An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

Order splitting in continuous review (Q,r) inventory models

A number of recent articles in the literature have argued the case, when lead time is variable, for splitting a replenishment order for Q between n suppliers by comparing this with the alternative of placing a single order for Q on one supplier. The split order compares favourably on the grounds that the arrival of the first component of a split order cannot be later than the arrival of an order from any one specified supplier. This note argues that an alternative comparison could be made with a policy of ordering Q/n from a single supplier (n times as often). It makes this comparison in the context of a continuous review (Q, r) inventory model but does so not by comparing aggregate costs but by fixing Q and the customer stock service level and comparing the average stock — an approach which is more appropriate to how many companies manage inventory in practice. We consider Poisson and deterministic demand processes, a general lead time distribution and both lost sales and backorder models.     

A multiple replenishment contract with ARIMA demand processes

This paper is concerned with a multiple replenishment contract with a purchase price discount in a supply chain. The chain is composed of one supplier, one buyer and consumers for a product. The replenishment contract is based upon the well-known (s, Q) policy, but allows us to contract replenishments at a future time with a price discount. Owing to the larger forecast error of future demand, the buyer should keep a higher level of safety stock to provide the same level of service as the usual (s, Q) policy. However, the buyer can reduce his purchase cost by ordering a larger quantity at a discounted price. Hence, there exists a trade-off between the price discount and the inventory holding cost. For the ARIMA demand processes, we present a model for the contract and an algorithm to find the number of the future replenishments. Computational experiments show that the algorithm finds the global optimum solution very quickly.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s,S) model with a maximum issue quantity restriction

The analysis of optimal inventory replenishment policies for items having lumpy demand patterns is difficult, and has not been studied extensively although these items constitute an appreciable portion of inventory populations in parts and supplies types of stockholdings. This paper studies the control of an inventory item when the demand is lumpy. A continuous review (s,S) policy with a maximum issue quantity restriction and with the possibility of opportunistic replenishment is proposed to avoid the stock of these items being depleted unduly when all the customer orders are satisfied from the available inventory and to reduce ordering cost by coordinating inventory replenishments. The nature of the customer demands is approximated by a compound Poisson distribution. When a customer order arrives, if the order size is greater than the maximum issue quantity w, the order is satisfied by placing a special replenishment order rather than from the available stock directly. In addition, if the current inventory position is equal to or below a critical level A when such an order arrives, an opportunistic replenishment order which combines the special replenishment order and the regular replenishment order will be placed, in order to satisfy the customer's demand and to bring the inventory position to S. In this paper, the properties of the cost function of such an inventory system with respect to the control parameters s, S and A are analysed in detail. An algorithm is developed to determine the global optimal values of the control parameters. Indeed, the incorporation of the maximum issue quantity and opportunistic replenishment into the (s,S) policy reduces the total operating cost of the inventory system.     

A Multiple Reorder Point Inventory Policy

This paper discusses the development and application of a multiple reorder inventory policy which can be stated as follows: reorder an optimal lot size Q when inventory (stock on hand) falls to R, R-Q, R-2Q,..., R-NQ; where R is the reorder level. If demands cause the inventory to fall below two reorder levels, say a jump from R+ ? to R-2Q+?′ where ? and ?′ < Q, an order for 2Q is placed. The policy is a form of (S,q) policy where the maximum stock level S = R + Q. The system is of particular value in cases where the coefficient of variation of lead time demand μ _l (μ _l = σ _l /λ _l )is large (say >0·5) and continuous inventory records are maintained. Tables, charts and nomographs to simplify clerical tasks can be obtained quite readily. In this formulation R and Q are not independent factors as in the usual Wilson formulation, but are obtained by minimizing a single cost functional subject to the constraint of a specified risk of out-of-stock condition or a specified level of service (Galliher and Simmond, 1957), (Morse et al., 1959). The particular application concerns the raw material inventories of a manufacturer of metal pressings who is required to offer “immediate service”. The demand distribution during the lead time closely approximates the exponential distribution, and lead times are constant for each raw material. The application of the multiple reorder policy results in a 30 to 35 per cent reduction in inventory for a 95 per cent service level. Measures of sensitivity and response are obtained, and the mean number of shortages is expressed in closed form. The policy is compared with the Wilson policy and shown to be more “effective” in that it results in lower inventories and a smaller number of orders for the case considered.     

VMI条件下具有复合二项随机需求的销售商库存策略研究

考虑一个典型的单一产品的二级供应链系统:单供应商对单销售商,假定系统中销售商的需求分布为复合二项分布,未满足的需求机会损失;补货间隔时间为一随机变量.本文采用概率方法对销售商的需求分布、期望缺货、期望库存周期及库存的稳定性分布进行研究的基础上,构建了使单位时间内销售商的期望库存成本费用最小的库存模型,由此模型便可确定VMI模式下供应商对销售商的库存补货参数s和S,并且给出了在补货响应时间为泊松分布的情况下模型的求解算法,还给出了及时补货响应情况下的5个算例.为补货策略的实施提供了一种简单易于控制的思路和方法.     

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.     

A weighted goal programming approach for replenishment planning and space allocation in a supermarket

We propose a mixed integer non-linear goal programming model for replenishment planning and space allocation in a supermarket in which some constraints on budget, space, holding times of perishable items, and number of replenishments are considered and weighted deviations from two conflicting objectives, namely profitability and customer service level, are minimized. We apply a minimum–maximum approach to introduce demand where the maximum demand is a function of price change and allocated space. Each item is presented in the form of multiple brands, probably exposed to price changes, competing to obtain more space. In addition to inventory investment costs, replenishment costs, and inventory holding costs we also include costs related to non-productive use of space. The order quantity, the amount of allocated showroom and backroom spaces, and the cycle time of joint replenishments are key decision variables. We also propose an extended model in which price is a decision variable. Finally we solve the model using LINGO software and provide computational results.     

Stock Rationing in a Continuous Review Two-Echelon Inventory Model

In this paper we consider a 1-warehouse, N-retailer inventory system where demand occurs at all locations. We introduce an inventory model which allows us to set different service levels for retailers and direct customer demand at the warehouse. For each retailer a critical level is defined, such that a retailer replenishment order is delivered from warehouse stock if and only if the stock level exceeds this critical level. It is assumed that retailer replenishment orders, which are not satisfied from warehouse stock, are delivered directly from the outside supplier, instead of being backlogged. We present an analytical upper bound on the total cost of the system, and develop a heuristic method to optimize the policy parameters. Numerical experiments indicate that our technique provides a very close approximation of the exact cost. Also, we show that differentiating among the retailers and direct customer demand can yield significant cost reductions.     

On a stochastic inventory model with a generalized holding costs

This paper is concerned with finding the optimal replenishment policy for an inventory model that minimizes the total expected discounted costs over an infinite planning horizon. The demand is assumed to be driven by a Brownian motion with drift and the holding costs (inventory and shortages) are assumed to take some general form. This generalizes the earlier work where holding costs were assumed linear. It turns out that problem of finding the optimal replenishment schedule reduces to the problem of solving a Quasi-Variational Inequality Problem (QVI). This QVI is then shown to lead to an (s, S) policy, where s and S are determined uniquely as a solution of some algebraic equations.     

Dynamic setup reduction in production lot sizing with nonconstant deterministic demand

We examine three production policies under nonconstant, deterministic demand and dynamic setup cost reduction, where a decision to invest in setup reduction is made at the beginning of each period of a planning horizon. The three production policies are the reorder point, order quantity (s, Q) policy; the fixed production cycle, variable order quantity (t, Q_i) policy; and the variable production cycle, variable order quantity (t_i, Q_i). We study the behavior of the total relevant cost and develop a lot sizing and an investment solution procedure. Numerical examples are provided and dynamic setup cost reduction is compared with static setup cost reduction, where the decision to invest in setup reduction is made only at the initial setup.     

On an (<Emphasis Type="Italic">s,Q</Emphasis>) Production Policy for an Integrated Inventory Model for a Single Product with Linearly Increasing Demand

We analyse an (s, Q) production policy for an inventory system consisting of a single finished product and the raw materials used for manufacturing it, and where the demand rate of the product increases linearly with time. We formulate a mathematical programming model with the objective of minimizing total inventory cost per unit time. The problem of grouping raw materials optimally so that common replenishment periods may be used is considered. Solution procedures are developed, and numerical examples are presented.     

Economic lot sizing problem with inventory bounds

This paper studies a economic lot sizing (ELS) problem with both upper and lower inventory bounds. Bounded ELS models address inventory control problems with time-varying inventory capacity and safety stock constraints. An O(n²) algorithm is found by using net cumulative demand (NCD) to measure the amount of replenishment requested to fulfill the cumulative demand till the end of the planning horizon. An O(n) algorithm is found for the special case, the bounded ELS problem with non-increasing marginal production cost.     

Optimal and near-optimal policies for lost sales inventory models with at most one replenishment order outstanding

In this paper we use policy-iteration to explore the behaviour of optimal control policies for lost sales inventory models with the constraint that not more than one replenishment order may be outstanding at any time. Continuous and periodic review, fixed and variable lead times, replenishment order sizes which are constrained to be an integral multiple of some fixed unit of transfer and service level constraint models are all considered. Demand is discrete and, for continuous review, assumed to derive from a compound Poisson process. It is demonstrated that, in general, neither the best (s, S) nor the best (r, Q) policy is optimal but that the best policy from within those classes will have a cost which is generally close to that of the optimal policy obtained by policy iteration. Finally, near-optimal computationally-efficient control procedures for finding (s, S) and (r, Q) policies are proposed and their performance illustrated.     

An order expediting policy for continuous review systems with manufacturing lead-time

Expedited shipments are often seen in practice. When the inventory level of an item gets dangerously low after an order has been placed, material managers are often willing to expedite the order at extra fixed and/or variable costs. This paper proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of ordinary (s,Q)$(s\text{,}Q)$ models. Besides the two usual operational parameters: reorder point s and order quantity Q, it consists of a third parameter called the expedite-up-to level R. If inventory falls below R at the end of the manufacturing lead-time, the buyer can request the upstream supplier to deliver part of an outstanding order via a fast transportation mode. The amount expedited will raise inventory to R, while the remaining order is delivered via a slow (regular) supply mode. Simple procedures are developed to obtain optimal operational parameters. Computational results show that the proposed policy can save large costs for a firm if service level is high, demand variability is large, the extra cost for expediting is small, or the manufacturing lead-time is long.     

Continuous-review inventory problem with random supply interruptions

This paper considers a continuous-review stochastic inventory problem with random demand and random lead-time where supply may be disrupted due to machine breakdowns, strikes or other randomly occurring events. The supplier availability is modelled as a semi-Markov process (more specifically, as an alternating renewal process). The standard (q, r) policy is used when the supplier is available (ON), i.e., when the inventory position reaches the reorder point r, q units are ordered to raise the inventory position to the target level of R = q + r. The form of the policy changes when the supplier becomes unavailable (OFF) in which case orders cannot be placed when the reorder point r is reached. However, as soon as the supplier becomes available again one orders enough to bring the inventory position up to the target level of R. The regenerative cycles are identified by observing the inventory position process. We construct the average cost per time objective function using the renewal reward theorem. It is assumed that the duration of the ON period is E_k (i.e., k-stage Erlangian) and the OFF period is general. In analogy with queuing notation we call this an E_k/G system. By employing the ‘method of stages’, we obtain a problem with a larger state space for the ON/OFF stochastic process; but the resulting ON process can now be analyzed using Markovian techniques. For asymptotic values of q, the objective function assumes a particularly simple form which is shown to be convex under mild restrictions on the density functions of demand. Numerical examples illustrate the results.     

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.     

Lost-sales inventory systems with a service level criterion

Competitive retail environments are characterized by service levels and lost sales in case of excess demand. We contribute to research on lost-sales models with a service level criterion in multiple ways. First, we study the optimal replenishment policy for this type of inventory system as well as base-stock policies and (R, s, S) policies. Furthermore, we derive lower and upper bounds on the order-up-to level, and we propose efficient approximation procedures to determine the order-up-to level. The procedures find values of the inventory control variables that are close to the best (R, s, S) policy and comply to the service level restriction for most of the instances, with an average cost increase of 2.3% and 1.2% for the case without and with fixed order costs, respectively.