A Stochastic Inventory Policy with Limited Transportation Capacity (original) (raw)
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A multi-item periodic replenishment policy with full truckloads
International Journal of Production Economics, 2009
In this paper we consider a stochastic multi item inventory problem. A retailer sells multiple products with stochastic demand and is replenished periodically from a supplier with ample stock. At each order instant it is decided which product to order and how much to order. For the delivery of the products trucks with a finite capacity are available. The dispatched trucks arrive at the retailer after a constant leadtime and with each truck fixed shipping costs are charged independent on the number of units shipped. Additionally, linear holding and backorder costs at the end of a review period are considered. Since fixed transportation costs are high coordination of orders and full truckload shipments can benefit from economies of scale. We propose a dynamic order-up-to policy where initial ordersizes can be reduced as well as enlarged to create full truckloads. We show how to compute the policy parameters and in a detailed numerical study we compare our policy with a lower bound and an uncoordinated periodic replenishment policy. An excellent cost performance of the proposed policy can be observed when average time between two shipments is not too large and fixed shipping costs are high.
European Journal of Operational Research, 2010
In this paper we consider the stochastic joint-replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of an QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximate formulas are based on renewal theoretic results and are tested in a simulation study, revealing a good performance. Second, we compare the QS policy with a simple allocation policy, where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level a replenishment order is triggered and the total vehicle capacity is allocated among all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy, standing for lower inventory levels to obtain the same service level. While for identical items the difference between the performance of both policies is negligible, differences can be large for different item characteristics.
Replenishment Planning for Stochastic Inventory Systems with Shortage Cost
Lecture Notes in Computer Science, 2007
One of the most important policies adopted in inventory control is the (R,S) policy (also known as the "replenishment cycle" policy). Under the non-stationary demand assumption the (R,S) policy takes the form (Rn,Sn) where Rn denotes the length of the n th replenishment cycle, and Sn the corresponding order-up-to-level. Such a policy provides an effective means of damping planning instability and coping with demand uncertainty. In this paper we develop a CP approach able to compute optimal (Rn,Sn) policy parameters under stochastic demand, ordering, holding and shortage costs. The convexity of the cost-function is exploited during the search to compute bounds. We use the optimal solutions to analyze the quality of the solutions provided by an approximate MIP approach that exploits a piecewise linear approximation for the cost function.
International Journal of Production Economics, 2007
We consider a model of single-item periodic-review inventory system with stochastic demand, linear ordering cost, where in each time period, the system must order either none or at least as much as a minimum order quantity (MOQ). Optimal inventory policies for such a system are typically too complicated to implement in practice. In fact, the ðs; SÞ type of policies are often utilized in the real world. We study the performance of a simple heuristic policy that is easily implementable because it is specified by only two parameters ðs; tÞ. We develop an algorithm to compute the optimal values for these parameters in the infinite time horizon under the average cost criterion. Through an extensive numerical study, we demonstrate that the best ðs; tÞ heuristic policy has performance close to that of the optimal policies when the coefficient of variation of the demand distribution is not very small. Furthermore, the best ðs; tÞ policy always outperforms the best feasible ðs; SÞ policies and on average the percentage differences are significant. Finally, we study the impact of MOQ on system performance. r
Analysis of an inventory system under supply uncertainty - Deterministic and Stochastic Models
International Journal of Production Economics, 1999
In this paper, we analyze a periodic review, single-item inventory model under supply uncertainty. The objective is to minimize expected holding and backorder costs over a finite planning horizon under the supply constraints. The uncertainty in supply is modeled using a three-point probability mass function. The supply is either completely available, partially available, or the supply is unavailable. Machine breakdowns, shortages in the capacity of the supplier, strikes, etc., are possible causes of uncertainty in supply. We demonstrate various properties of the expected cost function, and show the optimality of order-up-to type policies using a stochastic dynamic programming formulation. Under the assumption of a Bernoulli-type supply process, in which the supply is either completely available or unavailable, and when the demand is deterministic and dynamic, we provide a newsboy-like formula which explicitly characterizes the optimal order-up-to levels. An algorithm is given that computes the optimal inventory levels over the planning horizon. Extensions and computational analysis are presented for the case where the partial supply availability has positive probability of occurrence.
Inventory management in supply chain with stochastic inputs
2010
This thesis studies and proposes some new ways to manage inventory in supply chains with stochastic demand and lead time. In particular, it uses queuing principles to determine the parameters of supply chain stations with delayed differentiation (typical assemble-to-order systems) and went on to apply some previously known results of steady state of some queuing systems to the management of flow and work in process inventory in supply chain stations. Consideration was also given to the problem of joint replenishment in partially dependent demand conditions. The first chapter introduces the important concepts of supply chain, the role of inventory in a supply chain, and developing stochastic models for such system. It then went on to review the pertinent literature that has been contributed to the inventory management, especially using stochastic models. Chapter two presents a perishable inventory model with a multi-server system, where some services, having an exponentially distributed lead time, have to be done on the product before it is delivered to the customer. Customers whose demands are not met immediately are put in an orbit from where they send in random retrial requests for selection. The input stream follows a Markov Arrival Process, , and another flow of negative customers (typical of a competitive environment with customer poaching), also following an , takes customers away from the orbit. An (,) replenishment policy was used. The joint probability distribution of the number of busy servers, the inventory level and the number of customers in the orbit is obtained in the steady state. Various measures of ii stationary system performance are computed and the total expected cost per unit time is calculated. Numerical illustrations were made. Chapter three is also a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers are assumed to arrive following a quasi-random distribution. Items demanded are also made available after some service, exponentially distributed, has been done on the demanded item. Customers with unsatisfied orders join an orbit from where they can make retrials only if selected following a special rule. Replenishment follows an (,) policy and also has an exponentially distributed lead time. The intervals separating two successive repeated attempts are exponentially distributed with rate ߠ + ݅ߥ, when the orbit has ݅ customers ݅ ≥ 1. The joint probability distribution of the number of customers in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. Chapter four is a two-commodity continuous review inventory system, with three customer input flows, following the ; one for individual demand for product 1; another for bulk demand for product 2; and the third for a joint individual demand for product 1 and bulk demand for product 2. The ordering policy is to place orders for both commodities when the inventory levels are below prefixed levels for both commodities, using (,) replenishment. The replenishment lead time is assumed to have phase type distribution and the demands that occur during stock out period are assumed to be lost. The joint probability distribution for both commodities is obtained in the steady state case. Various measures of system performance and the total expected cost rate in the steady state are derived. Numerical illustrations were then done. Chapter five is a model that shows how the steady state parameters of a typical queuing system can be used in the dynamic management of flow and buffer in a Theory of Constraints () environment. This chapter is in two parts, and the typical ∞/1/ܯ/ܯ production environment with 0 < ߩ < 1 was assumed. The optimal feed rate for maximum profit was obtained. In the first part, the model was considered without consideration for shortage cost. This model was then extended in the second part to a case where a fixed cost is charged for every unit shortage from the desired production level. Part A result was iii shown to be a special case of part B result; the unit shortage cost has been implicitly taken to be zero in part A. Chapter six is the concluding chapter, where the various possible applications of the models developed and opportunities for possible future expansions of models and areas of research were highlighted. The main contributions of this work are in the Supply Chain area of delayed differentiation of products and service lead time. Others include management of joint replenishment and optimisation of flow in a TOC environment. The key contributions to knowledge made in this thesis include: • A model of a multi-server retrial queue with arrival and negative arrival, and deteriorating inventory system in which inventory items are made available only after some work has been done on the inventory item before it is delivered to the customer. No previous model is known to have considered any queuing system with such multi-server system ahead of this chapter. • A model of a retrial queuing system with multi-server rule based in which the arrival pattern is quasi-random, the calling population is finite, and an exponentially distributed system service is done on the inventory item before being delivered to the customer. It has not been found in literatures that such models have been developed elsewhere. • A stochastic model of joint replenishment of stocks in which two products are being ordered together; one of such is ordered in bulk and the other in single units, but both could be ordered together and unfilled order during the replenishment leadtime is lost. No published work is known to have also addressed such systems. • The management of flow in a theory of constraint environment, which explicitly utilises the holding cost, shortage cost, product margin, the level of utilisation of the resource and the effect of such on the stocks (inventory) build up in the system. Such flows are then explicitly considered in the process of buffering the system. Most works have been known to focus on buffer and not the flow of the products in order to optimise the system profit goal. iv Some of the insights derived include • An understanding of how the system cost rate is affected by the choice of the replenishment policy in systems with arrival pattern so that controlling policies (reorder point and capacity) could be chosen to optimise system profit • The effect of correlated arrival in input system on the cost rate of the system • How the nature of input pattern and their level of correlation affect the fraction of the retrials in a retrial queue in a competitive environment that are successful and how many of such customers are likely to be poached away by the randomly arriving competitors. This has direct effect on the future market size. • The nature of utilisation, blocking and idleness of servers in typical retrial queues, such that there could be yet-to-be-served customers in the orbit while there are still idle serves in such systems • Management of utilisation of resources in stochastic input and processing environment with respect to the throughput rate of such systems. It was shown that it may not be profitable to strive to always seek to fully utilise the full capacity of a Capacity Constrained Resource, even in the face of unmet demands. Increase in utilisation should always be considered in the light of the effect of such on the throughput time of the products and the consequence on the system's profit goal. This decision is also important in determining the necessity and level of buffers allowable in the production system. v ACKNOWLEDGEMENTS My profound gratitude goes to so many people that have made this study possible. But particular mention needs to be made of some very special people. First and foremost, I would like to thank Professor VSS Yadavalli, who is my promoter. He is actually more than just a promoter, but a reliable mentor, guide, instructor, teacher, listener and guardian, both in official and personal capacities. I am indebted to you. I would also like to thank my family members, especially my loving and understanding wife, Ireti, and my kids who have been denied many valuable moments to share, so that we can rejoice at the realisation of this dream. I thank my parents and siblings for the foundations you all provided for me. It still helps my development. I thank the entire staff members of the department of Industrial and Systems Engineering of the University of Pretoria, for giving me the opportunity to work with this great team, and doing that without prejudice or let. I have been much better with you in my life. I would like to appreciate the efforts of Pastor and Dr (Mrs) Akindele, who encouraged and supported me to quit my comfort zone in the office to pursue this course of life, which actually has become my passion. And most importantly, my Lord and Master, Jesus Christ, who has made a person out of a mere birth that would have been without direction or hope in life. vi
2010
The single-product, stationary inventory problem with set-up cost is one of the classical problems in stochastic operations research. Theories have been developed to cope with finite production capacity in periodic review systems, and it has been proved that optimal policies for these cases are not of the (modified) (s, S)-type in general, but more complex. In this paper we consider a production system such that the production rate is constrained, rather than the amount as is common in periodic review models. Thus, in our case the production rate is positive and finite when the system is on and zero when off, while a cost is incurred to switching on or off. We prove that a long-run optimal stationary policy exists for this single-item continuous review inventory problem with non-zero switching cost and finite production rate, and that this optimal policy has an (s, S)-structure. We also provide an efficient numerical procedure to compute the parameters of the optimal policy. Another, and perhaps more precise, way to include a capacity constraint is to constrain the production rate, rather then the production amount per review period. In this paper we follow this idea, and determine the structure of the optimal policy for the single-item inventory problem with set-up cost, backlogging, and finite production rate. Orders arrive according to a Poisson process and the i.i.d. demands follow some (rather) arbitrary distribution. The inventory is replenished at a constant rate only when production is on. It will be proved that a long-run optimal stationary policy exists and is described by two parameters: s and S. As soon as the inventory gets below s, production is switched on, while as soon as the inventory hits S, production is switched off. We also provide a dynamic-programming based numerical method to efficiently characterize the optimal policy. In passing we mention that in (s, S)-policy literature, see e.g. and references therein, an important objective of knowing that the optimal policy is of the (s, S)-type is to use this to find the optimal values for s and S. For our approach, however, it is not necessary to know the structure of the optimal policy; it is a corollary that the optimal stationary policy has an (s, S)-structure and can be found simply by iteration (bisection).
A stochastic inventory routing problem with stock-out
Transportation Research Part C: Emerging Technologies, 2013
In this paper, we study an inventory routing problem in which a supplier has to serve a set of retailers. For each retailer, a maximum inventory level is defined and a stochastic demand has to be satisfied over a given time horizon. An order-up-to level policy is applied to each retailer, i.e. the quantity sent to each retailer is such that its inventory level reaches the maximum level whenever the retailer is served. An inventory cost is applied to any positive inventory level, while a penalty cost is charged and the excess demand is not backlogged whenever the inventory level is negative. The problem is to determine a shipping strategy that minimizes the expected total cost, given by the sum of the expected total inventory and penalty cost at the retailers and of the expected routing cost. A hybrid rollout algorithm is proposed for the solution of the problem and its performance is evaluated on a large set of randomly generated problem instances.