Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging

In this paper, a deterministic inventory model for deteriorating items with price-dependent demand is developed. The demand and deterioration rates are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. Under these assumptions, for any given selling price, we first develop the criterion for the optimal solution for the replenishment schedule, and prove that the optimal replenishment policy not only exists but also is unique. If the criterion is not satisfied, the inventory system should not be operated. 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 numerical examples to illustrate the algorithm.     

Inventory models with ramp type demand rate,partial backlogging and Weibull deterioration rate

In this paper, an inventory model with general ramp type demand rate, time dependent (Weibull) deterioration rate and partial backlogging of unsatisfied demand is considered. The model is studied under the following different replenishment policies: (a) starting with no shortages and (b) starting with shortages. The model is fairly general as the demand rate, up to the time point of its stabilization, is a general function of time. The backlogging rate is any non-increasing function of the waiting time up to the next replenishment. The optimal replenishment policy for the model is derived for both the above mentioned policies.     

Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging

Inventory model for time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging is considered in this paper. The demand rate is defined as a continuous trapezoidal function of time, and the backlogging rate is a non-increasing exponential function of the waiting time up to the next replenishment. We proposed an optimal replenishment policy for such inventory model, numerical examples to illustrate the solution procedure.     

A deterministic EOQ model with delays in payments and price-discount offers

It is the purpose of this paper to model the retailer’s profit-maximizing strategy when confronted with supplier’s trade offer of credit and price-discount on the purchase of merchandise. Generally, retailers have to face many types of demands for different kinds of goods. In real situation, retailers have to correlate between the selling price and supplier’s trade offer, keeping in mind profit-maximization strategy. In the proposed model, all increasing deterministic demands are discussed analytically, numerically and graphically in the environment of permissible delay in payment and discount offer to the retailer.     

The stochastic EOQ model with random sales price

The article deals with a stochastic economic order quantity (EOQ) model over a finite time horizon where uniform demand over the replenishment period is price dependent. The selling price is assumed to be a random variable that follows a probability density function. As demand is probabilistic, stock out situation may occur. Based on the partial backlogging and lost sale cases during stock out period, the author develops the criterion for the optimal solution for the replenishment size such that the integrated expected profit is maximized. Moreover, the article suggests a new function regarding price dependent demand. Finally, numerical examples and its sensitivity analysis of key parameters are given to illustrate the proposed model.     

Optimal replenishment time for retailer under partial upstream prepayment and partial downstream overdue payment for quadratic demand

In competitive edge, the proffer of delay payment is of great consequence tool to boost the market demand by exerting influence of more customers. In opposition to, merchandiser may be asked to pay some per cent of purchasing cost before receiving products to reduce the risk of cancellation of order. In this article, optimal replenishment time for merchandiser under partial upstream prepayment and partial downstream overdue payment is presented under quadratic demand with three different models: (1) without deterioration, (2) with constant deterioration and (3) maximum fixed-lifetime deterioration as loss of utility is the real situation of products like fruits, vegetables, juices, etc., and to reduce deterioration of the product, merchandiser spends capital on preservation technology to preserve the item. Quadratic demand is suitable for the products for which demand increases initially and afterward it starts to decrease. The decision policies are analysed for the merchandiser. The objective is to minimize merchandiser’s total cost with respect to decision variables. The models are supported with numerical examples. Sensitivity analysis is carried out to derive insights for decision maker.     

Multi-period modeling of two-way price commitment under price-dependent demand

This paper examines the use of price-commitment policies in dynamic contracting in multiple-period, finite-time horizons. Two specific forms of price commitment are considered: one on the part of the retailer through a retail-fixed-markup contract and one on the part of the manufacturer through a price-protection contract. Optimal policies for each form of price commitment are analytically derived, as are optimal policies for the traditional price-only and centralized supply chain scenarios that we use as comparisons. We prove that optimal retail price and order size solutions exist in each period under the assumption of non-increasing price-dependent demand. We show that the existence of retailer inventory between periods causes the optimal policies to differ from a static single-period model. Further, we show that a supplier offers a price-protection policy as a signal to the retailer to resolve the gaming that naturally occurs under price-only; this effectively decouples the multi-period dynamic contracting setting into repeated single-period scenarios. However, the resulting behavior can actually inhibit supply chain performance. On the retail commitment side, we find that retail-fixed-markup policies are quite effective in improving supply chain efficiency. We show that such policies can lead to Pareto-improvement over price-only contracts and can even coordinate the supply chain in some situations.     

An economic lot-size model with partial backlogging hinging on waiting time and shortage period

We study a continuous review inventory control system over a infinite-horizon with deterministic demand where shortage is partially backlogged. The backlogging is characterized using an approach in which the collective behavior of customers hinges on the waiting time and on the shortage period. We assume that the fraction of the customers who are not willing to wait is proportional to the ratio between the waiting time and the length of the shortage cycle. Taking into account the above assumption we determine the optimal inventory policy. Several models studied by other authors result to be particular cases of the considered model.     

基于时变需求的集成多级供应链生产订货策略研究

考虑一个时变需求环境下集成多级供应链问题,在有限的规划时间内销售商以固定周期订货,而生产商以不同的周期生产,目的是寻找销售商最优的订货周期和生产商最佳的生产策略,从而使供应链系统的总运营成本最少.建立了该问题的混合整数非线性规划模型,求解该模型分为两步：先求对应一个订货周期的最佳生产策略,再求最优的订货周期,第一步用到了图论里求最短路方法.给出了两个步骤的算法和程序,实验证明它们是有效的.通过算例对模型进行了分析,研究了各参数对最优解及最小费用的影响.     

Optimal dynamic pricing and inventory control with stock deterioration and partial backordering

This paper studies the optimal dynamic pricing and inventory control policies in a periodic-review inventory system with fixed ordering cost and additive demand. The inventory may deteriorate over time and the unmet demand may be partially backlogged. We identify two sufficient conditions under which (s,S,p) policies are optimal.     

An EOQ model for perishable products with discounted selling price and stock dependent demand

A single item economic order quantity model is considered in which the demand is stock dependent. After a certain time the product starts to deteriorate and due to visualization effect and other aspects of deterioration the demand becomes constant. In that situation a discount on selling price provides significant increment in demand rate. In this paper we investigate how much discount on selling price may be given during deterioration to maximize the profit per unit time and whether a pre-deterioration discount affects the unit profit or not. A mathematical model is developed incorporating both pre- and post deterioration discounts on unit selling price, where analytical results reveal some important characteristics of discount structure. A numerical example is presented and sensitivity analysis of the model is carried out.     

An EOQ model for deteriorating items with time varying demand and partial backlogging

In the classical economic order quantity model, it is often assumed that the shortages are either completely backlogged or completely lost. However, in some inventory systems, it is more reasonable to assume that the backlogging rate is dependent on the length of the waiting time for the next replenishment. The longer the waiting time is, the smaller the backlogging rate would be. In this paper, we focus on the effect of the backlogging rate on the economic order quantity decision. Numerical examples are presented to illustrate the model.     

Instability results for slowly time varying linear dynamic systems on time scales

We develop eigenvalue criteria under which the solutions of a “slowly” time varying linear dynamic system of the form xΔ(t)=A(t)x(t) are unstable.     

Inventory returns and special sales in a lot-size system with constant rate of deterioration

The problem of returning or of selling the inventory excess to optimal stock level is considered for deteriorating items. Two inventory models, viz. the infinite and the finite horizon models, are developed, in which the deterioration is assumed to be a constant fraction of the on hand inventory. Both the models are developed under the assumptions of instantaneous delivery and no shortages. When there is no deterioration, the developed models are related to the corresponding inventory models for non-deteriorating items. Examples are given to illustrate the derived results.     

Modeling a multi-attribute utility newsvendor with partial backlogging

Traditional newsvendor models usually focus on single profit maximization or cost minimization approaches. However, making a monetary estimate of the consequences of lost sales as a result of shortages is often a difficult task for many practitioners. Besides, there is still a lack of an explicit account of decision-making judgments on the multiple consequences of making decisions with regard to order quantities. In order to deal with this problem, this paper presents a multi-attribute utility model for the newsvendor problem with regard to profit, the impacts of service level on corporate image and on customers’ goodwill, and the impact on the environment arising from the disposal of unsold products. Demand is partially backlogged according to a decreasing exponential function of the waiting time. The fundamental principles and limitations related to the application of the model built are also discussed.     

Modeling a stochastic multi-objective supplier quota allocation problem with price-dependent ordering

One of the interesting subjects in supply chain management is supply management, which generally relates to the activities regarding suppliers such as empowerment, evaluation, partnerships and so on. A major objective of supplier evaluation involves buyers determining the optimal quota allocated to each supplier when placing an order. In this paper, we propose a multi-objective model in which purchasing cost, rejected units, and late delivered units are minimized, while the obtained total score from the supplier evaluation process is maximized. We assume that the buyer obtains multiple products from a number of predetermined suppliers. The buyer faces a stochastic demand with a probability distribution of Poisson regarding each product type. A major assumption is that the supplier prices are linearly dependent on the order size of each product. Since demand is stochastic, the buyer may incur holding and stockout costs in addition to the regular purchasing cost. We use the well-known L-1 metric method to solve the supplier evaluation problem by utilizing two meta-heuristic algorithms to solve the corresponding mathematical problems.     

A production-inventory model with probabilistic deterioration in two-echelon supply chain management

In this study, a production-inventory model is developed for a deteriorating item in a two-echelon supply chain management (SCM). An algebraical approach is applied to find the minimum cost related to this entire SCM. We consider three types of continuous probabilistic deterioration function to find the associated cost. The purpose of this study is to obtain the minimum cost with integer number of deliveries and optimum lotsize for the three different models. Some numerical examples, sensitivity analysis and graphical representation are given to illustrate the model. A numerical comparison between the three models is also given.     

Optimal price and lot size determination for a perishable product under conditions of finite production, partial backordering and lost sale

The paper describes an EOQ model of a perishable product for the case of price dependent demand, partial backordering which depends on the length of the waiting time for the next replenishment, and lost sale. The model is solved analytically to obtain the optimal price and size of the replenishment. In the model, the customers are viewed to be impatient and a fraction of the demand is backlogged. This fraction is a function of the waiting time of the customers. In most of the inventory models developed so far, researchers considered that inventory accumulates at the early stage of the inventory and then shortage occurs. This type of inventory is called IFS (inventory followed by shortage) policy. In the present model we consider that shortage occurs before the starting of inventory. We have proved numerically that instead of taking IFS, if we consider SFI (shortage followed by inventory) policy, we would get better result, i.e., a higher profit. The model is extended to the case of non-perishable product also. The optimal solution of the model is illustrated with the help of a numerical example.     

An inventory system with unit demand and varying ordering levels

An inventory system with unit demand, varying ordering levels and random lead times is considered in this paper. Ordering level is determined by the number of demands during last lead time. The ordering quantity will be such as to bring back the inventory level to S at the ordering epoch. No backlog is permitted. The time dependent probability distribution of the inventory level is obtained. Correlation between the number of demands during a lead time and the length of the next inventory dry period is obtained and it is illustrated by an example.     

Optimal maintenance of systems with Markovian mission and deterioration

We consider the maintenance of a mission-based system that is designed to perform missions consisting of a random sequence of phases or stages with random durations. A finite state Markov process describes the mission process. The age or deterioration process of the system is described by another finite state Markov process whose generator depends on the phases of the mission. We discuss optimal repair and optimal replacement problems, and characterize the optimal policies under some monotonicity assumptions. We also provide numerical illustrations to demonstrate the structure of the optimal policies.