Multi-item EOQ inventory model with varying holding cost under two restrictions: A geometric programming approach

In this paper we provide a simple method to determine the inventory policy of multiple items having varying holding cost using a geometric programming approach. The varying holding cost is considered to be a continuous function of the order quantity. The EOQ inventory model with constant holding cost and the classical EOQ inventory model without constraints are derived.     

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.     

EOQ FORMULA: IS IT VALID UNDER INFLATIONARY CONDITIONS?

The classical analysis of the economic order quantity (EOQ) problem ignores the effect of inflation. When a firm's cost factors are expected to rise at an annual rate of 10 percent or more, what adjustments in order quantities should the firm make to control its lot-size inventory (or cycle stock)? Using a model that includes both inflationary trends and time discounting, it is concluded that inflation brings no incentive either to increase or to decrease order quantities. In addition, order quantities can be computed using the classical EOQ formula under inflationary conditions, provided that the cost of capital invested in inventory is interpreted as an inflation-free cost. This interpretation implies that changes in the inflation rate should not affect the cost of capital that is utilized in the EOQ formula for determining order quantities.     

An EOQ inventory model for items with Weibull distribution deterioration,ramp type demand rate and partial backlogging

In this paper, an EOQ inventory model is presented depleted not only by demand but also by Weibull distribution deterioration, in which the demand rate is assumed that with a ramp type function of time. In the model, shortages are allowed partial backlogging and the backlogging rate is variable and is dependent on waiting time for the next replenishment. The method is illustrated by three numerical examples, and sensitivity analysis of the optimal solution with respect to parameters of the system is carried out.     

EOQ with independent endogenous supply disruptions

We consider an inventory installation, controlled by the periodic review base stock (S, T) policy and facing a fixed-rate deterministic demand which, if unsatisfied, is backordered. The supply process is unreliable, so supply deliveries may fail according to an independent Bernoulli process; we refer to such failures reflecting the supply service quality and being internal to the supply chain, as endogenous disruptions. We seek to jointly determine the two policy variables, so to minimize long-run average cost. While an approximate model for this problem was recently analyzed, we present an exact analysis, valid for two common accounting schemes for inventory cost evaluation: continuous and end-of-cycle costing. After developing a unified (and exact) average cost model for both costing schemes, the cost for each scheme is analyzed. In both cases, the optimal policy variables and cost prevail in closed-form, having an identical structure to those of EOQ (with backorders). In fact, under continuous costing, the optimal solution reduces to EOQ for perfect supply. Analytical properties, demonstrating the impact of deteriorating supply quality on the optimal policy, are established. Moreover, computations reveal the cost impact of deploying heuristics that either ignore supply disruptions or rely on inaccurate costing information.     

Constrained production lot-size model with trade-credit policy: 'A comparison geometric programming approach via Lagrange'

In this paper our main objective is to investigate a deterministic inventory production lot-size model with a permissible delay in payment under a restriction. We analyse our deterministic inventory model under a restriction which will be assumed as the average inventory level. In fact we use in our analysis two approaches: the geometric programming approach; and the Lagrange method. Then a comparison between these two approaches is performed, which is our aim. Finally we deduce some previously published works of other researchers as special cases.     

Setups and effective capacity: the impact of lot sizing techniques in an MRP environment

This study investigates how lot sizing techniques influence the profit performance, inventory level, and order lardiness of an assembly job shop controlled by MRP. Four single-level lot sizing techniques are compared by simulation analysis under two levels of master schedule instability and two levels of end item demand. A second analysis investigates the influence of a multilevel lot sizing technique, the generalized constrained-K (GCK) cost modification, on the four single-level techniques at low demand and low nervousness. The analyses reveal a previously unreported phenomenon. Given the same inventory costs, the single-level lot sizing techniques generate substantially different average batch sizes. The lot sizing techniques maintain the following order of increasing average batch size (and decreasing total setup time):

economic order quantity (EOQ)

period order quantity (POQ)

least total cost (LTC)

Silver-Meal heuristic (SML)

The causes for different average batch sizes among the lot sizing techniques are analysed and explained. Demand lumpiness, inherent in multilevel manufacturing systems controlled by MRP, is found to be a major factor. The number of setups each lot sizing technique generates is the primary determinant of profit performance, inventory level, and order tardiness. EOQ, a fixed order quantity technique, is less sensitive to nervousness than the discrete lot sizing techniques. EOQ_, however, generates the smallest average batch size, and, therefore, the most setups. Since setups consume capacity, EOQ, is more sensitive to higher demand. SML generates the largest average batch sizes, and is, therefore, less sensitive to increased demand. At low demand, the other lot sizing techniques perform better on all criteria. They generate smaller batches and, therefore, shorter actual lead times. The GCK cost modification increases the average batch size generated by each lot sizing technique. GCK improves the profit and customer service level of EOQ the lot sizing technique with the smallest batches. GCK causes the other lot sizing techniques to generate excessively large batches and, therefore, excessively long actual lead times.     

Inventory Models with Continuous and Poisson Demands and Discounted and Average Costs

We develop a new, unified approach to treating continuous‐time stochastic inventory problems with both the average and discounted cost criteria. The approach involves the development of an adjusted discounted cycle cost formula, which has an appealing intuitive interpretation. We show for the first time that an (s, S) policy is optimal in the case of demand having a compound Poisson component as well as a constant rate component. Our demand structure simultaneously generalizes the classical EOQ model and the inventory models with Poisson demand, and we indicate the reasons why this task has been a difficult one. We do not require the surplus cost function to be convex or quasi‐convex as has been assumed in the literature. Finally, we show that the optimal s is unique, but we do not know if optimal S is unique.     

An EOQ model with deteriorating items in response to a temporary sale price

In economic order quantity models, it is often assumed that the unit purchase cost is constant. Such an assumption is usually not fulfilled in many practical situations. In practice, it is observed that suppliers sometimes offer temporary price discounts to stimulate demand, boost market share or decrease inventories of certain items. In this paper, a deteriorating inventory model with a temporary sale price has been developed. We shall be concerned with finding the optimal total cost saving for deteriorating items during the special replenishment period. Numerical examples are presented to illustrate the proposed model.     

A PROFIT-MAXIMIZATION MODEL FOR A RETAILER'S STOCKING DECISIONS ON PRODUCTS SUBJECT TO SUDDEN OBSOLESCENCE

A product has been formally denned as being subject to sudden obsolescence if its lifetime is negative exponentially distributed. Using an approximate model, Masters suggested that the traditional method of incorporating obsolescence cost as a component of inventory holding costs in the economic order quantity (EOQ) model was appropriate-for products subject to sudden obsolescence, provided that the obsolescence cost component was computed properly. Unfortunately, current practice of the EOQ model seriously underestimates the costs of sudden obsolescence. An exact model demonstrating that Masters' model also underestimated true lifetime costs and overestimated the optimal order quantity has been presented. Neither of these models addressed quantity discounts. Furthermore, with their cost-minimization focus, these models fail to identify situations when minimized costs exceed expected revenues. We extend Joglekar and Lee's model to focus on maximizing profits, rather than minimizing costs. This model answers such questions as whether to stock the product at all, whether to accept a quantity discount offer, and what order quantity to use. Numerical examples and sensitivity analyses suggest that Masters' model provides a significant improvement over the traditional model in moving toward true optimality. However, they also illustrate situations where both the traditional and the Masters' model accept a quantity discount that deserves to be rejected and stock a product that should not be stocked. In such situations, it seems important that a retailer uses the profit-maximization model presented here.     

Setup cost reduction a multi-stage order quantity model

In this paper, a single item, multi-stage serial order quantity (MSOQ) model with constant demand is discussed. The objective of the model is to minimize the total cost which includes the setup cost and the inventory holding cost. This paper examines and analyses the investment in a one-time cost to reduce the (current) setup level and adds a per unit item amortization of this cost to the other costs associated with the MSOQ model. We consider the setup cost to be decreased on each stage with the same rate and the cost of the joint setup cost reduction is a logarithmic cost function.     

一种需求和采购价均为时变的EOQ模型

本文提出了一种需求和采购价均为时变的EOQ模型,证明了该模型的总库存成本目标函数在给定条件下为凸函数,给出了寻求最佳采购次数及服务水平的算法,并对该模型进行了数值仿真和灵敏度分析。     

The effects of the warranty cost on the imperfect EMQ model with general discrete shift distribution

In this paper, we investigate the effect of the warranty cost on optimization of the economic manufacturing quality (EMQ). This is done for a deteriorating process where the production process shifts from the in-control state to the out-of-control state following a general discrete probability distribution. Once the production process goes out of control, the production process produces some defective items. The defective item cost includes reworking and warranty costs. Thus, in order to economically operate a production-inventory system with products sold under warranty, the tradeoffs among the production setup, inventory, and defective item cost, including the reworked cost before sale and the warranty cost after sale, needed to be analysed. This objective in this paper is to determine the production lot size while minimizing the total cost per unit of time per unit of time. Various special cases are presented. Two of them are extensions of results obtained previously in the literature. Finally, a numerical example is given which uses a discrete Weibull probability distribution. Sensitivity analysis of the model with respect to cost and time parameters is also performed.     

Economic Manufacturing Quantity and Its Integrating Implications

Several contradictions are noted among the Economic Order Quantity (EOQ), Just‐In‐Time (JIT), and Optimized Production Technology (OPT) approaches and the economic framework for profit maximization. A fundamental model referred to as the Economic Manufacturing Quantity (EMO) is developed and examined for its integrating implications for the three approaches. An implication for the classic EOQ approach is that the balance between setup and inventory carrying costs is valid when a production facility is operating at or below a certain critical level but not when operating above that level. An implication for the JIT approach is that one must reduce setup cost at non‐bottlenecks and setup time at bottlenecks to reduce inventory. An implication for the OPT approach is that trade‐offs between setup and inventory carrying costs may indeed be ignored while determining process batch sizes, provided each facility in a production system is operating at or above Its critical level. Economic theoretic analysis of the EMO model provides a basis for unification of JIT which advocates stability in operating level as a key to improved productivity and quality, and OPT that advocates maximizing operating level with resultant emphasis on bottlenecks as a key to increased profits. This unifying basis states that a profit‐maximizing production facility or system will operate at the full and stable level as long as market demand remains relatively sensitive to price and operating at the full (maximum) level provides positive unit contribution.     

有限时域与采购价格上升情形下的最优订货策略

研究了有限时域下采购商面对价格上升时的订货策略问题.在分析问题的基础上提出一种新的最优采购策略,并分析了价格上升幅度对订货量的影响,以经典EOQ模型的总成本为基准,比较了本文提出的策略与文献已有策略在成本节约上的差异.本文对库存总成本的计算方法更加精确;分析表明在有限时域背景下采购商的临时订货量决定于价格上涨的幅度、在库库存以及时段长度.     

Some Comments on the Validity of EOQ Formula under Inflationary Conditions

In an earlier issue of Decision Sciences, Jesse, Mitra, and Cox [1] examined the impact of inflationary conditions on the economic order quantity (EOQ) formula. Specifically, the authors analyzed the effect of inflation on order quantity decisions by means of a model that takes into account both inflationary trends and time discounting (over an infinite time horizon). In their analysis, the authors utilized two models: Current-dollars model and Constant-dollars model. These models were derived, of course, by setting up a total cost equation in the usual manner then finding the optimum order quantity that minimizes the total cost. Jesse, Mitra, and Cox [1] found that EOQ is approximately the same under both conditions; with or without inflation. However, we disagree with the conclusion drawn by [2] and show that EOQ will be different under inflationary conditions, provided that the inflationary conditions are properly accounted for in the formulation of the total cost model.     

The effects of component commonality in an infinite horizon inventory model

This paper studies the effects of component commonality in the context of an infinite horizon inventory model. Three models are proposed that are characterized by different degrees of component commonality. Assuming the three models all follow the same inventory policy, exact service level measures are derived and incorporated into cost optimization problems. With the infinite horizon assumption, potential setup cost reductions can be evaluated due to the inclusion of common components. The results indicate that, as expected, commonality incurs significant cost savings; what is new and unique is that setup cost may increase or decrease when commonality is present. In addition, when the behaviour of the optimal solutions is examined, it is found that some of the well-known properties suggested by the existing one-period models do not hold for this infinite horizon model.     

提前期、构建成本和短缺量拖后率均可控的EOQ模型

传统库存模型通常将提前期和构建成本视为不可控制。事实上可以通过追加投资缩短提前期和降低构建成本。缺货期间,为减少订单丢失量和补偿顾客的损失,供应商会给予一定的价格折扣。现实库存系统中,容易得到需求的期望值和标准差,但较难得到其分布规律。基于此,考虑短缺量拖后率与价格折扣和缺货期间库存水平相关,提出了一种需求为任意分布且提前期和构建成本均可控的EOQ模型,证明了模型存在唯一最优解,给出了一种寻优算法。数值仿真分析表明,一般情况下,压缩提前期和降低构建成本能降低订购批量和安全库存,降低库存总成本;短缺量拖后系数和缺货概率对库存总成本影响较大,企业应尽量降低缺货概率,尤其在短缺量拖后系数较小时。     

Note on inventory models with Weibull distribution deterioration

This article explores the inventory model with a general demand rate function in which both the Weibull distributed deterioration and partial backlogging are considered. The inventory model discussed here is based on the important finding by Wu [Wu, K.S., 2001. An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging. Production Planning and Control, 12, 787–793]. There are four parts in our research. First, we derive the analytical framework of the inventory model for a general demand rate function1 1.?Based on detailed suggestions of Professor Wafik H. Iskander, Email: whiskander@mail.wvu.edu . Second, for the ramp type demand, we improve Wu's model to find the criterion to guarantee the existence and uniqueness of the optimal solution. Third, we develop a new model to compensate for the missing case in Wu's article. Fourth, we combine our results to show that our findings are applicable to the ramp type demand inventory model, so that the optimal solution is independent of the demand function. Finally, some numerical examples and graphs are provided to illustrate our discovery and demonstrate the application of our analytical framework.     

A queuing/management consideration on Japanese production systems

A single-stage lot/cell production under a Poisson arrival and exponential service in a batch is considered. The three economic queuing models of push and pull types are presented, an economic comparison of push versus pull types is considered, and a strategic management/design consideration to the lot production is given. First, the total expected operating cost is given for the three queuing models including the Omote-Kanban type similar to VMI. Second, the push versus pull system is discussed from a view of setup time, inventory or operating cost, and it is ascertained that the three types are alternative. Finally, a strategic management basis for economic traffic, leadtime setting is given, and discussed by the introduction of production matrix on 2-stage design.