The multi-item single period problem with an initial stock of convertible units

This paper deals with the situation of a number of end items, each facing uncertain demand in a single period of interest. Besides being able to purchase units of the end items there is also available a stock of units that can be converted into end items but at unit costs that depend on the specific end item. Efficient solution procedures are presented for two situations: (i) where the end item demand distributions are assumed known (illustrated for the case of normally distributed demand) and (ii) a distribution free approach where only the first two moments of the distributions are assumed known. Computational results for a set of problems are presented.     

Analytical and computational solution of adaptive inventory processes

H. Scarf (Ann. Math. Statist.30 (2) (1959)) has discussed a technique for the computation of optimal inventory level in the case where the demand distribution contains an unknown statistical parameter. It was assumed that initially the parameter could be described by a priori distribution, which would be subsequently revised on the basis of additional demand information. By assuming that the demand distribution of a cumulative observed demand is a sufficient statistic for the unknown parameter, it was shown that the optimal inventory levels could be obtained by the recursive computation of a sequence of function of two variables. He also showed (N.R.L.A.7 (8) (1960)) that if the demand distributions are gamme distributions and if the holding and penalty costs are linear, then the sequence of function of two variables may be solved by a related set of equations requiring tabulation of functions of only one variable. The main part of the argument presented in this paper is that an analytical consideration and numerical examples of equation requiring tabulation of function of only one variable given by H. Scarf were given with special choices of the factors, and were compared with solution when demand distribution is assumed exactly known and unknown.     

Integrated inventory problem under trade credit in fuzzy random environment

This paper is concerned with an integrated inventory problem under trade credit where both the demand rate and deteriorating rate are assumed to be uncertain and characterized as fuzzy random variables with known distributions. The objective of this paper is to determine the optimal inventory policy by optimizing simultaneously the replenishment cycle length and trade credit period. At first, three decision criteria are given: (1) expected value criterion, (2) chance-constrained criterion and (3) chance maximization criterion. Then, after building the fuzzy random models based on the above decision criterion, a hybrid intelligent algorithm by integrating fuzzy random simulation and genetic algorithm is employed to deal with these models. At the end, three numerical examples are given to illustrate the benefits of the models and show the effectiveness of the algorithms.     

Analysis of two-sample truncated data using generalized logistic models

Parallel to Cox's [JRSS B34 (1972) 187-230] proportional hazards model, generalized logistic models have been discussed by Anderson [Bull. Int. Statist. Inst. 48 (1979) 35-53] and others. The essential assumption is that the two densities ratio has a known parametric form. A nice property of this model is that it naturally relates to the logistic regression model for categorical data. In astronomic, demographic, epidemiological, and other studies the variable of interest is often truncated by an associated variable. This paper studies generalized logistic models for the two-sample truncated data problem, where the two lifetime densities ratio is assumed to have the form exp{α+φ(x;β)}. Here φ is a known function of x and β, and the baseline density is unspecified. We develop a semiparametric maximum likelihood method for the case where the two samples have a common truncation distribution. It is shown that inferences for β do not depend the nonparametric components. We also derive an iterative algorithm to maximize the semiparametric likelihood for the general case where different truncation distributions are allowed. We further discuss how to check goodness of fit of the generalized logistic model. The developed methods are illustrated and evaluated using both simulated and real data.     

A stochastic approximation method for the single-leg revenue management problem with discrete demand distributions

We consider the problem of optimally allocating the seats on a single flight leg to the demands from multiple fare classes that arrive sequentially. It is well-known that the optimal policy for this problem is characterized by a set of protection levels. In this paper, we develop a new stochastic approximation method to compute the optimal protection levels under the assumption that the demand distributions are not known and we only have access to the samples from the demand distributions. The novel aspect of our method is that it works with the nonsmooth version of the problem where the capacity can only be allocated in integer quantities. We show that the sequence of protection levels generated by our method converges to a set of optimal protection levels with probability one. We discuss applications to the case where the demand information is censored by the seat availability. Computational experiments indicate that our method is especially advantageous when the total expected demand exceeds the capacity by a significant margin and we do not have good a priori estimates of the optimal protection levels.     

A Single-period Stochastic Inventory Model Under the Influence of Marketing Policies

A single-period stochastic inventory model is developed under different marketing policies using the concepts of fixed prices and fixed mark-up of prices where discounts are offered on bulk purchases. In the mathematical model shortages are allowed, and it is assumed that any unit unsold at the end of the period has no economic value. The model yields simple solutions for some particular demand distributions. It is illustrated with a suitable example.     

Modeling fuzzy multi-period production planning and sourcing problem with credibility service levels

A great deal of research has been done on production planning and sourcing problems, most of which concern deterministic or stochastic demand and cost situations and single period systems. In this paper, we consider a new class of multi-period production planning and sourcing problem with credibility service levels, in which a manufacturer has a number of plants and subcontractors and has to meet the product demand according to the credibility service levels set by its customers. In the proposed problem, demands and costs are uncertain and assumed to be fuzzy variables with known possibility distributions. The objective of the problem is to minimize the total expected cost, including the expected value of the sum of the inventory holding and production cost in the planning horizon. Because the proposed problem is too complex to apply conventional optimization algorithms, we suggest an approximation approach (AA) to evaluate the objective function. After that, two algorithms are designed to solve the proposed production planning problem. The first is a PSO algorithm combining the AA, and the second is a hybrid PSO algorithm integrating the AA, neural network (NN) and PSO. Finally, one numerical example is provided to compare the effectiveness of the proposed two algorithms.     

An inventory model: What is the influence of the shape of the lead time demand distribution?

In this paper the influence of the shape of the lead time demand distribution is studied for a specific inventory model which is described in a preceding paper by Heuts and van Lieshout [4]. This continuous review inventory model uses as lead time demand distribution a Schmeiser-Deutsch distribution (S-D distribution) [9]. In a previous paper [4] an algorithm was given to solve the decision problem.In the literature attention is given to the following problem: what information on the demand during the lead time is necessary and sufficient to obtain good decisions. Using a (s, S) policy; Naddor [8] concluded that thespecific form of the lead time demand distribution is negligible, and that only its first two moments are essential. For a simple (s, q) control system Fortuin [3] comes to the same conclusion. Both authors analysed the case with known lead times and with given demand distributions from the class of two parameter distributions. So in fact their results are obvious, as the lead time demand distributions resulting from their suppositions are all nearly symmetric. We shall demonstrate that the skewness of the lead time demand distribution in our inventory model is also an important measure, which should be taken into account, as the cost differences with regard to the case where this skewness measure is not used, can be considerable.     

Spare parts management: Linking distributional assumptions to demand classification

Spare parts are known to be associated with intermittent demand patterns and such patterns cause considerable problems with regards to forecasting and stock control due to their compound nature that renders the normality assumption invalid. Compound distributions have been used to model intermittent demand patterns; there is however a lack of theoretical analysis and little relevant empirical evidence in support of these distributions. In this paper, we conduct a detailed empirical investigation on the goodness of fit of various compound Poisson distributions and we develop a distribution-based demand classification scheme the validity of which is also assessed in empirical terms. Our empirical investigation provides evidence in support of certain demand distributions and the work described in this paper should facilitate the task of selecting such distributions in a real world spare parts inventory context. An extensive discussion on parameter estimation related difficulties in this area is also provided.     

Multi-resource investment strategies: Operational hedging under demand uncertainty

Consider a firm that markets multiple products, each manufactured using several resources representing various types of capital and labor, and a linear production technology. The firm faces uncertain product demand and has the option to dynamically readjust its resource investment levels, thereby changing the capacities of its linear manufacturing process. The cost to adjust a resource level either up or down is assumed to be linear. The model developed here explicitly incorporates both capacity investment decisions and production decisions, and is general enough to include reversible and irreversible investment. The product demand vectors for successive periods are assumed to be independent and identically distributed. The optimal investment strategy is determined with a multi-dimensional newsvendor model using demand distributions, a technology matrix, prices (product contribution margins), and marginal investment costs. Our analysis highlights an important conceptual distinction between deterministic and stochastic environments: the optimal investment strategy in our stochastic model typically involves some degree of capacity imbalance which can never be optimal when demand is known.     

Fuzzy periodic review system with fuzzy random variable demand

In this paper, a periodic review inventory system has been analyzed in a mixed imprecise and uncertain environment where fuzziness and randomness appear simultaneously. A model has been developed with customer demand assumed to be a fuzzy random variable. The lead-time has been assumed to be a constant. The lead-time demand and the lead-time plus one period’s demand have also been assumed to be fuzzy random variables. A methodology has been developed to determine the optimal inventory level and the optimal period of review such that the total expected annual cost in the fuzzy sense is minimized. A numerical example has been presented to illustrate the model.     

Lot sizing with random yield and different qualities

This paper considers a production/inventory system where items produced/purchased are of different qualities: Types A and B. Type A items are of perfect quality, and Type B items are of imperfect quality; but not necessarily defective; and have a lower selling price. The percentage of Type A (the yield rate) is assumed to be a random variable with known probability distribution. The electronics industry gives good examples of such situations. We extend the classical single period (newsvendor) and the economic order quantity (EOQ) models by accounting for random supply and for imperfect quality (Type B) items which are assumed to have their own demand and cost structure. We develop mathematical models and prove concavity of the expected profit function for both situations. We also present detailed analysis and numerical results. We focus on comparing the profitability of the novel proposed models with models from the literature (and derivatives of these models) that develop the optimal order quantity based on the properties of Type A items only (and ignore Type B items). We find that accounting for Type B items can significantly improve profitability.     

随机补货间隔且需求依赖库存水平的库存控制模型

定期补货库存模型在实践中被广泛使用,尤其是在单一供应商中购买多种不同产品的库存系统中更为常见.然而,大多数定期补货库存模型都假设补货的时间间隔是恒定不变的.但在实践中,补货的时间间隔也可能是一个随机的时间长度.提出了一个随机补货时间间隔和需求依赖于当前展示库存水平的库存控制模型,且补货间隔服从指数分布和均匀分布,同时允许短缺发生并且短缺量部分延期供给,并研究了模型最优解的存在性与唯一性.最后,给出了数值算例来说明模型在实际中的应用.     

Numerical Approach of Multi-Objective Optimal Control Problem in Imprecise Environment

In this paper, realistic production-inventory models without shortages for deteriorating items with imprecise holding and production costs for optimal production have been formulated. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is time dependent and known. The imprecise holding and production costs are assumed to be represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics, the objective function is changed to respective multi-objective functions and thus the single-objective problem is reduced to a multi-objective decision making(MODM) problem. The MODM problem is then again transformed to a single objective function with the help of weighted sum method and then solved using global criteria method, calculus method, the Kuhn–Tucker conditions and generalized reduced gradient(GRG) technique. The models have been illustrated by numerical data. The optimum results for different objectives are obtained for different types of production function. Numerical values of demand, production function and stock level are presented in both tabular and graphical forms     

Base stock policies for production/inventory problems with uncertain capacity levels

In this paper we consider a single item, stochastic demand production/inventory problem where the maximum amount that can be produced (or ordered) in any given period is assumed to be uncertain. Inventory levels are reviewed periodically. The system operates under a stationary modified base stock policy. The intent of our paper is to present a procedure for computing the optimal base stocl level of this policy under expected average cost per period criterion. This procedure would provide guidance as to the appropriate amount of capacity to store in the form of inventory in the face of stochastic demand and uncertain capacity. In achieving this goal, our main contribution is to establish the analogy between the class of base stock production/inventory policies that operate under demand/capacity uncertainty, and the G/G/1 queues and their associated random walks. We also present example derivations for some important capacity distributions.     

Determining a common production cycle time for an economic lot scheduling problem with deteriorating items

This paper considers the economic lot scheduling problem (ELSP) for a production-inventory system where items produced are subject to continuous deterioration. The problem is to schedule multiple products to be manufactured on a single machine repetitively over an infinite planning horizon. Each product is assumed to have a significant rate of deterioration. Only one product can be manufactured at a time. The demand rate for each product is constant, but an exponential distribution is used to represent the distribution of the time to deterioration. A common cycle time policy is assumed in the production process. A near optimal production cycle time is derived under conditions of continuous review, deterministic demand, and no shortage.     

On the demand pooling anomaly in inventory theory

The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order relation between the individual demands (assumed to be independent) and their average. In this paper, we present some sufficient conditions for the type F anomaly not to occur for two i.i.d. demands; furthermore we provide an asymptotic result showing whether this anomaly occurs for large n for a class of distributions containing all distributions with finite mean.     

The Use of Versatile Distribution Families in Some Stochastic Inventory Calculations

Previous works on stochastic inventory problems have often assumed that an item's lead time demand follows a "convenient" distribution such as the normal, the γ or the Weibull. First, this paper argues that these convenient distributions may be overly restrictive and unrealistic, and points out the versatility and realism of using four-parameter distributions of the Pearson's and the Schmeiser-Deutsch's systems. Second, using these four-parameter distributions, this paper presents practical `manual" methods for computing the stock-out probability, reorder level and expected lost sales of an inventory item and for solving the lot-size reorder-point model. Some of these methods are actually simpler than the ones developed previously for the more restrictive distributions.     

Alternative formulations of a combined trip generation,trip distribution,modal split,and trip assignment model

The traditional four-step model has been widely used in travel demand forecasting by considering trip generation, trip distribution, modal split and traffic assignment sequentially in a fixed order. However, this sequential approach suffers from the inconsistency among the level-of-service and flow values in each step of the procedure. In the last two decades, this problem has been addressed by many researchers who have sought to develop combined (or integrated) models that can consider travelers’ choice on different stages simultaneously and give consistent results. In this paper, alternative formulations, including mathematical programming (MP) formulation and variational inequality (VI) formulations, are provided for a combined travel demand model that integrates trip generation, trip distribution, modal split, and traffic assignment using the random utility theory framework. Thus, the proposed alternative formulations not only allow a systematic and consistent treatment of travel choice over different dimensions but also have behavioral richness. Qualitative properties of the formulations are also given to ensure the existence and uniqueness of the solution. Particularly, the model is analyzed for a special but useful case where the probabilistic travel choices are assumed to be a hierarchical logit model. Furthermore, a self-adaptive Goldstein–Levitin–Polyak (GLP) projection algorithm is adopted for solving this special case.