Fuzzy Solution of Z-Number-Based Multi-Objective Linear Programming Models with Different Membership Functions

Multiobjective linear programming model on injection oilfield recovery system

Multiobjective linear programming (MOLP) models are developed to evaluate total phosphorus (TP) load reduction strategies for an off-stream reservoir. Cost, water quality, and equity are the three major objectives considered. Watershed-based, subwatershed-based, and equal-transport-path-length-based strategies are the three zone control strategies analyzed and compared. The agricultural nonpoint-source pollution (AGNSP) model is used to simulate the non–point-source pollution loads on the reservoir. A twostep modeling procedure is applied to verify sediment mass balance and water quality of storm runoff. MOLP models are established based on the simulation results for the three control strategies and are solved using the ADBASE package to generate the noninferior solution sets of the models. The subwatershed-based control strategy is observed to be the most cost-effective one based on results obtained for TP concentration between 13 and 20 ppb. With respect to equity, however, the watershed-based control strategy is superior to the others. The proposed multiobjective analyses are intended to improve the comparison and evaluation of nonpoint source pollution control strategies.

In this paper, we consider interval multi-objective linear programming (IMOLP) models which are used to deal with uncertainties of real-world problems. So far, a variety of approaches for obtaining efficient solutions (ESs) of these problems have been developed. In this paper, we propose a new and two generalized methods. In the new method, converting IMOLP into an interval linear programming (ILP) and then obtaining its optimal solutions (OSs), ESs of the IMOLP are determined. This method has several advantages: (i) This method is the only method which obtains a solution box for IMOLP models. (ii) The solving process is not time consuming. (iii) The number of ESs is higher than for other methods. (V) The method is applicable for large-scale problems. Also, we generalize the $$\varepsilon$$-constraint and lexicographic methods which are used for obtaining ESs of the multi-objective linear programming (MOLP) models which do not have any problems such as lengthy and time-consuming and are applicable for large-scale problems. Some examples were solved to show the efficiency of the proposed methods. Finally, by the proposed method, we solve the IMOLP model corresponding to the problem of the facilities and non-return funds in a bank.

The present chapter proposes a Multi-Objective Linear Programming (MOLP) model for the optimal planning of the fuelwood supply chain management problem. The choice of the model was based on the fact that the nature of the problem is dual: economic and environmental. The sensitivity of the model to different demand scenarios is checked by the use of numerical simulations.

With the increasing pressure from global competition, manufacturers have realized that sustainable production is significant in supply chain management. Sustainable supplier selection and order allocation (SSS&OA) play a distinct and critical role for organizations to achieve sustainable development and build competitive advantage. In this paper, we aim to develop a novel SSS&OA model for selecting the most suitable sustainable suppliers and determining the optimal order sizes among them. First, double hierarchy hesitant linguistic term sets (DHHLTSs) are adopted to deal with uncertainty in evaluating the sustainable performance of alternative suppliers. Then, an extended decision field theory is proposed to choose efficient sustainable suppliers dynamically. Considering quantity discount, a multi-objective linear programming (MOLP) model is established to allocate reasonable order quantities among the selected suppliers. Finally, the applicability and effectiveness of the developed model are illustrated through its application in the electronic industry and through a comparative analysis with other methods.

Sustainable supply chain networks have attracted considerable attention in recent years as a means of dealing with a broad range of environmental and social issues. This paper reports a multi-objective mixed-integer linear programming (MILP) model for use in the design of a sustainable closed loop supply chain network under uncertain conditions. The proposed model aims to minimize total cost, optimize environmental impacts of establishment of facilities, processing and transportation between each level as well as social impacts including customer satisfaction. Due to changes in business environment the uncertainty existed in the research problem, in this paper the chance constrained fuzzy programming approach applied to cope with uncertainties in parameter of the proposed model. Then the proposed multi-objective model solves as single-objective model using LP-metric method.

Application of fuzzy optimization to a supply chain network design: A case study of an edible vegetable oils manufacturer

WITHDRAWN: Application of a fuzzy multi-objective defuzzification method to solve a transportation problem

This study presents a novel multi-objective linear fractional interval programming model for sustainable waste management under uncertainty, with a focus on the Melbourne metropolitan region. The model integrates three conflicting objectives: minimizing the cost per recycled ton, minimizing emissions per processed ton, and minimizing the energy-related carbon footprint. In the existing literature, most traditional models rely on fixed decision variables, which often fail to capture the inherent uncertainty present in real-world, data-driven problems. Thus, in this study, all key parameters, including transport costs, recycling efficiencies, emissions, and facility capacities, are represented as interval values to account for parameter variability. The optimal solution allocates waste primarily to residential and green streams, yielding a cost per recycled ton in the range [21.01, 37.44] AUD, emissions per processed ton of [8.88, 20.75] kg CO $$_2$$ , and an energy-related carbon footprint of [4.44, 9.00] kg CO $$_2$$ . All solutions satisfy the prescribed budget and emission caps, demonstrating operational feasibility. The results further indicate that environmental objectives exhibit greater robustness to parameter uncertainty than economic costs, highlighting the effectiveness of the proposed interval-based framework for supporting sustainable municipal waste management decisions.

Regional land use/cover change is one of the hot issues of the global research. Since the proposal of “land use/cover change” research program by the International Geosphere-Biosphere Program（IGBP）and International Human Dimensions Programme on Global Environmental Change（IHDP）in 1995, the research on the land use structure optimization has been increasingly important as the core issue of regional land use/cover change. The structural optimization of land use refers to the use of certain techniques and methods to make the regional future land use structure achieve the best economic, ecological and social benefits. Base on the multi-objective linear programming model and through the setting of 14 variables, this paper has proposed the land use structural optimization program for the rational land planning of Foshan, thus providing reference for the rational land planning of economically developed area.

This study focuses on solving multiobjective stochastic linear programming (MSLP) problems with partial information on probability distribution. The method presented in this paper utilises the concept of ranking function and linguistic hedges in the fuzzy transformation stage to take into account the internal values of the probability distribution. An adaptive arithmetic average approach is then used to convert the multiobjective problems into a single objective problem. Comparison of results with existing methods in the literature is presented which shows that the method presented performs as good as the existing methods in terms of solution quality but better in terms of computational effort.

In this paper, three criteria scheduling problem of n jobs on a single machine is considered. Each of these n jobs is to be processed without interruption and becomes available for processing at time zero. The problem is to minimize three objectives simultaneously, which are the completion time, maximum tardiness, and maximum earliness. Here, we develop a fuzzy multi-objective linear programming (FMOLP) model for solving multi-objective scheduling problem in a fuzzy environment by using piecewise linear membership function (PLMF). A numerical example demonstrates the feasibility of applying the proposed model to scheduling problem, and yields a compromised solution to help the decision maker’s overall levels of satisfaction. The algorithm is tested to show the ability of applying this model to three criteria.

Water allocation is an essential programming to support the sustainable development of Wuwei Basin, Gansu Province, China. To satisfy the demands of the decision makers (DMs) of each subarea and the total area, a bilevel multiobjective linear programming (BLMOLP) model is proposed. In the BLMOLP, DMs have a hierarchy of two levels—the upper level and the lower level DMs. In this paper, a fuzzy goal programming (FGP) approach is applied to solve the BLMOLP. Firstly, the upper level is solved and used as the tolerance for the lower level. Then the weights of each objective function in the lower level are evaluated. Finally, a satisfied optimization solution of the problem was calculated. The result suggests that the FGP is a simple and feasible approach to BLMOLP problems. The proposed method was applied to a case study for water resources allocation in Wuwei Basin. For four scenarios under consideration, the model can effectively balance the benefits among all regions and sections according to the priority of the upper level decision makers. The results indicate that comprehensive solutions have been obtained.

Smooth implementation and controlling conflicting goals of a project with the usage of all related resources through organization is inherently a complex task to management. At the same time deterministic models are never efficient in practical project management (PM) decision problems because the related parameters are frequently fuzzy in nature. The project execution time is a major concern of the involved stakeholders (client, contractors and consultants). For optimization of total project cost through time control, here crashing cost is considered as a critical factor in project management. The proposed approach aims to formulate a multi objective linear programming model to simultaneously minimize total project cost, completion time and crashing cost with reference to direct, indirect cost in the framework of the satisfaction level of decision maker with fuzzy goal and fuzzy cost coefficients.. To make such problems realistic, triangular fuzzy numbers and the concept of minimum accepted level method are employed to formulate the problem. The proposed model leads decision makers to choose the desired compromise solution under different risk levels and the project optimization problems have been solved under multiple uncertainty conditions. The Analytical Hierarchy Process is used to rank multiple objectives to make the problem realistic for the respective case. Here minimum operator and AHP based weighted average operator method is used to solved the model and the solution is obtained by using LINGO software

Cross-border e-commerce logistics supply chain alliances face the practical challenge of allocating profits in a way that is both fair and operationally controllable when member contributions and cooperation priorities jointly matter. This study proposes an integrated framework that first evaluates member contributions via a group decision-making (GDM) procedure to derive contribution weights, and then aggregates them to coalition-level importance scores to rank feasible cooperation structures. Building on these inputs, we formulate a Priority-Based Multiobjective Linear Programming (P-MOLP) model that performs tiered (priority) optimization under individual rationality and budget-balance constraints, thereby ensuring implementable allocations. Using alliance data, P-MOLP provides clearer structural differentiation than an ordinary goal-programming model: high-contribution, strategically central members receive larger shares, and low-contribution members receive less (“more-work-more-reward”). Unlike weighted-Shapley, which can violate individual rationality under extreme weights, P-MOLP is individually rational and budget-balanced, aligns better with observed practice, remains applicable when some coalition values are infeasible or missing, and attains ε-core near-stability. Priority weights serve as managerial levers to tune outcomes.