Distribution Function Of Demand Research Articles

Minimax Distribution-Free Procedure for Mixture Inventory Model with Variable Lead Time and a Service Level Constraint by Reducing Order Cost

SYNOPTIC ABSTRACTAs supply chains have become more extended in recent years, coordination and information sharing among supply chain members to improve system efficiency have attracted much interest. In order to achieve a higher degree of coordination and automation among the supply chain parties, the supply chain sometimes invests in reducing the ordering cost to streamline and speed up transactions via the application of information technology. If the ordering cost per order could be reduced effectively, then the total relevant cost per unit time could be automatically improved. In this study, we investigate a continuous review inventory model involving controllable lead time for the mixtures of distributions of the lead time demand, subject to a service level constraint in order to accommodate more practical features of the real inventory systems. Further, ordering cost can be reduced by an investment in which the optimal inventory control approach is shaped to account for the service level restriction.We assume that any mixture of distribution functions of the lead time demand gives only known finite, first two moments, but its probability distribution is unknown, and we apply a minimax distribution-free procedure to find the optimal solution. Moreover, a computational algorithm with the help of a computer code using the software MATLAB 7.0 is developed to solve the problem. A numerical example is presented to illustrate the theoretical results, and a sensitivity analysis is performed with respect to the various system parameters. Through sensitivity analysis, some observations and managerial implications are presented.

Benders’ decomposition for concurrent redesign of forward and closed-loop supply chain network with demand and return uncertainties

This paper attempts to design a reverse supply chain network (SCN), add it to an existing multi-product forward SCN and simultaneously redesign the existing forward supply chain (SC). The problem considers uncertainty on products demand and and also returned products in multi-period context. Benders’ decomposition is applied to solve the stochastic mixed-integer model to optimality. The scenarios are generated based on the demand distribution function using Cholesky’s factorization method to consider correlation among different products’ demands. To decrease the computational effort, the number of scenarios is reduced using k-means clustering algorithm. The method is tested on a cell phone SC.

A Simulation Approach to Determine the Probability of Demand during Lead-Time When Demand Distributed Normal and Lead-Time Distributed Gamma

Globalization and advances in information and production technologies make inventory management can be very difficult even for organizations with simple structures. The complexities of inventory management increase in multi-stage networks, where inventory appears in multiple tiers of locations. Due to massive practical applications in the reality of the world, an efficient inventory system policy whether single location or multi-stage location will avoid falling into overstock inventory or under stock inventory. However, the optimality of inventory and allocation policies in a supply chain is still unknown for most types of multi-stage systems. Hence, this paper aims to determine the probability distribution function of demand during lead-time by using a simulation model when the demand distributed normal and the lead-time distributed gamma. The simulation model showed a new probability distribution function of demand during lead-time in the considered inventory system, which is, Generalized Gamma distribution with 4 parameters. This probability distribution function makes the mathematical expression more difficult to build the inventory model especially in multistage or multi-echelon inventory model.

A continuous review production–inventory system in fuzzy random environment: Minmax distribution free procedure

In many manufacturing systems, the production process may take some time to start the initial phase due to various reasons such as delay in installation of machines, short supply of raw materials, unavailability of workers, etc. Thus, the organization should plan accordingly so that the manufacturing process can start at the desired time. In an economic production quantity (EPQ) model, lead-time plays a significant role in ensuring that the manufacturing process starts on time. As we know, when both lead-time and demand rate are deterministic and constant, then demand during the lead-time is constant, and is referred to as zero lead-time. Moreover, when either or both of them are random variables, then lead-time demand (LTD) is a random variable. In such a case, a crucial question is: “when should the order be placed?” On the other hand, the distributional information on demand may not always be available or there may be many distribution functions in the practice, which have same mean and variance, but their frequencies are different. In this study, we develop an EPQ model in stochastic framework, wherein the distribution function of demand is unknown, but the mean and variance are known. The inventory level is continuously reviewed, and an order is placed when it reaches the reorder level. The real-life business situations are so sophisticated and floating in nature that the consideration of ‘impreciseness’ along with ‘statistical variability’ in demand parameter is more preferable. To be a part of this contingency, we further extend the model in the fuzzy random environment by considering demand rate as a fuzzy random variable (FRV). Furthermore, we mathematically analyze the cost function and propose a heuristic procedure to find the global optimum. Numerical examples with sensitivity analysis are also provided for illustration purpose.

A general model for the exact computation of yield from a rainwater tank

The models presented in the literature for the computation by simulation of yield from a rainwater tank are either incorrect or approximate at best. In this paper a general, exact, continuous time model of the operation of rainwater tanks is presented. The operation of a tank is determined by the rainfall runoff and demand distribution functions and the capacity of the tank. The runoff and demand distributions are assumed to satisfy certain reasonable conditions which are satisfied in all cases of practical interest, e.g. when the distributions are described by piecewise constant functions. The yield, spill and volume functions are characterized by equations which, by means of a constructive proof involving mathematical induction, can be shown to have unique solutions. This model can be used to compute the exact yield and spill over any period and the volume at any time during the operation of the tank. Correct discrete time simulation models are derived in the special cases of demand before rain (DBR) or demand after rain (DAR).

Estimating Water Demands in Buildings

Abstract The 99th percentile of water demand during the peak period is often adopted to size the piping system in a building. Two new approaches for estimating peak water use are introduced. Both provide the cumulative distribution function (CDF) of water demand during the peak period. One approach uses full enumeration of all fixture options; the second employs Monte Carlo techniques. The Monte Carlo method generates a large ensemble of simulated design flows for a wide variety of fixture combinations. Results are displayed as a single curve on a universal dimensionless design chart applicable for any number and type of fixtures. An example demonstrates application of the new design chart.

The role of demand response in single and multi-objective wind-thermal generation scheduling: A stochastic programming

This paper focuses on using DR (Demand Response) as a means to provide reserve in order to cover uncertainty in wind power forecasting in SG (Smart Grid) environment. The proposed stochastic model schedules energy and reserves provided by both of generating units and responsive loads in power systems with high penetration of wind power. This model is formulated as a two-stage stochastic programming, where first-stage is associated with electricity market, its rules and constraints and the second-stage is related to actual operation of the power system and its physical limitations in each scenario. The discrete retail customer responses to incentive-based DR programs are aggregated by DRPs (Demand Response Providers) and are submitted as a load change price and amount offer package to ISO (Independent System Operator). Also, price-based DR program behavior and random nature of wind power are modeled by price elasticity concept of the demand and normal probability distribution function, respectively. In the proposed model, DRPs can participate in energy market as well as reserve market and submit their offers to the wholesale electricity market. This approach is implemented on a modified IEEE 30-bus test system over a daily time horizon. The simulation results are analyzed in six different case studies. The cost, emission and multiobjective functions are optimized in both without and with DR cases. The multiobjective generation scheduling model is solved using augmented epsilon constraint method and the best solution can be chosen by Entropy and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) methods. The results indicate demand side participation in energy and reserve scheduling reduces the total operation costs and emissions.

Optimizing a convex multi-constraint distribution-free inventory model with fixed and variable crash costs

This paper presents a multi-constraint stochastic inventory model in which the ordering cost, order quantity, reorder point and lead time are the decision variables. In order to fit some real and practical environment, this study considers the maximum permissible storage space and the budget is limited. The lead time crash cost is considered as a function of both order quantity and reduced lead time. This study relaxes the assumption about the form of the distribution function of the lead time demand by only assuming that the mean and standard deviation of the lead time demand are known. A minimax distribution-free procedure is developed for the constrained problem, a Lagrange multiplier method is applied in order to solve it, and a solution procedure is proposed to find the optimal values. This study also shows that the respective budget- and storage-space-constrained inventory model to be minimized is jointly convex in the decision variables. A numerical example is presented to illustrate the discussed model and a sensitivity analysis is conducted with respect to the important system parameters.

A minimax distribution-free procedure for a newsvendor problem with free shipping

In this paper, we study the optimal policies of retailers who operate their inventory with a single period model (i.e., newsvendor model) under a free shipping offer where a fixed shipping fee is exempted if an order quantity is greater than or equal to a given minimum quantity. Zhou et al. (2009) have explored this model, and we further investigate their analysis for the optimal ordering policies which they did not sufficiently develop. Based on the investigation, we extend the base model in order to deal with the practically important aspect of inventory management when the exact distribution function of demand is not available. We incorporate the aspect into the base model and present the optimal policies for the extended model with a numerical example. Finally, we conduct extensive numerical experiments to evaluate the performance of the extended model and analyze the impacts of minimum free shipping quantity and the fixed shipping fee on the performance.

Optimal Monetary Noise in an Economy with Bayesian Consumers and Risk-Averse Investors

Autonomous demand shock affects consumption spending. Variation in consumption spending contributes to the volatility in aggregate demand. As the investor is risk averse, volatility of aggregate demand reduces investment. Government injects monetary noise to reduce the volatility in aggregate demand and induce higher investment. The monetary noise clouds the observation of the autonomous aggregate demand by the consumer. The consumer forms Bayesian beliefs which are consistent with the equilibrium they supported for forecasting autonomous aggregate demand and monetary noise. With a greater monetary noise, the consumer relies less on the inaccurate observation of autonomous aggregate demand and more on the prior distribution functions of autonomous aggregate demand and monetary noise to decide on the level of consumption spending. Consumption spending therefore reflects less of the volatility in autonomous aggregate demand. Faced with a less volatile consumption spending, the investor increases investment.

English

  In this paper we developed an inventory model in mixed imprecise and uncertain environment. Presented model is the developed form of r,Q and is a multi-itemsmodel with two objectives as minimizing costs (holding and shortage) and risk level under constraints including available budgetary, the least service level, storage spaces and allowable quantities of shortage. Demand distribution functions are assumed to be exponential and extra demands are supposed in two situations as lost sales and backlogging. At first we develop crisp model then fuzzy stochastic model with fuzzy budgetary, allowable quantities of shortage and shortage spaces (that is stochastic with normal distribution function) parameter. All of fuzzy numbers are triangular type. In methodology of solution we change model to a crisp multi-objective by using difuzzification of fuzzy constraints and fuzzy chance-constrained programming methods, and then solve it by fuzzy logic method. Finally an illustrated example is taken and solved using LINGO package.   Key words: r,Q model, lost sales, backlogging, multi objective programming, fuzzy-chance constrained programming.

Supply Chain Contract Arrangements of Carbon Abatement with Consumer Preferences

With the global climate continues warming, carbon emissions become the global focus. How to design the contract of low-carbon supply chain is an important research direction. Our goal in this paper is to construct supply chain decision model under carbon tax. And we build the demand distribution function including consumer characters which is more in line with the reality of the actual retailer operations. Through numerical analysis using Matlab software， we find that carbon emissions allocation coefficient does not affect the optimal profits of suppliers and retailer under wholesale price contract; Supply chain enterprise could gain more profit through the carbon emissions collaborative, choosing reasonable carbon emissions levels.

Computational procedure of optimal inventory model involving controllable backorder rate and variable lead time with defective units

This article considers that the number of defective units in an arrival order is a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate be a control variable. In addition, we assume that the lead time demand follows a mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, three numerical examples are also given to illustrate the results.

Reducing lost-sales rate on the stochastic inventory model with defective goods for the mixtures of distributions

In this paper, we assume that the demands of different customers are not identical in the lead time. Thus, we investigate a continuous review inventory model involving controllable lead time and a random number of defective goods in buyer’s arriving order lot with partial lost sales for the mixtures of distributions of the lead time demand to accommodate more practical features of the real inventory systems. Moreover, we analyze the effects of increasing investment to reduce the lost sales rate when the order quantity, reorder point, lost sales rate and lead time are treated as decision variables. In our studies, we first assume that the lead time demand follows the mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. By analyzing the total expected cost function, we develop an algorithm to obtain the optimal ordering policy and the optimal investment strategy for each case. Finally, we provide numerical examples to illustrate the results.

Linear approximation approach for a stochastic seat allocation problem with cancellation & refund policy in airlines

This study is focused on the seat allocation problem with stochastic demands airlines to determine booking limits by fare classes in airlines. In this paper, we consider the additional constraints such as customers’ booking cancellations and airlines’ refund policy. Our problem can be modeled as a probabilistic non-linear programming model, which is difficult to solve optimally. Applying a linear approximation technique with some assumptions on the demand distribution functions, we can formulate the complex probabilistic non-linear programming model as a simple Integer Linear Programming model. Using randomly generated demands from a simulation program, the model demonstrates that revenue can be increased in a practical and effective manner.

A continuous review inventory system with controllable lead time and defective items in partial and perfect lead time demand distribution information environments

This paper presents the mixed inventory backorder and lost sales involving four variables; order quantity, lead time, safety factor and backorder rate. Optimal safety factor is obtained discretely. This article assumes that an order lot may contain some defective items and the number of defective items is random variable. Controllable negative exponential backorder rate is considered in the existing model. In the real market, as unsatisfied demands occur, the longer the length of lead time is, the smaller the proportion of backorder rate would be. Considering this reason, backorder rate is dependent on length of lead time through the amount of shortages. Negative exponential lead time crashing cost is considered in the presented model. This study, first assumes the lead time demand follows a normal distribution and then relax the assumption about the form of the distribution function of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, a numerical example is also given to illustrate the model.

Intermittently renewable energy, optimal capacity mix and prices in a deregulated electricity market

This paper assesses the effect of intermittently renewable energy on generation capacity mix and market prices. We consider two generating technologies: (1) conventional fossil-fueled technology such as combined cycle gas turbine (CCGT), and (2) sunshine-dependent renewable technology such as photovoltaic cells (PV). In the first stage of the model (game), when only the probability distribution functions of future daily electricity demand and sunshine are known, producers maximize their expected profits by determining the CCGT and PV capacity to be constructed. In the second stage, once daily demand and sunshine conditions become known, each producer selects the daily production by each technology, taking the capacities of both technologies as given, and subject to the availability of the PV capacity, which can be used only if the sun is shining. Using real-world data for Israel, we confirm that the introduction of PV technology amplifies price volatility. A large reduction in PV capacity cost increases PV adoption but may also raise the average price. Thus, when considering the promotion of renewable energy to reduce CO 2 emissions, regulators should assess the behavior of the electricity market, particularly with respect to characteristics of renewable technologies and demand and supply uncertainties.

Quantile and Probability Curves Without Crossing

This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone econometric function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural quantile functions using data on Vietnam veteran status and earnings.

The Newsvendor Problem with a General Price Dependent Demand Distribution

We consider a single product, single period inventory problem with stochastic price-dependent demand. The ordering and pricing decisions have to be made at the beginning of the period before demand is realized. Unsatisfied demand is lost and excess inventory has to be salvaged. The standard approach is to assume stochastic demand to be composed of deterministic functions decreasing in price and a stochastic error term. Instead of using deterministic demand functions we directly specify the distribution functions of demand depending on price. We provide analytical results concerning the optimality conditions and numerical examples based on specific demand distributions and discuss the relation of the findings to the standard approach. We show that the optimal inventory level is not a decreasing function over the whole price range, but it increases for low selling prices and decreases for higher prices, which is plausible from a managerial point of view.

A binary solution method for the multi-product newsboy problem with budget constraint

Multi-product newsboy problem (MPNP) with budget constraint is a classical inventory control/management problem. However, solution methods for MPNP under general demand distributions are limited in the current literature. In this paper, by analyzing properties of the optimal solution to the MPNP with a budget constraint, we develop a solution algorithm for the constrained MPNP. The proposed algorithm is binary in nature, and is applicable to general types of demand distribution functions, discrete as well as continuous. For continuous demand distribution function, our approach can obtain the optimal or near optimal solution to the constrained MPNP with polynomial computation complexity of the o( n) order. On the other hand, for discrete demand distribution functions, it can effectively provide good approximate solution. Numerical experiments are presented to show the performance of our method.