lagrangian multiplier method
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<div>The integration of Maritime Internet of Things (M-IoT) technology and unmanned aerial/surface vehicles (UAVs/USVs) has been emerging as a promising navigational information technique in intelligent ocean systems. With the unprecedented increase of computation-intensive yet latency sensitive marine mobile Internet services, mobile edge computing (MEC) and non-orthogonal multiple access (NOMA) have been envisioned as promising approaches to providing with the low-latency as well as reliable computing services and ultra-dense connectivity. In this paper, we investigate the energy consumption minimization based energy-efficient MEC via cooperative NOMA for the UAV-assisted M-IoT networks. We consider that USVs offload their computation-workload to the UAV equipped with the edge-computing server subject to the UAV mobility. To improve the energy efficiency of offloading transmission and workload computation, we focus on minimizing the total energy consumption by jointly optimizing the USVs’ offloaded workload, transmit power, computation resource allocation as well as the UAV trajectory subject to the USVs’ latency requirements. Despite the nature of mixed discrete and non-convex programming of the formulated problem, we exploit the vertical decomposition and propose a two-layered algorithm for solving it efficiently. Specifically, the top-layered algorithm is proposed to solve the problem of optimizing the UAV trajectory based on the idea of Deep Reinforcement Learning (DRL), and the underlying algorithm is proposed to optimize the underlying multi-domain resource allocation problem based on the idea of the Lagrangian multiplier method. Numerical results are provided to validate the effectiveness of our proposed algorithms as well as the performance advantage of NOMA-enabled computation offloading in terms of overall energy consumption.</div>

<div>The integration of Maritime Internet of Things (M-IoT) technology and unmanned aerial/surface vehicles (UAVs/USVs) has been emerging as a promising navigational information technique in intelligent ocean systems. With the unprecedented increase of computation-intensive yet latency sensitive marine mobile Internet services, mobile edge computing (MEC) and non-orthogonal multiple access (NOMA) have been envisioned as promising approaches to providing with the low-latency as well as reliable computing services and ultra-dense connectivity. In this paper, we investigate the energy consumption minimization based energy-efficient MEC via cooperative NOMA for the UAV-assisted M-IoT networks. We consider that USVs offload their computation-workload to the UAV equipped with the edge-computing server subject to the UAV mobility. To improve the energy efficiency of offloading transmission and workload computation, we focus on minimizing the total energy consumption by jointly optimizing the USVs’ offloaded workload, transmit power, computation resource allocation as well as the UAV trajectory subject to the USVs’ latency requirements. Despite the nature of mixed discrete and non-convex programming of the formulated problem, we exploit the vertical decomposition and propose a two-layered algorithm for solving it efficiently. Specifically, the top-layered algorithm is proposed to solve the problem of optimizing the UAV trajectory based on the idea of Deep Reinforcement Learning (DRL), and the underlying algorithm is proposed to optimize the underlying multi-domain resource allocation problem based on the idea of the Lagrangian multiplier method. Numerical results are provided to validate the effectiveness of our proposed algorithms as well as the performance advantage of NOMA-enabled computation offloading in terms of overall energy consumption.</div>

In response to the digital revolution, nowadays, many companies operate online and offline businesses in parallel to ensure their future competitiveness. This research examines the inventory strategy for multi-product vendor-buyer supply chain systems, considering space constraints and carbon emissions, in order to improve competence in managing online and offline integrated orders. We amalgamate costs and emissions in transport and storage. Here, we divide the warehouse of the buyer into two stages: one for satisfying online orders and the other for satisfying offline orders. We also assume that additional crashing costs reduce the lead times for receiving products in the buyer’s warehouse. This study demonstrates a mathematical model in the form of a constrained non-linear programme (NLP) and derives a Lagrangian multiplier method to solve it. An iterative solution procedure is designed in order to attain sustainable manufacturing decisions, which are illustrated numerically.

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

One long-standing open question remains regarding the theory of the generalized variational principle, that is, why can the stress-strain relation still be derived from the generalized variational principle while the method of Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional. In addition, that Chien's functional $\Pi_{Q}[u_i,\epsilon_{ij},\sigma_{ij},\lambda]$ is a much more general four-field functional and that the Hu-Washizu functional is its special case as $\lambda=0$ are confirmed.

This paper considers the source localization problem using time differences of arrival (TDOA) and frequency differences of arrival (FDOA) for multiple disjoint sources moving together with constraints on their distances and velocity correlation. To make full use of the synergistic improvement of multiple source localization, the constraints on all sources are combined together to obtain the optimal result. Unlike the existing methods that can achieve the normal Cramér-Rao lower bound (CRLB), our object is to further improve the accuracy of the estimation with constraints. On the basis of maximum likelihood criteria, a Lagrangian estimator is developed to solve the constrained optimization problem by iterative algorithm. Specifically, by transforming the inequality constraints into exponential functions, Lagrangian multipliers can be used to determine the source locations via Newton’s method. In addition, the constrained CRLB for source localization with distance and velocity correlation constraints is also derived. The estimated accuracy of the source positions and velocities is shown to achieve the constrained CRLB. Simulations are included to confirm the advantages of the proposed method over the existing methods.

Strategic workforce planning has become increasing essential for modern organisations. It is the key approach used by organisation to improve and maintain the capabilities of its workforce. However many experts distinguish between Training and Development, being that training tends to be more closely focused and adapted demands short-term performance concerns, while development tends to be adapted more towards expanding an individual skills for future responsibilities. Enhancement of skills through training programmes leads to different performance results. The success of any training organisation depends on the number of customers who have enrolled for the training programmes. This paper presents novel mathematical models and algorithms to accurately represent and efficiently solve workforce planning problems. The paper emphasizes on the optimum allocation of suitable trainer operational hours and minimise the cost of conducting the programme. A cost function derived is based on the assumption and the minimum cost is obtained satisfying the constraints. Lagrangian multiplier method is employed to find the minimum cost. In this paper, the efficiency of the organisation is analysed by using the relative service efficiency method and cost model is derived.

To stimulate purchases from consumers, retailers nowadays use the multiple retail prices strategy (MRPS), i.e., selling the products at multiple prices simultaneously. The paper extends the current newsboy model and proposes an optimal ordering model for MRPS corresponding to uncertain consumer demands. The Lagrangian multiplier method is applied to solve the problem, and an algorithm for finding the approximate optimal total order quantity is designed. Numerical results show that MRPS is better than the single retail price strategy (SRPS). It further reveals that when there is an order quantity constraint, the retailer needs to control the number of retail prices; that is, retailer’s MRPS is affected by order quantity constraint; sensitivity analysis demonstrates that MRPS is also affected by the price discount coefficient in the case of no order quantity constraint while it is not affected by demand volatility. The research work provides some useful managerial inspirations for retailers.