Issues in inventory control models with demand and supply uncertainty Thesis proposal

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1 Issues in inventory control models with demand and supply uncertainty Thesis proposal Abhijit B. Bendre August 8, 2008 CORAL Centre for Operations Research Applications in Logistics Dept. of Business Studies, Aarhus School of Business, University of Aarhus Fuglesangs Allé 4, DK-8210 Aarhus V, Denmark Introduction: If we observe closely, inventories can be found everywhere. We don t know since when ants and squirrels are keeping inventories of their food supplies. And we don t know how they learned to keep an account of these inventories. Not only wildlife but also humans have been smart enough to realize the benefits of inventories. Since stone-ages we have been carrying inventories and managing them. But, the development of modern inventory management principles began when Harris (1913) derived the Economic Order Quantity (EOQ) formula. EOQ assumes that demand occurs at known, constant rate and supply fulfills the replenishment order after a fixed lead time. Unfortunately, the real world is not as ideal as that. In reality, demand rate is rarely constant; hard-to-predict market is common in most practical situations. Also, unpredictable events in supply systems can cause unpredictable delays in replenishments. Moreover, in current times when outsourcing is at the centre stage, complex and longer supply chains magnify the length and variability of lead times (Welborn, 2008). Although in the early days researchers acknowledged the necessity for considering uncertainties present in the real world, the rigorous work on inventory control models with stochastic features really began in 1950s. The classic book by Hadley and Whitin (1963), comprehends the research work done in this field to that date. This fundamental research done in those early days has had a pivotal effect on the subsequent developments in the field of inventory theory. An article by Lee (2002) presents the uncertainty framework, which considers dimensions of demand and supply uncertainties. This framework can be a simple but powerful way to characterize a product; which can be useful in devising an appropriate supply chain strategy for that product. Uncertainties in demand and supply can result in excessive inventories and deteriorated customer service, indicating out of control supply chain. In the presence of uncertainties, it is difficult to foresee the final effects of the actions taken and hence to manage the inventories efficiently. In general, it is observed that stochastic lead times and demand have their greatest impact in combination (Zipkin, 2000). In this era of outsourcing and/or offshoring longer lead times are common, especially because the transportation time might be considerable. Usually, long lead times and uncertain demand hamper the performance of inventory control systems. Also, there are hardly any supply processes which have completely avoided the issues of limited capacity and unpredictable quality. These issues have even more pronounced effects in the presence of stochastic lead times. Hence, considering further stochastic features in inventory control models will bring our study closer to practical problems. 1 of 7

2 In inventory systems with stochastic elements, it is important to consider the effect of shortages and to trade off the cost of shortages against the cost of holding inventory. One way of dealing with shortages can be through backordering, where demand is backlogged if can not be satisfied immediately from the inventory. Lost sales represent another way in which shortages might result. Lost sales might be interpreted as a definite loss of a sales opportunity, and the case where the demand cannot be satisfied by the inventory system considered, but is eventually satisfied outside this system, e.g. by using expedited ordering or special supplies. The study by Corsten and Gruen (2004) shows that, in retail industry almost half of the cases of shortages result in lost sales. Lost sales also appear to be a typical mechanism for handling shortages in some spare parts industries. So far in inventory theory tremendous work has been done on policies with backorders. More sparsely studied is the case of lost sales. One reason is that fundamental results from the backorder case do not hold for the lost sales case, which makes the latter much more complicated to model exactly. Hence, it would be reasonable to say that with expanding retail and spare parts industries, the lost-sales case deserves more attention in research studies. We mainly consider single-item inventory systems. However, these systems may also be found embedded as building blocks in larger systems with multiple items and/or multi-echelon structures, commonly known as supply chains. We hope that our understanding of the smaller systems can be further exploited when studying more complicated supply networks. In this context, this PhD thesis focuses on studying a few of the particular problems concerning inventory control models with demand and/or supply uncertainty. The thesis is designed to consist of a collection of 4 or 5 papers. Although all problem statements are motivated by the above mentioned practical issues, not all of them are explicitly linked to each other. Hence, each paper is based on a separate research question of itself. The next section presents each project briefly as subprojects A-E, each with its problem specification, methodology, expected results, and if the project is already in progress, then also its current status. Description of subprojects: A. Base-stock policies for the lost-sales case under exogenous and endogenous, sequential supply systems. It is a challenge for any inventory policy to manage ubiquitous uncertain demand as well as supply with uncertain lead times, while achieving acceptable service levels at minimum total costs and it is particularly difficult for the lost-sales case. Hence, the purpose of this study is to obtain a better understanding of the performance of widely used base-stock policy for the lost-sales case under different stochastic leadtime regimes. The base-stock policy is sometimes also referred to as an (S-1, S) system with S-1 corresponding to a reorder point and S being the order-up-to level. In this subproject, we first classify different lead-time regimes. The matrix in Figure 1 aptly presents the classification of lead-time regimes under uncertainty. Then for this paper we focus on endogenous and exogenous regimes and specifically consider dependent sequential lead times (models IV and VI). 2 of 7

3 Lead-time crossover Sequential lead times (No lead-time crossover) Independent Dependent Independent Dependent Exogenous I II III IV supply system Endogenous (- - -) V (- - -) VI supply system Figure 1. Classification of lead-time regimes under uncertainty We consider a single-item inventory model managed by a continuously reviewed base-stock policy. Demand is Poisson and lead times are stochastically dependent under the realistic assumption that they are sequential. Also, we draw service times (for endogenous supply system) and quoted lead times (for exogenous supply system) from a Gamma(1/μ, r) distribution. These systems are modelled using discrete event simulation. During the experiments we adjust 1/μ and r to change the average lead time of the supply system and induce variability in service times and the quoted lead times. We run these experiments for a range of base-stock levels. For each experimental scenario, we observe effective lead-time characteristics and also study the effects on fill rate, average inventory and long-run average cost performance of the inventory system. The findings from our work so far indicate that, in the presence of lost sales an exogenous supply system is more effective than an endogenous system, especially in case of long lead times and high base-stock levels. Also, for an exogenous sequential supply system an interesting and subtle behaviour of average lead time was noticed for higher stock-out frequencies. We observed that average lead time decreases with increasing stock-out frequency. In the future it could be of interest to model sequential supply systems with other mechanisms than the one employed in this study. For such systems it could then be of particular interest to check for the leadtime decrease observed here for high stock-out frequencies. Moreover, it could be interesting to analyse the characteristics of realized lead times under a wider range of experimental scenarios. Further research could also involve analytical models for comparing the lead-time regimes considered in this paper. B. Base-stock policies for the lost-sales case under sequential and non-sequential supply systems. The subproject for this paper is inspired by the classification of lead-time regimes, presented in Figure 1. In reference to this figure, we plan to compare performance of the base-stock policy for models II vs. IV and models V vs. VI. Hence, we compare the performance of exogenous supply systems, which have sequential and nonsequential lead times. We also plan to conduct the same kind of comparison for endogenous supply systems. For this subproject we restrict our focus to dependent lead times and postpone work on independent lead times for future. In the literature, there is an ongoing debate on whether lead times with crossovers or sequential lead times represent the most realistic characteristic of real-life inventory systems under supply uncertainty. In practice both order crossovers and sequential 3 of 7

4 lead times can be found in different business settings. As discussed by Riezebos (2006), order crossovers are usually observed when there are multiple suppliers and it is most typical for the sole supplier to provide a sequential supply system. Hence, this study might help in understanding the implications of decisions regarding single and multiple suppliers, on the characteristics of replenishment lead times and performance of inventory systems. Two of the models for sequential lead times will be similar to those in Paper A, while new models have to be built for capturing the exact characteristics of order crossovers. In this subproject we intend to follow a hybrid approach, where both analytical and simulation models will be employed. For drawing conclusions from this study, we intend to study the effects of above mentioned regimes on lead-time characteristics and inventory performance measures such as fill rate and average inventory. C. Evaluation of performance approximations for (r, q) inventory policies in a lost-sales setting. The (r, q) inventory policy, in which the replenishment quantity q is ordered when the inventory position reaches the reorder point r, is one of the most widely practiced control policies for single-stage, single-item inventory systems. This policy has been thoroughly studied when demand is backordered, whereas more sparsely studied is the case of lost sales. We start our study by building a framework for studying models with lost sales and (r, q) policies. Also, we plot the (r, q) decision space for the case of lost sales, which helps in pin-pointing focus of this paper. The study of this decision space reveals the fact that no exact closed-form expressions are available for (r, 1 < q < r) inventory system, where more than one order might be outstanding. The purpose of this paper is to investigate the performance of an inventory system with lost sales controlled by the (r, q) policy under constant lead times and when several orders might be outstanding at a time. Demand is Poisson and lead times are assumed to be constant. In particular, we focus on long-run average performance of the fill rate, the inventory level and the ordering frequency. Although the system might appear simple, it is in fact well known to be rather difficult to model exactly. Until quite recently, no exact results have been available, except for some special cases. The exact results that are presented by Johansen and Thorstenson (2004) do not include closed-form expressions, but require rather elaborate computations. Hence, for practical applications there is still a need for simple approximations and to investigate their behaviour under different parameter settings. The results from our work so far, indicate that simple approximations suggested in the literature (Zipkin, 2000) can induce significant errors in the estimates of inventory performance. We evaluate the approximations by comparisons to results obtained from simulations and thus find that refinements are required. An appropriate choice of approximation may simplify the performance evaluation and be of use in guiding policy decisions of the inventory control system under consideration. As a first step towards developing such refined approximations, in a simulation study we test which distributions give the best fit to the simulated inventory position and inventory levels respectively. We conjecture that the knowledge obtained regarding 4 of 7

5 the representation of the inventory position, will be useful in developing improved simple approximations. In the future, we plan to extend the numerical experiments and analyses to complete the study and to test further parameter settings. Also, further processing and analysis of collected inventory level information can provide further insight in the inventory performance. This insight may enable us to formulate other simple approximations for performance measures. An interesting extension would be to investigate the problem setting when there is also supply uncertainty, for example stochastic lead times. D. Lost-sales case and base-stock policy with information about supply condition. During the last decade, flexible supply chains have attracted tremendous attention. In this era of globalisation, firms are serving a wide spectrum of market environments with diverse and uncertain demand patterns. The retail sector has always been in the trenches facing these diverse demands with low forecast accuracy, and at the same time trying to avoid lost sales while keeping the inventory levels low. Fisher (1997) presents the effect of such demand patterns on the retail sector and the significance of flexible supply chains for running an efficient business in such a market environment. One of the virtues of flexible supply chains, which have always been taken for granted, is the market responsive supply system. As discussed in Song et al. (1996), the supply condition responsive replenishment ordering systems can also contribute a great deal in achieving cost efficient flexibility in supply chains. Das and Abdel-Malek (2003) argue that in order to achieve such a responsive ordering system, suppliers also need to accommodate flexible order sizes. In Song et al. (1996) an inventory control model with backorders is presented, which includes a supply system that evolves over time, as do the replenishment lead times. Due to shared supply information, parameters of the inventory control model changes according to current supply conditions. The purpose of this subproject is to formulate a similar inventory model for the case of lost sales. In this subproject, we plan to focus on base-stock type policies. We assume that the supply system is exogenous and that it processes orders sequentially. Depending on the complexity of the problem and the availability of time, we hope to consider stochastic demand as well. As suggested by Song and Zipkin (1996), Markov decision processes can be an efficient way to model this problem. Obviously, as many fundamental relations from the backorder case do not apply in the case of lost sales, actual modelling of the problem could become quite complicated. As a result we hope to formulate a model, from which it is possible to compute a base-stock type of policy that gives the lowest average total cost. E. Flexible transportation options An article in Harvard Business Review by a senior vice president with the Boston Consulting Group (Stalk, 2006) rightly discusses the issue that when supplies are outsourced to suppliers situated halfway around the world, biggest concern is the long transportation times. These lengthier transportation routes have greater risk of disruptions and hence uncertainty. Obviously, such transportation times result in 5 of 7

6 longer and more uncertain order replenishment lead times. This can seriously affect the business performance by not only elevating the average inventory-in-transit level, but also by deteriorating the service levels when the delivery promises are not kept. Hence, flexible transportation may facilitate the optimization of total cost for outsourcing. To squeeze transportation time whenever it is necessary, Stalk (2006) also recognizes the need for appropriate selection of transportation options at the correct point of transit. Hence, it might be of great practical significance to formulate a formal model which helps in identifying an appropriate transportation option at each transit point. Basically we need to develop a model, which keeps track of an order and finds the best possible transportation alternative based on the information available about the current status of an order. This model should find best possible set of transportation alternatives, which can trade off inventory carrying and transportation costs with shortage cost and result in least possible total cost. This research idea is still in its infancy; hence at this stage we can not exactly identify the solution method for this problem. The order is going through the transition of states during transportation and decisions have to be made based on these states. So, we recon that Markov Decision Process might be one way to find the solution for this research problem. References Corsten, D and Gruen, T (2004). Stockouts cause walkouts, Harvard Business Review, 82(5): Das, S K and Abdel-Malek L (2003). Modeling the flexibility of order quantities and lead-times in supply chains, International Journal of Production Economics, 85(2): Fisher, M (1997). What is the right supply chain for your product?, Harvard Business Review, 75(2): Hadley, G and Whitin, T M (1963). Analysis of Inventory Systems, Prentice Hall, Inc., Englewood Cliffs, NJ Harris, F (1913). How many parts to make at once. Factory, The Magazine of Management, 10: , 152 Johansen, S G and Thorstenson, A (2004). The (r, q) policy for the lost-sales inventory system when more than one order may be outstanding, Working Paper L , Aarhus School of Business, University of Aarhus Lee H L (2002). Aligning supply chain strategies with product uncertainties, California Management Review, 44(3): Riezebos, J. (2006). Inventory order crossover, International Journal of Production Economics, 104(2): Song, J.-S. and Zipkin, P.H. (1996). Inventory control with information about supply conditions, Management Science, 42(10): Stalk, G Jr (2006). The costly secret of China sourcing, Harvard Business Review, 84(2): of 7

7 Welborn, C (2008). Strengthening supply chains, Operations Research and Management Science Today (ORMS Today by INFORMS), 35(3): Zipkin, P H (2000). Foundations of Inventory Management, McGraw-Hill, Boston 7 of 7