Set Partitioning Model Research Articles

Tools for primal degenerate linear programs: IPS, DCA, and PE

This paper describes three recent tools for dealing with primal degeneracy in linear programming. The first one is the improved primal simplex (IPS) algorithm which turns degeneracy into a possible advantage. The constraints of the original problem are dynamically partitioned based on the numerical values of the current basic variables. The idea is to work only with those constraints that correspond to nondegenerate basic variables. This leads to a row-reduced problem which decreases the size of the current working basis. The main feature of IPS is that it provides a nondegenerate pivot at every iteration of the solution process until optimality is reached. To achieve such a result, a negative reduced cost convex combination of the variables at their bounds is selected, if any. This pricing step provides a necessary and sufficient optimality condition for linear programming. The second tool is the dynamic constraint aggregation (DCA), a constructive strategy specifically designed for set partitioning constraints. It heuristically aims to achieve the properties provided by the IPS methodology. We bridge the similarities and differences of IPS and DCA on set partitioning models. The final tool is the positive edge (PE) rule. It capitalizes on the compatibility definition to determine the status of a column vector and the associated variable during the reduced cost computation. Within IPS, the selection of a compatible variable to enter the basis ensures a nondegenerate pivot, hence PE permits a trade-off between strict improvement and high, reduced cost degenerate pivots. This added value is obtained without explicitly computing the updated column components in the simplex tableau. Ultimately, we establish tight bonds between these three tools by going back to the linear algebra framework from which emanates the so-called concept of subspace basis.

An exact algorithm for the multi-trip vehicle routing and scheduling problem of pickup and delivery of customers to the airport

Door-to-Door service of Pickup and Delivery of Customers to the Airport (D2PDCA) is a new service provided by certain Airline Ticket Sales Agencies (ATSAs) in China. This new service provides an attractive alternative way by picking up customer at this/her specified position and at any time he/she preferred and delivering to the airport more conveniently than airport shuttle and thus earn high customer service quality. Compared with the single-trip mode, the multi-trip mode of D2PDCA (MTM-D2PDCA) service can reduce travel distances, the number of vehicles required and the operating cost. To obtain the exact solution of the MTM-D2PDCA problem, we propose a novel, exact algorithm based on the trip-chain-oriented set-partitioning (TCO-SP) model, where a trip-chain represents multiple trips made by a specific vehicle. In the exact algorithm, we propose an improved label-correcting method to remove infeasible trip-chains quickly and thus speed the search process. Based on the feasible trip-chains, the MTM-D2PDCA problem is formulated as the novel TCO-SP model, which can be solved exactly by the optimization software CPLEX. In addition, we present several mathematical insights into the relationship between the number of trip-chains and the number of local optimal trips that are applicable in both theory and practice. Extensive experiments are conducted to illustrate the application of the model and demonstrate the cost savings of the MTM-D2PDCA mode over the single-trip mode and provide managerial insights into successfully operating a MTM-D2PDCA service.

Multi-vehicle sequential resource allocation for a nonprofit distribution system

This article introduces a multi-vehicle sequential allocation problem that considers two critical objectives for nonprofit operations: providing equitable service and minimizing unused donations. This problem is motivated by an application in food redistribution from donors such as restaurants and grocery stores to agencies such as soup kitchens and homeless shelters. A set partitioning model is formulated that can be used to design vehicle routes; it primarily focuses on equity maximization and implicitly considers waste. The behavior of the model in clustering agencies and donors on routes is studied, and the impacts of demand variability and supply availability on route composition and solution performance are analyzed. A comprehensive numerical study is performed in order to develop insights on optimal solutions. Based on this study, an efficient decomposition-based heuristic for the problem that can handle an additional constraint on route length is developed and it is shown that the heuristic obtains high-quality solutions in terms of equity and waste.

A branch-cut-and-price algorithm for the vehicle routing problem with stochastic demands

This paper proposes a state-of-the-art branch-cut-and-price algorithm for the vehicle routing problem with stochastic demands (VRPSD). We adapt the model of Christiansen and Lysgaard [6] and formulate the VRPSD as a set partitioning model with additional constraints. Feasible routes are generated using a dynamic programming algorithm executed over a state-space graph. Our method combines 2-cycle elimination with ng-routes. In addition, our pricing problem is significantly accelerated by the introduction of a new aggregate dominance rule. To speed up the generation of negative reduced cost columns, we use a tabu search heuristic and a bidirectional labeling algorithm. We also add capacity and subset-row inequalities dynamically in order to strengthen the linear relaxation of the master problem. As extensive computational tests illustrate, our algorithm is very competitive with the one of [6]. We solve 20 additional instances from the 40-instance set considered by these authors and we considerably improve the computing times for instances already closed. We also solve 17 new instances from the literature.

Integrated Optimization of Airport Taxiway and Runway Scheduling

Capacity limitation of runway and taxiway in an airport is often the major limiting factor of air traffic operation. The congestions on the apron of an airport cause severe delay on aircraft schedule. This paper proposes an integrated algorithm to solve problems of runway scheduling and taxiway routing simultaneously for both departure and arrival aircrafts. A set partitioning model with side constraint is proposed in which each possible aircraft route in the taxiway and runway is regarded as a decision variable. Beside traditional set partitioning constraints, two types of constraints are proposed to maintain a minimum separation distance between aircrafts in the taxiway and runway. The preliminary results show that our integrated algorithm outperforms the sequential method. 

An employee transporting problem

Abstract An employee transporting problem is described and a set partitioning model is developed. An investigation of the model leads to a knapsack problem as a surrogate problem. Finding a partition corresponding to the knapsack problem provides a solution to the problem. An exact algorithm is proposed to obtain a partition (subset-vehicle combination) corresponding to the knapsack solution. It requires testing and matching too many alternatives to obtain a partition. The sweep algorithm is implemented in obtaining a partition (subset-vehicle combination) in an efficient manner. Illustrations are provided to show how the algorithms obtain solutions.

Balancing reconfigurable machining lines via a set partitioning model

We consider the problem of constructing an optimal machining line as a sequence of workstations performing specific sets of operations. The line is required to satisfy the given precedence order on operations, inclusion, exclusion and accessibility constraints. The cycle times of workstations are computed taking into account the processing and sequence dependent set-up times of operations and must not exceed the given bound. For solving the problem, we propose a reduction of this machining line design problem to a set partitioning type problem. This approach implies generating all possible workstations and, for each workstation, solving a scheduling problem in order to find the sequence of operations which minimises the total set-up time. To do this, a dynamic programming algorithm is developed. Several preprocessing procedures are suggested to reduce the number of workstations. A set of exact algorithms are proposed to solve the obtained set partitioning type problem: a constraint generation, a branch and cut, and a parallel branch and cut algorithms. Our experimental investigation demonstrated that the proposed method achieves a significant improvement in CPU times over those required by a previous study that is based on a different model formulation, and also, that the proposed method is able to solve large-sized problem instances.

Branch-and-price for staff rostering: An efficient implementation using generic programming and nested column generation

We present a novel generic programming implementation of a column-generation algorithm for the generalized staff rostering problem. The problem is represented as a generalized set partitioning model, which is able to capture commonly occurring problem characteristics given in the literature. Columns of the set partitioning problem are generated dynamically by solving a pricing subproblem, and constraint branching in a branch-and-bound framework is used to enforce integrality. The pricing problem is formulated as a novel three-stage nested shortest path problem with resource constraints that exploits the inherent problem structure. A very efficient implementation of this pricing problem is achieved by using generic programming principles in which careful use of the C++ pre-processor allows the generator to be customized for the target problem at compile-time. As well as decreasing run times, this new approach creates a more flexible modeling framework that is well suited to handling the variety of problems found in staff rostering. Comparison with a more-standard run-time customization approach shows that speedups of around a factor of 20 are achieved using our new approach. The adaption to a new problem is simple and the implementation is automatically adjusted internally according to the new definition. We present results for three practical rostering problems. The approach captures all features of each problem and is able to provide high-quality solutions in less than 15minutes. In two of the three instances, the optimal solution is found within this time frame.

A hybrid algorithm for a class of vehicle routing problems

In this work we propose a hybrid algorithm for a class of Vehicle Routing Problems with homogeneous fleet. A sequence of Set Partitioning (SP) models, with columns corresponding to routes found by a metaheuristic approach, are solved, not necessarily to optimality, using a Mixed Integer Programming (MIP) solver, that may interact with the metaheuristic during its execution. Moreover, we developed a reactive mechanism that dynamically controls the dimension of the SP models when dealing with large size instances. The algorithm was extensively tested on benchmark instances of the following Vechicle Routing Problem (VRP) variants: (i) Capacitated VRP; (ii) Asymmetric VRP; (iii) Open VRP; (iv) VRP with Simultaneous Pickup and Delivery; (v) VRP with Mixed Pickup and Delivery; (vi) Multi-depot VRP; (vii) Multi-depot VRP with Mixed Pickup and Delivery. The results obtained were quite competitive with those found by heuristics devoted to specific variants. A number of new best solutions were obtained.

A specialized column generation approach for a vehicle routing problem with demand allocation

Motivated by logistical operations for a food bank, this paper addresses a class of vehicle routing problems with demand allocation considerations over a network of partner agencies locations and candidate delivery sites. Any delivery tour starts at a central depot operated by the food bank and selected delivery sites are sequentially visited in order to supply goods to a set of partner agencies who travel from their respective locations to their assigned delivery sites. The problem is modelled as a mixed-integer programme with the objective of minimizing a weighted average of the distances travelled by delivery vehicles and partner agencies, and is tackled via two heuristics. First, a relax-and-fix heuristic is presented for the proposed model and is computationally enhanced using two symmetry-defeating strategies. Second, the problem is reformulated as a set partitioning model with side packing constraints that prompts a specialized column generation approach. Computational experience is provided using realistic data instances to demonstrate the usefulness of the proposed heuristics and the importance of integrated solution techniques for this class of problems.

A Fuel Distribution Problem – Application of New Multi-item Inventory Routing Formulation

This paper presents a new approach to solving the inventory routing problem. This approach could be classified as a first cluster -second route. In the first step a new formulation of mixed integer programming is used to determine the partitioning of customers as well as dates and quantities of deliveries. Then, using any algorithm for solving the traveling salesperson problem the optimal routes for each partition are determined. Although in the first step the set partitioning model is applied, one of the main components of the objective function is approximate routes length. While the routing and inventory costs are considered parallel, the formulation is not so complex compared to classical approaches with exact route length calculation. The presented formulation has been used to solve the problem of determining a delivery plan from a fuel base to 36 fuel stations of one network operating in Poland. In addition in this formulation more than one item is taken into consideration.

Relajacion lagrangeana para el problema de particionamiento de áreas geográficas

Entre las metodologías utilizadas en el particionamiento territorial, destacan los modelos de localización-asignación (“location-allocation”) y los de particionamiento de conjuntos ("set partitioning”), que agrupan pequeñas áreas geográficas llamadas unidades básicas en un número dado de grupos geográficos más grandes, denominados territorios. El problema de particionamiento territorial se modela como un problema de la p-mediana. Se utiliza un enfoque derelajación Lagrangeana para obtener cotas inferiores de la solución óptima y un procedimiento para la obtención de cotas superiores. Para evaluar el desempeño de la metología propuesta, se utilizan instancias de dos ciudades de México. Los resultados obtenidos se comparan con otros métodos de particionamiento de la literatura. De acuerdo con los resultados obtenidos para estas instancias, utilizando distintos números de grupos, se observa que se pueden obtener soluciones factibles de muy buena calidad con un esfuerzo computacional razonable.

Balancing reconfigurable machining lines by means of set partitioning model

We consider the problem of construction of an optimal reconfigurable machining line as a sequence of workstations performing specific sets of operations. The line is required to satisfy the given precedence order on operations, inclusion, exclusion and accessibility constraints. The cycle times of workstations are computed taking into account the processing and setup times of operations and must not exceed the given bound. For solving the problem, we propose a reduction of this machining line design problem to a set partitioning type problem. This approach implies generating all possible workstations and finding optimal sequence of operations in each workstation. To do this, a dynamic programming algorithm is developed. Several preprocessing procedures are suggested for reducing the number of workstations. For solving the obtained set partitioning type problem, a constraint generation algorithm based on mixed integer programming model is proposed.

A hybrid algorithm for the Heterogeneous Fleet Vehicle Routing Problem

This paper deals with the Heterogeneous Fleet Vehicle Routing Problem (HFVRP). The HFVRP generalizes the classical Capacitated Vehicle Routing Problem by considering the existence of different vehicle types, with distinct capacities and costs. The objective is to determine the best fleet composition as well as the set of routes that minimize the total costs. The proposed hybrid algorithm is composed by an Iterated Local Search (ILS) based heuristic and a Set Partitioning (SP) formulation. The SP model is solved by means of a Mixed Integer Programming solver that interactively calls the ILS heuristic during its execution. The developed algorithm was tested in benchmark instances with up to 360 customers. The results obtained are quite competitive with those found in the literature and new improved solutions are reported.

Conservative scales in packing problems

Packing problems (sometimes also called cutting problems) are combinatorial optimization problems concerned with placement of objects (items) in one or several containers. Some packing problems are special cases of several other problems such as resource-constrained scheduling, capacitated vehicle routing, etc. In this paper we consider a bounding technique for one- and higher-dimensional orthogonal packing problems, called conservative scales (CS) (in the scheduling terminology, redundant resources). CS are related to the possible structure of resource consumption: filling of a bin, distribution of the resource to the jobs, etc. In terms of packing, CS are modified item sizes such that the set of feasible packings is not reduced. In fact, every CS represents a valid inequality for a certain binary knapsack polyhedron. CS correspond to dual variables of the set-partitioning model of a special 1D cutting-stock problem. Some CS can be constructed by (data-dependent) dual-feasible functions ((D)DFFs). We discuss the relation of CS to DFFs: CS assume that at most one copy of each object can appear in a combination, whereas DFFs allow several copies. The literature has investigated the so-called extremal maximal DFFs (EMDFFs) which should provide very strong CS. Analogously, we introduce the notions of maximal CS (MCS) and extremal maximal CS (EMCS) and show that EMDFFs do not necessarily produce (E)MCS. We propose fast greedy methods to maximize a given CS. Using the fact that EMCS define facets of the binary knapsack polyhedron, we use lifted cover inequalities as EMCS. For higher-dimensional orthogonal packing, we propose a Sequential LP (SLP) method over the set of CS and investigate its convergence. Numerical results are presented.

Set partitioning and packing versus assignment formulations for subassembly matching problems

This paper addresses alternative formulations and model enhancements for two combinatorial optimization problems that arise in subassembly matching problems. The first problem seeks to minimize the total deviation in certain quality characteristics for the resulting final products from a vector of target values, whereas the second aims at maximizing the throughput under specified tolerance restrictions. We propose set partitioning and packing models in concert with a specialized column generation (CG) procedure that significantly outperform alternative assignment-based formulations presented in the literature, even when the latter are enhanced via tailored symmetry-defeating strategies. In particular, we emphasize the critical importance of incorporating a complementary CG feature to consistently produce near-optimal solutions to the proposed set partitioning and packing models. Extensive computational results are presented to demonstrate the relative effectiveness of the different proposed modelling and algorithmic strategies.

New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem

In this paper, we describe an effective exact method for solving both the capacitated vehicle routing problem (cvrp) and the vehicle routing problem with time windows (vrptw) that improves the method proposed by Baldacci et al. [Baldacci, R., N. Christofides, A. Mingozzi. 2008. An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Math. Programming 115(2) 351–385] for the cvrp. The proposed algorithm is based on the set partitioning (SP) formulation of the problem. We introduce a new route relaxation called ng-route, used by different dual ascent heuristics to find near-optimal dual solutions of the LP-relaxation of the SP model. We describe a column-and-cut generation algorithm strengthened by valid inequalities that uses a new strategy for solving the pricing problem. The new ng-route relaxation and the different dual solutions achieved allow us to generate a reduced SP problem containing all routes of any optimal solution that is finally solved by an integer programming solver. The proposed method solves four of the five open Solomon's vrptw instances and significantly improves the running times of state-of-the-art algorithms for both vrptw and cvrp.

A set partitioning approach to shunting

The Vehicle Positioning Problem (VPP), also known as the shunting problem, is a classical combinatorial optimization problem in public transport planning. It has been investigated using several models and approaches, which work well for small instances, but not for large ones. We propose in this article a novel set partitioning model and an associated column generation approach for the VPP and for a multi-period generalization. The main improvement of this model over previous ones is that it provides a tight linear description of the problem that can, in particular, produce non-trivial lower bounds. The pricing problem, and hence the LP relaxation itself, can be solved in polynomial, respectively, pseudo-polynomial time for some versions of the problem. Computational results for large-scale instances are reported.

A column generation heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint

This paper deals with the dynamic multi-item capacitated lot-sizing problem under random period demands (SCLSP). Unfilled demands are backordered and a fill rate constraint is in effect. It is assumed that, according to the static-uncertainty strategy of Bookbinder and Tan [1], all decisions concerning the time and the production quantities are made in advance for the entire planning horizon regardless of the realization of the demands. The problem is approximated with the set partitioning model and a heuristic solution procedure that combines column generation and the recently developed ABC β heuristic is proposed.

Models for the discrete berth allocation problem: A computational comparison

In this paper we consider the problem of allocating arriving ships to discrete berth locations at container terminals. This problem is recognized as one of the most important processes for any container terminal. We review and describe three main models of the discrete dynamic berth allocation problem, improve the performance of one model, and, through extensive numerical tests, compare all models from a computational perspective. The results indicate that a generalized set-partitioning model outperforms all other existing models.