Hands-on Tutorial on Optimization F. Eberle, R. Hoeksma, and N. Megow September 24, 2018 Modeling
What is a model?
What is a model? In general A representation of a real thing.
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original)
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (often larger than the original)
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (often larger than the original) ◮ Model of a human skeleton (same size as the original)
Example The real thing Models
Example The real thing Models
Example The real thing Models
Example The real thing Models
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation)
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation) ◮ All information necessary for some purpose
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation) ◮ All information necessary for some purpose Example: Small World factory produces globes and maps
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation) ◮ All information necessary for some purpose Example: Small World factory produces globes and maps ◮ List of materials: wood, metal, plastics, screws, paper, and glue
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation) ◮ All information necessary for some purpose Example: Small World factory produces globes and maps ◮ List of materials: wood, metal, plastics, screws, paper, and glue ◮ Prices, supply, demand
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation) ◮ All information necessary for some purpose Example: Small World factory produces globes and maps ◮ List of materials: wood, metal, plastics, screws, paper, and glue ◮ Prices, supply, demand ◮ Print patterns
What is a model? In general A representation of a real thing. ◮ Concrete models - models that you can touch ◮ Model car (smaller than the original) ◮ Model of an insect (larger than the original) ◮ Model of a human skeleton (same size as the original) ◮ Abstract models ◮ Description of a real thing (situation) ◮ All information necessary for some purpose Example: Small World factory produces globes and maps ◮ List of materials: wood, metal, plastics, screws, paper, and glue ◮ Prices, supply, demand ◮ Print patterns ◮ Assembly procedures
Why use models? Gain information by: ◮ examining concrete models
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them)
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them) ◮ analyzing abstract/mathematical models
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them) ◮ analyzing abstract/mathematical models Use the gained information to:
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them) ◮ analyzing abstract/mathematical models Use the gained information to: ◮ Cut costs
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them) ◮ analyzing abstract/mathematical models Use the gained information to: ◮ Cut costs ◮ Increase productivity
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them) ◮ analyzing abstract/mathematical models Use the gained information to: ◮ Cut costs ◮ Increase productivity ◮ Make predictions
Why use models? Gain information by: ◮ examining concrete models ◮ getting an overview of important properties ◮ constructing concrete/abstract models (think about them) ◮ analyzing abstract/mathematical models Use the gained information to: ◮ Cut costs ◮ Increase productivity ◮ Make predictions ◮ Automate decision processes
Modeling Real world situation
Modeling Real world situation Formulation Model
Modeling Real world situation Formulation Model Analysis/solver Mathematical solution
Modeling Real world situation Formulation Real solution Model Interpretation Analysis/solver Mathematical solution
Modeling Real world situation Evalutation Formulation Real solution Model Interpretation Analysis/solver Mathematical solution
Mathematical models Formulation ◮ Mathematical language ◮ Parameters: known data ◮ Variables: unknown (decisions) ◮ Operators: ◮ × , + , − , = , � = , <, ≥ , and , or , � , � Formulation should fit the available solution methods ◮ Mathematical theory ◮ Algorithms
Types of mathematical models Many types of mathematical models, e.g.: ◮ linear vs. non-linear ◮ stochastic vs. deterministic ◮ continuous vs. discrete ◮ dynamic vs. static
Types of mathematical models Many types of mathematical models, e.g.: ◮ linear vs. non-linear ◮ stochastic vs. deterministic ◮ continuous vs. discrete ◮ dynamic vs. static In this course: ◮ (integer) linear programming models (ILP/LP) ◮ mathematically assisted decision making ◮ optimization according to objectives
Linear programming Question → (linear) objective function Data → parameters Possible decisions → variables Conditions that have to be met → (linear) constraints
Linear programming Question → (linear) objective function Data → parameters Possible decisions → variables Conditions that have to be met → (linear) constraints Structure of an (I)LP min / max [linear objective function] s.t. [linear constraints]
Linear programming Question → (linear) objective function Data → parameters Possible decisions → variables Conditions that have to be met → (linear) constraints Example min x + 2 y x + 1 s.t. 2 y ≤ 7 x ≥ y x ≥ 0 y ≥ 0
Which expressions work, which do not? ◮ Variables: x , y ◮ Parameters: a , b , c Linear:
Which expressions work, which do not? ◮ Variables: x , y ◮ Parameters: a , b , c Linear: x + y = 9 ax − by ≤ c a b x = y ( x − 1) b ≤ a (1 − y )
Which expressions work, which do not? ◮ Variables: x , y ◮ Parameters: a , b , c Linear: x + y = 9 ax − by ≤ c a b x = y ( x − 1) b ≤ a (1 − y ) Non-linear: x a + y 2 ≥ 9 xy − ay ≤ c e x ≥ y
Example - Potato chips factory Client owns a potato chips factory
Example - Potato chips factory Client owns a potato chips factory Two types of chips ◮ Plain chips ◮ Mexican ribble chips
Example - Potato chips factory Client owns a potato chips factory Two types of chips ◮ Plain chips ◮ Mexican ribble chips Three processes: ◮ Slicing ◮ Frying ◮ Packaging
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