manufacturing company has a certain piece of equipment that is inspected at the end of each day and classified as just overhauled good, pair or inoperative. If the item is inoperative, it is overhauled, a procedure that takes one day. Let us denote the four classifications as state 1, 2, 3 and 4 respectively. Assume that the working condition of the equipment follows a Markov processes the following transition matrix: f it costs Rs. 125 to overhaul a machine (including lost time), on the average and Rs. 75 in production? a machine is found inoperative. Using steady state probabilities, compute the expected per day cost of maintenance.

To solve this problem, we'll apply the concepts of Markov processes and steady-state probabilities to compute the expected per day cost of maintenance. Here’s a step-by-step approach:

### Step 1: Define the Markov Process

The states of the equipment are as follows:
- **State 1**: Just overhauled
- **State 2**: Good
- **State 3**: Pair
- **State 4**: Inoperative

Let the transition matrix be denoted as \( P \). The matrix \( P \) captures the probabilities of transitioning from one state to another from one day to the next. It is structured as:

\[
P = \begin{pmatrix}
P_{11} & P_{12} & P_{13} & P_{14} \\
P_{21} & P_{22} & P_{23} & P_{24} \\
P_{31} & P_{32} & P_{33} & P_{34} \\
P_{41} & P_{42} & P_{43} & P_{44}
\end{pmatrix}
\]

### Step 2: Steady-State Probabilities

Steady-state probabilities \( \pi_i \) represent the long-term probabilities that the equipment will be in state \( i \).

The steady-state probabilities satisfy the following equation:

\[
\pi \cdot P = \pi
\]

This means:

\[
\pi_1 P_{11} + \pi_2 P_{21} + \pi_3 P_{31} + \pi_4 P_{41} = \pi_1
\]
\[
\pi_1 P_{12} + \pi_2 P_{22} + \pi_3 P_{32} + \pi_4 P_{42} = \pi_2
\]
\[
\pi_1 P_{13} + \pi_2 P_{23} + \pi_3 P_{33} + \pi_4 P_{43} = \pi_3
\]
\[
\pi_1 P_{14} + \pi_2 P_{24} + \pi_3 P_{34} + \pi_4 P_{44} = \pi_4
\]

Additionally, the probabilities must sum to 1:

\[
\pi_1 + \pi_2 + \pi_3 + \pi_4 = 1
\]

### Step 3: Cost Calculation

We are given:
- The cost to overhaul a machine is Rs. 125.
- The cost incurred in production when the machine is inoperative is Rs. 75.

The expected per day cost of maintenance is computed using the steady-state probability of being in the "inoperative" state (State 4) and multiplying it by the respective costs.

Let \( \pi_4 \) represent the steady-state probability of being in state 4 (inoperative):

The expected per day cost \( C \) is:

\[
C = \pi_4 \times 125 + \sum_{i \neq 4} \pi_i \times 75
\]

### Step 4: Solving for Steady-State Probabilities

To solve for the steady-state probabilities \( \pi_1, \pi_2, \pi_3, \pi_4 \), we need the transition matrix \( P \). Assuming the matrix is given (you didn't specify it), we solve the system of linear equations formed by the steady-state equation and the normalization condition.

Finally, plug the steady-state probabilities into the cost equation to get the expected per day cost.

---

### Example with Hypothetical Transition Matrix

If you provide the transition matrix, I can calculate the steady-state probabilities and the expected cost precisely. Otherwise, the above method describes the general process.

Would you like me to assist further in creating or solving for a specific matrix?

### Further Related Questions:
1. What happens if the cost to overhaul increases?
2. How does the transition matrix affect the steady-state probabilities?
3. What is the impact on costs if the equipment transitions more frequently to an inoperative state?
4. How to determine the most cost-effective maintenance schedule?
5. How can we reduce the transition to the inoperative state?
6. What if there are additional states, such as "repair pending"?
7. How does the daily production cost affect the overall expected cost?
8. What strategies could be implemented to minimize these costs?

### Tip:
Always verify the transition matrix to ensure it reflects realistic probabilities of transitioning between equipment states, as the accuracy of steady-state calculations directly depends on it.

Learn how to compute the expected per day cost of maintenance for a manufacturing company's equipment using Markov processes and steady-state probabilities. This problem involves a transition matrix representing equipment states and costs associated with overhauling and production downtime. Suitable for advanced learners and professionals in operations management and industrial engineering.

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