Simplex單純性算法的Python實現

單純性算法是解決線性規劃的經典方法，上世紀50年代就提出了，其基本思想是在可行域內沿着邊遍歷所有的頂點，找出最優值，即爲算法的最優值。

算法的執行過程如下：

1. 求出初始基向量
2. 構建單純性表格
3. 在所有非基向量對應的c中，找出一個最小的ci，若該ci大於0，則退出，輸出obj，否則將ai入基
4. 利用基向量組線性表示ai，得到該線性表示的係數向量Λ
5. 對於Λ中所有大於0的分量，求出minmj=1bjΛj對應的j，然後將aj出基
6. 問題矩陣旋轉，即通過高斯行變換，將ai變換成aj
7. 如果ci全大於0，則退出算法，輸出obj，否則，重複第3步

__author__ = 'xanxus'


class Problem:
    def __init__(self):
        self.obj = 0
        self.coMatrix = []
        self.b = []
        self.c = []

    def pivot(self, basic, l, e):
        # the l-th line
        self.b[l] /= self.coMatrix[e][l]
        origin = self.coMatrix[e][l]
        for i in range(len(self.coMatrix)):
            self.coMatrix[i][l] /= origin
        # the other lines
        for i in range(len(self.b)):
            if i != l:
                self.b[i] = self.b[i] - self.b[l] * self.coMatrix[e][i]
        for i in range(len(self.b)):
            if i != l:
                origin = self.coMatrix[e][i]
                for j in range(len(self.coMatrix)):
                    self.coMatrix[j][i] = self.coMatrix[j][i] - origin * self.coMatrix[j][l]
        origin = self.c[e]
        for i in range(len(self.coMatrix)):
            self.c[i] = self.c[i] - self.coMatrix[i][l] * origin
        self.obj = self.obj - self.b[l] * origin
        basic[l] = e

    def clone(self, another):
        self.obj = another.obj
        for i in another.b:
            self.b.append(i)
        for i in another.c:
            self.c.append(i)
        for v in another.coMatrix:
            newv = []
            for i in v:
                newv.append(i)
            self.coMatrix.append(newv)


basic = []
problem = Problem()


def readProblem(filename):
    with open(filename)as f:
        var_num = int(f.readline())
        constrait_num = int(f.readline())
        matrix = [([0] * var_num) for i in range(constrait_num)]
        for i in range(constrait_num):
            line = f.readline()
            tokens = line.split(' ')
            for j in range(var_num):
                matrix[i][j] = float(tokens[j])
            problem.b.append(float(tokens[-1]))
        for i in range(var_num):
            var = []
            for j in range(constrait_num):
                var.append(matrix[j][i])
            problem.coMatrix.append(var)
        line = f.readline()
        tokens = line.split(' ')
        for i in range(var_num):
            problem.c.append(float(tokens[i]))


def getMinCIndex(c):
    min, minIndex = 1, 0
    for i in range(len(c)):
        if c[i] < 0 and c[i] < min:
            min = c[i]
            minIndex = i
    if min > 0:
        return -1
    else:
        return minIndex


def getLamdaVector(evector):
    ld = []
    for i in range(len(evector)):
        ld.append(evector[i])
    return ld


def simplex(basic, problem):
    minCIndex = getMinCIndex(problem.c)
    while minCIndex != -1:
        ld = getLamdaVector(problem.coMatrix[minCIndex])
        # find the l line
        l, min = -1, 10000000000
        for i in range(len(problem.b)):
            if ld[i] > 0:
                if problem.b[i] / ld[i] < min:
                    l = i
                    min = problem.b[i] / ld[i]
        if l == -1:
            return False
        problem.pivot(basic, l, minCIndex)
        minCIndex = getMinCIndex(problem.c)
    return True


def initialSimplex(basic, problem):
    min, minIndex = 1000000000, -1
    for i in range(len(problem.b)):
        if problem.b[i] < min:
            min = problem.b[i]
            minIndex = i
    for i in range(len(problem.b)):
        basic.append(i+len(problem.b))
    if min >= 0:
        return True
    else:
        originC = problem.c
        newC = []
        for i in range(len(problem.c)):
            newC.append(0)
        newC.append(1)
        problem.c = newC
        x0 = []
        for i in range(len(problem.b)):
            x0.append(-1)
        problem.coMatrix.append(x0)
        problem.pivot(basic, minIndex, -1)
        res = simplex(basic, problem)
        if res == False or problem.obj != 0:
            return False
        else:
            problem.c = originC
            problem.coMatrix.pop()
            # Gaussian row operation
            counter = 0
            for i in basic:
                if problem.c[i] != 0:
                    origin = problem.c[i]
                    for j in range(len(problem.c)):
                        problem.c[j] -= problem.coMatrix[j][counter] * origin
                    problem.obj -= problem.b[counter] * origin
                counter += 1
            return True


filename = raw_input('please input the problem description file: ')
readProblem(filename)
if initialSimplex(basic, problem):
    res = simplex(basic, problem)
    if res:
        print 'the optimal obj is %.2f' % problem.obj
        index = ['0.00'] * len(problem.coMatrix)
        counter = 0
        for i in basic:
            index[i] = '%.2f' % problem.b[counter]
            counter += 1
        print 'the solution is {%s}' % ' '.join(index)
    else:
        print 'no feasible solution'
else:
    print 'no feasible solution'

raw_input('please input any key to quit.')

希望對大家有所幫助。