Replenishment Period Research Articles

Inventory Model for Ramp type items with Trade Credit under Extra ordinary purchase

We have endeavoured here to depict an interactive inventory model between producer (supplier) and prospective buyer (vendor) in the presence of an extraordinary purchase and credit period with two parameter weibull deterioration for items with ramp type demand. The case of shortages is also studied.Additonally, we deal with the problem of a supplier disposing a large inventory level through an offer to a prospective buyer in the form of a credit period. From our model we confirm that the behaviour of replenishment period and trade credit period and change of the total variable cost observed from numerical evidence should be interpreted with caution in any business organisation.

The stochastic EOQ model with random sales price

Abstract The article deals with a stochastic economic order quantity (EOQ) model over a finite time horizon where uniform demand over the replenishment period is price dependent. The selling price is assumed to be a random variable that follows a probability density function. As demand is probabilistic, stock out situation may occur. Based on the partial backlogging and lost sale cases during stock out period, the author develops the criterion for the optimal solution for the replenishment size such that the integrated expected profit is maximized. Moreover, the article suggests a new function regarding price dependent demand. Finally, numerical examples and its sensitivity analysis of key parameters are given to illustrate the proposed model.

A replenishment policy based on joint optimization in a downstream pharmaceutical supply chain: centralized vs. decentralized replenishment

This paper considers a number of problems in a pharmaceutical downstream supply chain under specific constraints related to pharmaceutical products (e.g., their expiry date and regulations) and to inventory control (e.g., low prices of products, gift products or prices near zero, shortages prohibited but excess inventory allowed). As traditional models of inventory control and replenishment cannot optimize the total cost of the system, it is very important to consider the transportation cost as well. In this paper, we take into account some of these constraints and propose two models for multi-product replenishment policies, namely centralized and decentralized models. We seek to identify the best quantity and period of replenishment of products for a joint optimization of inventory and transportation costs. The proposed models can be applied to a specific family of products with a stable demand and high turnover rate, low prices (or gift products for some of them), and without any shortage. These two models are compared and the global approach is illustrated by a numerical example taken from a real-case study.

An inventory model for deteriorating items with varying demand pattern and unknown time horizon

The primary assumptions with many multi-period inventory lot-sizing models are fixed time horizon and uniform demand variation within each period. In some real inventory situations, however, the time horizon may be unknown, uncertain or imprecise in nature and the demand pattern may vary within a given replenishment period. This paper presents an economic order quantity model for deteriorating items where demand has different pattern with unknown time horizon. The model generates optimal replenishment schedules, order quantity and costs using a general ramp-type demand pattern that allows three-phase variation in demand. Shortages are allowed with full backlogging of demand and all possible replenishment scenarios that can be encountered when shortages and demand pattern variation occur in multi-period inventory modeling are also considered. With the aid of numerical illustrations, the advantages of allowing for variation in demand pattern within replenishment periods, whenever they occur, are explored. The numerical examples show that the length of the replenishment period generated by the model varies with the changes in demand patterns.

A linear programming formulation for an inventory management decision problem with a service constraint

Inventory systems with uncertainty go hand in hand with the determination of a safety stock level. The decision on the safety stock level is based on a performance measure, for example the expected shortage per replenishment period or the probability of a stock-out per replenishment period. The performance measure assumes complete knowledge of the probability distribution during lead time, which might not be available. In case of incomplete information regarding the lead-time distribution of demand, no single figure for the safety stock can de determined in order to satisfy a performance measure. However, an optimisation model may be formulated in order to determine a safety stock level which guarantees the performance measure under the worst case of lead-time demand, of which the distribution is known in an incomplete way. It is shown that this optimisation problem can be formulated as a linear programming problem.

Leveraging World Bank Resources for the Poorest: IDA Blended Financing Facility Proposal

With the Millennium Development Goals deadline only five years away, the international donor community faces significant challenges due to the global economic crisis, record government deficits, and simultaneous funding requests from nearly every multilateral development institution. This paper proposes a new World Bank financing model for creditworthy emerging economies, such as India and Vietnam, which currently receive billions of dollars in IDA assistance. In contrast to the current IDA-centric financing model, the IBRD would provide the same loan volumes to qualifying emerging economies while IDA would provide grant subsidies to buy down the concessionality level of these IBRD loans. As such, these countries would be held harmless both in terms of aid volumes and lending terms. By better leveraging the IBRD’s balance sheet for loan capital, IDA then could re-allocate what it otherwise would have provided to emerging economies. For the current IDA-15 replenishment period, this would mean up to $7.5 billion in additional assistance for the world’s poorest, most vulnerable countries. In relative terms, this would entail a 30 percent increase over existing levels. Of this, African countries would have received an additional $5.5 billion in IDA assistance. If donor governments find a way to scrape together increased contributions to IDA, then the allocation pie would grow by an even larger margin. The Inter-American Development Bank already successfully utilizes a similar approach for its lower middle-income and low-income country clients. It is time for World Bank shareholders to seriously consider the same resource-maximizing model. With the IDA-16 replenishment and IBRD general capital increase negotiations currently underway, they have an excellent window of opportunity to implement this win-win-win approach.

Soil analysis (for plant nutrition) in Hungary: Practice and results

Agricultural goods obtained and produced in Hungary have played an important role in the markets of Western Europe. By utilizing the ecological potentials of the Carpathian Basin, local inhabitants are in the position to produce considerable food surpluses in addition to meeting their own demands. With agricultural production becoming more and more intensive in Hungary, the application of mineral fertilizers also started to increase slowly from the 1960’s. From the mid-1970’s a uniform sampling, soil testing and fertilization extension system was created together with its own institutional and laboratory testing network. The intensive use of mineral fertilizers in Hungary lasted from the mid-1970’s to the last quarter of the 1980’s, during which an average amount of 230 kg·ha -1 NPK fertilizer was applied. In this period the so-called “build-up” fertilization was applied in conformity with the improvement of all other elements involved in the production technology, which was also clearly expressed in the agro-political objectives of those days aiming to obtain higher yields. At that time the nutrient supply and nutrient base of soils in Hungary increased clearly, so the production technology could no longer limit higher yields. In 1990 agriculture changed fundamentally and radically in Hungary, and the same was valid for nutrient supplies as well. At the beginning of the 1990’s there was a sudden decrease in the level of mineral fertilizer application (to below 40 kg NPK active ingredients·ha -1 ), followed by a slow increase, which has reached the level of almost 70 kg·ha -1 by today. In the meantime the animal stock in Hungary has decreased and consequently the amount of manure has also fallen. All in all, the nutrient balance of Hungarian soils has always been negative since 1989. Due to the changes in its structure and ownership over the past twenty years or so, it has become very difficult to obtain reliable information about Hungarian agriculture. The Soil Resources Management General Partnership (in Hungarian: Talajerőgazdálkodás Kkt.) conducts extension work based on soil sampling and has a continuous flow of data on over thirty thousand hectares, beginning at the end of the 1970’s. Based on the analyses of these data it can be stated that the extra amount of nutrients over balance, applied during the period of replenishment (until the change in regimes) has been „removed” from the soil over the past fifteen years, consequently the Hungarian nutrient balance has become negative again. This kind of fertilization practice cannot be sustained in Hungary, as the maintenance of the production potential of Hungarian soils is far from being resolved at the moment; it poses risks to and questions sustainability, as well as it may cause a very serious competitive disadvantage to the country.

Inventory Management of a Fast-Fashion Retail Network

Working in collaboration with Spain-based retailer Zara, we address the problem of distributing, over time, a limited amount of inventory across all the stores in a fast-fashion retail network. Challenges specific to that environment include very short product life cycles, and store policies whereby an article is removed from display whenever one of its key sizes stocks out. To solve this problem, we first formulate and analyze a stochastic model predicting the sales of an article in a single store during a replenishment period as a function of demand forecasts, the inventory of each size initially available, and the store inventory management policy just stated. We then formulate a mixed-integer program embedding a piecewise-linear approximation of the first model applied to every store in the network, allowing us to compute store shipment quantities maximizing overall predicted sales, subject to inventory availability and other constraints. We report the implementation of this optimization model by Zara to support its inventory distribution process, and the ensuing controlled pilot experiment performed to assess the model's impact relative to the prior procedure used to determine weekly shipment quantities. The results of that experiment suggest that the new allocation process increases sales by 3% to 4%, which is equivalent to $275 M in additional revenues for 2007, reduces transshipments, and increases the proportion of time that Zara's products spend on display within their life cycle. Zara is currently using this process for all of its products worldwide.

A static-dynamic strategy for spare part inventory systems with nonstationary stochastic demand

Inventory control is especially difficult when demand is stochastic and nonstationary. We consider a spare part inventory control problem with multiple-period replenishment lead time, and describe a static-dynamic strategy for the problem. By solving a static-dynamic uncertainty model, the strategy first makes decisions on the replenishment periods and order-up-to-levels over the planning horizon, but implements only the decisions of the first period. It then uses the rolling horizon approach in the next period when the inventory status is revised, and the multi-period problem is updated as better forecasts become available. In light of structural property of the developed static-dynamic uncertainty model, the optimal solution to the model can be obtained without much computational effort and thus the strategy can be easily implemented. Computational experiments and result of a case study verify the efficacy of the proposed strategy.

Inventory policies in a fuzzy uncertain supply chain environment

Abstract Managers have begun to recognize that effectively managing risks in their business operations plays an important role in successfully managing their inventories. Accordingly, we develop a ( Q , r ) model based on fuzzy-set representations of various sources of uncertainty in the supply chain. Sources of risk and uncertainty in our model include demand, lead time, supplier yield, and penalty cost. The naturally imprecise nature of these risk factors in managing inventories is represented using triangular fuzzy numbers. In addition, we introduce a human risk attitude factor to quantify the decision maker’s attitude toward the risk of stocking out during the replenishment period. The total cost of the inventory system is computed using defuzzification methods built from techniques identified in the literature on fuzzy sets. Finally, we provide numerical examples to compare our fuzzy-set computations with those generated by more traditional models that assume full knowledge of the distributions of the stochastic parameters in the system.

On a Class of Optimization Problems for Finite Time Horizon Inventory Models

The paper proposes a general theory for the treatment of a class of optimization problems that arise as a consequence of a search for the optimal replenishment schedule for finite time horizon inventory models. The decision variables in these optimization problems consist of the number of replenishment periods and the lengths of the periods. When the number of replenishment periods is fixed, it is shown that the optimization problems have a unique solution under some partial differential inequalities. Furthermore, the minimal value is convex as a function of the number of replenishment periods. This leads to a simple procedure for determining the optimal control policy.

Situation reactive approach to Vendor Managed Inventory problem

In this research, we deal with VMI (Vendor Managed Inventory) problem where one supplier is responsible for managing a retailer’s inventory under unstable customer demand situation. To cope with the nonstationary demand situation, we develop a retrospective action-reward learning model, a kind of reinforcement learning techniques, which is faster in learning than conventional action-reward learning and more suitable to apply to the control domain where rewards for actions vary over time. The learning model enables the inventory control to become situation reactive in the sense that replenishment quantity for the retailer is automatically adjusted at each period by adapting to the change in customer demand. The replenishment quantity is a function of compensation factor that has an effect of increasing or decreasing the replenishment amount. At each replenishment period, a cost-minimizing compensation factor value is chosen in the candidate set. A simulation based experiment gave us encouraging results for the new approach.

Equilibrium replenishment in a supply chain with a single distributor and multiple retailers

This paper addresses a problem encountered by a large-scale health service supply chain operating in a periodic review mode. Due to the vital nature of the products it provides, the number and timing of urgent orders are not limited. As a result, increasingly high transportation costs are incurred and the problem is to select an inventory replenishment (review) period that minimizes the transportation cost. Moreover, the supply chain involves multiple retailers which inevitably and independently respond to any change in replenishment policy since it may affect their inventory costs. Such a relationship results in a game between a distribution centre and retailers. Since the problem is intractable due to its scale and stochastic nature, we combine a game theoretic approach with an empirical analysis. We show that this system is predictable using equilibria and that the current replenishment equilibrium of the health service supply chain is close to the Nash solution. Numerical analysis shows that the transportation costs are cut if the distribution centre implements in reality its formal (Stackelberg) leadership by reducing the replenishment period. However, this does not coordinate the supply chain and greater system-wide savings are possible by increasing the replenishment period if the supply chain is vertically integrated or the parties cooperate.

Single-supplier/multiple-buyer supply chain coordination: Incorporating buyers’ expectations under vertical information sharing

We address the coordination problem in a single-supplier/multiple-buyer supply chain. The supplier wishes to coordinate the supply chain by offering quantity discounts. To obtain their complete cost information, the supplier exchanges his own cost parameters with buyers leading to vertical information sharing. The supplier thinks that the buyers, as they have access to supplier’s setup and holding cost information, may demand a portion of the anticipated coordination savings based on the partial information they hold about the cost structure of the entire supply chain. We model each buyer’s expectations based on her limited view of the entire supply chain which consists of herself and the supplier only. These expectations are then incorporated into the modeling of the supply chain, which results in a generalization of the traditional Stackelberg type models. We discuss alternative efficiency sharing mechanisms, and propose methods to design the associated discount schemes that take buyers’ expectations into account. In designing the discount schemes, we consider both price discriminatory and non-price discriminatory approaches. The study adds to the existing body of work by incorporating buyers’ expectations into a constrained Stackelberg structure, and by achieving coordination without forcing buyers to explicitly comply with the supplier’s replenishment period in choosing their order quantities. The numerical analysis of the coordination efficiency and allocation of the net savings of the proposed discount schemes shows that the supplier is still able to coordinate the supply chain with high efficiency levels, and retain a significant portion of the net savings.

Retail replenishment models with display-space elastic demand

We study a single store multi-product inventory problem in which product sales are a composite function of shelf space. Since sales tend to deplete the amount of product on display, the effective shelf space assigned to the product diminishes with time unless replenishment occurs. We consider the problem of optimal replenishment times under these conditions. We assume a linear dependence of sales rate to effective shelf space in all our analysis. We present exact and approximate solutions for the single product and multi-product cases. For the single product case, we study the effect of space elasticity, cross elasticity and empty space elasticity on the optimal replenishment period. For the multi-product case we present a computationally attractive method using matrix exponentials and develop error bounds for this method.

An EOQ model with a varying demand followed by advertising expenditure and selling price under permissible delay in payments: for a retailer

This paper derives an Economic Order Quantity (EOQ)/Economic Production Quantity (EPQ) model with finite replenishment/production rate, taking into account the effect of the suppliers' trade credit and the retailers' promotional effort made by the advertising and the sales-team. The rate of demand in market is assumed to be a function of the selling price and the budget for the advertising and the sales-team. The objective of this paper is to determine the optimal selling price, replenishment period and the above budget so that the average net profit is maximised. The associated constrained maximisation problem is solved by Interior-Penalty-Function Method (Fiacco and Mc Cormick). The model is illustrated with a result, using numerical example and it provides some insights into the effect of the retailers' promotional effort.

Implementing a near-optimal solution for the multi-stage, multi-product capacitated lot-sizing problem by rolling out a cyclical production plan

This paper describes the implementation and operational results of a cyclical master production-scheduling model, steering the manufacturing process of a photographic film-producing company. The involved company has a number one or two selling position in the medical and graphical film market. In film production all items are produced within two stages. The first stage is process-oriented and produces a limited number of intermediate products. The second stage is flow-shop oriented and cuts and slices the intermediate products into a large number of end products. In its search for a global optimum, the company wanted to synchronise operations by establishing a common replenishment period as a co-ördination mechanism between the first and second production stage. The proposed cyclical model uses a level-by-level approach and solves the economic lot-scheduling problem (ELSP). The cyclical model proved to be very robust when aligning with dynamic demand, allowing a synchronisation of operations. Based on the resulting schedule stability management lowered the levels of safety stock, which resulted in significant savings. This paper compares the cyclical model with the mathematical optimum, generated by solving the corresponding capacitated lot-sizing problem (CLSP). The results prove the ability of the proposed method to achieve a near optimal solution. The cyclical model does not suffer from the NP hardness of the CLSP and is, therefore, able to solve large-scale models.

Downside risk-aversion analysis for a single-stage newsvendor problem

We extend the classical newsvendor problem by introducing a downside risk constraint from the perspective of inventory control. At the beginning of a replenishment period the newsvendor will place an order, then he will review the inventory level at the end of the period. If the inventory level is positive then he will bear the holding cost and if the inventory level is negative then he will bear the backorder cost. The optimal order quantity has a simple form. We analyze the form of the optimal order quantity when we restrict that the probability that the cost level is larger than or equal to a fixed cost constant is less than a fixed value of probability. At last, we analyze the case that the fixed cost constant is equal to the expected cost.

Buyer–vendor inventory coordination with credit period incentives

In this paper, a single-vendor, a single-buyer supply chain for a single product is considered and a model to study and analyze the benefit of coordinating supply chain inventories through the use of credit period is proposed. Under the proposed strategy, the vendor requests the buyer to alter his current order size such that the vendor can benefit from lower set up, ordering and inventory holding costs. To entice the buyer to accept this strategy, the vendor must compensate the buyer for his increased inventory cost, and possibly provide an additional saving by offering the buyer an order size dependent credit period such that the buyer is dispensed with interest on the money payable during the credit period. The optimal replenishment period of the vendor and the credit period are determined as a solution to the coordinating supply chain model. An optimization method and a detailed numerical study are used to evaluate the benefit of the proposed coordinated strategy.

Modelling and computing ( Rn, Sn) policies for inventory systems with non-stationary stochastic demand

This paper addresses the single-item, non-stationary stochastic demand inventory control problem under the non-stationary ( R, S) policy. In non-stationary ( R, S) policies two sets of control parameters—the review intervals, which are not necessarily equal, and the order-up-to-levels for replenishment periods—are fixed at the beginning of the planning horizon to minimize the expected total cost. It is assumed that the total cost is comprised of fixed ordering costs and proportional direct item, inventory holding and shortage costs. With the common assumption that the actual demand per period is a normally distributed random variable about some forecast value, a certainty equivalent mixed integer linear programming model is developed for computing policy parameters. The model is obtained by means of a piecewise linear approximation to the non-linear terms in the cost function. Numerical examples are provided.