A coordinated inventory control system for compound Poisson demand and zero lead time†

Often coordination of items for replenishment purposes can lead to significant savings in the costs of replenishment. This article presents a reasonable procedure for determining the values of the various control variables of an (S, c, e) control system, namely the Order-up-to-levels, the can-order points and the must-order points. It is assumed that demand is compound Poisson and the replenishment lead time is of negligible duration. Comparison with the best independent control strategy indicates that substantial savings (averaging 15-9% over some 64 examples) are possible through coordination of replenishments.     

Coordinated replenishment systems with discount opportunities

In many practical situations, coordination of replenishment orders for a family of items can lead to considerable cost savings. A well-known class of strategies for the case where cost savings are due to reduced joint ordering costs is the class of can-order strategies. However, these strategies, which are simple to implement in practice, do not take discount possibilities into account. We propose a method to incorporate discounts in the framework of can-order strategies. A continuous review multi-item inventory system is considered with independent compound Poisson demand processes for each of the individual items. Discounts are offered by the supplier as a percentage of the total dollar value whenever this value exceeds a given threshold. Starting from the can-order strategy as a basic decision rule, we develop a simple heuristic to evaluate these discount opportunities. The performance of the can-order strategy with discount evaluation is compared with that of another class of discount evaluation rules as proposed by Miltenburg and Silver.     

On the stability and bullwhip effect of a production and inventory control system

This paper focuses on the discrete-time automatic pipeline, inventory and order-based production control system (APIOBPCS), a well-established production and inventory control model. The feedback mechanism within the replenishment rule enables the model to mitigate the bullwhip effect, but introduces a stability problem. In this research, a comprehensive stability analysis is conducted for arbitrary lead times using difference equation theory. On the basis of stability, a state space approach is advocated to analyse the impact of replenishment parameters, demand processes, and lead times on the robustness of the bullwhip effect. The stability results demonstrate that the production control system can easily be destabilised without incorporating the work-in-progress (WIP) feedback loop. Furthermore, it reveals that the stability problem for long lead times can be simplified with the stability condition independent of the lead time. The results obtained in this study provide useful guidelines for the selection of replenishment parameters to guarantee stability and mitigate the bullwhip effect.     

A stochastic multi-item inventory model with unequal replenishment intervals and limited warehouse capacity

Managing inventory levels for multiple products that share warehouse capacity often presents significant coordination challenges. These coordination challenges are further compounded when multiple suppliers use different replenishment schedules for supplying the warehouse. To address these challenges, this paper studies stochastic inventory models for multiple items with both equal and unequal replenishment intervals under limited warehouse capacity. We propose three efficient and intuitively attractive heuristics. We show that these heuristics provide the optimal replenishment quantities in the case of equal replenishment intervals. For the general model, a numerical comparison of the heuristic solutions to the optimal solutions shows that the heuristics yield high quality solutions in very limited time.     

Inventory control of an item with a probabilistic replenishment lead time and a known supplier shutdown period

In this paper, we address an important practical situation, namely where the usual replenishment lead time (when the supplier's production facility is operating) is a random variable and the supplier shuts down for an interval of known duration (for maintenance, vacation, etc.) each year. The demand rate is constant and any demand when out of stock is assumed to be lost. Under such circumstances we develop a heuristic procedure to decide when to initiate replenishment as well as the associated order-up-to-levels. Through the use of simulation (which accurately estimates the average costs per unit time), the heuristic is shown to perform excellently in a selection of small size problems when one can find the optimal solution. For a large number of problems of more realistic size, the use of simulation reveals that the heuristic achieves substantial cost savings when compared with a simpler, baseline approach. The heuristic itself does not require the use of simulation. The sensitivity of total expected costs to various parameters (such as the length of the shutdown interval and characteristics of the lead time distribution) is discussed.     

基于协同补货期库存协调模型的扩展研究

在协同预测的库存补充策略和基于协同补给期的一个供应商与一个订货商之间的库存协调策略的研究基础之上,主要研究一个供应商与多个订货商库存协调策略,并与协调之前的库存成本相比,用数值实验的方法证明了合作能够降低供应商与订货商的库存成本,增加效益,给供应链带来了整体效益.     

A Modified Formula for Calculating Customer Service Under Continuous Inventory Review

This article shows that a formula commonly used for establishing safety stocks can lead to erroneous results, particularly when the magnitude of forecast errors over a replenishment lead time is large in comparison with the replenishment quantity. A correct procedure involving only slightly more computational effort is developed. Numerical results are provided for the case of normally distributed forecast errors.     

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.     

Optimal joint replenishment policy for multiple non-instantaneous deteriorating items

In this paper, we deal with the problem of determining the optimal economic operating policy when a number of non-instantaneous deteriorating items are jointly replenished. We establish a multi-item joint replenishment model for non-instantaneous deteriorating items under constant demand rate allowing full backlogging. This problem is challenging, in particular, the cost function is a piecewise function with exponential parts, which makes the problem more complicated. To solve this problem, an approximation method is used to simplify the objective function and a bound-based heuristic algorithm is developed to solve the model. Numerical examples illustrate the effectiveness of the proposed method and the quality of the approximation. Experimental results on a real-life case study show that the proposed model can achieve substantial cost savings compared to the individual replenishment policy for non-instantaneous deteriorating items. Furthermore, sensitivity analysis of key parameters is carried out and the implications are discussed in detail.     

The effectiveness of investment in lead time reduction for a make-to-stock product

This paper focuses on a firm selling a make-to-stock product with a constant customer demand rate. The firm follows an exact (Q, r) policy for raw material inventory control and faces a random replenishment lead time. Through this research, we wish to gain a better understanding of the impact of investing in reducing supply lead time when the investment costs have to be borne, partly or fully, by the firm. This work is motivated by the recognition that lead time reduction is now of strategic importance in the successful operation of a firm. We examine different types of investment schemes in replenishment lead time reduction and the different cost models they generate. We present analytical and numerical results and insights for each type of model, compute the optimal (Q, r) policy and the associated investment levels. The work presents new results, and sheds light on some apparently counter-intuitive observations.     

Forecasting replenishment orders in retail: value of modelling low and intermittent consumer demand with distributions

In retail, distribution centres can forecast the stores’ future replenishment orders by computing planned orders for each stock-keeping-unit. Planned orders are obtained by simulating the future replenishment ordering of each stock-keeping-unit based on information about the delivery schedules, the inventory levels, the order policies and the point-estimate forecasts of consumer demand. Point-estimate forecasts are commonly used because automated store ordering systems do not provide information on the demand distribution. However, it is not clear how accurate the resulting planned orders are in the case of products with low and intermittent demand, which make up large parts of the assortment in retail. This paper examines the added value of modelling consumer demand with distributions, when computing the planned orders of products with low and intermittent demand. We use real sales data to estimate two versions of a planned order model: One that uses point-estimates and another that uses distributions to model the consumer demand. We compare the forecasting accuracies of the two models and apply them to two example applications. Our results show that using distributions instead of point-estimates results in a significant improvement in the accuracy of replenishment order forecasts and offers potential for substantial cost savings.     

A stochastic model for joint inventory and outbound shipment decisions

In this paper, we consider a vendor realizing a sequence of random order arrivals in random sizes. The vendor has the autonomy to hold/consolidate small orders until an economical dispatch quantity accumulates. Consequently, the actual inventory requirements at the vendor are in part determined by the parameters of the shipment release policy in use. In this context, we investigate the impact of shipment consolidation on the expected long-run average cost by simultaneously computing the optimal order quantity for inventory replenishment at the vendor and the optimal dispatch quantity for outbound shipments. Since we consider the case where demand follows a general stochastic bulk arrival process, obtaining exact analytical expressions for some key operating characteristics of the cost function is intractable. Hence, we provide easy-to-compute approximations which enable efficient numerical solutions for the problem. We also investigate: (i) the cases where consolidated shipments are preferred over immediate deliveries; (ii) the sensitivity of optimal integrated policy variables to demand/cost parameters; (iii) the potential savings that can be obtained by shipment consolidation; and (iv) the tradeoffs between the waiting time induced by shipment consolidation and costs saved. Our results provide insights into the impact of outbound transportation operations on inventory replenishment decisions and outbound distribution system design. Moreover, numerical testing suggests that significant cost savings (up to 57%) are possible with shipment consolidation.     

A dynamic lot-sizing model with multi-mode replenishments: polynomial algorithms for special cases with dual and multiple modes

This paper generalizes the classical dynamic lot-sizing model to consider the case where replenishment orders may be delivered by multiple shipment modes. Each mode may have a different lead time and is characterized by a different cost function. The model represents those applications in which products can be purchased through various suppliers or delivered from a single source using various transportation modes with different lead times and costs. The problem is challenging due to the consideration of cargo capacity constraints, i.e., the multiple set-ups cost structure, associated with a replenishment mode. The paper presents several structural optimality properties of the problem and develops efficient algorithms, based on the dynamic programming approach, to find the optimal solution. The special, yet practical, cases of the two-mode replenishment problem analyzed in this paper are analytically tractable, and hence, the respective problems can be solved in polynomial time.     

On coordinating warehouse sizing, leasing and inventory policy

Organizations often fulfil their storage needs by supplementing their own warehouse with leased space, a scenario modeled here under the assumption of constant product demand. Closed-form formulae are obtained for the decision variables of interest, namely, the replenishment lot size and the warehouse size, as well as the amount of space to lease. Cost savings due to leasing are shown to have an upper bound of approximately 29% when the optimal warehouse capacity can be installed without restrictions (such as a budgetary constraint). A numerical example further indicates that leasing is significantly more beneficial when the warehouse size is tightly constrained, and that total costs are robust with respect to demand fluctuations.     

Application of control theoretic principles to manage inventory replenishment in a supply chain

Using uncertain real-time information to update supply chain operational policies creates a need for developing dynamic supply chain management capabilities that increase responsiveness to demand and decrease volatility of the replenishment process (popularly known as the Bullwhip Effect). To this end, we explore the use of control theoretic principles to manage the inventory replenishment process in a supply chain under different forecast situations. We study the use of proportional, proportional-integral and proportional-derivative control schemes to determine the conditions under which specific control actions are beneficial. Analytical models and simulation runs are used to study the trade-off between responsiveness to demand and volatility. Our analysis indicates that using proportional control to manage inventory replenishment is suitable for high forecast error situations. Proportional control along with integral control works well in situations where the forecast bias is relatively higher than the forecast error. Proportional control along with derivative control works best in situations with moderate forecast errors.     

Accounting for residual stock in continuous review coordinated control of a family of items

Inventory models which permit coordinated control (or joint replenishment) of families of items are of considerable interest in practice. A major complexity in coordinated control is that, typically, one item triggers the family order and the other items have stock remaining above their reorder points. Account should be taken of this so-called residual stock when establishing the reorder points. In this paper, modelling the inventory position as a diffusion process permits us to develop expressions for the components of the residual stock probability density functions. Heuristics are presented for quick calculation of these expressions. Then we show how residual stock is taken into account in determining the reorder points of the individual items. The relationship of these residual stock pdf expressions to other components in a new class of coordinated control models is also discussed. This class of models has been shown to outperform existing models (IBM's IMPACT Inventory Control package(.     

Coordination of replenishment activities†

Extensive research on the Economic Lot Scheduling Problem (ELSP), and on related topics, has produced powerful techniques originally designed to solve quite specific (coordination) problems. This paper emphasizes the omni-presence of coordination issues in multi-item inventories, characterizes them as problems of staggering and/or clustering replenishment activities or events, and shows how ELSP techniques combine to form a pertinent and fairly comprehensive methodology. A numerical example illustrates the use of one particularly powerful tool (powers of two) in stabilizing replenishment workloads.     

Rolling horizon production planning for probabilistic time-varying demands

This paper introduces and tests a production planning procedure to be applied in a rolling horizon with probabilistic demands, based on the work of Bookbinder and Tan (1983). First, the case of no lead time for replenishment is considered and trend-seasonal demands are studied. We vary the set-up cost, order cycle and number of periods in the future for which demand forecasts are available. The procedure is compared to that of Silver (1978) in terms of cost performance, percentage of demand short/period and percentage of periods with stock-outs. Each approach appears to have merit. The proposed procedure generally yields a lower solution cost, while Silver's procedure usually has a lower percentage of demand short/period and a smaller percentage of periods with stockouts. Differences between the two procedures were relatively insensitive to which of three lot-sizing rules was employed within them. Finally, we further extended our procedure to consider non-zero lead times for replenishment and different demand patterns, particularly the normal distribution.     

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.     

The diffusion process and residual stock in periodic review,coordinated control of families of items

Inventory control models which provide coordinated control ( or joint replenish ment) of families of items are very useful in practice. These models require expressions for the residual stock levels for each item in the family when a replenishment is placed. In this paper expressions are developed for the periodic review residual stock probability density function, when inventory position is modelled as a diffusion process, and heuristics are presented for quick calculation of these expressions. The importance of residual stock in setting inventory control model parameters is illustrated by showing how residual stock is taken into account in establishing item reorder points. The relationship of these residual stock pdf expressions to other components in a new class of coordinated control models is also discussed. This class of models has been shown to outperform existing models (IBM's IMPACT Inventory Control package).