Stockout Period Research Articles

Modelling and Analysis of Compliment Inventory System in Supply Chain with Partial Backlogging

This paper presents a continuous review two echelon inventory systems with two different items in stock, one is main product and other one is complement item for the main product. The operating policy at the lower echelon for the main product is (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. We assume that the demands accruing during the stock-out period are partially backlogged. The retailer replenishes the stock of main product from the supplier which adopts (0,M) policy. The complement product is replenished instantaneously from local supplier. The joint probability disruption of the inventory levels of main product, complement item at retailer and the main product 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.

Performance Analysis of Substitutable Perishable Inventory System with Direct Demand in Supply Chain

This paper presents continuous review inventory systems in a simple chain 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 common demand for both products at distributor follows Poisson distribution with rate λd .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,M) 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.

Continuous Review Two-Echelon (s,S) Inventory System with Partial Backlogging

Inventories exist throughout the supply chain in various forms for various reasons. This paper presents a continuous review two echelon inventory system. The operating policy at the lower echelon is (s, S) that is whenever the inventory level traps 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. We assume that the demands accruing during the stock-out period are partially backlogged. The retailer replenishes their stock from the regular supplier which adopts (0,M) policy, M = n1Q. When the regular supplier stock is empty the replacement of retailer stock made by the outside supplier who adopts (0,N) policy N = n2Q. The joint probability disruption of the inventory levels of retailer, regular supplier and the outside 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.

An economic order quantity model with nonlinear holding cost, partial backlogging and ramp-type demand

ABSTRACTIn this article, a deterministic inventory model with a ramp-type demand depending on price and time is developed. The cumulative holding cost is assumed to be a nonlinear function of time. Shortages are allowed and are partially backlogged. Thus, the fraction of backlogged demand depends on the waiting time and on the stock-out period. The aim is to maximize the total profit per unit time. To do this, a procedure that determines the economic lot size, the optimal inventory cycle and the maximum profit is presented. The inventory system studied here extends diverse inventory models proposed in the literature. Finally, some numerical examples are provided to illustrate the theoretical results previously propounded.

An optimal inventory model with partial backorders and diminishing demand rate

Most of the inventory models discuss two extreme situations with respect to the demand in stock-out period. In this research, the demand rate within stock-out period is assumed as a diminishing function of backorders. Shortage of inventory will lead to the opportunity cost of lost sales. This research considers backorders and lost sales to formulate an inventory model with partial backorders. The objective of the inventory model is to determine the time scales of inventory depletion and stock-out periods in order to minimise total relevant cost. An efficient algorithm is developed to find the optimal solution. Finally, a sensitivity analysis is performed to study the effects of the model parameters on the optimal solution. This research apparently makes an improvement compared to the previous works, and further fits the requirements of practical environment.

A closed-form solution for the distribution free continuous review integrated inventory model

In this paper, we develop an integrated inventory model in a single-vendor single-buyer supply chain under an unknown demand distribution at the buyer. It is assumed that each lot delivered to the buyer contains a random fraction of defective items, and lead time can be reduced at an extra crashing cost. We also consider that the unmet demand at the buyer during the stockout period is partially backordered. A model is formulated to minimize the total expected relevant costs of the system considering an exact expression of the service level constraint to ensure that a certain percentage of customer orders are filled by the buyer. Closed-form expressions are derived for the optimal order quantity and safety factor for given lead time and shipment frequency, and then an algorithm is proposed to find the global optimal solution. Finally, a numerical example is presented to illustrate the solution procedure and sensitivity analysis is carried out to analyze the proposed model.

Perishable inventory system with service interruptions, retrial demands and negative customers

In this paper, we consider a continuous review perishable inventory system wherein demands arrive according to the Poisson process, each demanding exactly one unit of inventory item and the life time of each item is assumed to be exponential. The operating policy is (s, S) policy, i.e., whenever the inventory level drops to s, an order for Q(=S − s) items is placed. The ordered items are received after a random time which is distributed as exponential. The service may be interrupted according to the Poisson process in which case it restarts after an exponentially distributed time. The demands that occur during the server breakdown period or stock-out period may turn out to be ordinary or a negative demand and then they enter into the orbit of infinite size. These orbiting demands send out a signal to compete for their demand which is distributed as exponential. The matrix analytic method is used for the steady state distribution of the model. Various performance measures and cost analysis are shown with numerical results.

Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost

In this paper, an economic order quantity inventory model is analyzed, considering that the unit cumulative holding cost has two significant components: a fixed cost which represents the cost of accommodating the item in the warehouse and a variable cost given by a potential function of the length of time over which the item is held in stock. Shortages are allowed and, during the stockout period, only a fraction of demand is partially backordered. The backordering cost includes a fixed cost and a cost linearly dependent on the length of time for which backorder exists. A solution procedure is developed for determining the optimal inventory policy. Moreover, to illustrate the effects of some parameters on the optimal policy and the minimum total inventory cost, a numerical study is developed.

Two substitutable perishable product disaster inventory systems

A disaster inventory system is considered in which two substitutable items are stored for disaster management. In the event of disaster management, a particular product may become stock-out and the situation warrants that a demand for the particular product during its stock-out period may be substituted with another available similar product in the inventory. From the utility point of view, continuous review inventory models are quite appropriate in disaster inventory management. In this paper, a continuous review two substitutable perishable product disaster inventory model is proposed and analyzed. Since the inventory is maintained for disaster management, an adjustable joint reordering policy for replenishment is adopted. There is no lead time and the replenishment is instantaneous. For this model, some measures of system performance are obtained. The stationary behavior of the model is also considered. Numerical examples are also provided to illustrate the results obtained.

Decaying inventory model with stock-dependent demand and shortages under two-level trade credit

In the present day, competitive marketplace, offering delay payments, has become a commonly adopted method. Previous inventory models under permissible delay in payments usually assumed that the demand of the items was either constant or merely dependent on the retailing price. This paper deals with a lot-sizing model for deteriorating items with stock-dependent demand and delay in payments. In the model, shortages are allowed to occur and demand during the stockout period is partially backlogged. In this study, not only the supplier would offer fixed credit period to the retailer, but retailer also adopt the trade credit policy to his customer in order to promote the market competition. The retailer can accumulate revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. That is, the retailer can accumulate revenue and earn interest during this period with rate Ie under the condition of trade credit. At the end of the trade credit period offered by the supplier, the account is settled and the retailer starts paying the capital opportunity cost for the items in stock with an annual rate Ip. The minimization of the total annual cost is taken as the objective function to study the retailer’s optimal ordering policy. Lemmas and Theorem to determine the criterion for the existence and uniqueness of the minimum solution are subsequently developed. We provide an easy and useful computational flowchart and with the help of a computer code using the software Matlab 7.0, the optimal results are determined. A numerical example is included to demonstrate the developed model and the solution procedure. To investigate the effect of changes in some main parameter values on the optimal solution, we conduct a sensitivity analysis and discuss some important managerial insights.

Retrial inventory system with negative customers and service interruptions

In this paper, we consider a single server queueing system wherein demands arrive according to Poisson process, each demanding exactly one unit of inventory item. The operating policy is (s, S) policy in which an order is placed for a quantity up to S whenever the inventory level falls to s. The ordered items are received after a random time which is exponentially distributed. The service may be interrupted according to Poisson process in which case it restarts after an exponentially distributed time. The demands that occur during the server breakdown period or stock-out period may turn out to be ordinary or negative and they enter into the orbit of infinite size. These demands retry after a random time, which is again exponentially distributed. Matrix geometric solution is obtained for the steady state distribution of the model. Various performance measures and cost analysis are shown with numerical results.

An inventory model for deteriorating items with stock-dependent demand, time-varying holding cost and shortages

In this article, an inventory model for deteriorating items with two components demand and time-varying holding cost has been developed. Demand is assumed to be stock-dependent up to the level of available stock. After which the demand is considered as constant i.e. during stock-out period. Shortages are allowed and are fully backlogged. Profits are maximized in both the infinite and finite time horizon cases. Some special cases are also derived from the main models. Two numerical examples are provided for both finite and infinite time horizon. Sensitivity analysis performed has shown successful effects of various model parameters on net profit.

Modeling and Analysis of Perishable Inventory System with Retrial demands in Supply Chain

In this article, we consider a continuous review perishable inventory system with poisson demands. The maximum storage capacity at lower echelon (retailer) is S and the upper Echelon (Distribution Center) is M (= nQ). The life time of each item is assumed to be exponential. The operating policy is (s, S) policy, that is, whenever the inventory level drops to s, an order for Q = (S - s > s) item is placed. The ordered items are received after a random time which is distributed as exponential. We assume that demands occurring during the stock-out period enter into the orbit. These orbiting demands send out signal to complete for their demand which is distributed as exponential. The joint probability distribution of the inventory level at retailer, inventory level at DC and the number of demands in the orbit are obtained in the steady state case. Various system performance measures are derived and the results are illustrated numerically.

A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume(s,S)ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is aboves, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool 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.

An optimal shipment strategy for imperfect items in a stock-out situation

The objective of this paper is to minimize the expected total cost by consolidating a number of batches of imperfect quality products of different cycles for a single shipment and maximize the expected average profit function. Shortages are allowed at the end of each ordering cycle and are partially backlogged. The stock-out period is assumed to be exponentially distributed. The ordered lot size and cycle length of all the ordering cycles of a shipping cycle are considered to be equal. The model is analyzed for both infinite and finite planning horizons. The percentage of defective items in a lot is assumed to follow a uniform distribution. All the fractions of imperfect quality items in the ordering cycles of each shipping phase are assumed to be independent and identically distributed. After the completion of a screening process, one portion of the perfect quality products is used to serve partial backlogging at a cost per unit and the rest of the stock is used to adjust the demand. Calculus method is used to obtain the optimal number of order cycles for shipment. Optimal shortage period, optimal lot size and expected average profit for the model are developed on both of the finite and infinite time horizons. A comparison between the our models and Maddah and Jaber’s (2008) [1] model is done through numerical studies which are also used to illustrate the models graphically.

A survey of deterministic models for the EOQ and EPQ with partial backordering

Models for the basic deterministic EOQ or EPQ problem with partial backordering or backlogging make all the assumptions of the classic EOQ or EPQ model with full backordering except that only a fraction of the demand during the stockout period is backordered. In this survey we review deterministic models that have been developed over the past 40 years that address the basic models and extensions that add other considerations, such as pricing, perishable or deteriorating inventory, time-varying or stock-dependent demand, quantity discounts, or multiple-warehouses.

A deterministic model for deteriorating items with displayed inventory level dependent demand rate incorporating marketing decisions with transportation cost

Article history: Received 06 January 2011 Accepted 16 March 2011 Available online 17 March 2011 This paper deals with an inventory model, which considers the impact of marketing strategies such as pricing and advertising as well as the displayed inventory level on the demand rate of the system. In addition, the demand rate during the stock-out period differs from that during the stock-in period by a function varied on the waiting time up to the beginning of the next cycle. Shortage are allowed and partially backlogged. Here, the deterioration rate is assumed to follow the Weibull distribution. Considering all these factors with others, different scenarios of the system are investigated. To obtain the solutions of these cases and to illustrate the model, an example is considered. Finally, to study the effects of changes of different parameters of the system, sensitivity analyses have been carried out with respect to the different parameters of the system. © 2011 Growing Science Ltd. All rights reserved

Multi-objective models for lot-sizing with supplier selection

In this paper, two multi-objective mixed integer non-linear models are developed for multi-period lot-sizing problems involving multiple products and multiple suppliers. Each model is constructed on the basis of three objective functions (cost, quality and service level) and a set of constraints. The total costs consist of purchasing, ordering, holding (and backordering) and transportation costs. Ordering cost is seen as an ‘ordering frequency’-dependent function, whereas total quality and service level are seen as time-dependent functions. The first model represents this problem in situations where shortage is not allowed while in the second model, all the demand during the stock-out period is backordered. Considering the complexity of these models on the one hand, and the ability of genetic algorithms to obtain a set of Pareto-optimal solutions, we apply a genetic algorithm in an innovative approach to solve the models. Comparison results indicate that, in a backordering situation, buyers are better able to optimize their objectives compared to situations where there is no shortage. If we take ordering frequency into account, the total costs are reduced significantly.

Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit

This study models a finite horizon inventory problem for deteriorating and fashion goods under trade credit and partial backlogging conditions. Demand may vary with price or time. The supplier can extend credit to the retailer. As a result, the retailer does not have to pay for goods immediately upon acquiring them, and can instead earn interest on the retail price of the goods between the time they are sold and the end of the credit period. The proposed model considers two-phase pricing and inventory decisions. In other words, it determines both the optimal prices and the lengths of the in-stock and stock-out period. This paper is the first to consider different price decisions for in-stock and stock-out periods under trade credit. We develop an algorithm to determine the optimal pricing and replenishment strategy while still maximizing the total profit. Further, this study shows that the proposed two-phase pricing strategy is superior to a one-phase pricing strategy in terms of profit maximization. Computational analysis illustrates the solution procedures and the impacts of the related parameters on decisions and profits. The results of this study can serve as references for business managers or administrators.

An inventory model for deteriorating items with inflation induced demand and exponential partial backorders—a discounted cash flow approach

Now-a-days the financial situation has changed at different countries due to inflation i.e. time value of money. As inflation increases the value of money goes down and it erodes the future worth of savings and forces one for more current spending. Usually, these spending are on peripherals and luxury items that give rise to demand of these items. In this paper, we intend to develop an inventory model for deteriorating items under inflationary conditions using a discounted cash flow (DCF) approach over a finite planning horizon. The demand rate is a function of inflation. In real life systems, some but not all customers will wait for the next order to satisfy their demands during stockout period. This situation is modeled by the consideration of partial backordering under negative exponential distribution of waiting time. An algorithm is developed to find optimal policy and the sensitivity analysis is studied with the help of numerical examples.