Optimisation on networks and congestion games
Week 2
EK Ch8 and HL Ch 9, 10
Transportation and assignment problems
Transportation problems
Terminology:
• Units of a commodity
• m sources FEASIBLESOLUTIONSPROPERTY
It
• n destinations dm libuted transportationproblemwillh ave
eachsource destinationhasafixedsupplydemand
• Supply from source i
• Demand Sid at destination j destinationsource
torecievedfromthe s olution
feasible
it Is d
• Cost per unit distributed from source i to j
ai I COST ASSUMPTION
ofdistributingunitsanounitsbeing
cost
distributed cost Cijxnounits
A problem is a transportation problem if it can be described completely in a parameter table like
below, it satis es the requirements assumption and cost assumption, and is a cost minimisation
1 LINEARPROGRAMMINGFORMULATION 2 PARAMETERTABLE
Min Z Icijxi
sit ai si forit
sci
sci Otis
E
diforget n
NETWORKREPRESENTATION
ADAPTATIONS
can
source betimeatwhichavailableand
astimew henrequiredcosts
destinations also
thenincludestoragecosts41327 3 29
EsEdcanintroducedummysourceand
t rack
keep ofexcesscapacity
Integer solutions property: For transportation problems where every
Sidhave an integer value, all
the basic variables (allocations) in every feasible (BF) solution, including the optimal one, also have
integer values
, Similar to transportation problems
Change transportation Problem
from problem type
transfer
Distributionpassest hrough enroute
points problem
transshipment
unit
withdifferent costsposttransfer
limito nquantity
upper toeachtransferpoint Minimum
cost flow
problem
Assignment problem
Assumptions:
• The number of assignees and the number of tasks are the same, n
• Each assignee is to be assigned exactly one task
• Each task is to be performed by exactly one assigned
• There is a cost associated with assignee i performing task j
Cij
• The objective is to determine how all n assignments should be made to minimise the total cost
1 LINEARPROGRAMMINGFORMULATION a NETWORKREPRESENTATION
Min Z Icijxi it
if performs
sitElegy forit m
Esci t forget n
sci 0 His
ADAPTATIONS
Assigneei can than
m ore
complete o netaskcapacity of a
Existforit m xp di for it im
Network optimisation models
More de nitions
• Flow through arcs
◦ Although links (undirected arcs) can have ow in either direction, this will not happen
simultaneously in opposite directions
‣ This would require a pair of directed arcs in opposite directions
‣ This can also be used to convert a network of directed and undirected arcs into a purely
directed network
◦ The actual ow will be the net ow
◦ Arc capacity - maximum ow through a directed arc
Week 2
EK Ch8 and HL Ch 9, 10
Transportation and assignment problems
Transportation problems
Terminology:
• Units of a commodity
• m sources FEASIBLESOLUTIONSPROPERTY
It
• n destinations dm libuted transportationproblemwillh ave
eachsource destinationhasafixedsupplydemand
• Supply from source i
• Demand Sid at destination j destinationsource
torecievedfromthe s olution
feasible
it Is d
• Cost per unit distributed from source i to j
ai I COST ASSUMPTION
ofdistributingunitsanounitsbeing
cost
distributed cost Cijxnounits
A problem is a transportation problem if it can be described completely in a parameter table like
below, it satis es the requirements assumption and cost assumption, and is a cost minimisation
1 LINEARPROGRAMMINGFORMULATION 2 PARAMETERTABLE
Min Z Icijxi
sit ai si forit
sci
sci Otis
E
diforget n
NETWORKREPRESENTATION
ADAPTATIONS
can
source betimeatwhichavailableand
astimew henrequiredcosts
destinations also
thenincludestoragecosts41327 3 29
EsEdcanintroducedummysourceand
t rack
keep ofexcesscapacity
Integer solutions property: For transportation problems where every
Sidhave an integer value, all
the basic variables (allocations) in every feasible (BF) solution, including the optimal one, also have
integer values
, Similar to transportation problems
Change transportation Problem
from problem type
transfer
Distributionpassest hrough enroute
points problem
transshipment
unit
withdifferent costsposttransfer
limito nquantity
upper toeachtransferpoint Minimum
cost flow
problem
Assignment problem
Assumptions:
• The number of assignees and the number of tasks are the same, n
• Each assignee is to be assigned exactly one task
• Each task is to be performed by exactly one assigned
• There is a cost associated with assignee i performing task j
Cij
• The objective is to determine how all n assignments should be made to minimise the total cost
1 LINEARPROGRAMMINGFORMULATION a NETWORKREPRESENTATION
Min Z Icijxi it
if performs
sitElegy forit m
Esci t forget n
sci 0 His
ADAPTATIONS
Assigneei can than
m ore
complete o netaskcapacity of a
Existforit m xp di for it im
Network optimisation models
More de nitions
• Flow through arcs
◦ Although links (undirected arcs) can have ow in either direction, this will not happen
simultaneously in opposite directions
‣ This would require a pair of directed arcs in opposite directions
‣ This can also be used to convert a network of directed and undirected arcs into a purely
directed network
◦ The actual ow will be the net ow
◦ Arc capacity - maximum ow through a directed arc
