局部搜索算法

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局部搜索算法是在一组可行解的基础上，在当前解的领域内进行局部搜索产生新的可行解的过程。

主要有路径内搜索和路径间搜索，以下都以VRP为例。

路径内搜索

2-opt

2-opt搜索算法由Lin S（1965）提的一种路径内改进方法。该方法通过改变一条路径中顾客的排序减少路径距离。如果该路径的成本减少，则改进的路径被保留。否则，路径返回修改前的情形。

以下是2-opt算法

这个是维基百科给出的2-opt路径改进图

以下是3-opt算法

Or-opt

Or-opt搜索算法是由Or（1976）提出的一种路径内改进方法。该方法是将一条路径中的m个连续的顾客节点在该路径中重新定位。

算法步骤（SVRP）

1. Let $k=3$ and $T$ be an initial route.
2. Compute the expected cost of route $T$ using the dynamic programming recursion. Also, for each $S$, a set of k successive nodes from $T$, compute:
   a. the approximate cost savings of removing the nodes in set $S$ from $T$ by using the approximation scheme discussed earlier.
   b. the cost of inserting $S$ back into any point along the reduced route excluding the place where it was taken off the route. Use the approximation scheme to compute this cost.
   c. the cost improvement—the difference of cost savings in (a) and additional cost in (b).
3. If none of the sets in Step 2 results in positive cost improvement, go to Step 5. Otherwise, go to Step 4.
4. Select the set $S$ that results in the maximum positive cost improvement in Step 2. Remove it from its current location and insert it at the point that results in this maximum savings. Go to Step 2.
5. If $k=1$, stop. Otherwise, decrease $k$ by 1, and go to Step 2.

路径间搜索

Swap/shift

Swap/shift是由Chen和Wu提出的一种路径间改进方法。该方法转移一条路径中的顾客到另一条路径上。

2-opt*

2-opt*是由Potvin和Rousseu提出的一种路径间改进方法。在该方法中两条路径分别取1个交换位置，将第一条路径位置之后的所有顾客节点与第2条路径交换位置之后的所有顾客节点互换。

参考文献

[1]Lin S . Computer solutions of the traveling salesman problem[J]. Bell Labs Technical Journal, 1965, 44(10):2245-2269.

[2]https://en.wikipedia.org/wiki/2-opt

[3]Or I. Traveling salesman-type combinatorial problems and their relation to the logistics of blood banking[J]. PhD thesis (Department of Industrial Engineering and Management Science, Northwestern University), 1976.

[4]Chen J F, Wu T H. Vehicle routing problem with simultaneous deliveries and pickups[J]. Journal of the Operational Research Society, 2006, 57(5): 579-587.

[5]Potvin J Y, Rousseau J M. An exchange heuristic for routeing problems with time windows[J]. Journal of the Operational Research Society, 1995, 46(12): 1433-1446.

[6]Yang W H, Mathur K, Ballou R H. Stochastic vehicle routing problem with restocking[J]. Transportation Science, 2000, 34(1): 99-112.