Mixed Integer Linear Mathematical Model Research Articles

A Fuzzy Programming Based Approach to a Multi-Objective Multi-Echelon Green Closed-Loop Supply Chain Problem

Abstract It has become a social and legal obligation to take into account environmental factors as well as economic factors when designing the supply chain network. Reducing the emission of harmful gases in supply chain operations, recycling and efficient use of resources must be taken into consideration for future generations. The supply chain created in this study, in addition to the abovementioned features, includes supplier selection, warehouse and distribution center setup, transportation amounts between facilities, and transportation modes of products to be transported. This model, which is a multi-objective multi-echelon green closed-loop supply chain, is a mixed integer linear mathematical model and tries to maximize the joint satisfaction of the objectives with the help of a fuzzy approach using Zimmermann’s minimum operator.

Designing a multi-period dynamic electric vehicle production-routing problem in a supply chain considering energy consumption

The coordinated decision-making approach for considering sequential activities of the supply chain results in additional benefits by optimizing production, inventory, and distribution operations. Accordingly, this paper proposes a mixed integer linear mathematical model to optimize a multi-period production routing problem utilizing electric vehicles. The proposed model optimizes the total cost associated with fixed and variable production, holding inventory, and routing, including the fixed cost of utilizing electric vehicles and travel time. However, mileage limitation is one of the main restrictions of utilizing electric vehicles in performing deliveries which is strongly affected by the consumed energy. Although optimization of the routes of vehicles can facilitate using them, considering the variation of travel speed network links during different times of the day because of traffic conditions can obviously affect the required energy to perform the assigned deliveries. To the best of our knowledge, this paper is the first to study simultaneous multi-period dynamic production routing problems using a set of heterogenous electric vehicles whose travel time of links can vary by dividing each production period into several hourly time intervals to capture different traffic conditions. Finally, a series of capable and hybrid metaheuristic algorithms are designed and implemented to solve this problem in a real-case dimension, and all proposed algorithms are compared.

Efficient multi-objective meta-heuristic algorithms for energy-aware non-permutation flow-shop scheduling problem

This study investigates the optimization of non-permutation flow-shop scheduling problems and lot-sizing simultaneously. Contrary to previous works, we first study the energy awareness of non-permutation flow-shop scheduling and lot-sizing using modified novel meta-heuristic algorithms. In this regard, first, a mixed-integer linear mathematical model is proposed. This model aimed to determine the size of each sub-category and determine each machine's speed within each sub-category to minimize makespan and total consumed energy simultaneously. In order to optimize this model, Multi-objective Ant Lion Optimizer (MOALO), Multi-objective Keshtel Algorithm (MOKA), and Multi-objective Keshtel and Social Engineering Optimizer (MOKSEA) are proposed. First, the validation of the mathematical model is evaluated by implementing it in a real case of the food industry using GAMS software. Next, the Taguchi design of the experiment is applied to adjust the meta-heuristic algorithms' parameters. Then the efficiency of these meta-heuristic algorithms is evaluated by comparing with Epsilon-constraint (EPC), Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-objective Particle Swarm Optimization (MOPSO) using several test problems. The results demonstrated that the MOALO, MOKA, and MOKSEO algorithms could find optimal solutions that can be viewed as a set of Pareto solutions, which means the used algorithm has the necessary validity. Moreover, the proposed hybrid algorithm can provide Pareto solutions in a shorter time than EPC and higher quality than NSGA-II and MOPSO. Finally, the model's key parameters were the subject of sensitivity analysis; the results showed a linear relationship between the processing time and the first and second objective functions.

A region-based model for optimizing charging station location problem of electric vehicles considering disruption - A case study

To reduce the negative impact of the conventional vehicles, governments and authorities try to provide facilities in order to encourage people to use electric vehicles. For this purpose, determining the location and the number of charging stations is of paramount importance. It becomes more challenging when some charging stations cannot provide service due to the disruption or breakdown. In this paper, a bi-objective mixed-integer linear mathematical model for a charging station location problem is developed and solved by Lagrangian relaxation method. Two different chargers that could be installed in the charging stations and budget constraint is considered in the model. To evaluate the applicability of the proposed model on real-world problems, a case study on Tehran is conducted. Based on computational experiments and numerical results, sensitivity analysis on key parameters of the problem is carried out and some helpful suggestions are presented for decision-makers.

Design of a distribution network for the school lunch program

The national school lunch program (NSLP) is crucial for providing healthy, inexpensive, or free lunches to children, thus benefiting society. Designing a distribution network for the program requires solving a location and routing problem. In this paper, first, we formulate a multi-objective non-linear integer programming formulation of the problem. Next, we develop a two-step approach since the problem is Np-hard. The first stage presents a K-mean clustering method that deals with routing decisions by determining the locations of food processing centers and allocating schools to these centers. The second stage offers a multi-objective mixed-integer linear mathematical model for finding the locations of distribution centers. Besides economic and environmental factors, we optimize travel time in the network as perishable items are involved. A weighted sum approach is presented for different weights of objectives. We provide a real case study in Turkey to demonstrate the applicability of the two-stage approach proposed in this study. The numerical results provide valuable information for decision-makers and authorities to prioritize and prepare action plans.

The integration of pelletized agricultural residues into electricity grid: Perspectives from the human, environmental and economic aspects

Agricultural residues are generated during the growing and harvesting of crops; consequently, its availability is linked with the local demand, socioeconomic status, climatic conditions as well as national policies. Only a fraction of these residues has a specific use, mainly in poultry and livestock breeding; however, there is potential to use them as bio-energy feedstock. A strong dependence on oil, as a primary energy source, has resulted in huge environmental problems, particularly climate change. Hence, the use of agricultural residues, particularly those that seem not to have any environmental or economic value; as a feedstock for electricity production, could be an attractive alternative in terms of economic, environmental and social benefits.Therefore, this paper is focused on evaluating these potential benefits from the production of fuel pellets using agricultural residues available from different crops; and their use as a renewable feedstock for electricity production in existing power plants. Life cycle assessment is included in the analysis to get the optimal integration between conventional and non-conventional feedstocks for generating electricity. In addition, the economic aspect was calculated taking under consideration all the activities included in the supply chain for the production of fuel pellets. Also, Human Development Index approach was considered for evaluating the social benefits achieved by the use of renewable fuel pellets. The proposed mathematical model includes all the material balances involved in the supply chain for the production of fuel pellets; also, binary variables were used as decision variables to evaluate the plant installation, giving a mixed integer linear mathematical model optimization. For this case study, parameters of the Mexican electric power grid were used; however, the model can be applied to any other case. The results show that it is possible to provide 200,899 GWh per year of the electricity to cover the demand in Mexico through pellets combustion obtained from agricultural residues. In addition, a carbon dioxide reduction of 20% and 11% increase in social benefit were achieved. Then, the agricultural residues revalorization as solid bio-fuel alternative for the electric power grid has positive economic, social and environmental impacts.

A Linearized Approach for the Electric Light Commercial Vehicle Routing Problem Combined with Charging Station Siting and Power Distribution Network Assessment

Transportation electrification has demonstrated a significant position on power utilities and logistic companies, in terms of assets operation and management. Under this context, this paper presents the problem of seeking feasible and good quality routes for electric light commercial vehicles considering battery capacity and charging station siting on the power distribution system. Different transportation patterns for goods delivery are included, such as the capacitated vehicle routing problem and the shortest path problem for the last mile delivery. To solve the problem framed within a mixed integer linear mathematical model, the GAMS software is used and validated on a test instance conformed by a 19-customer transportation network, spatially combined with the IEEE 34 nodes power distribution system. The sensitivity analysis, performed during the computational experiments, show the behavior of the variables involved in the logistics operation, i.e., routing cost for each transport pattern. The trade-off between the battery capacity, the cost of the charging station installation, and energy losses on the power distribution system is also shown, including the energy consumption cost created by the charging operation.

Sustainable supply chain network design using products' life cycle in the aluminum industry.

This study provides a three-objective mixed-integer linear mathematical model to design a sustainable closed-loop supply chain network in the aluminum industry. In this regard, the proposed model optimizes economic, social, and environmental objectives simultaneously. The main contribution of this research is to provide a framework for the sustainable aluminum supply chain in Iran by applying the life cycle assessment (LCA) to estimate the environmental impacts and using two novel meta-heuristic algorithms to optimize the proposed mathematical model. In this regard, the multi-objective gray wolf optimizer (MOGWO), the multi-objective red deer algorithm (MORDA), and augmented epsilon constraint (AEC) are used to achieve Pareto optimal solutions. Comparisons between solutions methods show that the MOGWO algorithm and MORDA have a very high advantage over the AEC method in terms of the scatter of Pareto solutions. Moreover, the statistical tests indicate that the MORDA has an advantage over MOGWO in terms of Pareto boundary diversification as well as the quality of solutions. On the other hand, results of the implementation in the aluminum industry show that increasing the coefficient of recycled materials' use in the production of secondary aluminum has a significant impact on the Pareto boundary and leads to reducing production costs and in particular the reduction of carbon dioxide emissions.

An artificial neural network based mathematical model for a stochastic health care facility location problem.

This research is conducted to investigate the problem of locating the trauma centers and helicopters' station in order to optimize the trauma care system. The stochastic characteristics of the system, such as stochastic transferring time of the patients, stochastic demand and stochastic servicing time of the patients in trauma centers are taken into account. The problem is first modeled as a stochastic mixed-integer linear mathematical model. In the proposed model, minimizing the total cost, minimizing the transferring time, and minimizing the waiting time inside the trauma center are considered as the three separate objectives. The third objective cannot be expressed by an analytical expression because of the complexity inside a trauma center. Therefore, an artificial neural network (ANN) is first trained by a simulation model and then is utilized to estimate the third objective function. A hybrid multi-objective algorithm is developed based on a non-dominated sorting water flow algorithm in order to search the solution space. Different numerical examples are applied to study the performance of the proposed method. The computational results show that the combination of simulation, ANN, and optimization technique provides an effective means for the highly complex optimization problems.

Optimisation of biochar-based supply chains for negative emissions and resource savings in carbon management networks

Climate change is a global issue, and the main reason causing climate change is carbon dioxide (CO2) emissions. Biochar, a stable form of carbon that can be obtained from pyrolysis of biomass, can potentially reduce CO2 emissions. By applying biochar into soil, a stable form of carbon is stored under the soil. Biochar application is able to achieve negative emissions because the stored carbon was previously fixed from CO2 via photosynthesis. Biochar application also brings additional benefits to soil as it helps to retain water and nutrients in soil. The benefits of biochar can be scaled up by developing biochar-based carbon management networks (CMNs). However, there are very limited decision support tools available for designing optimal biochar-based CMNs. This work presents a methodology to develop a decision support tool for biochar-based CMNs. Proposed methodology provides guidance on how a mixed-integer linear mathematical model can be formulated as a decision support tool to design biochar-based CMNs. The developed model has capabilities to determine the optimal network based on cost while considering several constraints. These constraints are CO2 emission reduction, water and nutrient retention and several land constraints. The network costs considered are capital cost, operating cost, transportation cost and transport fuel consumption cost. The model also factors transport routes, hiring cost and truck capacity into the network. A case study based on the palm oil industry is solved to demonstrate the use of the model. Results from the case study show that total biochar production, water savings, nutrient savings, cost savings and total network costs were 307 t/day, 7.1 M m3 water/year, 6.0 k t nutrients/year, RM 25.6 M/year and RM 17.9 M/year, respectively. Besides that, a sensitivity analysis is done to analyse the impact of CO2 emission reduction targets on the network costs. Aside from this, two alternative scenarios from the case study were investigated further.

The Potential Use of Drones for Tourism in Crises: A Facility Location Analysis Perspective

Considering the recent lockdowns and travel bans due to COVID-19, novel tourism strategies are necessary to face the increasing need for innovative products and services and to ensure long-term sustainable growth. This study looks into the potential use of drones in providing online virtual tours of open-space tourist attractions. To do so, a novel mixed-integer linear mathematical model is developed to optimally determine the number and location of required facilities and the number of drones assigned to each center. The model is applied to a case study of Rome by selecting six historic sites as the tourist attractions and considering several candidate locations for establishing the facilities. The results of different potential scenarios imply that the project is profitable, even if the demand for virtual tours is low.

Resolution for bounded-splitting jobs scheduling problem on a single machine in available time-windows

This paper studies a single machine scheduling problem which machine-availability constraint is imposed. The jobs are splittable into sub-jobs and the size of each sub-job has a common lower bound. The objective aims to find an optimal scheduling solution that minimizes the maximum completion time for the whole set jobs. The scheduling problem was proved to be strongly NP-hard by a reduction from 3-Partition problem. In this paper, a general mixed-integer linear mathematical model is constructed based on some structurally optimal properties in the literature. Besides that we propose some effective heuristics concerned about assignment strategy such as assignment heuristic, heuristic based on shortest/longest processing time rules, heuristic based on max flow resolution, matching and assignment approach. In order to improve solution quality, we also propose to apply a combination between these proposed heuristics and metaheuristics such as tabu search and genetic algorithm. In addition, in the hope of achieving the better solutions in acceptable time, we introduce another approach called exact for subset-jobs matheuristic which is combined between mathematical programming and heuristic derived from single-attribute priority rule. Experimental results show the performance of proposed approximating approaches in comparing between them and the exact method based on the MILP model which is implemented by CPLEX solver. Among proposed effective algorithms, the matheuristic showed the superiority in terms of solution quality and execution time.

Robotic disassembly line balancing problem: A mathematical model and ant colony optimization approach

The use of robots is significantly increasing day by day in manufacturing systems, and especially improving the efficiency of the lines. Robots can be used to complete disassembly tasks, and each of the robots can need different operation times to perform the tasks. In this paper, the balancing of the robotic disassembly line problem has been studied to develop efficient solution techniques. Firstly, a mixed-integer linear mathematical model is proposed to determine and solve the problem optimally. A case study from literature is addressed to assess and show the efficiency and effectiveness of the model to minimize cycle time. Secondly, a heuristic algorithm based on ant colony optimization is also proposed to discover a solution for especially the large-size test problems due to the complexity of the problem. The performance of the proposed heuristic algorithm is verified and compared with the different heuristic on data sets. The computational results indicate that the proposed mathematical model and the algorithms are promising for the small and large-size test problems, respectively. Finally, it should be stated that robots have great potential to use in the area of disassembly line and useful solutions provide according to test results.

Mathematical modeling and evolutionary generation of rule sets for energy-efficient flexible job shops

As environmental awareness grows, sustainable scheduling is attracting increasing attention. The purposes of this paper are obtain the lower bound of energy-efficient flexible job shops with machine selection, job sequencing, and machine on-off decision making via a new mathematical model and to discover more energy-efficient rules with easy implementation in real practice via an efficient Gene Expression Programming (eGEP) algorithm. This paper first formulates a novel mixed-integer linear mathematical model to achieve effective machine selection, job sequencing, and machine off-on decision making. Then for the purpose of avoiding the empirical combination, five attributes exerting direct influence on the total energy consumption are extracted and consequently involved in the evolutionary process of eGEP. Furthermore, diversified rule mining operations with multi-gene representation and self-study are designed to enhance the search space and solutions quality. And, unsupervised learning is utilized in which global best and current worst are set to guide evolution direction since the learning progress has no prior knowledge. Experimental results show that machine off-on decisions efficiently reduce the total energy consumption; and, the discovered rules reach the lower bound calculated by GAMS/CPLEX in small problems and have significant superiority over other dispatching rules in energy saving.

The impact of energy storage in power systems: The case of Brazil’s Northeastern grid

In this work, a mixed integer linear mathematical model for the energy storage devices dispatch in an interconnected power system is proposed. This model was implemented in a computer program written in C++ that outputs a linear program in LP format to be solved in any commercial or free solver. A case study of Brazil’s Northeastern Power System is presented to evaluate the impact of the use of energy storage devices on the operational cost of a system with renewable variable energy sources, such as wind and solar, as well as hydroelectricity. Alternative scenarios were considered, where the amount of wind energy was changed in order to assess the impact of wind energy growth on system performance. Simulation results show that up to 15% O&M cost reduction is attainable in this scenario.

Total completion time minimisation of no-wait flowshop group scheduling problem with sequence dependent setup times

In this research, the no-wait flowshop sequence dependent group scheduling problem with minimisation of total completion time, noted as Fm|nwt, fmls, Splk|ΣCj is approached. A mixed integer linear mathematical model is proposed for the research problem. Due to the NP-hard nature of the proposed problem, several metaheuristic algorithms based on particle swarm optimisation and variable neighbourhood search with different neighbourhood search structures are proposed to heuristically solve the problem. Also, an efficient heuristic algorithm is proposed to generate the initial feasible solutions for the proposed algorithms. The performances of the proposed algorithms are compared with each other and the best available one in the literature by using available test problems in the literature. Based on the results, the proposed metaheuristics with structure oriented initial solutions have a better performance than the algorithms with random-generated initial solutions. [Received 14 July 2015; Revised 22 September 2015; Accepted 30 May 2016]

Two-stage optimisation method for material flow and allocation management in cross-docking networks

Cross-docking is a relatively new logistics strategy in which items are moved from suppliers to customers through cross-docking centres without putting them into long-term storage. An important decision during the planning of cross-docking operations is related to the material flow management in the network, which has great potential to reduce transportation costs. However, until now, there has been a lack of studies regarding operations for both transportation of trucks between locations and trans-shipment of items in cross-docking centres. This study presents a novel two-stage mixed integer linear mathematical model for the transportation problem of cross-docking network design integrated with truck–door assignments to minimise total transportation costs from suppliers to customers. This model also considers incoming/outgoing truck-loading plans and product allocations in the cross-docking area with regard to the two-dimensional physical constraints. Due to the complexity of the problem, a genetic algorithm (GA) is proposed to solve large-sized problems. Computational studies are conducted to examine the validity of the two-stage model and performance of the GA. The computational studies show that the introduced model provides a comprehensive plan for material flow management in cross-docking networks and proposed GA is capable of obtaining effective results for the problem within a short computational time.

A hybrid meta-heuristic algorithm for vehicle routing and packing problem with cross-docking

Cross-docking is a relatively new logistics strategy that has a great potential to eliminate storage cost and speed up the product flows. This paper considers the vehicle routing and packing problem with cross-docking and presents a mixed integer linear mathematical model. In the model, a set of trucks are used to transport products from suppliers to customers through cross-docking centers. Each supplier and customer node can be visited only once and directly shipping is not allowed from suppliers to customers. Moreover, truck capacities are identified with physical dimensional limits on the contrary of weight or amount of load. The objective of the study is to determine the vehicle routes that minimize the total distance. Due to the complexity of the mathematical model, a hybrid meta-heuristic algorithm (HMA), which integrates tabu search (TS) algorithm within simulated annealing (SA) algorithm, is proposed to solve the problem. Proposed HMA is tested on a well-known benchmark problem data set and compared with the SA and TS solutions. Results show that proposed HMA can produce effective solutions and outperforms the SA and TS especially for the large-sized problems.

Minimizing total completion time in the flexible flowshop sequence-dependent group scheduling problem

This research considers a flexible flowshop sequence-dependent group scheduling problem with minimization of total completion time. A mixed integer linear mathematical model for the research problem is developed. Since the research problem is shown to be strongly NP-hard, a metaheuristic algorithm based on memetic algorithm (MA) is proposed. A lower bounding method based on the Branch and Price algorithm is also proposed to evaluate the quality of the MA. In order to evaluate the performance of the proposed algorithms, random test problems, ranging in size from small, medium, to large are generated and solved by the MA and the lower bounding method. The results show that the average percentage gap of the MA is 6.03 % compared to the result of the lower bounding method for randomly generated test problems.

Makespan minimisation in flexible flowshop sequence-dependent group scheduling problem

In this research, the flexible flowshop sequence-dependent group scheduling problem with minimisation of makespan as the criterion () is investigated. A mixed integer linear mathematical model for the research problem is developed. Since the research problem is shown to be NP-hard, a meta-heuristic algorithm based on memetic algorithm (MA) is developed to efficiently solve the problem. Also, a lower bounding technique based on the developed mathematical model is proposed to evaluate the quality of the proposed MA. The performance of the proposed MA is compared with the existing algorithm in the literature, i.e. tabu search (TS), by solving the available test problems in the literature. A comparison based on paired t-test shows that the average makespan of the proposed MA is 3% lower than the average makespan of the TS. The average percentage gap of MA for small-size problems comparing with the optimal solution is 0.8%. Also, the average percentage gap of the proposed MA compared to the proposed lower bound for medium-size test problems (problems up to 65 jobs in all groups) is 5%.