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数学代写|线性规划代写Linear Programming代考|MTH503 The Primal Network Simplex Method

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数学代写|线性规划代写Linear Programming代考|The Primal Network Simplex Method

Each of the variants of the simplex method presented in earlier chapters of this book can be applied to network flow problems. It would be overkill to describe them all here in the context of networks. However, they are all built on two simple algorithms: the primal simplex method (for problems that are primal feasible) and the dual simplex method (for problems that are dual feasible). We discuss them both in detail.
We shall describe the primal network simplex method by continuing with our example. As mentioned above, the tree shown in Figure 14.6 is primal feasible but not dual feasible. The basic idea that defines the primal simplex method is to pick a nontree arc that is dual infeasible and let it enter the tree (i.e., become basic) and then readjust everything so that we still have a tree solution.

The First Iteration. For our first pivot, we let arc (a,c) enter the tree using a primal pivot. In a primal pivot, we add flow to the entering variable, keeping all other nontree flows set to zero and adjusting the tree flows appropriately to maintain flow balance. Given any spanning tree, adding an extra arc must create a cycle (why?). Hence, the current spanning tree together with the entering arc must contain a cycle. The flows on the cycle must change to accommodate the increasing flow on the entering arc. The flows on the other tree arcs remain unchanged. In our example, the cycle is: “a”, “c”,”b”, “f”. This cycle is shown in Figure 14.7 with flows adjusted to take into account a flow of $t$ on the entering arc. As $t$ increases, eventually the flow on arc (f,b) decreases to zero. Hence, $\operatorname{arc}(\mathrm{f}, \mathrm{b})$ is the leaving arc. Updating the flows is easy; just take $t=3$ and adjust the flows appropriately.

数学代写|线性规划代写Linear Programming代考|The Dual Network Simplex Method

In the previous section, we developed simple rules for the primal network simplex method, which is used in situations where the tree solution is primal feasible but not dual feasible. When a tree solution is dual feasible but not primal feasible, then the dual network simplex method can be used. We shall define this method now. Consider the tree solution shown in Figure 14.13. It is dual feasible but not primal feasible (since $x_{\mathrm{db}}<0$ ). The basic idea that defines the dual simplex method is to pick a tree arc that is primal infeasible and let it leave the spanning tree (i.e., become nonbasic) and then readjust everything to preserve dual feasibility.

The First Iteration. For the first iteration, we need to let arc (d,b) leave the spanning tree using a dual pivot, which is defined as follows. Removing arc (d,b) disconnects the spanning tree into two disjoint subtrees. The entering arc must be one of the arcs that spans across the two subtrees so that it can reconnect them into a spanning tree. That is, it must be one of
$$(\mathrm{a}, \mathrm{e}), \quad(\mathrm{a}, \mathrm{d}), \quad(\mathrm{b}, \mathrm{e}), \quad \text { or } \quad(\mathrm{g}, \mathrm{e}) \text {. }$$
See Figure 14.14. To see how to decide which it must be, we need to consider carefully the impact of each possible choice.

To this end, let us consider the general situation. As mentioned above, the spanning tree with the leaving arc removed consists of two disjoint trees. The entering arc must reconnect these two trees.

数学代写|线性规划代写Linear Programming代考|The Dual Network Simplex Method

$$(\mathrm{a}, \mathrm{e}), \quad(\mathrm{a}, \mathrm{d}), \quad(\mathrm{b}, \mathrm{e}), \quad \text { or } \quad(\mathrm{g}, \mathrm{e}) \text {. }$$

MATLAB代写

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