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Optimization In Industry, Volume 1 Book

Optimization In Industry, Volume 1
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  • Optimization In Industry, Volume 1
  • Written by author T. A. J. Nicholson
  • Published by Transaction Publishers, October 2007
  • As optimization techniques have developed, a gap has arisen between the people devising the methods and the people who actually need to use them. Research into methods is necessarily long-term and located usually in academic establishments; whereas the ap
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Foreword     xi
Preface     xiii
The study of optimization     1
Mathematics and practitioners     1
Applying optimization methods     3
Mathematical background     5
Variables and functions     5
Differentiation and integration of functions     6
Vectors     8
Definition of regions     10
Matrices     11
Permutations     12
Set notation     12
Definition of a network     13
Probability distributions     14
Optimization problems and methods     17
Basic components     17
Problem variables     17
Objective functions     18
Constraints     19
Equality constraints     19
Inequality constraints     20
Discrete constraints     21
Disjoint alternative regions     22
The general optimization problem     23
Global optima     23
Local optima     23
The search for the optimum     24
Optimization methods     25
Calculus and Lagrange multipliers     29
The achievement of thecalculus     29
Restrictions on the use of the calculus     30
Problems in one variable     31
Unconstrained problems in two or more variables     33
Equality constraints: Lagrange multipliers     35
Sufficient conditions for local maxima and minima     37
Proof of the Lagrange multiplier method     39
Linear programming     42
Linear functions     42
The standard form     43
Converting to the standard form     43
The ideas of the simplex method     45
The canonical form     47
Improving the basis     49
Transformation to the new canonical form     51
Summary of the procedure for a change of basis     51
Obtaining an initial solution     54
The simplex tableau     56
The revised simplex method     58
Difficulties in special cases     61
The fundamental theorem of linear programming     61
The convex space of feasible solutions     63
Optimization of non-linear functions     65
Non-linear optimization techniques     65
Local and global optima     66
Finding an initial feasible solution      67
Unconstrained problems: the failure of the calculus     69
Steepest descent for unconstrained problems     69
More advanced descent methods for unconstrained problems     72
Projected gradient methods for linear constraints     75
Created response surface techniques for general constraints     80
Direct search procedures     84
Separable programming     88
Approximation programming     91
The direction of steepest descent     95
Construction of the projection matrix     96
Convergence of the response surface technique     97
Dynamic programming     101
The aim of dynamic programming     101
Multi-stage decision processes     102
The dynamic programming method: forwards calculation     103
The backwards calculation     105
Network problems     106
Allocation of a single resource to a number of activities     108
Reliability problems     113
Computational comments     114
Allocation processes involving two types of resources or two constraints     116
Problems with two neighbouring stages related: backward calculations     118
A stochastic problem      119
Branch and bound methods     123
Solutions as tree searches     123
Rules for the branch and bound method     125
The three-machine scheduling problem     127
The knapsack problem     131
Integer programming     134
Computational considerations     138
Suboptimization     138
Permutation procedures     141
The need to suboptimize     141
Permutation problems     141
Locally optimal permutations     143
Calculation of locally optimal permutations     145
Choosing sets of exchanges     149
Multi-permutation problems     149
Heuristic techniques     152
The need for skilful guessing     152
The nature of allocation problems     153
Problem specification and mathematical treatment     161
Flexibility in problem formulation     161
The choice of the problem variables     162
Transferring factors between the objective and constraint functions     162
Introducing auxiliary variables     164
Transformations     166
Duality in linear problems     168
Interpretation of the dual problem      172
Proof of the duality theorem     174
Answers to exercises     178


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