Formulating a Linear Programming Problem (Edexcel IAL Maths)

Introduction to Linear Programming

Spec point — 5.1

What is linear programming?

• Linear programming, often abbreviated to LP, is a means of solving problems that

  • involve working with a set of constraints
  • require a quantity to be maximised or minimised

• Typical uses of linear programming problems including finance and manufacturing

  • For example, a furniture manufacturer may make a mixture of chairs and tables

    • They would want to minimise their costs (of raw materials, manufacturing time)
    • They would want to maximise their profit (chairs may make more profit than tables)
    • They may have restrictions (constraints) such as build time and the amount of raw materials available

Decision Variables in Linear Programming

Spec point — 5.1

What are decision variables?

• In a linear programming problem, decision variables are the quantities that can be varied

  • $x,y,z$ are usually used as the decision variables
  • E.g. A furniture manufacturer could produce $x$ chairs and $y$ tables per week

• Varying the decision variables will vary the quantity that is to be maximised or minimised

  • E.g. 12 chairs and 3 tables may lead to a profit of £200
  • or 3 chairs and 12 tables may lead to a profit of £160

• The values that the decision variable may take will depend upon the constraints

  • A furniture manufacturer can't make endless chairs and tables to gain unlimited profit!

What is the objective function in a linear programming problem?

• The objective function is the quantity in a linear programming problem that requires optimising

  • The objective of a furniture manufacturer may be to maximise its profits
  • They may also wish to minimise their costs

• The objective function is the aim of a linear programming problem

  • It is a function of the decision variables
  • $P$ is usually used for maximising problems ($P$, profit)
  • $C$ is usually used for minimising problems ($C$, costs)

Constraints & Inequalities in Linear Programming

Spec point — 5.1

What are constraints in linear programming?

• The constraints in a linear programming problem are the restrictions the problem is contained within

  • Mathematically, constraints are represented by inequalities involving the decision variables
  • E.g. Constraints for a furniture manufacturer could include glue drying time or the amount of paint available

• Decision variables are usually zero or positive as they represent a 'number of things'

  • The constraint $x,y\geq0$ is usually included
  • This is called the non-negativity constraint

How do I formulate a linear programming problem?

• Formulating a linear programming problems involves

  • Defining the decision variables
  • Deducing the constraints as inequalities
  • Determining the objective function
  • Writing the problem out formally

• STEP 1 Define the decision variables

  • Read the question carefully to gather what quantities can be varied
  • Typically these will be a 'number of things'

• STEP 2 Write each constraint (given in words) as a mathematical inequality

  • Where possible, inequalities should be simplified, e.g. $2x+4y\leq8$ simplifies to $x+2y\leq4$
  • Each inequality will be in terms of the decision variables

• STEP 3 Determine the objective function

  • This will be the quantity that is required to be maximised or minimised
  • Typically this would be maximising profit or minimising costs

• STEP 4 Formulate the linear programming problem by writing it in a formal manner

  • This is of the form Maximise subject to
  • Include the non-negativity constraint, e.g. $x,y\geq0$

Exam Hint

• When writing a formal linear programming problem

  • whether specified or not, and where appropriate, always include the non-negativity constraint
  • $x,y\geq0$

Worked Example A furniture manufacturer makes chairs and tables.

Due to the availability of quality timber, on any particular day the manufacturer cannot make more than a total of 10 chairs and tables.

A chair takes an hour to produce whilst a table takes 2 hours to produce. The manufacturer's factory can produce items for a maximum of 18 hours per day.

The varnish on a chair takes 3 hours to dry whilst the varnish on a table takes 2 hours to dry. There are four drying zones within the factory, each able to provide 6 hours of drying time per day.

The manufacturer makes £30 profit on each chair it produces in a day, and £40 profit on each table. The manufacturer wants to maximise its daily profit.

Formulate the above as a linear programming problem, defining the decision variables, stating the objective function and listing the constraints.

Answer:

• STEP 1 Define the decision variables The 'things' that can be varied here are the number of chairs and the number of tables that are made per day

Let $x$ be the number of chairs made by the furniture manufacturer per day Let $y$ be the number of tables made by the furniture manufacturer per day

• STEP 2 Write each constraint (given in words) as an inequality The first constraint is the total amount of chairs and tables able to be made in a day

$x+y\leq10$

The second constraint is the production time Chairs take one hour, tables take two with a maximum of 18 hours available per day

$x+2y\leq18$

The third constraint is the varnish drying time This is 3 hours for a chair, 2 hours for a table The maximum drying time per day available is 4 x 6 = 24 hours

$3x+2y\leq24$

• STEP 3 Determine the objective function Profit is to be maximised

$P=30x+40y$

• STEP 4 Formulate the linear programming problem by writing it in a formal manner, including the non-negativity constraint

Final answer: Maximise

$P=30x+40y$

Final answer: subject to

$\begin{array}{cc} x + y & \le 10 \\ x + 2 y & \le 18 \\ 3 x + 2 y & \le 24 \\ x , y & \ge 0\end{array}$