Network Flow Model Designing Assignment

Network flow model Requirements:

Scenario: Let’s consider the metro line in a city where there are different sports events taking place. People are very much eager to attend the entire sports event since these events are taking place at different time slots. There is metro line in the city which also connects the sports complex. This system helps the passengers in transportation from the stadiums to the city. So, after every sports event, there will be peak demand of people arriving at the metro stations for travelling to another sports complex. For handling a high transport demand, there must be a temporary change in the capacity of the transport system. Also, the interactions with other transport services as well as traffic demand must be considered. There must be proper management for handling the traffic flow which takes place after the end of the sports event. For handling such scenario, a tailored metro time schedule must be implemented. Also, the passengers generally proceed to the metro station when different lines cross. Also, depending on the sports event schedule, there is a high pedestrian flow which is expected after the event. There must be some controlling measures which must be taken within the station as well as to the station. Also, the model of action of any such measures can also be analyzed by mathematical formulation. The approach used here is the Mathematical Programming. This can also be defined as the selecting the best elements with regards to certain criteria from some alternatives which are available.

Let us consider there is one sports event which is taking place at a given place. And there are different metro stations in between for reaching the destination.

The approach used here for solving the problem is Mathematical Programming. All the number of users which are expected as well as the capacity of each metro station is illustrated in the diagram. The main purpose of the given problem is for balancing the capacity as well as the demand for the metro stations.

9000 3000 0




4 8000


9000, 8000, 3000 are the expected number of metro users.

10000 will be the capacity per hour for each line.

The lines represent the metro line segment

The boxes represent the stations with station numbers. The destination is the place where the sports event is taking place.

Between station 2 and station 3 the demand is 20000 but the capacity is 10000. This is what is needed to be managed.

This can be mathematically solved.

The mathematical equation with the objective function as well as constrains are as follows.

Xtij.|T| <= F

The main purpose is to minimize the waiting time at the metro stations with respect to:

The capacity of the metro stations as well as the demands fulfilment.

T1,t2 =4500,4500 t1,t2=1500,1500 0




4 T1,t2=4000,4000


One way by which it can be achieved is by giving time slots for the different capacity of people. And dividing the number of passengers travelling into two slots.

Now if we check the capacity per hour between metro station 2 and 3 matches the capacity.

Next model which can be considered here will be the Network Flow model.

Here all the metro stations are connected as shown in the figure below.





Here, Xij = the mini max time

Suppose 5 people want to go from 1 to 4:

X13 = 5 and X34 = 5

And, when 2 people want to go from 1 to 3

X13 = 5 + 2 = 7

X34 = 5

For 3: we have

(X13 + X23 + X43) – (X31 + X32 + X34) = 2

For 2(from A) : (X12 + X32 + X42) – (X21 + X23 + X24) = 3

Here we can conclude the node 2 will demand 3 units from the good A which will pass through node 1.

For 2( from B) : ( X12+ X32 + X42) – (X21 + X23+ X24) = -1

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Here we can conclude the node 2 will supply 1 unit from the good B and will pass from node 3.

Xijs,L= starting from station s by using the line l and flowing between i and j passenger

= ∑ ∑ ∑ Xijs,L

The first ∑ will represent the line connected to node 2

The second ∑ represent the stations connecting line l

The third ∑ all the neighbouring stations of j


12 he network Flow model













For the network Flow model.

The formula to be used here will be:

Xtijs,L= the passengers flow which is entering in the station s at period t by using the line l between the station j and i.

The parameters uses here will be lime lag.

Considering the time lag between station 1 and 6 = 10

And the time lag between 7 and 12 = 4

To calculate the number of people who enter the node 7 we have the following formulae.

This must be less than the capacity of station number 7 or equal to the capacity of station number 7.

People entering the station number 7 at t = ∑ ∑ Xt7j 7,L

At t the inbound from the other station s to station 7 at time t = ∑ ∑ ∑ X(t-ds7),i,js,L

Adding both will give us the number of people entering through node 7.

  • ∑ ∑ Xt7j 7,L + ∑ ∑ ∑ X(t-ds7),i,js,L<= Capacity of 7

Now, the decision variable will be : Xtijl which is defined as the passenger flow at the time t by using the line l present between the stations i as well as j.

The Capacity constrain is given by ∑ ∑ ∑ Xtijl which is less than or equal to the capacity at l.

T = { t1,t2, tn}

=∑ (Xt,s1,j l + Xt,s2,j l  + Xt,sN,jl)

And S = { 1,2,3  N}

Sl = { S1l, S2lSNl}

Transport congestion – traffic congestion is a prerequisite in the transportation system where vehicles have a speed lower, longer driving time and an increased waiting list of vehicles.Traffic congestion or blockage occurs when traffic volume is available or when space demand exceeds available street capacity. The congestion is caused or exacerbated by so many particular situations. Rainfall lowers traffic capacity and operational speed in terms of road operation, leading to more congestion and a worse productivity of the road network. The capacity or the number of vehicles needed for a certain volume of persons or products grow by most at or over a specific point of time. Nearly half of the blockage is recurrent and due to the weight of traffic; most of the remainder is due to road accidents, road maintenance and weather occurrences

  • Following are the used Intelligent transportation system which guidetraffic

• The Parking Guidance and Information Systems should offer dynamic advise to motorists about free parking.

• Active traffic management system opens up the UK highway hard shoulder as additional traffic lanes, it utilises CCTV and VMS in order to regulate and traffic control the usage of an extra lane.

• Traffic counters have been placed permanently to give real-time traffic numbers.

• Road user recommendation or advise on road traffic reports through radio GPS and mobile app.

  • Variety of messages signs are installed along road side to advice or suggest the road user.
  • Example
  • Traffic problem create due to Event halls, (i.e. Music Festivals, Film Festivals, and Football matches) and downtown. Immediately after the end of a the event, apeak demand for traffic from the spectators is to beexpected.
  • Near the downtown from theTimemagazine it is declared thatSao Paulo has the world’s worst or bad traffic jamsandTraffic congestion on marginal Pinheiro’s Shown below figure
  • This

It’s obtainable. Some traffic engineers attempted to apply fluid dynamic principles to traffic flows by comparing them to the movement of liquid through a conduit. Congestion simulations and real-time observations have shown that in severe but free flow jams, minor events such as an abrupt steering change by a single engine driver may trigger spontaneously (a “butterfly effect.”). The fact that the supercooled fluid abruptly freezes is compared by traffic scientists.

Zhang et al. recently published an empirical study on traffic jam size distribution.

You discovered an approximate universal power law for the distribution of jam sizes. In real flow data during peak hours, G.Zeng et al. found evidence for numerous metastable situations and hysteresis. At crossroads, traffic flow is often affected by lights or other events, which interrupt traffic flow on a regular basis. Other math theories exist, such as Boris Kerner’s three-phase traffic theory.

The term is nearly identical to Edward Lorenz’s work as a meteorologist and mathematician. The butterfly effect is developed from the metaphorical example of him, according to the specifics of a tornado. (Real accurate creation time, actual right route followed) being affected by small disturbances such as a faraway butterfly fluttering its wings many weeks ago. When Lorenz examined runs of his weather model, he found the impact. using data from the original condition that was rounded in an apparently insignificant way He observed that the weather model will not reproduce the outcomes of runs using unrounded starting conditions data. A little difference in the starting circumstances has resulted in a noticeable difference in the outcome..

Between 1996 and 2002, Boris Kerne created a Three-Phase Traffic System Theory of Traffic Flow. It is primarily concerned with the mechanics of traffic breakdown and the resultant heavy traffic on roadways. Kerner proposed three phases of traffic, while traditional theories based on the basic diagram of traffic flow contain just two: free flow and congested traffic. According to Kerner’s, congested traffic was split into two different stages: synchronised flow and broad moving jam, bringing the total number of phases to three:

1. loose or unrestricted flow (F)

2. a speed flow that coexists (S)

3. Traffic snarl (J)

  • The observed statistics on looser traffic indicate a positive link between movement rate Q (in cars per unit time) and vehicle density K (in vehicles per unit distance).
  • when it reachs the maximum free flow the relationship stops {Qmax} With a corresponding critical density Kcrit.

When vehicles accelerate up to freeflow in “synchronised flow,” this distinctive feature of a big moving jam was not observed on the downstream front. The downstream synchronised flow front may be attached to a tailback in particular.

The term “synchronised flow” refers to traffic characteristics that are reversed:

If there is no major stoppage in the flow of traffic, a large amount of congestion may build within.As a result, the word “flow” refers to this feature.

(ii) In this flow, there is a propensity for vehicle speeds to be synchronised across several lanes. Furthermore, in a synchronised flow, there is an inclination for vehicle speeds in each lane to cohabit.

This was due to the low probability of passing.

This effect of speed synchronisation is rejected by the term “synchronised.”.

Any road around upstream may find the “wide-moving congestion.”

The average downstream speed of the front Vg is maintained throughout this process. This is a feature of the wide gridlock that characterises the period.

A jam’s capacity to extend beyond all other conditions of traffic and past any bottleneck was invented for the phrase “broad jam” to sustain downstream jam speed. “moving jam” means the expansion of a traffic jam as a completely localised road structure. The

If a moving jam is wider than the wide jam fronts and the car speed is none in the jam, the jam will always maintain its downstream jam speed. the word “broad” refers to the fact that the jam is nil while moving (in the longitudinal direction) Kerner used the word wide to differentiate between large moving jams and other moving jams which do no longer maintain the average downstream jam front speed.

  1. The metro transport system handles the passenger’s transportation to, from and between the Event halls, (i.e. Music Festivals, Film Festivals, and Football matches) and downtown. Immediately after the end of a the event, a peak demand for traffic from the spectators is to beexpected.
  2. In order to meet that demand, a high level of transport capacity must be provided on a temporary basis.The

interactions with other traffic demand and transport services must be considered.

  1. after an event has ended, traffic flows to another event hall, where a subsequent event is to take place, must bemanaged is a good example for this.
  1. To handle the different traffic demands requires a tailored designed metro train schedule. For passenger flows in the metro stations where several lines cross, depending on the events schedule, exceptionally high pedestrian flows are to beexpected.
  1. Appropriate controlling measures may have to be taken to the stations and within the stations. The appropriateness and mode of action of such measures shall be analyzed by means of mathmaticalmodelling.
  1. The metro stations with respect to interactions in the whole metronetwork of Management .
  2. Approach: MathematicalProgramming
  3. How to balance capacity and demand?




capacityperhour 0

number of metro users

10000 10000

1 2 3

demand: 20000 > 10000



metro line segment


metro station

catchment area metro users


Example Model

Equation system with an objective function andsome



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=1 ∀𝑖𝑗


𝑋𝑡𝑖𝑗. 𝑇

≤𝐹 ∀𝑡,𝑖𝑗

Minimize waiting times at metro stations with respect to

  • fulfillment ofdemands
  • capacity limits ofresources

෍ 𝑋𝑡𝑖𝑗.𝑑𝑖𝑗≤ 𝐶𝑟


∀𝑟 ∀𝑡

t=1 :4500

t=2 :4500

t=1 :1500

t=2 :1500

capacityperhour 0




2 3

demand per period 10000 <= 10000

t=1 : 4000

t=2:4000 4

metro station

metro line segment

catchment area metro users



  • Minimaxwaitingtimelastperiod𝑋𝑖𝑗
    • 5ppl wanttogofrom1to4-𝑋𝐴=5,𝑋𝐴=5

1,3 3,4

  • 2ppl wanttogofrom1to3-𝑋𝐴=5+2=7,𝑋𝐴=5

1,3 3,4

  • 1,3 2,3 4,3 3,1 3,2 3:(𝑋𝐴+𝑋𝐴+𝑋𝐴)−(𝑋𝐴+𝑋𝐴+𝑋𝐴)3=,42

7 0 0 0 0 5

  • 2:(𝑋𝐴+𝑋𝐴+𝑋𝐴)−(𝑋𝐴+𝑋𝐴+𝑋𝐴 )= 3

1,2 3,2 4,2 2,1 2,3 2,4

  • 2:(𝑋𝐵+𝑋𝐵+𝑋𝐵)−(𝑋𝐵+𝑋𝐵+𝑋𝐵)=−1

1,2 3,2 4,2 2,1 2,3 2,4

„node 2 demands 3 units fromgoodA“ pass from node1

„node2supplies1unitsfromgoodB“ pass from node3

All stations of line 𝑙(starting locationstation)

෍ ෍ ෍

𝑋 𝑠,𝑙




1,2 3,2 4,2

+ +𝑋2,𝑙1 𝑋2,𝑙1 𝑋2,𝑙1

1,2 3,2 4,2

+ +𝑋3,𝑙1 𝑋3,𝑙1 𝑋3,𝑙1

1,2 3,2 4,2

  • 𝑖𝑗 𝑋𝑠,𝐿:flowbetween𝑖and𝑗ofpassangersfromstation𝑠usingline𝑙.

Line connected to node 2



1,2 3,2 4,2

  • 𝑡,,𝑖𝑗 𝑋𝑠,𝑙 :flowofpassangers entering in station𝑠atperiod

𝑡using line 𝑙between station 𝑖and station 𝑗.



  • 𝑡,,𝑖𝑗 DecisionVariable:𝑋𝑙 :flowofpassangersatperiod𝑡usingline𝑙betweenstation𝑖andstation𝑗.
  • CapacityConstrain



𝑇={𝑡1,𝑡2,𝑡3,𝑡4,…,𝑡𝑁} →

≤ 𝐶𝑙

( 𝑡,s1 ,𝑗



+ ⋯+𝑋


) =𝑋𝑙𝑡1,s1,𝑗







𝑙 𝑙 𝑙 𝑙 𝑙 𝑙


+ 𝑋






1 2 3 4 5 𝑁




+𝑋 +𝑋 +⋯+𝑋 𝑙 𝑙 𝑙

𝑡n,s1,𝑗 𝑡n,s2,𝑗 𝑡,sN,𝑗

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