Optimization

CM_Opti MSE Course (lecture 1)

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Diese Lernkarten behandeln fortgeschrittene Konzepte der Optimierung im universitären Kontext, mit Fokus auf diskrete und lineare Optimierungsprobleme. Sie decken Themen wie Entscheidungsprobleme, den Simplex-Algorithmus, Anwendungen in Operations Research und die Unterschiede zwischen kontinuierlicher und diskreter Optimierung ab. Besonders nützlich sind sie für Studierende und Fachleute in Mathematik, Operations Management und Business Consulting, die quantitative Methoden zur Lösung komplexer Probleme in Produktion, Logistik und technischen Prozessen anwenden möchten.

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Langue

Allemand

Catégorie

Mathématiques

Niveau

Université

Créé / Mis à jour

06.06.2021 / 06.06.2021

Fichiers de cartes similaires

Which two main areas of application exist in optimization?

Commandes clavier:

= tourner,

= avant/arrière,

= faire défiler

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?

Commandes clavier:

= tourner,

= avant/arrière,

= faire défiler

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?

Commandes clavier:

= tourner,

= avant/arrière,

= faire défiler

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?

Commandes clavier:

= tourner,

= avant/arrière,

= faire défiler

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?

Commandes clavier:

= tourner,

= avant/arrière,

= faire défiler

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?

Commandes clavier:

= tourner,

= avant/arrière,

= faire défiler

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?

Commandes clavier:

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= avant/arrière,

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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?

Commandes clavier:

= tourner,

= avant/arrière,

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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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= tourner,

= avant/arrière,

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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"

1 / 9

Cartes-fiches

Which two main areas of application exist in optimization?

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?

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?

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?

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?

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?

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?

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?

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

What are Decisoin problems regarding otpimization?

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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