Demand Rate Function Research Articles

Inventory policies for seasonal items with logistic-growth demand rate under fully permissible delay in payment: a neutrosophic optimization approach

The present study investigates an inventory system for seasonal products under variable demand rate and partial backordering in a competitive market. Among various demand rate functions used in the existing literature of economic order quantity (EOQ) models, the logistic-growth function is best known to estimate market already captured and fraction of market remaining to be captured by new seasonal and technology driven items. Weibull distribution well represents the seasonality and versatility of these products. Due to retailers’ reluctance to purchase and store these perishable products, supplier offers the delay in payment. In view of the above, the proposed EOQ model suitable for those items considers the logistic-growth demand rate, Weibull distribution deterioration rate and partial backordering along with fully permissible delay in payment. Since the neutrosophic set quantifies the imprecise information in real-life scenarios, the proposed EOQ model is optimized in neutrosophic environment. A general unconstrained nonlinear mathematical model with neutrosophic coefficients is optimized using the weighted arithmetic mean function, subject to specified neutrosophic norm. A special case with the neutrosophic conjunction norm along with four lemmata is considered to minimize the cost functions with neutrosophic coefficients to proposed EOQ model across various trade-credit intervals. Here, the managerial insights identified through sensitivity analysis advocate to reduce the expenses on early promotions for foreshortening the demand at the beginning of cycle. Also, the present study demonstrates the optimal inventory depletion time to depend on the demand during the shortages in neutrosophic environment.

Optimal pricing and replenishment policy for non-instantaneous deteriorating items with varying rate of demand and partial backlogging

This paper proposes an inventory model for non-instantaneous deteriorating items with varying rate of demand and partial backlogging. The demand rate function has negative and positive exponential effect of price and promotional effort, respectively, when the product has no deterioration. Afterwards, the demand rate function assumes multiplicative exponential form of price-promotion and time wherein price-promotion and time each have exponential effect on demand rate. Under these settings, the criterion for optimal replenishment schedule, for any given selling price, is developed and deduced that optimal replenishment policy not only exist but also is unique. Next, it is shown that the total profit per unit time is concave function of price when optimal replenishment schedule is supplied. Numerical example is presented to demonstrate the applicability of the model accompanied by the sensitivity analysis. Results of numerical example indicates that granting promotional effort not only induces more demand but plays significant role to improve the profit.

Optimal Policies for Dynamic Pricing and Inventory Control with Nonparametric Censored Demands

We study the fundamental model in joint pricing and inventory replenishment control under the learning-while-doing framework, with T consecutive review periods and the firm not knowing the demand curve a priori. At the beginning of each period, the retailer makes both a price decision and an inventory order-up-to level decision, and collects revenues from consumers' realized demands while suffering costs from either holding unsold inventory items, or lost sales from unsatisfied customer demands. We make the following contributions to this fundamental problem as follows: 1. We propose a novel inversion method based on empirical measures to consistently estimate the of the instantaneous reward functions at two prices, directly tackling the fundamental challenge brought by censored demands, without raising the order-up-to levels to unnaturally high levels to collect more demand information. Based on this technical innovation, we design bisection and trisection search methods that attain an O(T^{1/2}) regret, assuming the reward function is concave and only twice continuously differentiable. 2. In the more general case of non-concave reward functions, we design an active tournament elimination method that attains O(T^{3/5}) regret, based also on the technical innovation of consistent estimates of reward differences at two prices. 3. We complement the O(T^{3/5}) regret upper bound with a matching \Omega(T^{3/5}) regret lower bound. The lower bound is established by a novel information-theoretical argument based on generalized squared Hellinger distance, which is significantly different from conventional arguments that are based on Kullback-Leibler divergence. This lower bound shows that no learning-while-doing algorithm could achieve O(T^{1/2}) regret without assuming the reward function is concave, even if the sales revenue as a function of demand rate or price is concave. Both the upper bound technique based on the difference estimator and the lower bound technique based on generalized Hellinger distance are new in the literature, and can be potentially applied to solve other inventory or censored demand type problems that involve learning.

Production Planning with Risk Hedging under a CVaR Objective

A central problem in planning production capacity is how to effectively manage demand risk. We develop a model that integrates capacity planning and risk hedging decisions under a popular risk measure, conditional value at risk (CVaR). The CVaR objective generalizes the usual risk-neutral objective (such as the expected payoff), and allows for explicit modeling of the degree of aversion to downside risk (associated with low demand). The starting point of our model is to incorporate the impact on demand from a financial asset (including, for instance, a tradable market index as a proxy for the general economy) via a demand rate function. This way, in addition to the capacity decision at the beginning of the planning horizon, there is also a dynamic hedging strategy throughout the horizon, the latter plays the role of both mitigating demand risk and supplementing the payoff. The hedging strategy is restricted to partial information along with a cap on loss (pathwise). To find the optimal hedging strategy, we construct and solve a dual problem to derive the optimal terminal wealth from hedging; the real-time hedging strategy is then mapped out via the martingale representation theorem. With the hedging strategy optimized, we show that optimizing the production quantity is a concave maximization problem. With both production and hedging (jointly) optimized, we provide a complete characterization of the efficient frontier, and quantify the improvement over the production-only approach. Furthermore, via sensitivity and asymptotic analyses, we spell out the impacts of the hedging budget and the risk aversion level, along with other qualitative insights.

Best pricing and optimal policy for an inventory system under time-and-price-dependent demand and backordering

In this paper, we study an inventory system for products where demand depends on time and price. Shortages are allowed and are fully backordered. We suppose that the demand rate is the product of a power time pattern and a three-parametric exponential price function. The objective is to determine the economic lot size, the optimal shortage level and the best selling price to maximize the total profit per unit time. We present an efficient procedure to determine the optimal solution of the inventory problem for all possible scenarios. This procedure is illustrated with several numerical examples. A sensitivity analysis of the optimal inventory policy with respect to the parameters of the demand rate function is also given. Finally, the main contributions of this paper are highlighted and future research directions are introduced.

A production inventory model with selling price and stock sensitive demand under partial backlogging

In the proposed paper we develop an inventory model for instantaneous deteriorating items with multi variate function of demand rate. Generally, we observe that the demand for any product depends on so many factors, out of which the available stock and selling price are the main factors. So in this paper we have assumed that the demand rate is a function of stock and selling price. The production rate is taken as a function of demand rate. The shortages are allowed and it is assumed that the occurring shortages will be backlogged for which the backlogging rate is a function of waiting time. The numerical example and sensitivity exploration with respect to various parameters are also cited to illustrate the study.

An EPQ model for deteriorating items with variable demand rate and allowable shortages

The fundamental assumption of an economic order quantity (EOQ) model is that 100% of items in an ordered lot are perfect. This assumption is not always pertinent for production processes because of process deterioration or other factors. This is an economic production quantity model for deteriorating items. To fulfil the market demands and expectations, the production rate is taken as a function of demand rate. The demand and deterioration of the products are time dependent function. Shortages are allowed and partially backlogged. The backlogging phenomenon in the literature is often modelled using backordering and lost sale costs. The backlogging option gets used only when it is economic to do so. The inventory policy proposed here considers the optimal production run time, production quantity and shortage period such that the total average cost can be minimised. Numerical examples are provided to illustrate and sensitivity analyses of optimal solutions are given for the proposed inventory model.

Robust Inventory Management: An Optimal Control Approach

We formulate and solve static and dynamic models of inventory management that lie at the intersection of robust optimization and optimal control theory. Our objective is to minimize cumulative ordering, holding, and shortage costs over a horizon [0, T], where the variable is a nonnegative ordering rate function q(t) ∈ ℒ2[0, T]. The demand rate function d(t) is unknown and is only assumed to belong to an uncertainty set [Formula: see text]; this set is motivated by the strong law of large numbers for stochastic processes [Formula: see text] where μ is the mean drift. We analyze a static model, where the ordering rate function must be fully specified at time zero, and three dynamic variants, where re-optimizations are allowed during the planning horizon [0, T] at prespecified review epochs. In the dynamic models, at review epoch τ ∈ [0, T], the past demand on [0, τ) is observable. In the first dynamic model, we ignore this information, and define a variant of Ω that is well formed for the remaining planning horizon [τ, T]. In the second model, we define a variant of Ω for [τ, T] that utilizes the past demand information, though we make a simplifying technical assumption about the consistency of the demand on [0, τ) and Ω. In the third dynamic model, we remove this assumption, and we remedy the arising complications using the Hilbert Projection Theorem. In all cases we derive optimal closed-form ordering rate functions that equal either the bounds a or b, or weighted averages of these bounds (sb + ha)/(s + h) or (sa + hb)/(s + h), where s and h are the shortage and holding costs, respectively. The strategies differ by when these four ordering rates are applied, which is determined by an uncertainty-set-dependent partition of the remaining planning horizon. Computational experiments, focused on studying the dynamic variants, supplement the analytical results, and demonstrate that (1) the three variants exhibit comparable performance under well-behaved stochastic demand and (2) the third variant has a significant advantage when demand is seasonal, especially when the review frequency is appropriately selected. Finally, computational comparisons with the omniscient strategy q(t) = d(t), for all t ∈ [0, T], are encouraging.

Optimal manufacturing batch size with rework for a finite-horizon and time-varying demand rates inventory model

This paper proposes a finite-horizon and time-varying demand rate function formulations for the optimal manufacturing batch size model with rework. The basic model is found in [Jamal et al. Comput. Ind. Eng. 47 (2004) 77–89.]. Two policies 1 and 2 are considered. In Policy 1 defective items produced in a given period are remanufactured within thesame period while Policy 2 accumulates the defective items until the last period. The search for the optimal manufacturing batch size for policies 1 and 2 is shown to reduce to the problem of determining the number of manufacturing-rework periods as well as their starting and finishing times. This leads to the examination of two mixed integer non-linear programming problem which are completely solved by appealing to some established techniques proposed in [Al-Khamis et al. Int. J. Syst. Sci. 45 (2014) 2196–2202]. Numerical results are also presented for illustration.

A Multi Item Integrated Inventory Model with Reparability and Manufacturing of Fresh Products

<p>In this paper a multi item integrated inventory model is presented with reparability of returned items. It is assumed here that only a certain ratio of returned items can be repaired and the remaining stock of returned items is salvaged. By using these returned items, the waste can be reduced, which pollute the environment. This is a green supply chain where the demand for the products is selling price dependent and production rate is taken as a function of demand rate. The shortages are allowed here. A numerical example and sensitivity analysis are also presented to illustrate the model.</p>

Optimal Control of Decoupling Point for Deteriorating Food with Time-Varying Demand

The position of decoupling point denotes the penetration degree of customer demand into supply chain. To optimize the performance of deteriorating food supply chain, we consider the decoupling point control in conjunction with food production and inventory management under time-varying demand over a finite time horizon. Using dynamic models, optimal position of decoupling point and production-inventory plan are simultaneously determined. The results show that the optimal decoupling point is not related to the change of food demand under zero-inventory policy and it is a monotonically ascending function of demand rate under production smoothing policy. The simulation illustrates a diagram depicting that the optimal decoupling point shifts to the upstream suppliers along with the increase of food deteriorating rate while shifts downstream to the end customers with the growth of the time elasticity of food demand.

A Supply Chain Production Inventory Model for Deteriorationg Product with Stock Dependent Demand under Inflationary Environment and Partial Backlogging

In this paper we have developed a two echelon supply chain production inventory model for deteriorating products having stock dependent demand under inflationary environment. This model is developed for finite time horizon. The shortages are allowed and partially backlogged. To make this study close to reality the production rate is assumed to be a function of demand rate. A numerical example and sensitivity analysis with respect to different associated parameter is also presented to illustrate the study.

Systematic Failures, Systematic Capability and Prior-Use

This paper is a discussion of the nature of systematic failures in Safety Instrumented Systems and the way that they can be managed. It identifies the upper bound to the probability of failure due to systematic failures as a function of demand rate and a systematic test interval. It discusses how the functional safety standards address systematic failures through systematic capability defences and how these considerations may be addressed when identifying a ‘Prior-use’ case for assessing fitness-for-purpose of system elements.

An EPQ model for non-instantaneous deteriorating item with time dependent holding cost and exponential demand rate

We consider the problem of a vendor that supplies an item to the buyer, which is a function of non-instantaneous deterioration. It is an integrated production inventory model of non-instantaneous deteriorating item in which demand rate is exponential and production rate is a function of demand rate. The holding cost varies with time. Partially deteriorated items are allowed to sale with a discount rate from original one, and the units which are completely deteriorated, are superfluous. Shortages are not allowed. The optimal cycle time is derived and the result is applied to numerical problems. A solution procedure is presented to determine an optimal replenishment cycle such that the total cost is maximised. Numerical example is cited to illustrate the results and its significant features. Finally, to study the effect of changes of demand parameters, deterioration, price discount and holding cost, on supplier a sensitivity analysis is presented numerically.

Optimal control of a stochastic production–inventory system under deteriorating items and environmental constraints

In this paper, we present an optimal control model of a stochastic production–inventory with deteriorating items, emission tax and pollution abatement investment. In our model, the emission tax is levied on the firm’s environmental obsolescence rate of technology rather than the total amount of the environmental externality. Our objective is to apply Hamilton–Jacobi–Bellman (HJB) equation to solve the stochastic production–inventory system with deteriorating items, emission tax and pollution abatement investment; and derive the optimal production rate and pollution abatement investment rate that maximise the objective function value. The results are discussed with some illustrative examples for different demand rate functions, and sensitivity analysis is conducted to study the effect of changing the parameters and coefficients on the objective function value.

Optimal policies for a finite-horizon batching inventory model

This paper is concerned with finding an optimal inventory policy for the integrated replenishment-production batching model of Omar and Smith (2002). Here, a company produces a single finished product which requires a single raw material and the objective is to minimise the total inventory costs over a finite planning horizon. Earlier work in the literature considered models with linear demand rate function of the finished product. This work proposes a general methodology for finding an optimal inventory policy for general demand rate functions. The proposed methodology is adapted from the recent work of Benkherouf and Gilding (2009).

An Integrated Model with Variable Production and Demand Rate under Inflation

In this article, an integrated model is developed in which a manufacturer purchases raw materials from a supplier, and then produces finished products/goods, after that delivers them to a buyer. In the intended model production rate is assumed as a function of demand rate and customer demand rate is time dependent. To make the model more realistic the effect of inflation and time value of money is also taken into consideration. The concept of the model is illustrated through the numerical example and sensitivity analysis with respect to the system parameters is also performed.

Note on inventory models with Weibull distribution deterioration

This article explores the inventory model with a general demand rate function in which both the Weibull distributed deterioration and partial backlogging are considered. The inventory model discussed here is based on the important finding by Wu [Wu, K.S., 2001. An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging. Production Planning and Control, 12, 787–793]. There are four parts in our research. First, we derive the analytical framework of the inventory model for a general demand rate function1. Second, for the ramp type demand, we improve Wu's model to find the criterion to guarantee the existence and uniqueness of the optimal solution. Third, we develop a new model to compensate for the missing case in Wu's article. Fourth, we combine our results to show that our findings are applicable to the ramp type demand inventory model, so that the optimal solution is independent of the demand function. Finally, some numerical examples and graphs are provided to illustrate our discovery and demonstrate the application of our analytical framework.

An Integrated Supplier‐Buyer Inventory Model with Conditionally Free Shipment under Permissible Delay in Payments

It is well known that production, distribution, marketing, inventory control, and financing all/each have a positive impact on the performance of a supply chain. Despite the growing interest in the development of integrated inventory models, the interactions between these elements of a supply chain may not be efficiently included, resulting in a restricted supply chain model presentation. To incorporate this phenomenon, a mathematical model that tackles the interdependent relationships between these aforementioned elements is developed in this paper. This study considers the determination of the optimal pricing, ordering, and delivery policies of a profit‐maximizing supply chain system, faced with (1) unit wholesale price of the supplier is set based on unit production cost, (2) unit production cost is taken as a function of demand rate and production rate, (3) the supplier′s production rate is adjusted according to market demand, (4) market demand depends upon buyer′s selling price, (5) a free freight is offered if the buyer′s order exceeds a certain minimum requirement, and (6) a constant credit period is offered by the supplier to stimulate the demand of the buyer. Algorithm for computing the optimal policies is derived. The sensitivity of the optimal results with respect to those parameters which directly influence the production and transportation costs is also examined.

THE PRICING STRATEGY OF COMPLETE PRE-ORDERED MERCHANDISE UNDER DISCOUNTED PROFIT FOR MAXIMIZING

When a customer steps into a complete pre-ordered store, he will review the merchandise and consider his demands based on the merchandise price levels and price variability at that point in time. However, after declaring his intention to purchase said merchandise, the store assistant informs him that the merchandise will not be available for a period of time. This is a typical stock-out merchandise scenario in which customers may only place an order for delivery at a later point in time. Therefore, whether or not customer purchases merchandise does not just depend on the price at that moment. It is also influenced by the expected future increase or decrease of the price of the merchandise and the length of time before the store can supply the merchandise. In this study, we will explore how to set price levels at each point in time during the stock-out period in order to maximize the discounted profit after considering the influence of the price level, price variability, and waiting time on customer demand. The main assumption in this study is that customers' potential demand rate function at a given point during the stock-out period is a linear function of the price level and price variability at that point. Also, the ratio function of customers willing to wait for the merchandise is an exponential function of the length of time before the merchandise will be delivered. Constructing a mathematical model that is concrete to discuss the above problem, to derive the optimal price function of the merchandise at each point in time, and to discuss the characteristics of this function are the main parts of this study.