A Model for Integrated Inventory and Assortment Planning

2021 ◽  
Author(s):  
Victor Martínez-de-Albéniz ◽  
Sumit Kunnumkal
Keyword(s):  
Closed Form Solution ◽  
Optimal Solution ◽  
Integer Program ◽  
Form Solution ◽  
Assortment Planning ◽  
Dimensional Parameter ◽  
Planning Decisions ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Random Instances

Integrating inventory and assortment planning decisions is a challenging task that requires comparing the value of demand expansion through broader choice for consumers with the value of higher in-stock availability. We develop a stockout-based substitution model for trading off these values in a setting with inventory replenishment, a feature missing in the literature. Using the closed form solution for the single-product case, we develop an accurate approximation for the multiproduct case. This approximated formulation allows us to optimize inventory decisions by solving a fractional integer program with a fixed point equation constraint. When products have equal margins, we solve the integer program exactly by bisection over a one-dimensional parameter. In contrast, when products have different margins, we propose a fractional relaxation that we can also solve by bisection and that results in near-optimal solutions. Overall, our approach provides solutions within 0.1% of the optimal policy and finds the optimal solution in 80% of the random instances we generate. This paper was accepted by David Simchi-Levi, optimization.

Capacity Analysis for Correlated Multi-Hop MIMO Channels under Colored Noise

2015 ◽  
Vol 1 ◽  
pp. 41
Author(s):  
Nguyen N. Tran ◽  
Ha X. Nguyen
Keyword(s):  
Computational Complexity ◽  
Mutual Information ◽  
Closed Form Solution ◽  
Optimal Solution ◽  
Form Solution ◽  
Maximization Problem ◽  
Capacity Analysis ◽  
Mimo Channels ◽  
Average Mutual Information ◽  
Channel Output

A capacity analysis for generally correlated wireless multi-hop multi-input multi-output (MIMO) channels is presented in this paper. The channel at each hop is spatially correlated, the source symbols are mutually correlated, and the additive Gaussian noises are colored. First, by invoking Karush-Kuhn-Tucker condition for the optimality of convex programming, we derive the optimal source symbol covariance for the maximum mutual information between the channel input and the channel output when having the full knowledge of channel at the transmitter. Secondly, we formulate the average mutual information maximization problem when having only the channel statistics at the transmitter. Since this problem is almost impossible to be solved analytically, the numerical interior-point-method is employed to obtain the optimal solution. Furthermore, to reduce the computational complexity, an asymptotic closed-form solution is derived by maximizing an upper bound of the objective function. Simulation results show that the average mutual information obtained by the asymptotic design is very closed to that obtained by the optimal design, while saving a huge computational complexity.

scholarly journals Best Proximity Point Results in Complex Valued Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Binayak S. Choudhury ◽  
Nikhilesh Metiya ◽  
Pranati Maity
Keyword(s):  
Minimum Distance ◽  
Metric Spaces ◽  
Optimal Solution ◽  
Global Optimal Solution ◽  
Complex Valued Metric Space ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Global Optimal ◽  
Metric Function ◽  
Complex Valued

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples.

scholarly journals Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks

Sensors ◽  
2020 ◽  
Vol 20 (21) ◽  
pp. 6284
Author(s):  
Donatella Darsena ◽  
Giacinto Gelli ◽  
Ivan Iudice ◽  
Francesco Verde
Keyword(s):  
Multiple Input Multiple Output ◽  
Closed Form Solution ◽  
Iterative Algorithms ◽  
Optimal Solution ◽  
Form Solution ◽  
5G Networks ◽  
Amplify And Forward ◽  
Input Multiple Output ◽  
Local Optimal Solution ◽  
Expected Benefits

While the combination of multi-antenna and relaying techniques has been extensively studied for Long Term Evolution Advanced (LTE-A) and Internet of Things (IoT) applications, it is expected to still play an important role in 5th Generation (5G) networks. However, the expected benefits of these technologies cannot be achieved without a proper system design. In this paper, we consider the problem of jointly optimizing terminal precoders/decoders and relay forwarding matrices on the basis of the sum mean square error (MSE) criterion in multiple-input multiple-output (MIMO) two-way relay systems, where two multi-antenna nodes mutually exchange information via multi-antenna amplify-and-forward relays. This problem is nonconvex and a local optimal solution is typically found by using iterative algorithms based on alternating optimization. We show how the constrained minimization of the sum-MSE can be relaxed to obtain two separated subproblems which, under mild conditions, admit a closed-form solution. Compared to iterative approaches, the proposed design is more suited to be integrated in 5G networks, since it is computationally more convenient and its performance exhibits a better scaling in the number of relays.

scholarly journals Data Reduction with Quantization Constraints for Decentralized Estimation in Wireless Sensor Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yang Weng
Keyword(s):  
Closed Form Solution ◽  
Optimal Solution ◽  
Form Solution ◽  
Estimation Problem ◽  
Sensor Nodes ◽  
Wireless Sensor ◽  
Fusion Center ◽  
Decentralized Estimation ◽  
Final Estimate ◽  
Unknown Vector

The unknown vector estimation problem with bandwidth constrained wireless sensor network is considered. In such networks, sensor nodes make distributed observations on the unknown vector and collaborate with a fusion center to generate a final estimate. Due to power and communication bandwidth limitations, each sensor node must compress its data and transmit to the fusion center. In this paper, both centralized and decentralized estimation frameworks are developed. The closed-form solution for the centralized estimation framework is proposed. The computational complexity of decentralized estimation problem is proven to be NP-hard and a Gauss-Seidel algorithm to search for an optimal solution is also proposed. Simulation results show the good performance of the proposed algorithms.

scholarly journals Similarity Measure Learning in Closed-Form Solution for Image Classification

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Jing Chen ◽  
Yuan Yan Tang ◽  
C. L. Philip Chen ◽  
Bin Fang ◽  
Zhaowei Shang ◽  
...  
Keyword(s):  
Closed Form ◽  
Similarity Measure ◽  
Dimensional Space ◽  
Closed Form Solution ◽  
Optimal Solution ◽  
Research Problem ◽  
Form Solution ◽  
Similarity Learning ◽  
Learning Tasks ◽  
Active Research

Adopting a measure is essential in many multimedia applications. Recently, distance learning is becoming an active research problem. In fact, the distance is the natural measure for dissimilarity. Generally, a pairwise relationship between two objects in learning tasks includes two aspects: similarity and dissimilarity. The similarity measure provides different information for pairwise relationships. However, similarity learning has been paid less attention in learning problems. In this work, firstly, we propose a general framework for similarity measure learning (SML). Additionally, we define a generalized type of correlation as a similarity measure. By a set of parameters, generalized correlation provides flexibility for learning tasks. Based on this similarity measure, we present a specific algorithm under the SML framework, called correlation similarity measure learning (CSML), to learn a parameterized similarity measure over input space. A nonlinear extension version of CSML, kernel CSML, is also proposed. Particularly, we give a closed-form solution avoiding iterative search for a local optimal solution in the high-dimensional space as the previous work did. Finally, classification experiments have been performed on face databases and a handwritten digits database to demonstrate the efficiency and reliability of CSML and KCSML.

scholarly journals A CLOSED-FORM SOLUTION FOR THE HEALTH CAPITAL MODEL

2015 ◽  
Vol 81 (3) ◽  
pp. 301-316 ◽  
Author(s):  
Holger Strulik
Keyword(s):  
Closed Form ◽  
Capital Stock ◽  
Closed Form Solution ◽  
Optimal Solution ◽  
Form Solution ◽  
Investment Model ◽  
Eternal Life ◽  
Health Capital ◽  
Comparative Dynamics ◽  
Health Demand

Abstract:This paper provides a closed-form solution for the health capital model of health demand. The results are exploited in order to prove analytically the comparative dynamics of the model. Results are derived for the so-called pure investment model, the pure consumption model and a combination of both types of models. Given the plausible assumptions that (i) health declines with age and that (ii) the health capital stock at death is lower than the health capital stock needed for eternal life, it is shown that the optimal solution implies eternal life.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

Plane Wave Resonance in the Tire Air Cavity as a Vehicle Interior Noise Source

1995 ◽  
Vol 23 (1) ◽  
pp. 2-10 ◽  
Author(s):  
J. K. Thompson
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Interior Noise ◽  
Cavity Resonance ◽  
Test Results ◽  
Vehicle Interior ◽  
Air Cavity ◽  
Vehicle Interior Noise ◽  
Wave Resonance ◽  
Resonance Frequencies

Abstract Vehicle interior noise is the result of numerous sources of excitation. One source involving tire pavement interaction is the tire air cavity resonance and the forcing it provides to the vehicle spindle: This paper applies fundamental principles combined with experimental verification to describe the tire cavity resonance. A closed form solution is developed to predict the resonance frequencies from geometric data. Tire test results are used to examine the accuracy of predictions of undeflected and deflected tire resonances. Errors in predicted and actual frequencies are shown to be less than 2%. The nature of the forcing this resonance as it applies to the vehicle spindle is also examined.

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