How MRT Routes and Transfers Are Optimised Using Mathematics

Open an MRT map and Singapore appears to become simple.

Stations are dots. Rail lines are coloured curves. Interchanges are places where the colours touch.

Choose where you are. Choose where you want to go. Trace a path between them.

But the route that looks shortest on the map may not be the fastest journey.

A path with fewer stations may require a long transfer walk. A path with one extra station may use a more frequent line. A direct train may be slower than a route with one well-timed interchange. A mathematically fast route may be crowded, unreliable or difficult for a passenger who needs lifts and barrier-free access. During a disruption, the route that was best five minutes ago may no longer exist as a usable path.

This means an MRT journey planner is not merely drawing a line between two points.

It is solving a changing graph in which every edge can carry time, waiting, walking, crowding, reliability, accessibility and transfer cost.

Singapore’s Land Transport Authority provides a journey planner for finding routes and calculating travel times and fares. Its MyTransport app also provides information such as train operating times and station exits. Underneath a passenger-facing answer such as “take the Downtown Line, transfer, then continue on the North-South Line” lies a family of mathematical problems involving graph theory, shortest-path algorithms, probability, behavioural modelling, queueing, optimisation and network resilience.

This article continues the eduKateSG MRT mathematics series. Begin with How MRT Timing Works Using Mathematics, then explore braking, headway, station dwell, delay recovery, timetable construction, energy optimisation, and passenger capacity.

Now we ask how the railway chooses a path.

The RFE — Why Does Route Optimisation Exist?

The weakest possible objective is:

find the geometrically shortest route

That is not the whole passenger job.

A useful route should help a real person reach a real destination under real constraints.

The Reason for Existence is therefore:

connect origin to destination
+
minimise avoidable journey burden
+
respect passenger abilities and preferences
+
respond to crowding and disruption
+
protect the capacity and resilience of the network

That word burden matters.

Passengers do not experience only in-train minutes. They experience:

walking to and through stations,
waiting on platforms,
uncertainty about whether a connection will be made,
crowding,
the mental effort of navigating an interchange,
stairs, escalators and lifts,
the risk of disruption,
and the inconvenience of changing trains.

So the real optimisation is not:

minimum map distance

It is closer to:

minimum generalised journey cost
subject to safety, service, capacity,
accessibility and information constraints

The best route is the route that performs the passenger’s whole journey, not merely the route that draws the shortest line.

Prompt 1 — How Does an MRT Map Become a Mathematical Graph?

Graph theory begins with two objects.

Vertices or nodes
Edges or links

For an MRT network:

station = node
rail connection = edge

The network can be written:

G = (V,E)

where V is the set of stations and E is the set of connections.

LTA currently describes Singapore’s MRT network as having more than 140 stations across six MRT lines, with LRT services adding more stations and links. A passenger sees a coloured map. A graph algorithm sees a structured set of possible movements.

Stations may need more than one node

A simple map may represent an interchange as one dot.

But a realistic route model may represent:

the North-South Line platform,
the Downtown Line platform,
the transfer walkway,
the lift route,
the escalator route,
and station entrances or exits

as separate nodes connected by walking edges.

Instead of:

Interchange X = one point

the model becomes:

X-Line-A platform
→ transfer corridor
→ escalator or lift
→ X-Line-B platform

The transfer is no longer free.

It has distance, time, uncertainty and accessibility properties.

Edges need weights

A graph tells us which stations connect.

A weighted graph tells us what each connection costs.

For edge e:

we = cost of using edge e

The weight might represent:

distance,
scheduled travel time,
expected travel time,
walking time,
fare,
crowding discomfort,
energy,
or disruption risk.

The cost of a route r containing edges e is:

Cr = Σe∈r we

The route problem is then:

find r* such that
Cr* = minr Cr

But the answer depends entirely on what we choose as the edge weight.

The graph can be directed

Travel from A to B need not have exactly the same cost as B to A.

Reasons include:

different platform layouts,
different transfer walking routes,
peak-direction crowding,
different waiting times,
one-way station circulation,
and asymmetric disruption.

So an edge can have direction:

wAB ≠ wBA

The graph changes with time

A route at 8am can differ from the same route at 11pm.

Train frequencies change. Crowding changes. First and last train constraints appear. Planned service adjustments can close or alter parts of the network.

Therefore edge weight is better represented as:

we(t)

and network availability as:

E(t)

A route planner is solving a time-dependent graph.

The MRT map looks static. The mathematical network is moving even while the paper map remains still.

Prompt 2 — How Does Mathematics Find the Shortest or Fastest Route?

Suppose a network has five stations:

A — B — D
 \  C —/

There are two possible paths from A to D:

Route 1: A → B → D
Route 2: A → C → D

Suppose travel-time weights are:

A→B = 6 min
B→D = 8 min

A→C = 5 min
C→D = 11 min

Then:

C1 = 6 + 8 = 14 min
C2 = 5 + 11 = 16 min

Route 1 is the shortest-time path.

Algorithms such as Dijkstra’s algorithm solve this problem efficiently when edge weights are non-negative. The algorithm repeatedly confirms the cheapest currently reachable node and relaxes neighbouring costs until the destination is settled.

Conceptually:

distance[source] = 0
all other distances = ∞

repeatedly:
choose unsettled node with smallest cost
update costs of its neighbours

For a simple static network, this works beautifully.

An MRT journey is more complicated.

Running time is only one component

Suppose Route 1 requires a transfer with:

4 min walking
3 min expected platform wait

Its full time is:

14 + 4 + 3 = 21 min

Suppose Route 2 is direct:

16 min onboard
+ 1 min initial expected wait
= 17 min

The path with the shortest train-running time is no longer the fastest passenger journey.

Time-dependent shortest paths

Suppose a train departs every two minutes on one line and every six minutes on another.

The cost of entering the line depends on arrival time at the platform.

If a passenger reaches Platform X at 08:10:50 and the train leaves at 08:11:00:

wait = 10 s

If the passenger reaches it at 08:11:10 and the next train is at 08:17:00:

wait = 5 min 50 s

The same platform edge has a very different weight ten seconds later.

A time-dependent route planner therefore uses:

arrival time at node
→ next available departure
→ resulting arrival at next node

The state must include time, not merely place.

Time-expanded graphs

One method creates a separate node for each event:

Station A at 08:00
Station A at 08:02
Station A at 08:04

Edges represent waiting, riding and transferring.

The ordinary geographic graph becomes a larger time-expanded graph.

This allows first trains, last trains, scheduled connections and temporary service patterns to be represented directly.

A journey planner does not merely ask “Where are you?” It asks “Where are you, at what time, and what movements remain possible from that state?”

Prompt 3 — What Is Generalised Journey Cost?

Passengers do not value every minute in the same way.

Five minutes sitting on a train may feel different from five minutes walking through a complex interchange.

Five minutes of predictable waiting may feel different from five minutes wondering whether a disrupted train will arrive.

Transport models therefore use generalised cost.

A conceptual route cost is:

Cr
=
βiv Tiv
+ βwait Twait
+ βwalk Twalk
+ βtr Ntransfers
+ βcrowd Xcrowd
+ βrel Xreliability
+ βaccess Xaccess
+ βfare F

where:

T_iv is in-vehicle time,
T_wait is waiting time,
T_walk is walking time,
N_transfers is the number of interchanges,
X_crowd measures crowding burden,
X_reliability measures uncertainty or disruption risk,
X_access represents accessibility burden,
F is fare or another monetary cost,
the β values convert unlike burdens into a common scale.

The coefficients are not universal constants.

They can differ among passengers, purposes and circumstances.

Transfer penalty

Research on public transport route choice commonly includes a transfer penalty beyond the measured walking and waiting time.

Why?

A transfer introduces:

navigation effort,
uncertainty,
the risk of missing a connection,
vertical movement,
crowding at interchange passages,
and interruption of a seated or settled journey.

So two routes with equal measured time may not be equally attractive.

Suppose Route A takes 30 minutes and has no transfer.

Route B takes 27 minutes but requires one transfer.

If the passenger’s transfer penalty is equivalent to five minutes:

Generalised cost A = 30
Generalised cost B = 27 + 5 = 32

The direct route is preferred even though it is physically slower.

Crowding cost

A minute in a lightly loaded carriage and a minute in severe crowding may not have the same perceived cost.

A simple crowding multiplier is:

Tperceived = mcrowd × Tiv

where m_crowd increases with load.

For illustration:

20 min uncrowded × 1.0 = 20 perceived min
20 min crowded   × 1.35 = 27 perceived min

The numerical factor is fictional.

The mathematical structure is widely used: crowding changes route utility and passenger behaviour.

Reliability cost

Suppose two routes have the same expected travel time.

Route A almost always takes 30–32 minutes.

Route B sometimes takes 24 minutes and sometimes 45.

A risk-sensitive passenger may prefer A.

A simple reliability-adjusted cost is:

Creliable = E[T] + λσT

where σ_T is travel-time standard deviation and λ describes the passenger’s aversion to uncertainty.

Another model may use a percentile:

C95 = 95th-percentile journey time

A traveller going to an examination, airport or medical appointment may care more about C₉₅ than the average.

Accessibility cost

A staircase edge may have finite cost for one passenger and be unusable for another.

For a wheelchair user:

wstairs = ∞

The algorithm must use a lift-accessible path.

LTA states that all MRT stations have barrier-free access, and its inclusive-transport pages describe priority access, lifts, tactile guidance and other support. But an accessible route still has its own geometry and walking time.

A personalised route model may therefore use:

Cr(user)
not merely
Cr(average passenger)

The shortest path for the average passenger can be the wrong path for the person actually asking.

Prompt 4 — What Is the Mathematics of an MRT Transfer?

A transfer is a small journey inside the larger journey.

alight
→ walk
→ change level
→ navigate
→ wait
→ board

Let:

A = arrival time of first train
W = transfer walking time
D = departure time of connecting train

The connection is feasible if:

A + W ≤ D

But W is not perfectly constant.

It depends on:

which carriage the passenger exits,
walking speed,
crowding,
escalator or lift availability,
wayfinding,
and platform position.

So W can be treated as a random variable.

Transfer success probability becomes:

Psuccess = P(W ≤ D − A)

If the available transfer window is five minutes and walking time has mean four minutes with meaningful variation, the connection is not guaranteed merely because the means fit.

Expected cost of missing a train

Suppose probability of missing the connection is p and the next-train wait is H.

A simple expected missed-connection penalty is:

Emiss = pH

At a two-minute service:

p = 0.25
H = 2 min
Emiss = 0.5 min

At a ten-minute service:

Emiss = 2.5 min

The same transfer geometry becomes more costly when the connecting frequency is lower.

This is why transfer cost is a function of both station design and timetable.

The right carriage changes the edge weight

A passenger exiting near the interchange escalator may save several minutes compared with someone at the far end of the platform.

LTA has highlighted apps that help passengers identify carriages near station exits, lifts, escalators and transfer points.

Mathematically, carriage choice changes the transfer walking distance:

W = W(carriage, platform, exit, passenger)

Route planning can therefore begin before the transfer station is reached.

Circle Line Stage 6: removing a transfer by adding an edge

Circle Line Stage 6 opened for passenger service on 12 July 2026, connecting HarbourFront and Marina Bay through Keppel, Cantonment and Prince Edward Road.

LTA states that the completed Circle Line spans 39 kilometres across 33 stations and has 12 interchange stations connecting to all existing MRT lines.

Before completion, LTA’s example journey from Telok Blangah to Marina Bay required transfers among the Circle, North East and North-South Lines. With the completed circle, the same journey can be made on one train, saving around ten minutes.

Graphically, the network gained new edges that closed a missing southern link.

Before:
origin → line → transfer → line → transfer → line → destination

After:
origin → continuous line → destination

The gain came from more than shorter track distance.

It removed:

transfer walking,
transfer waiting,
transfer uncertainty,
and two transfer penalties.

Adding one network connection can save time several times: on the train, in the walkway, on the platform and in the passenger’s uncertainty.

Prompt 5 — Why Do Different Passengers Choose Different Routes?

If one route has the lowest generalised cost, will every passenger choose it?

No.

Passengers differ.

Some dislike transfers strongly.
Some prefer a seat.
Some know one interchange better than another.
Some walk quickly.
Some need lifts.
Some are risk-averse.
Some follow habit.
Some respond to real-time crowding information.

Route choice is therefore probabilistic.

A logit route-choice model

Suppose route r has generalised cost C_r.

A common model gives route probability:

Pr
=
exp(−θCr)
/
Σk exp(−θCk)

where θ describes sensitivity to cost differences.

Suppose two routes have costs:

C1 = 30
C2 = 34
θ = 0.25

Then:

P1
= e−7.5 /(e−7.5 + e−8.5)
≈ 0.731

P2 ≈ 0.269

Most passengers choose Route 1, but some choose Route 2.

The model reflects unobserved preferences and imperfect knowledge.

Path overlap

Suppose Route A and Route B share 90 per cent of their journey.

Route C is genuinely different.

A basic route-choice model may mistakenly treat all three as equally independent alternatives.

Path-size and overlap corrections reduce this distortion.

One conceptual path-size term is:

PSr
=
Σe∈r (length share of e in r)
× 1/(number of candidate routes using e)

A route containing more unique edges receives a different correction from one that mostly duplicates another path.

Research comparing metro route-choice behaviour shows that passengers respond to multiple time components and transfer variables, not merely one shortest-time value.

Habit and bounded rationality

Passengers do not recalculate the entire graph perfectly every morning.

They may use a familiar route that is close enough.

This can be represented by a tolerance band:

choose any route r such that
Cr ≤ Cminimum + Δ

where Δ represents acceptable extra cost before the passenger changes habit.

This is bounded rationality.

The passenger is not irrational.

They are conserving cognitive effort.

A route can be mathematically optimal and behaviourally invisible if passengers do not know it, trust it or find it worth relearning.

Prompt 6 — How Do Crowding and Capacity Change the Best Route?

Suppose everyone is told to use the same fastest route.

That route becomes crowded.

Boarding slows. Some passengers may be unable to enter the first train. Dwell increases. Waiting rises. The route’s cost changes.

So edge weight depends on passenger flow:

we = we(xe)

where x_e is the passenger flow or load on edge e.

A simple crowding cost might be:

wcrowd(x)
=
0,                       x/c ≤ α
k(x/c − α)²,             x/c > α

where c is capacity and α is the load level at which crowding begins to impose substantial cost.

This fictional function illustrates a threshold effect.

At low load, another passenger adds little burden.

Near capacity, the same additional passenger can greatly increase discomfort, denied boarding and dwell instability.

Capacity-constrained assignment

Let route demand be x_r.

For each network edge e with passenger capacity c_e:

Σr containing e xr ≤ ce

If demand exceeds immediate capacity, passengers queue or shift routes.

Route planning therefore becomes an assignment problem:

distribute passenger demand among paths
while respecting capacity
and accounting for costs that change with load

Passenger equilibrium

At a user equilibrium, no passenger can improve their perceived cost by changing route alone, given everyone else’s choices.

For all used routes r between the same origin and destination:

Cr = Cminimum

while unused routes have:

Cr ≥ Cminimum

This is a version of Wardrop’s first principle.

But user equilibrium need not minimise total system cost.

Everyone may choose a route that is individually attractive and collectively overloaded.

System optimum

A system-optimal assignment minimises total passenger cost:

minimise Σr xr Cr(x)

This may ask some passengers to use a route that is slightly longer for them so that everyone’s combined crowding and delay falls.

That creates an ethical and practical problem.

Should a journey planner recommend the individually fastest route, or should it try to protect the network?

Research on passenger-oriented rail control integrates crowding and passenger route choice because train regulation can redistribute passengers across both time and routes.

Real-time crowding information

If passengers are shown crowding information, some may choose a different route, carriage or departure time.

Research modelling real-time crowding information finds that it can reduce unevenness in onboard crowding and improve passenger experience under the studied conditions.

But information also changes the system it describes.

system measures crowding
→ publishes information
→ passengers change routes
→ crowding changes
→ information becomes outdated

This is another feedback loop.

A route recommendation is not merely an observation of the network. At scale, it can become an intervention in the network.

Prompt 7 — How Do Alternative Routes Create MRT Resilience?

A network with only one path between important regions can be efficient during normal operation and fragile during disruption.

A network with multiple usable paths has redundancy.

For origin o and destination d, let K_od be the number of meaningfully different feasible paths.

A crude redundancy measure might be:

Rod = Kod

But route count alone is weak.

Ten alternatives that all share the same vulnerable interchange do not provide ten independent backups.

A stronger measure considers:

path overlap,
spare capacity,
extra travel time,
transfer burden,
and whether the alternatives remain available under the same disruption.

A weighted redundancy measure can be imagined:

Rod
=
Σr∈alternatives
uniquenessr
× capacityr
× exp(−θ extra_costr)

This is an educational formulation.

It captures the idea that an alternative is more valuable when it is independent, has spare capacity and does not impose excessive extra journey burden.

Structural and functional resilience

A network can remain physically connected while becoming functionally poor.

Suppose a failed link still leaves an alternative path.

Structurally:

origin and destination remain connected

Functionally:

the alternative may add 35 minutes,
require two transfers,
and already be near capacity

The graph is connected.

The passenger service has deteriorated sharply.

Research on urban rail resilience therefore distinguishes network connectivity from functional service performance.

Critical nodes and edges

A station or link is critical when its loss creates a large passenger impact.

Graph theory offers measures such as betweenness centrality.

For node v:

BC(v)
=
Σs≠v≠t
σst(v)/σst

where σ_st is the number of shortest paths from s to t and σ_st(v) is the number passing through v.

A station with high betweenness lies on many shortest paths.

But passenger-weighted criticality is more meaningful:

Criticality(v)
=
Σod Demandod
× increase in generalised cost
when v is unavailable

A visually modest station can be highly important if few substitutes exist.

A major interchange can be less catastrophic than expected if the surrounding network has strong alternatives and spare capacity.

New links change the entire graph

LTA states that the Downtown Line 2 Extension will provide alternative travel and transfer options and enhance network resilience in the north-west, including a future interchange with the North-South Line. It is expected to shorten Yew Tee–Chinatown journeys by about 20 minutes when completed.

The Downtown Line 3 Extension is designed to improve transfers among the Downtown, East-West and Thomson-East Coast Lines in the east.

Future Thomson-East Coast and Cross Island Line connections to Changi Airport and Terminal 5 will add more direct paths and transfer choices to major hubs.

Every new interchange or line extension changes:

shortest paths,
transfer counts,
passenger assignment,
crowding on old lines,
critical-node importance,
and disruption alternatives.

new edge
→ new route set
→ new passenger choices
→ new section loads
→ new crowding pattern
→ new resilience structure

The infrastructure is local.

The mathematical consequences can be network-wide.

A new MRT connection does not merely serve the stations it touches. It rewrites the route options of the entire graph.

Prompt 8 — How Should a Real-Time Journey Planner Choose the Best Route?

A modern journey planner can receive many inputs:

current time,
origin and destination,
train schedules and frequencies,
service adjustments or disruptions,
station exits,
walking links,
crowding information,
fare rules,
accessibility requirements,
and passenger preferences.

It then solves a multi-criteria problem.

A route cannot usually minimise everything simultaneously.

minimum time
minimum walking
minimum transfers
minimum crowding
minimum fare
maximum reliability
maximum accessibility

These objectives can conflict.

Pareto-optimal routes

A route is Pareto-dominated if another route is no worse in every relevant objective and better in at least one.

Suppose:

Route A: 28 min, 2 transfers, 900 m walk
Route B: 31 min, 1 transfer, 400 m walk
Route C: 38 min, 2 transfers, 1,100 m walk

Route C is dominated by A: it is slower, has the same number of transfers and requires more walking.

A and B are not directly comparable.

A is faster.

B is easier.

The planner can present both as meaningful alternatives.

Personalised objective weights

A commuter going to work may choose:

high weight on travel time

A parent travelling with a stroller may choose:

high weight on lifts and low transfer complexity

An elderly passenger may choose:

low walking, low crowding, high reliability

A traveller heading to Changi Airport may choose:

high penalty for uncertainty and missed connections

So:

best route
=
arg minr Cr(user, time, network state)

Dynamic replanning

Suppose a disruption removes edge e at time t.

we(t) = ∞

The current route becomes infeasible.

The planner must recompute from the passenger’s current state:

current station
+ current time
+ remaining network
+ current crowding
+ available alternatives

It should not continue optimising from the original journey origin because the passenger has already moved.

This is receding-horizon routing.

plan
→ travel
→ receive new information
→ replan from present state
→ continue

The ethical route-planner problem

A route planner knows that its recommendations can change crowding.

If it sends everyone to the same apparent shortcut, it may destroy the shortcut.

If it silently diverts some passengers to protect the network, it may impose a cost on those individuals.

A trustworthy system should therefore preserve:

accuracy,
accessibility,
clear explanation of trade-offs,
privacy,
fairness among passenger groups,
and a distinction between an individually fastest route and a less crowded or more reliable alternative.

The route planner should not pretend that one answer is objectively best when the result depends on human priorities.

Good routing does not merely calculate a path. It tells the truth about why that path was chosen.

A Complete Worked MRT Route-Choice Model

Let us compare three fictional routes from Home Station H to Destination D.

This is an educational model and does not represent actual Singapore journey times.

Route A — Direct but slower

in-vehicle time = 34 min
initial wait = 2 min
walking = 4 min
transfers = 0
crowding score = 2
travel-time standard deviation = 2 min

Route B — Fast with one transfer

in-vehicle time = 25 min
waiting = 5 min
walking = 7 min
transfers = 1
crowding score = 6
travel-time standard deviation = 5 min

Route C — Two transfers but low crowding

in-vehicle time = 27 min
waiting = 6 min
walking = 9 min
transfers = 2
crowding score = 1
travel-time standard deviation = 3 min

Use the fictional generalised-cost model:

C
=
1.0(Tin-vehicle)
+ 1.5(Twait)
+ 1.7(Twalk)
+ 4(Ntransfers)
+ 0.8(crowding score)
+ 0.6(standard deviation)

These weights are illustrative, not official values.

Calculate Route A

CA
= 34
+ 1.5(2)
+ 1.7(4)
+ 4(0)
+ 0.8(2)
+ 0.6(2)

= 34 + 3 + 6.8 + 0 + 1.6 + 1.2
= 46.6

Calculate Route B

CB
= 25
+ 1.5(5)
+ 1.7(7)
+ 4(1)
+ 0.8(6)
+ 0.6(5)

= 25 + 7.5 + 11.9 + 4 + 4.8 + 3
= 56.2

Calculate Route C

CC
= 27
+ 1.5(6)
+ 1.7(9)
+ 4(2)
+ 0.8(1)
+ 0.6(3)

= 27 + 9 + 15.3 + 8 + 0.8 + 1.8
= 61.9

The physically fastest in-vehicle route is B.

The lowest generalised-cost route is A.

The direct route wins because it avoids transfer, walking, crowding and uncertainty costs.

Change the passenger

Now consider a passenger who dislikes crowding intensely but is comfortable walking and transferring.

Increase crowding weight from 0.8 to 4.0 and reduce walking weight from 1.7 to 1.1.

Costs change.

Route C may become competitive despite having two transfers.

Now consider a wheelchair user for whom Route B contains a temporarily unavailable lift connection.

CB = ∞

Route B leaves the feasible set entirely.

Now consider a passenger late for an examination.

Increase the value of time and reduce the crowding penalty.

Route B may become optimal.

The network did not change. The correct route changed because the receiver changed.

Now add a disruption

Suppose the direct edge used by Route A is unavailable.

CA = ∞

Passengers shift to B and C.

But B becomes crowded.

Its crowding score rises from 6 to 9 and waiting rises by three minutes.

Its new cost becomes:

CB,new
= 25
+ 1.5(8)
+ 1.7(7)
+ 4
+ 0.8(9)
+ 0.6(5)

= 25 + 12 + 11.9 + 4 + 7.2 + 3
= 63.1

Route C at 61.9 is now slightly better.

The disruption did not merely remove Route A.

Passenger redistribution changed the costs of the remaining routes.

network failure
→ route removal
→ passenger rerouting
→ crowding redistribution
→ new travel times
→ new optimal route

This is why disruption routing must be dynamic and capacity-aware.

The Route Deletion Tests

We can understand the full problem by removing one component at a time.

Remove waiting time

The model can recommend an infrequent line as though a train appears instantly.

Remove transfer walking

Every interchange becomes a cost-free teleport.

Remove transfer penalty

The model underestimates navigation, uncertainty and interruption burden.

Remove crowding

The algorithm may send every passenger through a route whose practical cost rises sharply under load.

Remove capacity

Every passenger is assumed able to board every arriving train.

Remove accessibility

The mathematically recommended path may be physically unusable for the passenger.

Remove reliability

A volatile route can look identical to a predictable one with the same mean time.

Remove route redundancy

The model works in normal conditions but has nowhere to send passengers after a failure.

The deletion tests reveal the RFE again.

Route optimisation is not a line-drawing exercise.

It is the preservation of a complete human journey through a changing system.

The Route Paradoxes

Paradox 1 — The shortest route can take longer

Transfer walking and waiting can outweigh fewer kilometres.

Paradox 2 — The fastest route can feel worse

Crowding, uncertainty and transfer complexity add generalised cost.

Paradox 3 — A direct train can strengthen the whole network

It removes transfer burden and can redistribute passengers away from congested interchange paths.

Paradox 4 — More route choices can create more complexity

Passengers need information to understand the alternatives, and poorly communicated options may remain unused.

Paradox 5 — The individually best route can be collectively poor

If everyone selects the same shortcut, crowding and delay can destroy its advantage.

Paradox 6 — A connected network can be functionally disconnected

An alternative may exist on the map but be too slow, crowded or inaccessible to preserve a useful journey.

Paradox 7 — A longer route can be safer for an important appointment

A predictable path can dominate a shorter but volatile path when late arrival carries high cost.

The Route Planner as a Receiver-Aware System

A route does not exist as “best” by itself.

It is best for someone.

At a particular time.

Under a particular network state.

For a particular purpose.

A useful receiver packet might include:

origin
and destination
current time
desired arrival time
walking ability
lift requirement
transfer tolerance
crowding tolerance
reliability priority
fare sensitivity
luggage or stroller
knowledge of the network

The route engine then computes a projection:

shared network reality
+ passenger-specific constraints
+ current service state
= receiver-appropriate route

This avoids a common error in automated systems:

giving everyone the route optimised for an imaginary average user.

The same railway can legitimately produce different answers for:

a student late for school,
a wheelchair user,
a senior who wants fewer transfers,
a commuter who prefers a less crowded route,
and a traveller carrying luggage to the airport.

Difference is not inconsistency.

It is correct projection from the same network into different human requirements.

The World Return — When Passengers Change the Route Model

The planner predicts that 70 per cent of passengers will choose Route A.

Reality gives 45 per cent.

Why?

Perhaps transfer walking was underestimated.

Perhaps passengers dislike a crowded interchange.

Perhaps signage makes Route B easier to understand.

Perhaps habitual users have not adopted the new link.

Let predicted route share be p̂_r and observed share be p_r.

Residual is:

er = pr − p̂r

The model can recalibrate its coefficients:

transfer penalty
crowding sensitivity
walking-time value
reliability sensitivity
path-awareness assumptions

The operating loop becomes:

model route choices
→ publish information
→ passengers choose
→ crowding changes
→ observe actual journeys
→ compare
→ recalibrate
→ route again

A route planner is therefore not finished when the algorithm returns a path.

It must learn whether people could and did use that path as expected.

The map gives possible routes. Passenger behaviour reveals which routes are real.

RFE Return — What Is a Good MRT Route?

A good route is not automatically:

the shortest distance,
the fewest stations,
the lowest scheduled train time,
the fewest transfers,
or the route most passengers already use.

Each of these can be useful.

None is sufficient alone.

A good route should be:

feasible now
+
appropriate for the passenger
+
clear enough to follow
+
reliable enough for the journey purpose
+
compatible with capacity and crowding
+
resilient when the network changes

The route planner should therefore offer truth rather than false precision.

It may say:

Fastest: 31 minutes, two transfers

Simplest: 35 minutes, no transfer

Lower walking: 37 minutes, lift-accessible

Less crowded estimate: 34 minutes

More reliable for a fixed arrival: leave 8 minutes earlier

There may not be one best route.

There may be a small set of honest alternatives.

The RFE of route optimisation is not to tell every passenger where to go. It is to make the trade-offs visible enough that the right passenger can make the right journey.

Conclusion — Singapore’s MRT Map Is a Living Equation

The MRT map appears to be made from lines.

Mathematically, it is made from choices.

At every interchange, a passenger can continue, transfer, wait, walk, reroute or leave the rail network.

Each option has a cost.

That cost changes with time.

It changes with crowding.

It changes with the passenger.

It changes when a new line opens.

It changes when a disruption closes an edge.

And it changes again when thousands of passengers respond to the recommendation.

stations
→ nodes

rail links
→ edges

running, waiting and walking
→ weights

transfers
→ penalties and probabilities

passengers
→ route-choice distributions

crowding
→ load-dependent costs

new lines
→ new graph topology

disruption
→ edge removal and rerouting

observed behaviour
→ model correction

The completion of Circle Line Stage 6 shows the graph changing in the real city. Three new stations closed the southern gap, created direct journeys, removed transfers and gave the network new redundancy. Future Downtown, Thomson-East Coast and Cross Island connections will alter the graph again.

The passenger sees coloured lines.

The mathematician sees a weighted graph.

The behavioural scientist sees probabilities.

The station designer sees transfer geometry.

The operations researcher sees capacity-constrained assignment.

The resilience planner sees alternative paths.

The passenger sees one question:

How should I get there?

A good route engine answers with more than a path.

It answers with an understanding of time, transfer, uncertainty, ability, crowding and purpose.

That is how MRT routes and transfers are optimised using mathematics.

The shortest path is only one possibility.

The best journey is the path whose mathematics still makes sense after the human being enters it.

Key Equations

G = (V,E)
MRT network as nodes and edges

Cr = Σe∈r we
Cost of a route

r* = arg minr Cr
Minimum-cost route

we = we(t,xe,user)
Time-, load- and passenger-dependent edge weight

Cr = βivTiv + βwaitTwait + βwalkTwalk
     + βtrNtransfers + βcrowdXcrowd
     + βrelXreliability + βaccessXaccess + βfareF
Generalised journey cost

Creliable = E[T] + λσT
Reliability-adjusted cost

A + W ≤ D
Feasible transfer condition

Psuccess = P(W ≤ D − A)
Transfer success probability

Emiss = pH
Simple expected missed-connection penalty

Pr = exp(−θCr)/Σk exp(−θCk)
Logit route-choice probability

Cr ≤ Cminimum + Δ
Bounded-rational route set

Σr containing e xr ≤ ce
Capacity-constrained passenger assignment

Cr = Cminimum for used routes
Passenger user equilibrium

minimise Σr xrCr(x)
System-optimal assignment

BC(v) = Σs≠v≠t σst(v)/σst
Node betweenness centrality

Criticality(v) = Σod Demandod × ΔCostod(v removed)
Passenger-weighted network criticality

P(D ≤ C) ≥ 1 − ε
Illustrative reliability or capacity condition

er = pr − p̂r
Observed minus predicted route share

Reader-safety note: The equations, route examples and coefficients in this article are educational abstractions. They do not reproduce proprietary Singapore MRT journey-planning algorithms, restricted operating data, internal disruption rules or security-sensitive network controls. Actual routing systems use current service data, validated network models and operator-specific procedures.

Continue the MRT Mathematics Series

How MRT Timing Works Using Mathematics — the system-wide pillar.
How MRT Braking Works Using Mathematics — stopping distance and regenerative braking.
How MRT Headway Works Using Mathematics — train spacing and throughput.
How MRT Station Dwell Time Works Using Mathematics — queues, doors and platform flow.
How MRT Delays Propagate and Recover Using Mathematics — instability, regulation and resilience.
How an MRT Timetable Is Built Using Mathematics — demand, train cycles and robustness.
How MRT Energy Use Is Optimised Using Mathematics — traction, coasting and regenerative energy.
How MRT Passenger Capacity Is Calculated Using Mathematics — vehicle, line, station and delivered capacity.

The next natural leg is How MRT Network Resilience Is Measured Using Mathematics: redundancy, critical stations, cascading passenger overload, structural versus functional resilience, recovery curves and why an alternative route that exists on the map may still be too weak to protect the city.

How MRT Routes and Transfers Are Optimised Using Mathematics: Why the Shortest Path Is Not Always the Best Journey