Python 我如何在纸浆中使用整数规划指定多变量约束？

我试图用Python中的整数规划公式来解决装箱问题。该问题的模型如下所示：

_C5: 5 x11 + x12 + 2 x13 - 6 y1 <= 0
_C6: 5 x21 + x22 + 2 x23 - 4 y2 <= 0
_C7: 5 x31 + x32 + 2 x33 - 5 y3 <= 0

我已经使用纸浆库编写了以下Python代码

from pulp import *

#knapsack problem

def knapsolve(bins, binweight, items, weight):

    prob = LpProblem('BinPacking', LpMinimize)

    y = [LpVariable("y{0}".format(i+1), cat="Binary") for i in range(bins)]

    xs = [LpVariable("x{0}{1}".format(i+1, j+1), cat="Binary")
          for i in range(items) for j in range(bins)]

    #minimize objective
    nbins = sum(y)
    prob += nbins

    print(nbins)

    #constraints

    prob += nbins >= 1

    for i in range(items):
        con1 = sum(xs[(i * bins) + j] for j in range(bins))
        prob += con1 == 1
        print(con1)

    for k in range(bins):
        x = xs[k*bins : (k+1)*bins]
        con1 = sum([x1*y for x1, y in zip(x, weight)])
        prob += con1 <= binweight[k]
        print(con1)

    exec('prob')

    status = prob.solve()

    print(LpStatus[status])
    print("Objective value:", value(prob.objective))
    print ('\nThe values of the variables : \n')

    for v in prob.variables():
        print(v.name, "=", v.varValue)

    return

def knapsack():

    #bins

    bins = int(input ('Enter the upper bound on the number of bins:'))

    print ('\nEnter {0} bins\' capacities one by one'.format(bins))

    binweight = []

    for i in range(0, bins):
        print('Enter {0} bin capacity'.format(i+1))
        binweight.append(int(input()))

    for i in range(0, bins):
        print('The capacity at {0} is {1}'.format(i, binweight[i]))

    #items

    items = int(input('Enter the number of items:'))

    weight = []

    print ('\nEnter {0} items weights one by one'.format(items))

    for i in range(0, items):
        print('Enter {0} item weight'.format(i+1))
        weight.append(int(input()))

    for i in range(0, items):
        print('The weight at {0} is {1}'.format(i, weight[i]))

    knapsolve(bins, binweight, items, weight)

    return

knapsack()

---

输出不符合预期。如何正确指定上述约束以获得正确的输出？

生成问题后，您可以通过将结果LP/MIP模型写入文件来检查该模型：

...
prob.writeLP("binpacking")
status = prob.solve()
...

现在，如果您查看binpacking文件：

\* BinPacking *\
Minimize
OBJ: y1 + y2 + y3
Subject To
_C1: y1 + y2 + y3 >= 1
_C2: x11 + x12 + x13 = 1
_C3: x21 + x22 + x23 = 1
_C4: x31 + x32 + x33 = 1
_C5: 5 x11 + x12 + 2 x13 <= 6
_C6: 5 x21 + x22 + 2 x23 <= 4
_C7: 5 x31 + x32 + 2 x33 <= 5
Binaries
x11
x12
x13
x21
x22
x23
x31
x32
x33
y1
y2
y3
End

现在，它们将建模如下：

_C5: 5 x11 + x12 + 2 x13 - 6 y1 <= 0
_C6: 5 x21 + x22 + 2 x23 - 4 y2 <= 0
_C7: 5 x31 + x32 + 2 x33 - 5 y3 <= 0

这些限制将是：

_C2: x11 + x21 + x31 = 1
_C3: x12 + x22 + x32 = 1
_C4: x13 + x23 + x33 = 1

---

你能再解释一下项目索引约束吗？现在它工作正常，但我做错了什么？从

xij

的定义来看，如果项目

j

被放入bin

I

，那么

xij=1

。如果您将其建模为

x11+x12+x13=1

，则表示至少将一个项目放入箱子1中（与箱子2和箱子3相同-表示应使用所有箱子）。但您需要的是将物品

j

放入至少一个箱子中。这就是为什么您需要

x11+x21+x31=1

它意味着将第一项放入第1、第2或第3栏。与y[k]相乘的结果如何？它不存在于模型中，那么为什么会起作用呢？变量

yi

用于检查是否使用了bin i。现在，如果不将容量与

yi

相乘，则即使分配

yi=0

，容量仍然可用。它实际上是由第一个约束集右侧的

V*yi

在模型中表示的。哦，是的，我现在明白了

for i in range(items):
    con1 = sum(xs[(i + j*bins)] for j in range(bins))
    prob += con1 == 1
    print(con1)

_C2: x11 + x21 + x31 = 1
_C3: x12 + x22 + x32 = 1
_C4: x13 + x23 + x33 = 1