Mathematics Kenya
Linear Programming
A group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation
Linear Programming
Linear Programming
Formation of linear inequalities
Linear inequalities
- Normally any straight line drawn on xy – plane separates it into two disjoint sets. These sets are called half – planes
- Consider the equation y = 5 drawn on the xy plane as shown below.
From the figure above, all points above the line, that is all points in the half plane A which is above the line satisfy the relation y>5 and those lying in the half plane B which is below the given line, satisfy the relation y< 5
Shading of Regions
- In linear programming usually the region of interest is left clear that is we shade unwanted region(s).
NB:
When shading the half planes we consider the inequalities as the equations but dotted lines are used for the relations with > or < signs and normal lines are used for those with ≥ or ≤ signs.
Consider the inequalities x>0, y>0 and 2x + 3y >12 represented on the xy-plane In this case we draw the line x=0, y= 0 and 2x+3y=12 but the point about the inequality signs for each equation must be considered.
From the figure above, the clear region satisfy all the inequalitiesx>0, y>0 and 2x + 3y >12, these three lines are the boundaries of the region.
Analytical solutions of linear inequalities
Solutions of linear inequalities by graphs
Example 1
Draw and show the half plane represented by 8x + 2y ≥16
Feasible Region
Definition: In the xy plane the region that satisfies all the given inequalities is called the feasible region (F.R)
Example 2
Indicate the feasible region for the inequalities 2x+3y ≥ 12 and y-x ≤ 2.
Determine the solution set of the simultaneous inequalities y + x ≥3 and x-2y ≤ 9.
Example 3
Fatuma was given 30 shillings to buy oranges and mangoes. An orange costs 2shillings while a mango costs 3 shillings. If the number of oranges bought is at least twice the number of mangoes, show graphically the feasible region representing the number of ranges and mangoes she bought, assuming that no fraction of oranges and mangoes are sold at the market.
Solution:-
Le x be the number of oranges she bought and y the number of mangoes she bought. Now the cost of x and y together is 2x + 3y shillings which must not exceed 30 shillings. Inequalities:
2x + 3y ≤30 ……… (i) and x≥2y …………….. (ii),
Also because there is no negative oranges or mangoes that can be bought,
then x≥ and y≥0 ……….. (iii)
Now the line 2x + 3y ≤30 is the line passing through (0, 10) and (15,0) and the line x≥2y or x – 2y ≥ 0 is the line which passes through (0,0) and (2,1).
Exercise 1
For practice.
- Draw the graph of the equation 2x – y = 7 and show which half plane is represented by 2x – y >7 and the one represented by 2x – y <7
- On the same coordinate axes draw the graphs of the following inequalities: x + 2y ≤ 2, y-x ≤ 1 and y ≥ 0.
- Draw the graphs of y < 2x -1 and y > 3 – x on the same axes and indicate the feasible region.
- A post office has to transport 870 parcels using a lorry, which takes 150 parcels at a time and a van which can take 60 at a time. The cost of each journey is 350 shillings by lorry and 280 shillings by van. The van makes more trips than the lorry and the total cost should not exceed 3080 shillings. Show graphically the feasible region representing the number of trips that a lorry and a van can make.
Optimisation (include objective function)
Application to real life situations
Linear Programming
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