Finding the Optimal Integer Solution (Edexcel IAL Maths)

Integer Solutions

Spec point — 5.3

What is meant by integer solutions?

• The optimal solution to a linear programming problem lies on a vertex of the feasible region

  • The values of the decision variables at this vertex may not take integer values

• Sometimes the context of the problem may demand that the decision variables take integer values

  • Decision variables are often a 'number of things'

    • Is it possible for the furniture manufacturer to make 3.65 chairs per day?
    • For public health reasons, it would not be appropriate for a food factory to leave a tin of beans partially produced overnight!

• This is what is meant by the phrase integer solutions

How do I find the integer solutions to a linear programming problem?

• Find the optimal solution of the linear programming problem as usual

  • Use the objective line or vertex method

• Consider the four points with integer coordinates that surround the optimal solution

  • E.g.
  • For an optimal solution of $x=3.2,y=4.7$, the four surrounding points would be (3, 4), (3, 5), (4, 5) and (4, 4)

• Check whether each of these four points satisfies all of the constraints

  • It may be obvious that one (or more) do not but they should still be mentioned

• For those coordinates that do satisfy all the constraints

  • Evaluate the objective function ($P$) at each of the coordinates
  • The integer solution will be the point that maximises or minimises the objective function as required

• The integer solution may not be the optimal solution

  • Depending on the exact nature (gradient) of the objective line

    • The objective line 'moves away' from the boundary of the feasible region when an integer solution is found
    • So there could be another integer solution inside (or on the boundary of) the feasible region some way from the optimal solution
    • This other integer solution may be closer to the boundary of the feasible region than the one just found

  • You will not be expected to find this other integer solution

    • Just recognise that the integer solution found using the above process is not necessarily optimal

Exam Hint

• Questions won't necessarily indicate if integer solutions are required

  • Use common sense and think carefully about the context of the problem

Worked Example The linear programming problem formulated as

Maximise

$P=5x+10y$

subject to

$\begin{array}{cc}13 x + 22 y & \le 145 \\ 10 x - 20 y & \le 3 \\ 13 x - 8 y & \ge 4 \\ 6 x + 5 y & \le 50 \\ x , y & \ge 0\end{array}$

has optimal solution $x = 3.2 , y = 4.7 ( P = 63 )$.

However, the decision variables may only take integer values. Find the solution closest to the optimal solution, stating the values of the decision variables and the resulting value of $P$.

Answer:

The four surrounding integer coordinates to (3.2, 4.7) are

(3, 4), (3, 5), (4, 5), (4, 4)

Check that these satisfy all the constraints and if so, evaluate $P$ Once a point fails to satisfy an inequality we do not need to make any further checks

Final answer: The integer solution closest to the optimal solution is$x=4,y=4$and$P=60$