Page 68 - Mathematics_Form_3

Linear programming

while px qy+ ≥ r is a constraint for a Solution
minimization problem, where p, q and r The given information is summarized
are constants. in the following table.

Non-negativity constraints Resources
Profit
Since decision variables represent real Products Machine Labour (Tshs)
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objects/items. It is therefore important
to express them as non-negative values. P 1 1.5 2.5 1,600
That is, the values of the decision P 2 1 1,280
variables should be greater or equal to 2
zero. For instance, x ≥ 0 and y ≥ 0 . Available 300 240
hours
Objective function
An objective function of a linear Identify the decision variables as
programming problem is a linear function follows.
whose value is to be either minimized or Let: x be the number of units of
maximized subject to some constraints product of type P manufactured
defined over the set of points in the per month, and 1
region satisfying all the constraints.
For instance, in maximization problems, y be the number of units of
the objective function is written as: product of type P manufactured
2
Maximize f (x, y) = ax + by, while for per month.
minimization problems is written as:
Minimize f (x, y) = ax + by, where a and Since the profit for producing the
b are constants and x and y are decision two products is to be maximized,
variables.
then the objective function is given
Example 3.1 by;

Two products, P and P require Maximize f ( ,xy ) 1600x= + 1280y
2
1
machines and labour hours in order
to be produced from a manufacturing The constraints are:
industry. An item of product P requires 1.5x + 2y ≤ 300
1
1.5 machine hours and 2.5 labour hours, 2.5x +≤ 240
y
while that of product P requires 2 x ≥ 0, y ≥ 0 Mathematics for Secondary Schools
2
machine hours and 1 labour hour. The
available machine hours is 300 and Therefore, the linear programming
labour hours is 240 per month. The problem is;
profit per one unit of P item is 1,600
1
Tanzanian shillings per month and per Maximize f ( ,xy ) 1600x= + 1280y
one unit of P item is 1,280 Tanzanian Subject to: 1.5x + 2y ≤ 300
2
shillings per month. Formulate a 2.5x +≤ 240
y
linear programming problem for
maximization of profit. x ≥ 0, y ≥ 0

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MATHEMATIC F3 SB.indd 61