CMA Intermediate · Operations Management and Strategic Management

Simulation and Line Balancing: formula sheet

Full chapter guide

Key formulas

Cumulative probability: Cumulative probability = running total of the probabilities; Used in Monte Carlo simulation to build the table that links random numbers to values.
Random number range: Range for a value = (previous cumulative % + 1) to (its cumulative %); With two-digit random numbers, 00 is treated as 100 when the cumulative reaches 100. Always state your convention.
Expected value check: Expected value = Σ (value × probability); Compare the simulated average with this to judge how close the simulation is.
Simulated average: Average = total of simulated outcomes ÷ number of runs; A small number of runs gives only a rough estimate.
Probability from frequency: Probability = Frequency of outcome ÷ Total frequency; Use when the question gives counts or days instead of probabilities. Probabilities must add up to 1.
Cumulative probability: Cumulative probability of an outcome = Sum of probabilities up to and including that outcome; The last value must equal 1.00. Check this before moving on.
Random number range (two-digit): Lower limit = previous cumulative probability × 100; Upper limit = (cumulative probability × 100) − 1; The first range starts at 00. Example: cumulative 0.20 gives 00-19; next cumulative 0.50 gives 20-49.
Expected value (for comparison): Expected value = Σ (outcome × probability); Use it to check that the simulated average is close to the theoretical average.
Simulated average: Average = Total of simulated values ÷ Number of trials; Divide by the number of trials you actually ran.
Cumulative probability: Cumulative probability = sum of probabilities up to and including that outcome; Build this column first. The last value must be 1.00.
Random number range (two-digit): Range for an outcome = previous cumulative × 100 up to (current cumulative × 100) − 1; Example: cumulative 0.20 to 0.50 gives 20-49. The first range starts at 00, the last ends at 99.
Closing stock: Closing stock = Opening stock + Receipts − Demand (cannot go below zero); If demand is more than stock available, the extra is a shortage (lost sale or backorder, as the question states).
Average stock: Average stock = Σ closing stock of all days ÷ number of days; Use the basis the question gives (closing, or average of opening and closing).
Arrival time: Arrival time of a customer = arrival time of previous customer + inter-arrival time; The first customer usually arrives at time 0 unless told otherwise.
Service start and waiting time: Start = higher of (arrival time, previous customer's service end); Waiting time = Start − Arrival time; Service end = Start + Service time.
Average waiting time: Average waiting time = Σ waiting times ÷ number of customers; Divide by all customers, including those who did not wait.
Server idle time: Idle time = Start of service − previous service end (when positive); Add idle gaps across all customers.
Cycle time: Cycle time (C) = Available production time per period ÷ Required output per period; Keep time and output in the same period, such as minutes per day and units per day. Convert hours to minutes first if task times are in minutes.
Theoretical minimum number of stations: N(min) = Σt ÷ C, rounded up to the next whole number; Σt is the sum of all task times. Always round up, never down, even if the decimal is small, such as 3.1 becomes 4.
Maximum output rate from a given cycle time: Output per period = Available time per period ÷ Cycle time; This is the reverse of the cycle time formula. Use it when the cycle time is given.
Minimum possible cycle time: Minimum cycle time ≥ the longest single task time; A task cannot be split across stations in the basic method, so the cycle time cannot be shorter than the longest task.
Maximum possible cycle time: Maximum cycle time ≤ Σt; This happens when all tasks are done at one station.
Idle time at a station: Idle time = Cycle time − Station time; Station time is the sum of times of tasks assigned to that station.
Cycle time: Cycle time = Available production time per period ÷ Required output per period; Use the same time unit for both. Subtract breaks before computing available time.
Takt time: Takt time = Available time ÷ Customer demand; The pace set by demand. Cycle time should be less than or equal to takt time.
Theoretical minimum stations: N(min) = Σ task times ÷ Cycle time, rounded up to the next whole number; Always round up, never down.
Positional weight: RPW of a task = its own time + times of all tasks that follow it; Include every successor, direct and indirect, once each.
Station idle time: Idle time of a station = Cycle time − Sum of task times at that station; Total idle time = N × Cycle time − Σ task times.
Efficiency: Efficiency (%) = Σ task times ÷ (N × Cycle time) × 100; N is the actual number of stations used.
Balance delay: Balance delay (%) = 100 − Efficiency = Total idle time ÷ (N × Cycle time) × 100; Lower is better.

Quick revision

Simulation imitates a real system using a model, so you can test options without real risk or cost.
Monte Carlo simulation uses random numbers matched to probability distributions.
Steps: list the probabilities, find cumulative probabilities, assign random-number ranges, then pick values using the given random numbers.
Random-number ranges must cover the full set without gaps or overlaps.
Simulation gives an estimate for the trials run, not an exact or optimal answer.
In inventory simulation, closing stock = opening stock + receipts − demand, and shortages need a stated treatment.
In queuing simulation, waiting time = service start time − arrival time.
Line balancing assigns tasks to workstations so that workloads are even.
Precedence constraints decide which tasks must be done before others.
Cycle time is the time available at each station between successive units.
Line efficiency = total task time ÷ (number of stations × cycle time) × 100.
Idle time per cycle = (number of stations × cycle time) − total task time.

Common mistakes

Saying simulation always gives the optimal solution. Fix: Write that simulation evaluates chosen alternatives and gives estimates, not a guaranteed best answer.
Mixing up deterministic and stochastic models. Fix: Remember: deterministic means fixed inputs and the same result each run; stochastic means random inputs and varying results.
Starting the first random number range at 01 instead of 00 Fix: With two-digit numbers, 00 to 99 gives 100 numbers. Start at 00 and end at 99.
Using probability instead of cumulative probability to set the ranges Fix: Always build the cumulative column first. The range of each outcome ends one below its cumulative value times 100.
Wrong random number ranges, such as 00-20, 20-50 with overlap Fix: Start each range at the previous cumulative × 100 and end at current cumulative × 100 − 1. The last range always ends at 99.
Using random numbers out of order or for the wrong variable Fix: Use the numbers exactly as listed, one per trial. Label the arrival series and the service series separately.
Rounding the minimum number of stations down or to the nearest number Fix: Always round up. You cannot have 0.2 of a station, and 4 stations would not give enough capacity.
Mixing time units when finding cycle time Fix: Convert everything to minutes before dividing. Write the unit next to each number.
Rounding the minimum number of stations down Fix: Always round up. Three stations cannot hold 3.4 cycles of work.
Assigning a task before its predecessor Fix: Before each assignment, check that all predecessors are already placed. Only eligible tasks may be chosen.

Exam tips

Questions are mostly short notes or 'advantages and limitations', so keep balanced bullet lists.
When a numerical question appears, it is usually Monte Carlo; practise building cumulative probability tables.
Match the application to the case given in the question rather than listing generic uses.
In MCQs, watch for traps such as 'simulation guarantees the optimal solution' and mark them false.
Write a one-line definition first; it earns marks quickly in written answers.
Always show the three-column table of probability, cumulative probability and random number range. It carries step marks even if later arithmetic goes wrong.
Read how many digits the random numbers have. Two-digit numbers need ranges of 00-99. Three-digit numbers need 000-999.
Use the random numbers strictly in the given order, and say so in one line.