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Markov Chain Tyre Replacement Policy Problem: Expected Cost Comparison and Optimal Maintenance Strategy
Published on October 24, 2026
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A fleet maintenance problem modeling tyre condition transitions between classes A, B, C, and D as a Markov chain with monthly inspections. The solution compares the expected cost of waiting for road failures (USD 1000 each) against replacing class C tyres at inspection (USD 800 each) to find the most economical policy.
Question
Act as an expert maths tutor. Before you give me the final answer, please break it down into these four steps: 1. *Core Concept:* Explain the underlying principle 2. *Formula Needed:* State the exact formula in clean, simple terms. 3. *Step-by-Step Working:* Show the calculation step-by-step, explaining why you add, subtract 4. *Final Answer:* Clearly bold the final figure/answer
Answer
1. Core Concept
The tyre condition changes from one monthly inspection to the next with fixed probabilities, so this is a Markov chain. Each tyre can only move toward failure. Every tyre must pass through class C before it fails, so the question reduces to comparing the cost of two ways to end a class C episode:
- Policy 1 (wait): the tyre stays in C until it fails on the road, costing USD 1000.
- Policy 2 (replace at inspection): the tyre is replaced in the shop as soon as it is found in class C, costing USD 800.
2. Formula Needed
Expected cost of waiting per class C tyre:
Cost of replacing at inspection:
Saving per tyre:
3. Step-by-Step Working
Step 1: Set up the transition probabilities (one monthly inspection).
- From A: stays A with 0.7, moves to B with 0.2, moves to C with 0.1 (the remaining probability, since the row must sum to 1).
- From B: stays B with 0.8, moves to C with 0.2.
- From C: stays C with 0.2, fails with 0.8 (failures are replaced on the road).
Step 2: Show that every tyre ends in failure through C.
Tyres never move from A or B directly to D. A tyre can only fail from class C. Since the chain is absorbed only by failure, a tyre in class C eventually fails with probability 1 if nothing is done. So:
Step 3: Compare the costs per class C episode.
Under Policy 1, every class C episode ends with a road failure costing 1000. The 20% of tyres that stay in C each month are not avoided cost. They fail later at the same USD 1000 price.
Under Policy 2, each class C episode ends with a shop replacement costing 800:
Saving per tyre:
Step 4: Evaluate the foreman’s argument.
The foreman notes that 20% of replaced class C tyres would have stayed in class C. Those tyres were not wasted. Under the wait policy they would have failed later and cost USD 1000, so the replacement cost is paid either way, just earlier and at a lower price. The USD 200 saving per tyre more than compensates for the few tyres replaced “unnecessarily.”
4. Final Answer
The most economical policy is to replace every class C tyre at each monthly inspection (at USD 800 each), saving USD 200 per tyre compared with waiting for road failures at USD 1000 each. The financial manager’s argument is correct.