Multi-objective Linear Optimization Research Articles

Sustainable supply chain design using a multi-objective linear optimization approach for solid bulk products

This paper proposes a multi-objective optimization model for designing a sustainable logistics system that determines the ideal logistic flow of an ultrafine mineral residue product, tailings, from Brucutu/MG to multiple destinations over a capable multimodal system. These tailings are extracted from iron ore and can be used as a partial substitute for raw materials in construction materials. This substitution can help avoid dangerous dams of iron ore tailings and save natural resources and greenhouse gas emissions in the civil construction chain. However, the distribution incurs logistical costs and resulting socio-environmental impacts. Therefore, logistics plays a fundamental role in making this substitution economically viable. The proposed model aims to maximize total profit while minimizing negative socio-environmental impacts and maximizing the positives. The resulting ideal multimodal logistic system is obtained based on sustainable pillars, demonstrating that a small decrease in profits can yield significant reductions in negative impacts and increases in positive externalities.

An efficient algorithm for solving multilevel multi-objective linear fractional optimization problem with neutrosophic parameters

This work demonstrates the modeling of the multilevel-multiobjective linear fractional optimization (ML-MOLFO) problem in a neutrosophic decision environment. The primary driving force behind this work is to determine how to handle a ML-MOLFO problem while taking into account a real-life decision-making scenario. There is a hierarchy of decision-makers and many decision-making processes, such as agree, not-sure, and disagree, in a real-life decision-making setting. By taking into account such genuine situations, all resource, coefficient of objective, and constraint functions are represented as single-valued trapezoidal neutrosophic numbers. As far as we can tell, there is no established technique for resolving the ML-MOLFO problem with neutrosophic parameters in the literature review. Therefore, by outlining goal programming methods in the intuitionistic fuzzy environment, we create our new intuitionistic fuzzy Chebyshev goal programming (IFCGP) method to address the introduced problem. In the proposed method, first we used a modified ranking function to transform the problem into an analogous crisp ML-MOLFO problem, and then it was converted into a multilevel multiobjective linear optimization problem through linearized techniques. The linearization procedure in the proposed method, unlike the existing one, does not require derivatives, additional variables, or inequality constraints, which makes it simpler. To illustrate the adequacy and performance of the proposed new method, we considered numerical tests and application problems. We also compared the performance of the resulting optimal solution with existing approaches using the distance metric, which indicates our method is non-dominated.

Neutrosophic linear fractional optimization problem using fuzzy goal programming approach

This research paper develops an algorithm to solve a neutrosophic linear fractional optimization problem, where the cost of the objective functions, the technology coefficients, and the resources are neutrosophic numbers (triangular and trapezoidal numbers). Two defuzzification methods are used here to convert the neutrosophic linear fractional optimization problem into a multi-objective linear fractional optimization problem (MOLFOP). The fractional objective function is defuzzified using the component-wise optimization method, and the linear constraints are defuzzified using the aggregation ranking function strategy. Hence, the MOLFOP is then solved using the fuzzy goal programming method. In order to further explain our suggested algorithm, two numerical examples are solved at the end, and the outcomes obtained by our method are compared with the outcomes obtained using the other algorithm.

A note on “A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem”

Moges et al. (Inf. Sci. 625 (2023) 457–475) proposed a method to find a Pareto-optimal solution (POS) of intuitionistic fuzzy multi-objective linear fractional optimization problem (MOLFOP). In this method, firstly, an intuitionistic fuzzy MOLFOP is transformed into a crisp MOLFOP with the help of a ranking function. Then, using the existing method (Fuzzy Sets Syst. 125 (2002) 335–342), the transformed crisp MOLFOP is transformed into a crisp multi-objective linear optimization problem (MOLOP). Finally, it is assumed that to find a POS of the considered intuitionistic fuzzy MOLFOP is equivalent to find a POS of the transformed crisp MOLOP. It is pertinent to mention that this assumption is valid only if the transformed crisp MOLFOP is equivalent to the considered intuitionistic fuzzy MOLFOP as well as the transformed crisp MOLOP is equivalent to the transformed crisp MOLFOP. In this note, a numerical example is considered to show that the transformed crisp MOLFOP is not equivalent to the considered intuitionistic fuzzy MOLFOP. Also, on the basis of the existing results (Fuzzy Sets Syst. 246 (2014) 156–159), it is pointed out that the transformed crisp MOLOP is not equivalent to the transformed crisp MOLFOP. Therefore, Moges et al.’s method is not valid and hence, the results of agricultural planning problem of Ethopia for cultivation of different crops, obtained by Moges et al. with their proposed method, are not correct.

New Approach to Solving Fuzzy Multiobjective Linear Fractional Optimization Problems

In this paper, an iterative approach based on the use of fuzzy parametric functions is proposed to find the best preferred optimal solution to a fuzzy multiobjective linear fractional optimization problem. From this approach, the decision-maker imposes tolerance values or termination conditions for each parametric objective function. Indeed, the fuzzy parametric values are computed iteratively, and each fuzzy fractional objective is transformed into a fuzzy non-fractional parametric function using these values of parameters. The core value of fuzzy numbers is used to transform the fuzzy multiobjective non-fractional problem into a deterministic multiobjective non-fractional problem, and the ε-constraint approach is employed to obtain a linear single objective optimization problem. Finally, by setting the value of parameter ε, the Dangtzig simplex method is used to obtain an optimal solution. Therefore, the number of solutions is equal to the number of used values, and the optimal solution is chosen according to the preference of the decision-maker. We have provided a didactic example to highlight the step of our approach and its numerical performances.

Enhanced Solutions to Intuitionistic Fuzzy Multiobjective Linear Fractional Optimization Problems via Lexicographic Method

Optimization involving fuzzy numbers generally, and intuitionistic fuzzy numbers particularly is more and more an essential part of any computational intelligence method; and might be of interest in modeling fuzzy control systems, or carrying out a fuzzy sensitivity analysis. The recent literature includes many research papers related to both theoretical modelling, and practical implementation of fuzzy decision support systems. Fuzzy optimization is one of the main part in such decisionmaking tools. A realistic solution to a fuzzy optimization problem is always desired, and similarly, a Pareto optimal solution to a multiple objective optimization problem is always a need. In this paper we analyze the shortcomings; and eliminate the weaknesses of a solution approach from the literature proposed by Moges et al. in 2023. Firstly, Moges et al. used an accuracy function to de-fuzzify the original intuitionistic fuzzy optimization problem; then linearized the obtained crisp problem; and finally involved fuzzy goals to solve the derived multiple objective linear problem. We present two faulty key points of their approach, namely the de-fuzzification and linearization steps; prove the inappropriateness of their results; and propose some improvements. The final aim in addressing the first key point is to clarify the border between defuzzifications made in accordance to true theoretical statements, and those made for modeling reasons. The methodological improvement we propose is related to the second key point and it assures that Pareto optimal solutions - highly required in multiple objective optimization - are obtained. To illustrate our point of view, we use a numerical example from the literature, and report better numerical results derived by the improved methodology.

Comparison of Multi-Object Control Methods Using Multi-Objective Optimization

The aim of this work is to obtain multi-objective linear programming algorithms that can be used to solve the global problem of multi-object safety control processes in order to minimize the risk of collisions. In multi-objective linear optimization models, satisfactory trade-off assesses and resolves the conflict between different control objectives. A comparison of single-, bi-, and tri-objective linear programming algorithms allows us to adapt the appropriate optimization method to the conditions of the control process. An important outcome of the present research is the demonstration of the greater effectiveness of bi- and tri-objective optimization compared to single-objective optimization, reflecting the compromises taken into account when choosing between objects and achieving a minimum risk of collision when passing them.

Multi-objective fuzzy surgical scheduling during post-disaster with Nash equilibrium

The post-disaster surgical scheduling is a type of scheduling problem in healthcare system that occurs after a sudden disaster event. The rescue capacity of hospitals has faced an unprecedented challenge by increasing surgical loads due to the disaster occurrence. Hospital became excessively burdened by the influx of patients, who occupy inpatient beds and operation rooms (ORs), making the availability of medical resources uncertain and scarce. This paper focuses on the post-disaster surgical scheduling problem, considering uncertain demands such as the unknown surgical operation durations or random arrivals of emergency/urgent patients, as well as the interests of multiple stakeholders (patients and hospital managers). The objective of fuzzy post-disaster surgical scheduling (FPDSS) is to simultaneously minimize the expected recovery completion time for all patients (makespan), expected average makespan while prioritizing emergency/ urgent patients, as well as the expected total idle time of ORs. This problem can be formulated as a mixed integer linear multi-objective optimization model, and an improved hunger games search based on Nash equilibrium (IHGS-NE) is developed for FPDSS. Finally, experimental studies have been used to test the performance of the proposed approach, and the presented surgical scheduling scheme is validated. Results show that the IHGS-NE approach is an effective method on the research of the multi-objective FPDSS problem.

Minimizing the makespan and carbon emissions in the green flexible job shop scheduling problem with learning effects

One of the most difficult challenges for modern manufacturing is reducing carbon emissions. This paper focuses on the green scheduling problem in a flexible job shop system, taking into account energy consumption and worker learning effects. With the objective of simultaneously minimizing the makespan and total carbon emissions, the green flexible job shop scheduling problem (GFJSP) is formulated as a mixed integer linear multiobjective optimization model. Then, the improved multiobjective sparrow search algorithm (IMOSSA) is developed to find the optimal solution. Finally, we conduct computational experiments, including a comparison between IMOSSA and the nondominated sorting genetic algorithm II (NSGA-II), Jaya and the mixed integer linear programming (MILP) solver of CPLEX. The results demonstrate that IMOSSA has high precision, good convergence and excellent performance in solving the GFJSP in low-carbon manufacturing systems.

A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem

This paper presents a new method for solving an intuitionistic fuzzy multi-objective linear fractional optimization (IFMOLFO) problem with crisp and intuitionistic fuzzy constraints. Here, all uncertain parameters are represented as triangular intuitionistic fuzzy numbers. We used an accuracy ranking function and variable transformation in the proposed method to convert an IFMOLFO problem into a crisp multi-objective linear optimization problem. Then, we formulated the first phase of the weighted intuitionistic fuzzy goal programming (WIFGP) model to obtain an intuitionistic fuzzy non-dominant (IFND) solution for the IFMOLFO problem. Several strategies for obtaining an IFND solution to the IFMOLFO problem have been proposed in the literature. However, in addition to constructing the phase-I WIFGP model, this study shows that the IFND solution may not be Pareto-optimal when some of the under-deviation variables are zero. As a result, the second phase of the WIFGP model is applied to address this issue. The benefits of both models are merged to provide a novel method, unlike any other method in the literature, for producing optimal solutions that satisfy both IFND and Pareto-optimal requirements. The suggested algorithm’s efficiency and reliability are demonstrated by addressing a real-life case study of an agricultural production planning problem and followed by solving a numerical example from literature.

Multiobjective Optimization for Hurricane Retrofit to Improve Coastal Community Structural and Socioeconomic Resilience

With recent hurricanes causing devastating economic losses, structural damages, and societal impacts on coastal communities, residential building hurricane mitigation measures have gained increasing attention. However, retrofits are often costly and therefore beget inherent tradeoffs. In particular, the balancing of the postdisaster losses against upfront mitigation costs remains a constant challenge for community resilience planning. Existing mitigation optimization frameworks on the community scale have primarily focused on seismic hazards and typically have weak integration of the structural damages with the socioeconomic impacts. In the present study, finite element modeling of the residential buildings is applied to better incorporate the structural damages under combined wind and wave loadings. A multiobjective linear constrained optimization framework for improved residential building hurricane retrofit is developed to balance the structural damages, home displacements, morbidities, net direct economic losses, and initial mitigation expenditures under hurricane threats. For the present analysis, the mitigation actions of building elevation and improved construction practices are simulated. The framework is tested in a case study for an existing coastal community under 50-, 100- and 500-year hurricane events considering community-specific socioeconomic factors. The generalized pattern search algorithm is selected to determine the Pareto-optimal solutions. The results indicate large potential reductions in social, structural, and economic losses under optimized mitigation, but tradeoffs exist if the mitigation technique is found to be cost-ineffective, such as in the case of the improved construction practices. The findings highlight the importance of informed decision-making and home retrofits in improving community resilience.

Conic Duality for Multi-Objective Robust Optimization Problem

Duality theory is important in finding solutions to optimization problems. For example, in linear programming problems, the primal and dual problem pairs are closely related, i.e., if the optimal solution of one problem is known, then the optimal solution for the other problem can be obtained easily. In order for an optimization problem to be solved through the dual, the first step is to formulate its dual problem and analyze its characteristics. In this paper, we construct the dual model of an uncertain linear multi-objective optimization problem as well as its weak and strong duality criteria via conic duality. The multi-objective form of the problem is solved using the utility function method. In addition, the uncertainty is handled using robust optimization with ellipsoidal and polyhedral uncertainty sets. The robust counterpart formulation for the two uncertainty sets belongs to the conic optimization problem class; therefore, the dual problem can be built through conic duality. The results of the analysis show that the dual model obtained meets the weak duality, while the criteria for strong duality are identified based on the strict feasibility, boundedness, and solvability of the primal and dual problems.

An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Managing irrigation water resources with economic benefit and energy consumption: an interval linear multi-objective fractional optimization model under multiple uncertainties

At present, water shortage has increasingly become one of the key factors restricting the sustainable development of agriculture. And during the process of agricultural production, the energy consumption has also been paid more and more attention. Therefore, in order to deal with the above problems, an improved interval linear multi-objective fractional programming model (ILMOFPM) was developed, which has the characteristic is that it not only focuses on the optimal allocation of the amount of irrigation water resources, but also on improving irrigation water resources use efficiency. To demonstrate its effectiveness and applicability, ILMOFPM was applied to a case study in Jinghuiqu irrigation district, Shaanxi Province, China. Maximum economic benefit per unit of water resource and minimum energy consumption per unit of water resource were selected as the planning objectives. A series of irrigation water resources optimal allocation schemes were obtained, which could give the decision makers the desired optimal scheme according to the actual situation. Based on the results of the established model, economic benefit and energy consumption generally decreased as the representative hydrological years changed from wet to dry, and the total crop planting area also showed a decreased trend. Moreover, a lot of water resources was saved by introducing crop evapotranspiration and effective precipitation in the developed model, 57.45%, 44.17%, 39.33% of total water resources consumption were saved under different representative hydrological years, respectively.

Exact algorithms for multiobjective linear optimization problems with integer variables: A state of the art survey

AbstractWe provide a comprehensive overview of the literature of algorithmic approaches for multiobjective mixed‐integer and integer linear optimization problems. More precisely, we categorize and display exact methods for multiobjective linear problems with integer variables for computing the entire set of nondominated images. Our review lists 108 articles and is intended to serve as a reference for all researchers who are familiar with basic concepts of multiobjective optimization and who have an interest in getting a thorough view on the state‐of‐the‐art in multiobjective mixed‐integer programming.

Hesitant intuitionistic fuzzy algorithm for multiobjective optimization problem

Every real-life optimization problem with uncertainty and hesitation can not be with a single objective, and consequently, a class of multiobjective linear optimization problems (MOLOP) appears in the literature. Further, the experts assign values of uncertain parameters, and the expert’s opinions about the parameters are conflicting in nature. There are concerning methods based on fuzzy sets, or their other versions are available in the literature that only covers partial uncertainty and hesitation, but the hesitant intuitionistic fuzzy sets provides a collective understanding of the real-life MOLOP under uncertainty and hesitation, and it also reflects better practical aspects of decision-making of MOLOP. In this context, the paper defines the hesitant fuzzy membership function and nonmembership function to tackle the uncertainty and hesitation of the parameters. Here, a new solution called hesitant intuitionistic fuzzy Pareto optimal solution is defined, and some theorems are stated and proved. For the decision-making of MOLOP, we develop an iterative method, and an illustrative example shows the superiority of the proposed method. And lastly, the calculated results are compared with some popular methods.

A Novel Multi-Objective Linear Fractional Optimization Model in Intuitionistic Fuzzy Environment and its Application in Organization Planning

A Novel Multi-Objective Linear Fractional Optimization Model in Intuitionistic Fuzzy Environment and its Application in Organization Planning

Contribution to multi-level multi-objective linear optimization

In this paper, we present a new method for the resolution of a multi-level multi- objective linear programming problem with bounded variables based on the adaptive method for solving multi-level multi-objective linear programming with bounded variables. The peculiarity of this method is that it has the suboptimality criterion that allows us to find e - efficient solutions, and unlike to other methods, the method we are going to present has the advantage of solving exactly these kind of problems and manipulating the constraints as they are without any transformation. We will present the detailed algorithm of the method and give a numerical example to demonstrate its applicability.

Multi-Objective Linear Optimization Problem for Strategic Planning of Shared Autonomous Vehicle Operation and Infrastructure Design

This study proposes a unified optimization framework for strategic planning of shared autonomous vehicle (SAV) systems that explicitly and endogenously considers their operational aspects based on macroscopic dynamic traffic assignment. Specifically, the proposed model optimizes fleet size, road network design, and parking space allocation of an SAV system with optimized SAVs’ dynamic routing with passenger pickup/delivery and ridesharing. It is formulated as a multi-objective optimization problem that simultaneously minimizes total travel time of travelers, total distance traveled by SAVs, total number of SAVs, and infrastructure construction cost; thus, both the user-side cost and the system-side cost are taken into account, and their trade-off relations can be explicitly investigated. Furthermore, the problem is formulated as a linear programming problem, making it easy to solve. By leveraging the linearity, we mathematically derive a useful property of the problem: introduction of ridesharing can weakly monotonically and simultaneously decrease the user-side cost and system-side cost. The proposed model is evaluated by applying it to actual travel records obtained from New York City taxi data.

Stochastic energy management in a renewable energy-based microgrid considering demand response program

Climate changes increase concerns about global warming caused by greenhouse gases and have also increased the focus and implementation of renewable energy sources (RESs) planning. One of the important RESs is tidal energy or tidal power, which is a form of hydropower that converts the energy obtained from tides into the electrical power. Although tidal power is still not widely used, this energy resource has the potential for the future electricity generation. This paper addresses the stochastic energy management in a microgrid considering RESs such as solar, wind and tidal sources in the presence of the demand response program and storage devices. The uncertainty of the RESs, demand, and electricity price is handled by Monte Carlo simulation (MCS). The model is a linear multi-objective optimization which the first objective aims to reduce the cost and the second aims to reduce the emission. Augmented ε-constraint approach is applied to solve the problem in the CPLEX/GAMS software environment. The interactive fuzzy decision-making is applied to choose the best answer among the Pareto answers according to the planner criteria.