IB · BM HL

Business Management HL

Operations management (HL): decision trees and critical path analysis — Unit 5 HL

Name: ____________________Date: October 2, 2026

1.
A company is choosing between two investment options. Option A costs $10,000 and has a 0.6 probability of generating a profit of $100,000 and a 0.4 probability of generating a loss of $20,000. Option B costs $5,000 and has a 0.5 probability of generating a profit of $60,000 and a 0.5 probability of generating a profit of $20,000. (a) Calculate the expected value (net of cost) of Option A. (b) Calculate the expected value (net of cost) of Option B. (c) State which option has the higher expected value.
[5 marks]
Marking points
- Calculates the expected value of Option A's outcomes: (0.6 × 100,000) + (0.4 × −20,000) = 60,000 − 8,000 = $52,000.
- Subtracts the cost of Option A: 52,000 − 10,000 = $42,000.
- Calculates the expected value of Option B's outcomes: (0.5 × 60,000) + (0.5 × 20,000) = 30,000 + 10,000 = $40,000.
- Subtracts the cost of Option B: 40,000 − 5,000 = $35,000.
- States that Option A has the higher net expected value ($42,000 versus $35,000).
Examiner tip: Always subtract the initial cost from the expected value of the outcomes, not before — calculate the weighted average of the possible results first, then net out the cost to get a comparable final figure.
2.
Marking analysis: A learner attempts the following task: “A company is choosing between two investment options. Option A costs $10,000 and has a 0.6 probability of generating a profit of $100,000 and a 0.4 probability of generating a loss of $20,000. Option B costs $5,000 and has a 0.5 probability of generating a profit of $60,000 and a 0.5 probability of generating a profit of $20,000. (a) Calculate the expected value (net of cost) of Option A. (b) Calculate the expected value (net of cost) of Option B. (c) State which option has the higher expected value.” Their response addresses only this point: “Calculates the expected value of Option A's outcomes: (0.6 × 100,000) + (0.4 × −20,000) = 60,000 − 8,000 = $52,000.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Marking points
- Recognises credit for the stated point: Calculates the expected value of Option A's outcomes: (0.6 × 100,000) + (0.4 × −20,000) = 60,000 − 8,000 = $52,000.
- Identifies the missing requirement: Subtracts the cost of Option A: 52,000 − 10,000 = $42,000.
- Identifies the missing requirement: Calculates the expected value of Option B's outcomes: (0.5 × 60,000) + (0.5 × 20,000) = 30,000 + 10,000 = $40,000.
- Identifies the missing requirement: Subtracts the cost of Option B: 40,000 − 5,000 = $35,000.
- Identifies the missing requirement: States that Option A has the higher net expected value ($42,000 versus $35,000).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
3.
Using the decision tree from the previous question, explain one limitation of relying purely on expected value when choosing between Option A and Option B, given that Option A carries a real risk of an outright loss.
[2 marks] · no calculator
Marking points
- Explains that expected value is a long-run average outcome and does not capture a decision-maker's attitude toward risk, such as risk-aversion.
- Explains that even though Option A has a higher expected value, a risk-averse manager might prefer Option B because it guarantees a profit in both possible outcomes, with no chance of an outright loss, unlike Option A's 40% chance of losing $20,000.
Examiner tip: Expected value is a useful starting point for decision-making but should never be the only factor considered — the spread/range of possible outcomes and the decision-maker's risk tolerance both matter in practice.
4.
Marking analysis: A learner attempts the following task: “Using the decision tree from the previous question, explain one limitation of relying purely on expected value when choosing between Option A and Option B, given that Option A carries a real risk of an outright loss.” Their response addresses only this point: “Explains that expected value is a long-run average outcome and does not capture a decision-maker's attitude toward risk, such as risk-aversion.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains that expected value is a long-run average outcome and does not capture a decision-maker's attitude toward risk, such as risk-aversion.
- Identifies the missing requirement: Explains that even though Option A has a higher expected value, a risk-averse manager might prefer Option B because it guarantees a profit in both possible outcomes, with no chance of an outright loss, unlike Option A's 40% chance of losing $20,000.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
5.
Define 'critical path' in the context of critical path analysis (network planning), and explain why identifying it is important for project managers.
[3 marks] · no calculator
Marking points
- Defines the critical path as the longest sequence of dependent activities through a project network, determining the minimum possible time needed to complete the whole project.
- Explains that any delay to an activity on the critical path directly delays the entire project's completion date, since there is no spare time (float) built into these activities.
- Explains that project managers should therefore prioritize monitoring and resourcing critical path activities most closely, since they have the greatest impact on whether the project finishes on time.
Examiner tip: Activities not on the critical path have 'float' (spare time) and can be delayed slightly without affecting the overall project finish date — this is the key distinction examiners expect between critical and non-critical activities.
6.
Marking analysis: A learner attempts the following task: “Define 'critical path' in the context of critical path analysis (network planning), and explain why identifying it is important for project managers.” Their response addresses only this point: “Defines the critical path as the longest sequence of dependent activities through a project network, determining the minimum possible time needed to complete the whole project.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Marking points
- Recognises credit for the stated point: Defines the critical path as the longest sequence of dependent activities through a project network, determining the minimum possible time needed to complete the whole project.
- Identifies the missing requirement: Explains that any delay to an activity on the critical path directly delays the entire project's completion date, since there is no spare time (float) built into these activities.
- Identifies the missing requirement: Explains that project managers should therefore prioritize monitoring and resourcing critical path activities most closely, since they have the greatest impact on whether the project finishes on time.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
7.
A project has the following activities: Activity A (2 days) must be completed first. Activity B (3 days) and Activity C (4 days) can both start once A is finished. Activity D (2 days) can only start once both B and C are finished. (a) Calculate the total duration of path A-B-D. (b) Calculate the total duration of path A-C-D. (c) Identify the critical path and state the minimum project duration. (d) Calculate the float (spare time) available on activity B.
[4 marks]
Marking points
- Calculates the duration of path A-B-D as 2 + 3 + 2 = 7 days.
- Calculates the duration of path A-C-D as 2 + 4 + 2 = 8 days.
- Identifies A-C-D as the critical path, since it is the longer of the two paths, and states the minimum project duration as 8 days.
- Calculates the float on activity B as the difference between the critical path duration and the duration of the path containing B: 8 − 7 = 1 day.
Examiner tip: Compare every possible path from start to finish and take the longest one as the critical path — any activity on a shorter path has float equal to the difference between its path's duration and the critical path's duration.
8.
Marking analysis: A learner attempts the following task: “A project has the following activities: Activity A (2 days) must be completed first. Activity B (3 days) and Activity C (4 days) can both start once A is finished. Activity D (2 days) can only start once both B and C are finished. (a) Calculate the total duration of path A-B-D. (b) Calculate the total duration of path A-C-D. (c) Identify the critical path and state the minimum project duration. (d) Calculate the float (spare time) available on activity B.” Their response addresses only this point: “Calculates the duration of path A-B-D as 2 + 3 + 2 = 7 days.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Marking points
- Recognises credit for the stated point: Calculates the duration of path A-B-D as 2 + 3 + 2 = 7 days.
- Identifies the missing requirement: Calculates the duration of path A-C-D as 2 + 4 + 2 = 8 days.
- Identifies the missing requirement: Identifies A-C-D as the critical path, since it is the longer of the two paths, and states the minimum project duration as 8 days.
- Identifies the missing requirement: Calculates the float on activity B as the difference between the critical path duration and the duration of the path containing B: 8 − 7 = 1 day.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
9.
Explain two benefits to a project manager of using critical path analysis before starting a complex project.
[2 marks] · no calculator
Marking points
- Explains that critical path analysis identifies the minimum possible time to complete the project, allowing for realistic scheduling and deadline-setting before work begins.
- Explains that it identifies which activities have float (spare time) and which do not, helping managers allocate resources (staff, equipment, attention) more efficiently by prioritizing critical activities.
Examiner tip: Critical path analysis is fundamentally a planning and resource-allocation tool — its value comes from identifying constraints and flexibility before problems occur, not from solving problems after they arise.
10.
Marking analysis: A learner attempts the following task: “Explain two benefits to a project manager of using critical path analysis before starting a complex project.” Their response addresses only this point: “Explains that critical path analysis identifies the minimum possible time to complete the project, allowing for realistic scheduling and deadline-setting before work begins.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains that critical path analysis identifies the minimum possible time to complete the project, allowing for realistic scheduling and deadline-setting before work begins.
- Identifies the missing requirement: Explains that it identifies which activities have float (spare time) and which do not, helping managers allocate resources (staff, equipment, attention) more efficiently by prioritizing critical activities.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
11.
State one limitation of critical path analysis as a planning tool.
[1 mark] · no calculator
Marking points
- States a valid limitation, such as the technique relying on activity duration estimates that may be inaccurate, or that it does not account for resource constraints (e.g., the same staff being needed for two simultaneous activities).
Examiner tip: Like any quantitative planning model, critical path analysis is only as reliable as the time estimates fed into it — inaccurate estimates for individual activities will produce an inaccurate critical path and project duration.
12.
Marking analysis: A learner attempts the following task: “State one limitation of critical path analysis as a planning tool.” Their response addresses only this point: “States a valid limitation, such as the technique relying on activity duration estimates that may be inaccurate, or that it does not account for resource constraints (e.g., the same staff being needed for two simultaneous activities).” Evaluate the response against the complete 1-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[1 mark] · no calculator
Marking points
- Recognises credit for the stated point: States a valid limitation, such as the technique relying on activity duration estimates that may be inaccurate, or that it does not account for resource constraints (e.g., the same staff being needed for two simultaneous activities).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
13.
A project has activities: A (3 days) first; B (4 days) and C (2 days) both follow A; D (5 days) follows both B and C; E (2 days) follows D. (a) Calculate the length of path A-B-D-E and path A-C-D-E. (b) State the critical path and minimum duration. (c) Calculate the float on activity C.
[4 marks]
Marking points
- Calculates path A-B-D-E as 3 + 4 + 5 + 2 = 14 days.
- Calculates path A-C-D-E as 3 + 2 + 5 + 2 = 12 days.
- States the critical path is A-B-D-E with a minimum duration of 14 days.
- Calculates the float on C as 14 − 12 = 2 days.
Examiner tip: The critical path is the longest path; float on a non-critical activity equals the critical length minus the length of its path.
14.
Marking analysis: A learner attempts the following task: “A project has activities: A (3 days) first; B (4 days) and C (2 days) both follow A; D (5 days) follows both B and C; E (2 days) follows D. (a) Calculate the length of path A-B-D-E and path A-C-D-E. (b) State the critical path and minimum duration. (c) Calculate the float on activity C.” Their response addresses only this point: “Calculates path A-B-D-E as 3 + 4 + 5 + 2 = 14 days.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Marking points
- Recognises credit for the stated point: Calculates path A-B-D-E as 3 + 4 + 5 + 2 = 14 days.
- Identifies the missing requirement: Calculates path A-C-D-E as 3 + 2 + 5 + 2 = 12 days.
- Identifies the missing requirement: States the critical path is A-B-D-E with a minimum duration of 14 days.
- Identifies the missing requirement: Calculates the float on C as 14 − 12 = 2 days.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
15.
Explain how a project manager could shorten the duration of a project shown by a network diagram, and state one cost of doing so.
[3 marks] · no calculator
Marking points
- Explains that only shortening activities on the critical path reduces total duration, e.g. by adding labour or equipment to them.
- Notes that once a path is shortened, another path may become critical, limiting further savings.
- States a cost, such as extra labour or equipment costs, or overtime payments.
Examiner tip: Spending money on non-critical activities does not shorten the project.
16.
Marking analysis: A learner attempts the following task: “Explain how a project manager could shorten the duration of a project shown by a network diagram, and state one cost of doing so.” Their response addresses only this point: “Explains that only shortening activities on the critical path reduces total duration, e.g. by adding labour or equipment to them.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains that only shortening activities on the critical path reduces total duration, e.g. by adding labour or equipment to them.
- Identifies the missing requirement: Notes that once a path is shortened, another path may become critical, limiting further savings.
- Identifies the missing requirement: States a cost, such as extra labour or equipment costs, or overtime payments.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
17.
A firm sells a product for $25 with variable cost of $15 per unit; fixed costs are $40,000. (a) Calculate the break-even quantity. (b) Calculate the sales needed to earn a target profit of $20,000. (c) Calculate the margin of safety, as a percentage of actual sales, if 7,000 units are sold.
[4 marks]
Marking points
- Calculates contribution per unit as 25 − 15 = $10.
- Calculates break-even as 40,000 / 10 = 4,000 units.
- Calculates target-profit output as (40,000 + 20,000) / 10 = 6,000 units.
- Calculates margin of safety as (7,000 − 4,000) / 7,000 × 100 ≈ 42.9%.
Examiner tip: For a profit target, add it to fixed costs before dividing by contribution per unit.
18.
Marking analysis: A learner attempts the following task: “A firm sells a product for $25 with variable cost of $15 per unit; fixed costs are $40,000. (a) Calculate the break-even quantity. (b) Calculate the sales needed to earn a target profit of $20,000. (c) Calculate the margin of safety, as a percentage of actual sales, if 7,000 units are sold.” Their response addresses only this point: “Calculates contribution per unit as 25 − 15 = $10.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Marking points
- Recognises credit for the stated point: Calculates contribution per unit as 25 − 15 = $10.
- Identifies the missing requirement: Calculates break-even as 40,000 / 10 = 4,000 units.
- Identifies the missing requirement: Calculates target-profit output as (40,000 + 20,000) / 10 = 6,000 units.
- Identifies the missing requirement: Calculates margin of safety as (7,000 − 4,000) / 7,000 × 100 ≈ 42.9%.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
19.
A firm produced 12,000 units with 40 workers at a total cost of $96,000. Calculate (a) labour productivity and (b) unit cost, and (c) explain one way the firm could lower unit cost without cutting wages.
[3 marks]
Marking points
- Calculates labour productivity as 12,000 / 40 = 300 units per worker.
- Calculates unit cost as 96,000 / 12,000 = $8.
- Explains a method such as raising productivity through training or better equipment so more units are produced from the same cost base.
Examiner tip: Unit cost falls when output per worker rises while total cost stays roughly constant.
20.
Marking analysis: A learner attempts the following task: “A firm produced 12,000 units with 40 workers at a total cost of $96,000. Calculate (a) labour productivity and (b) unit cost, and (c) explain one way the firm could lower unit cost without cutting wages.” Their response addresses only this point: “Calculates labour productivity as 12,000 / 40 = 300 units per worker.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Marking points
- Recognises credit for the stated point: Calculates labour productivity as 12,000 / 40 = 300 units per worker.
- Identifies the missing requirement: Calculates unit cost as 96,000 / 12,000 = $8.
- Identifies the missing requirement: Explains a method such as raising productivity through training or better equipment so more units are produced from the same cost base.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
21.
Evaluate the use of just-in-time (JIT) production compared with just-in-case stock holding for a supermarket.
[4 marks] · no calculator
Marking points
- Explains the JIT benefit: lower storage costs, less waste of perishable goods and less cash tied up in stock.
- Explains the JIT risk: stock-outs and lost sales if deliveries are late or demand spikes.
- Explains that just-in-case holding protects availability but raises storage and wastage costs.
- Reaches a justified judgement, e.g. a hybrid: JIT for perishables with reliable suppliers, buffer stock for high-demand staples.
Examiner tip: Evaluation needs a conclusion that depends on context: product type, supplier reliability and cost of a stock-out.
22.
Marking analysis: A learner attempts the following task: “Evaluate the use of just-in-time (JIT) production compared with just-in-case stock holding for a supermarket.” Their response addresses only this point: “Explains the JIT benefit: lower storage costs, less waste of perishable goods and less cash tied up in stock.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains the JIT benefit: lower storage costs, less waste of perishable goods and less cash tied up in stock.
- Identifies the missing requirement: Explains the JIT risk: stock-outs and lost sales if deliveries are late or demand spikes.
- Identifies the missing requirement: Explains that just-in-case holding protects availability but raises storage and wastage costs.
- Identifies the missing requirement: Reaches a justified judgement, e.g. a hybrid: JIT for perishables with reliable suppliers, buffer stock for high-demand staples.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
23.
Explain how a business adopting total quality management (TQM) may reduce its costs in the long run despite higher short-run costs.
[3 marks] · no calculator
Marking points
- Explains that TQM requires short-run spending on training, process redesign and monitoring.
- Explains that fewer defects reduce scrap, rework, returns and warranty claims over time.
- Explains that improved quality strengthens reputation and customer loyalty, supporting sales.
Examiner tip: The cost of quality: prevention costs now are usually cheaper than failure costs later.
24.
Marking analysis: A learner attempts the following task: “Explain how a business adopting total quality management (TQM) may reduce its costs in the long run despite higher short-run costs.” Their response addresses only this point: “Explains that TQM requires short-run spending on training, process redesign and monitoring.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains that TQM requires short-run spending on training, process redesign and monitoring.
- Identifies the missing requirement: Explains that fewer defects reduce scrap, rework, returns and warranty claims over time.
- Identifies the missing requirement: Explains that improved quality strengthens reputation and customer loyalty, supporting sales.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
25.
Explain two potential benefits and one drawback to a manufacturer of introducing greater automation.
[3 marks] · no calculator
Marking points
- Explains a benefit such as higher productivity and consistency of output with fewer errors.
- Explains a second benefit such as lower long-run unit labour costs and the ability to run continuously.
- Explains a drawback such as high capital cost, job losses and lower flexibility for small or customised orders.
Examiner tip: Automation trades flexibility and up-front capital for productivity and consistency, so it suits high-volume standardized output.
26.
Marking analysis: A learner attempts the following task: “Explain two potential benefits and one drawback to a manufacturer of introducing greater automation.” Their response addresses only this point: “Explains a benefit such as higher productivity and consistency of output with fewer errors.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains a benefit such as higher productivity and consistency of output with fewer errors.
- Identifies the missing requirement: Explains a second benefit such as lower long-run unit labour costs and the ability to run continuously.
- Identifies the missing requirement: Explains a drawback such as high capital cost, job losses and lower flexibility for small or customised orders.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
27.
Explain two limitations of break-even analysis as a decision-making tool.
[2 marks] · no calculator
Marking points
- Explains that it assumes a constant selling price and linear costs, whereas prices may fall with volume and variable costs may change with bulk buying.
- Explains that it assumes everything produced is sold and ignores demand uncertainty and changes in fixed costs at higher output.
Examiner tip: Break-even is a simplified model; its answers are only as reliable as its assumptions.
28.
Marking analysis: A learner attempts the following task: “Explain two limitations of break-even analysis as a decision-making tool.” Their response addresses only this point: “Explains that it assumes a constant selling price and linear costs, whereas prices may fall with volume and variable costs may change with bulk buying.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Marking points
- Recognises credit for the stated point: Explains that it assumes a constant selling price and linear costs, whereas prices may fall with volume and variable costs may change with bulk buying.
- Identifies the missing requirement: Explains that it assumes everything produced is sold and ignores demand uncertainty and changes in fixed costs at higher output.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
29.
A decision tree shows an investment costing $20,000 with a 0.7 probability of a $80,000 return and a 0.3 probability of a $10,000 return. Calculate the expected net value and state whether the firm should proceed if the alternative is to do nothing (net value $0).
[3 marks]
Marking points
- Calculates expected return as (0.7 × 80,000) + (0.3 × 10,000) = 56,000 + 3,000 = $59,000.
- Subtracts the cost: 59,000 − 20,000 = $39,000.
- States that the firm should proceed, as $39,000 exceeds the $0 alternative, noting the risk of the lower outcome.
Examiner tip: Weight each outcome by its probability, add them, then subtract the cost of the decision.
30.
Marking analysis: A learner attempts the following task: “A decision tree shows an investment costing $20,000 with a 0.7 probability of a $80,000 return and a 0.3 probability of a $10,000 return. Calculate the expected net value and state whether the firm should proceed if the alternative is to do nothing (net value $0).” Their response addresses only this point: “Calculates expected return as (0.7 × 80,000) + (0.3 × 10,000) = 56,000 + 3,000 = $59,000.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Marking points
- Recognises credit for the stated point: Calculates expected return as (0.7 × 80,000) + (0.3 × 10,000) = 56,000 + 3,000 = $59,000.
- Identifies the missing requirement: Subtracts the cost: 59,000 − 20,000 = $39,000.
- Identifies the missing requirement: States that the firm should proceed, as $39,000 exceeds the $0 alternative, noting the risk of the lower outcome.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.