我有一个项目列表a, b, c,...，每个项目都有权重和值。
'普通'背包算法将找到最大化所选项目价值的项目选择，同时确保权重低于给定约束。
我遇到的问题略有不同。我希望最小化该值（通过使用值的倒数很容易），同时确保权重至少给定约束的值，不小于或等于约束。
我尝试通过普通的Knapsack算法重新路由这个想法，但是这不能做到。我希望有另一种组合算法，我不知道这是做到这一点。
答案 0 :(得分：1)
在德语wiki中，它被正式化为：
finite set of objects U
w: weight-function
v: value-function
w: U -> R
v: U -> R
B in R # constraint rhs
Find subset K in U subject to:
sum( w(u) <= B ) | all w in K
such that:
max sum( v(u) ) | all u in K
因此没有像非负性这样的限制。
只需使用负权重，负值和负数B. 基本概念是：
sum( w(u) ) <= B | all w in K
<->
-sum( w(u) ) >= -B | all w in K
所以在你的情况下：
classic constraint: x0 + x1 <= B | 3 + 7 <= 12 Y | 3 + 10 <= 12 N
becomes: -x0 - x1 <= -B |-3 - 7 <=-12 N |-3 - 10 <=-12 Y
因此，对于给定的实现，如果允许，则取决于软件。就优化问题而言，没有问题。您的案例的整数编程公式与经典案例（并且有界）一样自然。
import numpy as np
import scipy.sparse as sp
from cylp.cy import CyClpSimplex
np.random.seed(1)
""" INSTANCE """
weight = np.random.randint(50, size = 5)
value = np.random.randint(50, size = 5)
capacity = 50
""" SOLVE """
n = weight.shape[0]
model = CyClpSimplex()
x = model.addVariable('x', n, isInt=True)
model.objective = value # MODIFICATION: default = minimize!
model += sp.eye(n) * x >= np.zeros(n) # could be improved
model += sp.eye(n) * x <= np.ones(n) # """
model += np.matrix(-weight) * x <= -capacity # MODIFICATION
cbcModel = model.getCbcModel()
cbcModel.logLevel = True
status = cbcModel.solve()
x_sol = np.array(cbcModel.primalVariableSolution['x'].round()).astype(int) # assumes existence
print("INSTANCE")
print(" weights: ", weight)
print(" values: ", value)
print(" capacity: ", capacity)
print("Solution")
print(x_sol)
print("sum weight: ", x_sol.dot(weight))
print("value: ", x_sol.dot(value))
小看核心代码：
# create model
model = CyClpSimplex()
# create one variable for each how-often-do-i-pick-this-item decision
# variable needs to be integer (or binary for 0-1 knapsack)
x = model.addVariable('x', n, isInt=True)
# the objective value of our IP: a linear-function
# cylp only needs the coefficients of this function: c0*x0 + c1*x1 + c2*x2...
# we only need our value vector
model.objective = value # MODIFICATION: default = minimize!
# WARNING: typically one should always use variable-bounds
# (cylp problems...)
# workaround: express bounds lower_bound <= var <= upper_bound as two constraints
# a constraint is an affine-expression
# sp.eye creates a sparse-diagonal with 1's
# example: sp.eye(3) * x >= 5
# 1 0 0 -> 1 * x0 + 0 * x1 + 0 * x2 >= 5
# 0 1 0 -> 0 * x0 + 1 * x1 + 0 * x2 >= 5
# 0 0 1 -> 0 * x0 + 0 * x1 + 1 * x2 >= 5
model += sp.eye(n) * x >= np.zeros(n) # could be improved
model += sp.eye(n) * x <= np.ones(n) # """
# cylp somewhat outdated: need numpy's matrix class
# apart from that it's just the weight-constraint as defined at wiki
# same affine-expression as above (but only a row-vector-like matrix)
model += np.matrix(-weight) * x <= -capacity # MODIFICATION
# internal conversion of type neeeded to treat it as IP (or else it would be
LP)
cbcModel = model.getCbcModel()
cbcModel.logLevel = True
status = cbcModel.solve()
# type-casting
x_sol = np.array(cbcModel.primalVariableSolution['x'].round()).astype(int)
Welcome to the CBC MILP Solver
Version: 2.9.9
Build Date: Jan 15 2018
command line - ICbcModel -solve -quit (default strategy 1)
Continuous objective value is 4.88372 - 0.00 seconds
Cgl0004I processed model has 1 rows, 4 columns (4 integer (4 of which binary)) and 4 elements
Cutoff increment increased from 1e-05 to 0.9999
Cbc0038I Initial state - 0 integers unsatisfied sum - 0
Cbc0038I Solution found of 5
Cbc0038I Before mini branch and bound, 4 integers at bound fixed and 0 continuous
Cbc0038I Mini branch and bound did not improve solution (0.00 seconds)
Cbc0038I After 0.00 seconds - Feasibility pump exiting with objective of 5 - took 0.00 seconds
Cbc0012I Integer solution of 5 found by feasibility pump after 0 iterations and 0 nodes (0.00 seconds)
Cbc0001I Search completed - best objective 5, took 0 iterations and 0 nodes (0.00 seconds)
Cbc0035I Maximum depth 0, 0 variables fixed on reduced cost
Cuts at root node changed objective from 5 to 5
Probing was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Gomory was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Knapsack was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Clique was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
MixedIntegerRounding2 was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
FlowCover was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
TwoMirCuts was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Result - Optimal solution found
Objective value: 5.00000000
Enumerated nodes: 0
Total iterations: 0
Time (CPU seconds): 0.00
Time (Wallclock seconds): 0.00
Total time (CPU seconds): 0.00 (Wallclock seconds): 0.00
INSTANCE
weights: [37 43 12 8 9]
values: [11 5 15 0 16]
capacity: 50
Solution
[0 1 0 1 0]
sum weight: 51
value: 5