Define linear programming .

Linear programming is a way of finding the greatest or least possible value of a quantity in a situation that is restricted by a set of constraints . It is used in problems such as maximising a manufacturer's profit or minimising its costs.

True or False? Two linear programming problems can have exactly the same constraints and still have different optimal solutions.

True. The constraints fix which points are allowed , but they say nothing about which of those points is best. That is decided by the objective function , so changing it, for instance from maximising profit to minimising cost, can move the answer to a completely different point.

International A Level (IAL)MathsEdexcelDecision 1FlashcardsLinear ProgrammingLinear Programming (LP) Problems

Linear Programming (LP) Problems (Edexcel International A Level (IAL) Maths: Decision 1): Flashcards

Exam code: YMA01

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FrontFormulating a Linear Programming Problem
Define linear programming.

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Define linear programming.
Linear programming is a way of finding the greatest or least possible value of a quantity in a situation that is restricted by a set of constraints.
It is used in problems such as maximising a manufacturer's profit or minimising its costs.
Define the decision variables of a linear programming problem.
The decision variables are the quantities in the problem that can be varied, usually written as $x$ , $y$ and $z$ .
They are typically a 'number of things', such as the number of chairs and the number of tables a manufacturer makes in a day, and the values they may take are limited by the constraints.
What is the objective function of a linear programming problem, and what do the letters $P$ and $C$ usually stand for?
The objective function is the quantity the problem asks you to maximise or minimise, written as a function of the decision variables.
The letter $P$ is used in maximising problems, standing for profit, and $C$ in minimising problems, standing for costs.
Why must the decision variables be defined before the constraints of a linear programming problem can be written down?
Because each constraint is an inequality in the decision variables, so until they have been named and given a meaning there is nothing for an inequality to be written in terms of.
Defining them first also fixes what an answer will mean, for example $x$ chairs and $y$ tables per day.
A furniture manufacturer makes $x$ chairs and $y$ tables per day. A chair takes 1 hour to make and a table takes 2 hours, and the factory can run for at most 18 hours a day. Complete the constraint:
$x +______y______18$
The completed constraint is:
$x + 2 y \leq 18$
Each chair uses 1 hour and each table uses 2, and 'at most 18 hours' makes the sign less than or equal to.
Why is the constraint $x, y \geq 0$ nearly always included in a linear programming problem?
Because the decision variables are almost always a number of things, and a negative number of chairs or hours has no meaning.
It is known as the non-negativity constraint.
True or False?
Two linear programming problems can have exactly the same constraints and still have different optimal solutions.
True.
The constraints fix which points are allowed, but they say nothing about which of those points is best.
That is decided by the objective function, so changing it, for instance from maximising profit to minimising cost, can move the answer to a completely different point.
A constraint has come out as $2 x + 4 y \leq 8$ . How can it be simplified, and why is that worth doing?
Divide every term by the common factor 2, giving $x + 2 y \leq 4$ .
Simplifying leaves exactly the same set of points satisfying the constraint, but the numbers are smaller and the line is easier to work with.

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