Inventory and sales effort management under unobservable lost sales

A retailer needs to make decisions regarding how much to order and how much sales effort to exert in an environment with uncertain demand. One intrinsic complexity in a typical retail environment is caused by the fact that the retailer can obtain information about demand only based on sales, as demand itself is unobservable. Taking a Bayesian approach, Lariviere and Porteus (1999) show that in such a setting a retailer should stock more to increase the probability of an exact demand observation. In this article, we extend their work by allowing the retailer to control both the stocking quantity and sales effort, which can be used to affect demand. We show that their insights with respect to information stalking carry over to this setting. In addition, our model allows gaining a better understanding of optimal sales effort strategies. We find that demand management has a dual role in supporting information gathering: while at the beginning of a product life cycle it is optimal to support learning effects by sharply reducing sales effort, at later stages of the product life cycle an aggressive strategy of increased promotional activities can be used to harvest the information gathered in earlier periods.     

-symmetry for (r,s)-hypergeometric series

Contiguous operators for a two-parameter analogue of hypergeometric series are constructed. These represent a two-parameter quantum enveloping algebra introduced by Takeuchi.     

On the law S ( S ( x , y ), T ( x , y )) = S ( x , y ) of fuzzy logic

Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.     

Optimal policies for inventory systems with discretionary sales, random yield and lost sales

We determine replenishment and sales decisions jointly for an inventory system with random demand, lost sales and random yield. Demands in consecutive periods are independent random variables and their distributions are known. We incorporate discretionary sales, when inventory may be set aside to satisfy future demand even if some present demand may be lost. Our objective is to minimize the total discounted cost over the problem horizon by choosing an optimal replenishment and discretionary sales policy. We obtain the structure of the optimal replenishment and discretionary sales policy and show that the optimal policy for finite horizon problem converges to that of the infinite horizon problem. Moreover, we compare the optimal policy under random yield with that under certain yield, and show that the optimal order quantity （sales quantity） under random yield is more （less） than that under certain yield.     

Some Arithmetic on the Hardy Sums s2（h, κ） and s3（h, κ）

In this work, Rademacher‘s two questions about Dedekind sums are applied to the Hardy sums s2(h,κ) and s3(h,κ). Some relations between these sums are obtained and some inequalities are given.     

Variations on (0, s)-Sequences

In the first part of this paper, after a survey on the general theory of (t, s)-sequences, we review the main constructions of (0, s)-sequences and point out some relations between them. From these comparisons and from examples we have met in another context, we have found new (0, s)-sequences in prime base b to which the second part is devoted; the proofs are based on the general framework of formal Laurent series introduced by Niederreiter; the construction can be randomized and should have numerical applications.     

Inventory models with a mixture of backorders and lost sales under fuzzy cost

Continuous review and periodic review inventory models in which a fraction of demand is backordered and the remaining fraction is lost during the stockout period are considered under fuzzy environment. Fuzziness is introduced by allowing the cost components imprecise and vague to certain extent. Trapezoidal fuzzy numbers are used to represent these characteristics. The optimum policies of these models under fuzzy costs are derived. Numerical results highlighting the sensitivity in the decision variables are also described.     

On two-echelon inventory systems with Poisson demand and lost sales

We consider a two-echelon, continuous review inventory system under Poisson demand and a one-for-one replenishment policy. Demand is lost if no items are available at the local warehouse, the central depot, or in the pipeline in between. We give a simple, fast and accurate approach to approximate the service levels in this system. In contrast to other methods, we do not need an iterative analysis scheme. Our method works very well for a broad set of cases, with deviations to simulation below 0.1% on average and below 0.36% for 95% of all test instances.     

Topological singularities in W S,P ( S N , S 1)

We are interested in the location of the singularities of maps u ∈ W ^s,p (S ^N , S ¹) when 1 ≤ sp and 1 < sp < 2. To this end, we consider the distributional Jacobian. We show that the range of this operator on W ^s,p (S ^N , S ¹) is the closure in W ^s−2,p ⋂ W ^−1,sp of the set of N − 2-currents defined as the integration on smooth oriented N − 2-dimensional boundaryless submanifolds.     

The Microstructure of (t, m, s)-Nets

In an article of A. B. Owen (1998, J. Complexity14, 466–489) the question about the distribution properties of digital (t, m, s)-nets in small intervals was raised. We give upper and lower bounds for the maximum number of points of a (t, m, s)-net in these intervals and also provide a way of improving the distribution properties in some cases.     

On exhibiting inventory systems with Erlangian lifetimes under renewal demands

A continuous revies (s, S) inventory system with renewal demand in which one item is put into operation as an exhibiting piece is analyzed. The lifetime of any operating unit has Erlangian distribution, and on failure is replaced by another one from the stock and the failed item is disposed of. Replenishment of stock is instantaneous. The transient and stationary values of inventory level distribution and the mean reorder rate are obtained using the techniques of semi-regenerative processes. Decision rules for optimums andS that minimize the long-run expected cost rate are derived. The solution for a dual model with the distribution of lifetimes and inter-demand times interchanged is also given.     

Advanced supply chain planning with mixtures of backorders,lost sales,and lost contract

This paper investigates a supply chain system in which a supplier prepares for the selling season by building stock levels prior to the beginning of the season and shortages realized at the beginning of the season are represented as mixtures of backorders and lost sales. Backlogged items are replenished as soon as possible through an emergency procurement as opposed to waiting until the next scheduled delivery as in many continuous review scenarios, and the backorder rate is modeled as a piecewise linear function of the magnitude of the shortage. The often intangible cost associated with lost sales and customer goodwill is also quantified. In particular, the buyer and supplier are engaged in a contractual agreement and the loss of customer goodwill from the supplier’s perspective is represented as the expected cost associated with violating the conditions of the contract. The likelihood of contract cancellation is also represented as a function of the magnitude of shortage. The optimal solution is derived in closed form for the case of exponential demand distribution, and an example problem is illustrated with numerical data in order to demonstrate calculation of the optimal solution and corresponding sensitivity analysis for demand distributions in which the solution cannot be expressed in closed form.     

Performance analysis for assemble-to-order systems with general renewal arrivals and random batch demands

We study multi-product and multi-item assemble-to-order systems under general assumptions on demand patterns and replenish leadtime distributions. We only assume that the demand process of each product being a renewal process, and the replenish leadtimes follow general distributions. Based upon techniques from renewal theory, we developed procedures for approximating key performance measures of these inventory systems, such as average inventory and immediate order fill rate. We also obtain qualitative results that reveal the impacts of changes in demand patterns and leadtime variability upon the performance of the systems.     

The Number of Linearly Independent Binary Vectors with Applications to the Construction of Hypercubes and Orthogonal Arrays, Pseudo ( t, m, s)-Nets and Linear Codes

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1kn. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes.     

The nonexistence of a certain Steiner system S(3, 12, 112)

Although the automorphism group of a projective plane of order 10, if one exists, must be very small, such a plane could be the derived design of a Steiner system S(3, 12, 112) with a larger group. There are several reasons why the Frobenius group of order 56 is a promising candidate for the latter group. However, in this paper it is shown that there is no S(3, 12, 112) which is fixed by this Frobenius group.     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.     

S(Hn)上的满数值半径等距

本文研究了矩阵空间到自身的满数值半径等距问题. 利用等距嵌入方法, 获得了自共轭矩阵空间单位球面到自身的满数值半径等距可实线性延拓至全空间上的满数值半径等距, 为Tingley等距延拓问题提供了一种方法.