SINGLE-STAGE, MULTI-PRODUCT PRODUCTION/INVENTORY SYSTEMS WITH BACKORDERS

A production/inventory system consisting of a single-stage facility producing multiple product types and a warehouse is considered. The demand process is assumed to be Poisson and the excess demand is backordered. Products have a priority structure and the processor is shared according to a switching rule. (R, r) continuous-review inventory control policies are used to start and stop production of each product. Production of a product continues until its target level is reached. Then, a switch-over takes place to the highest priority product among the ones that need the processor's attention. A set-up takes place every time a switch-over occurs. Both processing and set-up times are assumed to have arbitrary distributions. The number of switch-overs among the high priority products is restricted in case low priority products need attention. The objective is to obtain performance measures such as the average inventory and backorder levels as well as die probability of having a backorder upon arrival for each product. We develop an approximation procedure which emphasizes the characterization of the delay each product type experiences in receiving the processor's attention, once they place a request for it. The approximating arguments are based on independence assumptions.     

Multi-item inventory model with variable backorder and price discount under trade credit policy in stochastic demand

In this paper, a two echelon supply chain with one manufacturer and one retailer is developed for multi products. The retailer faced with the uncertain demand for all products which follows a normal distribution. The production process is assumed to be imperfect, and the defectiveness is assumed to follow a beta distribution. The manufacturer produces and delivers the products in a number of equal-sized batches to the manufacturer's warehouse, and thereby it is delivers in a number of equal batches to the retailer's warehouse. Shortages are allowed to occur, at the retailer side, and it is backordered partially. The retailer offers a price discount for backordered items to his customers. Both the lead time crashing cost and the partial backorder ratio are considered as the inverse function of lead time. Under these assumptions, there are three inventory models proposed in this paper, one with non-integrated approach, the other with an integrated approach without trade credit and finally an integrated approach with trade credit. A new iterative algorithmic procedure has been developed to minimise the total cost. Finally, numerical examples are given to illustrate the models and the sensitivity analysis is conducted over various model parameters.     

Ordering Cost Depletion in Inventory Policy with Imperfect Products and Backorder Rebate

This study presents an inventory model for imperfect products with depletion in ordering costs and constant lead time where the price discount in the backorder is permitted. The imperfect products are refused or modified or if they reached to the customer, returned and thus some extra costs are experienced. Lately some of the researchers explicitly present on the significant association between size of lot and quality imperfection. In practical situations, the unsatisfied demands increase the period of lead time and decrease the backorders. To control customers' problems and losses, the supplier provides a price discount in backorders during shortages. Also, an order’s policies may result in including some imperfect products in arrival lots. A discount on price may be offered by the supplier on the out-of-stock products to manage the backorder problems. The study aims to develop a model with imperfect products by permitting the price discount in backorders, and the cost of ordering is considered a decision variable. First, it is assumed that the demand for lead time is followed by a normal distribution and then stops it and assumed that the first two moments of demand for lead time are known. Further, the minimax distribution method is used to solve this model, and a separate algorithm is designed. In this study, two models are discussed with and without a normally distributed rate of demand. The current study verified with the help of some numerical examples over various model parameters.     

An Inventory System with Periodic Regular Review and Flexible Emergency Review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two,⃛, up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.     

A single-vendor single-buyer production-inventory model with controllable lead time and service level constraint for decaying items

This paper presents a production-inventory model in a single-vendor single-buyer system for decaying items which deplete with constant decay rate. It is assumed that the lead time demand follows a normal distribution and shortages are backordered. Also, buyer's lead time can be reduced at an added cost. A model has been formulated to find the optimal order quantity, lead time and the number of shipments from the vendor to the buyer during one production cycle while minimising the total expected cost per unit time of the vendor-buyer integrated system. It is often difficult to estimate the stock-out cost in inventory systems. Therefore, instead of having a stock-out term in the objective function, a service level constraint is included in the model. Due to mathematical complexities involved, it is difficult to establish the solutions analytically. An iterative procedure is developed to find the optimal solution using LINGO 10, and a numerical example is given to illustrate the results of the proposed model. Furthermore, the effects of key parameters are also studied to provide the insight into the structure of the solution. In addition, the savings of buyer and vendor are investigated from implementation of joint optimisation model over the model in which they minimise their own cost independently.     

On offering economic incentives to backorder

We study a strategy to manage demands that occur when an inventory system is temporarily out of stock: offer the customer facing the unsatisfied demand an economic incentive to backorder. We explore the benefits of this inventory management strategy by analyzing a model of an inventory system with stochastic demand and random supply disruptions, where the probability that a customer facing an unfilled demand will backorder (as opposed to becoming a lost sale) can be influenced by an economic incentive. Our results provide several insights regarding this inventory management strategy and suggest that the benefits of offering backorder incentives can be significant.     

One-machine, N-product-type, continuous-time scheduling with a common due date: a polynomially solvable case

The paper analyzes a manufacturing system made up of one-machine which produces TV-product-types with controllable production rates in response to product demands. The demands are characterized by different amounts of each product-type to be produced and a common due date. The objective is to minimize inventory and backlog costs which are incurred when meeting the due date results in inventory surpluses and shortages. With the aid of the maximum principle, the continuous-time scheduling problem is studied as an optimal control model and is reduced to a combinatorial one, polynomially solvable when the costs are either “agreeable” or when the number of the non-agreeable costs is limited.     

DETERMINING THE NUMBER OF KANBANS IN MULTI-ITEM JUST-IN-TIME SYSTEMS

Kanbans (cards) are one means of controlling Just-in-time production systems. The kanbans serve as production authorization orders and essentially become the information system. The presence of a production ordering kanban for a specific part type constitutes an instruction to the workcenter to produce a “container” of that part. In this paper we assume workcenters produce multiple part types and address the issue of the number of kanbans needed for each part type. Our objective is to minimize the sum of inventory holding and backorder cost. A stochastic model is formulated. Steady-state results are derived for the cases of a few and many part types. Modifications for the case of expediting of backorders are also provided. Simulation results demonstrate the accuracy of the model for a large number of kanbans, but an investigation of a finite population queueing model for small numbers of kanbans is suggested as a worthwhile extension.     

A joint replenishment policy with individual control and constant size orders

We consider inventory systems with multiple items under stochastic demand and jointly incurred order setup costs. The problem is to determine the replenishment policy that minimises the total expected ordering, inventory holding, and backordering costs–the so-called stochastic joint replenishment problem. In particular, we study the settings in which order setup costs reflect the transportation costs and have a step-wise cost structure, each step corresponding to an additional transportation vehicle. For this setting, we propose a new policy that we call the (s, 𝒬) policy, under which a replenishment order of constant size 𝒬 is triggered whenever the inventory position of one of the items drops to its reorder point s. The replenishment order is allocated to multiple items so that the inventory positions are equalised as much as possible. The policy is designed for settings in which backorder and setup costs are high, as it allows the items to independently trigger replenishment orders and fully exploits the economies of scale by consistently ordering the same quantity. A numerical study is conducted to show that the proposed (s, 𝒬) policy outperforms the well-known (𝒬, S) policy when backorder costs are high and lead times are small.     

A trade off between emergency repair and inventory investment

In this paper we present an emergency repair model. We model a situation where failed parts arrive at a repair shop according to a Poisson process. If the stock on hand of serviceable spare parts exceeds a given emergency trigger level, the failed part is sent into normal repair. Otherwise the failed part is sent into an alternative emergency repair channel. Upon arrival failed parts are exchanged for serviceable spare parts, if available. Otherwise the demand is backordered. A backorder is filled with the first part that becomes available from either one of the repair processes. The repair times (including transportation time) are assumed to be exponentially distributed. We analyze the impact of using emergency repair on two service measures: the fraction of demand that is satisfied from stock on hand and the expected duration of a backorder. We calculate the initial stock level and the emergency trigger level that minimize the total cost for a given emergency repair rate. The proposed policy is compared with two other policies: (1) emergency repair is not used and (2) emergency trigger level is zero. We find that significant service improvements and cost reductions can be obtained by using our emergency repair policy. We also present simulation results showing that the distribution of the repair times has a negligible effect on the relevant service measures. Finally, we compare our results with the results of an emergency supply model by Muckstadt and Thomas [1] Again we observe significant cost reductions when using our policy.     

Continuous review (s,S) policies for inventory systems incorporating a cutoff transaction size

In this paper, a mathematical model is developed for the analysis of optimal replenishment policies for items which experience lumpy demands. In order to avoid disrupting the inventory system, a cutoff transaction size of w units is specified such that customer demands with sizes exceeding w units will be filtered out of the inventory system and treated as special orders to be satisfied by special deliveries. Customer demands with sizes > units will be met From stock. The control discipline is the (s, S) inventory policy with continuous review and the nature of the demands is approximated by a stuttering Poisson distribution. Recursive formulae are developed for the determination of the probabilities of the lead time demand and the available inventory position. The optimal values of the control parameters, s and S, are determined. The theoretical results obtained are illustrated with a numerical example.     

A membership function approach to lot size re-order point inventory problems in fuzzy environments

Inventory models play an important role in logistics and supply chain management for reducing cost and increasing customer satisfaction. This paper develops an approach to derive the fuzzy objective value and decision variables of the fuzzy lot size re-order point inventory problem with parameters being fuzzy numbers and the shortages are backordered with extra cost incurred. Different from the existing studies, the idea is based on Zadeh's extension principle. A pair of mixed integer nonlinear programs (MINLP) parameterised by the possibility level α is formulated to calculate the lower and upper bounds of the minimal total cost per unit time at α, through which the membership function of the minimal total cost per unit time is constructed. At the same time the membership functions of the optimal order quantity and the optimal re-order point are also provided. A numerical example studied by previous studies is solved successfully to demonstrate the validity of the proposed method. Compared with previous studies, the obtained results which precisely and completely conserve the fuzziness of the input information are more informative for finding the best inventory policy since they are expressed by membership functions rather than by crisp ones. Moreover, to provide representative crisp solutions for designing inventory systems, the Yager's ranking index method is adopted to defuzzify the obtained membership functions. The successful extension of inventory models to fuzzy environments permits inventory models to have wider applications in practice.     

ECONOMIC PRODUCTION QUOTAS FOR PULL MANUFACTURING SYSTEMS

We consider the problem of using “safety capacity” to ensure due date integrity in a pull manufacturing system and quantify the basic tradeoff between lost revenue opportunity and overtime costs. In this context, we address the question of when it is economically attractive to use “under capacity scheduling” and the problem of setting economic production quotas.

We develop four models for addressing the quota setting problem. The first three assume that quota shortfalls cannot be carried over to the next regular time production period. Models 1 and 3 assume that these shortages are made up on overtime and incur fixed or fixed plus variable costs. Model 2 does not use a capacity buffer and treats shortages as lost sales. Finally, Model 4 assumes that shortages can be backlogged to the next regular time production period at a cost. For this model, we compute both an optimal quota and an overtime “trigger,” which represents the minimum shortage for which overtime is used. We give computational results that illustrate and contrast the various models.     

基于供需一体化协同控制的供应链库存模型

根据供需一体化协同库存控制策略，构建了一个一体化库存模型。在模型里，假定提前期需求是随机的且服从正态分布，允许在提前期内发生缺货，并产生缺货成本；假定提前期是可控的，能够通过增加成本缩短提前期。该模型通过同时计算最优订货量、再订货点、提前期以及运送批次．使供需一体化库存总成本最小。实例分析证明该模型能够达到节约成本的目的。     

二维保证下汽车维修备件动态库存控制策略

以汽车产品为例,针对失效率以时间或里程为计量单位的单位价值高、需求率低的不可修零部件,以未来销售预测为起点,依据二维保证政策的特点确定某时点处于保证时效内的汽车数量,基于产品可靠性对保证备件的需求进行预测,针对保证备件需求的特点,提出动态分段(Q,r)库存控制策略,通过案例验证了该库存策略可有效降低库存成本支出。     

Comparison of DRP and TOC financial performance within a multi-product,multi-echelon physical distribution environment

This paper propositioned and tested whether a heuristic based on Theory of Constraints logic could improve system financial performance beyond traditional supply chain methods in a multi-product, multi-echelon physical distribution environment exhibiting seasonality and partial lost sales. A review of the literature was conducted about current distribution management and Theory of Constraints techniques. Next, field research was conducted with a major US manufacturer in order to capture the structure of its multi-product, multi-echelon physical distribution system. The field research facilitated the development of a baseline computer simulation of a fully distributed inventory system with orders planned by Distribution Resource Planning. That model served as the basis for development of comparative multi-echelon distribution models, one employing partial centralization of inventory with orders planned by Distribution Resource Planning, the other two models employing a Theory of Constraints-based heuristic for buffering and inventory replenishment. Simulation results show the Theory of Constraints-based systems are more effective on a financial basis when considering inventory carrying costs, retail-level transhipment and obsolescence expenses than either the existing distributed inventory system or the partially centralized system when orders are based on Distribution Resource Planning logic.     

Spare parts inventory management with demand lead times and rationing

We study an inventory system that consists of two demand classes. The orders in the first class need to be satisfied immediately, whereas the orders in the second class are to be filled in a given demand lead time. The two classes are also of different criticality. For this system, we propose a policy that rations the non-critical orders. Under a one-for-one replenishment policy with backordering and for Poisson demand arrivals for both classes, we first derive expressions for the service levels of both classes. The service level for the critical class is an approximation, whereas the service level for the non-critical class is exact. We then conduct a computational study to show that our approximation works reasonably, the benefits of rationing can be substantial, and the incorporation of demand lead time provides more value when the demand class with demand lead time is the critical class. The research is motivated by the spare parts service system of a major capital equipment manufacturer that faces two types of demand. For this company, the critical down orders need to be satisfied immediately, while the less critical maintenance orders can be satisfied after a fixed demand lead time. We conduct a case study with 64 representative parts and show that significant savings (as much as 14% on inventory on hand) are possible through incorporation of demand lead times and rationing.     

Justifying New Technology Acquisition Through its Impact on the Cost of Running an Inventory Policy

This paper presents a model for justification of new technology acquisition derived from its effect on the inventory setup costs. Recently, Porteus [10] considered a situation where investment in the new technology (Just-in-Time) is evaluated on the basis of its impact on reducing the setup costs in the EOQ model. This paper extends the Porteus work to the situation where demand during lead time is probabilistic. In this case, the justification of new technology acquisition is made on the basis of its impact on reducing the setup costs in the lot-size reorder-point, (Q,r), model. Explicit solutions are obtained for two specific demand distributions and two general cost functions. The paper also compares the proposed model with that of Porteus and presents a qualitative observation for situations where the application of EOQ and (Q),r systems will result in significant differences in the decision to acquire new technology.     

Inventory control of raw materials under stochastic and seasonal lead times

Most inventory modelling has assumed stochastic demands and constant lead times. However, here we consider a problem for which the opposite situation holds; namely, there is a known constant demand rate, but lead times are random variables. Moreover, the probability distributions of the lead times change in a seasonal fashion. Also, shortages of raw materials result in lost sales. The goal of this paper is to propose heuristic methods for minimizing the expected costs in such a situation. This study was motivated by a problem of management of raw material at a sawmill.     

Ex-post inventory control

Classical inventory control, using an ex-ante optimization approach and tending to recommend large stockholdings, has been criticized over recent decades. As a result, just in time (JIT) based inventory systems, which exhibit sensitivity to customer requests (rather than production plans based on demand forecasts) are increasingly recommended. In such inventory control schemes, orders are made once the demand is revealed. As a result, ex-ante classical and ex-post JIT inventory schemes can be compared and analysed through: (a) the valuation and the management of demand uncertainty and (b) the time phasing of information and its relationship to the (inventory) control policy. The purpose of this paper is to elaborate and compare such approaches and devise an approach to inventory control based on regret decision making. Examples are used to show how we may be able to value flexibility explicitly in supplies and construct robust inventory policies which can respond to predictable and unpredictable demands (such as using options in inventory control and increased reactivity through JIT controls).