Impacts of two-stage deterioration on an integrated inventory model under trade credit and variable capacity utilization

The joint economic lot sizing problem (JELP) model provides a global view to facilitate the development of a production-inventory policy for an integrated system. However, when a deteriorating item is involved, previous studies have neglected the following two important issues: (1) the deterioration quantity increases the demand for the supplier's capacity, which consequently requires the supplier to recalculate the corresponding average cost and reevaluate the capacity utilization, and (2) given the supplier's production rate, in-transit deterioration imposes restrictions on the delivery distance or in-transit time. Therefore, the existing integrated policies may lead to infeasible solutions for the distribution channel when a deterioration item is included. In view of these two issues, a generalized JELP model under delay in payments is formulated to investigate the integrated production-inventory policy for an item with two-stage deterioration (in-transit and retail deterioration) while incorporating both transportation time and capacity utilization. By developing the average cost functions of the supply chain members and employing several new definitions (e.g., variable capacity utilization), this paper provides a mechanism for measuring the influence of two-stage deterioration on the supplier's capacity utilization for the JELP. Three algorithms are proposed to obtain optimal decisions based on the theoretical results. This paper demonstrates that the supplier's variable capacity utilization is relevant to transportation time and two-stage deterioration, which can be applied to evaluate the feasibility of the integrated production-inventory policy for the deteriorating item. Furthermore, there is a maximum allowable value for the retailer's order cycle.     

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.     

Determining the optimal run time for EPQ model with scrap,rework, and stochastic breakdowns

This paper is concerned with determination of optimal run time for an economic production quantity (EPQ) model with scrap, rework, and stochastic machine breakdowns. In real life manufacturing systems, generation of defective items and random breakdown of production equipment are inevitable. In this study, a portion of the defective items is considered to be scrap, while the other is assumed to be repairable. Total production-inventory cost functions are derived respectively for both EPQ models with breakdown (no-resumption policy is adopted) and without breakdown taking place. These cost functions are integrated and the renewal reward theorem is used to cope with the variable cycle length. Theorems on conditional convexity of the integrated overall costs and bounds of the production run time are proposed and proved. We conclude that the optimal run time falls within the range of bounds and it can be pinpointed by the use of the bisection method based on the intermediate value theorem. Numerical example is provided to demonstrate its practical usages.     

The optimal production cycle time in an integrated production-inventory model for decaying raw materials

In 1983, Park presented an integrated production-inventory model for decaying raw materials. This paper discusses the same model as that of Park. The main purpose is threefold:

(1)

This paper shows that the total cost function per unit time is convex by a rigorous proof.     

Single-vendor multi-buyer integrated production-inventory model with controllable lead time and service level constraints

This paper presents an integrated production-inventory model where a vendor produces an item in a batch production environment and supplies it to a set of buyers. The buyer level demand is assumed to be independent normally distributed and lead time of every buyer can be reduced at an added crash cost. The buyers review their inventory using continuous review policy, and the unsatisfied demand at the buyers is completely backordered. A model is formulated to minimize the joint total expected cost of the vendor–buyers system to determine the optimal production-inventory policy. Since it is often difficult to estimate the stock-out cost in inventory systems, and so instead of having stock-out cost component in the objective function, a service level constraint (SLC) corresponding to each buyer is included in the model. A Lagrangian multiplier technique based algorithmic approach is proposed, which evaluates a very limited number of combinations of lead time of the buyers to find simultaneously the optimal lead time, order quantity and safety factor of the buyers and the number of shipments between the vendor and the buyers in a production cycle. Finally, a numerical example and effects of the key parameters are included to illustrate the results of the proposed model.     

The single-vendor single-buyer integrated production-inventory model with a generalised policy

This note considers a more general type of policy for the single-vendor, single-buyer integrated production-inventory model than has hitherto been presented, based on successive shipments to the buyer, within a single production batch, increasing by a fixed factor.     

Integrated Production Inventory Policies for Multistage Multiproduct Batch Production Systems

An integrated production inventory model is considered in this paper, for a flow shop type multiproduct batch production system, with a multifacility structure. Instantaneous production is allowed in each facility. The model aims to determine simultaneously the optimal manufacturing cycle for the multiple products and the corresponding optimal procurement policies for the raw material. The cycle concept of multiproduct batch processing is extended to multifacility system and is integrated with the concept of production-inventory system for a single product, single facility system.     

Integrated multi-item production-inventory systems

We study an integrated multi-item production-inventory system with stochastic demands and capacitated production. The problem is to find the base stock levels which minimize expected inventory costs per unit time, for an infinite time horizon. When unit manufacturing times are deterministic or exponentially distributed, we derive analytical expressions that lead to the optimal base stock levels. Our results provide several implications about the interaction of inventories, capacity utilization, and variation in the production environment.     

Robust optimization of uncertain multistage inventory systems with inexact data in decision rules

In production-inventory problems customer demand is often subject to uncertainty. Therefore, it is challenging to design production plans that satisfy both demand and a set of constraints on e.g. production capacity and required inventory levels. Adjustable robust optimization (ARO) is a technique to solve these dynamic (multistage) production-inventory problems. In ARO, the decision in each stage is a function of the data on the realizations of the uncertain demand gathered from the previous periods. These data, however, are often inaccurate; there is much evidence in the information management literature that data quality in inventory systems is often poor. Reliance on data “as is” may then lead to poor performance of “data-driven” methods such as ARO. In this paper, we remedy this weakness of ARO by introducing a model that treats past data itself as an uncertain model parameter. We show that computational tractability of the robust counterparts associated with this extension of ARO is still maintained. The benefits of the new model are demonstrated by a numerical test case of a well-studied production-inventory problem. Our approach is also applicable to other ARO models outside the realm of production-inventory planning.     

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.     

Possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem

In this paper, possibility and necessity representations of fuzzy inequality constraints are presented and then crisp versions of the constraints are derived. Here analogous to chance constraints, real-life necessity and possibility constraints in the context of two warehouse multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic two warehouse multi-item production-inventory model with fuzzy constraints has been formulated for a finite period of time and solved for optimal production with the objective of having maximum profit. The rate of production is unknown, assumed to be a function of time and considered as a control variable. Also the present system produces some defective units alongwith the perfect ones and the rate of produced defective units is stochastic in nature. Demand of the good units is stock dependent and known and the defective units are sold at a reduced price. The space required per unit item and available storage space are assumed to be imprecise. The inequality of budget constraints is also imprecise. The space and budget constraints are expressed as necessity and/or possibility types. The model is reduced to an equivalent deterministic model using fuzzy relations 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 pictorial forms.     

存贮时间水平有限且费用参数可变的最佳生产—库存模型

本文研究时间周期有限，费用参数呈阶梯变化的生产－库存模型的最佳策略及参数的确定途径。     

Control of a production-inventory system with returns under imperfect advance return information

We consider a production-inventory system with product returns that are announced in advance by the customers. Demands and announcements of returns occur according to independent Poisson processes. An announced return is either actually returned or cancelled after a random return lead time. We consider both lost sale and backorder situations. Using a Markov decision formulation, the optimal production policy, with respect to the discounted cost over an infinite horizon, is characterized for situations with and without advance return information. We give insights in the potential value of this information. Also some attention is paid to combining advance return and advance demand information. Further applications of the model as well as topics for further research are indicated.     

客户耐烦期相同的生产——库存系统最优模型

在常数需求率以及有限生产率条件下研究了订货客户耐烦期相同的一类新的生产——库存模型 ,在每一个周期内考虑了延期交货时间超过耐烦期的短缺费用和销售机会损失等因素 ,给出了相应的最优生产时间和周期的确定方法 ,利用数学软件 Matlab及计算机为工具给出了数字例子进行说明 ,其方法和结果为库存系统的管理决策提供了理论依据     

碳限额与碳交易约束下制造商生产-库存控制策略

针对制造商订货、储存、生产过程中的碳排放问题,探讨了碳限额与碳交易约束下制造商生产-库存控制策略,在对碳限额与碳交易进行数学度量的基础上构建了碳限额与碳交易约束下制造商生产-库存成本模型,并通过模型分析得出有碳约束且成本最优生产量及原材料最大订货倍数以及碳限额与碳交易约束下制造商最优生产-库存策略。计算实验与算例分析表明:相比于无碳约束情形,碳限额与碳交易约束下制造商订货数量更高,而订货频率及生产批量更低,并得出三种交易价格之下,决策变量的变化趋势。     

A note on a single-vendor and multiple-buyers production-inventory policy for a deteriorating item

In this note we examine the total cost function of a single-vendor multiple-buyers production-inventory policy for a deteriorating item by Yang and Wee [P.C. Yang and H.M. Wee, A single-vendor multiple-buyers production-inventory policy for a deteriorating item, European Journal of Operational Research 143 (2002) 570–581]. Two possible flaws in the cost function of Wee and Yang’s model are pointed out. A proposal to eradicate the flaws is given.     

Two-warehouse production model for deteriorating inventory items with stock-dependent demand under inflation over a random planning horizon

This paper develops a production-inventory model for a deteriorating item with stock-dependent demand under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. The effects of learning in set-up, production, selling and reduced selling is incorporated. Different inflation rates for various inventory costs and time value of money are also considered. A hybrid genetic algorithm is designed to solve the optimization problem which is hard to solve with existing algorithms due to the complexity of the decision variable. To illustrate the model and to show the effectiveness of the proposed approach a numerical example is provided. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.     

生产率、需求率、变质率及损失率均随时间变化的生产库存模型

建立了生产率、需求率、变质率及损失率均随时间变化的生产库存模型,在假定了(I)变质率线性依赖于时间及库存开始点;(II)损失率线性依赖于时间及短缺量拖后供货的结束点的前提下,给出了寻求最优策略的方法,指出了文[10]中不合理的假设,并以实际的例子与已有结果作比较,其结果为生产管理部门的决策提供了理论依据.     

Optimal production stopping and restarting times for an EOQ model with deteriorating items

A perishable single item production-inventory system is studied in this paper. The objective is to describe a general model in which the production rate, the product demand rate, and the item deterioration rate are all considered as functions of time, and to discuss the optimal production stopping and restarting times which minimise the total relevant cost per unit time. In the general model, demand shortage is allowed, where some of the demand is lost and the rest is backlogged. Popular models, such as the pure inventory system and the zero shortage system, are shown to be special cases of our model. The conditions for a feasible stationary point to be optimal are given. The simplest cases with constant rates of production, demand and deterioration are discussed and shown as illustrative examples.     

Determining the optimal procurement policy and maximum allowable lifespan for machining tools with stochastically distributed toollife

For high-value added products, machining tools’ lifespan significantly influences the quantity of procurement in machining process. Preemption of tools from the workpiece while processing is continuing is sometime beneficial to safeguard the product from the damage due to tool failure or its malfunction. Also an early discard of a tool is costly for the manufacturing operation. Therefore an optimal strategy for the tool life is sought here to determine the maximum allowable tool lifespan to preempt from the workpiece and to have an appropriate amount of tool stock in the crib to ascertain the proper running of the production schedule and tool inventory. Therefore, an impact of the machining tool lifespan on the production-inventory policy of the system is investigated in this paper. An integrated lifespan related inventory model for machining tools is developed to meet the responding accurate requirement of procurement and inventory. Two numerical examples are presented to illustrate the integrated model. The results show that the practical lifespan adoption of machining tools has significant impact on the whole quantity of procurement, and eventually influences the coordinating economic decision making.