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.     

Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.     

Optimal production lot size and reorder point of a two-stage supply chain while random demand is sensitive with sales teams' initiatives

The paper develops a production-inventory model of a two-stage supply chain consisting of one manufacturer and one retailer to study production lot size/order quantity, reorder point sales teams’ initiatives where demand of the end customers is dependent on random variable and sales teams’ initiatives simultaneously. The manufacturer produces the order quantity of the retailer at one lot in which the procurement cost per unit quantity follows a realistic convex function of production lot size. In the chain, the cost of sales team's initiatives/promotion efforts and wholesale price of the manufacturer are negotiated at the points such that their optimum profits reached nearer to their target profits. This study suggests to the management of firms to determine the optimal order quantity/production quantity, reorder point and sales teams’ initiatives/promotional effort in order to achieve their maximum profits. An analytical method is applied to determine the optimal values of the decision variables. Finally, numerical examples with its graphical presentation and sensitivity analysis of the key parameters are presented to illustrate more insights of the model.     

Some remarks on an optimal order quantity and reorder point when supply and demand are uncertain

In the reports in the literature on inventory control, the effects of the random capacity on an order quantity and reorder point inventory control model have been integrated with lead time demand following general distribution. An iterative solution procedure has been proposed for obtaining the optimal solution. However, the resulting solution may not exist or it may not guarantee to give a minimum to the objective cost function, the expected cost per unit time. The aim of this study was to introduce a complete solution of the order quantity/reorder point problem, optimality, properties and bounds on the optimal order quantity and reorder point. The two most appealing distributions of lead time demand, normal and uniform distributions, in conjunction with an exponentially distributed capacity, are used to illustrate our findings in determining the optimal order quantity and reorder point.     

(Q,r,L) Inventory model with defective items

This paper assumes that an arrival order lot may contain some defective items, and the number of defective items is a random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity, the reorder point and the lead time are decision variables. In our studies, we first assume that the lead time demand follows a normal distribution, and then relax the assumption about the form of the distribution function of the lead time demand and apply the minimax distribution-free procedure to solve the problem. We develop an algorithm procedure to obtain the optimal ordering strategy. Furthermore, the effects of parameters are also included.     

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.     

Implementing an optimal policy for set-up time reduction in an economic production quantity model

An economic production quantity (EPQ) system consisting of single and multiple items with an optimal policy of set-up time reduction and a fixed increment cost are discussed in the present study. The set-up time reduction ratio as a decision variable under various cases of demand in the EPQ model is considered. It is assumed that the set-up cost is linearly related to the set-up time. The set-up time reduction ratio and the lot size are solved simultaneously to obtain an optimal value of the total annual cost. Numerical examples are presented to demonstrate the accuracy of the proposed method.     

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.     

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.     

Multi-Objective Optimization of Slow Moving Inventory System Using Cuckoo Search

This paper focuses on the development of a multi-objective lot size–reorder point backorder inventory model for a slow moving item. The three objectives are the minimization of (1) the total annual relevant cost, (2) the expected number of stocked out units incurred annually and (3) the expected frequency of stockout occasions annually. Laplace distribution is used to model the variability of lead time demand. The multi-objective Cuckoo Search (MOCS) algorithm is proposed to solve the model. Pareto curves are generated between cost and service levels for decision-makers. A numerical problem is considered on a slow moving item to illustrate the results. Furthermore, the performance of the MOCS algorithm is evaluated in comparison to multi-objective particle swarm optimization (MOPSO) using metrics, such as error ratio, maximum spread and spacing.     

Optimal policy for set-up time reduction in a multistage production inventory system

A batch production-inventory system consisting of multiple stages with an optimal policy of set-up time reduction and a fixed increment cost are discussed. The ratio of set-up time reduction as a decision variable under various cases of demand in the batch production-inventory model is considered. The ratio of set-up time reduction and lot size are solved simultaneously to obtain an optimal value of the total annual cost. A numerical example is presented to demonstrate the accuracy of the proposed method.     

Coordinating supplier׳s reorder point: A coordination mechanism for supply chains with long supplier lead time

In this paper, the issue of the upstream stochastic lead time in supply chain (SC) is investigated. A coordination mechanism is developed for reducing the harmful effect of upstream lead time. The supplier stochastic lead time can substantially harm the whole supply chain service level, especially when it is accumulated with downstream stochastic lead times. In this study, aggregation of both the supplier and the retailer stochastic lead time is analyzed in a two-stage supply chain (SC). To dampen harmful effects of a long aggregate lead time, a ‘per order extra payment’ model is developed for convincing the supplier to increase its reorder point. Numerical experiments show that coordinating the supplier׳s reorder point creates a significant profit for the whole supply chain. In addition, the proposed model is capable of optimizing the supplier׳s reorder point and fairly sharing the extra benefits. Some conditions are also extracted, under which the proposed model shows good performance.     

Optimal reorder decision utilizing centralized stock information in a two-echelon distribution system

The objective of this paper is to develop an optimal reorder policy for a two-echelon distribution system with one central warehouse and multiple retailers. We assume the warehouse has centralized stock information and each facility uses continuous-review batch ordering policy. Since echelon stock policies may show poor performance for distribution systems, we propose a new type of policy that utilizes the centralized stock information more effectively. We define the order risk policy, which decides reorder time based on the order risk which represents the relative cost increase due to immediate order compared to delayed order. We formulate the order risk and prove the optimality of the order risk policy under the system assumption that the warehouse guarantees delivery within the fixed lead time. The order risk is derived from the marginal analysis. Since exact calculation of the order risk is complex, an approximation method is provided. Computational experiment that compares our policy with existing policies shows that a significant cost savings is obtained. The concept of the order risk can be extended to the other models.Scope and purposeDue to the improvement of modern information technologies, many companies start tracking the real-time stock information of the supply chain members. Thus, the reorder policy based on the real-time centralized stock information becomes very important. In this paper, we consider the reorder policy for a continuous-review batch-ordering two-echelon distribution system, utilizing the centralized stock information. Existing reorder policies are classified into installation stock policies and echelon stock policies. Since installation stock policies consider only local stock information, echelon stock policies have been used when the centralized stock information is available. For serial and assembly systems, it has been known that the echelon stock policies are superior to the installation stock policies. However, for distribution systems, both policies may outperform each other in different situations. The purpose of this study is to develop the optimal reorder policy for a distribution system with one-warehouse and multiple retailers, where the real-time stock information is centralized at the warehouse. We define a new type of reorder policy of which the reorder decision is based on the ‘order risk’. The order risk is defined as the relative cost increase due to immediate order compared to delayed order. We formulate the order risk and prove the optimality. For computational simplicity, we provide an approximation method to calculate the order risk. Computational experiment shows that a significant cost savings is obtained.     

A manufacturer–buyer integrated inventory model with stochastic lead times for delivering equal- and/or unequal-sized batches of a lot

Although the subject of manufacturer–buyer integrated inventory management with deterministic lead times has received a lot of attention from researchers, the corresponding problem with stochastic lead times has been given comparatively little consideration. Recently, it has been treated in the case of an exponential distribution of lead times with the lot transferred in equal-sized batches (sub-lots). In this treatment the buyer orders the next batch when his/her stock level falls to a certain reorder point, allowing for shortages and complete backordering. The total cost benefit of solving the problem using an integrated inventory system instead of independent ones had been demonstrated. However, rather than an exponential distribution, a normal distribution of lead times seems to provide a better fit to the problem. Moreover, synchronization of the integrated production flow by generalizing the method of transferring batches of the lot might lead to a lower total cost. Based on these notions, we develop here a manufacturer–buyer integrated inventory model with a normal distribution of lead times for delivering equal- and/or unequal-sized batches of a lot. Then a solution technique to the model and hence a solution algorithm are presented. The potential benefit of the present method is illustrated with solutions of some numerical problems. The sensitivities of the solutions to variations in the parameter values are also studied.     

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.     

A mathematical model for order splitting in a multiple supplier single-item inventory system

In this paper we develop a mathematical model which considers multiple-supplier single-item inventory systems. The lead times of the suppliers and demand arrival rate are random variables. All shortages are backordered. Continuous review (s, Q) policy has been assumed. When the inventory level hits the reorder level, the total order is split among n suppliers. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost and shortage cost is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared to the relevant models in the literature. In addition, some managerial insights are observed.     

Sole versus dual sourcing under order dependent lead times and prices

In this paper, we consider a dual-sourcing model with constant demand and stochastic lead times. Two suppliers may be different in terms of purchasing prices and lead-time parameters. The ordering takes place when the inventory level depletes to a reorder level, and the order is split among two suppliers. Unlike previous works in the order splitting literature, the supply lead time between vendor and buyer as well as unit purchasing prices is considered to be order quantity dependent. The proposed model finds out the optimal reorder point, order quantity and splitting proportion, using a solution procedure. Numerical results show that neglecting the relationship between ordering batch size and lead times is a shortcoming that hides one of order splitting advantages. Moreover, connecting unit prices to order quantity can decrease the percentage saving from dual sourcing compared to sole sourcing. Furthermore, sensitivity analysis shows some managerial insights.     

Simultaneous coordination of order quantity and reorder point in a two-stage supply chain

In this paper, we investigate the simultaneous coordination of order quantity and reorder point in a two-stage supply chain (SC). While coordination of order quantity has received much attention in the supply chain management literature, coordination of the reorder point has been less-studied. The retailer's reorder point has a direct impact on product availability and customer service level (CSL) and therefore has a great impact on SC profitability. Our proposed model adopts a two-stage SC with stochastic demand and lead times over multiple periods. The proposed coordination model assures global optimization of order quantity–reorder point decisions. Using a pricing scheme with a discount factor, we extract conditions in which both downstream and upstream members have sufficient motivation to participate in the coordination scheme. Numerical experiments demonstrate that the proposed model can achieve channel coordination. Results of the modeling and analyses show that coordination of both reorder point and order quantity can lead to increased SC profitability as well as CSL improvement.     

On the distribution free continuous-review inventory model with a service level constraint

We consider a continuous-review (Q, r) inventory model with a fill rate service constraint and relax the assumption that the distribution of lead time demand is known. We adopt a distribution free approach: We assume that only the first two moments of the lead time demand distribution are known, and then, optimize the policy parameters against the worst possible distribution. We are able to derive closed-form expressions for the optimal order quantity and reorder point.     

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.