具有两类需求服务的随机库存系统控制策略研究

研究具有两类顾客排队需求服务的随机库存系统.系统采取(s,Q)补货策略且当库存水平下降到安全库存s时,到达的第二类顾客以概率P得到服务.首先,建立库存水平状态转移方程并通过递推算法求解获得库存水平稳态概率分布和系统稳态指标;接下来,构建库存成本函数;最后,采用数值试验的方法研究该库存系统的最优控制策略并考察系统参数的敏感性.     

Optimal production control of a dynamic two-product manufacturing system with setup costs and setup times

This paper deals with the optimal control of a one-machine two-product manufacturing system with setup changes, operating in a continuous time dynamic environment. The system is deterministic. When production is switched from one product to the other, a known constant setup time and a setup cost are incurred. Each product has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a feedback control problem. The objective is to minimize the total backlog, inventory and setup costs incurred over a finite horizon. The optimal solution provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady state, the optimal cyclic schedule is determined. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region, the optimal control policy is determined analytically.     

Stability analysis and optimization of an inventory system with bounded orders

This paper addresses the control of a one-item inventory system subject to random order lead time and random demand. The key parameter of the control policy is the objective inventory. In each period, the order to be placed brings the inventory position as close as possible to the objective inventory. The order of each period is kept between a lower bound and an upper bound. We show that the distribution of the inventory level converges to its stationary distribution provided that the lower bound is smaller than the average demand, the upper bound is greater than the average demand and some regularity conditions hold. The average inventory cost is shown to be a convex function of the objective inventory level. A simulation-based approach is proposed for the determination of the optimal objective inventory. A method of bisection with derivative is then used to determine the optimal objective inventory. The derivatives needed in various iterations of this method are estimated using a single sample path with respect to a given objective inventory. Numerical results are provided.     

Optimal pricing and inventory policies: Centralized and decentralized decision making

The paper studies an optimal control problem of pricing and inventory replenishment in a system with serial inventories. Consumer demand for a specific product at a retail outlet depends on price as well as the in-store stock of the product. The hypothesis is that for certain consumer products, a large volume of displayed goods leads consumers to buy more than if the stock is small. In addition to the stock that is on display in the store, there is an inventory of the product in a central warehouse. First we consider a setup in which management of the two stocks is decentralized such that pricing decisions are made by the store manager who also decides on the level of in-store inventory. The warehouse manager makes the replenishment decisions concerning the stock in the warehouse. Next we study the problem where stock management and pricing decisions are centralized. Optimal trajectories for inventories, replenishment rates, and retail price are derived by using phase diagrams and a formal synthesizing procedure.     

Transient and steady-state analysis of a manufacturing system with setup changes

This paper deals with the optimal scheduling of a one-machine two-product manufacturing system with setup, operating in a continuous time dynamic environment. The machine is reliable. A known constant setup time is incurred when switching over from a part to the other. Each part has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a production flow control problem. The objective is to minimize the sum of the backlog and inventory costs incurred over a finite planning horizon. The global optimal solution, expressed as an optimal feedback control law, provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady-state, the optimal cyclic schedule (Limit Cycle) is determined. This is equivalent to solving a one-machine two-product Lot Scheduling Problem. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region is associated an optimal control policy. A novel algorithm (Direction Sweeping Algorithm) is developed to obtain the optimal state trajectory (optimal policy that minimizes the sum of inventory and backlog costs) for this last case.     

Inverse optimization: towards the optimal parameter set of inverse LP with interval coefficients

We study the inverse optimization problem in the following formulation: given a family of parametrized optimization problems and a real number called demand, determine for which values of parameters the optimal value of the objective function equals to the demand. We formulate general questions and problems about the optimal parameter set and the optimal value function. Then we turn our attention to the case of linear programming, when parameters can be selected from given intervals (“inverse interval LP”). We prove that the problem is NP-hard not only in general, but even in a very special case. We inspect three special cases—the case when parameters appear in the right-hand sides, the case when parameters appear in the objective function, and the case when parameters appear in both the right-hand sides and the objective function. We design a technique based on parametric programming, which allows us to inspect the optimal parameter set. We illustrate the theory by examples.     

Analysis of deteriorating inventory/production systems using a linear quadratic regulator

We consider an inventory-production system where items deteriorate at a constant rate. The objective is to develop an optimal production policy that minimizes the cost associated with inventory and production rate. The inventory problem is first modeled as a linear optimal control problem. Then linear quadratic regulator (LQR) technique is applied to the control problem in order to determine the optimal production policy. Examples are solved for three different demand functions. Sensitivity analysis is then conducted to study the effect of changing the cost parameters on the objective function.     

替代产品的动态库存决策与控制

考虑了替代产品的动态库存决策与控制问题,建立了替代产品的多周期动态库存决策与控制模型.得到了目标函数的一些重要性质,给出了系统最优参数的求解算法,利用动态规划方法对系统的库存参数进行了优化求解.     

Layoff costs and underutilization of labour in fisheries

The cost of reducing the labour force during a transition from an overexploited fishery to a bionomic fishery is taken into account. This affects both the long run steady state and the optimal approach to steady state. These effects are illustrated using the case of the NortheastArctic cod stock as a stylized example. The method outlined represents an operational way to assess harvest quotas as well as effort quotas both in the steady state and not least on the path to steady state. In the steady state analysis completely general functional forms are used, whereas in the optimal path analysis the objective function is required to be quadratic in the control variable. This requirement, however, incorporates the most important sources of nonlinearities such as downward sloping demand and increasing marginal costs.     

Robust distribution planning for supplier-managed inventory agreements when demand rates and travel times are stationary

Supplier-managed inventory (SMI) is a partnering agreement between a supplier and his customers. Under this SMI agreement, inventory monitoring and ordering responsibilities are entirely transferred to the supplier. Subsequently, the supplier decides both the quantity and timing of his customer deliveries. The inventory routing problem is an underlying optimization model for SMI partnerships to cost-effectively coordinate and manage customer inventories and related replenishments logistics. This paper discusses the case where customer demand rates and travel times are stochastic but stationary, and proposes a version of the inventory routing optimization model that generates optimal robust distribution plans. The approach proposed to obtain and deploy these robust plans combines optimization and Monte Carlo simulation. Optimization is used to determine the robust distribution plan and simulation is used to fine-tune the plan's critical parameters such as replenishment cycle times and safety stock levels. Results of a simplified real-life case implementing the proposed optimization-simulation approach are shown and discussed in detail.     

Zero-Inventory Conditions for a Two-Part-Type Make-to-Stock Production System

We consider the dynamic scheduling of a two-part-type make-to-stock production system using the model of Wein [12]. Exogenous demand for each part type is met from finished goods inventory; unmet demand is backordered. The control policy determines which part type, if any, to produce at each moment; complete flexibility is assumed. The objective is to minimize average holding and backorder costs. For exponentially distributed interarrival and production times, necessary and sufficient conditions are found for a zero-inventory policy to be optimal. This result indicates the economic and production conditions under which a simple make-to-order control is optimal. Weaker results are given for the case of general production times.     

随机需求下考虑半成品库存的多周期生产决策优化

为了更好地应对需求的不确定性,在需求实现之前,企业既可以生产成品直接满足需求,亦可生产部分半成品,在观察到实际需求之后短时间内迅速完成剩余生产环节以满足需求。未加工的半成品和未售出的成品可用于满足后续周期的需求。作为一种提高生产灵活性的手段,分阶段生产的方式会产生更高的成本。企业需要在成本和灵活性之间作出权衡,优化生产决策。模型通过动态规划的方法,研究需求不确定情况下考虑半成品库存的多周期生产决策问题,通过分析目标函数以及最优值函数的结构性质,推导出最优的多周期生产策略为修正的目标库存策略,并且分析了不同参数对最优策略的影响。     

Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns

This paper deals with an economic production quantity inventory model for non-instantaneous deteriorating items under inflationary conditions considering customer returns. We adopt a price- and time-dependent demand function. Also, the customer returns are considered as a function of both price and demand. The effects of time value of money are studied using the Discounted Cash Flow approach. The main objective is to determine the optimal selling price, the optimal replenishment cycles, and the optimal production quantity simultaneously such that the present value of total profit is maximized. An efficient algorithm is presented to find the optimal solution. Finally, numerical examples are provided to solve the presented inventory model using our proposed algorithm, which is further clarified through a sensitivity analysis. The results of analysing customer returns provide important suggestions to financial managers who use price as a control to match the quantity sold to inventory while maximizing revenues. The paper ends with a conclusion and an outlook to future studies.     

Optimizing delivery lead time/inventory placement in a two-stage production/distribution system

In this paper we study a system composed of a supplier and buyer(s). We assume that the buyer faces random demand with a known distribution function. The supplier faces a known production lead time. The main objective of this study is to determine the optimal delivery lead time and the resulting location of the system inventory. In a system with a single-supplier and a single-buyer it is shown that system inventory should not be split between a buyer and supplier. Based on system parameters of shortage and holding costs, production lead times, and standard deviations of demand distributions, conditions indicating when the supplier or buyer(s) should keep the system inventory are derived. The impact of changes to these parameters on the location of system inventory is examined. For the case with multiple buyers, it is found that the supplier holds inventory for the buyers with the smallest standard deviations, while the buyers with the largest standard deviations hold their own inventory.     

Simple Models and Insights for Warehouse Sizing

This study is concerned with minimizing the total discounted cost of operating an inventory system and providing the warehouse space necessary to accommodate the replenishment lots, under the assumption of constant product demand. The use of an approximation objective function for the single-item case allows the optimal warehouse size as well as the ratio of relevant investment costs to relevant inventory costs to be written in closed-form. Based upon the value of this ratio, circumstances are identified under which an integrated approach is justified, and others under which the inventory policy and storage capacity can be determined sequentially. The multi-item version of the problem under study is solved by the Lagrangian multiplier method, given that no coordination takes place between the items. Finding the optimal Lagrange multiplier can be accomplished efficiently by the Newton–Raphson method.     

On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time

This paper considers the impact of random machine breakdowns on the classical Economic Production Quantity (EPQ) model for a product subject to exponential decay and under a no-resumption (NR) inventory control policy. A product is manufactured in batches on a machine that is subject to random breakdowns in order to meet a constant demand over an infinite planning horizon. The product is assumed to have a significant rate of deterioration and time to deterioration is described by an exponential distribution. Also, the time-to-breakdown is a random variable following an exponential distribution. Under the NR policy, when a breakdown occurs during a production run, the run is immediately aborted. A new run will not be started until all available inventories are depleted. Corrective maintenance of the production system is carried out immediately after a breakdown and it takes a fixed period of time to complete such an activity. The objective is to determine the optimal production uptime that minimizes the expected total cost per unit time consisting of setup, corrective maintenance, inventory carrying, deterioration, and lost sales costs. A near optimal production uptime is derived under conditions of continuous review, deterministic demand, and no shortages.     

Optimal production,purchasing and pricing: A differential game approach

This paper deals with a problem of determining optimal production and pricing policies of a manufacturing firm which is supplying a retailer, The latter faces a price-dependent demand function towards the final consumers and wishes to determine optimal purchasing and pricing policies. Both firms carry inventories and backlogging is permitted. The problem is modelled as a two-player nonzero-sum differential game with the inventory levels as the state variables. The controls are the rates of production and purchasing as well as the prices. Assuming linear production costs, but convex ordering and holding costs, open-loop Nash controls are characterized by using switching point analysis as well as phase diagrams.     

An Incomplete Information Inventory Model with Presence of Inventories or Backorders as Only Observations

In many real-life contexts, inventory levels are only incompletely observed due to non-observation of demand, discrepancies in transmitting sales data, transaction errors, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system, where the demand is not observed and the unmet demand is backordered. As a result, the inventory manager cannot tell the exact quantities of inventories or backorders. However, by looking at the shelf, he knows whether the inventory is positive or nonpositive. Only with this information, the inventory manager must determine the order quantity in each period that would minimize the expected total discounted cost over an infinite-horizon. The dynamic programming formulation of this problem has an infinite-dimensional state space. We use the concept of unnormalized probability to establish the existence of an optimal feedback policy and the uniqueness of the solution of the dynamic programming equation when the periodic cost has linear growth.     

A Semi-Markov decision problem for proactive and reactive transshipments between multiple warehouses

Lateral transshipments are an effective strategy to pool inventories. We present a Semi-Markov decision problem formulation for proactive and reactive transshipments in a multi-location continuous review distribution inventory system with Poisson demand and one-for-one replenishment policy. For a two-location model we state the monotonicity of an optimal policy. In a numerical study, we compare the benefits of proactive and different reactive transshipment rules. The benefits of proactive transshipments are the largest for networks with intermediate opportunities of demand pooling and the difference between alternative reactive transshipment rules is negligible.     

Optimal control policy for a standing order inventory system

In this paper, we consider a standing order inventory system in which an order of fixed size arrives in each period. Since demand is stochastic, such a system must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory. Assuming the average-cost criterion, Rosenshine and Obee (Operations Research 24 (1976) 1143–1155) first studied such a system and devised a 4-parameter inventory control policy that is not generally optimal. The current paper uses dynamic programming to determine the optimal control policy for a standing order system, which consists of only two operational parameters: the dispose-down-to level and order-up-to level. Either the average-cost or discounted-cost criterion can be assumed in the proposed model. Also, both the backlogged and lost-sales problems are investigated in this paper. By using a convergence theorem, we stop the dynamic programming computation and obtain the two optimal parameters.