Questions: A jetliner transfers passengers among the cities of Rome, Frankfort, Amsterdam, and London. The jetliner wants to maximize the number of passengers en route from Frankfort to Rome. Capacities: From To Rome Frankfort Amsterdam London Rome 0 250 0 400 Frankfort MAX 0 315 0 Amsterdam 200 0 0 100 London 0 220 0 0 What is the net flow for Frankfort? (XFR+XFA)-(XFR+XFA+XFL)>0 (XFR+XFA)+(XFR+XFA+XFL)=0 (XFR+XFA)-(XFR+XFA+XFL)<0 (XFR+XFA)=(XFR+XFA+XFL)

A jetliner transfers passengers among the cities of Rome, Frankfort, Amsterdam, and London. The jetliner wants to maximize the number of passengers en route from Frankfort to Rome. Capacities: From To Rome Frankfort Amsterdam London Rome 0 250 0 400 Frankfort MAX 0 315 0 Amsterdam 200 0 0 100 London 0 220 0 0 What is the net flow for Frankfort? (XFR+XFA)-(XFR+XFA+XFL)>0 (XFR+XFA)+(XFR+XFA+XFL)=0 (XFR+XFA)-(XFR+XFA+XFL)<0 (XFR+XFA)=(XFR+XFA+XFL)
Transcript text: A jetliner transfers passengers among the cities of Rome, Frankfort, Amsterdam, and London. The jetliner wants to maximize the number of passengers enroute from Frankfort to Rome. Capacities: From To Rome Frankfort Amsterdam London Rome 0 250 0 400 Frankfort MAX 0 315 0 Amsterdam 200 0 0 100 London 0 220 0 0 Question 2 What is the net flow for Frankfort? $\left(X_{F R}+X_{F A}\right)-\left(X_{F R}+X_{F A}+X_{F L}\right)>0$ $\left(X_{F R}+X_{F A}\right)+\left(X_{F R}+X_{F A}+X_{F L}\right)=0$ $\left(X_{F R}+X_{F A}\right)-\left(X_{F R}+X_{F A}+X_{F L}\right)<0$ $\left(X_{F R}+X_{F A}\right)=\left(X_{F R}+X_{F A}+X_{F L}\right)$
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Solution

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Solution Steps

To solve the problem of maximizing the number of passengers enroute from Frankfort to Rome, we need to consider the capacity constraints given in the table. The goal is to determine the maximum flow from Frankfort to Rome while respecting the capacities of the other routes. For the second question, we need to calculate the net flow for Frankfort by considering the inflow and outflow of passengers.

Step 1: Understanding the Problem

We are given a table of capacities for passenger transfers between four cities: Rome, Frankfort, Amsterdam, and London. The goal is to determine the net flow for Frankfort based on the given options.

Step 2: Analyzing the Flow

The net flow for Frankfort can be calculated by considering the passengers arriving at and departing from Frankfort. The flow into Frankfort is from Rome and London, while the flow out of Frankfort is to Rome and Amsterdam.

  • Flow into Frankfort:

    • From Rome: 250 passengers
    • From London: 220 passengers

  • Flow out of Frankfort:

    • To Rome: MAX (we are trying to maximize this)
    • To Amsterdam: 315 passengers
Step 3: Calculating the Net Flow

The net flow for Frankfort is calculated as the difference between the total inflow and the total outflow:

\[ \text{Net Flow} = (\text{Inflow from Rome} + \text{Inflow from London}) - (\text{Outflow to Rome} + \text{Outflow to Amsterdam}) \]

Substituting the given values:

\[ \text{Net Flow} = (250 + 220) - (\text{MAX} + 315) \]

Step 4: Evaluating the Options

We need to determine which of the given options correctly describes the net flow for Frankfort:

  1. \((X_{FR} + X_{FA}) - (X_{FR} + X_{FA} + X_{FL}) > 0\)
  2. \((X_{FR} + X_{FA}) + (X_{FR} + X_{FA} + X_{FL}) = 0\)
  3. \((X_{FR} + X_{FA}) - (X_{FR} + X_{FA} + X_{FL}) < 0\)
  4. \((X_{FR} + X_{FA}) = (X_{FR} + X_{FA} + X_{FL})\)

Given the net flow calculation, the inflow is less than the outflow due to the MAX term, which implies that the net flow is negative. Therefore, the correct option is:

\[ (X_{FR} + X_{FA}) - (X_{FR} + X_{FA} + X_{FL}) < 0 \]

Final Answer

\[ \boxed{(X_{FR} + X_{FA}) - (X_{FR} + X_{FA} + X_{FL}) < 0} \]

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