Inventory policy of a deteriorating item with variable demand under trade credit period

In this paper, an inventory model of a deteriorating item with stock and selling price dependent demand under two-level credit period has been developed. Here, the retailer enjoys a price discount if he pays normal purchase cost on or before the first level of credit period, or an interest is charged for the delay of payments. In return, retailer also offers a fixed credit period to his customers to boost the demand. In this regard, the authors develop an EOQ model incorporating the effect of inflation and time value of money over all the costs. Keeping the business of seasonal products in mind, it is assumed that planning horizon of business is random and follows a normal distribution with a known mean and standard deviation. The model is formulated as retailer’s profit maximization problem for both crisp and fuzzy inventory costs and solved using a modified Genetic Algorithm (MGA). This algorithm is developed following fuzzy age based selection process for crossover and gradually reducing mutation parameter. For different values of MGA parameters, optimum results are obtained. Numerical experiments are performed to illustrate the model.     

A finite time horizon inventory model with deterioration and time-value of money under the conditions of permissible delay in payments

A varying deterioration rate, time-value of money and the condition of permissible delay in payments used in conjunction with the EOQ model are the focus of discussion. The replenishment number and fraction of each cycle in which there is no shortage are both determined to minimize the present value of inventory cost over a finite planning horizon. Two special cases and numerical examples are presented to illustrate the model.     

Economic production quantity with rework process at a single-stage manufacturing system with planned backorders

In traditional inventory models such as the economic order quantity (EOQ) and the economic production quantity (EPQ) the sole objective is to minimize the total inventory-related costs, typically holding cost and ordering cost. These models do not consider the presence of defective products in the lot or rework of them. Recently, Jamal, Sarker, and Mondal (Jamal, A. A. M., Sarker, B. R., & Mondal, S., (2004). Optimal manufacturing batch size with rework process at single-stage production system. Computers and Industrial Engineering, 47(1), 77–89) proposed a model, which dealt with the optimum batch quantity in a single-stage system in which rework is done by addressing two different operational policies to minimize the total system cost, but their models do not consider planned backorders. In this direction, this paper develops an EPQ type inventory model with planned backorders for determining the economic production quantity for a single product, which is manufactured in a single-stage manufacturing system that generates imperfect quality products, and all these defective products are reworked in the same cycle. We also establish the range of real values of the proportion of defective products for which there is an optimal solution, and the close form for the total cost of inventory system. The use of the inventory model is illustrated with numerical examples. The classical EOQ, EPQ inventory models with or without planned backorders and Jamal, Sarker and Mondal’s model (Jamal, A. A. M., Sarker, B. R., & Mondal, S., (2004). Optimal manufacturing batch size with rework process at single-stage production system. Computers and Industrial Engineering, 47(1), 77–89) are shown to be special cases of the EPQ inventory model presented in this paper.     

Finite horizon EOQ model for non-instantaneous deteriorating items with price and advertisement dependent demand and partial backlogging under inflation

This paper deals with an economic order quantity (EOQ) model for non-instantaneous deteriorating items with price and advertisement dependent demand pattern under the effect of inflation and time value of money over a finite planning horizon. In this model, shortages are allowed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. This paper aids the retailer in minimising the total inventory cost by finding the optimal interval and the optimal order quantity. An algorithm is designed to find the optimum solution of the proposed model. Numerical examples are given to demonstrate the results. Also, the effect of changes in the different parameters on the optimal total cost is graphically presented and the implications are discussed in detail.     

Multi-item partial backlogging inventory models over random planninghorizon in random fuzzy environment

In this paper, some multi-item inventory models for deteriorating items are developed in a random planning horizon under inflation and time value money with space and budget constraints. The proposed models allow stock dependent consumption rate and partially backlogged shortages. Here the time horizon is a random variable with exponential distribution. The inventory parameters other than planning horizon are deterministic in one model and in the other, the deterioration and net value of the money are fuzzy, available budget and space are fuzzy and random fuzzy respectively. Fuzzy and random fuzzy constraints have been defuzzified using possibility and possibility–probability chance constraint techniques. The fuzzy objective function also has been defuzzified using possibility chance constraint against a goal. Both deterministic optimization problems are formulated for maximization of profit and solved using genetic algorithm (GA) and fuzzy simulation based genetic algorithm (FAGA). The models are illustrated with some numerical data. Results for different achievement levels are obtained and sensitivity analysis on expected profit function is also presented.Scope and purposeThe traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However for more sale, inventory should be maintained at a higher level. Of course, this would result in higher holding or procurement cost, etc. Also, in many real situations, during a shortage period, the longer the waiting time is, the smaller the backlogging rate would be. For instance, for fashionable commodities and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging diminishes with the length of the waiting time. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But at present, the economic situation of most of the countries has been much deteriorated due to large scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any further. The purpose of this article is to maximize the expected profit of two inventory control systems in the random planning horizon.     

Lot-sizing decisions for material requirements planning with hybrid uncertainties in a smart factory

Material requirements planning (MRP) is a kind of medium-term production planning, which aims to plan the end item requirements of the master production schedule over a finite planning horizon. In a smart factory, the customer requirements and the production status are varying in time, which increases the uncertainties in making lot-sizing decisions for MRP. In this study, a hybrid chance-constrained programming (HCCP) model is developed for solving an MRP problem with hybrid uncertainties, in which both randomness and fuzziness exist in a lot-sizing decision process. The objective of the HCCP model is to determine the lot sizes of all items while satisfying the stochastic demands and the fuzzy capacity constraints. The credibility and probability are incorporated into the proposed model to measure the fuzziness and randomness, respectively. In order to solve the model, relevant approaches for converting the probability-based and credibility-based constraints into the equivalent deterministic forms are proposed. Decision makers can set different confidence levels according to their own risk preferences to get different results. Finally, an example is presented to verify that the approach proposed in this paper is feasible for solving MRP problems with hybrid uncertainties.     

Solution of an imperfect‐quality EOQ model with backorder under fuzzy lock leadership game approach

This paper considers an economic order quantity (EOQ) inventory model for items with imperfect quality and shortage backordering under several styles of managerial leadership via lock fuzzy game theoretic approach. The decision maker (DM) controls several cost components by playing as Player 1 on the one side and the consumers who may accept/reject those items (unwilling to buy those commodities) stands as Player 2 on the other side. First of all, we develop a profit maximization backlogging EOQ model where the imperfect items are screened out batchwise. Because of the fuzzy flexibility of the model parameters we also develop a fuzzy mathematical model by considering the demand, all cost parameters, and other input parameters of the inventory system as triangular lock fuzzy numbers. Then we develop a 3 × 3 matrix game by applying a five‐stage leadership theory employing several key vectors in the model itself. The problem has been solved for crisp, general fuzzy models of several leadership styles also. Numerical results show that for a cooperative game, inventory profit function reaches its maximum rather than the noncooperative game by the use of proper keys. Finally, comparative study, sensitivity analysis, and graphical illustrations are made to justify the new approach.     

An EOQ model for deteriorating items with price- and stock-dependent selling rates under inflation and time value of money

This study applies the discounted cash flow (DCF) approach for the analysis of a replenishment problem over a finite planning horizon. Thus, a deterministic economic order quantity (EOQ) inventory model taking into account inflation and time value of money is developed for deteriorating items with price- and stock-dependent selling rates. An efficient solution procedure is presented to determine the optimal number of replenishment, the cycle time and selling price. Then the optimal order quantity and the total present value of profits are obtained. Numerical examples are presented to illustrate the proposed model and particular cases of the model are also discussed.     

Comparison of fuzzy inference system (FIS), FIS with artificial neural networks (FIS + ANN) and FIS with adaptive neuro-fuzzy inference system (FIS + ANFIS) for inventory control

Conventional inventory models mostly cope with a known demand and adequate supply, but are not realistic for many industries. In this research, the fuzzy inference system (FIS) model, FIS with artificial neural network (ANN) model and FIS with adaptive neuro-fuzzy inference system (ANFIS) model in which both supply and demand are uncertain were applied for the inventory system. For FIS model, the generated fuzzy rules were applied to draw out the fuzzy order quantity continuously. The order quantity was adjusted according to the FIS model with the evaluation algorithm for the inventory model. The output of FIS model was also used as data for FIS + ANN and FIS + ANFIS models. The FIS + ANFIS model was studied with three membership functions; trapezoidal and triangular (Trap), Gaussian and bell shape. Inventory costs of the proposed models were compared with the stochastic economic order quantity (EOQ) models based on previous data of a case study factory. The results showed that the FIS + ANFIS_Gauss model gave the best performance of total inventory cost saving by more than 75 % compared to stochastic EOQ model.     

Fuzzy opportunity cost for EOQ model with quality improvement investment

The cost of capital (i.e. opportunity cost) is one of the key factors that will influence the inventory and investment decisions. Previously, the classical EOQ model has been extended to include an imperfect production process and quality improvement investment, where the opportunity cost rate (interest rate or discounted rate) for evaluating the cost of capital investment is known with certainty. However, in some practical situations, the opportunity cost rate probably incurs disturbance due to the unstable environments. To capture this reality, this paper attempts to combine the statistical technique and fuzzy sets concept to deal with the unstable opportunity cost rate, so as to modify the aforementioned inventory/investment model. We derive the optimal lot size and the optimal process quality level in the fuzzy sense utilizing the logarithmic investment cost function. A numerical example is provided, and the results are compared with those obtained from a crisp opportunity cost rate model.     

Improving production planning by utilizing stochastic programming

The effects of fluctuating demand on production and inventory levels are important in manufacturing resource planning. Thus, the focus of this presentation is on aggregate production planning of manufacturing resources in order to satisfy stochastic demand for a family of products to minimize total costs that include production and inventory holding costs over a rolling horizon.

If it is assumed that, in a commercial setting, the demands are fixed, then the production plans generated by a mathematical programming procedure are not responsive to the actual fluctuations of stochastic demand in each time period.

The situation discussed here considers the case where demands are normally distributed with means and variances that are sequentially revised as new observations of demand are received over time. This assumption allows the probabilistic constraint to be converted to an equivalent linearly-constrained deterministic model. Extensions to the normality assumption are discussed. Also other ideas such as optimal control theory, learning and adaptive signal processing extensions are discussed as well.     

模糊随机环境下考虑残次品处理的库存配送问题研究

针对考虑残次品的多生产商选择多商品多阶段库存配送问题,建立了一个基于动态规划的双层库存配送模型。高端物流服务集成商以整个供应链网络成本最小为目标制定采购决策;库存配送服务商以运营成本最小为目标,在集成商决策下制定库存和配送决策。设计了模糊随机环境下基于动态规划的双层全局-局部-邻域粒子群算法（Bi-DPGLNPSO）对模型进行求解。并通过算例验证模型和算法的有效性和合理性。通过参数测试和算法对比检验算法的优越性。     

An economic production quantity model with random defective rate,rework process and backorders for a single stage production system

This paper revisits the economic production quantity (EPQ) model with rework process at a single-stage manufacturing system with planned backorders. It is well known that any imperfect production system of real life has random defective rates. In this direction, this paper extends an inventory model to allow random defective rates. Basically, three different inventory models are developed for three different distribution density functions such as uniform, triangular, and beta. The analytical derivation provides closed-form solution for each inventory model. We have made comparison tables of optimal results among the distribution functions. Some numerical examples and sensitivity analysis are given to illustrate the inventory models.     

A study on an inventory model for items with weibull ameliorating

This paper is concerned with the development of ameliorating inventory models. The ameliorating inventory is the inventory of goods whose utility increases over the time by ameliorating activation. The term ameliorating inventory is used in this paper at least, since the terminology is not standard well known. This study is performed according to areas; one is an economic order quantity(EOQ) model for the items whose utility is ameliorating in accordance with Weibull distribution, and the other is a partial selling quantity(PSQ) model developed for selling the surplus inventory accumulated by ameliorating activation. The proposed models cannot be solved directly in a closed form, thus we used a computer program and a graphical solution method to obtain the optimal ordering and selling quantity in this paper. Numerical examples to illustrate the effect of ameliorating rate on inventory polices are shown at the end of this paper.     

An EOQ model with pricing consideration

This paper presents an economic order quantity (EOQ) model that integrates the product pricing and order sizing decisions to maximize profit. The Kuhn-Tucker conditions are used to determine the optimal solution under conditions of storage space and inventory investment limitations. The aim of this paper is to explore the effect of relating the pricing and order sizing decisions on the optimal solution. Such a relationship, although it exists in real situations, has long been ignored in the classical EOQ model.     

An integrated multi-stage supply chain inventory model under an infinite planning horizon and continuous price decrease

This paper studies an integrated inventory model in a supply chain that involves procurement, production and delivery activities. The model is studied in an environment where products experience continuous price decrease and planning is performed in an infinite time horizon. In this model, a manufacturing facility purchases a fixed-quantity of raw materials from an outside supplier, processes the materials, and delivers a fixed-quantity of finished products to a customer periodically. In order to take advantage of the decreasing price trend, customers demand frequent deliveries of small lots of finished products, and this inventory management strategy has been used by many successful companies in technology-related industries. Therefore, the ultimate intention of this research is to study and model the inventory system for high-tech companies whose products are experiencing continuous price decrease. This model is used to determine an optimal economic lot size model for raw material procurement, production setup and finished goods delivering under an infinite planning horizon. Two efficient algorithms are developed in this paper to solve this nonlinear model and the test results consistently indicate that ordering of raw materials and delivery of finished goods should be frequent in small lots for low ordering and shipment costs. Finally an operational schedule is provided to show the implementation procedure of the model.     

Balancing stocks and backlogs with stochastic lead times: A special case

The paper gives compact, finite horizon and asymptotic results concerning an inventory system, that allows backlogging, with constant demand and costs but stochastic lead times, not necessarily i.i.d., under two special conditions. The special conditions are: (1) no crossovers of orders; (2) there are both shortage and stock build-up between any two consecutive orders. It is shown that the minimal total expected cost, namely, of holding inventory, shortages and replenishments, as well as the optimal replenishment times depend, among other parameters, only on the lead time variances as representatives of the stochastic component of the model. The results are direct extensions of the deterministic EOQ model and lend themselves to tractable sensitivity analysis. Further possible developments are discussed.     

An easy method to derive EOQ and EPQ inventory models with backorders

Recently, a cost minimization method to determine the lot size for the EOQ/EPQ models with backorders was published. This method is based on the well-known arithmetic–geometric mean inequality. Although the cost minimization method is correct and interesting, it does not focus on deriving the backorders level. This paper proposes another simple approach. The proposed method finds both the lot size and the backorders level.     

A fuzzy framework for coordinating pricing and inventory policies for deteriorating items under retailer partial trade credit financing

In this paper, we proposed a generalized economic order quantity (EOQ) – based inventory model using a trade credit policy in a fuzzy sense. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer, in turn, partially provides customers a permissible delay period N. This study considers fuzzy EOQ model to allow for: (1) selling price dependent demand rate which is imprecise in nature, (2) a profit maximization objective and (3) an imprecise holding cost, ordering cost, purchasing cost, interest earned and interest charged rate. Besides, the cases N ? M and N ? M are explored thoroughly. The objective function for the retailer in fuzzy sense is defuzzified using Modified Graded Mean Integration Representation Method. For the defuzzified objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.     

Inventory lot-sizing with supplier selection

This paper presents a multi-period inventory lot-sizing scenario, where there are multiple products and multiple suppliers. We consider a situation where the demand of multiple discrete products is known over a planning horizon. Each of these products can be sourced from a set of approved suppliers, a supplier-dependent transaction cost applying for each period in which an order is placed on a supplier. A product-dependent holding cost per period applies for each product in the inventory that is carried across a period in the planning horizon. The decision maker needs to decide what products to order in what quantities with which suppliers in which periods. An enumerative search algorithm and a heuristic are presented to address the problem.