Optimal inventory control of lumpy demand items using (s,S) policies with a maximum issue quantity restriction and opportunistic replenishments

This paper presents a modified (s, S) inventory model which describes the characteristics of an inventory system with lumpy demand items. A maximum issue quantity restriction of w units and a critical inventory position of A units are incorporated into the inventory control policy. Customer orders with demand sizes larger than the maximum issue quantity will be filtered out from the inventory system and satisfied by using special replenishment orders in order to avoid disruption to the inventory system. The option of opportunistic replenishments is introduced to further minimize the total replenishment cost. An opportunistic replenishment is initiated if the level of the current inventory position is equal to or below the critical level when a customer demand with a size exceeding the maximum issue quantity arrives, which does not only initiate a direct shipment to the customer, but also raises the inventory position to S. Two effective algorithms are developed to determine the optimal values of w, A, s and S simultaneously. The first algorithm is based on the branch-and-bound tree search technique, and the second one is based on the concept of genetic algorithms. Numerical examples are used to illustrate the effectiveness of the algorithms developed. The effects of changes in the cost and system parameters on the optimal inventory control policy are also studied by using sensitivity analysis.     

Simulation of an integrated age replacement and spare provisioning policy using SLAM

This paper presents a SLAM simulation model for determining a jointly optimal age replacement and spare part provisioning policy. The policy, referred to as a stocking policy, is formulated by combining age replacement policy with a continuous review (s, S) type inventory policy, where s is the stock reorder level and S is the maximum stock level. The optimal values of the decision variables are obtained by minimizing the total cost of replacement and inventory. The simulation procedure outlined in the paper can be used to model any operating situation having either a single item or a number of identical items. Results from a number of case problems specifically constructed by 5-factor second order rotatory design have been presented and the effects of different cost elements, item failure characteristics and lead time characteristics have been highlighted. For all case problems, optimal (s, S) policies to support the Barlow-Proschan age policy have also been determined. Simulation results clearly indicate the separate optimizations of replacement and spare provisioning policies do not ensure global optimality when total system cost has to be minimized.     

A new algorithm for computing optimal (s, S) policies in a stochastic single item/location inventory system

It has been shown that a class of (s, S) policies is optimal to the single item/location inventory system. However, the computational complexity of finding the optimal (s, S) policy has restricted applications of this inventory system. This paper proposes a new algorithm to search for the optimal pair of s and S. We introduce a dummy cost factor and an auxiliary function into our algorithm. The algorithm searches for the optimal dummy cost through continuously evaluating the auxiliary function. It differs from the approach of Zheng and Federgruen (1991) in several aspects and has certain advantages. First, as it revises the dummy cost based on the sign of the auxiliary function, the primary goal of the search is not to compute the optimal s and S during each iteration. Second, by identifying the non-prospective sets of S, the algorithm further reduces the search effort. Numerical tests show that on the average, the proposed algorithm saves more than 30% of evaluation effort compared with Zheng and Federgruen's method.     

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.     

An optimal constant level rationing policy under service level constraints

We analyze a periodic review inventory system where inventory of a single item is used to serve high and low priority customers. High priority customers demand a higher service level than low priority customers and both types of customers can have quite general discrete demand distributions. The inventory is controlled by a (S, CL)-policy, where S denotes the base stock level and CL specifies the critical level, i.e., the amount of inventory reserved for high priority demand. We consider a policy, where the critical level is constant over time, which is not optimal, but makes the model analytically tractable. Unlike previous research, we clear backorders optimally and find the optimal solution. We also derive structural results.     

The trans-shipment problem in a two-echelon,multi-location inventory system with lost sales

In this paper, we address a two-echelon, multi-location pooling inventory system that consists of an outside supplier, a warehouse and two retailers. To control their inventories, both warehouse and retailers use (R,?s,?S) policy. The retailers face stochastic customer demands for a single product and the warehouse receives only the replenishment orders of retailers. In case of stock-out at retailers, emergency trans-shipments are used to satisfy the unmet demand at one retailer with the surplus from the other retailer. When the stock is insufficient at the warehouse, we propose two rationing policies to allocate on-hand stock between retailers. The demand that cannot be satisfied neither by stock on-hand nor by trans-shipment from retailers is considered lost. Our work has two objectives. First, we propose an inventory model based on three components: the optimisation inventory model, the trans-shipment policy and the rationing policies for determining the best values of (s,?S) at each location that minimise total system cost. Second, we validate this model via an empirical simulation study that allows us to identify the influential parameters on trans-shipment benefits.     

Inventory management in a lateral collaborative manufacturing supply chain: a simulation study

Due to global competition, firms are seeking more effective supply chain (SC) collaboration in order to provide quality products with less cost, at the right time and in the right quantity. The present study examines manufacturing SC collaboration on the basis of holding cost, backorder cost and ordering cost. The types of collaboration examined are vertical, horizontal and lateral collaboration. This research emphasises lateral collaboration by determining the impact of inventory policies ((s, S) and (s, Q) inventory policies) on SC performance. For better understanding, a conceptual model is provided that is supported by a numerical example. As the study of SCs is complex in nature, a simulation approach has been employed to show the impact of lateral collaboration on performance measures such as the total cost, which is the sum of several cost components: inventory holding cost, backorder cost and ordering cost. The research is based on two manufacturing SCs where the manufacturer is taken as the collaborative node. To allow more clarity, a separate study on each cost component has been conducted. The laterally collaborative SC was simulated on ARENA 9.0, a simulation package. The results show that the efficacy of lateral collaboration outperforms horizontal collaboration due to having the individual SC members at more liberty to make decisions.     

Joint hybrid repair and remanufacturing systems and supply control

The control of a stochastic manufacturing system that executes capital asset repairs and remanufacturing in an integrated system is examined. The remanufacturing resources respond to planned returns of worn-out equipment at the end of their expected life and unplanned returns triggered by major equipment failures. Remanufacturing operations for planned demand can be executed at different rates and costs corresponding to different replacement and repair modes. The replacement components inventory is provided by an upstream supply with random lead times. The objective is to determine a control policy for both the supply and remanufacturing activities that minimises the average repair/replacement, acquisition and inventory/shortage total cost over an infinite horizon. We propose a suboptimal joint remanufacturing and supply control policy, composed of a multi-hedging point policy (MHPP) for the remanufacturing stage and an (s, Q) policy for the replacement parts supply. The MHPP is based on two inventory thresholds that trigger the use of predefined remanufacturing modes. Control policy parameters are obtained combining analytical modelling, simulation experiments and response surface methodology. The effects of the distribution, mean and variability of the lead time are tested and a sensitivity analysis of cost parameters is conducted to validate the proposed control policy. We also show that our policy leads to a significant cost reduction as compared to a combination of a hedging point policy (HPP) and an (s, Q) policy.     

Nervousness in inventory control: Analytical results

Using a rolling horizon planning framework in inventory control leads to nervousness in the planning system caused by instability of order release decisions in successive planning cycles. For a single-stage inventory system with stochastic demand it is shown analytically, how planning stability is affected by policy parameters if (s, n Q) and (s, S) control rules are applied. It turns out that the reorder points does not influence stability whereas the lot sizeQ and minimum reorder quantityS-s can have a considerable impact. Under the aspect of nervousness the (s, n Q) policy can be assessed to be less risky than the (s, S) rule.     

Modelling and solving spare parts supply chain network design problems

Spare parts are key operational assets in order to minimise unexpected equipment downtimes that may significantly impact a company’s results. The spare parts supply chain network supports the entire spare parts operations management and it is essential to achieve the planned goals. However, most of the traditional literature on spare parts management has not focused on the underlying supply chain network. Thus, this paper studies the integration of supply chain network design and control with traditional spare parts management. In particular, a generic network optimisation modelling structure is proposed, with simultaneous optimisation of warehouse locations and inventory control decisions, allowing minimising the total costs associated with the spare parts supply chain network. The generic model is specified based on three inventory control policies widely employed in the industry, which are suitable for managing a great variety of spare parts, i.e. (s, Q), (R, s, S) and (S-1, S). Furthermore, a solution approach is proposed based on Generalised Benders Decomposition. Finally, numerical results from a real-world application case in the process industry are shown and discussed.     

On the Bullwhip Avoidance Phase: supply chain collaboration and order smoothing

This paper provides an assessment of the impact of collaboration and smoothing replenishment rules on supply chain operational performance and customer service level. Three supply chain configurations (i.e. Traditional, Information Exchange and Synchronised) in which orders are generated by smoothing (S, R) inventory control policies are studied for different proportional controllers. A supply chain stress test is performed through a sudden and intense change in demand. A structured and extended supply chain assessment framework is adopted. The main conclusions of this paper are the following. (i) The impact of Supply Chain Collaboration on overall supply chain performance is greater than that of order smoothing. Order smoothing mitigates the bullwhip effect, but it may have a negative impact on customer service. Supply Chain Collaboration mitigates the bullwhip effect, provides inventory stability, limits lumpy orders and enhances customer service level. (ii) The negative effect on customer service level of order smoothing is almost eliminated in synchronised supply chains.     

Optimal reordering policies for the AHM inventory model with two supply modes

Summary We characterize the optimal reordering policies for the stationary periodic review AHM inventory model when there is a quick and a — generally cheaper — slow supplier, whose delivery lags differ by one period and whose fixed and unit ordering costs are arbitrary, as generalized (s,S) policies described by four parameters (s ₁,S ₁,s ₂,S ₂). Computational results using Howard's policy iteration algorithm yield four possible types of such policies depending on the model parameters.     

Planning stability under (s,S) inventory control rules

The problem of planning instability caused by new and actualized data in material coordination systems with periodic review and replanning activities has been discussed from several viewpoints in literature and it is of great practical importance. The application of a rolling horizon planning framework in forecast-dependent inventory control leads to nervousness in the planning system caused by different order release decisions in successive planning cycles. In this paper a setup-oriented planning stability concerning deviations of planned orders in all periods of a given stability horizon in a single-stage inventory system with arbitrary stochastic demand is analyzed. The influence of an (s, S) control rule on planning stability is derived analytically. It is shown that the reorder points does not affect stability whereas the minimum reorder quantityS-s has a considerable impact. Especially, it is analyzed how planning stability depends on the length of the stability horizon, on period-specific weights, and on the accuracy of demand forecasts.     

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.     

The use of rush deliveries in periodic review assemble-to-order systems

We calculate optimal safety stock in a periodic review (T,S) assemble-to-order system having multiple components and multiple finished goods (FGs). Customer orders for FGs arrive according to independent Poisson processes, and cannot be neither backlogged nor lost. In case of potential component stock-out, the studied system uses rush deliveries from suppliers. For this setting, approximate expressions of the optimal safety stock that minimise the sum of inventory holding and rush ordering costs are developed. Exact optimal safety stocks are calculated using Discrete Event Simulation, and compared numerically to the approximate expressions. The model is applied to a first-tier automotive supplier and yields to a significant reduction in terms of inventory holding and rush ordering costs. A sensitivity analysis on relevant system parameters such as components demand, assembly coefficients and unit rush ordering cost is conducted.     

Evaluation of divergent N-echelon (s,nQ)-policies under compound renewal demand

In this paper, decentralized inventory control in a divergent multi-echelon network is considered. All stockpoints are controlled by continuous review (s,nQ)-installation stock policies. Due to the fact that orders from a stockpoint to its predecessors may have to wait, this delay must be incorporated into the lead time of an arbitrary replenishment. We develop analytical approximations for the first two moments of this waiting time. These results enable us to accurately approximate the relevant performance characteristics at each stockpoint, given its (s,nQ)-policy. The performance measure considered in this paper are fill rate and average physical inventory. The analysis can easily be extended for other performance measures. The main contribution of this paper is the fact that the results apply to arbitrary divergent multi-echelon systems, with stochastic transportation times, arbitrary lot sizes and compound renewal demand. Correspondence to: Gudrun P. Kiesmüller     

Optimal policies in multiproduct inventory models

Summary We consider a period review, dynamic multiproduct inventory model with stochastic demand, in which the cost of ordering consists of linear portions for each product as well as a nonlinear term. This nonlinear term depends for example, on the products we order. We prove the existence of an optimal policy and characterize such a policy. We show that an optimal policy reduces to the well-known (, S)-policy, if we spezialise the nonlinear ordering cost in a proper way. Finite horizon results are given.

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Zusammenfassung Wir betrachten ein periodisch kontrolliertes dynamisches Mehrprodukt-Lagerhaltungsmodell mit stochastischem Bedarf, bei dem sich die Bestellkosten aus einem linearen und einem nichtlinearen Anteil zusammensetzen. Der nichtlineare Anteil kann z. B. davon abhängen, welche Produkte bestellt werden. Es wird die Existenz einer optimalen Bestellpolitik gezeigt. Außerdem werden Eigenschaften einer solchen Politik nachgewiesen. Wir zeigen, daß bei Spezialisierung der nichtlinearen Bestellkosten eine optimale Politik eine (, S)-Politik ist. Die Ergebnisse beziehen sich auf den endlichen Planungshorizont.

     

An (s,S) inventory system with poisson demands and exponential lead time

Summary This paper considers a stochastic inventory system with Poisson demand and exponentialy distributed delivery time. Unsatisfied demand is lost. Optimality of ans, S policy andD=S–ss are assumed. The steady state probabilities of the system are calculated, average costs are obtained and the minimized. Two limiting cases may be solved in closed form: rate of demand much larger or much smaller than rate of delivery. In the last case the Wilson lot size formula is regained.Stochastic inventory systems have been studied extensively in the literature (see, for example, Arrow, Karlin and Scarf 1958 [1]. Hadley and Within 1963 [3], Veinott and Wagner 1966 [9], Beckmann 1961 [2], Kaplan 1970 [4], Simon 1971 [5], Sivazlian 1974 [6]).

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Zusammenfassung Ein stochastisches Lagerhaltungssystem mit Poisson-Nachfrage und exponentieller Lieferzeit wird betrachtet. Nichtbefriedigte Nachfrage geht verloren. Die Optimalität einers, S Politik wird vorausgesetzt und die Annahme gemacht, daß die Bestellmenge größer ist als der Wiederbestellpunkt. Die Zustandswahrscheinlichkeiten des Systems werden berechnet und die Durchschnittskosten bestimmt und minimiert. Zwei Grenzfälle werden näher untersucht: die durchschnittlichen Intervalle zwischen den Nachfragen sind entweder sehr groß oder sehr klein im Vergleich zur mittleren Lieferzeit. Im letzten Fall erhält man wieder die Wilsonsche Losgrößenformel.

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This work has been supported by a grant from the Deutscher Akademischer Austausch Dienst (DAAD). We should like to thank Dr. Joachim Fischer for several corrections, for drawing Fig. 1 and for computing Table 1     

Setting planned orders in master production scheduling under demand uncertainty

For single end-product master production scheduling with time-varying demand uncertainty and supply capacity, we study approaches to set replenishment quantities over the planning horizon. We present a stochastic programming model along with a simulation-based optimisation and two traditional approaches for setting order quantities. We compare these approaches to two new methods: gamma approximation and safety stock search. Computational experiments show that the gamma approximation and safety stock search perform well in terms of holding and shortage costs, with expected total cost on average, respectively, within 0.06% and 0.66% of the optimal from the stochastic program. On average, the two traditional approaches incur 12% and 45% higher cost than optimal. We provide managerial insights on the effects of parameters such as demand coefficient of variation (cv), utilisation, and target service level on the optimal total cost, the corresponding fill rate, and the relative performance of the approaches. We find that, for finite-normal demand, on average, the impact of target service level on cost is larger than that of demand cv, whose impact is larger than utilisation, except at high utilisation. We illustrate that, when demand is not normal, the gamma approximation significantly outperforms the existing normal approximation from Bollapragada and Rao (2006 Bollapragada, R and Rao, US. 2006. Replenishment planning in discrete-time, capacitated, non-stationary, stochastic inventory systems. IIE Transactions, 38(7): 583–595.  [Google Scholar]).