Identify pivot and justify

A question is this type if and only if it asks to identify which element should be chosen as the pivot for the next iteration and explain the reasoning for this choice.

2 questions · Standard +0.0

7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations
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AQA D2 2012 January Q4
13 marks Standard +0.3
4 A linear programming problem consists of maximising an objective function \(P\) involving three variables, \(x , y\) and \(z\), subject to constraints given by three inequalities other than \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\). Slack variables \(s , t\) and \(u\) are introduced and the Simplex method is used to solve the problem. One iteration of the method leads to the following tableau.
\(\boldsymbol { P }\)\(\boldsymbol { x }\)\(\boldsymbol { y }\)\(\boldsymbol { Z }\)\(\boldsymbol { s }\)\(\boldsymbol { t }\)\(\boldsymbol { u }\)value
1-21103006
02311002
06-300-6103
0-1-90-3014
    1. State the column from which the pivot for the next iteration should be chosen. Identify this pivot and explain the reason for your choice.
    2. Perform the next iteration of the Simplex method.
    1. Explain why you know that the maximum value of \(P\) has been achieved.
    2. State how many of the three original inequalities still have slack.
    1. State the maximum value of \(P\) and the values of \(x , y\) and \(z\) that produce this maximum value.
    2. The objective function for this problem is \(P = k x - 2 y + 3 z\), where \(k\) is a constant. Find the value of \(k\).
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AQA D2 2008 June Q4
11 marks Moderate -0.3
4 A linear programming problem consists of maximising an objective function \(P\) involving three variables \(x , y\) and \(z\). Slack variables \(s , t , u\) and \(v\) are introduced and the Simplex method is used to solve the problem. Several iterations of the method lead to the following tableau.
\(\boldsymbol { P }\)\(x\)\(y\)\(\boldsymbol { Z }\)\(\boldsymbol { s }\)\(\boldsymbol { t }\)\(\boldsymbol { u }\)\(v\)value
10-1205-30037
01-80120016
0040030120
0020-321014
001125008
    1. The pivot for the next iteration is chosen from the \(\boldsymbol { y }\)-column. State which value should be chosen and explain the reason for your choice.
    2. Perform the next iteration of the Simplex method.
  2. Explain why your new tableau solves the original problem.
  3. State the maximum value of \(P\) and the values of \(x , y\) and \(z\) that produce this maximum value.
  4. State the values of the slack variables at the optimum point. Hence determine how many of the original inequalities still have some slack when the optimum is reached.