Optimal vertex with additional constraint

A question is this type if and only if it requires determining how an additional constraint affects the optimal solution or finding the critical value where optimality changes.

3 questions · Standard +0.3

7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations
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OCR MEI D1 2009 June Q3
8 marks Standard +0.3
3 Consider the following linear programming problem:
Maximise \(\quad 3 x + 4 y\) subject to \(\quad 2 x + 5 y \leqslant 60\) \(x + 2 y \leqslant 25\) \(x + y \leqslant 18\)
  1. Graph the inequalities and hence solve the LP.
  2. The right-hand side of the second inequality is increased from 25 . At what new value will this inequality become redundant?
Edexcel D1 2022 June Q7
10 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{27296f39-bd03-47ff-9a5e-c2212d0c68ed-09_956_1290_212_383} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines and the three intersection points, \(A\), \(B\) and \(C\), are shown. The coordinates of \(C\) are \(\left( \frac { 35 } { 4 } , \frac { 15 } { 4 } \right)\) The objective function is \(P = x + 3 y\) When the objective is to maximise \(x + 3 y\), the value of \(P\) is 24
When the objective is to minimise \(x + 3 y\), the value of \(P\) is 10
    1. Find the coordinates of \(A\) and \(B\).
    2. Determine the inequalities that define \(R\). An additional constraint, \(y \geqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
  2. Determine the greatest value of \(k\) for which this additional constraint does not affect the feasible region.
Edexcel D1 2015 June Q6
13 marks Standard +0.3
6. A linear programming problem in \(x\) and \(y\) is described as follows. Minimise \(C = 2 x + 3 y\) subject to $$\begin{aligned} x + y & \geqslant 8 \\ x & < 8 \\ 4 y & \geqslant x \\ 3 y & \leqslant 9 + 2 x \end{aligned}$$
  1. Add lines and shading to Diagram 1 in the answer book to represent these constraints.
  2. Hence determine the feasible region and label it R .
  3. Use the objective line (ruler) method to find the exact coordinates of the optimal vertex, V, of the feasible region. You must draw and label your objective line clearly.
  4. Calculate the corresponding value of \(C\) at V . The objective is now to maximise \(2 x + 3 y\), where \(x\) and \(y\) are integers.
  5. Write down the optimal values of \(x\) and \(y\) and the corresponding maximum value of \(2 x + 3 y\). A further constraint, \(y \leqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
  6. Determine the least value of \(k\) for which this additional constraint does not affect the feasible region.