Tracing a loop, Operation Management

Tracing a loop: When a closed loop is to be traced start with the empty cell which is to be evaluated ( or to be included in the solution ). Then moving clockwise draw an arrow from this cell to an occupied cell in the same row or column as the case may be after that move vertically( or horizontally) to another occupied cell and draw an arrow follow the same procedure to other occupied cells before returning to the original empty cell. In the process of moving from one occupied cell to another (A) move only horizontally or vertically but never diagonally an d(b) step over empty or occupied cells if the need be without changing them. Thus a loop would always have right angled turns with corners only on the occupied cells.

Having traces the path place plus and minus signs alternately in the cells on each turn of the loop beginning with a plus (+) sign in the empty cell. An important restriction is that there must be exactly one cell with a plus sign and one cell with a minus sign in any row or column in which the loop takes a turn. This restriction ensures that the rim requirements would not be violated when units are shifted among cells.

The followings points may also be noted in connection with the closed loops:

a. An even number of at least four cells must participate in a closed loop and an occupied cell be considered only once and not more.

b. If there exists a basic feasible solution with m + n- 1 positive variable then there would be one and only one closed loop for each cell. This is irrespective of the size of the matrix given.

c. All cells that receive a plus or a minus sign except the starting empty cell must be the occupied cells.

d. Closed loops may or may not be square or rectangular in shape. In larger transportation tables the closed loop have particular configuration and a loop may cross over itself.

e. Although as mentioned earlier movement on the path set by the loop is generally clockwise even if the progression on the path is anticlockwise it would not affect the result.

Posted Date: 4/5/2013 2:18:46 AM | Location : United States

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