimal solution values, when available, or to best known solution values obtained by state-of-the-art algorithms. The method comes out to be, on average, quite effective allowing to improve the best know values for 25 test instances.     

GRASP-based heuristic algorithm for the multi-product multi-vehicle inventory routing problem

In this paper, we introduce an improved Greedy Randomized Adaptive Search Procedure (GRASP) based heuristic for the multi-product multi-vehicle inventory routing problem (MMIRP). The inventory routing problem, which combines the vehicle-routing problem and the inventory control decisions, is one of the most important problems in combinatorial optimization field. To deal with the MMIRP, we develop a GRASP-based heuristic (GBH). Each GBH iteration consists of two sequential phases; the first phase is a Greedy Randomized Procedure, in which, the best tradeoff between the inventory holding cost and routing cost is looked. Then, in the second phase, as local search for the GRASP, we use the Tabu search (TS) meta-heuristic to improve the solution found in the first phase. The GBH two phases are repeated until some stopped criterion is met. Our proposed method is evaluated on two benchmark data sets, and successfully compared with two state-of-the-art algorithms.     

Strategies for designing energy-efficient clusters-based WSN topologies

Wireless Sensor Networks are used in several practical applications such as environmental monitoring and risk detection. In this work, we deal with the problem of organizing the network topology into clusters in order to minimize the total energy consumption. The problem is modeled as an Independent Dominating Problem with Connecting requirements. We first present a state-of-the-art on the problems to optimize energy consumption in WSN. Then, we propose a mixed integer linear programming formulation, constructive heuristics, a local search procedure, and a GRASP-based metaheuristic. Results are provided for large scale WSN instances.     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.     

A General Approach to the Physician Rostering Problem

This paper presents a hybridization of a Constraint Programming (CP) model and search techniques with Local Search (LS) and some ideas borrowed from Genetic Algorithms (GA). The context is the physician rostering problem, whose instances can vary greatly and for which almost no general tool has been developed. It is hoped that the combination of the three techniques will lead to an algorithm that has sufficient flexibility to solve most instances with a small amount of customization. To achieve this goal we also introduce Generic constraints: these constraints are used to model several types of ergonomic constraints that are found amongst physician rostering problems.     

Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs

The asymmetric travelling salesman problem with replenishment arcs (RATSP), arising from work related to aircraft routing, is a generalisation of the well-known ATSP. In this paper, we introduce a polynomial size mixed-integer linear programming (MILP) formulation for the RATSP, and improve an existing exponential size ILP formulation of Zhu [The aircraft rotation problem, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, 1994] by proposing two classes of stronger cuts. We present results that under certain conditions, these two classes of stronger cuts are facet-defining for the RATS polytope, and that ATSP facets can be lifted, to give RATSP facets. We implement our polyhedral findings and develop a Lagrangean relaxation (LR)-based branch-and-bound (BNB) algorithm for the RATSP, and compare this method with solving the polynomial size formulation using ILOG Cplex 9.0, using both randomly generated problems and aircraft routing problems. Finally we compare our methods with the existing method of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408-427]. It turns out that both of our methods are much faster than that of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408-427], and that the LR-based BNB method is more efficient for problems that resemble the aircraft rotation problems.     

Cutting plane versus compact formulations for uncertain (integer) linear programs

Robustness is about reducing the feasible set of a given nominal optimization problem by cutting ??risky?? solutions away. To this end, the most popular approach in the literature is to extend the nominal model with a polynomial number of additional variables and constraints, so as to obtain its robust counterpart. Robustness can also be enforced by adding a possibly exponential family of cutting planes, which typically leads to an exponential formulation where cuts have to be generated at run time. Both approaches have pros and cons, and it is not clear which is the best one when approaching a specific problem. In this paper we computationally compare the two options on some prototype problems with different characteristics. We first address robust optimization à la Bertsimas and Sim for linear programs, and show through computational experiments that a considerable speedup (up to 2 orders of magnitude) can be achieved by exploiting a dynamic cut generation scheme. For integer linear problems, instead, the compact formulation exhibits a typically better performance. We then move to a probabilistic setting and introduce the uncertain set covering problem where each column has a certain probability of disappearing, and each row has to be covered with high probability. A related uncertain graph connectivity problem is also investigated, where edges have a certain probability of failure. For both problems, compact ILP models and cutting plane solution schemes are presented and compared through extensive computational tests. The outcome is that a compact ILP formulation (if available) can be preferable because it allows for a better use of the rich arsenal of preprocessing/cut generation tools available in modern ILP solvers. For the cases where such a compact ILP formulation is not available, as in the uncertain connectivity problem, we propose a restart solution strategy and computationally show its practical effectiveness.     

On the capacitated vehicle routing problem

 We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY. Received: October 30, 2000 / Accepted: December 19, 2001 Published online: September 5, 2002 Key words. vehicle routing problem – integer programming – decomposition algorithm – separation algorithm – branch and cut Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

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.     

A unified exact method for solving different classes of vehicle routing problems

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.      

Ants can orienteer a thief in their robbery

The Thief Orienteering Problem (ThOP) is a multi-component problem that combines features of two classic combinatorial optimization problems: Orienteering Problem and Knapsack Problem. The ThOP is challenging due to the given time constraint and the interaction between its components. We propose an Ant Colony Optimization algorithm together with a new packing heuristic to deal individually and interactively with problem components. Our approach outperforms existing work on more than 90% of the benchmarking instances, with an average improvement of over 300%.     

A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem

The multiple-choice multidimensional knapsack problem (MMKP) is a well-known NP-hard combinatorial optimization problem with a number of important applications. In this paper, we present a “reduce and solve” heuristic approach which combines problem reduction techniques with an Integer Linear Programming (ILP) solver (CPLEX). The key ingredient of the proposed approach is a set of group fixing and variable fixing rules. These fixing rules rely mainly on information from the linear relaxation of the given problem and aim to generate reduced critical subproblem to be solved by the ILP solver. Additional strategies are used to explore the space of the reduced problems. Extensive experimental studies over two sets of 37 MMKP benchmark instances in the literature show that our approach competes favorably with the most recent state-of-the-art algorithms. In particular, for the set of 27 conventional benchmarks, the proposed approach finds an improved best lower bound for 11 instances and as a by-product improves all the previous best upper bounds. For the 10 additional instances with irregular structures, the method improves 7 best known results.     

The nurse rostering problem: A critical appraisal of the problem structure

This paper is concerned with the problem of nurse rostering within hospitals. We analyse a class of four benchmark instances from the nurse rostering literature to provide insight into the nature of the problem. By highlighting the structure of the problem we are able to reduce the relevant solution space. A mixed integer linear programme is then able to find optimal solutions to all four instances of this class of benchmark problems, each within half an hour. Our second contribution is to extend current mathematical approaches to nurse rostering to take better account of the practical considerations. We provide a methodology for handling rostering constraints and preferences arising from the continuity from one scheduling period to the next.     

Multi-depot Multiple TSP: a polyhedral study and computational results

We study the Multi-Depot Multiple Traveling Salesman Problem (MDMTSP), which is a variant of the very well-known Traveling Salesman Problem (TSP). In the MDMTSP an unlimited number of salesmen have to visit a set of customers using routes that can be based on a subset of available depots. The MDMTSP is an NP-hard problem because it includes the TSP as a particular case when the distances satisfy the triangular inequality. The problem has some real applications and is closely related to other important multi-depot routing problems, like the Multi-Depot Vehicle Routing Problem and the Location Routing Problem. We present an integer linear formulation for the MDMTSP and strengthen it with the introduction of several families of valid inequalities. Certain facet-inducing inequalities for the TSP polyhedron can be used to derive facet-inducing inequalities for the MDMTSP. Furthermore, several inequalities that are specific to the MDMTSP are also studied and proved to be facet-inducing. The partial knowledge of the polyhedron has been used to implement a Branch-and-Cut algorithm in which the new inequalities have been shown to be very effective. Computational results show that instances involving up to 255 customers and 25 possible depots can be solved optimally using the proposed methodology.     

A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint

In this paper we consider a variation of the bin packing problem in which bins of different types have different costs and capacities. Furthermore, each bin has to be filled at least to a certain level, which depends on its type. We present a set partitioning formulation and an exact optimization algorithm which exploits column generation and specialized heuristics. We compare our algorithm with the general purpose solver ILOG CPLEX, running on two compact ILP formulations and we report on experimental results on instances we have generated from data-sets for the variable size bin packing problem.