Replenishment Quantity Research Articles

A multiple-retailer replenishment model under VMI: Accounting for the retailer heterogeneity

Vendor managed inventory (VMI) is a well-established supply chain practice where the supplier is responsible for managing inventory at the retail point. In particular, the supplier takes care of when to order and how much to order on behalf of the retailer. This paper considers a single supplier – multiple retailer setting where the supplier takes inventory replenishment decisions for retailers such that the replenishment quantity for each retailer is within an upper bound that is mutually agreed upon in the VMI contract. We develop a nonlinear mixed-integer programming model to compute the optimal replenishment frequency and quantity for each of the retailer, such that the total system cost is minimized. A conceptual and numerical comparison is made with the existing models in the VMI literature. The proposed model is found to perform better for all levels of retailer heterogeneity, thereby establishing generalization among the class of models. We also propose an efficient heuristic for solving the proposed model by utilizing the concept of cycle ratio (setup cost/holding cost∗demand), thus reducing the computational time drastically. Lastly, through a numerical analysis, we find that the proposed model with integer ratio policy structure is more stable as compared to the existing alternative replenishment models.

Optimal Multi-Products Dynamic Pricing and Inventory Policies for Coordinating Production and Marketing under Fuzzy Environment: Fuzzy Expansion Methods

This paper investigates the optimal multi-products dynamic pricing and inventory policies over a multi-period planning horizon with deteriorating products and a fuzzy demand function. The objective is maximization of the discount profit. A dynamic programming model is presented to determine retail price and replenishment quantities. Also, due to the existence of uncertainties in the parameter values such as cost, deterioration rate, and the optimal strategies in general, it cannot be obtained with high feasibility. So, the concept of fuzzy set theory can be applied to cope whit this issue. Since the presented model is a fuzzy partial differential equation, three novel fuzzy expansion methods including Jacobi polynomials, airfoil polynomials, and fuzzy collocation methods are proposed for solving this problem. Finally, this paper carries out various computational experiments to assess the proposed model and solution approaches.

Impacts of decision sequences on a random yield supply chain with a service level requirement

The last decade has witnessed a rising trend of social network structures where decision making may be influenced by members’ channel power. This paper considers a decentralized supply chain that comprises a random yield supplier and a manufacturer with a service level requirement. The supplier determines the production quantity while the manufacturer decides on the replenishment quantity. We investigate several game-theoretic models with respect to decision sequences of the supplier and manufacturer with a service level requirement. We compare firms’ equilibrium production/order quantity and profits under these models. Both firms achieve a higher payoff in the manufacturer-leader game than in the supplier-leader and simultaneous move games, which indicates that the manufacturer-leader game is Pareto efficient from the channel structure perspective. A critical value is observed at the service level, beyond which the supplier’s profit decreases, while the manufacturer’s profit increases in the service level requirement. Interestingly, the critical value is independent of the models and it just depends on the wholesale price and production cost. We also examine the decentralized random yield supply chain under the vendor-managed inventory arrangement. Finally, numerical computations are conducted to evaluate supply chain performance with respect to various parameters.

Optimal replenishment quantity for new products and return rate of used products for a retailer

Recently, due to increasing environmental concerns, the collection of used products for remanufacturing, reusing, or recycling in a closed-loop supply chain have attracted much attention from the academic and industrial communities. In this study, the decision maker is a retailer who follows the rule of the economic order quantity (EOQ) model. The EOQ decision-making retailer is assumed to sell the products to public consumers, as well as collect the used sold products from public consumers. The proposed problem for the retailer is formulated as a cost-minimization problem, where the replenishment quantity of new products and the return rate of used products are the decision variables. The optimal solution of the replenishment quantity and the return rate are obtained in closed-form under cost minimization. Furthermore, the cost-minimizing model is extended to consider the pricing decision so the total relevant profit for the retailer is maximized and an efficient iterative algorithm is provided to search for the solution. Several numerical examples are provided to illustrate the features of the proposed models and the iterative algorithm.

Inventory management for a remanufacture-to-order production with multi-components (parts)

In remanufacturing industries, inventory is significant because remanufacturing industries includes the stock of recycled products, stock of components (parts) after disassembly and refurbishing operation, stock of new parts purchased from manufacturers and stock of remanufactured products. Therefore, current research presents an approach to search an appropriate inventory policy for single product with multi-components for a third party remanufacturing company cooperated with the original equipment manufacturer (OEM). The third party for remanufacturing working in close cooperation with the OEM (such as dealer) can acquire the regional protection rules and can gain the technical support from the OEM. A general mathematical model is developed to describe the multiple components and multi-level inventory. Moreover, a case company problem is solved to validate the proposed model. Base on considered problem, the proposed model provided an efficient inventory management method to solve the situation in which remanufacturing company only recycled products but not components (parts). Furthermore, the presented model is significant to help the case company to set the best resource allocation by analyzing the impact on inventory policy from different replenishment quantities in remanufacturing process (recycling, disassembling and refurbishing). At last the proposed model is proved by a numerical experiment which used genetic algorithm (GA) to solve the inventory policy, and the result discusses the sensitivity analysis for model parameters in different replenishment quantities scenarios.

Optimal Policies for a Two‐Product Inventory System under a Flexible Substitution Scheme

We study a two‐product inventory model that allows substitution. Both products can be used to supply demand over a selling season of N periods, with a one‐time replenishment opportunity at the beginning of the season. A substitution could be offered even when the demanded product is available. The substitution rule is flexible in the sense that the seller can choose whether or not to offer substitution and at what price or discount level, and the customer may or may not accept the offer, with the acceptance probability being a decreasing function of the substitution price. The decisions are the replenishment quantities at the beginning of the season, and the dynamic substitution‐pricing policy in each period of the season. Using a stochastic dynamic programming approach, we present a complete solution to the problem. Furthermore, we show that the objective function is concave and submodular in the inventory levels—structural properties that facilitate the solution procedure and help identify threshold policies for the optimal substitution/pricing decisions. Finally, with a state transformation, we also show that the objective function is [Formula: see text]‐concave, which allows us to derive similar structural properties of the optimal policy for multiple‐season problems.

A robust inventory routing policy under inventory inaccuracy and replenishment lead-time

In this paper, we study an inventory routing problem under replenishment lead-time and inventory inaccuracy, which exist extensively in distribution systems for fresh products, but are often ignored in existing research. To solve the problem, a robust inventory routing policy is developed in three steps. At first, we propose the methods of updating the probability of the current net inventory and predicting those in future periods. For each candidate route, we develop a Robust TQL Algorithm to optimize the replenishment time, replenishment quantity and replenishment stage length. Finally, a genetic algorithm-based method is developed to optimize the delivery route.

An Optimal Replenishment Policy for the EPQ Model with Permissible Delay in Payments

Most of the past researches on economic replenishment quantity did not consider the trade credit period. In practice, however, companies in Taiwan have shown that taking the trade credit period into account when they are making inventory decisions decreases costs substantially. The aim of this paper is to develop an inventory model that incorporates a delay in payments to minimize the total inventory-relevant cost. The optimal replenishment policy of the proposed model is identified by utilizing the global optimum conditions and non-constrained quadratic nonlinear programming. The results show that the existence of global optimal inventory cycle under the condition of total cost function is convex. Accordingly, the proposed model can assist managers in effectively addressing EPQ issue under competitive market.

A genetic algorithm for the vendor-managed inventory routing problem with lost sales

This paper proposes a genetic algorithm (GA) for the inventory routing problem with lost sales under a vendor-managed inventory strategy in a two-echelon supply chain comprised of a single manufacturer and multiple retailers. The proposed GA is inspired by the solving mechanism of CPLEX for the optimization model of the problem. The proposed GA determines replenishment times and quantities and vehicle routes in a decoupled manner, while maximizing supply chain profits. The proposed GA is compared with the optimization model with respect to the effectiveness and efficiency in various test problems. The proposed GA finds solutions in a short computational time that are very close to those obtained with the optimization model for small problems and solutions that are within 3.2% of those for large problems. Furthermore, sensitivity analysis is conducted to investigate the effects of several problem parameters on the performance of the proposed GA and total profits.

Centralised and decentralised three echelon inventory model for optimal inventory decisions under price dependent demand

In this paper, a three echelon inventory system with a single manufacturer supplying a single kind of a product to a single distributor and then to a single retailer, is considered. Mathematical model is developed for annual total relevant costs of the coordinated and non-coordinated supply chain. The novel idea of the proposed model is that the demand is a function of retailer's unit selling price, which in turn is a function of retailer's replenishment quantity with dependence factor. The objective of the model is to demonstrate the optimality of inventory decisions, shipment frequencies and total relevant costs with respect to dependence factor. Numerical illustration is carried out to demonstrate the optimality of replenishment decisions. Finally, managerial insights are inferred in selecting the optimal replenishment quantity and shipment frequency that minimises the total relevant cost of the supply chain.

Joint Pricing and Inventory Management with Strategic Customers

We consider a joint pricing and inventory management problem wherein a seller sells a single product over an infinite horizon via dynamically determining anonymous posted prices and inventory replenishment quantities. Customers arrive over time with a deterministic arrival rate but heterogeneous product valuations. Unlike typical inventory models, we assume that customers are forward-looking, who can strategize their times of purchases. A customer incurs waiting and monitoring cost if he delays his time of purchase anticipating a lower future price. The seller incurs fixed ordering cost and inventory holding cost. The seller seeks a joint pricing and inventory policy that maximizes her long-run average profit. We show that the optimal policy is cyclic, i.e., the seller repeats the pricing and ordering decisions over cycles of the same length. Under this optimal policy, strategic customer equilibrium behaviors are proven to be myopic. The seller's optimal long-run average profit in the presence of strategic customers is the same as her optimal profit in an auxiliary classical backlogging model wherein customers are myopic that they make their purchasing decisions immediately upon their arrivals. Remarkably, this implies that solving the (seemingly very hard) model with strategic customers is as easy and tractable as solving the above auxiliary model with myopic customers. Our results settle the open debate whether incorporating strategic customer behaviors significantly complicates the model. The answer is surprisingly appealing. Our results allow firms to incorporate strategic customer behaviors with little modeling and computational overhead. We develop a novel approach to prove the optimality of our proposed policy in the presence of strategic customers. In particular, we establish a benchmark profit that is the seller's optimal profit in the joint mechanism design and inventory control problem. We show that this benchmark serves as an upper bound of all joint pricing and inventory policies. We then show that this benchmark is upper bounded by the seller's profit under our proposed policy, which then implies the optimality of our policy.

Coordinating Pricing and Inventory Replenishment with Nonparametric Demand Learning

We consider a firm (e.g., retailer) selling a single nonperishable product over a finite-period planning horizon. Demand in each period is stochastic and price-dependent, and unsatisfied demands are backlogged. At the beginning of each period, the firm determines its selling price and inventory replenishment quantity, but it knows neither the form of demand dependency on selling price nor the distribution of demand uncertainty a priori, hence it has to make pricing and ordering decisions based on historical demand data. We propose a nonparametric data-driven policy that learns about the demand on the fly and, concurrently, applies learned information to determine replenishment and pricing decisions. The policy integrates learning and action in a sense that the firm actively experiments on pricing and inventory levels to collect demand information with the least possible profit loss. Besides convergence of optimal policies, we show that the regret, defined as the average profit loss compared with that of the optimal solution when the firm has complete information about the underlying demand, vanishes at the fastest possible rate as the planning horizon increases.

Demand-pull replenishment model for hospital inventory management: a dynamic buffer-adjustment approach

Hospital inventory management faces a continually increasing challenge to ensure the availability of medical and surgical supplies at the lowest inventory cost. For overcoming the drawbacks of existing re-order point approaches commonly applied in hospital materials replenishment management, this research presents an innovative demand-pull replenishment approach named the dynamic drum-buffer-rope (DDBR) replenishment model. The DDBR model is implemented using a system dynamics approach in which two essential mechanisms – the demand-pull characteristics and dynamic buffer-adjustment activities – are simulated and experimented on. To determine appropriate buffer sizes and replenishment quantities, this research adopted Powell search algorithm to achieve the objective of no stock-out occurrence and low inventory cost. The evaluation of the proposed DDBR model in a real hospital case through a series of comparisons shows that the DDBR model can determine optimal replenishment timing and quantity for total inventory cost with no stock-out occurrence.

Multiple-vendor, multiple-retailer based vendor-managed inventory

Vendor-managed inventory is a coordinated system where a vendor decides about the retailer’s replenishment quantity and time. This work studies a two-echelon distribution work composed of multiple vendors and retailers in traditional and vendor-managed inventory systems where unsatisfied demands are lost sales. We also consider that the retailers’ demand is stochastic following a uniform distribution. The mathematical models are developed and applied in vendor-managed inventory and traditional systems. Under the traditional supply chain, the vendor incurs the cost of holding and production setup, while a retailer incurs expenses for inventory holding, ordering, transportation and lost sales. In a vendor-managed inventory system, as the vendor is responsible for his retailers, the retailer’s costs are transferred to the vendor. We aim to identify benefits of vendor-managed inventory. The total cost per unit time is used as a comparable measure between vendor-managed inventory and traditional systems. Numerical examples and a sensitivity analysis of key parameters include the vendor’s setup cost and holding cost; the retailer’s transportation and ordering costs are presented in both vendor-managed inventory and traditional systems. The results illustrate that vendor-managed inventory total system inventory cost is lower than a traditional system where shortage is allowed.

Order quantities for perishable inventory control with non-stationary demand and a fill rate constraint

We study the practical decision problem of fresh food production with a long production lead time to decide every period (e.g. week) how many items to produce. When a batch is ready for use, its items have a fixed shelf life, after which the items become waste in the sense that they cannot be sold anymore. The demand for (fresh) food products is uncertain and highly fluctuating, mainly caused by price promotions of retail organisations. We focus on cases where a so-called cycle fill rate service level requirement applies. We investigate the generation of a production plan that fixes the timing and quantity of the production for a finite time horizon. To minimise waste, one issues the oldest items first, i.e. a FIFO issuing policy. In case of out-of-stock, sales are lost.We model this decision problem as a Stochastic Programming (SP) model. The objective of our study is to find order quantities for the SP model, that approximately meet cycle fill rate service level requirements while keeping outdating low. To find approximate solutions for the SP model, an MILP model is developed. The MILP model is a deterministic approximation that generates feasible replenishment quantities in less than a second. With a scenario-based MINLP approach, optimal solutions are generated for a large sample of demand paths as a benchmark for the MILP solutions. We show that the MILP model is suitable for practical use if the setup cost is such that the replenishment cycles in the production plan are close to or of the same length as the maximum shelf life. In those cases, the expected total costs are close to the costs of the optimal solution and the average fill rate is close to the required one.

Dynamic pricing in a dual‐market environment

AbstractThis article is concerned with the determination of pricing strategies for a firm that in each period of a finite horizon receives replenishment quantities of a single product which it sells in two markets, for example, a long‐distance market and an on‐site market. The key difference between the two markets is that the long‐distance market provides for a one period delay in demand fulfillment. In contrast, on‐site orders must be filled immediately as the customer is at the physical on‐site location. We model the demands in consecutive periods as independent random variables and their distributions depend on the item's price in accordance with two general stochastic demand functions: additive or multiplicative. The firm uses a single pool of inventory to fulfill demands from both markets. We investigate properties of the structure of the dynamic pricing strategy that maximizes the total expected discounted profit over the finite time horizon, under fixed or controlled replenishment conditions. Further, we provide conditions under which one market may be the preferred outlet to sale over the other. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 531–549, 2015

Buyer–supplier relationship and optimisation model in a dynamic collaborative network with shortages allowed

To remain competitive, enterprises must build and manage product/service customisation and dynamic collaborative networks to respond to market needs in a flexible manner with competitive price and high product/service quality. This study deals with the emerging research problem of optimally dynamic combined decisions: retail price, stock depletion time/service level (SL) and replenishment schedule/quantity in a decentralised two-echelon perishable product collaborative network comprising a single supplier and a single retailer (buyer) over a finite multi-period planning horizon. Analytical solutions are derived using calculus with dynamic programming under two different trading policies, namely retailer-managed inventory with price-only (RMIPO) mode and vendor-managed inventory with consignment contract (VMICC) mode. In both models, shortages are allowed and are fully backlogged. The validity of the proposed approach is demonstrated using the case of a simple supply chain comprising a regional seafood supplier and a local branch store of a national retail chain. The results show that the VMICC policy yields lower retail price, larger replenishment quantity, higher SL and greater channel-wide profit than the RMIPO policy and achieves a win-win situation for both parties in the supply chain. Additionally, consumers benefit from lower retail price and higher SL in the VMICC policy.

Pricing and inventory decisions for deteriorating products under shipment consolidation

In this article, we consider a three-echelon supply chain including a manufacturer, a vendor and multiple retailers. That is a single-item inventory system which only produces deteriorating items with constant deteriorating rate. Demand in this chain is deterministic and price-sensitive, and lead time for replenishment is considered to be zero. The aim of this article is to determine the selling price, dispatch cycle length and replenishment quantity such that the total profit is maximized. In order to reduce transportation costs we use a kind of shipment group dispatching policy (time-based consolidation policy) in which shipments are consolidated to decrease the total costs. At the end, we present a numerical example to show the applicability of developed model and proposed solution method.

Optimal lot sizing and inventory decisions in a centralised and decentralised two echelon inventory system with price dependent demand

In the present-day scenario of global competitive environment, the consideration of the elasticity of demand has become one of the significant factors for designing effective supply chain coordination. In this paper, an integrated inventory model for optimal lot sizing and inventory decisions for centralised and decentralised supply chain with a single customer and single supplier considering ordering/setup costs, carrying costs and transportation costs of respective entities has been developed. The main contribution of this work is to demonstrate the implications of a dependence factor over the optimality of inventory levels, number of shipments and the total variable costs under price dependent demand which in turn is a function of replenishment quantity with dependence factor. A computer program is written in MATLAB as per the solution procedure developed to solve the model. Numerical example is devised to illustrate the optimality of decision variables, and objective function and also the sensitivity analysis is carried out.

Inland river terminal lake preservation: determining basin scale and the ecological water requirement

Inland lakes are a major water resource for arid regions. Determining basin scale is therefore an important indicator of variability for water resources in such regions. The goal of this study was to ascertain the appropriate Juyan Lake basin scale and its minimum ecological water requirement for the lake’s continued preservation. This was accomplished in three parts. First, an analysis was carried out on the inflow of the lower reach of Heihe River and any subsequent water surface area change to Juyan Lake. Second, an optimum Juyan Lake basin scale was determined by establishing existing relationships between reservoir capacity and lake area, water level and lake area, and variation in lake area during different times of the year. Third, the ecological water requirement of the lake was determined. Results showed that the decrease in surface runoff and ceased flow of the channel resulted in the drying up of the terminal lake of Heihe River. It was determined that 35.6 km2 is the most stable Juyan Lake surface area, and 0.55 × 108 m3 is the minimum replenishment quantity in which to maintain Juyan Lake throughout the year without it drying up.