Arc Declaration and Attributes

Arc attributes

Arcs are used to represent the possible flows between nodes in a network. From these flows, balancing constraints can be generated by AIMMS for every node in the network. The possible attributes of an arc are given in this table.

Attribute	Value-type	See also page
IndexDomain	index-domain	The IndexDomain attribute
Range	range	The Range attribute
Default	constant-expression	The Default attribute, The Default attribute
From	node-reference
FromMultiplier	expression
To	node-reference
ToMultiplier	expression
Cost	expression
Unit	unit-valued expression	The Unit attribute
Priority	expression	The Priority attribute
NonvarStatus	expression	The NonvarStatus attribute
RelaxStatus	expression	The RelaxStatus attribute
Property	NoSave, numeric-storage-property,	The Property attribute, The Property attribute, Properties of variables
	Inline, SemiContinuous, ReducedCost,
	ValueRange, CoefficientRange
Text	string	The Text and Comment attributes, The Text attribute
Comment	comment string	The Text and Comment attributes

Arcs are like variables

Arcs play the role of variables in a network problem, but have some extra attributes compared to ordinary variables, namely the From, To, FromMultiplier, ToMultiplier, and Cost attributes. Arcs do not have a Definition attribute because they are implicitly defined by the From and To attributes.

The From and To attributes

For each arc, the From attribute is used to specify the starting node, and the To attribute to specify the end node. The value of both attributes must be a reference to a declared node.

The Multiplier attributes

With the FromMultiplier and ToMultiplier attributes you can specify whether the flow along an arc has a gain or loss factor. Their value must be an expression defined over some or all of the indices of the index domain of the arc. The result of the expression must be positive. If you do not specify a Multiplier attribute, AIMMS assumes a default of one. Network problems with non unit-valued Multipliers are called generalized networks.

FromMultiplier and ToMultiplier

The FromMultiplier is the conversion factor of the flow at the source node, while the ToMultiplier is the conversion factor at the destination node. Having both multipliers offers you the freedom to specify the network in its most natural way.

The Cost attribute

You can use the Cost attribute to specify the cost associated with the transport of one unit of flow across the arc. Its value is used in the computation of the special variable FlowCost, which is the accumulated cost over all arcs. In the computation of the FlowCost variable the component of an arc is computed as the product of the unit cost and the level value of the flow.

Graphically illustrated

In the presence of FromMultiplier and ToMultipliers, the drawing in Fig. 11 illustrates

• the level value of the flow,

• its associated cost component in the predefined FlowCost variable, and

• the flows as they enter into the flow balances at the source and destination nodes (denoted by SBF and DBF, respectively).

Fig. 11 Flow levels and cost from node \(i\) to node \(j\)

Semi-continuous arcs

You can only use the SemiContinuous property for arcs if you use an LP solver to find the solution. If you use the pure network solver integrated in AIMMS, AIMMS will issue an error message.

Example

Using the declaration of nodes from the previous section, an example of a valid arc declaration is given by

Arc Transport {
    IndexDomain  :  (i,j,p) | Distance(i,j);
    Range        :  nonnegative;
    From         :  DepotStockSupplyNode(i,p);
    To           :  CustomerDemandNode(j,p);
    Cost         :  UnitTransportCost(i,j);
}

Note that this arc declaration declares flows between nodes i and j for multiple products p.