THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

https://bura.brunel.ac.uk/bitstream/2438/17829/4/FullText.pdf

Big data analytics in supply chain management: A state-of-the-art literature review

Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.

Applications in Statistics to Industrial Experimentation.

We address a vendor-managed inventory-routing problem where a supplier (vendor) receives a given amount of a single product each period and distributes it to multiple retailers over a finite time horizon using a capacitated vehicle. Each retailer faces external dynamic demand and is controlled by a deterministic order-up-to level policy requiring that the supplier raise the retailer's inventory level to a predetermined maximum in each replenishment. The problem is deciding on when and in what sequence to visit the retailers such that systemwide inventory holding and routing costs are minimized. We propose a branch-and-cut algorithm and a heuristic based on an a priori tour using a strong formulation. To the best of our knowledge, this study is the first to consider a strong formulation for the inventory replenishment part of inventory-routing problems. Computational results reveal that the new branch-and-cut algorithm and heuristic perform better than those noted in the literature.

A Branch-and-Cut Algorithm Using a Strong Formulation and an A Priori Tour-Based Heuristic for an Inventory-Routing Problem

Twin support vector machine (TSVM) is a novel machine learning algorithm, which aims at finding two nonparallel planes for each class. In order to do so, one needs to resolve a pair of smaller-sized quadratic programming problems rather than a single large one. Classical TSVM is proposed for the binary classification problem. However, multi-class classification problem is often met in our real world. For this problem, a new multi-class classification algorithm, called Twin-KSVC, is proposed in this paper. It takes the advantages of both TSVM and K-SVCR (support vector classification-regression machine for k-class classification) and evaluates all the training points into a “1-versus-1-versus-rest” structure, so it generates ternary outputs { −1, 0, +1}. As all the samples are utilized in constructing the classification hyper-plane, our proposed algorithm yields higher classification accuracy in comparison with other two algorithms. Experimental results on eleven benchmark datasets demonstrate the feasibility and validity of our proposed algorithm.

A Twin Multi-Class Classification Support Vector Machine

In this paper, we introduce a class of two-person finite discounted AR–AT (Additive Reward–Additive Transition) semi-Markov games (SMGs). We provide counterexamples to show that AR–AT and AR–AT–PT (Additive Reward–Additive Transition Probability and Time) SMGs do not satisfy the ordered field property. Some results on AR–AT–AITT (Additive Reward–Additive Transition and Action Independent Transition Time) and AR–AIT–ATT (Additive Reward–Action Independent Transition and Additive Transition Time) games are obtained in this paper. For the zero-sum games, we prove the ordered field property and the existence of pure stationary optimals for the players. Moreover, such games are formulated as a vertical linear complementarity problem (VLCP) and have been solved by Cottle-Dantzig’s algorithm under a mild assumption. We illustrate that the nonzero-sum case of such games do not necessarily have pure stationary equilibria. However, there exists a stationary equilibria which has at most two pure actions in each state for each player.

On discounted AR–AT semi-Markov games and its complementarity formulations

In this paper, we propose an iterative and descent type interior point method to compute solution of linear complementarity problem LCP(q, A) given that A is real square matrix and q is a real vector. The linear complementarity problem includes many of the optimization problems and applications. In this context, we consider the class of generalized positive subdefinite matrices (GPSBD) which is a generalization of the class of positive subdefinite (PSBD) matrices. Though Lemke’s algorithm is frequently used to solve small and medium-size LCP(q, A), Lemke’s algorithm does not compute solution of all problems. It is known that Lemke’s algorithm is not a polynomial time bound algorithm. We show that the proposed algorithm converges to the solution of LCP(q, A) where A belongs to GPSBD class. We provide the complexity analysis of the proposed algorithm. A numerical example is illustrated to show the performance of the proposed algorithm.

On Generalized Positive Subdefinite Matrices and Interior Point Algorithm

Principal pivot transforms of some classes of matrices

On a Mixture Class of Stochastic Game with Ordered Field Property

In this article, we present a weaker version of the class of generalized positive subdefinite matrices introduced by Crouzeix and Komlosi [J.P. Crouzeix and S. Komlosi, The Linear Complementarity Problem and the Class of Generalized Positive Subdefinite Matrices, Applied Optimization, Vol. 59, Kluwer, Dordrecht, 2001, pp. 45–63], which is new in the literature, and obtain some properties of weak generalized positive subdefinite (WGPSBD) matrices. We show that this weaker class of matrices is also captured by row-sufficient matrices introduced by Cottle et al. [R.W. Cottle, J.S. Pang, and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra Appl. 114/115 (1989), pp. 231–249] and show that for WGPSBD matrices under appropriate assumptions, the solution set of a linear complementarity problem is the same as the set of Karush–Kuhn–Tucker-stationary points of the corresponding quadratic programming problem. This further extends the results obtained in an earlier paper by...

A. K. Das

Papers

On discounted AR–AT semi-Markov games and its complementarity formulations

On Generalized Positive Subdefinite Matrices and Interior Point Algorithm

Principal pivot transforms of some classes of matrices

On a Mixture Class of Stochastic Game with Ordered Field Property

On weak generalized positive subdefinite matrices and the linear complementarity problem