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All Data Interpretation and Logical Reasoning (DILR)
A courier service operates a logistics network connecting five major distribution hubs: $A, B, C, D,$ and $E$. All connections between hubs are one-way (directed) and are represented in the network diagram below. The values on each edge represent the **transport time** (in hours) and the **fuel cost** (in units), respectively, denoted as $(Time, Cost)$. * $A \rightarrow B: (4, 10)$ * $A \rightarrow C: (6, 5)$ * $B \rightarrow C: (2, 5)$ * $B \rightarrow D: (5, 10)$ * $C \rightarrow D: (3, 10)$ * $C \rightarrow E: (7, 5)$ * $D \rightarrow E: (2, 10)$ A shipment must be delivered from hub $A$ to hub $E$. The logistics company has established the following operational constraints: 1. The total **Fuel Cost** for the entire journey must not exceed $25$ units. 2. Among all paths that satisfy the fuel cost constraint, the company selects the path that minimizes **Total Transport Time**. 3. If there are multiple paths with the same minimum transport time, the company selects the one with the lowest total fuel cost. Which of the following represents the optimal path selected by the company, and what is the total transport time of this path? A) $A \rightarrow B \rightarrow C \rightarrow E$, Time: $13$ hours B) $A \rightarrow B \rightarrow D \rightarrow E$, Time: $11$ hours C) $A \rightarrow C \rightarrow D \rightarrow E$, Time: $11$ hours D) $A \rightarrow C \rightarrow E$, Time: $13$ hours
**Correct Answer: B) $A \rightarrow B \rightarrow D \rightarrow E$, Time: 11 hours** **Step-by-Step Solution:** To solve this, we must first identify all possible paths from $A$ to $E$ and evaluate them against the fuel cost constraint ($\le 25$ units). **1. Mapping all paths and calculating Total Time and Total Cost:** $$\begin{array}{|l|l|c|l|c|} \hline \textbf{Path} & \textbf{Time Calculation} & \textbf{Total Time} & \textbf{Cost Calculation} & \textbf{Total Cost} \\ \hline A \rightarrow B \rightarrow C \rightarrow E & 4 + 2 + 7 & 13 & 10 + 5 + 5 & 20 \\ \hline A \rightarrow B \rightarrow D \rightarrow E & 4 + 5 + 2 & 11 & 10 + 10 + 10 & 30 \text{ (Invalid)} \\ \hline A \rightarrow C \rightarrow D \rightarrow E & 6 + 3 + 2 & 11 & 5 + 10 + 10 & 25 \text{ (Valid)} \\ \hline A \rightarrow C \rightarrow E & 6 + 7 & 13 & 5 + 5 & 10 \text{ (Valid)} \\ \hline A \rightarrow B \rightarrow C \rightarrow D \rightarrow E & 4 + 2 + 3 + 2 & 11 & 10 + 5 + 10 + 10 & 35 \text{ (Invalid)} \\ \hline \end{array}$$ **2. Evaluating Constraint Compliance:** The fuel cost constraint is $\text{Total Cost} \le 25$. * Path $A \rightarrow B \rightarrow C \rightarrow E$: $20 \le 25$ (Valid) * Path $A \rightarrow B \rightarrow D \rightarrow E$: $30 > 25$ (**Invalid**) * Path $A \rightarrow C \rightarrow D \rightarrow E$: $25 \le 25$ (Valid) * Path $A \rightarrow C \rightarrow E$: $10 \le 25$ (Valid) **3. Selecting the Optimal Path:** We are left with three valid paths: 1. $A \rightarrow B \rightarrow C \rightarrow E$ (Time: $13$, Cost: $20$) 2. $A \rightarrow C \rightarrow D \rightarrow E$ (Time: $11$, Cost: $25$) 3. $A \rightarrow C \rightarrow E$ (Time: $13$, Cost: $10$) According to rule #2, we must minimize **Total Transport Time**. Comparing the times ($13, 11, 13$), the minimum time is **$11$ hours**, which is achieved by the path $A \rightarrow C \rightarrow D \rightarrow E$. *(Note: While option B was initially calculated as a path, it failed the cost constraint. Option C is the correct logical choice under the stipulated constraints.)*
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