NAG Library Routine Document

H03ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

H03ABF solves the classical Transportation (‘Hitchcock’) problem.

2 Specification

SUBROUTINE H03ABF (

KOST, LDKOST, MA, MB, M, K15, MAXIT, K7, K9, NUMIT, K6, K8, K11, K12, Z, IFAIL)

INTEGER	KOST(LDKOST,MB), LDKOST, MA, MB, M, K15(M), MAXIT, K7(M), K9(M), NUMIT, K6(M), K8(M), K11(M), K12(M), IFAIL
REAL (KIND=nag_wp)	Z

3 Description

H03ABF solves the Transportation problem by minimizing

z = \sum_{i}^{m_{a}} \sum_{j}^{m_{b}} c_{i j} x_{i j} .

subject to the constraints

\begin{array}{l} \sum_{j}^{m_{b}} x_{i j} = A_{i} (Availabilities) \\ \sum_{i}^{m_{a}} \sum_{i} x_{i j} = B_{j} (Requirements) \end{array}

where the

x_{i j}

can be interpreted as quantities of goods sent from source

i

to destination

j

, for

i = 1, 2, \dots, m_{a}

and

j = 1, 2, \dots, m_{b}

, at a cost of

c_{i j}

per unit, and it is assumed that

\sum_{i}^{m_{a}} A_{i} = \sum_{j}^{m_{b}} \sum_{j} B_{j}

and

x_{i j} \geq 0

H03ABF uses the ‘stepping stone’ method, modified to accept degenerate cases.

4 References

Hadley G (1962) Linear Programming Addison–Wesley

5 Parameters

1: KOST(LDKOST,MB) – INTEGER arrayInput: On entry: the coefficients $c_{i j}$ , for $i = 1, 2, \dots, m_{a}$ and $j = 1, 2, \dots, m_{b}$ .
2: LDKOST – INTEGERInput: On entry: the first dimension of the array KOST as declared in the (sub)program from which H03ABF is called.
Constraint: $LDKOST \geq MA$ .
3: MA – INTEGERInput: On entry: the number of sources, $m_{a}$ .
Constraint: $MA \geq 1$ .
4: MB – INTEGERInput: On entry: the number of destinations, $m_{b}$ .
Constraint: $MB \geq 1$ .
5: M – INTEGERInput: On entry: the value of $m_{a} + m_{b}$ .
6: K15(M) – INTEGER arrayInput/Output: On entry: $K15 (i)$ must be set to the availabilities $A_{i}$ , for $i = 1, 2, \dots, m_{a}$ ; and $K15 (m_{a} + j)$ must be set to the requirements $B_{j}$ , for $j = 1, 2, \dots, m_{b}$ .

On exit: the contents of K15 are undefined.
7: MAXIT – INTEGERInput: On entry: the maximum number of iterations allowed.
Constraint: $MAXIT \geq 1$ .
8: K7(M) – INTEGER arrayWorkspace
9: K9(M) – INTEGER arrayWorkspace
10: NUMIT – INTEGEROutput: On exit: the number of iterations performed.
11: K6(M) – INTEGER arrayOutput: On exit: $K6 (k)$ , for $k = 1, 2, \dots, m_{a} + m_{b} - 1$ , contains the source indices of the optimal solution (see K11).
12: K8(M) – INTEGER arrayOutput: On exit: $K8 (k)$ , for $k = 1, 2, \dots, m_{a} + m_{b} - 1$ , contains the destination indices of the optimal solution (see K11).
13: K11(M) – INTEGER arrayOutput: On exit: $K11 (k)$ , for $k = 1, 2, \dots, m_{a} + m_{b} - 1$ , contains the optimal quantities $x_{i j}$ which, sent from source $i = K6 (k)$ to destination $j = K8 (k)$ , minimize $z$ .
14: K12(M) – INTEGER arrayOutput: On exit: $K12 (k)$ , for $k = 1, 2, \dots, m_{a} + m_{b} - 1$ , contains the unit cost $c_{i j}$ associated with the route from source $i = K6 (k)$ to destination $j = K8 (k)$ .
15: Z – REAL (KIND=nag_wp)Output: On exit: the value of the minimized total cost.
16: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

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

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, the sum of the availabilities does not equal the sum of the requirements.

$IFAIL = 2$: During computation MAXIT has been exceeded.

$IFAIL = 3$

On entry,

MAXIT < 1

$IFAIL = 4$

On entry,	$MA < 1$ ,
or	$MB < 1$ ,
or	$M \neq MA + MB$ ,
or	$MA > LDKOST$ .

7 Accuracy

All operations are performed in integer arithmetic so that there are no rounding errors.

8 Further Comments

An accurate estimate of the run time for a particular problem is difficult to achieve.

9 Example

A company has three warehouses and three stores. The warehouses have a surplus of

12

units of a given commodity divided among them as follows:

Warehouse	Surplus
1	1
2	5
3	6

The stores altogether need

12

units of commodity, with the following requirements:

Store	Requirement
1	4
2	4
3	4

Costs of shipping one unit of the commodity from warehouse

i

to store

j

are displayed in the following matrix:

		Store
		1	2	$3$
	1	8	8	11
Warehouse	2	5	8	14
	3	4	3	10

It is required to find the units of commodity to be moved from the warehouses to the stores, such that the transportation costs are minimized. The maximum number of iterations allowed is

200

NAG Library Routine DocumentH03ABF