Linear optimization: Mathematical formulations of complex problems (How to) 1

Theory

In contradiction to simple optimization problems, complex problems are not to solve graphically in 2D because of containing more than 2 variables. The complexity rises with incerasing number of variables and interdependance between them.

How to formulate complex problems mathematically

First of all the problem should be regarded properly and the information should be structured. This helps not to lose the overview caused of the huge amount of information.

In the next step it could be useful to transform the text-based information into a graph if it is possible. So you can easily comprehend the processes and go through the problem step by step.

After that you should define the variables. Clear assignments like input and output factors can be set directly as well as all other arrows of the graph can be equipped with variables. Some of these variables could lead to equations in the formulation and will be cleared while setting up the restrictions.

Now you can formulate the restrictions. You have to use the information in the graph in combination with further information given in the text or additional tables.

Example: Beverage Factory

In a beverage factory are delivered concentrate for the production of coke and orangelemonade soft drinks every day. The storage capacity is limited to 3500 l each. Furthermore, 1000 l of whisky are delivered and can be stored each day.

The concentrates are transferred into barrels with a capacity of 100 l for further transportation. The Whisky is transferred into 50 l barrels. Beneath the use in production the "coke-concentrate" is also sold to a fast-food restaurant for 1500€ per barrel. The factory can sell up to 6 barrels to the fast-food restaurant every day. Only full barrels are sold.

In the soft drink plant the concentrate, water and CO2 are mixed to coke and lemonade. The CO2 bottles have a capacity of 20 l each. Water is available to infinity. The capacity of the production is limited to 20000 l of soft drinks each day. The mixture of 1 l of lemonade consists of 0,3 l orange lemonade concentrate, 0,68 l water and 0,0002 l CO2. The mixture of 1 l of coke consists of 0,35 l coke concentrate, 0,63 l water and 0,0002 l CO2.

A part of the produced coke soft drink is transferred to the long drink plant where it is mixed with whisky. Each liter of whisky-coke consists of 40% of whisky and 60% of coke

After the production the drinks are filled into cans with a capacity of 0,3 l. The material cost for each can are 0,05€.

"A=π" "r" ^"2"

20000 cans of orange lemonade have to be sold to a supermarket due to a delivery contract.

The purchase prices for the other ressources are: - orange concentrate: 12 €/l - coke concentrate: 10 €/l - water: 0,2 €/l - CO2: 70 €/bottle - whisky: 20 €/l

The cans are sold for: - lemonade: 1,50€ - coke: 1,7€ - whisky-coke: 4,50€

Every can produced is sold.

How many cans of each product have to be produced and sold to maximize the contribution margin?