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.     

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.     

Discounted multi-item inventory model via genetic algorithm with Roulette wheel selection,arithmetic crossover and uniform mutation in constraints bounded domains

Genetic Algorithm (GA) with different logic structures for price breaks has been developed and implemented for a multi-item inventory control system of breakable items like the items made of glass, mud, porcelain, etc. with all unit discount (AUD), incremental quantity discount (IQD) and a combination of these discounts. Here, AUD and IQD on purchasing price with two price breaks are allowed. Also, demand and breakability of the items are stock-dependent. Shortages are not allowed. Replenishment is instantaneous. Selling price is a mark-up of the purchasing cost. For storage, warehouse capacity is limited. For the present model, GA has been developed in real code representation using Roulette wheel selection, arithmetic crossover and uniform mutation. This algorithm has been implemented successfully to find the optimum order quantities for the above inventory control system to achieve the maximum possible profit. The algorithm and the inventory model have been illustrated numerically and some sensitivity analyses with respect to breakability and demand are presented.     

A breakable multi-item multi stage solid transportation problem under budget with Gaussian type-2 fuzzy parameters

This paper is based on two mathematical models for multi-item multi-stage solid transportation problem with budget on total transportation cost in Gaussian type-2 fuzzy environment considering the fixed opening charge and operating cost in distribution center. The first model is about transportation of breakable/damageable items, and the second one considers non breakable/damageable items. The main aspect here is to develop the mathematical formulation of multi stage related solid transportation problem where several items are available for transportation. In order to deal with the Gaussian type-2 fuzziness, two chance-constrained programming models are developed based on generalized credibility measures for the objective function as well as the constraints sets with the help of the CV-based reductions method. Finally the reduced model is turned into its equivalent parametric programming problem. The problem is of high complexity and is difficult to find the optimal solution by any classical method and hence a time and space based meta-heuristic Genetic Algorithm has been proposed. Also the equivalent crisp models are solved using GA and LINGO 13.0 and after comparison, GA results are better. The proposed models and techniques are finally illustrated by providing numerical examples. Some sensitivity analysis and particular cases are presented and discussed. Degrees of efficiency is also evaluated for both the techniques.     

A multi-item transportation problem with fuzzy tolerance

This paper presents the recently introduced modified subgradient method for optimization and its effectiveness in a fuzzy transportation model. Here a multi-item balanced transportation problem (MIBTP) is formulated where unit transportation costs are imprecise. Also available spaces and budgets at destinations are limited but imprecise. The objective is to find a shipment schedule for the items that minimizes the total cost subjected to imprecise warehouse and budget constraints at destinations. The proposed model is reduced to a multi-objective optimization problem using tolerances, then to a crisp single-objective one using fuzzy non-linear programming (FNLP) technique and Zimmermann's method. The above fuzzy MIBTP is also reduced to another form of deterministic one using modified sub-gradient method (MSM). These two crisp optimization problems are solved by Genetic Algorithm (GA). As an extension, fuzzy multi-item balanced solid transportation problems (STPs) with and without restrictions on some routes and items are formulated and reduced to deterministic ones following FNLP and Zimmermann's methods. These models are also solved by GA. Models are illustrated numerically, optimum results of fuzzy MIBTP from two deductions are compared. Results are also presented for different GA parameters.     

The (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">l</Emphasis>) model for stochastic demand under Intuitionistic fuzzy aggregation with Bonferroni mean

This paper investigates a hill type economic production-inventory quantity (EPIQ) model with variable lead-time, order size and reorder point for uncertain demand. The average expected cost function is formulated by trading off costs of lead-time, inventory, lost sale and partial backordering. Due to the nature of the demand function, the frequent peak (maximum) and valley (minimum) of the expected cost function occur within a specific range of lead time. The aim of this paper is to search the lowest valley of all the valley points (minimum objective values) under fuzzy stochastic demand rate. We consider Intuitionistic fuzzy sets for the parameters and used Intuitionistic Fuzzy Aggregation Bonferroni mean for the defuzzification of the hill type EPIQ model. Finally, numerical examples and graphical illustrations are made to justify the model.     

Inventory of damageable items with variable replenishment and unit production cost via simulated annealing method

Items made of glass, ceramic, etc. are normally stored in stacks and get damaged during the storage due to the accumulated stress of heaped stock. The researchers have overlooked the inventory problems for this type of items. Again the classical iterative optimization techniques very often stuck to the local optimum present in the search space. This is one of the hindrances in optimizing the non-linear problems.Annealing is the physical process of heating up a solid until it melts followed by cooling it down until it crystallizes into a state with perfect lattice. Following this physical phenomenon, recently a soft computing method, Simulated Annealing (SA), has been developed to find the global optimum for a complex cost surface through stochastic search process.In this paper, a deterministic inventory model of a damageable item is developed with variable replenishment rate and unit production cost. Here replenishment rate and unit production cost are dependent on demand. Demand and damageability are stock dependent. This dependency may be linear or non-linear. The optimum inventory level is evaluated by the profit maximization principle through SA algorithm. The model is illustrated numerically with different forms of demand and damage functions.     

Robust fuzzy control for uncertain discrete-time nonlinear Markovian jump systems without mode observations

This paper studies the robust fuzzy control problem of uncertain discrete-time nonlinear Markovian jump systems without mode observations. The Takagi and Sugeno (T-S) fuzzy model is employed to represent a discrete-time nonlinear system with norm-bounded parameter uncertainties and Markovian jump parameters. As a result, an uncertain Markovian jump fuzzy system (MJFS) is obtained. A stochastic fuzzy Lyapunov function (FLF) is employed to analyze the robust stability of the uncertain MJFS, which not only is dependent on the operation modes of the system, but also directly includes the membership functions. Then, based on this stochastic FLF and a non-parallel distributed compensation (non-PDC) scheme, a mode-independent state-feedback control design is developed to guarantee that the closed-loop MJFS is stochastically stable for all admissible parameter uncertainties. The proposed sufficient conditions for the robust stability and mode-independent robust stabilization are formulated as a set of coupled linear matrix inequalities (LMIs), which can be solved efficiently by using existing LMI optimization techniques. Finally, it is also demonstrated, via a simulation example, that the proposed design method is effective.     

需求随机的变质性产品生产计划模型

研究了能力约束的有限计划展望期生产计划问题,各周期的需求随机,库存产品存在变质且变质率为常数。建立了问题的期望值模型,目标函数为极小化生产准备成本、生产成本、库存成本的期望值。提出了随机模拟、遗传算法和启发式算法相结合的求解算法。用数值实例对模型和算法进行了验证,优化结果表明模型和算法是有效的。     

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.     

A multiple-depot,multiple-vehicle,location-routing problem with stochastically processed demands

We formulate a multiple-depot, multiple-vehicle, location-routing problem with stochastically processed demands, which are defined as demands that are generated upon completing site-specific service on their predecessors. When a factory is re-supplied with manufacturing materials, for example, demand for raw materials surfaces only after the existing inventory has been exhausted. A special separable case of the problem was solved, wherein probable demands are estimated by stochastic processes at the demand nodes (the factories) before the vehicle location-routing decisions. Posterior solutions to the complete 90-day instances of the problem help to gauge the performance of the a priori stochastic model. The 90 day-by-day instances also provide researchers with a benchmark data-set for future experimentation. It was shown that the a priori optimization solution provides a robust location-routing strategy for real-time decision-making in a medical-evacuation case study of the U.S. Air Force. Given this modest success, the same methodology can possibly be applied toward “pure” just-in-time deliveries in supply-chain management, where inventory storage is totally eliminated.Scope and purposeIn resupplying a factory from a central depot, there are several considerations. First, one has to locate the distribution centers. Second, a delivery-vehicle fleet of the right size has to be stationed at each of these centers. Third, deliveries need to be made on a timely basis in response to demands for raw materials. While this problem is well known, only part of it is solved satisfactorily to date. We propose a comprehensive model and a solution method for this class of problems. A unique feature of this model is that we recognize that in today's just-in-time deliveries, one wishes to order exactly what is anticipated to avoid surplus inventory or stock out. However, demands are often highly uncertain since the manufacturing process is plagued with uncertainty. This paper has an innovative way of representing this phenomenon that is a departure from the traditional literature.     

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.     

Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP,SFS, SA,and WCA

A multi-product economic production quantity model with several real-world technical and physical constraints is developed in this paper. The cost function includes ordering, holding, backordering, lost sale, and the cost caused by unused space in the warehouse. The goal is to minimize the total inventory cost, where shortages are allowed and partially backordered with fixed and linear costs. The aim is to determine the length of the inventory cycle, the length of positive inventory period, and the backordering rates of the products during the shortage period in order to minimize the total inventory costs while satisfying all constraints. Due to complexity and non-linearity of the proposed model, sequential quadratic programming (SQP), stochastic fractal search (SFS), simulated annealing (SA), and water cycle algorithm (WCA) are utilized for solution. Ninety numerical examples in small, medium, and large sizes are solved to evaluate the efficiency of the solution methods. The performances of the solution methods are compared statistically. Besides, sensitivity analysis is performed to determine the effect of change in the main parameters of the problem on the objective function value and decision variables.     

A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment

An “economic production lot size” (EPLS) model for an item with imperfect quality is developed by considering random machine failure. Breakdown of the manufacturing machines is taken into account by considering its failure rate to be random (continuous). The production rate is treated as a decision variable. It is assumed that some defective units are produced during the production process. Machine breakdown resulting in idle time of the respective machine which leads to additional cost for loss of manpower is taken into account. It is assumed that the production of the imperfect quality units is a random variable and all these units are treated as scrap items that are completely wasted. The models have been formulated as profit maximization problems in stochastic and fuzzy-stochastic environments by considering some inventory parameters as imprecise in nature. In a fuzzy-stochastic environment, using interval arithmetic technique, the interval objective function has been transformed into an equivalent deterministic multi-objective problem. Finally, multi-objective problem is solved by Global Criteria Method (GCM). Stochastic and fuzzy-stochastic problems and their significant features are illustrated by numerical examples. Using the result of the stochastic model, sensitivity of the nearer optimal solution due to changes of some key parameters are analysed.     

带有时限的多点应急系统库存策略优化研究

主要研究有时限约束的应急供应系统的库存服务水平问题。考虑随机需求下的包含一个中央供应点和多个不同地方供应点的二级系统,当缺货发生时,基于就近原则的转运模式建立了有时限约束的地方供应点库存策略优化模型,在此基础上分析了随机转运和不转运两种模式对模型的影响,最后采用粒子群算法对模型进行求解和分析。结果表明:在有时限约束时,就近原则模式下的库存策略具有较高的服务水平,并且时限约束越紧,其优势越显著。     

Location based treatment activities for end of life products network design under uncertainty by a robust multi-objective memetic-based heuristic approach

Rapid growth in world population and recourse limitations necessitate remanufacturing of products and their parts/modules. Managing these processes requires special activities such as inspection, disassembly, and sorting activities known as treatment activities. This paper proposes a capacitated multi-echelon, multi-product reverse logistic network design with fuzzy returned products in which both locations of the treatment activities and facilities are decision variables. As the obtained nonlinear mixed integer programming model is a combinatorial problem, a memetic-based heuristic approach is presented to solve the resulted model. To validate the proposed memetic-based heuristic method, the obtained results are compared with the results of the linear approximation of the model, which is obtained by a commercial optimization package. Moreover, due to inherent uncertainty in return products, demands of these products are considered as uncertain parameters and therefore a fuzzy approach is employed to tackle this matter. In order to deal with the uncertainty, a stochastic simulation approach is employed to defuzzify the demands, where extra costs due to opening new centers or extra transportation costs may be imposed to the system. These costs are considered as penalty in the objective function. To minimize the resulting penalties during simulation's iterations, the average of penalties is added to the objective function of the deterministic model considered as the primary objective function and variance of penalties are considered as the secondary objective function to make a robust solution. The resulted bi-objective model is solved through goal programming method to minimizing the objectives, simultaneously.     

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.     

Enforcing Service-Level Constraints in Supply Chains With Assembly Operations

We study a feedforward supply network that involves assembly operations. We compute optimal stock levels which minimize inventory costs and maintain stockout probabilities below given desirable levels (service-level constraints). To that end, we develop large deviations approximations for inventory costs and service level constraints and formulate the stock level selection problem as a nonlinear programming problem which can be solved using standard techniques. This results in significant computational savings when compared to exhaustive search using simulation. Our distributional assumptions are general enough to include temporal dependencies in the demand and production processes. We leverage the solution of the inventory control problem in the design of supply contracts under explicit service-level constraints     

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.     

A supplier-selection model with classification and joint replenishment of inventory items

Since inventory costs are closely related to suppliers, many models in the literature have selected the suppliers and also allocated orders, simultaneously. Such models usually consider either a single inventory item or multiple inventory items which have independent holding and ordering costs. However, in practice, ordering multiple items from the same supplier leads to a reduction in ordering costs. This paper presents a model in capacity-constrained supplier-selection and order-allocation problem, which considers the joint replenishment of inventory items with a direct grouping approach. In such supplier-selection problems, the following items are considered: a fixed major ordering cost to each supplier, which is independent from the items in the order; a minor ordering cost for each item ordered to each supplier; and the inventory holding and purchasing costs. To solve the developed NP-hard problem, a simulated annealing algorithm was proposed and then compared to a modified genetic algorithm of the literature. The numerical example represented that the number of groups and selected suppliers were reduced when the major ordering cost increased in comparison to other costs. There were also more savings when the number of groups was determined by the model in comparison to predetermined number of groups or no grouping scenarios.