我正在尝试使用额外的约束来解决TSP问题 - 时间窗口。
所有标准假设均适用：
- 我们从给定的城市开始和结束。
- 每个城市只访问一次。
- 我们试图在旅行成本(这里是旅行时间(方面找到最佳路径。
此外，每个城市都有自己的时间窗口格式，这限制了何时可以访问城市：
- 我们不能在关闭时间后访问城市。
- 我们可以在开放时间之前到达任何城市并等待开放。如果我们这样做，等待时间将添加到经过的总时间中，但不会添加到旅行所花费的时间中。因此，time_spent_travelling和total_time_passed是我们需要跟踪的两件不同的事情。
我设法编写了在total_time_passed方面找到最佳解决方案的约束，但我需要找到最佳time_spent_travelling。
这是我的逻辑：
% THE TRAVELING SALESMAN WITH TIME WINDOWS
% DESC ------------------------------------------------------------------------------------
% visited(city, arrive_step)
% travel(dest_city, depart_step)
% location(city, arrive_step, when_visited (summary travel time))
% place(name, opening_time, closing_time)
% path(from, to, travel_cost)
% Warunki ----------------------------------------------------------------------------------
% Start and end must be in the same city
:- not location(Place, t, _), location(Place, 0, _).
% Paths are symmetrical
path(A, B, COST) :- path(B, A, COST).
% In each step, there can be only one travel from one city to another
{ travel(Place, T) : place(Place, _, _) } 1 :- T = 0..t-1.
% If there was a travel to a city, that city has been visited (this way starting city is not visited at the beginning)
visited(Place, T) :- travel(Place, T).
% We cannot visit a city, we've been to before
:- travel(Place, T1), visited(Place, T2), T1 > T2.
% We cannot travel to city, we are staying right now
:- travel(Place, T), location(Place, T, _).
% We cannot go to somewhere, to where leads no path
:- travel(To, T), location(From, T, _), not path(From, To, _).
% We cannot travel to city if we arrive after it's closing time
:- travel(TO, T), location(From, T, TOTAL_TIME), path(From, TO, TRAVEL_TIME), place(TO, OPENED_FROM, OPENED_TO), TOTAL_TIME + TRAVEL_TIME > OPENED_TO.
% If we started travel to city A at step T, we must have reached it at step T + 1
% City might me not opened yet, so our travel time is MAX of (CITY_OPENED_TIME, ARRIVAL_TIME)
% max = ((a+b)+|a-b|)/2
% min = ((a+b)-|a-b|)/2
location(To, T + 1, ((ARRIVAL+OPENED_FROM)+|ARRIVAL-OPENED_FROM|)/2) :- travel(To, T), location(From, T, C1) , path(From, To, C2), place(To, OPENED_FROM, _), ARRIVAL = C1+C2.
% There isn't a single city, we haven't visited
:- place(Place, _, _), not visited(Place, _).
% Find minimal travel time (Arrival time at starting city)
result(C) :- location(_, t, C).
#minimize{C : result(C)}.
#show location/3.
下面是示例数据(使用 clingo 运行它需要 ~ 30 秒(：
% Cities count
#const t=21.
% Starting point
location(city_0, 0, 0).
% City list in format (name, opening_time, closing_time)
place(city_0, 0, 408).
place(city_1, 62, 68).
place(city_2, 181, 205).
place(city_3, 306, 324).
place(city_4, 214, 217).
place(city_5, 51, 61).
place(city_6, 102, 129).
place(city_7, 175, 186).
place(city_8, 250, 263).
place(city_9, 3, 23).
place(city_10, 21, 49).
place(city_11, 79, 90).
place(city_12, 78, 96).
place(city_13, 140, 154).
place(city_14, 354, 386).
place(city_15, 42, 63).
place(city_16, 2, 13).
place(city_17, 24, 42).
place(city_18, 20, 33).
place(city_19, 9, 21).
place(city_20, 275, 300).
% Distance between cities (from, to, travel_cost)
path(city_0, city_1, 19).
path(city_0, city_2, 17).
path(city_0, city_3, 34).
path(city_0, city_4, 7).
path(city_0, city_5, 20).
path(city_0, city_6, 10).
path(city_0, city_7, 17).
path(city_0, city_8, 28).
path(city_0, city_9, 15).
path(city_0, city_10, 23).
path(city_0, city_11, 29).
path(city_0, city_12, 23).
path(city_0, city_13, 29).
path(city_0, city_14, 21).
path(city_0, city_15, 20).
path(city_0, city_16, 9).
path(city_0, city_17, 16).
path(city_0, city_18, 21).
path(city_0, city_19, 13).
path(city_0, city_20, 12).
path(city_1, city_2, 10).
path(city_1, city_3, 41).
path(city_1, city_4, 26).
path(city_1, city_5, 3).
path(city_1, city_6, 27).
path(city_1, city_7, 25).
path(city_1, city_8, 15).
path(city_1, city_9, 17).
path(city_1, city_10, 17).
path(city_1, city_11, 14).
path(city_1, city_12, 18).
path(city_1, city_13, 48).
path(city_1, city_14, 17).
path(city_1, city_15, 6).
path(city_1, city_16, 21).
path(city_1, city_17, 14).
path(city_1, city_18, 17).
path(city_1, city_19, 13).
path(city_1, city_20, 31).
path(city_2, city_3, 47).
path(city_2, city_4, 23).
path(city_2, city_5, 13).
path(city_2, city_6, 26).
path(city_2, city_7, 15).
path(city_2, city_8, 25).
path(city_2, city_9, 22).
path(city_2, city_10, 26).
path(city_2, city_11, 24).
path(city_2, city_12, 27).
path(city_2, city_13, 44).
path(city_2, city_14, 7).
path(city_2, city_15, 5).
path(city_2, city_16, 23).
path(city_2, city_17, 21).
path(city_2, city_18, 25).
path(city_2, city_19, 18).
path(city_2, city_20, 29).
path(city_3, city_4, 36).
path(city_3, city_5, 39).
path(city_3, city_6, 25).
path(city_3, city_7, 51).
path(city_3, city_8, 36).
path(city_3, city_9, 24).
path(city_3, city_10, 27).
path(city_3, city_11, 38).
path(city_3, city_12, 25).
path(city_3, city_13, 44).
path(city_3, city_14, 54).
path(city_3, city_15, 45).
path(city_3, city_16, 25).
path(city_3, city_17, 28).
path(city_3, city_18, 26).
path(city_3, city_19, 28).
path(city_3, city_20, 27).
path(city_4, city_5, 27).
path(city_4, city_6, 11).
path(city_4, city_7, 17).
path(city_4, city_8, 35).
path(city_4, city_9, 22).
path(city_4, city_10, 30).
path(city_4, city_11, 36).
path(city_4, city_12, 30).
path(city_4, city_13, 22).
path(city_4, city_14, 25).
path(city_4, city_15, 26).
path(city_4, city_16, 14).
path(city_4, city_17, 23).
path(city_4, city_18, 28).
path(city_4, city_19, 20).
path(city_4, city_20, 10).
path(city_5, city_6, 26).
path(city_5, city_7, 27).
path(city_5, city_8, 12).
path(city_5, city_9, 15).
path(city_5, city_10, 14).
path(city_5, city_11, 11).
path(city_5, city_12, 15).
path(city_5, city_13, 49).
path(city_5, city_14, 20).
path(city_5, city_15, 9).
path(city_5, city_16, 20).
path(city_5, city_17, 11).
path(city_5, city_18, 14).
path(city_5, city_19, 11).
path(city_5, city_20, 30).
path(city_6, city_7, 26).
path(city_6, city_8, 31).
path(city_6, city_9, 14).
path(city_6, city_10, 23).
path(city_6, city_11, 32).
path(city_6, city_12, 22).
path(city_6, city_13, 25).
path(city_6, city_14, 31).
path(city_6, city_15, 28).
path(city_6, city_16, 6).
path(city_6, city_17, 17).
path(city_6, city_18, 21).
path(city_6, city_19, 15).
path(city_6, city_20, 4).
path(city_7, city_8, 39).
path(city_7, city_9, 31).
path(city_7, city_10, 38).
path(city_7, city_11, 38).
path(city_7, city_12, 38).
path(city_7, city_13, 34).
path(city_7, city_14, 13).
path(city_7, city_15, 20).
path(city_7, city_16, 26).
path(city_7, city_17, 31).
path(city_7, city_18, 36).
path(city_7, city_19, 28).
path(city_7, city_20, 27).
path(city_8, city_9, 17).
path(city_8, city_10, 9).
path(city_8, city_11, 2).
path(city_8, city_12, 11).
path(city_8, city_13, 56).
path(city_8, city_14, 32).
path(city_8, city_15, 21).
path(city_8, city_16, 24).
path(city_8, city_17, 13).
path(city_8, city_18, 11).
path(city_8, city_19, 15).
path(city_8, city_20, 35).
path(city_9, city_10, 9).
path(city_9, city_11, 18).
path(city_9, city_12, 8).
path(city_9, city_13, 39).
path(city_9, city_14, 29).
path(city_9, city_15, 21).
path(city_9, city_16, 8).
path(city_9, city_17, 4).
path(city_9, city_18, 7).
path(city_9, city_19, 4).
path(city_9, city_20, 18).
path(city_10, city_11, 11).
path(city_10, city_12, 2).
path(city_10, city_13, 48).
path(city_10, city_14, 33).
path(city_10, city_15, 23).
path(city_10, city_16, 17).
path(city_10, city_17, 7).
path(city_10, city_18, 2).
path(city_10, city_19, 10).
path(city_10, city_20, 27).
path(city_11, city_12, 13).
path(city_11, city_13, 57).
path(city_11, city_14, 31).
path(city_11, city_15, 20).
path(city_11, city_16, 25).
path(city_11, city_17, 14).
path(city_11, city_18, 13).
path(city_11, city_19, 17).
path(city_11, city_20, 36).
path(city_12, city_13, 47).
path(city_12, city_14, 34).
path(city_12, city_15, 24).
path(city_12, city_16, 16).
path(city_12, city_17, 7).
path(city_12, city_18, 2).
path(city_12, city_19, 10).
path(city_12, city_20, 26).
path(city_13, city_14, 46).
path(city_13, city_15, 48).
path(city_13, city_16, 31).
path(city_13, city_17, 42).
path(city_13, city_18, 46).
path(city_13, city_19, 40).
path(city_13, city_20, 21).
path(city_14, city_15, 11).
path(city_14, city_16, 29).
path(city_14, city_17, 28).
path(city_14, city_18, 32).
path(city_14, city_19, 25).
path(city_14, city_20, 33).
path(city_15, city_16, 23).
path(city_15, city_17, 19).
path(city_15, city_18, 22).
path(city_15, city_19, 17).
path(city_15, city_20, 32).
path(city_16, city_17, 11).
path(city_16, city_18, 15).
path(city_16, city_19, 9).
path(city_16, city_20, 10).
path(city_17, city_18, 5).
path(city_17, city_19, 3).
path(city_17, city_20, 21).
path(city_18, city_19, 8).
path(city_18, city_20, 25).
path(city_19, city_20, 19).
我使用MAX函数通过选择实际到达时间或城市的开放时间来计算给定城市的到达时间 - 以较晚者为准。它工作得很好，所以我的第一个想法是向位置事实添加额外的字段，更改此行，如下所示：
%Before:
location(To, T + 1, ((ARRIVAL+OPENED_FROM)+|ARRIVAL-OPENED_FROM|)/2) :- travel(To, T), location(From, T, C1) , path(From, To, C2), place(To, OPENED_FROM, _), ARRIVAL = C1+C2.
%After:
location(To, T + 1, ((ARRIVAL+OPENED_FROM)+|ARRIVAL-OPENED_FROM|)/2, TRAVEL_TIME + C2) :- travel(To, T), location(From, T, C1, TRAVEL_TIME) , path(From, To, C2), place(To, OPENED_FROM, _), ARRIVAL = C1+C2.
这样，位置保存有关time_spent_travelling和total_time_passed的信息。虽然这适用于 5 个城市，但对于 20 个城市，它计算的时间太长(我在 15 分钟后放弃了( - 我希望程序在两种情况下运行大致相同的时间，但显然这里有一些我不明白的地方。
我还试图将等待时间存储为单独的事实，但它似乎以同样的方式影响计算时间，并引入了另一个问题，即在 #minimize 函数中考虑它，我无法解决。
所以这是我的问题：
- 我该怎么做才能计算time_spent_travelling的最佳值，同时正确考虑等待时间？
- 为什么我上面描述的代码中的一个小变化对求解过程有如此大的计算影响？
我最近开始使用clingo，很有可能我没有看到这个问题的简单解决方案。很难改变你编写程序的方式，习惯于声明式编程。
我提供的代码可以使用 clingo 简单运行：clingo logic data
我的输出：
Solving...
Answer: 1
location(city_0,0,0) location(city_16,1,9) location(city_9,2,17) location(city_19,3,21) location(city_17,4,24) location(city_10,5,31) location(city_18,6,33) location(city_5,7,51) location(city_15,8,60) location(city_1,9,66) location(city_11,10,80) location(city_12,11,93) location(city_6,12,115) location(city_13,13,140) location(city_7,14,175) location(city_2,15,190) location(city_4,16,214) location(city_8,17,250) location(city_20,18,285) location(city_3,19,312) location(city_14,20,366) location(city_0,21,387)
Optimization: 387
OPTIMUM FOUND
Models : 1
Optimum : yes
Optimization : 387
Calls : 1
Time : 27.654s (Solving: 0.10s 1st Model: 0.04s Unsat: 0.06s)
CPU Time : 27.651s
(base) igor@i:~/projects/transInfo/TSPTW/src$ clingo dane logika
clingo version 5.4.0
Reading from dane ...
Solving...
Answer: 1
location(city_0,0,0) location(city_16,1,9) location(city_9,2,17) location(city_19,3,21) location(city_17,4,24) location(city_10,5,31) location(city_18,6,33) location(city_5,7,51) location(city_15,8,60) location(city_1,9,66) location(city_11,10,80) location(city_12,11,93) location(city_6,12,115) location(city_13,13,140) location(city_7,14,175) location(city_2,15,190) location(city_4,16,214) location(city_8,17,250) location(city_20,18,285) location(city_3,19,312) location(city_14,20,366) location(city_0,21,387)
Optimization: 387
OPTIMUM FOUND
Models : 1
Optimum : yes
Optimization : 387
Calls : 1
Time : 29.682s (Solving: 0.09s 1st Model: 0.03s Unsat: 0.06s)
CPU Time : 29.680s
这里的结果考虑了等待时间，在这个特定示例中为 9。(378 是只花在旅行上的时间(。
我已经设法解决了这个问题。 最后几行的简单更改就可以了：
% Find minimal travel time (Arrival time at starting city)
#minimize{C, From, To, T : travel(To, T), location(From, T, _), path(From, To, C)}.