3D.5.4 Two-stage Simplex and big-M methods Quiz: Pearson Edexcel Further Maths, Unit 39

The 20 questions

1. In the constraint x + y - s + t = 5, what is the purpose of the artificial variable t?

   • Objective maximiser
   • Slack variable
   • Surplus measure
   • Starting basic variable

2. When writing a greater-than-or-equal constraint in standard form, what sign does the surplus variable have?

   • Divided
   • Added
   • Multiplied
   • Subtracted

3. In the big-M method for a maximisation problem, what coefficient does an artificial variable receive?

   • Zero
   • One
   • Positive M
   • Negative M

4. What quantity is minimised in stage one of the two-stage Simplex method?

   • Decision variables
   • Slack variables
   • Objective function
   • Artificial variables

5. At the end of stage one, the minimum sum of artificial variables is zero. What does this indicate?

   • Optimal solution reached
   • No solution exists
   • Problem is unbounded
   • Feasible solution found

6. Why does the standard Simplex method fail for a greater-than-or-equal-to constraint with a positive right-hand side?

   • Origin is infeasible
   • Objective is unbounded
   • Origin is optimal
   • Slack is negative

7. In the big-M method, what is the purpose of the parameter M?

   • Very large penalty
   • Maximum objective bound
   • Minimum constraint value
   • Multiplier for slack

8. Minimise C = 2x + 3y subject to x + y ≥ 4 and x + 3y ≥ 6 with x, y ≥ 0. What is the minimum value of C?

   • 12
   • 9
   • 10
   • 8

9. For minimise C = 2x + 3y subject to x + y ≥ 4 and x + 3y ≥ 6 with x, y ≥ 0, at which point is the minimum attained?

   • (3, 1)
   • (6, 0)
   • (4, 0)
   • (0, 4)

10. What is the maximum of P = x + 2y subject to x + y ≤ 5 and x + y ≥ 1 with x, y ≥ 0?

    • 5
    • 10
    • 12
    • 8

11. What is the maximum of P = x + y subject to x + 2y ≥ 4 and 3x + y ≤ 9 with x, y ≥ 0?

    • 9
    • 3.4
    • 6
    • 4

12. A maximisation problem has two ≥ constraints and one ≤ constraint. How many artificial variables are needed?

    • 0
    • 1
    • 2
    • 3

13. For x + y ≥ 4 and x + y ≤ 3 with x, y ≥ 0, what is the minimum possible sum of the artificial variables in stage one?

    • 4
    • 3
    • 0
    • 1

14. Write 2x + 3y ≥ 6 in standard form with surplus variable s and artificial variable t.

    • 2x + 3y - s + t = 6
    • 2x + 3y + s - t = 6
    • 2x + 3y + s + t = 6
    • 2x + 3y - s - t = 6

15. Maximise P = 2x + y subject to x + y ≥ 1 and x + y ≤ 4 with x, y ≥ 0. What is the maximum?

    • 4
    • 6
    • 2
    • 8

16. In the big-M method, what is the purpose of choosing M as a very large positive number?

    • Eliminate slack variables
    • Increase objective value
    • Maximise profit function
    • Penalise artificial variables

17. What does a positive artificial variable remaining in the optimal big-M tableau indicate?

    • Alternative optima
    • Degenerate solution
    • Unbounded solution
    • No feasible solution

18. After artificial variables reach zero in stage one, what happens to their columns in stage two?

    • Inverted in tableau
    • Dropped completely
    • Doubled in size
    • Set as basic

19. What error occurs if a student incorrectly uses positive M instead of negative M for a maximisation problem?

    • Causes immediate cycling
    • Rewards artificial variables
    • Makes problem unbounded
    • Forces zero objective

20. What does substituting the origin into a greater-than-or-equal-to constraint with a positive RHS yield?

    • Undefined value
    • True statement
    • Zero remainder
    • False statement

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