Convex hull formulations for mixed-integer multilinear functions

We study the adaptive distributionally robust hub location problem with multiple commodities under demand and cost uncertainty in both uncapacitated and capacitated cases. The hub location decision anticipates the worst-case expected cost over an ambiguity set of possible distributions of the uncertain demand and cost, and the routing policy, being adaptive to the uncertainty realization, ships commodities through selected hubs. We investigate the adaptivity and tractability of the distributionally robust model under different distributional information about uncertainty. In the uncapacitated case in which demand and cost are independent and costs of different commodities are also mutually independent, the adaptive distributionally robust model is equivalent to a nonadaptive classical robust model and the second-stage routing decision follows an optimal static policy. We then relax the independence assumption and show that the second-stage routing decision follows an optimal scenario-wise policy if either the demand or the cost is supported on a convex hull of given scenarios. We extend our analysis to the capacitated case and show that the second-stage routing decision still follows an optimal scenario-wise policy if the demand is supported on the convex hull of given scenarios. In terms of tractability, for all mentioned cases, we reformulate the distributionally robust model as a moderate-sized mixed-integer linear program, and we recover the associated worst-case distribution by solving a collection of linear programs. Through numerical studies using the Civil Aeronautics Board data set, we demonstrate the advantages of the distributionally robust model by examining its superior out-of-sample performance against the classical robust model and the stochastic model.

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Near optimal minimal convex hulls of disks

Journal of Global Optimization ◽

10.1007/s10898-021-01002-5 ◽

2021 ◽

Author(s):

Josef Kallrath ◽

Joonghyun Ryu ◽

Chanyoung Song ◽

Mokwon Lee ◽

Deok-Soo Kim

Keyword(s):

Linear Programming ◽

Convex Hull ◽

Programming Model ◽

Linear Programming Model ◽

Mixed Integer ◽

Convex Hulls ◽

Non Linear Programming ◽

Minimal Convex ◽

Non Linear ◽

Problem Instances

AbstractThe minimal convex hulls of disks problem is to find such arrangements of circular disks in the plane that minimize the length of the convex hull boundary. The mixed-integer non-linear programming model, named [17], works only for small to moderate-sized problems. Here we propose a polylithic framework of the problem for big problem instances by combining the following algorithms and models: (i) A fast disk-packing algorithm based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model taking the output of as its input. We present complete analytic solutions for small problems up to four disks and a semi-analytic mixed-integer linear programming model which yields exact solutions for strip packing problems with up to one thousand congruent disks. It turns out that the proposed polylithic approach works fine for large problem instances containing up to 1,000 disks. Monolithic and polylithic solutions using usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

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An Innovative Formulation Tightening Approach for Job-Shop Scheduling

10.36227/techrxiv.12783893 ◽

2020 ◽

Author(s):

Bing Yan ◽

Mikhail Bragin ◽

Peter Luh

Keyword(s):

Linear Programming ◽

Convex Hull ◽

Integer Linear Programming ◽

Branch And Cut ◽

Mixed Integer ◽

Job Shops ◽

Production Environment ◽

Processing Stage ◽

Binary Variables ◽

Integer Variables

Job shops are an important production environment for low-volume high-variety manufacturing. Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data pre-processing stage, then a solution can be obtained by using linear programming methods without much difficulty. The tightening process, however, is NP hard because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data pre-processing stage. It is a major extension from our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by binary variables. With binary variables and continuous only, the vertices of the convex hull can be obtained based on the vertices of the linear problem after relaxing binary requirements with proved tightness. These vertices are then converted to tight constraints for general use. This approach significantly improves and extends our previous results on tightening single-operation parts without actually achieving tightness. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

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An Innovative Formulation Tightening Approach for Job-Shop Scheduling

10.36227/techrxiv.12783893.v2 ◽

2021 ◽

Author(s):

Bing Yan ◽

Mikhail Bragin ◽

Peter Luh

Keyword(s):

Linear Programming ◽

Convex Hull ◽

Integer Linear Programming ◽

Data Preprocessing ◽

Mixed Integer ◽

Continuous Variables ◽

Job Shops ◽

Production Environment ◽

Binary Variables ◽

Integer Variables

Job shops are an important production environment for low-volume high-variety manufacturing. Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data preprocessing stage, then a solution can be obtained by using linear programming methods without much difficulty. The tightening process, however, is fundamentally challenging because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data preprocessing stage. It is a major advancement of our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by the binary variables. With binary and continuous variables only, it is proved that the vertices of the convex hull can be obtained based on vertices of the linear problem after relaxing binary requirements. These vertices are then converted to tight constraints for general use. This approach significantly improves our previous results on tightening individual operations. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

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An Innovative Formulation Tightening Approach for Job-Shop Scheduling

10.36227/techrxiv.12783893.v1 ◽

2020 ◽

Author(s):

Bing Yan ◽

Mikhail Bragin ◽

Peter Luh

Keyword(s):

Linear Programming ◽

Convex Hull ◽

Integer Linear Programming ◽

Branch And Cut ◽

Mixed Integer ◽

Job Shops ◽

Production Environment ◽

Processing Stage ◽

Binary Variables ◽

Integer Variables

Job shops are an important production environment for low-volume high-variety manufacturing. Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data pre-processing stage, then a solution can be obtained by using linear programming methods without much difficulty. The tightening process, however, is NP hard because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data pre-processing stage. It is a major extension from our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by binary variables. With binary variables and continuous only, the vertices of the convex hull can be obtained based on the vertices of the linear problem after relaxing binary requirements with proved tightness. These vertices are then converted to tight constraints for general use. This approach significantly improves and extends our previous results on tightening single-operation parts without actually achieving tightness. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

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Optimal Test Assembly in Practice

Zeitschrift für Psychologie ◽

10.1027/2151-2604/a000146 ◽

2013 ◽

Vol 221 (3) ◽

pp. 190-200 ◽

Cited By ~ 9

Author(s):

Jörg-Tobias Kuhn ◽

Thomas Kiefer

Keyword(s):

Particle Physics ◽

Large Scale ◽

Block Design ◽

Linear Optimization ◽

Mixed Integer ◽

Optimal Test ◽

Automated Test Assembly ◽

Test Assembly ◽

Block Level ◽

Mixed Integer Linear Optimization

Several techniques have been developed in recent years to generate optimal large-scale assessments (LSAs) of student achievement. These techniques often represent a blend of procedures from such diverse fields as experimental design, combinatorial optimization, particle physics, or neural networks. However, despite the theoretical advances in the field, there still exists a surprising scarcity of well-documented test designs in which all factors that have guided design decisions are explicitly and clearly communicated. This paper therefore has two goals. First, a brief summary of relevant key terms, as well as experimental designs and automated test assembly routines in LSA, is given. Second, conceptual and methodological steps in designing the assessment of the Austrian educational standards in mathematics are described in detail. The test design was generated using a two-step procedure, starting at the item block level and continuing at the item level. Initially, a partially balanced incomplete item block design was generated using simulated annealing, whereas in a second step, items were assigned to the item blocks using mixed-integer linear optimization in combination with a shadow-test approach.

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