This section provides an overview of the network that we use for modeling the highway construction project, as well as the main notation that we adopt in the remaining of the chapter. To account for the fact that the MPS bounds each activity of the project in time by a time window, we consider an environment in which the overall horizon is discretized into a set of uniform periods (weeks in our study), T = {1, 2, . . . , Tmax}. The
network is represented by a graph G = (V, A) defined by a collection of nodes V and arcs A, and is schematized in Figure 5.2.
Figure 5.2: Network representation.
Among all the nodes in V, we identify a set H of nodes representing material locations, that is, cut and fill areas where different types of materials can be dug or filled in (e.g., nodes 1, . . . , 12 in Figure 5.2. In other words, H represents the discretization of the highway being built. The material locations might need more than one digging/filling of one or more materials to be completed in different time windows. We thus need to introduce a new set of nodes (in addition to the material location ones) to deal with the digging and filling activities of the same material with overlapping time windows to be performed on the material location. Thus, we consider a set I of nodes representing tasks, that is, digging or filling activities required for a certain material in a given area. In particular, in I we identify If
as the set of filling tasks (e.g., nodes 17, 19, 20, and 21) and Id as the set
of digging tasks (e.g., nodes 13, 14, 15, 16, and 18), with If ∪ Id = I
and If ∩ Id = ∅. Each task i ∈ I is associated with a unique location
h(i) ∈ H and a unique material m(i) ∈ M, being M the set of all earth materials involved in the construction process. Moreover, each task i has its own quantity di, that must be fully dug/filled in a time window [ts(i), te(i)]
The network also includes a set D of dump sites, representing locations where materials can be disposed of (e.g., nodes 31 and 32), and a set S of quarries, representing sources where materials can be acquired to respond to filling requests (e.g., nodes 28 and 29). Note that in our model it is allowed to dispose of some material in a dump site and acquire the same material from a quarry if this leads to decreased costs. Dug materials can also be stored in temporary depots (e.g., nodes 25 and 26), whose set is denoted by L. Facilities which process materials for recycling are represented by a set of nodes F. We distinguish among separation plants (Fr), mixing plants
for asphalt (Fa), and mixing plants for concrete (Fc). The union of all the
dump sites, quarries, temporary depots and recycling facilities is denoted by N .
The highway being built is accessible from the outside through a set E of access points. Access points (pictured by diamond nodes in Figure 5.2, see, e.g., nodes 24, 27, 30, and 33) represent the passage between the existing public road network and the network that is still private. The public network is simply schematized by dashed lines in Figure 5.2, but it is, in general, quite complex. It is composed by a subnetwork (represented by dashed lines in in Figure5.1) including a set of nodes, denoted by B, and a set of arcs connecting these nodes. This information is needed to represent the different capacitated paths that can be used to reach the set N of outside facilities.
Connections among the nodes are represented by the overall set A of arcs. Note that in general there is not an arc for each pair of nodes. Indeed, movement of any material into a quarry is not permitted, therefore only leaving arcs from S exist and they are headed to the access points or to the public network. Similarly, movement of materials out of a dump is not permitted, thus only arcs entering in D exist and they come from the access points or from the public network. Plants and temporary depots are also only connected to access points and/or to the public network, but in this case the arcs are in both directions. Access points are only connected to nodes in N , B, and H. All the arcs incident the access points are in both directions but those entering in D or leaving S. The nodes of the public network (set B) can only be connected to the nodes of E, N , and to each other. Each material location h ∈ H, is connected to other material locations by arcs in both directions. Material locations can also be connected to access points with arcs in both directions and each material location is connected to at least one filling and/or digging task. No other arcs are incident to the material locations. Each filling task i ∈ If is connected only to its material
location h(i) ∈ H by an entering arc, while each digging task i ∈ Id is
connected only to its material location h(i) ∈ H by a leaving arc. To each arc (i, j) ∈ A we associate a maximum flow capacity Ut
ij for each period
t ∈ T.
terials M, the subset Ms of materials that can be stored into temporary
depots, constituted by rawI and rawII, the subset Mr of materials exiting
the separation plants, constituted by recI, recII, and recWaste, the subset Maof materials entering the asphalt mixing plants, constituted by recI and
bitumen, and the subset Mc of materials entering concrete mixing plants,
constituted by recII and cement. We denote by ϕm
r the quantity of recy-
cled aggregate m ∈ Mr that can be obtained from a unit quantity of rawI
material at plant k ∈ Fr.Similarly, the quantity of m ∈ Ma necessary for
producing a unit quantity of asphalt at plant k ∈ Fais denoted by ϕma,and
the quantity of m ∈ Mcnecessary for producing a unit quantity of concrete
at plant k ∈ Fc is denoted by ϕmc .
To move the materials, different types of capacitated trucks can be used. We consider four types, that can load, respectively, bitumen, concrete, ce- ment, and all the other materials. We denote by W the set of types of truck, by M(w) the set of materials that a truck of type w ∈ W can load, and by Vcap(w) the overall transportation capacity of the fleet of trucks of type w.