Economic order quantity models with nonlinear holding costs

We consider a generalization of the classical Economic Order Quantity Model. The traditional parameters of unit cost, selling price, demand rate and set-up cost are constant but the holding cost per unit is a non-linear function of the length of time the item is held in stock. The application is to any inventory system where the value of the item decreases non-linearily the longer it is held in stock. For the case of deterministic demands we present the cost formula and the optimal order quantity for both finite and infinite horizons. For the case of stochastic demands the cost function is examined and the optimal order amount is presented. Computational results are presented indicating the effect of the non-linearity in holding costs.     

Forecasting demand variation when there are stockouts

This paper addresses the common problem of forecasting demand when there are a large number of stockouts. The well-known single period stochastic inventory (or ‘newsboy’) problem provides the optimum, single period, stocking level for a product subject to stochastic demand. There are many situations where repetitive ‘newsboy’ solutions are implemented to guide stocking of repeat, but related, products, such as newspapers, magazines, or perishable groceries. Implementation of the ‘newsboy’ solution requires forecasts of the distribution of demand, although there are many plausible cost parameters that lead to optimum stocking policies where there is a high probability of a stockout. The company is, therefore, faced with the problem of attempting to forecast demand when a high percentage of the available sales data reflects the stock available for sale, rather than the true demand.A procedure has been developed¹ to improve estimates of the mean and variance of the distribution of demand when there are stockouts, but this procedure fails when the percentage of stockouts increases above 50%. A modified stockout adjustment procedure is presented in this paper, and it is shown that use of this new procedure can lead to greatly improved estimates of demand parameters, and greatly improved profitability, when there are a high percentage of stockouts.     

Concepts for safety stock determination under stochastic demand and different types of random production yield

We consider a manufacturer’s stochastic production/inventory problem under periodic review and present methods for safety stock determination to cope with uncertainties that are caused by stochastic demand and different types of yield randomness. Following well-proven inventory control concepts for this problem type, we focus on a critical stock policy with a linear order release rule. A central parameter of this type of policy is given by the safety stock value. When non-zero manufacturing lead times are taken into account in the random yield context, it turns out that safety stocks have to be determined that vary from period to period. We present a simple approach for calculating these dynamic safety stocks for different yield models. Additionally, we suggest approaches for determining appropriate static safety stocks that are easier to apply in practice. In a simulation study we investigate the performance of the proposed safety stock variants.     

Inventory control based on advanced probability theory,an application

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.     

An alternative to safety stock policies for multi-level rolling schedule MRP problems

We study the impact of positive lead times on the multi-level lot-sizing problem in a rolling schedule environment. We show how stockout situations may arise even in a context of deterministic demand. We therefore develop a procedure to avoid such stockouts and we compare its performance through a simulation study to a safety stock strategy. Simulation results show the superiority of the proposed procedure.     

An EOQ model for perishable products with discounted selling price and stock dependent demand

A single item economic order quantity model is considered in which the demand is stock dependent. After a certain time the product starts to deteriorate and due to visualization effect and other aspects of deterioration the demand becomes constant. In that situation a discount on selling price provides significant increment in demand rate. In this paper we investigate how much discount on selling price may be given during deterioration to maximize the profit per unit time and whether a pre-deterioration discount affects the unit profit or not. A mathematical model is developed incorporating both pre- and post deterioration discounts on unit selling price, where analytical results reveal some important characteristics of discount structure. A numerical example is presented and sensitivity analysis of the model is carried out.     

Optimal ordering and transfer policy for an inventory with stock dependent demand

This paper deals with an ordering-transfer inventory model to determine the retailer’s optimal order quantity and the number of transfers per order from the warehouse to the display area. It is assumed that the amount of display space is limited and the demand rate depends on the display stock level. The objective is to maximize the average profit per unit time yielded by the retailer. The proposed models and algorithms are developed to find the optimal strategy by retailer. Numerical examples are presented to illustrate the models developed and the sensitivity analysis is also reported.     

Joint determination of optimal safety stocks and production policy for an imperfect production system

In this article, we develop an imperfect economic manufacturing quantity (EMQ) model for an unreliable production system subject to process deterioration, machine breakdown and repair and buffer stock. The basic model is developed under general process shift, machine breakdown and repair time distributions. We suggest a computational algorithm for determination of the optimal safety stock and production run time which minimize the expected cost per unit time in the steady state. For a numerical example, we illustrate the outcome of the proposed model and perform a sensitivity analysis with respect to the model-parameters which have direct influence on the optimal decisions.     

需求随价格变化的具有折扣的易变质物品的库存模型

本文研究了在需求随价格变化及物品易变质的条件下，当供应商给予数量折扣时的库存问题。证明了当供应商给予数量折扣时，零售商的需求量是增大的，并给出了供应商给予数量折扣时零售商的订货量和订货周期的计算方法。对物品变质率和需求价格敏感系数对零售商的订货量、订货周期、出售价格和单位时间利润的影响进行了数值分析，并给出了数值算例。     

Continuous-review,lost-sales inventory models with Poisson demand,a fixed lead time and no fixed order cost

This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.     

The Numerical Generation of Lead Time Demand Distributions for Inventory Models

This paper presents a general method for computing a lead time demand distribution from the basic distributions of order quantity, orders per unit time, and lead time. These distributions are specified numerically and can therefore be derived directly from historical data.     

When supplier’s availability affects the replenishment lead time—An extension of the supply-interruption problem

This paper presents an exact treatment of a continuous-review inventory system with compound Poisson demand, Erlang-distributed lead times and random supply interruptions. In contrast with the existing models in the literature, we take into account the supplier’s availability in characterizing the lead time of a replenishment order. Assuming that the supplier’s availability can be described by a continuous-time homogeneous Markov chain with two states (on and off) and that stockouts are lost, we derive the stationary distribution of the inventory level (stock-on-hand) under an (s, Q)-type control policy. This probability distribution is then used to formulate an exact expression for the long-run average cost per unit time of operating the inventory system. Some numerical results are also provided.     

The deterministic EOQ with partial backordering: A new approach

Previous authors have shown that if demand that cannot be filled from stock is partially backordered, then using the full-backordering model or assuming that all stockouts will result in lost sales can lead to substantial increases in cost relative to using a model that specifically recognizes the percentage of the stockouts that will be backordered. The models that these authors developed resulted in procedures or equations that are relatively difficult to use. In this paper we take a different approach to modeling the deterministic EOQ with partial backordering that results in equations that are more like the comparable equations for the basic EOQ and its full-backordering extension.     

THE EFFECTS OF ITQ IMPLEMENTATION: A DYNAMIC APPROACH

ABSTRACT. This paper investigates the intertemporal effects of introducing Individual Transferable Quota, ITQ, fishery management programs on stock size, fleet size and composition, and returns to quota holders and to vessel operators. Theoretical analysis is conducted using a specific version of a general dynamic model of a regulated fishery. It is demonstrated that the effects will differ depending upon the prevailing regulation program, current stock size, and existing fleet size, composition and mobility and upon how the stock and fleet change over time after the switch to ITQs. The paper expands upon previous works by modeling the dynamics of change in fleet and stock size and by allowing for changes in the TAC as stock size changes, by comparing ITQs to different regulations, and by allowing the status quo before ITQ implementation to be something other than a bioeconomic equilibrium. Specific cases are analyzed using a simulation model. The analysis shows that the annual return per unit harvest to quota owners can increase or decrease over the transition period due to counteracting effects of changes in stock and fleet size. With ITQs denominated as a percentage of the TAC, the current annual value of a quota share depends upon the annual return per unit of harvest and the annual amount of harvest rights. Because the per unit value can increase or decrease over time, it is also possible that the total value can do the same. Distribution effects are also studied and it is shown that while the gains from quota share received are the present value of a potentially infinite stream of returns, potential losses are the present value of a finite stream, the length of which depends upon the remaining life of the vessel and the expected time it will continue to operate.     

Assortment and inventory decisions with multiple quality levels

We consider the assortment and inventory decisions of a retailer under a locational consumer choice model where products can be differentiated both horizontally (e.g., color of a product) and vertically (e.g., quality of a product). The assortment and quantity decisions affect customer choice and, hence, the demand and sales for each product. In this paper, we investigate two different environments where product availability and assortment affect consumer choice and demand in different ways: make-to-order (MTO) and make-to-stock (MTS). In the MTO environment, customers order and purchase their most preferred product; that is, stockouts do not occur. In the MTS model, customers buy their most preferred product if it is in stock or do not buy if it is out of stock. In both environments we find conditions under which it is optimal to carry assortments of only a single quality level. In the MTS case, we show that an assortment of mixed quality levels can be optimal only within a narrow range of parameters.     

A multi-item mixture inventory model involving random lead time and demand with budget constraint and surprise function

This study deals with a multi-item mixture inventory model in which both demand and lead time are random. A budget constraint is also added to this model. The optimization problem with budget constraint is then transformed into a multi-objective optimization problem with the help of fuzzy chance-constrained programming technique and surprise function. In our studies, we relax the assumption about the demand, lead time and demand during lead time that follows a known distribution and then apply the minimax distribution free procedure to solve the problem. We develop an algorithm procedure to find the optimal order quantity and optimal value of the safety factor. Finally, the model is illustrated by a numerical example.     

Prioritizing replenishments of the piece picking area

Having sufficient inventories in the forward or piece picking area of a warehouse is an essential condition for warehouse operations. As pickers consume the inventory in the piece racks, there is a risk of stockout. This can be reduced by the timely replenishment of products from the bulk reserve area to the forward area. We develop and compare three policies for prioritizing replenishments for the case where order picking and replenishments occur concurrently because of time restrictions. The first policy, based on the ratio of available inventory to wave demand, reduces the number of stockouts considerably. The other two more sophisticated policies reduce the number of stockouts even more but require much more computation time, and are more costly in terms of implementation, maintenance and software updates. We present the results of implementing one of these policies in the warehouse of a large cosmetics firm.     

Calculating safety stocks for assembly systems with random component procurement lead times: A branch and bound algorithm

In this paper, a discrete single-level multi-component inventory control model for assembly systems with random component procurement lead times is considered. The economic order quantity (EOQ) policy is used for a type of finished product. The requirements of the components are constant and cyclic (periodic), and their values per period are deduced from the EOQ for the finished product. The paper focuses on the components safety stock calculation. The objective is to minimise the average holding cost of the components while keeping the desired service level for the finished product. For this, an upper bound, two lower bounds, two dominance properties and an efficient branch and bound algorithm are suggested. Several tests are executed and conclusions are drawn. The proposed model provides a substantial saving for assembly systems with a large number and unreliable delivery of components as in semi-conductor and automotive industries.     

Multi-Item Inventory Aggregation into Groups

The problem of grouping the many items in stock into a few groups with the provision of a common order cycle or common order quantity per group is modelled as a dynamic program. It is shown that by a simple reorganization of the stock items, the computation time of the dynamic program can be reduced to a manageable size. Further the order cycle grouping is shown to be, usually, superior to the order quantity grouping scheme. Also, the composition of the order cycle grouping is found to be remarkably close to that of the well known A/B/C system of stock control therby giving theoretical credence to that system.     

A two-echelon inventory model with lost sales

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 − 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.