Replenishment Interval Research Articles

Optimization of the Replenishment Strategy for the Supplier Based on Genetic Algorithm

In the supply chain management, the supplier can reduce its total cost by providing price discounts to restrict theretailers’ replenishment intervals to multiples of a common replenishment interval. In order to accomplish thispurpose, supplier often determined a price discount that all the retailers can accept. After the cost of the supplierwas further considered, a better price discount was designed, which can meet all the retailers’ requirements tomaximize their benefits. And a new supplier’s minimization problem model based on stackelberg game wasdesigned also. Then a genetic algorithm is used to solve the supplier’s replenishment model. Experiment resultsof the genetic algorithm demonstrated the feasibility and the effectiveness of the strategy.

A hybrid of monopoly and perfect competition model for hi-tech products

For Hi-tech products, the demand rate, the component cost as well as the selling price usually decline significantly with time. In the case of perfect competition, shortages usually result in lost sales; while in a monopoly, shortages will be completely backordered. However, neither perfect competition nor monopoly exists. Therefore, there is a need to develop a replenishment model considering a hybrid of perfect competition and monopoly when the cost, price and demand are decreasing simultaneously. A numerical example and sensitivity analysis are carried out to illustrate this model. The results show that a higher decline-rate in the component cost leads to a smaller service level and a larger replenishment interval. When the component cost decline rate increases and the selling price decline rate decreases simultaneously, the replenishment interval decreases. In perfect competition it is better to have a high service level, while for the case with monopoly, keeping a low service level is better due to complete backordering.

A genetic algorithm to optimize multiproduct multiconstraint inventory control systems with stochastic replenishment intervals and discount

There are two main assumptions in multiperiodic inventory control problems. The first is the continuous review, where, depending on the inventory level, orders can happen at any time, and the other is the periodic review, where orders can only happen at the beginning of each period. In this paper, these assumptions are relaxed, and the periods between two replenishments are assumed independent and identically distributed random variables. Furthermore, the decision variables are assumed integer-type and that there are two kinds of space and budget constraints. The incremental discounts to purchase products are considered, and a combination of backorder and lost sales are taken into account for the shortages. The model of this problem is shown to be a mixed integer-nonlinear programming type, and in order to solve it, both genetic algorithm and simulated annealing approaches are employed. At the end, two numerical examples are given to demonstrate the applicability of the proposed methodologies in which genetic algorithm method performs better than simulated annealing in terms of objective function values.

Optimal Policy for a Multiechelon Inventory System with Batch Ordering and Fixed Replenishment Intervals

In many production/distribution systems, materials flow in fixed lot sizes (e.g., in full truckloads or full containers) and under regular schedules (e.g., delivery every week). In this paper, we study a multiechelon serial system with batch ordering and fixed replenishment intervals. We derive the optimal inventory control policy, provide a distribution-function solution for its optimal control parameters, and present an efficient algorithm for computing those parameters. Further, we show that the optimal expected system cost is minimized when the ordering times for all stages are synchronized. In contrast to the known approach in the literature that develops a lower bound for the average cost of a given period for the classical serial system, we develop a lower bound for the average total cost over an appropriately defined cycle and then construct a policy that reaches the lower bound. We also discuss its extension to the nonlinear shortage cost case (i.e., the nonlinear cost case). This paper generalizes several recent results on the analysis of multiechelon systems.

A practical solution approach for the cyclic inventory routing problem

Abstract Vendor managed inventory (VMI) is an example of effective cooperation and partnering practices between up- and downstream stages in a supply chain. In VMI, the supplier takes the responsibility for replenishing his customers’ inventories based on their consumption data, with the aim of optimizing the over all distribution and inventory costs throughout the supply chain. This paper discusses the challenging optimization problem that arises in this context, known as the inventory routing problem (IRP). The objective of this IRP problem is to determine a distribution plan that minimizes average distribution and inventory costs without causing any stock-out at the customers. Deterministic constant customer demand rates are assumed and therefore, a long-term cyclical approach is adopted, integrating fleet sizing, vehicle routing, and inventory management. Further, realistic side-constraints such as limited storage capacities, driving time restrictions and constant replenishment intervals are taken into account. A heuristic solution approach is proposed, analyzed and evaluated against a comparable state-of-the-art heuristic.

Multi-item, economic order quantity model with permissible delay in payments and a budget constraint

This paper investigates the optimal replenishment policy for multi-item ordering under the conditions of permissible delay in payment, a budget constraint and permissible partial payment at a penalty. Such conditions exist in the use of purchase cards to order merchandise directly from the supplier. The paper presents mathematical models and closed form solutions for three situations ? permissible delay is negative (payment made before receipt of good), is less than the replenishment interval and is greater than the replenishment interval. The analyses indicate that it is more economical for the purchaser to make entire payments even when the interest earned is greater than the interest charged. [Received 06 April 2007; Revised 02 December 2007; Accepted 23 January 2008]

Conservation biological control of arthropods using artificial food sprays: Current status and future challenges

The use of artificial food sprays to increase the abundance and impact of natural enemies of arthropod pests has been recognized for over 40 years. Nevertheless, artificial food sprays are applied in relatively few conservation biological control programs, possibly because of inconsistent performance. To consider this explanation, using a hierarchical classification approach ranging from altered foraging behavior at the simplest level to profitability of farm production at the highest level, we quantitatively reviewed 234 trials from 77 publications. The levels of assessment of food sprays most commonly found in the publications were the densities of arthropod pests (59 trials) and their natural enemies (124). Although the density of natural enemies increased in 108 of the 124 trials (or 87% of cases) and pest populations declined in 28 of the 59 trials (or 47%), increased profit was not demonstrated in the five trials where it was examined. The most commonly studied natural enemies belonged to the order Neuroptera (104 trials). Nevertheless, the parasitic Hymenoptera had the highest proportion of positive successes (56 of the 69 trials or 81%). Contrary to predictions, the likelihood of a successful result was not affected by the duration of food spray provision, management intensity, application frequency, or replenishment interval. However, success was more likely as spray concentration increased. When analyzed separately, the Coleoptera showed the greatest response to carbohydrates alone, such as honey and sucrose, compared with a mixture of carbohydrates and proteins. To achieve greater success with artificial food sprays, and therefore overcome a major impediment to their adoption, we contend that researchers should: (i) demonstrate greater success across several levels in the hierarchy, especially profit; (ii) consistently demonstrate success in field trials replicated across regions and years; and (iii) combine artificial food sprays with other compatible elements of an integrated pest management strategy. Until these outcomes are demonstrated, it is envisaged that artificial food sprays will form only a small part of future conservation biological control programs.

Optimal Control of Serial Inventory Systems with Fixed Replenishment Intervals

We consider a single-item, periodic-review, serial inventory/production system, with linear inventory-holding and penalty costs. To facilitate shipment consolidation and capacity planning, we assume that the system has implemented fixed replenishment intervals; each stage is allowed to order only at given equidistant times. Further, for each stage except the most downstream one, the replenishment interval is assumed to be an integer multiple of the replenishment interval of the next downstream stage. This reflects the fact that the further upstream in a supply chain, the higher setup times and costs tend to be, and thus larger batches are desired. Our model with fixed replenishment intervals is a direct generalization of the serial model of Clark and Scarf (1960). For this generalized model, we prove the optimality of base-stock policies, we derive newsboy equations for the optimal base-stock levels, and we describe an efficient exact solution procedure for the case with mixed Erlang demands. Finally, we present extensions to assembly systems and to systems with a modified fill-rate constraint instead of backorder costs.

Collaborative vendor–buyer inventory system with declining market

Due to rapid technological innovation, shortening product life cycle and severe competition, product demand rate will decrease. In this study, we develop a collaborative vendor–buyer inventory system with exponentially decreasing market, finite horizon and constant replenishment interval. The result of this study shows an impressive cost reduction as a result of collaboration. The result is even more significant for products with decreasing demand. High-tech products (e.g., microcomputer chip), which is being substituted by another, has the characteristic of declining market demand with negatively exponential distribution. A numerical example and sensitivity analysis are provided to illustrate the theory and to derive the percentage total cost reduction of the collaborative vendor–buyer system.

A partition approach to the inventory/routing problem

In this study we focus on the integration of inventory control and vehicle routing schedules for a distribution system in which the warehouse is responsible for the replenishment of a single item to the retailers with demands occurring at a specific constant (but retailer-dependent) rate, combining deliveries into efficient routes. This research proposes a fixed partition policy for this type of problem, in which the replenishment interval of each of the retailers’ partition region as well as the warehouse is accorded the power of two (POT) principle. A lower bound of the long-run average cost of any feasible strategy for the considered distribution system is drawn. And a tabu search algorithm is designed to find the retailers’ optimal partition regions under the fixed partition policy proposed. Computational results reveal the effectiveness of the policy as well as of the algorithm.

A joint economic lot-size model for an integrated supply network using genetic algorithm

Competitive environment calls for cost reduction in tree-like inventory systems. One of the key elements to cost reduction of a supply chain is integration of inventory system. In this paper, we consider an integrated four-stage supply chain system, incorporating one supplier, multiple producers, multiple distributors multiple retailers. The aim of this model is to determine order quantity of each stage (from its upstream) and shortage level of each stage (for its downstream) such that the total cost of the supply chain to be minimized. Thus, for three possible situations of relation between supplier and producers replenishment intervals, an integrated inventory model, making joint economic lot-size of shortage, holding, ordering and setup costs in all stages is developed. Then a heuristic approach based on genetic algorithm for solving this problem is presented.

A stochastic multi-item inventory model with unequal replenishment intervals and limited warehouse capacity

Managing inventory levels for multiple products that share warehouse capacity often presents significant coordination challenges. These coordination challenges are further compounded when multiple suppliers use different replenishment schedules for supplying the warehouse. To address these challenges, this paper studies stochastic inventory models for multiple items with both equal and unequal replenishment intervals under limited warehouse capacity. We propose three efficient and intuitively attractive heuristics. We show that these heuristics provide the optimal replenishment quantities in the case of equal replenishment intervals. For the general model, a numerical comparison of the heuristic solutions to the optimal solutions shows that the heuristics yield high quality solutions in very limited time.

Staggering Periodic Replenishment in Multivendor JIT Environments

The delivery scheduling problem studied in this paper was motivated by the operation in a large personal computer assembly plant, which was using multisourcing for some of its materials. The company’s objective was to design a delivery schedule so that the average inventory level in the factory was minimized. We show that the problem is intimately related to a classical inventory staggering problem, where the focus is on the computation of the peak inventory level associated with the replenishment policy. This connection allows us to show that the delivery scheduling problem is NP-hard. For the two-vendor case with integral replenishment intervals, we propose a generalized form of Homer’s scheduling heuristic and obtain performance bounds for the classical inventory staggering problem. Our analysis uses the Chinese remainder theorem in an interesting way. The approach can be generalized to the case with more than two vendors, leading to a strong linear-programming-based lower bound for the inventory staggering problem. We illustrate this technique for the case in which all the replenishment intervals are relatively prime, establishing a bound that is not greater than 140% of the optimal. We examine the implications of these results to the delivery scheduling problem.

Optimal pricing and ordering policy under permissible delay in payments

Many researchers have assumed that the selling price is the same as the purchase cost, and developed various EOQ models for a retailer when the supplier offers a permissible delay in payments. In this paper, we complement the shortcoming of their models by considering the difference between the selling price and the purchase cost. We then develop an algorithm for a retailer to determine its optimal price and lot size simultaneously when the supplier offers a permissible delay in payments. Our theoretical result contradicts to their conclusion that the economic replenishment interval and order quantity generally increases marginally under the permissible delay in payments.

Replenish-up-to inventory control policy with random replenishment intervals

The multi period inventory problems have been studied under two main assumptions. Continuous review assumption where an order can be made at any time depending on inventory position and periodic review assumption where an order can be initiated only at discrete time epochs. In this study, we analyze a multi period inventory problem that falls under neither of these two assumptions. In the case we consider, there are periodic replenishments but the replenishment intervals are taken to be i.i.d. random variables. This setting represents the real life cases where a supplier visits a retailer with random inter arrival times and the retailer replenishes his inventories based on a replenish-up-to-level inventory control policy. We also assume that only a certain fraction of unmet demand is backordered and the rest of it is lost.In this setting under general distribution between replenishment epochs, we show the concavity of the expected profit function and give the condition that must hold for the optimal replenish-up-to-level. We also present the specific solutions and analysis under two different distributions, namely, uniform and exponential distributions, together with some numerical examples.

Selective discount for supplier–buyer coordination using common replenishment epochs

A supplier may reduce its order costs by providing an incentive in the form of price discounts to buyers to restrict their replenishment intervals to multiples of a common replenishment epoch (CRE). This coordination mechanism was studied by Viswanathan and Piplani who suggested that the supplier offer a discount that is the maximum of the discount required by all buyers. We generalize their model to allow for a selective discount policy that, if beneficial, excludes some buyers to minimize the supplier’s total cost. Using a computational study, we observe that offering discounts to buyers selectively, if necessary, by segmenting them and offering multiple CRE, reduces the supplier’s cost in many scenarios.

Leveraging information in multi-echelon inventory systems

Decision-making in multi-echelon inventory systems is often facilitated by the availability of information. In this paper, we examine the effect of utilising demand information in a one-warehouse two-retailer system operating under periodic-review. A common and fixed replenishment interval is assumed for all stages. The lead times are deterministic, and are small compared to the replenishment interval. The warehouse does not maintain any cycle stock, i.e., as soon as replenishment arrives at the warehouse, retailers’ demands are fulfilled from the available stock. In the event of a shortage at the warehouse, two cases have been considered––emergency shipment and allocation. For both the cases, expected total cost (ETC) models have been developed, and it has been shown through numerical examples that the optimal ETC in the presence of demand information is always lower than that in the absence of demand information. It is also noted that the order-up-to level at the warehouse has to be dynamic to reap the benefits of availability of demand information.

A simple rule for determining replenishment intervals of an inventory item with linear decreasing demand rate

Considerable amount of research efforts have not so far been devoted to inventory replenishment problems where the rate of demand decreases with time in comparison to increasing pattern of demand rate. This article presents a simple rule for determining the replenishment intervals of an inventory item having a linear decreasing demand rate over a finite planning horizon in heuristic approach. The proposed method is applied to the well-known Silver's heuristic, continuous time part period heuristic and continuous least unit cost method and its performance is examined with a numerical example.

Optimal replenishment scheduling for inventory items with Weibull distributed deterioration and time-varying demand

This paper proposes a dynamic programming model for solving replenishment scheduling problems. The model is developed based on the following assumptions: the inventory items are subject to deterioration of two-parameter Weibull distribution, the demand rate is exponentially distributed over time, shortages are allowed, and the effects of inflation and time-value of money are taken into consideration. The solutions of the model determine the optimal replenishment schedule over a finite horizon so that the present worth of the total cost is minimized. The solution procedure is illustrated by two numerical examples. The strength of the proposed model is supported by carrying out a comparison study with an existing model. The proposed model permits variation in both the replenishment intervals and the service levels between order cycles. As a result, it generates a better solution than other optimization models having fixed order intervals and fixed service levels. In addition, the Weibull deterioration used in the model is a generalized form of exponentially decaying functions, and hence it is applicable to the management of a wider range of deteriorating items in practice, like agricultural products, blood, drugs, fashion goods, high-tech and electronic components, volatile liquids, and radioactive substances

Properties of the Periodic Review (R, T) Inventory Control Policy for Stationary, Stochastic Demand

This paper compares the commonly used periodic review, replenishment interval, order-up-to (R, T ) policy to the continuous review, reorder point, order quantity, (Q, r) model. We show that long-run average cost function for the single-product (R, T ) policy has a structure similar to that of the (Q, r) model. Consequently, many of the useful properties of the latter model are applicable. In particular, the optimal cost is insensitive to the choice of the replenishment interval, T, provided the optimal order-up-to level, R, corresponding to T is used. For instance, a suboptimal T obtained from a deterministic analysis increases costs by no more than 6.125%. For continuous demand, we analytically prove that use of a (R, T)policy instead of the optimal policy increases costs by at most 41.42% in the worst case. Computational experiments on Poisson demand demonstrate that the average-case relative error of using a (R,T)policy is under 7.5%. This relative error is lower when the demand rate and leadtime are high and the fixed order costs are either very low or very high. When coordination of order placement epochs is desirable, the (R,T) policy may sometimes be preferred to the (Q, r) policy. In this context, we illustrate application of our single-product results to more complex systems. In particular, we show that a simple power-of-two, (R,T) based heuristic for the stochastic multiproduct joint replenishment problem has a worst-case performance guarantee of 1.5. A similar result is explored for a special case of a two-echelon serial inventory system