Lecture 19: Maximum Flow Problem

General Description

For a network modeled as a directed graph \(G=(N,A)\), with \(N\) and \(A\) representing the set of nodes and arcs, respectively, the objective of a Maximum Flow Problem is to find the maximum amount of flow that can be transported this network from a source node \(r\) to a sink node \(s\), given that each arc \((i,j) \in A\) has a capacity \(q_{ij}\).

Objective:

\[ \max \ x_{sr} \]

Subject to:

\[\begin{split} \begin{aligned} \sum_{j \in H(r)} x_{rj} & = x_{sr} \\ \sum_{i \in T(s)} x_{is} & = x_{sr} \\ \sum_{i \in T(j)} x_{ij} & = \sum_{k \in H(j)} x_{jk} & \ \forall \ j \in N \\ x_{ij} & \in [0,q_{ij}] & \ \forall \ (i,j) \in A \end{aligned} \end{split}\]

Here, \(T(i)\) represents the set of predecessor (tail) nodes of node \(i\), while \(H(i)\) represents the set of successor (head) nodes, defined as follows,

\[\begin{split} \begin{aligned} T(i) & = \{j; \ (j,i) \in A\} \\ H(i) & = \{j; \ (i,j) \in A\} \end{aligned} \end{split}\]

Note, \(x_{ij}\) represents the amount of flow from node \(i\) to node \(j\).

Example

The city of El Dorado is planning to optimize the passenger flow on its transportation network during peak hours. The city has a complex transportation network connecting residential areas, office districts, and commercial hubs. The goal is to maximize the passenger flow between a major residential area and a key office district while considering the network capacity.

Figure 1. El Dorado City Network

From	To	Capacity (passengers/hour)
1	2	700
1	3	300
2	3	100
2	5	600
3	4	800
4	5	200
4	6	800
5	6	200

1. Formulate a linear optimisation model for this problem.

   Objective:

   \[ \max \ x_{61} \]

   Subject to:

   \[\begin{split} \begin{aligned} x_{61} & = x_{12} + x_{13} \\ x_{12} & = x_{23} + x_{25} \\ x_{13} + x_{23} & = x_{34} \\ x_{34} & = x_{45} + x_{46} \\ x_{25} + x_{45} & = x_{56} \\ x_{46} + x_{56} & = x_{61} \\ x_{12} & \in [0, 700] \\ x_{13} & \in [0, 300] \\ x_{23} & \in [0, 100] \\ x_{25} & \in [0, 600] \\ x_{34} & \in [0, 800] \\ x_{45} & \in [0, 200] \\ x_{46} & \in [0, 800] \\ x_{56} & \in [0, 200] \end{aligned} \end{split}\]

2. Solve the above linear optimisation model using a spreadsheet to find the optimal solution.

   From	To	Flow
   1	2	300
   1	3	300
   2	3	100
   2	5	200
   3	4	400
   4	5	0
   4	6	400
   5	6	200

   The overall network flow is 600 passengers/hour.

3. As part of future proofing, El Dorado city wants to upgrade its network. Which streets should it invest in?

   Streets 1-3, 2-3, and 5-6 could render improvements in network flow since these streets are capacitated.

4. How does an investment increasing capacity of these streets by 100 passengers/hour impact the overall network flow?

   Naturally, in each of these cases, the overall network flow increases by 100, to 700 passengers/hour.

Instead of augmenting capacity to these arcs with additional lanes, El Dorado city added a street connecting node 5 with node 3, having a capacity of 300 passengers/hour.

Figure 2. Updated El Dorado City Network

5. Reformulate the linear optimisation problem.

   Objective:

   \[ \max \ x_{61} \]

   Subject to:

   \[\begin{split} \begin{aligned} x_{61} & = x_{12} + x_{13} \\ x_{12} & = x_{23} + x_{25} \\ x_{13} + x_{23} + x_{53} & = x_{34} \\ x_{34} & = x_{45} + x_{46} \\ x_{25} + x_{45} & = x_{53} + x_{56} \\ x_{46} + x_{56} & = x_{61} \\ x_{12} & \in [0, 700] \\ x_{13} & \in [0, 300] \\ x_{23} & \in [0, 100] \\ x_{25} & \in [0, 600] \\ x_{34} & \in [0, 800] \\ x_{45} & \in [0, 200] \\ x_{46} & \in [0, 800] \\ x_{53} & \in [0, 300] \\ x_{56} & \in [0, 200] \end{aligned} \end{split}\]

6. Solve the above linear optimisation model using a spreadsheet to find the optimal solution.

   From	To	Flow
   1	2	700
   1	3	300
   2	3	100
   2	5	600
   3	4	800
   4	5	0
   4	6	800
   5	3	300
   5	6	200

   The overall network flow increases by 300, to 9000 passengers/hour.