A two-echelon inventory model with lost sales

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R_i) control policy. The warehouse operates an (SQ, (S − 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L_i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.     

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.     

Dynamic pricing inventory control under fixed cost and lost sales

This paper studies a periodic review pricing and inventory replenishment problem which encounters stochastic demands in multiple periods. In many inventory control problems, the unsatisfied demand is traditionally assumed to be backlogged but in this paper is assumed to be lost. In many practical problems, a consumer who could not buy what he/she wants in one store is not willing to wait until that store restocks it but tries to buy alternatives in other stores. Also, in this paper, the random variable for the demand function is assumed to be general, which means that any probability function for the random variable can be applied to our result. Cost terms consist of the holding cost by the leftover, the shortage cost by lost sales, and the strictly positive fixed ordering cost. The objective of this paper is to dynamically and simultaneously decide the optimal selling price and replenishment in each period by maximizing the expected profit over the finite selling horizon. We show that, under the general assumption on the random variable for the demand, the objective function is K$K$-concave, an (s,S)$(s\text{,}S)$ policy is optimal for the replenishment and the optimal price is determined based on the inventory level after the replenishment in each period.     

An exact solution procedure for multi-item two-echelon spare parts inventory control problem with batch ordering in the central warehouse

We consider a multi-item two-echelon inventory system in which the central warehouse operates under a (Q,R) policy, and the local warehouses implement basestock policy. An exact solution procedure is proposed to find the inventory control policy parameters that minimize the system-wide inventory holding and fixed ordering cost subject to an aggregate mean response time constraint at each facility.     

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.     

Research note on the criteria for the optimal solution of the inventory model with a mixture of partial backordering and lost sales

Perishable products are commonly seen in inventory management. By allowing shortages and backlogging, the impact on the cost from the decay of the products can be balanced out. In a recent paper published in Computers and Industrial Engineering [P.L. Abad, Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale, Comput. Ind. Eng. 38 (2000) 457–465] considered a problem in such context. However, his algorithm was incomplete due to flaws in his solution procedure. The purpose of this note is to explore the same production inventory models with a mixture of partial backordering and lost sales for deteriorated items. We find the criteria for the optimal solution for different cases and derive a formulated minimum value. By theoretical analysis, we develop a few lemmas to reveal parameter effects and optimal solution procedure. The solutions are illustrated by solving the same examples from Abad’s paper to illustrate the accuracy and completeness of our procedure.     

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.     

Determining the fill rate for a periodic review inventory policy with capacitated replenishments,lost sales and zero lead time

In this paper we consider a periodic review order-up-to inventory system with capacitated replenishments, lost sales and zero lead time. We consider discrete demand. It is shown that the initial stock levels of the different review periods form a Markov chain and we determine the transition matrix. Furthermore we study for what probability mass functions of the review period demand the Markov chain has a unique stationary distribution. Finally, we present a method to determine the fill rate.     

Continuous-review,lost-sales inventory models with Poisson demand,a fixed lead time and no fixed order cost

This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.     

A two-echelon base-stock inventory model with Poisson demand and the sequential processing of orders at the upper echelon

We consider a two-echelon inventory system with a number of non-identical, independent ‘retailers’ at the lower echelon and a single ‘supplier’ at the upper echelon. Each retailer experiences Poisson demand and operates a base stock policy with backorders. The supplier manufactures to order and holds no stock. Orders are produced, in first-come first-served sequence, with a fixed production time. The supplier therefore functions as an M/D/1 queue. We are interested in the performance characteristics (average inventory, average backorder level) at each retailer. By finding the distribution of order lead time and hence the distribution of demand during order lead time, we find the steady state inventory and backorder levels based on the assumption that order lead times are independent of demand during order lead time at a retailer. We also propose two alternative approximation procedures based on assumed forms for the order lead time distribution. Finally we provide a derivation of the steady state inventory and backorder levels which will be exact as long as there is no transportation time on orders between the supplier and retailers. A numerical comparison is made between the exact and approximate measures. We conclude by recommending an approach which is intuitive and computationally straightforward.     

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.     

Dynamic inventory rationing strategies for inventory systems with two demand classes,Poisson demand and backordering

We study inventory systems with two demand classes (critical and non-critical), Poisson demand and backordering. We analyze dynamic rationing strategies where the number of items reserved for critical demand depends on the remaining time until the next order arrives. Different from results in the literature, we do not discretize demand but derive a set of formulae that determine the optimal rationing level for any possible value of the remaining time. Moreover, we show that the cost parameters can be captured in a single relevant dimension, which allows us to present the optimal rationing levels in charts and lookup tables that are easy to implement. Numerical examples illustrate that the optimal dynamic rationing strategy outperforms all static strategies with fixed rationing levels.     

Optimal control of a finite-capacity inventory system with setup cost and lost sales

One of the most fundamental results in inventory theoryis the optimality of (s, S) policy for inventory systems withsetup cost. This result is established based on a key assumptionof infinite production/ordering capacity. Several studies haveshown that, when there is a finite production/ordering capacity,the optimal policy for the inventory system is very complicatedand indeed, only partial characterization for the optimal policyis possible. In this paper, we consider a continuous reviewinventory system with finite production/ordering capacity andsetup cost, and show that the optimal control policy for thissystem has a very simple structure. We also develop efficientalgorithms to compute the optimal control parameters.     

Lost-sales inventory systems with a service level criterion

Competitive retail environments are characterized by service levels and lost sales in case of excess demand. We contribute to research on lost-sales models with a service level criterion in multiple ways. First, we study the optimal replenishment policy for this type of inventory system as well as base-stock policies and (R, s, S) policies. Furthermore, we derive lower and upper bounds on the order-up-to level, and we propose efficient approximation procedures to determine the order-up-to level. The procedures find values of the inventory control variables that are close to the best (R, s, S) policy and comply to the service level restriction for most of the instances, with an average cost increase of 2.3% and 1.2% for the case without and with fixed order costs, respectively.     

Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are expensive, ready–for–use spare parts are kept in stock to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival. Each stockpoint faces demands from multiple demand classes. We are interested in the optimal lateral transshipment policy. There are three ways in which a demand can by satisfied: from own stock, via a lateral transshipment, or via an emergency procedure. Using stochastic dynamic programming, we characterize and prove the structure of the optimal policy, that is, the policy for satisfying the demands which minimizes the average operating costs of the system. This optimal policy is a threshold type policy, with state-dependent thresholds at each stockpoint for every demand class. We show a partial ordering in these thresholds in the demand classes. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, two policies that are often assumed in the literature. Furthermore, we study several model extensions which fit in the same modeling framework.     

On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns

The primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.     

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.     

A two-echelon inventory model for a deteriorating item with stock-dependent demand,partial backlogging and capacity constraints

This study investigates a two-echelon supply chain model for deteriorating inventory in which the retailer’s warehouse has a limited capacity. The system includes one wholesaler and one retailer and aims to minimise the total cost. The demand rate in retailer is stock-dependent and in case of any shortages, the demand is partially backlogged. The warehouse capacity in the retailer (OW) is limited; therefore the retailer can rent a warehouse (RW) if needed with a higher cost compared to OW. The optimisation is done from both the wholesaler’s and retailer’s perspectives simultaneously. In order to solve the problem a genetic algorithm is devised. After developing a heuristic a numerical example together with sensitivity analysis are presented. Finally, some recommendations for future research are presented.