# (Tim Lauck) Several bike paths connect The University of British Columbia to nearby residential…

(Tim Lauck) Several bike paths connect The University of British Columbia to nearby residential areas. These bike paths deteriorate because of heavy wear and harsh weather. As an illustration of the methodology in Sec. 1.4, we provide the following model to determine optimal maintenance schedules.

Assume that 200-m segments of the roadway are classified into four states on the basis of wear at annual inspection. State 3 denotes heavily worn, state 2 denotes moderate wear, state 1 slight wear, and state 0 denotes no wear. On the basis of these observations, three maintenance actions can be chosen; action 0 denotes routine maintenance, action 1 corresponds to applying a thin overcoat of paving material, and action 2 corresponds to applying a thick overcoat. The effect of these actions are described in terms of the following transition probabilities. They give the probability that a segment of the bike path is in a particular state next year given the specified maintenance action is chosen in the current state this year:

Assume that action 0 costs 0, action 1 costs \$1 per segment, and action 2 costs \$2 per segment.

a. Verify that this model is unichain.

b. Show that if p(0 10,2) = 1 instead of 0.95, that the model would be multichain and communicating.

c. Find a maintenance policy that minimizes long run average cost, subject to a constraint that no more than 10% of the segments can be in the heavily worn state.

d. Suppose that the maintenance department wishes to spend at most \$SO per segment per year. Find an optimal policy under this constraint and comment on its effect on the quality of the bike paths.

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