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.     

Optimal replenishment policies with allowable shortages for a product life cycle

In this paper, we investigate replenishment policies with allowable shortages by considering a general, time-varying, continuous, and deterministic demand function for a product life cycle. The objective is to optimally determine the number of inventory replenishments, the inventory replenishment time points, and the beginning time points of shortages within the product life cycle by minimizing the total relevant costs of the inventory replenishment system. The proposed problem is mathematically formulated as a mixed-integer nonlinear programming model. A complete search procedure is developed to find the optimal solution by employing the properties derived in this paper and the well-known Nelder–Mead algorithm. Also, several numerical examples and the corresponding sensitivity analyses are carried out to illustrate the features of our model by utilizing the search procedure developed in this paper.     

Modelling of the inventory replenishment problem with a two-segment piecewise linear demand

The classical inventory replenishment problem with a linear function in demand uses a ‘single-segment’ linear function as its demand and can be modelled by a simple algorithm. Moreover, this article extends the algorithm to provide a heuristic solution for the inventory replenishment model with a two-segment linear function in demand called the ‘two-segment piecewise linear demand model’. In addition, this article proposes a general procedure for solving both models. Meanwhile, several examples taken from the literature illustrate our algorithm for these two models with convincing results. Furthermore, this study shows that when the demand is a two-segment piecewise linear function over time, it is better to use the proposed algorithm rather than devising a decoupled solution approach by treating segments separately. Finally, a sensitivity analysis of two factors, demand and cost, is performed. The model is highly extensible and applicable, so it can serve as an inventory planning tool to solve the replenishment problem.     

Stackelberg–Nash equilibrium of pricing and inventory decisions in duopoly supply chains using a nested evolutionary algorithm

Pricing and inventory control in a competing environment, as separate entities, have attracted much attention from academics and practitioners. However, integrating these decisions in a competitive setting has not been significantly analyzed by academics, but is of great significance to practitioners. In this study, the joint decision on price and inventory control of a deterioration product is investigated in a duopoly setting. We consider two competing supply chains, each consisting of one manufacturer and one retailer. Each manufacturer, as the leader of their supply chain determines the wholesale price to maximize their profit, while the retailer as the follower should determine the retail price and inventory cycle to maximize his or her profit. Using a game theoretic approach, we formulate in-chain, and chain-to-chain competition as a bi-level programming problem, and analyze Stackelberg–Nash equilibrium of the problem. Furthermore, two versions of a nested algorithm are proposed to obtain the equilibrium. Both versions employ a modified threshold-accepting (TA) algorithm to solve the first level of the problem. However, while the first version utilizes the modified TA algorithm to deal with the second level of the problem, the second version applies a differential evolution (DE) approach. Eventually, a numerical study is carried out not only to compare two developed versions of the algorithm, but also to implement the sensitivity analysis of main parameters. Based on numerical experiments, although the accuracy of both versions of algorithm are alike, using TA is more computationally efficient than using DE. Furthermore, despite the permissibility of partial backlogging, it has never occurred in equilibrium points due to in-chain and chain-to-chain competition.     

Pricing for seasonal deteriorating products with price- and ramp-type time-dependent demand

A production–inventory problem for a seasonal deteriorating product is considered. It is assumed that the demand is price- and ramp-type time-dependent. The selling season for the deteriorating product is fixed. The decision maker needs to set up the price and the production schedule at the beginning of the season. Although the profit function is not concave in general, the optimal price can be determined efficiently through a simple algorithm.     

Deterministic inventory model with two levels of storage,a linear trend in demand and a fixed time horizon

A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a fixed finite time horizon. The model is formulated by assuming that the rate of replenishment is infinite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples.Scope and purposeThroughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted.In this paper, a two-storage inventory model with time-dependent demand and fixed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time.     

Scheduling concurrent production over a finite planning horizon: polynomially solvable cases

The paper analyzes a manufacturing system made up of one workstation which is able to produce concurrently a number of product types with controllable production rates in response to time-dependent product demands. Given a finite planning horizon, the objective is to minimize production cost, which is incurred when the workstation is not idle and inventory and backlog costs, which are incurred when the meeting of demand results in inventory surpluses and shortages. With the aid of the maximum principle, optimal production regimes are derived and continuous-time scheduling is reduced to a combinatorial problem of sequencing and timing the regimes. The problem is proved to be polynomially solvable if demand does not exceed the capacity of the workstation or it is steadily pressing and the costs are “agreeable”.

Scope and purpose

Efficient utilization of modern flexible manufacturing systems is heavily dependent on proper scheduling of products throughout the available facilities. Scheduling of a workstation which produces concurrently a number of product types with controllable production rates in response to continuous, time-dependent demand is under consideration. Similar to the systems considered by many authors in recent years, a buffer with unlimited capacity is placed after the workstation for each product type. The objective is to minimize inventory storage, backlog and production costs over a finite planning horizon. Numerical approaches are commonly used to approximate the optimal solution for similar problems. The key contribution of this work is that the continuous-time scheduling problem is reduced to a combinatorial problem, exactly solvable in polynomial time if demand does not exceed the capacity of the workstation or the manufacturing system is organized such that the early production and storage of a product to reduce later backlogs are justified.     

基于马尔科夫决策过程的ATO系统独立组件与产品双需求最优决策研究

研究多维组件, 单一产品的双需求型面向订单装配(Assemble-to-order, ATO)系统. 产品需求为延期交货型, 当其不被满足时将产生缺货等待成本; 而独立组件需求为销售损失型, 其不被满足时将产生缺货损失成本. 该问题可以抽象成一个动态马尔科夫决策过程(Markov decision process, MDP), 通过对双需求模型求解得到状态依赖型最优策略, 即任一组件的最优生产--库存策略由系统内其他组件的库存水平决定. 研究解决了多需求复杂ATO系统的生产和库存优化控制问题. 提出在一定条件下, 组件的基础库存值可以等价于最终产品需求的库存配给值. 组件的基础库存值与库存配给值随系统内其他组件库存的增加而增加, 而产品需求的库存配给值随系统组件库存和产品缺货量的增加而减少. 最后通过数值实验分析缺货量及组件库存对最优策略结构的影响, 并得到了相应的企业生产实践的管理启示.     

Multiproduct production/inventory control under random demands

Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the “norm-based” approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time     

Analyzing a lost-sale stochastic inventory model with Markov-modulated demands: A simulation-based optimization study

A consumer demand that presents auto-correlated components is a class of demand commonly found in competitive markets in which consumers may develop preferences for certain products which influence their willingness to purchase them again. This behavior may be observed in inventory systems whose products are subject to promotion plans in which mechanisms that incentivize the demand are implemented. Inventory systems that ignore these dependency components may severely impair their performance. This paper analyzes a stochastic inventory model where the control review system is periodic, is categorized as a lost-sale case, and is exposed to this class of auto-correlated demand pattern. The demand for products is characterized as a discrete Markov-modulated demand in which product quantities of the same item may relate to one another according to an empirical probability distribution. A simulation-based optimization that combines simulated annealing, pattern search, and ranking and selection (SAPS&RS) methods to approximate near-optimal solutions to this problem is employed. Lower and upper bounds for a range of near-optimal solutions are determined by the pattern search step enhanced by ranking and selection—indifferent zone. Results indicate that inventory performance significantly declines as the autocorrelation increases and is disregarded.     

价格波动下的生产-库存控制研究

研究原材料价格波动下多级生产-库存系统的控制问题.所有的原材料价格、半成品加工成本、成品的生产成本、库存费用率和产品的需求率都随时间变动,为此,分析了最优采购、加工、生产决策的必要和充分条件,得到了在某些假设条件下的最优生产-库存策略为JIT(Just-in-time)采购、加工、生产策略,或者为在最开始阶段以最大能力进行采购、加工、生产活动的Bang-Bang策略.     

Optimal policies for a finite-horizon batching inventory model

This paper is concerned with finding an optimal inventory policy for the integrated replenishment-production batching model of Omar and Smith (2002). Here, a company produces a single finished product which requires a single raw material and the objective is to minimise the total inventory costs over a finite planning horizon. Earlier work in the literature considered models with linear demand rate function of the finished product. This work proposes a general methodology for finding an optimal inventory policy for general demand rate functions. The proposed methodology is adapted from the recent work of Benkherouf and Gilding (2009).     

Inventory systems for deteriorating items with shortages and a linear trend in demand-taking account of time value

This paper derives an inventory model for deteriorating items with the demand of linear trend and shortages during the finite planning horizon considering the time value of money. A simple solution algorithm using a line search is presented to determine the optimal interval which has positive inventories. Numerical examples are given to explain the solution algorithm. Sensitivity analysis is performed to study the effect of changes in the system parameters.Scope and purpose The traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However, in real-life situations there is inventory loss due to deterioration. In a realistic product life cycle, demand is increasing with time and eventually reaching zero. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But in the past, the economic situation of most of the countries has changed to such an extent due to large scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any further. The purpose of this article is to present a solution procedure for the inventory problem of deteriorating items with shortages and a linear trend in demand taking account of time value.     

Lot sizing under (Q, R) policy in a capcity constrained manufacturing facility

Traditional approaches to lot sizing and inventory control consider uncertainties in either the demand or manufacturing process but not both. In these models it is usually assumed that the lead times are independent and identically distributed, but this is not realistic in many practical instances. In this paper we consider the lot sizing problem for items with stochastic demands and manufacturing lead times. It is assumed that the inventory of the finished product is controlled by continuous review policy of Q,R type—order quantity, order point system—and the problem is to determine optimal Q and R. We examine the decision parameters under a variety of conditions using exact and approximate methods.     

The vendor location problem

The vendor location problem is the problem of locating a given number of vendors and determining the number of vehicles and the service zones necessary for each vendor to achieve at least a given profit. We consider two versions of the problem with different objectives: maximizing the total profit and maximizing the demand covered. The demand and profit generated by a demand point are functions of the distance to the vendor. We propose integer programming models for both versions of the vendor location problem. We then prove that both are strongly NP-hard and we derive several families of valid inequalities to strengthen our formulations. We report the outcomes of a computational study where we investigate the effect of valid inequalities in reducing the duality gaps and the solution times for the vendor location problem.     

基于购买行为的随机生命周期易逝品库存策略

研究需求和商品生命周期均为随机的零售商库存管理问题,提出按照购买行为特征对需求进行分类,在不同类型的客户之间进行库存分配并允许缺货的库存策略.通过构建动态规划模型,求解出零售商最优库存策略,包括补货策略和库存分配策略.与先到先服务策略相比,该策略能显著提升零售商的利润,减少商品损坏的损失.     

Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale

Perishable products constitute a sizable component of inventories. A common question in a reselling situation involving a perishable (or a non-perishable) product is: what should be the size of the replenishment? When a product is highly perishable, the demand may need to be backlogged to contain costs due to deterioration. In this sense, perishability and backlogging are complementary conditions. In this paper we consider the problem of determining the lot size for a perishable good under finite production, exponential decay and partial backordering and lost sale. The problem is complex because it involves exponential and logarithmic expressions. We offer some new insights to this important managerial problem in inventory control. We show that as a constrained non-linear problem, it does have the convexity characteristics that make the problem easy to solve. Practitioners can solve this problem without resorting to Taylor's series approximation. They can solve the original problem using common non-linear programming software such as the Solver in Excel and be confident that the solution returned is the global minimum.     

An integrated local search method for inventory and routing decisions

The present article studies an inventory routing model which integrates two important components of the supply chain: transportation logistics and inventory control. The distribution system examined consists of customers that face product demand at a deterministic and constant rate. Customer demand is satisfied by a fixed vehicle fleet located at the central depot. The aim of the problem is to determine the timing and size of the replenishment services together with the vehicle routes, so that the total transportation and inventory holding cost of the system is minimized. In methodological terms, we propose a solution approach applying two innovative local search operators for jointly dealing with the inventory and routing aspects of the examined problem, and Tabu Search for further reducing the transportation costs. The proposed algorithmic framework was tested on a set of new benchmark instances of various scales. It produced satisfactory results both in terms of effectiveness and robustness.     

Regression approximation for a partially centralized inventory system considering transportation costs

Inventory centralization for multiple stores with stochastic demands reduces costs by establishing and maintaining a central ordering/distribution point. However the inventory centralization may increase the transportation costs since either the customer must travel more to reach the product, or the central warehouse must ship the product over longer distance to reach the customer. In this paper, we study a partially centralized inventory system where multiple central warehouses exist and a central warehouse fulfills the aggregated demand of stores. We want to determine the number, the location of central warehouses and an assignment of central warehouses and a set of stores. The objective is the minimization of the sum of warehouse costs and transportation cost. With the help of the regression approximation of cost function, we transform the original problem to more manageable facility location problems. Regression analysis shows that the approximated cost function is close to the original one for normally distributed demands.     

Complexity of DNA sequencing by hybridization

In the paper, the question of the complexity of the combinatorial part of the DNA sequencing by hybridization, is analyzed. Subproblems of the general problem, depending on the type of error (positive, negative), are distinguished. Since decision versions of the subproblems assuming only one type of error are trivial, complexities of the search counterparts are studied. Both search subproblems are proved to be strongly NP-hard, as well as their uniquely promised versions.