264
A. Riccardi et al.
potential routes between these cities and c a set of distances. Typically, network
problems are bound by both flow constraints and flow bounds. The upper flow
bound or ‘capacity’ of an arc is denoted commonly by k ij , describing the maximum
possible quantity of material that can be moved across each node.
l ij ≤ f ij ≤ k ij
(7.28)
For any given problem, there is a ‘balance’ constraint for each node, where
basically the net flow from this node (i.e. outflow-inflow) will be equal to the
‘supply’ of this node. The supply b i of node i is either positive (e.g. if this node
is a location providing entries into the network), negative (e.g. if this node is a client
with a demand) or zero (if the node plays a location for transhipment). Then, the
balance constraint will be in the following form [102]:
j ∈N
f ij −
j ∈N
f ji = b i Flow Balance Equations
(7.29)
Provided the network follows these bounds and constraints, and the supplies of the
nodes are balanced, the network will be valid.
7.4 Summary
This chapter gives a brief introduction to optimisation problem formulations and
solution methods. After a general overview of different problems, the chapter is
mainly divided in two main sections: continuous problems and methods and discrete
problems and methods.
The first section on continuous problems is further divided into four parts, related
to local methods, optimal control, global methods and multi-objective optimisation.
On the other hand, the section on discrete problems is composed by three parts on
pure integer optimisation, mixed-integer optimisation and network optimisation.
This chapter is meant to give an accessible introduction to formulation and
solving methods. The reader is kindly invited to use the list of references and read
the following chapters, to know more about the methods and to see what are the
most recent advances in the field.
References
1. D.H. Wolpert, W.G. Macready, No Free Lunch Theorems for Optimization (IEEE, Piscataway,
1997)
2. J. Nocedal, S.J. Wright, Numerical Optimisation (Springer, Berlin, 1999)
A. Riccardi et al.
potential routes between these cities and c a set of distances. Typically, network
problems are bound by both flow constraints and flow bounds. The upper flow
bound or ‘capacity’ of an arc is denoted commonly by k ij , describing the maximum
possible quantity of material that can be moved across each node.
l ij ≤ f ij ≤ k ij
(7.28)
For any given problem, there is a ‘balance’ constraint for each node, where
basically the net flow from this node (i.e. outflow-inflow) will be equal to the
‘supply’ of this node. The supply b i of node i is either positive (e.g. if this node
is a location providing entries into the network), negative (e.g. if this node is a client
with a demand) or zero (if the node plays a location for transhipment). Then, the
balance constraint will be in the following form [102]:
j ∈N
f ij −
j ∈N
f ji = b i Flow Balance Equations
(7.29)
Provided the network follows these bounds and constraints, and the supplies of the
nodes are balanced, the network will be valid.
7.4 Summary
This chapter gives a brief introduction to optimisation problem formulations and
solution methods. After a general overview of different problems, the chapter is
mainly divided in two main sections: continuous problems and methods and discrete
problems and methods.
The first section on continuous problems is further divided into four parts, related
to local methods, optimal control, global methods and multi-objective optimisation.
On the other hand, the section on discrete problems is composed by three parts on
pure integer optimisation, mixed-integer optimisation and network optimisation.
This chapter is meant to give an accessible introduction to formulation and
solving methods. The reader is kindly invited to use the list of references and read
the following chapters, to know more about the methods and to see what are the
most recent advances in the field.
References
1. D.H. Wolpert, W.G. Macready, No Free Lunch Theorems for Optimization (IEEE, Piscataway,
1997)
2. J. Nocedal, S.J. Wright, Numerical Optimisation (Springer, Berlin, 1999)