A single period inventory model with a truncated normally distributed fuzzy random variable demand

In this article, a single period inventory model has been considered in the mixed fuzzy random environment by assuming the annual customer demand to be a fuzzy random variable. Since assuming demand to be normally distributed implies that some amount of demand information is being automatically taken to be negative, the model has been developed for two cases, using the non-truncated and the truncated normal distributions. The problem has been developed to represent scenarios where the aim of the decision-maker is to determine the optimal order quantity such that the expected profit is greater than or equal to a predetermined target. This ‘greater than or equal to’ inequality has been modelled as a fuzzy inequality and a methodology has been developed to this effect. This methodology has been illustrated through a numerical example.     

Mixture inventory model involving variable lead time and controllable backorder rate

This paper allows the backorder rate as a control variable to widen applications of Ouyang et al.'s model [J. Oper. Res. Soc. 47 (1996) 829]. In this study, we assume that the backorder rate is dependent on the length of lead time through the amount of shortages. We discuss two models that are perfect and partial information about the lead time demand distribution, that is, we first assume that the lead time demand follows a normal distribution, and then remove this assumption by only assuming that the first and second moments of the probability distribution of lead time demand are known. For each case, we develop an algorithm to find the optimal ordering strategy. Three numerical examples are given to illustrate solution procedure.     

A discrete-in-time deteriorating inventory model with time-varying demand,variable deterioration rate and waiting-time-dependent partial backlogging

A new inventory system is considered for deteriorating items in which unsatisfied demands are partially backlogged depending on the waiting time until the next replenishment and deterioration rates are constant, but different from period to period. Time is considered as a discrete variable, which better represents some real-life scenarios. An optimal solution can be easily derived. A sensitivity study is provided through substantial numerical experiments to illustrate the use and robustness of the proposed model. The sensitivity analysis reveals some useful and significant insights.     

Two-warehouse partial backlogging inventory model for deteriorating items with linear trend in demand under inflationary conditions

In today's business transactions, there are various reasons, namely, bulk purchase discounts, re-ordering costs, seasonality of products, inflation induced demand, etc., which force the buyer to order more than the warehouse capacity. Such situations call for additional storage space to store the excess units purchased. This additional storage space is typically a rented warehouse. Inflation plays a very interesting and significant role here: It increases the cost of goods. To safeguard from the rising prices, during the inflation regime, the organisation prefers to keep a higher inventory, thereby increasing the aggregate demand. This additional inventory needs additional storage space, which is facilitated by a rented warehouse. Ignoring the effects of the time value of money and inflation might yield misleading results. In this study, a two-warehouse inventory model with linear trend in demand under inflationary conditions having different rates of deterioration has been developed. Shortages at the owned warehouse are also allowed subject to partial backlogging. The solution methodology provided in the model helps to decide on the feasibility of renting a warehouse. Finally, findings have been illustrated with the help of numerical examples. Comprehensive sensitivity analysis has also been provided.     

A periodic review inventory model involving variable lead time with a service level constraint

In this study, we consider a mixture periodic review inventory model in which both the lead time and the review period are considered as decision variables. Instead of having a stock-out term in the objective function, a service level constraint is added to the model. In our paper, we first assume that the protection interval (i.e. the review period plus the lead time) demand follows a normal distribution, and then we relax this assumption and only assume that the first two moments of the protection interval demand are given. For each case, we develop an algorithm to find the optimal review period and optimal lead time. Furthermore, a sensitivity analysis is also performed.     

A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand

In a recent paper, Ouyang and Wu applied the minimax decision approach to solve a continuous review mixed inventory model in which the lead time demand distribution information is unknown but the annual demand is fixed and given. However, in the practical situation, the annual demand probably incurs disturbance due to various uncertainties. In this article, we attempt to modify Ouyang and Wu's model by considering two fuzziness of annual demand (i.e., fuzzy number of annual demand and statistic-fuzzy number of annual demand) and to investigate a computing schema for the continuous review inventory model in the fuzzy sense. We give an algorithm procedure to obtain the optimal ordering strategy for each case.Scope and purposeIn most of the early literature dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Recently, some researchers (e.g., Liao and Shyu, Ben-Daya and Raouf, and Ouyang and Wu) incorporated the crashing lead time idea to continuous review inventory models, in which the annual demand is given and fixed. However, in the real situation, the annual demand will probably have a little disturbance due to various uncertainties. The purpose of this article is to modify the Ouyang and Wu's model to accommodate this reality, specifically, we apply the fuzzy set concepts to deal with the uncertain annual demand. We first consider a case where the annual demand is treated as the triangular fuzzy number. Then, we employ the statistical method to construct a confidence interval for the annual demand, and through it to establish the corresponding fuzzy number (namely, the statistic-fuzzy number). For each fuzzy case, we investigate a computing schema for the new model and develop an algorithm to find the optimal ordering strategy.     

Two parameter tuned multi-objective evolutionary algorithms for a bi-objective vendor managed inventory model with trapezoidal fuzzy demand

This paper presents a bi-objective vendor managed inventory (BOVMI) model for a supply chain problem with a single vendor and multiple retailers, in which the demand is fuzzy and the vendor manages the retailers’ inventory in a central warehouse. The vendor confronts two constraints: number of orders and available budget. In this model, the fuzzy demand is formulated using trapezoidal fuzzy number (TrFN) where the centroid defuzzification method is employed to defuzzify fuzzy output functions. Minimizing both the total inventory cost and the warehouse space are the two objectives of the model. Since the proposed model is formulated into a bi-objective integer nonlinear programming (INLP) problem, the multi-objective evolutionary algorithm (MOEA) of non-dominated sorting genetic algorithm-II (NSGA-II) is developed to find Pareto front solutions. Besides, since there is no benchmark available in the literature to validate the solutions obtained, another MOEA, namely the non-dominated ranking genetic algorithms (NRGA), is developed to solve the problem as well. To improve the performances of both algorithms, their parameters are calibrated using the Taguchi method. Finally, conclusions are made and future research works are recommended.     

An alternative simple solution algorithm of an inventory model with fixed and variable lead time crash costs under unknown demand distribution

Inventory models with controllable lead time both for known and unknown demand distributions have been proposed in the literature. A model is useless unless it is formulated correctly and feasible. A simple solution procedure of a model also plays an important role in its application. This article highlights an erroneous formulation of an inventory model developed with fixed and variable lead time crash costs under unknown demand distribution, and also demonstrates its infeasibility. To attain feasibility we extend the model to include a constraint. Then, we present an alternative simple solution technique of the modified model and carry out a comparative study on a numerical example to show its potential significance.     

A mixed inventory model with variable lead time and random back-order rate

In this paper, we study the determination of the optimal lead time, reorder point and order quantity considering that the back-order probability of a demand made during a stock-out period depends on the interval from the moment in which the order is placed until the next replenishment. Two models are analysed for the specification of the back-order probability: exponential functions and piecewise constant functions. The distribution of the lead time demand is assumed to be Poisson. An algorithm for the determination of the optimal order quantity, reorder point and lead time is given. A numerical example is presented to illustrate the results.     

Two-warehouse inventory model with ramp-type demand and partially backlogged shortages

In this article, we consider an inventory model for items that are stored in two-warehouses when demand is a general ramp-type function of time. Shortages are allowed and a constant fraction of shortages is backlogged. The existence and uniqueness of optimal solution is proved for both – the single-warehouse and the two-warehouse models. An algorithm is developed to facilitate the choice between the two-warehouse and the single-warehouse systems and hence to obtain the optimal replenishment policy. Numerical examples are presented. Sensitivity analysis with respect to the parameters of the model is performed.     

Computational procedure of optimal inventory model involving controllable backorder rate and variable lead time with defective units

This article considers that the number of defective units in an arrival order is a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate be a control variable. In addition, we assume that the lead time demand follows a mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, three numerical examples are also given to illustrate the results.     

Optimal pricing and lot-sizing for perishable inventory with price and time dependent ramp-type demand

Product perishability is an important aspect of inventory control. To minimise the effect of deterioration, retailers in supermarkets, departmental store managers, etc. always want higher inventory depletion rate. In this article, we propose a dynamic pre- and post-deterioration cumulative discount policy to enhance inventory depletion rate resulting low volume of deterioration cost, holding cost and hence higher profit. It is assumed that demand is a price and time dependent ramp-type function and the product starts to deteriorate after certain amount of time. Unlike the conventional inventory models with pricing strategies, which are restricted to a fixed number of price changes and to a fixed cycle length, we allow the number of price changes before as well as after the start of deterioration and the replenishment cycle length to be the decision variables. Before start of deterioration, discounts on unit selling price are provided cumulatively in successive pricing cycles. After the start of deterioration, discounts on reduced unit selling price are also provided in a cumulative way. A mathematical model is developed and the existence of the optimal solution is verified. A numerical example is presented, which indicates that under the cumulative effect of price discounting, dynamic pricing policy outperforms static pricing strategy. Sensitivity analysis of the model is carried out.     

Quality improvement and backorder price discount under controllable lead time in an inventory model

The proposed study investigates a continuous review inventory model with order quantity, reorder point, backorder price discount, process quality, and lead time as decision variables. An investment function is used to improve the process quality. Two models are developed based on the probability distribution of lead time demand. The lead time demand follows a normal distribution in the first model and in the second model it does not follow any specific distribution but mean and standard deviation are known. We prove two lemmas to obtain optimal solutions for the normal distribution model and distribution free model. Finally, some numerical examples are given to illustrate the model.     

Continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand and different decision situations

In this paper, continuous review inventory models in which a fraction of demand is backordered and the remaining fraction is lost during the stock out period are considered under fuzzy demands. In order to find the optimal decision under different situations, two decision methods are proposed. The first one is finding a minimum value of the expected annual total cost, and the second one is maximizing the credibility of an event that the total cost in the planning periods does not exceed a certain budget level. For the first decision method, an approach of ranking fuzzy numbers by their possibilistic mean value is adopted to achieve the optimal solution. For the second one, the technique of fuzzy simulation and differential evolution algorithms are integrated to design hybrid intelligent algorithms to solve the fuzzy models. Subsequently, the two decision models are compared and some advices about inventory cash flow management are given. Further, sensitivity analysis is conducted to give more general situations to illustrate the rationality of the management advices.     

Two-echelon multi-product multi-constraint product returns inventory model with permissible delay in payments and variable lead time

Supply chain is not limited to delivering products to the end-costumers since the defective products that are returned back to the producers by the consumers. The producers should be superior knowledge to utilize the return products effectively so as to maintain our natural resources and to provide better service to customers. In this paper, a distributor and a warehouse consisting of a serviceable part and a recoverable part supply chain problem is considered in which there are several products, the distributor has limited space capacity and budget to purchase all products. In this supply chain, the defective products are returned back to the warehouse by the distributor and the warehouse recovered those defective products into perfect products having the same value as the procured products. The lead-time of receiving products from a warehouse to a distributor is a variable which is controllable by adding extra crashing cost. For each product, a fraction of the shortage is backordered and the rest are lost. A mathematical model is employed in this study for optimizing the order quantity, lead time and total number of deliveries with the objective of minimizing system total cost. We show that the model of this problem is a constrained non-linear programme and present a simple Lagrangian multiplier technique to solve it. Numerical and sensitivity analysis are given to show the applicability of the proposed model in real-world product returns inventory problems.     

Performance evaluation of a two‐echelon supply chain with stochastic demand,lost sales,and Coxian‐2 phase replenishment times

This paper deals with a two‐echelon supply chain comprising a retailer and manufacturer. The retailer faces Poisson demand and follows a (S, s) continuous review inventory policy. The manufacturer produces and ships the retailer's orders with random delay that follows the Coxian‐2 distribution. Assuming lost sales at the retailer and infinite capacity at the manufacturer, we try to explore the performance of the supply chain system. The system is modeled as a continuous‐time Markov process with discrete space. The structure of the transition matrices of these specific systems is categorized as block‐partitioned, and a computational algorithm generates the matrices for different values of system characteristics. The proposed algorithm allows the calculation of performance measures—fill rate, cycle times, average inventory (work in progress [WIP])—from the derivation of the steady‐state probabilities. Moreover, expressions for the holding costs and shortage costs are derived.     

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.     

随机需求下提前期可控的生产-库存联合优化模型

考虑单供应商和单采购商的生产-库存联合优化问题.假设采购商面临正态需求,供应商的提前期可以控制,并基于此建立供应商与采购商联合期望总成本最小化模型.在所建立的模型中允许采购商缺货,且部分缺货可延期交付,部分缺货发生销售损失;同时考虑运输成本,并假设运输成本依赖于订货量和提前期.给出了求解最优生产批量、最优提前期、最优再订货点和订货量的算法,并通过数值算例进行了说明.     

模糊需求下时间依赖型车辆路径优化

针对客户需求模糊且有时间窗约束的时间依赖型车辆路径问题(TDVRP),基于先预优化后重调度的思想构建模型.在预优化阶段,依据可信性理论构建模糊机会约束优化模型处理客户点模糊需求;针对不同时间段道路的交通情况,采用Ichoua速度时间依赖函数表征车辆的行驶速度,并设计自适应大规模邻域搜索算法(ALNS)对其求解.在重调度阶段,应用随机模拟算法模拟客户点的真实需求,采用点重调度策略对预优化方案进行调整.通过改进的Solomon算例实验验证模型和算法的有效性.研究成果可丰富TDVRP问题的相关研究,为现实配送方案的优化决策提供理论依据.     

Mixture inventory model with back orders and lost sales for variable lead time demand with the mixtures of normal distribution

In recent papers by Ben-Daya and Raouf and by Ouyang et al. a continuous review inventory model is presented in which they considered both the lead time and the order quantity as decision variables. When the demands of the different customers do not have identical lead times, then we cannot use only a distribution (such as Ouyang et al. who used a normal distribution) to describe the demand of the lead time. Hence, we have extended the model of Ouyang et al. by considering the mixtures of normal distribution (see the book by Everitt and Hand). In addition, we also still assume that shortages are allowed. Moreover, the total amount of stock-out is considered as a mixture of back orders and lost sales during the stock-out period. Moreover, we also develop an algorithmic procedure to find the optimal order quantity and optimal lead time; the effects of parameters are also studied.