Math 3c/d Linear programming

Modelling of real world problems Maximizing/Minimising functions (objective function) subjected to linear

constraints

Graphical procedure to solve linear programming models

Determine the objective functions and associated variables Write down objective function in terms of the variables Determine constraints and equivalent inequalities Sketch the feasible region – check if integer solutions are required Minimise or maximize the objective function by determining and

comparing the values of the objective function at each vertices (extreme point theorem)

E.g. – OT Lee Pg. 169

Nick has a block of land with an area of 1000 m2 which Nick hopes to turn into small vegetable garden. Fertilizer, watering and maintenance costs are $1.50/m2

and $2/m2 to plant a crop of tomatoes and lettuce respectively. Nick intends to utilize not more than twice as much land for tomatoes than lettuces. Nick estimates that he and his family will spend 5 hours/m2 and 3-hours/m2 respectively for a crop of tomatoes and lettuces. This includes time for preparing the ground, maintenance and harvesting. Nick and his family have 4000 hours of time available for this project.

Construct a linear programming model to determine how much land should be allocated to tomatoes and lettuces such that the project cost is minimised.

Solution

Determine the objective function and variables

Constraints

Express constraints as inequalities

Sketch the feasible region

Minimise/Maximize the objective function

Min Cost: $1538.46