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Let's break down and solve this problem in parts:

### Part (a): What is the current capacity of the entire system?

#### Step 1: Identify the bottleneck in each path.

- **Path 1**: Steps 1 → 2 → 3 → 7 → 8
  - Step 1: 11 units/hour
  - Step 2: 10 units/hour
  - Step 3: 19 units/hour
  - Step 7: 34 units/hour
  - Step 8: 34 units/hour
  - **Bottleneck in Path 1**: 10 units/hour (since Step 2 is the slowest step).

- **Path 2**: Steps 4 → 5 → 6 → 7 → 8
  - Step 4: 12 units/hour
  - Step 5: 11 units/hour
  - Step 6: 7 units/hour
  - Step 7: 34 units/hour
  - Step 8: 34 units/hour
  - **Bottleneck in Path 2**: 7 units/hour (since Step 6 is the slowest step).

#### Step 2: The overall system capacity is determined by the sum of the capacities of the two parallel paths, taking into account their respective bottlenecks.

- **Path 1 Capacity**: 10 units/hour
- **Path 2 Capacity**: 7 units/hour

Since the two paths are parallel, the overall capacity is the sum of these bottlenecks:
\[
\text{Overall capacity} = 10 \, \text{units/hour} + 7 \, \text{units/hour} = 17 \, \text{units/hour}.
\]

### Part (b): If you could increase the capacity of only one operation, which one would you choose?

Since Path 2 has the lower capacity (7 units/hour), improving the bottleneck in **Path 2** would be the best choice to improve the system's overall capacity. 

- The bottleneck in Path 2 is **Step 6** (7 units/hour).
- If we could increase the capacity of Step 6, we would want it to match the capacity of the next limiting step, which is **Step 5** (11 units/hour).

Thus, if we improve Step 6's capacity to **11 units/hour**, the new system capacities would be:
- **Path 1**: 10 units/hour (unchanged)
- **Path 2**: 11 units/hour (new bottleneck after improvement).

The new overall system capacity would be:
\[
\text{New overall capacity} = 10 \, \text{units/hour} + 11 \, \text{units/hour} = 21 \, \text{units/hour}.
\]

### Final Answers:
- **Current capacity**: 17 units/hour
- **Operation to improve**: Step 6
- **How much additional capacity to strive for**: 4 units/hour (to increase from 7 units/hour to 11 units/hour)
- **Resulting capacity of the process**: 21 units/hour

---

Let me know if you'd like further clarification on any part of the problem! 

Here are some follow-up questions for practice:

1. How would the system capacity change if Step 2's capacity were increased to 15 units/hour?
2. What if Step 7 became the bottleneck with a capacity of 20 units/hour? What would be the new system capacity?
3. If Step 4's capacity was reduced to 10 units/hour, how would the overall capacity be affected?
4. What would happen if both Step 6 and Step 2 were improved by 5 units/hour each?
5. If the two parallel paths had different demand rates, how would you adjust the capacities to balance them?

**Tip**: When dealing with parallel systems, always focus on improving the bottleneck first to maximize overall capacity.

The diagram describes a service process where customers go through either of two parallel three-step processes and then merge into a single line for two final steps. Capacities of each step are shown on the diagram. a. What is the current capacity of the entire system? b. If you could increase the capacity of only one operation through process improvement efforts, which operation would you select, how much additional capacity would you strive for, and what would the resulting capacity of the process be?

Learn how to calculate the overall capacity of a service process involving parallel paths with bottlenecks. Explore ways to improve bottlenecks to increase the overall system capacity, using detailed steps. This problem is suitable for students in operations management.

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