Stockout Period Research Articles

Dynamic pricing and replenishment policy under price, time, and service level-dependent demand and preservation investment

Time, price, and servicing are three important factors which affect the demand for any product. In a multi-period inventory model, different pricing in different periods may have a positive impact on demand. Offering discounts to customers during a shortage period is an efficient way to reduce lost sales. To address all these issues, a multi-period inventory model is developed and analyzed in this paper to obtain optimal pricing and replenishment periods where the market demand depends on price, time and service level provided by the retailer. The time horizon is fixed; the replenishment cycles are of equal length, but the shortage period may vary in different cycles. The retailer is allowed to charge different prices in different periods and provide discounts if someone places demand during a shortage period. Learning effects in holding and ordering costs are considered. Under a dynamic pricing framework, this study examines service improvement through learning and price discounts during shortages as an incentive for customers. It demonstrates that discounting indeed helps to produce more revenue when the back-order rate is low, and the service level is critical in generating demand when the stock-out period is lengthy, revealing a critical relationship between shortage periods and service level. A numerical experiment is performed to investigate the optimal policy in different situations and obtain some managerial insights.

On the Control Policy of a Queuing–Inventory System with Variable Inventory Replenishment Speed

This paper considers a make-to-order production–inventory system that comprises a production facility and an inventory warehouse. Customers arrive at the facility to place an order, and the orders are processed using the first-come-first-served (FCFS) discipline. The warehouse supplies inventory items (raw materials) for the production process, and the warehouse inventory is replenished by internal production. The speed of internal production can be controlled through additional costs. If the inventory level drops to zero, the unmet demand waits in the facility until the inventory is replenished. During the stockout period, newly arriving demand is lost. The stationary joint probability of unmet demands and inventory items is derived, and a cost model is constructed. The optimal control policy for internal production is investigated to minimize the cost per unit time of the system. The experimental results show that such a production speed adjustment could reduce costs by up to 42% compared to the cases without the adjustment.

AN EOQ MODEL WITH TRADE CREDIT BASED DEMAND UNDER INFLATION

In this work, an EOQ model is presented with trade credit period and time dependent demand under inflation and delay in payments for deteriorating items. Shortages are permitted and partially backlogged that depends on the waiting time of next replenishment cycle. The holding cost and deterioration rate is considered constant. The aim of this study is to maximize the total prot. An algorithm is presented to get the optimal values of total profit, total inventory and stock-out period. To illustrate theoretical model numerical assessments, graphical representation and sensitivity analysis is also discussed.

Effect of Inflation and Permitted Three-Slot Payment on Two-Warehouse Inventory System with Stock-Dependent Demand and Partial Backlogging

This study examines the effect of monetary inflation for a two-warehouse single-product inventory system, in which items are stored in a limited capacity Own Warehouse (OW) and an unlimited capacity Rental Warehouse (RW). Demand for an item is considered stock dependent. Items may deteriorate at a different constant rate in both warehouses. Shortages are allowed in the stock-out period and are partially backlogged and satisfied in the next replenishment point. The supplier permits flexible payment options for the retailer to pay the amount in three equal payments at different time points. The retailers’ preferred payment option is as follows: the first payment is prior to the replenishment point with some discount; the second payment is one-third of the total purchasing cost, which is paid at the time of the replenishment epoch; and the third payment is after the replenishment point and before the start of the next cycle, with some penalty. The influence of inflation on the cost calculation is considered, and an analytic expression for optimal minimal cost is explicitly derived from this. We performed arrived sensitivity analysis to discern the effects of the inflation and backlogging rates, as well as the effects of the discount rate on purchasing cost, and the effects of penalties upon the late payment of purchasing costs in optimizing the total cost.

An inventory system with time-dependent demand and partial backordering under return on inventory investment maximization

In this article, we study an inventory system for items that have a power demand pattern and where shortages are allowed. We suppose that only a fixed proportion of demand during the stock-out period is backordered. The decision variables are the inventory cycle and the ratio between the initial stock and the total quantity demanded throughout the inventory cycle. The objective is to maximize the Return on Inventory Investment (ROII) defined as the ratio of the profit per unit time over the average inventory cost. After analysing the objective function, the optimal global solutions for all the possible cases of the inventory problem are determined. These optimal policies that maximize the ROII are, in general, different from those that minimize the total inventory cost per unit time. Finally, a numerical sensitivity analysis of the optimal inventory policy with respect to the system input parameters and some useful managerial insights derived from the results are presented.

Availability of Essential Medicines in Ethiopia: A Systematic Review

Objective: Essential Medicines (EMs) are those that satisfy the priority health care needs of the population. Access to health care including EMs is a fundamental human right. In Ethiopia, studies designate the frequent drug shortages in public health facilities. Hence, this review aimed to assess and evaluate the national availability of essential medicines in Ethiopia. Method: A systematic searching for all study designs which are published and/or reported from inception to August 27/2019 strategy was conducted in two literature databases (Google Scholar and PubMed. Additionally, references of related articles were searched using different search engines. Data qualities were assessed by abstracting the quality procedures taken in each individual’s studies and cross checked with Joanna Briggs Institute (JBI) quality appraisal criteria. Results: A total of 256 studies, including duplicates, were primarily searched. Nine studies were met current study inclusion criteria. All included studies were published from 2003 to 2019 G.C. Consequently, reviewed studies conducted a total of 356 public and private drug outlets. All studies used cross-sectional study design. The most frequent concepts emerged in these documents were regarding availability and affordability of EMs (5 Articles), availability and pharmaceutical inventory management (3 Articles), one document on availability and price. The average national availability of EMs was precisely in public and private facilities scored about 70.16% and 70.1%, respectively. Results from the sub-national study indicate that, the national average stock out duration in primary health care facilities was 99.2 days, highest in Benishangul Gumuz (139 days). The most frequently reported stocked out drugs within the past 6-12 months at the time of study were; ORS, Amoxicillin syrup 125mg/5ml, Tetracycline eye ointment and tablet, Ergometrine injection in Adama and chloroquine syrup in Gondar. Whereas, carbamazepine 100mg/5ml syrup, diazepam 5mg/ml ampoule, gentamicin 20mg/2ml ampoule and ibuprofen 100mg/5ml suspension were not found in any drug outlets in both public and private sectors at the day of visit in West Wollega. Conclusion: The finding of this review indicates the average national availability of essential medicines in Ethiopia is presently under the requirements of WHO recommendations. Moreover, the longer stockout period was also perceived in healthcare facilities. Thus, might need further attention of the respective stakeholders.

Profit maximization in an inventory system with time-varying demand, partial backordering and discrete inventory cycle

In this paper, an inventory problem where the inventory cycle must be an integer multiple of a known basic period is considered. Furthermore, the demand rate in each basic period is a power time-dependent function. Shortages are allowed but, taking necessities or interests of the customers into account, only a fixed proportion of the demand during the stock-out period is satisfied with the arrival of the next replenishment. The costs related to the management of the inventory system are the ordering cost, the purchasing cost, the holding cost, the backordering cost and the lost sale cost. The problem is to determine the best inventory policy that maximizes the profit per unit time, which is the difference between the income obtained from the sales of the product and the sum of the previous costs. The modeling of the inventory problem leads to an integer nonlinear mathematical programming problem. To solve this problem, a new and efficient algorithm to calculate the optimal inventory cycle and the economic order quantity is proposed. Numerical examples are presented to illustrate how the algorithm works to determine the best inventory policies. A sensitivity analysis of the optimal policy with respect to some parameters of the inventory system is developed. Finally, conclusions and suggestions for future research lines are given.

A Finite Source Two Commodity Inventory System with Retrial Demands and Multiple Server Vacation

Consider a two commodity continuous review inventory system with a finite number of homogeneous sources generating demands. The maximum storage capacity for the commodity - i is S i (i = 1, 2). We adopt a joint reordering policy for placing orders. The lead time for the delivery of orders is assumed to have an exponential distribution. An arriving demand receives either commodity - 1 or commodity - 2 according to a Bernoulli trial. The commodities are assumed to be substitutable in the sense that during the zero stock of any commodity, the demand for that commodity can be met by the unit of the other commodity. The server goes for a vacation of an exponentially distributed duration whenever the inventory level of both commodities reaches zero. If the server finds empty stock when he returns to the system, he immediately takes another vacation. The demands that occur during stock-out period or during the server vacation period enter into the orbit of finite size. These orbiting demands retry for their demand after a random time, which is assumed to be an exponential distribution. The joint probability distribution of the inventory level and the number of demands in the orbit is obtained in the steady state case. Various system performance measures in the steady state are derived and the long-run total expected cost rate is calculated.

Price determination of a non-instantaneous deteriorating EOQ model with shortage and inflation under delay in payment

ABSTRACT This article analyses, Economic Order Quantity (EOQ) based inventory model for non-instantaneous deteriorating items with inflation under permissible delay in payment. Demand is a linear decreasing function of selling price. Shortages are allowed which are partially backlogged. An algorithm is proposed to find the optimal selling price, optimal stock-out period, optimal replenishment cycle time and the optimal ending inventory level. To ascertain the optimal solution exists, the conditions for the total profit in the inventory model which attains its global maximum are provided. Comparisons with the present model without inflation, without delay in payment, complete backlogging and instantaneous deterioration are presented. The numerical examples, graphical representation, and sensitivity analysis are given to illustrate the practical application of the proposed model.

Two warehouse dispatching policies for perishable items with freshness efforts, inflationary conditions and partial backlogging

Application of freshness preservation technologies in case of perishable inventory management can play a vital role in mitigating the losses that occur due to deterioration of perishable inventory, typical perishable inventory includes Agri-fresh products like fresh fruits, fresh meat, fresh vegetables, seafood and packed foods. In case of perishable inventory price as well as freshness both decide the demand, so supplying a fresh product at a competitive price creates more customer satisfaction and makes a firm profitable. This paper develops two warehouse dispatching policies, first in, first out (FIFO) and last in, first out (LIFO) for perishable items taking deterioration into account. Demand varies as a function of price and freshness keeping effort. As the price increases and freshness keeping efforts decreases, demand decreases, and vice versa. Stockout is allowed, and during the stockout period as the waiting time increases backlogging rate decreases exponentially. Inflation is considered while calculating different costs. The objective of this paper is to determine the optimum price per unit item and lot size to maximize the profit. The sensitivity of the models is examined through the design of experiment. The present models are applicable for perishable items that start losing their quality immediately after arriving into the system.

A production inventory model for deteriorating items with backlog-dependent demand

This paper investigates a production inventory model under classical EPQ framework with the assumption that the customer demand during the stock-out period is affected by the accumulated back-orders. The backlog rate is not fixed; instead, the demand during stock-out is assumed to decrease proportionally to the existing backlog which is thereby approximated by a piecewise constant function. Deteriorating items are taken into consideration in this work. For better illustration of the theoretical results and to highlight managerial insights, numerical examples are presented which are then compared to the results obtained by considering an exact (non-approximated) backlogging rate (from literature). The comparisons indicate high quality results for the approximated model.

Continuous review inventory model with lost sale reduction and ordering cost dependent on lead time for the mixtures of distributions

We consider a continuous review inventory system for inventory model involving lost sales reduction through capital investment cost function and the reduction of lead time further which reduces the ordering cost. To reduce the lost sales rate, two forms of capital investment cost function, viz. logarithmic and power are employed. Two relationships between ordering cost and lead time, viz. linear and logarithmic are considered. We develop four inventory models by taking different combinations of capital investment cost function and ordering cost lead time relationship. Objective of the study is to reduce the total related cost by simultaneously optimizing the order quantity, safety factor, fraction of the shortages during the stock-out period that will be lost and length of lead time. The lead time demand is assumed to follow a mixture of normal distributions. The optimal solution is derived by developing computer programs using the software MATLAB. We also provide four numerical examples to illustrate the models.

Impact of delay in payment, shortage and inflation on an EOQ model with bivariate demand

The major challenge of inventory decision makers is to determine an inventory optimization strategy that ensures the right balance between keeping abundant on hand inventory to meet the demand of the customers and optimizing costs related to holding inventory. This article analyzes on providing a general deterministic inventory model in which the rate of demand is determined by price and time over the ordering cycle time. The traditional assumption of zero ending invento ry level is relaxed to a non-zero ending inventory level. Shortages are allowed which are partially backlogged. We develop models with partial backlogging and without backlogging. The aim is to maximize the profit per unit time, assuming delay in payment and inflation. An algorithm is proposed to find the optimal selling price, optimal stockout period, optimal replenishment cycle time and the optimal ending inventory level. All the possible special cases of these two models are also discussed. The numerical examples, graphical representation, and sensitivity analysis are given to illustrate the practical application of the proposed model.

A simulation-optimisation genetic algorithm approach to product allocation in vending machine systems

In recent years, vending machines have seen increasing levels of popularity. In a fast-paced world where convenience and accessibility of products is highly sought after the vending industry has provided a suitable solution. Although the economic impact of the vending industry is indisputable, it is not without challenges, especially when it comes to the efficiency of the vending logistics operations.The optimisation of logistic vending machine systems is decidedly complex. Product allocation to columns in a vending machine, replenishment points of products, product thresholds at vending machines, and vehicle routes for inventory replenishments are all essential challenges in vending machine system management and operation. If all facets of the problem were to be addressed, it would require techniques such as forecasting, machine learning, data mining, combinatorial optimization and vehicle routing, among others. In the past, these approaches have been explored individually despite their intrinsic interdependence within the problem. This paper aims to help to fill in this gap and proposes a model for the optimisation of product allocation within a vending machine under the constraint of fixed restocking instances. The optimal product allocation is based on the definition of product profitability which accounts for the net revenue earned after the cost of restock, as opposed to the revenue earned until first stock-out to prevent arbitrary extension of the stock-out period. The whole approach is encompassed in the simulation-optimisation framework that utilises a Genetic Algorithm, with fitness evaluated as simulated revenue, to determine the optimal product allocation. The acceptable threshold of missed sales for a machine is also determined as a means to make intelligent restocking decisions. Overall, the proposed approach allows the strengths of mathematically robust optimization algorithms and the implementation of analytic solutions to be combined and applied to realistic scenarios where uncertainty may rule out some high quality analytic solutions. It respects problem intricacies proper to vending and addresses the interdependence between routing and portfolio optimisation.The proposal is application-driven and stems from a collaboration with an industry partner. The model is validated against an authentic data set supplied by the partner. The case study results revealed a network-wide improvement in net revenue of approximately 3.4%, with varied efficacy based on machine popularity. The method of optimisation was found to be significantly more effective for higher performing machines, with median improvements as high as 6%.Our framework based on the optimization-simulation model yields clear benefits to vending logistics operations management. The simulation component provides the decision maker with a more comprehensive view on the actual implementation of the solution. Effectively, the joint use of simulation and optimization methods provides managers with enhanced information to help decide on both: (i) the most beneficial product portfolio, and (ii) the quality of the proposed restocking schedules. Simulation-optimisation based approach is a powerful technique used to address stochastic problems. However, it was yet to be applied specifically to logistic vending machine systems.

Analysis of an inventory system with discrete scheduling period, time-dependent demand and backlogged shortages

A deterministic inventory model for goods with a power demand pattern, allowing backlogged shortages, is analyzed. The inventory cycle must be an integer multiple of a fixed time period (called basic period). Demand follows a power demand pattern during this basic time period. Demand not satisfied along the inventory cycle is fully backlogged and the stock-out period must be an integer multiple of the basic period. The goal of the inventory management is to minimize the inventory cost per unit time, considering that this inventory cost is the sum of the costs of holding, backlogging and ordering. Thus, the objective function of the inventory problem depends on two integer decision variables: the number of basic periods contained in an inventory cycle and the number of basic periods that constitute the stock-in period. The optimal inventory policy is found through a two-dimensional search method. This procedure is based on the properties of the inventory cost function. An algorithmic approach to calculate the economic order quantity and the optimal inventory cycle that minimize the total cost per unit time is proposed. The theoretical results are discussed and illustrated with some numerical examples. Finally, a sensitivity analysis of the optimal policy with respect to some parameters of the inventory model is developed.

Comparison of three vacation policies in discrete time inventory system with postponed demands

In this paper, we consider a single server discrete time inventory system with postponed demands and server vacation. Arrival process follows DMAP and ordering policy follows policy with geometrically distributed lead time. Due to depletion of inventory in the stock to zero, server may avail vacation immediately or after some random time. Here, we proposed three different vacation policies, namely modified multiple vacation (random idle time before each vacation), single (only one vacation) and multiple (one after another vacation). The demand who arrives during the stock-out period or server vacation period enters the pool of infinite size. These pooled demands are selected one-by-one according to a rule. The vacation time, the server idle time and the inter-selection time follow independent DPH distribution. The joint probability distribution of the number of demands in the pool, the server status, the on-hand inventory level is obtained in the steady-state case. We derived the total expected cost function in the steady-state case for each vacation model. We present some numerical examples and try to justify the best vacation policy for different parameter rates. Also, we analyse the impact of system costs on the total optimal cost of modified multiple vacation policy.

A two-echelon integrated inventory model under generalized lead time distribution with variable backordering rate

This paper investigates a generalized lead time distribution with a variable backordering rate in a two-echelon supply chain system. The vendor produces a single product and delivers to the buyer in equal sized batches. The delivery lead time follows a generalized stochastic variable. Shortages are allowed to occur and backordered partially. The backorder rate depends on the demand on stock-out period. Based on this notion, we formulate a mixed integer non-linear cost function which needs to be minimized with respect to reorder point, number of deliveries and lot size from the vendor to the buyer, to operate cooperatively in the integrated model. Analytically we proved the convexity of the generalized lead time distribution cost function with respect to the control parameters. Further, the uniqueness of optimality has been proved. To validate the proposed model, uniform, exponential and normal distributed lead times are presented in numerical example section. Sensitivity analysis also performed to the values of the parameters.

Earthquakes and Brand Loyalty: Beyond the Short-Term Effects of Stockouts

We exploit a natural experiment that exogenously removed the top leading brands of beer from the retail stores for several weeks to test whether prolonged stockouts can erode market shares persistently and study the potential mechanisms at play. Using panel data of consumer purchases before and after the product shortage, we observe that the top brands only partially recovered their pre-stockout market shares, especially among their most frequent buyers. Moreover, we identify a sizable portion of consumers who tried some small brands for the first time during the stockout period and remained buying those products persistently. We estimate a choice model with heterogeneous preferences to control for prices, state dependence, and product availability and find that the exposure to stockouts has long-run effects in purchase behavior. We interpret our estimates as evidence that consumers facing a restricted choice set may become aware or learn about competing products with long-lasting consequences in terms of their preferences.

Analysis of Two-Echelon Inventory System with Two Demand Classes with Partial Backlogging

This paper deals with a continuous review two-echelon inventory system with two demand classes. Two echelon inventory systems consist of one retailer (lower echelon) and one distributor (upper echelon) handling a single finished product. The demand at retailer is of two types. The First type of demand is usual single unit and the second type is of bulk or packet demand. The arrival distribution for single and packet demands are assumed to be independent Poisson with rates λ1(>0) and λd (>0) respectively. The operating policy at the lower echelon for the (s, S) that is whenever the inventory level drops to ‘s’ on order for Q = (S-s) items is placed, the ordered items are received after a random time which is distributed as exponential with rate µ>0. We assume that the demands occurring during the stock-out period are lost. The retailer replenishes the stock from the distributor which adopts (0, M) policy. The objective is to minimize the anticipated total cost rate by simultaneously optimizing the inventory level. The joint probability disruption of the inventory levels at retailer and the distributor are obtained in the steady state case. Various system performance measures are derived and the long run total expected inventory cost rate is calculated. Several instances of numerical examples, which provide insight into the behavior of the system, are presented.

A Two Echelon Subsutable Inventory Model for Deteriorating Items

This paper presents continuous review two-echelon inventory systems with two different substitutable items in stock. The demand for the products follows independent Poisson with rates λ1 and λ2 respectively for product and A and B. The operating policy at the lower echelon for the products are (si , Si) that is whenever the inventory level drops to ‘si’ on order for Qi = (Si-si) items is placed, the ordered items are received after a random time which is distributed as exponential. We assume that the demands accruing during the stock-out period are lost. The retailer replenishes the stock of products from the supplier which adopts (0,Mi) policy. The joint probability disruption of the inventory levels of the products, at retailer and the products at supplier are obtained in the steady state case. Various system performance measures are derived and the long run total expected inventory cost rate is calculated. Several instances of numerical examples, which provide insight into the behavior of the system, are presented.