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.