This paper investigates higher-order optimality conditions for properly efficient solutions of $C^{1}$ multiobjective optimization problems with inequality constraints. By employing generalized radial directional derivatives, we introduce a higher-order Robinson-type constraint qualification and discuss its relationships with the Mangasarian-Fromovitz and Kurcyusz-Robinson-Zowe constraint qualifications. These conditions are then employed to derive strong Karush-Kuhn-Tucker optimality conditions for Borwein properly efficient solutions, featuring \textit{strictly positive multipliers} for the objective functions. Under relaxed higher-order generalized convexity assumptions, we establish sufficient conditions for Geoffrion properly efficient solutions. Furthermore, we develop weak and strong duality results within the Mond-Weir and Wolfe duality frameworks, thereby extending classical duality theory to a higher-order setting. Several examples are provided to illustrate the advantages of our results over existing ones.
This paper develops a C2M supply chain model involving a C2M manufacturer, a brand manufacturer, and an e-commerce platform. The platform collects consumer data and provides demand information to the C2M manufacturer, who adjusts production accordingly. We incorporate overconfidence biases into this framework, where the C2M manufacturer overestimates the effectiveness of data services and the brand manufacturer overestimates market demand. Four scenarios are considered: rr (both manufacturers are rational), or (C2M manufacturer is overconfident), ro (brand manufacturer is overconfident), and oo (both manufacturers are overconfident). The results show that overconfidence has non-monotonic and asymmetric effects on supply chain outcomes. First, the C2M manufacturer’s overconfidence intensifies price competition and may reduce overall supply chain profit; only moderate overconfidence within a certain threshold improves its performance. Second, the brand manufacturer’s overconfidence exhibits an inverted U-shaped effect: while excessive overconfidence harms its own profit, moderate overconfidence increases not only its profitability but also that of the C2M manufacturer and the e-commerce platform, thereby promoting a “win-win-win” outcome. Third, a higher data service level provided by the platform raises the retail prices of both products, enabling the platform to obtain higher data service fees and commission revenue. Overall, our findings highlight the important role of behavioral bias and platform strategy in shaping equilibrium decisions and profit distribution in C2M supply chains.
A linear system is a pair ( P , ℒ ) where ℒ is a family of subsets of a ground finite set P such that | l ∩ l ′|≤1, for every pair of different subsets l , l ′∈ ℒ . The linear system ( P , ℒ ) is intersecting if | l ∩ l ′|=1 for every pair of distinct subsets l , l ′∈ ℒ . If every member of ℒ has k elements, the linear system is called k -uniform linear system. The transversal number of a linear system ( P , ℒ ), τ ( P , ℒ ), is the smallest cardinality of a subset T ⊆ P satisfying l ∩ T ≠∅, for every l ∈ ℒ . Let I k be the class of all intersecting k -uniform linear systems. In this note, we deal with the problem of estimating the constant i k (which only depends of k ) such that τ ( P , ℒ )≤ i k (| P |+| ℒ |), for all ( P , ℒ )∈ I k . We establish an upper bound for i k , for all k ≥2 integer. We show that i k < 2/k+3.
Modern supply chains are increasingly required to balance multiple, often conflicting objectives such as economic efficiency, environmental responsibility, blended-product quality, and delivery performance, all under significant uncertainty. Ensuring the consistent quality of blended products is particularly critical in sectors such as energy, food processing, and chemicals, where multiple raw materials must be combined in precise proportions to satisfy regulatory and performance requirements. Achieving high-quality blending while minimizing transportation costs and environmental impact becomes considerably more complex when products flow through multiple stages and transport modes. This paper proposes a comprehensive optimization framework for a multi-stage green transportation problem with product blending, spanning the flow from suppliers through manufacturers and retailers to end consumers. The model incorporates alternative transportation routes and modes—such as trucks, rail, and maritime transport—each exhibiting distinct cost, time, and emission characteristics. By jointly integrating blending decisions with route and mode selection, the framework captures the real trade-offs among economic, environmental, and quality-oriented objectives. The formulation seeks to ensure that the final blended product satisfies required specifications while simultaneously minimizing (i) total transportation cost, (ii) carbon emissions, (iii) product degradation, and (iv) delivery time. Uncertainty associated with transportation and blending operations is represented using a Fermatean fuzzy framework, where key parameters—including transportation costs, emissions, blending quality, travel times, and the total transported or blended amount are modeled as Triangular Fermatean Fuzzy Numbers (TrFFNs). To enhance tractability, a ranking index is developed to transform the fuzzy multi-objective problem into an equivalent deterministic formulation, enabling the use of efficient optimization algorithms while preserving the inherent uncertainty in the system. To generate Pareto-optimal solutions, several multi-objective optimization techniques are employed, including fuzzy TOPSIS, the ε-constraint method, the augmented Tchebycheff approach, and weighted Tchebycheff metric programming. A TOPSIS-based ranking algorithm is then applied solely to rank and compare the four resulting solution methods. A real-worldinspired case study demonstrates the applicability of the proposed framework for identifying energy-efficient routes, optimizing blending strategies, and enhancing operational performance under uncertainty. The results confirm that the proposed methodology constitutes a robust decision-support tool for advancing sustainable and resilient supply chain management.
The carbon cap-and-trade mechanism is increasingly influencing the online selling channel selection and pricing strategy in an e-commerce platform supply chain with two different sales channels (agency selling and reselling channels). This paper examines how two carbon allowance allocation rules (grandfathering and benchmarking) influence channel selection, pricing and profitability in a platform supply chain consisting of one supplier and one e-commerce platform. Using game-theoretic models under four scenarios arising from two carbon allowance allocation rules and two sales channels, we find that the benchmarking rule consistently yields lower retail prices, higher demand, and greater consumer surplus compared to grandfathering. However, the preferred carbon allowance allocation rule differs across stakeholders. In agency channel, the platform’s profit is lower (higher) under benchmarking with the higher (lower) supplier’s carbon allowance compared to grandfathering. In reselling channel, the platform consistently benefits more from benchmarking. Under agency channel with a high (low) platform fee, the government prefers the grandfathering (benchmarking) when the grandfathering allowance is high (low). Under reselling channel, the government prefers the benchmarking (grandfathering) when the benchmarking allowance is high (low).
Cross-border e-commerce has shifted product sales from traditional offline retail to a dual-channel model combining an offline retailer and a cross-border e-commerce platform, intensifying competition between channels and raising issues of incomplete consumer trust in product quality. We develop a game model where an overseas manufacturer distributes products through both channels while competing with a competitive manufacturer. With varying consumer trust in product quality across multiple channels, we investigate the overseas manufacturer’s blockchain adoption strategy and the value of blockchain. First, we find that when the overseas manufacturer’s combined tax and shipping cost is high, blockchain adoption depends on a low blockchain cost. Otherwise, a moderate blockchain cost is necessary. Second, the offline retailer benefits from blockchain if and only if there is a high channel quality variation level, a high blockchain cost, or a low overseas manufacturer’s combined tax and shipping cost, while the competitive manufacturer benefits if and only if the channel quality variation level is low and either the blockchain cost is high or the overseas manufacturer’s combined tax and shipping cost is low. Finally, rising blockchain cost decreases the value of blockchain for the overseas manufacturer yet enhances it for the offline retailer and the competitive manufacturer. Interestingly, as the product quality variation level decreases, the value of blockchain for an offline retailer reduces under a high platform commission rate but increases otherwise. Our analysis provides insights into when the overseas manufacturer should adopt blockchain and the value of blockchain considering competition and consumer trust.
We propose two different models of government subsidy mechanisms in a scenario that considers consumers’ fairness concerns: the government only subsidizes enterprises for their corporate social responsibility efforts (E-strategy), and the government not only subsidizes enterprises for their social responsibility efforts but also subsidizes consumers with consumption coupons (CE-strategy). We introduce the government as the decision-making subject to analyze and compare the impact of two different subsidy strategies on the supply chain. The results indicate that (1) government subsidies are effective only if the enterprises’ CSR efforts satisfy certain conditions; (2) when the coefficient of consumers’ fairness concerns is small, the government tends to adopt the E-strategy, consumers’ fairness concerns always exert a detrimental impact on the supply chain, and government subsidies can adjust this impact to enterprises; (3) when the coefficient of consumers’ fairness concerns is larger, the overall social benefit brought by CE-strategy is more significant; (4) different subsidy strategies bring different regulatory effects, the government can choose its subsidy strategy according to the focus of the overall social utility or the regulatory effect on the market.
The encroachment of a remanufacturer not only intensifies market competition but also affects the manufacturer's adoption of Design for the Environment (DfE) in its product manufacturing process. Although the manufacturer typically holds an information advantage regarding market demand and is therefore expected to respond readily to the remanufacturer's encroachment, the specific interaction mechanisms remain unclear. Motivated by this observation, we examine the implications of remanufacturer encroachment in the presence of both DfE and asymmetric demand information. By developing a signaling game model, we derive several insightful findings. First, contributing to the ongoing debate on the impact of remanufacturing, this study supports the negative view by demonstrating that encroachment by an independent remanufacturer (IR) consistently reduces the profit of the independent manufacturer (IM), irrespective of the information structure. Second, contrary to conventional wisdom, the IR may benefit from being informationally disadvantaged, achieving higher profits than under a full-information scenario. This finding offers a novel perspective for the informed IM: by sharing information, the IM can prevent the IR from making quantity decisions that would inadvertently intensify competition. Third, through numerical analysis with data from setting values that are in line with reality to parameters, we find that the dual signal mechanism may be more beneficial for the IM than the single signal mechanism. Taken together, these findings underscore the need for remanufacturing stakeholders to carefully evaluate the multifaceted effects of DfE on IR behavior and to fully account for the critical role of information structure in strategic interactions.
Medical diagnosis problems often involve uncertainty, vagueness, and linguistic evaluations provided by experts, which makes accurate decision-making challenging. Therefore, developing advanced fuzzy decision-making frameworks is important for improving diagnostic reliability in healthcare systems. In this study, the q-rung orthopair hexagonal fuzzy set (q-ROHxFS), which generalizes intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and Fermatean fuzzy sets (FFS), is employed to address multi-attribute group decision-making (MAGDM) problems in medical diagnosis. The evaluations of decision makers regarding lung disease patients are expressed using linguistic variables and subsequently transformed into q-ROHxF numbers. These assessments are aggregated to construct a decision matrix for determining the optimal alternative. Based on the proposed fuzzy environment, three modified multi-criteria decision-making (MCDM) methods, namely q-ROHxF TOPSIS, q-ROHxF COPRAS, and q-ROHxF VIKOR, are developed. In addition, a novel defuzzification technique and a distance measure for q-ROHxF numbers are introduced to effectively rank the alternatives. To validate the effectiveness of the proposed framework, the obtained results are compared with four existing q-ROF based MCDM methods. Furthermore, rank correlation analysis is performed to examine the consistency and reliability of the proposed algorithms. Sensitivity analysis with different values of q is also conducted to demonstrate the stability of the optimal solution. The results indicate that the proposed q-ROHxF MCDM framework provides a flexible and reliable approach for medical diagnosis decision-making under uncertainty.
Due to data source and forecast competency differences, an increasing number of enterprises in hybrid platform supply chains implement forecast signal sharing to eliminate information asymmetry. Existing studies mostly examine vertical-sharing incentive of the platform retailer with suppliers, while neglecting horizontal sharing among suppliers. This paper explores the optimal bidirection signal-sharing strategy including vertical sharing of platform retailer and horizontal sharing between two suppliers adopting reselling channel and agency channel respectively to improve supply chain performance. By analyzing the established Bayesian Stackelberg game models, the study finds that the two types of suppliers are always willing to implement horizontal sharing, while the vertical sharing strategy of platform retailers depends on channel competition and signal correlation. Specially, when channel competition intensity and signal correlations are sufficiently high, the platform retailer would like to share signal only with the supplier adopting reselling channel, which is different from previous studies. Moreover, it is found that horizontal sharing weakens the vertical-sharing incentive of the platform retailer and reduces the benefits of vertical sharing for two suppliers. This paper enriches the existing literature by combining the research on horizontal and vertical sharing within the context of hybrid-channel platform supply chains, and systematically clarifies the impacts of channel competition and signal correlation on the optimal strategy. Some managerial insights and practical schemes derived from findings may guide enterprises in making decisions on effective bi-direction information cooperation to improve supply chain performance.
In modern commercial transactions, the use of permissible delay payment options is widespread. Furthermore, the implementation of more rigorous carbon restrictions is compelling enterprises to seek more effective methods of inventory management in order to minimize carbon emissions resulting from their operations. This study advances an inventory model that comprises trade credit resulting default risk for non-instantaneous deteriorated Items (NIDIs) under preservation technology investment (PTI), while also considering partial backlogging and carbon emissions. The model aims to optimize inventory decisions by balancing economic benefits and environmental responsibilities. Trade credit terms are used to stimulate demand but introduce default risk, which is managed alongside the challenges posed by non-instantaneous deterioration of items. PTI is explored as a means to (1) extend the non-deterioration period and (2) reduce the deterioration rate. Partial backlogging is included to handle shortages, while carbon emissions are accounted for under carbon cap-and-trade (CCAT) and carbon tax (CT) regulations. The proposed model provides a comprehensive framework for making informed inventory decisions that maximize profit while adhering to environmental regulations. Numerical examples are used to validate the model, offering managerial insights into the effective integration of these factors in inventory management.
This study develops an economic production quantity (EPQ) model for a single-item inventory with deterioration in a fuzzy environment. Conventional stochastic inventory models considered uncertainty in the demand functions; however, the majority of manufacturing systems experience supply (production) unavailability from time to time. The proposed Fuzzy Inventory Control (FIC) system considers both demand and supply as uncertain variables. As a result, the inventory level also remains uncertain. Uncertainty in production rate and inventory level is expressed by fuzzy valued functions of time. The unit density-dependent demand is taken as a fuzzy variable. The unit holding cost is considered as an uncertain variable and expressed by a fuzzy number. The total cost of production takes a known form, where the unit cost of production is also given by a fuzzy number. The deterioration rate obeys the generalized Pareto distribution with a negative shape parameter, which is given by a crisp function of time. The proposed fuzzy production inventory model has been solved by applying Pontryagin's fuzzy maximum principle. The optimal inventory level and optimal production level are determined by closed bounded intervals of real numbers where the left and right-hand functions of their α-levels are the parameters. Now, under a special choice of the demand, holding cost, and production cost, left-hand and right-hand functions of the α-levels of the optimal inventory and of the production level are determined. Graphical representations of both types of functions are obtained separately for different a-levels for a = 0.25, 0.5. 0.75, 1. Detailed explanations of graphical interpretations are provided in every instance.
Against the background of the low-carbon transition and rapid electrification of road transport, this study examines strategic interaction between an electric vehicle (EV) manufacturer and a fuel vehicle (FV) manufacturer under alternative government subsidy mechanisms. We compare three policy scenarios, i.e., no subsidy, a fixed lump-sum subsidy, and a quantity subsidy. The findings show that the fixed subsidy neither affects equilibrium prices nor quality investment, it only transfers profit to the EV manufacturer. The quantity subsidy increases quality investment and can improve profits for both manufacturers when the subsidy intensity is moderate. It further shows that subsidies become inefficient when the investment coefficient is low, and the effectiveness depends on the EV-FV cost difference. These results reveal analytical and managerial insights for the design of EV subsidy policies.
Given the uncertainty in market demand and the complexity of supply chain management for perishable vegetable products, traditional demand forecasting and pricing strategies face substantial challenges in practical applications. To address these issues, this study proposes a KAN-LSTM-based framework for demand forecasting and ordering optimization. By integrating the nonlinear representation capability of KAN with the long-term dependency modeling ability of LSTM, the proposed framework provides an effective approach for forecasting perishable vegetable sales and procurement prices. In addition, a multivariate stepwise regression model is employed to estimate price elasticity and support pricing decisions. The ordering strategy is then optimized using the McCormick envelope method combined with the SLSQP algorithm. Experimental results show that the KAN-LSTM model achieves high forecasting accuracy and stable performance in both sales and price prediction tasks. Compared with xLSTM, TCN, and Transformer models, KAN-LSTM demonstrates favorable predictive performance for the dataset considered in this study. The optimization results further indicate that the proposed framework is effective in improving profit and cost control, highlighting its practical value for perishable goods supply chain management in farmers' markets.
This study provides a systematic examination of the evolution and application of fuzzy multi-criteria decision-making (MCDM) methods in green supply chain management (GSCM) from 2011 to 2024. A dataset of 3,850 publications was initially retrieved, from which 104 relevant articles were selected and analyzed through a combination of bibliometric mapping and qualitative content analysis. The study aims to address the lack of an integrated synthesis that combines trend analysis with critical evaluation of fuzzy MCDM applications in GSCM. The analysis highlights an increasing reliance on fuzzy MCDM techniques—such as Fuzzy AHP, Fuzzy TOPSIS, DEMATEL, and hybrid approaches—for supporting decisions related to supplier selection, sustainability assessment, and risk mitigation. Key findings indicate that while methodological diversity is expanding, empirical validation and real-world impact assessments remain limited. Furthermore, a small number of influential authors and publication clusters dominate the discourse. By identifying methodological gaps and emerging patterns, this study contributes a structured framework for understanding the current state of research and offers clear directions for future studies aiming to enhance the practical effectiveness of fuzzy MCDM in sustainable supply chain design.
In this paper, using the properties of Clarke subdifferential, we derive some characterizations of locally Lipschitz geodesic pseudolinear functions on Hadamard manifolds. Moreover, employing these characterizations, we derive some properties of the solution sets of pseudolinear programming problems on the Hadamard manifold. The results established in this paper generalize and extend several known results in the literature. We furnish several examples to illustrate these results. Furthermore, to demonstrate the practical applications of our outcomes, we solve a specific form of the Karcher mean problem by employing the results established in this paper.
In the stochastic obstacle scene (SOS) problem (a continuous counterpart of the Canadian traveler’s problem), a navigating agent seeks a low-cost route from a source to a target in the presence of obstacles whose true/false status is initially uncertain. We study the discretized SOS setting and develop a family of penalty-based navigation heuristics that incorporate both sensor-derived obstacle probabilities and a disambiguation cost incurred upon querying an obstacle at its boundary. Our main methodological contribution is a parametric unification of classical penalty rules via the ACS( k ) family, together with norm-based approximations that relate ACS( k ) to existing penalties. We evaluate the resulting heuristics through extensive Monte Carlo experiments across sensor-accuracy regimes, disambiguation costs, and obstacle densities, and we summarize which penalty-growth behaviors are preferable under each regime. The results provide practical guidance for selecting a penalty-based navigation rule for a given sensor and operational cost configuration.
A vertex subset S of a graph G = ( V , E ) is said to be geodetic set if every vertex in V − S lies in some geodesic of any two vertices in S . The minimum cardinality of a geodetic set of G is called as geodetic number of G , and is denoted by g ( G ). A geodetic coloring of a graph G is a proper vertex coloring of G in which every vertex in at least one geodetic set of G receives a different color. The minimum number of colors used in a geodetic coloring of G is called the geodetic chromatic number of G , and is denoted by χ g ( G ). In this paper, we conduct a detailed study of χ g ( G ), presenting existence results, bounds in relation to other graph parameters, and characterizations of graphs for which χ g ( G ) = 2 and χ g ( G ) = n , where n is the order of the graph. Furthermore, we prove that for every tree T , χ g ( T ) equals the number of pendant vertices in T , unless T is a star or a path on odd vertices.
This study investigates an integrated production planning and scheduling problem within a complex industrial environment featuring non-identical parallel production lines. It focuses on reducing the impact of downtime, which incurs substantial setup costs. To this end, we introduce advanced transition mechanisms between production periods, motivated by real-world industrial practices, such as setup carryover and setup crossover, in order to enhance inter-period coordination, reduce operational disruptions, and improve overall system efficiency. A mixed-integer mathematical formulation is developed and solved using CPLEX for medium-scale instances, while a Simulated Annealing metaheuristic is designed to handle large-scale cases. The proposed solution framework yields realistic and cost-efficient plans that jointly minimize downtime-related costs, setup costs, production costs, and inventory holding costs.
This paper builds on our previous work by conducting a comprehensive bibliometric anal- ysis of Varroa mite research, addressing one of the most critical challenges in apiculture. Leveraging datasets and tools presented in our earlier conference paper, this study maps research trends, collab- oration networks, and technological advancements in Varroa mite management. Through an in-depth exploration of co-authorship patterns, keyword co-occurrence, and bibliographic coupling, the analysis uncovers the field's intellectual structure and collaborative landscape. Key findings highlight dominant themes such as chemical control methods, biological interventions, and the integration of advanced technologies like Artificial Intelligence (AI) and Internet of Things (IoT) for precise hive monitoring. The analysis also identifies emerging research regions and underrepresented countries, highlighting gaps in global collaboration and the need for a more inclusive research agenda. Furthermore, it underscores the ecological and economic urgency of combating Varroa infestations, especially in light of climate change, pollution, and habitat degradation. As a major conclusion, we propose the development of a new dataset with specific characteristics designed to enhance the detection and prediction of Varroa infestations. This dataset would support more precise monitoring through advanced Al models, en- abling proactive and sustainable control strategies. By offering a roadmap for integrating innovative technologies into apicultural practices, this study contributes to safeguarding pollinators, preserving biodiversity, and ensuring global food security.
