Page 67 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes Maximise Z = C X
B B
= (10 × 77.85) + (12 × 10.387) + (15 × 180) + 0 + 0 + 0 + 0
= 778.5 + 124.644 + 2,700 + 0 + 0 + 0 + 0
= 3,603.144
Therefore, Minimise Z = –3,603.144
Steps to Sum-up
1. Prepare the Table No. 1 and find the cost coefficients (Z ) for different columns of A, i.e., for
j
Y , by multiplying C column with entries of Y and adding the products, i.e., Z - C Y
i B j j Bi ij
and C is the most of coefficients for Z in objective function. Then, find the difference of Z
j j j
and C (Z – C) for different columns of A.
j j j
th
(a) Incoming Vector: The j vector, i.e., y enters the basis if Z – C is minimum where y
j j j j
is the j column of the coefficient matrix ‘A’.
th
(b) Outgoing Vector: Find the ratio of XB /y (for all Y > 0) for all the elements of the
i ij ij
incoming vector. Then, the vector attached to the row having minimum ratio would
be removed from the basis (if y is greater than ‘0’, otherwise that item should be
ij
neglected).
(c) Key Element or Pivotal Element: The element common to the incoming and outgoing
vector is called key element or pivotal element.
2. The incoming vector has the coefficient of objective function in CB. Hence, make the
pivotal element as “1” by dividing that row completely by the pivotal element. The other
elements of the incoming vector other than pivotal element must be made “0”/“Zero”.
This can be done by deducting the elements of the respective rows by “K” times the
adjusted pivotal row elements completely. The constant ‘K’ is chosen such that the pivotal
columns element(s) is made “0”/“Zero”. Then find Z and C in the usual manner of matrix
j j
method and if Z – C is greater than or equal to zero for all columns, then the basic feasible
j j
solution is optimum otherwise the same procedure is to be continued.

Notes Brief Steps of the simplex method:

1. Convert the inequalities into equalities.
2. Identify the coefficients of equalities & put them into a matrix form.
3. Tabulate the data into 1st iteration of simplex method.

4. Reinstate entries in the 2nd iteration.
5. Find the 'Z' value.

Self Assessment

Fill in the blanks:
3. The element common to the incoming and outgoing vector is called ……………….. .
4. A ………………. variable represents unused resources and are added to original objective
function with zero coefficients.
5. A ………………. Variable represents amount by which solution value exceed a resource.

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