【算法项目】Capacitated Facility Location Problem

项目地址：https://github.com/GostBop/CFLP

Capacitated Facility Location Problem

问题描述

Suppose there are n facilities and m customers. We wish to choose:

1. which of the n facilities to open
2. the assignment of customers to facilities

Note：

1. The objective is to minimize the sum of the opening cost and the assignment cost.
2. The total demand assigned to a facility must not exceed its capacity.

问题建模：

Let
$I$={1,…, $I$}：the set of $facilities$
$J$={1,…, $J$}：the set of $customers$
$d_j$： $customer$ $j$ 's $Demand$， $j∈J$
$V_i$： $facility$ $i$ 's $Capacity$， $i∈I$
$C_i$： $facility$ $i$ 's $OpeningCost$， $i∈I$
$A_{ij}$: $facility$ $i$ 's $AssignmentCost$ for $customer$ $j$， $i∈I$， $j∈J$

For every $facility$ $i$， $i∈I$， define the following notations：
$y_i = \begin{cases} 1 & if.facility.i.is.opened \\ 0 & otherwise \end{cases}$

For every $facility$ $i$， $i∈I$； $customer$ $j$， $j∈J$；
$x_{ij} = \begin{cases} 1 & if.facility.i.is.assigned.for.customer.j\\ 0 & otherwise \end{cases}$

Then the Capacitated Facility Location Problem may be written：
$\sum_{j∈J}d_jx_{ij} \leq V_iy_i，i∈I$
$min\lbrace \sum_{i∈I}c_iy_i+\sum_{i∈I}\sum_{j∈J}A_{ij}x_{ij}\rbrace$
$\sum_{i∈I}x_{ij}=1，j∈J$
$x_{ij}，y_i∈\lbrace0, 1\rbrace，i∈I，j∈J$

贪心算法

CFLP问题是NP问题，很难设计一种直接求出最优解的算法，但是我们可以用贪心算法来接近最优解。

贪心算法的策略是：

1. 遍历每一个 $customer$ $j$， $j∈J$
2. 让 $customer$ $j$逐个选择cost最小的 $facility$ $i$，即：
   $min\lbrace \sum_{i∈I}c_iy_i+\sum_{i∈I}A_{ij}x_{ij}\rbrace$

代码实现

# coding:utf-8

import numpy as np
import time
import pandas as pd

Capacity =[]
Open = []
OpenCost = []
Demand = []
AssignCost = [[]]
Assign = [[]]

def readFile(file_name):
  global Capacity, Assign, Demand, OpenCost, AssignCost, Open
  fr = open('../Instances/' + file_name)
  firstArr = fr.readline().strip().split()
  fac_num = int(firstArr[0])
  cus_num = int(firstArr[1])
  Capacity = [0] * fac_num
  OpenCost = [0] * fac_num
  Demand = [0] * cus_num
  Open = [0] * fac_num
  AssignCost = [[0 for _ in range(cus_num)] for _ in range(fac_num)]
  Assign = [[0 for _ in range(cus_num)] for _ in range(fac_num)]
  for i in range(fac_num):
    lineArr = fr.readline().strip().split()
    Capacity[i] = float(lineArr[0])
    OpenCost[i] = float(lineArr[1])
  for i in range(cus_num / 10):
    lineArr = fr.readline().strip().split()
    for j in range(10):
        Demand[i * 10 + j] = float(lineArr[j])
  for i in range(fac_num):
    for j in range(cus_num / 10):
        lineArr = fr.readline().strip().split()
        for k in range(10):
            AssignCost[i][j * 10 + k] = float(lineArr[k])
  fr.close()


def greedy():
    #print len(Assign[0])
    i = len(Assign[0]) - 1
    while i >= 0:
    #for i in range(len(Assign[0])):
      min = 99999
      fac = 0
      cus = 0
      for j in range(len(Assign)):
          Assign[j][i] = 1
          if capcaityLimit(j) == 1:   
              Assign[j][i] = 0 
              continue
          Assign[j][i] = 0
          cost = getOneCost(j, i)
          if cost < min:
            min = cost
            fac = j
            cus = i
      Assign[fac][cus] = 1
      i -= 1
    #print Assign

def getOneCost(fac, cus):
  cost = OpenCost[fac] + AssignCost[fac][cus]
  return cost

def getcost():
    global Open
    cost = 0
    Open = [0] * len(Capacity)
    for i in range(len(Demand)):
        for j in range(len(Open)):
            if Assign[j][i] == 1:
                Open[j] = 1
    for i in range(len(OpenCost)):
        cost += OpenCost[i] * Open[i]
    for i in range(len(AssignCost)):
        for j in range(len(AssignCost[0])):
            cost += AssignCost[i][j] * Assign[i][j]
    return cost

def capcaityLimit(fac):
    cap = 0
    flag = 0
    for i in range(len(Assign[fac])):
        cap += Assign[fac][i] * Demand[i]
        if cap > Capacity[fac]:
            flag = 1
            break
    return flag

def capcaityNow(fac):
    cap = 0
    for i in range(len(Assign[fac])):
        cap += Assign[fac][i] * Demand[i]
    return cap

def main(file_name):
    readFile(file_name)

    start = time.time()
    greedy()
    
    cost = getcost() 

    end = time.time()
    t = end - start
    print cost

    cus_fac = [0] * len(Demand)
    for i in range(len(Demand)):
        for j in range(len(Open)):
            if Assign[j][i] == 1:
                cus_fac[i] = j + 1
                Open[j] = 1
    print Open
    print cus_fac

    f = open('result.txt', 'a')
    f.write(file_name + '\n')
    f.write(str(cost) + '\n')
    for i in range(len(Open)):
        f.write(str(Open[i]) + ' ')
    f.write('\n')
    for i in range(len(cus_fac)):
        f.write(str(cus_fac[i]) + ' ')
    f.write('\n\n')
    f.close()
    return cost, t

if __name__ == "__main__":
    file_name = []
    for i in range(71):
        file_name.append("p" + str(i + 1))
    result = []
    times = []
    for i in range(71):
        r, t = main(file_name[i])
        result.append(r)
        times.append(t)
    #print result ,times
    #字典中的key值即为csv中列名
    dataframe = pd.DataFrame({'file': file_name, 'result': result,'time':times})

    #将DataFrame存储为csv,index表示是否显示行名，default=True
    dataframe.to_csv("Greedy.csv",index=False,sep=',')

运算结果

file	result	time
p1	12581.0	0.016000032424926758
p2	8010.0	0.016000032424926758
p3	10010.0	0.014999866485595703
p4	12010.0	0.016000032424926758
p5	12128.0	0.0
p6	8329.0	0.0
p7	10329.0	0.0
p8	12329.0	0.016000032424926758
p9	10776.0	0.0
p10	7726.0	0.0
p11	9726.0	0.016000032424926758
p12	11726.0	0.0
p13	10913.0	0.015000104904174805
p14	9180.0	0.016000032424926758
p15	13180.0	0.014999866485595703
p16	17180.0	0.016000032424926758
p17	10104.0	0.014999866485595703
p18	9180.0	0.015000104904174805
p19	13180.0	0.032000064849853516
p20	17180.0	0.014999866485595703
p21	9845.0	0.016000032424926758
p22	9180.0	0.0
p23	13180.0	0.015000104904174805
p24	17180.0	0.0
p25	18186.0	0.14499998092651367
p26	16095.0	0.1419999599456787
p27	21495.0	0.127000093460083
p28	26895.0	0.127000093460083
p29	18225.0	0.18000006675720215
p30	16173.0	0.14300012588500977
p31	21573.0	0.12800002098083496
p32	26973.0	0.1530001163482666
p33	17364.0	0.1550002098083496
p34	15989.0	0.14599990844726562
p35	21389.0	0.14400005340576172
p36	26789.0	0.15900015830993652
p37	16825.0	0.20200014114379883
p38	15989.0	0.15900015830993652
p39	21389.0	0.127000093460083
p40	26789.0	0.14300012588500977
p41	9276.0	0.016000032424926758
p42	9267.0	0.03099989891052246
p43	7629.0	0.031000137329101562
p44	8833.0	0.02499985694885254
p45	8331.0	0.03299999237060547
p46	8581.0	0.1099998950958252
p47	9203.0	0.046000003814697266
p48	9262.0	0.04699993133544922
p49	7766.0	0.03099989891052246
p50	11502.0	0.015000104904174805
p51	10099.0	0.06299996376037598
p52	13352.0	0.031000137329101562
p53	13680.0	0.04699993133544922
p54	13598.0	0.016000032424926758
p55	11411.0	0.04699993133544922
p56	23882.0	0.2760000228881836
p57	32882.0	0.33500003814697266
p58	53882.0	0.3280000686645508
p59	43697.0	0.24000000953674316
p60	23882.0	0.2519998550415039
p61	32882.0	0.2369999885559082
p62	53882.0	0.2519998550415039
p63	39308.0	0.23600006103515625
p64	23882.0	0.252000093460083
p65	32882.0	0.2519998550415039
p66	53882.0	0.34800004959106445
p67	42079.0	0.252000093460083
p68	23882.0	0.23799991607666016
p69	32882.0	0.26799988746643066
p70	53882.0	0.25299978256225586
p71	42449.0	0.2519998550415039

每一个测例的具体结果
https://gostbop.github.io/CFLP/

模拟退火

模拟退火是一种启发式搜索，该算法每次从当前解的临近解空间中选择一个最优解作为当前解，直到达到一个局部最优解。

模拟退火其实也是一种贪心算法，但是它的搜索过程引入了随机因素。模拟退火算法以一定的概率来接受一个比当前解要差的解，因此有可能会跳出这个局部的最优解，达到全局的最优解。

初始解

可以使用贪心算法的解作为初始解

邻域搜索

这里使用一种简单的邻域搜索。

1. 随机选择一个 $facility$和一个 $customer$： $a$， $b$
2. 如果该 $facility$ $a$已经 $Open$并且 $Assign$ $for$ $customer$ $b$
3. 那么随机选择另一个 $facility$ $c$
4. 否则重复选择 $a$和 $b$
5. 将 $customer$ $b$从 $facility$ $a$中撤出给 $facility$ $c$
6. 如果没有超出 $facility$ $c$的 $Capacity$限制，则操作成功
7. 否则从1开始重复该操作

初始参数设置

初温：100℃
内循环迭代次数：200
退火速率： $t = t *0.9$
退火停止点： $t < 1$

代码实现

# coding:utf-8

import numpy as np
import time as ti
import pandas as pd

Capacity =[]
Open = []
OpenCost = []
Demand = []
AssignCost = [[]]
Assign = [[]]

def readFile(file_name):
  global Capacity, Assign, Demand, OpenCost, AssignCost, Open

  fr = open('Instances/' + file_name)
  firstArr = fr.readline().strip().split()
  fac_num = int(firstArr[0])
  cus_num = int(firstArr[1])

  Capacity = [0] * fac_num
  OpenCost = [0] * fac_num
  Demand = [0] * cus_num
  Open = [0] * fac_num
  AssignCost = [[0 for _ in range(cus_num)] for _ in range(fac_num)]
  Assign = [[0 for _ in range(cus_num)] for _ in range(fac_num)]

  for i in range(fac_num):
    lineArr = fr.readline().strip().split()
    Capacity[i] = float(lineArr[0])
    OpenCost[i] = float(lineArr[1])

  for i in range(cus_num / 10):
    lineArr = fr.readline().strip().split()
    for j in range(10):
        Demand[i * 10 + j] = float(lineArr[j])

  for i in range(fac_num):
    for j in range(cus_num / 10):
        lineArr = fr.readline().strip().split()
        for k in range(10):
            AssignCost[i][j * 10 + k] = float(lineArr[k])
  fr.close()

def greedy():
    i = len(Assign[0]) - 1
    while i >= 0:
      min = 99999
      fac = 0
      cus = 0
      for j in range(len(Assign)):
          Assign[j][i] = 1
          if capcaityLimit(j) == 1:   
              Assign[j][i] = 0 
              continue
          Assign[j][i] = 0
          cost = getOneCost(j, i)
          if cost < min:
            min = cost
            fac = j
            cus = i
      Assign[fac][cus] = 1
      i -= 1

def getOneCost(fac, cus):
  cost = OpenCost[fac] + AssignCost[fac][cus]
  return cost

def getcost():
    global Open
    cost = 0
    Open = [0] * len(Capacity)

    for i in range(len(Demand)):
        for j in range(len(Open)):
            if Assign[j][i] == 1:
                Open[j] = 1

    for i in range(len(OpenCost)):
        cost += OpenCost[i] * Open[i]

    for i in range(len(AssignCost)):
        for j in range(len(AssignCost[0])):
            cost += AssignCost[i][j] * Assign[i][j]
    return cost

def capcaityLimit(fac):
    cap = 0
    flag = 0
    for i in range(len(Assign[fac])):
        cap += Assign[fac][i] * Demand[i]
        if cap > Capacity[fac]:
            flag = 1
            break
    return flag

def exchange():
    while True:  
        while True:
            fac1 = np.int(np.ceil(np.random.rand()*(len(Open)-1)))
            cus = np.int(np.ceil(np.random.rand()*(len(Assign[0])-1)))
            if Assign[fac1][cus] == 1:
                break

        fac2 = np.int(np.ceil(np.random.rand()*(len(Open)-1)))
        Assign[fac1][cus], Assign[fac2][cus] = Assign[fac2][cus], Assign[fac1][cus]

        if capcaityLimit(fac1) == 0 and capcaityLimit(fac2) == 0:
            break
        else:
            Assign[fac1][cus], Assign[fac2][cus] = Assign[fac2][cus], Assign[fac1][cus]
            
    return fac1, fac2, cus


def initpara():
    alpha = 0.9
    t = (1,100)
    markovlen = 200
 
    return alpha,t,markovlen

def main(file_name):
    readFile(file_name)

    start = ti.time()
    greedy()
    
    costcurrent = getcost() 
    alpha,t2,markovlen = initpara()
    t = t2[1]

    costbest = costcurrent
    while t > t2[0]:
        for i in np.arange(markovlen):
            fac1, fac2, cus = exchange()
            costnew = getcost()
            if costnew < costcurrent: 
                costcurrent = costnew
            else:
                if np.random.rand() < np.exp(-(costnew-costcurrent)/t):
                    costcurrent = costnew
                    if costcurrent < costbest:
                        costbest = costcurrent
                else:
                    Assign[fac1][cus], Assign[fac2][cus] = Assign[fac2][cus], Assign[fac1][cus]

        t = alpha*t
    end = ti.time()
    time = end-start
    print costbest

    cus_fac = [0] * len(Demand)
    for i in range(len(Demand)):
        for j in range(len(Open)):
            if Assign[j][i] == 1:
                cus_fac[i] = j + 1
                Open[j] = 1
    print Open
    print cus_fac
    f = open('result.txt', 'a')
    f.write(file_name + '\n')
    f.write(str(costbest) + '\n')
    for i in range(len(Open)):
        f.write(str(Open[i]) + ' ')
    f.write('\n')
    for i in range(len(cus_fac)):
        f.write(str(cus_fac[i]) + ' ')
    f.write('\n\n')
    f.close()
    return costbest, time

if __name__ == "__main__":
    file_name = []
    for i in range(71):
        file_name.append("p" + str(i + 1))
    result = []
    times = []
    for i in range(71):
        r, t = main(file_name[i])
        result.append(r)
        times.append(t)
    dataframe = pd.DataFrame({' ': file_name, 'result': result,'time':times})
    dataframe.to_csv("result.csv",index=False,sep=',')

运算结果

file	result	time
p1	9678.0	2.75600004196167
p2	7990.0	2.744999885559082
p3	9603.0	2.640000104904175
p4	11591.0	2.743000030517578
p5	9605.0	3.25600004196167
p6	8138.0	2.810999870300293
p7	10329.0	2.6430001258850098
p8	12329.0	2.49399995803833
p9	9148.0	2.372000217437744
p10	7704.0	2.4840002059936523
p11	9504.0	2.305000066757202
p12	11455.0	2.312000036239624
p13	8452.0	4.171000003814697
p14	7560.0	4.086000204086304
p15	10662.0	4.01200008392334
p16	11602.0	4.312000036239624
p17	9014.0	4.2099997997283936
p18	8003.0	5.271000146865845
p19	9775.0	5.880000114440918
p20	11640.0	5.6559998989105225
p21	8349.0	7.17300009727478
p22	7831.0	6.1540000438690186
p23	9877.0	7.026000022888184
p24	12410.0	6.108999967575073
p25	14339.0	20.210000038146973
p26	13302.0	19.980000019073486
p27	15251.0	18.19099998474121
p28	19461.0	23.40000009536743
p29	15079.0	19.846999883651733
p30	13307.0	20.64199995994568
p31	16281.0	20.246000051498413
p32	17362.0	19.317000150680542
p33	13992.0	20.42799997329712
p34	13222.0	19.358999967575073
p35	15299.0	21.421000003814697
p36	17986.0	18.194000005722046
p37	13301.0	18.980000019073486
p38	12688.0	20.819000005722046
p39	15036.0	21.80900001525879
p40	18436.0	20.05400013923645
p41	7366.0	6.083000183105469
p42	7391.0	8.682999849319458
p43	6614.0	11.13700008392334
p44	7577.0	6.707000017166138
p45	7425.0	10.75600004196167
p46	6829.0	13.08899998664856
p47	7678.0	6.708999872207642
p48	6772.0	9.803999900817871
p49	6676.0	13.911999940872192
p50	9516.0	7.547000169754028
p51	8883.0	11.312000036239624
p52	9972.0	6.828999996185303
p53	10456.0	11.537000179290771
p54	11990.0	5.876000165939331
p55	9105.0	11.332000017166138
p56	23882.0	29.610000133514404
p57	32882.0	27.73200011253357
p58	53882.0	26.651000022888184
p59	32983.0	30.92199993133545
p60	23882.0	29.806999921798706
p61	32882.0	27.8989999294281
p62	53736.0	28.141000032424927
p63	29141.0	32.33500003814697
p64	23882.0	29.52300000190735
p65	32882.0	30.270999908447266
p66	53882.0	28.205999851226807
p67	29236.0	29.211999893188477
p68	23882.0	30.06599998474121
p69	32882.0	26.61400008201599
p70	53882.0	29.68400001525879
p71	31321.0	29.894999980926514

每一个测例的具体结果
https://gostbop.github.io/CFLP/