Evaluation of a multi-item inventory system with coupled arrivals and returns (original) (raw)
Related papers
Annals of Operations Research, 2015
Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a constant replenishment rate. Consider a single-product inventory system under continuous review with constant replenishment and compound Poisson demands subject to lost-sales. The system incurs inventory carrying costs and lost-sales penalties, where the carrying cost is a linear function of on-hand inventory and a lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until and including the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions: constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform and Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes,
Transient Analysis of a Stochastic Inventory System for Serving Eligible Customers With Reworks
Research article, 2018
This paper presents an inventory system where eligible customers are screen out at the first stage of servicing. The arrival of demand for fresh items and for rework items follows Poisson process with parameter λ and β . From fresh items store, items will be provided to the arrival customer within a negligible service time. We assume that a certain portion of arriving customers will get service with rate αλ and rest of the arriving customer will be rejected to serve with a rate (1-α)λ. when inventory level for fresh items reaches to the reorder level s an order takes place which follows exponential distribution with parameter γ. The defective items will dysfunction before expire date, a service will be provided once it returns to the service center with parameter μ. If the store of rework items is full then the next case will be served at home as early as possible. We considered two stores in the system one for fresh items and another for returned items. When inventory level is zero then arrival customer will be lost forever. A suitable mathematical model is developed and the solution of the developed model using Markov process with Rate-matrix is derived. Also the systems characteristics are numerically illustrated. The validation of the result in this model was coded in Mathematica 11.
A decision model for an inventory system with two compound Poisson demands
Uncertain Supply Chain Management, 2020
A real inventory system for single item with specific demand characteristics motivates this works. The demand can be seen as two types of independent demand, where compound Poisson process describes the characteristics of each demand. The first type of demand is rarely occurred with relatively large size, while the second type of demand is often happened with relatively small size. In order to maintain inventory level, every time the first type of demand occurs a replenishment of stock is conducted which follows order-up-to-level inventory policy. In order to find the optimal inventory decision for that system, a mathematical model of the system is developed with the objective to minimize expected total inventory cost. Some of model assumptions are infinite replenishment, deterministic lead time, and completely backlogged shortages. To solve the model, it is then divided into two sub-problems and classical optimization technique is employed to help find the solution of each sub problem. .
In this paper we extend earlier work that analyzes a single echelon single item base-stock inventory system where Demand is modeled as a compound Poisson process and the lead-time is stochastic. The extension consists in considering a cost oriented system where unfilled demands are lost. The case of partial lost sales is assumed. We first model the inventory system as a Makovian M/G/∞ queue then we propose a method to calculate numerically the optimal base-stock level. A preliminary numerical investigation is also conducted to show the performance of our solution.
An analysis of single item inventory systems with returns
Naval Research Logistics Quarterly, 1981
Inventory systems with returns are systems in which there are units returned in a repairable state, as well as demands for units in a serviceable state, where the return and demand processes are independent. We begin by examining the control of a single item at a single location in which the stationary return rate is less than the stationary demand rate. This necessitates an occasional procurement of units from an outside source. We present a cost model of this system, which we assume is managed under a continuous review procurement policy, and develop a solution method for finding the policy parameter values. The key to the analysis is the use of a normally distributed random variable to approximate the steady-state distribution of net inventory. Next, we study a single item, two echelon system in which a warehouse (the upper echelon) supports N (N 2 1) retailers (the lower echelon). In this case, customers return units in a repairable state as well as demand units in a serviceable state at the retailer level only. We assume the constant system return rate is less than the constant system demand rate so that a procurement is required at certain times from an outside supplier. We develop a cost model of this two echelon system assuming that each location follows a continuous review procurement policy. We also present an algorithm for finding the policy parameter values at each location that is based on the method used to solve the single location problem. *This research was supported in part by the Office of Naval Research under Contract N00014-75-C-1172. 'A repairable item is an item which fails, but which can be repaired and subsequently made available to satisfy a future demand or an existing backorder.
Comparison of Inventory Systems with Service, Positive Lead-Time, Loss, and Retrial of Customers
Journal of Applied Mathematics and Stochastic Analysis, 2007
We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an o...
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
Analysis of order-up-to-level inventory systems with compound Poisson demand
European Journal of Operational Research, 2011
We analyse a single echelon single item inventory system where the demand and the lead time are stochastic. Demand is modelled as a compound Poisson process and the stock is controlled according to a continuous time order-up-to level policy. We propose a new method for determining the optimal order-up-to level for a cost oriented inventory systems where unfilled demands are backordered. The conditions under which the system behaves like a Make-To-Order setting are also discussed. By means of a numerical investigation, we show that the proposed method provides very good results. It is also shown to outperform another approximate solution provided in the literature. Our work allows insights to be gained on stock control related issues for both fast and slow moving Stock Keeping Units.
Mathematics
This paper deals with an integrated and interconnected stochastic queuing-inventory system with a fresh item, a returned item, and a refurbished item. This system provides a multi-type service facility to an arriving multi-class customer through a dedicated channel. It sells fresh and refurbished items, buys used items from customers, refurbishes the used items for resale, and provides a repair service for defective items. The assumption of purchasing a used item from the customer and allowing them to buy a fresh item is a new idea in stochastic queuing-inventory modeling. To do so, this system has four parallel queues to receive four classes of customers and five dedicated servers to provide a multi-type service facility. Customers are classified according to the type of service they require. Each class of arrival follows an independent Poisson process. The service time of each dedicated server is assumed to be exponentially distributed and independent. This system assumes an insta...
A lost sales inventory model with a compound poisson demand pattern
2005
In this paper, we study the decision problem of a retailer, who wants to optimize the amount of shelf inventory of a particular product, given that the demand for the product is stochastic and replenishment lead times (from the store's stockroom to the shelf) are negligible. The shelf inventory is managed according to a (0, B * )-inventory policy: when the shelf inventory is sold out, the retailer gets a fixed amount of B * units from the central stockroom to replenish the shelf inventory.