The divergence measures play a crucial role in information theory and reliability theory by quantifying the difference between probability distributions, thereby capturing the underlying uncertainty and structural variation. Among the broad class of the phi-divergence measures, the relative entropy (or Kullback-Leibler divergence) has been extensively studied and widely applied in various disciplines. However, the Pearson chi 2-divergence measure though a notable member of this family has received comparatively less attention in both theoretical and applied contexts. In this study, we propose and investigate a weighted version of the Pearson chi 2-divergence for doubly truncated random variables. We establish conditions under which the proposed divergence measure uniquely identifies probability distributions and derive several useful bounds. We further explore how monotone transformations influence the proposed measure. A simulation study based on exponential distributions is conducted to examine the behavior of the weighted interval Pearson chi 2-divergence measure. The results highlight the utility of this measure in scenarios involving truncation and weighting. Further, we illustrate the practical relevance of the proposed measure through a real-world data set. Finally, we conclude by summarizing the key findings and outlining directions for future research.
This paper examines a two-player zero-sum discrete-time stochastic dynamic game distinguished by its hybrid structure. The system dynamics exhibit a Markovian-like property and is decomposed into two interacting sub-dynamics: a regular component and an impulsive component, each contributing to the evolution of the system state. Both players have access to two types of actions (regular and impulsive) and the activation of each sub-dynamic is governed by the type of action selected. Player 2 seeks to maximize an infinite-horizon discounted payoff, where the discount factor is allowed to depend on the entire history of state-action pairs. Conversely, Player 1 aims to minimize this payoff, thereby formulating a zero-sum game framework. The strategic interaction between the players leads to the notion of a zero-sum Nash equilibrium. Our analysis leverages dynamic programming techniques to (i) characterize the value of the game as a solution to the dynamic programming equation (DPE) associated to the game, and (ii) establish the existence of a zero-sum Nash equilibrium under suitable regularity conditions through the DPE. To demonstrate the applicability of the theoretical results, we conclude with a case study in the context of pollution management, illustrating the role of hybrid games in environmental policy design.
This study investigates how risk aversion characteristics of manufacturers in exporting countries and retailers in importing countries influence pricing, carbon reduction rates, order quantities, and utility within supply chains. Employing a combination of analytical modeling and numerical simulation, we establish baseline models under risk-neutral decentralized and centralized decision-making frameworks. Furthermore, three distinct scenarios are formulated: exclusive risk aversion by the manufacturer, exclusive risk aversion by the retailer, and mutual risk aversion by both entities. The findings indicate that centralized decision-making yields superior outcomes compared to its decentralized counterpart. The adoption of risk-averse tactics by manufacturers is associated with heightened carbon reduction rates and improved utility for the manufacturers themselves; however, this comes at the expense of retailer utility, which suffers from compressed profit margins. In contrast, risk-averse conduct by retailers mitigates utility risk for both parties but concurrently suppresses carbon reduction rates. When both parties act in a risk-averse manner, a decline is observed in both carbon reduction rates and the utility of manufacturers and retailers. While a stronger consumer inclination toward low-carbon products can boost market demand and enhance overall supply chain value, risk-averse postures adopted by manufacturers undermine their own profitability and negotiating leverage. Retailer risk aversion demonstrates a pattern of asymmetric risk propagation. Additionally, the analysis elucidates how carbon tariffs affect supply chain profitability and pricing dynamics in environments characterized by risk aversion.
We study observer-hider games for expansive dynamical systems. Given an expansive homeomorphism f : X -> X and an expansive constant delta, the observer chooses a set of observation times A subset of Z, while the hider chooses two distinct initial states. The observer wins if the two orbits are delta-separated at some time in A. We introduce the Banach-density observation cost beta(delta)(f) = inf{_d(B)(A) : A is an element of Sep delta (f)}, where Sep delta (f) is the family of schedules that detect every distinct pair at scale delta. Our main result shows that doubly asymptotic pairs give intrinsic obstructions to sparse observation. If F = W delta(x, y) is the finite witness pattern of such a pair, then every winning schedule must meet every translate of F, and beta(delta)(f) >= kappa(F) >= 1 / |f|>= 1/ diam(F) + 1, where kappa(F) is a computable finite-pattern covering density. We also develop finite-horizon games with observation costs, prove a mixed minimax theorem for a softened version, and obtain finite linear-programming and graph criteria for shifts of finite type. For the full shift, the exact value is beta(delta)(sigma) = 1/(2R+1).
In the e-commerce environment, the design of closed-loop supply chain networks faces the challenges of spatio-temp oral heterogeneity of demand and complexity of return flow. Considering at this problem, this paper proposes an innovative framework integrating mixed integer programming and intelligent optimization algorithms. The framework accurately predicts dynamic requirements through the GCN-GRU model, combines distributed robust optimization to deal with uncertainties, and introduces a cooperative game mechanism to achieve fair cost sharing. Moreover, it adopts a two-stage adaptive optimization strategy. The initial network scheme is generated in the first stage, and the ALNS algorithm is dynamically adjusted in the second stage. The experimental results show that the average total cost of the IMIP-IOA model proposed in this paper is 680.4 thousand yuan under the scale of 200 nodes, which is 5-10% lower than that of the comparison algorithm. When the demand fluctuates 20%, the cost volatility is only 4.7%, and the solution success rate still maintains 95% in a 2000-node large-scale network, which significantly improves the solution quality, robustness, and scalability. This study provides an effective decision support tool for e-commerce closed-loop supply chain, and its architecture integrating prediction, optimization and game provides a new idea for the optimization of complex logistics system.
This paper presents a green approach to optimizing freight train routes in rail networks, taking into account time windows, capacity limits, and operational realities. Our goal is to reduce total transportation and operating costs, minimize CO2 emissions, and enhance delivery reliability, which we see as a key social benefit. We handle uncertainties in things like demand, costs, capacities, and emission rates with a fuzzy robust method that keeps solutions stable even in tough scenarios. To solve this tricky, large-scale problem, we use multi-choice goal programming with utility functions (MCGP-UF) to balance conflicting objectives. For better results, we developed a hybrid algorithm, RDGA, combining red deer and genetic algorithms. We tested our MCGP-UF-RDGA method on simulated cases and a real-world rail freight network, including sensitivity and robustness analyses. The results show flexible routes that cut costs and emissions while improving on-time deliveries, giving rail planners a valuable tool for more sustainable and resilient systems.
The scope of this paper is the prediction of extreme stopping-time behaviour in the Collatz system, especially the statistical identification of integers whose normalized stopping time is unusually large. We study this problem from a probabilistic and statistical perspective. We first relate the phenomenon to the Collatz lattice, to a modified Collatz system with more regular convergence properties, and to the empirical density of long-orbit integers in finite windows. We then construct two logit models for extreme stopping-time behaviour using the modified-system stopping time and local spatial lags as predictors. The first model gives stable predictive performance near seventy percent across several scales, while the enhanced quadratic model improves both accuracy and specificity. We also introduce an interpretable decision-tree classifier for a moderately large stopping-time event, showing that lagged local information and normalized self-descent time contain substantial predictive signal. These results do not prove the Collatz conjecture, but they indicate that rare and moderately large stopping-time events have measurable local and dynamical structure that can be exploited statistically.
Intoday's large open-pit mines, the truck-shovel system sits at the heart of the upstream supply chain, directly feeding crushers, processors, stockpiles, and ultimately customers. Any delay or imbalance here quickly ripples downstream. To address this, we developed a high-fidelity, data-driven Agent-Based Simulation (ABS) framework that works hand-in-hand with simulation-optimization. The model treats trucks, shovels, and loading/dumping points as intelligent agents that react in real time to congestion, equipment readiness, changing mine geometry, and stochastic failures. By feeding the simulation with actual reliability data from Golgohar Iron Ore Mine No. 1, we captured real-world variability far more accurately than traditional deterministic or discrete-event approaches. The results are clear and actionable: the supply chain is mainly bottlenecked by truck availability. A coordinated strategy-raising truck availability by 30 % and shovel availability by 10 %-delivers a 31.4 % jump in total material moved, cuts average queue length by 13 %, and keeps the loading and hauling subsystems nicely balanced. Monte Carlo validation across 1,000 runs confirms these gains are statistically robust. In short, blending decentralized agent behavior, real operational data, and joint optimization gives mine managers a practical, reliable tool for strengthening the entire mining supply chain.
. In this paper we propose a system of ordinary differential equations to mathematically model the dynamics of the gender gap in labor markets. The model represents gender preferences and market dynamism as parameters that influence employment disparities within a population with multiple genders, not necessarily two. Using Brouwer's fixed point theorem and the PerronFrobenius theorem, we prove the system has a unique nontrivial equilibrium point in the region where the proportions of occupied jobs make sense. For cases where the market dynamism is the same for the entire population, using Gershgorin's circle theorem, we prove that this point is globally asymptotically stable. For cases involving two genders, using the Poincare-Bendixon theorem, we prove that for any set of parameters this point is globally asymptotically stable. This last result allows us to identify the conditions under which the system reaches asymptotic equality for two genders (see Equation (4)). Finally, we numerically approximate the parameters of a particular system using the number of people of working age, the number of employed people, and the number of job abandonment cases recorded by the INEGI 1rom the first quarter of 2009 to the first quarter of 2020.
. On April 2, 2025, U.S. President Donald Trump unilaterally announced the imposition of tariffs on countries exporting goods to the U.S. market. This paper examines whether the adoption of reciprocal tariffs constitutes an effective strategic response. Beyond political or diplomatic considerations, we address the issue from an economic perspective, focusing on welfare implications for consumers and firms in the responding country. We develop a game-theoretic model of bilateral trade in which countries choose between retaliatory and non-retaliatory trade policies. The static analysis characterizes Nash equilibria and their welfare properties, while the dynamic analysis-based on replicator dynamics-examines the stability and robustness of these equilibria under policy shocks. The evolutionary framework allows us to assess whether reciprocal tariffs emerge as persistent outcomes or whether economies converge toward cooperative trade regimes. Our results show that reciprocal tariffs do not necessarily constitute either a welfare-improving or a dynamically stable response, even when they appear strategically justified. Retaliation may therefore amplify welfare losses rather than mitigate them, highlighting the limitations of reciprocal trade policies.
. We investigate a location-price game between two firms competing in equilibrium within a two-dimensional Euclidean space modeled as an annulus. In this framework, the firms simultaneously choose their locations and set prices to attract a continuum of customers uniformly distributed over the annular market. We then extend the analysis to the case where customer distribution becomes non-uniform. Furthermore, we explore the problem of determining the optimal locations for the two firms under both uniform and non-uniform customer distributions. Our results show that under a uniform distribution, the firms optimally locate at opposite points on the outer boundary of the market to maximize their payoffs. However, under a non-uniform distribution where consumers are concentrated near the inner boundary, the firms' optimal locations shift inward depending on the radius of the inner circle.
. To address the quality deterioration and cost increase of fresh products during cold chain logistics distribution due to fluctuations in temperature and humidity and uncertainties in traffic conditions, this study constructs a multi-objective collaborative optimization model. First, the influence mechanism of the coupling effect of temperature and humidity on the spoilage rate of fresh products was analyzed, and a dynamic model was introduced to quantify the quality loss. Second, based on real-time traffic data, the driving speed was made dynamic, and a traffic condition influence function based on time windows was constructed. On this basis, a mixed integer programming model with a soft time window was established with the goals of the lowest total cost and the highest average product freshness. To solve this NP-hard problem, an adaptive large neighborhood search algorithm integrating the simulated annealing mechanism was designed. Finally, it was verified through standard example simulation and an actual case of a certain fresh food e-commerce enterprise. The results show that compared with the traditional genetic algorithm and the standard ant colony algorithm, the algorithm proposed in this paper improves the solution accuracy and stability by an average of approximately 12.5% and 18.7%, respectively. Meanwhile, the optimization plan considering the dynamic constraints of temperature, humidity, and traffic can reduce the total cost by 15.3% and lower the average cargo damage rate by 4.8 percentage points, providing a theoretical basis and decision support for the refined and intelligent management of fresh cold chain logistics.
. In recent periods, the focus on tourism economics' issues has covered environmental issues as one of the most relevant sources that concern the tourism-growth nexus. The substitution of energy sources and the mitigation of greenhouse gas emissions are the main objectives on the sustainable development path. It has been observed that an increase in pollution is detrimental to growth in tourism-specialized economies. However, the literature [21] shows that adequate technological shocks can alleviate or even eliminate this effect, leading to sustainable growth. This paper aims to model a tourism-specialized economy which internalizes environmental dynamics within the tourism-growth path on a DSGE framework, with a stochastic path for technology implementation to validate the latter empirical result. Results demonstrate that when pollution dynamics are governed by adaptive regimes and complemented by cumulative technological progress, tourism can support sustained economic growth without compromising environmental integrity, empowering public policymakers to align tourism with a sustainable development agenda.
