- Identify all 6 possible routes for the circuit visiting Tel Aviv, Holon, and Ra'anana starting and ending in Jerusalem.
- Calculate the total cost for each route by summing the individual travel costs between consecutive cities.
- Compare the total costs: \(998, 840, 778, 960, 928, 932\).
- Determine the minimum cost to be \(\boxed{778 \text{ NIS}}\).
Explanation
- Problem Analysis and Strategy
The task is to find the minimum travel cost to visit three cities—Tel Aviv, Holon, and Ra'anana—starting and ending at Jerusalem, visiting each city exactly once. This is a classic Traveling Salesperson Problem (TSP) with four cities: Jerusalem (J), Tel Aviv (T), Holon (H), and Ra'anana (R). There are \(3! = 6\) possible routes for this journey, starting and ending at Jerusalem. Each route follows the structure \(J \rightarrow A \rightarrow B \rightarrow C \rightarrow J\), where \(A, B, C\) are permutations of \(T, H, R\).
- Calculating Total Costs for All Possible Routes
To determine the most cost-effective path, we evaluate the six possible routes using the provided cost data:
- Route 1: \(J \rightarrow T \rightarrow H \rightarrow R \rightarrow J\)
Cost: \(269 (J-T) + 263 (T-H) + 290 (H-R) + 176 (R-J) = \boxed{998 \text{ NIS}}\)
- Route 2: \(J \rightarrow T \rightarrow R \rightarrow H \rightarrow J\)
Cost: \(269 (J-T) + 203 (T-R) + 218 (R-H) + 150 (H-J) = \boxed{840 \text{ NIS}}\)
- Route 3: \(J \rightarrow H \rightarrow T \rightarrow R \rightarrow J\)
Cost: \(201 (J-H) + 198 (H-T) + 203 (T-R) + 176 (R-J) = \boxed{778 \text{ NIS}}\)
- Route 4: \(J \rightarrow H \rightarrow R \rightarrow T \rightarrow J\)
Cost: \(201 (J-H) + 290 (H-R) + 234 (R-T) + 235 (T-J) = \boxed{960 \text{ NIS}}\)
- Route 5: \(J \rightarrow R \rightarrow T \rightarrow H \rightarrow J\)
Cost: \(281 (J-R) + 234 (R-T) + 263 (T-H) + 150 (H-J) = \boxed{928 \text{ NIS}}\)
- Route 6: \(J \rightarrow R \rightarrow H \rightarrow T \rightarrow J\)
Cost: \(281 (J-R) + 218 (R-H) + 198 (H-T) + 235 (T-J) = \boxed{932 \text{ NIS}}\)
- Identifying the Minimum Cost Route
Comparing the total costs calculated for all possible routes, we have the following values in NIS: \(998, 840, 778, 960, 928, 932\). The smallest cost among these is \(778 \text{ NIS}\). Therefore, the route \(J \rightarrow H \rightarrow T \rightarrow R \rightarrow J\) is the most efficient path for the trip. The minimum total cost is \(\boxed{778 \text{ NIS}}\).
Examples
This problem is a practical application of the Traveling Salesperson Problem (TSP). In logistics and delivery services, companies often need to visit multiple locations while minimizing fuel consumption and time. For example, a delivery driver in a city needs to pick up parcels from different neighborhood hubs before returning to their base. By calculating the total travel costs for each possible sequence, the driver can determine the most efficient route, saving both time and fuel, much like calculating the minimum cost path between Jerusalem, Tel Aviv, Holon, and Ra'anana.