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.     

The production–inventory problem of a product with time varying demand,production and deterioration rates

In this paper, we consider the production–inventory problem in which the demand, production and deterioration rates of a product are assumed to vary with time. Shortages of a cycle are allowed to be backlogged partially. Two models are developed for the problem by employing different modeling approaches over an infinite planning horizon. Solution procedures are derived for determining the optimal replenishment policies. A procedure to find the near-optimal operating policy of the problem over a finite time horizon is also suggested.     

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 with two-parameter Weibull distribution deterioration and price-dependent demand

An Inventory replenishment policy is developed for a deteriorating item and price-dependent demand. The rate of deterioration is taken to be time-proportional and the time to deterioration is assumed to follow a two-parameter Weibull distribution. A power law form of the price dependence of demand is considered. The model is solved analytically and is illustrated with a numerical example.     

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.     

Optimal EOQ Models for Deteriorating Items with Time-Varying Demand

In this paper, optimal inventory lot-sizing models are developed for deteriorating items with general continuous time-varying demand over a finite planning horizon and under three replenishment policies. The deterioration rate is assumed to be a constant fraction of the on-hand inventory. Shortages are permitted and are completely backordered. The proposed solution procedures are shown to generate global minimum replenishment schedules for both general increasing and decreasing demand patterns. An extensive empirical comparison using randomly generated linear and exponential demands revealed that the replenishment policy which starts with shortages in every cycle is the least cost policy and the replenishment policy which prohibits shortages in the last cycle exhibited the best service level effectiveness. An optimal procedure for the same problem with trended inventory subject to a single constraint on the minimum service level (maximum fraction of time the inventory system is out of stock during the planning horizon) is also proposed in this paper.     

An EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment

In this paper, an EOQ (Economic Order Quantity) model is developed for a deteriorating item having time dependent demand when delay in payment is permissible. The deterioration rate is assumed to be constant and the time varying demand rate is taken to be a quadratic function of time. Mathematical models are also derived under two different circumstances, i.e. Case I: The credit period is less than or equal to the cycle time for settling the account and Case II: The credit period is greater than the cycle time for settling the account. The results are illustrated with numerical examples. Justification for considering a time quadratic demand and permissible delay in payment are discussed.     

Optimal selling price and lotsize with time varying deterioration and partial backlogging

The article deals with an EOQ (economic order quantity) model over an infinite time horizon for perishable items where demand is price dependent and partial backorder is permitted. The rate of deterioration is taken to be time proportional and it is assumed that shortage occurs at starting of the inventory cycle. Based on the partial backlogging and lost sale cases, the author develops the criterion for the optimal solution for the replenishment schedule, and proves the optimal ordering policy is unique. Moreover, the article suggests to new functions regarding price-dependent demand and time varying deterioration rate. Finally, numerical examples are illustrated to test the model in various issues.     

An EOQ Model for Deteriorating Items with Time Varying Demand and Costs

A generalized EOQ model for deteriorating items is considered here in which the demand rate, deterioration rate, holding cost and ordering cost are all assumed to be continuous functions of time. Shortages are also allowed and are completely backlogged. The planning horizon is finite and the replenishment periods are assumed to be constant. The optimal replenishment policy and the decision rule which minimizes the total system cost are derived. A numerical example is given to illustrate the developed model. Sensitivity analysis is also presented for the given model.     

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.     

Integrated production and distribution scheduling with lifespan constraints

We consider an integrated production and distribution scheduling problem in a make-to-order business scenario. A product with a short lifespan (e.g., perishable or seasonal) is produced at a single production facility with a limited production rate. This means that the product expires in a constant time after its production is finished. Orders are received from a set of geographically dispersed customers, where a demand for the product and a time window for the delivery is associated with each customer for the planning period. A single vehicle with non-negligible traveling times between the locations is responsible for the deliveries. Due to the limited production and distribution resources, possibly not all customers may be supplied within their time windows or the lifespan. The problem consists in finding a selection of customers to be supplied such that the total satisfied demand is maximized. We extend the work by Armstrong et al. (Annals of Operations Research 159(1):395–414, 2008) on the problem for fixed delivery sequences by pointing out an error in their branch and bound algorithm and presenting a corrected variant. Furthermore, we introduce model extensions for handling delays of the production start as well as for variable production and distribution sequences. Efficient heuristic solution algorithms and computational results for randomly generated instances are presented.     

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.     

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.     

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．     

An Economic order quantity model for Items with Three-parameter Weibull distribution Deterioration,Ramp-type Demand and Shortages

In this paper, we develop an economic order quantity inventory model for items with three-parameter Weibull distribution deterioration and ramp-type demand. Shortages are allowed in the inventory system and are completely backlogged. The demand rate is deterministic and varies with time up to a certain point and eventually stabilized and becomes constant. The instantaneous rate of deterioration is an increasing function of time. We provide simple analytical tractable procedures for deriving the model and give numerical examples to illustrate the solution procedure. Our adoption of ramp-type demand reflects a real market demand for newly launched product.     

Dynamic economic lot size model with perishable inventory and capacity constraints

In this study, we consider a dynamic economic lot sizing problem for a single perishable item under production capacities. We aim to identify the production, inventory and backlogging decisions over the planning horizon, where (i) the parameters of the problem are deterministic but changing over time, and (ii) producer has a constant production capacity that limits the production amount at each period and is allowed to backorder the unmet demand later on. All cost functions are assumed to be concave. A similar problem without production capacities was studied in the literature and a polynomial time algorithm was suggested (Hsu, 2003 [1]). We assume age-dependent holding cost functions and the deterioration rates, which are more realistic for perishable items. Backordering cost functions are period-pair dependent. We prove the NP-hardness of the problem even with zero inventory holding and backlogging costs under our assumptions. We show the structural properties of the optimal solution and suggest a heuristic that finds a good production and distribution plan when the production periods are given. We discuss the performance of the heuristic. We also give a Dynamic Programing-based heuristic for the solution of the overall problem.     

An analysis of production-inventory models with deteriorating items in a two-echelon supply chain

This paper revisits two previous studies that addressed the integrated production–inventory problem for deteriorating items in a two-echelon supply chain, where the item’s deterioration rate is a constant or follows a continuous probability distribution function. The aim of this study is to present an improved solution procedure to determine the delivery lot size and the number of deliveries per production batch cycle that minimizes the total cost of the entire supply chain. The performance of the proposed methodology is illustrated analytically and numerically.     

(<Emphasis Type="Italic">T</Emphasis>, <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">i</Emphasis></Subscript>) Policy Inventory Model for Deteriorating Items with Time Proportional Demand

An EOQ model is reconsidered here in which the demand rate is changing linearly with time and the deterioration is assumed to be a constant fraction of the onhand inventory. The planning horizon is finite and known and the replenishment periods are assumed to be constant. The problem is to find the optimal number of replenishments, which are instantaneous. When there is no deterioration, the model developed is related to the corresponding model for nondeteriorating items. An example followed by sensitivity analysis is given to illustrate the derived results.     

Stock replenishment policies for a stochastic exponentially-declining demand process

This paper addresses the problem of determining stock replenishment policies to meet the demand for spare parts for items of equipment which are no longer manufactured. The assumptions that the number of items still in use is decreasing and that parts fail randomly lend credence to a Poisson demand process with an underlying mean which is decreasing exponentially. We use a dynamic programming formulation in continuous time to determine that replenishment policy which minimises the mean total discounted cost of set-up/order, unit production/purchase, unsatisfied demand and stock left over at the end of the time horizon.     

Optimal pricing and quantity of products with two offerings

This paper analyzes the decision of a firm offering two versions of a product, a deluxe and a regular. While both products satisfy the same market, the deluxe version is sold at a high price relative to its cost and is aimed at the high end of the demand curve. The regular version is sold at a low price relative to its cost and is targeted to customers at the low end of the demand curve. This two-offering strategy is especially popular with book publishers where a paperback book is introduced some time after the hardbound version is introduced. The time between the introduction of the two versions of the product is accompanied by a downward shift in the demand curve due to customers losing interest in the product or satisfying their demand from a secondary used market. We solve a profit maximization model for a firm using a two-offering strategy. The model is solved for linear and exponential deterioration in demand, which is assumed to be deterministic. Also, a model with linear deterioration in demand, which is assumed to be stochastic, is solved. The results indicate that substantial improvements in profit can be obtained by using the two-offering strategy. Numerical sensitivity analysis and examples are used to illustrate the results.