Compact Mixed Integer Programming Research Articles

The Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station: Compact formulation and a General Variable Neighborhood Search metaheuristic

In this article, we introduce the Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station (VRPCDSD), which integrates pickup and delivery routing decisions with the cross-docking scheduling of the unloading and reloading of requests. We present a compact Mixed Integer Programming (MIP) formulation and a General Variable Neighborhood Search (GVNS) metaheuristic for the problem. The proposed GVNS employs several routing and scheduling neighborhoods that require problem-specific adjustments due to the integrated characteristic of the problem. Computational experiments indicate that the MIP formulation can solve instances with up to 31 nodes (15 requests) within the specified time limit when using a standard commercial solver. Furthermore, the GVNS metaheuristic can obtain high-quality solutions that at least match those encountered by the solver for the tested instances, achieving considerably large improvements over the latter especially as the number of requests increases. In fact, GVNS is able to find the optimal solution for all the instances for which the optimal is known. In addition, GVNS is a viable alternative for tackling larger instances of VRPCDSD and is very robust in that it obtains low deviations from the best-encountered solutions when multiple executions are performed. We also present some insights on how the different cross-docking operation costs impact the optimal solutions.

Exact and heuristic approaches for maximizing flows in UAV-enabled wireless cellular networks with multi-hop backhauls

This paper investigates the problem of data routing in backhaul networks using Unmanned Aerial Vehicles (UAVs) to relay data from Small Cells (SCs) to the core network. The objective is to maximize the total fulfilled demand of data to be routed, while ensuring technical requirements such as hop constraints and edge capacity. The problem is formulated using a compact mixed-integer programming model, which can solve small- and medium-sized topologies. In addition, a fast constructive heuristic based on a maximal tree is developed to solve large-scale topologies, resulting in a significant reduction in CPU time. The quality of the heuristic is evaluated by using column generation for solving the linear programming relaxation of an exponential formulation. The computational study shows the effectiveness and value of the proposed compact model and constructive heuristic for various topology sizes. Furthermore, experiments demonstrate that by keeping the network setup constant and updating the demand vector only, the computational time of the compact model can be drastically reduced for all topology sizes.

Two extended formulations for the virtual network function placement and routing problem

AbstractGiven a bi‐directed graph modeling a telecommunication network, and a set of origin‐destination pairs representing traffic requests (commodities) along with their associated Service Function Chains (SFCs), the Virtual Network Function Placement and Routing Problem (VNFPRP) aims to find, for each commodity, one latency‐constrained routing path that visits the required Virtual Network Functions in a specific order. The function installation costs together with the node activation costs have to be minimized. In this paper, we present two extended Mixed Integer Programming (MIP) formulations to model the VNFPRP. For each formulation we define the master problem, the pricing problem, the associated Lagrangian bound and a specific branching scheme, in order to derive an efficient Branch‐and‐Price algorithm. We also provide several families of valid inequalities to strengthen the LP‐relaxation bounds. Computational results are reported comparing the performance of the two Branch‐and‐Price algorithms with a compact MIP formulation and its Branch‐and‐Benders‐cut implementation on a set of SNDlib instances representing telecommunication networks.

The Backhaul Profit Maximization Problem: Optimization Models and Solution Procedures

We present a compact mixed integer program (MIP) for the backhaul profit maximization problem in which a freight carrier seeks to generate profit from an empty delivery vehicle’s backhaul trip from its last scheduled delivery to its depot by allowing it to deviate from the least expensive (or fastest) route to accept pickup-and-delivery requests between various points on the route as allowed by its capacity and required return time. The MIP is inspired by a novel representation of multicommodity flow that significantly reduces the size of the constraint matrix compared with a formulation based on the classical node-arc representation. This, in turn, leads to faster solution times when using a state-of-the-art MIP solver. In an empirical study of both formulations, problem instances with 10 potential pickup/drop-off locations and up to 72 pickup-and-delivery requests were solved an average 1.44 times faster in real time with our formulation, whereas instances with 20 locations and up to 332 pickup-and-delivery requests were solved an average of 11.88 times faster. The largest instances in the comparative study had 60 locations and up to 3,267 pickup-and-delivery requests; these instances required an average of more than 54 hours of real time to solve with the node-arc–based formulation but were solved in an average of under two hours of real time using our compact formulation. We also present a heuristic algorithm based on our compact formulation that finds near optimal solutions to each of the 60-location instances within 22 minutes of real time and near optimal solutions to instances with up to 80 locations within four and a half hours of real time.

Compact mixed-integer programming formulations in quadratic optimization

Compact mixed-integer programming formulations in quadratic optimization

A branch-and-price algorithm for location-routing problems with pick-up stations in the last-mile distribution system

In catering to the needs of the growing e-commerce demand and environmentally friendly urban delivery solutions, package delivery companies are increasingly focusing on user experience and sustainable operations, such as alternative delivery methods (e.g., pick-up stations) and green delivery vehicles (e.g., electric vehicles). We consider designing a location-routing problem with pick-up stations (LRP-PS) where the location of pick-up stations to open and the delivery plan of green vehicles (GVs) are optimized simultaneously. The purpose is to satisfy the total demand while minimizing the sum of opening cost and handling cost of pick-up stations and the fixed cost and routing cost of GVs. Then, we present a compact mixed-integer programming formulation to define the LRP-PS and reveal some interesting properties of the optimal solution. We propose an effective branch-and-price (B&P) algorithm to solve the LRP-PS, which is demonstrated to outperform greatly commercial branch-and-cut solvers such as CPLEX in the computational study. Finally, through a comprehensive analysis of several key parameters (e.g., coverage ranges of pick-up stations and battery driving ranges of GVs), we assess the impact of pick-up stations on this last-mile distribution system and provide some helpful business insights for package delivery companies.

Recoverable robust representatives selection problems with discrete budgeted uncertainty

Recoverable robust optimization is a multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We analyze this approach for a class of selection problems. The aim is to choose a fixed number of items from several disjoint sets, such that the worst-case costs after taking a recovery action are as small as possible. The uncertainty is modeled as a discrete budgeted set, where the adversary can increase the costs of a fixed number of items. While special cases of this problem have been studied before, its complexity has remained open. In this work we make several contributions towards closing this gap. We show that the problem is NP-hard and identify a special case that remains solvable in polynomial time. We provide a compact mixed-integer programming formulation and two additional extended formulations. Finally, computational results are provided that compare the efficiency of different exact solution approaches.

Dominance inequalities for scheduling around an unrestrictive common due date

The problem considered in this work consists in scheduling a set of tasks on a single machine, around an unrestrictive common due date to minimize the weighted sum of earliness and tardiness. This problem can be formulated as a compact mixed integer program (MIP). In this article, we focus on neighborhood-based dominance properties, where the neighborhood is associated to insert and swap operations. We derive from these properties a local search procedure providing a very good heuristic solution. The main contribution of this work stands in an exact solving context: we derive constraints eliminating the non locally optimal solutions with respect to the insert and swap operations. We propose linear inequalities translating these constraints to strengthen the MIP compact formulation. These inequalities, called dominance inequalities, are different from standard reinforcement inequalities. We provide a numerical analysis which shows that adding these inequalities significantly reduces the computation time required for solving the scheduling problem using a standard solver.

Mallows-Smoothed Distribution over Rankings Approach for Modeling Choice

Mallows-Smoothed Distribution over Rankings Approach for Modeling Choice

Compact formulations and an iterated local search-based matheuristic for the minimum weighted feedback vertex set problem

Given a weighted graph G=(V,E), the minimum weighted feedback vertex set problem consists in obtaining a minimum weight subset F⊆V of the vertex set whose removal makes the graph acyclic. Differently from other approaches in the literature, in this work we tackle this problem via the maximum weighted induced forest problem. First, we propose two new compact mixed integer programming (MIP) formulations, using a polynomial number of variables and constraints. Next, we develop a matheuristic that hybridizes a multi-start iterated local search heuristic with a MIP-based local search procedure. Extensive computational experiments carried out on a set of benchmark instances show that the newly proposed MILS+-mtz matheuristic is extremely effective and outperforms or at least match the best heuristics available in the literature in terms of solution quality for 79 out of 99 (79.80%) large instance groups, with 55 (55.56%) of them being strictly better than the previously best known. Furthermore, the compact formulations were able to optimally solve 474 out of 810 test instances in less than 3600 seconds of running time.

A* Search for Prize-Collecting Job Sequencing with One Common and Multiple Secondary Resources

We consider a sequencing problem with time windows, in which a subset of a given set of jobs shall be scheduled. A scheduled job has to execute without preemption and during this time, the job needs both a common resource for a part of the execution as well as a secondary resource for the whole execution time. The common resource is shared by all jobs while a secondary resource is shared only by a subset of the jobs. Each job has one or more time windows and due to these, it is not possible to schedule all jobs. Instead, each job is associated with a prize and the task is to select a subset of jobs which yields a feasible schedule with a maximum sum of prizes. First, we argue that the problem is NP-hard. Then, we present an exact A* algorithm and derive different upper bounds for the total prize; these bounds are based on constraint and Lagrangian relaxations of a linear programming relaxation of a multidimensional knapsack problem. For comparison, a compact mixed integer programming (MIP) model and a constraint programming model are also presented. An extensive experimental evaluation on three types of problem instances shows that the A* algorithm outperforms the other approaches and is able to solve small to medium size instances with up to about 40 jobs to proven optimality. In cases where A* does not prove that an optimal solution is found, the obtained upper bounds are stronger than those of the MIP model.

The two-echelon vehicle routing problem with covering options: City logistics with cargo bikes and parcel lockers

We introduce the two-echelon vehicle routing problem with covering options (2E-VRP-CO). This problem arises in sustainable applications for e-commerce and city distribution. In the first echelon, trucks depart from a single depot and transport goods to two types of locations. At covering locations, such as parcel lockers, customers can pick up goods themselves. At satellite locations, goods are transferred to zero-emission vehicles (such as cargo bikes) that deliver to customers. If desired, customers can indicate their choice for delivery. The 2E-VRP-CO aims at finding cost-minimizing solutions by selecting locations and routes to serve all customers. We present a compact mixed integer programming formulation and an efficient and tailored adaptive large neighborhood search heuristic that provides high-quality, and often optimal, solutions to the 2E-VRP-CO. The 2E-VRP-CO has as special cases the two-echelon vehicle routing problem, and the simultaneous facility location and vehicle routing problem without duration constraints. On these special cases, for which our heuristic predominantly solves the established benchmark instances either to optimality or to the best-known solution, our heuristic finds three new best-known solutions. Moreover, we introduce a new set of benchmark instances for the 2E-VRP-CO and provide managerial insights when distribution via both satellite and covering locations is most beneficial. Our results indicate that customers in the same area are best-served either via cargo-bikes or parcel lockers (i.e., not both), and that the use of parcel lockers has a great potential to reduce driving distance.

Branch-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows

The split delivery vehicle routing problem with time windows (SDVRPTW) is a notoriously hard combinatorial optimization problem. First, it is hard to find a useful compact mixed-integer programming (MIP) formulation for the SDVRPTW. Standard modeling approaches either suffer from inherent symmetries (mixed-integer programs with a vehicle index) or cannot exactly capture all aspects of feasibility. Because of the possibility to visit customers more than once, the standard mechanisms to propagate load and time along the routes fail. Second, the lack of useful formulations has rendered any direct MIP-based approach impossible. Up to now, the most effective exact algorithms for the SDVRPTW have been branch-and-price-and-cut approaches using path-based formulations. In this paper, we propose a new and tailored branch-and-cut algorithm to solve the SDVRPTW. It is based on a new, relaxed compact model, in which some integer solutions are infeasible for the SDVRPTW. We use known and introduce some new classes of valid inequalities to cut off such infeasible solutions. One new class is path-matching constraints that generalize infeasible-path constraints. However, even with the valid inequalities, some integer solutions to the new compact formulation remain to be tested for feasibility. For a given integer solution, we build a generally sparse subnetwork of the original instance. On this subnetwork, all time-window-feasible routes can be enumerated, and a path-based residual problem then solved to decide on the selection of routes, the delivery quantities, and thereby the overall feasibility. All infeasible solutions need to be cut off. For this reason, we derive some strengthened feasibility cuts exploiting the fact that solutions often decompose into clusters. Computational experiments show that the new approach is able to prove optimality for several previously unsolved instances from the literature. The e-companion is available at https://doi.org/10.1287/trsc.2018.0825 .

Probabilistic Partial Set Covering with an Oracle for Chance Constraints

We consider a class of chance-constrained combinatorial optimization problems, which we refer to as probabilistic partial set-covering (PPSC) problems. Given a prespecified risk level $\epsilon\in[0,1]$, the PPSC problem aims to find the minimum cost selection of subsets of items such that a target number of items is covered with probability at least $1-\epsilon$. We show that PPSC admits an efficient probability oracle that computes the coverage probability exactly, under certain distributions of the random variables representing the coverage relation. Using this oracle, we give a compact mixed-integer program that solves the PPSC problem for a special case. For large-scale instances for which an exact oracle-based method exhibits slow convergence, we propose a sampling-based approach that exploits the special structure of PPSC. The oracle can be used as a tool for checking and fixing the feasibility of the solution given by this approach. In particular, we introduce a new class of facet-defining inequalities for a submodular substructure of PPSC, and show that a sampling-based algorithm coupled with the probability oracle effectively provides high-quality feasible solutions to the large-scale test instances.

Exact methods for the quay crane scheduling problem when tasks are modeled at the single container level

The scheduling of quay cranes (QCs) to minimize the handling time of a berthed vessel is one of the most important operations in container terminals as it impacts the terminal’s overall productivity. In this paper, we propose two exact methods to solve the quay crane scheduling problem (QCSP) where a task is defined as handling a single container and subject to different technical constraints including QCs’ safety margin, non-crossing, initial position, and nonzero traveling time. The first method is based on two versions of a compact mixed-integer programming formulation that can solve large problem instances using a general purpose solver. The second is a combination of some constraints of the proposed mathematical model and the binary search algorithm to reduce the CPU time, and solve more efficiently large-sized problems. Unlike existing studies, the computational study demonstrates that both methods can reach optimal solutions for large-sized instances and validates their dominance compared to an exact model proposed in the literature which finds solutions only for small problems.

Mallows-Smoothed Distribution Over Rankings Approach for Modeling Choice

Assortment optimization is an important problem that arises in many applications including retailing and online advertising. The goal in such problems is to determine a revenue/profit maximizing subset of products to offer from a large universe of products when customers exhibit stochastic substitution behavior. We consider a mixture of Mallows model for demand, which can be viewed as a “smoothed” generalization of the class of sparse rank-based choice models, designed to overcome some of its key limitations. In spite of these advantages, the Mallows distribution has an exponential support size and does not admit a closed-form expression for choice probabilities. We first conduct a case study using a publicly available data set involving real-world preferences on sushi types to show the benefits of Mallows-based smoothing. We show that smoothing significantly improves both the prediction and the decision accuracy on this data set. We then present an efficient procedure to compute the choice probabilities for any assortment under the mixture of Mallows model. Surprisingly, this finding allows to formulate a compact mixed integer program (MIP) that leads to a practical approach for solving the constrained assortment optimization problem under a general mixture of Mallows model. To complement this MIP formulation, we also exploit additional structural properties of the underlying distribution to propose several polynomial-time approximation schemes. These are the first efficient algorithms with provably near-optimal performance guarantees for the assortment optimization problem under the Mallows or the mixture of Mallows model in such generality.

Assortment Optimization Under a Single Transition Model

In this paper, we consider a Markov chain choice model with single transition. In this model, customers arrive at each product with a certain probability. If the arrived product is unavailable, then the seller can recommend a subset of available products to the customer and the customer will purchase one of the recommended products or choose not to purchase with certain transition probabilities. The distinguishing features of the model are that the seller can control which products to recommend depending on the arrived product and that each customer either purchases a product or leaves the market after one transition. We study the assortment optimization problem under this model. Particularly, we show that this problem is generally NP-Hard even if each product could only transit to at most two products. Despite the complexity of the problem, we provide polynomial time algorithms for several special cases, such as when the transition probabilities are homogeneous with respect to the starting point, or when each product can only transit to one other product. We also provide a tight performance bound for revenue-ordered assortments. In addition, we propose a compact mixed integer program formulation that can solve this problem of large size. Through extensive numerical experiments, we show that the proposed algorithms can solve the problem efficiently and the obtained assortments could significantly improve the revenue of the seller than under the Markov chain choice model.

Projected mixed integer programming formulations for unit commitment problem

Abstract The thermal unit commitment (UC) problem is a large-scale mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale instances. This paper presents a projected reformulation for UC problem. After projecting the power output of unit onto [0,1], a novel MIQP reformulation, denoted as P-MIQP, can be formed. The obtained P-MIQP is tighter than traditional MIQP formulation of UC problem. And the reduced problem of P-MIQP, which is eventually solved by solvers such as CPLEX, is compacter than that of traditional MIQP. In addition, two mixed integer linear programming (MILP) formulations can be obtained from traditional MIQP and our P-MIQP of UC by replacing the quadratic terms in the objective functions with a sequence of piece-wise perspective-cuts. Projected MILP is also tighter and compacter than the traditional MILP due to the same reason of MIQP. The simulation results for realistic instances that range in size from 10 to 200 units over a scheduling period of 24 h show that the projected reformulation yields tight and compact mixed integer programming UC formulations, which are competitive with currently traditional ones.

Efficient MIP formulation for optimal 1-for-N diversity coding

Abstract In the paper we consider a communication network that uses diversity coding in order to achieve reliability. Having a set of demands and a network topology we face a problem of optimal routing of the demands and backup trees, and associations between the demands and the backup trees. We present a compact mixed integer programming (MIP) formulation for the optimization problem, which proves to be more efficient than other approaches that can be found in the literature.

MIP models for minimizing total tardiness in a two-machine flow shop

We propose compact mixed-integer programming models for the -hard problem of minimizing tardiness in a two-machine flow shop. Also, we propose valid inequalities that aim at tightening the models’ representations. We provide empirical evidence that the linear programming relaxation of an enhanced formulation yields an excellent lower bound that consistently outperforms the best bound from the literature. We further provide the results of extensive computational experiments that attest to the efficacy of the proposed MIP models. In particular, our computational study reveals that most of the 30-job hard instances are optimally solved using the proposed MIP models. Furthermore, we found that even much larger instances, with up to 70 jobs, can be solved for several problem classes.