optimization.org

-*- mode: Org; fill-column: 90; coding: utf-8; -*-

task

Разбить предложенный массив из весов рулонов на машины двух типов (22.2 и 27.6, превышать эту нагрузку нельзя), таким образом, чтобы удельная загрузка была максимальной и все рулоны были разложены по машинам.

В качестве подсказки укажем, что стоит попробовать использовать солверы SCIP, GUROBI, CPLEX и тд.

libraries

SCIP - Apache 2.0

• https://github.com/scipopt/scip
  • reuirement:
  • Apache License 2.0 https://github.com/scipopt/soplex
  • MIT License https://github.com/scipopt/PySCIPOpt
  • (optional) https://github.com/scipopt/papilo

GUROBI - proprietary?

CPLEX - Proprietary

COIN-OR - GPL and Eclipse Public License v 2.0 https://github.com/coin-or/CyLP https://en.wikipedia.org/wiki/COIN-OR

OR-Tools - Apache License 2.0 https://github.com/google/or-tools (uses SCIP)

table

weight
8.78
8.77
8.77
8.76
8.74
8.73
8.72
8.7
8.63
8.63
8.62
8.62
8.62
8.61
8.6
8.57
8.57
8.53
8.53
8.5
8.49
8.49
8.47
8.44
8.42
8.41
8.39
8.38
8.37
8.33
8.31
8.31
8.28
8.27
8.27
8.26
8.2
8.15
8.12
8.12
7.85
7.14
7.13
7.06
6.31
6.3
6.03
6.02
5.81
5.78
5.76
5.76
5.75
5.74
5.64
5.31
5.31

import pandas as pd

def pd2org(df_to_table):
    return tabulate(df_to_table, headers=df_to_table.columns, tablefmt='orgtbl')

df = pd.DataFrame(data, columns=['col'])
# df['acidity'] = df.acidity.str.extract('(?P<digit>([-+])?\d+(.\d+)?)')['digit'].astype(float)
import numpy as np
weights = np.array(data).flatten()
pd2org(df.describe())

	col
count	57
mean	7.79263
std	1.15286
min	5.31
25%	7.13
50%	8.37
75%	8.6
max	8.78

классификация задачи

Является задачей по упаковке в контейнеры. NP-трудна.

use as few boxes as possible

cutting stock problem

links

• https://en.wikipedia.org/wiki/Bin_packing_problem
• https://scipbook.readthedocs.io/en/latest/bpp.html

math_my

min (i)sum((j)sum(xij)/(j)sum(xij))

si for xi

• (i)sum(s_i*x_ij) <= b_j*y_j, for every j. where y_j = (j)sum(xij)/(j)sum(xij)
• (j)sum(xij) = 1 for every i - only in one container

math1

• x_ij - a boolean variable that indicates whether item i is packaed in bin j
• y_j - a boolean variable if bin j is used
• s_i - size of i item
• b_j - capacity of j
• i = 1..n
• j = 1..u

minimize: sum(y_j) - minimization of the number of bins used

subject to:

1. (j)sum(x_ij) = 1, for every i.
2. (i)sum(s_i*x_ij) <= b_j*y_j, for every j. where y_j = (j)sum(xij)/(j)sum(xij)
3. x_ij <= y_j, for every i and j. - ????????
4. x_ij ∈ {0,1}
5. y_ ∈ {0,1}

where

1. force the placement of each item in one bin
2. the upper limit on the bins contents, as well as the fact that items cannot be packed in a bin that is not in use

math2

• xi - a boolean variable that indicates whether item is included in the knapsack
• n - the total number of items,
• si - the size of item
• C - the capacity of the knapsack

The problem:

• max (i..n)∑

? = 22.2 27.6

x1 + x2 + x3 < n

sum(x1 + x2 + x3 + …) = count(n)

f= sum( x1 + x2 + x3)

python

def BinPackingExample():
    B = 9
    w = [2,3,4,5,6,7,8]
    q = [4,2,6,6,2,2,2]
    s=[]
    for j in range(len(w)):
        for i in range(q[j]):
            s.append(w[j])
    return s,B


def FFD(s, B):
    """heuristic - pack to containers
    :return [[8], [8], [7, 2], [7, 2], [6, 3], ...]"""
    remain = [B]
    sol = [[]]
    for item in sorted(s, reverse=True):
        for j,free in enumerate(remain):
            if free >= item:
                remain[j] -= item
                sol[j].append(item)
                break
        else:
            sol.append([item])
            remain.append(B-item)
    return sol

s, B = BinPackingExample()
print(s, B)
print(FFD(s, B))

[2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8] 9
[[8], [8], [7, 2], [7, 2], [6, 3], [6, 3], [5, 4], [5, 4], [5, 4], [5, 4], [5, 4], [5, 4], [2, 2]]

from pyscipopt import Model
from pyscipopt import Model, quicksum, multidict

def bpp(s,B):
    " "
    n = len(s)
    U = len(FFD(s,B))
    model = Model("bpp")
    x,y = {},{}
    # Variables - binary
    for i in range(n):
        for j in range(U):
            x[i,j] = model.addVar(vtype="B", name="x(%s,%s)"%(i,j))
    for j in range(U):
        y[j] = model.addVar(vtype="B", name="y(%s)"%j)
    # Constraints
    # Each item must be in exactly one bin.
    for i in range(n):
        model.addCons(quicksum(x[i,j] for j in range(U)) == 1, "Assign(%s)"%i)
    # The amount packed in each bin cannot exceed its capacity.
    for j in range(U):
        model.addCons(quicksum(s[i]*x[i,j] for i in range(n)) <= B*y[j], "Capac(%s)"%j)
    for j in range(U):
        for i in range(n):
            model.addCons(x[i,j] <= y[j], "Strong(%s,%s)"%(i,j))
    model.setObjective(quicksum(y[j] for j in range(U)), "minimize")
    model.data = x,y
    return model

def solveBinPacking(s,B):
    n = len(s)
    U = len(FFD(s,B))
    model = bpp(s,B)
    x,y = model.data
    model.optimize()
    bins = [[] for i in range(U)]
    for (i,j) in x:
        if model.getVal(x[i,j]) > .5:
            bins[j].append(s[i])
    for i in range(bins.count([])):
        bins.remove([])
    for b in bins:
        b.sort()
    bins.sort()
    return bins

print(solveBinPacking(s, B))

presolving:
(round 1, exhaustive) 0 del vars, 0 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 0 chg coeffs, 349 upgd conss, 0 impls, 505 clqs
(round 2, fast)       0 del vars, 0 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 26 chg coeffs, 349 upgd conss, 0 impls, 505 clqs
   (0.0s) probing: 51/325 (15.7%) - 0 fixings, 0 aggregations, 0 implications, 0 bound changes
   (0.0s) probing aborted: 50/50 successive totally useless probings
   (0.0s) symmetry computation started: requiring (bin +, int -, cont +), (fixed: bin -, int +, cont -)
   (0.0s) symmetry computation finished: 29 generators found (max: 1500, log10 of symmetry group size: 18.1) (symcode time: 0.00)
(round 3, exhaustive) 0 del vars, 0 del conss, 51 add conss, 0 chg bounds, 0 chg sides, 26 chg coeffs, 349 upgd conss, 0 impls, 573 clqs
(round 4, exhaustive) 0 del vars, 0 del conss, 51 add conss, 0 chg bounds, 0 chg sides, 26 chg coeffs, 397 upgd conss, 0 impls, 573 clqs
presolving (5 rounds: 5 fast, 4 medium, 4 exhaustive):
 0 deleted vars, 0 deleted constraints, 51 added constraints, 0 tightened bounds, 0 added holes, 0 changed sides, 26 changed coefficients
 0 implications, 573 cliques
presolved problem has 325 variables (325 bin, 0 int, 0 impl, 0 cont) and 400 constraints
     13 constraints of type <knapsack>
    384 constraints of type <setppc>
      3 constraints of type <orbitope>
transformed objective value is always integral (scale: 1)
Presolving Time: 0.02

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
  0.0s|     1 |     0 |   605 |     - |  7312k |   0 | 325 | 431 | 397 |   0 |  0 |  42 |   0 | 1.244444e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   671 |     - |  7360k |   0 | 325 | 431 |  38 |   0 |  0 |  42 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   700 |     - |  7432k |   0 | 325 |  75 |  44 |   8 |  1 |  45 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   731 |     - |  7556k |   0 | 325 |  73 |  53 |  17 |  2 |  45 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   751 |     - |  7765k |   0 | 325 |  73 |  60 |  24 |  3 |  45 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   764 |     - |  8055k |   0 | 325 |  73 |  67 |  31 |  4 |  45 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   780 |     - |  8453k |   0 | 325 |  76 |  75 |  39 |  5 |  48 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   796 |     - |  9233k |   0 | 325 |  76 |  81 |  45 |  6 |  48 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   809 |     - |  9528k |   0 | 325 |  76 |  85 |  49 |  7 |  48 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   823 |     - |    10M |   0 | 325 |  78 |  90 |  54 |  8 |  50 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.1s|     1 |     0 |   837 |     - |    11M |   0 | 325 |  78 |  93 |  57 |  9 |  50 |   0 | 1.300000e+01 |      --      |    Inf | unknown
  0.2s|     1 |     0 |   851 |     - |    11M |   0 | 325 |  79 |  98 |  62 | 10 |  51 |   0 | 1.300000e+01 |      --      |    Inf | unknown
r 0.2s|     1 |     0 |   851 |     - |shifting|   0 | 325 |  79 |  98 |  62 | 10 |  51 |   0 | 1.300000e+01 | 1.300000e+01 |   0.00%| unknown
  0.2s|     1 |     0 |   851 |     - |    11M |   0 | 325 |  79 |  98 |  62 | 10 |  51 |   0 | 1.300000e+01 | 1.300000e+01 |   0.00%| unknown

SCIP Status        : problem is solved [optimal solution found]
Solving Time (sec) : 0.16
Solving Nodes      : 1
Primal Bound       : +1.30000000000000e+01 (1 solutions)
Dual Bound         : +1.30000000000000e+01
Gap                : 0.00 %
[[2, 3, 4], [2, 6], [2, 7], [2, 7], [3, 5], [4, 5], [4, 5], [4, 5], [4, 5], [4, 5], [6], [8], [8]]

python_2bins

import numpy as np

def BinPackingExample():
    B = 9
    w = [2,3,4,5,6,7,8]
    q = [4,2,6,6,2,2,2]
    s=[]
    for j in range(len(w)):
        for i in range(q[j]):
            s.append(w[j])
    return s,B


def FFD(s, B):
    """heuristic - pack to containers
    :return [[8], [8], [7, 2], [7, 2], [6, 3], ...]"""
    remain = [B]
    sol = [[]]
    for item in sorted(s, reverse=True):
        for j,free in enumerate(remain):
            if free >= item:
                remain[j] -= item
                sol[j].append(item)
                break
        else:
            sol.append([item])
            remain.append(B-item)
    return sol

# s, B = BinPackingExample()
# print(s, B)
# print(FFD(s, B))

from pyscipopt import Model
from pyscipopt import Model, quicksum, multidict

def bpp(s,bes):
    " "
    # - patch
    B=np.min(bes) # min
    # print("Bmin", B)

    n = len(s)
    U = len(FFD(s,B))

    # - patch
    b_per = U //len(bes)
    Bs = np.array([[x]*b_per for x in bes]).flatten()
    if U > Bs.shape[0]:
        Bs = np.append(Bs, (U - Bs.shape[0])*[bes[-1]])
    # print("Bs, U", Bs, U)
    # quit()

    model = Model("bpp")
    x,y = {},{}
    # Variables - binary
    for i in range(n):
        for j in range(U):
            x[i,j] = model.addVar(vtype="B", name="x(%s,%s)"%(i,j))
    for j in range(U):
        y[j] = model.addVar(vtype="B", name="y(%s)"%j)
    # Constraints
    # Each item must be in exactly one bin.
    for i in range(n):
        model.addCons(quicksum(x[i,j] for j in range(U)) == 1, "Assign(%s)"%i)
    # The amount packed in each bin cannot exceed its capacity.
    for j in range(U):
        B = Bs[j-1]
        model.addCons(quicksum(s[i]*x[i,j] for i in range(n)) <= B*y[j], "Capac(%s)"%j)
    for j in range(U):
        for i in range(n):
            model.addCons(x[i,j] <= y[j], "Strong(%s,%s)"%(i,j))
    model.setObjective(quicksum(y[j] for j in range(U)), "minimize")
    model.data = x,y
    return model, U,

def solveBinPacking(s,bes):
    n = len(s)
    # U = len(FFD(s,B))
    model, U = bpp(s,bes)
    x,y = model.data
    # quit()

    # print("y",y)
    model.optimize()
    # quit()
    # print("x",[print(model.getVal(x[v])) for v in x])
    bins = [[] for i in range(U)]
    for (i,j) in x:
        if model.getVal(x[i,j]) > .5:
            bins[j].append(s[i])
    # -- counts bins
    # for x in bes

    # b1 = 0
    # b2 = 0
    # for i, b in enumerate(bins):
    #     if len(b) > 0:
    #         if i <= U/2:
    #             b1+=1
    #         else:
    #             b2+=1
    # print(b1,b2)
    # for i in range(bins.count([])):
    #     bins.remove([])
    # for b in bins:
        # b.sort()
    # bins.sort()
    return bins



weights = np.array(data).flatten()
print(weights)
print(solveBinPacking(weights, [22.2, 27.6]))

presolving:
(round 1, exhaustive) 0 del vars, 0 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 0 chg coeffs, 798 upgd conss, 0 impls, 798 clqs
   (0.0s) probing: 51/754 (6.8%) - 0 fixings, 0 aggregations, 0 implications, 0 bound changes
   (0.0s) probing aborted: 50/50 successive totally useless probings
   (0.0s) symmetry computation started: requiring (bin +, int -, cont +), (fixed: bin -, int +, cont -)
   (0.0s) symmetry computation finished: 23 generators found (max: 1500, log10 of symmetry group size: 12.5) (symcode time: 0.01)
presolving (2 rounds: 2 fast, 2 medium, 2 exhaustive):
 0 deleted vars, 0 deleted constraints, 0 added constraints, 0 tightened bounds, 0 added holes, 0 changed sides, 0 changed coefficients
 0 implications, 798 cliques
presolved problem has 754 variables (754 bin, 0 int, 0 impl, 0 cont) and 811 constraints
    798 constraints of type <setppc>
     13 constraints of type <linear>
transformed objective value is always integral (scale: 1)
Presolving Time: 0.03

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
p 0.0s|     1 |     0 |     0 |     - |  clique|   0 | 754 | 811 | 811 |   0 |  0 |   0 |   0 | 0.000000e+00 | 1.000000e+01 |    Inf | unknown
  0.1s|     1 |     0 |  1785 |     - |    14M |   0 | 754 | 847 | 811 |   0 |  0 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.2s|     1 |     0 |  1785 |     - |    14M |   0 | 754 | 847 | 785 |   0 |  0 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.2s|     1 |     0 |  1886 |     - |    15M |   0 | 754 | 821 | 787 |   9 |  1 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.3s|     1 |     0 |  1964 |     - |    17M |   0 | 754 | 814 | 795 |  17 |  2 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.4s|     1 |     0 |  2020 |     - |    19M |   0 | 754 | 814 | 799 |  21 |  3 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.5s|     1 |     0 |  2043 |     - |    21M |   0 | 754 | 812 | 804 |  26 |  4 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.7s|     1 |     0 |  2072 |     - |    24M |   0 | 754 | 812 | 808 |  30 |  5 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.9s|     1 |     0 |  2509 |     - |    26M |   0 | 754 | 812 | 815 |  37 |  6 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  0.9s|     1 |     0 |  2571 |     - |    27M |   0 | 754 | 812 | 820 |  42 |  7 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  1.1s|     1 |     0 |  2629 |     - |    30M |   0 | 754 | 812 | 828 |  50 |  8 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  1.2s|     1 |     0 |  2659 |     - |    32M |   0 | 754 | 812 | 835 |  57 |  9 |  36 |   0 | 8.883600e+00 | 1.000000e+01 |  12.57%| unknown
  1.3s|     1 |     0 |  2686 |     - |    33M |   0 | 754 | 812 | 838 |  60 | 10 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.4s|     1 |     0 |  2743 |     - |    33M |   0 | 754 | 812 | 843 |  65 | 11 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.4s|     1 |     0 |  2798 |     - |    33M |   0 | 754 | 812 | 852 |  74 | 12 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
  1.4s|     1 |     0 |  2850 |     - |    34M |   0 | 754 | 812 | 860 |  82 | 13 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.5s|     1 |     0 |  2906 |     - |    34M |   0 | 754 | 812 | 870 |  92 | 14 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.5s|     1 |     0 |  2933 |     - |    34M |   0 | 754 | 812 | 878 | 100 | 15 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.6s|     1 |     0 |  2990 |     - |    34M |   0 | 754 | 812 | 856 | 110 | 16 |  36 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.6s|     1 |     0 |  3043 |     - |    34M |   0 | 754 | 813 | 866 | 120 | 17 |  37 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.6s|     1 |     0 |  3083 |     - |    35M |   0 | 754 | 813 | 873 | 127 | 18 |  37 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.6s|     1 |     0 |  3112 |     - |    35M |   0 | 754 | 813 | 877 | 131 | 19 |  37 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.6s|     1 |     0 |  3134 |     - |    35M |   0 | 754 | 813 | 882 | 136 | 20 |  37 |   0 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  1.8s|     1 |     2 |  3963 |     - |    35M |   0 | 754 | 791 | 882 | 136 | 20 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
(run 1, node 1) restarting after 58 global fixings of integer variables

(restart) converted 97 cuts from the global cut pool into linear constraints

presolving:
(round 1, fast)       58 del vars, 13 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 80 chg coeffs, 0 upgd conss, 0 impls, 741 clqs
(round 2, exhaustive) 58 del vars, 13 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 80 chg coeffs, 65 upgd conss, 0 impls, 741 clqs
(round 3, fast)       58 del vars, 13 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 86 chg coeffs, 65 upgd conss, 0 impls, 741 clqs
(round 4, exhaustive) 58 del vars, 16 del conss, 0 add conss, 0 chg bounds, 0 chg sides, 113 chg coeffs, 65 upgd conss, 0 impls, 741 clqs
presolving (5 rounds: 5 fast, 3 medium, 3 exhaustive):
 58 deleted vars, 16 deleted constraints, 0 added constraints, 0 tightened bounds, 0 added holes, 0 changed sides, 113 changed coefficients
 0 implications, 741 cliques
presolved problem has 696 variables (696 bin, 0 int, 0 impl, 0 cont) and 872 constraints
     61 constraints of type <knapsack>
    741 constraints of type <setppc>
     44 constraints of type <linear>
     26 constraints of type <logicor>
transformed objective value is always integral (scale: 1)
Presolving Time: 0.06
transformed 6/7 original solutions to the transformed problem space

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
  1.9s|     1 |     0 |  4445 |     - |    33M |   0 | 696 | 872 | 847 |   0 |  0 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.0s|     1 |     0 |  4472 |     - |    33M |   0 | 696 | 872 | 855 |   8 |  1 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.0s|     1 |     0 |  4573 |     - |    34M |   0 | 696 | 872 | 863 |  16 |  2 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.1s|     1 |     0 |  4657 |     - |    34M |   0 | 696 | 872 | 870 |  23 |  3 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.2s|     1 |     0 |  4722 |     - |    34M |   0 | 696 | 872 | 880 |  33 |  4 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.4s|     1 |     0 |  4799 |     - |    35M |   0 | 696 | 872 | 890 |  43 |  5 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.5s|     1 |     0 |  4842 |     - |    37M |   0 | 696 | 872 | 898 |  51 |  6 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.7s|     1 |     0 |  4891 |     - |    37M |   0 | 696 | 872 | 906 |  59 |  7 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  2.9s|     1 |     0 |  4959 |     - |    38M |   0 | 696 | 872 | 915 |  68 |  8 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  3.1s|     1 |     0 |  4984 |     - |    39M |   0 | 696 | 872 | 922 |  75 |  9 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  3.3s|     1 |     0 |  5040 |     - |    43M |   0 | 696 | 872 | 928 |  81 | 10 |  40 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  3.4s|     1 |     0 |  5118 |     - |    43M |   0 | 696 | 873 | 939 |  92 | 11 |  41 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  3.4s|     1 |     0 |  5199 |     - |    43M |   0 | 696 | 873 | 947 | 100 | 12 |  41 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  3.5s|     1 |     0 |  5240 |     - |    43M |   0 | 696 | 873 | 956 | 109 | 13 |  41 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  3.5s|     1 |     0 |  5280 |     - |    43M |   0 | 696 | 873 | 962 | 115 | 14 |  41 |  12 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
  3.6s|     1 |     2 |  5781 |     - |    43M |   0 | 696 | 876 | 962 | 115 | 14 |  44 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%| unknown
  5.1s|   100 |    91 | 16341 | 119.2 |    44M |  36 | 696 | 899 | 776 | 138 |  1 |  71 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  5.8s|   200 |   191 | 20660 |  81.2 |    48M |  44 | 696 | 901 | 781 | 218 |  2 |  73 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  6.4s|   300 |   291 | 23821 |  64.7 |    51M |  44 | 696 | 903 | 786 | 271 |  1 |  75 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  7.2s|   400 |   391 | 27312 |  57.2 |    55M |  45 | 696 | 906 | 776 | 390 |  1 |  78 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  7.7s|   500 |   491 | 29384 |  49.9 |    55M |  55 | 696 | 907 | 784 | 515 |  1 |  79 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  8.1s|   600 |   589 | 31396 |  45.0 |    56M |  55 | 696 | 908 | 788 | 622 |  1 |  80 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  8.5s|   700 |   687 | 32905 |  40.7 |    56M |  72 | 696 | 909 | 797 | 709 |  2 |  81 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  8.9s|   800 |   739 | 33606 |  36.5 |    56M |  91 | 696 | 913 | 801 | 727 |  0 |  85 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|   1.22%
  9.1s|   900 |   771 | 34236 |  33.1 |    57M |  98 | 696 | 861 | 796 | 749 |  1 |  90 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
  9.6s|  1000 |   828 | 34686 |  30.3 |    56M |  98 | 696 | 911 | 796 | 754 |  0 | 140 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 10.0s|  1100 |   860 | 35096 |  27.9 |    56M |  98 | 696 | 980 | 796 | 756 |  1 | 209 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 10.2s|  1200 |   887 | 35593 |  26.0 |    57M |  98 | 696 |1000 |   0 | 760 |  0 | 229 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 10.5s|  1300 |   933 | 36238 |  24.5 |    57M |  98 | 696 |1031 | 797 | 778 |  1 | 260 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 11.1s|  1400 |  1001 | 37036 |  23.3 |    57M |  98 | 696 |1097 | 797 | 806 |  1 | 326 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
 11.4s|  1500 |  1057 | 37638 |  22.1 |    57M |  98 | 696 |1101 | 800 | 821 |  0 | 330 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 11.6s|  1600 |  1093 | 38161 |  21.1 |    57M |  98 | 696 |1104 | 799 | 823 |  0 | 333 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 11.8s|  1700 |  1109 | 38649 |  20.1 |    57M |  98 | 696 |1115 |   0 | 826 |  0 | 344 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 12.3s|  1800 |  1158 | 39293 |  19.4 |    57M |  98 | 696 |1134 | 799 | 840 |  1 | 363 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 12.6s|  1900 |  1202 | 39843 |  18.6 |    57M |  98 | 696 |1147 | 801 | 855 |  2 | 376 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 12.9s|  2000 |  1232 | 40375 |  18.0 |    57M |  98 | 696 |1158 | 796 | 858 |  1 | 387 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 13.3s|  2100 |  1252 | 40835 |  17.3 |    57M |  98 | 696 |1208 | 800 | 865 |  0 | 437 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 13.6s|  2200 |  1280 | 41372 |  16.8 |    57M |  98 | 696 |1227 | 800 | 869 |  0 | 456 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 13.9s|  2300 |  1298 | 41860 |  16.3 |    57M |  98 | 696 |1245 | 796 | 871 |  1 | 475 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 14.3s|  2400 |  1329 | 42310 |  15.8 |    57M |  98 | 696 |1288 |   0 | 882 |  0 | 518 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 14.7s|  2500 |  1357 | 42577 |  15.3 |    57M |  98 | 696 |1318 | 801 | 892 |  0 | 548 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 15.0s|  2600 |  1375 | 43073 |  14.9 |    57M |  98 | 696 |1358 | 798 | 897 |  1 | 589 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 15.3s|  2700 |  1387 | 43830 |  14.6 |    58M |  98 | 696 |1364 | 799 | 900 |  1 | 595 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 15.4s|  2800 |  1399 | 44344 |  14.3 |    58M |  99 | 696 |1380 | 797 | 904 |  1 | 611 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 15.6s|  2900 |  1403 | 44786 |  13.9 |    58M |  99 | 696 |1459 | 800 | 904 |  1 | 690 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
 15.8s|  3000 |  1426 | 45314 |  13.6 |    58M |  99 | 696 |1463 | 801 | 911 |  0 | 694 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 16.0s|  3100 |  1460 | 46031 |  13.4 |    58M |  99 | 696 |1471 | 800 | 915 |  1 | 702 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 16.1s|  3200 |  1475 | 46509 |  13.2 |    58M | 102 | 696 |1553 | 800 | 918 |  1 | 784 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 16.3s|  3300 |  1500 | 47027 |  12.9 |    59M | 102 | 696 |1625 | 798 | 928 |  1 | 856 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 16.5s|  3400 |  1509 | 47486 |  12.7 |    59M | 102 | 696 |1660 | 800 | 934 |  1 | 891 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 16.7s|  3500 |  1513 | 47924 |  12.4 |    59M | 102 | 696 |1691 | 801 | 936 |  0 | 923 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 16.9s|  3600 |  1533 | 48534 |  12.3 |    59M | 102 | 696 |1711 | 800 | 944 |  1 | 943 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 17.1s|  3700 |  1556 | 49109 |  12.1 |    59M | 102 | 696 |1749 | 800 | 951 |  1 | 981 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 17.4s|  3800 |  1570 | 49616 |  11.9 |    59M | 102 | 696 |1798 |   0 | 957 |  0 |1030 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 17.6s|  3900 |  1585 | 50243 |  11.7 |    59M | 102 | 696 |1826 | 801 | 962 |  0 |1058 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 17.8s|  4000 |  1591 | 50734 |  11.6 |    59M | 102 | 696 |1859 | 800 | 969 |  1 |1091 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 18.1s|  4100 |  1608 | 51436 |  11.5 |    59M | 102 | 696 |1906 | 797 | 975 |  1 |1139 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 18.2s|  4200 |  1610 | 52020 |  11.3 |    59M | 102 | 696 |1919 | 799 | 981 |  1 |1152 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 18.4s|  4300 |  1644 | 52467 |  11.2 |    59M | 102 | 696 |1928 | 799 | 990 |  1 |1162 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 18.7s|  4400 |  1665 | 52952 |  11.0 |    59M | 102 | 696 |1958 | 799 |1000 |  1 |1194 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
 19.1s|  4500 |  1689 | 53404 |  10.9 |    59M | 102 | 696 |1986 | 798 |1010 |  0 |1222 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 19.2s|  4600 |  1704 | 53890 |  10.8 |    59M | 102 | 696 |2009 | 783 |1011 |  1 |1247 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 19.6s|  4700 |  1804 | 55880 |  10.9 |    59M | 102 | 696 |2006 | 790 |1063 |  1 |1251 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 19.8s|  4800 |  1902 | 57452 |  11.0 |    59M | 102 | 696 |2017 | 787 |1093 |  1 |1265 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 20.1s|  4900 |  1996 | 58609 |  11.1 |    60M | 102 | 696 |2032 | 789 |1126 |  1 |1282 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 20.2s|  5000 |  2071 | 59468 |  11.0 |    60M | 102 | 696 |2055 | 790 |1147 |  0 |1306 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 20.4s|  5100 |  2108 | 60009 |  10.9 |    60M | 102 | 696 |2070 | 793 |1166 |  0 |1321 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 20.6s|  5200 |  2166 | 60842 |  10.9 |    60M | 102 | 696 |2084 | 786 |1186 |  1 |1339 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 20.8s|  5300 |  2240 | 61934 |  10.9 |    60M | 102 | 696 |2081 | 791 |1225 |  1 |1340 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 20.9s|  5400 |  2309 | 62399 |  10.7 |    60M | 102 | 696 |2162 | 792 |1226 |  1 |1422 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 21.1s|  5500 |  2360 | 62953 |  10.6 |    60M | 102 | 696 |2218 |   0 |1236 |  0 |1481 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 21.2s|  5600 |  2396 | 63346 |  10.5 |    60M | 102 | 696 |2283 | 793 |1241 |  1 |1546 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 21.4s|  5700 |  2457 | 64154 |  10.5 |    60M | 102 | 696 |2286 | 791 |1260 |  1 |1554 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 21.6s|  5800 |  2539 | 64861 |  10.4 |    60M | 102 | 696 |2312 | 792 |1273 |  0 |1580 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 21.7s|  5900 |  2613 | 65589 |  10.4 |    60M | 102 | 696 |2306 | 792 |1288 |  0 |1580 |  13 | 9.000000e+00 | 1.000000e+01 |  11.11%|  51.22%
 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
r21.9s|  5979 |     0 | 66116 |  10.3 |randroun| 102 | 696 |2304 | 788 |1298 |  1 |1604 |  13 | 9.000000e+00 | 9.000000e+00 |   0.00%| 100.00%

SCIP Status        : problem is solved [optimal solution found]
Solving Time (sec) : 21.90
Solving Nodes      : 5979 (total of 5980 nodes in 2 runs)
Primal Bound       : +9.00000000000000e+00 (119 solutions)
Dual Bound         : +9.00000000000000e+00
Gap                : 0.00 %
[[], [8.62, 8.57, 8.31, 7.14, 6.03, 5.75, 5.31], [], [8.5, 8.47, 8.2, 8.12, 5.74, 5.64, 5.31], [], [], [8.63, 8.63, 8.49, 8.26, 8.12, 7.85], [8.77, 8.72, 8.53, 8.44, 8.27, 7.06], [8.7, 8.62, 8.6, 8.42, 8.33, 5.81], [8.74, 8.73, 8.61, 8.37, 8.31, 6.3], [8.41, 8.39, 8.38, 8.28, 8.27, 8.15], [8.77, 8.62, 8.53, 8.49, 7.13, 6.31], [8.78, 8.76, 8.57, 6.02, 5.78, 5.76, 5.76]]

python_my

import pandas as pd

def pd2org(df_to_table):
    return tabulate(df_to_table, headers=df_to_table.columns, tablefmt='orgtbl')

df = pd.DataFrame(data, columns=['weight'])
pd2org(df.describe())

	weight
count	57
mean	7.79263
std	1.15286
min	5.31
25%	7.13
50%	8.37
75%	8.6
max	8.78

import numpy as np
weights = np.array(data).flatten()
print(weights)

[8.78 8.77 8.77 8.76 8.74 8.73 8.72 8.7  8.63 8.63 8.62 8.62 8.62 8.61
 8.6  8.57 8.57 8.53 8.53 8.5  8.49 8.49 8.47 8.44 8.42 8.41 8.39 8.38
 8.37 8.33 8.31 8.31 8.28 8.27 8.27 8.26 8.2  8.15 8.12 8.12 7.85 7.14
 7.13 7.06 6.31 6.3  6.03 6.02 5.81 5.78 5.76 5.76 5.75 5.74 5.64 5.31
 5.31]

import numpy as np
from scipy import optimize
sizes = weights
bounds = optimize.Bounds(0, 1) # 0 <= x_i <= 1
integrality = np.full_like(values, True)  # x_i are integers

constraints = optimize.LinearConstraint(A=sizes, lb=0, ub=capacity)

python3 - final solution

import random
import numpy as np
import time

def FFD(s, bs):
    """heuristic - pack to containers
    :return [[8], [8], [7, 2], [7, 2], [6, 3], ...]"""
    bin_sizes=[]
    r = random.randint(0,len(bs)-1)
    remain = [bs[r]]
    bin_sizes.append(bs[r])
    sol = [[]]
    for item in sorted(s, reverse=True):
        for j,free in enumerate(remain):
            if free >= item:
                remain[j] -= item
                sol[j].append(item)
                break
        else:
            sol.append([item])
            r = random.randint(0,len(bs)-1)
            remain.append(bs[r]-item)
            bin_sizes.append(bs[r])
    return sol, bin_sizes


start_time = time.time()
# print(weight)
# weight = np.array(data).flatten()
weight = [8.78, 8.77, 8.77, 8.76, 8.74, 8.73, 8.72, 8.7, 8.63, 8.63, 8.62, 8.62, 8.62, 8.61, 8.6, 8.57, 8.57, 8.53, 8.53, 8.5, 8.49, 8.49, 8.47, 8.44, 8.42, 8.41, 8.39, 8.38, 8.37, 8.33, 8.31, 8.31, 8.28, 8.27, 8.27, 8.26, 8.2, 8.15, 8.12, 8.12, 7.85, 7.14, 7.13, 7.06, 6.31, 6.3, 6.03, 6.02, 5.81, 5.78, 5.76, 5.76, 5.75, 5.74, 5.64, 5.31, 5.31]
bins_s = [22.2, 27.6]
c = 12
# min_bins = len(weight)
# min_mass = max(bins_s)
mer = 99999999999999999
s,b = None, None
for _ in range(10000):
    sol, bs = FFD(weight,bins_s)
    # if len(sol) < min_bins or sum(bs) < min_mass:
    if (len(sol) + sum(bs)) < mer:
        s = sol
        b = bs
        # min_bins=len(sol)
        mer = len(sol) + sum(bs)

import collections
print(s)
print(collections.Counter(b))
print("time seconds:", round(time.time() - start_time))

[[8.78, 8.77, 8.77], [8.76, 8.74, 8.73], [8.72, 8.7, 8.63], [8.63, 8.62, 8.62], [8.62, 8.61, 8.6], [8.57, 8.57, 8.53], [8.53, 8.5, 8.49], [8.49, 8.47, 8.44], [8.42, 8.41, 8.39], [8.38, 8.37, 5.31], [8.33, 8.31, 8.31], [8.28, 8.27, 8.27], [8.26, 8.2, 5.74], [8.15, 8.12, 5.81], [8.12, 7.85, 6.03], [7.14, 7.13, 7.06, 6.02], [6.31, 6.3, 5.78, 5.76], [5.76, 5.75, 5.64, 5.31]]
Counter({27.6: 14, 22.2: 4})
time seconds: 1