A survey of deterministic models for the EOQ and EPQ with partial backordering

Models for the basic deterministic EOQ or EPQ problem with partial backordering or backlogging make all the assumptions of the classic EOQ or EPQ model with full backordering except that only a fraction of the demand during the stockout period is backordered. In this survey we review deterministic models that have been developed over the past 40 years that address the basic models and extensions that add other considerations, such as pricing, perishable or deteriorating inventory, time-varying or stock-dependent demand, quantity discounts, or multiple-warehouses.     

Optimal advertising and pricing in a class of general new-product adoption models

In [21], Sethi et al. introduced a particular new-product adoption model. They determine optimal advertising and pricing policies of an associated deterministic infinite horizon discounted control problem. Their analysis is based on the fact that the corresponding Hamilton–Jacobi–Bellman (HJB) equation is an ordinary non-linear differential equation which has an analytical solution. In this paper, generalizations of their model are considered. We take arbitrary adoption and saturation effects into account, and solve finite and infinite horizon discounted variations of associated control problems. If the horizon is finite, the HJB-equation is a 1st order non-linear partial differential equation with specific boundary conditions. For a fairly general class of models we show that these partial differential equations have analytical solutions. Explicit formulas of the value function and the optimal policies are derived. The controlled Bass model with isoelastic demand is a special example of the class of controlled adoption models to be examined and will be analyzed in some detail.     

On an equilibrium problem with complementarity constraints formulation of pay-as-clear electricity market with demand elasticity

We consider a model of pay-as-clear electricity market based on a Equilibrium Problem with Complementarity Constraints approach where the producers are playing a noncooperative game parameterized by the decisions of regulator of the market (ISO). In the proposed approach the bids are assumed to be convex quadratic functions of the production quantity. The demand is endogenously determined. The ISO problem aims to maximize the total welfare of the market. The demand being elastic, this total welfare take into account at the same time the willingness to pay of the aggregated consumer, as well as the cost of transactions. The market clearing will determine the market price in a pay-as-clear way. An explicit formula for the optimal solution of the ISO problem is obtained and the optimal price is proved to be unique. We also state some conditions for the existence of equilibria for this electricity market with elastic demand. Some numerical experiments on a simplified market model are also provided.     

A simulation of the impacts of dynamic price management for perishable foods on retailer performance in the presence of need-driven purchasing consumers

Numerous studies have investigated dynamic pricing for perishable products. The models have been designed to determine an optimal pricing structure and improve retailer performance. Previous studies on pricing models for perishable products have considered various assumptions of consumer demand and purchasing behaviour from deterministic and stochastic price-dependent demands to myopic and strategic consumer purchasing behaviour. They have not, however, considered consumer demand in reaction to a situation where the display stock of a particular product has different qualities (such as shelf-life) and prices available at the same time. This is particularly applicable in the analysis of dynamic pricing models for perishable foods. In this paper, we investigate the impact of frequency of discount during a product’s selling period on retailer performance, by considering changes in consumer purchasing behaviour in response to the display stock of a particular food product having different remaining shelf-life and prices. On the basis of a literature review and data obtained from interviews with food retailers, a simulation study is performed to compare the performance of different pricing policies. The results demonstrate the benefits gained by adopting more dynamic price policies.     

Pricing American options with uncertain volatility through stochastic linear complementarity models

We consider the problem of pricing American options with uncertain volatility and propose two deterministic formulations based on the expected value method and the expected residual minimization method for a stochastic complementarity problem. We give sufficient conditions that ensure the existence of a solution of those deterministic formulations. Furthermore we show numerical results and discuss the usefulness of the proposed approach.     

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.     

Efficient formulations for pricing under attraction demand models

We propose a modeling and optimization framework to cast a broad range of fundamental multi-product pricing problems as tractable convex optimization problems. We consider a retailer offering an assortment of differentiated substitutable products to a population of customers that are price-sensitive. The retailer selects prices to maximize profits, subject to constraints on sales arising from inventory and capacity availability, market share goals, bounds on allowable prices and other considerations. Consumers’ response to price changes is represented by attraction demand models, which subsume the well known multinomial logit (MNL) and multiplicative competitive interaction demand models. Our approach transforms seemingly non-convex pricing problems (both in the objective function and constraints) into convex optimization problems that can be solved efficiently with commercial software. We establish a condition which ensures that the resulting problem is convex, prove that it can be solved in polynomial time under MNL demand, and show computationally that our new formulations reduce the solution time from days to seconds. We also propose an approximation of demand models with multiple overlapping customer segments, and show that it falls within the class of demand models we are able to solve. Such mixed demand models are highly desirable in practice, but yield a pricing problem which appears computationally challenging to solve exactly.     

A joint optimal pricing,rebate value,and lot sizing model

Delayed incentives in the form of mail-in cash rebates are very popular among manufacturers, and more recently, among retailers. One of the main advantages of rebates is that while they increase demand, a small proportion of consumers redeem them. In this paper, we formulate and solve models for jointly determining the optimal price, rebate face value, and the optimal order quantity for a price and rebate sensitive deterministic demand. The models show that under realistic conditions, offering rebates can have significant pricing and inventory policy implications and can lead to a significant increase in profit.     

Revenue management for low-cost providers

Low-cost providers have emerged as important players in many service industries, the most predominant being low-cost, or the so-called discount airlines. This paper presents models and results leading toward understanding the revenue management outlook for a discount pricing firm. A framework and model is formulated specifically for the airline industry, but is generalizable to low-cost providers in similar revenue management settings. We formulate an optimal pricing control model for a firm that must underprice to capture a segment of exogenous demand. Two specific model formulations are considered: a continuous deterministic version, and a discrete stochastic version. Structural results are derived for the deterministic case, providing insight into the general form of optimal underpricing policies. The stochastic results support the structural insight from the deterministic solution, and illuminate the effect of randomness on the underpricing policies.     

模糊需求下订购批量问题的研究

库存管理模型在现实生活中有着广泛的运用,它为管理决策者有效地确定最佳订购批量提供帮助.然而,由于历史数据的缺乏,需求量在很多情况下往往被主观地确定,因而带有一定的模糊性.本文针对两种不同类型的模糊需求:离散型与连续型,运用模糊理论分别建立了相应的模糊库存模型.该模型不同于已有的模糊库存模型如下:在现有的模糊库存的文献中,大多采用的是利用模糊集的知识对确定EOQ模型加以研究,而本文从模糊理论的角度对报童问题进行研究.