Linear Optimization Model Research Articles

Scale-specific effects of urban landscape pattern on the COVID-19 epidemic in Hangzhou, China

Abstract Context Understanding the scale-specific effects of different landscape variables on the COVID-19 epidemics is critical for developing the precise and effective prevention and control strategies within urban areas. Objective Based on the landscape epidemiology framework, we analyzed the scale-specific effects of urban landscape pattern on COVID-19 epidemics in Hangzhou, China. Methods We collected COVID-19 cases in Hangzhou from 2020‒2022 and combined the datasets of land use and land cover (LULC) and social gathering point (SGP) to quantify the urban landscape pattern. Optimal general linear model with stepwise regression was applied to explore the dominant landscape factors driving the COVID-19 transmission in the city. Furthermore, multi-scale geographically weighted regression illustrated the spatial heterogeneity and scale specificity of these landscape variables’ effects to COVID-19 epidemic. Results Eight landscape variables of LULC and SGP patterns were identified which explained 68.5% of the variance in spatial risk of COVID-19. Different optimal bandwidths across these variables in MGWR indicated their scale-specific effects. LSI of green space enhanced the spatial risk across the entire region. The effects of landscape contagion, the number of water bodies, LSI of cropland and built-up areas, and the density of commercial houses were detected to vary between urban and suburban areas. The effects of LSI of water bodies and the density of shopping malls were found to vary among different districts. Conclusions In this study, we firstly discriminated the scale-specific effects of different landscape variables on the COVID-19 epidemic in the urban region. These findings can help to optimize the differentiated zoning prevention and control strategies for COVID-19 in cities and guide policy-making and urban planning at a multi-scale hierarchical perspective to improve public health and urban sustainability.

Low-Carbon Water–Rail–Road Multimodal Routing Problem with Hard Time Windows for Time-Sensitive Goods Under Uncertainty: A Chance-Constrained Programming Approach

In this study, a low-carbon freight routing problem for time-sensitive goods is investigated in the context of water–rail–road multimodal transportation. To enhance the on-time transportation of time-sensitive goods, hard time windows are employed to regulate both pickup and delivery services at the start and end of their transportation. The uncertainty of both the demand for time-sensitive goods and the capacity of the transportation network are modeled using L-R triangular fuzzy numbers in the routing process to make the advanced routing more feasible in the actual transportation. Based on the carbon tax policy, a fuzzy linear optimization model is established to address the proposed problem, and an equivalent chance-constrained programming formulation is then obtained to make the solution to the problem attainable. A numerical experiment is carried out to verify the feasibility of incorporating the carbon tax policy, uncertainty, and water–rail–road multimodal transportation to optimize the low-carbon freight routing problem for time-sensitive goods. Furthermore, a multi-objective optimization is used to reveal that lowering the transportation costs, reducing the carbon emissions, and avoiding the risk are in conflict with each in the routing. We also analyze the sensitivity of the optimization results concerning the confidence level of the chance constraints and the uncertainty degree of the uncertain demand and capacity. Based on the numerical experiment, we draw several conclusions to help the shipper, receiver, and multimodal transportation operator to organize efficient water–rail–road multimodal transportation for time-sensitive goods.

Planning municipal drainage infrastructure maintenance operations with finite available crews: pragmatic optimization approach

This paper proposes a streamlined approach to addressing the problem of allocating finite crew resources to concurrent jobs in the context of municipal drainage infrastructure maintenance. The problem was defined from the perspective of a project manager involved in planning such operations on a day-by-day basis. The problem statement was then transformed into a simplified Integer Linear Programming optimization model. Performance metrics were devised to evaluate the optimization model’s effectiveness. A heuristic algorithm representing the decision-making process by a seasoned planner in the partner company was also developed. Both methods were applied to a case study and contrasted based on the same performance metrics. The findings underscored substantial optimization benefits in rendering decision support in resource-constrained drainage construction operations planning. In conclusion, this research presents an alternative strategy for navigating the complexities inherent in finite crew resource allocation on multiple concurrent drainage projects; lends a cost-effective optimization solution to improving the utilization of finite available crews while satisfying service demands from multiple clients to the largest extent possible.

Economic viability of decentralised battery storage systems for single-family buildings up to cross-building utilisation

Due to the great cost decrease of photovoltaics as well as battery storage, especially in the segment of decentralised home storage, the number of grid-connected battery-supported photovoltaic systems being installed in recent years is steadily increasing. However, the scientific community has intensively discussed that lithium-based battery storage systems cannot yet be operated economically in most cases. This paper addresses the level to which the cost of lithium battery storage needs to decrease in order to be economically viable. For this purpose, the economic viability of battery storage systems in single-family buildings, multi-apartment buildings and across-buildings is analysed on the basis of a linear optimisation model and the method of the internal rate of return. The utilisation of the storage system is optimised for different battery and photovoltaic capacities on the basis of generation and consumption. The internal rate of return method is used to compare the savings resulting from the reduced consumption from the electricity grid with the investment costs and the operation and maintenance costs. In order to be able to estimate the influence of the most important parameters a sensitivity analysis is also carried out. The analysis concludes that, depending on the combination of capacities of photovoltaics, battery storage and in relation to the load profile, the battery storage costs would have to drop by at least 85% in order to generate a certain predefined return over a depreciation period of 25 years. Furthermore, the more different load profiles can be covered directly with photovoltaic electricity, e.g. in a multi-apartment building or across buildings, the less electricity needs to be stored and this reduces the benefit and the utilisation of the battery storage and therefore the specific investment costs must further decrease. Another conclusion that emerges from the sensitivity analysis is that the electricity price and the spread between the electricity price and the feed-in tariff have the greatest influence on the investment costs and profitability. Due to limited space for photovoltaics and simultaneously high consumption, self-consumption is already quite high with cross-building utilisation and can no longer be increased to the necessary extent by the battery storage system, which is why the investment costs must also be lower. The novelty of this paper lies in particular in the fact that it deals with the target costs of battery storage systems in various scenarios for certain rates of return. The analyses in this paper are intended to provide a deeper understanding of the framework conditions for the economic operation of a battery storage system in the aforementioned scenarios. However, this paper does not take into account alternative sources of income other than savings on grid consumption. The possibility of time-variable (grid) tariffs, for example, is also not considered in detail in this paper and should be analysed further in future work.

Resolving the Paradox: How mobile money drives economic growth through financial inclusion

The primary aim of this study is to identify the most relevant financial inclusion variables influencing gross domestic product (GDP) and to choose the optimal multiple linear regression model to quantify their contribution to economic growth. Finance is essential for every economic activity and financial inclusion is closely related to societal empowerment. It is observed in different nations that financial inclusion variables have a valuable impact on economic growth. For this, secondary data were gathered for this study from the IMF (International Monetary Fund) from 2011 to 2020. Graphical presentations were used to visualize the relationship between financial inclusion variables and economic growth. The most eligible variables and the best model among a collection of models based on empirical data were chosen using a stepwise backward elimination technique using multiple linear regression and model selection criteria. From the graph, it is seen that financial inclusion variables are proportionately related to GDP and the results demonstrate that the most eligible variables of financial inclusion that affect GDP are (1) the number of registered mobile money agent outlets, (2) the number of active mobile money accounts, (3) the number of mobile money transactions, and (4) outstanding balances on active mobile money accounts. These variables have a positive impact on economic growth, and the fitted model measures the more significant contribution to economic growth. As a result, it may be summarized that financial enclosure has to be enhanced for the development of rural and border areas of Bangladesh. This study will assist policymakers, development organizations, and scholars and make knowledgeable findings that will guide long-term improvements to minimize the gap between rich and poor, leading to the improvement of the community's wealth and welfare.

Power system benefits of simultaneous domestic transport and heating demand flexibility in Great Britain’s energy transition

The increasing share of variable renewables in power generation leads to a shortage of affordable and carbon neutral options for grid balancing. This research assesses the potential of demand flexibility in Great Britain to fill this gap using a novel linear optimisation model PyPSA-FES, designed to simulate optimistic and pessimistic transition pathways in National Grid ESO Future Energy Scenarios. PyPSA-FES models the future power system in Great Britain at high spatiotemporal resolution and integrates demand flexibility from both smart charging electric vehicles and thermal storage-coupled heat pumps. The model then optimises the trade-off between reinforcing the grid to align charging and heating profiles with renewable generation versus expanding dispatchable generation capacity. The results show that from 2030, under optimistic transition assumptions, domestic demand flexibility can enable an additional 20–30 TWh of renewable generation annually and reduce dispatchable generation and distribution network capacity by approximately 20 GW each, resulting in a total cost reduction of around £5bn yearly. However, our experiments suggest that half of the total system cost reduction is already achieved by only 25% of electric vehicles alone. Further, the findings indicate that once smart electric vehicle charging reaches this 25% penetration rate in households, minimal benefits are observed for implementing smart 12-hour thermal storages for heating flexibility at the national level. Additionally, smart heating benefits decrease by 90% across all metrics when only pre-heating (without thermal storages) is considered. Spatially, demand flexibility is often considered to alleviate the need for north–south transmission grid expansion. While neither confirmed nor opposed here, the results show a more nuanced dynamic where generation capacities are moved closer to demand centres, enhancing connectivity within UK sub-regions through around 1000 GWkm of additional transmission capacity.

Food system adaptation and maintaining trade could mitigate global famine in abrupt sunlight reduction scenarios

After a major nuclear war, volcanic eruption or asteroid or comet impact that causes an abrupt sunlight reduction scenario, agricultural yields would plummet. We analyzed a nuclear winter scenario involving the injection of 150 Tg of soot in the stratosphere using a linear optimization model with and without global food trade. We investigated the effects of loss of global food trade, some simple adaptations like rationing and storage of excess food for the coldest years, and rapid, large-scale deployment of food sources which are less dependent on present day climate (so called resilient foods) including cool tolerant crops, methane single cell protein, lignocellulosic sugar, greenhouse crops, and seaweed. In the worst case of no global food trade and no adaptations, the model predicts a global famine. However, scaling up resilient foods quickly could mitigate this for many countries. Maintaining global food trade would further alleviate pressure on local food systems, unlocking the potential to feed the entire global population. However, insufficient preparation, post-disaster conflict, or economic collapse would worsen outcomes and hinder adaptation.

Investigation of Railway Network Capacity by Means of Dynamic Flows

Capacity calculations are essential for the long-term planning of railway infrastructure. Many of the methods currently used in practice calculate characteristic capacity values separately for the single elements (mainly lines and nodes) of a railway network. Approaches that consider the entire network to account for interactions between the elements often rely on many assumptions, which renders a direct practical application difficult. This paper therefore introduces a linear optimization model that comprises railway-specific constraints such as minimum headway times, line capacities, and route conflicts. Under the consideration of these constraints, the developed model permits the calculation of a network-wide capacity. The model is validated on a sample network that is based on a real network of the German railway infrastructure.

Design of closed-loop manufacturing-remanufacturing system network considering uncertain defect and waste rate: Scenario-based chance-constrained programming

To facilitate sustainability by extending the life-cycle and residual value of production materials, a collaborate scheduling scheme for closed-loop manufacturing system with disturbance uncertainty is studied. Based on the features of the closed-loop manufacturing system under multi-product and multi-period scenarios, a mixed-integer linear optimization model is developed, where manufacturing and remanufacturing units are coordinated by recycling the production materials from work-in-process and final products. Based on this, a novel closed-loop manufacturing-remanufacturing scheduling framework is proposed. Additionally, in response to variations in product quality and processing capabilities, a scenario-based chance-constrained programming is introduced to address potential sources of uncertainty. Subsequently, a sequential optimization method is devised to reduce computational complexity. Finally, an industrial case from the electronic factory is investigated to demonstrate its applicability by implementing the proposed approach with comparison to several other strategies.

Deceleration Capacity During Directional Change as a Time-Efficient (Ecological) Prescreening of Hip Adductor Force Status in Amateur Soccer Players.

Miralles-Iborra, A, Del Coso, J, De Los Ríos-Calonge, J, Elvira, JLL, Barbado, D, Urban, T, and Moreno-Pérez, V. Deceleration capacity during directional change as a time-efficient (ecological) prescreening of hip adductor force status in amateur soccer players. J Strength Cond Res XX(X): 000-000, 2024-Reduced isometric adductor muscle strength has been identified as a modifiable risk factor contributing to injury in soccer players. However, the measurement of hip adductor muscle strength is habitually laboratory-based, with isolated hip movements that do not reflect soccer-specific movements that induce groin injury during match play. This study aimed to determine the usefulness of deceleration capacity during a change of direction (COD) as a time-efficient (ecological) prescreening of hip adductor force status in soccer players. Nineteen amateur soccer players completed unilateral isometric hip adductor strength assessments and a 180° COD test. Isometric hip strength assessment included the maximum peak torque (PT) and maximum rate of torque development (RTDmax) relative to players' body mass. Players' deceleration capacity during the COD test was determined for each leg through maximum deceleration normalized to the linear momentum. A linear regression analysis was performed to associate isometric hip strength variables with the deceleration capacity during the COD test at each leg. There was not a statistically significant association between deceleration capacity and hip isometric maximum PT of the dominant and nondominant legs (r ≤ 0.14, p > 0.05). Nevertheless, a moderate association was found between deceleration capacity and RTDmax for both legs (r ≥ 0.58, p < 0.05). The optimal linear regression model suggests that measuring deceleration capacity during a directional change test could explain RTDmax by 33 and 43% for the dominant and nondominant legs, respectively. During a 180° COD test, the deceleration capacity captured through GPS-accelerometer device was limited as a prescreening tool to evaluate hip adductor force status in soccer players.

Human-in-the-Loop Behavior Modeling via an Integral Concurrent Adaptive Inverse Reinforcement Learning.

One goal of artificial intelligence (AI) research is to teach machines how to learn from humans, such that they can perform a certain task in a natural human-like way. In this article, an online adaptive inverse reinforcement learning (IRL) approach to human behavior modeling is proposed to enhance machine intelligence for a class of linear human-in-the-loop (HiTL) systems using the state data only, where the human behavior is described by a linear quadratic optimal control model with an unknown weighting matrix for the quadratic cost function. First, an integral concurrent adaptive law is developed to learn the human feedback gain matrix online using the demonstrated state data only, which removes the persistent excitation (PE) conditions required by traditional adaptive estimation approaches and thus is more in line with real applications. Then, with the learned feedback gain matrix, the IRL problem is formulated as a linear matrix inequality (LMI) optimization problem, which can be efficiently solved to retrieve the weighting matrix of the human cost function. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approach.

Modeling of a “Hydrogen Valley” to investigate the impact of a regional pipeline for hydrogen supply

Introduction: The transition towards electrolysis-produced hydrogen in refineries and chemical industries is expected to have a potent impact on the local energy system of which these industries are part. In this study, three urban areas with hydrogen-intense industries are studied regarding how the energy system configuration is affected if the expected future hydrogen demand is met in each node individually, as compared to forming a “Hydrogen Valley,” in which a pipeline can be used to trade hydrogen between the nodes.Method: A technoeconomic, mixed-integer, linear optimization model is used to study the investments in and dispatch of the included technologies with an hourly time resolution, while minimizing the total system cost. Four cases are investigated based on the availability of offshore wind power and the possibility to invest in a pipeline.Results: The results show that investments in a pipeline reduces by 4%–7% the total system cost of meeting the demands for electricity, heating, and hydrogen in the cases investigated. Furthermore, investments in a pipeline result in greater utilization of local variable renewable electricity resources, as compared to the cases without the possibility to invest in a pipeline.Discussion: The different characteristics of the local energy systems of the three nodes in local availability of variable renewable electricity, grid capacity and available storage options compared to local demands of electricity, heating and hydrogen, are found to be the driving forces for forming a Hydrogen Valley.

Integrated biomarker-based ecological risks assessment of tadpole responses to tris(2-chloroethyl) phosphate, tris(1-chloro-2-propyl) phosphate, and their combined environmental exposure

Tris(2-chloroethyl) phosphate (TCEP) and tris(1-chloro-2-propyl) phosphate (TCPP) are common chlorinated organophosphorus flame retardants (OPFRs) used in industry. They have been frequently detected together in aquatic environments and associated with various hazardous effects. However, the ecological risks of prolonged exposure to these OPFRs at environmentally relevant concentrations in non-model aquatic organisms remain unexplored. This study investigated the effects of long-term exposure (up to 25 days) to TCEP and TCPP on metamorphosis, hepatic antioxidants, and endocrine function in Polypedates megacephalus tadpoles. Exposure concentrations were set at 3, 30, and 90 μg/L for each substance, conducted independently and in equal-concentration combinations, with a control group included for comparison. The integrated biomarker response (IBR) method developed an optimal linear model for predicting the overall ecological risks of TCEP and TCPP to tadpoles in potential distribution areas of Polypedates species. Results showed that: (1) Exposure to environmentally relevant concentrations of TCEP and TCPP elicited variable adverse effects on tadpole metamorphosis time, hepatic antioxidant enzyme activity and related gene expression, and endocrine-related gene expression, with their combined exposure exacerbating these effects. (2) The IBR value of TCEP was consistently greater than that of TCPP at each concentration, with an additive effect observed under their combined exposure. (3) The ecological risk of tadpoles exposed to the combined presence of TCEP and TCPP was highest in China's Taihu Lake and Vietnam's Hanoi than in other distribution locations. In summary, prolonged exposure to environmentally relevant concentrations of TCEP and TCPP presents potential ecological risks to amphibian tadpoles, offering insights for the development of policies and strategies to control TCEP and TCPP pollution in aquatic ecosystems. Furthermore, the methodology employed in establishing the IBR prediction model provides a methodological framework for assessing the overall ecological risks of multiple OPFRs.

Linear optimal power flow model for modern distribution systems: Management of normally closed loop grids and on-load tap changers

Modern distribution systems face new challenges due to the rise of small-scale renewable energy resources and the electrification of energy demand. Additionally, the complexity of network management increases with the exchange of flexibility services between distribution and higher voltage levels. The system operator's role involves optimizing the planning and operation of the distribution network. In current scenarios, this includes managing flexibility resources (such as generators, demand response, and on-load tap changers) and implementing new operating strategies (such as closed-loop topologies and dynamic reconfiguration). This paper proposes a reformulation of one of the most used linear approximations of the Optimal Power Flow model for distribution networks. The novelty of this reformulation lies in integrating equations to handle tap-changing transformers with closed-loop/meshed network operation, while preserving the original linear formulation and computational tractability.

Linear Programming Model for Sunflower Oil Supply Chain Network

Food supply networks are vital to the global economy, producing and consuming products worldwide. Supply chain management (SCM) has emerged as a crucial technique to optimize the integrated network of suppliers, factories, warehouses, customers, and distribution channels. SCM aims to ensure efficient raw material sources, transformation, transportation, and timely customer delivery. Edible oil, a crucial ingredient for cooking and food preparation, is utilized by people from all social classes. Vegetable oils, particularly sunflower, soybean, peanut, and others, are the main source of edible oils, which have less unsaturated fatty acids compared to animal oils. The production and consumption of edible oils have substantial environmental impacts, necessitating increased research to address these implications. Sunflower, a valuable oilseed crop, is grown worldwide due to its short growing season and adaptability to various climates and soil types. Sunflower seeds have high oil and protein content, making them popular for their health benefits. This article uses a linear optimization model for the sunflower oil supply chain from the farmer to the end user. The objective is to minimize the overall transportation cost subject to capacity, demand fulfillment, and waste management.

Fleet Repositioning, Flag Switching, Transportation Scheduling, and Speed Optimization for Tanker Shipping Firms

In response to the European Union (EU)’s sanctions on Russian oil products, tanker shipping firms may adopt two strategies to reoptimize their shipping networks. The first strategy is to switch the flag states of tankers that are not eligible to operate on certain routes. The second strategy is to reposition tankers based on their flag states, i.e., moving those tankers that are eligible from other groups to specified routes. To help tanker shipping firms minimize the total operating cost during the planning horizon in the context of EU oil sanctions, including costs of fleet repositioning, flag switching, and fuel, this study investigates an integrated problem of fleet repositioning, flag switching, transportation scheduling, and speed optimization considering the dynamic relationships among fuel consumption, speed, and load. By formulating the problem as a nonlinear integer programming model and applying various linearization techniques to convert the nonlinear model into a linear optimization model solvable by off-the-shelf linear optimization solvers, this study demonstrates the practical application potential of the proposed model, with the longest solution time of less than two hours for a numerical instance with seven routes. Furthermore, through sensitivity analyses on important factors including unit fuel prices, crude oil transportation demand, and the tanker repositioning cost, this study provides managerial insights into the operations management of tanker shipping firms.

A machine learning approach to analysis of Canadian provincial power system decarbonization

Canada committed to reducing national greenhouse gas emissions by 40–45 % by 2030 and achieving net-zero by 2050, relying on low-carbon electrification across economic sectors. This study investigates the machine learning’s potential to analyze least-cost optimization of Canadian provincial power systems to 2050, to identify robust pathways for net-zero electricity system greenhouse gas emissions, and to provide policy insights respecting wind and solar generation, reducing emissions from natural gas generation, and expanding interprovincial transmission. Surrogate modeling is applied as sensitivities to generation technology capital costs, electricity demand, and carbon pricing to results from an existing capacity expansion mixed-integer linear optimization model. Under almost all future scenarios, six provinces develop new wind generation, with substantial deployment in Alberta (mean 27.3 GW, max. 62.7 GW), Ontario (mean 6.8 GW, max. 19.3 GW), Saskatchewan (mean 3.6 GW, max 8.9 GW) and Manitoba (mean 1.5 GW, max 3.5 GW). With the potential exception of BC, where higher-cost wind generation results in solar generation becoming the primary low-carbon electrical energy source (mean 25.2 GW), future solar generation across Canada is more limited (mean 37.1 GW) and less certain than wind generation in all other Canadian provinces. Existing natural gas generation is maintained in almost all scenarios in all provinces, providing support for maintaining these resources for reliability and reserve requirements. Very limited natural gas is developed under any scenarios, with the exception of British Columbia (mean 4.3 GW) where more than 80 % of scenarios include new natural gas capacity. Additional transmission transfer capacity is developed in most scenarios between BC-Alberta (∼60 % of scenarios), Saskatchewan-Manitoba (∼85 %), and Ontario-Quebec (∼75 %). Carbon prices, along with wind and solar capital costs, act as primary determinants of least-cost decarbonization pathways. These findings offer insights for Canadian policymakers into net-zero pathways for provincial electricity systems that remain robust despite variability in inputs.

Optimal product quantity linear programming model for profit maximization in plastic small medium manufacturing firms

Planning production and maximizing profits can be difficult tasks for small and medium-sized manufacturing businesses (SMEs). Owing to their difficulties implementing the newest and most advanced Industry 4.0 technologies, small and medium-sized enterprises (SMEs) generally depend more on expert guidance. Thus this study focuses on the development of a Linear programming model for determining the optimum quantities of product in order to maximize profit profits in manufacturing Small medium enterprises. Data spanning over a period of forty five months was obtained from a manufacturing small and medium-sized enterprise. An optimization was developed in the study to determine the optimal product quantity for profit maximization. Based on the developed model and its optimization, it was determined that the optimal production quantity for the plastic factory to be 12000cups resulting in a maximum profit of N456000. This optimization approach provides valuable insights for the plastic factory’s production planning.

Research on Capacity Optimization Configuration of Renewable Energy Off Grid Hydrogen Production System Considering Collaborative Electrolysis

This study proposes a multitype electrolytic collaborative hydrogen production model for optimizing the capacity configuration of renewable energy off grid hydrogen production systems. The electrolytic hydrogen production process utilizes the synergistic electrolysis of an alkaline electrolyzer (AEL) and proton exchange membrane electrolyzer (PEMEL), fully leveraging the advantages of the low cost of the AEL and strong regulation characteristics of the PEMEL. For the convenience of the optimization solution, the article constructs a mixed linear optimization model that considers the constraints during system operation, with the objective function of minimizing total costs while meeting industrial production requirements. Gurobi is used for the optimal solution to obtain the optimal configuration of a renewable energy off grid hydrogen production system. By comparing and analyzing the optimal configuration under conventional load and high-load conditions, it is concluded that collaborative electrolysis has advantages in improving resource consumption and reducing hydrogen production costs. This is of great significance for optimizing the capacity configuration of off grid hydrogen production systems and improving the overall economic benefits of the system.

Social versus individual work preferences: implications for optimal linear income taxation

We extend the standard optimal linear income taxation model to allow for differences in social and individual work preferences while still maintaining the assumption that individuals are rational. The theoretical and simulation analyses show that when the government places a higher social weight on work than do individuals, the optimal marginal income tax rate (MIT) becomes lower. This implies lower revenue, income guarantee, and overall progressivity. The case for lower MIT is reinforced when the government places a relatively higher weight on work for low earners. Combining our analysis with that of An and Coady (2022), we, on the one hand, agree with previous studies that the optimal nonlinear income tax schedule would be close to the optimal linear one but, on the other, show that the degree of closeness would depend on preference differences. Our work contributes to the burgeoning field of non-welfarist economics.