Optimization

CM_Opti MSE Course (lecture 1)

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Cartes-fiches	9
Langue	Deutsch
Catégorie	Mathématiques
Niveau	Université
Crée / Actualisé	06.06.2021 / 06.06.2021
Lien de web	https://card2brain.ch/box/20210606_optimization
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Which two main areas of application exist in optimization?

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Optimization of business processes (production, logistics, services, operations management, ...) -> main focus in course:
1. Optimize aircraft assignment in flight operations
2. Optimize production planning in complex manufacturing plants
3. Optimize machine utilizatino in shop floor scheduling
Optimization of technical processes (engineering):
1. Optimize machining conditions in metal-cutting (speed, pressure, angle, ...)
2. Optimize parameters of chemical or physical experiments

What is the difference betwen qualitative vs. quntitative optimization?

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Quantitative analysis and optimization:
- Bases of quantifiable information and knowledge:
  - Numerical, measurable data, mathematical models and algorithms
Qualitative analysis and optimization
- Bases on non-quantifiable information and knowledge:
  - "informal" facts, verbal descriptions of processes and procedures, unstructered information, experience, implicit know-how
Example of typically quantitative optimization problems:
- Finding "the best" equipment (machines, tools, etc.) for a cerain task
- Finding "optimal" locations for facilities (plants, warehouses, ...) in a supply network
- Improving business processes

What do we typically have in Business Consulting projects?

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Phase 1: Qualitative analysis (very important, often up to 80%)
- Often unclear problem descriptions, mess of information, contradictory opinions
Phse 2:
- Either qualitative improvements: e.g. definition and implementation of new processes
- Or quantitative, e.g.:
  - Identification of certain subsystems crucial for performance
  - Having quantitative characteristics, not solvable by "common sense" (intuition, ...)
  - Need for "decision support" fromo methematical models

What should you alway remember when working on optimizatin problems?

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In general, before applying quantitative methos:
- Huge amount of challenging and crucial prelininary qualitative work is necessary:
  - Finding out, what the real problem is, defining scope and project boundaries, ...
  - Understanding all necessary details of the business areas involved, ...
  - Often this is the most difficult part of the project
  - Solving the wrong problem or solving the right problem inadequately
  - Qualitative analysis and initial, conceptual design crucial for project success
  - Often more challenging than development of quantitative methods
  - Matter of experience, intuition, dialogue with business partners

What are the two main types of optimization?

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Continuous optimization:
- Infinite number of solutions represented by continuous variables
- Graph of the objective function is arbitrary "landscape"
- Main part of the theory: local optimization
- Methods mainly bases on differentiability information (1st and 2nd derivatives)
- Reminder: gradient (vector of 1st partial derivatieves) pointing the direction of the steepest increase of the objective function
- Very diffucult in case of non-differentiable of even non-continuous functions
- Constrained optimization (with constraints) are more difficult than optimization without constraints
- Global optimization: mostly "stochastic search"
Discrete optimization
- Number of solutions is finite (or countable): represented by integer variables
- Not interesting for mathematics theory until ca. 1930
- There exists a trivial and finite algorithm: Enumeration
- Invention of computers -> possibility to solve real life problems
- Major part can be fomulated as discrete problems
- Goal: to solve (exactly or approximately) a real life problem efficiently
- Question: What is a good (efficient) algorithm for a given problem?

What means linearity in discrete optimization?

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Discrete Optimization: finite number of solutions
Every solution represented by a set of variables
Solution set is a finite set of point in n-dimensional space
"Contour" of these points has a "linear" shape (with corners): caused by the finiteness
Mathematically: "Convex hull (Hülle) of the points is a cnves polyhedron"

Where is discrete optimization used mostly in and what are its main topics?

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Central methodology of Operations Research (OR)
Great importance for Operations Management:
- Applications: Production, Logistics, Services, ...
- Many large and complex problems from industry solved successfully
Main Topics
- Linear Programming (LP)
- Integer Linear Programming (ILP), Mixed Integer Programming (MIP)

When was the Simplex algorithm for Linear Programming (LP) invented and what was the biggest OR success of it?

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1947 by Dantzig
1992: American Airlines saved 1'400 Mio. $ over 3 years by yield management

What are Decisoin problems regarding otpimization?

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Decision support:
- By quantitative models
- qualitative (business Process Engineering, ...)
Decision maker
Existence of several possible action alternatives
Every alternative has consequences
- Functional connection
- Choice of relevant consequences
Consequences of deterministic or stochastic nature
- -> only deterministic choices in this course
Evaluation of alternatives regarding their consequences
- Satisfaction
  - Consequence has to satisfy certain restrictions (constraints) so that the alternative is feasible
- Optimization
  - Consequence has to achieve the best possible characteristics, i.e. it has to be optimal regarding all feasible alternatives
Connection between alternatives and consequences:
- analytical: "explicit mathematical formula"
- algorithmic: no "explicit mathematical formula"

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