Figure 7.1: Chain routes. The Chain model allows a route to visit multiple ports twice.
In this chapter we present another formula-tion to enable complex route structures than the one presented in Chapter 6.
The two models are fundamentally different, both in which types of complex route struc-tures they allow, and also how they are for-mulated.
The model presented in this chapter is called the Chain model because it facilitates the modelling of ”chain-like”-routes. Figure 7.1 shows examples of routes that can be con-structed with the Chain model.
The Chain model allows a route to visit sev-eral ports more than once. However, each port can only be visited at most twice. It enables a route to traverse the same arc twice. An arc that is traversed twice is denoted a butterarc.
Figure 7.2 shows an example of a butterarc and how it can be realized with the Chain model.
A virtual double network, called the twin network, forms the foundation of the formulation of the Chain model. The properties and advantages of this network is explained in Section 7.1. In Section 7.2 the mathematical formulation of the Chain model is given.
Chapter 7. Chain model
(a) Butter arc (b) Butter arc realization in a twin network
Figure 7.2: Figure 7.2a shows an example of a butterarc - an arc that is traversed twice in the same route. Figure 7.2b shows how the butterarc can be realized in a twin network. The Flower model does not facilitate butter arcs.
7.1 Twin network
The basis of the Chain model is a virtual double network - called the twin network. Figure 7.3 illustrates the concept of the twin network used in the Chain model, and shows how complex route structures can be enabled by utilizing the network.
In the twin network all the ports are duplicated, each port getting an associated twin port.
Doubling the network makes it possible to visit the same port twice by visiting the original port and its twin, and count this as two visits to the same port. This way, we do not have to alter the flow constraints that apply for each node in the network.
The set of original ports and twin ports is called nodes and denoted N . Original ports are indexed from 1 to N , while the twin ports are indexed from N + 1 to 2N . Port i’s twin is indexed i + N . This makes it possible to refer to the original port and its twin port together.
The twin network has the following properties:
• The distance between port i and its twin port i + N is 0, i.e. Ti,i+N = Ti+N,i= 0.
• The distance between twin port i+N and twin port j +N is equal to the distance between port i and j, i.e. Ti+N,j+N = Tij.
• The transhipment cost of twin port i + N is equal to the transhipment cost of port i, i.e.
Ci+NT = CiT. The same applies for the visiting cost.
Chapter 7. Chain model
(a) Twin network
(b) Butterfly route
(c) Twin network with a route
Figure 7.3: Twin network. Figure 7.3a shows the twin network structure in the Chain model. All ports are duplicated getting a corresponding twin node. The white nodes represent the original ports while the grey nodes represent the twin ports. Figure 7.3b shows a butterfly route, and Figure 7.3c shows how this butterfly route is realized in the Chain network structure.
Chapter 7. Chain model
7.2 Mathematical formulation of the Chain model
In this section the additional notation and changes necessary in the Basic model to enable the described properties of the Chain model.
Unlike the Flower model, the realization of the Chain model requires few new variables and constraint formulations. On the other hand, in the Chain model we need a new set, and make some assumptions regarding the combinations of indices of which the variables are defined.
Sets and indices
N - Nodes, i,j i, j ∈ {1..2N }
We introduce the set of nodes N which consists of both original ports and twin ports. The original ports are indexed from 1 to N , while the twin ports are indexed from N + 1 to 2N . Port i’s twin is indexed i + N . With regards to the set of arcs A this is not defined for any tuples of ports and twin ports, except between a port and its own twin, i.e. (i,i + N ).
Constraints Number of visits
We have to make a slight change in constraints (5.15) from the Basic model because a move between a port and its twin port should not be counted as a visit. This is because they are in reality the same port, and we have to avoid double counting the visits. So, constraints (5.15) have to be modified to:
It is not necessary to do any changes in regards to the ship flow xijkr - we still need ship flow conservation in every node. On the other hand, when it comes to the demand flow fijodkr we have to do some changes. This is to enable transhipment in all ports. If one route visits port i and leaves some units of demand (o,d) we want a route that visits port i’s twin port, port i + N , to be able to pick up these units without visiting port i. In order to allow this we redefine the demand flow conservation in a node to be a joint demand flow conservation for twin ports. This
Chapter 7. Chain model
of the nodes, but not necessarily for each node. The same concept applies in origin ports and destination ports the joint outgoing/incoming flow has to equal the demand (o,d).
Therefore we have to substitute constraints (5.6-5.8) from the Basic model with the correspond-ing constraints (7.2-7.4).
In the Chain model the calculation of the internal transhipment does not require any extra consideration. The internal transhipment is taken care of by constraints (5.10) because we, unlike what we do with the demand flow conservation constraints do not calculate the flow balance for the port and its twin port in total. Thus, constraints (5.10) will track a change in the flow for each node separately, hence discovering any transhipment that happens in each node.