Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost

This paper deals with an extended EOQ-type inventory model for a perishable product where the demand rate is a function of the on-hand inventory. The traditional parameters of unit item cost and ordering cost are kept constant; but the holding cost is treated as (i) a nonlinear function of the length of time for which the item is held in stock, and (ii) a functional form of the amount of the on-hand inventory. The approximate optimal solution in both the cases are derived. Computational results are presented indicating the effects of nonlinearity in holding costs.     

变库存费的变质性物品的最优订货策略

经典的 EOQ模型所解决的问题都是视库存费为常数 ,在这篇文章中 ,我们研究了库存费的二种可能的变化 :(A)库存费的变化率为存贮时间的函数 ;(B)库存费的变化率为库存量的函数 .揭示了变库存费对库存系统最优订货策略的影响     

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.     

Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost

This paper studies a deterministic inventory model with a stock-dependent demand pattern where the cumulative holding cost is a non-linear function of both time and stock level. When the monetary resources are limited and the inventory manager can invest his/her money in buying different products, it seems reasonable to select the ones that provide a higher profitability. Thus, a new approach with the aim of maximizing the profitability ratio (defined as the profit/cost quotient) is considered in this paper. We prove that the profitability ratio maximization is equivalent to minimizing the inventory cost per unit of an item. The optimal policy is obtained in a closed form, whose general expression is a generalization of the classical EOQ formula for inventory models with a stock-dependent demand rate and a non-linear holding cost. This optimal solution is different from the other policies proposed for the problems of minimum cost or maximum profit per unit time. A complete sensitivity analysis of the optimal solution with respect to all the parameters of the model is developed. Finally, numerical examples are solved to illustrate the theoretical results and the solution methodology.     

An integrated inventory model under conditions of order processing cost reduction and permissible delay in payments

This article develops an integrated inventory model to determine the optimal policy under conditions of order processing cost reduction and permissible delay in payments. Both the vendor and the buyer participate in order processing cost reduction by applying information technologies. The order processing cost can be reduced by certain expenditures and will affect lot-size decisions. Simultaneously, the existence of the credit period serves to reduce the cost of holding stock to the buyer, because it reduces the amount of capital invested in stock for the duration of the credit period. The article derives the total cost function and shows that the function possesses some kinds of convexities. A solution procedure is provided to determine the optimal order policy. Finally, numerical examples are presented to illustrate the solution procedure.     

需求率依赖于库存水平的离散性库存费的库存模型

传统的库存控制模型都视需求率为固定不变的,放松了这个假定,通过考虑库存费为存储时间的阶梯函数的情形:(1)全单位库存费用,(2)增量库存费用,并且在需求率依赖于库存水平,当库存水平下降到一定程度时,需求率变为常数的形式下,把变化的订购费引入,发展了两个离散性库存费的变质物品的库存控制模型。在模型中允许周期末库存水平不为零,并且提出了最优解的算法。     

Continuous-review,lost-sales inventory models with Poisson demand,a fixed lead time and no fixed order cost

This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.     

Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

带有可变库存费用和短缺的变质性物品的经济批量模型

传统的经济批量模型通常都假定物品的库存费用是固定不变的.放松了这个假定,通过考虑库存费用的两种可能变化情形即(A)库存费的变化率为存储时间的函数;(B)库存费的变化率为库存量的函数,并在需求线性依赖于库存水平的形式下,发展了两个变库存费的变质性物品的经济批量模型.在模型中允许短缺发生且假定短缺完全拖后,理论上证明了模型具有唯一的整体最优解,揭示了库存费的变化对库存系统最优订货策略的影响.     

STOPPING RULES AND THE CONTROL OF STOCK POLLUTANTS

A stock pollutant is defined as a residual waste that might accumulate over time. This paper examines some of the important distinctions between degradable and nondegradable stock pollutants and between nondegradable stock pollutants with known versus uncertain environmental cost. The latter case is examined using the more recent literature on stochastic control with Brownian motion. The presence of irreversibility and uncertainty is known to lead to more conservative investment rules and places a value on the preservation of options. In the case of a nondegradable stock pollutant with Brownian environmental cost, options are preserved by stopping accumulation at a lower level than in the corresponding certainty-equivalent problem. The model presented in this paper permits the derivation of closed-form stopping rules. For a simple numerical problem, the optimal nondegradable stock with Brownian environmental cost was 20 to 45 percent lower than the optimal level with known environmental cost. The empirical study of an actual nondegradable stock pollutant will require time series data on private and social cost in order to estimate drift and variance parameters which will influence the actual extent to which the optimal stock is less than the certainty-equivalent stock.     

A Deterministic Inventory System with an Inventory-Level-Dependent Demand Rate

This analysis is concerned with the continuous, deterministic case of an inventory system in which the demand rate of an item is of a polynomial functional form, dependent on the inventory level. Differential and integral calculus are used to find the inventory function with respect to time. From this, the objective function (to maximize average profit per unit time) is developed. For the continuous, multiperiod situation, a non-linear programming algorithm—separable programming—is utilized to determine the optimal order level (the quantity to order up to) and the order point (the quantity at which an order is placed). A numeric example and a sensitivity analysis are also presented.     

优化交货期窗口的两阶段供应链排序问题

研究一类优化交货期窗口的两阶段供应链排序问题. 优化交货期窗口是指交货期窗口的开始与结束时刻是决策变量, 不是输入常量. 两阶段是指工件先加工, 后运输: 加工阶段是一台加工机器逐个加工工件;运输阶段是无限台车辆分批运输完工的工件. 工件的开始运输时刻与完工时刻之差定义为工件的储存时间, 且有相应的储存费用. 若工件的运输完成时刻早于(晚于)交货期窗口的开始(结束)时刻, 则有相应的提前(延误)惩罚费用. 目标是极小化总提前惩罚费用、总延误惩罚费用、总储存费用、总运输费用以及与交货期窗口有关的费用之和. 针对单位时间的延误惩罚费用不超过单位时间的储存费用、单位时间的储存费用不超过单位时间的提前惩罚费用的情形, 给出了时间复杂性为O(n^{8})的动态规划算法.     

Dynamic lot-sizing with price changes and price-dependent holding costs

This paper proposes a new formulation of the dynamic lot-sizing problem with price changes which considers the unit inventory holding costs in a period as a function of the procurement decisions made in previous periods. In Section 1, the problem is defined and some of its fundamental properties are identified. A dynamic programming approach is developed to solve it when solutions are restricted to sequential extreme flows, and results from location theory are used to derive an O(T²) algorithm which provides a provably optimal solution of an integer linear programming formulation of the general problem. In Section 2, a heuristic is developed for the case where the inventory carrying rates and the order costs are constant, and where the item price can change once during the planning horizon. Permanent price increases, permanent price decreases and temporary price reductions are considered. In Section 3, extensive testing of the various optimal and heuristic algorithms is reported. Our results show that, in this context, the two following intuitive actions usually lead to near optimal solutions: accumulate stock at the lower price just prior to price increase and cut short on orders when a price decrease is imminent.     

A stocking policy for spare part provisioning under age based preventive replacement

A new policy, called stocking policy for ease of reference, has been advanced for joint optimization of age replacement and spare provisioning. It combines age replacement policy with continuous review (s, S) type inventory policy, where s is the stock reorder level and S is the maximum stock level. The policy is generally applicable to any operating situation having either a single item or a number of identical items. A simulation model has been developed to determine the optimal values of the decision variables by minimizing the total cost of replacement and inventory. The behaviour of the stocking policy has been studied for a number of case problems specifically constructed by 5-factor second order rotatory design and the effects of different cost elements and item failure characteristics have been highlighted. For all case problems, optimal (s, S) policies to-support the Barlow-Proschan age policy have also been determined. Simulation results clearly indicate that the optimal stocking policy is, in general, more cost-effective than the Barlow-Proschan policy.     

Optimal production and shipment models for a single-vendor–single-buyer integrated system

This paper develops a more general production-inventory model for a single-vendor–single-buyer integrated system. Unlike the hitherto existing production-inventory models for the vendor–buyer system, the present model neither requires the buyer’s unit holding cost greater than the vendor’s nor assumes the structure of shipment policy. Secondly, the model is extended to the situation with shortages permitted, based on shortages being allowed to occur only for the buyer. Thirdly, the paper also presents a corresponding production-inventory model for a deteriorating item for the integrated system. The solution procedures are provided for finding the optimal production and shipment policies and illustrated with numerical examples. Three significant insights are shown: (1) no matter whether the buyer’s unit holding cost is greater than the vendor’s or not, the present model always performs best in reducing the average total cost as compared to the hitherto existing models; (2) if the buyer’s unit holding cost is less than the vendor’s, the optimal shipment policy for the integrated system will only comprise shipments with successive shipment sizes increasing by a fixed factor. It is different from that obtained by Hill [Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem. International Journal of Production Research 37, 2463–2475] for the opposite case; (3) when designing a single-vendor–single-buyer integrated system, making the buyer’s unit holding cost lower than the vendor’s is more beneficial to the system if shortages are not permitted to occur; otherwise it just reverses.     

储存时间有上限的两阶段供应链排序问题

研究一类储存时间有上限的两阶段供应链排序问题.两阶段是指工件先加工,后运输:加工阶段是一台加工机器逐个加工工件;运输阶段是无限台车辆分批运输完工的工件.工件的运输完成时刻与完工时刻之差定义为工件的储存时间,且有相应的储存费用,且任意工件的储存时间都不超过某一常数.若工件的运输完成时刻早于(晚于)交货期窗口的开始(结束)时刻,则有相应的提前(延误)惩罚费用.目标是极小化总提前惩罚费用、总延误惩罚费用、总储存费用、总运输费用以及与交货期窗口有关的费用之和.先证明该问题是NP-难的,后对单位时间的储存费用不超过单位时间的延误惩罚费用的情形给出了伪多项式时间算法.     

Possibility and necessity constraints and their defuzzification—A multi-item production-inventory scenario via optimal control theory

In this paper, analogous to chance constraints, real-life necessity and possibility constraints in the context of a multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic multi-item production-inventory model with shortages and fuzzy constraints has been formulated and solved for optimal production with the objective of having minimum cost. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the present system produces some defective units along with the perfect ones and the rate of produced defective units is constant. Here demand of the good units is time dependent and known and the defective units are of no use. The space required per unit item, available storage space and investment capital are assumed to be imprecise. The space and budget constraints are of necessity and/or possibility types. The model is formulated as an optimal control problem and solved for optimum production function using Pontryagin’s optimal control policy, the Kuhn–Tucker conditions and generalized reduced gradient (GRG) technique. The model is illustrated numerically and values of demand, optimal production function and stock level are presented in both tabular and graphical forms. The sensitivity of the cost functional due to the changes in confidence level of imprecise constraints is also presented.     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

A Recursive Algorithm for Order Cycle-time that Minimizes Logistics Cost

A firm's logistics cost, including shipping and inventory-carrying, is a substantial percentage of its sales. Nevertheless, typical inventory-control methods ignore or insufficiently represent the shipping cost. This paper describes a recursive algorithm that determines the reorder cycle-time that minimizes total logistics cost. It allows for a realistic accounting of shipping cost, which is modelled here as a function of shipping distance and weight. The algorithm uses a relaxation procedure to identify a suitable initial approximation to the optimal order cycle-time and then, through a series of recursive steps, moves to the optimal result. We demonstrate the algorithm with a single item, with a group of items that share a common order cycle, and with multi-items when item demands are random variables. Experience with this algorithm indicates that it converges to the optimal result in a very few steps.     

需求依赖于服务水平的易变质品库存策略研究

本文研究需求依赖于上一周期服务水平、缺货时订单部分损失的两周期易变质品库存问题。分别考虑一次订货和多次订货两种情况,以平均利润最大化为目标构建库存模型,证明了模型解的存在性和唯一性,得到了最优库存服务水平和最优补货策略。最后,通过算例给出两个模型的应用,对重要参数进行了灵敏度分析,并且将两种模型的结果进行了对比分析。结果表明:订单损失率的增加会提高服务水平,但会使得利润降低;顾客期望服务水平的提高会降低第一阶段的服务水平,同时使利润减少;单位库存持有成本或变质率的增加会降低服务水平和平均利润。通常情况,企业通过多次订货能获得更大的利润,而只有当库存持有成本极小时,一次订购才能够获得更大的利润。同时,结果也表明:服务水平对库存策略有较大的影响,因此在进行库存决策时考虑服务水平具有重要的作用。