Lead Time Distribution Research Articles

Inventory rationing on a one-for-one inventory model for two priority customer classes with backorders and lost sales

In this study, we are primarily motivated by the research problem of recognizing heterogeneous customer behavior towards waiting for order fulfillment under the threshold rationing policy (also known as the critical level policy), and aim to find its effect on system stock levels and performance measures. We assume a continuous review one-for-one ordering policy with generally distributed lead times. In the first model, we consider the case in which the low-priority customer class exhibits zero patience for waiting if the demand is not satisfied immediately (a lost sale), whereas the demand of the high-priority customer class can be backordered. This is the first study in the literature to consider this model. We provide an exact analysis for the derivation of the steady-state probability distribution and the average infinite horizon cost per unit time. We then develop an efficient optimization procedure to minimize the average expected cost rate. We also determine the forms of the optimal solutions for the two service level optimization models that are common in practice. In the second model, we study the opposite case in which the high-priority customer class exhibits zero patience for waiting. We establish a theoretical basis for the rationale of using the Continuous-Time Markov Chain (CTMC) approach as an approximation. We show that under certain assumptions, the steady-state probabilities of the system with generally distributed lead times are identical to the steady-state probabilities of the CTMC system with the same mean. This result enables us to link the dynamics of the studied model to the CTMC model, which may open new doors for future research.

A Discrete Event Simulation Analysis of the Bullwhip Effect in a Multi-Product and Multi-Echelon Supply Chain of Fast Moving Consumer Goods

Timely delivery is the major issue in Fast Moving Consumer Good (FMCG) since it depends on the lead time which is stochastic and long due to several reasons; e.g., delay in processing orders and transportation. Stochastic lead time can cause inventory inaccuracy where echelons have to keep high product stocks. Such performance inefficiency reflects the existence of the bullwhip effect (BWE), which is a common challenge in supply chain networks. Thus, this paper studies the impact of stochastic lead time on the BWE in a multi-product and multi-echelon supply chain of FMCG industries under two information-sharing strategies; i.e., decentralized and centralized. The impact was measured using a discrete event simulation approach, where a simulation model of a four-tier supply chain whose echelons adopt the same lead time distribution and continuous review inventory policy was developed and simulated. Different lead time cases under the information-sharing strategies were experimented and the BWE was measured using the standard deviation of demand ratios between echelons. The results show that the BWE cannot be eliminated but can be reduced under centralized information sharing. All the research analyses help the practitioners in FMCG industries get insight into the impact of sharing demand information on the performance of a supply chain when lead time is stochastic.

The residual time approach for (Q, r) model under perishability, general lead times, and lost sales

We consider a (Q, r) perishable inventory system with state-dependent compound Poisson demands with a random batch size, general lead times, exponential shelf times, and lost sales. We assume $$r<Q$$ and analyze the system using an embedded Markov process at the replenishment points. Using the queueing and Markov chain decomposition approach, we characterize the distribution of the residual lead time and derive the stationary distribution of the inventory level. We construct closed-form expressions for the expected total long-run average cost function. The closed form allows us to efficiently obtain, numerically, the optimal Q and r parameters. Numerical study provides several guidelines for the optimal control. For example, we show that approximating the lead time distribution by an exponential one only works when the optimal reorder point of the approximation is very small; in other cases the usage of the exact distribution can lead to substantial cost savings (up to 14%). We further provide intuition insight on the optimal controls and how they depend on different factors, e.g., the lead time variability, and the demand features (arrival rate, size and variability).

A manufacturer-buyers integrated inventory model with generic distribution of lead times to deliver equal and/or unequal batch sizes

Although lead time variation is common in practice, integrated single-manufacturer multi-buyer model considering this factor is unavailable in the extant literature. This article considers generic distribution of lead times of delivering equal and/or unequal batch (sub-lot) sizes of a lot in developing a synchronised integrated single-manufacturer multi-buyer model. The batch sizes are assumed to be in geometric series. The variables considered in the model are the smallest batch size, total number of batches and number of unequal batch sizes delivered from the manufacturer to buyers. The smallest batch sizes delivered to the buyers are bounded below by 1 and above by the capacity of the transport vehicle. The minimal total cost solution technique to the model is derived by the method of differentiation. Significant minimal total cost reductions by the synchronised flow is illustrated through solutions to some numerical example problems. Sensitivity analyses on increasing costs of transportation, shortage, inventory and increasing mean lead times upon the optimal solution have been performed.

A manufacturer-buyers integrated inventory model with various distributions of lead times of delivering equal- sized batches of a lot

The single manufacturer multi-buyer integrated inventory supply chain problem with deterministic lead time of delivering batches (sub-lot) of a lot has received considerable attention. Because of influences by various factors, usually deviations occur in the times of setup of machines, inspection, loading, transportation, unloading etc., which result deviation in the lead time of delivering batches of a lot. Although the single-manufacturer single-buyer integrated supply chain problem with a particular distribution of lead times has received some attention, a model considering various distributions of lead times is unavailable. Also, the single-manufacturer multi-buyer integrated supply chain problem with stochastic lead time has received no attention from researchers. To fill up these gaps, here we develop a single-manufacturer multi-buyer integrated inventory supply chain model with various distributions of lead times of delivering equal-sized batches of a lot. Then an optimal solution technique to the model is developed, and illustrated with numerical examples. The significant potential benefit is highlighted by comparative study on solutions to numerical problems. Sensitivity analyses on the effects of various parameter values upon the minimal total cost solutions are carried out. The shortcomings of the current solution method are highlighted along with the indication of future research scope.

A note on the stochastic nature of lead times in multi-echelon supply chains

Recent supply chain research has concluded that stochastic lead times are a significant source of bullwhip effect in reorder-based supply chains. Despite this, the study of lead times and their nature has received remarkably little attention in literature. In this research we prove there exists an interdependency between lead time and service level measure, and thus the magnitude of the bullwhip effect. In a multi-echelon supply chain setting, the decision maker can experience disproportionate bullwhip effect. One cause of this is the fact that lead time distribution may likely stem from skewed and peaked distributions. As a member of a multi-echelon supply chain, a firm can experience an order variance that depend on the total number of echelon and information about their lead-time dependencies. These results correspond well with the few contributions found in literature showing actual lead times.

A Stochastic Queueing - Inventory System with Renewal Demands and Positive Lead Time

This article analyses a stochastic inventory system with a service facility. This is an extended work of Yadavalli et al. (2008) by including the positive lead time. The customer arrives according to a renewal process and demanded item is delivered to the customer after performing an exponentially distributed service time. An (s, S) type ordering policy is adopted with exponentially distributed lead times. The stationary probability distribution for number of customers in the system and inventory level at arrival epoch and at arbitrary time point are derived. Some system performance measures in the steady state are computed and using these system performance measures the long-run expected cost rate is calculated. Since the long run expected cost rate is highly complex, the mixed integer distributed ant colony optimisation is used to obtain the optimal values. A sensitivity analysis to illustrate the effects of parameters and cost on the optimal values is also carried out in this work. [Received: 13 December 2018; Accepted: 10 October 2019]

An inventory system with retrial demands, multiple vacations and two supply modes

This paper analyses a continuous review inventory system with single server, multiple vacations, Poisson demand, retrial demand, exponential distributed lead time and two supply modes for replenishment with one having a shorter lead time. To derive the stationary distribution of the system, we employ the Gaver's method. After computing various system performance measures, some cost minimisation numerical results are presented.

A stochastic queueing - inventory system with renewal demands and positive lead time

This article analyses a stochastic inventory system with a service facility. This is an extended work of Yadavalli et al. (2008) by including the positive lead time. The customer arrives according to a renewal process and demanded item is delivered to the customer after performing an exponentially distributed service time. An (s, S) type ordering policy is adopted with exponentially distributed lead times. The stationary probability distribution for number of customers in the system and inventory level at arrival epoch and at arbitrary time point are derived. Some system performance measures in the steady state are computed and using these system performance measures the long-run expected cost rate is calculated. Since the long run expected cost rate is highly complex, the mixed integer distributed ant colony optimisation is used to obtain the optimal values. A sensitivity analysis to illustrate the effects of parameters and cost on the optimal values is also carried out in this work. [Received: 13 December 2018; Accepted: 10 October 2019]

Numerical Validation of the Continuous Time Markov Chain Approximation as a Service Level Estimation Method for Heterogeneous Customer Demand Classes

In this article, we study a threshold level based inventory allocation and replenishment problem for two priority-demand classes under the lot-for-lot replenishment policy. Demands of low-priority class are backordered when on-hand stock is at or below a certain threshold level, while high-priority class demands are lost in case of stock-out situations. We assume a continuous review lot-for-lot ordering policy with both deterministic and stochastic replenishment lead times. Both priority classes exhibit mutually independent, stationary, Poisson demand processes. There is no exact solution yet in the literature except for the special case of exponentially distributed lead times. In this complementary study, we aim to numerically validate the Continuous-Time Markov Chain (CTMC) approximation as a service level approximation method by thoroughly analyzing the effect of the type of lead time distribution, lead time variability, and value of system parameters on the quality of the class-specific service level estimations. As long as expected lead times are identical, we also show empirically the near-insensitivity of the effect of the type of lead time distribution and the lead time variability on the achieved service levels. The CTMC approximation works well under variety of system parameters and lead time distributions. For the cases considered in the constant lead time scenarios with service levels which we expect to see in practice (with fill rate requirements at least 60 % and 90 % for the low-priority and high priority customer classes, respectively), the average absolute errors for the CTMC approximations are calculated as 0.09 % (for the low-priority class fill rate) and 0.05 % (for the high-priority class fill rate); while the maximum absolute errors for the CTMC approximations are calculated as 0.37 % (for the low-priority fill rate) and 0.18 % (for the high-priority fill rate). The CTMC approach also provides quality approximations for the Erlang, lognormal and gamma distributed lead times.

Inventory Rationing on a One-for-One Inventory Model with Backorders and Lost Sales

In this study, we are primarily motivated by the research problem of recognizing heterogeneous customer behavior towards waiting for order fulfillment under the 'threshold rationing policy' (also known as the 'critical level policy'), and aim to find its effect on system stock levels and performance measures. We assume a continuous review one-for-one ordering policy with generally distributed lead times. In the first model, we consider the case in which low-priority customer class exhibits zero patience for waiting if the demand is not satisfied immediately (a lost sale), while the demand of high-priority customer class can be backordered. This is the first study in the literature to consider this model. We provide an exact analysis for the derivation of the steady-state probability distribution and the average infinite horizon cost per unit time. We then develop an efficient procedure to minimize the average expected cost rate by deriving bounds on the optimal cost, which reduces the enumeration space considerably. In the numerical study, we show the relative savings achieved by implementing the proposed threshold rationing policy over some of the traditional industry practices. In the second model, we study the opposite case in which the high-priority customer class exhibits zero patience for waiting. The existing study in the literature uses simulation to demonstrate that the performance of the critical level policy is robust with respect to the form and variability of the lead time distribution provided the mean lead times are identical. Then it proposes using the Continuous-Time Markov Chain (CTMC) equivalent of the model as an approximation. In this study, we establish a theoretical basis for the rationale of using the CTMC approach as an approximation. We show that under certain assumptions the steady-state probabilities of the system with generally distributed lead times are identical to the steady-state probabilities of the CTMC system with the same mean. This result enables us to link the dynamics of the studied model to the CTMC model, which then allows us to provide a theoretical explanation to the empirically observed phenomenon that why the steady-state probabilities and performance measures are near-insensitive to the form and variability of the lead time distribution as long as mean lead times are identical.

A Simulation-Based Model for Determining Safety Inventory at a Fixed-time Period System

This paper develops a simulation model for determining safety inventory associated with a certain value of cycle service level in a fixed-time period system. The model takes into account actual amount of materials received from suppliers, and deviation from probability distribution of daily forecast demand. Constraints on order size are also embodied into the model. This model was constructed by using Visual Basic Application added in Microsoft Excel. After developing the model, hypotheses testing is employed to verify the model. This model allows identifying safety inventory under uncertain conditions which prohibits from the use of ordinary mathematical formula. The model was locally verified. Stochastic variables including customer demand and supplier’s lead time are assumed to be normally distributed. Independent demand items are considered and backorders are not allowed. Under specific conditions, such as distributions of demand and lead time are normally distributed, and fixed-time period system is being used. This model allows materials planner promptly identifies safety inventory associated with a certain level of cycle service level. Furthermore, planner can perceive the affects of changing input parameters on the amount of safety inventory required. There were very few researches focus on variations of demand and lead time at the same time. In reality, this case usually happens, thus the firms have been facing highly variations form both supplier and customers. Therefore, this paper intends to close this gap by simulating these factors and taken into account for determining safety inventory.

QMCD approach for perishability models: The (S, s) control policy with lead time

We consider cost minimization for an (S, s) continuous-review perishable inventory system with random lead times and times to perishability, and a state-dependent Poisson demand. We derive the stationary distributions for the inventory level using the Queueing and Markov Chain Decomposition (QMCD) methodology. Applying QMCD, we develop an intuitive approach to characterizing the distribution of the residual time for the next event in different states of the system. We provide comprehensive analysis of two main models. The first model assumes a general random lifetime and an exponential distributed lead time. The second model assumes an exponential distributed lifetime and a general lead time. Each model is analyzed under both backordering and lost sales assumptions. We consider a fixed cost for each order, a purchase cost, a holding cost, a cost for perished items, and a penalty cost in the case of shortage. Numerical examples are provided and show that variability of lead time is more costly than that of perishability time. Therefore, after reducing lead time and increasing perishability time, managers should focus on reducing variability of lead time.

Optimization of multi-period supply planning under stochastic lead times and a dynamic demand

Supply planning and inventory control are vital in Supply Chain (SC) optimization processes for companies that aim to produce highest-quality finished product at lowest cost and right on time. Optimization means planners’ need to reduce average stock levels and determine the optimal safety lead-times. This study deals with a multi-period production planning problem with a known dynamic demand. Order lead-times are independent, discrete random variables with known and bounded probability distributions. A general probabilistic model, including a recursive procedure to calculate the expected total cost (ETC), is derived. A genetic algorithm (GA) is developed for this model to determine planned lead-times and safety stock level while minimizing the ETC, where ETC is the sum of expected backlogging cost (ETBC) and expected inventory holding cost (ETHC). This approach is then compared to three other classical models to test its efficiency. The results prove that, under certain assumptions, it could be better to optimize planned lead-times rather than implement safety stocks. To understand the effect of lead-time dispersion on solution robustness, different levels of variance and different shapes of lead-time distributions are studied. Analysis proves that the lead-time variability has little effect on ETC when unit inventory holding cost is close to unit backlogging cost.

Continuous tumour growth models, lead time estimation and length bias in breast cancer screening studies.

Comparisons of survival times between screen-detected and symptomatically detected breast cancer cases are subject to lead time and length biases. Whilst the existence of these biases is well known, correction procedures for these are not always clear, as are not the interpretation of these biases. In this paper we derive, based on a recently developed continuous tumour growth model, conditional lead time distributions, using information on each individual's tumour size, screening history and percent mammographic density. We show how these distributions can be used to obtain an individual-based (conditional) procedure for correcting survival comparisons. In stratified analyses, our correction procedure works markedly better than a previously used unconditional lead time correction, based on multi-state Markov modelling. In a study of postmenopausal invasive breast cancer patients, we estimate that, in large (>12 mm) tumours, the multi-state Markov model correction over-corrects five-year survival by 2-3 percentage points. The traditional view of length bias is that tumours being present in a woman's breast for a long time, due to being slow-growing, have a greater chance of being screen-detected. This gives a survival advantage for screening cases which is not due to the earlier detection by screening. We use simulated data to share the new insight that, not only the tumour growth rate but also the symptomatic tumour size will affect the sampling procedure, and thus be a part of the length bias through any link between tumour size and survival. We explain how this has a bearing on how observable breast cancer-specific survival curves should be interpreted. We also propose an approach for correcting survival comparisons for the length bias.

Comparison of three vacation policies in discrete time inventory system with postponed demands

In this paper, we consider a single server discrete time inventory system with postponed demands and server vacation. Arrival process follows DMAP and ordering policy follows policy with geometrically distributed lead time. Due to depletion of inventory in the stock to zero, server may avail vacation immediately or after some random time. Here, we proposed three different vacation policies, namely modified multiple vacation (random idle time before each vacation), single (only one vacation) and multiple (one after another vacation). The demand who arrives during the stock-out period or server vacation period enters the pool of infinite size. These pooled demands are selected one-by-one according to a rule. The vacation time, the server idle time and the inter-selection time follow independent DPH distribution. The joint probability distribution of the number of demands in the pool, the server status, the on-hand inventory level is obtained in the steady-state case. We derived the total expected cost function in the steady-state case for each vacation model. We present some numerical examples and try to justify the best vacation policy for different parameter rates. Also, we analyse the impact of system costs on the total optimal cost of modified multiple vacation policy.

A two-echelon integrated inventory model under generalized lead time distribution with variable backordering rate

This paper investigates a generalized lead time distribution with a variable backordering rate in a two-echelon supply chain system. The vendor produces a single product and delivers to the buyer in equal sized batches. The delivery lead time follows a generalized stochastic variable. Shortages are allowed to occur and backordered partially. The backorder rate depends on the demand on stock-out period. Based on this notion, we formulate a mixed integer non-linear cost function which needs to be minimized with respect to reorder point, number of deliveries and lot size from the vendor to the buyer, to operate cooperatively in the integrated model. Analytically we proved the convexity of the generalized lead time distribution cost function with respect to the control parameters. Further, the uniqueness of optimality has been proved. To validate the proposed model, uniform, exponential and normal distributed lead times are presented in numerical example section. Sensitivity analysis also performed to the values of the parameters.

A note on the stochastic nature of lead times in multi-echelon supply chains

Recent supply chain research has concluded that stochastic lead times are a significant source of bullwhip effect in reorder-based supply chains. Despite this, the study of lead times and their nature has received remarkably little attention in literature. In this research we prove there exists an interdependency between lead time and service level measure, and thus the magnitude of the bullwhip effect. In a multi-echelon supply chain setting, the decision maker can experience disproportionate bullwhip effect. One cause of this is the fact that lead time distribution may likely stem from skewed and peaked distributions. As a member of a multi-echelon supply chain, a firm can experience an order variance that depend on the total number of echelon and information about their lead-time dependencies. These results correspond well with the few contributions found in literature showing actual lead times.

Bullwhip Effect of Weighted Moving Average Forecast under Stochastic Lead Time

The bullwhip effect is well-known as the propagation of fluctuation from downstream to upstream under stochastic demand. Researches in literature assume deterministic lead time, but it may be stochastic because of disruption of production. In this article we analyze the bullwhip effect of weighted moving average forecast under stochastic lead time. Under a general setting we express a performance measure on bullwhip effect theoretically. Numerical results give effects of parameters of forecasting and the lead time distribution on the bullwhip effect.

An Inventory Model for Deteriorating Drugs with Stochastic Lead Time

Inventory management of deteriorating drugs has attracted considerable attention recently in hospitals. Drugs are a kind of special product. Two characteristics of some drugs are the shorter shelf life and high service level. This causes hospitals a great deal of difficulty in inventory management of perishable drugs. On one hand, hospitals should increase the drug inventory to achieve a higher service level. On the other hand, hospitals should decrease the drug inventory because of the short shelf life of drugs. An effective management of pharmaceuticals is required to ensure 100% product availability at the right time, at the right cost, in good conditions to the right customers. This requires a trade-off between shelf-life and service level. In addition, many uncontrollable factors can lead to random lead time of drugs. This paper focuses on deteriorating drugs with stochastic lead time. We have established a stochastic lead time inventory model for deteriorating drugs with fixed demand. The lead time obeyed a certain distribution function and shortages were allowed. This model also considered constraints on service level, stock space and drug shelf life. Through the analysis of the model, the shelf life of drugs and service level were weighted in different lead time distributions. Empirical analysis and sensitivity analysis were given to get reach important conclusions and enlightenment.