Optimal single-replacement for repairable products with different failure rate under a finite planning horizon

This paper provides a replacement policy for repairable products with free-repair warranty (FRW) under a finite planning horizon from the consumer's viewpoint. Assume that the product is replaced once within a finite planning horizon, and the failure rate of the second product is lower than the failure rate of the first product. Within FRW, the failed product is corrected by minimal repair without any cost to the consumers. After FRW, the failed product is repaired with a fixed repair cost to the consumers. However, each failure incurs a fixed downtime cost to the consumers over a finite planning horizon. In this paper, we derive the three models of the expected total disbursement cost within a finite planning horizon and some properties of the optimal replacement policy under some reasonable conditions are obtained. Finally, numerical examples are given to illustrate the features of the optimal replacement policy under various maintenance costs.     

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.     

Periodic and sequential preventive maintenance policies over a finite planning horizon with a dynamic failure law

The purpose of this study is to propose and model periodic and sequential preventive maintenance policies for a system that performs various missions over a finite planning horizon. Each mission can have different characteristics that depend on operational and environmental conditions. These proposed preventive maintenance policies are defined and modeled mathematically. The study of these two policies is based on a dynamic system failure law that takes into account the different missions performed. The first step is to determine the optimal business plan to achieve, i.e. the set of missions to perform in order to maximize the profit of missions minus maintenance costs. Thus, for each plan, we determine the maintenance planning considering two policies. The first preventive maintenance policy is periodic and the objective is to determine the optimal number of preventive maintenance to achieve. For the second policy, namely sequential, we calculate the optimal number of preventive maintenance intervals and the duration of these different intervals.     

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.     

需求依赖于变质时间的非立即变质品联合补货优化

基于产品非立即变质的特征,构建需求依赖于变质时间的多品种联合补货库存模型,目标是使单位时间内的总成本最小.由于联合补货问题是NP难题,且考虑变质使问题变得更加复杂,针对这一难点,采用一种截断泰勒级数的方法对目标函数的指数项进行简化,提出一种基于定界的启发式算法求解模型,并通过数值案例验证算法的有效性和实用性.最后对主要参数的敏感性进行分析,为非立即变质品的零售商在实施联合补货时提供有益的管理建议.     

Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand

An inventory system for perishable items with limited replenishment capacity is introduced in this paper. The demand rate depends on the stock quantity displayed in the store as well as the sales price. With the goal to realise profit maximisation, an optimisation problem is addressed to seek for the optimal joint dynamic pricing and replenishment policy which is obtained by solving the optimisation problem with Pontryagin’s maximum principle. A joint mixed policy, in which the sales price is a static decision variable and the replenishment rate remains to be a dynamic decision variable, is presented to compare with the joint dynamic policy. Numerical results demonstrate the advantages of the joint dynamic one, and further show the effects of different system parameters on the optimal joint dynamic policy and the maximal total profit.     

Potential-based reward shaping for finite horizon online POMDP planning

In this paper, we address the problem of suboptimal behavior during online partially observable Markov decision process (POMDP) planning caused by time constraints on planning. Taking inspiration from the related field of reinforcement learning (RL), our solution is to shape the agent’s reward function in order to lead the agent to large future rewards without having to spend as much time explicitly estimating cumulative future rewards, enabling the agent to save time to improve the breadth planning and build higher quality plans. Specifically, we extend potential-based reward shaping (PBRS) from RL to online POMDP planning. In our extension, information about belief states is added to the function optimized by the agent during planning. This information provides hints of where the agent might find high future rewards beyond its planning horizon, and thus achieve greater cumulative rewards. We develop novel potential functions measuring information useful to agent metareasoning in POMDPs (reflecting on agent knowledge and/or histories of experience with the environment), theoretically prove several important properties and benefits of using PBRS for online POMDP planning, and empirically demonstrate these results in a range of classic benchmark POMDP planning problems.     

Optimal joint replenishment,delivery and inventory management policies for perishable products

In this paper we analyze the optimal joint decisions of when, how and how much to replenish customers with products of varying ages. We discuss the main features of the problem arising in the joint replenishment and delivery of perishable products, and we model them under general assumptions. We then solve the problem by means of an exact branch-and-cut algorithm, and we test its performance on a set of randomly generated instances. Our algorithm is capable of computing optimal solutions for instances with up to 30 customers, three periods, and a maximum age of two periods for the perishable product. For the unsolved instances the optimality gap is always small, less than 1.5% on average for instances with up to 50 customers. We also implement and compare two suboptimal selling priority policies with an optimized policy: always sell the oldest available items first to avoid spoilage, and always sell the fresher items first to increase revenue.     

Combined optimal price and optimal inventory replenishment policies when a sale results in increase in demand

When a supplier reduces the price of a product temporarily a buyer might place a large order and offer a sale on these units to its customers. In most cases a price discount results in an increase in demand. In this paper we relax the constant demand assumption made in most studies of inventory systems with price changes. We analyze the options available to a buyer and develop profit functions for different combinations of sales period and replenishment time and present optimal ordering policies. The paper also presents a procedure to include any relationship between price and demand to determine the combined optimal price and optimal order quantity.     

合作连锁营销下随机需求的分销系统双层补货优化控制模型

研究在每一补货间隔内需求为服从有限区间上渐变对称分布的随机变量的各零售商集体与一个地区分销中心组成的地区分销系统.分销系统中各零售商可独立决定所经营卖场的补到水平,补货周期是固定值;分销中心给各零售商送货,送货循环周期固定,各零售商补货周期等于分销中心送货循环周期.假定零售商顾客需求是均匀发生的,给出了零售商补到水平优化控制策略和分销中心进货控制策略.最后,通过一个算例验证了该模型的有效性.     

Optimal production planning for a multi-product closed loop system with uncertain demand and return

We study the production planning problem for a multi-product closed loop system, in which the manufacturer has two channels for supplying products: producing brand-new products and remanufacturing returns into as-new ones. In the remanufacturing process, used products are bought back and remanufactured into as-new products which are sold together with the brand-new ones. The demands for all the products are uncertain, and their returns are uncertain and price-sensitive. The problem is to maximize the manufacturer's expected profit by jointly determining the production quantities of brand-new products, the quantities of remanufactured products and the acquisition prices of the used products, subject to a capacity constraint. A mathematical model is presented to formulate the problem and a Lagrangian relaxation based approach is developed to solve the problem. Numerical examples are presented to illustrate the model and test the solution approach. Computational results show that the proposed approach is highly promising for solving the problems. The sensitivity analysis is also conducted to generate managerial insights.     

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.     

随机需求随机补货间隔零售商补货控制策略研究

研究分销系统中零售商的补货控制策略．分销系统中各零售商可独立决定自己的补到水平．零售商需求率是随机变量．服从某一泊松分布;分销中心循环为各零售商送货．送货间隔是随机变量．认为所有未满足的需求销售机会都丢失,零售商既要支付库存持有费用．又要支付缺货损失费用．给出了收益数学期望值函数,求出了送货间隔是均匀分布随机变量时使收益数学期望值最大化的零售商补到水平控制策略．     

考虑异质需求的再制造系统最优产量与定价研究

考虑消费者对再制造品和新产品基于价格敏感的异质性需求特征,引入再制造品和新产品的需求函数,进而构建一个考虑单一再制造商并以该再制造商利润最大化为目标的最优产量及定价模型,分别求出了各期的再制造品与新产品的最优产量及价格策略。最后通过算例分析表明,随着消费者偏好系数的增加,企业越来越趋向于生产再制造品,再制造品与新产品的价格逐渐接近,企业实施再制造不仅能减少生产成本,同时企业的总利润也得到大幅提升。     

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.     

Optimal motion planning for parallel robots via convex optimization and receding horizon

In this paper, the problem of motion planning for parallel robots in the presence of static and dynamic obstacles has been investigated. The proposed algorithm can be regarded as a synergy of convex optimization with discrete optimization and receding horizon. This algorithm has several advantages, including absence of trapping in local optimums and a high computational speed. This problem has been fully analyzed for two three-DOF parallel robots, ie 3s-RPR parallel mechanism and the so-called Tripteron, while the shortest path is selected as the objective function. It should be noted that the first case study is a parallel mechanism with complex singularity loci expression from a convex optimization problem standpoint, while the second case is a parallel manipulator for which each limb has two links, an issue which increases the complexity of the optimization problem. Since some of the constraints are non-convex, two approaches are introduced in order to convexify them: (1) A McCormick-based relaxation merged with a branch-and-prune algorithm to prevent it from becoming too loose and (2) a first-order approximation which linearizes the non-convex quadratic constraints. The computational time for the approaches presented in this paper is considerably low, which will pave the way for online applications.     

Effective replenishment policies for the multi-item dynamic lot-sizing problem with storage capacities

We address the dynamic lot-sizing problem considering multiple items and storage capacity. Despite we can easily characterize a subset of optimal solutions just extending the properties of the single-item case, these results are not helpful to design an efficient algorithm. Accordingly, heuristics are appropriate approaches to obtain near-optimal solutions for this NP-hard problem. Thus, we propose a heuristic procedure based on the smoothing technique, which is tested on a large set of randomly generated instances. The computational results show that the method is able to build policies that are both easily implemented and very effective, since they are on average 5% above the best solution reported by CPLEX. Moreover, an additional computational experiment is carried out to show that the performance of this new heuristic is on average better and more robust than other methods previously proposed for this problem.     

A partial backlogging production-inventory lot-size model for deteriorating items with time-varying production and demand rate over a finite time horizon

In this article, I extend Balkhi ((2001), ‘On a Finite Horizon Production Lot Size Inventory Model for Deteriorating Items: An Optimal Solution’, European Journal of Operational Research, 132, 210–223), by considering a generalised mathematical production-inventory model for deteriorating items with partial backlogging. The demand, production and backlogging rates are assumed to be continuous and varying with time. The objective is to find the optimal production restarting and stopping time to keep the total relevant cost as low as possible. To ascertain the optimal solution exists, the conditions for the total relevant cost in the system which attains its global minimum are provided. In addition, based on the minimum total relevant cost, an alternative among the proposed four cases is also suggested. Finally, a numerical example and sensitivity analysis is illustrated and some management insights are presented.     

Dynamic reorder point replenishment strategies for a capacitated supply chain with seasonal demand

Managing inventory and service levels in a capacitated supply chain environment with seasonal demand requires appropriate selection and readjustment of replenishment decision variables. This study focuses on the dynamic adjustment of decision variables within supply chains using continuous-review reorder point (ROP) replenishment. A framework is proposed to adjust reorder points and lot sizes based on optimal settings within different regions of a seasonal demand cycle. This framework also includes the optimal timing of adjustments defining these regions. A discrete-event simulation model of a simple, capacity-constrained supply chain is developed and simulation–optimization experiments are performed, the objective being to minimize the total supply chain inventory subject to a target delivery service level. The performance of ROP systems with optimal static and optimal dynamic decision variable settings are compared using two different seasonal demand patterns. The results confirm that performance with dynamic decision variable adjustment is better. For a given delivery service level, average work-in-process inventory levels are almost the same for both systems. However average finished goods inventory levels decrease significantly and are more stable under dynamic adjustment. The practical implication is that both finished goods holding costs and maximum storage capacity requirements are reduced.     

Determination of replenishment dates for restricted-storage,static demand,cyclic replenishment schedule

This study focuses on scheduling replenishment lots of multiple products in a warehouse with restricted storage space where the replenishment cycle time of each product is given and is an integer multiple of some basic period. Assuming that the warehouse replenishes at the beginning of some basic period, this study proposes a new heuristic that utilizes two simple procedures to generate a replenishment schedule that minimizes the maximum warehouse-space requirement. By including a re-optimization mechanism and a Boltzmann function, the proposed heuristic obtains solutions very close to the global optimum within a reasonable run time. Using randomly generated instances, this study shows that the proposed heuristic significantly outperforms a previously published heuristic.