On-line Non-stationary Inventory Control using Champion Competition

The commonly adopted assumption of stationary demands cannot actually reflect fluctuating demands and will weaken solution effectiveness in real practice. We consider an On-line Non-stationary Inventory Control Problem (ONICP), in which no specific assumption is imposed on demands and their probability distributions are allowed to vary over periods and correlate with each other. The nature of non-stationary demands disables the optimality of static (s,S) policies and the applicability of its corresponding algorithms. The ONICP becomes computationally intractable by using general Simulation-based Optimization (SO) methods, especially under an on-line decision-making environment with no luxury of time and computing resources to afford the huge computational burden. We develop a new SO method, termed "Champion Competition" (CC), which provides a different framework and bypasses the time-consuming sample average routine adopted in general SO methods. An alternate type of optimal solution, termed "Champion Solution", is pursued in the CC framework, which coincides the traditional optimality sense under certain conditions and serves as a near-optimal solution for general cases. The CC can reduce the complexity of general SO methods by orders of magnitude in solving a class of SO problems, including the ONICP. A polynomial algorithm, termed "Renewal Cycle Algorithm" (RCA), is further developed to fulfill an important procedure of the CC framework in solving this ONICP. Numerical examples are included to demonstrate the performance of the CC framework with the RCA embedded.Comment: I just identified a flaw in the paper. It may take me some time to fix it. I would like to withdraw the article and update it once I finished. Thank you for your kind suppor

The Multi-Location Transshipment Problem with Positive Replenishment Lead Times

Transshipments, monitored movements of material at the same echelon of a supply chain, represent an effective pooling mechanism. With a single exception, research on transshipments overlooks replenishment lead times. The only approach for two-location inventory systems with non-negligible lead times could not be generalized to a multi-location setting, and the proposed heuristic method cannot guarantee to provide optimal solutions. This paper uses simulation optimization by combining an LP/network flow formulation with infinitesimal perturbation analysis to examine the multi-location transshipment problem with positive replenishment lead times, and demonstrates the computation of the optimal base stock quantities through sample path optimization. From a methodological perspective, this paper deploys an elegant duality-based gradient computation method to improve computational efficiency. In test problems, our algorithm was also able to achieve better objective values than an existing algorithm.Transshipment;Infinitesimal Perturbation Analysis (IPA);Simulation Optimization

Three Essays on Stochastic Optimization Applied in Financial Engineering and Inventory Management

Stochastic optimization methods are now being widely used in a multitude of applications. This dissertation includes three essays on applying stochastic optimization methods to solve problems in inventory management and financial engineering. Essay one addresses the problem of simultaneous price determination and inventory management. Demand depends explicitly on the product price p, and the inventory control system operates under a periodic review (s, S) ordering policy. To minimize the long-run average loss, we derive sample path derivatives that can be used in a gradient-based algorithm for determining the optimal values of the three parameters (s, S, p) in a simulation-based optimization procedure. Numerical results for several optimization examples via different stochastic algorithms are presented, and consistency proofs for the estimators are provided. Essay two considers the application of stochastic optimization methods to American-style option pricing. We apply a randomized optimization algorithm called Model Reference Adaptive Search (MRAS) to pricing American-style options through parameterizing the early exercise boundary. Numerical results are provided for pricing American-style call and put options written on underlying assets following geometric Brownian motion and Merton jump-diffusion processes. We also price American-style Asian options written on underlying assets following geometric Brownian motion. The results from the MRAS algorithm are compared with the cross-entropy (CE) method, and MRAS is found to be an efficient method. Essay three addresses the problem of finding the optimal importance sampling measure when simulating portfolios of credit risky assets. We apply a gradient-based stochastic approximation method to find the parameters in the minimum variance problem when importance sampling is used. The gradient estimator is obtained under the original measure. We also employ the CE method to solve the same variance minimization problem. Numerical results illustrating the variance reduction are presented for the estimation of the portfolios' expected loss, unexpected loss and quantiles

The Multi-Location Transshipment Problem with Positive Replenishment Lead Times

Transshipments, monitored movements of material at the same echelon of a supply chain, represent an effective pooling mechanism. With a single exception, research on transshipments overlooks replenishment lead times. The only approach for two-location inventory systems with non-negligible lead times could not be generalized to a multi-location setting, and the proposed heuristic method cannot guarantee to provide optimal solutions. This paper uses simulation optimization by combining an LP/network flow formulation with infinitesimal perturbation analysis to examine the multi-location transshipment problem with positive replenishment lead times, and demonstrates the computation of the optimal base stock quantities through sample path optimization. From a methodological perspective, this paper deploys an elegant duality-based gradient computation method to improve computational efficiency. In test problems, our algorithm was also able to achieve better objective values than an existing algorithm

Generalized Likelihood Ratio Method for Stochastic Models with Uniform Random Numbers As Inputs

We propose a new unbiased stochastic gradient estimator for a family of stochastic models with uniform random numbers as inputs. By extending the generalized likelihood ratio (GLR) method, the proposed estimator applies to discontinuous sample performances with structural parameters without requiring that the tails of the density of the input random variables go down to zero smoothly, an assumption in Peng et al. (2018) and Peng et al. (2020a) that precludes a direct formulation in terms of uniform random numbers as inputs. By overcoming this limitation, our new estimator greatly expands the applicability of the GLR method, which we demonstrate for several general classes of uniform input random numbers, including independent inverse transform random variates and dependent input random variables governed by an Archimedean copula. We show how the new derivative estimator works in specific settings such as density estimation, distribution sensitivity for quantiles, and sensitivity analysis for Markov chain stopping time problems, which we illustrate with applications to statistical quality control, stochastic activity networks, and credit risk derivatives. Numerical experiments substantiate broad applicability and flexibility in dealing with discontinuities in sample performance

Finite perturbation analysis methods for optimization of periodic (s, S) inventory control systems

Ankara : The Deaprtment of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2008.Thesis (Master's) -- Bilkent University, 2008.Includes bibliographical references leaves 52-54.We are dealing with single item inventory systems where the period time is constant and the unsatisfied demands are backordered. The demands are independent and identically distributed random variables, but the distribution of those variables are not known. The total cost of a period consists of; ordering cost "K" which is independent of the ordering quantity, holding cost "h" for each item that remains in stock, and penalty cost "p" for the each backordered item. In the considered system, it is known that when the parameters of an (s,S) inventory policy are chosen appropriate, then the expected period cost can be minimized. There are some exact methods or heuristics for finding the optimal s and S parameters in the literature for the case where the demand distribution is known. In our study, we introduce a perturbation analysis based method for finding the optimal s and S parameters where the demand distribution is not known. Our method anticipates the sensitivity of (s,S) parameters to the period cost for the observed demand quantities. This method's performance is compared with a method that uses Integer Programming with the past data and with a method that calculates the mean and standard variation values with the past data and feeds them to the Ehrhardt's Heuristic.Mert, ErdinçM.S

Simulation-based Methods for Stochastic Control and Global Optimization

Ideas of stochastic control have found applications in a variety of areas. A subclass of the problems with parameterized policies (including some stochastic impulse control problems) has received significant attention recently because of emerging applications in the areas of engineering, management, and mathematical finance. However, explicit solutions for this type of stochastic control problems only exist for some special cases, and effective numerical methods are relatively rare. Deriving efficient stochastic derivative estimators for payoff functions with discontinuities arising in many problems of practical interest is very challenging. Global optimization problems are extremely hard to solve due to the typical multimodal properties of objective functions. With the increasing availability of computing power and memory, there is a rapid development in the merging of simulation and optimization techniques. Developing new and efficient simulation-based optimization algorithms for solving stochastic control and global optimization problems is the primary goal of this thesis. First we develop a new simulation-based optimization algorithm to solve a stochastic control problem with a parameterized policy that arises in the setting of dynamic pricing and inventory control. We consider a joint dynamic pricing and inventory control problem with continuous stochastic demand and model the problem as a stochastic control problem. An explicit solution is given when a special demand model is considered. For general demand models with a parameterized policy, we develop a new simulation-based method to solve this stochastic control problem. We prove the convergence of the algorithm and show the effectiveness of the algorithm by numerical experiments. In the second part of this thesis, we focus on the problem of estimating the derivatives for a class of discontinuous payoff functions, for which existing methods are either not valid or not efficient. We derive a new unbiased stochastic derivative estimator for performance functions containing indicator functions. One important feature of this new estimator is that it can be computed from a single sample path or simulation, whereas existing estimators in the literature require additional simulations. Finally we propose a new framework for solving global optimization problems by establishing a connection with evolutionary games, and show that a particular equilibrium set of the evolutionary game is asymptotically stable. Based on this connection, we propose a Model-based Evolutionary Optimization (MEO) algorithm, which uses probabilistic models to generate new candidate solutions and uses dynamics from evolutionary game theory to govern the evolution of the probabilistic models. MEO gives new insight into the mechanism of model updating in model-based global optimization algorithms from the perspective of evolutionary game theory. Furthermore, it opens the door to developing new algorithms by using various learning algorithms and analysis techniques from evolutionary game theory

Applications of robust optimization to queueing and inventory systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 105-111).This thesis investigates the application of robust optimization in the performance analysis of queueing and inventory systems. In the first part of the thesis, we propose a new approach for performance analysis of queueing systems based on robust optimization. We first derive explicit upper bounds on performance for tandem single class, multiclass single server, and single class multi-server queueing systems by solving appropriate robust optimization problems. We then show that these bounds derived by solving deterministic optimization problems translate to upper bounds on the expected steady-state performance for a variety of widely used performance measures such as waiting times and queue lengths. Additionally, these explicit bounds agree qualitatively with known results. In the second part of the thesis, we propose methods to compute (s,S) policies in supply chain networks using robust and stochastic optimization and compare their performance. Our algorithms handle general uncertainty sets, arbitrary network topologies, and flexible cost functions including the presence of fixed costs. The algorithms exhibit empirically practical running times. We contrast the performance of robust and stochastic (s,S) policies in a numerical study, and we find that the robust policy is comparable to the average performance of the stochastic policy, but has a considerably lower standard deviation across a variety of networks and realized demand distributions. Additionally, we identify regimes when the robust policy exhibits particular strengths even in average performance and tail behavior as compared with the stochastic policy.by Alexander Anatolyevich Rikun.Ph.D