Form 5 Additional Mathematics Bab 7: Linear Programming
7.1 Linear Programming Model
Linear programming problems distribute limited resources (money, manpower, raw materials) to minimise costs or maximise profits. (p.234)
Steps to formulate a model: identify the decision variables (x and y); identify the objective function; identify the constraints as linear equations or inequalities. (p.234)
Constraints use =, <, ≤, > and/or ≥, and are written in terms of the decision variables. (p.234)
The graphical method is the most common method for solving linear programming problems. (p.234)
Use a solid line for ≤ or ≥; a dotted line for < or >. (p.235)
For ax + by = c with b > 0, the region above the line satisfies ax + by ≥ c (or >); the region below satisfies ax + by ≤ c (or <). (p.235)
The region to the right of ax = c satisfies ax ≥ c; the region to the left satisfies ax ≤ c. (p.235)
Check which side to shade by substituting a test point into the inequality. (p.236)
An objective function has the form k = ax + by. (p.237)
The optimum value is found by moving the objective function line parallel to itself towards and into the region that satisfies all constraints. (p.237)
Substitute the coordinates of the maximum or minimum vertex into the objective function to get the optimum value. (p.238)
To choose a suitable k for k = ax + by: take a common multiple of a and b. (p.238)
7.2 Linear Programming Applications
A feasible value satisfies all constraints; an infeasible value does not. (p.240)
The common region satisfying all constraints is the feasible region; a solution in it is a feasible solution. (p.240)
Optimum points are at the vertices of the feasible region. (p.241)
Steps: interpret the problem and variables; write the inequalities; draw the graphs and find the feasible region R; write the objective function k = ax + by; draw the line k and find the optimum point. (p.242)
A table of the data can help simplify the problem. (p.242)
To find a range (e.g. number of tables when y = 10), draw the line y = 10 and read the values of x across region R. (p.243)