Simulation, Decision Theory, Learning Curve and Network Analysis
This closing chapter of Strategic Cost Management's quantitative-tools section covers four techniques for decision-making under conditions the earlier, more deterministic chapters largely assumed away: genuine uncertainty about which future will actually occur.
1. Monte Carlo simulation
Monte Carlo simulation models a real-world uncertain process by repeatedly generating random outcomes according to a known probability distribution, and observing the pattern of results across many repeated trials, when the underlying problem is too complex to solve with a single neat formula.
The method proceeds in a fixed sequence: establish the probability distribution of the uncertain variable (such as daily customer demand); assign a range of random numbers to each possible outcome, proportional to its probability (an outcome with 30% probability is assigned 30% of the available random-number range).
The remaining steps are to generate a random number and read off the outcome it corresponds to; repeat this process for a large number of trials; and finally, analyse the resulting distribution of simulated outcomes — the average, the spread, the frequency of a particular result — to draw a conclusion.
Simulation is used precisely where a problem's complexity or interdependence between random variables makes an exact analytical (formula-based) solution impractical — it substitutes repeated experimentation with random numbers for a mathematical proof, and its result is only as reliable as the underlying probability distribution assumed and the number of trials run, which is why a small number of simulated trials is explicitly flagged as a limitation whenever it is examined.
2. Decision theory — deciding without certainty
Decision theory classifies a decision problem by how much is known about which future state of the world will actually occur, and a different decision rule applies to each classification.
Decision-making under risk (where probabilities of each possible future state are known) uses Expected Monetary Value (EMV) — the probability-weighted average of the outcomes under each possible course of action — selecting the action with the highest EMV (for a profit-maximisation problem) or lowest EMV (for a cost-minimisation problem):
Decision-making under uncertainty (where even the probabilities of future states are unknown) uses one of several named, competing rules, each reflecting a different attitude to risk:
| Rule | Logic | Attitude reflected |
|---|---|---|
| Maximax | Choose the action with the best possible best-case outcome | Optimistic |
| Maximin | Choose the action with the best possible worst-case outcome | Pessimistic (cautious) |
| Minimax Regret | Choose the action minimising the maximum possible regret (the gap between what was chosen and what would have been best in hindsight, for each state) | Regret-averse |
A decision tree represents a sequence of decisions and chance events graphically, as a branching structure, and is solved by working backward from the final outcomes ("rolling back" the tree) — computing the expected value at each chance node and choosing the better branch at each decision node — which is simply EMV logic applied systematically to a multi-stage decision problem rather than a single-stage one.
3. Learning curve — a brief cross-reference
The learning curve (average time per unit falling by a constant percentage each time cumulative output doubles) was introduced in the Management Accounting paper at Intermediate level, and the same doubling-rule mechanics apply here — this paper's quantitative-tools syllabus lists it again specifically because a learning-curve effect can itself be a source of uncertainty worth simulating.
For instance, a firm might simulate a range of possible learning rates when the exact rate is not known with certainty, connecting this earlier deterministic technique to this chapter's uncertainty-focused toolkit rather than introducing new mechanics of its own.
4. Network analysis — crashing a project
PERT and CPM's critical-path logic was covered at Intermediate level; the specific extension examined here is project crashing — the decision of how much to pay to shorten a project's duration, and which activities to shorten first. Every activity that can be shortened ("crashed") has a normal time and cost, and a crash time and cost (the fastest it can be completed, at a higher cost), giving a crash cost per unit time:
Only critical-path activities are worth crashing — exactly as established at Intermediate level, since shortening a non-critical activity does not shorten the overall project until its float is fully used up — and among critical-path activities, the one with the lowest crash cost per unit time should be crashed first, since it buys the required time reduction most cheaply.
After crashing the cheapest critical activity, the critical path itself must be re-checked, since shortening one critical activity enough can cause a previously non-critical path to become the new critical path (or a second path to become critical alongside the first), at which point activities on both critical paths may need to be crashed simultaneously to shorten the project further.
Worked Examples
Example 1. Daily demand for a product follows this distribution: 10 units (30% probability), 20 units (50% probability), 30 units (20% probability). Assign random number ranges (00-99) to each outcome for a Monte Carlo simulation.
10 units: 00-29 (30%). 20 units: 30-79 (50%). 30 units: 80-99 (20%).
Example 2. A firm can choose Strategy A (payoff ₹80,000 if demand is high, probability 0.4; payoff ₹20,000 if demand is low, probability 0.6) or Strategy B (payoff ₹50,000 regardless of demand). Compute the EMV of each strategy and recommend a choice.
EMV(A) = (0.4 × 80,000) + (0.6 × 20,000) = 32,000 + 12,000 = ₹44,000. EMV(B) = ₹50,000 (certain). Since EMV(B) > EMV(A), Strategy B should be chosen on an EMV basis.
Example 3. A decision-maker facing genuine uncertainty (no known probabilities) has three strategies with worst-case outcomes of ₹10,000, ₹15,000 and ₹8,000 respectively. Applying the Maximin rule, which strategy is chosen?
The Maximin rule chooses the best of the worst-case outcomes: max(10,000, 15,000, 8,000) = ₹15,000, so the strategy offering that worst-case outcome is selected.
Example 4. An activity has a normal time of 10 days at a normal cost of ₹20,000, and a crash time of 6 days at a crash cost of ₹32,000. Compute the crash cost per day.
Crash Cost per Day = (32,000 − 20,000) ÷ (10 − 6) = 12,000 ÷ 4 = ₹3,000 per day.
Example 5. Two critical-path activities have crash costs per day of ₹2,500 and ₹4,000 respectively. If only one day of project time needs to be saved, which activity should be crashed?
The activity with the crash cost per day of ₹2,500, since it achieves the required one-day reduction most cheaply.
Example 6. Explain why crashing a non-critical-path activity would not shorten a project's overall duration, even if it is technically possible to crash it.
A non-critical activity has positive float (spare time before it would start delaying the project), so shortening its duration simply increases its float further without changing the length of the critical path, which is what actually determines the project's minimum overall duration — the project finish date does not move until the critical path itself is shortened.
Example 7. After crashing the cheapest critical-path activity by two days, a second, previously non-critical path in the network now has the same length as the (now-shortened) critical path. Explain what this means for further crashing decisions.
This means there are now two critical paths of equal length running in parallel. To shorten the project further, activities on BOTH critical paths must be shortened simultaneously (either a single activity common to both paths, if one exists, or one activity on each path), since shortening only one of the two equal-length critical paths would leave the other one still governing the project's overall duration unchanged.
Summary
Monte Carlo simulation models a complex, uncertain process by repeatedly generating outcomes from an assigned random-number range proportional to each outcome's probability, analysing the resulting distribution across many trials when an exact formula-based solution is impractical.
Decision theory applies EMV (probability-weighted average payoff) under risk, and one of several named rules — Maximax (optimistic), Maximin (cautious), Minimax Regret (regret-averse) — under genuine uncertainty where probabilities are unknown, with decision trees applying the same EMV logic systematically across multi-stage decisions by rolling back from the final outcomes.
The learning curve's doubling-rule mechanics (introduced at Intermediate level) connect here to broader uncertainty analysis, while network analysis extends CPM/PERT into project crashing — shortening only critical-path activities, cheapest crash-cost-per-day first, and re-checking the critical path after each crash since a second path can become critical alongside the first.
