Methodology for Solving Multi-Objective Quadratic Programming Problems in a Fuzzy Stochastic Environment

This paper presents a fuzzy goal programming approach for modelling and solving multi-objective decision making problem having fuzzy random variables and fuzzy numbers associated with the system constraints. In the model formulation process, the problem is converted into an equivalent fuzzy programming problem by using chance constrained programming technique. The problem is then decomposed into sub problems by considering the tolerance limits of fuzzy numbers relating to the system constraints. The individual optimal solution of each objective is found to construct the membership goals. A two-phase fuzzy goal programming model is developed to achieve the highest degree of each of the defined membership goals of the objectives to the extent possible by minimising under deviational variables and thereby obtaining most satisfactory solution in the decision making environment. A numerical example is solved to illustrate the proposed approach and the achieved solutions are compared with other existing methodologies.

In this paper, we propose an alternate modified solution procedure based on fuzzy goal programming (FGP) approach for solving bi-level multiobjective linear fractional programming problems (BL-MOLFPPs) with some major modifications in the algorithm suggested by Abo-Sinna and Baky (2010). The proposed modified technique is simpler and requires less computational works than that of earlier technique. In the formulation of FGP models, each objective function (at both the levels - leader's level and follower's level) as well as control variables (of leader's level) are transformed into fuzzy goals with modified membership functions. Then FGP approach is used for achieving the highest degree of each of membership goals. To avoid decision deadlock situations and difficulties associated with bi-level programming problems, solution preferences by decision makers (at leader's and follower's levels) are not taken into account in proposed algorithm. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed algorithm over earlier technique.

Optimizing sustainable and renewable energy portfolios using a fuzzy interval goal programming approach

• Propose novel behavioral membership functions within the fuzzy goal programming approach. • Model decision-makers’ behavioral aspects and risk perceptions. • Develop a behavior-based fuzzy goal programming approach. • Enhance utility functions within the fuzzy goal programming methodology. • Demonstrate applicability and efficacy through a venture capital investment problem. A relevant extension of traditional goal programming (GP), fuzzy goal programming (FGP) can handle uncertainty and imprecision in multi-objective optimization problems. Based on fuzzy set theory, the notion of membership functions has been introduced to consider the fuzziness related to objectives and constraints. These membership functions are mainly intended for fuzziness in the GP rather than modeling the decision-maker's (DM's) preferences and his/her attitude toward risk in the decision-making process. In the satisfying philosophy of FGP, little attention has been given to how preferences evolve in terms of the behavior of the decision-maker (DM) and how these preferences may affect decisions in risky scenarios. To address this issue, we suggest novel behavioral-type utility functions for the FGP approach by introducing the concept of behavioral membership functions. This concept offers an innovative procedure for simulating the DM's behavioral preferences in the FGP approach. First, two main categories of objectives in relation to the DM's behavioral preferences are distinguished in this work. A risk aversion parameter is integrated into membership functions according to the nature of each objective type, obtaining the so-called behavioral membership functions. A behavioral FGP approach is subsequently formulated. Finally, an illustrative example of venture capital investments, a sensitivity analysis, and comparisons with other FGP approaches are provided to demonstrate the validity and practicality of our proposed approach.

In this paper, we present modified fuzzy goal programming (FGP) approach and generalized MATLAB program for solving multi-level linear fractional programming problems (ML-LFPPs) based on with some major modifications in earlier FGP algorithms. In proposed modified FGP approach, solution preferences by the decision makers at each level are not considered and fuzzy goal for the decision vectors is defined using individual best solutions. The proposed modified algorithm as well as MATLAB program simplifies the earlier algorithm on ML-LFPP by eliminating solution preferences by the decision makers at each level, thereby avoiding difficulties associate with multi-level programming problems and decision deadlock situation. The proposed modified technique is simple, efficient and requires less computational efforts in comparison of earlier FGP techniques. Also, the proposed coding of generalized MATLAB program based on this modified approach for solving ML-LFPPs is the unique programming tool toward dealing with such complex mathematical problems with MATLAB. This software based program is useful and user can directly obtain compromise optimal solution of ML-LFPPs with it. The aim of this paper is to present modified FGP technique and generalized MATLAB program to obtain compromise optimal solution of ML-LFP problems in simple and efficient manner. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed modified approach and to demonstrate functionality of MATLAB program.

Multi-item inventory models with two storage facility and bulk release pattern are developed with linearly time dependent demand in a finite time horizon under crisp, stochastic and fuzzy-stochastic environments. Here different inventory parameters--holding costs, ordering costs, purchase costs, etc.--are assumed as probabilistic or fuzzy in nature. In particular cases stochastic and crisp models are derived. Models are formulated as profit maximization principle and three different approaches are proposed for solution. In the first approach, fuzzy extension principle is used to find membership function of the objective function and then it's Graded Mean Integration Value (GMIV) for different optimistic levels are taken as equivalent stochastic objectives. Then the stochastic model is transformed to a constraint multi-objective programming problem using Stochastic Non-linear Programming (SNLP) technique. The multi-objective problems are transferred to single objective problems using Interactive Fuzzy Satisfising (IFS) technique. Finally, a Region Reducing Genetic Algorithm (RRGA) based on entropy has been developed and implemented to solve the single objective problems. In the second approach, the above GMIV (which is stochastic in nature) is optimized with some degree of probability and using SNLP technique model is transferred to an equivalent single objective crisp problem and solved using RRGA. In the third approach, objective function is optimized with some degree of possibility/necessity and following this approach model is transformed to an equivalent constrained stochastic programming problem. Then it is transformed to an equivalent single objective crisp problem using SNLP technique and solved via RRGA. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.

A fuzzy goal programming approach for vendor selection problem in a supply chain

The hybrid metaheuristics algorithms (HMHAs) have gained a considerable attention for their capability to solve difficult problems in different fields of science. This chapter introduces some applications of HMHAs in solving inventory theory problems. Three basic inventory problems, joint replenishment EOQ problem, newsboy problem, and stochastic review problem, in certain and uncertain environments such as stochastic, rough, and fuzzy environments with six different applications, are considered. Several HMHAs such as genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO), harmony search (HS), variable neighborhood search (VNS), and bees colony optimization (BCO) methods are used to solve the inventory problems. The proposed metaheuristics algorithms also are combined with fuzzy simulation, rough simulation, Pareto selecting and goal programming approaches. The computational performance of all of them, on solving these three optimization problems, is compared together.

Application of fuzzy goal programming and genetic algorithm is considered in this paper. We extend a multi-objective model for integrated inventory-production-distribution planning in supply chain (SC). We consider a supply chain network which consists of a manufacture, with multiple plants, multiple distribution centers (DCs), multiple retailers and multiple customers. The manufacturer produces several items. Decision maker’s imprecise aspiration levels of goals are incorporated into model using fuzzy goal programming approach. Due to the complexity of problem in large size and in order to get a satisfactory near optimal solution with great speed, a new genetic algorithm is proposed to solve constrained problems. To show the efficiency of the used model and fuzzy goal programming approach and genetic algorithm for the collaborative inventory-production-distribution problem, computational experiments are performed on a hypothetically constructed case problem.

In countries around the world, whether developed or developing, there is increasing competition in the industry to strengthen their economies and waste. Managing the supply chain of many goods becomes an essential part of transportation. Along with this, in inventory management, products are most likely to require maintenance, and the production and transportation work necessitate different handling procedures. When a product is manufactured using raw materials in a production house, the manufacturer checks the inventory by producing for a certain time interval. Only some unusable products or products sent by the retailer that cannot be used are sent to the production house again for new products. This study develops an integrated green design production model for deteriorating items with reverse logistics and preservation technology investment under inflation. Firstly, the model is developed in a crisp environment, and then it is extended to a fuzzy environment. Optimal solutions are established numerically, and comparative analysis is performed to validate the result. A comparison study is provided. Finally, sensitivity analysis is done for some influential parameters. It is observed that in each part of the supply chain, the deterioration rate for perishable items is almost uncertain due to several reasons. For that, special methods are used here to reduce the total cost. Results show that the preservation model under a fuzzy environment provides 0.76% less cost than a crisp environment. The cycle time of the retailer and the production time of the producer is less in a fuzzy environment than in a crisp environment.

This chapter describes a Genetic Algorithm (GA) based Fuzzy Goal Programming (FGP) model to solve a Multiobjective Bilevel Programming Problem (MOBLPP) with a set of chance constraints within a structure of decentralized decision problems. To formulate the model, the chance constraints are converted first to their crisp equivalents to employ FGP methodology. Then, the tolerance membership functions associated with fuzzily described goals of the objective functions are defined to measure the degree of satisfaction of Decision Makers (DMs) with achievement of objective function values and also to obtain the degree of optimality of vector of decision variables controlled by upper-level DM in the decision system. In decision-making process, a GA scheme is adopted to solve the problem and thereby to obtain a proper solution for balancing execution powers of DMs in uncertain environment. A numerical example is provided to illustrate the method.

ABSTRACTA stochastic and fuzzy chance-constrained programming (SFCCP) model was developed in this study for supporting energy–environment management in the city of Beijing, China. SFCCP was capable of tackling the variables in constraints as fuzzy random variables, which were integration of randomness and vagueness. SFCCP was applied to an energy–environment management system in the city of Beijing. The study results indicated that SFCCP was useful in helping decision makers gain in-depth insights into proposed management system and establish environment-friendly energy allocation alternatives. The application of SFCCP is expected to provide a good demonstration to energy–environment management problems under complex uncertainties.

To tackle the issue of operational flexibility degradation of integrated energy system (IES) due to the random charging behavior of massive electric vehicles (EVs), a chance constrained programming (CCP)-based optimal dispatch model of IES with EVs is proposed to ensure the economical and efficient operation. First, an optimal dispatch model with the objective of minimizing system operation cost and environmental protection cost is constructed, and the system operation constraints and the chance constraint that the random charging power of EVs does not exceed their standby capacity are integrated to improve the economic and environmental benefits of IES. Then, the above chance constraint is transformed into a deterministic equivalence class according to a predefined confidence level based on the deterministic equivalence method of chance constraint. Further, to overcome the problem of high complexity of the chance constrained transformation programming model, the alternating directional multiplier method (ADMM) with good convergence is used to obtain high-quality solutions. Finally, simulation results show that the method enable effective response to the randomness of EVs charging behavior and improve the economy and flexibility of system operation.

This paper explores the application of fuzzy goal programming (FGP) for modeling and solving land allocation problems with chance constraints, aimed at optimizing the production of seasonal crops within agricultural systems in uncertain environments. The decision-making environment in agricultural systems is often characterized by imprecision, particularly due to unpredictable rainfall patterns and limited availability of irrigation water, which are influenced by socio-economic factors. These uncertainties pose significant challenges in real-world agricultural scenarios. The proposed model incorporates fuzzy descriptions for the utilization of cultivable land, farming resources, and the achievement of production goals for seasonal crops. Water supply, as a critical resource, and socio-economic constraints are represented probabilistically within this uncertain decision-making framework. The model addresses land-use planning for the cultivation of five major crops—Paddy, Wheat, Mustard, Potato, and Pulses—across three crop cycles: Pre-Kharif, Kharif, and Rabi, throughout the planning year for the Bardhaman district of West Bengal, India. The solution process focuses on maximizing the membership value (unity) of the fuzzy goals, reflecting the aspirations of the decision maker (DM) as closely as possible, based on their needs and objectives. The potential of this approach is demonstrated through a case study of Bardhaman district, West Bengal, India, illustrating how FGP can be used to address agricultural planning challenges under uncertain conditions.

In this paper we propose a mathematical model for diabetes and pre-diabetes in fuzzy environment. Now due to the presence of fuzzy variable, the model is transformed to a fuzzy differential equatio...