A partial backorder control for continuous review (r, Q) inventory system with poisson demand and constant lead time

doi:10.1016/0305-0548(94)00062-D

Computers & Operations Research

Volume 22, Issue 7, August 1995, Pages 689-700

https://doi.org/10.1016/0305-0548(94)00062-D Get rights and content

Abstract

In this paper, we consider the partial backorder policy used in conjunction with the traditional stochastic (r, Q) inventory system. The partial backorder control is modeled using a control variable, b, which limits the maximum number of backorders allowed to be accumulated in any given cycle. In traditional inventory control policies, the unsatisfied demands are either completely backordered or completely lost. The new control variable b provides an alternative to dealing with shortages, which is different from the two extremes policies: the pure backorder policy (which corresponds to b=∞) and the lost sales policy (b=0). We obtain the expected annual cost of the model and give a procedure to compute the optimal parameters, $r^{∗}, b^{∗}$ and $Q^{∗}$ . Numerical examples which demonstrate the advantage of implementing the partial backorder policy are presented and the percentage of cost savings depends on the fill rate.

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Determining optimal uptime considering an unreliable machine, a maximum permitted backorder level, a multi-delivery plan, and disposal/rework of imperfect items
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Citation Excerpt :
Consequently, for different demand rates and cost variables, the authors found approximations of the optimal (s, S) values. Rabinowitz et al. (1995) explored an inventory system with Poisson-distributed demand, continuous review, partial backlogging, and a (r, Q) order strategy. They used a maximum allowable level of shortage with backordering ‘b’ as the control variable for a given cycle time.
To remain competitive in unstable business environments, production managers of contemporary manufacturing firms must simultaneously achieve some important operating goals while planning batch fabrication. These goals may include ensuring high product quality, satisfying buyer demand with timely deliveries and acceptable stock availability, lowering cost, and keeping the batch fabrication process undisruptive. In production planning, permitting shortage backordering is an effective policy to reduce stock holding cost, given that stock availability meets an acceptable level of customer’s satisfaction. In addition, in an actual manufacturing environment, random equipment breakdowns and imperfect items are inevitable due to unpredictable issues in manufacturing processes. This study develops a decision support model to explore a fabrication uptime problem considering an unreliable machine, a maximum permitted backorder level, a multi-delivery plan, and disposal/rework of imperfect items to assist manages with the aforementioned operating goals. With the help of mathematical modeling, an optimization method, and a solution-seeking algorithm, the proposed precise model is capable of not only determining the optimal fabrication uptime policy, but also revealing the individual and joint impact(s) of various key system parameters on the problem, thus facilitating managerial decision-making.
An enhanced L-Shaped method for optimizing periodic-review inventory control problems modeled via two-stage stochastic programming
2019, European Journal of Operational Research
This paper presents the development of an enhanced L-Shaped method applied to an inventory management problem that considers a replenishment control system based on the periodic review (R, S) policy. We consider single-item one-echelon problems with uncertain demands and partial backorder that are modeled using two-stage stochastic programming. To enable the consideration of large-scale problems, the classical single-cut L-Shaped method and its extended multi-cut form were initially applied. Preliminary computational results indicated that the classical L-Shaped method outperformed its multi-cut counterpart, even though the former required more iterations to converge to the optimal solution. This observation inspired the development of the techniques presented for enhancing the L-Shape method, which consist of the combination of a novel acceleration technique with an efficient formulation and valid inequalities for the proposed model. Numerical experiments suggest that the proposed approach significantly reduced the computational time required to solve large-scale problems.
Optimizing stock levels for rental systems with a support warehouse and partial backordering
2018, European Journal of Operational Research
Citation Excerpt :
Several authors set a partial backorder level as in our work. For a single warehouse with Poisson demand and (R, Q) policies, Rabinowitz, Mehrez, Chu, and Patuwo (1995) show that, for certain ranges of the backorder cost per time unit, partial backordering yields significant cost savings over no and full backordering. Essentially, this same result is shown by Thangam and Uthayakumar (2008) for a two-echelon system with Poisson demand and (R, Q) policies.
Various rental systems, such as public libraries and tool rental companies, have a support warehouse for storing rental products at low cost and carrying out expedited shipments in response to stock-outs at local warehouses. In case of stock-outs, rental systems commonly employ partial backordering, i.e., a limited number of demands is backordered while additional demand is lost. We include partial backordering in our models and demonstrate its relevance. We fully characterize costs and optimal base stock policies in cases with one support and one local warehouse and provide upper bounds for base stock levels in the general system. By enumerating base stock policies and evaluating exact costs numerically, we determine optimal base stock policies for small systems with up to six local warehouses. For larger systems we develop a greedy algorithm based on approximate cost evaluations. The resulting base stock policies have costs within 0.2% from optimality on average in experiments. Among other things, experiments indicate that existing methods with no backordering may lead to poor solutions with partial backordering and that partial backordering is an effective means to reduce lost demand even with limited partial backorder levels.
A finite-horizon inventory system with partial backorders and inventory holdback
2017, Operations Research Letters
Citation Excerpt :
One limitation of these works is that to elicit structural properties of the underlying system, the authors have to assume that restock lead times are zero. Finally our work complements the literature on continuous review inventory systems, including the works by Rabinowitz et al. [8] and Decroix and Arreola-Risa [5]; and, extensions by Cheung [4], Zhang et al. [13],
We consider a periodic-review base stock inventory system with partial backorders. At the beginning of each period $t$ of a $T$ period problem, an order of size $q_{t}$ is placed for delivery one period later. As the stochastic demand is realized, as much as possible of it is filled immediately from the inventory on-hand. If the realized demand exceeds the inventory on-hand, up to $q_{t}$ – $k_{t}$ units of excess demand are backordered to be filled from the pipeline inventory or future orders. The on-order quantity $k_{t}$ denotes the reservation quantity held back for use in subsequent periods. The case $k_{t} = q_{t}$ yields the full lost sales model while the case $k_{t} = - \infty$ yields the full backorder model.
Dynamic inventory rationing with mixed backorders and lost sales
2014, International Journal of Production Economics
Citation Excerpt :
They consider the time duration of stockout (which is ignored in Montgomery et al.,1973) and obtain the exact optimal solution of the model. Rabinowitz et al. (1995) study another kind of mixed inventory model. Initially, all demand is backlogged when it runs out of stock.
Customers may react differently when stockouts occur. In this paper we investigate the rationing policy for an inventory system with a mixture of demand classes of backorder type and lost sales type. Since the penalty cost of backorders varies with time, the priorities of demand classes also alter with time. This totally changes the problem structure compared with the classic rationing models. A dynamic rationing policy is studied in this paper by considering the dynamics of demand priorities. A Markov decision model is developed to obtain the optimal dynamic rationing levels for multiple demand classes. The results indicate that between the priority switching points, rationing levels often exhibit different patterns. For lost sales demand classes, the rationing levels always decrease as the remaining time approaches to zero. For backorder demand classes, the rationing levels increase in some parts due to declining of the priorities. The rationing levels of all demand classes finally decline to zero to reduce the inventory holding cost. The application of dynamic rationing is further extended from a single period model to a multi-period (S,T) model where unit cost has to be included. The optimal ordering policy is proved to be a myopic base stock policy and the dynamic rationing policy in the single period model can still be applied with modified time-independent penalty costs for lost sales classes. To overcome the computational complexity, a heuristic dynamic rationing policy is introduced. Due to its good outcome, implementing such a heuristic dynamic rationing policy can be a practical solution for inventory system with mixed backorders and lost sales, in order to enhance the system performance.

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^†: Gad Rabinowitz earned his Ph.D. from the Department of Operations Research at Case Western Reserve University. His research interest is in the analysis of production operations. He has research papers published in Management Science, European Journal of Operational Research, OMEGA and other journals. He holds an appointment with Ben-Gurion University, Israel.

^∥: Ching-Wu Chu is a Ph.D. student at Kent State University. His research is on the analysis of inventory system.

^¶: B. Eddy Patuwo is an Assistant Professor in the Department of Administrative Sciences at Kent State University. He earned his Ph.D. in IEOR from Virginia Polytechnic Institute and State University. His research interests are in the study of stochastic systems (queueing, inventory, manufacturing) and neural networks.

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A partial backorder control for continuous review (r, Q) inventory system with poisson demand and constant lead time

Abstract

Eur. J. opl Res.

Inventory models with a mixture of backorders and lost sales

Naval Res. logist. Q.

A new analysis of lot-size model with partial backlogging

Naval Res. logist. Q.

Class of inventory models with customer impatience

Naval Res. logist. Q.

The (S−1, S) inventory model under time limit on backorders

Ops Res.

Operating characteristics for the (S−1, S) inventory system with partial backorders and constant resupply time

Mgmt Sci.

A note on the operating characteristics of the (S−1, S) inventory systems with partial backorders and constant resupply times

Mgmt Sci.

(Q, R) Inventory model with a mixture of lost sales and time weighted backorders

J opl Res. Soc.

(Q, R) Inventory models with partial backorders, Poisson demands, and exponential lead times