C OMPETENCIAS ESPECÍFICAS DEL DOMINIO NUMÉRICO

In the basic model, two decisions are made simultaneously to minimize the overall cost: (1) the location of the hubs and (2) the allocation of non-hub nodes to “home” hubs. We follow the three conventional assumptions in HLPs listed in Sec.2.2.5: (1) a fully interconnected backbone network, (2) a fixed discount factor



on hub arcs and (3) a strict and restrictive H/S network structure. Other assumptions are listed as follows. Most of them are also applicable to the extension models. If relaxed, they are pointed out accordingly.

(1) All non-hub cities are directly connected to their unique “home” hubs. That is, the tributary network is “star” shaped with the single-allocation criterion.

(2) Hubs are fully linked by direct flight. That is, the backbone network is fully interconnected.

(3) The time window for the earliest departure and the latest arrival of aircraft is the same for all potential hubs.

(4) Air cost is linearly dependent on distance and traffic volume. (5) There are no capacity constraints on hubs or arcs.

186 _{For next morning EDS, parcels arrive before 12:00 next business day. For next day EDS, parcels arrive before 18:00 next business day. Also see}

Tab.1-2 in Sec.1.3.3.

(6) Demand volume between all O-D pairs is deterministic.

(7) Time window for air transportation satisfies the longest direct flight between hubs. (8) All the distance applies to the triangle inequality.

It is also to underline that there is no assumption on symmetric demand volume, which is an assumption quite commonly applied by previous studies188_.

3.2.2. Model formulation

In this section, we propose a 0-1 integer programming model for our network planning problem. HLPs can be formulated in a variety of ways, depending on the form of decision variables. Different formulation methods for the same problem result in different numbers of variables and constraints189_{. In this dissertation, we use the} four-subscript formulation method introduced by Campbell for a linear model190_.

The model is based on a network, in which the demand node set N and the potential hub set H (HN) are

identified. The model is to locate hubs in the potential hub set H and allocate the remaining demand nodes in N to the located hubs under the constraints of the maximum hub coverage radius with the objective of mini- mizing the total cost. The hub coverage radius is considered in the form of distance by transforming the time window for the feeder transportation into a distance bound with an average speed of truck on highway. The decision variables and parameters for the basic model are listed in Tab.3-3 and 3-4.

Decision variables Description ik

x

(iN k, H) Refer to both location and allocation variable. xik 1if node i is allocated to hub k, otherwise 0. When x_kk 1, k is a hub.

ijkm

y

( ,i jN k m, ,

H)

Refer to path variable. yijkm 1, if flow from node i to j via hubs k and m in

that order. Otherwise 0. Note yijkm xik xjm.

Table 3-3: Decision variables in the basic model

Parameters Description

188 _{See e.g. Kuby /Gray (1993), pp. 1-12; Lin et al. (2003), pp.255-265.}

189 _{Campbell et al (2005b) listed three different integer programming formulations appeared in former studies: binary allocation variable by O’Kelly,}

four-subscript formulation by Campbell (1994) and flow tracking variable by Ernst and Krishnamoorthy (1996, 1998a). The first one is for single allocation problems. Because of the quadratic terms in the objective function, it was proved to be a poor choice for solution. The second and third ones are applicable to multiple-allocation problems. See Campbell et al.(2005b), p.1557; O’Kelly (1987), pp. 394-404; Campbell (1994), pp. 387-405; Ernst/Krishnamoorthy (1996), pp.139-154; Ernst/Krishnamoorthy (1998a), pp.100-112.

N Set of demand nodes

H

Set of potential hubs, HN

ij

w

Demand volume from node (i iN)to node (j jN) k

fh

Fixed cost of opening a new hub or expanding an existing hub at potential hub node

(

)

k kH

km



Cost rate by air in backbone network (per kilo- kilometer),

k m,

H

ik

 _{Cost rate by truck in feeder network (per kilo- kilometer),}

_iN

_,

_kH

ik

d

Distance by highway between non-hub node

i i(

N)

and its “home” hub

k k(

H)

km

d

Distance by air from hub node k to m ,

k m,

H

, dkm dmk

k

D _{Distance bound as hub coverage radius of potential hub node k,}

_k_H

Table 3-4: Parameters in the basic model

With the defined parameters and decision variables, we formulate the objective function as follows:

Minimize k kk ij ijkm



ik ik km km mj mj



k H i N j N k H m H fh x w y



d



d



d        



  

(3-1)

In the objective function, the first term sums the fixed costs of hubs either for new establishment or expansion. The second term calculates the total transportation cost. The demand volume and costs are calculated on daily basis. S. T. _ik 1 k H x  



 i N (3-2) ik kk

x

x

   i

N,

k

H

(3-3)

 0,1

ik

x



   i

N,

k

H

(3-4) 0dik xik Dk

   i

N,

k

H

(3-5) ijkm ik m H

y

x







   i j,

N,

k

H

(3-6) ijkm jm k H

y

x







i j,

N, m

H

(3-7)

1

ijkm m H k H

y

 



 

,

i j

N





(3-8)

 0,1

ijkm

y



i j,

N,k m,

H

(3-9)

Constraints (3-2) and (3-4) ensure that every non-hub node is allocated to exactly one hub in conformity to the single-allocation criterion. Constraints (3-3) state that non-hub node cannot be allocated to another node un- less that node is a hub. And when a node is hub itself, it is allocated to itself. Constraints (3-5) make sure that all non-hub nodes are allocated to their “home” hubs subject to the distance criteria. Constraints (3-6) state

that if node i is allocated to hub k, all the demand from node i to any other node j must go through some hub m. If node i and j are in the same hub region, m=k. Constraints (3-7) have a similar interpretation involving the demand from any node i to node j, which is assigned to hub m. Constraints (3-8) guarantee that there is one path for each O-D pair. Note that constraints (3-8), together with constraints (3-6) and (3-7), ensure that every node is allocated to only one hub. Constraints (3-9) are the constraints of path variables.

In the basic model, when the hub locations are fixed, the allocation sub-problem is the same as the allocation sub-problem of the single allocation p-hub median problem, which has been proved to be NP-hard by Kara191_. So the basic model here belongs to the class of NP-hard problems. If we set

n



N

_and

h

H

, the model has 2 2

(n h

nh)

_{binary variables and}

(2n h

2

n

2

2nhn)

linear constraints.

3.3. Flow-dependent air cost

In document DOCTORADO EN ENSEÑANZA DE LAS CIENCIAS MENCIÓN MATEMÁTICA. Universidad Nacional del Centro de la Provincia de Buenos Aires (página 55-60)