In the following we describe the overall aggregate LP model that we devel- oped to optimize the earthwork activities.
Variables.
The decision variables required to formulate the problem are denoted by xmt ij
and represent the flow (expressed in cubic meters) of material m ∈ M in period t ∈ T on arc (i, j) ∈ A, and fmt
k , giving, respectively, the quantity of
material m stored in a temporary depot k ∈ L, carried to a dump site k ∈ D, or bought from a quarry k ∈ S, at period t. Non-negativity constraints are imposed on both variables, that is:
xmtij ≥ 0 (i, j) ∈ A, m ∈ M, t ∈ T , (5.1) fkmt≥ 0 k ∈ L ∪ D ∪ S, m ∈ M, t ∈ T . (5.2)
Objective Function.
Our aim is to provide an optimal material distribution to complete the project within the specified time, minimizing the overall costs. To each arc (i, j) ∈ A is associated a transportation cost cij. Moreover let cmk denote
the cost for storing, loading, or acquiring a unit of material m ∈ M into a temporary depot k ∈ L, into a dump site k ∈ D, or from a quarry k ∈ S, respectively. The objective function is then:
min zaggr = X (i,j)∈A X m∈M X t∈T cijxmtij + X k∈L∪D∪S X m∈M X t∈T cmkfkmt. (5.3)
Arc capacity constraints.
Every arc (i, j) ∈ A has a maximum capacity Ut
ij (in cubic meters) that may
vary according to the period t ∈ T . This is enforced by:
X
m∈M
xmtij ≤ Uijt (i, j) ∈ A, t ∈ T . (5.4)
To account for the fact that some nodes can be unaccessible during some periods (e.g., because they can only be reached after the erection of an access road, or because some construction activities temporary block part of the network) a capacity zero is associated to the arcs connecting those nodes in those periods.
Vehicle capacity constraints.
Materials are moved by trucks, so the quantity of materials moved along the network is constrained by the quantity of trucks available. The fleet is composed by a set W of types of trucks, each of those may transport a set M(w) of materials and is assigned an aggregate estimation Vcap(w) of its
transportation capacity, expressed in km · m3. In practice, V
cap(w) gives
the total amount of material that this type of truck can move in a period, and is estimated on the basis of past activities performed by Strabag. To each arc (i, j) ∈ A is assigned a non-negative length `ij in km. The overall
transportation capacity is imposed by:
X (i,j)∈A X m∈M(w) `ijxmtij ≤ Vcap(w) t ∈ T , w ∈ W. (5.5) Digging constraints.
Recall that each task i ∈ Id is assigned to a digging location h(i) and
requires a certain amount di of material m(i) to be dug in the time window
[ts(i),te(i)]. Therefore it must be:
xmti,h(i)= 0 i ∈ Id, m ∈ M: m 6= m(i), (5.6)
xm(i),ti,h(i) = 0 i ∈ Id, t ∈ T : t < ts(i) or t > te(i), (5.7) te(i)
X
t=ts(i)
xm(i),ti,h(i) = di i ∈ Id. (5.8)
Filling constraints.
As for digging, each task i ∈ If is assigned to location h(i)and has its own
therefore:
xmth(i),i = 0 i ∈ If, m ∈ M: m 6= m(i), (5.9)
xm(i),th(i),i = 0 i ∈ If, t ∈ T : t < ts(i) or t > te(i), (5.10) te(i)
X
t=ts(i)
xm(i),th(i),i = di i ∈ If. (5.11)
Flow conservation at material locations, access points, and exter- nal nodes.
To ensure the conservation flow for each material m and period t we need to impose the following constraints for material locations, access points, and external nodes: X (i,j)∈A xmtij = X (j,i)∈A xmtji j ∈ H ∪ E ∪ B, m ∈ M, t ∈ T . (5.12)
Temporary depots constraints.
Recall that Ms= {rawI, rawII} gives the set of materials that can be stored
into temporary depots during the construction process. Each depot k ∈ L has a maximum storing capacity Qkthat must be respected at each period,
thus:
X
m∈Ms
fkmt≤ Qk k ∈ L, t ∈ T . (5.13)
In a rolling horizon perspective, perhaps, we might consider just a set of periods of the entire construction process, thus having an initial level of storage imposed by the evolution of the construction activities and/or a final desired level of storage. Let therefore fm0
k be the initial stock of material
m ∈ Ms in depot k at the beginning of the first period (i.e., the initial
available volume), and fm,Tmax+1
k the upper bound on the closing stock of
min k at the end of the last period. We have the following material balance constraints: X (i,k)∈A xmtik + fkm,t−1= X (k,i)∈A xmtki + fkmt k ∈ L, m ∈ Ms, t ∈ T . (5.14)
The final storage volumes are constrained by:
fm,Tmax
k ≤ f
m,Tmax+1
k k ∈ L, m ∈ Ms. (5.15)
To assume, for example, that depots must be empty before and after the completion of the overall project, it is enough to set fm0
k and f
m,Tmax+1
k to
Dump sites constraints.
The amount of waste deposited in a dump site k ∈ D is limited by the total disposal capacity Ck, therefore:
X (i,k)∈A X m∈M X t∈T xmtik ≤ Ck k ∈ D. (5.16)
Moreover, in each period there is a maximum allowable quantity of material m,say Qm
k,that can be disposed of in the dump site, and hence:
X
(i,k)∈A
xmtik ≤ Qmk k ∈ D, m ∈ M, t ∈ T . (5.17)
Quarries constraints.
As for the dump sites, each quarry k ∈ S has an overall capacity Ck and a
maximum capacity Qm
k for each material m in each period, therefore:
X (k,i)∈A X m∈M X t∈T xmtki ≤ Ck k ∈ S, (5.18) X (k,i)∈A xmtki ≤ Qm k k ∈ S, m ∈ M, t ∈ T . (5.19)
Separation plants constraints.
Recall only rawI material may be carried to the separation plants (see Figure 5.1), and the exiting set of materials is Mr= {recI, recII, recWaste}.
It follows that the flows entering nodes k ∈ Fr for all materials m ∈ M
other than rawI, as well as the flows leaving the node for all materials other than Mr must be zero, that is:
xmtik = 0 (i, k) ∈ A : k ∈ Fr, m ∈ M \ {rawI}, t ∈ T , (5.20)
xmtki = 0 (k, i) ∈ A : k ∈ Fr, m ∈ M \ Mr, t ∈ T . (5.21)
The capacity of any facility represents a physical limitation that must be considered. The accumulated material inflow should not exceed the max- imum allowable operating capacity Qk of each separation plant k ∈ Fr.
Thus, we get:
X
(i,k)∈A
xrawI,tik ≤ Qk k ∈ Fr, t ∈ T . (5.22)
Each separation plant has its own and specified recipe, that states the quan- tities of exiting materials obtained from a unit of rawI. Formally, recall ϕm r
a unit of rawI at plant k ∈ Fr, and note that, for feasibility, we must have
P
m∈Mrϕ
m
r = 1. The conservation flow at each separation plant is stated
by: X (k,i)∈A xmtki = ϕmr X (i,k)∈A xrawI,tik k ∈ Fr, m ∈ Mr, t ∈ T . (5.23)
Asphalt mixing plants constraints.
Each asphalt mixing plant k ∈ Fareceives in input only Ma= {recI, bitumen},
and gives in output only asphalt, thus:
xmtki = 0 (k, i) ∈ A : k ∈ Fa, m ∈ M \ {asphalt}, t ∈ T , (5.24)
xmtik = 0 (i, k) ∈ A : k ∈ Fa, m ∈ M \ Ma, t ∈ T . (5.25)
The plant has a total production capacity Qk for each period. Its recipe is
expressed by the ϕm
a input, which gives the quantity of material m ∈ Ma
necessary for producing a unit quantity of asphalt at plant k ∈ Fa. For
feasibility, the values of ϕm
a satisfy P m∈Maϕ m a = 1. We thus get: X (k,i)∈A xasphalt,tki ≤ Qk k ∈ Fa, t ∈ T , (5.26) X (i,k)∈A xmtik = ϕma X (k,i)∈A xasphalt,tki k ∈ Fa, m ∈ Ma, t ∈ T . (5.27)
Concrete mixing plants constraints.
These constraints are equivalent to the ones for the asphalt mixing plants. Each concrete mixing plant k ∈ Fc receives in input materials Mc =
{recII, cement}, gives in output concrete, and has total processing capacity Qkfor each period. Its recipe is expressed by ϕmc , that gives the quantity of
material m ∈ Mc necessary for producing a unit quantity of concrete, and
is such that P m∈Mcϕ m c = 1. We thus obtain: xmtki = 0 (k, i) ∈ A : k ∈ Fc, m ∈ M \ {concrete}, t ∈ T , (5.28) xmtik = 0 (i, k) ∈ A : k ∈ Fc, m ∈ M \ Mc, t ∈ T , (5.29) X (k,i)∈A xconcrete,tki ≤ Qk k ∈ Fc, t ∈ T , (5.30) X (i,k)∈A xmtik = ϕmc X (k,i)∈A xasphalt,tki k ∈ Fc, m ∈ Mc, t ∈ T . (5.31)
The resulting aggregate model is thus to minimize (6.3), subject to (6.1)– (6.2), (6.4)–(5.31).