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数学代写|运筹学代写Operations Research代考|Matrix Form of the Current Set of Equations

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数学代写|运筹学代写Operations Research代考|Matrix Form of the Current Set of Equations

Thus far we have presented the details of the simplex method under the assumptions that the problem is in our standard form (maximize $Z$ subject to functional constraints in $\leq$ form and nonnegativity constraints on all variables) and that $b_i \geq 0$ for all $i=1,2, \ldots, m$. In this section we point out how to make the adjustments required for other legitimate forms of the linear programming model. You will see that all these adjustments can be made during the initialization, so the rest of the simplex method can then be applied just as you have learned it already.

The only serious problem introduced by the other forms for functional constraints (the $=$ or $\geq$ forms, or having a negative right-hand side) lies in identifying an initial $B F$ solution. Before, this initial solution was found very conveniently by letting the slack variables be the initial basic variables, so that each one just equals the nonnegative right-hand side of its equation. Now, something else must be done. The standard approach that is used for all these cases is the artificial-variable technique. This technique constructs a more convenient artificial problem by introducing a dummy variable (called an artificial variable) into each constraint that needs one. This new variable is introduced just for the purpose of being the initial basic variable for that equation. The usual nonnegativity constraints are placed on these variables, and the objective function also is modified to impose an exorbitant penalty on their having values larger than zero. The iterations of the simplex method then automatically force the artificial variables to disappear (become zero), one at a time, until they are all gone, after which the real problem is solved.

To illustrate the artificial-variable technique, first we consider the case where the only nonstandard form in the problem is the presence of one or more equality constraints.
Equality Constraints
Any equality constraint
$$a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n=b_i$$
actually is equivalent to a pair of inequality constraints:
\begin{aligned} & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \leq b_i \ & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \geq b_i . \end{aligned}

数学代写|运筹学代写Operations Research代考|The Overall Procedure

There are two key implications from the matrix form of the current set of equations shown at the bottom of Table 5.8. The first is that only $\mathbf{B}^{-1}$ needs to be derived to be able to calculate all the numbers in the simplex tableau from the original parameters $\left(\mathbf{A}, \mathbf{b}, \mathbf{c}_B\right)$ of the problem. (This implication is the essence of the fundamental insight described in the next section.) The second is that any one of these numbers can be obtained individually, usually by performing only a vector multiplication (one row times one column) instead of a complete matrix multiplication. Therefore, the required numbers to perform an iteration of the simplex method can be obtained as needed without expending the computational effort to obtain all the numbers. These two key implications are incorporated into the following summary of the overall procedure.
Summary of the Revised Simplex Method.

1. Initialization: Same as for the original simplex method.
2. Iteration:
Step 1 Determine the entering basic variable: Same as for the original simplex method.

Step 2 Determine the leaving basic variable: Same as for the original simplex method, except calculate only the numbers required to do this [the coefficients of the entering basic variable in every equation but Eq. (0), and then, for each strictly positive coefficient, the right-hand side of that equation]. ${ }^1$
Step 3 Determine the new BF solution: Derive $\mathbf{B}^{-1}$ and set $\mathbf{x}_B=\mathbf{B}^{-1} \mathbf{b}$.

1. Optimality test: Same as for the original simplex method, except calculate only the numbers required to do this test, i.e., the coefficients of the nonbasic variables in Eq. $(0)$.

In step 3 of an iteration, $\mathbf{B}^{-1}$ could be derived each time by using a standard computer routine for inverting a matrix. However, since $\mathbf{B}$ (and therefore $\mathbf{B}^{-1}$ ) changes so little from one iteration to the next, it is much more efficient to derive the new $\mathbf{B}^{-1}$ (denote it by $\mathbf{B}{\text {new }}^{-1}$ ) from the $\mathbf{B}^{-1}$ at the preceding iteration (denote it by $\mathbf{B}{\text {old }}^{-1}$ ). (For the initial BF solution,$\mathbf{B}=\mathbf{I}=\mathbf{B}^{-1}$.) One method for doing this derivation is based directly upon the interpretation of the elements of $\mathbf{B}^{-1}$ [the coefficients of the slack variables in the current Eqs. (1), $(2), \ldots,(m)$ ] presented in the next section, as well as upon the procedure used by the original simplex method to obtain the new set of equations from the preceding set.
To describe this method formally, let
\begin{aligned} x_k & =\text { entering basic variable }, \ a_{i k}^{\prime} & =\text { coefficient of } x_k \text { in current Eq. (i), for } i=1,2, \ldots, m \text { (calculated in step } 2 \text { of } \ & \text { an iteration) } \ r & =\text { number of equation containing the leaving basic variable. } \end{aligned}

数学代写|运筹学代写Operations Research代考|Matrix Form of the Current Set of Equations

$$a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n=b_i$$

\begin{aligned} & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \leq b_i \ & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \geq b_i . \end{aligned}

数学代写|运筹学代写Operations Research代考|The Overall Procedure

\begin{aligned} x_k & =\text { entering basic variable }, \ a_{i k}^{\prime} & =\text { coefficient of } x_k \text { in current Eq. (i), for } i=1,2, \ldots, m \text { (calculated in step } 2 \text { of } \ & \text { an iteration) } \ r & =\text { number of equation containing the leaving basic variable. } \end{aligned}

MATLAB代写

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