Page 77 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes 3.3.1 Change in Objective Function Coefficients and Effect on Optimal
Solution

Referring to the current objective coefficient of Figure 2.4, if the values of the objective function
coefficient decrease by 16 (Min. obj. coefficient) and increase by 20 (Max. obj. coefficient) there
will not be any change in the optimal values of x = 12.50 and x = 25.00. But there will be a change
1 2
in the optimal solution, i.e. Z .
max

Notes This applies only when there is a change in any one of the coefficients of variables
i.e., x or x . Simultaneous changes in values of the coefficients need to apply for 100
1 2
Percent Rule for objective function coefficients.

For x , Allowable decrease = Current value – Min. Obj. coefficient
1
= 40 – 24
= ` 16 (a)
Allowable increase = Max. Obj. coefficient – Current value

= 60 – 40
= ` 20.00 (b)
Similarly, For x , Allowable decrease = ` 10.00 (c)
2
Allowable increase = ` 20.00 (d)
For example, if coefficient of x is increased to 48, then the increase is 48 – 40 = ` 8.00 From (b), the
1
allowable increase is 20, thus the increase in x coefficient is 8/20 = 0.40 or 40%.
1
Similarly,
If coefficient of x is decreased to 27, then the decrease is 30 – 27= ` 3.00.
2
From (c), the allowable decrease is 10, thus the decrease in x coefficient is 3/10 = 0.30
2
or 30%.
Therefore, the percentage of increase in x and the percentage of decrease in x is 40 and 30
1 2
respectively.
i.e. 40% + 30% = 70%
For all the objective function coefficients that are changed, sum the percentage of the allowable
increase and allowable decrease. If the sum of the percentages is less than or equal to 100%, the
optimal solution does not change, i.e., x and x values will not change.
1 2
But Z will change, i.e.,
max
= 48(12.50) + 27(25)

= ` 1275.00
If the sum of the percentages of increase and decrease is greater than 100%, a different optimal
solution exists. A revised problem must be solved in order to determine the new optimal
values.

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