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数学代写|运筹学代写Operations Research代考|OPR561 THE PRIMAL DUAL ALGORITHM

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数学代写|运筹学代写Operations Research代考|THE PRIMAL DUAL ALGORITHM

Consider the Linear Programming illustration given below
ILlUSTRATION $6.5$
Minimize $Z=3 X_1+4 X_2$
Subject to
\begin{aligned} 2 X_1+3 X_2 & \geq 8 \ 5 X_1+2 X_2 & \geq 12 \ X_1, X_2 & \geq 0 \end{aligned}
This can be solved using the two phase method or the Big-M method. We have already seen that both these would require two iterations to get to the optimal solution for this instance. Both the methods would require at least as many iterations as the number of artificial variables to replace them in the basis.

We have also solved this problem using the dual simplex algorithm and this would also require as many iterations as the number of infeasible basic variables in the initial basis. We solve this problem using a new method called the Primal Dual Algorithm.
We add slack variables $X_3$ and $X_4$ and rewrite the problem as
Minimize $Z=3 X_1+4 X_2$
Subject to
\begin{aligned} 2 X_1+3 X_2-X_3 &=8 \ 5 X_1+2 X_2-X_4 &=12 \ X_1, X_2, X_3, X_4 & \geq 0 \end{aligned}
The dual of this problem is
Maximize $8 y_1+12 y_2$
Subject to
$$\begin{array}{r} 2 y_1+5 y_2 \leq 3 \ 3 y_1+2 y_2 \leq 4 \ -y_1 \leq 0 \ -y_2 \leq 0 \end{array}$$
$y_1, y_2$ unrestricted in sign
The dual solution $y=\left[\begin{array}{ll}0 & 0\end{array}\right]$ is feasible to the dual. Also, this satisfies the third and fourth constraint of the dual as an equation.

数学代写|运筹学代写Operations Research代考|GOAL PROGRAMMING

So far, in all our problems encountered in linear programming, we considered a single linear objective function and a set of well defined and rigid linear constraints. When even one of the constraints is violated, we encountered an infeasible solution to the problem.

In practice, we could have multiple objectives when addressing real-life situations. Some constraints may be rigid while some may be flexible and would have targets or goals rather than rigid constraints. We model such situations using a technique called goal programming (Charnes et al., 1960).

Goal programming is one of the ways to address multiple objectives. The easier way is to convert a multiple objective problem into a single objective problem by considering weights for each of the objectives. Another way is to rank the objectives and solve it as a sequence of single objective problems, considering the ranked objectives one at a time. This method should ensure that the solution obtained for a particular objective does not worsen the solution (objective function) of an earlier solved higher ranked objective. This is called lexicographic minimization (optimization) and when applied to goal programming, it is called lexicographic goal programming.

In this section, we illustrate some aspects of lexicographic goal programming through few examples and solve them using graphical and simplex algorithms.

数学代写|运筹学代写Operations Research代考|THE PRIMAL DUAL ALGORITHM

$$2 X_1+3 X_2 \geq 85 X_1+2 X_2 \quad \geq 12 X_1, X_2 \geq 0$$

$$2 X_1+3 X_2-X_3=85 X_1+2 X_2-X_4 \quad=12 X_1, X_2, X_3, X_4 \geq 0$$

$$2 y_1+5 y_2 \leq 33 y_1+2 y_2 \leq 4-y_1 \leq 0-y_2 \leq 0$$
$y_1, y_2$ 不绶限制的符昊

MATLAB代写

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