Optimal price and order size for a reseller under partial backordering

The problem of determining the optimal price and lot size for a reseller is considered in this paper. It is assumed that demand can be backlogged and that the selling price is constant within the inventory cycle. The backlogging phenomenon is modeled without using the backorder cost and the lost sale cost since these costs are not easy to estimate in practice. The case in which the selling price is fixed and therefore, demand is a known constant is also considered. Given the new way of modeling the backlogging phenomenon, the results for the case of constant demand are developed. Analysis is also presented for the reselling situation in which a nonperishable product is sold.Scope and purposePerishable products constitute a sizable component of inventories. A common question in a reselling situation involving a perishable (or a nonperishable) product is: What should be the size of the replenishment? If demand for the product is sensitive to price, then another question is: What should be the selling price? Although the ability to vary price within an inventory cycle is important, in many cases, the reseller may opt for a policy of constant selling price for administrative convenience. In this paper the pricing and/or lot sizing problem faced by a reseller is modeled assuming a general deterioration rate and a general demand function. The model allows for backlogging of demand. When a product is highly perishable, the reseller may need to backlog demand to contain costs due to deterioration. In this sense, perishability and backlogging are complementary conditions. Given that the problem entails revenue and costs, a natural objective function for the model is profit per period. The conventional approach to modeling the backlogging phenomenon requires the use of the backorder cost and the lost sale cost. These costs, however, are difficult to estimate in practice. A new approach is used in which customers are considered impatient. Hence the fraction of demand that gets backlogged at a given point in time is a decreasing function of waiting time. First the subproblem in which price is fixed is solved to determine the optimal inventory policy. The subproblem represents the important case in which the reseller has no flexibility to change the selling price. Then a procedure is developed for determining the optimal quantity and the selling price for the broader problem. The procedure can be implemented on a spreadsheet.     

Pricing and inventory carryover strategy considering cost learning effect and strategic consumers

Anticipating the markdown in the future price due to the cost reduction of the firm, more and more consumers tend to delay their purchasing to the later period. Incorporating the strategic behavior of consumers, this paper establishes a two-period production and selling model under dynamic pricing strategy and price commitment strategy respectively, with considering stochastic learning effect in which the firm may or may not have the inventory carryover option. The results show that the firm may hold inventory under dynamic pricing while no inventory is kept under price commitment. Additionally, consumers' patience level enhances the benefit of inventory carryover. Furthermore, for the firm with relatively high farsighted level and low inventory holding cost, the dynamic pricing equipped with inventory carryover outperforms the price commitment strategy. Finally, numerical examples are conducted to analyze the impacts of important parameters on the firm's choice of pricing strategies.     

Joint pricing and replenishment policies for non-instantaneous deteriorating items with imprecise deterioration free time and credibility constraint

This study develops an inventory model for non-instantaneous deteriorating items with imprecise deterioration free time and credibility constraint. The model assumes price sensitive demand when the product has no deterioration and price and time dependent demand when the product has deterioration. Under these considerations, the study attempts to offer best policy for selling price and replenishment cycle for the retailer that aims at maximizing the total profit per unit time. Making use of nearest interval approximation and interval arithmetic, the single objective problem is transformed to multi objective problem. Employing Weighted Sum Method, an analytical approach along with simple algorithm is developed to identify Pareto optimal solution. Finally, the behavior of the model with varied parameters is illustrated in numerical examples.     

Investigating imperfect process and demand effects on inspection scheduling and supply chain replenishment policy

The challenge of supply chain management (SCM) is to have a supervision in managing operations that is critical to coordinate with partners and have high quality product. Global companies are now striving to engage in production improvement and product design efficiency to provide a good support for competition in any parts of the world. Recently, the development and improved capability of production and information technology have changed the ways in which companies and manufacturers operate. With the assist of advanced production philosophy and collaborative design and manufacturing (CDM), the quality of product and manufacturing has been notably enhanced. However, a traditional single-stage inventory model considers a case in which depletion of inventory is merely caused by a constant rare. In practice, there is an inventory loss due to deterioration and imperfect production. The purpose of this paper is to investigate the effects of stock- and warranty-dependent demand, and take the relationship among imperfect, warranty policy and inspection scheduling into consideration. An integrated two-stage production inventory deteriorating model for replenishment policy and inspection plan is developed utilizing time-weighted inventory approach. A numerical example is presented to demonstrate the theory. The results show that fixed selling rate, the holding cost and the unit inspection cost are the critical factors in applying the deteriorating inventory model.     

The value of modeling with reference effects in stochastic inventory and pricing problems

We analyze a periodic review inventory system in which the random demand is contingent on the current price and the reference price. The reference price captures the price history and acts as a benchmark against which the current price is compared. The randomness is due to additive and multiplicative random terms. The objective is to maximize the discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We study three key issues using numerical computation and simulation. First, we study the effects of reference price mechanism on the total expected profit. It is shown that high dependence on a good history increases the profit. Second, we investigate the value of dynamic programming and show that the firm that ignores the dynamic structure suffers from the revenue. Third, we analyze the value of estimating the correct demand model with reference effects. We observe that this value is significant when the inventory related costs are low.     

Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach

In this article, we consider an infinite horizon, single product economic order quantity where demand and deterioration rate are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and completely backlogged. The objective is to find the optimal inventory and pricing strategies maximizing the net present value of total profit over the infinite horizon. For any given selling price, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use a couple of numerical examples to illustrate the algorithm.     

Pricing and inventory policies for Hi-tech products under replacement warranty

Companies, especially in the Hi-tech (high-technology) industry (such as computer, communication and consumer electronic products), often provide a replacement warranty period for purchased items. In reality, simultaneously determining the price and inventory decisions under warranty policy is an important issue. The objective of this paper is to develop a joint pricing and inventory model for Hi-tech products under replacement warranty policy. In the first model, we consider a Hi-tech product feature in which the selling price is declining in a trend. We determine the optimal inventory level for each period and retail price for the first period while maximising the total profit. In the second model, we further determine the optimal retail price and inventory level for each period in the dynamic demand market. This study develops solution approaches to solve the problems described above. Numerical analysis discusses the influence of system parameters on the company's decisions and behaviours. The results of this study could serve as a reference for business managers or administrators.     

A single-period perishable inventory model where deterioration begins at a random point in time

This paper develops a single-period inventory model where the stock begins to deteriorate at a random point in time within the period. Once this deterioration begins, the selling price can be adjusted once at that point. Optimum order size and pricing policies are determined for such an inventory model through a numerical illustration.     

Supplier selection and procurement decisions with uncertain demand,fixed selection costs and quantity discounts

In this paper, we study the supplier selection and procurement decision problem with uncertain demand, quantity discounts and fixed selection costs. In addition, a holding cost is incurred for the excess inventory if the buyer orders more than the realized demand and the shortage must be satisfied by an emergent purchase at a higher price otherwise. The objective is to select the suppliers and to allocate the ordering quantity among them to minimize the total cost (including selecting, procurement, holding and shortage costs, etc.). The problem is modeled as a Mixed Integer Programming (MIP) and is shown to be NP-hard. Some properties of the optimal policy are provided and an optimal algorithm is proposed based on the generalized Bender's decomposition. Numerical experiments are conducted to show the efficiency of the algorithm and to obtain some managerial insights.     

Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price-sensitive demand

This article will formulate and solve an inventory system with non-instantaneous deteriorating items and price-sensitive demand. The purpose of this study is to determine the optimal selling price and the length of replenishment cycle such that the total profit per unit time has a maximum value for the retailer. We first establish a proper model for a mathematical formulation. Then we develop several theoretical results and provide the decision-maker with an algorithm to find the optimal solution. Finally, two numerical examples are provided to illustrate the solution procedure, and a sensitivity analysis of the optimal solution with respect to major parameters is carried out.     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

Dynamic pricing,promotion and replenishment policies for a deteriorating item under permissible delay in payments

We study the problem of dynamic pricing, promotion and replenishment for a deteriorating item subject to the supplier's trade credit and retailer's promotional effort. In this paper we adopt a price- and time-dependent demand function to model the finite time horizon inventory for deteriorating items. The objective of this paper is to determine the optimal retail price, the promotional effort and the replenishment quantity so that the net profit is maximized. We discuss the properties and develop an algorithm for solving the problem described. The numerical analyses show that an appropriate promotion policy can benefit the retailer and that the promotion policy is important, especially for deteriorating items. Furthermore dynamic decision-making is shown to be superior to fixed decision-making in terms of profit maximization. Some special cases, such as with no credit period and for non-deteriorating items, are discussed as is the influence of the time-varying demand, the rate of deterioration and the credit period on the retailer behavior.     

Optimal inventory control policy and supply chain coordination problem with carbon footprint constraints

Carbon footprint constraints exert pressure on supply chains to reexamine decisions. In this paper, we consider carbon transfer cost and carbon holding cost in a supply chain. A multiperiod dynamic programming model with carbon footprint constraints is presented to investigate the impact of carbon transfer cost and carbon holding cost on inventory control policy as well as the supply chain coordination problem. A two‐control limit inventory control policy is proved to be optimal and a contract with wholesale price, subsidy, and fixed setup cost is verified analytically to coordinate the supply chain. Finally, a numerical study is conducted to reveal managerial insights. We find that when the supply chain is coordinated, the chain's profit is more sensitive to carbon transfer cost while inventory level is more sensitive to carbon holding cost. Additionally, because of the complexity of the coordinated contract, when it is not easy to coordinate the supply chain, it is better to keep the values of wholesale price, subsidy, and fixed setup cost below the corresponding values for the coordinated supply chain.     

动态定价策略下的精确库存成本建模与优化

提出一种更接近实际的需求率公式,在式中同时考虑了价格和出厂时间对客户需求率的影响.基于新的需求率公式,建立了动态定价策略下的精确库存持有成本模型和库存商品的利润函数.注意到利润函数的复杂性,使用遗传算法分析了利润函数的性质,得出最优定价时间、定价价格和最大利润的关系,并分析了库存持有成本变化和消费者购买欲望变化对各定价参数的影响.     

A fuzzy framework for coordinating pricing and inventory policies for deteriorating items under retailer partial trade credit financing

In this paper, we proposed a generalized economic order quantity (EOQ) – based inventory model using a trade credit policy in a fuzzy sense. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer, in turn, partially provides customers a permissible delay period N. This study considers fuzzy EOQ model to allow for: (1) selling price dependent demand rate which is imprecise in nature, (2) a profit maximization objective and (3) an imprecise holding cost, ordering cost, purchasing cost, interest earned and interest charged rate. Besides, the cases N ? M and N ? M are explored thoroughly. The objective function for the retailer in fuzzy sense is defuzzified using Modified Graded Mean Integration Representation Method. For the defuzzified objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.     

价格影响需求的易变质产品动态定价模型

在产品销售价格影响需求的条件下,利用最优控制理论建立了易变质产品的动态定价模型,目标是最大化产品销售周期内总的销售利润。利用Pontryagin 最大值原理得到了产品销售价格的最优性条件。通过对模型的理论分析得出如果产品销售价格介于单位产品购买费用和产品销售价格上限之间,且产品库存在销售周期结束之前始终为正时,销售周期内各时刻的产品最优销售价格一定大于与相应时刻变质率和产品单位库存成本有关的一个下界,销售周期内各时刻的产品最优库存水平一定小于与相应时刻变质率和产品单位库存成本有关的一个上界。     

A return-on-inventory-investment maximization model for intermediate firms

In a channel distribution system, the function of an intermediate firm is to purchase products and to sell those purchased products to the public or to other firms. In this paper, for the intermediate firms, an inventory model under return-on-inventory-investment maximization is proposed to determine optimally the quality level, the selling quantity and the purchasing price of a product. The selling price and the supply rate of the product as well as the fixed selling cost are assumed to be power functions of one or more of the decision variables. Under this assumption, the global optimal closed-form solution is derived for the inventory model by utilizing geometric programming techniques. In addition, sensitivity analyses on primal and dual geometric programming problems for the inventory model are presented.     

需求依赖库存且短缺量部分拖后的促销商品库存模型

促销商品是商场吸引顾客前往购买消费的一种重要手段,它可以有效带动其他商品的销售从而提高商场销售收入。考虑促销商品在缺货期间价格和时间对顾客等待行为的影响,构造了一个与销售价和等待时间相关的短缺量拖后率,建立了多次订货下两阶段存货影响需求和顾客等待的促销商品库存模型,并利用仿真方法分析价格和时间敏感因子、存货影响需求临界点、销售价格对销售商订货策略和系统总利润的影响。结果表明：价格和时间敏感因子对各周期服务水平影响显著,存货影响需求临界点对订货次数影响较大;当销售价在一定范围时,销售商只需调整各周期服务水平,而当销售价过高或过低时,销售商则需同时调整各周期服务水平和订货次数     

The joint replenishment problem involving multiple suppliers offering quantity discounts

In this article, we deal with the problem of determining the economic operating policy when a number of items are to be procured from a number of suppliers offering different quantity discounts schedules. In such inventory problems, a fixed cost is incurred with each replenishment order, independent of the suppliers as well as the items involved in the order. Further, the item involves a minor fixed cost. In such a system, it includes the supplier selection problem when considering the quantity discounts as well as the general joint replenishment problem. We develop a hybrid genetic algorithm for this NP-hard decision problem and extend it to systems with resource restrictions.     

Competitive Delivered Pricing by Mail-Order and Internet Retailers

Mail-order and internet sellers must decide how customers pay shipping charges. Typically, these sellers choose between two pricing policies: either “uniform pricing,” where the firm delivers to any customer at a fixed delivery charge (that may be volume dependent), or “mill pricing,” where the firm bills the customer a distance-related shipping charge. This paper studies price competition between a mail-order (or internet) seller and local retailers, and the mail-order firm’s choice of pricing policy. The price policy choice is studied when retailers do not change price in reaction to the mail-order firm’s policy choice, and when they do. In the second case, a two-stage non-cooperative game is used and it is found that for low customer willingness to pay, mill pricing is favored but as willingness to pay rises, uniform pricing becomes more attractive. These results are generalized showing that larger markets, higher transportation rates, higher unit production cost, and greater competition between retailers all increase profit under mill pricing relative to uniform pricing (and vice versa). On the other hand, cost asymmetries that favor the mail-order firm will tend to induce uniform rather than mill pricing. Some empirical data on retail and mail-order sales that confirm these results are presented.