Continuous Review Two-Echelon (s,S) Inventory System with Partial Backlogging (original) (raw)

Modelling and Analysis of Compliment Inventory System in Supply Chain with Partial Backlogging

International Journal of Fuzzy Mathematical Archive

This paper presents a continuous review two echelon inventory systems with two different items in stock, one is main product and other one is complement item for the main product. The operating policy at the lower echelon for the main product is (s, S) that is whenever the inventory level drops to ‘s’ on order for Q = (S-s) items is placed, the ordered items are received after a random time which is distributed as exponential. We assume that the demands accruing during the stock-out period are partially backlogged. The retailer replenishes the stock of main product from the supplier which adopts (0,M) policy. The complement product is replenished instantaneously from local supplier. The joint probability disruption of the inventory levels of main product, complement item at retailer and the main product at supplier are obtained in the steady state case. Various system performance measures are derived and the long run total expected inventory cost rate is calculated. Several instance...

Cost Optimization of Inventory System in Two-Echelon with Two Suppliers

Zenodo (CERN European Organization for Nuclear Research), 2017

This paper presents a continuous review two echelon inventory system. The operating policy at the lower echelon is (s, S) that is whenever the inventory level traps to s on order for Q = (S-s) items is placed, the ordered items are received after a random time which is distributed as exponential. We assume that the demands accruing during the stock-out period are lost. The retailer replenishes their stock from the regular supplier which adopts (0, M) policy, M = nQ. When the regular supplier stock is empty the replacement of retailer stock made by the outside supplier who adopts (0, N) policy N = nQ. The joint probability disruption of the inventory levels of retailer, regular supplier and the outside supplier are obtained in the steady state case. Various system performance measures are derived and the long run total expected inventory cost rate is calculated. Several instances of numerical examples, which provide insight into the behavior of the system are presented.

ANALYSIS OF TWO ECHELON INVENTORY SYSTEM WITH JOINT ORDERING POLICY

Inventory is essential for the efficient running of any business. Single location inventory system is considered by many researchers. This paper deals with two echelon inventory system with handling two products having joint ordering policy. The demand for the products follows independent poison distributions at retailer and distributor node. The items are supplied to the retailers from the distribution center (DC) administrated with exponential lead time having parameter µ (>0). The joint probability disruption of the inventory levels of two products at retailer and the supplier are obtained in the steady state case. Various system performance measures are derived and the long run total expected inventory cost rate is calculated. Several instances of numerical examples, which provide insight into the behavior of the system, are presented.

Analysis of a Two-Echelon Inventory System With Two Supply Modes

2010

In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We use results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and show that it is useful to decrease upstream stock levels. In cases with high demand uncertainty, high backlogging cost or long lead times, we conclude that dual-sourcing can lead to significant savings.

A Continuous Review Model for an Inventory System with Two Supply Modes

Management Science, 1988

In this paper we develop an approximate model of an inventory control system in which there exist two options for resupply, with one having a shorter lead time. Because the optimal policy appears to be extremely complex, we consider a reasonable extension of the standard (Q, R) policy to allow for two different lot sizes QI and Q2, and two different reorder levels, R I and R2. Expressions for the expected on hand inventory and the expected backorders are developed and a procedure for determining the policy parameters is given. The model is validated by simulation, and calculations are included which compare the average annual cost with and without emergency ordering. (INVENTORY PRODUCTION; OPERATING CHARACTERISTICS; LEADTIME POL-ICY; ORDERING POLICY; STOCHASTIC MODELS)

A two-echelon inventory model with lost sales

International Journal of Production Economics, 2001

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R i ) control policy. The warehouse operates an (SQ, (S À 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.

Analysis of Two-Commodity Continuous Review Inventory System in Supply Chain

This paper considers a two commodity continuous review inventory system. Continuous review inventory control of a single item at a single location had been considered by many researchers in past. We extend this inventory control strategy to two-echelon system, which is a building block for serial supply chain. The inventory control system consists of two warehouse (WH i

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

Inventory control based on advanced probability theory, an application

European Journal of Operational Research, 2005

Whenever stock is placed as a buffer between consumption and supply the decision when to replenish the stock is based on uncertain values of future demand and supply variables. Uncertainty exists about the replenishment lead time, about the number of demands and the quantities demanded during this period. We develop a new analytical expression for the reorder point, which is based on the desired service level and three distributions: the distribution of the quantity of single demands during lead time, the distribution of the lengths of time intervals between successive demands, and the distribution of the lead time itself. The distribution of lead time demand is derived from the distributions of individual demand quantities and not from the demand per period. It is not surprising that the resulting formulae for the mean and variance are different from those currently used. The theory developed is also applicable to periodic review systems. The system has been implemented at CERN and enables a significant enhancement of the service level, while reducing the average stock.