Deep Dive: Decision Trees Cpa and Contribution

#CODEX Deep-Dive: Decision Trees, Critical Path Analysis & Contribution (WBS13 §3.3.3.3–3.3.3.5)

Provenance — written 2026-09-28. Built from a direct read of research/veridian/WBS13-verified-facts.md (2,057 lines; grepped and read in full at the relevant sections — table of contents, the quantitative-formulas table, the full §3.3.3 topic write-up, and the flagged-issues list), the existing lesson file content/veridian/curriculum/wbs13/decision-trees-cpa-and-contribution.ts (read in full, including its own provenance header comment), and the three companion technique-hub documents in this folder — codex-definitions-and-application-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md — plus content/veridian/question-bank/wbs13.ts (checked for overlap; it currently holds no decision-tree, CPA or contribution question). No new primary-source PDF was fetched for this document; every real-exam fact below is drawn from citations already verified in the sources just listed, not independently re-checked against a fresh Pearson download.

#What this document is, and isn't

This is a single-topic reference for spec points 3.3.3.3–3.3.3.5 only (decision trees, critical path analysis, contribution) — not 3.3.3.1–2 (forecasting, investment appraisal), which the curriculum splits into the same lesson file but this document doesn't cover. It does not repeat the lesson's full worked chains (Center Parcs, Thornfield Furniture Co., Solmere Outdoors, Aldergate Print Co., Kestwick Furnishings, Marrow & Vale) stage-by-stage — those already exist, are pedagogically paced, and re-typing them here would be restatement, not addition. What follows instead is: the theory stated as a compact reference rather than a teaching narrative; short (2–5 line) illustrative micro-examples, clearly marked [illustrative], built with fresh scenarios and numbers so they don't duplicate the lesson's own exemplars; and a consolidated synthesis of every real WBS13 exam appearance of this topic this course's research has actually found, with nothing beyond that claimed.

Two labelling conventions carried over from the lesson, used consistently below:

• [real] — a verified fact from a genuine Pearson mark scheme, examiner report or question paper, with series and question number given.
• [illustrative] — an invented scenario built to demonstrate a mechanism, with no claim to be, or resemble, a specific real Pearson question.

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#Part 1 — Decision trees (3.3.3.3)

#1.1 Anatomy and construction rules

A decision tree is drawn left to right and solved right to left. Two node types:

• Decision node (square) — a point where the firm itself chooses between mutually exclusive options. Each option is a labelled branch carrying that option's own cost.
• Chance node (circle) — a point where an outcome is decided by the market or wider world, not the firm. Each outcome branch carries a probability and a payoff. At any one chance node, the outcome probabilities must sum to 1.

Expected value (EV) at a chance node = Σ(probability × payoff) — the probability-weighted average of every outcome branching from that node. Net gain for an option = EV of its chance node minus that option's own cost. The tree is solved by computing EV then net gain for every option, then selecting the higher net gain and marking the rejected branch (conventionally a double line or a cross) — the finished diagram shows the working, not just the answer.

#1.2 Why net gain, not raw EV, is the decision criterion

EV alone tells you the average payoff a chance node would produce if the same gamble were run repeatedly; it says nothing about what that option cost to enter. Two options can have EV figures that rank one way and net-gain figures that rank the other way whenever their costs differ — which is the normal case, since options being compared rarely cost the same amount. Comparing raw EV and stopping there is the single most consequential mechanical error the technique invites; the lesson's own Option P vs Option Q diagram demonstrates the reversal directly, and is not repeated here.

#1.3 The one real sub-case this paper has actually tested

Every real WBS13 decision-tree question found so far — one, see §1.6 — is the same structural shape: two mutually exclusive options, each leading to exactly one chance node with exactly two outcome branches (success/failure). There is no confirmed WBS13 example of a multi-stage tree (a chance node whose "success" branch itself leads to a further decision node, i.e. a sequential decision under compounding uncertainty), and the spec's own wording for this sub-topic — "construct/interpret" and "calculation/interpretation" — names no such variant explicitly. This document does not claim multi-stage trees are untested on this paper generally, only that this course's research has not found one; treat a multi-stage tree as a plausible but unevidenced possibility, not a confirmed exam pattern.

#1.4 Extension: how sensitive is a recommendation to the estimated probability? [illustrative]

The real Center Parcs mark scheme (§1.6 below) explicitly cautions that a recommendation shouldn't rest on the size of the best-case payoff alone, given a stated failure probability. A natural way to make that caution numerically concrete — not itself confirmed as a tested WBS13 question type, offered here as a worked extension of "interpretation" (3.3.3.3b) — is to ask: how far would the success probability have to fall before an option's net gain turns negative?

Vantage Kitchens is deciding whether to launch a new product line: cost £60,000; success pays £150,000, failure books a £20,000 write-off. Net gain = [p × £150,000 + (1−p) × (−£20,000)] − £60,000 = £170,000p − £80,000. Setting this to zero: p = £80,000 ÷ £170,000 ≈ 0.47. Below roughly a 47% chance of success, launching becomes net-gain-negative on these figures; above it, positive. This single number — the break-even probability — is a genuinely stronger evaluative point than restating "it depends on the probability," because it states exactly how much the estimate would have to be wrong before the recommendation flips, which is precisely the kind of quantified caution a Level 4 Evaluate answer is rewarded for (see the paper's general level descriptors in the SIGNAL/FORGE documents in this folder).

#1.5 Limitations (3.3.3.3c) — condensed, sourced

• Ignores attitudes to risk. EV is the correct benchmark only under risk-neutrality — indifference between a certain sum and a gamble with the same average value. A risk-averse board can rationally reject the higher-net-gain option specifically because its outcomes are more spread out. This is core, spec-named content, not enrichment — the lesson's mechanism block derives it in full; the formal economic theory behind it (Bernoulli's expected-utility resolution of the St Petersburg paradox; Kahneman and Tversky's loss-aversion finding) is flagged there as genuinely beyond-spec background, not repeated here.
• Inputs are estimates, not certainties. Every probability and payoff on the diagram is a forecast, usually built from market research or past experience — the diagram's own precision (two decimal places of probability, exact payoff figures) can overstate how confidently those numbers are actually known.
• Reduces a genuinely qualitative decision to two numbers. A tree captures cost and probability-weighted payoff; it says nothing about brand reputation, staff morale, or strategic fit, which a real board would also weigh.

#1.6 Real exam angle [real]

Center Parcs, growth-strategy choice — Oct 2024 Q2, Publications Code WBS13_01_2410_MS, Section B, 20-mark Evaluate. The question asked students to use decision-tree-analysis calculations, alongside two data extracts, to evaluate how far DTA would help Center Parcs decide its growth strategy. Two options: build a new holiday village (cost €520m; 0.8 probability of €780m success, 0.2 probability of a €120m loss) versus take over an existing one (cost €100m; 0.7 probability of €220m success, 0.3 probability of a €70m loss). Working the figures: new village EV = (0.8 × €780m) + (0.2 × −€120m) = €600m, net gain = €600m − €520m = €80m; takeover EV = (0.7 × €220m) + (0.3 × −€70m) = €133m, net gain = €133m − €100m = €33m. The mark scheme recommends the new village on the strength of its higher net gain, while separately crediting the caution that the €780m best-case figure alone shouldn't drive the decision given the accompanying 20% failure probability — independently recomputed here against the source figures and confirmed correct. This is the only WBS13 decision-tree question with a named real company this course's research has found, after three separate mining passes and nine series checked specifically for this sub-topic before the hit (see Part 4 for the fuller series count). This document does not reproduce Extract F's own hand-drawn diagram; only the figures and the EMV outcome are used, as they appear in the mark scheme's own indicative content.

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#Part 2 — Critical path analysis (3.3.3.4)

#2.1 Anatomy and vocabulary

A network diagram is built from nodes (numbered circles marking a point in time — an "event" — reached once every activity leading into it is complete) and activities (arrows between two nodes, each labelled with a letter and a duration). Two figures sit at every node: EST (earliest start time — found working forward from the start, taking the later of any routes converging on that node) and LFT (latest finish time — found working backward from the finish, taking the earlier of any routes that node feeds into). Total float for an activity = LFT(the node it ends at) − EST(the node it starts from) − its own duration: the spare time that activity carries without delaying the project. The critical path is the unbroken chain of zero-float activities from the first node to the last — the full derivation of why zero float is exactly what defines it, rather than "longest path" being a separate rule to memorise, is in the lesson's own mechanism block and isn't repeated here.

#2.2 A network can have more than one critical path [illustrative — general technique fact, not

confirmed against a WBS13 extract specifically]

Nothing about the mechanism above guarantees a unique longest route: two structurally different chains through the same network can legitimately tie for the longest total duration, in which case both carry zero float and both are critical.

Small network: from a shared start node 1, chain M (4 weeks) → N (3 weeks) reaches node 4 via node 2 in 4 + 3 = 7 weeks; a separate chain O (2 weeks) → P (5 weeks) reaches the same node 4 via node 3 in 2 + 5 = 7 weeks. Both chains total 7 weeks, so EST(node 4) = max(7, 7) = 7 and LFT(node 4) = 7 either way. Working back: Float(M) = Float(N) = Float(O) = Float(P) = 0 — every activity on both chains is critical. A delay to either chain delays the finish date; a completion task that identifies only one of the two as "the" critical path has under-marked the network. This case is not confirmed against a real WBS13 question — neither the Coca-Cola nor Burger King networks below have their full activity lists captured verbatim in this course's sources (§2.6) — so it is offered here purely as a genuine feature of the technique a completion-style question could test, not as evidence Pearson has tested it this way.

#2.3 Benefits (3.3.3.4a) — condensed, sourced to the Burger King mark scheme

Beyond the core purpose (finding the minimum project duration and which activities control it), the real Summer 2024 mark scheme names two further genuine benefits worth stating precisely:

• Building the network forces a careful, activity-by-activity assessment of what a project actually requires before work starts, which the mark scheme credits directly as reducing the project's own risk and cost — not merely a side effect of finding the finish date.
• Once a network shows exactly when each activity is genuinely needed, deliveries (the mark scheme's own example: furniture and kitchen equipment) can be scheduled just-in-time rather than stored early, keeping capital from being tied up in stock ahead of when the project actually needs it.
• Float, once identified, is a genuine resource-reallocation signal — not just a reason not to waste effort accelerating a non-critical activity, but grounds to actively move staff or budget off a floating activity and onto a behind-schedule critical one.

#2.4 Limitations (3.3.3.4d) — condensed, sourced

From the real June 2022 (Coca-Cola) and Summer 2024 (Burger King) mark schemes and examiner reports, five separable limitations:

1. Durations are estimated from past activities. A genuinely novel activity with no real precedent is a weaker input than the diagram's single neat number suggests.
2. External shocks are invisible to the diagram. A supplier failure, strike, or extreme weather isn't modelled anywhere in the network unless a manager manually redraws it afterward.
3. Building and maintaining an accurate network is itself a real ongoing cost in specialist planning time for a genuinely large project — not a free-to-produce add-on.
4. One completed network doesn't generalise across every site of a multi-location rollout. The Burger King mark scheme states plainly: "the CPA may not be effective for all restaurants" — site layout and local contractor availability genuinely differ location to location, so an accurate network for site 1 can still mislead at site 47.
5. The zero-lag assumption between activities. The network assumes a successor activity starts the instant its predecessor's node is reached, with no handover delay, inspection, or sign-off modelled — a limitation about the technique's structure, not about any one duration being wrong.

#2.5 Real exam angle [real]

Coca-Cola Zero Sugar relaunch — June 2022 Q2, Section B, 20-mark Evaluate. Correct project duration, confirmed against the mark scheme: 32 weeks. The underlying network's own activity-level data (the specific letters, durations and precedence) was not captured verbatim in this course's research — only the 32-week total and the examiner report's summary of a genuine 20/20 exemplar's balanced argument (CPA speeds project completion and clarifies float, against CPA relying on past data and ignoring external delay, and being costly to produce). No EST/LFT/float table for this series exists in the sources behind this document.

Burger King restaurant-renovation programme — Summer 2024 Q3, Publications Code WBS13_01_2406_MS, Section C, 20-mark Evaluate. Correct project duration: 11 weeks. Activity C's float is confirmed as 1 week, and Activity B (a 3-week estimate) sits on the critical path. The mark scheme's own indicative content runs to 16 bullets; four of the genuinely distinct ones are the benefits/limitations already folded into §2.3–2.4 above. The examiner report confirms candidates who both calculated the correct 11-week duration and evaluated CPA specifically as applied to Burger King's renovation — not as a generic "CPA is useful because…" essay — reached the higher mark bands; a genuine 18/20 exemplar is recorded as losing marks on an underdeveloped conclusion, not on the calculation itself. As with Coca-Cola, the underlying network's activity list and precedence data are presented in the mark scheme as a diagram image, not machine-readable text, and were not captured verbatim in this course's sources.

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#Part 3 — Contribution (3.3.3.5)

#3.1 Definitions

Contribution per unit = selling price per unit − variable cost per unit. Total contribution = contribution per unit × units sold. Contribution measures what each unit adds toward covering fixed costs and then profit — it is not the same figure as profit per unit, which would additionally deduct a share of fixed cost.

#3.2 Why contribution, not full average cost, drives a short-run decision

Comparing "accept" against "reject" for a single order or product-line decision, every fixed cost the firm is committed to paying regardless of which option is chosen cancels out of the comparison exactly — it appears identically on both sides. What's left to compare is only what genuinely differs between the two options: the order's or product's own revenue, variable cost, and any extra fixed cost the decision specifically causes. Whenever a decision uses only spare capacity that's already being paid for, that extra-fixed-cost term is zero, and the decision collapses to a single test: is contribution positive? Full average cost, by contrast, already has a share of the same unavoidable fixed costs baked into it — comparing a price to average cost silently reintroduces a cost the decision doesn't actually depend on. The lesson's mechanism block derives this algebraically in full; it is stated here as the underlying reason for all three decision-uses below, not re-derived.

#3.3 Three decision-uses, compared

Spec point 3.3.3.5c names contribution's use "as a decision-making technique," and three genuinely distinct real business decisions fall under it:

#3.4 Micro-examples [illustrative]

• Accept/reject. Harlow Bakery sells bread rolls at £1.20; variable cost £0.50 (normal contribution £0.70). A canteen offers a one-off bulk order of 2,000 rolls at £0.80, and Harlow has spare oven capacity this week. Special-order contribution = £0.80 − £0.50 = £0.30 per roll — positive, even well below normal price — so accepting adds £0.30 × 2,000 = £600 to profit that rejecting would simply leave unclaimed.
• Discontinuation. Fenwick Tonics has £0.90 contribution per unit and sells 8,000 units — £7,200 total contribution — but is allocated £9,000 of shared warehouse overhead under absorption costing, so it reports a £1,800 accounting loss. Because that overhead is shared and largely survives the line's removal, dropping Fenwick Tonics would remove a real £7,200 of contribution while leaving most of the £9,000 of cost still to be covered by what remains — the opposite of the intended effect.
• Scarce-resource ranking. Product X earns £12 contribution per unit using 3 skilled-labour hours; Product Y earns £20 per unit using 6 hours. Per labour hour — the genuinely limiting resource — X earns £4.00 and Y earns £3.33. X is the better use of each scarce hour despite Y's higher per-unit figure, exactly the reversal a naive per-unit ranking would miss.

#3.5 Real exam angle [real]

Lush bath-soap range — June 2023 Q1(b), 4-mark Calculate. Selling price £5.50 per unit, variable cost £1.00 per unit, so contribution per unit = £4.50; on 3,300,000 units sold in a month, total contribution = £14,850,000. The examiner report confirms two independently valid working methods both reached full marks: contribution-per-unit × units, and total revenue minus total variable cost. This is the only real WBS13 contribution question this course's research has found with a full worked figure; no primary-source question testing contribution specifically as a decision-making technique (3.3.3.5c — accept/reject, discontinuation, or scarce-resource ranking) has been located, which is why §3.4 above is entirely illustrative rather than real-exam-sourced.

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#Part 4 — Cross-topic exam-angle synthesis

Series counts and tariffs above are taken directly from signal-frequency-and-boundaries.md's own frequency table in this folder, which tallies every individually dated, sourced real question verified in WBS13-verified-facts.md across twelve located WBS13 series (Oct 2021 through Oct 2025). Following that document's own explicit discipline: these counts are not converted into a percentage or a "likely to appear" claim anywhere in this document, and no grade-boundary data exists in any source behind this document for this paper (see SIGNAL §3) — nothing here should be read as implying one.

Two structural observations that do follow directly from the figures above, without extrapolating beyond them: first, this whole three-technique group supplies only Section B/C 20-mark Evaluate content for decision trees and CPA, and Section A 4-mark Calculate content for contribution — no confirmed Discuss (8- mark) or Assess (12-mark) question has been found testing any of these three sub-topics specifically. Second, CPA is the only one of the three confirmed more than once, and both of its confirmed appearances independently produced a full-marks-or-near-full-marks exemplar quoted in the examiner report — Coca-Cola's genuine 20/20 answer and Burger King's genuine 18/20 answer — making CPA the sub-topic in this group with the clearest evidence of what a top-band answer actually contains.

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#Part 5 — What remains unverified (do not overstate these)

• EST/LFT/float formula wording has never been captured verbatim from a real WBS13 mark scheme in this course's sources; the mechanism in the lesson (and summarised in §2.1 above) is derived from first principles, not quoted from a primary document.
• Multi-stage decision trees and networks with more than one critical path (§1.3, §2.2) are not confirmed as tested WBS13 content — they are named here as genuine features of the underlying technique, clearly marked illustrative, not as evidenced exam patterns.
• Contribution as a decision-making technique (accept/reject, discontinuation, scarce-resource ranking) has no real WBS13 question located testing it directly — only the Lush calculation itself (§3.5) is real; every example under §3.4 is illustrative.
• The full activity lists for the Coca-Cola and Burger King CPA networks are not available as machine-readable data in this course's sources (both are presented as diagram images in their mark schemes) — only the project-duration totals, named critical/floating activities, and prose indicative content are verified.
• No likelihood, percentage, or grade-boundary claim is made anywhere in this document; every frequency statement above is stated exactly as "seen in N of 12 series checked," per the sourcing discipline in signal-frequency-and-boundaries.md.

#Sources consulted

• research/veridian/WBS13-verified-facts.md (table of contents/spec mapping, quantitative-formulas table, full §3.3.3 topic section, flagged-issues list)
• content/veridian/curriculum/wbs13/decision-trees-cpa-and-contribution.ts (lesson content and its own provenance header, read in full — not duplicated)
• content/veridian/technique-hub/wbs13/codex-definitions-and-application-bank.md
• content/veridian/technique-hub/wbs13/forge-exam-technique-guide.md
• content/veridian/technique-hub/wbs13/signal-frequency-and-boundaries.md
• content/veridian/question-bank/wbs13.ts (checked for overlap; none found for this topic)