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# 数学代写|线性规划代写Linear Programming代考|Two-Phase Simplex Methods

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## 数学代写|线性规划代写Linear Programming代考|Two-Phase Simplex Methods

The two-phase simplex method consists of two phases, phase I and phase II. Phase I is trying to find someone initial basic feasible solution. When the initial basic permissible solution is found, then Phase II is applied to find the optimal solution. A simplex method is an iterative procedure whose each iteration is characterized by the determination of $m$ basis variables $y_{B, 1}, \ldots, y_{B, m}$ and $n$ non-base variables $y_{N, 1}, \ldots, y_{N, n}$.

Geometrically, the simplex method moves from one extreme point (angle) to the set of admissible solutions in the second, while improving the values of the objective function in each iteration. The twophase simplex method goes through two phases, phase I and phase II. Phase I attempts an extremely extreme point from above. Once the initial extreme point is found once, phase II is applied to resolve the original LP.
Example 2.6.1. For
\begin{aligned} & -y_1-y_2 \ & 2 y_1+3 y_2 \leq 24,2 y_1-y_2 \leq 8, y_1-2 y_2 \leq 2, \ & -y_1+2 y_2 \leq 8, y_1+3 y_2 \geq 6,3 y_1-y_2 \geq 3 \ & 0 \leq y_1 \leq 7,0 \leq y_2 \leq 7 \end{aligned}
Phase I of the simplex algorithm ends at the extreme point indicated by (a) in the following figure. Then Phase II follows a sequence of extreme points marked by arrows along the edges of a set of admissible solutions. The optimum extreme point is indicated by (d).

If $\beta \geq 0$ and if all nonbasic variables $y_{N, 1}, \ldots, y_{N, n}$ are equal to zero, then $y_{B, 1}=\beta_1, \ldots, y_{B, m}=\beta_m$ base admissible solution. If the condition $\beta \geq 0$ is not met, it is necessary to find an initial basic admissible solution or to determine that it does not exist. There are several strategies for Phase I.

## 数学代写|线性规划代写Linear Programming代考|A Two-Phase Simplex Method That Uses Artificial Variables

The classic approach is to associate a linear program in standard form the so-called widespread problem [3, 13].

Let the linear programming problem be given in standard form:
$$\begin{array}{ll} & \gamma^T x, \ \text { subj. } & A y=\beta, \ & y \geq 0 . \end{array}$$
$\mathrm{J}$ it is only fair that we can assume that $\mathrm{u}$ the standard form $\beta \geq 0$ (otherwise we multiply the corresponding equations by -1 ). We attach to the problem (2.6.1.1) an auxiliary linear programming problem:
$$\begin{array}{cl} \min & e^T w, \ \text { subj. } & A y+w=\beta, \ & y \geq 0, w \geq 0, \end{array}$$
where $e=(1, \ldots, 1) \in \mathbb{R}^m$ and $w \in \mathbb{R}^m$ is a vector of so-called. artificial variables. The important fact is that the set of admissible solutions to the problem (2.6.1.2) is empty because its gur no belongs to it point $(y=0, w=\beta)$. It is also clear that the target function is on that set bottom bounded by zero. The permissible base consists of columns that correspond to the variables $w_1, \ldots, w_m$, a the canonical form of the problem (2.6.1.2) is obtained by eliminating $w$ from the objective function using Eq $w=\beta-A y$. Problems (2.6.1.1) and (2.6.1.2) are related by the following theorem:

Theorem 2.6.1. The set of admissible solutions to the problem (2.6.1.1) is non-empty if and only if the optimal value of the objective function problems (2.6.1.2) equal to zero.

Proof. Let $\bar{y}$ be a valid solution (2.6.1.1). Then $(\bar{y}, 0)$ is an admissible solution to the problem (2.6.1.2), with the value of the objective function is zero. Since zero is the lower bound for the objective function of the problem (2.6.1.2), it follows that $(\bar{y}, 0)$ is optimal solution and that zero is the optimal value of the objective function of the problem. Suppose now that $(\bar{y}, \bar{w})$ is the optimal solution to the problem (2.6.1.2) and suppose $e^\tau \bar{w}=0$. From $e>0, \bar{w} \geq 0$ follows $\bar{w}=0$, so we have $A \bar{y}=\beta$, i.e., $\bar{y}$ is a permissible solution to the problem $(2.6 .1 .2)$.

The previous theorem is based on the so-called two-phase modification of simplex methods.

## 数学代写|线性规划代写Linear Programming代考|Two-Phase Simplex Methods

\begin{aligned} & -y_1-y_2 \ & 2 y_1+3 y_2 \leq 24,2 y_1-y_2 \leq 8, y_1-2 y_2 \leq 2, \ & -y_1+2 y_2 \leq 8, y_1+3 y_2 \geq 6,3 y_1-y_2 \geq 3 \ & 0 \leq y_1 \leq 7,0 \leq y_2 \leq 7 \end{aligned}

## 数学代写|线性规划代写Linear Programming代考|A Two-Phase Simplex Method That Uses Artificial Variables

$$\begin{array}{ll} & \gamma^T x, \ \text { subj. } & A y=\beta, \ & y \geq 0 . \end{array}$$
$\mathrm{J}$我们只能假设$\mathrm{u}$是标准形式$\beta \geq 0$(否则我们将相应的方程乘以-1)。我们将(2.6.1.1)问题附加一个辅助线性规划问题:
$$\begin{array}{cl} \min & e^T w, \ \text { subj. } & A y+w=\beta, \ & y \geq 0, w \geq 0, \end{array}$$

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