Cutting Stock with Binary Patterns: Arc-flow Formulation with Graph Compression

Minimal proper non-IRUP instances of the one-dimensional Cutting Stock Problem

We consider the well-known one dimensional cutting stock problem (1CSP). Based on the pattern structure of the classical ILP formulation of Gilmore and Gomory, we can decompose the infinite set of 1CSP instances, with a fixed demand n, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property we computationally show that all 1CSP instances with $n\le 9$ are proper IRUP, while we give examples of a proper non-IRUP instances with $n=10$. A gap larger than 1 occurs for $n=11$. The worst known gap is raised from 1.003…

对双目标切割问题中嵌入标量化方法的列生成方法的研究

A study of column generation embedded in scalarization methods for the bi-objective cutting stock problem

本文研究了一维和二维切割问题的双目标问题，采用动态列生成嵌入标量化方法，发现其能更准确地逼近帕累托前沿，而标量化方法则互补性更强。

将多阶段切割下料问题表述为QUBO形式

Formulating Multistage Cutting Stock Problems as QUBO

本文提出多阶段二维切割下料问题的线性规划表述，并推导其QUBO非精确重构，通过模拟退火和迭代增广拉格朗日方法求解受限问题，并探讨量子退火的应用前景。

A Reinforcement Learning Approach to the Stochastic Cutting Stock Problem

We propose a formulation of the stochastic cutting stock problem as a discounted infinite-horizon Markov decision process. At each decision epoch, given current inventory of items, an agent chooses in which patterns to cut objects in stock in anticipation of the unknown demand. An optimal solution corresponds to a policy that associates each state with a decision and minimizes the expected total cost. Since exact algorithms scale exponentially with the state-space dimension, we develop a heuristic solution approach based on reinforcement learning. We propose an approximate policy iteration alg…

Set-Packing 与 Sequence-Pair QUBO 用于量子退火硬件上的二维下料问题

Set-Packing and Sequence-Pair QUBOs for the 2D Cutting Stock Problem on Quantum Annealing Hardware

本研究针对二维下料问题，比较了坐标基集合打包与无坐标序列对两种QUBO编码，证明后者存在结构性限制，并给出选择实用规则。

Counterexamples in the CSP

We present a collection of examples in various aspects of the one-dimensional cutting stock problem ('1D-CSP'). Based on our 37+ year experience in this area, we discuss some modelling pitfalls and counterexamples.

SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning

We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an au…

Solving the Team Orienteering Problem with Cutting Planes

The Team Orienteering Problem (TOP) is an attractive variant of the Vehicle Routing Problem (VRP). The aim is to select customers and at the same time organize the visits for a vehicle fleet so as to maximize the collected profits and subject to a travel time restriction on each vehicle. In this paper, we investigate the effective use of a linear formulation with polynomial number of variables to solve TOP. Cutting planes are the core components of our solving algorithm. It is first used to solve smaller and intermediate models of the original problem by considering fewer vehicles. Useful info…

An exact separation algorithm for unsplittable flow capacitated network design arc-set polyhedron

In this paper, we concentrate on generating cutting planes for the unsplittable capacitated network design problem. We use the unsplittable flow arc-set polyhedron of the considered problem as a substructure and generate cutting planes by solving the separation problem over it. To relieve the computational burden, we show that, in some special cases, a closed form of the separation problem can be derived. For the general case, a brute-force algorithm, called exact separation algorithm, is employed in solving the separation problem of the considered polyhedron such that the constructed inequali…

混合整数线性规划求解中学习选择割平面

Learning to Select Cutting Planes in Mixed Integer Linear Programming Solving

针对现有MILP割平面选择的序列学习模型忽略LP松弛、受输入位置影响的问题，提出基于异质三部图编码的HGTSM模型，在多数据集上性能优于启发式与基线方法，且具备稳定性与泛化性。