Linear Mathematical Programming Research Articles

Trapping Sets Search Using the Method of Mixed Integer Linear Programming with a Priori List of Variable Nodes

Purpose of research is to develop a new high-speed method for searching trappin sets in graph codes, ensuring the completeness of the search.Methods. There are two approaches to finding trappin sets. The first, based on the Monte Carlo method with a biased probability estimation using Importance Sampling, involves the use of a decoder. The advantage of this approach is its high performance. The disadvantages are the dependence on decoder parameters and channel characteristics and the finite probability of missing trappin sets. The second approach is based on the use of linear programming methods. The advantage of this approach is the completeness of the resulting list of trappin sets, due to its independence from the decoder parameters and channel characteristics. The disadvantage of this approach is its high computational complexity. In the article, within the framework of the second approach, a new method for searching trappin sets with less computational complexity is proposed. The method involves solving a mixed integer linear programming problem using an a priori list of code vertices participating in the shortest cycles in the code graph. Results. Using the proposed method, a search for trappin sets was performed for several low-density codes. For this purpose, the mathematical linear programming package IBM CPLEX version 12.8 was used, which was run on 32 threads of a 16-core AMD Ryzen 3950X processor with 32GB of RAM (DDR4). In the Margulis code (2640, 1320), using the proposed method, the trappin set TS(6,6) was found in a time of 0.53 s. The speedup provided by the method proposed in the paper compared to the Velazquez-Subramani method is 8252.415 times. Thanks to the high speed and completeness of the search, trappin sets were found for the first time TS(62,16) and TS(52,14) in the Margulis code (4896, 2474 ).Conclusion. The paper proposes a new method for searching trapping sets by solving a mixed integer linear programming problem with an a priori list of code. The method is fast and provides completeness of the search.

Just‐in‐time scheduling problem with due windows and release dates for precast bridge girders

AbstractAs the prefabrication construction method plays an increasingly important role in the construction of cross‐sea bridges, coordinating the schedule between off‐site prefabrication and on‐site assembly becomes crucial and challenging. This paper studies a just‐in‐time single‐machine scheduling problem with due windows and release dates, which originates from the production and supply process of large segments of precast steel box girders in bridge construction. To solve this problem, we equivalently decompose it into the independent and same type of subproblems and formulate a mixed‐integer linear mathematical programming model. Furthermore, we propose a branch‐and‐bound algorithm with the initialization of a novel linear early and tardy dispatching rule with due windows to solve the problem exactly. The dispatching rule, based on the local dominance criteria with due windows, is designed and used to improve the upper bound. Moreover, a lower bound is obtained by further decomposing and relaxing the subproblems. Finally, the parameter of the dispatching rule is calibrated, and the best search strategy of the branch‐and‐bound algorithm is determined. The comprehensive computational experiments carried out show the efficiency and effectiveness of the proposed branch‐and‐bound algorithm.

Optimizing Chance Constraint Multiple-Objective Fractional Mathematical Programming Problem Involving Dependent Random Variable

This manuscript suggests a methodology to solve chance constraint multiple-objective linear fractional mathematical programming problem in which the parameters are dependent random variables to each other. The proposed problem is formulated by taking few of the parameters as continuous dependent random variables. The proposed model cannot be solved directly by using existing methodology. Thus in order to solve the proposed model, an equivalent deterministic model is derived. The procedure to solve the proposed model is accomplished in two main steps. Initially, the proposed multiple-objective chance constraint linear fractional mathematical problem is transformed to deterministic equivalent multiple-objective linear fractional mathematical programming by the help of chance constrained method. In the second step, multiple-objective functions, which consist of fractional functions is solved by using lexicographic programming approach. Finally, an example is mentioned to illustrate the methodology.

On-site workshop investment problem: A novel mathematical approach and solution procedure

In real-world construction sites, On-Site Workshops (OSW) are installed to accelerate construction activities and facilitate the material handling process. These temporary OSWs are cost-effective, leading to decreasing the material handling cost and project makespan, which indicates their important role as a part of a construction project. However, considering the OSW, which has not been addressed in the project scheduling problems, requires the construction site to have a space capacity constraint while considering the workshop size, availability level, and other project-related constraints. In the present work, by considering the OSWs, a real construction project scheduling problem is studied as a Multi-Mode On-Site Workshop Investment Problem with Tardiness (MOSWIPT) while finding the installation/dismantling time of the OSWs. Two new (linear) mathematical programming models are proposed for MOSWIPT. Next, due to the NP-hardness of the problem, an enhanced Genetic Algorithm (GA)-based metaheuristic with efficient problem-specific improvement rules as local search and effective crossover and mutation operators is proposed. Computational experiments show that the proposed method has solved most of the instances of the addressed problem to optimality and outperformed the existing metaheuristics, e.g., Simulated Annealing (SA) and Particle Swarm Optimization (PSO). Finally, conclusions and suggestions for future studies are stated.

Kemampuan Verbal Matematis dalam Menyelesaikan Soal Cerita Program Linier Pada Siswa Kelas XI IPA MAN 2 Kota Bima

This study aims to k now verbal ability in mathematical linear program solving word problems in class XI IPA MAN 2 Kota Bima spread over 5 classes. The subject determination technique used purposive sampling with a total of 52 students out of a total of 211 students. The criteria for determining the subject with the following considerations: 1) Students have received learning about linear programming; 2) Students have potential verbal abilities which are assessed based on the results of the verbal ability test in the linear program as a requirement in determining the subject group. The analysis technique in this study uses descriptive analysis. Based on the results of the study, it was found that the level of mathematical verbal ability in solving story problems in the linear program of class XI IPA MAN 2 Kota Bima was in the medium category. Students' mathematical verbal ability shows that the higher the level of students' verbal ability, the easier it is for students to understand and solve math problems or questions with the right answers.

A Mathematical Programming Approach for Multi-Objective Optimization in Designing Flexible Manufacturing Cells

The study aims to design flexible manufacturing cells with routing flexibility. A weighted mixed-integer linear mathematical programming model that aims to find optimal routing of parts in flexible manufacturing cells under constraints such as minimum machine utilization rate, maximum machine utilization rate, tool capacities, the utilization rate of workers, and labor-system unbalance is developed. The mathematical programming model aims to minimize the weighted sum of five objective functions: (1) the total number of intracellular movements; (2) the total number of intercellular movements; (3) the total workload unbalance of the machine system; (4) the total number of tools in all machines in the cells; and (5) the total labor-system workload unbalance. The main contribution of this study is to obtain these five objectives simultaneously, which have not been encountered to handle together before. By integrating these factors, the study presents a comprehensive approach to optimizing the design of flexible manufacturing cells. This study also has the potential to enhance system performance by addressing these factors. An illustrative problem tests the developed model, and the LINGO 17.0 optimization program is used to solve the generated mathematical programming model. Moreover, the related sensitivity analysis is performed with some parameters to examine the obtained results.

Dynamic Multi-Compartment Vehicle Routing Problem for Smart Waste Collection

The rapid increase in urbanization results in an increase in the volume of municipal solid waste produced every day, causing overflow of the garbage cans and thus distorting the city’s appearance; for this and environmental reasons, smart cities involve the use of modern technologies for intelligent and efficient waste management. Smart bins in urban environments contain sensors that measure the status of containers in real-time and trigger wireless alarms if the container reaches a predetermined threshold, and then communicate the information to the operations center, which then sends vehicles to collect the waste from the selected stations in order to collect a significant waste amount and reduce transportation costs. In this article, we will address the issue of the Dynamic Multi-Compartmental Vehicle Routing Problem (DM-CVRP) for selective and intelligent waste collection. This problem is summarized as a linear mathematical programming model to define optimal dynamic routes to minimize the total cost, which are the transportation costs and the penalty costs caused by exceeding the bin capacity. The hybridized genetic algorithm (GA) is proposed to solve this problem, and the effectiveness of the proposed approach is verified by extensive numerical experiments on instances given by Valorsul, with some modifications to adapt these data to our problem. Then we were able to ensure the effectiveness of our approach based on the results in the static and dynamic cases, which are very encouraging.

Experimental research of transition processes of asynchronous motors of belt conveyors on mathematical modeling

The purpose of the study. increasing the performance of belt conveyors and reducing transportation costs based on the developed mathematical modeling of transient processes of asynchronous motors of open-pit conveyor transport. The object of the research is the operating modes of asynchronous motors of belt conveyors of mining enterprises. The subject of research is the starting and braking modes of operation of asynchronous motors of belt conveyors of mining enterprises. Research methods are based on the modern theory of electrical machines, methods of linear and nonlinear mathematical programming, mathematical statistics, as well as methods of system analysis. The practical results of the study are as follows: the actual coefficient of resistance to the rotation of the conveyor rollers and the coefficient of resistance to the movement of the belt, leading to a decrease in the tractive effort, were determined; a method was developed for determining the additional power consumption of electric motors of belt conveyors, taking into account the overall volume, weight and speed of transported lumpy cargo; a mathematical model of braking and starting modes of induction motors of belt conveyors of mining enterprises was developed, taking into account the resistance coefficient of the belt and rollers.

Lower Bound Limit Analysis of Non-Persistent Jointed Rock Masses Using Mixed Numerical Discretization

The bearing capacity of a non-persistent jointed rock mass containing a rock bridge is investigated by combining the lower bound limit analysis theory, a mixed numerical discrete method, and linear mathematical programming. A mixed numerical discrete method is proposed to divide non-persistent jointed rock masses in which rigid block elements are used to simulate the rock blocks, whereas the finite element method is used to simulate the intact rock bridges. A linear mathematical programming model for the ultimate bearing capacity is constructed and solved using the interior point algorithm. The proposed formulation is validated by application to three rock slopes.

Parallel assembly lines worker assignment and balancing problem: A mathematical model and an artificial bee colony algorithm

In today’s competitive world, assembly lines are one of the most important industrial manufacturing systems, which can accommodate the massive increase in customer demands and product range, for the production industry. One possible way to increase the capacity and efficiency of assembly lines is to design a parallel assembly line system, consisting of two or more assembly lines placed next to each other. In the literature, the assembly lines are usually balanced according to tasks with fixed processing times without considering worker–task compatibility. In real-life applications, however, since the processing time of each task may vary according to the capability of each worker, the assignment of tasks to stations in the assembly line depends on the skills of the worker assigned to the related station. Considering the worker and task assignments together for each assembly line will provide a more realistic perspective. This paper, to the best of our knowledge, is the first study that considers the minimization of joint cycle time for the worker assignment and assembly line balancing problem in parallel assembly lines (PALWABP-II). Accordingly, a binary linear mathematical programming (BLP) model is developed and an artificial bee colony (ABC)-based solution approach is proposed for medium and large-sized problems. A new set of test problems is presented to the literature for a computational experiment. In addition, a summary of the literature is presented. An example problem is solved using the BLP model. In order to evaluate the effectiveness of the proposed model and approach, they are compared with the classical particle swarm optimization (PSO) algorithm and ten different priority rules. The computational results on test problems prove the effectiveness of the proposed ABC algorithm for the PALWABP-II.

Container Ship Planning Optimization Using Intelligent Meta-heuristic Algorithms

The timing of container ships is one of the more attractive optimization issues. Because container ships carry 80% of the volume of cargo worldwide. Hence, maritime transport and container trade are known as the backbone of global trade. Today, with the development of shipbuilding technology and increasing the frequency of maritime transport, determining the optimal route for container ships has become more and more important for shipping lines and port managers. Accordingly, the present study intends investigate the problem of optimizing the trajectory of ships using an integer linear mathematical programming model. In order to solve the proposed model, the meta-heuristic algorithms of differential evolution and gray wolf in MATLAB software are utilized. Finally, it was shown that the proposed mathematical model and solution approaches can assist ship operators and terminal managers in making sensible optimization decisions.

Managing change in nature-based tourism: A decision-making model using linear programming

In conditions of forced isolation, nature-based tourism meets the needs of safe and comfortable recreation and travel combined with the solution of acute issues of medical treatment and rehabilitation during the pandemic and post-pandemic periods. This study aims to develop a model for decision-making on change management in nature tourism based on the approach of linear economic and mathematical programming. The paper formalized changes in the variability of objective function parameters of the model and the system of its restrictions, following the structure of assets of nature-based tourism, balanced by the sustainability principle. The algorithm for implementing the model includes four stages: collection and processing of relevant data on nature-based tourism; considering changes in the objective function and the system of its limitations; linear programming with variability tests using the simplex method; defining ranges/limits in which decisions are made. The initial data are summarized and averaged based on the primary data analysis on the functioning of sanatoriums and other tourist and recreational facilities in Ukraine. Short-term nature-based tourism is considered, the services of which are classified according to the criterion of the primary purpose of travel: “wow-effect” tourism, sports tourism, health tourism, traditional recreation, and green tourism. The results make it possible to substantiate decisions on changes in recreational land areas and human resources, on the limits of changes in income due to the dynamics of service prices, as well as determine the price range while maintaining income structure and sustainability limits for natural and human assets of nature-based tourism. AcknowledgmentThe paper contains the results of a study conducted under the National Academy of Science of Ukraine’s grant Formation and Use of Natural-Resource Assets of the Recreational and Tourism Sphere (0120U100159) and the Nominal Scholarship of the Verkhovna Rada of Ukraine for Young Scientists-Doctors of Sciences for 2021 (0121U113482).

Machining Parameters Optimization Based on Objective Function Linearization

Manufacturing process optimization is an ever-actual goal. Within this goal, machining parameters optimization is a very important task. Machining parameters strongly influence the manufacturing costs, process productivity and piece quality. Literature presents a series of optimization methods. The results supplied by these methods are comparable and it is difficult to establish which method is the best. For machining parameters optimization, this paper proposes a novel, simple and efficient method, without additional costs related to new software packages. This approach is based on linear mathematical programming. The optimization mathematical models are, however, nonlinear. Therefore, mathematical model linearization is required. The major and difficult problem is the linearization of the objective function. This represents the key element of the proposed optimization method. In this respect, the paper proposes an original mathematical procedure for calculating the part of the objective function that refers to the analytical integration of the tool life into the model. This calculus procedure was transposed into an original software tool. For demonstrating the validity of the method, a comparison is presented among the results obtained by certain optimization techniques. It results that the proposed method is simple and as good as those presented by the literature.

Mysterious Neutrosophic Linear Models

Operations research often shortened to the initialism O.R., is a discipline that deals with the development and application of advanced analytical methods to improve decision-making. It is sometimes considered to be a subfield of mathematical sciences. The term management science is occasionally used as a synonym. It has the ability to express the concepts of efficiency and scarcity in a well-defined mathematical model for a specific issue. It has the ability to use scientific methods to solve complex problems in managing large scale systems for factories, institutions, and companies, and enables them to make optimal scientific decisions for the functioning of Its work. Employing techniques from other mathematical sciences, such as modelling, statistics, and optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems. Because of its emphasis on practical applications, operations research has overlapped with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in military efforts before World War II, its techniques have grown to concern problems in a variety of industries. The mathematical model is the simplified image of expressing a practical system from a real life problem or an idea put forward for an executable system, as the mathematical models consist of a goal function through which we search for the maximum or minimum value subject to restrictions. Linear mathematical programming is one of the most important topics in the field of operations research due to their frequent use in most areas of life. When studying linear programming, the first step is to identify the various types of linear models and how to transition from one to the next. We realize that the ideal solution of the linear model is influenced by the coefficients of the variables of objective function that describes a profit if the model is a maximizing model or represents a cost if the model is a minimization model, which is affected by environmental conditions. The fixed values that represent the right side of the inequalities (constraints), which express the available capital, time, raw resources, and so on, have an impact on the optimum solution. They are also affected by environmental conditions. We used to take these values as fixed values in classical logic, which does not correspond to reality and leads to erroneous solutions to the problems described by the linear model. As a result, it was essential to reformulate the classical linear models' problems, taking into consideration all probable scenarios and changes in the work environment. In this study, we will look into linear models and their kinds in view of neutrosophic logic, which takes into account all of the data and all of the changes that may occur in the issue under investigation, as well as the uncertainty that is encountered in the problem's data. We'll also look at it if the coefficients of the variables in the objective function are neutrosophic values, and the accessible options are neutrosophic values because we'll reformulate the existing linear mathematical models using neutrosophic logic, and show how to convert from one to another using some examples.

Technique for optimizing the adaptive control of the output of an enterprise based on a dynamical economic and mathematical model

The article is devoted to the application of dynamic economic and mathematical models for managing the production of an enterprise based on the use of the feedback principle. Formation of a discrete controllable dynamical system is given, which describes the process of production output by a manufacturing enterprise in the presence of a predictable demand function for products. The phase vector of a dynamical system describes the main parameters of production, and the control action vector (control vector) describes the intensity of the use of technological methods of production that are available to the subject of control. It is assumed that in each period of time the subject of control knows the vector function that describes the volume of demand for the company's products in subsequent periods of time, and the given geometric restrictions on the implementation of the phase vector, control vector and demand vector are also known. As the target function of the problem, the value of the discrepancy between the volumes of output by the enterprise relative to the given predicted value of the demand function in the subsequent control period is considered. Using the generated dynamical system, the paper proposes an economic-mathematical model of the studied problem of optimizing the adaptive control of the enterprise's output, which includes a class of admissible strategies for adaptive control and the formulation of the problem. The paper proposes a method for solving the formulated problem of optimizing the adaptive control of the output of an enterprise, which is implemented as a finite sequence of one-step algebraic operations on vectors of a finite-dimensional vector space, a finite set of solutions to problems of linear and convex mathematical programming. The results obtained can be used in the development of intelligent decision support systems for the actual tasks of managing the production of products at industrial enterprises.

Optimization of regional transport networks based on a mathematical model of passenger preferences

This article presents the results of developing a model for a regional passenger transport network aimed at solving the logistical problem of constructing rational intermodal routes in a defined closed loop. The tools of graph theory and linear mathematical programming have been applied to build the model algorithm, which allows finding solutions for weighted graphs, in the absence of negative weight links, while keeping information about the sequence of hub points on the selected paths. The proposed solutions are versatile enough to be scalable for regions with different network topologies. The model is adapted to dynamically changing and extensible systems, allowing it to be practically applied to justify options for the future development of infrastructure in different modes of transport.

Cultural Challenge: Teaching Mathematics to Non-mathematicians

A “mathematics for non-mathematicians” course, commonly known as a “service” course is an undergraduate mathematics course developed for students who are not (going to become) mathematics majors. Besides calculus, such courses may include linear algebra, mathematical reasoning, differential equations, mathematical programming and modeling, discrete mathematics, mathematics for teachers, and so on. In this article we argue that a good, productive curricular design and teaching of service courses happen through a meaningful collaboration between a mathematics instructor and the department whose students are taking the course. This collaboration ensures that “non-mathematicians” see the relevance of learning mathematics for their discipline (say, by discussing authentic problems and examples), but also appreciate the relevance and benefits which mathematics brings to their overall education and skills set.

Exact optimization and decomposition approaches for shelf space allocation

Shelf space is one of the scarcest resources, and its effective management to maximize profits has become essential to gain a competitive advantage for retailers. We consider the shelf space allocation problem with additional features (e.g., integer facings, rectangular arrangement restrictions) motivated by literature and our interactions with a local bookstore. We determine optimal number of facings of all products in two aspects (width and height of a rectangular arrangement space for each product), and allocate them as contiguous rectangles to maximize profit. We first develop a mixed-integer linear mathematical programming model (MIP) for our problem and propose a solution method based on logic-based Benders decomposition (LBBD). Next, we construct an exact 2-stage algorithm (IP1/IP2), inspired by LBBD, which can handle larger and real-world size instances. To compare performances of our methods, we generate 100 test instances inspired by real-world applications and benchmarks from the literature. We observe that IP1/IP2 finds optimal solutions for real-world instances efficiently and can increase the local bookstore’s profit up to 16.56%. IP1/IP2 can provide optimal solutions for instances with 100 products in minutes and optimally solve up to 250 products (assigned to 8 rows x 160 columns) within a time limit of 1800 s. This exact 2-stage IP1/IP2 solution approach can be effective in solving similar problems such as display problem of webpage design, allocation of product families in grocery stores, and flyer advertising.

Application of linear and nonlinear mathematical programming to retrofit hydrogen networks

Hydrogen network management has economic appeal due to its importance in oil refineries. It has become genuinely relevant due to the restrictions of sulfur content in fuels, which need hydrogen to be removed. Mathematical programming can be used as a tool for optimizing hydrogen networks, and the efficient management of hydrogen within the refineries can be achieved through a material balance of the units that make up the hydrogen network. In this work, an optimization model Mixed-Integer Linear Programming (MILP) and Mixed-Integer Nonlinear Programming (MINLP) for hydrogen networks was applied to minimize the operating costs. The optimization model was developed in GAMS, and it was validated using a literature case study and a real case study from a Brazilian Refinery. The operation cost was reduced by 10% and 19.6% with MILP and 9.7% and 31.5% with MINLP, for example 1 and 2, respectively. Comparing the results, both achieve significant savings in operating costs. The MILP model, which is easier to solve, has proved to be an efficient tool for optimizing hydrogen networks. However, optimization via MINLP, although not guaranteeing the optimal solution, resulted in lower operating and capital costs. The design of the optimized hydrogen networks was also detailed, and other extra restrictions were imposed on the problem.

Strategic planning of new product introductions: Integrated planning of products and modules in the automotive industry

Timing the introduction of new products to the market is an important strategic decision in the automotive industry. For several reasons, it is also a complex decision problem. First, the use of platforms creates many interactions between different vehicles via shared modules (e.g. engines). Second, new and existing products rely on various shared resources (e.g. development resources or production capacities). Furthermore, different conflicting objectives must be considered. In this paper, we develop a mathematical linear programming model describing the decision problem based on the resource-constrained project scheduling problem. It simultaneously decides on the start of production date for vehicle models, variants, and engines as well as on the assignment of engines to the given variants. Integrating a multi-criteria approach, the model helps to analyze trade-offs between important managerial objectives related to resource utilization and fleet emission metrics. Using realistic data from a major European automotive company, we demonstrate that our model enables the efficient evaluation of various courses of action. Such capabilities are especially relevant in times of rapid technological change, such as the current transition towards electrified vehicle portfolios.