A single-period inventory model with fuzzy demand

The single-period inventory models have wide applications in the real world in assisting the decision maker to determine the optimal quantity to order. Due to lack of historical data, the demand has to be subjectively determined in many cases. In this paper, a single-period inventory model for cases of fuzzy demand is constructed. The costs considered include the procurement cost, shortage cost, and holding cost. For different fuzzy total cost resulted from different order quantity, a method for ranking fuzzy numbers is adopted to find the optimal order quantity in terms of the cost. When the profit gained from selling one item is less (greater) than the loss incurred due to one unsold item, the optimal order quantity lies in the range defined for the left-shape (right-shape) function of the fuzzy demand. If the unit profit is equal to the unit loss, then all quantities with a membership grade 1 are optimal to be ordered. The methodology of this paper can be applied to construct other inventory models with fuzzy demand.     

Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables

This paper considers inventory models for items with imperfect quality and shortage backordering in fuzzy environments by employing two types of fuzzy numbers, which are trapezoidal and triangular. Two fuzzy models are developed. In the first model the input parameters are fuzzified, while the decision variables are treated as crisp variables. In the second model, not only the input parameters but also the decision variables are fuzzified. For each fuzzy model, a method of defuzzification, namely the graded mean integration method, is employed to find the estimate of the profit function in the fuzzy sense, and then the optimal policy for the each model is determined. The optimal policy for the second model is determined by using the Kuhn–Tucker conditions after the defuzzification of the profit function. Numerical examples are provided in order to ascertain the sensitiveness in the decision variables with respect to fuzziness in the components.     

Periodic review fuzzy inventory model with variable lead time and fuzzy demand

This paper investigates a periodic review fuzzy inventory model with lead time, reorder point, and cycle length as decision variables. The main goal of this study is to minimize the expected total annual cost by simultaneously optimizing cycle length, reorder point, and lead time for the whole system based on fuzzy demand. Two models are considered in this paper: one with normal demand distribution and another with a distribution‐free approach. The model assumes a logarithmic investment function for lost‐sale rate reduction. Furthermore, two separate efficient computational algorithms are explained to obtain the optimal solution. Some numerical examples are given to illustrate the model.     

Optimization of variable demand fuzzy economic order quantity inventory models without and with backordering

A variable demand inventory model was developed for minimizing inventory cost, treating the holding and ordering costs and demand as independent fuzzy variables. Thereafter, backordering cost was also considered as an independent fuzzy variable. Fuzzy expected value model and fuzzy dependent chance programming model were constructed to find the optimal economic order quantity, which would minimize the fuzzy expected value of the total cost, so that the credibility of the total cost not exceeding a certain budget level was maximized. Optimization was carried out using genetic algorithms and particle swarm optimization algorithm, and their performances were compared. The developed model was found to be efficient not only in one artificial case study but also in two data sets collected from the industries. Therefore, this model could solve real-world problems, too.     

A production inventory model with fuzzy random demand and with flexibility and reliability considerations

The classical inventory control models assume that items are produced by perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be increased without a price, its set-up cost can be reduced with investment in flexibility improvement. In this paper, a production inventory model with flexibility and reliability (of production process) consideration is developed in an imprecise and uncertain mixed environment. The aim of this paper is to introduce demand as a fuzzy random variable in an imperfect production process. Here, the set-up cost and the reliability of the production process along with the production period are the decision variables. Due to fuzzy-randomness of the demand, expected average profit of the model is a fuzzy quantity and its graded mean integration value (GMIV) is optimized using unconstraint signomial geometric programming to determine optimal decision for the decision maker (DM). A numerical example has been considered to illustrate the model.     

Joint optimization of preventive maintenance and inventory policies for multi-unit systems subject to deteriorating spare part inventory

The interconnection of maintenance and spare part inventory management often puzzles managers and researchers. The deterioration of the inventory affects decision-making and increases losses. Block replacement and periodic review inventory policies were here used to evaluate a joint optimization problem for multi-unit systems in the presence of inventory deterioration. The deterministic deteriorating inventory (DDI) model was used to describe deteriorating inventory when deteriorating inventory data were available and the stochastic deteriorating inventory (SDI) model was used when they were not. Analytical joint optimization models were established, and the preventive replacement interval and the maximum inventory level served as decision variables to minimize the expected system total cost rate. This work proved the existence of the optimal maximum inventory level and gave the uniqueness condition under the DDI model. Numerical experiments based on the electric locomotives in Slovenian Railways were performed to confirm the effectiveness of the proposed models. Results showed the total cost rate to be sensitive to the maximum inventory level, which indicates that the research of this work is necessary. Further, the optimal preventive replacement interval was reduced and the optimal maximum inventory level was increased to balance the influence of deteriorating inventory. Monte Carlo experiments were used to show that the proposed policy is better than policies that do not take deteriorating inventory into account.     

Developing a multi-objective,multi-item inventory model and three algorithms for its solution

We develop a multi-objective model in a multi-product inventory system.The proposed model is a joint replenishment problem(JRP) that has two objective functions.The first one is minimization of total ordering and inventory holding costs,which is the same objective function as the classic JRP.To increase the applicability of the proposed model,we suppose that transportation cost is independent of time,is not a part of holding cost,and is calculated based on the maximum of stored inventory,as is the case in many real inventory problems.Thus,the second objective function is minimization of total transportation cost.To solve this problem three efficient algorithms are proposed.First,the RAND algorithm,called the best heuristic algorithm for solving the JRP,is modified to be applicable for the proposed problem.A multi-objective genetic algorithm(MOGA) is developed as the second algorithm to solve the problem.Finally,the model is solved by a new algorithm that is a combination of the RAND algorithm and MOGA.The performances of these algorithms are then compared with those of the previous approaches and with each other,and the findings imply their ability in finding Pareto optimal solutions to 3200 randomly produced problems.     

A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate

The purpose of this paper is to extend [Ouyang, L. Y., Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13] periodic review inventory model with variable lead time by considering the fuzziness of expected demand shortage and backorder rate. We fuzzify the expected shortage quantity at the end of cycle and the backorder (or lost sales) rate, and then obtain the fuzzy total expected annual cost. Using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. For the proposed model, we provide a solution procedure to find the optimal review period and optimal lead time in the fuzzy sense so that the total expected annual cost in the fuzzy sense has a minimum value. Furthermore, a numerical example is provided and the results of fuzzy and crisp models are compared.     

A new multiple criteria decision making method based on intuitionistic fuzzy information

The risk attitude of a decision maker is considered in the decision process. Inspired by mean-variance type utility functions in the financial risk management, a new class of decision functions are defined based on the weighted score function and the weighted accuracy function in the intuitionistic fuzzy setting. By choosing a suitable parameter value, the decision maker’s risk attitude can be flexibly reflected by our decision function. The new method can be applied for both the exactly known and partly known criteria weight situations. For the latter case, it is only necessary to solve one linear programming problem. The developed models and algorithms are then extended to multiple criteria decision making problems with the interval-valued intuitionistic fuzzy information. Numerical examples are provided to illustrate the practicality, flexibility and efficiency of our new algorithms.     

Fuzzy inference systems and inventory allocation decisions: Exploring the impact of priority rules on total costs and service levels

Inventory allocation decisions in a distribution system concern issues such as how much and where stock should be assigned to orders in a supply chain. When the inventory level of an inventory point is lower than the total number of items ordered by lower echelons in the chain, the decision of how many items to allocate to each ``competing'' order must take into consideration the trade-off between cost and service level. This paper proposes a decision-support system that makes use of fuzzy logic to consider inventory carrying, shortage and ordering costs as well as transportation costs. The proposed system is compared through simulation with three other inventory allocation decision support models in terms of cost and service levels achieved. Conclusions are then drawn.     

Application of fuzzy sets to manufacturing/distribution planning decisions with multi-product and multi-time period in supply chains

This work applies fuzzy sets to integrating manufacturing/distribution planning decision (MDPD) problems with multi-product and multi-time period in supply chains by considering time value of money for each of the operating cost categories. The proposed fuzzy multi-objective linear programming model (FMOLP) attempts to simultaneously minimize total costs and total delivery time with reference to inventory levels, available machine capacity and labor levels at each source, as well as market demand and available warehouse space at each destination, and the constraint on total budget. An industrial case demonstrates the feasibility of applying the proposed model to a realistic MDPD problem and several significant management implications are presented based on computational analysis and comparisons with the existing MDPD methods. The main advantage of the proposed model is that it presents a systematic framework that facilitates fuzzy decision-making for solving the multi-objective MDPD problems with multi-product and multi-time period in supply chains under an uncertain environment, enabling the decision maker to adjust the search direction during the solution procedure to obtain a preferred satisfactory solution.     

Financial cost inclusive reformulations of inventory lot size models

Inventory systems for deterministic demand have been extensively discussed in the literature. Generally, lot size models have been developed to minimize per-period total inventory costs. Financial management theory, however, strongly suggests that the fundamental objective of management is to maximize shareholder wealth. Thus, in theory, inventory policy decisions should be made within a net present value, wealth maximization context. This paper reformulates the uniform replenishing rate inventory model in a present value framework under two cash-flow scenarios. In the first scenario, which is shown to be equivalent to the classical EOQ model, it is demonstrated that the classical EOQ methodology is consistent with the present value reformulation. In the second scenario, which is consistent with the classical uniform replenishing rate model, the present value reformulation recommends substantially higher optimal order quantities than the classical model and provides insight about both the traditional methodology and future uses of the present value methodology.     

Application of particle swarm optimisation for solving deteriorating inventory model with fluctuating demand and controllable deterioration rate

In most of the inventory models in the literature, the deterioration rate of goods is viewed as an exogenous variable, which is not subject to control. In the real market, the retailer can reduce the deterioration rate of product by making effective capital investment in storehouse equipments. In this study, we formulate a deteriorating inventory model with time-varying demand by allowing preservation technology cost as a decision variable in conjunction with replacement policy. The objective is to find the optimal replenishment and preservation technology investment strategies while minimising the total cost over the planning horizon. For any given feasible replenishment scheme, we first prove that the optimal preservation technology investment strategy not only exists but is also unique. Then, a particle swarm optimisation is coded and used to solve the nonlinear programming problem by employing the properties derived from this article. Some numerical examples are used to illustrate the features of the proposed model.     

可用度约束条件下的设备不完美预防维修策略

针对2M1B生产系统,基于设备实际维修情况,提出了故障设备不完美预防维修策略。首先,考虑设备随时间不断劣化的情况,基于准更新过程,建立了生产周期内设备随机故障次数的表达式,计算了设备维修总费用。其次,通过分析缓冲区内库存的变化轨迹,以生产周期内设备随机故障次数为基础,计算了缓冲区库存费用,综合设备维修费用和缓冲区库存费用,构建了周期内生产总成本模型。以满足系统最低要求的可用度水平为约束条件,以预防维修周期和缓冲区库存量为决策变量,以生产周期内单位时间总成本为目标函数,通过离散迭代算法获得最优预防维修周期和此周期下的最佳缓冲区库存量。最后,通过数值分析验证了模型的有效性,为制定最佳维修策略提供了有效依据。     

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.     

An application of fuzzy set theory to inventory control models

A method for solving an inventory control problem, of which input data are described by triangular fuzzy numbers will be presented here. The continuous review model of the inventory control problem with fuzzy input data will be focused in, and a new solution method will be presented. For the reason that the result should be a fuzzy number because of fuzzy input data, and the certain number about order quantity is prefered in the real-world, it is necessary to transform the fuzzy result to crisp one. The interval mean value concept is used here to help to solve this problem. Under the condition of total cost minimum, the interval order quantity maximum can be obtained.     

一类最小代价模糊决策系统及其算法

针对模糊决策系统在应用中的实际问题,提出一类最小代价模糊决策系统模型,定义了最优决策约简和最优决策代价,并对其性质进行分析。求解最优决策约简和最优决策代价是NP完全问题,为此给出基本算法、贪婪算法和基于拉格朗日松弛的子梯度优化算法,并进行实验分析。     

A simulation-based decision support system for a multi-echelon inventory problem with service level constraints

In this paper, we present a simulation-based decision support system for solving the multi-echelon constrained inventory problem. The goal is to determine the optimal setting of stocking levels to minimize the total inventory investment costs while satisfying the expected response time targets for each field depot. We derive new decision support algorithms to be applied in different scenarios, including small-sample and large-sample cases. The first case requires that the set of alternative solutions is known at the beginning of the experiment, and the number of evaluated solutions may depend on the simulation budget (i.e., the time available to solve the problem). In the second case, the alternative solutions are generated sequentially during the searching process, and we may terminate the algorithm when the specified sampling budget is exhausted. Empirical studies are conducted to compare the performance of the proposed algorithms with other conventional optimization approaches.     

An operational policy for an integrated inventory system under consignment stock policy with controllable lead time and buyers’ space limitation

This paper aims to enable the decision maker of an integrated vendor–buyer system, under Consignment Stock (CS) policy, to make the optimal/sub-optimal production/replenishment decisions where the buyer places a space limitation to the vendor and the lead-time is controllable with an extra investment. Within any production cycle, the vendor produces at a finite rate and ships the outputs to the buyer with a number of equal-sized lots. With a long-term consignment stock agreement, the vendor takes responsibility to maintain a certain inventory level in the buyer's warehouse. Some of the shipments are delayed so that the buyer's inventory does not go beyond the capacity limitation. The buyer compensates the vendor after the complete consumption of the products. The holding cost consists of a storage component and a financial component. Two constraint four-variable non-linear integer optimization models are established wherein the buyer space limitation is considered. Because the developed models are mathematically very difficult to solve, three doubly hybrid meta-heuristic algorithms are employed to solve the models. The computational results show that one of these three algorithms works very well both in the sense of the success rate and the mean CPU time. The analysis of the computational example also reveals the quantitative effects the buyer space limitation may have to the annual joint total expected cost (JTEC) of the integrated system.     

Cost allocation model for optimizing supply chain inventory with controllable lead time

The coordination issue of a decentralized supply chain composed of a vendor and a buyer is considered in this paper. The vendor offers a single product to the buyer and the lead time can be controllable with adding crashing cost. Two supply chain inventory models with controllable lead time under different decision modes are considered, one is proposed under decentralized model based on Stackelberg model, the other is proposed under centralized model of the integrated supply chain. The solution procedures are also suggested to get the optimal solutions of these two models. In addition, an asymmetric Nash bargaining model based on satisfaction level is also developed to get the best cost allocation ratio between the vendor and the buyer by taking their individual rationalities into consideration. The results of numerical example show that shortening lead time reasonably can reduce inventory cost and the cost allocation model based on satisfaction level developed in this paper is effective.