A two warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time

An optimization inventory policy for a deteriorating item with imprecise lead-time, partially/fully backlogged shortages and price dependent demand is developed under two-warehouse system. For display and storage, the retailer hires one warehouse of finite capacity at market place, treated as own warehouse (OW) and another warehouse of large capacity as it may be required at a distance place from the market, treated as rented warehouse (RW). Holding cost at RW decreases with the increase of distance from the market place. Units are transferred from RW to OW in bulk release pattern and sold from OW. Using the nearest interval approximation method the estimated fuzzy average profit function is defuzzified and transformed to multiple crisp objective functions which are solved by Global Criteria Method. The models are illustrated numerically. Sensitivity of the inventory costs on the location of RW has been depicted graphically. Also loss in profit due to deteriorations for both models have been presented.     

A two warehouse inventory model for deteriorating items with a linear trend in demand and shortages

In this paper, we develop a deterministic inventory model with two warehouses (one is the existing storage known as own warehouse (OW) and the other is hired on rental basis known as rented warehouse (RW). The model allows different levels of item deterioration in both warehouses. The demand rate is supposed to be a linear (increasing) function of time and the replenishment rate is infinite. The stock is transferred from RW to OW in continuous release pattern and the associated transportation cost is taken into account. Shortages in OW are allowed and excess demand is backlogged. For the general model, we give the equations for the optimal policy and cost function and we discuss some special cases. A numerical example is given to illustrate the solution procedure of the model. Finally, based on this example, we conduct a sensitivity analysis of the model.     

Investigation of two-warehouse inventory problems in interval environment under inflation via particle swarm optimization

ABSTRACT

In this paper, a two-warehouse inventory problem has been investigated under inflation with different deterioration effects in two separate warehouses (rented warehouse, RW, and owned warehouse, OW). The objective of this investigation is to determine the lot-size of the cycle of the two-warehouse inventory system by minimizing the average cost of the system. Considering different inventory policies, the corresponding models have been formulated for linear trend in demand and interval valued cost parameters. In OW, shortages, if any, are allowed and partially backlogged with a variable rate dependent on the duration of the waiting time up to the arrival of the next lot. The corresponding optimization problems have been formulated as non-linear constrained optimization problems with interval parameters. These problems have been solved by an efficient soft computing method, viz. practical swarm optimization. To illustrate the model, a numerical example has been solved with different partially backlogging rates. Then to study the effect of changes of different system parameters on the optimal policy, sensitivity analyses have been carried out graphically by changing one parameter at a time and keeping the others at their original values. Finally, a fruitful conclusion has been reached regarding the selection of an appropriate inventory policy of the two-warehouse system.     

Inventory model with deteriorating items,ramp-type demand and partially backlogged shortages for a two warehouse system

In this paper, we consider a two warehouse inventory model, an owned one (OW) and a rented one (RW). Inventory deteriorates in the two warehouses at different constant rates, demand rate is a general ramp-type function of time and shortages are partially backlogged at a constant rate. Existence and uniqueness of the optimal solution is discussed. An algorithm is developed to obtain the overall optimal replenishment policy, which would enable the manager to decide upon the feasibility of renting a warehouse. The dynamics of the model and application of the algorithm are demonstrated through numerical examples. Sensitivity analysis is conducted with respect to model parameters and some important observations are drawn.     

A multiobjective model of wholesaler-retailers' problem via genetic algorithm

In the existing literature, most of the purchasing models were developed only for retailers problem ignoring the constraint of storage capacity of retailers shop/showroom. In this paper, we have developed a deterministic model of wholesaler-retailers' problem of single product. The storage capacity of wholesaler's warehouse/showroom and retailers' showroom/shop are assumed to be finite. The items are transported from wholesaler's warehouse to retailers' Own Warehouse (OW) in a lot. The customer's demand is assumed to be displayed inventory level dependent. Demands are met from OW and that spaces of OW will immediately be filled by shifting the same amount from the Rented Warehouse (RW) till the RW is empty. The time duration between selling from OW and filling up its space by new ones from RW is negligible. According to relative size of the retailers' existing (own) warehouse capacity and the demand factors, different scenarios are identified. Our objectives are to optimize the cost functions of wholesaler and two retailers separately. To solve this problem, a real coded Genetic Algorithm (GA) with roulette wheel selection/reproduction, whole arithmetic crossover and non-uniform mutation is developed. Finally a numerical example is presented to illustrate the results for different scenarios. To compare the results of GA, Generalised Reduced Gradient Method has been used for the problem. Also, a sensitivity analysis has been performed to study the variations of the optimal average cost with respect to the different parameters.     

一个带有增加量价格折扣的两货栈库存模型

已有的确定性两货栈(其中一个是自己货栈(OW);另一个是租用货栈(RW))库存模型通常不考虑增加量价格折扣,然而在实际生活中,增加量价格折扣却是促使库存管理者加大订货量的一个重要原因.本文通过考虑增加量价格折扣而将两货栈系统作了进一步扩展,在采用间隔式运输模式运送RW的物品到OW的情形下,建立了一个带有增加量价格折扣并允许短缺的两货栈库存模型,提供了一种寻求最优库存策略的简单方法.     

A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment

For the capacity of any warehouse is limited, it has to rent warehouse (RW) for storing the excess units over the fixed capacity W of the own warehouse (OW) in practice. The RW is assumed to offer better preserving facilities than the OW resulting in a lower rate of deterioration and is assumed to charge higher holding cost than the OW. In this paper, a two-warehouse inventory model for deteriorating items is considered with constant demand under conditionally permissible delay in payment. The purpose of this study is to find the optimal replenishment policies for minimizing the total relevant inventory costs. Useful theorems to characterize the optimal solutions have been derived. Furthermore, numerical examples are provided to illustrate the proposed model, sensitivity analysis of the optimal solutions with respect to major parameters is carried out and some managerial inferences are obtained.     

An application of Genetic Algorithm in solving an inventory model with advance payment and interval valued inventory costs

The purpose of this research is to solve the mixed integer constrained optimization problem with interval coefficient by a real-coded genetic algorithm (RCGA) with ranking selection, whole arithmetical crossover and non-uniform mutation for non-integer decision variables. In the ranking selection, as well as in finding the best solution in each generation of RCGA, recently developed modified definitions of order relations between interval numbers with respect to decision-making are used. Also, for integer decision variables, new types of crossover and mutation are introduced. This methodology is applied to solve a finite time horizon inventory model with constant lead-time, uniform demand rate and a discount by paying an amount of money in advance. Moreover, different inventory costs are considered to be interval valued. According to the consumption of items during lead-time and reorder level, two cases may arise. For each case, the mathematical model becomes a constrained nonlinear mixed integer problem with interval objective. Our objective is to determine the optimal number of cycles in the finite time horizon, lot-size in each cycle and optimal profit. The model is illustrated with some numerical examples and sensitivity analysis has been done graphically with the variation of different inventory parameters.     

A two-echelon inventory model for a deteriorating item with stock-dependent demand,partial backlogging and capacity constraints

This study investigates a two-echelon supply chain model for deteriorating inventory in which the retailer’s warehouse has a limited capacity. The system includes one wholesaler and one retailer and aims to minimise the total cost. The demand rate in retailer is stock-dependent and in case of any shortages, the demand is partially backlogged. The warehouse capacity in the retailer (OW) is limited; therefore the retailer can rent a warehouse (RW) if needed with a higher cost compared to OW. The optimisation is done from both the wholesaler’s and retailer’s perspectives simultaneously. In order to solve the problem a genetic algorithm is devised. After developing a heuristic a numerical example together with sensitivity analysis are presented. Finally, some recommendations for future research are presented.     

The production–inventory problem of a product with time varying demand,production and deterioration rates

In this paper, we consider the production–inventory problem in which the demand, production and deterioration rates of a product are assumed to vary with time. Shortages of a cycle are allowed to be backlogged partially. Two models are developed for the problem by employing different modeling approaches over an infinite planning horizon. Solution procedures are derived for determining the optimal replenishment policies. A procedure to find the near-optimal operating policy of the problem over a finite time horizon is also suggested.     

An inventory model of deteriorating items with lot-size dependent replenishment cost and a linear trend in demand

This paper presents an inventory model for deteriorating items over a finite time horizon where the demand increases linearly with time. The method is developed by assuming that the successive replenishment cycle lengths are the same. Many O.R. scientists/researchers obtained an optimal replenishment schedule where the replenishment cost is constant in each cycle length over the finite time horizon. In this paper, we relax the assumption of fixed replenishment cost. The replenishment cost per replenishment is taken to be linearly dependent on the lot-size of that replenishment. Shortages are allowed and are fully backlogged. As a special case, the results for the model without shortages are derived. Finally, two numerical examples are presented to illustrate the model.     

Credit financing in a two-warehouse environment for deteriorating items with price-sensitive demand and fully backlogged shortages

In the current global market, organizations use many promotional tools in order to increase their sales. One such tool is permissible delay in payments, i.e., the buyer does not have to pay for the goods purchased immediately rather can defer the payment for a prescribed period given by the supplier. This phenomenon motivates the retailer/buyer to order a large inventory lot so as to take full benefit of credit period. But the well decorated showroom (OW) with modern facilities has a limited storage capacity. Thus the retailer has to hire a rented warehouse to store the excess units. In this scenario, retailer usually adopts two types of dispatch policy: FIFO & LIFO, depending upon the situation, e.g., nature of items/deteriorating items, location of warehouse. Further in order to survive in the market, the retailer dynamically adjusts the prices of the goods to boost the demand and enhance the revenues.In the light of these facts, this paper develops an inventory model for deteriorating items with price-sensitive demand under permissible delay in payment in a two warehouse environment. Shortages are allowed and fully backlogged. The objective of this study is to find the optimal inventory and pricing policies so as to maximize the total average profit. Further, the different trade credit scenario has been exhibited with the help of a numerical example. A comprehensive sensitivity analysis has also been carried out to advocate the implication of FIFO and LIFO dispatch policy.     

An inventory sharing and allocation method for a multi-location service parts logistics network with time-based service levels

We consider a model to allocate stock levels at warehouses in a service parts logistics network. The network is a two-echelon distribution system with one central warehouse with infinite capacity and a number of local warehouses, each facing Poisson demands from geographically dispersed customers. Each local warehouse uses a potentially different base stock policy. The warehouses are collectively required to satisfy time-based service targets: Certain percentages of overall demand need to be satisfied from facilities within specified time windows. These service levels not only depend on the distance between customers and the warehouses, but also depend on the part availabilities at the warehouses. Moreover, the warehouses share their inventory as a way to increase achieved service levels, i.e., when a local warehouse is out of stock, demand is satisfied with an emergency shipment from another close-by warehouse. Observing that the problem of finding minimum-cost stock levels is an integer non-linear program, we develop an implicit enumeration-based method which adapts an existing inventory sharing model from the literature, prioritizes the warehouses for emergency shipments, and makes use of a lower bound. The results show that the proposed inventory sharing strategy results in considerable cost reduction when compared to the no-sharing case and the method is quite efficient for the considered test problems.     

A displayed inventory model with L–R fuzzy number

In this study, we formulate a multi-item displayed inventory model under shelf-space constraint in fuzzy environment. Here demand rate of an item is considered as a function of the displayed inventory level. The problem is formulated to maximize average profit. In real life situation, the goals and inventory parameters are may not precise. Such type of uncertainty may be characterized by fuzzy numbers. Here, the constraint goal and the inventory cost parameters are assumed to be triangular shaped fuzzy numbers with different types of left and right membership functions. The fuzzy numbers are then approximated to a nearest interval number. Using arithmetic of interval numbers, the problem is described as a multi-objective inventory problem. The problem is then solved by fuzzy geometric programming approach. Finally a numerical example is given to illustrate the problem.     

On a finite horizon production lot size inventory model for deteriorating items: An optimal solution

A generalized production lot size inventory model for deteriorating items over a finite planning horizon is considered. The demand, production, and deteriorating rates are assumed to be known and continuous functions of time. Shortages are allowed and completely backlogged. The conditions under which the system total cost attains its (unique) global minimum are derived. An example which illustrate the applicability of theoretical results is also introduced.     

销售率线性依赖瞬时库存水平的两货栈决策模型

在成熟期的存货影响销售环境下,考虑销售率线性依赖瞬时库存水平,不允许缺货,研究了一类非变质性物品的两货栈库存决策问题.建立了以系统平均总利润最大为目标的决策模型,分析了系统最优库存策略的存在性和唯一性,并给出了求解模型的有效方法.分析结果表明,库存管理者利用租用货栈进行订货决策时,除了要充分考虑企业自身的库存容量外,还取决于自有货栈产品相关参数对库存系统绩效的边际贡献率.     

An interactive method for inventory control with fuzzy lead-time and dynamic demand

Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise demand, lead-time and inventory costs have been formulated and an inventory policy is proposed to minimize the cost using man–machine interaction. Here, demand increases with time at a decreasing rate. The imprecise parameters of lead-time, inventory costs and demand are expressed through linear/non-linear membership functions. These are represented by different types of membership functions, linear or quadratic, depending upon the prevailing supply condition and marketing environment. The imprecise parameters are first transformed into corresponding interval numbers and then following the interval mathematics, the objective function for average cost is changed into respective multi-objective functions. These functions are minimized and solved for a Pareto-optimum solution by interactive fuzzy decision-making procedure. This process leads to man–machine interaction for optimum and appropriate decision acceptable to the decision maker’s firm. The model is illustrated numerically and the results are presented in tabular forms.     

Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate

In this paper, a deterministic inventory model for deteriorating items with two warehouses is developed. A rented warehouse is used when the ordering quantity exceeds the limited capacity of the owned warehouse, and it is assumed that deterioration rates of items in the two warehouses may be different. In addition, we allow for shortages in the owned warehouse and assume that the backlogging demand rate is dependent on the duration of the stockout. We obtain the condition when to rent the warehouse and provide simple solution procedures for finding the maximum total profit per unit time. Further, we use a numerical example to illustrate the model and conclude the paper with suggestions for possible future research.     

A deterministic order level inventory model for deteriorating items with two storage facilities

In this paper a deterministic inventory model is developed for a single deteriorating item which is stored in two different warehouses. A rented warehouse is used to store the excess units over the fixed capacity W of the own warehouse. The rented warehouse is assumed to charge higher unit holding cost than the own warehouse, but to offer a better preserving facility resulting in a lower rate of deterioration for the goods than the own warehouse. The optimal stock level for the beginning of the period is found and the model developed is shown to agree with the order level model for non deteriorating items with a single storage facility. An illustration to show the applicability of the model is also presented.     

A multi-objective wholesaler–retailers inventory-distribution model with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number

This paper develops a single wholesaler and multi retailers mixture inventory distribution model for a single item involving controllable lead-time with backorder and lost sales. The retailers purchase their items from the wholesaler in lots at some intervals throughout the year to meet the customers’ demand. Not to loose the demands, the retailers offer a price discount to the customers on the stock-out items. Here, it is assumed that the lead-time demands of retailers are uncertain in both stochastic and fuzzy sense, i.e., these are simultaneously random and imprecise. To implement this behavior of the lead-time demands, at first, these demands are assumed to be random, say following a normal distribution. With these random demands, the expected total cost for each retailer is obtained. Now, the mean lead-time demands (which are crisp ones) of the retailers are fuzzified. This fuzzy nature of the lead-time demands implies that the annual average demands of the retailers must be fuzzy numbers, suppose these are triangular fuzzy numbers. Using signed distance technique for defuzzification, the estimate of total costs for each retailer is derived. Therefore, the problem is reduced to optimize the crisp annual costs of wholesaler and retailers separately. The multi-objective model is solved using Global Criteria method. Numerical illustrations have been made with the help of an example taking two retailers into consideration. Mathematical analyses have been made for global pareto-optimal solutions of the multi-objective optimization problem. Sensitivity analyses have been made on backorder ratio and pareto-optimal solutions for wholesaler and different retailers are compared graphically.