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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Optimal price and lot size determination for a perishable product under conditions of finite production, partial backordering and lost sale

The paper describes an EOQ model of a perishable product for the case of price dependent demand, partial backordering which depends on the length of the waiting time for the next replenishment, and lost sale. The model is solved analytically to obtain the optimal price and size of the replenishment. In the model, the customers are viewed to be impatient and a fraction of the demand is backlogged. This fraction is a function of the waiting time of the customers. In most of the inventory models developed so far, researchers considered that inventory accumulates at the early stage of the inventory and then shortage occurs. This type of inventory is called IFS (inventory followed by shortage) policy. In the present model we consider that shortage occurs before the starting of inventory. We have proved numerically that instead of taking IFS, if we consider SFI (shortage followed by inventory) policy, we would get better result, i.e., a higher profit. The model is extended to the case of non-perishable product also. The optimal solution of the model is illustrated with the help of a numerical example.     

Note on inventory model with a mixture of back orders and lost sales

Competitiveness is an important means of determining whether a company will prosper. Business organizations compete with one another in a variety of ways. Among these competitive methods are time and cost factors. The purpose of this paper is to examine the inventory models presented by Padmanabhan and Vrat [International Journal of Systems Sciences 21 (1990) 1721] with a mixture of back orders and lost sales. We develop the criterion for the optimal solution for the total cost function. If the criterion is not satisfied, this model will degenerate into one cycle inventory model with a finite inventory period. This implies an extension of shortage period as long as possible to produce lower cost. However, we know that time is another important factor in company competitiveness. Customers will not indefinitely wait for back orders. A tradeoff will be made between the two most important factors; time and cost. The minimum total cost is evaluated under the diversity cycle time and illustrations are applied to explain the calculation process. This work provides a reference for decision-makers.     

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.     

An inventory model with partial backordering and unit backorder cost linearly increasing with the waiting time

As the implementation of JIT practice becomes increasingly popular, each echelon in a supply chain tends to carry fewer inventories, and thus the whole supply chain is made more vulnerable to lost sales and/or backorders. The purpose of this paper is to recast the inventory model to be more relevant to current situations, where the penalty cost for a shortage occurrence at a downstream stage in a supply chain is continually transmitted to the upstream stages. The supplier, in this case, at the upstream of the supply chain is responsible for all the downstream shortages due to the chain reaction of its backlog. The current paper proposes a model in which the backorder cost per unit time is a linearly increasing function of shortage time, and it claims that the optimal policy for the supplier is setting the optimal shortage time per inventory cycle to minimize its total relevant cost in a JIT environment.     

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 dynamic pricing and inventory control with stock deterioration and partial backordering

This paper studies the optimal dynamic pricing and inventory control policies in a periodic-review inventory system with fixed ordering cost and additive demand. The inventory may deteriorate over time and the unmet demand may be partially backlogged. We identify two sufficient conditions under which (s,S,p) policies are optimal.     

Reducing lost-sales rate on the stochastic inventory model with defective goods for the mixtures of distributions

In this paper, we assume that the demands of different customers are not identical in the lead time. Thus, we investigate a continuous review inventory model involving controllable lead time and a random number of defective goods in buyer’s arriving order lot with partial lost sales for the mixtures of distributions of the lead time demand to accommodate more practical features of the real inventory systems. Moreover, we analyze the effects of increasing investment to reduce the lost sales rate when the order quantity, reorder point, lost sales rate and lead time are treated as decision variables. In our studies, we first assume that the lead time demand follows the mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. By analyzing the total expected cost function, we develop an algorithm to obtain the optimal ordering policy and the optimal investment strategy for each case. Finally, we provide numerical examples to illustrate the results.     

On an inventory model for deteriorating items and time-varying demand

In this paper, we present an optimal procedure for finding the replenishment schedule for the inventory system in which items deteriorate over time and demand rates are increasing over a known and finite planning horizon.     

Single item lot-sizing problem for a warm/cold process with immediate lost sales

We consider the dynamic lot-sizing problem with finite capacity and possible lost sales for a process that could be kept warm at a unit variable cost for the next period t + 1 only if more than a threshold value Q_t has been produced and would be cold, otherwise. Production with a cold process incurs a fixed positive setup cost, K_t and setup time, S_t, which may be positive. Setup costs and times for a warm process are negligible. We develop a dynamic programming formulation of the problem, establish theoretical results on the structure of the optimal production plan in the presence of zero and positive setup times with Wagner–Whitin-type cost structures. We also show that the solution to the dynamic lot-sizing problem with lost sales are generated from the full commitment production series improved via lost sales decisions in the presence of a warm/cold process.     

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.