In this paper, we investigate inventory and order strategies of a two-echelon supply chain, which is composed of two unreliable suppliers that are subject to random disruption. We develop the gross weighted profit benchmark model and the service level constrained model of the supply chain, respectively. We derive the retailer’s optimal order quantity and analyze the retailer’ optimal order policy and also obtain the analytical closed-form solutions. In addition, some numerical examples are provided to illustrate the effect of disruption time, disruption probability and fill rate on the optimal decisions and expected profit.

Based on two substitute products, we study the inventory and contract coordination strategy of a three-echelon supply chain, which consists of two suppliers, a manufacturer and a retailer, under supply disruption and stochastic demand. We investigate the channel gross profit model of the centralized supply chain and obtain a unique optimal order quantity. Under a decentralized decision, we find that the commonly-used wholesale price contracts cannot coordinate the system. Then, we propose a buy-back contract and prove that this contract can more efficiently coordinate the system than the former. At last, we show that the integrated performance of the decentralized system can be maximized through choosing the buy-back parameter by the manufacturer and reveal that the effects of supply uncertainty and the substitution behavior of customers on the optimal decision by numerical examples.

This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

This paper deals with stochastic comparisons of two parallel (series) systems with Type II half logistic-resilience scale (TIIHL-RS) distribution components with different baseline distribution functions. Under the conditions of interdependency and independency, the research shows that the system performance is better (worse) with the stronger component heterogeneity in the parallel (series) system under the usual stochastic order and the (reversed) hazard rate order.

In recent years, the study of variability in uncertainty measures has gained significant interest in information theory. In this context, the concept of varextropy has been introduced and investigated by several researchers. This paper focuses on the study of weighted varextropy and introduces weighted residual varextropy, presenting a thorough examination of their theoretical properties. Specifically, we investigate the impact of monotonic transformations on these measures and derive various bounds. Additionally, we analyze the application of weighted varextropy within coherent systems and proportional hazard rates models. A non-parametric kernel-based method is introduced to estimate weighted (residual) varextropy, and the estimators' consistency and asymptotic normality are given. Finally, the effectiveness of this estimation method is demonstrated through simulations as well as analysis of a real-world data set, illustrating its potential applications in uncertainty analysis.

In this paper, we investigate the problem of stochastically comparing the second-order statistics from dependent and heterogeneous samples following modified proportional hazard rates scale (MPHRS) and modified proportional reversed hazard rates scale (MPRHRS) models under Archimedean copula. We built some sufficient conditions for the usual stochastic order whenever the samples have different parameter vectors. Finally, some numerical examples were provided to illustrate the theoretical results.