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Gibbs energy against a reaction coordinate is where x ≥ 0,
known as a Gibbs-energy profile; such plots,
unlikepotential-energyprofiles, aretemperature- {x(i, j) : j in T }≤ s(i) for all i in S,
j
dependent.

In principle, the expressions for k above must {x(i, j) : i in S}≥ d(j) for allj in T.
i
be multiplied by a transmission coefficient, κ,
which is the probability that an activated com- The decision variables (x) are called flows, and
plex forms a particular set of products rather than the two classes of constraints are called supply
revertingtoreactantsorformingalternativeprod- limits and demand requirements, respectively.
ucts. (Some authors use equality constraints, rather
‡
0
‡ 0
It is to be emphasized that S , H , and than the inequalities shown.) An extension is
‡
0
G occurring in the former three equations the capacitated transportation problem, where
are not ordinary thermodynamic quantities, since the flows have bounds x ≤ U.
one degree of freedom in the activated complex
is ignored. transpose (of a matrix [a ]) The matrix
ij
Transition-state theory has also been known [a ].
ji
as the absolute rate theory, and as activated-
complex theory, but these terms are no longer transposition theorem Same as a theorem
recommended. of the alternative.
transition structure A saddle point on a
transshipment problem This is an exten-
potential-energy surface. It has one negative
force constant in the harmonic force constant sion of the transportation problem whereby the
network is not bipartite. Additional nodes serve
matrix. See also transition state.
as transshipment points, rather than providing
supply or final consumption. The network is
translation The action of a group (V, +)
N = [V, A], where V is an arbitrary set of nodes,
regarded as a group of transformations onV itself
except that it contains a nonempty subset of sup-
through the action T : V × V → V given by
ply nodes (where there is external supply) and a
T : (T, v) → v + T. nonempty subset of demand nodes (where there
is external demand). A is an arbitrary set of arcs,
n
transport equation Let U ⊂ R be open and there could also be capacity constraints.
and u : U × R → R. The (linear) transport
equation for u is traveling salesman problem (TSP) Given
n points and a cost matrix, [c(i, j)], a tour is
n
i
u + b u = 0. a permutation of the n points. The points can
t x i
i=1 be cities, and the permutation the visitation of
each city exactly once, then returning to the
transportation problem Find a flow of first city (called home). The cost of a tour,
least cost that ships from supply sources to con- /i ,i , ..., i n−1 ,i ,i 0, is the sum of its costs:
n
1
1
2
sumer destinations. This is a bipartite network,
∗
N = [S T, A], where S is the set of sources, T is c(i ,i ) + c(i ,i ) + ... + c(i n−1 ,i ) + c(i ,i ),
2
2
1
1
n
3
n
the set of destinations, and A is the set of arcs. In
the standard form, N is bi-complete (A contains where (i ,i , ..., i ) is apermutation of {1, ..., n}.
1
n
2
all arcs from S to T ), but in practice networks The TSP is to find a tour of minimum total cost.
tend to be sparsely linked. Let c(i, j) be the unit The two common integer programming formula-
cost of flow from i in S to j in T , s(i) = supply tions are:
at ith source, and d(j) = demand at jth destina- ILP: min ij c x : x ∈ P, x ∈{0, 1}
ij ij
ij
tion. Then, the problem is the linear program Subtour elimination constraints:
i,j∈V
x
ij ≤|V |− 1 for ∅ = V ⊂{1,...,n} (V =
Minimize {c(i, j)x(i, j) : i in S, j inT } {1,...,n})
ij
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC