Capacity Inequalities Research Articles

A Neural Separation Algorithm for the Rounded Capacity Inequalities

The cutting plane method is a key technique for successful branch-and-cut and branch-price-and-cut algorithms that find the exact optimal solutions for various vehicle routing problems (VRPs). Among various cuts, the rounded capacity inequalities (RCIs) are the most fundamental. To generate RCIs, we need to solve the separation problem, whose exact solution takes a long time to obtain; therefore, heuristic methods are widely used. We design a learning-based separation heuristic algorithm with graph coarsening that learns the solutions of the exact separation problem with a graph neural network (GNN), which is trained with small instances of 50 to 100 customers. We embed our separation algorithm within the cutting plane method to find a lower bound for the capacitated VRP (CVRP) with up to 1,000 customers. We compare the performance of our approach with CVRPSEP, a popular separation software package for various cuts used in solving VRPs. Our computational results show that our approach finds better lower bounds than CVRPSEP for large-scale problems with 400 or more customers, whereas CVRPSEP shows strong competency for problems with less than 400 customers. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) [RS-2023-00259550] and the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) [2022-0-01032, Development of Collective Collaboration Intelligence Framework for Internet of Autonomous Things]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0310 .

Capacity of rings and mappings generate embeddings of Sobolev spaces

In this paper we give characterizations of mappings generate embeddings of Sobolev spaces in the terms of ring capacity inequalities. In addition we prove that such mappings are Lipschitz mappings in the sub-hyperbolic type capacitary metrics.

Exact separation of the rounded capacity inequalities for the capacitated vehicle routing problem

AbstractThe family of Rounded Capacity (RC) inequalities is one of the most important sets of valid inequalities for the Capacitated Vehicle Routing Problem (CVRP). This paper considers the problem of separation of violated RC inequalities and develops an exact procedure employing mixed integer linear programming. The developed routine is demonstrated to be very efficient for small and medium‐sized problem instances. For larger‐scale problem instances, an iterative approach for exact separation of RC inequalities is developed, based upon a selective variable pricing strategy. The approach combines column and row generation and allows us to introduce variables only when they are needed, which is essential when dealing with large‐scale problem instances. A computational study demonstrates scalability of the proposed separation routines and provides exact RC‐based lower bounds to some of the publicly available unsolved CVRP instances. The same computational study provides RC‐based lower bounds for very large‐scale CVRP instances with more than 3000 locations obtained within appropriate computational time limits.

Hybrid branch-and-price-and-cut algorithm for the two-dimensional vector packing problem with time windows

This paper introduces the two-dimensional vector packing problem with time windows (2DVPPTW). It packs all items into identical bins to minimize the number of used bins. Items are characterized by their weights, volumes, and time windows. Items have different time requirements for delivery in many practical settings. For example, items have their time windows for packing and delivery. The 2DVPPTW is subjected to three constraints: weight, volume, and time windows. We present an integer programming (IP) model for the proposed 2DVPPTW. The IP model is reformulated into the master problem and the subproblem (SP). A hybrid branch-and-price-and-cut (H-BPC) algorithm is developed to solve the 2DVPPTW. On the basis of extensive computational experiments, the proposed H-BPC is significantly effective in comparison with the commercial branch-and-bound/cut solvers, such as CPLEX, and two exact methods. The primary reason for the computational efficiency of the H-BPC is the development of a heuristic algorithm, namely, adaptive large neighborhood search (ALNS), to solve the SP, which is usually computationally expensive. Developing the heuristic algorithm extends the works on the solution to the SPs of bin packing problems and two-dimensional vector packing problems. Two dynamic programming (DP) algorithms are also proposed to solve the SP optimally, namely, the label-setting algorithm and the maximal clique-based DP (MC_DP) algorithm. Moreover, two cooperative schemes are proposed and compared. One is composed of the ALNS and label-setting algorithm, which has a shorter computational time, and the other is composed of the ALNS and MC_DP, which can obtain more optimal solutions. Furthermore, subset-row inequalities, rounded capacity inequalities, multidimensional dual-feasible function, and incompatibility preprocessing policy help to decrease the computational burden dramatically.

Life-course inequalities in intrinsic capacity and healthy ageing, China.

To investigate the contribution of early-life factors on intrinsic capacity of Chinese adults older than 45 years. We used data on 21 783 participants from waves 1 (2011) and 2 (2013) of the China Health and Retirement Longitudinal Study (CHARLS), who also participated in the 2014 CHARLS Life History Survey to calculate a previously validated measure of intrinsic capacity. We considered 11 early-life factors and investigated their direct association with participants' intrinsic capacity later in life, as well as their indirect association through four current socioeconomic factors. We used multivariable linear regression and the decomposition of the concentration index to investigate the contribution of each determinant to intrinsic capacity inequalities. Participants with a favourable environment in early life (that is, parental education, childhood health and neighbourhood environment) had a significantly higher intrinsic capacity score in later life. For example, participants with a literate father recorded a 0.040 (95% confidence interval, CI: 0.020 to 0.051) higher intrinsic capacity score than those with an illiterate father. This inequality was greater for cognitive, sensory and psychological capacities than locomotion and vitality. Overall, early-life factors directly explained 13.92% (95% CI: 12.07 to 15.77) of intrinsic capacity inequalities, and a further 28.57% (95% CI: 28.19 to 28.95) of these inequalities through their influence on current socioeconomic inequalities. Unfavourable early-life factors appear to decrease late-life health status in China, particularly cognitive, sensory and psychological capacities, and these effects are exacerbated by cumulative socioeconomic inequalities over a person's life course.

Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations

In this work, a generalised concept of dependent random variables, i.e., m-widely acceptable random variables, is put forward and some inequalities such as exponential inequality and Rosenthal-type inequality are established under sub-linear expectations. By virtue of these inequalities, we obtain a general result on complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-widely acceptable random variables, which extend and improve some existing ones in classical probability space as well as in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression model under sub-linear expectations are also investigated.

An initialization-free distributed algorithm for dynamic economic dispatch problems in microgrid: Modeling, optimization and analysis

In this paper, a distributed optimization algorithm is designed for a hybrid microgrid network to minimize the total generation cost in a dynamic economic dispatch problem (DEDP). The hybrid microgrid model is constructed with different types of traditional power resources, renewable energy and energy storage batteries, which are subject to the supply–demand balance, capacity, and ramp-rate constraints of the generation facilities. Meantime, from the perspective of environment protection, the pollutant emissions from traditional generators are considered to reduce its impact on environment. Firstly, we transform the multi-objective optimization problem to a single objective optimization problem through the weight-sum method. Then, compared to the most existing centralized algorithms, we propose a fully distributed algorithm that does not depend on the initialization process to solve the dynamic dispatch problem. Moreover, we assume that the optimization objective functions are convex functions rather than a strictly standard quadratic function, and the convergence of the proposed algorithm is analyzed through convex analysis and a Lyapunov function method. Finally, some experiments with quadratic or non-quadratic cost functions and comparison examples are simulated, the experimental results verify that the optimal solution satisfies the constraints of the supply–demand constraints and capacity inequalities in each time slot.

A branch-and-cut algorithm for the vehicle routing problem with two-dimensional loading constraints

In this study, we develop a branch-and-cut algorithm for the vehicle routing problem with two-dimensional loading constraints. A separation algorithm is proposed to simultaneously identify infeasible set inequalities and weak capacity inequalities for fractional solutions. Classic capacitated vehicle routing problem (CVRP) inequalities are adopted as well. A branch-and-cut (B&C) algorithm is built to solve the problem. Experimental results illustrate that the algorithm is competitive. In particular, we solve 6 instances to optimality for the first time. Extensive computational analysis is also conducted to reveal the impact of infeasible set inequalities, the branching strategy, and the packing algorithms on the B&C algorithm.

Robust vehicle routing under uncertainty via branch-price-and-cut

This paper contemplates how branch-price-and-cut solvers can be employed along with the robust optimization paradigm to address parametric uncertainty in the context of vehicle routing problems. In this setting, given postulated uncertainty sets for customer demands and vehicle travel times, one aims to identify a set of cost-effective routes for vehicles to traverse, such that the vehicle capacities and customer time window constraints are respected under any anticipated demand and travel time realization, respectively. To tackle such problems, we propose a novel approach that combines cutting-plane techniques with an advanced branch-price-and-cut algorithm. Specifically, we use deterministic pricing procedures to generate “partially robust” vehicle routes and then utilize robust versions of rounded capacity inequalities and infeasible path elimination constraints to guarantee complete robust feasibility of routing designs against demand and travel time uncertainty. In contrast to recent approaches that modify the pricing algorithm, our approach is both modular and versatile. It permits the use of advanced branch-price-and-cut technologies without significant modification, while it can admit a variety of uncertainty sets that are commonly used in robust optimization but could not be previously employed in a branch-price-and-cut setting.

The multi-trip vehicle routing problem with time windows and unloading queue at depot

Motivated by the practice of urban waste collection, we introduce a new variant of the multi-trip vehicle routing problem with time windows, where vehicles unload cargos collected from customers at a depot that owns a limited unloading capacity, and therefore some vehicles need to wait at a queue once the unloading capacity is fully occupied. Unloading queue at depot significantly compicates this problem since it makes a trip involve traveling, waiting, and unloading stages. We first formulate this problem into a trip-based set partitioning model, which has two sets of mutual exclusion constraints. To solve the model, we propose a branch-and-price-and-cut algorithm (BPC), in which the linear relaxation of the model is solved by column generation, and moreover, the relaxation gap is tightened by incorporating rounded capacity inequalities. Particularly, the mutual exclusion constraints induce a new and challening pricing problem, and we design a novel label-setting algorithm with tailored label structure to solve it. We conduct compuational experiments on a set of random instances generated from the Solomon’s benchmark, and the results demonstrate the effectiveness of the proposed model and algorithm.

Distributed algorithm based on consensus control strategy for dynamic economic dispatch problem

In this paper, the dynamic economic dispatch problem (DEDP) of a hybrid power network is studied. The micro-grid model is constructed with three types of non-renewable power sources, renewable energy and energy storage batteries. Meanwhile, the pollutant emission costs and benefit function are considered here to maximize the total welfare. Firstly, we transform the benefit-maximum problem into its equivalent minimization problem which the generators connected each other in a directed graph. Then, we introduced a distributed algorithm based on alternating direction method of multipliers (ADMM) to find the optimal solution of different generators within 24 h. To solve the sub-problem, the bisection method and the finite-time distributed algorithm are adopted here. Finally, the convergence of the distributed algorithm is analyzed. Moreover, our experimental results verify that the optimal solution satisfies the constraints of supply–demand balance equality constraint and capacity inequalities well in each time slot, and meanwhile satisfies the constraints of output threshold between different time slots, furthermore, we utilized a practical example to make a comparison with the existed literature, and the superiority of the algorithm is verified to a certain extent.

A Branch-and-Price-and-Cut Algorithm for the Cable-Routing Problem in Solar Power Plants

A solar power plant is a large-scale photovoltaic (PV) system designed to supply usable solar power to the electricity grid. Building a solar power plant needs consideration of arrangements of several important components, such as PV arrays, solar inverters, combiner boxes, cables, and other electrical accessories. The design of solar power plants is very complex because of various optimization parameters and design regulations. In this study, we address the cable-routing problem arising in the planning of large-scale solar power plants, which aims to determine the partition of the PV arrays, the location of combiner boxes, and cable routing such that the installation cost of the cables connecting the components is minimized. We formulate the problem as a mathematical programming problem, which can be viewed as a generalized capacitated minimum spanning tree (CMST) problem, and then devise a branch-and-price-and-cut (BPC) algorithm to solve it. The BPC algorithm uses two important valid inequalities, namely the capacity inequalities and the subset-row inequalities, to tighten the lower bounds. We also adopt several acceleration strategies to speed up the algorithm. Using real-world data sets, we show by numerical experiments that our BPC algorithm is superior to the typical manual-based planning approach used by many electric power planning companies. In addition, when solving the CMST problem with unitary demands, our algorithm is highly competitive compared with the best exact algorithm in the literature.

A Branch-and-Cut-and-Price algorithm for the Multi-trip Separate Pickup and Delivery Problem with Time Windows at Customers and Facilities

We study the Multi-trip Separate Pickup and Delivery Problem with Time Windows at Customers and Facilities (MT-PDTWCF), arising in two-tiered city logistics systems. The first tier refers to the transportation between the city distribution centers, in the outskirts of the city, and intermediate facilities, while the second tier refers to the transportation of goods between the intermediate facilities and the (pickup and delivery) customers. We focus on the second tier, and consider that customers and facilities have time windows in which they can be visited. Waiting is possible at waiting stations for free or at customers and facilities at a given cost or penalty. Therefore, it is relevant to coordinate the arrivals of vehicles at facilities and customers with the corresponding time windows. The MT-PDTWCF calls for determining minimum (fixed, routing and waiting) cost multi-trip routes, for a given fleet of vehicles, to service separately pickup and delivery customers, while taking into account vehicle capacity and time windows both at customers and facilities. We propose the first exact algorithm for MT-PDTWCF, namely a Branch-and-Cut-and-Price algorithm. It is based on column generation, where the pricing problem is solved by a bi-directional dynamic programming algorithm designed to cope with the features of the problem. Subset-row and rounded capacity inequalities are adapted to deal with MT-PDTWCF and inserted in the Branch-and-Cut-and-Price algorithm. The performance of the proposed algorithm is tested on benchmark instances with up to 200 customers, showing its effectiveness.

The robust vehicle routing problem with time windows: Solution by branch and price and cut

We study the vehicle routing problem with time windows under demand uncertainty. Such a problem arises in fuel delivery to gas stations, to farms, or to production plants. We suggest a robust formulation where uncertain demand has support defined by cardinality constrained sets. The model ensures the feasibility of each route when demand values on the route change in predefined set. The range is customer specific, does not assume any probability distribution of demand, and may be estimated using historical demand data. We develop a branch-and-price-and-cut algorithm where the pricing problem is a robust resource constrained shortest path problem. We extend the rounded capacity inequalities to the robust case and suggest a separation procedure that solves a robust bin packing problem. The bound is further strengthened using 2-path and subset row inequalities and the full algorithm is tested on an extension of the benchmark Solomon instances. The robust schedules are simulated and compared to schedules obtained by solving the deterministic problem under varying levels and distributions of uncertainty. Simulation results show that the robust model provides superior protection against demand uncertainty. Over all scenarios, only 0.57% of the robust solutions turn out to be infeasible compared to 41.62% of the deterministic ones. Such protection against infeasibility comes with an 11.22% increase in routing costs on average.

Capacity inequalities and rigidity of cornered/conical manifolds

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Polia–Szego-, Alexandrov–Fenchel- and Penrose-type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.

Network Loading Problem: Valid inequalities from 5- and higher partitions

This paper addresses the well-known Network Loading Problem, and develops several large new classes of valid inequalities based on p-partitions of the graph. The total capacity inequalities are obtained by recursively applying a simple C-G procedure to the cut inequalities, and can be efficiently computed for p-partitions with p ≤ 10. Another class of inequalities called one-two inequalities are separated by solving a sequence of LPs, heuristically modifying the LHS coefficients of the inequality until a violated inequality is found. The spanning tree inequalities are based on a simple observation that if the demand graph of a problem is connected, the solution must be at least a tree, i.e. have (n−1) or more edges. The conditions under which these inequalities are facet-defining are analyzed. The effectiveness of these inequalities in obtaining substantially tighter bounds and reduced solutions times is demonstrated via several computational experiments. More than 10-fold reduction in the solution time is obtained on several benchmark problems.

On the complexity of the separation problem for rounded capacity inequalities

In this paper, we are interested in the separation problem for the so-called rounded capacity inequalities which are valid for the CVRP (Capacitated Vehicle Routing Problem) polytope. Rounded capacity inequalities, as well as the associated separation problem, are of particular interest for solving the CVRP, especially when dealing with the two-index formulation of the CVRP and Branch-and-Cut algorithms based on this formulation. To the best of our knowledge, it is not known in the literature whether this separation problem is NP-hard or polynomial (see for instance Augerat et al. (1998) and Letchford and Salazar (2015)), and it has been conjectured that the problem is NP-hard. In this paper, we prove this conjecture. We also show that the separation problem for rounded capacity inequalities is strongly NP-hard when the considered solution belongs to a particular relaxation of the problem and the clients have all the same demand.

The separation problem of rounded capacity inequalities

In this paper we are interested in the separation problem of the so-called rounded capacity inequalities which are involved in the two-index formulation of the CVRP (Capacitated Vehicle Routing Pro...

The separation problem of rounded capacity inequalities: Some polynomial cases

In this paper we are interested in the separation problem of the so-called rounded capacity inequalities which are involved in the two-index formulation of the CVRP (Capacitated Vehicle Routing Problem) polytope. In a recent work (Diarrassouba, 2015, [1]), we have investigated the theoretical complexity of that problem in the general case. Several authors have devised heuristic separation algorithms for rounded capacity inequalities for solving the CVRP. In this paper, we investigate the conditions under which this separation problem can be solved in polynomial time, and this, in the context of the CVRP or for solving other combinatorial optimization problems in which rounded capacity inequalities are involved. We present four cases in which they can be separated in polynomial time and reduce the problem to O(n2) maximum flow computations.

A Branch-Price-and-Cut Algorithm for the Inventory-Routing Problem

The inventory-routing problem (IRP) integrates two well-studied problems, namely, inventory management and vehicle routing. Given a set of customers to service over a multiperiod horizon, the IRP consists of determining when to visit each customer, which quantity to deliver in each visit, and how to combine the visits in each period into feasible routes such that the total routing and inventory costs are minimized. In this paper, we propose an innovative mathematical formulation for the IRP and develop a state-of-the-art branch-price-and-cut algorithm for solving it. This algorithm incorporates known and new families of valid inequalities, including an adaptation of the well-known capacity inequalities, as well as an ad hoc labeling algorithm for solving the column generation subproblems. Through extensive computational experiments on a widely used set of 640 benchmark instances involving between two and five vehicles, we show that our branch-price-and-cut algorithm clearly outperforms a state-of-the-art branch-and-cut algorithm on the instances with four and five vehicles. In this instance set, 238 were still open before this work and we proved optimality for 54 of them.