An adaptive large neighborhood search algorithm for a selective and periodic inventory routing problem

We study a selective and periodic inventory routing problem (SPIRP) and develop an Adaptive Large Neighborhood Search (ALNS) algorithm for its solution. The problem concerns a biodiesel production facility collecting used vegetable oil from sources, such as restaurants, catering companies and hotels that produce waste vegetable oil in considerable amounts. The facility reuses the collected waste oil as raw material to produce biodiesel. It has to meet certain raw material requirements either from daily collection, or from its inventory, or by purchasing virgin oil. SPIRP involves decisions about which of the present source nodes to include in the collection program, and which periodic (weekly) routing schedule to repeat over an infinite planning horizon. The objective is to minimize the total collection, inventory and purchasing costs while meeting the raw material requirements and operational constraints. A single-commodity flow-based mixed integer linear programming (MILP) model was proposed for this problem in an earlier study. The model was solved with 25 source nodes on a 7-day cyclic planning horizon. In order to tackle larger instances, we develop an ALNS algorithm that is based on a rich neighborhood structure with 11 distinct moves tailored to this problem. We demonstrate the performance of the ALNS, and compare it with the MILP model on test instances containing up to 100 source nodes.     

A comparison of reduced and unreduced KKT systems arising from interior point methods

This paper addresses the solution of a cardinality Boolean quadratic programming problem using three different approaches. The first transforms the original problem into six mixed-integer linear programming (MILP) formulations. The second approach takes one of the MILP formulations and relies on the specific features of an MILP solver, namely using starting incumbents, polishing, and callbacks. The last involves the direct solution of the original problem by solvers that can accomodate the nonlinear combinatorial problem. Particular emphasis is placed on the definition of the MILP reformulations and their comparison with the other approaches. The results indicate that the data of the problem has a strong influence on the performance of the different approaches, and that there are clear-cut approaches that are better for some instances of the data. A detailed analysis of the results is made to identify the most effective approaches for specific instances of the data.     

A mixed integer linear programming model for reliability optimisation in the component deployment problem

Component deployment is a combinatorial optimisation problem in software engineering that aims at finding the best allocation of software components to hardware resources in order to optimise quality attributes, such as reliability. The problem is often constrained because of the limited hardware resources, and the communication network, which may connect only certain resources. Owing to the non-linear nature of the reliability function, current optimisation methods have focused mainly on heuristic or metaheuristic algorithms. These are approximate methods, which find near-optimal solutions in a reasonable amount of time. In this paper, we present a mixed integer linear programming (MILP) formulation of the component deployment problem. We design a set of experiments where we compare the MILP solver to methods previously used to solve this problem. Results show that the MILP solver is efficient in finding feasible solutions even where other methods fail, or prove infeasibility where feasible solutions do not exist.     

Finished-vehicle transporter routing problem solved by loading pattern discovery

This work addresses a new transportation problem in outbound logistics in the automobile industry: the finished-vehicle transporter routing problem (FVTRP). The FVTRP is a practical routing problem with loading constraints, and it assumes that dealers have deterministic demands for finished vehicles that have three-dimensional irregular shapes. The problem solution will identify optimal routes while satisfying demands. In terms of complex packing, finished vehicles are not directly loaded into the spaces of transporters; instead, loading patterns matching finished vehicles with transporters are identified first by mining successful loading records through virtual and manual loading test procedures, such that the packing problem is practically solved with the help of a procedure to discover loading patterns. This work proposes a mixed-integer linear programming (MILP) model for the FVTRP considering loading patterns. As a special class of routing models, the FVTRP is typically difficult to solve within a manageable computing time. Thus, an evolutionary algorithm is designed to solve the FVTRP. Comparisons of the proposed algorithm and a commercial MILP solver demonstrate that the proposed algorithm is more effective in solving medium- and large-scale problems. The proposed scheme for addressing the FVTRP is illustrated with an example and tested with benchmark instances that are derived from well-studied vehicle routing datasets.     

Selective and periodic inventory routing problem for waste vegetable oil collection

We consider a biodiesel production company that collects waste vegetable oil from source points that generate waste in large amounts. The company uses the collected waste as raw material for biodiesel production. The manager of this company needs to decide which of the present source points to include in the collection program, which of them to visit on each day, which periodic routing schedule to repeat over an infinite horizon and how many vehicles to operate such that the total collection, inventory and purchasing costs are minimized while the production requirements and operational constraints are met. For this selective and periodic inventory routing problem, we propose two different formulations, compare them and apply the better performing one on a real-world problem with 36 scenarios. We generate lower bounds using a partial linear relaxation model, and observe that the solutions obtained through our model are within 3.28% of optimality on the average. Several insights regarding the customer selection, routing and purchasing decisions are acquired with sensitivity analysis.     

A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows

This paper presents a novel three-phase heuristic/algorithmic approach for the multi-depot routing problem with time windows and heterogeneous vehicles. It has been derived from embedding a heuristic-based clustering algorithm within a VRPTW optimization framework. To this purpose, a rigorous MILP mathematical model for the VRPTW problem is first introduced. Likewise other optimization approaches, the new formulation can efficiently solve case studies involving at most 25 nodes to optimality. To overcome this limitation, a preprocessing stage clustering nodes together is initially performed to yield a more compact cluster-based MILP problem formulation. In this way, a hierarchical hybrid procedure involving one heuristic and two algorithmic phases was developed. Phase I aims to identifying a set of cost-effective feasible clusters while Phase II assigns clusters to vehicles and sequences them on each tour by using the cluster-based MILP formulation. Ordering nodes within clusters and scheduling vehicle arrival times at customer locations for each tour through solving a small MILP model is finally performed at Phase III. Numerous benchmark problems featuring different sizes, clustered/random customer locations and time window distributions have been solved at acceptable CPU times.     

Mixed-Integer Linear Programming Formulations for the Software Clustering Problem

The clustering problem has an important application in software engineering, which usually deals with large software systems with complex structures. To facilitate the work of software maintainers, components of the system are divided into groups in such a way that the groups formed contain highly-interdependent modules and the independent modules are placed in different groups. The measure used to analyze the quality of the system partition is called Modularization Quality (MQ). Designers represent the software system as a graph where modules are represented by nodes and relationships between modules are represented by edges. This graph is referred in the literature as Module Dependency Graph (MDG). The Software Clustering Problem (SCP) consists in finding the partition of the MDG that maximizes the MQ. In this paper we present three new mathematical programming formulations for the SCP. Firstly, we formulate the SCP as a sum of linear fractional functions problem and then we apply two different linearization procedures to reformulate the problem as Mixed-Integer Linear Programming (MILP) problems. We discuss a preprocessing technique that reduces the size of the original problem and develop valid inequalities that have been shown to be very effective in tightening the formulations. We present numerical results that compare the formulations proposed and compare our results with the solutions obtained by the exhaustive algorithm supported by the freely available Bunch clustering tool, for benchmark problems.     

A new mixed integer linear programming model for the multi level uncapacitated facility location problem

This paper considers the multi level uncapacitated facility location problem (MLUFLP). A new mixed integer linear programming (MILP) formulation is presented and validity of this formulation is given. Experimental results are performed on instances known from literature. The results achieved by CPLEX and Gurobi solvers, based on the proposed MILP formulation, are compared to the results obtained by the same solvers on the already known formulations. The results show that CPLEX and Gurobi can optimally solve all small and medium sized instances and even some large-scale instances using the new formulation.     

The complexity of branch-and-price algorithms for the capacitated vehicle routing problem with stochastic demands

The capacitated vehicle routing problem with stochastic demands (CVRPSD) is a variant of the deterministic capacitated vehicle routing problem where customer demands are random variables. While the most successful formulations for several deterministic vehicle-routing problem variants are based on a set-partitioning formulation, adapting such formulations for the CVRPSD under mild assumptions on the demands remains challenging. In this work we provide an explanation to such challenge, by proving that when demands are given as a finite set of scenarios, solving the LP relaxation of such formulation is strongly NP-Hard. We also prove a hardness result for the case of independent normal demands.     

Robust delay-constrained routing in telecommunications

In telecommunications, operators usually use market surveys and statistical models to estimate traffic evolution in networks or to approximate queuing delay functions in routing strategies. Many research activities concentrated on handling traffic uncertainty in network design. Measurements on real world networks have shown significant errors in delay approximations, leading to weak management decisions in network planning. In this work, we introduce elements of robust optimization theory for delay modeling in routing problems. Different types of data uncertainty are considered and linked to corresponding robust models. We study a special case of constraints featuring separable additive functions. Specifically, we consider that each term of the sum is disturbed by a random parameter. These constraints are frequent in network based problems, where functions reflecting real world measurements on links are summed up over end-to-end paths. While classical robust formulations have to deal with the introduction of new variables, we show that, under specific hypotheses, the deterministic robust counterpart can be formulated in the space of original variables. This offers the possibility of constructing tractable robust models. Starting from Soyster’s conservative model, we write and compare different uncertainty sets and formulations offering each a different protection level for the delay constrained routing problem. Computational experiments are developed in order to evaluate the “price of robustness” and to assess the quality of the new formulations.     

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearizations are tight, but rarely exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation (Eur. J. Oper. Res. 2(3):204–211, 1978) is the smallest of these linearizations, but very weak. In this paper, we analyze how the Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As shown in our numerical experiments, these tightened formulations remain small but computationally effective to solve the QAP by means of general purpose MILP solvers.     

A global MILP model for FMS scheduling

Flexible manufacturing systems (FMS) require intelligent scheduling strategies to achieve their principal benefit — combining high flexibility with high productivity. A mixed-integer linear programming model (MILP) is presented here for FMS scheduling. The model takes a global view of the problem and specifically takes into account constraints on storage and transportation. Both of these constrained resources are critical for practical FMS scheduling problems and are difficult to model. The MILP model is explained and justified and its complexity is discussed. Two heuristic procedures are developed, based on an analysis of the global MILP model. Computational results are presented comparing the performance of the different solution strategies. The development of iterative global heuristics based on mathematical programming formulations is advocated for a wide class of FMS scheduling problems.     

Relaxations and approximations of chance constraints under finite distributions

Optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications. Such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity. In the setting of a finite distribution of the stochastic parameters, an optimization problem with linear chance constraints can be formulated as a mixed integer linear program (MILP). The natural MILP formulation has a weak relaxation bound and is quite difficult to solve. In this paper, we review some recent results on improving the relaxation bounds and constructing approximate solutions for MILP formulations of chance constraints. We also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems. This algorithm uses a relaxation to construct a solution whose (constraint violation) risk level may be larger than the pre-specified threshold, but is within a constant factor of it, and whose objective value is also within a constant factor of the true optimal value. Finally, we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios.     

A parallel clustering technique for the vehicle routing problem with split deliveries and pickups

In this paper, a parallel clustering technique and route construction heuristic have been developed for the vehicle routing problem (VRP) with split deliveries and pickups. An MILP formulation for determining the exact solution to the problem has also been included. It has been shown through extensive experimentation that the algorithm proposed in this paper statistically produces better results than the only heuristic existing for this class of problems in literature. We also form a basis of comparison between this class of problems and the VRP with simultaneous deliveries and pickups. We note that while heuristics for simultaneous deliveries and pickups cannot be applied in situations where customers' delivery or pickup demands exceed the vehicle capacity, heuristics allowing split deliveries and pickups can, in fact, be applied in every situation, even producing superior results under the combined objective of minimization of the fixed charge and mileage associated with vehicle routes. A guideline as to which heuristic could be used under what parametric conditions and objective functions, has also been provided.     

Piecewise-Linear Approximations of Multidimensional Functions

We develop explicit, piecewise-linear formulations of functions f(x):ℝ ⁿ ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

Integer programming formulations of vehicle routing problems

Various mathematical formulations are available for situations represented by vehicle routing problems. The assignment-based integer programming formulations of these problems are more common and easy to understand. Such formulations are discussed in this paper and a much simpler formulation for the vechicle routing problem is presented for the case, when all the vehicles have the same load capacity and maximum allowable cost per route.     

Robust scheduling of parallel machines with sequence-dependent set-up costs

In this paper we propose a robust approach for solving the scheduling problem of parallel machines with sequence-dependent set-up costs. In the literature, several mathematical models and solution methods have been proposed to solve such scheduling problems, but most of which are based on the strong assumption that input data are known in a deterministic way. In this paper, a fuzzy mathematical programming model is formulated by taking into account the uncertainty in processing times to provide the optimal solution as a trade-off between total set-up cost and robustness in demand satisfaction. The proposed approach requires the solution of a non-linear mixed integer programming (NLMIP), that can be formulated as an equivalent mixed integer linear programming (MILP) model. The resulting MILP model in real applications could be intractable due to its NP-hardness. Therefore, we propose a solution method technique, based on the solution of an approximated model, whose dimension is remarkably reduced with respect to the original counterpart. Numerical experiments conducted on the basis of data taken from a real application show that the average deviation of the reduced model solution over the optimum is less than 1.5%.     

Modeling demand management strategies for evacuations

Evacuations are massive operations that create heavy travel demand on road networks some of which are experiencing major congestions even with regular traffic demand. Congestion in traffic networks during evacuations, can be eased either by supply or demand management actions. This study focuses on modeling demand management strategies of optimal departure time, optimal destination choice and optimal zone evacuation scheduling (also known as staggered evacuation) under a given fixed evacuation time assumption. The analytical models are developed for a system optimal dynamic traffic assignment problem, so that their characteristics can be studied to produce insights to be used for large-scale solution algorithms. While the first two strategies were represented in a linear programming (LP) model, evacuation zone scheduling problem inevitable included integers and resulted in a mixed integer LP (MILP) one. The dual of the LP produced an optimal assignment principle, and the nature of the MILP formulations revealed clues about more efficient heuristics. The discussed properties of the models are also supported via numerical results from a hypothetical network example.     

2-Facility manifold location routing problem

The location routing problem (LRP), known to be the combination of the facility location and vehicle routing problems, is solved in the literature by either assuming planar or spherical surfaces. In this work, the manifold location routing problem (MLRP), that is an LRP on Riemannian manifold surfaces, is explained for the 2-facility (2-MLRP) case with the corresponding heuristic algorithm solution. The 2-MLRP problem is a mixed integer non-linear programming problem that is determined to be NP-hard. Special cases of MLRP include LRP on planar surfaces, when the manifold’s curvature is 0, and LRP on spherical surfaces when the curvature of the manifold is 1.