# 数学代写|运筹学作业代写operational research代考|Preemptive Goal Programming

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## 数学代写|运筹学作业代写operational research代考|Preemptive Goal Programming

In the preceding example we assume that all the goals are of roughly comparable importance. Now consider the case of preemptive goal programming, where there is a hierarchy of priority levels for the goals. Such a case arises when one or more of the goals clearly are far more important than the others. Thus, the initial focus should be on achieving as closely as possible these first-priority goals. The other goals also might naturally divide further into second-priority goals, third-priority goals, and so on. After we find an optimal solution with respect to the first-priority goals, we can break any ties for the optimal solution by considering the second-priority goals. Any ties that remain after this reoptimization can be broken by considering the third-priority goals, and so on.

When we deal with goals on the same priority level, our approach is just like the one described for nonpreemptive goal programming. Any of the same three types of goals (lower one-sided, two-sided, upper one-sided) can arise. Different penalty weights for deviations from different goals still can be included, if desired. The same formulation technique of introducing auxiliary variables again is used to reformulate this portion of the problem to fit the linear programming format.

There are two basic methods based on linear programming for solving preemptive goal programming problems. One is called the sequential procedure, and the other is the streamlined procedure. We shall illustrate these procedures in turn by solving the following example.

## 数学代写|运筹学作业代写operational research代考|The Sequential Procedure for Preemptive Goal Programming

The sequential procedure solves a preemptive goal programming problem by solving a sequence of linear programming models.

At the first stage of the sequential procedure, the only goals included in the linear programming model are the first-priority goals, and the simplex method is applied in the usual way. If the resulting optimal solution is unique, we adopt it immediately without considering any additional goals.

However, if there are multiple optimal solutions with the same optimal value of $Z$ (call it $Z^$ ), we prepare to break the tie among these solutions by moving to the second stage and adding the second-priority goals to the model. If $Z^=0$, all the auxiliary variables representing the deviations from first-priority goals must equal zero (full achievement of these goals) for the solutions remaining under consideration. Thus, in this case, all these auxiliary variables now can be completely deleted from the model, where the equality constraints that contain these variables are replaced by the mathematical expressions (inequalities or equations) for these first-priority goals, to ensure that they continue to be fully achieved. On the other hand, if $Z^>0$, the second-stage model simply adds the second-priority goals to the first-stage model (as if these additional goals actually were first-priority goals), but then it also adds the constraint that the first-stage objective function equals $Z^$ (which enables us again to delete the terms involving first-priority goals from the second-stage objective function). After we apply the simplex method again, if there still are multiple optimal solutions, we repeat the same process for any lowerpriority goals.

Example. We now illustrate this procedure by applying it to the example summarized in Table 7.6.

At the first stage, only the two first-priority goals are included in the linear programming model. Therefore, we can drop the common factor $M$ for their penalty weights, shown in Table 7.6. By proceeding just as for the nonpreemptive model if these were the only goals, the resulting linear programming model is
\begin{aligned} & \text { Minimize } Z=2 y_2^{+}+3 y_3^{+}, \ & \text {subject to } \ & 5 x_1+3 x_2+4 x_3-\left(y_2^{+}-y_2^{-}\right)=40 \ & 5 x_1+7 x_2+8 x_3-\left(y_3^{+}-y_3^{-}\right)=55 \end{aligned}
and
$$x_j \geq 0, \quad y_k^{+} \geq 0, \quad y_k^{-} \geq 0 \quad(j=1,2,3 ; k=2,3) .$$
(For ease of comparison with the nonpreemptive model with all four goals, we have kept the same subscripts on the auxiliary variables.)

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|The Sequential Procedure for Preemptive Goal Programming

\begin{aligned} & \text { Minimize } Z=2 y_2^{+}+3 y_3^{+}, \ & \text {subject to } \ & 5 x_1+3 x_2+4 x_3-\left(y_2^{+}-y_2^{-}\right)=40 \ & 5 x_1+7 x_2+8 x_3-\left(y_3^{+}-y_3^{-}\right)=55 \end{aligned}

$$x_j \geq 0, \quad y_k^{+} \geq 0, \quad y_k^{-} \geq 0 \quad(j=1,2,3 ; k=2,3) .$$
(为了便于与具有所有四个目标的非抢占模型进行比较，我们在辅助变量上保留了相同的下标。)

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