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Optimization

CM_Opti MSE Course (lecture 1)

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
Attribution de licence Non précisé
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Which two main areas of application exist in optimization?

  1. 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
  2. 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