Illustration of Marginal profit
To illustrate the computations, suppose that the marginal profit or XE in our model is changed from 3 to 3 + δ1, where δ represents either positive or negative change. This means that the objective function reads as: Z = (3 + δ1) X_{E} + 2X_{I}.
If we use this information in the starting tableau and carry out the same arithmetic operations used to produce the optimum tableau, the optimum Z equation will appear as:
This equation is the same as the optimum Z equation before the change δ_{1} is effected, modified by terms of δ_{1}. The coefficients of δ_{1} are essentially those in the X_{E} equation of the optimum tableau, which are
Pivot Element
We choose the X_{E} equation because XE is the variable whose objective coefficient is being changed by δ_{1}.
The change δ_{1} will not affect the optimality of the problem as long as all the Z equation coefficients of the non-basic variables remain non-negative (maximization) that is:
1/3 - δ_{1}/3 ≥0
4/3 + 2δ_{1}/3 ≥ 0
First Relationship shows that δ_{1} ≥ 1 and the relation second yields δ_{1} ≥ -2. Both relations limit δ_{1 }by -2 ≤ δ_{1} ≤ 1. This means that the coefficient of XE can be as small as 3 + (-2) = 1 or as large as 3 + 1 = 4 without causing any change in the optimal values of the variables. The optimal value of Z, however, will change according to the expression 12 2/3 + 10/3 δ_{1} where -2 ≤ δ ≤ 1.
The foregoing discussion assumed that the variable whose coefficient is being changed has an equation in the constraints. This is true only if the variable is basic (such as X_{E} and X_{I} above). If it is non-basic, it will not appear in the basic column.
The treatment of non-basic variable is straight forward. Any change in the objective coefficient of a non-basic variable will affect only that coefficient in the optimal tableau. To illustrate this point, consider changing the coefficient of SI (the first slack variable) from 0 to 0 + δ_{3}. If you carry out the arithmetic operations leading to the optimum tableau, the resulting Z equation becomes:
It shows that the only change occurs in the coefficient of SI, where it is decreased by δ3. As a general rule, then, all we have to do in the case of a non-basic variable is to decrease the Z coefficient of the non-basic variable by the amount by which the original coefficient of the variable is increased.