Newsvendor with multiple options of expediting

We consider a manufacturer facing single period inventory planning problem with uncertain demand and multiple options of expediting. The demand comes at a certain time in the future. The manufacturer may order the product in advance with a relatively low cost. She can order additional amount by expediting after the demand is realized. There are a number of expediting options, each of which corresponds to a certain delivery lead time and a unit procurement price. The unit procurement price is decreasing over delivery lead time. The selling price is also decreasing over time. In this paper, we assume that the manufacturer must deliver all products to the customer in a single shipment. The problem can be formulated as a profit maximization problem. We develop structural properties and show how the optimal solution can be identified efficiently. In addition, we compare our model with the classical newsvendor model and obtain a number of managerial insights.     

A fuzzy optimization approach for procurement transport operational planning in an automobile supply chain

We consider a real-world automobile supply chain in which a first-tier supplier serves an assembler and determines its procurement transport planning for a second-tier supplier by using the automobile assembler’s demand information, the available capacity of trucks and inventory levels. The proposed fuzzy multi-objective integer linear programming model (FMOILP) improves the transport planning process for material procurement at the first-tier supplier level, which is subject to product groups composed of items that must be ordered together, order lot sizes, fuzzy aspiration levels for inventory and used trucks and uncertain truck maximum available capacities and minimum percentages of demand in stock. Regarding the defuzzification process, we apply two existing methods based on the weighted average method to convert the FMOILP into a crisp MOILP to then apply two different aggregation functions, which we compare, to transform this crisp MOILP into a single objective MILP model. A sensitivity analysis is included to show the impact of the objectives weight vector on the final solutions. The model, based on the full truck load material pick method, provides the quantity of products and number of containers to be loaded per truck and period. An industrial automobile supply chain case study demonstrates the feasibility of applying the proposed model and the solution methodology to a realistic procurement transport planning problem. The results provide lower stock levels and higher occupation of the trucks used to fulfill both demand and minimum inventory requirements than those obtained by the manual spreadsheet-based method.     

Optimal dynamic pricing of inventories with stochastic demand and discounted criterion

We consider a continuous time dynamic pricing problem for selling a given number of items over a finite or infinite time horizon. The demand is price sensitive and follows a non-homogeneous Poisson process. We formulate this problem as to maximize the expected discounted revenue and obtain the structural properties of the optimal revenue function and optimal price policy by the Hamilton-Jacobi-Bellman (HJB) equation. Moreover, we study the impact of the discount rate on the optimal revenue function and the optimal price. Further, we extend the problem to the case with discounting and time-varying demand, the infinite time horizon problem. Numerical examples are used to illustrate our analytical results.     

Dynamic Pricing via Dynamic Programming<Superscript>1</Superscript>

This article specifies an efficient numerical scheme for computing optimal dynamic prices in a setting where the demand in a given period depends on the price in that period, cumulative sales up to the current period, and remaining market potential. The problem is studied in a deterministic and monopolistic context with a general form of the demand function. While traditional approaches produce closed-form equations that are difficult to solve due to the boundary conditions, we specify a computationally tractable numerical procedure by converting the problem to an initial-value problem based on a dynamic programming formulation. We find also that the optimal price dynamics preserves certain properties over the planning horizon: the unit revenue is linearly proportional to the demand elasticity of price; the unit revenue is constant over time when the demand elasticity is constant; and the sales rate is constant over time when the demand elasticity is linear in the price. ¹We acknowledge professor robert e. kalaba for initiating this work and suggesting solution methods.     

On optimal bidding and inventory control in sequential procurement auctions: the multi period case

We consider the problem of a firm that in each cycle of a planning horizon builds inventory of identical items that it acquires by participating in auctions in order to satisfy its own market demand. The firm’s objective is to have a procurement strategy that maximizes the expected present value of the profit for an infinite planning horizon of identical cycles. We formulate this problem as a Markov decision process. We establish monotonicity properties of the value function and of the optimal bidding rule.     

On the structural properties of a discrete-time single product revenue management problem

We consider a multi-period revenue management problem in which multiple classes of demand arrive over time for the common inventory. The demand classes are differentiated by their revenues and their arrival distributions. We investigate monotonicity properties of varying problem parameters on the optimal reward and the policy.     

The price-setting newsvendor with nonlinear salvage revenue and shortage cost

We consider the price-setting newsvendor problems with linear additive demand, nonlinear salvage revenue and nonlinear shortage penalty costs. We prove existence–uniqueness of the optimal solution and investigate the equivalence of our model to related models in the literature.     

The impact of delivery lags on irreversible investment under uncertainty

We consider the valuation and rational exercise of irreversible investment opportunities in the presence of revenue uncertainty and delivery lags. In order to capture supply side market imperfections in the markets for investment goods, we assume that the lag depends on the revenue process faced by the investor. We show that such imperfections have a pronounced decelerating impact on rational investment demand as they may increase the value of waiting in excess of the exercise payoff even for projects which otherwise would be perceived as highly remunerative. We also consider the comparative static properties of the optimal investment policy and its value, and demonstrate that typically increased uncertainty decreases the investment incentives by increasing the value of waiting.     

Target market selection and marketing effort under uncertainty: The selective newsvendor

We consider a firm that markets, procures, and delivers a good with a single selling season in a number of different markets. The price for the good is market-dependent, and each market has an associated demand distribution, with parameters that depend on the amount of marketing effort applied. Given long procurement lead-times, the firm must decide which markets it will serve prior to procuring the good. We develop a profit maximizing model to address the firm’s integrated market selection, marketing effort, and procurement decisions. The model implicitly accounts for inventory pooling across markets, which reduces safety stock costs but increases model complexity. The resulting model is a nonlinear integer optimization problem, for which we develop specialized solution methods. For the case in which budget constraints exist, we provide a novel solution approach that uses a tailored branch-and-bound algorithm. Our approach solves a broad range of 3000 test instances in an average of less than 2 seconds, significantly outperforming a leading commercial global optimization solver.     

Optimizing product assortment under customer-driven demand substitution

The problem of product assortment and inventory planning under customer-driven demand substitution is analyzed and a mathematical model for this problem is provided in this paper. Realistic issues in a retail context such as supplier selection, shelf space constraints, and poor quality procurement are also taken into account. The performance of three modified models, one that neglects customers’ substitution behavior, another that excludes supplier selection decision, and one that ignores shelf space limitations, are analyzed separately with computational experiments. The results of the analysis demonstrate that neglecting customer-driven substitution or excluding supplier selection or ignoring shelf space limitations may lead to significantly inefficient assortments. The effects of demand variability and substitution cost on optimal assortment and supplier selection decisions as well as on the optimal revenue are also investigated. The main contribution of this paper is the development of a practical and flexible model to aid retailers in finding optimal assortments to maximize the expected profit.     

Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem

This paper studies the analytical and managerial implications of product substitutability on the joint pricing and procurement decisions. We consider a single-period model with two products: an existing product and an improved new product that can substitute the demand for the existing product in case of a shortage. Demand for each product follows a general distribution with an expected value that is a linear function of the price of the new product. While the price of the existing product is determined by the market, it is necessary to determine the new product’s price and the procurement quantities of both products so as to maximize the profits. We analytically show that the expected profit function is unimodal and in the existence of substitution: the expected total profit is higher; the optimal price and the safety stock of the new product are higher; and the optimal safety stock of the existing product is less. Using these properties an efficient algorithm is developed. We also provide a numerical analysis to demonstrate that considering substitution in advance could increase the profitability by 58% and the new product price by 5% while decreasing the total procurement quantity by 15%.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

Furniture supply chain tactical planning optimization using a time decomposition approach

We study the supply chain tactical planning problem of an integrated furniture company located in the Province of Québec, Canada. The paper presents a mathematical model for tactical planning of a subset of the supply chain. The decisions concern procurement, inventory, outsourcing and demand allocation policies. The goal is to define manufacturing and logistics policies that will allow the furniture company to have a competitive level of service at minimum cost. We consider planning horizon of 1 year and the time periods are based on weeks. We assume that customer’s demand is known and dynamic over the planning horizon. Supply chain planning is formulated as a large mixed integer programming model. We developed a heuristic using a time decomposition approach in order to obtain good solutions within reasonable time limit for large size problems. Computational results of the heuristic are reported. We also present the quantitative and qualitative results of the application of the mathematical model to a real industrial case.     

Planning and approximation models for delivery route based services with price-sensitive demands

Classical vehicle routing problems typically do not consider the impact of delivery price on the demand for delivery services. Existing models seek the minimum sum of tour lengths in order to serve the demands of a given set of customers. This paper proposes approximation models to estimate the impacts of price on delivery services when demand for delivery service is price dependent. Such models can serve as useful tools in the planning phase for delivery service providers and can assist in understanding the economics of delivery services. These models seek to maximize profit from delivery service, where price determines demand for deliveries as well as the total revenue generated by satisfying demand. We consider a variant of the model in which each customer’s delivery volume is price sensitive, as well as the case in which customer delivery volumes are fixed, but the total number of customers who select the delivery service provider is price sensitive. A third model variant allows the delivery service provider to select a subset of delivery requests at the offered price in order to maximize profit.     

基于收益共享的多周期易变质产品供应链协调模型

研究了单个供应商与单个零售商组成的易变质产品供应链系统在有限计划期内的协调问题。考虑到市场需求会同时受到多种因素的影响,构建了市场需求率依赖于销售价格与当前库存水平且随时间呈现一般连续变化的需求函数。将零售商的订购次数、销售价格以及供应商的批发价格作为决策变量,分别求解了集中式与分散式供应链系统下的最优策略。通过与分散式决策的利润值进行比较,得出集中式决策能够使得供应链系统的利润值至少增加1/3。利用收益共享契约机制协调此系统,求解了供应链实现完美协调时收益共享因子的取值区间。最后通过算例验证了理论结果并分析了相关参数变化对系统协调的影响。     

时变需求等量进货情形下集成供应链生产订货策略研究

构建了一个包含原料采购、生产和销售过程的集成供应链模型,研究了由原料、生产商和销售商产品构成的三层库存系统的生产订货问题。在有限的规划期内,销售商每次进货量相同,生产商按照EOQ模型采购原材料。以最小化供应链系统的总运营成本为目标,构建一个混合整数非线性规划模型,寻找销售商最优订货方案和生产商最佳生产策略。首先利用网络优化方法求解生产商的最优生产计划,其次利用定界穷举法寻求销售商最优的订货周期,给出了具体的计算方法和Matlab程序。通过算例分析验证了算法的有效性,并研究了各参数对最小费用及最优解的影响。     

Integrated revenue management approaches for capacity control with planned upgrades

In many service industries, firms offer a portfolio of similar products based on different types of resources. Mismatches between demand and capacity can therefore often be managed by using product upgrades. Clearly, it is desirable to consider this possibility in the revenue management systems that are used to decide on the acceptance of requests. To incorporate upgrades, we build upon different dynamic programming formulations from the literature and gain several new structural insights that facilitate the control process under certain conditions. We then propose two dynamic programming decomposition approaches that extend the traditional decomposition for capacity control by simultaneously considering upgrades as well as capacity control decisions. While the first approach is specifically suited for the multi-day capacity control problem faced, for example, by hotels and car rental companies, the second one is more general and can be applied in arbitrary network revenue management settings that allow upgrading. Both approaches are formally derived and analytically related to each other. It is shown that they give tighter upper bounds on the optimal solution of the original dynamic program than the well-known deterministic linear program. Using data from a major car rental company, we perform computational experiments that show that the proposed approaches are tractable for real-world problem sizes and outperform those disaggregated, successive planning approaches that are used in revenue management practice today.     

A study on the effect of yield uncertainty in price-setting newsvendor models with additive-multiplicative demand

Random yield and uncertain demand usually both exist in many industries, such as the semiconductor industry. In this paper, the price-setting newsvendor model is studied which involves a single manufacturer and a single retailer with random yield and uncertain demand respectively. Under the condition of additive-multiplicative demand, we investigate the varying effects of random yield on the optimal price, order quantity, and expected profit in two situations with different cost structures. The first case is an in-house production case where the firm pays for the raw material quantity it has ordered, and the second one is a procurement case where the firm pays for the real product quantity it receives only. By using the theory of stochastic comparisons, we find that a less variable and a stochastically larger yield rate both lead to a lower optimal price and a higher expected profit for the in-house production case. Moreover, a less variable yield rate also results in a lower optimal price and a higher profit for the procurement case, but this is not true for a stochastically larger yield rate. Numerical examples illustrate that the effect of yield randomness on the optimal order quantity is not general.     

Optimal production plans and shipment schedules in a supply-chain system with multiple suppliers and multiple buyers

This research addresses an optimal policy for production and procurement in a supply-chain system with multiple non-competing suppliers, a manufacturer and multiple non-identical buyers. The manufacturer procures raw materials from suppliers, converts them to finished products and ships the products to each buyer at a fixed-interval of time over a finite planning horizon. The demand of finished product is given by buyers and the shipment size to each buyer is fixed. The problem is to determine the production start time, the initial and ending inventory, the cycle beginning and ending time, the number of orders of raw materials in each cycle, and the number of cycles for a finite planning horizon so as to minimize the system cost. A surrogate network representation of the problem developed to obtain an efficient, optimal solution to determine the production cycle and cycle costs with predetermined shipment schedules in the planning horizon. This research prescribes optimal policies for a multi-stage production and procurements for all shipments scheduled over the planning horizon. Numerical examples are also provided to illustrate the system.     

收益管理中单产品动态定价的稳健模型研究

在收益管理的动态定价模型的研究中,由传统的确定性模型和随机模型所得到的定价策略常常受限制于需求估计的准确性,当对需求的估计出现偏差时定价策略可能达不到最大化收益的目的,因此定价策略即最优解的稳健性越来越受到研究者的重视。针对需求函数系数的不确定性,在未知需求分布的条件下,应用稳健最优化思想,提出了一种稳健的动态定价模型,并对模型的最优解和最大收益进行了数值模拟分析。