ACCA PM · Chapter 6
Limiting factors and linear programming MCQs with Answers
10 multiple-choice questions on Limiting factors and linear programming for ACCA PM Performance Management. Try each one before revealing the answer and explanation.
Practise this chapter interactivelyQuestion 1
Tau Co makes three products. Labour is limited to 12,000 hours next period. Product X: contribution $24 per unit, 3 labour hours per unit, maximum demand 2,000 units Product Y: contribution $30 per unit, 5 labour hours per unit, maximum demand 1,500 units Product Z: contribution $21 per unit, 2 labour hours per unit, maximum demand 1,000 units In what order should the products be ranked to maximise contribution?
- A) Y, X, Z
- B) X, Z, Y
- C) Y, Z, X
- D) Z, X, Y
Show answer & explanation
Answer: D) Z, X, Y
Contribution per labour hour: X = $24 / 3 = $8.00; Y = $30 / 5 = $6.00; Z = $21 / 2 = $10.50. With labour as the limiting factor, products are ranked by contribution per hour: Z, X, then Y. Ranking by contribution per unit would wrongly put Y first.
Question 2
Tau Co makes three products. Labour is limited to 12,000 hours next period. Product X: contribution $24 per unit, 3 labour hours per unit, maximum demand 2,000 units Product Y: contribution $30 per unit, 5 labour hours per unit, maximum demand 1,500 units Product Z: contribution $21 per unit, 2 labour hours per unit, maximum demand 1,000 units How many units of product Y will be made in the optimal plan?
- A) 800 units
- B) 1,500 units
- C) 2,400 units
- D) 1,200 units
Show answer & explanation
Answer: A) 800 units
Make Z first: 1,000 units x 2 hours = 2,000 hours. Then X: 2,000 units x 3 hours = 6,000 hours. Hours remaining = 12,000 - 2,000 - 6,000 = 4,000. Units of Y = 4,000 / 5 = 800 units. Omitting Z and making only X before Y would wrongly leave 6,000 hours for 1,200 units of Y.
Question 3
Tau Co makes three products. Labour is limited to 12,000 hours next period. Product X: contribution $24 per unit, 3 labour hours per unit, maximum demand 2,000 units Product Y: contribution $30 per unit, 5 labour hours per unit, maximum demand 1,500 units Product Z: contribution $21 per unit, 2 labour hours per unit, maximum demand 1,000 units What is the maximum contribution that Tau Co can earn next period?
- A) $114,000
- B) $93,000
- C) $81,000
- D) $126,000
Show answer & explanation
Answer: B) $93,000
Optimal plan: 1,000 Z, 2,000 X and 800 Y. Contribution = (1,000 x $21) + (2,000 x $24) + (800 x $30) = $21,000 + $48,000 + $24,000 = $93,000.
Question 4
Omega Co makes products X and Y. Contribution is $40 per unit of X and $50 per unit of Y. Constraints are: Labour: 2x + 4y <= 4,000 hours Machine: 3x + 2y <= 3,600 hours Demand for X: x <= 1,000 units where x and y are the units of X and Y produced. What is the optimal production plan?
- A) 1,000 units of X and 300 units of Y
- B) 0 units of X and 1,000 units of Y
- C) 800 units of X and 600 units of Y
- D) 1,000 units of X and 500 units of Y
Show answer & explanation
Answer: C) 800 units of X and 600 units of Y
Solving the binding labour and machine constraints simultaneously: from 2x + 4y = 4,000, x = 2,000 - 2y; substituting into 3x + 2y = 3,600 gives 6,000 - 4y = 3,600, so y = 600 and x = 800. Contribution at the vertices is: (0, 1,000) $50,000; (800, 600) $62,000; (1,000, 300) $55,000; (1,000, 0) $40,000. The optimum is 800 X and 600 Y. The combination 1,000 X and 500 Y breaches the machine constraint.
Question 5
Omega Co makes products X and Y. Contribution is $40 per unit of X and $50 per unit of Y. Constraints are: Labour: 2x + 4y <= 4,000 hours Machine: 3x + 2y <= 3,600 hours Demand for X: x <= 1,000 units where x and y are the units of X and Y produced. What is the shadow price of one labour hour?
- A) $7.50
- B) $12.50
- C) $8.75
- D) $20.00
Show answer & explanation
Answer: C) $8.75
Adding one labour hour and solving 2x + 4y = 4,001 with 3x + 2y = 3,600 gives x = 799.75, y = 600.375. New contribution = (40 x 799.75) + (50 x 600.375) = $62,008.75, an increase of $8.75. Equivalently, solving 2L + 3M = 40 and 4L + 2M = 50 gives L = $8.75 and M = $7.50.
Question 6
In linear programming, what does the shadow price of a scarce resource represent?
- A) The normal purchase price of one unit of the resource
- B) The contribution earned per unit of the product that uses most of the resource
- C) The amount of the resource that remains unused in the optimal solution
- D) The increase in total contribution if one extra unit of the resource were available, which is the maximum premium worth paying above its normal cost
Show answer & explanation
Answer: D) The increase in total contribution if one extra unit of the resource were available, which is the maximum premium worth paying above its normal cost
The shadow price (dual price) is the change in the objective function from one additional unit of a binding constraint. It is the most a business should pay over and above the normal cost for an extra unit. The unused amount of a resource is called slack.
Question 7
In the optimal solution to a linear programming problem, machine hours are not fully used. Which of the following is true of the machine-hour constraint?
- A) It has slack and its shadow price is zero
- B) It has slack and a positive shadow price
- C) It has no slack and its shadow price is zero
- D) It is binding and must be relaxed to increase contribution
Show answer & explanation
Answer: A) It has slack and its shadow price is zero
A constraint that is not fully used is non-binding: it has slack (unused capacity). Obtaining more of a resource that is already in surplus cannot increase contribution, so its shadow price is zero.
Question 8
When solving a two-product linear programming problem graphically, how is the optimal point identified?
- A) By choosing the vertex of the feasible region closest to the origin
- B) By finding where the iso-contribution line crosses the horizontal axis
- C) By selecting the point where the demand constraint meets the vertical axis
- D) By moving the iso-contribution line away from the origin until it last touches the feasible region
Show answer & explanation
Answer: D) By moving the iso-contribution line away from the origin until it last touches the feasible region
The iso-contribution line joins all combinations giving the same contribution and has a slope set by the products' relative contributions. Moving it outwards, parallel to itself, increases contribution; the last point of the feasible region it touches is the optimal solution, usually a vertex.
Question 9
Omega Co makes products X and Y. Contribution is $40 per unit of X and $50 per unit of Y. Constraints are: Labour: 2x + 4y <= 4,000 hours Machine: 3x + 2y <= 3,600 hours Demand for X: x <= 1,000 units where x and y are the units of X and Y produced. The shadow price of labour has been calculated as $8.75 per hour. Additional labour hours could be obtained by paying an overtime premium of $10 per hour above the normal rate. What should Omega Co do?
- A) Buy the extra hours, because labour is a binding constraint
- B) Not buy the extra hours, because the premium exceeds the extra contribution each hour would generate
- C) Buy the extra hours, because the premium is less than the contribution per unit of Y
- D) Not buy the extra hours, because labour has slack in the optimal solution
Show answer & explanation
Answer: B) Not buy the extra hours, because the premium exceeds the extra contribution each hour would generate
Each additional labour hour increases contribution by the shadow price of $8.75 (before the premium). Paying a premium of $10 would reduce profit by $1.25 per hour, so it is not worthwhile even though labour is binding. Comparing the premium with contribution per unit rather than per hour is a common error.
Question 10
Psi Co makes products X and Y. Each unit of X uses 2 kg of material M and each unit of Y uses 5 kg. Only 10,000 kg of material M are available. Each unit of X earns contribution of $12 and each unit of Y $25. If x and y are the units produced, which of the following correctly states the material constraint?
- A) 2x + 5y <= 10,000
- B) 12x + 25y <= 10,000
- C) 5x + 2y <= 10,000
- D) 2x + 5y >= 10,000
Show answer & explanation
Answer: A) 2x + 5y <= 10,000
Each unit of X uses 2 kg and each unit of Y uses 5 kg, so total material used is 2x + 5y, which cannot exceed the 10,000 kg available. Contribution figures belong in the objective function (maximise 12x + 25y), not in the resource constraint.