## Location-Routing for Distribution Centers

Read this article. One objective of this paper is to determine distribution center locations. Compare and contrast the two cases presented.

### Problem formulation

The following notations are used to describe the problem.

#### Sets

: Transportation modes,

: Set of intermediate nodes (between supplier and DCs) crossed by only one transportation mode

: Set of intermediate nodes (between supplier and DCs) crossed by road and railway transportation modes only

: Set of intermediate nodes (between supplier and DCs) crossed by road and seaway transportation modes only

: Set of intermediate nodes (between supplier and DCs) crossed by railway and seaway transportation modes only

: Set of intermediate nodes (between supplier and DCs) crossed by more than one transportation modes,

: Set of nodes including supplier node as well as the nodes between supplier and potential

: Set of arcs for each transportation mode from supplier to DCs,

: Set of potential DCs

: Set of potential DCs and retailers

: Set of vehicles to be used to deliver products from DCs to retailers

#### Parameters

: Distance between nodes and

: Distance between nodes and

: Raised demand by retailer

: Unit product distribution cost per unit distance from DCs to retailers by vehicle

: Unit product transportation cost per unit distance for each mode

: Fixed cost of providing a mode changing facility at node ,

: Fixed cost of establishing a DC at node

#### Decision variables

: Amount of products passing, on the transportation mode m, through link to

Moreover, is
a slack variable used in sub-tour elimination constraint, and is
large enough constant parameter. Note that, if three different modes
cross over one another a node, a combination of the modes is accounted
for and the node will be taken as a member of all three sets, namely
.

##### The problem formulation is as follows:

(1)

(2)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Objective
function (1) seeks to minimize costs including multimodal
transportation costs and location-routing costs. First and second
sentences are about location-routing costs which calculate cost of
routing from DCs to retailers and DCs establishment fixed cost,
respectively. Third and fourth sentences, respectively, calculate cost
of transportation from supplier to DC on multimodal network, and cost of
providing mode changing facilities as multimodal transportation costs.
Constraint set (2) defines flow conservation on multimodal network and
states that product flow inters to a node should be to equal product
flow exits it. Mode changing can be happened only on multimodal
terminals (nodes that more than one mode inters or exits the node). For
each transportation mode, if products inter to a multimodal network and
leave it with the same mode, mode changing will not happen and the
corresponding variable will be zero. Same logic uses for
changing modes. Constraint set (3) illustrate this for each node .
As accounted for by constraint set (4), sum of products arriving at a
DC should be equal to total demands raised by the assigned retailers to
that DC. Based on the constraint set (5), each retailer is assigned to
exactly one DC. Constraint set (6) ensures that no product can be
dispatched to a non-established DC and if DC establishment variable
takes zero, all the product flows ending to that DC take zero as well.
Constraint set (7) assigns each retailer to exactly one vehicle, while
the constraint set (8) limits the capacity of the vehicles and ensures
that summation of assigned customers demand does not pass vehicle
capacity. In each product delivery tour on road network, every vehicle
once entered a node will definitely exit that node and constraint set
(9) guarantees that. Constraint set (10) is to ensure that each and
every vehicle is allocated to at most one DC. The absence of common
links between DCs is promised by constraint set (11). The constraint set
(12) assures that a retailer is assigned to a DC if and only if there
is a route from the DC to that retailer. Constraint sets (13) and (14)
state that vehicles can be dispatched only from established DCs.
Constraint set (15) relates to sub-tour elimination, and constraints
(16), (17), and (18) define variables. The last three constraints
determine model variables signs.