F O R M U L A T I O N S

DOOR TO DOOR FREIGHT TRANSPORTATION

PROJET RESPET

• Aims to develop quantitative approaches to door to

door freight transportation

• Members:

• LAAS-CNRS

• INRIA

• LIA

• DHL

• JASSP

MAIN GOALS

• Model door to door network operation.

• Take into account conflicting objectives related to the subject

(economical, environmental, QoS, etc).

• Develop a methodology based on exact/hybrid algorithms.

• First year main focus:

• ILP modeling.

SCENARIO

Schedule transportation over a network using consolidation

terminals

MODEL 1

Assume containers are already assembled and ready to

be transported.

• 𝐺 = 𝑁, 𝐴 - graph representing

network;

• 𝑁 – set of terminals 𝑖, 𝑗, … , 𝑘 ;

• 𝐴 – set of routes 𝑖, 𝑗 , … , 𝑗, 𝑘 ;

• 𝑃 – set of containers;

• 𝑇 = *1,… , T+ – set of periods.

MODEL 1 - PARAMETERS

• Terminals:

• 𝑆𝑖 - Storage capacity of terminal 𝑖.

• 𝐶𝑖 - Storage cost of terminal 𝑖.

• 𝛿+ 𝑖 = 𝑗 𝑖, 𝑗 ∈ 𝐴+ – Set of terminals which 𝑖 has a direct

route to.

• 𝛿− 𝑖 = 𝑗 𝑗, 𝑖 ∈ 𝐴+ – Set of terminals that have a direct

route to 𝑖.

• Routes:

• Δ𝑖𝑗 - transportation time between terminals 𝑖 and 𝑗.

• 𝑄𝑖𝑗 - Capacity of route (𝑖, 𝑗).

• 𝐶𝑖𝑗 - Transportation cost of route 𝑖, 𝑗 .

MODEL 1 - PARAMETERS

• Containers:

• 𝜙𝑝 - Release period of container 𝑝.

• 𝜔𝑝 - Deadline period of container 𝑝.

• 𝑜𝑝 - Origin of container 𝑝.

• 𝑑𝑝 - Destination of container 𝑝.

𝜙𝑝 𝜔𝑝

MODEL 1 - OBJECTIVE

• Decision variables:

• 𝑥𝑖𝑗𝑝𝑡 =

1, 𝑖𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝 𝑖𝑠 𝑠𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑖 𝑡𝑜 𝑗 𝑎𝑡 𝑡 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• 𝑠𝑖𝑝𝑡 = 1, 𝑖𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝 𝑖𝑠 𝑠𝑡𝑜𝑟𝑒𝑑 𝑎𝑡 𝑖 𝑎𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• Objective:

• minimize:

( 𝐶𝑖𝑠𝑖𝑝𝑡 + 𝐶𝑖𝑗𝑥𝑖𝑗𝑝

𝑡 )

𝑖,𝑗 ∈𝐴𝑖∈𝑁𝑝∈𝑃𝑡∈𝑇

MODEL 1 - CONSTRAINTS

• Capacity constraints

𝑠𝑖𝑝𝑡

𝑝∈𝑃

≤ 𝑆𝑖 , ∀𝑖 ∈ 𝑁, ∀𝑡 ∈ 𝑇

𝑥𝑖𝑗𝑝𝑡

𝑝∈𝑃

≤ 𝑄𝑖𝑗 , ∀ 𝑖, 𝑗 ∈ 𝐴, ∀𝑡 ∈ 𝑇

MODEL 1 - CONSTRAINTS

• Departure and arrival constraints

𝑥𝑖𝑗𝑝𝑡 = 1

𝑗∈𝛿+(𝑖)

, ∀𝑝 ∈ 𝑃, 𝑖 = 𝑜𝑝

𝜔𝑝

𝑡=𝜙𝑝

𝑥𝑗𝑖𝑝𝑡 = 1

𝑗∈𝛿−(𝑖)

, ∀𝑝 ∈ 𝑃

𝜔𝑡

𝑡=𝜙𝑝

, 𝑖 = 𝑑𝑝

MODEL 1 - CONSTRAINTS

• Flow conservation constraint

𝑠𝑖𝑝𝑡−1 + 𝑥

𝑗𝑖𝑝

𝑡−Δ𝑗𝑖

𝑗∈𝛿−(𝑖)

= 𝑠𝑖𝑝𝑡 + 𝑥𝑖𝑗𝑝

𝑡

𝑗∈𝛿+(𝑖)

,

∀𝑝 ∈ 𝑃, ∀𝑡 ∈ 𝑇, 𝑖 ≠ 𝑜𝑝 ≠ 𝑑𝑝

MODEL 2

Assign orders to containers.

• 𝐿 – set of orders;

• Period of assignment is

not taken into account.

• 𝑃 – set of containers;

• Assume there are as

many containers as

orders ( 𝑃 = |𝐿|);

MODEL 2 - PARAMETERS

• Containers:

• 𝑉𝑝 - Storage capacity of container 𝑝.

• Orders:

• 𝑣𝑙 - weight of order 𝑙;

• 𝜙𝑙 - Release period of order 𝑙;

• 𝜔𝑙 - Deadline period of order 𝑙;

• 𝑜𝑙 - Origin of order 𝑙;

• 𝑑𝑙 - Destination of order 𝑙.

MODEL 2 - OBJECTIVE

• Decision variables:

• 𝑥𝑖𝑗𝑝𝑡 =

1, 𝑖𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝 𝑖𝑠 𝑠𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑖 𝑡𝑜 𝑗 𝑎𝑡 𝑡 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• 𝑠𝑖𝑝𝑡 = 1, 𝑖𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝 𝑖𝑠 𝑠𝑡𝑜𝑟𝑒𝑑 𝑎𝑡 𝑖 𝑎𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• 𝑦𝑙𝑝 = 1, 𝑖𝑓 𝑜𝑟𝑑𝑒𝑟 𝑙 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑜 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝. 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• Objective: • minimize:

( 𝐶𝑖𝑠𝑖𝑝𝑡 + 𝐶𝑖𝑗𝑥𝑖𝑗𝑝

𝑡 )

𝑖,𝑗 ∈𝐴𝑖∈𝑁𝑝∈𝑃𝑡∈𝑇

MODEL 2 - CONSTRAINTS

• Capacity constraints

𝑠𝑖𝑝𝑡

𝑝∈𝑃

≤ 𝑆𝑖 , ∀𝑖 ∈ 𝑁, ∀𝑡 ∈ 𝑇

𝑥𝑖𝑗𝑝𝑡

𝑝∈𝑃

≤ 𝑄𝑖𝑗 , ∀ 𝑖, 𝑗 ∈ 𝐴, ∀𝑡 ∈ 𝑇

𝑣𝑙𝑦𝑙𝑝𝑙∈𝐿

≤ 𝑉𝑝, ∀𝑝 ∈ 𝑃

MODEL 2 - CONSTRAINTS

• Assignment constraints

𝑦𝑙𝑝𝑝∈𝑃

= 1, ∀𝑙 ∈ 𝐿

𝑦𝑙𝑝 + 𝑦𝑚𝑝 ≤ 1, ∀𝑝 ∈ 𝑃, ∀𝑙,𝑚 ∈ 𝐿, 𝑑𝑙 ≠ 𝑑𝑚

MODEL 2 - CONSTRAINTS

Origin and destination of each container is unknown apriori.

• Departure and arrival constraints

𝑥𝑖𝑗𝑝𝑡

𝑗∈𝛿+(𝑖)

≥ 𝑦𝑙𝑝, ∀𝑙 ∈ 𝐿, ∀𝑝 ∈ 𝑃, 𝑖 = 𝑜𝑙

𝜔𝑙

𝑡=𝜙𝑙

𝑥𝑗𝑖𝑝𝑡

𝑗∈𝛿−(𝑖)

≥ 𝑦𝑙𝑝, ∀𝑖 ∈ 𝐿, ∀𝑝 ∈ 𝑃, 𝑖 = 𝑑𝑙

𝜔𝑙

𝑡=𝜙𝑙

MODEL 2 - CONSTRAINTS

Origin and destination of each container is unknown apriori.

• Flow conservation constraints

𝑠𝑖𝑝𝑡−1 + 𝑥𝑗𝑖𝑝

𝑡−Δ𝑗𝑖

𝑗∈𝛿−(𝑖)

≤ 𝑠𝑖𝑝𝑡 + 𝑥𝑖𝑗𝑝

𝑡

𝑗∈𝛿+ 𝑖

+ 𝒚𝒍𝒑𝒍∈𝑳𝒅𝒍=𝒊

, ∀𝑝 ∈ 𝑃, ∀𝑖 ∈ 𝑁, ∀𝑡 ∈ 𝑇

𝑠𝑖𝑝𝑡−1 + 𝑥𝑗𝑖𝑝

𝑡−Δ𝑗𝑖

𝑗∈𝛿−(𝑖)

+ 𝒚𝒍𝒑𝒍∈𝑳𝒐𝒍=𝒊

≥ 𝑠𝑖𝑝𝑡 + 𝑥𝑖𝑗𝑝

𝑡

𝑗∈𝛿+ 𝑖

, ∀𝑝 ∈ 𝑃, ∀𝑖 ∈ 𝑁, ∀𝑡 ∈ 𝑇

−1 ≤ 𝑥𝑖𝑗𝑝𝑡

𝑗∈𝛿+ 𝑖

− 𝑥𝑗𝑖𝑝𝑡−Δ𝑗𝑖

𝑗∈𝛿− 𝑖

≤ 1, ∀𝑝 ∈ 𝑃, ∀𝑖 ∈ 𝑁, ∀𝑡 ∈ 𝑇

MODEL 3

Take into account storage of orders

• Pick-up and delivery time

windows for each order;

• Time windows for containers

transportation.

• Additional cost if order is shipped

or arrives outside its time window

MODEL 3 - PARAMETERS

• 𝜙𝑙 - Time window for picking up order 𝑙 or

shipping container 𝑝;

• 𝜔𝑙 - Time window for delivery of order 𝑙 or

arrival of container 𝑝;

• 𝐶𝑙 - Storage cost of order 𝑙.

𝜙𝑙+ 𝜔𝑙

− 𝜔𝑙+ 𝜙𝑙

−

𝜙𝑝+ 𝜔𝑝

− 𝜔𝑝+ 𝜙𝑝

−

MODEL 3 - OBJECTIVE

• Decision variables:

• 𝑥𝑖𝑗𝑝𝑡 =

1, 𝑖𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝 𝑖𝑠 𝑠𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑖 𝑡𝑜 𝑗 𝑎𝑡 𝑡 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• 𝑠𝑖𝑝𝑡 = 1, 𝑖𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝 𝑖𝑠 𝑠𝑡𝑜𝑟𝑒𝑑 𝑎𝑡 𝑖 𝑎𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• 𝑦𝑙𝑝 = 1, 𝑖𝑓 𝑜𝑟𝑑𝑒𝑟 𝑙 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑜 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑝. 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

• 𝑧𝑙𝑡 = 1, 𝑖𝑓 𝑜𝑟𝑑𝑒𝑟 𝑙 𝑖𝑠 𝑠𝑡𝑜𝑟𝑒𝑑 𝑎𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡. 0, 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

MODEL 3 - OBJECTIVE

• Objective:

• minimize:

( 𝐶𝑖𝑠𝑖𝑝𝑡 + 𝐶𝑖𝑗𝑥𝑖𝑗𝑝

𝑡 )

𝑖,𝑗 ∈𝐴𝑖∈𝑁𝑝∈𝑃𝑡∈𝑇

+ 𝐶𝑙𝑧𝑙𝑡

𝑡∈𝑇𝑙∈𝐿

MODEL 3 - CONSTRAINTS

• Departure and arrival constraints

𝑥𝑖𝑗𝑝𝑡

𝑗∈𝛿+(𝑖)

≥ 𝑦𝑙𝑝, ∀𝑖 ∈ 𝑁, ∀𝑝 ∈ 𝑃

𝜙𝑝+

𝑡=max (𝜙𝑙−,𝜙𝑝−)

𝑥𝑗𝑖𝑝

𝑡 −Δ𝑖𝑗

𝑗∈𝛿−(𝑖)

≥ 𝑦𝑙𝑝, ∀𝑖 ∈ 𝑁, ∀𝑝 ∈ 𝑃

min(𝜔𝑙+,𝜔𝑝+)

𝑡=𝜔𝑝−

𝜙𝑙+ 𝜔𝑙

− 𝜔𝑙+ 𝜙𝑙

−

𝜙𝑝+ 𝜔𝑝

− 𝜔𝑝+ 𝜙𝑝

−

MODEL 3 - CONSTRAINTS

• Order storage constraints

𝑧𝑙𝑡 ≥ 𝑦𝑙𝑝 − 𝑥𝑖𝑗𝑝

𝑡′𝑡

𝑡′=𝜙𝑙+𝑗∈𝛿+ 𝑖

, ∀𝑙 ∈ 𝐿, ∀𝑝 ∈ 𝑃, 𝑡 ∈ 𝜙𝑙+, 𝜙𝑝+ , 𝑖 = 𝑜𝑙

𝑧𝑙𝑡 ≥ 𝑦𝑙𝑝 + 𝑥𝑗𝑖𝑝

𝑡′𝜔𝑙−

𝑡′=𝑡𝑗∈𝛿− 𝑖

− 1, ∀𝑙 ∈ 𝐿, ∀𝑝 ∈ 𝑃, 𝑡 ∈ 𝜔𝑝−, 𝜔𝑙− , 𝑖 = 𝑑𝑙

𝜙𝑙+ 𝜔𝑙

− 𝜔𝑙+ 𝜙𝑙

−

𝜙𝑝+ 𝜔𝑝

− 𝜔𝑝+ 𝜙𝑝

−

MODEL 4

Take into account different transportation modes and

vehicles

• V1 = A – B – C

• V2 = A – B – D

• V3 = B - C.

• Different mode terminals

and mode transfer arcs

A

B

C

D

A B

C

D

MODEL 4 – TIME SPACE NETWORK

• 𝑹 − 𝒔𝒆𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒗𝒆𝒉𝒊𝒄𝒍𝒆𝒔:

• Each vehicle v is represented by a different network.

• 𝐺𝑟 = (𝑉𝑟 , 𝐴𝑟) - time space network of vehicle 𝑟.

• Transport network is the union of all vehicles

• 𝐺 = (𝑉, 𝐴).

• 𝑉 = 𝑉𝑟𝑟∈𝑅 - All vehicle terminals;

• 𝐴 = 𝐴𝑡 ∪ 𝐴𝑠 ∪ 𝐴𝑚;

• 𝐴 = 𝐴𝑟𝑟∈𝑅 - All vehicle routes;

PERSPECTIVES

• Take into account conflicting objectives related to

the subject (economical, environmental, QoS, etc).

• Develop a methodology based on exact/hybrid

algorithms.

Thank you!