Week 3 - MCF

6 Managing freight transport6.1 Introduction6.2 Freight traffic assignment problems6.3 Service network design problems6.4 Vehicle allocation problems6.5 A dynamic driver assignment problem6.6 Fleet composition6.7 Shipment consolidation6.8 Vehicle routing problems6.9 Real-time vehicle routing problems

6.10 Integrated location and routing problems6.11 Vendor-managed inventory routing6.12 Case study: Air network design at Intexpress6.13 Case study: Meter reader routing and scheduling at Socal6.14 Case study: Dynamic vehicle-dispatching problem with pickups

and deliveries at eCourier6.15 Questions and problems

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 1 / 58

6 Managing freight transport Freight traffic assignment problems

Freight TAPs

- Amount to determining a least-cost routing of goods over anexisting network of transport services from their origins (e.g.manufacturing plants) to their destinations (e.g. retailoutlets);

- can be casted as network flow (NF) problems, including, asspecial cases, the shortest path problem and thetransportation problem;

- can be classified as static or dynamic.



Freight TAPs






Freight TAPs






Static models

- Suitable when the decisions are not affected by time;- formulated on a directed graph (or multigraph) G = (V ,A):

> V : set of facilities (terminals, plants and warehouses);> A : set of arcs representing possible transport services

linking the facilities;> vertices represent origins of transport demand for one

or several products, destinations, or transshipmentpoints;

> K : set of traffic classes (commodities);> with each arc is associated a cost (possibly dependent

on the amount of freight flow on the arc) and a capacity;> cost functions may represent both monetary costs and

congestion effects arising at terminals.



Static models










Static models










Static models










Static models










Static models










Static models










Static models










Dynamic models (1/4)- Transport services over a given planning horizon modelled

through a time-expanded directed graph;- planning horizon divided into time periods t1, t2, . . . ;- physical network replicated in each time period;- an arc connecting two representations of the same terminal

at two different time periods represents freight waiting to beloaded onto an incoming vehicle, or the time required forfreight classification at the terminal;

- an arc connecting two representations of different terminalsdescribes a transport service;

- further vertices and arcs models the arrival of commoditiesat destinations and impose penalties in case of delays;

- with each link may be associated a capacity and a cost (asin static formulations).



















































Dynamic models (2/4)

Example- Static network: see Figure 1;- time-expanded network: see Figure 2;- case of planning horizon of four days:

> four vertices for each terminal (Ai , i = 1, . . . ,4, describesterminal A at the ith day);

> some arcs (such as (A1, B3)) represent transportservices; others (such as (B2, B3)) describecommodities standing idle at terminals;

> supersinks (such as C). Costs on the arcs entering thesupersinks represent economic sanctions andpenalties in case of transport service failure.














































A

B C

Figure 1: A static representation of a three-terminal transport system.




Supersink forterminal C

Time

A1

A2

A3

A4

B1 C1

C2

C3

C4

B2

B3

B4

Figure 2: Dynamic network representation of the transport systemillustrated in Figure 1.



Multicommodity minimum-cost flowformulation (1/3)

- O(k ), k ∈K : set of origins of commodity k ;- D(k ), k ∈K : set of destinations of commodity k ;- T(k ), k ∈K : set of transshipment points with respect to

commodity k ;- ok

i , i ∈O(k ), k ∈K : supply of commodity k of vertex i;- dk

i , i ∈D(k ), k ∈K : demand of commodity k of vertex i;- uij, (i, j) ∈A : capacity of arc (i, j);- uk

ij , (i, j) ∈A , k ∈K : maximum flow of commodity k on arc(i, j);

- xkij ,(i, j) ∈A ,k ∈K ; decision variables representing the flow

of commodity k on arc (i, j);- Ck

ij (xkij ), (i, j) ∈A , k ∈K : cost for transporting xk

ij flow units ofcommodity k on arc (i, j).





commodity k ;- ok












commodity k ;- ok












commodity k ;- ok












commodity k ;- ok












commodity k ;- ok












commodity k ;- ok












commodity k ;- ok












commodity k ;- ok











Minimize∑

k∈K

∑

(i,j)∈ACk

ij (xkij ) (1)

subject to

∑

{j∈V :(i,j)∈A }

xkij −

∑

{j∈V :(j,i)∈A }

xkji =

oki , if i ∈O(k )−dk

i , if i ∈D(k )0, if i ∈T(k )

i ∈V ,

k ∈K(2)

xkij ≤ uk

ij , (i, j) ∈A , k ∈K (3)∑

k∈Kxk

ij ≤ uij ,(i, j) ∈A (4)

xkij ≥ 0, (i, j) ∈A , k ∈K




- Objective function (1): total cost;- constraints (2): flow conservation constraints holding at

each vertex i ∈V for each commodity k ∈K ;- constraints (3): flow of each commodity k ∈K does not

exceed capacity ukij on each arc (i, j) ∈A ;

- constraints (4) (bundle constraints): for each (i, j) ∈A , thetotal flow on arc (i, j) is not greater than the capacity uij.

Note. oki , k ∈K , i ∈O(k ) and dk

i , k ∈K , i ∈D(k ), mustsatisfy:

∑

i∈O(k )ok

i =∑

i∈D(k )dk

i , k ∈K ,

otherwise the problem is infeasible.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 10 / 58









∑

i∈O(k )ok

i =∑

i∈D(k )dk

i , k ∈K ,










∑

i∈O(k )ok

i =∑

i∈D(k )dk

i , k ∈K ,










∑

i∈O(k )ok

i =∑

i∈D(k )dk

i , k ∈K ,










∑

i∈O(k )ok

i =∑

i∈D(k )dk

i , k ∈K ,



Linear single-commodity minimum-cost flowproblem (LMCFP)

Minimize∑

(i,j)∈Acijxij (5)

subject to

∑

{j∈V :(i,j)∈A }

xij−∑

{j∈V :(j,i)∈A }

xji =

oi , if i ∈O−di , if i ∈D,

0, if i ∈Ti ∈V (6)

xij ≤ uij , (i, j) ∈A (7)xij ≥ 0, (i, j) ∈A (8)


Documents

Week 3 - MCF