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International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals.org/systems_and_measurements/
Case 2:
and
. It implies that the
transportation models under both scenarios have similar
constraints for some ports
. Similarly,
based on the model under the scenario-I (scenario-II), we
increase (decrease) the RHS values of ports that are
(
) up (down) to zero. Then, the
minimum objective cost of the new model-I (model-II),
denoted by ( ), should be not greater (less) than that of
the original model, i.e.,
(
). For the new model-I and model-II, they
have similar constraints for ports
. Hence,
as that in case 1, we have . Finally, we get that
.
Similarly, when in the case of
,
we can get that
.
APPENDIX B
A numerical example is firstly presented to show that
how the perturbation in the supply or demand of one binding
port, such as , will change the net actual imported ECs of a
pair of ports. Then, a modified stepping stone (MSS) method
is proposed to find .
Consider a numerical example with three surplus ports
and two deficit ports. ECs are repositioned from the surplus
ports to the deficit ports. Assume the total number of EC
supplies is greater than the total number of EC demands.
And the transportation tableau is presented in Figure 7. Note
that one dummy node is created in the transportation tableau
to ensure that the total number of demands is equal to the
total number of supplies. For illustration, the value of
cell represents the number of ECs repositioned from
t
the surplus port to the deficit port. Two sub-examples
are investigated.
Surplus port
1
2
3
Change of demand
1 2
Deficit port
3(dummy)
Figure 7. The transportation tableau
● Basic cell
Change of supply
Sub-example 1: Perturb the number of EC supply of the
first surplus port by
Consider the balance of the transportation tableau.
Perturbing the number of EC supply of the first surplus port
by implies that the number of EC demand of the dummy
deficit port will be perturbed by . However, since
cell , is a non-basic cell (see Figure 7), its value should
be kept as zero and cannot be changed by the perturbation.
To track the changes on the basic variables, a loop is
developed, which begins at cell , and is back to this cell
(see Figure 8).
Surplus port
Change of demand
Deficit port
1 2 3(dummy) Change of supply
Figure 8. Perturb the number of EC supply of the first surplus port by
◎ Beginning cell
The loop in Figure 8 consists of successive horizontal
and vertical segments whose end nodes must be basic
variables, except for the two segments starting and ending at
the non-basic variable.
More specifically, the changes on the basic variables can
be obtained as follows: first increase the value of cell ,
by ; then go around the loop, alternately decrease and then
increase basic variables in the loop by , i.e. decrease the
value of cell , and increase the value of
cell , by . It is observed that the total numbers of
repositioning out ECs at the first and the second surplus
ports will be increased and decreased by , respectively.
Sub-example 2: Perturb the number of EC demand of the
first deficit port by
Perturbing the number of EC demand of a deficit port
(not the dummy deficit port) by will result in the total
number of demands from deficit ports (excluding the dummy
deficit port) increasing by . To maintain the balance of the
transportation tableau, a hypothetic surplus port is introduced
and its supply is assumed to increase by . Besides, to
make the transportation tableau non-degeneracy, the cell (the
hypothetic surplus port, the dummy deficit port), i.e., cell (4,
3) in Figure 9, is set as a basic cell.
To track the changes on the basic variables by the
perturbation of on the number of EC demand of the first
deficit port, similar loop is developed in Figure 9. It is
observed that the total number of repositioning in ECs at the
first deficit port and the total number of repositioning out
ECs at the second surplus port will be both increased by .
Just as the loop using in the typical stepping stone
method, the loop in either Figure 8 or Figure 9 is unique. The
uniqueness of the loop guarantees that there will be only a
pair of ports, whose net actual imported ECs will be affected
by perturbing the supply or demand of one binding port.
2012, © Copyright by authors, Published under agreement with IARIA - www.iaria.org
1
2
3
Surplus port
1
2
3
4 (hypothetic)
Change of demand
1 2
Deficit port
3(dummy)
Change of supply
Figure 9. Perturb the number of EC demand of the first deficit port by
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