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Chapter 4: Problem 53

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

Short Answer

Expert verified
In the Simplex method, the value of the objective function should not decrease from one tableau to the next, as the algorithm is designed to improve the objective function value at each step until the optimal solution is reached. However, slight decreases might occur due to possible numerical errors like rounding, but this is not intended in a correctly implemented algorithm.

Step by step solution

01

Simplex Method Overview

The Simplex Method is an optimization algorithm used to solve linear programming problems. The algorithm iteratively refines a solution by moving from one vertex of the feasible region to another, improving the objective function value at each step. A tableau is a matrix representation of each iteration, including information about the objective function, decision variables, and slack variables.
02

Objective Function

The objective function represents the goal of the linear programming problem. We either want to minimize or maximize the value of the objective function depending on the problem. During the Simplex method, the algorithm stops when there are no more better vertices to move to, i.e., when the optimal solution has been reached.
03

Iterations in Simplex Method

In each iteration of the Simplex method, we aim to improve (maximize for a maximization problem, minimize for a minimization problem) the value of the objective function. We achieve this by selecting a variable to enter the basis (called the entering variable) and a variable to leave the basis (called the departing variable). The chosen entering variable should have a positive coefficient in the objective function.
04

Can the Objective Function Value Decrease?

In the Simplex method, the value of the objective function is designed to improve in each iteration for maximization problems and not decrease. However, it is theoretically possible that due to numerical errors (rounding errors, for example), the value might appear to decrease slightly in certain cases. But as long as the algorithm is implemented correctly, the objective function value should not decrease. In summary, the value of the objective function should not decrease in passing from one tableau to the next when using the Simplex method, except in the case of possible numerical errors. The algorithm is designed to improve the objective function value at each step until the optimal solution is reached.

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Most popular questions from this chapter

If a linear programming problem has an unbounded, nonempty feasible region, then optimal solutions (A) must exist (B) may or may not exist (C) cannot exist

We suggest the use of technology. Round all answers to two decimal places. \(\begin{array}{ll}\operatorname{Maximize} & p=2.2 x+2 y+1.1 z+2 w \\ \text { subject to } & x+1.5 y+1.5 z+\quad w \leq 50.5 \\ 2 x+1.5 y-\quad z-\quad w \geq 10 \\ & x+1.5 y+\quad z+1.5 w \geq 21 \\ x & \geq 0, y \geq 0, z \geq 0, w \geq 0\end{array}\)

$$ \begin{array}{ll} \text { Maximize } & p=x+y+z+w \\ \text { subject to } & x+y+z \leq 3 \\ & y+z+w \leq 4 \\ & x+z+w \leq 5 \\ & x+y+w \leq 6 \\ & x \geq 0, y \geq 0, z \geq 0, w \geq 0 \end{array} $$

$$ \begin{aligned} \text { Minimize } & c=s+t \\ \text { subject to } & s+2 t \geq 6 \\ & 2 s+t \geq 6 \\ & s \geq 0, t \geq 0 . \end{aligned} $$

Give one possible advantage of using duality to solve a standard minimization problem.
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