CMA Final · Strategic Cost Management

Simulation: formula sheet

Full chapter guide

Key formulas

Cumulative probability: Cumulative probability = running total of probabilities of the values, ending at 1.00; Used to build the table for assigning random numbers.
Random number allocation: Range for a value = (previous cumulative % + 1) to its own cumulative %; With two-digit numbers, 00 to 99 gives 100 numbers. A probability of 0.25 gets 25 numbers.
Average of simulated results: Average = Σ simulated values ÷ number of trials; Results from a few trials are only indicative; more trials improve reliability.
Probability from frequency: Probability = Frequency of value ÷ Total frequency; Use it to turn past data into a probability distribution. Probabilities must total 1.
Cumulative probability: Cumulative probability = running total of probabilities; The last cumulative value must equal 1.00 (or 100).
Random number range: Range for a value = (previous cumulative probability, this cumulative probability], on a 00–99 scale for two-digit numbers; Example: probabilities 0.20 and 0.30 give ranges 00–19 and 20–49. Allot ranges exactly as many numbers as the percentage.
Average from simulation: Average = Σ outcomes of all trials ÷ Number of trials; Use it for average demand, cost or profit per period.
Probability from frequency: Probability = Frequency of the value ÷ Total frequency; Use this when the question gives counts or days instead of probabilities. All probabilities must add up to 1.
Cumulative probability: Cumulative probability of a value = Sum of probabilities up to and including that value; The last cumulative probability must be 1.00. If not, recheck your addition.
Number of random numbers per value: Count of random numbers = Probability × 100 (for 2-digit numbers); Use × 1,000 if probabilities have three decimals and 3-digit random numbers are given.
Random number range (2-digit): Start = previous cumulative × 100 (first range starts at 00); End = own cumulative × 100 − 1; Example: cumulative 0.35 to 0.60 gives range 35 to 59. Last range ends at 99.
Expected value check: Expected value = Σ (value × probability); Use it to check whether the simulated average is reasonable. With few trials, it will not match exactly.
Probability from frequency: Probability = Frequency of outcome ÷ Total frequency; Use this when the question gives counts instead of probabilities.
Cumulative probability: Cumulative probability = Sum of probabilities up to and including that outcome; The last value must be 1.00. If not, recheck.
Random number range (two-digit): Range runs from (previous cumulative × 100) to (current cumulative × 100 − 1); Example: cumulative 0.20 gives 00-19; cumulative 0.50 gives 20-49. Use 0-9 for one-digit and 000-999 for three-digit numbers.
Closing stock: Closing stock = Opening stock + Receipts − Demand (not below zero); If demand is more than stock available, the excess is shortage. State whether it is lost or backordered, as the question directs.
Queue timings: Arrival time = Previous arrival + Inter-arrival time; Service start = Higher of arrival time and previous service end; Service end = Start + Service time; Use these three lines for every customer.
Waiting and idle time: Customer waiting time = Service start − Arrival time; Server idle time = Service start − Previous service end (when positive); Time in system = Service end − Arrival time.
Expected value (check): Expected value = Σ (outcome × probability); Use it to compare your short simulation with the theoretical average.
Probability to cumulative probability: Cumulative probability = running total of the probabilities; Build this column first for every variable.
Random number range: Range = (previous cumulative % + 1) to current cumulative %; With two-digit random numbers, 00 to 99 covers 100 numbers. A 0.30 probability gets 30 numbers, e.g. 00-29 or 01-30 depending on the method; follow the question's convention.
Profit per trial: Profit = (Selling price − Variable cost per unit) × Units sold − Fixed cost; Adjust if the question gives total cost or other items.
Simulated average profit: Average profit = Σ (profit of each trial) ÷ Number of trials; Divide by the number of trials actually run.
Exact expected value (for comparison): E(X) = Σ (value × probability); Simulation result is an estimate of this and usually differs slightly.

Quick revision

Simulation imitates a real system with a model to study outcomes under uncertainty.
Monte Carlo simulation draws values from a probability distribution using random numbers.
Steps: define the problem, set objectives, build the model, collect data, assign random numbers, run trials, analyse and decide.
Cumulative probability is the running total of probabilities and ends at 1.
Random number ranges follow the cumulative probabilities, so 2-digit numbers cover 00 to 99.
A probability of 0.25 takes 25 random numbers, for example 00 to 24.
Use the random numbers given in the question, in the order given, one for each trial.
Simulation gives an estimate, not an exact or optimal answer, and more trials improve reliability.
In inventory problems, track opening stock, demand, closing stock, shortage and cost for each period.
In queuing problems, waiting time is service start minus arrival, and idle time is the gap before the next customer begins service.
Average the results over all trials before comparing alternatives.
End every decision problem with a clear recommendation and the figure behind it.

Common mistakes

Saying simulation gives the optimal solution Fix: Write that simulation evaluates alternatives and shows likely outcomes; it does not guarantee the best answer.
Calling every simulation Monte Carlo Fix: Monte Carlo is the stochastic type that samples random numbers. A model with fixed inputs is deterministic.
Listing the steps in the wrong order, such as running trials before assigning random number ranges. Fix: Remember the chain: problem, variables, model, distributions, ranges, run, validate, decide. Each step needs the output of the one before.
Overlapping or gapped random number ranges, for example 0–20 and 20–50. Fix: Start each range one above the previous end. With 0.20 and 0.30 use 00–19 and 20–49.
Starting a range at the cumulative probability figure of the same row, or ending it one too high, so ranges overlap. Fix: Start at the previous cumulative figure and end at own cumulative minus 1. First range starts at 00; last ends at 99.
Using the probabilities directly as random number ranges without cumulating. Fix: Always build the cumulative column first. Ranges come only from cumulative values.
Wrong random number ranges, such as 00-20 and 20-49 overlapping. Fix: Upper limit = cumulative × 100 − 1. Check that the last range ends at 99.
Using the same random number for two variables or skipping numbers. Fix: Tick each random number as you use it. If the question gives separate lists for demand and lead time, use each list only for its own variable.
Overlapping or incomplete random number ranges Fix: Always write the cumulative column first. Each range starts right after the previous one ends. The last range must end at 99 (or 00).
Using random numbers in the wrong order or for the wrong variable Fix: Fix the variable order from the question (e.g. sales first, then price, then cost) and mark each random number as you use it.

Exam tips

Theory questions often ask for meaning, advantages and limitations together; give each its own short heading or bullet group.
In cases, tie the use of simulation to the uncertainty in the scenario rather than quoting a generic list.
If asked to compare with optimisation, state the key point: simulation describes outcomes, optimisation prescribes the best choice.
When allocating random numbers, check that ranges cover all numbers without overlap.
Write a concluding line telling the manager what the simulation outcome suggests.
Theory questions on this topic usually ask for the steps, so write them in order with one line of explanation each.
In numerical questions, show the table of probabilities, cumulative probabilities and ranges. Marks are often given for it even if the final figure goes wrong.
Always state the number of random digits you use (two digits for whole-percentage probabilities).