3D.5.3 The Simplex algorithm for maximising and minimising Quiz: Pearson Edexcel Further Maths, Unit 39

The 20 questions

1. Which type of constraint allows the basic Simplex method to start directly from the origin?

   • Strict inequality
   • Equality constraint
   • Less-than-or-equal
   • Greater-than-or-equal

2. In a maximisation Simplex tableau, which objective row entry determines the entering variable?

   • Smallest value entry
   • Most positive entry
   • Most negative entry
   • Zero entry

3. In the constraint 3x + 2y + s1 = 20, what does the slack variable s1 represent?

   • Objective value
   • Artificial variable
   • Surplus capacity
   • Unused resource

4. How is the leaving variable chosen in a Simplex iteration?

   • Most negative ratio
   • Largest positive ratio
   • Smallest positive ratio
   • Zero ratio

5. A maximisation problem has three ≤ constraints. How many slack variables are introduced to write it in standard form?

   • 1
   • 3
   • 2
   • 6

6. What objective row condition shows a maximisation Simplex tableau has reached optimality?

   • Mixed signs
   • All zero
   • All non-negative
   • All negative

7. When minimising an objective function using Simplex, what optimality condition is required in the objective row?

   • All entries equal zero
   • All entries strictly positive
   • All entries non-negative
   • All entries non-positive

8. What is the maximum value of P = 3x + 2y subject to x + y ≤ 4, x + 3y ≤ 6 and x, y ≥ 0?

   • 12
   • 10
   • 11
   • 14

9. For the problem maximise P = 3x + 2y subject to x + y ≤ 4, x + 3y ≤ 6, at which point is the maximum attained?

   • (4, 0)
   • (0, 2)
   • (3, 1)
   • (2, 2)

10. What is the maximum of P = 2x + 5y subject to x + 2y ≤ 10, 3x + y ≤ 15 and x, y ≥ 0?

    • 23
    • 27
    • 20
    • 25

11. The initial tableau for maximise P = 3x + 2y with slack variables s1 and s2 has which objective row (columns x, y, s1, s2, Value)?

    • -3, -2, 0, 0, 4
    • -3, -2, 1, 1, 0
    • 3, 2, 0, 0, 0
    • -3, -2, 0, 0, 0

12. What is the maximum of P = 4x + 3y subject to 2x + y ≤ 8 and x + 2y ≤ 8 with x, y ≥ 0?

    • 56/3
    • 16
    • 64/3
    • 18

13. What is the maximum of P = 5x + 4y subject to x + y ≤ 5, 2x + y ≤ 8 and x, y ≥ 0?

    • 21
    • 25
    • 23
    • 20

14. If the constraint x + 3y ≤ 6 in maximise P = 3x + 2y subject to x + y ≤ 4 is relaxed to x + 3y ≤ 9, what is the new maximum of P?

    • 9.5
    • 13
    • 11
    • 12

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

    • 10
    • 11
    • 8
    • 12

16. Why are artificial variables unnecessary when all constraints are less-than-or-equal with non-negative right-hand sides?

    • Variables are integers
    • Objective is zero
    • Origin is feasible
    • Constraints are redundant

17. What does a negative objective entry with no positive constraint entries indicate in a tableau?

    • Infeasible region
    • Degenerate pivot
    • Optimal solution
    • Unbounded objective

18. What is the maximum value of P = x + y subject to x + 2y ≤ 6 and 2x + y ≤ 6?

    • 6
    • 3
    • 2
    • 4

19. A pivot element is 4 and the Value entry in the pivot row is 16. What is the new value of the entering variable after the pivot?

    • 12
    • 4
    • 16
    • 1/4

20. In a final objective row reading P + 2 s1 = 18, what does the coefficient 2 represent?

    • Slack capacity
    • Shadow price
    • Objective penalty
    • Pivot value

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