For an integer k >= 0, a connected graph G is called a k-cactus graph if each edge e is an element of E(G) is contained in at most k cycles of G. Inspired by the Brualdi-Solheid problem, in this paper, we address the problem of determining the maximum spectral radius of k-cactus graphs. Lovasz and Pelikan [Period. Math. Hungar.3 (1973) 175-182], Borovicanin and Petrovic [Publ. Inst. Math. (Beograd) (N.S.)79 (2006) 13-18] resolved the cases of k = 0 and k = 1, respectively. We solve this problem for the cases of k = 2 and k = 3 that is the graphs with the maximum spectral radius among all 2-cactus graphs and 3-cactus graphs are determined, respectively.
We study a mathematical program with switching constraints (MPSC). By introducing a suitable constraint qualification and an alternative stationarity concept, we derive necessary optimality conditions using D-directional upper convexificators and D-directional upper semi-regular convexificators, where D is a nonempty closed cone within the set of continuity directions for functions that may lack continuity. An illustrative example is provided.
Minimizing maximum cost and makespan simultaneously on a bounded series-batching machine is considered in the paper, in which each job contains a specific component and a standard component. Specific components are scheduled separately, while standard components are scheduled in batches. Completing a job means that its two components have been completed. Additionally, the two components nents of a job can be scheduled in any order. We present an O ( n 4 )-time algorithm for the simultaneous optimization scheduling problem. When the maximum cost is the maximum lateness, time complexity of the algorithm is O ( n 3 log n ) time.
The goal of this article is to obtain closed formulas for the outer-independent total domina- tion number and the outer-independent total Roman domination number of the following well-known graph operators defined from a connected graph G : the central graph C(G) the middle graph M(G) the graph R(G) and the total graph T(G) . We show that these formulas can be obtained for the last three graph operators and for the outer-independent total domination number of C(G) . The picture is quite different when it concerns the outer-independent total Roman domination number of C(G) . In this case, we obtain tight bounds and, imposing some restrictions on G , we obtain closed formulas.
This study examines the steady-state dynamics of an M/G /1 retrial queue that incorpo- rates features such as balking, optional service, negative customer, and general retrial time under a constant retrial policy. The probability generating functions for the orbit length with server status are obtained using the supplementary variable technique at various time epochs. We further extract the orbit size probabilities from these generating functions using the roots method. Additionally, the tail distribution of the orbit length is also presented for different time epochs. We conduct a comprehensive cost analysis to minimize the overall system operation cost. Advanced metaheuristic techniques such as particle swarm optimization and genetic algorithm are employed to determine the optimal system parameters that minimize the total cost. The impact of various system parameters on performance measures is demonstrated through detailed numerical examples.
After the COVID-19 pandemic, all organizations are involved in tackling the challenges of the disease, which continues after vaccination. it has affected various sectors of organizations including the supply chain, and has made coordination along the supply chain extremely difficult. The present study purpose to analyze the challenges of green supply chain in post- COVID-19 conditions. This study has an applied and developmental objective with a hybrid approach. This study was conducted in two stages and each stage had several consecutive steps. In the first stage, based on the results of thematic analysis, 21 challenges affecting the green supply chain in post- COVID-19 conditions were identified which were classified into 10 main challenges and 11 sub-challenges. In the second stage, the main challenges were analyzed based on hesitant fuzzy DEMATEL technique. Four challenges in an optimistic mode and five in a pessimistic mode were known as the causes. resulting in affecting all of the three levels in the supply chain and its disruption at the national and global level.
Nowadays, determining optimal lead time, inventory, and pricing decisions has become a critical challenge in competitive markets, particularly when demand exhibits sensitivity to both price and product availability. This study develops a game-theoretic model in which demand depends jointly on price and inventory levels for complementary products. The analysis considers both single-firm and duopoly settings under alternative competitive regimes, including Nash, Stackelberg, and cooperative strategies. The results show that cooperative behavior yields the highest profits, while pricing and inventory decisions exert stronger impacts on performance than lead-time adjustments. From a managerial perspective, the findings provide guidance on when firms facing complementary demand should compete or cooperate, how coordination can reduce lead times and holding costs, and how sensitivity insights support more effective pricing and inventory decisions.
The central operator C(G) is a structural transformation that combines edge subdivision with the complementation on the original vertex set. In this paper, we characterize the behavior of the differential partial derivative(C(G)), an invariant that measures the maximum influence potential of a network. We establish sharp bounds for partial derivative(C(G)) in terms of the order n and the maximum degree Delta(G) of the base graph, proving a guaranteed growth property: for any connected graph of order n >= 4, partial derivative(C(G)) >= partial derivative((G) over bar)+1. Our structural analysis reveals a parsimony property of optimal sets. Specifically, Theorem 3.16 shows that minimum differential sets in C(G) are primarily supported on the original vertex set V, effectively reducing the search space for optimization algorithms. Furthermore, in Theorem 3.22 we establish a notable theoretical convergence by identifying conditions under which the central, subdivision, and R(G) operators yield the same differential value, namely m + n - 4. Finally, we provide exact evaluations for several fundamental graph families and derive Nordhaus-Gaddum type inequalities (Prop. 3.24 and Cor. 3.25) for the class of trees. These results clarify how the central operator enhances the diffusion capacity of networks, thereby bridging the gap between topological transformations and practical influence maximization.
The Russell Measure Model (RM), a nonlinear data envelopment analysis (DEA) model for evaluating decision-making units, allows for independent and disproportionate inputs and outputs, which makes it superior and more accurate than the radial models. The model is formulated as a second-order cone programming (SOCP) problem, and its dual is derived using SOCP duality. Previous studies have noted the complexity and limited interpretability of this dual formulation and have proposed an alternative using semidefinite programming (SDP) problem. This paper demonstrates the equivalence of these dual formulations through variable transformations. In addition, a new SOCP formulation of the dual RM model is introduced, which is in the usual form of multiplier models without any variable transformations. It is shown that this new formulation is equivalent to the SDP model. Moreover, using the conic model, a new approach is proposed to identify the unique maximal reference set and projection by solving one model, thereby improving upon the existing two-stage approach. Two examples demonstrate the advantages of the proposed models.
Fractional programming is an extensively used technique to simultaneously deal with two conflicting objectives, by optimizing their ratio. This paper focuses on fully intuitionistic fuzzy fractional programming to address real-world problems. Firstly, the intuitionistic fuzzy model is converted into a crisp multi-objective problem with fractional objectives. Subsequently, a novel intuitionistic fuzzy programming approach is proposed, offering an insightful method for selecting least acceptable values. The existing literature on intuitionistic fuzzy programming employs linear, exponential, hyperbolic, or parabolic membership and non-membership functions. This article presents that these functions result in a highly restrictive feasibility region. Thus, a family of parameterized membership and non-membership functions is introduced, which overcomes the limitations of conventionally used functions and effectively captures both the acceptance as well as rejection degrees. In this article, theoretical foundations are established for the proposed technique through several theorems which have been constructed and proved. Additionally, the proposed technique is illustrated through a numerical example and applied to a real-life portfolio optimization problem, demonstrating its effectiveness. Finally, a comparative analysis with prevalent studies is also conducted to emphasize the versatile nature of the proposed functions.
The rapid advancement of wireless mobile technology enables user equipment (UE) to process vast amount of data transmission. However, since wireless communication UEs are battery-powered, they require an extended operational lifespan. These battery-operated devices gradually deplete their energy with prolonged usage. To optimize energy consumption, the Long-Term Evaluation (LTE) standard incorporates a discontinuous reception (DRX) mechanism, significantly extending UE battery life in modern fifth-generation (5G) wireless networks. 5G-DRX networks utilize millimeter-wave (mm Wave) frequency bands to achieve high-speed data transmission. However, mm Wave transmissions require beamforming techniques to compensate for high isotropic path loss. This article proposes a Markovian queue tailored for the 5G-DRX wireless deployment, integrating a beam search-based policy to accommodate dynamic short and long sleep cycles. By incorporating queueing analysis with sleep cycles in the 5G-DRX beam searching wireless system, both UE energy efficiency and data packet delay are effectively assessed. Analytical results are derived for the transient and steady-state probabilities of UE status, considering the 5G-DRX beam alignment tracking capability and the average number of data packets queued in the buffer at the evolved Node B (eNB). Several essential queueing performance metrics are determined. Additionally, UE energy efficiency and consumption in a stationary regime are analyzed to provide valuable numerical insights.
To propose a conjugate gradient (CG) scheme with a promising structure, it has been observed that Newton's direction is optimal when the current iteration is near the solution, and the objective function behaves like a quadratic. However, for large-scale problems, a method that does not require second-derivative information is often necessary. Therefore, to develop a more effective scheme for handling complex problems, we apply the standard secant equation to construct a combination of three-term CG search directions using the beta kRMIL beta kRMIL$ \beta_{k}<^>{ ext{RMIL}} $ method. This combination approximates the quasi-Newton direction and ensures sufficient descent. Furthermore, we establish the global convergence of the scheme under mild assumptions, demonstrating that the algorithm is robust and reliable compared to earlier CG methods. Finally, we showcase the efficiency of the proposed scheme by applying it to solve a three degrees of freedom (3DOF) motion control model.
Blockchain traceability technology enhances supply chain transparency and data security in cloud manufacturing supply chain, thereby affecting the choice of charging models. This paper innovatively integrates blockchain traceability technology with cloud data security management to construct a cloud manufacturing supply chain model that includes operators and suppliers. It conducts an in-depth analysis of equilibrium decisions under different charging models across varying levels of blockchain traceability technology, and designs a cost-sharing and revenue-sharing contract to coordinate the interests of all parties in the supply chain. It identifies four important results. First, as the level of blockchain traceability technology transitions from weak to strong, the service price, market demand, and profit levels for all parties in the cloud manufacturing supply chain show significant improvement. Second, the sensitivity coefficient of blockchain traceability technology and the elasticity coefficient of cloud data security positively influence the cloud data security level, blockchain traceability technology level, service price, demand, and profit, whereas the associated cost coefficients exert a negative influence. Third, he combined cost-sharing-revenue-sharing contract exhibits strong robustness and can effectively coordinate the cloud manufacturing supply chain, achieving Pareto improvement. Fourth, the strategic choices of the operator and the supplier are highly dependent on the revenue-sharing ratio. Both excessively high and low ratios lead to preference misalignment, which is further exacerbated by the cost coefficients of blockchain traceability technology and cloud data security management. However, when the ratio lies within a specific intermediate range, both parties consistently favor the revenue-sharing model. Increases in the sensitivity coefficient of blockchain traceability technology and the elasticity coefficient of cloud data security further widen this cooperative range.
To solve the problems caused by the increasing information credibility issue on platforms, we consider the situation in which a cross-channel supplier determines BT adoption strategies and a platform chooses sales modes. Then, we examine the influence of the cross-channel spillover effect on interaction decisions. We show that the supplier chooses BT to disclose more product information under the agency mode, even if the disclosure cost coefficient of BT is high. Namely, a positive spillover effect leads to higher profit margins and demand in dual channels. However, when the negative spillover effect is salient and the disclosure cost coefficient of BT is low, the wholesale mode improves the position of the supplier that adopts BT, but the supplier discloses less product information to reduce costs. As the product market size and/or spillover effect increase(s), the agency mode not only increases the information disclosure level but also decreases the retail price, which leads to win-win outcomes for firms and multi-win outcomes for firms and consumers. In contrast, since a negative spillover effect and BT exacerbate the DMP (double marginalization problem), firms and consumers achieve a multi-win situation under the wholesale format.
Cancer microarray datasets are distinguished by their high dimensionality and a relatively small sample sizes, which presents significant challenges for accurate cancer classification. Gene selection therefore becomes essential to eliminate irrelevant genes and improve classification accuracy. This paper presents a hybrid approach combining filter and wrapper techniques for gene selection, integrating an improved binary firefly algorithm and the support vector machine classifier. The objective is to select the most cancer-related genes to decrease computation time and enhance classification model performance. Three filter methods (Information Gain Ratio, ReliefF, and Correlation-based Feature Selection) are used in ensemble with the enhanced binary firefly algorithm. The firefly algorithm's exploration and exploitation capabilities are improved through opposition-based learning during initialization and movement of the fireflies. Additionally, a mutation step is added to improve the diversity of solutions. To validate our approach, we conducted an experimental study on twelve public benchmark datasets and compared it to several recent gene selection methods used for cancer gene expression data classification. The results reveal that the suggested methodology enhances classifier performance while reducing data volume by finding a limited group of genes with strong predictive power for cancer classification.
Risk response is an important measure for ensuring success of a Project Portfolio (PP). To address the limitations of the existing centralized decision-making for PP risk response, a new distributed decision-making method for PP risk response is proposed. Initially, a comprehensive risk interaction evaluation method is proposed integrating Interval Neutrosophic Sets (INSs) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) along with Decision-making Trial and Evaluation Laboratory (DEMATEL). Subsequently, project risk interaction evaluation in different scenarios of Project Decision-Maker (PDM) is defined. The methods for the comprehensive evaluation of risks from the perspectives of PDM and of PP Decision-Maker (PPDM) are also given. Following this, a bi-level programming model is constructed in which the decision-maker of the upper layer is PPDM and that of the lower layer is PDM. A case study is conducted on a display Research and Development (R&D) PP to validate the feasibility of the method. The results of the case study show that PP Risk Response Decision (PPRRD) can be affected by different scenarios of PDM and the hierarchical model can effectively resolve conflicts caused by differences in decision-making perspectives between PPDM and PDM.
The real world optimization problems in hierarchical decision making systems, often encounter multiple functions in fractional forms with uncertain parameters. To tackle such situation of uncertainty, this paper proposes a novel methodology to find a compromise solution of a bi-level multi-objective linear fractional programming problem which is designed in a fuzzy environment with its parameters expressed as intuitionistic triangular fuzzy numbers. Based on the concept of intuitionistic fuzzy (alpha, beta)-cuts and some theoretical aspects, the bi-level intuitionistic fuzzy model is formulated into an equivalent bi-level optimization with multiple interval valued fractional functions. The method proposed by Chakraborty and Gupta, is utilized to compute the individual compromise solution of each interval valued fractional objective function. Subsequently, the upper and lower level compromise solutions are computed to ascertain the aspiration values of the multiple interval valued fractional functions and the decision variables controlled at the upper level. Goal programming approach using a proposed modified linearization technique for fractional functions, is implemented to derive the compromise solution of the bi-level fuzzy optimization model. An existing numerical example, a practical problem in production sector are solved and the comparative discussion on result analysis is incorporated to demonstrate the feasibility and efficiency of the proposed approach.
This study introduces a novel ceiling constraint on clean energy advertisement input and examines its interaction with two carbon regulation policies carbon tax (CT) and carbon allowance mechanism (CAM) in shaping operational decisions. The key findings are as follows: (1) Under a low ceiling, both policies lead to identical advertisement input, yet CT results in lower profits for chain members. Under a moderate or high ceiling, CAM induces higher advertisement input than CT. (2) Regardless of the ceiling level, CAM consistently leads to higher power demand, higher total emissions, and lower conventional power prices compared to CT. (3) When the ceiling constraint is binding, raising it reduces conventional power prices, power demand, and emissions under both policies, while increasing the equilibrium values for other decision variables. Theoretical and managerial contributions: Theoretically, this study advances low-carbon operations modeling by incorporating a regulatory ceiling on advertisement input, offering a refined framework for evaluating carbon policies. It further identifies critical threshold levels (e.g., of the ceiling) that dictate the relative efficacy of CT versus CAM in promoting green input and shaping profitability. From a managerial perspective, the findings offer clear guidance; policymakers can use them to design balanced regulations, while power generators can better select operational strategies under different policy regimes.
The joint optimization of transportation and inventory replenishment related decisions promises to yield significant cost savings coupled with higher customer satisfaction levels. This paper investigates a supply chain system consisting of a single supplier replenishing a single retailer, where the primary focus is on the retailer's decision-making process, aimed at determining the most efficient operational policy. This includes identifying the optimal replenishment quantity from the supplier and selecting the appropriate mix and size of the truck fleet under different situations. At first, the scenario whereby the retailer exclusively operates its limited fleet of trucks for inbound transportation is considered. An efficient solution procedure along with closed form expressions for the optimal ordering quantity and the number of trucks are devised. Subsequently, the problem is extended to incorporate environmental considerations under carbon tax and carbon cap policies. We propose a computationally efficient algorithm for generating the optimal operational policy following the carbon cap policy. Finally, to better resemble reality, the scope of the operational optimization model is extended via allowing the retailer the option to lease trucks from the external market. The conducted numerical experiments demonstrate that this flexibility can lead to significant cost reductions that are increasing with demand.
If G = (V(G), E(G)) is a simple connected graph with vertex set V(G) and edge set E(G), we say that a subset D subset of V(G) is a strong defensive alliance if for every vertex v is an element of D the condition delta D(v) >= delta D(v) holds. The strong defensive alliance number alpha(G) is defined as the minimum cardinality among all the strong defensive alliances. A unitary operator of graphs & Oscr; assigns to each graph G a graph & Oscr;(G). A few examples of unitary operators of graphs are: Subdivision S(G), R(G), Middle Q(G), Total T(G), and Central & Cscr;(G). In this paper we determine the exact values of alpha(S(G)) and alpha(R(G)). We also characterize the graphs G for which the number of strong defensive alliances is 1, 2, or 3 in Q(G) and T(G). We also we give tight bounds for alpha(S(G))alpha(S(G)& strns;)$ \alpha(\overline{S(G)}) $, alpha(Q(G)), alpha(Q(G))alpha(Q(G)& strns;)$ \alpha(\overline{Q(G)}) $, alpha(T(G)), and alpha(& Cscr;(G)).
