Scilab provides a high-level matrix language and allows to define complex mathematical models and to easily connect to existing libraries. That is why optimization is an important and practical topic in Scilab, which provides tools to solve linear and nonlinear optimization problems by a large collection of tools.

Overview of the industrial-grade solvers available in Scilab and the type of optimization problems which can be solved by Scilab. Objective Linear Bounds y Equality l Inequalities l Problem size m m l l y Nonlinear s Gradient needed y n Solver linpro quapro qld qpsolve optim neldermead optim_ga fminsearch optim_sa lsqrsolve leastsq optim/"nd" optim_moga semidef lmisolve

Quadratic

y

l

l

s Nonlinear Least Squares Min-Max Multi-Obj. Semi-Def. y y l*

n

l m s l* l* l l

optional y n n

For the constraint columns, the letter "l" means linear, the letter "n" means nonlinear and "l*" means linear constraints in spectral sense. For the problem size column, the letters "s", "m" and "l" respectively mean small, medium and large.

Focus on nonlinear optimization
w The optim function solves optimization problems with nonlinear objectives, with or without bound constraints on the unknowns. The quasi-Newton method optim/"qn" uses a Broyden-Fletcher-Goldfarb-Shanno formula to update the approximate Hessian matrix. The quasi-Newton method has a O(n²) memory requirement. The limited memory BFGS algorithm optim/"gc" is efficient for large size problems due to its memory requirement in O(n). Finally, the optim/"nd" algorithm is a bundle method which may be used to solve unconstrained, non-differentiable problems. For all these solvers, a function that computes the gradient g must be provided. That gradient can be computed using finite differences based on an optimal step with the derivative function, for example. w The fminsearch function is based on the simplex algorithm of Nelder and Mead (not...

...Introduction
Linear optimization is a mathematical method for determining a way to achieve the best outcome such as maximum profit or lowest cost in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming
The Primary Purpose of the present investigation is to develop an interactive spreadsheet tool to aid in determining a maximum return function in 401K plan. In this paper, we discuss how the Excel spreadsheet can be used to solve a linear optimization in retirement industry to get maximum returns in 401 k plan
Organize information
There are four categories of information needed for solving an optimization problem in Excel: an Objective Function, Decision Variables, Constraints, and write the objective function and constraints as mathematical expressions. The model consists of following components
• Decision variables: The decisions of the problem are represented using symbols such as X1, X2, and X3…Xn. These variables represent unknown quantities such as number of items to produce or amounts of money to invest in and so on.
• Objective function: The objective of the problem is expressed as a mathematical expression in decision variables. The objective for the current analysis is maximizing return in 401k Plan.
• Constraints: The limitations or requirements of the problem are expressed as inequalities or equations in...

...a corporation faces today is whether optimization, simulation, or a hybrid model (combination of optimization and simulation) is a better option to pursue.
In this paper, we fundamentally distinguish the two modeling approaches – Supply Chain Optimization vs. Supply Chain Simulation, and the scenarios where the each option should be employed.
Overview
Optimization focuses on finding the optimal solution from millions of possible alternatives while meeting the given constraints of the supply chain. Optimization utilizes mixed integer programming (MIP) or linear programming (LP) to obtain the optimal solution. Optimization models are used for network optimization, allocation management (refinery and terminals), route optimization (retail logistics) and vendor-managed inventory (retail network management).
Simulation identifies the impact of different variables on an organization’s entire supply chain. It answers the fundamental question – what will happen to the cost and service levels associated with a Supply Chain if an ‘X’ factor is manipulated The tool however does not drive to an optimal solution. Simulation also enables a user to visualize real world behavior of an “optimal solution” derived from the optimization. Simulation models are best suited for decision analysis, diagnostic evaluation, and project planning.
Supply Chain...

...models which try to minimize the cost of shipments between designated sources and destinations and assignment models which assign specific "jobs" to specific "candidates", such as jobs to machines, in an optimal way.
3. Integer programming models where some or all of the decision recommendations can be integer values.
4. Nonlinear programming models allow the functions making up the problem to be nonlinear, i.e., quadratic, cubic, or even a higher exponential for the decision variables involved (Chase, Aquilano, and Jacobs, 2001).
5. Game theory models imbed players in a simulated business environment where decisions at one time (quarter) affect the conditions under which the subsequent decisions are made (Watson, 1981).
6. Network optimization models employ graphical descriptions of problems employing specialized solution procedures which optimize interrelated activities and their completion or flow from start to finish. Applications include areas such as transportation system design, information system design, and project scheduling (Anderson, Sweeney and Williams, 2002). Project management models monitor the execution of projects in terms of time, effort and cost. PERT/CPM (Program Evaluation and Review Technique/Critical Path Method) are the two techniques typically employed for this class of models (Chase, Aquilano, and Jacobs, 2001).
7. Inventory models monitor the cost of carrying, ordering, and purchasing inventory (Chase, Aquilano, and Jacobs,...

...Convex Optimization
Convex Optimization
Stephen Boyd
Department of Electrical Engineering
Stanford University
Lieven Vandenberghe
Electrical Engineering Department
University of California, Los Angeles
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜o Paolo, Delhi
a
Cambridge University Press
The Edinburgh Building, Cambridge, CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
http://www.cambridge.org
Information on this title: www.cambridge.org/9780521833783
c Cambridge University Press 2004
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2004
Seventh printing with corrections 2009
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing-in-Publication data
Boyd, Stephen P.
Convex Optimization / Stephen Boyd & Lieven Vandenberghe
p. cm.
Includes bibliographical references and index.
ISBN 0 521 83378 7
1. Mathematical optimization. 2. Convex functions. I. Vandenberghe, Lieven. II. Title.
QA402.5.B69 2004
519.6–dc22
2003063284
ISBN 978-0-521-83378-3 hardback
Cambridge...

...Answers are hi-lighted yellow.
Company A's nationally advertised brand is Brand X. Contribution to profit with Brand X is $40 per case.
Company A's re-proportioned formula is sold under a private label Brand Y. Contribution to profit with Brand Y is $30 per case.
Company A's objective is to maximize the total contribution to profit.
Three constraints limit the number of cases of Brand X and Brand Y that can be produced.
Constraint 1: The available units of nutrient C (n) is 30.
Constraint 2: The available units of flavor additive (f) is 72.
Constraint 3: The available units of color additive (c) is 90.
Material units per case of Brand X and Brand Y:
Product
Brand X Brand Y Formula for a case of Brand X = 4n+12f+6c
Nutrient C: 4 4 Formula for a case of Brand Y = 4n+6f+15c
Flavor Additive: 12 6
Color Additive: 6 15
Objective Function:
Max 40X + 30Y = total profit contribution
Constraint 1: Units of nutrient C used < units of nutrient C available
Units of nutrient C used = 4X + 4Y
30 units of nutrient C are available so the mathematical equation of constraint 1 is
4X + 4Y < 30
Constraint 2: Units of flavor additive used < units of flavor additive available
Units of flavor additive used = 12X + 6Y
72 units of flavor additive are available so the mathematical equation of constraint 2 is
12X + 6Y < 72
Constraint 3: Units of color additive used < units of color additive available
Units of color additive used = 6X + 15Y90 units of color additive are...

...Data Depth and Optimization
Komei Fukuda
fukuda@ifor.math.ethz.ch
Vera Rosta
rosta@renyi.hu
In this short article, we consider the notion of data depth which generalizes the median to higher dimensions. Our main objective is to present a snapshot of the data depth,
several closely related notions, associated optimization problems and algorithms. In particular, we brieﬂy touch on our recent approaches to compute the data depth using linear
and integeroptimization programming. Although the problem is NP-hard, there are ways
to compute nontrivial lower and upper bounds of the depth.
The notion of data depth has been studied independently in statistics, discrete geometry, political science and optimization. The motivation and the necessity in statistics to
generalize the median and the rank is very natural, as the mean is not considered to be a
robust measure of central location. It is enough to place one outlier to change the mean.
In contrast, the median in one dimension is very robust as half of the observations need
to be changed to corrupt the value of the median.
In nonparametric statistics, several data depth measures were introduced as multivariate generalizations of ranks to complement classical multivariate analysis, ﬁrst by Tukey
(1975), then followed by Oja (1983), Liu (1990), Donoho and Gasko (1992), Singh (1992),
Rousseeuw and Hubert (1999) among others. These measures, though seemingly...

...A Survey of Optimization Models for Train Routing and Scheduling1
JEAN-FRANCOIS CORDEAU ¸
´ GERAD and Ecole Polytechnique de Montreal, 3000, chemin de la Cote-Sainte-Catherine, Montreal H3T 2A7, Canada ´ ˆ ´
PAOLO TOTH and DANIELE VIGO
DEIS, Universita di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italia `
The aim of this paper is to present a survey of recent optimization models for the most commonly studied rail transportation problems. For each group of problems, we propose a classification of models and describe their important characteristics by focusing on model structure and algorithmic aspects. The review mainly concentrates on routing and scheduling problems since they represent the most important portion of the planning activities performed by railways. Routing models surveyed concern the operating policies for freight transportation and railcar fleet management, whereas scheduling models address the dispatching of trains and the assignment of locomotives and cars. A brief discussion of analytical yard and line models is also presented. The emphasis is on recent contributions, but several older yet important works are also cited.
he rail transportation industry is very rich in terms of problems that can be modeled and solved using mathematical optimization techniques. However, the related literature has experienced a slow growth and, until recently, most contributions were dealing with simplified models or...

...Optimization Modeling for Inventory Logistics
Engineering & Technology Management
ETM 540 – Operations Research in
Engineering and Technology Management
Fall 2013
Portland State University
Dr. Tim Anderson
Team: Logistics
Noppadon Vannaprapa
Philip Bottjen
Rodney Danskin
Srujana Penmetsa
Joseph Lethlean
Optimization Modeling for Inventory Logistics
Contents
Abstract ......................................................................................................................................................... 2
Literature review........................................................................................................................................... 2
Review of Optimization Modeling ............................................................................................................ 3
Logistics Characterization ......................................................................................................................... 4
Methodology............................................................................................................................................. 6
Problem Background..................................................................................................................................... 6
Industry...