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Say the change in the coefficient is within the allowable increase or decrease. Can the objective value change? I'm reading that it doesn't change. But, say the simplex solves the problem and one of the coefficients, x, is 5 and its value is 100, and say obj value = 500 and the allowable increase for x is 1. If I increase x to 6. (6-5)(100) + 500 = 600. The obj value change. I'm confused.

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The objective value does change.

The optimal basis doesn't have to change, that is the set of independent column doesn't change. Let $B$ be the index for basic variable.

$$\min c_B^Tx_B$$ $$A_Bx_B=b$$

$$x_B \ge 0$$

We have $$x_B = A_B^{-1}b$$

and the previous objective value is $$c_B^TA_B^{-1}x_B$$

changing $c_B$ would change the cost. If you are changing the value that correspond to a non-basic variable, then it remains the same.

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