NAG Library Routine Document

h03bbf (tsp_simann)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{nc}$ – IntegerInput

On entry: the number of cities. In the trivial cases $\mathbf{nc}=1$, $2$ or $3$, the routine returns the optimal solution immediately with $\mathbf{tmode}=0$ (provided the relevant distance matrix entries are not negative).

Constraint: $\mathbf{nc}\ge 1$.

2:     $\mathbf{dm}\left(\mathbf{nc},\mathbf{nc}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the distance matrix; each $\mathbf{dm}\left(\mathit{i},\mathit{j}\right)$ is the effective cost or weight between nodes $\mathit{i}$ and $\mathit{j}$. Only the strictly upper half of the matrix is referenced.

Constraint: $\mathbf{dm}\left(\mathit{i},\mathit{j}\right)\ge 0.0$, for $\mathit{j}=2,3,\dots ,\mathbf{nc}$ and $\mathit{i}=1,2,\dots ,\mathit{j}-1$.

3:     $\mathbf{bound}$ – Real (Kind=nag_wp)Input

On entry: a lower bound on the solution. If the optimum is unknown set bound to zero or a negative value; the routine will then calculate the minimum spanning tree for dm and use this as a lower bound (returned in $\mathbf{alg\_stats}\left(6\right)$). If an optimal value for the cost is known then this should be used for the lower bound. A detailed discussion of relaxations for lower bounds, including the minimal spanning tree, can be found in Reinelt (1994).

4:     $\mathbf{targc}$ – Real (Kind=nag_wp)Input

On entry: a measure of how close an approximation needs to be to the lower bound. The routine terminates when a cost is found less than or equal to $\mathbf{bound}+\mathbf{targc}$. This argument is useful when an optimal value for the cost is known and supplied in bound. It may be sufficient to obtain a path that is close enough (in terms of cost) to the optimal path; this allows the algorithm to terminate at that point and avoid further computation in attempting to find a better path.

If $\mathbf{targc}<0$, $\mathbf{targc}=0$ is assumed.

5:     $\mathbf{path}\left(\mathbf{nc}\right)$ – Integer arrayOutput

On exit: the best path discovered by the simulation. That is, path contains the city indices in path order. If $\mathbf{ifail}\ne 0$ on exit, path contains the indices $1$ to nc.

6:     $\mathbf{cost}$ – Real (Kind=nag_wp)Output

On exit: the cost or weight of path. If $\mathbf{ifail}\ne 0$ on exit, cost contains the largest model real number (see x02blf).

7:     $\mathbf{tmode}$ – IntegerOutput

On exit: the termination mode of the routine (if $\mathbf{ifail}\ne 0$ on exit, tmode is set to $-1$):

$\mathbf{tmode}=0$

Optimal solution found, $\mathbf{cost}=\mathbf{bound}$.

$\mathbf{tmode}=1$

System temperature cooled. The algorithm returns a path and associated cost that does not attain, nor lie within targc of, the bound. This could be a sufficiently good approximation to the optimal path, particularly when $\mathbf{bound}+\mathbf{targc}$ lies below the optimal cost.

$\mathbf{tmode}=2$

Halted by cost falling within the desired targc range of the bound.

$\mathbf{tmode}=3$

System stalled following lack of improvement.

$\mathbf{tmode}=4$

Initial search failed to find a single improvement (the solution could be optimal).

8:     $\mathbf{alg\_stats}\left(6\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: an array of metrics collected during the initial search. These could be used as a basis for future optimization. If $\mathbf{ifail}\ne 0$ on exit, the elements of alg_stats are set to zero; the first five elements are also set to zero in the trival cases $\mathbf{nc}=1$, $2$ or $3$.

$\mathbf{alg\_stats}\left(1\right)$

Mean delta.

$\mathbf{alg\_stats}\left(2\right)$

Standard deviation of deltas.

$\mathbf{alg\_stats}\left(3\right)$

Cost at end of initial search phase.

$\mathbf{alg\_stats}\left(4\right)$

Best cost encountered during search phase.

$\mathbf{alg\_stats}\left(5\right)$

Initial system temperature. At the end of stage 1 of the algorithm, this is a function of the mean and variance of the deltas, and of the distance from best cost to the lower bound. It is a measure of the initial acceptance criteria for stage $2$. The larger this value, the more iterations it will take to geometrically reduce it during stage 2 until the system is cooled (below a threshold value).

$\mathbf{alg\_stats}\left(6\right)$

The lower bound used, which will be that computed internally when $\mathbf{bound}\le 0$ on input. Subsequent calls with different random states can set bound to the value returned in $\mathbf{alg\_stats}\left(6\right)$ to avoid recomputation of the minimal spanning tree.

9:     $\mathbf{state}\left(*\right)$ – Integer arrayCommunication Array

Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.

The actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.

On entry: a valid RNG state initialized by g05kff or g05kgf. Since the algorithm used is stochastic, a random number generator is employed; if the generator is initialized to a non-repeatable sequence (g05kgf) then different solution paths will be taken on successive runs, returning possibly different final approximate solutions.

On exit: contains updated information on the state of the generator.

10:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{nc}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nc}\ge 1$.

$\mathbf{ifail}=2$

On entry, the strictly upper triangle of dm had a negative element.

$\mathbf{ifail}=9$

On entry, state vector has been corrupted or not initialized.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (h03bbfe.f90)

10.2

Program Data

Program Data (h03bbfe.d)

10.3

Program Results

Program Results (h03bbfe.r)