An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

Optimal business policies for a supplier–transporter–buyer channel with a price-sensitive demand

As the third party logistics partners (carriers) taking a more and more significant role in supply chain practices and customer service performance improvement, there is an emerging need for the studies on optimal channel coordination policies for business processes involving not only supplier and buyer (retailer), but also transportation partners. In this paper, we explicitly add a transportation partner with concave cost functions into the analysis for supplier–buyer channel coordination policies, and analyse the impact of coordination and pricing policies on supply chain profitability. The market demand is assumed to be a decreasing convex function of buyer's selling price (x), D(x)=d/x². Under this assumption, we quantify the improvement on total supply chain profitability when moving from a non-cooperative environment to a fully cooperative environment, and show that the joint annual profit of three partners in a cooperative environment can be at least twice of what may be achieved by three independently operated companies in a leader–follower business game. While in a real-world business environment, a perfect collaboration is hard to achieve, this result can be used to provide a quick estimation on the upper bound on the budget for profit sharing or discount offers among the supply chain partners.     

Optimum preventive maintenance for a 2-unit priority-standby system with patience–time for repair

This paper deals with the effect of preventive maintenance (PM) on the reliability measurcs for a 2-unit priority standby system with patience-time for repair. Four types of PM (type (a),(b), (c), and(d)) are considcrcd. Failurc, repair, PM, and replacement time disiributiotls are general whereas delivery time distribution is negative exponential. Regenerative technique in Markov renewal is applied to obtain several reliability characteristics of interest to designers. Finally numerical calculations are given to illustrate the theoretical results     

Optimal control of a nonlinear parabolic–elliptic system

In this paper, we shall study the problem of optimal control of the parabolic–elliptic system

_ut+_(f(t,x,u))x+g(t,x,u)+_Px−_(a(t,x)ux)x=_f0+B^∗ν$u_t+{\left(f(t,x,u)\right)}_x+g(t,x,u)+P_x-{\left(a(t,x)u_x\right)}_x=f_0+B^{\ast }\nu$

Traveling wave solutions for a predator–prey system with two predators and one prey

We study a predator–prey model with two alien predators and one aborigine prey in which the net growth rates of both predators are negative. We characterize the invading speed of these two predators by the minimal wave speed of traveling wave solutions connecting the predator-free state to the co-existence state. The proof of the existence of traveling waves is based on a standard method by constructing (generalized) upper-lower-solutions with the help of Schauder’s fixed point theorem. However, in this three species model, we are able to construct some suitable pairs of upper-lower-solutions not only for the super-critical speeds but also for the critical speed. Moreover, a new form of shrinking rectangles is introduced to derive the right-hand tail limit of wave profile.     

Optimal relativities and transition rules of a bonus–malus system

When a bonus–malus system with a single set of optimal relativities and a set of simple transition rules is implemented, two inadequacy scenarios are induced because all policyholders are subject to the same a posteriori premium relativities (level transitions) independent of their a priori characteristics (current levels occupied). In this paper we propose a new objective function in the determination of optimal relativities that directly incorporates the a priori expected claim frequencies to partially address one of the inadequacy scenarios. We derive the analytical solution for the optimal relativities under a financial equilibrium constraint. Furthermore, we introduce a metric called effectiveness of transition rules to compare the different specifications of transition rules. We also argue that varying transition rules which are more flexible in addressing the other inadequacy scenario may be more effective than their corresponding simple rules.     

Permanence of periodic predator–prey system with two predators and stage structure for prey

A stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response is proposed and analyzed. Based on the comparison theorem, some sufficient and necessary conditions are derived for permanence of the system. Finally, two examples are presented to illustrate the application of our main results.     

Discrete traveling waves in a relay system of Mackey–Glass equations with two delays

Theoretical and Mathematical Physics - We propose a model of a ring circuit of $$m$$ generators that is a relay analog of a circuit of Mackey–Glass generators. In this model, each of the...     

Optimal boundary control with critical penalization for a PDE model of fluid–solid interactions

We study the finite-horizon optimal control problem with quadratic functionals for an established fluid-structure interaction model. The coupled PDE system under investigation comprises a parabolic (the fluid) and a hyperbolic (the solid) dynamics; the coupling occurs at the interface between the regions occupied by the fluid and the solid. We establish several trace regularity results for the fluid component of the system, which are then applied to show well-posedness of the Differential Riccati Equations arising in the optimization problem. This yields the feedback synthesis of the unique optimal control, under a very weak constraint on the observation operator; in particular, the present analysis allows general functionals, such as the integral of the natural energy of the physical system. Furthermore, this work confirms that the theory developed in Acquistapace et al. (Adv Diff Eq, [2])—crucially utilized here—encompasses widely differing PDE problems, from thermoelastic systems to models of acoustic-structure and, now, fluid-structure interactions.     

Optimal birth control of free horizon problems for predator-prey system with age-structure

We study optimal birth policies for three age-dependent populations in a predator-prey system, which is controlled by fertility. New results on problems with free final time and integral phase constraints are presented, the approximate controllability of system is discussed.     

Dynamics of a reaction–diffusion–ODE system with quiescence

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic solutions. A new method to obtain global stability and dissipative structure is also explored by constructing Lyapunov functionals to overcome the loss of compactness.     

Topological multivortex solutions for the Chern–Simons system with two Higgs particles

In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.     

Existence of positive solution for a sixth–order differential system with variable parameters

This paper investigates the existence of positive solutions for a sixth-order differential system with three variable parameters. Using a fixed point theorem and an operator spectral theorem we give some new existence results.     

Multiple fixed-sign solutions for a system of difference equations with Sturm–Liouville conditions

We consider the following system of difference equations:${\mathrm{\Delta }}^m{u}_i(k)+P_i(k,u_1(k),u_2(k),\dots ,u_n(k))=0,k\in \{0,1,\dots ,N\},i=1,2,\dots ,n$together with Sturm–Liouville boundary conditions${\mathrm{\Delta }}^j{u}_i(0)=0,0⩽j⩽m-3\text{,}$$\zeta {\mathrm{\Delta }}^{m-2}{u}_i(0)-\eta {\mathrm{\Delta }}^{m-1}{u}_i(0)=0,\omega {\mathrm{\Delta }}^{m-2}{u}_i(N+1)+\delta {\mathrm{\Delta }}^{m-1}{u}_i(N+1)=0\text{,}$where $m⩾2,N⩾m-1,\zeta >0,\omega >0,\eta ⩾0,\delta ⩾\omega ,\zeta \omega (N+1)+\zeta \delta +\eta \omega >0$. By using two different fixed point theorems, we develop criteria for the existence of three solutions of the system which are of fixed signs on $\{0,1,\dots ,N+m\}$. Examples are also included to illustrate the results obtained.     

Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion

In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.