Class Of Linear Programs Research Articles

Multiple-Depot Vehicle Scheduling Problem Heuristics

Abstract The Multiple-Depot Vehicle Scheduling Problem (MDVSP) is very important in the planning process of transport systems. It consists in assigning a set of trips to a set of vehicles in order to minimize a certain total cost. We introduce three fast and reliable heuristics for MDVSP based on a classical integer linear programming formulation and on graph theoretic methods of fixing the infeasible subtours gathered from an integer solution. Extensive experimentations using a large set of benchmark instances show that our heuristics are faster and give good or even better results compared to other existing heuristics.

A Deep Learning Approach for Maximum Activity Links in D2D Communications.

Mobile cellular communications are experiencing an exponential growth in traffic load on Long Term Evolution (LTE) eNode B (eNB) components. Such load can be significantly contained by directly sharing content among nearby users through device-to-device (D2D) communications, so that repeated downloads of the same data can be avoided as much as possible. Accordingly, for the purpose of improving the efficiency of content sharing and decreasing the load on the eNB, it is important to maximize the number of simultaneous D2D transmissions. Specially, maximizing the number of D2D links can not only improve spectrum and energy efficiency but can also reduce transmission delay. However, enabling maximum D2D links in a cellular network poses two major challenges. First, the interference between the D2D and cellular communications could critically affect their performance. Second, the minimum quality of service (QoS) requirement of cellular and D2D communication must be guaranteed. Therefore, a selection of active links is critical to gain the maximum number of D2D links. This can be formulated as a classical integer linear programming problem (link scheduling) that is known to be NP-hard. This paper proposes to obtain a set of network features via deep learning for solving this challenging problem. The idea is to optimize the D2D link schedule problem with a deep neural network (DNN). This makes a significant time reduction for delay-sensitive operations, since the computational overhead is mainly spent in the training process of the model. The simulation performed on a randomly generated link schedule problem showed that our algorithm is capable of finding satisfactory D2D link scheduling solutions by reducing computation time up to 90% without significantly affecting their accuracy.

Solving Linear Programming Problem in ℤn by Graphical Method

The aim of this paper is to find a new type of linear programming problems depending on the set ℤn (the set of integers modulo n) by using the graphical method. We gave a new definitions that is characterized by the set ℤn. We found that in this set many basic concepts in classic linear programming changed.

A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach

ABSTRACT Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decision-making problems are generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper, we concentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, a numerical example is included to illustrate the mentioned solving process.

Interval linear programming under transformations: optimal solutions and optimal value range

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interval program cannot always be converted into a desired form without affecting its properties, due to the so-called dependency problem. In this paper, we discuss the common transformations used in linear programming, such as imposing non-negativity on free variables or splitting equations into inequalities, and their effects on interval programs. Specifically, we examine changes in the set of all optimal solutions, optimal values and the optimal value range. Since some of the considered properties do not holds in the general case, we also study a special class of interval programs, in which uncertainty only affects the objective function and the right-hand-side vector. For this class, we obtain stronger results.

An interleaved depth-first search method for the linear optimization problem with disjunctive constraints

Being an extension of classical linear programming, disjunctive programming has the ability to express the problem constraints as combinations of linear equalities and inequalities linked with logic AND and OR operations. All the existing theories such as generalized disjunctive programming, optimization modulo theories, linear optimization over arithmetic constraint formula, and mixed logical linear programming pose one commonality of branching among different solving techniques. However, branching constructs a depth-first search which may traverse a whole bad subtree when the branching makes a mistake ordering a bad successor. In this paper, we propose the interleaved depth-first search with stochastic local optimal increasing (IDFS-SLOI) method for solving the linear optimization problem with disjunctive constraints. Our technique searches depth-first several subtrees in turn, accelerates the search by subtree splitting, and uses efficient backtracking and pruning among the subtrees. Additionally, the local optimal solution is improved iteratively by constructing and solving a stochastic linear programming problem. We evaluate our approach against existing counterparts on the rate-monotonic optimization problem (RM-OPT) and the linear optimization with fuzzy relation inequalities problem (LOFRI). Experimental results show that for the tested instances, the IDFS-SLOI method performs better from performance perspective, especially promising results have been obtained for the larger three groups where the execution time is reduced by 85.6 and 51.6% for RM-OPT and LOFRI, respectively.

Solving binary cutting stock with matheuristics using particle swarm optimization and simulated annealing

In last decade, researchers have focused on improving existing methodologies through hybrid algorithms; these are a combination of algorithms between a metaheuristic with other metaheuristic and an exact method, to solve combinatorial optimization problems in the best possible way. This work presents a benchmark of different methodologies to solve the binary cutting stock problem using a column generation framework, this framework is divided into master and subproblem, master problem is solved using a classical integer linear programming, and the subproblem is solved using metaheuristic algorithms (genetic algorithm, simulated annealing and particle swarm optimization). This benchmark analysis is aimed to compare hybrid metaheuristics results with an exact methodology.

Solving linear optimization over arithmetic constraint formula

Since Balas extended the classical linear programming problem to the disjunctive programming (DP) problem where the constraints are combinations of both logic AND and OR, many researchers explored this optimization problem under various theoretical or application scenarios such as generalized disjunctive programming (GDP), optimization modulo theories (OMT), robot path planning, real-time systems, etc. However, the possibility of combining these differently-described but form-equivalent problems into a single expression remains overlooked. The contribution of this paper is two folded. First, we convert the linear DP/GDP model, linear-arithmetic OMT problem and related application problems into an equivalent form, referred to as the linear optimization over arithmetic constraint formula (LOACF). Second, a tree-search-based algorithm named RS-LPT is proposed to solve LOACF. RS-LPT exploits the techniques of interval analysis and nonparametric estimation for reducing the search tree and lowering the number of visited nodes. Also, RS-LPT alleviates bad construction of search tree by backtracking and pruning dynamically. We evaluate RS-LPT against two most common DP/GDP methods, three state-of-the-art OMT solvers and the disjunctive transformation based method on optimization benchmarks with different types and scales. Our results favor RS-LPT as compared to existing competing methods, especially for large scale cases.

Extension of Wolfe Method for Solving Quadratic Programming with Interval Coefficients

Quadratic programming with interval coefficients developed to overcome cases in classic quadratic programming where the coefficient value is unknown and must be estimated. This paper discusses the extension of Wolfe method. The extended Wolfe method can be used to solve quadratic programming with interval coefficients. The extension process of Wolfe method involves the transformation of the quadratic programming with interval coefficients model into linear programming with interval coefficients model. The next step is transforming linear programming with interval coefficients model into two classic linear programming models with special characteristics, namely, the optimum best and the worst optimum problem.

Problema de Programação Linear da Dieta Aplicado à Nutrição de Suínos

A Programação Linear caracteriza-se como uma técnica de otimização voltada para a solução de modelos descritos por equações lineares, beneficiando-se da capacidade crescente de processamento dos computadores. O artigo em tela representa um problema de programação linear clássico, formulado a partir de informações do setor suinocultor brasileiro. Nesse sentido, o objetivo do presente estudo é propor uma ração com valores nutricionais adequados e de reduzido custo de produção. Por ocasião da implementação da referida dieta, o tomador de decisão deverá ter em mente de que está diante de um sistema dinâmico, em que os insumos são afetados o tempo todo por distintas variáveis, de difícil modelagem, o que certamente demandará do gestor criatividade e flexibilidade.

Interval-valued intuitionistic fuzzy programming technique for multicriteria group decision making based on Shapley values and incomplete preference information

In the classical LINear programming technique for Multidimensional Analysis of Preference (LINMAP), the criteria are assumed to be independent, which may cause distorted results. The aim of this paper is to employ the LINMAP to develop an interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for multicriteria group decision making considering not only the importance of criteria and that of their ordered positions but also the interactions among criteria and those among their ordered positions. In the spirit of LINMAP, new group consistency and group inconsistency indices based on cross-entropy and Shapley values are firstly defined by considering the interactions among criteria and among their ordered positions. To simultaneously determine the fuzzy measures on criteria set and on ordered set, a bi-objective IVIF programming model is then constructed by minimizing the inconsistency index and maximizing the consistency index. The proposed IVIF programming model is subsequently solved by transforming it into a multiobjective programming model. Furthermore, some generalizations of the proposed programming model are investigated. Finally, two examples are given to illustrate the application of the proposed method, and its superiority is demonstrated by comparing the performance of the proposed method with that of the existing methods.

Metaheuristic-based simulation optimization approach to network revenue management with an improved self-adjusting bid price function

ABSTRACTMaking accurate accept/reject decisions on dynamically arriving customer requests for different combinations of resources is a challenging task under uncertainty of competitors' pricing strategies. Because customer demand may be affected by a competitor's pricing action, changes in customer interarrival times should also be considered in capacity control procedures. In this article, a simulation model is developed for a bid price–based capacity control problem of an airline network revenue management system by considering the uncertain nature of booking cancellations and competitors' pricing strategy. An improved bid price function is proposed by considering competitors' different pricing scenarios that occur with different probabilities and their effects on the customers' demands. The classical deterministic linear program (DLP) is reformulated to determine the initial base bid prices that are utilized as control parameters in the proposed self-adjusting bid price function. Furthermore, a simulation optimization approach is applied in order to determine the appropriate values of the coefficients in the bid price function. Different evolutionary computation techniques such as differential evolution (DE), particle swarm optimization (PSO), and seeker optimization algorithm (SOA), are utilized to determine these coefficients along with comparisons. The computational experiments show that promising results can be obtained by making use of the proposed metaheuristic-based simulation optimization approach.

A new data envelopment analysis in fully fuzzy environment on the base of the degree of certainty of information

Data Envelopment Analysis (DEA) is a linear programming (LP) based technique for evaluating the comparative efficiency of decision-making units (DMUs) based on multiple inputs and outputs. Conventional DEA forms require precise data. However in real problems, it is not easy to measure inputs and ou tputs in an exact way to find precise data. Although, fuzzy set theory has been introduced as a powerful approach to quantify vague data and several authors have suggested various fuzzy DEA models, there is a key flaw in previous approaches. When coping with real information, fuzziness is not sufficient to consider and a reliability of the information is very vital too. A Z-number has extra ability to depict the uncertain information. This concept relates to the topic of reliability of information. Z-number can portray fuzziness and reliability of information concurrently. In this paper, we consider a different fuzzy DEA form, where all the inputs and outputs and also their weights are Z-numbers. This Z-numbers DEA model turned into fully fuzzy LP on the basis of fuzzy expectation. Finally, we transform the fully fuzzified DEA model to the classical LP model. This method has very straightforward calculations and the key benefit of the proposed method is its low computational intricacy.

Production control system specified quality sausage products

The problem of management of production target in technological system of production of sausages of the set quality is considered in article. Decomposition of technological system is considered. Functions of management are allocated: formation of an optimum compounding of forcemeat, expert analysis of a compounding, laboratory analysis of a compounding and its statement. Information technology of interaction of these functions is offered. The mathematical problem definition of finding of an optimum compounding meat product with use of possible substitutes of ingredients is presented. This mathematical problem is a classical linear programming problem whose solution has the standard program. Since the manufacture of the finished product are various nonlinear effects are taken into account at the present time it is practically impossible, the methodology provided in this operation "Expert analysis of the formulation" and "Laboratory analysis of the finished product." An example of calculating the optimum alternative base recipe "Sausages “Viennese with cheese”" TS 9213-010-40155161-2002. For an alternative formulation demands were made at a cost of meat, the ingredient composition, as well as the final product organoleptic and physic-chemical indicators should comply with regulatory requirements "Sausages “Viennese with cheese”" TS 9213-010-40155161-2002. Indicator acid activity (pH) calculated stuffing formulation should be in the range 6.0-6.3. As a partial replacement for the main raw material have been proposed acceptable substitutes. It was necessary to calculate on the basis of the formulation "Sausages “Viennese with cheese”" TS 9213-010-40155161-2002 optimal price and quality alternative formulation. As a result of depreciation of the value of alternative stuffing recipe was 14,5 % when all of the restrictions on the consumer properties. The proposed information technology implemented in the software package "MultiMeat Expert" and a decision support system for the technology of meat processing enterprise.

A Single Machine Capacitated Production Planning Problem Under Uncertainty: A Grey Linear Programming Approach

The production planning is an important problem in most of manufacturing systems in practice. Unlike many researches existing in literature, this problem encounters with great uncertainties in parameters and input data. In this paper, a single machine capacitated production planning problem is considered and a linear programming formulation is presented. The production costs are assumed to be uncertain parameters. To handle the uncertainties in the model, the grey systems theory is employed and the concept of grey numbers is incorporated into an optimization framework. In such systems, the uncertain parameters with unknown distributions can be handled by grey numbers. The grey linear programming (GLP) is a development of the classical linear programming which allows uncertainty to be effectively communicated into the optimization process. Finally, the uncertain problem is transformed into a GLP, and is solved by two linear deterministic sub-models.

On Solving the Fuzzy Customer Information Problem in Multicommodity Multimodal Routing with Schedule-Based Services

In this study, we combine the fuzzy customer information problem with the multicommodity multimodal routing with schedule-based services which was explored in our previous study [1]. The fuzzy characteristics of the customer information are embodied in the demanded volumes of the multiple commodities and the time windows of their due dates. When the schedule-based services are considered in the routing, schedule constraints emerge because the operations of block container trains should follow their predetermined schedules. This will restrict the routes selection from space-time feasibility. To solve this combinatorial optimization problem, we first build a fuzzy chance-constrained nonlinear programming model based on fuzzy possibility theory. We then use a crisp equivalent method and a linearization method to transform the proposed model into the classical linear programming model that can be effectively solved by the standard mathematical programming software. Finally, a numerical case is presented to demonstrate the feasibility of the proposed method. The sensitivity of the best solution with respect to the values of the confidence levels is also examined.

Data envelopment analysis and its related linear programming models

We provide a unifying framework synthesizing the dual spaces of production and value used in DEA efficiency measurement with some well-known linear programming (LP) problems. Specifically, we make use of the technology matrix to map intensity variables into input–output space, and the adjoint transformation of the technology matrix to map input–output prices into prices of intensity variables. We show how the diet problem, a classical LP problem, is related to DEA and also use the adjoint matrix to demonstrate a procedure for pricing efficient decision-making units. We further illustrate the relationship between benefit-of-the-doubt aggregation and the diet problem.

Optimization instances for deterministic and stochastic problems on energy efficient investments planning at the building level.

Optimization instances relate to the input and output data stemming from optimization problems in general. Typically, an optimization problem consists of an objective function to be optimized (either minimized or maximized) and a set of constraints. Thus, objective and constraints are jointly a set of equations in the optimization model. Such equations are a combination of decision variables and known parameters, which are usually related to a set domain. When this combination is a linear combination, we are facing a classical Linear Programming (LP) problem. An optimization instance is related to an optimization model. We refer to that model as the Symbolic Model Specification (SMS) containing all the sets, variables, and parameters symbols and relations. Thus, a whole instance is composed by the SMS, the elements in each set, the data values for all the parameters, and, eventually, the optimal decisions resulting from the optimization solution. This data article contains several optimization instances from a real-world optimization problem relating to investment planning on energy efficient technologies at the building level.

Financial economics without probabilistic prior assumptions

The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison–Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem.

Optimal distribution (re)planning in a centralized multi-stage supply network under conditions of the ripple effect and structure dynamics

In this paper, an original approach to formulate and solve a multi-period and multi-commodity distribution (re)planning problem for a multi-stage centralized upstream network with structure dynamics considerations is proposed. First original idea of this study is description of the supply chain as a non-stationary dynamic system along with a linear programming (LP) model. This allows distribute design and control variables between dynamic and static models. Second original idea is to transit from the classical LP model to maximal flow problem by excluding demand constraint from the LP model. The first contribution of this study is multi-objective problem formulation that opens additional perspectives for decision-making beyond cost-oriented optimization. Second, the maximal flow LP model allows the finding of a feasible solution even for unbalanced supply and demand cases without relaxing hard capacity constraints. Third, this allows improve service level at the strategic inventory holding point. Fourth, structure dynamics and ripple effect can be taken into account. Structure dynamics allows considering different execution scenarios and developing suggestions on replanning in the case of disturbances. The graph of structural reliability allows identify the optimistic and pessimistic scenarios. These scenarios are used for computational experiments with the developed model and the industrial models. With the developed model, the practical issues of scenario-based risk identification strategy and operational distribution planning can be interlinked.