On a finite horizon production lot size inventory model for deteriorating items: An optimal solution

A generalized production lot size inventory model for deteriorating items over a finite planning horizon is considered. The demand, production, and deteriorating rates are assumed to be known and continuous functions of time. Shortages are allowed and completely backlogged. The conditions under which the system total cost attains its (unique) global minimum are derived. An example which illustrate the applicability of theoretical results is also introduced.     

Joint determination of production run length and number of standbys in a deteriorating production process

Two economic manufacturing quantity models with unrepairable and repairable standby key modules are proposed in this study that determine the economic production run length and the economic number of standbys in a deteriorating production process, where the key module of the production unit deteriorates over time and incurs some portion of defective items. For the model with unrepairable standbys, the active key module, once deteriorating, is replaced by a standby and the module itself is disposed. For the model with repairable standbys, the deteriorating key module is replaced by a standby and the module is then sent to the service center for maintenance. When completing the maintenance, it then joins the standbys for later production use. By minimizing the annual cost, which takes into account setup cost, holding cost, costs due to standbys and defective items, the economic production run length and the economic number of standbys are obtained for each of the proposed models.     

Optimal production run time for a deteriorating production system under an extended inspection policy

A deteriorating production system is subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. In order to reduce the defective items, part inspection policy, under which production inspections are performed only at the end of the production run, and full inspection policy are both considered in the literature. Moreover, the former dominates the latter. Since the product produced towards the end of a production cycle are more likely to be defective, it can further economize the inspection costs that they are directly reworked without inspection. In this paper, we propose an extended product inspection policy for a deteriorating production system. Product inspections are performed in the middle of a production cycle, and after the inspection, all products produced until the end of the production run are fully reworked. Based on the model, we show that there exists a production run time and a corresponding unique inspection policy such that the expected total cost per item per cycle is minimized. Finally, numerical examples are provided to illustrate our extended inspection policy, and indicate that such product inspection model will reduce the quality-related cost than part inspection does.     

Determining a common production cycle time for an economic lot scheduling problem with deteriorating items

This paper considers the economic lot scheduling problem (ELSP) for a production-inventory system where items produced are subject to continuous deterioration. The problem is to schedule multiple products to be manufactured on a single machine repetitively over an infinite planning horizon. Each product is assumed to have a significant rate of deterioration. Only one product can be manufactured at a time. The demand rate for each product is constant, but an exponential distribution is used to represent the distribution of the time to deterioration. A common cycle time policy is assumed in the production process. A near optimal production cycle time is derived under conditions of continuous review, deterministic demand, and no shortage.     

A Two-Warehouse Probabilistic Order-Level Inventory Model for Deteriorating Items

In this paper an order-level probabilistic inventory model for deteriorating items with two warehouses is developed. It is assumed that the production rate is infinite and shortages are allowed. The rates of deterioration of items in the two warehouses are different. Also, a single-warehouse version for deteriorating items is discussed. This paper also considers a two-warehouse model for non-deteriorating items. An illustrative example offers a comparative study of the optimum cost for varying shortage cost.     

Numerical Approach of Multi-Objective Optimal Control Problem in Imprecise Environment

In this paper, realistic production-inventory models without shortages for deteriorating items with imprecise holding and production costs for optimal production have been formulated. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is time dependent and known. The imprecise holding and production costs are assumed to be represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics, the objective function is changed to respective multi-objective functions and thus the single-objective problem is reduced to a multi-objective decision making(MODM) problem. The MODM problem is then again transformed to a single objective function with the help of weighted sum method and then solved using global criteria method, calculus method, the Kuhn–Tucker conditions and generalized reduced gradient(GRG) technique. The models have been illustrated by numerical data. The optimum results for different objectives are obtained for different types of production function. Numerical values of demand, production function and stock level are presented in both tabular and graphical forms     

带时变生产成本的易变质经济批量模型的最优策略分析

考虑了具有时变生产成本的易变质产品经济批量模型．有限计划期内,单位生产成本、生产率以及需求率假定为时间的连续函数,生产固定成本则具有遗忘效应现象．当不允许缺货时,建立了以总成本最小为目标的混合整数优化模型并证明了此问题最优解的相关性质．对于此问题的特殊情形,将成本函数中的离散型变量松弛为连续型变量,通过分析其最优解的存在性及唯一性,求解了此最优解,将其作为初始值设计了求取一般情形最优解的有效算法．最后通过算例验证了理论结果的有效性．     

A probabilistic inventory model for deteriorating items with lead time equal to one scheduling period

A probabilistic inventory model for continuously deteriorating items is developed, in which the lead time is assumed to be of one scheduling period, shortages are not allowed, time is treated as discrete variable, and deterioration is assumed to be a constant fraction of the on hand inventory. The developed model is related to the EOQ model for deteriorating items. An example is given to illustrate the derived results.     

An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments

In the classic economic order quantity model the purchasing cost of an order should be paid at the time of its receipt. Sometimes, retailers ask purchasers to pay all or a fraction of the purchasing cost in advance and sometimes allow them to divide the prepayment into several equal-sized parts. In this paper, economic order quantity model for a deteriorating product with and without shortage under consecutive prepayments are developed. Numerical examples illustrate the proposed models and solution method and sensitivity analysis provides some managerial insights for managers.     

A collaborative inventory system with permissible delay in payment for deteriorating items

A collaborative inventory system of single vendor and single buyer is developed to maximize the total profit of the whole system. However, the optimal solution for the whole system is not always beneficial to both players. To ensure mutual benefit, a negotiation factor is incorporated to share the profit between the two players according to their contributions. The permissible delay in payment is a win–win strategy for sharing profit in the collaborative system. A deteriorating inventory model with finite replenishment rate and price sensitive demand is assumed to occur in a high-tech, short life cycle and perishable electronic product. A numerical example is provided to illustrate our models. The sensitivity analysis of the demand rate, replenishment rate, deterioration factor, and other related parameters shows that the percentage extra total profit is significant when both the collaboration strategy and the deterioration factor are considered.     

A leader-follower model and analysis for a two-stage network of oligopolies

This paper is concerned with the existence, uniqueness and computation of leader-follower equilibrium solutions for an industry involved with two major stages of production. We assume that there exists one set of firms performing the first stage of production, which produces a semi-finished product. This semi-finished product is converted to a final good by a second set of firms performing the second stage of production. Furthermore, also competing in the final product market is a third set of firms, which are vertically integrated through the two stages of production and which are assumed to lead the second set of firms by explicitly considering the reaction or response of these latter firms to their own outputs. We model such an industry as a two-stage network of oligopolies, and define equilibrium solutions based on assumed market structures. Our analysis examines the existence and uniqueness of such equilibrium solutions, characterizes the nature of the production strategies of the various firms at an equilibrium, and prescribes algorithms to compute such solutions. This provides the machinery required to perform sensitivity analyses for studying the effects of various mergers or integrations on individual firm profits, and on the industry outputs and prices at equilibrium. The presentation is self-contained, and does not necessarily require any significant prior preparation in economic theory on the part of the reader.This paper is based on work done for the Minerals and Mining Resources Research Institute, under the sponsorship of the Bureau of Mines, Department of the Interior.     

On the global optimal solution to an integrated inventory system with general time varying demand,production and deterioration rates

In this paper a unified inventory model for integrated production system with a single product is presented. The production, demand and deterioration rates for the finished product and the deterioration rates for raw materials are assumed to be functions of time. A rigorous mathematical proof which shows the global optimality of the solution to the considered inventory system is introduced. A numerical example that illustrates the solution procedure is included.     

An EOQ model for deteriorating inventory with alternating demand rates

The present paper deals with an economic order quantity model for items deteriorating at some constant rate with demand changing at a known and at a random point of time in the fixed production cycle.     

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.     

First-off inspection of capable manufacturing processes

This paper considers the problem of determining how much inspection should be performed before deciding whether a process ‘setup’ may be accepted and routine production operations commenced. A number of separate cases are considered, which together account for different practical circumstances which arise. The objective is to optimise the economic performance of the process. Separate objective functions are constructed for the different cases. General properties of these functions are established, on the basis of which easy-to-use algorithms for their optimisation are developed. It is assumed that the process is inherently capable of producing conforming product.     

Multi-resource investment strategies: Operational hedging under demand uncertainty

Consider a firm that markets multiple products, each manufactured using several resources representing various types of capital and labor, and a linear production technology. The firm faces uncertain product demand and has the option to dynamically readjust its resource investment levels, thereby changing the capacities of its linear manufacturing process. The cost to adjust a resource level either up or down is assumed to be linear. The model developed here explicitly incorporates both capacity investment decisions and production decisions, and is general enough to include reversible and irreversible investment. The product demand vectors for successive periods are assumed to be independent and identically distributed. The optimal investment strategy is determined with a multi-dimensional newsvendor model using demand distributions, a technology matrix, prices (product contribution margins), and marginal investment costs. Our analysis highlights an important conceptual distinction between deterministic and stochastic environments: the optimal investment strategy in our stochastic model typically involves some degree of capacity imbalance which can never be optimal when demand is known.     

An integrated production-inventory model for ameliorating and deteriorating items taking account of time discounting

This study develops an integrated production-inventory model from the perspectives of both the manufacturer and the retailer. The model considered both ameliorating and deteriorating effects taking account of multiple deliveries, partial backordering and time discounting. The amelioration and deterioration are assumed to follow the Weibull distribution. The discounted cash flow and optimization technique are used to derive the optimal solution. A numerical example and the sensitivity analysis are given to illustrate the theory of the integrated production-inventory model. This model is particularly useful for items that ameliorate and deteriorate at the same time.     

The economic production and pricing model with lot-size-dependent production cost

In this paper, the economic production quantity problem for a single-product single-machine system is extended. It is assumed that annual demand of the product is a function of price set by manufacturer. This extension considers sales revenue, inventory and setup costs as well as a variable cost of production which is a function of the lot size. Several linear and non-linear functions of demand and variable cost are considered in this paper and a global solution methodology is presented for the models developed. Newton??s method is used to find local optima and asymptotic convergence of the solution algorithm to a global optimum is proved. Numerical studies followed by a discussion provide additional insights into the problem.     

Optimal production stopping time for perishable products with ramp-type quadratic demand dependent production and setup cost

A single item economic production quantity (EPQ) model is discussed to analyse the behaviour of the inventory level after it’s introduction to the market. It is assumed that demand is time dependent accelerated growth-effect of accelerated growth-steady type. Unlike the conventional EPQ models, which are restricted to general production cycle over the finite or infinite time horizon, we consider the production sale scenario of the very first production cycle for newly introduced perishable product. Shortage is not allowed. Set up cost of an order cycle depends on the total amount of inventory produced. The finite production rate is proportional to demand rate. Optimal production stopping time is determined to maximize total unit profit of the system. A numerical example is presented to illustrate the development of the model. Sensitivity analysis of the model is carried out.     

An EOQ model for deteriorating items under supplier credits linked to ordering quantity

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the purchaser must pay for the items received immediately. However, in practices, the supplier usually is willing to provide the purchaser a permissible delay of payments if the purchaser orders a large quantity. As a result, in this paper, we establish an EOQ model for deteriorating items, in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. We then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Finally, several numerical examples are given to illustrate the theoretical results.