我想我的问题与背包问题的变体有关，但我无法为此提出解决方案：
假设您在五金店，需要购买21颗螺丝。 他们只提供袋装：
现在你必须找出你应该购买哪些包以获得最低价格的21个螺丝（或更多！）。
所以我得到的是一张包含所有行李的表格和一个用于定义所需金额的变量。因此，我需要的是一个包含行李名称和所需金额的表格。
不幸的是，sqlfiddle已经关闭..但至少在这里是示例数据：
declare @bags table (id int, qty int, price decimal(19,4))
insert into @bags values
(10, 16, 25.00)
,(20, 8, 18.00)
,(30, 4, 7.00)
declare @ReqQty int = 21
非常感谢你的帮助！希望我们能够解决这个问题，因为我需要使用这个重要的功能来定制我们公司的ERP系统。
提前谢谢！
编辑： 我阅读了关于背包的整篇维基百科文章，并说： 溢出近似算法 有可能生成一个近似算法，我们可以略微溢出允许的重量限制。您希望获得至少与给定界限B一样高的总值，但您可以超过重量限制...... 目前，这种近似算法的解决方案尚不清楚。
所以看起来我最好使用贪心算法而不是发明轮子？ ;）
答案 0 :(得分：2)
这是一个可能的解决方案。我明天是否可以完成，因为现在差不多已经凌晨3点了。那里有主要的逻辑。剩下要做的就是使用prev_w值追溯。只需跳回（从best_price行开始），直到到达w=0行。当前行和前一行的w之间的差异为您提供了每步所需购买的行李的大小。
在您的示例中，解决方案路线显然是：
“w = 24，w = 8，w = 4，w = 0”，翻译为“购买行李：16,4,4”。
这3个袋子售价39美元。
此解决方案假定此人不打算购买 超过1000个螺丝（这是@limit的用途）。
脚本草稿：
-- use TEST;
declare @limit decimal(19,4);
set @limit = 1000;
create table #bags
(
id int primary key,
qty int,
price decimal(19,4),
unit_price decimal(19,4),
w int, -- weight
v decimal(19,4) -- value
);
insert into #bags(id, qty, price)
values
(10, 16, 25.00)
,(20, 8, 18.00)
,(30, 4, 7.00);
declare @ReqQty int;
set @ReqQty = 21;
update #bags set unit_price = price / ( 1.0 * qty );
update #bags set w = qty;
update #bags set v = -price;
select * From #bags;
create table #m(w int primary key, m int, prev_w int);
declare @w int;
set @w = 0;
while (@w<=@limit)
begin
insert into #m(w) values (@w);
set @w = @w + 1;
end;
update #m
set m = 0;
set @w = 1;
declare @x decimal(19,4);
declare @y decimal(19,4);
update m1
set
m1.m = 0
from #m m1
where
m1.w = 0;
while (@w<=@limit)
begin
select
@x = max(b.v + m2.m)
from
#m m1
join #bags b on m1.w >= b.w and m1.w = @w
join #m m2 on m2.w = m1.w-b.w;
select @y = min(m22.w) from
#m m11
join #bags bb on m11.w >= bb.w and m11.w = @w
join #m m22 on m22.w = m11.w-bb.w
where
(bb.v + m22.m) = ( @x );
update m1
set
m1.m = @x,
m1.prev_w = @y
from #m m1
where
m1.w = @w;
set @w = @w + 1;
end;
select * from #m;
select
-m1.m as best_price
from
#m m1
where
m1.w = (select min(m2.w) from #m m2 where m2.w >= @ReqQty and (m2.m is not null));
drop table #bags;
drop table #m;
脚本最终版本：
-- use TEST;
declare @limit decimal(19,4);
set @limit = 1000;
declare @ReqQty int;
set @ReqQty = 21;
create table #bags
(
id int primary key,
qty int,
price decimal(19,4),
unit_price decimal(19,4),
w int, -- weight
v decimal(19,4), -- value
reqAmount int,
CONSTRAINT UNQ_qty UNIQUE(qty)
);
insert into #bags(id, qty, price)
values
(10, 16, 25.00)
,(20, 7, 14.00)
,(30, 4, 7.00);
update #bags set unit_price = price / ( 1.0 * qty );
update #bags set w = qty;
update #bags set v = -price;
update #bags set reqAmount = 0;
-- Uncomment the next line when debugging!
-- select * From #bags;
create table #m(w int primary key, m int, prev_w int);
declare @w int;
set @w = 0;
while (@w<=@limit)
begin
insert into #m(w) values (@w);
set @w = @w + 1;
end;
update #m
set m = 0;
set @w = 1;
declare @x decimal(19,4);
declare @y decimal(19,4);
update m1
set
m1.m = 0
from #m m1
where
m1.w = 0;
while (@w<=@limit)
begin
select
@x = max(b.v + m2.m)
from
#m m1
join #bags b on m1.w >= b.w and m1.w = @w
join #m m2 on m2.w = m1.w-b.w;
select @y = min(m22.w) from
#m m11
join #bags bb on m11.w >= bb.w and m11.w = @w
join #m m22 on m22.w = m11.w-bb.w
where
(bb.v + m22.m) = ( @x );
update m1
set
m1.m = @x,
m1.prev_w = @y
from #m m1
where
m1.w = @w;
set @w = @w + 1;
end;
-- Uncomment the next line when debugging!
-- select * from #m;
declare @z int;
set @z = -1;
select
@x = -m1.m,
@y = m1.w ,
@z = m1.prev_w
from
#m m1
where
m1.w =
-- The next line contained a bug. It's fixed now.
-- (select min(m2.w) from #m m2 where m2.w >= @ReqQty and (m2.m is not null));
(
select top 1 best.w from
(
select m1.m, max(m1.w) as w
from
#m m1
where
m1.m is not null
group by m1.m
) best where best.w >= @ReqQty and best.w < 2 * @ReqQty
order by best.m desc
)
-- Uncomment the next line when debugging!
-- select * From #m m1 where m1.w = @y;
while (@y > 0)
begin
update #bags
set reqAmount = reqAmount + 1
where
qty = @y-@z;
select
@x = -m1.m,
@y = m1.w ,
@z = m1.prev_w
from
#m m1
where
m1.w = @z;
end;
select * from #bags;
select sum(price * reqAmount) as best_price
from #bags;
drop table #bags;
drop table #m;
答案 1 :(得分：2)
我决定采用略微不同的方法。这个是基于设置的，一般的想法是找到符合所需条件的所有可能的袋数量组合，然后选择最便宜的一个。
步骤：
@ReqQty，对于每种袋子，找到有多少这样的袋子有意义包含在表达中（那就是如果包包含5件，我们想要购买12件，这是有意义的考虑1,2或3袋，但4袋显然太多了）A，以及金额为1和2的行类B可以尝试： A * 1 + B * 1，A * 2 + B * 1，A * 3 + B * 1，A * 1 + B * 2，A * 2 + B * 2，A * 3 + B * 2）这是提供样本数据OP的完整解决方案：
（解决方案已修改，新版本可在下方找到！）
-- sample data
declare @ReqQty int = 21
declare @Bags table (Code nvarchar(1), Quantity int, Price decimal(10,2))
insert into @Bags
select 'X', 16, 25.00
union
select 'Y', 8, 18.00
union
select 'Z', 4, 7
; with
-- helper table: all possible integer numbers <= @ReqQty
Nums (I) as
(
select 1
union all
select I + 1
from Nums
where I < @ReqQty
),
-- possible amounts of each kind bag that make sense
-- i.e. with 3-piece bag and 5-piece requirement it
-- is worth checking 1 (x3 = 3) or 2 (x2 = 6) bags, but
-- 3, 4... would be definitely too much
Vars (Code, Amount) as
(
select B.Code, Nums.I
from @Bags as B
inner join Nums on B.Quantity * I - @ReqQty < B.Quantity
),
Sums (Expr, Amount, TotalQuantity, TotalPrice) as
(
-- take each kind of bag with every amount as recursion root
select
convert(nvarchar(100), V.Code + '(' + convert(nvarchar(100), Amount) + ')'),
Amount,
B.Quantity * Amount,
convert(decimal(10, 2), B.Price * Amount)
from Vars as V
inner join @Bags as B on V.Code = B.Code
union all
-- add different kind of bag to the summary
-- 'Sums.Amount >= V.Amount' is to eliminate at least some duplicates
select
convert(nvarchar(100), Expr + ' + ' + V.Code + '(' + convert(nvarchar(100), V.Amount) + ')'),
V.Amount,
Sums.TotalQuantity + B.Quantity * V.Amount,
convert(decimal(10, 2), Sums.TotalPrice + B.Price * V.Amount)
from Vars as V
inner join @Bags as B on V.Code = B.Code
inner join Sums on (charindex(V.Code, Expr) = 0) and Sums.Amount >= V.Amount
)
-- now find lowest price that matches required quantity
-- remove 'top 1' to see all combinations
select top 1 Expr, TotalQuantity, TotalPrice from Sums
where TotalQuantity >= @ReqQty
order by TotalPrice asc
对于给定的样本数据，结果如下：
Expr TotalQuantity TotalPrice
Z(2) + X(1) 24 39.00
解决方案绝对不是完美的：
charindex来消除相同类型的行李但我缺乏时间或技巧来提出更聪明的想法。我认为很好的是它是纯粹基于集合的声明性解决方案。
修改
我已经稍微修改了解决方案以摆脱charindex（从而摆脱了基于文本的包标识符的依赖性）。不幸的是，我不得不为每种包装添加0金额，这使得更多组合，但似乎对性能没有明显影响。同样价格也显示了更多件的组合。 ： - ）
-- sample data
declare @ReqQty int = 21
declare @Bags table (Code nvarchar(1), Quantity int, Price decimal(10,2))
insert into @Bags
select 'X', 16, 25.00
union
select 'Y', 8, 18.00
union
select 'Z', 4, 7.00
; with
-- helper table to apply order to bag types
Bags (Code, Quantity, Price, BI) as
(
select Code, Quantity, Price, ROW_NUMBER() over (order by Code)
from @Bags
),
-- helper table: all possible integer numbers <= @ReqQty
Nums (I) as
(
select 0
union all
select I + 1
from Nums
where I < @ReqQty
),
-- possible amounts of each kind bag that make sense
-- i.e. with 3-piece bag and 5-piece requirement it
-- is worth checking 1 (x3 = 3) or 2 (x2 = 6) bags, but
-- 3, 4... would be definitely too much
Vars (Code, Amount) as
(
select B.Code, Nums.I
from Bags as B
inner join Nums on B.Quantity * I - @ReqQty < B.Quantity
),
Sums (Expr, Amount, BI, TotalQuantity, TotalPrice) as
(
-- take first kind of bag with every amount as recursion root
select
convert(nvarchar(100), V.Code + '(' + convert(nvarchar(100), Amount) + ')'),
Amount, B.BI,
B.Quantity * Amount,
convert(decimal(10, 2), B.Price * Amount)
from Vars as V
inner join Bags as B on V.Code = B.Code
where B.BI = 1
union all
-- add different kind of bag to the summary
select
convert(nvarchar(100), Expr + ' + ' + V.Code + '(' + convert(nvarchar(100), V.Amount) + ')'),
V.Amount, B.BI,
Sums.TotalQuantity + B.Quantity * V.Amount,
convert(decimal(10, 2), Sums.TotalPrice + B.Price * V.Amount)
from Vars as V
inner join Bags as B on V.Code = B.Code
-- take next bag kind according to order
inner join Sums on B.BI = Sums.BI + 1
and Sums.TotalQuantity + B.Quantity * V.Amount - @ReqQty < B.Quantity
)
-- now find lowest price that matches required quantity
-- remove 'top 1' to see all combinations
select top 1 Expr, TotalQuantity, TotalPrice from Sums
where TotalQuantity >= @ReqQty
order by TotalPrice asc, TotalQuantity desc, Expr asc