Monotonicity properties of a class of stochastic inventory systems

We consider inventory systems which are governed by an (r,q) or (r,nq) policy. We derive general conditions for monotonicity of the three optimal policy parameters, i.e., the optimal reorder level, order quantity and order-up-to level, as well as the optimal cost value, as a function of the various model primitives, be it cost parameters or complete cost rate functions or characteristics of the demand and leadtime processes. These results are obtained as corollaries from a few general theorems, with separate treatment given to the case where the policy parameters are continuous variables and that where they need to be restricted to integer values. The results are applied both to standard inventory models and to those with general shelf age and delay dependent inventory costs.     

On the unimodality of discrete probability measures

This work was supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec, Canada     

Some characterizations of discrete unimodality

Let F be a discrete distribution function on $\text{|Z}$. This paper gives a characterization of discrete unimodal distribution functions (Theorem 5.1) and a representation theorem for those distribution functions (Theorem 6.3), both in terms of their Lévy concentration functions.     

Strong unimodality and scale mixtures

Summary It is known that the asymptotically optimalL-estimator has non-negative weights if and only if the underlying density is strongly unimodea. Since the Tukey model is often used to model data containing gross errors, it is of interest to know when such densities are strongly unimodal. This paper examines the question more generally for a large class of scale mixtures. A simple necessary and sufficient condition is given in the case of the Tukey model. This research was supported in part by National Science Foundation Grants MCS78-25301 and MCS79-03716.     

A note on unimodality and asymptotic normality

From asymptotic normality of a unimodal sequence (p_n) estimates are obtained for the values N such that p_N ? p_n for all n.     

On inventory problems with arbitrary cost pattern,demand pattern and demand distribution

A detailed analysis of inventory models without setup costs, arbitrary demand distribution and arbitrary demand and cost pattern is given. First it is shown that the corresponding one-period model without ordering costs may be reduced to another simpler one with appropriately modified demand distribution. Several representations are given for the modified demand distribution. As one of the main results this reduction turns out to be robust in most cases. In a final chapter the results are applied to the determination of an optimal policy for a class of N-period inventory models with convex holding-and shortage costs and without setup costs.     

Compositions inside a rectangle and unimodality

Let c ^k,l (n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k×l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the sequence c ^k,l (0),c ^k,l (1),…,c ^k,l (kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open.     

Optimal inventory policy under power demand pattern and partial backlogging

In this paper, we study an inventory model with a power demand pattern that allows shortages. It is assumed that only a fraction of demand is backlogged during the shortage period and the remainder is considered lost sales. The aim of the paper is to determine the lot size and the length of the inventory cycle that maximize the total inventory profit per unit time. A general approach to obtain the optimal solution of the inventory problem and the maximum associated profit is developed. Some inventory models proposed in the literature are particular cases of the model analyzed here. Numerical examples are included to complement the theoretical results.     

Log-concavity of stirling numbers and unimodality of stirling distributions

A series of inequalities involving Stirling numbers of the first and second kinds with adjacent indices are obtained. Some of them show log-concavity of Stirling numbers in three different directions. The inequalities are used to prove unimodality or strong unimodality of all the subfamilies of Stirling probability functions. Some additional applications are also presented.     

Monotonicity and the principle of optimality

This paper explores some of the theoretical and algorithmic implications of the fact that the Monotonicity Assumption does not ensure either the validity of the Principle of Optimality or the discovery of all optimal solutions in finite dynamic programs, even though it is sufficient to ensure the validity of the functional equations. A slightly stronger assumption is introduced to resolve these problems. Our analysis is illustrated with some extremely simple examples.     

On the unimodality of average edge cover polynomials

The edge cover polynomial of a graph G is the function $E(G,x)={\sum }_{i\ge 1}e(G,i)x^i$, where $e(G,i)$ is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph $K_n$, based on which we establish that the average edge cover polynomial of order n is unimodal and has at least $n?3$ non-real roots.     

On strong unimodality of multivariate discrete distributions

A discrete function f defined on Zⁿ is said to be logconcave if for , , . A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189-216] a discrete function is called strongly unimodal if there exists a convex function such that  if . In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented.     

On the unimodality of spherically symmetric stable distribution functions

Several theorems are obtained concerning the unimodality of spherically symmetric distribution functions. These theorems are used to show that a class of spherically symmetric infinitely divisible distribution functions that contains the class of spherically symmetric stable distribution functions is unimodal.     

On the unimodality of power transformations of positive stable densities

Let Z_α be a positive α‐stable random variable and $r\in {\mathbb {R}}.Let Z_α be a positive α‐stable random variable and $r\in {\mathbb {R}}.$ We show the existence of an unbounded open domain D in [1/2, 1] × ( ? ∞, ?1/2] with a cusp at (1/2, ?1/2), characterized by the complete monotonicity of the function $F_{\alpha ,r}(\lambda ) = (\alpha \lambda ^\alpha -r)e^{-\lambda ^\alpha }\!\! ,$ such that $Z_\alpha ^r$ is unimodal if and only if (α, r)?D.