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.     

具单周期特性的短生命周期产品随机IRP及算法

为优化企业物流系统,针对单周期,短生命周期产品的特点,将库存控制与配送路径安排决策集成,考虑随机需求、缺货成本、积压贬值成本、配送成本等,建立一个具有单周期特性的短生命周期产品随机IRP离散模型,目标是合理确定各零售门店的订购数量及配送路线使得系统成本最小。该问题属于NP-hard问题。对此,采用“报童模型”和差分法求解最佳订购量,将模型予以转化,并设计了一种遗传算法进行求解。算例结果表明所提算法能在较短时间内求解出不同客户数目组合的满意解。结论是：门店订购量宜采用组合选择方式;系统成本与单位行程运价正相关;车容量增大有助于降低系统成本。     

A software cost model with imperfect debugging,random life cycle and penalty cost

The paper develops a cost model with an imperfect debugging and random life cycle as well as a penalty cost that is used to determine the optimal release policies for a software system. The software reliability model, based on the nonhomogeneous Poisson process, allows for three different error types: critical, major and minor errors. The model also allows for the introduction of any of these errors during the removal of an error. Using the software reliability model presented, the cost model with multiple error types and imperfect debugging is developed. This cost also considers the penalty cost due to delay for a scheduled delivery time and the length of the software life cycle is random with a known distribution. The optimal software release policies that minimize the expected software system costs (subject to the various constraints) or maximize the software reliability subject to a cost constraint, are then determined. Numerical examples are provided to illustrate the results.     

Mathematical programming approach to inventory control

In many production systems inventory cost is the major element of investment. It is essential in these systems to hold inventory at a near optimal level to minimize the total costs of the system. This paper presents a procedure for developing an optimal inventory control plan which can be solved by mathematical programming. Input to the model includes demand forecasts, ordering costs, carrying costs, shortage costs, price break data, space constraints, and probability of usage. Most of the factors affecting this plan are investigated in order to simplify the way of handling all the input values.     

随机时变需求下有阶段和总量约束的多源采购优化

针对有阶段采购量区间约束及最小总量要求的多供应商多阶段订单分配问题,以采购方采购、库存、缺货成本总和与期末库存残值之差的期望值最小为目标,建立随机非线性规划模型.考虑采购量约束及多供应商情形,扩展报童模型得出单阶段最优采购策略.以此为基础建立多阶段启发式算法,通过仿真分析各参数对最优采购策略及其总成本的影响.将最小总量承诺契约扩展到时变需求和多供应商情形,可为采购方降低成本.     

A Study of the infinite horizon ‘solution’ of inventory lot size models with a linear demand function

The article studies the infinite time horizon inventory lot size model with a linear demand function. The pertinent costs are the carrying and order costs, shortages are prohibited, and replenishments are instantaneous. The problem is to minimize the total cost by exhibiting a schedule of optimal replenishment times.The computerized asymptotic solution suggested here is a generalization of the classical square root law and the more recent cubic root law. Due to its broader usefulness, it should preponderate previously known results.     

基于二阶段时间延迟的多产品生产系统生产与维修联合优化研究

针对多产品生产部件串联系统的生产和维修问题进行了研究,提出了基于二阶段时间延迟的联合优化模型。首先,基于生产周期分段理论,将整个周期等分成若干单位时间段,生产与维修共用每段时间,且若干时间段后采取一次预防维修。其次,考虑生产系统的实际生产时间、可用生产时间和维修耗费时间,建立了生产计划与维修计划总成本模型。其中,维修计划考虑缺陷和故障维修费用、维修检查费用,以及非正常状态下设备运行可能产生的不合格产品损失费用;生产计划考虑生产成本、库存成本、延期未交货成本和维修停机后恢复生产的设备启动成本。最后,通过算例分析,计算最优预防维修周期和各单位时间段各产品产量,验证了模型的有效性。     

Capacitated lot sizing by using multi-chromosome crossover strategy

The capacitated lot sizing problem (CLSP) arises when, under capacity constraints, the decision maker has to determine the production schedule and lot sizes that will minimize the total costs involved. The costs considered in this article are order, inventory carrying, and labor costs. The fitness function for the chromosome is computed using these cost elements. Next, the chromosomes are partitioned into good and poor segments based on the individual product chromosomes. This information is later used during crossover operation and results in crossover among multiple chromosomes. Product chromosomes are grouped into three groups, group 1 (top X%), group 2 (next Y%), and group 3 (last Z%). Product chromosomes from Groups 1, 2 and 3 can only form pairs with chromosomes from group 1. Besides, different crossover and mutation probabilities are applied for each group. The results of the experimentation showed that the different strategies of the proposed approach produced much better results than the classical genetic algorithm. This article has been submitted to the Journal of Intelligent Manufacturing for publication.     

Multiproduct multiple-buyer single-vendor supply chain problem with stochastic demand, variable lead-time, and multi-chance constraint

In this paper, a multi-product multi-chance constraint joint single-vendor multi-buyers inventory problem is considered in which the demand follows a uniform distribution, the lead-time is assumed to vary linearly with respect to the lot size, and the shortage in combination of backorder and lost-sale is assumed. Furthermore, the orders are placed in multiple of packets, there is a limited space available for the vendor, there are chance constraints on the vendor service rate to supply the products, and there is a limited budget for each buyer to purchase the products. While the elements of the buyers’ cost function are holding, shortage, order and transportation costs, the set up and holding costs are assumed for the vendor. The goal is to determine the re-order point and the order quantity of each product for each buyer such that the chain total cost is minimized. We show the model of this problem to be a mixed integer nonlinear programming type and in order to solve it a particle swarm optimization (PSO) approach is used. To justify the results of the proposed PSO algorithm, a genetic algorithm (GA) is applied as well to solve the problem. Then, the quality of the results and the CPU times of reaching the solution are compared through three numerical examples that are given to demonstrate the applicability of the proposed methodology in real world inventory control problems. The comparison results show the PSO approach has better performances than the GA method.     

差分进化算法在模糊多资源约束联合补货模型中的应用

本文用三角模糊数表示不确定的资金约束,用梯形模糊数表示不确定的存储空间约束,构建了模糊规划联合补货模型,目标函数为最小化订货成本、库存持有成本和运输成本,决策变量为基本补充周期和每种产品的补充周期。通过对变异算子与选择操作进行变化,设计了改进的差分进化算法对模型进行求解,并通过实例证实了模型与算法的科学合理性。     

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.     

Joint determination of order quantity and reorder point of continuous review model under quantity and freight rate discounts

The increased emphasis on transportation costs has enhanced the need to develop models with transportation consideration explicitly. However, in stochastic inventory models, the transportation cost is considered implicitly as part of fixed ordering cost and thus is assumed to be independent of the size of the shipment. As such, the effect of the transportation and purchasing costs are not adequately reflected in final planning decisions. In this paper, transportation and purchasing considerations are integrated with continuous review inventory model. The objective is to view the system as an integrated whole and determine the lot size and reorder point which minimize the expected total cost per unit time. In addition, procedures are developed to solve the proposed models. Numerical experiments are also performed to explore the effect of key parameters on lot size, reorder point and expected total cost. The new models have a significant impact on lot size, reorder point and expected total cost. Savings up to 17.15% of the expected total cost are realized when using the proposed models.     

Two-echelon supply chain inventory model with controllable lead time and service level constraint

This paper considers a two-echelon supply chain inventory problem consisting of a single-vendor and a single-buyer. In the system under study, a vendor produces a product in a batch production environment and supplies it to a buyer facing a stochastic demand, which is assumed to be normally distributed. Also, buyer’s lead time is controllable which can be shortened at an added cost and all shortages are backordered. A model has been formulated for an integrated vendor–buyer problem to jointly determine the optimal order quantity, lead time and the number of shipments from the vendor to the buyer during a production cycle while minimizing the total expected cost of the vendor–buyer integrated system. It is often difficult to estimate the shortage cost in inventory systems. Therefore, instead of having a shortage cost term in the objective function, a service level constraint (SLC) is included in the model that requires a certain proportion of demands to be met in each cycle. An efficient procedure has been suggested to find the bounds on number of shipments and then, an algorithm is developed to obtain the optimal solution of the proposed model. A numerical example is included to illustrate the algorithmic procedure and the effects of key parameters are studied to analyze the behavior of the model. Finally, the savings of buyer and vendor are investigated from implementation of joint optimization model over the model in which they minimize their own cost independently.     

Procurement scheduling for complex projects with fuzzy activity durations and lead times

Material procurement is a vital activity for manufacturing complex products like ships and aircrafts with long manufacturing cycle times. The project activities are interlinked through numerous precedence and succession rules. Given these interdependencies, it is difficult to ascertain the durations of the activities precisely at planning stage and creates uncertainty in the requirement dates of the items. The lead times of items are also not known accurately and result in uncertainty in the availability dates. Consequently, inventory holding and shortage costs incurred cannot be crisply defined. The objective of this paper is to develop a procurement scheduling model considering the above uncertainties.We have developed a method to calculate fuzzy holding and shortage costs and used those as fuzzy cost coefficients in the procurement scheduling model. It minimizes the sum of the above costs under budget constraints and generates optimal ordering schedule. It is applied for procurement scheduling of a real ship building project. Two types of sensitivity analyses were performed: first to understand the effect of variation of degree of uncertainty on total cost and on stage budget requirements and second to study the effect of changes in allocated stage budget parameters on total cost. The results indicate that total cost can be reduced significantly if stage wise budgets are determined considering the uncertainties rather than allocating budget upfront and treating them as constraints. The sensitivity analyses performed, helps in identifying the most sensitive stage of the project and determine the ranges in which stage-wise budgets can be varied.     

The lot size-reorder level inventory system with customers impatience functions

In this paper, we study a continuous review inventory model with deterministic demand. The model allows shortages, which are partially backlogged. The backlogging is characterized using an approach in which customers are considered impatient. Total profit function is developed using three general costs: holding cost, order cost and shortage cost. Holding cost is based on average stocks and order cost is fixed per replenishment. In shortage cost, we include three significant costs: the unit backorder cost (depending on the shortage time), the goodwill cost (constant) and the opportunity cost. A general approach is presented to determine the economic lot size, the reorder level and the minimum total inventory cost. We consider two customers impatience functions to illustrate the application of the procedure. This paper extends several models studied by other authors.     

Integrated optimization of inventory-distribution systems by random local search and a genetic algorithm

A new model and its solution procedure for the commodity distribution system consisting of distribution centers and consumer points are discussed. Demand is assumed to be a random variable that obeys a known, stationary probability distribution. An integrated optimization model is built where both the order-up-to-R policy, which is one of the typical inventory policies for periodic review models, and the transportation problem are considered simultaneously. The assignment of consumer points to distribution centers is not fixed. The problem is to determine the target inventory and the transportation quantity in order to minimize the expectation of the sum of inventory related costs and transportation costs. Simulation and linear programming are used to calculate the expected costs, and a random local search method is developed in order to determine the optimum target inventory. A genetic algorithm is also tested and compared with the proposed random local search method. The model and effectiveness of the proposed solution procedure are clarified by computational experiments.     

Multi-product production quantity model with repair failure and partial backordering

In this paper, a production quantity model with random defective items, service level constraints and repair failure is studied. The existence of only one machine results in limited production capacity and partial backordering. The aim of this research is to determine the optimal cycle length, optimal production quantity and optimal backordered quantity of each product such that the expected total cost (holding, shortage, production, setup, defective items and repair costs) is minimized. Two numerical examples and sensitivity analysis are provided to illustrate the practical usage of the proposed method.     

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.     

An integrated replenishment model with quantity discounts, reentry and downward substitution for control wafers

This paper considers control wafers replenishment problem in wafer fabrication factories. A dynamic lot-sizing replenishment problem with reentry and downward substitution is examined in a pulling control production environment. The objective is to set the inventory level so as to minimize the total cost of control wafers, where the costs include order cost, purchase cost, setup cost, production cost and holding cost, while maintaining the same level of production throughput. In addition, purchase quantity discounts and precise inventory level are considered in the replenishment model. The control wafers replenishment problem is first constructed as a network, and is then transformed into a mixed integer programming model. Lastly, an efficient heuristic algorithm is proposed for solving large-scale problems. A numerical example is given to illustrate the practicality for empirical investigation. The results demonstrate that the proposed mixed integer programming model and the heuristic algorithm are effective tools for determining the inventory level of control wafers for multi-grades in multi-periods.     

Optimal replenishment policies with allowable shortages for a product life cycle

In this paper, we investigate replenishment policies with allowable shortages by considering a general, time-varying, continuous, and deterministic demand function for a product life cycle. The objective is to optimally determine the number of inventory replenishments, the inventory replenishment time points, and the beginning time points of shortages within the product life cycle by minimizing the total relevant costs of the inventory replenishment system. The proposed problem is mathematically formulated as a mixed-integer nonlinear programming model. A complete search procedure is developed to find the optimal solution by employing the properties derived in this paper and the well-known Nelder–Mead algorithm. Also, several numerical examples and the corresponding sensitivity analyses are carried out to illustrate the features of our model by utilizing the search procedure developed in this paper.