In this section, a generic description of Distributed Logistic Chains (DLC) is provided with the aim of pointing out the transportation problems underlying such systems [152,153].
In doing so, it is worth underlining that a DLC not only has a logic structure consisting of the relations among the different nodes (each node can be a supplier and a receiver of other nodes) but also has a physical structure consisting of the geographical positions of each node and, in particular, of the transportation networks connecting them.
Then, formally, the logic structure of a DLC can be defined as the four-tuple
DLC= {S, NDLC, T, E} (5.20)
where
S = {S1, S2,. . ., SS}is the set of the suppliers that feed the network, such as, for instance, sea ports or production sites.
NDLC= {n1, n2,. . .nN}is the set of DLC internal nodes which have upstream (resp., downstream) nodes to be supplied from (resp., to supply). Such nodes can perform different operations on goods and can be manufacturers, assemblers/disassemblers or, in a more general framework, intermodal nodes in which consolidation/deconsolidation operations, mode change, and so on, are performed.
T = {t1, t2,. . ., tT}is the set of the transportation networks underlying the logistic chain, whose model has been already described in Section 5.2.2. In particular, each element of the set T is a transportation supply system. Hence, it can be represented by a graph ti = {Ni, Li}with the relevant link and path cost models, as described in Section 5.2.2. In this definition, the subscript i allows to distinguish among the different networks of the DLC.
E= {e1, e,. . ., eE}is the set of end users who generate orders and represent the final receivers of goods. The elements of such a set are retailers, households, and so on.
With this definition, each generic node of the logistic chain in the set NDLC, or in the subset Niof the nodes of the transportation network ti, can be either the input or the output for other nodes.
Coming back to the physical structure of the DLC, it is worth saying that the elements of the set T consist of both
Logic network
s
n2 n1
t1 t2
e1 n4
n3
e3 e2
s t1 t2
n1
n2
e1
n4 n3
e3 e2
Physical network FIGURE 5.9 Logic and physical structure of a DLC.
1. The transportation network (such as road networks, railways and maritime links)
2. The transportation operators (such as truckers, rail companies and shipping companies that agree with the other nodes of the DLC with the aim of scheduling the goods dispatching and then choosing the departing and delivery times of shipments, the transportation mode, and so on
Therefore, in the considered model, from the logic point of view, the elements of the set T are logistic nodes as well as those in the other sets. Hence, they have internal optimization problems, aiming to minimize costs and to maximize performances, and receive inputs from and provide outputs to the other nodes of the DLC, as described in the following section.
An example of the general scheme of DLC is presented in Figure 5.9, where both the logical interactions among the nodes and the physical transportation networks are depicted. In the generic example reported in such a figure, it is assumed that the end users e2 and e3 are in the same geographical sites of the nodes n3and n4, respectively; on the contrary, e1 is connected to the DLC via the transportation network t2.
With reference to the definitions provided in Section 5.2, and in particular to the general scheme of Figure 5.2, the geographical locations of the elements of the sets S, N and E, and the relevant activities, represent the activity system of the DLC that generates the freight transportation demand.
On the contrary, the elements of the set T represent the transportation supply systems providing infrastructures and services for their mobility.
In the following section, a general model of DLC management is proposed with the aim of pointing out the variables that influence the freight transportation choices, with particular reference to the path choice.
5.4.1 DISTRIBUTEDLOGISTICMANAGEMENT AND THETRANSPORTATIONPROBLEM
The earlier proposed definition of distributed logistic chain allows to easily define, at a very general level, the management problem characterizing it.
In doing so, it is possible to state that the decisions on the production and on the dispatching of goods are usually taken by the different elements of the DLC, with the aim of optimizing their costs (i.e., maximizing their profit) both in collaborative or competitive environments. In the first case, some elements of the supply can solve a common optimization problem, even if they are geographically separated. In the second case, each element of the supply chain has its own internal
ni–1 t ni+1
rddi–1,tj rddt,i+1j
tddi–1,tj tddt,i+1j FIGURE 5.10 Example of a three-node DLC.
optimization problem, and in some cases, two nodes can compete for providing the same kind of goods or service.
Moreover, it can be noted that the proposed representation of DLC is modular, that is, it allows to consider large network models or modifying existing ones by simply adding or deleting nodes and/or links.
Finally, it is worth saying that, for the sake of simplicity, only deterministic variables and user behaviours are considered. With such an assumption, users of the transportation systems connecting nodes of the DLC always choose the shortest path.
Then, as regards the logistic problem underlying DLC, consider the three-node scheme depicted in Figure 5.10, where the logic links among a transportation node and two generic nodes are reported.
With reference to such a scheme, it is possible to define the following variables:
• rddt,ij+1, that is, the due date time of the jthfreight shipment requested to the node t by the node ni+1
• tddt,ij+1, that is, the forecasted dispatching time of the jthshipment estimated by the node t, based on the cost and travel times due to the link traffic flows
• rddij−1,t, that is, the due date of the jthshipment requested by the node t to the node ni−1, so as to be able to deliver it on time
• tddij−1,t, that is, the delivery time of the jthshipment proposed to the node t by the node ni−1, depending on the production constraints of the node ni−1itself
These definitions provide some of the information that all the nodes of the DLC have to take into account, or provide, when they solve their internal optimization problems. In particular, with reference to the transportation node t, the terms rddt,ij+1, ∀j, represent the due date for the dispatching time. Hence, it is a constraint to fulfill. On the other hand, the terms rddij−1,tand tddij−1, t,∀j, represent, for it, variables to optimize: in particular, the first is imposed, as a due date constraint, to the upstream node ni−1; the second represents the optimal delivery times of the shipments to ni−1 and has to be compared with the requested due date rddt,ij+1.
For what concerns the optimization problem solved by the transportation node t, it can consist of the minimization of the differences, in terms of tardiness, between the requested and the proposed delivery times of all the shipments. In solving its optimization problem, the node t determines the best path in the graph and the optimal requested due date, for all the shipments to be proposed to the node ni−1. Note that a strict definition of the previous variables requires the statement of the constraints tddt,i+1j ≤ rddt,i+1j for all the shipments. Nevertheless, in general, these constraints are relaxed and considered, suitably weighted, as terms of the cost function. Then, the optimization problem results into the sum of two contributions that aim to
A. Minimize the internal costs: For transportation nodes, such a term of the cost function consists, for instance, of the minimization of the travel costs by means of the travel choices.
As said, such costs depend on the flows on the links of the transportation network and are usually determined not only by the shipment flows but also by passenger flows sharing
the same transportation supply system. By means of this term, the choices taken by freight transportation operators are influenced by the transportation system performances and, in turn, by the choices of passengers, with whom they share the same transportation supply system
B. Minimize the differences between the tardiness of all the shipments: Such a term introduces the logistic terms in the local optimization problem and also provides the link between the optimization problems of different nodes
As regards, with particular reference to the path choice problem, the optimization variables consist of
• The requested due dates rddij−1,tof all the shipments with respect to the upstream node ni−1
• The generic binary variables xl,hj indicating whether the links(l, h)∈Lt, of the transportation network represented by the graph t, belong to the optimal path of the jthshipment (in this case, xl,hj =1)or not (in this case, xl,hj =0)
For what concerns the optimization problem, it can be formally written as min
xl,hj
j
gj+
j
max{0, rddt,ij+1−tddt,ij+1} (5.21)
subject to
h
xl,hj −
h
xh,lj =
⎧⎨
⎩
1 l≡i−1
−1 l≡i+1 0 otherwise
∀j (5.22)
tj =
xl,hj∈Lt
xl,hj tl,h ∀j (5.23)
gj =
xl,hj∈Lt
xl,hj cl,h ∀j (5.24)
tddt,ij+1=rddij−1,t+tj ∀j (5.25)
∀j
xl,hj ≤Cl,h ∀(l, h)∈Lt (5.26)
xl,hj = {0, 1}∀(l, h)∈Lt, ∀j (5.27)
rddij−1,t≥0 ∀j (5.28)
where the first term of the cost function in Equation 5.21 represents the sum of the path costs for all the shipments (A) and the second term represents logistic costs (B) previously described.
In addition,
• The constraints in Equation 5.22 define the graph structure
• The constraints in Equations 5.23 and 5.24 determine the travel time and the generalized cost of the jthshipment
• The constraints in Equation 5.25 defines the tentative due date of the jthshipment
• The constraints in Equation 5.26 limit the capacity of links
• Finally, the last two constraints in Equations 5.27 and 5.28 define the admissible values of the variables
Note that the capacity constraints expressed by Equation 5.26, that for instance represent the limited space on trains or ships, may make the choice of the shortest path not always possible for all the shipments at a time.
Moreover, it is interesting to note that the set of constraints in Equation 5.22 may be different for any shipment j, thus allowing to consider a different transportation network for each of them.
Such a modelling feature permits to take into account some specificities of some kind of freight, enabling and disabling particular transportation opportunities, depending on their characteristics.
An important application of such a feature is the possibility of forbidding some paths, or modes, to dangerous goods [154,155].
The resulting problem can be thought of as a generalized shortest path problem and results to be a linear mixed continuous-binary optimization problem, whose solution can be obtained easily via well-known optimization algorithms.
It is worth remarking that the problem defined by Equations 5.21 through 5.28 represents an example of a decision a transportation operator of a DLC has to take. Nevertheless, its definition is helpful for pointing out the connections rising between the generic logistic problems of a DLC and the transportation problems underlying them.
Note that such a problem can be easily extended to the integrated mode/path choice, if the links of the transportation network represents different transportation modes, each with the relevant cost and performance characteristics.
In particular, it is interesting to note that the variables xjl,h,∀j, ∀(i, l)∈Lt represent, in practice, the elementary path choices taken by the operator, for each shipment. The values assumed by these variables are, for the considered freight transportation problem, the result of an optimization process analogous to the one virtually solved by passengers, which, as mentioned, take decisions aiming to minimize their own disutilities, although not mathematically formalized.
The sequence of choices taken by all the transportation operator operating on the transportation network t, for all the shipments, constitutes the freight path flows in the transportation network and, due to the feedback dynamics described in Section 5.2.4, the relevant travel costs.