In many industrial practices, market demand is likely affected by the trade credit policy. This study aims to develop a retailersupplier non-cooperative replenishment model with credit length dependent demand and default risk to determine the best credit period in a supplier-Stackelberg game. The perks of credit policy include attracting new buyers and avoiding long-term price rivalry. This study was conducted under three distinct decision-making scenarios. The first section demonstrates the optimal outcomes for decentralized and centralized decisions when trade credit is unavailable. Second, using a supplier Stackelberg method, we construct the existence and uniqueness requirements for the retailer and the supplier optimal solutions. In this supplier Stackelberg strategy, the retailer sets the best lot-size, while the supplier chooses the best credit length. A series of theorems and corollaries were designed to determine the best response under these conditions. In order to demonstrate the proposed method and the anticipated outcomes, the sensitivity analyses of few examples has been carried out. Also the decision variable’s characteristics, obtained from the sensitivity analysis, have been figure out. Additionally, this research shows a high correlation between the demand, credit length and supply chain’s profit.

The flow shop scheduling problem is traditionally studied under the assumption that the optimal solution is a permutation schedule. However, for many practical problems, this may not be the case. Since finding a flow shop schedule is significantly simpler with the permutation consideration restrection, it becomes interesting to know a priori at what extent relaxing this constraint could be beneficial. In this paper, we study the conditions under which this assumption holds, particularly in flow shops with missing operations. We focus our analysis on the makespan and total completion time. Preliminary results show that, in certain cases, non-permutation schedules can yield significantly better outcomes.

This paper compares three classical lower bounds for the Parallel Machines Scheduling Problem (PMSP). The first lower bound considered is the makespan of Jackson’s Pseudo Preemptive Schedule (JPPS). The second is the preemptive bound, which is obtained by relaxing the non-preemption constraint, and the third is based on Energetic Reasoning (ER). To compare these three bounds, we first provide a rigorous mathematical characterization. Specifically, we introduce a mathematical definition of the ER-derived lower bound as the smallest value of the trial makespan that results in a successful ER feasibility test. This definition relies on the fundamental concepts of critical intervals and specific operations that we called crossing operations, for which we give some theoretical properties. We establish an Energy theorem, which provides an analytical formulation of the energetic lower bound. In an extension of the JPPS and ER lower bounds, we give an analytical formulation of the preemptive bound. From the analytical characterization of these three bounds, we attempt to analyze their relative performances. In particular, we find that the ER lower bound differs from JPPS because on some machines at least two operations must be scheduled for ER. The preemptive bound differs from JPPS because idle periods may be required in an optimal preemptive schedule. Interestingly, these three characterizations are quite similar, and so can provide insight into the ways in which the bounds differ. Our results can be directly extended to the cumulative scheduling problem, and we report some experimental experiments that compare these bounds and confirm the theoretical results stated above.

A graph G has a P≥k-factor if G has a spanning subgraph H such that every component of H is a path of order at least k. A graph G is P≥k-factor deleted if G − e has a P≥k-factor for each edge e ∈ E(G). In this paper, we study the P≥2-factor deleted graphs by their Q-index (the largest eigenvalue of the signless Laplacian matrix) and D-index (the largest eigenvalue of the distance matrix). Sufficient conditions (in terms of Q-index and D-index) to guarantee that a graph G is a P≥2-factor deleted graph are established.

Let G = (V, E) be a graph of order n. For S ⊆ V (G), the set Np(S) is defined as the perfect neighborhood of S such that all vertices in V (G)∖S have exactly one neighbor in S. The perfect differential of S is defined to be ∂p(S) = |Np(S)| − |S| and the perfect differential of a graph is defined as ∂p(G) = max{∂p(S) : S ⊆ V (G)}. A perfect Roman dominating function is defined as a Roman dominating function f satisfying the condition that every vertex u for which f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The perfect Roman domination number, denoted by γpR(G), is the minimum weight among all perfect Roman dominating functions on G, that is γpR(G) = min{w(f) : f is a perfect Roman dominating function on G}. Let G̅ be the complement of a Graph G. The complementary prism GG̅ of G is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. This paper is devoted to the computation of perfect differentials of complementary prisms G G̅ and perfect Roman domination numbers of complementary prisms G G̅ by the use of the Gallai-type result proven before. Particular attention is given to the complementary prims of special types of graphs. Furthermore, a sharp lower bound on the perfect differential of the complementary prism G G̅ of a graph G in terms of the order of G is presented and the graphs attaining this lower bound are characterized. Finally, the graphs are characterized for which ∂p(GG̅) and γpR(GG̅) are small.

Narushima et al. [SIAM J. Optim. 21(1), 212-230, 2011] proposed a three-term search direction that ensures the sufficient descent condition. In this study, we introduce a control factor to enhance the search direction, which not only possesses the sufficient descent property but also automatically incorporates the trust region feature, independent of the choice of line search. We then extend the modified search direction to solve constrained nonlinear equations using hyperplane projection technique. Moreover, we incorporate an inertial acceleration technique to develop an inertial version of the proposed method. The global convergence of the inertial method is theoretically established, even in the absence of Lipschitz continuity in the underlying mapping. Additionally, under suitable conditions, we prove its linear convergence rate. Numerical experiments demonstrate the excellent performance of our proposed methods in solving constrained nonlinear equations and restoring blurred images contaminated by Gaussian noise.

This work carried out an empirical study with the objective of determining the most efficient strategy in guaranteeing the radiality of the distribution network during the reconfiguration. The reconfiguration problem in a distribution network aims to reduce technical losses by changing the paths that energy takes through the network until it reaches consumers. In this problem, a mandatory constraint is to ensure that the network operates radially. However, the literature does not present a consensus on the best way to guarantee this restriction. Computational experiments were carried out using the two main methodologies in the literature: limiting the number of input edges; and search with removal of cycles in each partial solution. A third and new strategy was proposed as an alternative to the literature using the concept of minimal cuts in graphs. Computational tests showed that the strategy of limiting the number of input edges was the most efficient in ensuring radiality. We also point out a drawback in the applicability of this strategy to networks with high penetrations of distributed generation (DG). For this reason, our experiments also analyze the other strategies that are suitable for DG power flows. For this case, we show that the methodology proposed in this paper provides a better way to tackle modern networks arising from applications of smart grids.

This paper combines product design and patent protection and construct a dynamic decision-making model consisting of a single original manufacturer and a single remanufacturer based on Stackelberg game. We study the impact of different product design strategies on remanufacturing decisions in the presence of patent protection and explore the influencing factors of the original manufacturer’s product design level. The results show that: (1) the original manufacturer’s product design level is negatively correlated with the coefficient of R&D cost impact and the coefficient of remanufacturing cost, and positively correlated with the coefficient of consumer utility impact; (2) the unit patent licensing fee is positively correlated with the coefficient of R&D cost impact and negatively correlated with the coefficient of remanufacturing cost and the coefficient of consumer utility impact; (3) under different product design strategies, product design strategies that consider both eco-orientation and remanufacturing-orientation are most conducive to the stable development of the closed-loop supply chain.

‎A subset $S$ of vertices of a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$‎. ‎A dominating set $S$ is a restrained dominating set if for each vertex $x\in V(G)-S$ there is a vertex $y\in V(G)-S$ such that $xy\in E(G)$‎. ‎The restrained domination number of $G$‎, ‎denoted by $\gamma_r(G)$ is the minimum cardinality of a restrained dominating set of $G$‎. ‎The restrained domination stability number of $G$‎, ‎denoted by $st_{\gamma_r}(G)$‎, ‎is the minimum number of vertices whose‎ ‎removal changes the restrained domination number of $G$‎. ‎In this paper we study the restrained domination stability number of a graph and determine it for several families of graphs‎. ‎We present bounds for the restrained domination stability number of a graph‎. ‎We also prove that determining the Restrained domination stability number is NP-hard even when restricted to planar bipartite graphs‎.

An r-dominating set (r-total dominating set) of G is a subset S of V (G) for which Nr(u)∩S is non-empty for all u not in S (for all u in V (G)). An r-locating-dominating set (r-locating-total dominating set) of G is an r-dominating set (r-total dominating set) S of G for which Nr(u) ∩ S is different from Nr(v) ∩ S for all u and v not in S. This paper presents an extension of the locating-total dominating set of G. Further, we establish a lower bound on r-locating-dominating set and r-locating-total dominating set for k-regular graphs, as well as demonstrate that r-locating-total dominating set is an NP-complete problem. Furthermore, the r-locating-dominating set and r-locating-total dominating set problems are discussed for some standard graphs.