Question

A facility has a waste storage tank with a capacity of 40 cubic feet. Each week the tank produces either 0, 10, 20, or 30 cubic feet of waste with respective probabilities of 0.1, 0.4, 0.3, and 0.2. If the amount of waste produced in a week creates a situation where the tank would overflow, the amount exceeding the tank’s capacity can be removed at a cost of $3 per cubic foot. At the end of each week, a contracted service is available to remove waste. The service costs $40 for each visit plus $1 per cubic foot of waste removed. The facility manager decides to adopt a policy where, if the tank contains more than 20 cubic feet of waste, the contract service comes at the end of the week and removes all of the waste in the tank. Otherwise, the service does not come, and no waste is removed. Model the amount of waste in the tank as a Markov chain. Pay particular attention to when (at what point in the week) the amount of waste is measured or recorded

Answer 1

Since, each week the tank produces either 0, 10, 20, or 30 cubic feet of waste with respective probabilities of 0.1, 0.4, 0.3, and 0.2, and the the contract service comes at the end of the week and removes all of the waste in the tank if it exceeds 20 cubic fet, the amount of waste in the tank at the end of the week can be 0, 10, 20 cubic feet at the end of the week. We can model the amount of waste in the tank at the end of the week as a Markov chain with 3 states - S0, S10, S20 denoting the amount of waste in the tank as 0, 10, 20 cubic feet respectively.

From State S0, (Amount of waste in the tank = 0 cubic feet)

When 0 cubic feet of waste produced, Transition probability to State S0 = 0.1

When 10 cubic feet of waste produced, Transition probability to State S10 = 0.4

When 20 cubic feet of waste produced, Transition probability to State S20 = 0.3

When 30 cubic feet of waste produced, Transition probability to State S0 = 0.2 (In this case, the amount of waste becomes 30 cubic feet and contract service removes all of the waste in the tank)

Thus, Transition probability to State S0 = 0.1 + 0.2 = 0.3

From State S10, (Amount of waste in the tank = 10 cubic feet)

When 0 cubic feet of waste produced, Transition probability to State S10 = 0.1

When 10 cubic feet of waste produced, Transition probability to State S20 = 0.4

When 20 cubic feet of waste produced, Transition probability to State S0 = 0.3 (In this case, the amount of waste becomes 30 cubic feet and contract service removes all of the waste in the tank)

When 30 cubic feet of waste produced, Transition probability to State S0 = 0.2 (In this case, the amount of waste becomes 40 cubic feet and contract service removes all of the waste in the tank)

Thus, Transition probability to State S0 = 0.3 + 0.2 = 0.5

From State S20, (Amount of waste in the tank = 20 cubic feet)

When 0 cubic feet of waste produced, Transition probability to State S20 = 0.1

When 10 cubic feet of waste produced, Transition probability to State S0 = 0.4  (In this case, the amount of waste becomes 30 cubic feet and contract service removes all of the waste in the tank)

When 20 cubic feet of waste produced, Transition probability to State S0 = 0.3  (In this case, the amount of waste becomes 40 cubic feet and contract service removes all of the waste in the tank)

When 30 cubic feet of waste produced, Transition probability to State S0 = 0.2 (In this case, the amount of waste becomes more than 20 cubic feet and contract service removes all of the waste in the tank)

Thus, Transition probability to State S0 = 0.4 + 0.3 + 0.2 = 0.9

The Transition probability matrix is,

and the transition probability diagram is,