Unreliable newsboy problem with a forecast update

In this paper, we derive an optimal ordering policy for an unreliable newsboy who can place two sequential orders before the start of a single selling season by using a demand forecast update. Supply yield is modeled using a uniform distribution considering both the minimum order guarantee and the maximum yield. Our results indicate that a firm should focus on increasing the minimum order guarantee from a first stage supplier to reduce its total supply chain cost.     

The distribution-free newsboy problem under the worst-case and best-case scenarios

New theoretical foundations for analyzing the newsboy problem under incomplete information about the probability distribution of random demand are presented. Firstly, we reveal that the distribution-free newsboy problem under the worst-case and best-case demand scenarios actually reduces to the standard newsboy problem with demand distributions that bound the allowable distributions in the sense of increasing concave order. Secondly, we provide a theoretical tool for seeking the best-case and worst-case order quantities when merely the support and the first k moments of the demand are known. Using this tool we derive closed form formulas for such quantities in the case of known support, mean and variance, i.e. k = 2. Consequently, we generalize all results presented so far in literature for the worst-case and best-case scenarios, and present some new ones. Extensions of our findings to the cases of the known mode of a unimodal demand distribution, the known median, and to other stochastic inventory problems are indicated.     

Supply chain coordination for fuzzy random newsboy problem with imperfect quality

Facing to imperfect quality and fuzzy random market demand in the real-life inventory management, a two-echelon supply chain system with one retailer and one manufacturer for perishable products is considered. Two fuzzy random models for the newsboy problem with imperfect quality in the decentralized and centralized systems are presented. The expectation theory and signed distance are employed to transform the fuzzy random model into crisp model. The optimal policies in the two decision-making systems are derived and analyzed contrastively. The theoretical analysis shows that manufacturer’s repurchase strategy can achieve the increase in the whole supply chain profit. The influence of the fuzzy randomness of the demand and the defective rate on the optimal order quantity, the whole supply chain profit and the repurchasing price is analyzed via numerical examples.     

Multi-product newsboy problem with limited capacity and outsourcing

This paper considers a multi-product newsboy system that produces multiple products for fulfilling independently uncertain demands, which share the same production capacity. To deal with possible shortage of limited capacity, productions can be outsourced. We consider two outsourcing strategies: zero lead time outsourcing, and nonzero lead time outsourcing. The structural properties and solution procedures for the profit-maximization models are developed. Numerical results are provided for obtaining some managerial insights.     

The single-item newsboy problem with dual performance measures and quantity discounts

This paper considers models for the single-item newsboy problem with quantity discount and the following dual performance measure: “maximize the expected profit subject to a constraint that the probability of achieving a target profit level is no less than a predetermined risk level.” We also consider two types of quantity discount: all-unit and incremental. For our models with zero shortage cost, a closed-form solution for determining the optimal order quantity is derived. However, models with positive shortage cost can only be solved numerically.     

Modeling customer impatience in a newsboy problem with time-sensitive shortages

Customers across all stages of the supply chain often respond negatively to inventory shortages. One approach to modeling customer responses to shortages in the inventory control literature is time-dependent partial backlogging. Partial backlogging refers to the case in which a customer will backorder shortages with some probability, or will otherwise solicit the supplier’s competitors to fulfill outstanding shortages. If the backorder rate (i.e., the probability that a customer elects to backorder shortages) is assumed to be dependent on the supplier’s backorder replenishment lead-time, then shortages are said to be represented as time-dependent partial backlogging. This paper explores various backorder rate functions in a single period stochastic inventory problem in an effort to characterize a diversity of customer responses to shortages. We use concepts from utility theory to formally classify customers in terms of their willingness to wait for the supplier to replenish shortages. Under mild assumptions, we verify the existence of a unique optimal solution that corresponds to each customer type. Sensitivity analysis experiments are conducted in order to compare the optimal actions associated with each customer type under a variety of conditions. Additionally, we introduce the notion of expected value of customer patience information (EVCPI), and then conduct additional sensitivity analyses to determine the most and least opportune conditions for distinguishing between customer behaviors.     

The multi-product newsboy problem with supplier quantity discounts and a budget constraint

This paper considers the multi-product newsboy problem with both supplier quantity discounts and a budget constraint, while each feature has been addressed separately in the literature. Different from most previous nonlinear optimization models on the topic, the problem is formulated as a mixed integer nonlinear programming model due to price discounts. A Lagrangian relaxation approach is presented to solve the problem. Computational results on both small and large-scale test instances indicate that the proposed algorithm is extremely effective for the problem. An extension to multiple constraints and preliminary computational results are also reported.     

Solving the knapsack problem with imprecise weight coefficients using genetic algorithms

This paper investigates solving the knapsack problem with imprecise weight coefficients using genetic algorithms. This work is based on the assumption that each weight coefficient is imprecise due to decimal truncation or coefficient rough estimation by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. We investigate the possibility of using genetic algorithms in solving the fuzzy knapsack problem without defining membership functions for each imprecise weight coefficient. The proposed approach simulates a fuzzy number by distributing it into some partition points. We use genetic algorithms to evolve the values in each partition point so that the final values represent the membership grade of a fuzzy number. The empirical results show that the proposed approach can obtain very good solutions within the given bound of each imprecise weight coefficient than the fuzzy knapsack approach. The fuzzy genetic algorithm concept approach is different, but gives better results than the traditional fuzzy approach.     

The Newsboy problem when customer demand is a compound renewal process

The Newsboy (Newsvendor) problem is probably the simplest of all stochastic inventory problems, involving a one-time purchase decision and a stochastic sales outcome. As an investment, it can be interpreted as the simplest stochastic version of the point-in, point-out investment problem of Jevons [Jevons, W.S., Theory of Political Economy, Macmillan, London 1871].     

Inventory control problem with freight cost and stochastic demand

This paper generalizes the standard newsboy model to the case including freight cost, in which the capacity of one container is the limit and the freight cost is proportional to the number of the containers used. We show that the optimal ordering quantity is either the newsboy solution or some multiple of the container’s capacity. We also propose an algorithm to compute the optimal policy. Furthermore, we generalize these results to the case in which the inventory and the price are determined jointly with emergency purchase.     

Bayesian demand updating in the lost sales newsvendor problem: A two-moment approximation

We consider Bayesian updating of demand in a lost sales newsvendor model with censored observations. In a lost sales environment, where the arrival process is not recorded, the exact demand is not observed if it exceeds the beginning stock level, resulting in censored observations. Adopting a Bayesian approach for updating the demand distribution, we develop expressions for the exact posteriors starting with conjugate priors, for negative binomial, gamma, Poisson and normal distributions. Having shown that non-informative priors result in degenerate predictive densities except for negative binomial demand, we propose an approximation within the conjugate family by matching the first two moments of the posterior distribution. The conjugacy property of the priors also ensure analytical tractability and ease of computation in successive updates. In our numerical study, we show that the posteriors and the predictive demand distributions obtained exactly and with the approximation are very close to each other, and that the approximation works very well from both probabilistic and operational perspectives in a sequential updating setting as well.     

Inventory models with managerial policy independent of demand

This paper is an extension of two papers. The first of these, published in European Journal of Operational Research, 2007, 112-120 is by Deng et al. (2007) and concerns inventory models for deteriorating items with ramp type demand. The second, published in Computer & Industrial Engineering, 2009, 1296-1300 is by Cheng and Wang (2009) and concerns inventory models for deteriorating items with trapezoidal type demand. The purpose of this paper is threefold. First, this paper will show that the optimal solution is independent of the demand considered in the two previous papers. Second, several replenishment cycles were considered during the finite time horizon, to balance the set-up cost with the sum of the deteriorated cost, holding cost, and shortage cost. Third, this paper will examine the same numerical example in Cheng and Wang (2009) to show that this new approach will result in the saving of 84.39%.     

Optimal replenishment policy for product with season pattern demand

This paper investigates a deterministic inventory model in which demand follows a seasonal pattern that repeats itself after a short time interval. An algorithm is developed for determining an optimal replenishment cycle, a shortage length and an order quantity such that the total profit per unit time is maximized.     

Dynamic demand fulfillment in spare parts networks with multiple customer classes

We study real-time demand fulfillment for networks consisting of multiple local warehouses, where spare parts of expensive technical systems are kept on stock for customers with different service contracts. Each service contract specifies a maximum response time in case of a failure and hourly penalty costs for contract violations. Part requests can be fulfilled from multiple local warehouses via a regular delivery, or from an external source with ample capacity via an expensive emergency delivery. The objective is to minimize delivery cost and penalty cost by smartly allocating items from the available network stock to arriving part requests. We propose a dynamic allocation rule that belongs to the class of one-step lookahead policies. To approximate the optimal relative cost, we develop an iterative calculation scheme that estimates the expected total cost over an infinite time horizon, assuming that future demands are fulfilled according to a simple static allocation rule. In a series of numerical experiments, we compare our dynamic allocation rule with the optimal allocation rule, and a simple but widely used static allocation rule. We show that the dynamic allocation rule has a small optimality gap and that it achieves an average cost reduction of 7.9% compared to the static allocation rule on a large test bed containing problem instances of real-life size.     

Optimal ordering and transfer policy for an inventory with stock dependent demand

This paper deals with an ordering-transfer inventory model to determine the retailer’s optimal order quantity and the number of transfers per order from the warehouse to the display area. It is assumed that the amount of display space is limited and the demand rate depends on the display stock level. The objective is to maximize the average profit per unit time yielded by the retailer. The proposed models and algorithms are developed to find the optimal strategy by retailer. Numerical examples are presented to illustrate the models developed and the sensitivity analysis is also reported.     

Coordinated control of distribution supply chains in the presence of fuzzy customer demand

This paper considers a single product inventory control in a Distribution Supply Chain (DSC). The DSC operates in the presence of uncertainty in customer demands. The demands are described by imprecise linguistic expressions that are modelled by discrete fuzzy sets. Inventories at each facility within the DSC are replenished by applying periodic review policies with optimal order up-to-quantities. Fuzzy customer demands imply fuzziness in inventory positions at the end of review intervals and in incurred relevant costs per unit time interval. The determination of the minimum of defuzzified total cost of the DSC is a complex problem which is solved by applying decomposition; the original problem is decomposed into a number of simpler independent optimisation subproblems, where each retailer and the warehouse determine their optimum periodic reviews and order up-to-quantities. An iterative coordination mechanism is proposed for changing the review periods and order up-to-quantities for each retailer and the warehouse in such a way that all parties within the DSC are satisfied with respect to total incurred costs per unit time interval. Coordination is performed by introducing fuzzy constraints on review periods and fuzzy tolerances on retailers and warehouse costs in local optimisation subproblems.     

Ordering and pricing decisions in a two-echelon supply chain with asymmetric demand information

This article examines coordinated decisions in a decentralized supply chain that consists of one supplier and one retailer, and faces random demand of a single product with a short life cycle. We consider a setting where the retailer has accurate demand information while the supplier does not. Such a problem with asymmetric demand information can be viewed as an extension of the newsboy problem in which both the supplier and the retailer possess the same demand information. Combining the mechanism of sharing demand information and that of quantity discount and return policy enables us to develop three coordinated models in contrast with the basic and uncoordinated model. We are able to show the ordinal relationship among the retailer’s optimal order quantities in these four models under a general form of random demand, and compare the supply chain profits and conduct sensitivity analysis analytically in four models under uniform random demand. We also provide numerical results under normal random demand that bear a resemblance to those under uniform random demand.     

A fuzzy genetic algorithm with varying population size to solve an inventory model with credit-linked promotional demand in an imprecise planning horizon

A genetic algorithm (GA) with varying population size is developed where crossover probability is a function of parents’ age-type (young, middle-aged, old, etc.) and is obtained using a fuzzy rule base and possibility theory. It is an improved GA where a subset of better children is included with the parent population for next generation and size of this subset is a percentage of the size of its parent set. This GA is used to make managerial decision for an inventory model of a newly launched product. It is assumed that lifetime of the product is finite and imprecise (fuzzy) in nature. Here wholesaler/producer offers a delay period of payment to its retailers to capture the market. Due to this facility retailer also offers a fixed credit-period to its customers for some cycles to boost the demand. During these cycles demand of the item increases with time at a decreasing rate depending upon the duration of customers’ credit-period. Models are formulated for both the crisp and fuzzy inventory parameters to maximize the present value of total possible profit from the whole planning horizon under inflation and time value of money. Fuzzy models are transferred to deterministic ones following possibility/necessity measure on fuzzy goal and necessity measure on imprecise constraints. Finally optimal decision is made using above mentioned GA. Performance of the proposed GA on the model with respect to some other GAs are compared.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging

Inventory model for time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging is considered in this paper. The demand rate is defined as a continuous trapezoidal function of time, and the backlogging rate is a non-increasing exponential function of the waiting time up to the next replenishment. We proposed an optimal replenishment policy for such inventory model, numerical examples to illustrate the solution procedure.