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Chapter 4: Problem 54

Can the value of the objective function remain unchanged in passing from one tableau to the next? Explain.

Short Answer

Expert verified
In general, the value of the objective function is expected to change during a valid pivot operation in the Simplex method, as it is designed to improve the solution when progressing from one tableau to the next. However, occasional cases may arise where the value of the objective function remains unchanged after a pivot operation, typically when the pivot element's contribution to the objective function is null. Though not common, it is possible for the objective function value to remain unchanged in passing from one tableau to the next.

Step by step solution

01

Understanding the pivot operation

The pivot operation is applied to a tableau representing a linear programming problem. The purpose of this operation is to move from one basic feasible solution to another while improving (either maximizing or minimizing, depending on the problem) the value of the objective function. The pivot operation involves selecting a pivot column and a pivot row, then performing row operations to achieve a new tableau, which represents a new basic feasible solution.
02

Considering the objective function

The objective function of a linear programming problem represents the quantity we are trying to optimize (either maximize or minimize) - typically a profit, cost, or utility function. In a tableau, the objective function is represented in the bottom row, with the coefficient of the variable we're optimizing in the Objective Function column.
03

The pivot operation's effect on the objective function

When we perform a pivot operation, we change the current basic solution by identifying the new entering and leaving variables, attempting to improve the objective function value. This is done by choosing the most negative value in the bottom row (for maximization problems) or the most positive value in the bottom row (for minimization problems) as the pivot column. This process guarantees that (assuming a feasible solution exists), the objective function will be improved in the next tableau.
04

Answering the question

In conclusion, it is generally expected that the value of the objective function changes during a valid pivot operation, improving the solution as we progress from one tableau to the next. However, there might be occasional cases where the value of the objective function remains unchanged after a pivot operation, which might happen when the pivot element's contribution to the objective function is null. But in practice, the Simplex method is designed to select those pivots that lead to improvements in the objective function, and therefore it is not common for the value of the objective function to remain unchanged in passing from one tableau to the next.

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