Summary: Lineaire Optimalisatie

Read the summary and the most important questions on Lineaire Optimalisatie

• 1 Introduction

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• How do you solve an optimization problem?

  - Build the mathematical model, which involved a set of mathematical relationships: equations, inequalities, logical dependencies, etc.  
  
  Determine the: decision variables, parameters, constraints, and objective function.

• What are the feasible set, solution and region?

  A feasible solution is any values of the variables that satisfy all of the constraints
  
  The feasible set is the set of all feasible solutions. 
  
  The objective is to find a solution that yields the best objective function value. Such is called an optimal solution. The objective function value at said point is called the optimal value

• What assumptions can be made in Linear Programming

  1) Proportionality & Additivity:
  The contribution of each decision variable to the objective function or to each constraint is proportional to the value of the variable. The contributions from decision variables are independent of one another and the total contribution is the sum of the individual contributions. (Linear)
  
  2) Divisibility: 
  Each decision variable is allowed to take fractional values. (continuous) 
  
  3) Certainty: 
  All input parameters are known with certainty (Deterministic)

• 2 Geometry of linear programming

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• How does the graphical solution of a 2-variable LP problem work

  The restrictions create an area which is the feasible solution. Next we can create lines with the objective function and move em up/down for the max or minimum value.

• What is a line called on which all the points have the same z-value?

  An isoprofit line for maximization problems
  An isocost line for minimization problems

• When is a set of point convex?

  A set of points S is a convex set if for any two points x and y in S, their context combination a*x+(1-a)y is also in S for all a in [0,1]
  
  In other words, a set is a convex set if the line segment joining any pair of points in S is wholly contained in S

• What are unbounded LP problems?

  The objective function for this maximization problem can be increased by moving in the improving direction c as much as we want while still staying in the feasible region. In this case we say that the LP is unbounded. For unbounded problems, there is no optimal solution and the optimal value is defined to be infinite. 
  
  For minimization problems we say it is unbounded if we can decrease the objective function value as much as we want while still staying in the feasible region. In this case the optimal value is defined to be -infinit

• Every LP problems falls into which 4 cases?

  1: the LP problem has a unique optimal solution
  2: the LP problem has alternative or multiple optimal solution. There are infinitely many optimal solutions
  3: The LP problem is unbounded, there is no optimal solution
  4: The LP problem is infeasible, there is no feasible solution

• How do you graphically solve an LP problem?

  1: Graph the feasible region
  2: Draw an iso line for max or min problem
  3: Move the line in the improving direction. The last point in the feasible region that is on the line is an optimal solution

• 3 Modelling Exercises

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• When is a constraint binding at a solution?

  A constraint is binding at a solution if the left-hand and right-hand sides of the constraint are equal at this solution. Otherwise the constraint is nonbinding at this solution