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.     

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.     

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 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.     

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.     

Two storage inventory problem with dynamic demand and interval valued lead-time over finite time horizon under inflation and time-value of money

A finite time horizon inventory problem for a deteriorating item having two separate warehouses, one is a own warehouse (OW) of finite dimension and other a rented warehouse (RW), is developed with interval-valued lead-time under inflation and time value of money. Due to different preserving facilities and storage environment, inventory holding cost is considered to be different in different warehouses. The demand rate of item is increasing with time at a decreasing rate. Shortages are allowed in each cycle and backlogged them partially. Shortages may or may not be allowed in the last cycle and under this circumstance, there may be three different types of model. Here it is assumed that the replenishment cycle lengths are of equal length and the stocks of RW are transported to OW in continuous release pattern. For each model, different scenarios are depicted depending upon the re-order point for the next lot. Representing the lead-time by an interval number and using the interval arithmetic, the single objective function for profit is changed to corresponding multi-objective functions. These functions are maximized and solved by Fast and Elitist Multi-objective Genetic Algorithm (FEMGA). The models are illustrated numerically and the results are presented in tabular form.     

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.     

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.     

Two-warehouse inventory model with deterioration under FIFO dispatching policy

In most of the literatures on two-warehouse inventory decision models, the last-in-first-out (LIFO) dispatching policy has always been assumed. This presumption, however, is not in line with the actual practice of most business entities. To enhance the freshness of merchandise or goods, businesses commonly follow the first-in-first-out procedure (FIFO). This inconsistency forms the base and main motivation for our research. In this paper, Pakkala and Achary’s two-warehouse LIFO model is first modified and then a FIFO dispatching two-warehouse model with deterioration is proposed. Comparison of the two models indicated that the FIFO model is less expensive to operate than LIFO, if the mixed effects of deterioration and holding cost in RW are less than that of OW. Finally, numerical examples are provided to investigate and examine the impact that various parameters have on policy choice.     

The continuous resupply policy for deteriorating items with stock-dependent observable demand in a two-warehouse and two-echelon supply chain

This paper expands previous work on stock-dependent demand for a retailer with a two-warehouse (OW/RW) situation to the case of deteriorating items and where the retailer seeks to obtain the integrated optimal distribution policy from collaboration with a supplier. Motivated by practical applications and recent literature, a policy is considered whereby products in good order from the retailer’s back-room (RW) are frequently transferred to its capacitated main store OW. Because the demand depends on the stock of good products in the OW, the aim is to keep this stock at its full capacity with products in good condition, and this can be done for as long as the RW stock of good products is positive. A firm’s objective function is the Net Present Value (NPV) of the firm’s future cash-flows. The profit functions are developed for both this continuous resupply policy and the commonly used policy in the OW/RW literature. Numerical examples are included and have been solved with grid search methods. The examples illustrate the benefits of adopting the continuous resupply policy, and also collaboration between the retailer and the wholesaler. Moreover, it is shown how these benefits can be shared by small adjustments to the product’s unit price between the firms.     

The EOQ problem with decidable warehouse capacity: Analysis, solution approaches and applications

The Economic Order Quantity (EOQ) problem is a fundamental problem in supply and inventory management. In its classical setting, solutions are not affected by the warehouse capacity. We study a type of EOQ problem where the (maximum) warehouse capacity is a decision variable. Furthermore, we assume that the warehouse cost dominates all the other inventory holding costs. We call this the EOQ-Max problem and the D-EOQ-Max problem, if the product is continuously divisible and discrete, respectively. The EOQ-Max problem admits a closed form optimal solution, while the D-EOQ-Max problem does not because its objective function may have several local minima. We present an optimal polynomial time algorithm for the discrete problem. Construction of this algorithm is supported by the fact that continuous relaxation of the D-EOQ-Max problem provides a solution that can be up to 50% worse than the optimal solution, and this worst-case error bound is tight. Applications of the D-EOQ-Max problem include supply and inventory management, logistics and scheduling.     

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.     

需求依赖于展示区库存水平的易腐品订购和转运策略

本文主要研究易腐品零售商的订货和转运策略。零售商的库存分为两部分,即展示区/货架库存和仓库库存。零售商定期向供应商订货,零售商收到订购的商品首先将其中一部分商品存放在展示区中,余下的部分储存在仓库。展示区的空间是有限的,并且需求依赖于展示区商品的库存量。本文首先建立了以平均利润最大化为目标的库存优化模型并对模型最优解的存在性进行了分析,然后得到了求解最优订购量、转运量、转运时间间隔以及再订购点的算法,最后给出了不同参数条件下的算例。     

An integrated model for warehouse and inventory planning

The purpose of this article is to evaluate the value of integrating tactical warehouse and inventory decisions. Therefore, a global warehouse and inventory model is presented and solved. In order to solve this mathematical model, two solution methodologies are developed which offer different level of integration of warehouse and inventory decisions. Computational tests are performed on a real world database using multiple scenarios differing by the warehouse capacity limits and the warehouse and inventory costs. Our observation is that the total cost of the inventory and warehouse systems can be reduced drastically by taking into account the warehouse capacity restrictions in the inventory planning decisions, in an aggregate way. Moreover additional inventory and warehouse savings can be achieved by using more sophisticated integration methods for inventory and warehouse decisions.     

A periodic-review inventory control policy for a two-level supply chain with multiple retailers and stochastic demand

We consider a time-based inventory control policy for a two-level supply chain with one warehouse and multiple retailers in this paper. Let the warehouse order in a fixed base replenishment interval. The retailers are required to order in intervals that are integer-ratio multiples of the base replenishment interval at the warehouse. The warehouse and the retailers each adopt an order-up-to policy, i.e. order the needed stock at a review point to raise the inventory position to a fixed order-up-to level. It is assumed that the retailers face independent Poisson demand processes and no transshipments between them are allowed. The contribution of the study is threefold. First, we assume that when facing a shortage the warehouse allocates the remaining stock to the retailers optimally to minimize system cost in the last minute before delivery and provide an approach to evaluate the exact system cost. Second, we characterize the structural properties and develop an exact optimal solution for the inventory control system. Finally, we demonstrate that the last minute optimal warehouse stock allocation rule we adopt dominates the virtual allocation rule in which warehouse stock is allocated to meet retailer demand on a first-come first-served basis with significant cost benefits. Moreover, the proposed time-based inventory control policy can perform equally well or better than the commonly used stock-based batch-ordering policy for distribution systems with multiple retailers.     

Modelling and analysis for capacity expansion planning in warehousing

In this paper, a dynamic programming model is developed for the purpose of establishing a warehouse capacity expansion schedule and underlying multi-item inventory policy that are jointly optimal. The optimal warehouse size over any segment of the planning horizon is obtained by solving a nonlinear optimisation problem, this being accomplished efficiently by exploiting the Kuhn–Tucker conditions. Repeating this procedure between each pair of time periods results in a discrete state space, so that the optimal capacity expansion schedule corresponds to a shortest path in a network. Managerial insights are provided through experimentation with the model.     

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.     

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.     

Stock Rationing in a Continuous Review Two-Echelon Inventory Model

In this paper we consider a 1-warehouse, N-retailer inventory system where demand occurs at all locations. We introduce an inventory model which allows us to set different service levels for retailers and direct customer demand at the warehouse. For each retailer a critical level is defined, such that a retailer replenishment order is delivered from warehouse stock if and only if the stock level exceeds this critical level. It is assumed that retailer replenishment orders, which are not satisfied from warehouse stock, are delivered directly from the outside supplier, instead of being backlogged. We present an analytical upper bound on the total cost of the system, and develop a heuristic method to optimize the policy parameters. Numerical experiments indicate that our technique provides a very close approximation of the exact cost. Also, we show that differentiating among the retailers and direct customer demand can yield significant cost reductions.     

Allocation of warehouse inventory with electronic data interchange and fixed order intervals

This paper considers the problem of allocating warehouse inventory to retailers where retailer orders and the replenishment of warehouse inventory occur periodically on a fixed schedule. We assume that the warehouse and the retailers have the opportunity to exchange demand information through Electronic Data Interchange (EDI). At the warehouse level, for instance, the available information on the retailer's demand may be utilized in determining the shipment quantities needed to meet the desired service level to the retailers. Unlike similar models focusing primarily on optimizing systems wide performance measures, in this paper we focus on the service level furnished to the retailers by the warehouse. To this end, three different allocation policies are considered: static, myopic, and dynamic rules characterizing the impact of available demand information on the resulting service levels. Numerical illustrations exemplify the allocation rules considered. An interesting though counter intuitive observation is that the existence of additional demand information cannot, a prior, be assumed superior.