Linear optimization: Mathematical formulations of problems presented in the course 1

Mathematical formulations of problems presented in the course

Example:

A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B.

At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours.

The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximise the combined sum of the units of X and the units of Y in stock at the end of the week.

x = number of units of X produced in the current week

y = number of units of Y produced in the current week

Constraints:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 50x + 24y \leq 40*60\rightarrow

machine time A: 50 minutes per product x and 24 minutes per product y has to be lower than the capacity of 40 hours of machine A. 

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 30x + 33y \leq 35*60\rightarrow

machine time B: 30 minutes per product x and 33 minutes per product y has to be lower than the capacity of 35 hours of machine B. 

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x \geq (75 - 30)

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow x \geq  45
so production of X Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \geq
 demand (75) - initial stock (30)  , which ensures we meet demand

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y \geq 95 - 90

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow y \geq  5
so production of Y Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \geq 
demand (95) - initial stock (90)  , which ensures we meet demand

objective function: Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): max (x+30-75) + (y+90-95)

i.e. to maximise the number of units left in stock at the end of the week