Mixed Integer Nonlinear Mathematical Model Research Articles

Designing a Fuzzy Mathematical Model for a Two-Echelon Allocation-Routing Problem by Applying Route Conditions: A New Interactive Fuzzy Approach

In vehicle routing problems (VRP), the optimal allocation of transportation by considering factors such as route hardness, driver experience and vehicle worn-out has a significant effect on costs reduction and approaching real-world conditions. In this paper, a novel fuzzy mixed integer non-linear mathematical model to address the two-echelon allocation-routing problem under uncertainty is proposed by applying route and fleet conditions. The cost of allocating drivers to diverse vehicles is computed at the first echelon of the problem, considering factors such as vehicle type, vehicle wear-out, and driver experience. Additionally, different routes are defused with varying levels of hardness. The goal of the second echelon of the model is to improve reliability by defining the reliability of routes within each section. To solve the model, the Torabi and Hessini (TH), the Selimi and Ozkarahan (SO) methods, and a newly proposed approach (PIA) were utilized to transform the multi-objective model into a single-objective one. Numerical tests and performance indicators were used to validate the effectiveness of both the multi-objective mathematical model and the proposed solution method. The validation computation results indicate that the proposed solution approach outperforms both the TH and SO approaches.

Dynamic opportunistic maintenance grouping in a lot streaming based job-shop scheduling problem

This paper aims to model maintenance planning with a dynamic opportunistic approach for a job-shop production system. One issue in such a system is the positive or negative economic dependency. That is grouping maintenance activities may decrease or increase system costs. Furthermore, many maintenance models consider the planning of maintenance only based on a long-term horizon. While short-term and real circumstances such as system characteristics and constraints, workload, number of available maintenance teams, and variable maintenance cost and time are almost ignored. To address these issues, a rolling-horizon approach based on a long-term maintenance plan is proposed so that subsequent scheduling of maintenance and production activities are performed as events unfold through the time. Hence, we have developed a mixed-integer nonlinear mathematical model to simultaneously make decisions on maintenance selection, maintenance grouping, lot sizing and production scheduling. The objective function includes the costs of preventive and corrective maintenance activities as well as various production costs such as production and setup, tardiness penalty, and safety stock penalty. A self-adaptive Cuckoo Optimization Algorithm has been used to solve the proposed model. Numerical experiments were conducted to demonstrate the validity of the model and investigate the efficiency and effectiveness of the optimization algorithm.

Capturing spatial, time-wise and technological detail in hydrogen supply chains: A bi-level multi-objective optimization approach

This study presents a methodological framework for designing a hydrogen supply chain (HSC) utilizing a superstructure that incorporates various alternatives for feedstock, production, storage, transportation, and distribution. The framework features a mixed-integer non-linear mathematical model that includes innovative factors such as power efficiencies, continuous sizing capacities, and time-varying costs of different energy feedstocks related to the learning rate. Furthermore, emerging production technologies such as alkaline electrolysis and proton exchange membrane water electrolysis, along with steam methane reforming with carbon capture, utilization, and storage, are considered. To address the resulting bi-objective optimization problem that aims to minimize both total daily costs and greenhouse gas emissions, a specialized solution strategy based on bi-level decomposition and a matheuristic algorithm is developed. This methodology for designing HSCs is applied to a case study in southern France, demonstrating the effectiveness of the solution technique in approximating the Pareto frontier. By analyzing in detail selected relevant solutions along the Pareto front, insights into the spatial, temporal, and technological deployment of the HSC are gained, enabling decision-makers to make informed choices based on economic and environmental criteria.

Optimal dispatch of nearly-zero carbon integrated energy system considering waste incineration plant-carbon capture system and market mechanisms

Considering the waste incineration plant-carbon capture (WIP-CC) system and market mechanisms, the optimal dispatching strategy of nearly-zero carbon integrated energy system (NZC-IES) is proposed. The incineration of large amounts of Municipal Solid Waste can result in significant carbon emission and air pollution. However, lack of consideration of mechanisms for carbon processing and environmental protection will prevent the development of NZC-IES. Accordingly, a novel mixed integer nonlinear mathematical model, NZC-IES, is established that minimizes the total cost and controls carbon emission and air pollution. Unlike previous studies of optimal dispatching for WIP-CC system, the climate and health costs of WIP and the social costs of energy sources for CC are considered. A model of adjustable thermoelectric ratios for WIP and a model for CC with the storage tanks are established to enhance the flexibility of the thermoelectric output. Also, demand response model and ladder-type carbon trading model are developed to serve for NZC-IES. Case studies reveal that the proposed optimal dispatching strategy can realize the waste to energy utilization and low-carbon emission with economic performance.

A discrete differential evolution with local search particle swarm optimization to direct angle and aperture optimization in IMRT treatment planning problem

Intensity-modulated radiation therapy is a well-known technique for treating cancer patients worldwide. A treatment plan in this technique requires decision-making for three main problems: selection of beam angles, intensity map calculation, and leaf sequencing. Previous works investigated these problems sequentially. We present a new integrated framework for simultaneous decision-making of directions, intensities, and aperture shape, called direct angle and aperture optimization, and develop a mixed-integer nonlinear mathematical model for the problem. Due to the nonlinearity and the dimension of the problem, three efficient metaheuristics based on differential evolution (DE) called classic differential evolution (cDE), discrete differential evolution (dDE), and adaptive hybrid discrete differential evolution-particle swarm optimization (ahdDE-PSO) algorithms are designed to solve the problem. Parameters calibration is performed using the Taguchi design of experiments. The performance of the algorithms is evaluated by solving the problem for ten real cases of liver cancer disease from the TROTS data set. The performed ablation study and statistical analysis of computational results demonstrate that ahdDE-PSO is capable of finding high-quality treatment plans.

Optimal Planning of Secondary Power Distribution Systems Considering Renewable and Storage Sources: An Energy Management Approach

This study focuses on the optimal planning of secondary power distribution systems considering distributed renewable generators (DG) and energy storage systems (ESS) to minimize expansion costs. The methodology solves a mixed integer non-linear mathematical model that describes the planning problem, including the operating and technical aspects of the secondary power distribution system. Such methodology uses an iterated local search algorithm and a two-stage load flow decomposition method to solve said problem. The two-stage load flow decomposition method finds the optimal operation of the storage devices and the low-voltage distribution system for each solution proposed by the iterated local search algorithm; thus, optimal energy management is achieved for the best solution. The proposed methodology was tested on a real medium-sized secondary power distribution system to establish its effectiveness. The results obtained show a reduction of 51.97 % in the total energy purchase cost of the system and a decrease of 3.02 % in the installation costs of the secondary circuits and distribution transformers when DG and ESS are considered. In conclusion, the results show that the integration of these distributed energy resources into the distribution system planning problem increases the profits of distribution companies from energy purchase and sale and reduces their fixed costs.

A novel mathematical optimization model for a preemptive multi-priority M/M/C queueing system of emergency department’s patients, a real case study in Iran

The Covid-19 pandemic crisis has caused many difficulties worldwide. One of the most critical problems is that the emergency departments (EDs) have become overcrowded. Because this problem can increase patients’ queue length and waiting times in EDs, this paper provides a mixed-integer non-linear mathematical model considering a preemptive M/M/C queueing system to solve the problem and optimize a benefit function concerning the number of servers and treatment rate. In this model, different patient priorities, which are modified according to Covid-19 patients, are considered. This model is then solved using an exact approach and a meta-heuristic algorithm, the grasshopper optimization algorithm, for two shifts of the ED of a hospital in order to consider non stationery arrival rate in Varamin, Iran. The results of both algorithms confirmed the effectiveness of the proposed model. Moreover, to justify using the preemptive model, a comparison between the preemptive and non-preemptive models is conducted. An extensive sensitivity analysis is presented, and finally, a list of managerial insights is provided for managers to improve their service system further.

Location-allocation problem for resource distribution under uncertainty in disaster relief operations

Disaster disrupts society to lead a normal life by causing huge casualties and damages or loss of properties, environment, or essential services of a society or a nation. In the case of a catastrophe, a suitable resource management program is needed to distribute resources efficiently among the people of the areas affected by the calamity. In this work, we propose a mixed-integer non-linear mathematical model related to resource management. We consider a multi-objective optimization problem to provide the maximum services to the people by determining the positions of the distribution centers. This humanitarian logistic model optimizes three criteria: (i) minimizing the cost, (ii) minimizing the time, (iii) maximizing the affected coverage area. Due to the impreciseness of the disastrous environment, some parameters of the model are chosen as a triangular type-2 fuzzy number in order to deal with the uncertainty. An improved neutrosophic compromise programming is incorporated to solve the stated problem. Therefore the solutions are compared with the other two existing techniques, such as the novel method and fuzzy TOPSIS. Three numerical examples are considered to explain the model properly. After that, a sensitivity analysis is performed to visualize the impact of demand, supply and capacity on the objective functions. Finally, conclusions and future scopes related to this study are drawn.

Design of a Distribution Network in a Multi-product, Multi-period Green Supply Chain System Under Demand Uncertainty

This paper proposes a novel fuzzy mathematical model for a distribution network design problem in a multi-product, multi-period, multi-echelon, multi-plant, multi-retailer, multi-mode of transportation green supply chain system. The three purposes of the model are to minimise total network cost, maximise net profit per capita for each human resource, and diminish CO2 emission throughout the network. P-hub median location with multiple allocations is used for locating the distribution centres. One scenario is designed for fuzzy customer demands with a trapezoidal membership function. Furthermore, the model determines the design of the network (selecting the optimum numbers, locations of plants, and distribution centres to open), finding the best strategy for material transportation through the network with the availability of different transportation modes, the capacities level of the facilities (plants or distribution centres (DCs)), and the number of outsourced products. Finally, all uncertain customer demands for all product types can be satisfied based on the methods mentioned above. This multi-objective mixed-integer non-linear mathematical model is solved by NSGA-II, MOPSO and a hybrid meta-heuristic algorithm. The results show that NSGA-II is the exclusive algorithm that obtains the best result according to the evaluation criteria.

Making an integrated decision in a three-stage supply chain along with cellular manufacturing under uncertain environments: A queueing-based analysis

Today’s complicated business environment has underscored the importance of integrated decision-making in supply chains. In this paper, a novel mixed-integer nonlinear mathematical model is proposed to integrate cellular manufacturing systems into a three-stage supply chain to deal with customers’ changing demands, which has been little explored in the literature. This model determines the types of vehicles to transport raw materials and final parts, the suppliers to procure, the priorities of parts to be processed, and the cell formation to configure work centers. In addition, queueing theory is used to formulate the uncertainties in demands, processing times, and transportation times in the model more realistically. A linearization method is employed to facilitate the tractability of the model. A genetic algorithm is also developed to deal with the NP-hardness of the problem. Numerous instances are used to validate the effectiveness of the modeling and the efficiency of solution procedures. Finally, a sensitivity analysis and a real case study are discussed to provide important management insights and evaluate the applicability of the proposed model.

Designing a new mathematical model for optimising a multi-product RFID-based closed-loop food supply chain with a green entrepreneurial orientation

PurposeEnvironmental awareness is increasing among people in developing countries. In this regard, companies should consider ecological goals in addition to financial goals. Since the food industry is recognised as one of the largest emitters of CO2, profit and ecological objectives are optimised in radio-frequency identification (RFID) based closed-loop supply chain in the food industry in this paper.Design/methodology/approachBased on the literature, companies with a green entrepreneurial orientation (GEO) can turn ecological problems into opportunities using their proactiveness. In this regard, a new mixed-integer non-linear mathematical model is presented for optimising a new multi-product RFID-based closed-loop supply chain with a GEO in the food industry. The case study in this paper is Ofogh-e Kourosh company which is located in Iran. The GAMS software is used to code this model.FindingsThe optimum number of new products and materials flow was found among the closed-loop supply chain entities. Some factors as price, quality and warranty of products were considered, and the number of reopening of facilities if needed was set. The optimum node for RFID installation was found.Originality/valueThe paper presents a multi-objective mathematical model for optimising a multi-product RFID-based closed-loop supply chain with a GEO in the food industry. In addition, this paper gives insights into how can model this type of supply chain considering ecological and financial attributes.

Incorporating sales and marketing considerations into a competitive multi-echelon distribution network design problem with pricing strategy in a stochastic environment

This paper presents a multi-echelon distribution network design problem with pricing strategy in a stochastic environment, where location, inventory, and pricing decisions in retail and wholesale channels are made simultaneously. The considered network is comprised of a central warehouse, a set of distribution centers, and a set of retailers and wholesalers. The objective of the problem is to maximize the supply chain profit. The decisions include the location of distribution centers and the allocation of retailers and wholesalers to them, order-size for each distribution center and product price at retail and wholesale channels for different payment conditions. A mixed-integer nonlinear mathematical model was formulated and solved using the Lagrangian relaxation method and a Genetic algorithm. Computational results indicate that Lagrangian relaxation algorithm has good performance in terms of objective function value and runtime, even in large sized problems.

Dynamic Vehicle Routing Problem with Cooperative Strategy in Disaster Relief

In recent years, the growth of technology has brought about a new range of problems, referred to as dynamic vehicle routing problems. In these problems, a part of the orders are received in advance before departure of vehicles from depots, but some new orders will come in after the vehicle's departure. Although cooperative strategy has not received considerable attention in the literature, it could be a possibility in practice which can help reduce costs. Multiple vehicles in this strategy are allowed to travel and they can transfer goods between one another when they meet in demand points so as to better satisfy the late demands. A mixed integer nonlinear mathematical model is proposed for multi-vehicle routing problem considering product transshipment between vehicles in dynamic situations. A genetic algorithm is then developed to deal with the complexity of the problem. The experimental results show that cooperative strategy is an attractive possibility to reduce unsatisfied late received demands and costs.

A Vessel Schedule Recovery Problem at the Liner Shipping Route with Emission Control Areas

Liner shipping is a vital component of the world trade. Liner shipping companies usually operate fixed routes and announce their schedules. However, disruptions in sea and/or at ports affect the planned vessel schedules. Moreover, some liner shipping routes pass through the areas, designated by the International Maritime Organization (IMO) as emission control areas (ECAs). IMO imposes restrictions on the type of fuel that can be used by vessels within ECAs. The vessel schedule recovery problem becomes more complex when disruptions occur at such liner shipping routes, as liner shipping companies must comply with the IMO regulations. This study presents a novel mixed-integer nonlinear mathematical model for the green vessel schedule recovery problem, which considers two recovery strategies, including vessel sailing speed adjustment and port skipping. The objective aims to minimize the total profit loss, endured by a given liner shipping company due to disruptions in the planned operations. The nonlinear model is linearized and solved using CPLEX. A number of computational experiments are conducted for the liner shipping route, passing through ECAs. Important managerial insights reveal that the proposed methodology can assist liner shipping companies with efficient vessel schedule recovery, minimize the monetary losses due to disruptions in vessel schedules, and improve energy efficiency as well as environmental sustainability.

Mathematical Model for Locating a Pre-Positioned Warehouse and for Calculating Inventory Levels

A mixed integer non-linear mathematical model is proposed to efficiently provision necessary supplies for the dignified survival of people affected by hydrometeorological disasters that are usually experienced in the state of Veracruz, Mexico. In the context of damages to property and infrastructure in the municipalities affected by such natural disasters and to develop resilience, a model that combines the problem of locating facilities with the problem of determining inventory levels for uncertain demand is developed, facilitating the timely supply of survival kits. The number of survival kits required is determined according to the needs of the affected people in terms of food products, equipment, water, and medicine. Requirements for food products and equipment are based on the human life cycle, from the stages of early childhood until old age; those for water and medicine on total demand. The model is tested in the Capital Region, one of the ten regions within Veracruz, because the municipalities that comprise the region have experienced numerous declarations of emergency registered in the Natural Disasters Fund. The Capital Region comprises 33 municipalities, 17 of which have experienced this type of declaration in 2016. Therefore, our model has been developed to determine the optimal location of a pre-positioned warehouse in the Naolinco municipality, including the optimal levels of inventory that would satisfy 99% of the affected population in this region. Our model can serve as a tool through which individuals involved in the decision-making process can select the locations that have not been impacted by the hydrometeorological events, including pre-positioned warehouses that in turn safely store the survival kits. This model can be utilized not only in Mexico but in any part of the world by individuals involved in the decision-making process associated with providing relief to affected people.

Location and Optimal Sizing of Distributed Generators and Banks Capacitors in Distribution Systems

This paper presents the optimal location and sizing of distributed generators and capacitor banks in radial distribution networks to minimize the active and reactive power losses. A mixed-integer nonlinear mathematical model is proposed to formulate the problem. This mathematical model is solved through general algebraic modeling system (GAMS) and the CONOPT solver. GAMS implementation is presented with 10 nodes test system classical used in the specialized literature. As test system, the 33 nodes system is used under three simulation scenarios. The first simulating scenario consists only in the location of distributed generators, while the second scenario evaluates only the location of capacitor banks and the third scenario combines both elements. The simulation results obtained show that the integration of both elements improve significantly the operating features of the system by comparison with the first two scenarios analyzed.

Solving an Integrated Sales-Leasing Problem With Remanufacturing and Inventory Shortage Using Differential Evolution

In this article, an integrated sales and leasing company is considered. This company remanufactures leased products at the end of operating lease contracts to make them as good as new ones and sell them to the customers. In order to satisfy customers' demand, required products are provided from a third-party when the company meets inventory shortage. Non-linear competitive demand functions are used which are sensitive to manufacturer suggested retail price (MSRP) and inflation rate. A mixed integer non-linear mathematical model (MINLP) is developed to determine optimal price of selling products, optimal amount of monthly payments in leasing contracts, and optimal inventory control planning, i.e. the optimal amount of manufacturing and remanufacturing products and optimal inventory levels. The main objective is to maximize net profit of the company. Small, medium and large-scale sizes of the model are solved to show the applicability of the model. To solve the large-scale problem, differential evolution (DE) algorithm is applied as a meta-heuristic solution approach. Numerical results show high sensitivity of model to demands. Also, optimal trend behaviors of some main variables of the problem seem similar to the competitive behavior of demands.

Integrating a wind- and solar-powered hybrid to the power system by coupling it with a hydroelectric power station with pumping installation

The high variability of solar and wind energy sources makes their integration into power systems complicated and in some cases unnecessarily delays their transition from centralised to dispersed energy sources. In this paper, a mixed-integer non-linear mathematical model has been developed for simulating the integrated operation of a novel hybrid involving wind- and solar power and a hydroelectric power station with pumping installation. This hydropower plant is a special case of pumped storage hydroelectricity which to some extent utilises the available flow of the river on which it is located. It is thereby able to compensate for the varying energy output from the wind turbines (WT) and photovoltaics (PV) by discharging water previously pumped to the upper reservoir (or held back from the available flow) when a surplus from WTs and/or PVs occurs. The impact on the national power system (NPS) has been investigated based on the energy exchange values (unexpectedly occurring deficits and surpluses) between the considered hybrid energy source and the grid. The obtained results indicate that such a hybrid energy source not only significantly reduces the total volume of the energy exchange with the grid but also minimizes the ramp rate of those values. Accumulating water from available flow rate minimizes the need for oversizing the capacity installed in PVs and WTs. However, the inherent variability and typically low heads of existing run-off-river power plants with pondage lead to the size of the upper reservoir being prohibitive. The conclusions show that such a type of pumped storage hydroelectricity should mainly be used on a small scale.

Economic and environmental analysis of a hybrid solar, wind and pumped storage hydroelectric energy source: a Polish perspective

AbstractThis paper introduces a mixed integer non-linear mathematical model for a simulation of a hybrid energy source consisting of photovoltaics (PV), wind turbines (WT) and pumped storage hydroelectricity (PSH). The concept of PV–WT–PSH has been well described and evaluated for sparsely populated or remote areas such as islands. Here, due to the rapid development of renewable energy sources and most importantly the variable (non-dispatchable) energy sources such as wind and solar, the idea of wind and solar powered PSHs has been investigated in the context of the national energy system. The economic and environmental impact of the proposed hybrid has been assessed. The results reveal that to cover almost 40% of the energy demand one should expect the energy cost to increase by 25%.

Green vessel scheduling in liner shipping: Modeling carbon dioxide emission costs in sea and at ports of call

Abstract Considering a substantial increase in volumes of the international seaborne trade and drastic climate changes due to carbon dioxide emissions, liner shipping companies have to improve planning of their vessel schedules and improve energy efficiency. This paper presents a novel mixed integer non-linear mathematical model for the green vessel scheduling problem, which directly accounts for the carbon dioxide emission costs in sea and at ports of call. The original non-linear model is linearized and then solved using CPLEX. A set of numerical experiments are conducted for a real-life liner shipping route to reveal managerial insights that can be of importance to liner shipping companies. Results indicate that the proposed mathematical model can serve as an efficient planning tool for liner shipping companies and may assist with evaluation of various carbon dioxide taxation schemes. Increasing carbon dioxide tax may substantially change the design of vessel schedules, incur additional route service costs, and improve the environmental sustainability. However, the effects from increasing carbon dioxide tax on the marine container terminal operations are found to be very limited.