On some optimality conditions and their equivalence in time-dependent inventory models

The paper considers a deterministic, time-and-review-continuous, finite horizon inventory model with general time-dependent demand, holding cost, shortage cost and replenishment rate functions. It gives conditions for the optimal ordering schedule, for a fixed number of replenishments, which reduce the effort for its finding to a simple one-dimensional search. The paper also proves equivalence relationships between the derived optimality conditions and the corresponding ones for special cases of the general model where some of its component functions tend to infinity. It concludes with a discussion concerning the uniqueness of the optimal number of replenishments, and the convergence of the optimal ordering schedule for an infinite time horizon.     

A Study of the infinite horizon ‘solution’ of inventory lot size models with a linear demand function

The article studies the infinite time horizon inventory lot size model with a linear demand function. The pertinent costs are the carrying and order costs, shortages are prohibited, and replenishments are instantaneous. The problem is to minimize the total cost by exhibiting a schedule of optimal replenishment times.The computerized asymptotic solution suggested here is a generalization of the classical square root law and the more recent cubic root law. Due to its broader usefulness, it should preponderate previously known results.     

Optimal replenishment policies for the case of a demand function with product-life-cycle shape in a finite planning horizon

In this paper, we explore the demand function follows the product-life-cycle shape for the decision maker to determine the optimal number of inventory replenishments and the corresponding optimal inventory replenishment time points in the finite planning horizon. The objective function of the total relevant costs considered in our model is mathematically formulated as a mixed-integer nonlinear programming problem. A complete search procedure is provided to find the optimal solution by employing the properties derived in this paper and the Nelder–Mead algorithm. Also, based on the search procedure developed in this paper, a decision support system is implemented on a personal computer to solve the proposed problem.     

An EOQ model for deteriorating items with time-varying demand and shortages

The study explores the inventory replenishment policies for the cases with time-varying demand, linearly increasing deterioration rate, partial back-ordering, constant service level and equal replenishment intervals over a fixed planning horizon. Since it is difficult to solve the problem directly, we derive the upper bound of replenishment number for a specific planning horizon and find the solution of service level under a given number of replenishment. The optimal solutions of replenishment number and service level are then determined. Numerical examples and sensitivity analyses are also provided to illustrate the solution procedure.     

Economic Order Quantity model with Weibull deterioration distribution,shortage and ramp-type demand

A single-item single-period Economic Order Quantity model for deteriorating items with a ramp-type demand and Weibull deterioration distribution is considered. The shortages in inventory are allowed and backlogged completely. The model is developed over an infinite planning horizon and the optimal replenishment policy is derived by minimizing the total inventory cost per unit time. The numerical solution of the model is obtained, and the sensitivity of the parameters involved in the model is also examined.     

On inventory replenishment with non-linear increasing demand

We present a consecutive improvement approach to solve the problem of determining the timing and size of replenishments in order to minimize the total inventory costs in the case of a non-linear increasing demand pattern. The proposed procedure is computationally simple compared to other methods. It considers the replenishment in reverse time sequence with the time horizon being exactly the ending point and it thus guarantees to meet the demand over the preestablished time frame. The concept of inventory correspondence is introduced. Numerical examples are also presented and compared with other heuristics using an analytic procedure for replenishment with non-linear increasing demand.Scope and purposeYang et al. [2] is one of the best heuristic algorithms known today for determining the optimal replenishment policy when demand is non-linear increasing. This article proposes a consecutive improvement method which is effective, computationally simple, and intuitively appealing from the graphical illustration. Numerical tests show the proposed method even outperforms Yang's procedure.     

A production lot size inventory model for deteriorating items with time-varying demand

This paper presents a production lot size model for deteriorating items with time-varying demand. The replenishment cycle and deterioration rates are allowed to vary over a finite planning horizon. A solution procedure for this model is developed to provide the optimal replenishment cycle number. Also three numerical examples are solved for different deterioration rates. Finally, a sensitivity analysis is conducted to study the effect of changes in the related parameters on the objective function.     

Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach

In this article, we consider an infinite horizon, single product economic order quantity where demand and deterioration rate are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and completely backlogged. The objective is to find the optimal inventory and pricing strategies maximizing the net present value of total profit over the infinite horizon. For any given selling price, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use a couple of numerical examples to illustrate the algorithm.     

Finite horizon EOQ model for non-instantaneous deteriorating items with price and advertisement dependent demand and partial backlogging under inflation

This paper deals with an economic order quantity (EOQ) model for non-instantaneous deteriorating items with price and advertisement dependent demand pattern under the effect of inflation and time value of money over a finite planning horizon. In this model, shortages are allowed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. This paper aids the retailer in minimising the total inventory cost by finding the optimal interval and the optimal order quantity. An algorithm is designed to find the optimum solution of the proposed model. Numerical examples are given to demonstrate the results. Also, the effect of changes in the different parameters on the optimal total cost is graphically presented and the implications are discussed in detail.     

Optimal replenishment policies with allowable shortages for a product life cycle

In this paper, we investigate replenishment policies with allowable shortages by considering a general, time-varying, continuous, and deterministic demand function for a product life cycle. The objective is to optimally determine the number of inventory replenishments, the inventory replenishment time points, and the beginning time points of shortages within the product life cycle by minimizing the total relevant costs of the inventory replenishment system. The proposed problem is mathematically formulated as a mixed-integer nonlinear programming model. A complete search procedure is developed to find the optimal solution by employing the properties derived in this paper and the well-known Nelder–Mead algorithm. Also, several numerical examples and the corresponding sensitivity analyses are carried out to illustrate the features of our model by utilizing the search procedure developed in this paper.     

An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate

In this paper, we formulate a deteriorating inventory model with stock-dependent demand by allowing preservation technology cost as a decision variable in conjunction with replacement policy. Moreover, it is assumed that the shortages are allowed and partially backlogged, depending on the length of the waiting time for the next replenishment. The objective is to find the optimal replenishment and preservation technology investment strategies while maximizing the total profit per unit time. For any given preservation technology cost, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of preservation technology cost when the replenishment schedule is given. We then provide a simple algorithm to find the optimal preservation technology cost and replenishment schedule for the proposed model. Finally, we use some numerical examples to illustrate the model.     

Balancing stocks and backlogs with stochastic lead times: A special case

The paper gives compact, finite horizon and asymptotic results concerning an inventory system, that allows backlogging, with constant demand and costs but stochastic lead times, not necessarily i.i.d., under two special conditions. The special conditions are: (1) no crossovers of orders; (2) there are both shortage and stock build-up between any two consecutive orders. It is shown that the minimal total expected cost, namely, of holding inventory, shortages and replenishments, as well as the optimal replenishment times depend, among other parameters, only on the lead time variances as representatives of the stochastic component of the model. The results are direct extensions of the deterministic EOQ model and lend themselves to tractable sensitivity analysis. Further possible developments are discussed.     

The replenishment policy of agri-products with stochastic demand in integrated agricultural supply chains

Market demand of agri-products is influenced by uncertain factors, such as weather, temperature, and customer preferences. In integrated agricultural supply chains, traditional inventory models are useless because of the stochastic demand and deteriorative characteristic of agri-products. This paper provides a method to determine the optimal replenishment policy of integrated agricultural supply chains with stochastic demand. In these EOQ/EPQ models, shortages are allowed and are backlogged if market demand is stochastic. The objective function is to minimize the total cost of the supply chain in the planning horizon. The total cost includes the ordering cost, the holding cost, the shortage cost and the purchasing cost. Thinking of the nonlinear relationship and dynamic forces in models, a system dynamic (SD) simulation model is constructed to find the optimal lot size and replenishment interval. Finally, an example is given to make a sensitivity analysis of the simulation model. Compared to traditional methods (such as equalize stochastic demand), the total cost decreases by 16.27% if the supply chains adopt the new replenishment policy. The results illustrated that the new replenishment policy (with intelligent method) is beneficial to help supply chain make decision scientifically. Moreover, the intelligent method can simulate stochastic demand perfectly, and it is effectively for solving the complicated and mathematically intractable replenishment problem.     

A general and optimal approach for three inventory models with a linear trend in demand

This study provides a general and simple algorithm to obtain an optimal solution for three inventory models with a replenishment batching policy, production batching policy, and an integrated replenishment/production batching policy in a manufacturing system, under a finite time horizon and linear trend in demand. This study determines the replenishment or production schedule with one general equation for these three problems and provides fully theoretical proofs for relaxing some of the conjectures in previous studies. A general and explicit procedure to derive the optimal solution for these three inventory models is presented, while considering both linearly increasing and decreasing demands. In addition, demonstrations of applicability are performed.     

On the optimal replenishment schedule for an inventory system with deteriorating items and time-varying demand and production rates

In this paper, we present an exact method for finding the optimal replenishment schedule for an inventory system with deteriorating items, and in which demand and production rates are allowed to vary over a finite planning horizon. A numerical example is presented to illustrate the method.     

Supplier–retailer inventory coordination with credit term for inventory‐dependent and linear‐trend demand

In this paper, a nonintegrated and collaborative replenishment policy is considered, respectively, which incorporates varying demand depending on both inventory level and time during the finite planning horizon. For additional cost savings realized from coordination, the paper adopts trade credit as a cost‐saving shift means and introduces a brand new parameter, that is, credit period rate. Then, the equitable credit period rate is determined, and different values of the credit period rate reflect the allocation of additional cost savings between the supplier and retailer. Furthermore, the conditions for the existence and uniqueness of an optimal solution are proved for the nonintegrated and collaborative replenishment policy, and an efficient solution procedure is developed to determine the optimal results and coordination of the inventory model. Finally, several numerical examples are provided to illustrate the proposed strategy and algorithm, and the sensitivity analysis of the optimal solution with respect to each parameter is presented. The sensitivity analysis suggests that the size of the credit period rate has a strong relationship with the supplier's and retailer's inventory cost (including capital cost) and setup cost. In real‐life situations, this proposed strategy may be applied to some consumer products in the growth phase or best‐selling consumer goods, etc.     

An economic production lot size model with increasing demand, shortages and partial backlogging

In this paper, an economic production quantity model is developed for a production–inventory system where the demand rate increases with time, the production rate is finite and adjustable in each cycle over an infinite planning horizon and shortages are permitted. The cost of adjusting the production rate depends linearly on the magnitude of the change in the production rate. During the stock‐out period, a known fraction of the unsatisfied demands is backordered while the remaining fraction is lost. The model is formulated taking the demand rate as a general increasing function of time and the optimal production policy is obtained for the special case of a linearly increasing demand rate. The proposed model is also shown to be suitable for a prescribed time horizon. A procedure to find approximately the minimum total cost of the system over a finite time horizon is suggested. A numerical example is taken to illustrate the solution procedure of the developed model.     

An EOQ model for deteriorating items with price- and stock-dependent selling rates under inflation and time value of money

This study applies the discounted cash flow (DCF) approach for the analysis of a replenishment problem over a finite planning horizon. Thus, a deterministic economic order quantity (EOQ) inventory model taking into account inflation and time value of money is developed for deteriorating items with price- and stock-dependent selling rates. An efficient solution procedure is presented to determine the optimal number of replenishment, the cycle time and selling price. Then the optimal order quantity and the total present value of profits are obtained. Numerical examples are presented to illustrate the proposed model and particular cases of the model are also discussed.     

An inventory replenishment policy for deteriorating items with shortages and partial backlogging

It has long been assumed that the shortages in inventory systems are either completely backlogged or totally lost. However, it is more reasonable to characterize that the longer the waiting time for the next replenishment, the smaller the backlogging rate would be. Moreover, the opportunity cost due to lost sales should be considered since some customers would not like to wait for backlogging during the shortage periods. Without considering these two realistic conditions, study on the inventory modeling for deteriorating items with shortages and partial backlogging cannot be complete and general. In the present article we define an appropriate time-dependent partial backlogging rate and introduce the opportunity cost due to lost sales. Numerical examples are also presented to illustrate the effects of changes in backlogging parameter and unit opportunity cost on total cost and the optimal number of replenishments.Scope and purposeIn a recent article published in this Journal, Giri et al. (Comput. Oper. Res. 27 (2000) 495–505) implemented an existing procedure to the inventory problem of Hariga and Al-Alyan (Comput. Oper. Res. 24 (1997) 1075–83) which concerns with lot-sizing heuristic for deteriorating items with shortages allowed in all cycles except the last one. Giri et al. deviated from the traditional practice and suggested a new policy allowing shortages in all cycles over a finite planning horizon. Their numerical results indicated the proposed policy is cheaper to operate with a cost reduction up to 15%. However, they did not consider the opportunity cost due to lost sales that happen because customers would not like to wait for backlogging. Moreover, for many products with growing sales, the length of the waiting time for the next replenishment is the main factor for determining whether the backlogging will be accepted or not, and the backlogging rate is expected to be time-dependent. Thus the assumption made in Giri et al. that the backlogging rate is a fixed fraction of the total amount of shortages is not reasonable.The purpose of this paper is to present a more realistic discussion of the inventory problem for deteriorating items with time-varying demands and shortages over a finite planning horizon. In contrast to the model by Giri et al., we define an appropriate partial backlogging rate and introduce the opportunity cost due to lost sales. We attempt to complement their model as a practical and general solution for inventory replenishment problems. With these extensions, the scope of applications of the present results is expanded.     

价格影响需求的易变质产品动态定价模型

在产品销售价格影响需求的条件下,利用最优控制理论建立了易变质产品的动态定价模型,目标是最大化产品销售周期内总的销售利润。利用Pontryagin 最大值原理得到了产品销售价格的最优性条件。通过对模型的理论分析得出如果产品销售价格介于单位产品购买费用和产品销售价格上限之间,且产品库存在销售周期结束之前始终为正时,销售周期内各时刻的产品最优销售价格一定大于与相应时刻变质率和产品单位库存成本有关的一个下界,销售周期内各时刻的产品最优库存水平一定小于与相应时刻变质率和产品单位库存成本有关的一个上界。