Search results for: multi-objective integer linear programming

Authors: Ebrahim Asadi-Gangraj

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Hybrid flow shop scheduling problem (HFS) contains sequencing in a flow shop where, at any stage, there exist one or more related or unrelated parallel machines. This production system is a common manufacturing environment in many real industries, such as the steel manufacturing, ceramic tile manufacturing, and car assembly industries. In this research, a mixed integer linear programming (MILP) model is presented for the hybrid flow shop scheduling problem, in which, the objective consists of minimizing the maximum completion time (makespan). For this purpose, a Benders Decomposition (BD) method is developed to solve the research problem. The proposed approach is tested on some test problems, small to moderate scale. The experimental results show that the Benders decomposition approach can solve the hybrid flow shop scheduling problem in a reasonable time, especially for small and moderate-size test problems.

Keywords: hybrid flow shop, mixed integer linear programming, Benders decomposition, makespan

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Authors: Muhammed Ordu, Eren Demir, Chris Tofallis

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The number of inpatient admissions in the UK has been significantly increasing over the past decade. These increases cause bed occupancy rates to exceed the target level (85%) set by the Department of Health in England. Therefore, hospital service managers are struggling to better manage key resource such as beds. On the other hand, this severe demand pressure might lead to confusion in wards. For example, patients can be admitted to the ward of another inpatient specialty due to lack of resources (i.e., bed). This study aims to develop a simulation-optimization model to reallocate the available number of beds in a mid-sized hospital in the UK. A hospital simulation model was developed to capture the stochastic behaviours of the hospital by taking into account the accident and emergency department, all outpatient and inpatient services, and the interactions between each other. A couple of outputs of the simulation model (e.g., average length of stay and revenue) were generated as inputs to be used in the optimization model. An integer linear programming was developed under a number of constraints (financial, demand, target level of bed occupancy rate and staffing level) with the aims of maximizing number of admitted patients. In addition, a sensitivity analysis was carried out by taking into account unexpected increases on inpatient demand over the next 12 months. As a result, the major findings of the approach proposed in this study optimally reallocate the available number of beds for each inpatient speciality and reveal that 74 beds are idle. In addition, the findings of the study indicate that the hospital wards will be able to cope with 14% demand increase at most in the projected year. In conclusion, this paper sheds a new light on how best to reallocate beds in order to cope with current and future demand for healthcare services.

Keywords: bed occupancy rate, bed reallocation, discrete event simulation, inpatient admissions, integer linear programming, projected usage

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Authors: Stojan Kravanja, Andrej Ivanič, Tomaž Žula

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This contribution focuses on structural optimization in civil engineering using mixed integer non-linear programming (MINLP). MINLP is characterized as a versatile method that can handle both continuous and discrete optimization variables simultaneously. Continuous variables are used to optimize parameters such as dimensions, stresses, masses, or costs, while discrete variables represent binary decisions to determine the presence or absence of structural elements within a structure while also calculating discrete materials and standard sections. The optimization process is divided into three main steps. First, a mechanical superstructure with a variety of different topology-, material- and dimensional alternatives. Next, a MINLP model is formulated to encapsulate the optimization problem. Finally, an optimal solution is searched in the direction of the defined objective function while respecting the structural constraints. The economic or mass objective function of the material and labor costs of a structure is subjected to the constraints known from structural analysis. These constraints include equations for the calculation of internal forces and deflections, as well as equations for the dimensioning of structural components (in accordance with the Eurocode standards). Given the complex, non-convex and highly non-linear nature of optimization problems in civil engineering, the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm is applied. This algorithm alternately solves subproblems of non-linear programming (NLP) and main problems of mixed-integer linear programming (MILP), in this way gradually refines the solution space up to the optimal solution. The NLP corresponds to the continuous optimization of parameters (with fixed topology, discrete materials and standard dimensions, all determined in the previous MILP), while the MILP involves a global approximation to the superstructure of alternatives, where a new topology, materials, standard dimensions are determined. The optimization of a convex problem is stopped when the MILP solution becomes better than the best NLP solution. Otherwise, it is terminated when the NLP solution can no longer be improved. While the OA/ER algorithm, like all other algorithms, does not guarantee global optimality due to the presence of non-convex functions, various modifications, including convexity tests, are implemented in OA/ER to mitigate these difficulties. The effectiveness of the proposed MINLP approach is demonstrated by its application to various structural optimization tasks, such as mass optimization of steel buildings, cost optimization of timber halls, composite floor systems, etc. Special optimization models have been developed for the optimization of these structures. The MINLP optimizations, facilitated by the user-friendly software package MIPSYN, provide insights into a mass or cost-optimal solutions, optimal structural topologies, optimal material and standard cross-section choices, confirming MINLP as a valuable method for the optimization of structures in civil engineering.

Keywords: MINLP, mixed-integer non-linear programming, optimization, structures

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Authors: Sangchul Kim, Dosaeng Park

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It is one of the important design issues to provide various levels of difficulty in order to suit the skillfulness of an individual. In this paper we propose an integer programming-based method for selecting a mixture of challenges for a racing game that meet a given degree of difficulty. The proposed method can also be used to dynamically adjust the difficulty of the game during the progression of playing. By experiments, it is shown that our method performs well enough to generate games with various degrees of difficulty that match the perception of players.

Keywords: level generation, level adjustment, racing game, ip

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Authors: Mohammad Pourmohammadi Fallah, Maziar Salahi

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In the literature, a mixed-integer linear programming model for the integrated production planning and warehouse layout problem is proposed. To solve the model, the authors proposed a Lagrangian relax-and-fix heuristic that takes a significant amount of time to stop with gaps above 5$\%$ for large-scale instances. Here, we present two heuristic algorithms to solve the problem. In the first one, we use a greedy approach by allocating warehouse locations with less reservation costs and also less transportation costs from the production area to locations and from locations to the output point to items with higher demands. Then a smaller model is solved. In the second heuristic, first, we sort items in descending order according to the fraction of the sum of the demands for that item in the time horizon plus the maximum demand for that item in the time horizon and the sum of all its demands in the time horizon. Then we categorize the sorted items into groups of 3, 4, or 5 and solve a small-scale optimization problem for each group, hoping to improve the solution of the first heuristic. Our preliminary numerical results show the effectiveness of the proposed heuristics.

Keywords: capacitated lot-sizing, warehouse layout, mixed-integer linear programming, heuristics algorithm

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Authors: Mai A. Ibrahim, Mohammed El-Beltagy, Motaz Khorshid

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Emerging Markets return distributions have shown significance departure from normality were they are characterized by fatter tails relative to the normal distribution and exhibit levels of skewness and kurtosis that constitute a significant departure from normality. Therefore, the classical Markowitz Mean-Variance is not applicable for emerging markets since it assumes normally-distributed returns (with zero skewness and kurtosis) and a quadratic utility function. Moreover, the Markowitz mean-variance analysis can be used in cases of moderate non-normality and it still provides a good approximation of the expected utility, but it may be ineffective under large departure from normality. Higher moments models and median models have been suggested in the literature for asset allocation in this case. Higher moments models have been introduced to account for the insufficiency of the description of a portfolio by only its first two moments while the median model has been introduced as a robust statistic which is less affected by outliers than the mean. Tail risk measures such as Value-at Risk (VaR) and Conditional Value-at-Risk (CVaR) have been introduced instead of Variance to capture the effect of risk. In this research, higher moment models including the Mean-Variance-Skewness (MVS) and Mean-Variance-Skewness-Kurtosis (MVSK) are formulated as single-objective non-linear programming problems (NLP) and median models including the Median-Value at Risk (MedVaR) and Median-Mean Absolute Deviation (MedMAD) are formulated as a single-objective mixed-integer linear programming (MILP) problems. The higher moment models and median models are compared to some benchmark portfolios and tested on real financial data in the Egyptian main Index EGX30. The results show that all the median models outperform the higher moment models were they provide higher final wealth for the investor over the entire period of study. In addition, the results have confirmed the inapplicability of the classical Markowitz Mean-Variance to the Egyptian stock market as it resulted in very low realized profits.

Keywords: Egyptian stock exchange, emerging markets, higher moment models, median models, mixed-integer linear programming, non-linear programming

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Authors: Harshith Gowda K. S, Tejaskumar N, Shubhanga R. B, Gowtham N, Deekshith Gowda H. S

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Phasor measurement units (PMU) are playing an important role in the current power system for state estimation. It is necessary to have complete observability of the power system while minimizing the cost. For this purpose, the optimal location of the phasor measurement units in the power system is essential. In a bus system, zero injection buses need to be evaluated to minimize the number of PMUs. In this paper, the optimization problem is formulated using mixed integer programming to obtain the optimal location of the PMUs with increased observability. The formulation consists of with and without zero injection bus as constraints. The formulated problem is simulated using a CPLEX solver in the GAMS software package. The proposed method is tested on IEEE 30, IEEE 39, IEEE 57, and IEEE 118 bus systems. The results obtained show that the number of PMUs required is minimal with increased observability.

Keywords: PMU, observability, mixed integer programming (MIP), zero injection buses (ZIB)

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Authors: Cem Isik Dogru, Salih Tekin, Kursad Derinkuyu

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Electric vehicle (EV) carsharing is an alternative method to tackle urban transportation problems. This method can be applied by several options. One of the options is the one-way carsharing, which allow an EV to be taken at a designated location and leaving it on another specified location customer desires. Although it may increase users’ satisfaction, the issues, namely, demand dissatisfaction, relocation of EVs and charging schedules, must be dealt with. Also, excessive electricity has to be stored in batteries of EVs. To cope with aforementioned issues, two-step mixed integer programming (MIP) model is proposed. In first step, the integer programming model is used to determine amount of electricity to be sold to the grid in terms of time periods for extra profit. Determined amounts are provided from the batteries of EVs. Also, this step works in day-ahead electricity markets with forecast of periodical electricity prices. In second step, other MIP model optimizes daily operations of one-way carsharing: charging-discharging schedules, relocation of EVs to serve more demand and renting to maximize the profit of EV fleet owner. Due to complexity of the models, heuristic methods are introduced to attain a feasible solution and different price information scenarios are compared.

Keywords: electric vehicles, forecasting, mixed integer programming, one-way carsharing

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Authors: Alexandre Barbosa de Almeida, Telma Woerle de Lima Soares

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Protein structure prediction is a challenging task in the bioinformatics field. The biological function of all proteins majorly relies on the shape of their three-dimensional conformational structure, but less than 1% of all known proteins in the world have their structure solved. This work proposes a deep learning model to address this problem, attempting to predict some aspects of the protein conformations. Throughout a process of multiobjective dominance, a recurrent neural network was trained to abstract the particular bias of each individual multiobjective algorithm, generating a heuristic that could be useful to predict some of the relevant aspects of the three-dimensional conformation process formation, known as protein folding.

Keywords: Ab initio heuristic modeling, multiobjective optimization, protein structure prediction, recurrent neural network

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Authors: Marin Mandić, Elis Sutlović, Tonći Modrić, Luka Stanić

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With the restructuring and deregulation of the power system, storage owners, generation companies or private producers can offer their multiple services on various power markets and earn income in different types of markets, such as the day-ahead, real-time, ancillary services market, etc. During the COVID-19 pandemic, electricity prices, as well as ancillary services prices, increased significantly. The optimization of the energy storage operation was performed using a suitable model for simulating the operation of a pumped storage hydropower plant under market conditions. The objective function maximizes the income earned through energy arbitration, regulation-up, regulation-down and spinning reserve services. The optimization technique used for solving the objective function is mixed integer linear programming (MILP). In numerical examples, the pumped storage hydropower plant operation has been optimized considering the already achieved hourly electricity market prices from Nord Pool for the pre-pandemic (2019) and the pandemic (2020 and 2021) years. The impact of the electricity market prices during the COVID-19 pandemic on energy storage operation is shown through the analysis of income, operating hours, reserved capacity and consumed energy for each service. The results indicate the role of energy storage during a significant fluctuation in electricity and services prices.

Keywords: electrical market prices, electricity market, energy storage optimization, mixed integer linear programming (MILP) optimization

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Authors: Onur Kaya, Ilknur Tukenmez

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In this study, we consider vehicle routing problems on networks with alternative direct links between nodes, and we analyze a multi-objective problem considering the financial, environmental and social objectives in this context. In real life, there might exist several alternative direct roads between two nodes, and these roads might have differences in terms of their lengths and durations. For example, a road might be shorter than another but might require longer time due to traffic and speed limits. Similarly, some toll roads might be shorter or faster but require additional payment, leading to higher costs. We consider such alternative links in our problem and develop a mixed integer linear programming model that determines which alternative link to use between two nodes, in addition to determining the optimal routes for different vehicles, depending on the model objectives and constraints. We consider the minimum cost routing as the financial objective for the company, minimizing the CO2 emissions and gas usage as the environmental objectives, and optimizing the driver working conditions/working hours, and minimizing the risks of accidents as the social objectives. With these objective functions, we aim to determine which routes, and which alternative links should be used in addition to the speed choices on each link. We discuss the results of the developed vehicle routing models and compare their results depending on the system parameters.

Keywords: vehicle routing, alternative links between nodes, mixed integer linear programming, triple bottom line accounting

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Authors: K. -H. Yang

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Supplier selection problem is one of the important issues of supply chain problems. Two categories of methodologies include qualitative and quantitative approaches which can be applied to supplier selection problems. However, due to the complexities of the problem and lacking of reliable and quantitative data, qualitative approaches are more than quantitative approaches. This study considers operational cost and supplier’s reliability factor and solves the problem by using a quantitative approach. A mixed integer programming model is the primary analytic tool. Analyses of different scenarios with variable cost and reliability structures show that the effectiveness of this approach to the supplier selection problem.

Keywords: mixed integer programming, quantitative approach, supplier’s reliability, supplier selection

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Authors: P. Priyanka, S. Shruthi, N. Guruprasad

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Health is a common theme in most cultures. In fact all communities have their concepts of health, as part of their culture. Health continues to be a neglected entity. Planning of Human diet should be done very careful by selecting the food items or groups of food items also the composition involved. Low price and good taste of foods are regarded as two major factors for optimal human nutrition. Linear programming techniques have been extensively used for human diet formulation for quiet good number of years. Through the process, we mainly apply “The Simplex Method” which is a very useful statistical tool based on the theorem of Elementary Row Operation from Linear Algebra and also incorporate some other necessary rules set by the Simplex Method to help solve the problem. The study done by us is an attempt to develop a programming model for optimal planning and best use of nutrient ingredients.

Keywords: diet formulation, linear programming, nutrient ingredients, optimization, simplex method

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Authors: Jean Berger, Nassirou Lo, Martin Noel

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Perfectly suited for natural or man-made emergency and disaster management situations such as flood, earthquakes, tornadoes, or tsunami, multi-target search path planning for a team of rescue agents is known to be computationally hard, and most techniques developed so far come short to successfully estimate optimality gap. A novel mixed-integer linear programming (MIP) formulation is proposed to optimally solve the multi-target multi-agent discrete search and rescue (SAR) path planning problem. Aimed at maximizing cumulative probability of successful target detection, it captures anticipated feedback information associated with possible observation outcomes resulting from projected path execution, while modeling agent discrete actions over all possible moving directions. Problem modeling further takes advantage of network representation to encompass decision variables, expedite compact constraint specification, and lead to substantial problem-solving speed-up. The proposed MIP approach uses CPLEX optimization machinery, efficiently computing near-optimal solutions for practical size problems, while giving a robust upper bound obtained from Lagrangean integrality constraint relaxation. Should eventually a target be positively detected during plan execution, a new problem instance would simply be reformulated from the current state, and then solved over the next decision cycle. A computational experiment shows the feasibility and the value of the proposed approach.

Keywords: search path planning, search and rescue, multi-agent, mixed-integer linear programming, optimization

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Authors: Mahnaz Hosseinzadeh, Aliyeh Kazemi

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In this paper, a method is proposed for solving Fuzzy Multi-Objective Linear Programming problems (FMOLPP) with fuzzy right hand side and fuzzy decision variables. To illustrate the proposed method, it is applied to the problem of selecting suppliers for an automotive parts producer company in Iran in order to find the number of optimal orders allocated to each supplier considering the conflicting objectives. Finally, the obtained results are discussed.

Keywords: fuzzy multi-objective linear programming problems, triangular fuzzy numbers, fuzzy ranking, supplier selection problem

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Authors: Yixun Shi

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Many interior point methods for convex programming solve an (n+m)x(n+m)linear system in each iteration. Many implementations solve this system in each iteration by considering an equivalent mXm system (4) as listed in the paper, and thus the job is reduced into solving the system (4). However, the system(4) has to be solved exactly since otherwise the error would be entirely passed onto the last m equations of the original system. Often the Cholesky factorization is computed to obtain the exact solution of (4). One Cholesky factorization is to be done in every iteration, resulting in higher computational costs. In this paper, two iterative methods for solving linear systems using vector division are combined together and embedded into interior point methods. Instead of computing one Cholesky factorization in each iteration, it requires only one Cholesky factorization in the entire procedure, thus significantly reduces the amount of computation needed for solving the problem. Based on that, a hybrid algorithm for solving convex programming problems is proposed.

Keywords: convex programming, interior point method, linear systems, vector division

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Authors: Houda Abadlia, Nadia Smairi, Khaled Ghedira

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Multiobjective Particle Swarm Optimization (MOPSO) has shown an effective performance for solving test functions and real-world optimization problems. However, this method has a premature convergence problem, which may lead to lack of diversity. In order to improve its performance, this paper presents a hybrid approach which embedded the MOPSO into the island model and integrated a local search technique, Variable Neighborhood Search, to enhance the diversity into the swarm. Experiments on two series of test functions have shown the effectiveness of the proposed approach. A comparison with other evolutionary algorithms shows that the proposed approach presented a good performance in solving multiobjective optimization problems.

Keywords: particle swarm optimization, migration, variable neighborhood search, multiobjective optimization

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Authors: Amir Azaron

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In this research, a stochastic programming approach is developed for designing supply chains with uncertain parameters. Demands and selling prices of products at markets are considered as the uncertain parameters. The proposed mathematical model will be multi-period two-stage stochastic programming, which takes into account the selection of retailer sites, suppliers, production levels, inventory levels, transportation modes to be used for shipping goods, and shipping quantities among the entities of the supply chain network. The objective function is to maximize the chain’s net present value. In order to maximize the chain’s NPV, the sum of first-stage investment costs on retailers, and the expected second-stage processing, inventory-holding and transportation costs should be kept as low as possible over multiple periods. The effects of supply uncertainty where suppliers are unreliable will also be investigated on the efficiency of the supply chain.

Keywords: supply chain management, stochastic programming, multiobjective programming, inventory control

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Authors: D. Calogine, O. Chau, S. Dotti, O. Ramiarinjanahary, P. Rasoavonjy, F. Tovondahiniriko

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Mafate is a natural circus in the north-western part of Reunion Island, without an electrical grid and road network. A micro-grid concept is being experimented in this area, composed of a photovoltaic production combined with electrochemical batteries, in order to meet the local population for self-consumption of electricity demands. This work develops a discrete model as well as a stochastic model in order to reach an optimal equilibrium between production and consumptions for a cluster of houses. The management of the energy power leads to a large linearized programming system, where the time interval of interest is 24 hours The experimental data are solar production, storage energy, and the parameters of the different electrical devices and batteries. The unknown variables to evaluate are the consumptions of the various electrical services, the energy drawn from and stored in the batteries, and the inhabitants’ planning wishes. The objective is to fit the solar production to the electrical consumption of the inhabitants, with an optimal use of the energies in the batteries by satisfying as widely as possible the users' planning requirements. In the discrete model, the different parameters and solutions of the linear programming system are deterministic scalars. Whereas in the stochastic approach, the data parameters and the linear programming solutions become random variables, then the distributions of which could be imposed or established by estimation from samples of real observations or from samples of optimal discrete equilibrium solutions.

Keywords: photovoltaic production, power consumption, battery storage resources, random variables, stochastic modeling, estimations of probability distributions, mixed integer linear programming, smart micro-grid, self-consumption of electricity.

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Authors: Omar Dib, Alexandre Caminada, Marie-Ange Manier

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The purpose of this study is to introduce a novel approach to solve the one-to-one shortest path problem. A directed connected graph is assumed in which all edges’ weights are positive. Our method is based on a memetic algorithm in which we combine a genetic algorithm (GA) and a variable neighborhood search method (VNS). We compare our approximate method with two exact algorithms Dijkstra and Integer Programming (IP). We made experimentations using random generated, complete and real graph instances. In most case studies, numerical results show that our method outperforms exact methods with 5% average gap to the optimality. Our algorithm’s average speed is 20-times faster than Dijkstra and more than 1000-times compared to IP. The details of the experimental results are also discussed and presented in the paper.

Keywords: shortest path problem, Dijkstra’s algorithm, integer programming, memetic algorithm

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Authors: Luciano Oliveira, Elsa Vásquez-Alvarez

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The increase in rail freight transport in Brazil in recent years requires new railway lines and the maintenance of existing ones, which generates high costs for concessionaires. It is in this context that this work is inserted, whose objective is to propose a method that uses Binary Linear Programming for the choice of sleeper and rail fastening, from various options, including the way to apply these materials, with focus to minimize costs. Unit value information, the life cycle each of material type, and service expenses are considered. The model was implemented in commercial software using real data for its validation. The formulated model can be replicated to support decision-making for other railway projects in the choice of sleepers and rail fastening with lowest cost.

Keywords: linear programming, rail fastening, rail sleeper, railway

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Authors: Y. Long, L. Liu, K. V. Branin

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One challenge faced by procurement decision-maker during the acquisition process is how to compare similar products from different suppliers and allocate orders among different products or services. This work focuses on allocating orders among multiple suppliers considering rebate. The objective function is to minimize the total acquisition cost including purchasing cost and rebate benefit. Rebate benefit is complex and difficult to estimate at the ordering step. Rebate rules vary for different suppliers and usually change over time. In this work, we developed a system to collect the rebate policies, standardized the rebate policies and developed two-stage optimization models for ordering allocation. Rebate policy with multi-tiers is considered in modeling. The discontinuous cost function of rebate benefit is formulated for different scenarios. A piecewise linear function is used to approximate the discontinuous cost function of rebate benefit. And a Mixed Integer Programing (MIP) model is built for order allocation problem with multi-tier rebate. A case study is presented and it shows that our optimization model can reduce the total acquisition cost by considering rebate rules.

Keywords: discontinuous cost function, mixed integer programming, optimization, procurement, rebate

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Authors: Mohammad Mehdi Mazarei, Ali Asghar Behroozpoor

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In this paper, we define an optimization problem corresponding to smooth and nonsmooth functions which its optimal solution is the first derivative of these functions in a domain. For this purpose, a linear programming problem corresponding to optimization problem is obtained. The optimal solution of this linear programming problem is the approximate generalized first derivative. In fact, we approximate generalized first derivative of nonsmooth functions as tailor series. We show the efficiency of our approach by some smooth and nonsmooth functions in some examples.

Keywords: general derivative, linear programming, optimization problem, smooth and nonsmooth functions

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Authors: Homa Ghave, Parmis Shahmaleki

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This paper presents a new multi-criteria stochastic mathematical model for a single machine scheduling with outsourcing allowed. There are multiple jobs processing in batch. For each batch, all of job or a quantity of it can be outsourced. The jobs have stochastic processing time and lead time and deterministic due dates arrive randomly. Because of the stochastic inherent of processing time and lead time, we use the chance constrained programming for modeling the problem. First, the problem is formulated in form of stochastic programming and then prepared in a form of deterministic mixed integer linear programming. The objectives are considered in the model to minimize the maximum tardiness and outsourcing cost simultaneously. Several procedures have been developed to deal with the multi-criteria problem. In this paper, we utilize the concept of satisfaction functions to increases the manager’s preference. The proposed approach is tested on instances where the random variables are normally distributed.

Keywords: single machine scheduling, multi-criteria mathematical model, outsourcing strategy, uncertain lead times and processing times, chance constrained programming, satisfaction function

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Authors: Farzad Pargar

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We introduce the preventive maintenance and renewal scheduling problem for a multi-unit system over a finite and discretized time horizon. Given the latest possible time for carrying out the next maintenance and renewal projects after the previous ones and considering several common set-up costs, the introduced scheduling model tries to minimize the cost of projects by grouping them and simultaneously finding the optimal balance between doing maintenance and renewal. We present a 0-1 pure integer linear programming that determines which projects should be performed together on which location and in which period (e.g., week or month). We consider railway track as a case for our study and test the performance of the proposed model on a set of test problems. The experimental results show that the proposed approach performs well.

Keywords: maintenance, renewal, scheduling, mathematical programming model

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Authors: Darwin Castillo Huamaní, Francisco A. M. Gomes

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The aim of the multi-material topology optimization (MTO) is to obtain the optimal topology of structures composed by many materials, according to a given set of constraints and cost criteria. In this work, we seek the optimal distribution of materials in a domain, such that the flexibility of the structure is minimized, under certain boundary conditions and the intervention of external forces. In the case we have only one material, each point of the discretized domain is represented by two values from a function, where the value of the function is 1 if the element belongs to the structure or 0 if the element is empty. A common way to avoid the high computational cost of solving integer variable optimization problems is to adopt the Solid Isotropic Material with Penalization (SIMP) method. This method relies on the continuous interpolation function, power function, where the base variable represents a pseudo density at each point of domain. For proper exponent values, the SIMP method reduces intermediate densities, since values other than 0 or 1 usually does not have a physical meaning for the problem. Several extension of the SIMP method were proposed for the multi-material case. The one that we explore here is the ordered SIMP method, that has the advantage of not being based on the addition of variables to represent material selection, so the computational cost is independent of the number of materials considered. Although the number of variables is not increased by this algorithm, the optimization subproblems that are generated at each iteration cannot be solved by methods that rely on second derivatives, due to the cost of calculating the second derivatives. To overcome this, we apply a globally convergent version of the sequential linear programming method, which solves a linear approximation sequence of optimization problems.

Keywords: globally convergence, multi-material design ordered simp, sequential linear programming, topology optimization

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Authors: Mahsa Oroojeni Mohammad Javad

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Globalization of economic activity and rapid growth of information technology has resulted in shorter product lifecycles, reduced transport capacity, dynamic and changing customer behaviors, and an increased focus on supply chain design in recent years. The design of the supply chain network is one of the most important supply chain management decisions. These decisions will have a long-term impact on the efficacy and efficiency of the supply chain. In this paper, a two-objective mixed-integer linear programming (MILP) model is developed for designing and optimizing a closed-loop green supply chain network that, to the greatest extent possible, includes all real-world assumptions such as multi-level supply chain, the multiplicity of production technologies, and multiple modes of transportation, with the goals of minimizing the total cost of the chain (first objective) and minimizing total emissions of emissions (second objective). The ε-constraint and CPLEX Solver have been used to solve the problem as a single-objective problem and validate the problem. Finally, the sensitivity analysis is applied to study the effect of the real-world parameters’ changes on the objective function. The optimal management suggestions and policies are presented.

Keywords: closed-loop supply chain, multi-level green supply chain, mixed-integer programming, transportation modes

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Authors: J. Satya Eswari, Ch. Venkateswarlu

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The formulation of a commercial pharmaceutical product involves several composition factors and response characteristics. When the formulation requires to satisfy multiple response characteristics which are conflicting, an optimal solution requires the need for an efficient multiobjective optimization technique. In this work, a regression is combined with a non-dominated sorting differential evolution (NSDE) involving Naïve & Slow and ε constraint techniques to derive different multiobjective optimization strategies, which are then evaluated by means of a trapidil pharmaceutical formulation. The analysis of the results show the effectiveness of the strategy that combines the regression model and NSDE with the integration of both Naïve & Slow and ε constraint techniques for Pareto optimization of trapidil formulation. With this strategy, the optimal formulation at pH=6.8 is obtained with the decision variables of micro crystalline cellulose, hydroxypropyl methylcellulose and compression pressure. The corresponding response characteristics of rate constant and release order are also noted down. The comparison of these results with the experimental data and with those of other multiple regression model based multiobjective evolutionary optimization strategies signify the better performance for optimal trapidil formulation.

Keywords: pharmaceutical formulation, multiple regression model, response surface method, radial basis function network, differential evolution, multiobjective optimization

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Authors: Christian Liebchen

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Both from an environmental perspective and with respect to road traffic flow quality, planning so-called green waves along major road axes is a well-established target for traffic engineers. For one-way road axes (e.g. the Avenues in Manhattan), this is a trivial downstream task. For bidirectional arterials, the well-known necessary condition for establishing a green wave in both directions is that the driving times between two subsequent crossings must be an integer multiple of half of the cycle time of the signal programs at the nodes. In this paper, we propose an integer linear optimization model to establish fixed-time green waves in both directions that are as long and as wide as possible, even in the situation where the driving time condition is not fulfilled. In particular, we are considering an arterial along whose nodes separate left-turn signal groups are realized. In our computational results, we show that scheduling left-turn phases before or after the straight phases can reduce waiting times along the arterial. Moreover, we show that there is always a solution with green waves in both directions that are as long and as wide as possible, where absolute priority is put on just one direction. Compared to optimizing both directions together, establishing an ideal green wave into one direction can only provide suboptimal quality when considering prioritized parts of a green band (e.g., first few seconds).

Keywords: traffic light coordination, synchronization, phase sequencing, green waves, integer programming

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Authors: Zahra Majd, Mohsen Kalantar

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Power management and voltage regulation is one of the most important issues in microgrid (MG) control and scheduling. This paper proposes a multiobjective scheduling formulation that consists of active power costs, voltage fluctuations summation, and technical constraints of MG. Furthermore, load flow and reserve constraints are considered to achieve proper voltage regulation. A modified Jacobian matrix is presented for calculating voltage variations and Mont Carlo simulation is used for generating and reducing scenarios. To convert the problem to a mixed integer linear program, a linearization procedure for nonlinear equations is presented. The proposed model is applied to a typical low-voltage MG and two different cases are investigated. The results show the effectiveness of the proposed model.

Keywords: microgrid, energy management system, voltage fluctuations, modified Jacobian matrix

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