scholarly journals MIPaaL: Mixed Integer Program as a Layer

2020 ◽  
Vol 34 (02) ◽  
pp. 1504-1511 ◽  
Author(s):  
Aaron Ferber ◽  
Bryan Wilder ◽  
Bistra Dilkina ◽  
Milind Tambe
Keyword(s):  
Broad Class ◽  
Linear Constraints ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Linear Programs ◽  
Continuous Variables ◽  
Two Phase ◽  
Mixed Integer Linear Program ◽  
Submodular Optimization ◽  
Solution Approach

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures average accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a mixed integer linear program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, an algorithm that iteratively tightens the continuous relaxation by adding constraints removing fractional solutions. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and optimization separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP. Lastly, we demonstrate generalization performance in several transfer learning tasks.

scholarly journals A game-theoretic optimisation approach to fair customer allocation in oligopolies

2020 ◽  
Vol 21 (4) ◽  
pp. 1459-1486
Author(s):  
Vassilis M. Charitopoulos ◽  
Vivek Dua ◽  
Jose M. Pinto ◽  
Lazaros G. Papageorgiou
Keyword(s):  
Integer Program ◽  
Linear Constraints ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Programming Approach ◽  
Theoretic Approach ◽  
Separable Programming ◽  
Process Industries ◽  
Game Theoretic

Abstract Under the ever-increasing capital intensive environment that contemporary process industries face, oligopolies begin to form in mature markets where a small number of companies regulate and serve the customer base. Strategic and operational decisions are highly dependent on the firms’ customer portfolio and conventional modelling approaches neglect the rational behaviour of the decision makers, with regards to the problem of customer allocation, by assuming either static competition or a leader-follower structure. In this article, we address the fair customer allocation within oligopolies by employing the Nash bargaining approach. The overall problem is formulated as mixed integer program with linear constraints and a nonlinear objective function which is further linearised following a separable programming approach. Case studies from the industrial liquid market highlight the importance and benefits of the proposed game theoretic approach.

A Mathematical Programming-Based Approach for Architecture Selection

2012 ◽  
Author(s):  
Aleksandr A. Kerzhner ◽  
Christiaan J. J. Paredis
Keyword(s):  
Mathematical Programming ◽  
Optimization Technique ◽  
Problem Formulation ◽  
Mixed Integer ◽  
Model Transformations ◽  
Continuous Variables ◽  
Solution Approach ◽  
Computational Support ◽  
Algebraic Constraints ◽  
Selection Of

Modern systems are difficult to design because there are a significant number of potential alternatives to consider. The specification of an alternative includes an architecture (which describes the components and connections of the system) and component sizings (the sizing parameter for each component). In current practice, designers rely mainly on their experience and intuition to select a desired architecture without much computational support and then spend most of their effort on optimizing component sizings. In this paper, an approach for representing an architecture selection as a mixed-integer linear programming optimization is presented; existing solvers are then used to identify promising candidate architectures at early stages of the design process. Mathematical programming is a common optimization technique, but it is rarely used for architecture selection because of the difficulty of manually formulating an architecture selection as a mathematical program. In this paper, the formulation is presented in a modular fashion so that model transformations can be applied to transform a problem formulation that is convenient for designers into the mathematical programming optimization. A modular superstructure representation is used to model the design space; in a superstructure a union of all potential architectures is represented as a set of discrete and continuous variables. Algebraic constraints are added to describe both acceptable variable combinations and system behavior to allow the solver to eliminate clearly poor alternatives and identify promising alternatives. The framework is demonstrated on the selection of an actuation subsystem for a hydraulic excavator, although the solution approach would be similar for most mechanical systems.

scholarly journals Application of Two-Phase Fuzzy Optimization Approach to Multiproduct Multistage Integrated Production Planning with Linguistic Preference under Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-20 ◽  
Author(s):  
Shan Lu ◽  
Hongye Su ◽  
Lian Xiao ◽  
Li Zhu
Keyword(s):  
Production Planning ◽  
Programming Model ◽  
Fuzzy Optimization ◽  
Service Level ◽  
Mixed Integer ◽  
Planning Problem ◽  
Optimization Approach ◽  
Two Phase ◽  
Steel Rolling ◽  
Solution Approach

This paper tackles the challenges for a production planning problem with linguistic preference on the objectives in an uncertain multiproduct multistage manufacturing environment. The uncertain sources are modelled by fuzzy sets and involve those induced by both the epistemic factors of process and external factors from customers and suppliers. A fuzzy multiobjective mixed integer programming model with different objective priorities is proposed to address the problem which attempts to simultaneously minimize the relevant operations cost and maximize the average safety stock holding level and the average service level. The epistemic and external uncertainty is simultaneously considered and formulated as flexible constraints. By defining the priority levels, a two-phase fuzzy optimization approach is used to manage the preference extent and convert the original model into an auxiliary crisp one. Then a novel interactive solution approach is proposed to solve this problem. An industrial case originating from a steel rolling plant is applied to implement the proposed approach. The numerical results demonstrate the efficiency and feasibility to handle the linguistic preference and provide a compromised solution in an uncertain environment.

scholarly journals A Model-Based Optimisation Approach for Process Synthesis of Olefins from Petroleum with Application to the Malaysian Petrochemical Industry

2019 ◽  
Vol 12 ◽  
pp. 1-15
Author(s):  
Khor Cheng Seong
Keyword(s):  
Practical Knowledge ◽  
Optimal Solution ◽  
Raw Material ◽  
Mixed Integer ◽  
Hydrocarbon Mixture ◽  
Continuous Variables ◽  
Mixed Integer Linear Program ◽  
Actual Case ◽  
Model Based ◽  
Recent Developments

The shale gas revolution has rekindled interest in olefins production due to the abundance of ethane as a raw material resource. However, the main technology still revolves around the cost-intensive distillation operation. Hence this work aims to investigate the economic optimisation of olefins synthesis from petroleum in the light of recent developments. A model-based approach is applied to determine the optimal sequencing of separation and reaction processes for a multi-component hydrocarbon mixture feed to produce mainly ethylene and propylene. a mixed-integer linear program (MILP) is formulated based on a superstructure that captures numerous plausible synthesis alternatives. The model comprises linear mass balance reactor representation and simple sharp distillation based on split fractions for product recovery. Integer binary variablesis used for selecting the task for equipment and continuous variables for representing the flowrate of each task. To expedite converging to an optimal solution of a least total annualised cost configuration, the formulation is appended with logical constraints on the design and structural specifications derived from heuristics based on practical knowledge and experience. The modelling approach on actual case studies based on two such petrochemical facilities operating in Malaysia is implemented. Additionally, the solution analysis is enriched with the investigation on a second- and third-best (suboptimal) configurations obtained through appropriate integer cuts as constraints to the model. The results show good agreement with existing plant configurations, thus substantiating the value and verification of the proposed model-based optimisation approach.

HIGH FIDELITY INTERVAL ASSIGNMENT

2000 ◽  
Vol 10 (04) ◽  
pp. 399-415 ◽  
Author(s):  
SCOTT A. MITCHELL
Keyword(s):  
Linear Program ◽  
Integer Linear Program ◽  
Mixed Integer ◽  
Linear Programs ◽  
Soft Sets ◽  
Mixed Integer Linear Program ◽  
All Solutions ◽  
Quadrilateral Meshing ◽  
A Minor ◽  
Assigned Number

Quadrilateral meshing algorithms impose certain constraints on the number of intervals or mesh edges of the curves bounding a surface. When constructing a conformal mesh of a collection of adjoining surfaces, the constraints for all of the surfaces must be simultaneously satisfied. These constraints can be formulated as an integer linear program. Not all solutions to this problem are equally desirable, however. The user typically indicates a goal (soft-set) or required (hard-set) number of intervals for each curve. The hard-sets constrain the problem further, while the soft-sets influence the objective function. This paper describes an algorithm for solving this interval assignment problem. In a good solution, for each curve the positive or negative difference between its goal and assigned number of intervals is small relative to its number of goal intervals. The algorithm solves a series of linear programs, and comes close to minimizing the maximum lexicographic vector of these weighted differences. Then the algorithm solves a nearby mixed-integer linear program to satisfy certain "sum-even" constraints. The algorithm reliably produces numbers of intervals that are very close to the user's desires and is easily extendible to new constraints. Earlier versions of the algorithm1 were slower than alternative algorithms, but this is no longer a significant issue; in practice, the running time is a minor fraction of the time to mesh, even in models composed of thousands of curves.

scholarly journals Bi-objective design-for-control of water distribution networks with global bounds

2021 ◽  
Author(s):  
Aly-Joy Ulusoy ◽  
Filippo Pecci ◽  
Ivan Stoianov
Keyword(s):  
Water Distribution ◽  
Distribution Networks ◽  
Water Distribution Networks ◽  
Mixed Integer ◽  
Global Optimality ◽  
Linear Programs ◽  
Design For Control ◽  
Non Linear ◽  
Feasible Solutions

AbstractThis manuscript investigates the design-for-control (DfC) problem of minimizing pressure induced leakage and maximizing resilience in existing water distribution networks. The problem consists in simultaneously selecting locations for the installation of new valves and/or pipes, and optimizing valve control settings. This results in a challenging optimization problem belonging to the class of non-convex bi-objective mixed-integer non-linear programs (BOMINLP). In this manuscript, we propose and investigate a method to approximate the non-dominated set of the DfC problem with guarantees of global non-dominance. The BOMINLP is first scalarized using the method of $$\epsilon $$ ϵ -constraints. Feasible solutions with global optimality bounds are then computed for the resulting sequence of single-objective mixed-integer non-linear programs, using a tailored spatial branch-and-bound (sBB) method. In particular, we propose an equivalent reformulation of the non-linear resilience objective function to enable the computation of global optimality bounds. We show that our approach returns a set of potentially non-dominated solutions along with guarantees of their non-dominance in the form of a superset of the true non-dominated set of the BOMINLP. Finally, we evaluate the method on two case study networks and show that the tailored sBB method outperforms state-of-the-art global optimization solvers.

scholarly journals Investment Incentives in Competitive Electricity Markets

2018 ◽  
Vol 8 (10) ◽  
pp. 1978 ◽  
Author(s):  
Jaber Valinejad ◽  
Taghi Barforoshi ◽  
Mousa Marzband ◽  
Edris Pouresmaeil ◽  
Radu Godina ◽  
...  
Keyword(s):  
Electricity Markets ◽  
Electricity Market ◽  
Planning Horizon ◽  
Optimization Method ◽  
Design Criterion ◽  
Mixed Integer ◽  
Investment Incentives ◽  
Mixed Integer Linear Program ◽  
Interconnected Power System ◽  
The Impact

This paper presents the analysis of a novel framework of study and the impact of different market design criterion for the generation expansion planning (GEP) in competitive electricity market incentives, under variable uncertainties in a single year horizon. As investment incentives conventionally consist of firm contracts and capacity payments, in this study, the electricity generation investment problem is considered from a strategic generation company (GENCO) ′ s perspective, modelled as a bi-level optimization method. The first-level includes decision steps related to investment incentives to maximize the total profit in the planning horizon. The second-level includes optimization steps focusing on maximizing social welfare when the electricity market is regulated for the current horizon. In addition, variable uncertainties, on offering and investment, are modelled using set of different scenarios. The bi-level optimization problem is then converted to a single-level problem and then represented as a mixed integer linear program (MILP) after linearization. The efficiency of the proposed framework is assessed on the MAZANDARAN regional electric company (MREC) transmission network, integral to IRAN interconnected power system for both elastic and inelastic demands. Simulations show the significance of optimizing the firm contract and the capacity payment that encourages the generation investment for peak technology and improves long-term stability of electricity markets.

A stochastic mixed integer program to model spatial wildfire behavior and suppression placement decisions with uncertain weather

2016 ◽  
Vol 46 (2) ◽  
pp. 234-248 ◽  
Author(s):  
Erin J. Belval ◽  
Yu Wei ◽  
Michael Bevers
Keyword(s):  
Wildland Fire ◽  
Fire Behavior ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Weather Forecasts ◽  
Placement Decisions ◽  
Weather Events ◽  
Weather Changes ◽  
Nonanticipativity Constraints

Wildfire behavior is a complex and stochastic phenomenon that can present unique tactical management challenges. This paper investigates a multistage stochastic mixed integer program with full recourse to model spatially explicit fire behavior and to select suppression locations for a wildland fire. Simplified suppression decisions take the form of “suppression nodes”, which are placed on a raster landscape for multiple decision stages. Weather scenarios are used to represent a distribution of probable changes in fire behavior in response to random weather changes, modeled using probabilistic weather trees. Multistage suppression decisions and fire behavior respond to these weather events and to each other. Nonanticipativity constraints ensure that suppression decisions account for uncertainty in weather forecasts. Test cases for this model provide examples of fire behavior interacting with suppression to achieve a minimum expected area impacted by fire and suppression.

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