Nonlinear Opt.: Examples and Modeling 1
Nonlinear Optimization Problems – Examples and Modeling
Inhaltsverzeichnis
Theory
Nonlinear Optimization implies a special case of linear optimization. A Nonlinear Optimization problem is characterized by
- a nonlinear objective function and/or
- at least one nonlinear constraint
In the following you will get some application examples:
- with a nonlinear objective function and linear constraint
- with a linear objective function and nonlinear constraint
- with nonlinear objective function and nonlinear constraint
Nonlinear objective function and linear constraint
It is going to be summer outside and the Barbeque season starts. Because the soccer world championship starts next week as well, you decide to set up a TV outside to watch the matches in the garden with your friends. You connect the TV with a cable from your living room and try to minimize the resistance R in the cable for getting the best possible signal. The resistance is according to the length l, the radius r and the specific resistance (which is a constant). Therefore you drill a hole into the wall with a diameter of the length of a. The length of the cable must be minimum b meters.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): R= \rho \frac{l}{A} with A = \pi r^2
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): R=\rho \frac{l}{\pi r^2} \rightarrow min!
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): l\geq b
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): r\leq a
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): l\geq 0
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): r\geq 0
Linear objective function and nonlinear constraint
A gardener plants onions and beans. For the next period he expects a higher demand. Therefore he uses an extra property (150sqm), which he gets from his father for free, to satisfy the higher demand. The father loves onions, so in return his son has to seed at least 60sqm of the property for onions. His supplier offers him onions seed for per sqm and grain of beans for per sqm. The gardener can sell the onions for 21€ per sqm and the beans for 18€ per sqm. The budget for this project is 1800€ and the revenue should be maximized.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 21x_1 + 18x_2 \rightarrow max!
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1 + x_2 \leq 150 sqm (red)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1 \geq 60 sqm (green)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{20}{1,005^{x_1}}x_1 + \frac{15}{1,002^{x_2}} x_2 \leq 1800 (blue)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1, x_2 \geq 0
Nonlinear Objective function and nonlinear constraint
Given is a rectangular and a quadratic piece of iron sheet. You want to build a lidless barrel for your garden to collect rainwater. It should catch up 1000 liter of water. Because the iron sheet is very expensive you will try to use as little material as possible.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): (2r)^2 + 2\pi r h \rightarrow min!
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): V=\pi r^2 h = 1000
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): r\geq 0
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h\geq 0
Sources
www.cs.hs-rm.de/~weber/or/weber/orfolien5.pdf
Literature
- Prof. Dr. Oliver Wendt: Operations Research Script, Summer Term 2013
- Prof. Dr. Hans Corsten: Übungsbuch zur Produktionswirtschaft 4. Auflage
Authors
382710, 382059
