The general routing problem polyhedron: Facets from the RPP and GTSP polyhedra

In this paper we study the polyhedron associated with the General Routing Problem (GRP). This problem, first introduced by Orloff in 1974, is a generalization of both the Rural Postman Problem (RPP) and the Graphical Traveling Salesman Problem (GTSP) and, thus, is NP -hard. We describe a formulation of the problem such that from every non-trivial facet-inducing inequality for the RPP and GTSP polyhedra, we obtain facet-inducing inequalities for the GRP polyhedron. We describe a new family of facet-inducing inequalities for the GRP, the honeycomb constraints, which seem to be very useful for solving GRP and RPP instances. Finally, new classes of facets obtained by composition of facet-inducing inequalities are presented.     

The distance constrained multiple vehicle traveling purchaser problem

In the Distance Constrained Multiple Vehicle Traveling Purchaser Problem (DC-MVTPP) a fleet of vehicles is available to visit suppliers offering products at different prices and with different quantity availabilities. The DC-MVTPP consists in selecting a subset of suppliers so to satisfy products demand at the minimum traveling and purchasing costs, while ensuring that the distance traveled by each vehicle does not exceed a predefined upper bound. The problem generalizes the classical Traveling Purchaser Problem (TPP) and adds new realistic features to the decision problem. In this paper we present different mathematical programming formulations for the problem. A branch-and-price algorithm is also proposed to solve a set partitioning formulation where columns represent feasible routes for the vehicles. At each node of the branch-and-bound tree, the linear relaxation of the set partitioning formulation, augmented by the branching constraints, is solved through column generation. The pricing problem is solved using dynamic programming. A set of instances has been derived from benchmark instances for the asymmetric TPP. Instances with up to 100 suppliers and 200 products have been solved to optimality.     

Minimizing the completion time of a project under resource constraints and feeding precedence relations: a Lagrangian relaxation based lower bound

In this paper we study the Resource Constrained Project Scheduling Problem (RCPSP) with “Feeding Precedence” (FP) constraints and minimum makespan objective. This problem typically arises in production planning environment, like make-to-order manufacturing, where the effort associated with the execution of an activity is not univocally related to its duration percentage and the traditional finish-to-start precedence constraints or the generalized precedence relations cannot completely represent the overlapping among activities. In this context, we need to introduce in the RCPSP the FP constraints. For this problem we propose a new mathematical formulation and define a lower bound based on the Lagrangian relaxation of the resource constraints. A computational experimentation on randomly generated instances of sizes of up to 100 activities shows a better performance of this lower bound when compared to other lower bounds. Moreover, for the optimally solved instances, its value is very close to the optimal one. Furthermore, in order to show the effectiveness of the proposed lower bound on large instances for which the optimal solution is known, we adapted our approach to solve the benchmarks of the basic RCPSP from the PSLIB with 120 activities.     

A node rooted flow-based model for the local access network expansion problem

In this paper, we present a new formulation for the local access network expansion problem. Previously, we have shown that this problem can be seen as an extension of the well-known Capacitated Minimum Spanning Tree Problem and have presented and tested two flow-based models. By including additional information on the definition of the variables, we propose a new flow-based model that permits us to use effectively variable eliminations tests as well as coefficient reduction on some of the constraints. We present computational results for instances with up to 500 nodes in order to show the advantages of the new model in comparison with the others.     

A compact optimization model for the tail assignment problem

This paper investigates a new model for the so-called Tail Assignment Problem, which consists in assigning a well-identified airplane to each flight leg of a given flight schedule, in order to minimize total cost (cost of operating the flights and possible maintenance costs) while complying with a number of operational constraints. The mathematical programming formulation proposed is compact (i.e., involves a number of $0?1$ decision variables and constraints polynomial in the problem size parameters) and is shown to be of significantly reduced dimension as compared with previously known compact models. Computational experiments on series of realistic problem instances (obtained by random sampling from real-world data set) are reported. It is shown that with the proposed model, current state-of-the art MIP solvers can efficiently solve to exact optimality large instances representing 30-day flight schedules with typically up to 40 airplanes and 1500 flight legs connecting as many as 21 airports. The model also includes the main existing types of maintenance constraints, and extensive computational experiments are reported on problem instances of size typical of practical applications.     

A Bus Driver Scheduling Problem: a?new mathematical model and a GRASP approximate solution

This paper addresses the problem of determining the best scheduling for Bus Drivers, a $\mathcal{NP}$ -hard problem consisting of finding the minimum number of drivers to cover a set of Pieces-Of-Work (POWs) subject to a variety of rules and regulations that must be enforced such as spreadover and working time. This problem is known in literature as Crew Scheduling Problem and, in particular in public transportation, it is designated as Bus Driver Scheduling Problem. We propose a new mathematical formulation of a Bus Driver Scheduling Problem under special constraints imposed by Italian transportation rules. Unfortunately, this model can only be usefully applied to small or medium size problem instances. For large instances, a Greedy Randomized Adaptive Search Procedure (GRASP) is proposed. Results are reported for a set of real-word problems and comparison is made with an exact method. Moreover, we report a comparison of the computational results obtained with our GRASP procedure with the results obtained by Huisman et al. (Transp. Sci. 39(4):491?C502, 2005).     

Improving computational capabilities for addressing volume constraints in forest harvest scheduling problems

Forest Harvest Scheduling problems incorporating area-based restrictions have been of great practical interest for several years, but only recently have advances been made that allow them to be efficiently solved. One significant development has made use of formulation strengthening using the Cluster Packing Problem. This improved formulation has allowed medium sized problems to be easily solved, but when restrictions on volume production over time are added, problem difficulty increases substantially. In this paper, we study the degrading effect of certain types of volume constraints and propose methods for reducing this effect. Developed methods include the use of constraint branching, the use of elastic constraints with dynamic penalty adjustment and a simple integer allocation heuristic. Application results are presented to illustrate the computational improvement afforded by the use of these methods.     

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.     

On the Solution of the Inverse Eigenvalue Complementarity Problem

In this paper, we discuss the solution of an Inverse Eigenvalue Complementarity Problem. Two nonlinear formulations are presented for this problem. A necessary and sufficient condition for a stationary point of the first of these formulations to be a solution of the problem is established. On the other hand, to assure global convergence to a solution of this problem when it exists, an enumerative algorithm is designed by exploiting the structure of the second formulation. The use of additional implied constraints for enhancing the efficiency of the algorithm is also discussed. Computational results are provided to highlight the performance of the algorithm.     

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.     

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     

Quartic Formulation of Standard Quadratic Optimization Problems

A standard quadratic optimization problem (StQP) consists of finding the largest or smallest value of a (possibly indefinite) quadratic form over the standard simplex which is the intersection of a hyperplane with the positive orthant. This NP-hard problem has several immediate real-world applications like the Maximum-Clique Problem, and it also occurs in a natural way as a subproblem in quadratic programming with linear constraints. To get rid of the (sign) constraints, we propose a quartic reformulation of StQPs, which is a special case (degree four) of a homogeneous program over the unit sphere. It turns out that while KKT points are not exactly corresponding to each other, there is a one-to-one correspondence between feasible points of the StQP satisfying second-order necessary optimality conditions, to the counterparts in the quartic homogeneous formulation. We supplement this study by showing how exact penalty approaches can be used for finding local solutions satisfying second-order necessary optimality conditions to the quartic problem: we show that the level sets of the penalty function are bounded for a finite value of the penalty parameter which can be fixed in advance, thus establishing exact equivalence of the constrained quartic problem with the unconstrained penalized version.     

New results for the Directed Profitable Rural Postman Problem

We study a generalization of the Directed Rural Postman Problem where not all arcs requiring a service have to be visited provided that a penalty cost is paid if a service arc is not crossed. The problem, known as Directed Profitable Rural Postman Problem, looks for a tour visiting the selected set of service arcs while minimizing both traveling and penalty costs. We add different valid inequalities to a known mathematical formulation of the problem and develop a branch-and-cut algorithm that introduces connectivity constraints both in a “lazy” and in a standard way. We also propose a matheuristic followed by an improvement heuristic (final refinement). The matheuristic exploits information provided by a problem relaxation to select promising service arcs used to solve optimally Directed Rural Postman problems. The ex-post refinement tries to improve the solution provided by the matheuristic using a branch-and-cut algorithm. The method gets a quick convergence through the introduction of connectivity cuts that are not guaranteed to be valid inequalities, and thus may exclude integer feasible solutions.     

A matheuristic approach for solving a home health care problem

We deal with a Home Health Care Problem (HHCP) which objective consists in constructing the optimal routes and rosters for the health care staffs. The challenge lies in combining aspects of vehicle routing and staff rostering which are two well known hard combinatorial optimization problems. To solve this problem, we initially propose an integer linear programming formulation (ILP) and we tested this model on small instances. To deal with larger instances we develop a matheuristic based on the decomposition of the ILP formulation into two problems. The first one is a set partitioning like problem and it represents the rostering part. The second problem consists in the routing part. This latter is equivalent to a Multi-depot Traveling Salesman Problem with Time Windows (MTSPTW).     

Extensions to a Lagrangean relaxation approach for the capacitated warehouse location problem

In this paper we present a lower bound for the capacitated warehouse location problem based upon the Lagrangean relaxation of a mixed-integer formulation of the problem, where we use subgradient optimisation in an attempt to maximise this lower bound. Problem reduction tests based upon this lower bound and the original problem are given. Incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers.     

The Multi-Story Space Assignment Problem

The Multi-Story Space Assignment Problem (MSAP) is an innovative formulation of the multi-story facility assignment problem that allows one to model the location of departments of unequal size within multi-story facilities as a Generalized Quadratic 3-dimensional Assignment Problem (GQ3AP). Not only can the MSAP generate the design of the location of the departments in the facility, the MSAP also includes the evacuation planning for the facility. The formulation, background mathematical development, and computational experience with a branch and bound algorithm for the MSAP are also presented.     

A partial dual algorithm for the capacitated warehouse location problem

The Capacitated Warehouse Location Problem (CWLP) consists of the ordinary transportation problem with the additional feature of a fixed cost associated with each supplier. A supplier can be used towards meeting the demands of the customers only if the corresponding fixed cost is incurred. The problem is to determine which suppliers to use and how the customer demands should be met, so that total cost is minimised.Most of the recently published algorithms for CWLP use branch and bound based on a Lagrangian relaxation of demand constraints. Here, a partial dual of a tight LP formulation is used in order to take advantage of the properties of transportation problems. Computational results are given which show good overall performance of the algorithm, with the size of the tree search being reduced compared with previous published results.     

A review of urban transportation network design problems

This paper presents a comprehensive review of the definitions, classifications, objectives, constraints, network topology decision variables, and solution methods of the Urban Transportation Network Design Problem (UTNDP), which includes both the Road Network Design Problem (RNDP) and the Public Transit Network Design Problem (PTNDP). The current trends and gaps in each class of the problem are discussed and future directions in terms of both modeling and solution approaches are given. This review intends to provide a bigger picture of transportation network design problems, allow comparisons of formulation approaches and solution methods of different problems in various classes of UTNDP, and encourage cross-fertilization between the RNDP and PTNDP research.     

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.     

Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem

The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) combines the Generalized Traveling Salesman Problem (GTSP) and the Sequential Ordering Problem (SOP). We present a novel branching technique for the GTSP which enables the extension of a powerful pruning technique. This is combined with some modifications of known bounding methods for related problems. The algorithm manages to solve problem instances with 12–26 groups within a minute, and instances with around 50 groups which are denser with precedence constraints within 24 h.