Interval Order Relations Research Articles

A comparative analysis of metaheuristic algorithms in interval-valued sustainable economic production quantity inventory models using center-radius optimization

The real-world production inventory systems involve uncertainties surrounding demand, production, defectiveness, and costs, which pose significant challenges. Various methodologies, including interval, fuzzy, stochastic, and fuzzy-stochastic approaches, have been developed to address these challenges. Among these, the interval approach offers a realistic representation of uncertainty. This study develops a green production model within an interval-based framework, incorporating interval-valued representations of defective rates and demand, which is also stochastic in nature. Differential equations governing inventory levels are formulated in an interval format and solved using advanced parametric techniques. The study extends to profit optimization within this interval-based framework, with the profit maximization problem transformed into a crisp form using interval order relations and center-radius optimization. The optimized solution is obtained through various metaheuristic algorithms. A numerical example validates the proposed model, and sensitivity analyses explore variations in different algorithms and inventory parameters. Additionally, statistical analysis using ANOVA tests is performed. This research contributes to production inventory management by providing a robust framework for handling uncertainty and optimizing performance in real-world scenarios.

Collaborative Planning of Distribution Network, Data Centres and Renewable Energy in the Power Distribution IoT via Interval Optimization

With the development of the power distribution Internet of Things (IoT), the escalating power demand of data centers (DCs) poses a formidable challenge to the operation of distribution networks (DNs). To address this, the present study considers the operational flexibility of DCs and its impact on DNs and constructs a collaborative planning framework of DCs, renewable energy sources (RESs), and DNs. This framework employs the interval optimization method to mitigate uncertainties associated with RES output, wholesale market prices, carbon emission factors, power demand, and workloads, and the collaborative planning model is transformed into an interval optimization problem (IOP). On this basis, a novel hybrid solution method is developed to solve the IOP, where an interval order relation and interval possibility method are employed to transform the IOP into a deterministic optimization problem, and an improved integrated particle swarm optimization algorithm and gravitational search algorithm (IIPSOA-GSA) is presented to solve it. Finally, the proposed planning framework and solution algorithm are directly integrated into an actual integrated system with a distribution network and DC to verify the effectiveness of the proposed method.

Impact of Carbon Emissions and Advance Payment on Optimal Decisions for Perishable Products via Parametric Approach of Interval

Under the heading of sustainability challenges, this study incorporates payment procedures and inventory selections. The article’s goal is to acquire insight into how the methods of payment affect perishable product inventory selections under the widely utilised carbon tax regime in view of policies to reduce emissions. Uncertainty arises as a natural consequence of the unpredictable behaviour of customers. Keeping these impacts in mind, an interval-valued inventory model is introduced, where all the related inventory parameters are considered interval-valued. In addition, the vendor offers an interval-valued discount rate to customers against advance payment. Due to the inventory parameters being chosen as interval-valued, the objective function is changed into an interval-valued form, and the corresponding differential equation is changed into an interval differential equation. To solve the interval-valued differential equation, a parametric approach to interval is introduced, and the corresponding interval-valued objective function is constructed. Interval order relations and the MATHEMATICA software are used to solve the objective function with interval values. To assess the validity of the proposed model, one numerical example is solved, and a sensitivity analysis of the ideal course of action is performed.

Some well known inequalities on two dimensional convex mappings by means of Pseudo $ \mathcal{L-R} $ interval order relations via fractional integral operators having non-singular kernel

<abstract><p>Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fejér, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines.</p></abstract>

OPTIMAL POLICY OF THE INTERVAL EPQ MODEL USING C-L INTERVAL INEQUALITY

The objective of this work is to study the optimal policy of the classical economic production quantity (EPQ) model under interval uncertainty using interval inequality. To serve this purpose existing arithmetic mean-geometric mean (AM-GM) inequality is extended for interval numbers using c-L interval order relation. Then, using the said AM-GM interval inequality, the optimal policy of the classical EPQ model in the interval environment is developed. Thereafter, the optimality policy of the classical EPQ model in a crisp environment is obtained as a special case of that of the interval environment. Finally, all the optimality results are illustrated with the help of some numerical examples.

Interval valued inventory model for deterioration, carbon emissions and selling price dependent demand considering buy now and pay later facility

Uncertainty refers to a lack of precise knowledge or information about a particular event, situation, or outcome. It is an inherent characteristic of many real-world phenomena and is often associated with the presence of risk or ambiguity. Uncertainty can arise due to various factors, such as incomplete information, unpredictability, complexity, or variability of a system or process. In many cases, uncertainty can lead to difficulties in decision-making and pose challenges in planning, forecasting, and managing risks. There are different types of uncertainty, including epistemic uncertainty, which arises from the inherent randomness or variability of a system or process. This study encompasses the inclusion of all inventory parameters as uncertainty parameters, which are expressed in the form of intervals. Furthermore, demand for the product is taken into account in interval form. The conversion of the differential equation into an interval differential equation is carried out in order to account for the variability in demand within a certain interval. The utilisation of the centre-radius technique leads to the derivation of the related optimisation problem under a ‘buy now, pay later’ (BNPL) payment scheme. In order to demonstrate the concept, a numerical case is selected and solved with the MATHEMATICA software and interval order relations. In the end, a comprehensive sensitivity analysis has been conducted to derive insightful conclusions pertaining to this study.

Inventory model for perishable items for interval-valued price dependent demand and advance payment policy with parametric approach via particle swarm optimization

ABSTRACT Advance payment for replenishment and the lifetime of an item are the most vital issues in the business sector. With the consideration of these issues, the formulation of a non-deterministic inventory model in the interval environment is of great importance. In this work, two different non-deterministic inventory models are formulated along with a variable holding cost, variable demand, and advertisement frequency under an advance payment policy in the interval environment. In the first inventory model, a no-shortage situation is discussed, whereas in the second model, a stock-out scenario is adopted by partially backordering during modelling. The modelling framework for both problems is accomplished based on interval differential equations. The interval optimization problems related to both models are obtained utilizing parametric approach and interval mathematics. Then, using interval order relations and different types of quantum particle swarm optimization, both interval optimization problems are solved. By means of two examples, the authentication of both models is examined and the benefits of the proposed approach are demonstrated. The impact of deviations of various parameters on optimality is studied graphically through the performance of sensitivity analyses, and the work is concluded by providing potential management insights and future scopes.

An interval particle swarm optimization method for interval nonlinear uncertain optimization problems

For nonlinear optimization problems involving interval variables or parameters, the interval possibility degree or interval order relation is conventionally adopted to convert them into a deterministic corresponding counterpart and then can be solved with linear programing approaches, through which computational resources and costs have been preserved to a certain extent. However, some information and computational accuracy will be discounted during the model transformation. In this paper, an interval optimization algorithm based on particle swarm optimization is proposed aiming to cut down the loss of useful information by means of straight optimization instead of model deterministic converting so as to enhance calculation accuracy with few growths of computing expenses. The proposed method firstly employs an interval satisfaction value to cope with the constraints. Consequently, the particle swarm optimization method is used to seek the optimum solution sets without the model deterministic converting. And a criterion based on the satisfaction value model of interval possibility degree plays the role in individual selecting out of the above solution sets. Finally, the effectiveness of the proposed method is verified by investigating four examples.

Optimal configuration planning of vehicle sharing station-based electro-hydrogen micro-energy systems for transportation decarbonization

With the popularization of electric vehicles and hydrogen fuel vehicles, establishing an eco-friendly and economic-affordable fuel supply system turns out to be critical for the decarbonization of transportation sector. To achieve this, this paper proposes a systematic methodological framework to facilitate optimal configuration planning of an electric-hydrogen micro-energy system with vehicle sharing stations (EHVS). The proposed approach is based on the notion of integrated resource planning, which combines supply side with demand side simultaneously. The problem is formulated as a two-stage hybrid stochastic-interval programming, where the first-stage is aimed at minimizing the investment cost by determining the optimal configuration planning, while the second-stage is aimed at maximizing the expected benefits through optimal operation simulation of the system. The potential flexibility of vehicular demands in response to the rental price-setting has been explicitly considered in this work and represented using an evolutionary game model. Through this, the substitutability between different vehicle demands and its impact on the long-term planning of electric-hydrogen integrated energy system can be properly captured. A hybrid interval-stochastic method is presented to handle different types of uncertainties involved in the discussed problem, wherein the scenario analysis, the interval order relation and the fuzzy preference methods are jointly utilized. The resultant problem is solved with a genetic-algorithm based framework, and implemented in the MATLAB environment. The simulation results show that, compared with conventional independent planning, there could be an improvement in the carbon emission reduction by 24.74% and an additional cost saving of 11.97% achievable using the proposed approach.

Necessary optimality conditions for approximate Pareto efficient solutions of nonsmooth fractional interval-valued multiobjective optimization problems

This paper deals with approximate Pareto efficient solutions of a nonsmooth fractional interval-valued multiobjective optimization. We first introduce some types of approximate Pareto efficient solutions of the considered problem by considering the lower-upper interval order relation. Then we apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions of Karush–Kuhn–Tucker (KKT)-type for these solutions.

An application of real coded Self-organizing Migrating Genetic Algorithm on a two-warehouse inventory problem with Type-2 interval valued inventory costs via mean bounds optimization technique

Hybrid meta-heuristic algorithms play significant role for solving highly non-linear and non-convex optimization problems in uncertain environment because any hybrid algorithm is more efficient and robust as well as takes less computational cost than the basic algorithms. The objective of this work is to implement hybrid real coded Self-organizing Migrating Genetic Algorithm (RCSOMGA) in inventory control problems with Type-2 interval uncertainty. To achieve this goal, in this work, a two-warehouse inventory problem is modelled for deteriorating commodities under preservation technology with Type-2 interval valued inventory costs. Then the corresponding Type-2 interval valued average profit of the proposed model is obtained using interval mathematics. Thereafter, using Type-2 interval order relations, mean bounds optimization technique is established to maximize the average profit. To illustrate the proposed model, six numerical examples are considered and solved by the hybrid algorithm RCSOMGA. Also, the same numerical examples are solved by some of the state-of-the-art meta-heuristic algorithms and the obtained results are compared with the results obtained by RCSOMGA. Then ANOVA and non-parametric statistical tests are carried out to show the significance of hybrid RCSOMGA. Numerical comparison and statistical analysis of the computational results demonstrate that hybrid RCSOMGA outperforms than other existing algorithms. Finally, sensitivity analyses are carried out to establish the impacts of the inventory parameters on the optimal policy of the proposed model and the work is concluded with some managerial insights.

Theoretical developments and application of variational principle in a production inventory problem with interval uncertainty

The goal of this work is two folded: (i) theoretical development of optimality condition of a variational problem with interval uncertainty; (ii) application of the established results in a production inventory model with interval uncertainty. To serve the purpose of first fold, the definitions of optimizers of an interval valued variational problem (IVVP) are proposed by using interval order relations. Then both the necessary and sufficient optimality conditions of an IVVP with single as well as several variables are derived. In this derivation, the obtained necessary conditions are named as c-r Euler equations and sufficient conditions are named c-r as Legendre conditions. To achieve the purpose of second fold, a production inventory model is formulated by considering interval valued time dependent production and demand rates. Then the corresponding IVVP of the proposed model is obtained and its optimal policy is studied with the help of established c-r Euler’s equations and c-r Legendre conditions. Finally, to illustrate the optimal policy of the proposed model, a numerical example is considered and optimality of its obtained results are verified graphically.

Optimality theory of an unconstrained interval optimization problem in parametric form: Its application in inventory control

The aim of this work is to establish the optimality conditions of an unconstrained interval optimization problem in parametric form. For this purpose, a new definition of interval order relation is proposed. With the help of the proposed order relation, the definition of maximizer and minimizer of an unconstrained interval optimization problem are introduced. After those the necessary and sufficient conditions for the optimality of both the interval maximization and interval minimization problems are derived separately. Then the established results are applied in the area of inventory control to solve a production inventory problem with time dependent demand in interval environment. Finally, the established results are justified by solving the said inventory problem numerically and the work is concluded with some important findings.

On approximate quasi Pareto solutions in nonsmooth semi-infinite interval-valued vector optimization problems

This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approximate (strictly) pseudo-quasi generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Mond–Weir type dual model in approximate form is formulated, and weak, strong and converse-like duality relations are proposed.

Robust optimization of EMU brake module based on interval analysis

PurposeThis study aims to research the influence of non-probabilistic design variables on interval robust optimization of electric multiple units (EMU) brake module, therefore obtain the reasonable of design variables of the EMU brake module.Design/methodology/approachA robust optimization model of the EMU brake module based on interval analysis is established. This model also considers the dimension tolerance of design variables, and it uses symmetric tolerance to describe the uncertainty of design variables. The interval order relation and possibility degree of interval number are employed to deal with the uncertainty of objective function and constraint condition, respectively. On this basis, a multiobjective robust optimization model in view of interval analysis is established and applied to the robust optimization of the EMU brake module.FindingsCompared with the traditional method and the method proposed in the reference, the maximum stress fluctuation of the EMU brake module structure is smaller after using the method proposed in this paper, which indicates that the robustness of the maximum stress of the structure has been improved. In addition, the weight and strength of the structure meet the design requirements. It shows that this method and model introduced in this research have certain feasibility.Originality/valueThis study is the first attempt to apply the robust optimization model based on interval analysis to the optimization of EMU structure and obtain the optimal solution set that meets the design requirements. Therefore, this study provides an idea for nonprobabilistic robust optimization of the EMU structure.

A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.

Decision support system for customers during availability of trade credit financing with different pricing situations

This study introduces an inventory system with a non-instantaneous deteriorating product with credit facility and variable demand depending on the selling price. Two different selling prices are considered in the deterioration and non-deterioration periods. Shortages are partially backlogged and dependent on the length of the customers’ waiting time upto the arrival of fresh lot. Alternative trade-credit policy is applied herein, and several cases, sub-cases and situations are investigated. The corresponding optimization problems of different cases, sub-cases and situations are solved using an interval-oriented multi-section technique with the help of interval mathematics and interval order relations. A numerical example with three different credit periods is studied and solved to validate the said problem. Also, two different case studies are investigated. Then to investigate the effect of changes of several system parameters on the optimal policy, post optimality analyses are performed.

Optimal Planning Design of a District-Level Integrated Energy System Considering the Impacts of Multi-Dimensional Uncertainties: A Multi-Objective Interval Optimization Method

Improving the utilization efficiency of renewable energy sources (RES) is an important task for the development of an integrated energy system (IES). To address this challenge, this paper proposes a novel multi-objective interval optimization framework for the energy hub (EH) planning problem from the perspective of the source load synergy, while considering the impacts of both supply- and demand-side uncertainties. For this aim, based on an in-depth analysis of the adjustable characteristics of various loads in EH and their effect on RES absorption, an interval model is first established to describe the responsiveness of users’ load demand to real-time energy price variations and its associated uncertainties. In view of the natural contradiction between the system’s economic and environmental benefits, a multi-objective interval optimization model for the EH planning problem is developed, wherein the minimization of the system’s economic costs and the maximization of the RES utilization rate are considered as the dual objectives to be optimized simultaneously. Moreover, this study takes into account the uncertainties of RES availability and demand-side behaviors by using interval numbers and properly considering their impacts in the context of long-term planning. According to the features of the proposed model, the interval order relation and possible degree method are jointly used to transform the model into a deterministic optimization problem first, and then an improved non-dominant sorting genetic algorithm is used to derive the optimal solution to the problem. The results show that the proposed method can effectively improve the economy of EH and the utilization efficiency of RES and flexibly meet different planning requirements, giving better engineering practicability.

Necessary and sufficient optimality conditions for non-linear unconstrained and constrained optimization problem with interval valued objective function

In the theory of non-linear constrained optimization problem, the most familiar Karush Kuhn Tucker (KKT) conditions play an important role. Basically, these conditions are necessary optimality conditions of a constrained optimization problem with equality and inequality constraints. The goal of this paper is to introduce the necessary optimality conditions (named as equivalent KKT conditions) of a constrained optimization problem with interval-valued objective function. Depending on the type of objective function and inequality constraints, all possible cases (i.e., objective function interval-valued and constraints are real-valued, objective function and constraints both are interval-valued, the objective function is real-valued and constraints are interval-valued) have been investigated to find the necessary optimality conditions. For this purpose, this paper deals with the necessary and sufficient conditions of unconstrained optimization problem with interval-valued objective function. Also, with the help of interval order relations the definitions of local and global optima of interval-valued function have been proposed. Finally, in order to illustrate the equivalent KKT conditions of the interval-valued constrained optimization problem and also the necessary as well as sufficient conditions of an unconstrained optimization problem, some numerical examples have been considered and solved.

An application of interval differential equation on a production inventory model with interval‐valued demand via center‐radius optimization technique and particle swarm optimization

Due to the fluctuation of market economy and uncertainty of customers' demand, it is quite difficult to develop an appropriate inventory model under uncertain situations. To overcome this difficulty, a food production model with preservation technology and credit-linked demand under default risk of capital in uncertain environment is developed with the help of parametric approach and interval mathematics. In this proposed model, all the inventory parameters, including demand rate, production rate and deterioration rate are considered as interval-valued. Because of the consideration of demand rate, production rate as well as deterioration rate as interval-valued, to represent the proposed model mathematically, the interval differential equations have been used. Solving these differential equations by using parametric approach, all the cost components and the corresponding average profit are obtained in the form of intervals. Therefore, the optimization problem of this model becomes interval-optimization problem. Then, to solve the interval-optimization problem, the center-radius optimization technique is established with the help of interval order relations. With the help of this technique, the interval-optimization problem is converted into crisp problem and then it is solved numerically by using different variants (Gaussian Quantum-behaved Particle Swarm Optimization, Weighted Quantum-behaved Particle Swarm Optimization, and Adaptive Quantum-behaved Particle Swarm Optimization) of Quantum-behaved Particle Swarm Optimization technique. Some real-life problems are considered and solved to justify the validity of the proposed model. The same model is also analyzed in crisp environment to verify the result of interval environment. Finally, the sensitivity analyses of both crisp and interval environments are performed separately with respect to the different system parameters.