VIPSolutionsGive step-by-step solution with explanation and final answer:LACAN [20 Sra RTI -_ SEE entifed 103primary pickup and drop locations in Downtown Co v. To minimise, travel time an improve customer service and the u Bes po , management would like the drivers to take the shortest % os le for these pickup and drop-off locations. E nia 12 a8 BN @ a @ EL EB . 3 E 2 3 Ee a BANG oe Fs = Be & = Re Ne E po=——_o, : ree eee NE a) Identify the shortest route from Node 1 to each other node by the diagram method. Note ¢ "those with single arrow are paths with one-way direction. Otherwise they arc 2-way streets. [1 marks] 3 b) Construct a Linear Programming model o solve the shortest oute from node 1 t0 10. Solve problem using Excel. ok [marks] aa pees snd TE —— de if need 7 A
Question:
Give step-by-step solution with explanation and final answer:
LACAN
[20 Sra RTI
-_ SEE
entifed 103primary pickup and drop locations in Downtown Co
v. To minimise, travel time an improve customer service and the u
Bes po , management would like the drivers to take the shortest
% os le for these pickup and drop-off locations. E
nia 12 a8
BN @ a @ EL
EB . 3 E
2 3 Ee
a BANG oe
Fs = Be & =
Re Ne E
po=——_o, :
ree eee NE
a) Identify the shortest route from Node 1 to each other node by the diagram method. Note ¢
"those with single arrow are paths with one-way direction. Otherwise they arc 2-way streets.
[1 marks] 3
b) Construct a Linear Programming model o solve the shortest oute from node 1 t0 10. Solve
problem using Excel. ok
[marks] aa
pees snd
TE —— de if need
7 A
This is AI generated Response.
Asked by: gvr
Created at: 2025-07-26 20:23:57
Asked by: gvr
Created at: 2025-07-26 20:23:57
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---
### **Part (a): Identify the shortest route from Node 1 to each other node by the diagram method**
We'll use **Dijkstra's Algorithm** to find the shortest path from Node 1 to all other nodes.
#### **Step 1: Initialization**
- Set distance to Node 1 as 0.
- Set distances to all other nodes as infinity.
- Mark all nodes as unvisited.
- Set the current node as Node 1.
#### **Step 2: Iterative Calculation**
| Node | Distance from 1 | Previous Node |
|------|-----------------|--------------|
| 1 | 0 | - |
| 2 | ∞ | |
| 3 | ∞ | |
| 4 | ∞ | |
| 5 | ∞ | |
| 6 | ∞ | |
| 7 | ∞ | |
| 8 | ∞ | |
| 9 | ∞ | |
| 10 | ∞ | |
##### **From Node 1:**
- 1 → 2: 8
- 1 → 3: 13
- 1 → 4: 14
- 1 → 5: 9
Update:
| Node | Distance | Prev |
|------|----------|------|
| 1 | 0 | - |
| 2 | 8 | 1 |
| 3 | 13 | 1 |
| 4 | 14 | 1 |
| 5 | 9 | 1 |
| others|∞ | |
Mark Node 1 as visited.
##### **Pick Node 2 (smallest unvisited, dist=8):**
- 2 → 3: 8 + 4 = 12
- 2 → 4: 8 + 3 = 11
- 2 → 7: 8 + 12 = 20
Update:
| Node | Distance | Prev |
|------|----------|------|
| 3 | 12 | 2 |
| 4 | 11 | 2 |
| 5 | 9 | 1 |
| 7 | 20 | 2 |
##### **Pick Node 5 (dist=9):**
- 5 → 4: 9 + 3 = 12 (current best for 4 is 11, so keep 11)
- 5 → 6: 9 + 11 = 20
- 5 → 9: 9 + 9 = 18
Update:
| Node | Distance | Prev |
|------|----------|------|
| 6 | 20 | 5 |
| 9 | 18 | 5 |
##### **Pick Node 4 (dist=11):**
- 4 → 3: 11 + 1 = 12 (already 12, no change)
- 4 → 6: 11 + 9 = 20 (already 20, no change)
##### **Pick Node 3 (dist=12):**
- 3 → 7: 12 + 9 = 21
- 3 → 6: 12 + 3 = 15 (update 6)
- 3 → 4: 12 + 1 = 13 (already 11, keep 11)
Update:
| Node | Distance | Prev |
|------|----------|------|
| 6 | 15 | 3 |
| 7 | 20 | 2 |
##### **Pick Node 6 (dist=15):**
- 6 → 7: 15 + 3 = 18 (update 7)
- 6 → 8: 15 + 1 = 16 (update 8)
Update:
| Node | Distance | Prev |
|------|----------|------|
| 7 | 18 | 6 |
| 8 | 16 | 6 |
##### **Pick Node 7 (dist=18):**
- 7 → 8: 18 + 1 = 19 (8 is 16, keep 16)
- 7 → 10: 18 + 6 = 24
##### **Pick Node 8 (dist=16):**
- 8 → 9: 16 + 4 = 20 (update 9)
- 8 → 10: 16 + 6 = 22 (update 10)
Update:
| Node | Distance | Prev |
|------|----------|------|
| 9 | 20 | 8 |
| 10 | 22 | 8 |
##### **Pick Node 9 (dist=20):**
- 9 → 10: 20 + 4 = 24 (already 22, keep 22)
##### **Pick Node 10 (dist=22):**
- No outgoing edges to unvisited nodes.
---
### **Final Shortest Distances and Paths from Node 1:**
| Node | Shortest Distance | Path |
|------|-------------------|-----------------|
| 1 | 0 | 1 |
| 2 | 8 | 1 → 2 |
| 3 | 12 | 1 → 2 → 3 |
| 4 | 11 | 1 → 2 → 4 |
| 5 | 9 | 1 → 5 |
| 6 | 15 | 1 → 2 → 3 → 6 |
| 7 | 18 | 1 → 2 → 3 → 6 → 7 |
| 8 | 16 | 1 → 2 → 3 → 6 → 8 |
| 9 | 20 | 1 → 2 → 3 → 6 → 8 → 9 |
| 10 | 22 | 1 → 2 → 3 → 6 → 8 → 10 |
---
## **Part (b): Linear Programming Model**
**Let \( x_{ij} \) be a decision variable for each edge, where:**
- \( x_{ij} = 1 \) if the path from node \( i \) to node \( j \) is used in the shortest path, \( 0 \) otherwise.
### **Objective Function:**
Minimize the total path length from node 1 to node 10:
\[
\text{Minimize } Z = \sum_{(i,j)} c_{ij} x_{ij}
\]
Where \( c_{ij} \) is the distance from node \( i \) to node \( j \).
### **Constraints:**
- **Flow Conservation:**
- For **start node (1):** \(\sum_{j} x_{1j} - \sum_{i} x_{i1} = 1\)
- For **end node (10):** \(\sum_{j} x_{10j} - \sum_{i} x_{i10} = -1\)
- For **all other nodes \( k \):** \(\sum_{j} x_{kj} - \sum_{i} x_{ik} = 0\)
- **Binary Variables:** \( x_{ij} \in \{0, 1\} \)
### **Set up in Excel:**
- List all possible edges with their costs.
- Create variables for each edge.
- Set up the objective function and constraints as above.
- Use Solver to minimize the total cost.
---
## **Summary Table for Shortest Paths (From Node 1):**
| To Node | Shortest Distance | Path |
|---------|-------------------|-----------------------------|
| 1 | 0 | 1 |
| 2 | 8 | 1 → 2 |
| 3 | 12 | 1 → 2 → 3 |
| 4 | 11 | 1 → 2 → 4 |
| 5 | 9 | 1 → 5 |
| 6 | 15 | 1 → 2 → 3 → 6 |
| 7 | 18 | 1 → 2 → 3 → 6 → 7 |
| 8 | 16 | 1 → 2 → 3 → 6 → 8 |
| 9 | 20 | 1 → 2 → 3 → 6 → 8 → 9 |
| 10 | 22 | 1 → 2 → 3 → 6 → 8 → 10 |
---
**If you need the actual Excel setup or further breakdown for the LP, let me know!**
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