Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by $K(\mathrm{R}\text{-}\mathrm{Proj})$ the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that $K(\mathrm{R}\text{-}\mathrm{Proj})$ is ${\mathrm{?}}_1$-compactly generated, with the category $K^{+}(\mathrm{R}\text{-}\mathrm{proj})$ of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K(\mathrm{R}\text{-}\mathrm{Proj})$ vanishes in the Bousfield localization $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.     

On a theorem by Brewer

One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theorem 21 that if M is a non-SFT maximal ideal of a commutative ring R with identity, then there exists an infinite ascending chain of prime ideals in the power series ring $R?X?$, $Q_0?Q_1???Q_n??$ such that $Q_n\cap R=M$ for each n. Moreover, the height of $M?X?$ is infinite. In this paper, we show that the above theorem is false by presenting two counter examples. The first counter example shows that the height of $M?X?$ can be zero (and hence there is no chain $Q_0?Q_1???Q_n??$ of prime ideals in $R?X?$ satisfying $Q_n\cap R=M$ for each n). In this example, the ring R is one-dimensional. In the second counter example, we prove that even if the height of $M?X?$ is uncountably infinite, there may be no infinite chain $\{Q_n\}$ of prime ideals in $R?X?$ satisfying $Q_n\cap R=M$ for each n.     

Global and local behavior of zeros of nonpositive type

A generalized Nevanlinna function Q(z)$Q(z)$ with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by _Qτ(z)=(Q(z)−τ)/(1+τQ(z))$Q_{\tau }(z)=(Q(z)-\tau )/(1+\tau Q(z))$, τ∈R∪{∞}$\tau \in \mathbb{R}\cup \{\infty \}$, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ)$\alpha (\tau )$ as a function of τ is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.     

A new characterization of comonotonicity and its application in behavioral finance

It is well-known that an Rⁿ${\mathbb{R}}^n$-valued random vector (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ is comonotonic if and only if (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ and (_Q1(U),_Q2(U),?,_Qn(U))$(Q_1(U),Q_2(U),?,Q_n(U))$ coincide in distribution, for any random variable U   uniformly distributed on the unit interval (0,1)$(0,1)$, where _Qk(⋅)$Q_k(\cdot )$ are the quantile functions of _Xk$X_k$, k=1,2,?,n$k=1,2,?,n$. It is natural to ask whether (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ and (_Q1(U),_Q2(U),?,_Qn(U))$(Q_1(U),Q_2(U),?,Q_n(U))$ can coincide almost surely for some special U. In this paper, we give a positive answer to this question by construction. We then apply this result to a general behavioral investment model with a law-invariant preference measure and develop a universal framework to link the problem to its quantile formulation. We show that any optimal investment output should be anti-comonotonic with the market pricing kernel. Unlike previous studies, our approach avoids making the assumption that the pricing kernel is atomless, and consequently, we overcome one of the major difficulties encountered when one considers behavioral economic equilibrium models in which the pricing kernel is a yet-to-be-determined unknown random variable. The method is applicable to general models such as risk sharing model.     

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u   is a solution of (−Δ)^su=g${(-\mathrm{\Delta })}^{s}u=g$ in Ω  , u≡0$u\equiv 0$ in Rⁿ\Ω${\mathbb{R}}^n\backslash \Omega$, for some s∈(0,1)$s\in (0,1)$ and g∈L^∞(Ω)$g\in L^{\infty }(\Omega )$, then u   is C^s(Rⁿ)$C^s({\mathbb{R}}^n)$ and u/δ^s_|Ω$u/{\delta }^s{|}_{\Omega }$ is C^α$C^{\alpha }$ up to the boundary ∂Ω   for some α∈(0,1)$\alpha \in (0,1)$, where δ(x)=dist(x,∂Ω)$\delta (x)=\mathrm{dist}(x,\partial \Omega )$. For this, we develop a fractional analog of the Krylov boundary Harnack method.     

Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations

This paper provides results on the correct simulation, when using continuous Runge–Kutta methods, of certain stability properties of nonlinear neutral delay-differential equations (NDDEs) y^′(t)=f(t,y(t),y(t-τ(t)),y^′(t-τ(t)))(t?_t0)$y^{\prime }(t)=f(t\text{,}y(t)\text{,}y(t-\tau (t))\text{,}{y}^{\prime }(t-\tau (t)))(t?t_0)$. In particular, it is shown that certain continuous Runge–Kutta methods based upon the backward Euler method or the 2-stage Lobatto IIIC method, combined with linear interpolation, are GRN$\mathit{GRN}$-stable and asymptotically stable for NDDEs.     

Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$