The Gap Function: Evaluating Integer Programming Models over Multiple Right-Hand Sides

2021 ◽  
Author(s):  
Temitayo Ajayi ◽  
Christopher Thomas ◽  
Andrew J. Schaefer
Keyword(s):  
Linear Programming ◽  
Integer Programming ◽  
Optimization Problems ◽  
Programming Model ◽  
Programming Models ◽  
Linear Programming Relaxation ◽  
Gap Functions ◽  
Model Quality ◽  
Right Hand ◽  
The Right

For an integer programming model with fixed data, the linear programming relaxation gap is considered one of the most important measures of model quality. There is no consensus, however, on appropriate measures of model quality that account for data variation. In particular, when the right-hand side is not known exactly, one must assess a model based on its behavior over many right-hand sides. Gap functions are the linear programming relaxation gaps parametrized by the right-hand side. Despite drawing research interest in the early days of integer programming, the properties and applications of these functions have been little studied. In this paper, we construct measures of integer programming model quality over sets of right-hand sides based on the absolute and relative gap functions. In particular, we formulate optimization problems to compute the expectation and extrema of gap functions over finite discrete sets and bounded hyperrectangles. These optimization problems are linear programs (albeit of an exponentially large size) that contain at most one special ordered-set constraint. These measures for integer programming models, along with their associated formulations, provide a framework for determining a model’s quality over a range of right-hand sides.

scholarly journals Applying Simulation Modelling in Quantifying Optimization Results

2021 ◽  
Vol 15 (4) ◽  
pp. 518-523
Author(s):  
Ratko Stanković ◽  
Diana Božić
Keyword(s):  
Linear Programming ◽  
Simulation Model ◽  
Input Data ◽  
Optimization Problems ◽  
Programming Model ◽  
Optimal Solution ◽  
Simulation Modelling ◽  
Simulation Software ◽  
Programming Models ◽  
Freight Forwarding

Improvements achieved by applying linear programming models in solving optimization problems in logistics cannot always be expressed by physically measurable values (dimensions), but in non-dimensional values. Therefore, it may be difficult to present the actual benefits of the improvements to the stake holders of the system being optimized. In this article, a possibility of applying simulation modelling in quantifying results of optimizing cross dock terminal gates allocation is outlined. Optimal solution is obtained on the linear programming model by using MS Excel spreadsheet optimizer, while the results are quantified on the simulation model, by using Rockwell Automation simulation software. Input data are collected from a freight forwarding company in Zagreb, specialized in groupage transport (Less Than Truckload - LTL).

scholarly journals Postoptimization of the model of water supply for urban and industrial agglomeration

2018 ◽  
Vol 23 ◽  
pp. 00035
Author(s):  
Jacek Wawrzosek ◽  
Szymon Ignaciuk
Keyword(s):  
Linear Programming ◽  
Water Supply ◽  
Programming Models ◽  
Industrial Agglomeration ◽  
Infinitely Many Solutions ◽  
Nodal Solutions ◽  
Parametric Linear Programming ◽  
Right Hand ◽  
The Right

A case study of the tools used by an analyst of the economic aspects of the operation of the water supply network has been undertaken in this paper. All issues discussed here are formulated by using degenerated linear programming models ( PL ). Below, it is noted that the linear dependence of binding constraints ( CO ) distorts standard postoptimization procedures in PL. This observed fact makes postoptimization analysis mostly unhelpful for an average analyst due to problems with the int erpretation of ambiguous sensitivity reports which are obtained from popular computer packages. In standard postoptimization methods, changes to single parameters of the right-hand vector CO are analyzed or referred to parametric linear programming that unfortunately requires prior knowledge of mathematically and economically justified vectors of changes of right-hand sides CO. Therefore, it is suggested that modifications are introduced to some of the postoptimization procedures in this work. For issues in the field of hydrology, the following were presented: interpretation and methods of generating justified vectors of changes of right-hand sides of limiting conditions. And so, the procedure of generating infinitely many solutions of the dual issue based on certain vectors orthogonal to the vector of right-hand sides of constraint conditions was demonstrated. Furthermore, the same orthogonal vectors were used to obtain nodal solutions of the dua0l model and the corresponding vectors of changes of the entire right-hand sides of the constraint conditions. Then, managerial interpretation was applied to this way of proceeding. The methods presented in the work serve to improve the functioning of the system of water supply.

A method for solving linear programming with interval-valued trapezoidal fuzzy variables

2018 ◽  
Vol 52 (3) ◽  
pp. 955-979 ◽  
Author(s):  
Ali Ebrahimnejad
Keyword(s):  
Linear Programming ◽  
Fuzzy Numbers ◽  
Auxiliary Problem ◽  
Coefficient Matrix ◽  
Uncertain Parameters ◽  
Fuzzy Variables ◽  
Right Hand ◽  
Fuzzy Cost ◽  
The Right ◽  
Interval Valued

An efficient method to handle the uncertain parameters of a linear programming (LP) problem is to express the uncertain parameters by fuzzy numbers which are more realistic, and create a conceptual and theoretical framework for dealing with imprecision and vagueness. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand side, and/or the elements of the coefficient matrix. The aim of this article is to introduce a formulation of FLP problems involving interval-valued trapezoidal fuzzy numbers for the decision variables and the right-hand-side of the constraints. We propose a new method for solving this kind of FLP problems based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To do this, we first define an auxiliary problem, having only interval-valued trapezoidal fuzzy cost coefficients, and then study the relationships between these problems leading to a solution for the primary problem. It is demonstrated that study of LP problems with interval-valued trapezoidal fuzzy variables gives rise to the same expected results as those obtained for LP with trapezoidal fuzzy variables.

scholarly journals Task-based programming in COMPSs to converge from HPC to big data

2017 ◽  
Vol 32 (1) ◽  
pp. 45-60 ◽  
Author(s):  
Javier Conejero ◽  
Sandra Corella ◽  
Rosa M Badia ◽  
Jesus Labarta
Keyword(s):  
Big Data ◽  
High Performance ◽  
Programming Model ◽  
Good Alternative ◽  
Programming Models ◽  
Suitable Model ◽  
Advantages And Disadvantages ◽  
Big Data Applications ◽  
And Performance ◽  
The Right

Task-based programming has proven to be a suitable model for high-performance computing (HPC) applications. Different implementations have been good demonstrators of this fact and have promoted the acceptance of task-based programming in the OpenMP standard. Furthermore, in recent years, Apache Spark has gained wide popularity in business and research environments as a programming model for addressing emerging big data problems. COMP Superscalar (COMPSs) is a task-based environment that tackles distributed computing (including Clouds) and is a good alternative for a task-based programming model for big data applications. This article describes why we consider that task-based programming models are a good approach for big data applications. The article includes a comparison of Spark and COMPSs in terms of architecture, programming model, and performance. It focuses on the differences that both frameworks have in structural terms, on their programmability interface, and in terms of their efficiency by means of three widely known benchmarking kernels: Wordcount, Kmeans, and Terasort. These kernels enable the evaluation of the more important functionalities of both programming models and analyze different work flows and conditions. The main results achieved from this comparison are (1) COMPSs is able to extract the inherent parallelism from the user code with minimal coding effort as opposed to Spark, which requires the existing algorithms to be adapted and rewritten by explicitly using their predefined functions, (2) it is an improvement in terms of performance when compared with Spark, and (3) COMPSs has shown to scale better than Spark in most cases. Finally, we discuss the advantages and disadvantages of both frameworks, highlighting the differences that make them unique, thereby helping to choose the right framework for each particular objective.

Parametric nonlinear integer programming: The right-hand side case

1991 ◽  
Vol 54 (2) ◽  
pp. 237-255 ◽  
Author(s):  
Maw-Sheng Chern ◽  
Rong-Hong Jan ◽  
Ren-Jer Chern
Keyword(s):  
Integer Programming ◽  
Nonlinear Integer Programming ◽  
Right Hand ◽  
The Right

scholarly journals Aggregation of spatial units in linear programming models to explore land use options

1996 ◽  
Vol 44 (2) ◽  
pp. 145-162 ◽  
Author(s):  
R.J. Hijmans ◽  
M.K. Van Ittersum
Keyword(s):  
Land Use ◽  
Linear Programming ◽  
Programming Model ◽  
Programming Models ◽  
The European Union ◽  
Objective Functions ◽  
Land Use Allocation ◽  
Policy Level ◽  
Ecological Criteria ◽  
Use Studies

Consequences of aggregating spatial units in an interactive multiple-goal linear programming (IMGLP) model are analysed for a schematized and an existing IMGLP model (GOAL) exploring land use options for the European Union. A discrimination was made between effects on objective functions for the system as a whole, and effects on related optimum land use allocation within the system. In GOAL, effects on land use allocation are more important than effects on the value of objective functions. Several rules or factors were identified that determine the effect of aggregation, among which the degree in curvilinearity in input-output relations and the method of aggregation are important ones. However, because of complicated interacting effects, the aggregation error is difficult to predict. Therefore, in land use studies using IMGLP it is important to first optimize the linear programming model at the non-aggregated level and then aggregate to the appropriate policy level. If aggregation is inevitable because LP models become too big, aggregation according to agro-ecological criteria, i.e., aggregation of units with similar output-input ratios and constraints, results in the smallest errors.

Sign in / Sign up

Export Citation Format

Share Document