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 can the basic Simplex method handle directly, starting from the origin?

   • Equality (=) constraints only
   • Both ≤ and ≥ constraints with any sign of right-hand side
   • Less-than-or-equal (≤) constraints with non-negative right-hand sides
   • Greater-than-or-equal (≥) constraints only

2. In a maximisation Simplex tableau, which entry of the objective row is used to choose the entering variable?

   • The largest coefficient in the first constraint row
   • The largest positive entry in the Value column
   • The most negative entry in the objective (P) row
   • The smallest positive entry in the objective row

3. What does the slack variable s1 represent in the constraint 3x + 2y ≤ 20?

   • The profit contribution made by the first constraint
   • The coefficient that turns the inequality into an equation
   • The amount by which 3x + 2y exceeds the limit of 20
   • The unused amount of the resource, s1 = 20 - 3x - 2y, which must be non-negative

4. How is the leaving variable chosen in the Simplex method?

   • Pick the row with the largest Value entry
   • Divide each Value entry by the negative entry in the pivot column and pick the largest ratio
   • Divide each Value entry by the positive entry in the pivot column and pick the smallest such ratio
   • Pick the row containing the most negative objective coefficient

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

   • 1
   • 3
   • 2
   • 6

6. Which condition shows that a Simplex tableau for a maximisation problem has reached its optimal solution?

   • Every basic variable appears with coefficient 2 in its row
   • The Value column contains no zero entries
   • All entries in the objective row are zero or positive
   • All entries in the objective row are zero or negative

7. Within the specification's limits, how many decision variables and constraints can a basic Simplex problem have at most?

   • Two decision variables and six constraints
   • Five decision variables and five constraints
   • Three decision variables and three constraints
   • Four decision variables and four constraints

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 does the basic Simplex method need no artificial variables when all constraints are ≤ with non-negative right-hand sides?

    • The objective function is always negative at the origin.
    • Slack variables make every constraint an equality, so artificial variables are never needed.
    • The origin is feasible, so the slack variables give a starting basic feasible solution.
    • Artificial variables are only needed for maximisation problems.

17. A column of the tableau has a negative objective-row entry but no positive entries in the constraint rows. What does this indicate?

    • The solution is optimal but degenerate.
    • The slack variable in that column must be zero.
    • The problem has no feasible solution at all.
    • The objective is unbounded in that direction, so there is no finite maximum.

18. Subject to x + 2y ≤ 6 and 2x + y ≤ 6 with x, y ≥ 0, a student says the maximum of P = x + y is 4. Is this correct?

    • No, since the maximum is 5 at (1, 4).
    • No, since the intersection gives P = 3.5.
    • No, since the maximum is 6 at the origin.
    • Yes, since the intersection (2, 2) gives P = 4, which is larger than the axis points, each giving 3.

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 maximisation problem the final objective row reads P + 2 s1 = 18 with s1 non-basic. What does the coefficient 2 mean?

    • Resource 1 is fully used and has no effect on P.
    • Each extra unit of the first resource would increase the maximum P by 2.
    • P falls by 18 for each unit of s1 used.
    • Each unit of unused resource 1 increases P by 2.

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