A comparison between the order and the volume fill rate for a base-stock inventory control system under a compound renewal demand process

The order fill rate (OFR) is sometimes suggested as an alternative to the volume fill rate (VFR) (most often just denoted fill rate) as a performance measure for inventory control systems. We consider a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered. For this policy, we develop exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process. We also elaborate on when the OFR can be interpreted as the (extended) ready rate. For the case when customer orders are generated by a negative binomial distribution, we show that it is the size of the shape parameter of this distribution that determines the relative magnitude of the two fill rates. In particular, we show that when customer orders are generated by a geometric distribution, the OFR and the VFR are equal.     

Approximating the Mean Waiting Time in the <Emphasis Type="Italic">GI/G/s</Emphasis> Queue

We provide two distribution-dependent approximations for the mean waiting time in a GI/G/s queue. Both approximations are weighted combinations of the exact mean waiting times for the GI/M/s and M/D/s queues each of which has the same mean service time and traffic intensity as in the approximating GI/G/s queue. The weights in the approximations are expressed by the service-time c.d.f. and the first two moments of interarrival and service times. To examine the performance of our approximations, they are numerically compared with exact solutions and previous two-moment approximations for various cases. Extensive numerical comparisons indicate that the relative percentage errors of the approximations are of the order of 5% in moderate traffic and 1% in heavy traffic, except for extreme cases.     

New approximations to the principal real-valued branch of the Lambert <Emphasis Type="Italic">W</Emphasis>-function

The Lambert W-function is the solution to the transcendental equation W(x)e ^W(x) = x. It has two real branches, one of which, for x ∈ [?1/e, ∞], is usually denoted as the principal branch. On this branch, the function grows from ? 1 to infinity, logarithmically at large x. The present work is devoted to the construction of accurate approximations for the principal branch of the W-function. In particular, a simple, global analytic approximation is derived that covers the whole branch with a maximum relative error smaller than 5 × 10^?3. Starting from it, machine precision accuracy is reached everywhere with only three steps of a quadratically convergent iterative scheme, here examined for the first time, which is more efficient than standard Newton’s iteration at large x. Analytic bounds for W are also constructed, for x > e, which are much tighter than those currently available. It is noted that the exponential of the upper bounding function yields an upper bound for the prime counting function π(n) that is better than the well-known Chebyshev’s estimates at large n. Finally, the construction of accurate approximations to W based on Chebyshev spectral theory is discussed; the difficulties involved are highlighted, and methods to overcome them are presented.     

Two new heuristics for the <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">n</Emphasis>/0 queueing loss system with examples based on the two-phase Coxian distribution

In this paper, we introduce a new heuristic approach for the numerical analysis of queueing systems. In particular, we study the general, multi-server queueing loss system, the GI/G/n/0 queue, with an emphasis on the calculation of steady-state loss probabilities. Two new heuristics are developed, called the GM Heuristic and the MG Heuristic, both of which make use of an exact analysis of the corresponding single-server GI/G/1/0 queue. The GM Heuristic also uses an exact analysis of the GI/M/n/0 queue, while the MG Heuristic uses an exact analysis of the M/G/n/0 queue. Experimental results are based on the use of two-phase Coxian distributions for both the inter-arrival time and the service time; these include an error analysis for each heuristic and the derivation of experimental probability bounds for the loss probability. For the class of problems studied, it is concluded that there are likely to be many situations where the accuracy of the GM Heuristic is adequate for practical purposes. Methods are also developed for combining the GM and MG Heuristics. In some cases, this leads to approximations that are significantly more accurate than those obtained by the individual heuristics.     

Periodic review (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) inventory model with permissible delay in payments

This paper investigates the effect of permissible delay in payments on ordering policies in a periodic review (s, S) inventory model with stochastic demand. A new mathematical model is developed, which is an extension to that of Veinott and Wagner (Mngt Sci 1965; 11: 525) who applied renewal theory and stationary probabilistic analysis to determine the equivalent average cost per review period. The performance of the model is validated using a custom-built simulation programme. In addition, two distribution-free heuristic methods of reasonable accuracy develop approximate optimal policies for practical purposes based only on the mean and the standard deviation of the demand. Numerical examples are presented with results discussed.     

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L _∞(0, T; L ₂(Ω))-norm and to ?u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h ^k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h ^k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.     

Analysis of Local Search Landscapes for <Emphasis Type="Italic">k</Emphasis>-SAT Instances

Stochastic local search is a successful technique in diverse areas of combinatorial optimisation and is predominantly applied to hard problems. When dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase is helpful for an appropriate parameter adjustment of local search-based procedures. In the present paper, we address parameter estimations in the context of landscapes induced by k-SAT instances: at first, we utilise a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood relation. The objective function is given by the number of satisfied clauses. The procedure provides good approximations of the actual number of local maxima, with a deviation typically around 10%. Secondly, we provide a method for obtaining upper bounds for the average number of local maxima in k-SAT instances. The method allows us to obtain the upper bound \(2^{n-O(\sqrt{n/k})}\) for the average number of local maxima, if m is in the region of 2 ^k · n/k.     

Multi-item two-echelon spare parts inventory control problem with batch ordering in the central warehouse under compound Poisson demand

We consider a multi-item two-echelon spare part inventory system in which the central warehouse operates under an (nQ,?R) policy and the local warehouses implement order-up-to S policy, each facing a compound Poisson demand. The objective is to find the policy parameters minimizing expected system-wide inventory holding and fixed ordering costs subject to an aggregate mean response time constraint at each warehouse. In this paper, we propose four alternative approximations for the steady state performance of the system; and extend a heuristic and a lower bound proposed under Poisson demand assumption to the compound Poisson setting. In a computational study, we show that the performances of the approximations, the heuristic, and the lower bound are quite satisfactory; and the relative cost saving of setting an aggregate service level rather than individually for each part is quite high.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

Analyzing the Steady-State Queue <Emphasis Type="Italic">GI</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript><Emphasis Type="Italic">/G/</Emphasis>1

It is proved in this note that the delay in the queue GI ^X /G/1 can be expressed as the sum of two independent components, such that known results of the queue GI/G/1 (e.g. approximations) can be readily applied. Based on this result, closed-form expressions are also derived for other performance measures of interest.     

<Emphasis Type="Italic">k</Emphasis>-Trails: recognition,complexity, and approximations

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail, and every graph containing a k-trail is a \((k+1)\)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a \((2k-1)\)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnár, Newman, and Seb?.     

Randers change of <Emphasis Type="Italic">m</Emphasis><Superscript>th</Superscript> root metric

The present paper deals with a Randers metric that has been derived after a particular β-change in the mth root metric. Various geometers such as [7], [9], [10] etc. have studied the mth root metric and its transformations. We have obtained some tensors and theorems holding the relation between the Finsler space equipped with the mth root metric and the one obtained after its Randers change.     

<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and _S V _R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom _R (I _V (R),?) and Hom _R (?,I _V (R)) exact exact complex \( \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots \) of V-injective modules I _i and I ⁱ , i ∈ N₀, such that N ? Im(I ₀ → I ⁰). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A _V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I _V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if \(Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0\) for all modules I with finite I _V (R)-injective dimension.     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

Decay Properties of <Emphasis Type="Italic">n</Emphasis>-type Markov Branching Processes with Disasters

In this paper, we consider the decay properties of n-type Markov branching processes with disasters, including the decay parameter and invariant measures. It is first proved that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest. Then, the exact value of the decay parameter \(\lambda _C\) is obtained. We prove that the decay parameter can be easily expressed explicitly. Furthermore, we prove that the Markov branching process with disasters is always \(\lambda _C\)-positive. The invariant vectors and the invariant measures are given explicitly.     

Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method

Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h~(1+min){α,1}) is established for both the displacement approximation in H~1-norm and the stress approximation in L~2-norm under a mesh assumption, where α 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.     

Effects of Centralization on Expected Costs in a Multi-location Newsboy Problem

This is a single-period, single-product inventory model with several individual sources of demand. It is a multi-location problem with an opportunity for centralization. The holding and penalty cost functions at each location are assumed to be identical. Two types of inventory system are considered in this paper: the decentralized system and the centralized system. The decentralized system is a system in which a separate inventory is kept to satisfy the demand at each source of demand. The centralized system is a system in which all demands are satisfied from one central warehouse. This paper demonstrates that, for any probability distribution of a location's demands, the following properties are always true: given that the holding and penalty cost functions are identical at all locations, (1) if the holding and penalty cost functions are concave functions, then the expected holding and penalty costs in a decentralized system exceed those in a centralized system, except that (2) if the holding and penalty cost functions are linear functions, and for any i≠j, P_ij, the coefficient of correlation between the ith location's demand and the jth location's demand is equal to 1, then the expected holding and penalty costs in a decentralized system are equal to those in a centralized system.     

The embedding and approximation for classes of functions with a dominant mixed difference

We obtain a criterion for embedding the class SH _p ^Ω into that SB _q,? ^Ω* (1 < p ≤ q < ∞). We also determine the exact order of the best approximations of functions from classes SB _p,? ^Ω by trigonometric polynomials whose harmonics belong to sets generated by level surfaces of the majorant Λ (t).     

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H _m+H_om+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H _m, H _om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.