Question 8 of 10: Monte Carlo Simulation — Machine Repair Workload
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — May 2019 — 17-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 175 marks across 10 questions and only 100 marks are required, so a candidate would normally answer a subset — all ten are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming & the simplex method (ch. 3–4), duality & sensitivity analysis (ch. 6), dynamic programming (ch. 11), network optimization & CPM/PERT project crashing (ch. 9–10), queueing theory incl. finite-source (machine-repair) models (ch. 17), decision analysis & the value of information (ch. 15–16), Markov chains (ch. 16), Monte Carlo simulation (ch. 20). Nahmias, Production and Operations Analysis — deterministic EOQ inventory models with and without planned shortages.
Question 8: Monte Carlo Simulation — Machine Repair Workload (20 marks)
Given data — repair-time and breakdown-count distributions
Repair time (hr)
1
2
3
Probability
0.30
0.30
0.40
Breakdowns/day
0
1
2
Probability
0.50
0.30
0.20
Given. Repair-time distribution and daily-breakdown-count distribution as tabulated; a 34-digit random-number string to drive a hand simulation.
Find. (a) A flowchart for the simulation procedure. (b) Simulate 2 days using the given random numbers; compare the simulated and theoretical average daily repair work.
Approach. Split the digit string into 2-digit random numbers (00–99), map each to a cumulative-probability interval, and for each day first draw the breakdown COUNT, then draw one repair-time number per breakdown that day.
Flowchart (part a): for each of the 2 days, draw one random number for breakdown count, then one further random number PER breakdown for its repair time.
Assign 2-digit random-number ranges (00–99) by cumulative probability:
RN assignment (00–99)
Breakdowns/day
0 → 00–49
1 → 50–79
2 → 80–99
Repair time (hr)
1 → 00–29
2 → 30–59
3 → 60–99
Splitting the given 34-digit string into 2-digit numbers: 13, 51, 60, 48, 66, 29, 61, 14, 28, 04, 22, 36, 66, 65, 43, 99, 75 (17 numbers — only as many as needed for 2 days are used below).
Day 1. Draw RN = 13 → falls in 00–49 → $b_1=0$ breakdowns. No repair-time draw needed.
$$\text{Day 1 work}=0\text{ hr}$$
Day 2. Draw RN = 51 → falls in 50–79 → $b_2=1$ breakdown. Draw one repair-time RN = 60 → falls in 60–99 → repair time = 3 hr.
$$\text{Day 2 work}=3\text{ hr}$$
(3 random numbers — 13, 51, 60 — fully determine both simulated days; the remaining 14 numbers in the list are unused.)
Simulated 2-day average vs. theoretical average ($E[\text{breakdowns/day}]\times E[\text{repair time}]$, since repair time per breakdown is independent of the day's breakdown count):
$$\text{sim. avg}=\frac{0+3}{2}=1.5\text{ hr/day}$$
$$E[\text{breakdowns}]=0(.5)+1(.3)+2(.2)=0.7;\quad E[\text{repair time}]=1(.3)+2(.3)+3(.4)=2.1$$
$$\text{theoretical avg}=0.7\times 2.1=\boxed{1.47\text{ hr/day}}$$
The 2-day simulated average (1.50 hr/day) is close to the theoretical long-run average (1.47 hr/day) — the small gap is ordinary sampling variability from using only 2 simulated days.
