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我正在尝试为酒店收入管理系统构建关于动态房间定价模型的白皮书的实施。万一此链接将来失效，我将在此处粘贴相关部分： 在此处输入图像描述

到目前为止，我目前的实现被严重破坏，因为我真的不完全理解如何求解非线性最大化方程。

# magical lookup table that returns demand based on those inputs
# this will eventually be a db lookup against past years rental activity and not hardcoded to a specific value
def demand(dateFirstNight, duration):
    return 1

# magical function that fetches the price we have allocated for a room on this date to existing customers
# this should be a db lookup against previous stays, and not hardcoded to a specific value
def getPrice(date):
    return 75

# Typical room base price
# Defined as: Nominal price of the hotel (usually the average historical price)
nominalPrice = 89

# from the white paper, but perhaps needs to be adjusted in the future using the methods they explain
priceElasticity = 2

# this is an adjustable constant it depends how far forward we want to look into the future when optimizing the prices
# likely this will effect how long this will take to run, so it will be a balancing game with regards to accuracy vs runtime
numberOfDays = 30

def roomsAlocated(dateFirstNight, duration)
    roomPriceSum = 0.0
    for date in range(dateFirstNight, dateFirstNight+duration-1):
        roomPriceSum += getPrice(date)

    return demand(dateFirstNight, duration) * (roomPriceSum/(nominalPrice*duration))**priceElasticity


def roomsReserved(date):
    # find all stays that contain this date, this 


def maximizeRevenue(dateFirstNight):
    # we are inverting the price sum which is to be maximized because mystic only does minimization
    # and when you minimize the inverse you are maximizing!
    return (sum([getPrice(date)*roomsReserved(date) for date in range(dateFirstNight, dateFirstNight+numberOfDays)]))**-1


def constraint(x): # Ol - totalNumberOfRoomsInHotel <= 0
    return roomsReserved(date) - totalNumberOfRoomsInHotel

from mystic.penalty import quadratic_inequality
@quadratic_inequality(constraint, k=1e4)
def penalty(x):
  return 0.0

from mystic.solvers import diffev2
from mystic.monitors import VerboseMonitor
mon = VerboseMonitor(10)

bounds = [0,1e4]*numberOfDays
result = diffev2(maximizeRevenue, x0=bounds, penalty=penalty, npop=10, gtol=200, disp=False, full_output=True, itermon=mon, maxiter=M*N*100)

任何熟悉 mystic 的人都可以给我一些关于如何实现它的建议吗？

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2 回答 2

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虽然您要求使用该库mystic，但在开始使用非线性优化时，您可能不需要这种细粒度的控制。模块scipy应该足够了。接下来是一个或多或少完整的解决方案，纠正了我认为原始白皮书中关于定价范围的错字：

import numpy as np
from scipy.optimize import minimize

P_nom = 89
P_max = None
price_elasticity = 2
number_of_days = 7
demand = lambda a, L: 1./L
total_rooms = [5]*number_of_days

def objective(P, *args):
    return -np.dot(P, O(P, *args))

def worst_leftover(P, C, *args):
    return min(np.subtract(C, O(P, *args)))

def X(P, a, L, d, e, P_nom):
    return d(a, L)*(sum(P[a:a+L])/P_nom/L)**e

def d(a, L):
    return 1.

def O_l(P, l, l_max, *args):
    return sum([X(P, a, L, *args)
                for a in xrange(0, l)
                for L in xrange(l-a+1, l_max+1)])

def O(P, *args):
    return [O_l(P, l, *args) for l in xrange(len(P))]

result = minimize(
    objective,
    [P_nom]*number_of_days,
    args=(number_of_days-1, demand, price_elasticity, P_nom),
    method='SLSQP',
    bounds=[(0, P_max)]*number_of_days,
    constraints={
        'type': 'ineq',
        'fun': worst_leftover,
        'args': (total_rooms, number_of_days-1, demand, price_elasticity, P_nom)
    },
    tol=1e-1,
    options={'maxiter': 10**3}
)

print result.x

有几点值得一提：

  1. 目标函数添加了一个减号，用于 scipy 的minimize()例程，与原始白皮书中引用的最大化形成对比。这将导致result.fun为负数，而不是表示总收入。

  2. 该公式似乎对参数有点敏感。最小化是正确的（至少，当它说它正确执行时它是正确的——检查result.success），但如果输入与现实相差太远，那么您可能会发现价格远高于预期。您可能还想使用比我在以下示例中更多的天数。您的白皮书似乎会引起周期性效应。

  3. 我不太喜欢白皮书的命名方案，因为它与可读代码有关。我改变了一些东西，但有些东西真的很糟糕，应该被替换，比如小写的 l 很容易与数字 1 混淆。

  4. 我确实设置了界限，以便价格是正数而不是负数。凭借您的领域专业知识，您应该验证这是正确的决定。

  5. 您可能更喜欢比我指定的更严格的公差。这在某种程度上取决于您希望运行时是什么。随意使用tol参数。此外，对于更严格的公差，您可能会发现必须'maxiter'增加options参数才能minimize()正确收敛。

  6. 我很确定total_rooms应该是酒店中尚未预订的房间数量，因为白皮书用字母 l 对其进行索引，而不是像原始代码中那样保持不变。我将其设置为用于测试目的的常量列表。

  7. 该方法必须是“SLSQP”才能处理价格界限和房间数量界限。注意不要改变这个。

  8. 计算方式存在巨大的低效率O_l()。如果运行时是一个问题，我会采取的第一步是弄清楚如何缓存/记忆对X(). 所有这一切实际上只是第一次通过，概念验证。它应该是合理的无错误和正确的，但它几乎是直接从白皮书中提取的，并且可以进行一些重构。

任何人，玩得开心，如有任何进一步的问题，请随时发表评论/PM/等。

于 2018-01-04T05:18:25.467 回答
3

抱歉，我回答这个问题迟到了，但我认为接受的答案并没有解决完整的问题，而且还没有正确解决问题。请注意，在局部最小化中，解决接近名义价格的问题并不能提供最佳解决方案。

让我们首先构建一个hotel类：

"""
This file is 'hotel.py'
"""
import math

class hotel(object):
    def __init__(self, rooms, price_ave, price_elastic):
        self.rooms = rooms
        self.price_ave = price_ave
        self.price_elastic = price_elastic

    def profit(self, P):
        # assert len(P) == len(self.rooms)
        return sum(j * self._reserved(P, i) for i,j in enumerate(P))

    def remaining(self, P): # >= 0
        C = self.rooms
        # assert len(P) == C
        return [C[i] - self._reserved(P, i) for i,j in enumerate(P)]

    def _reserved(self, P, day):
        max_days = len(self.rooms)
        As = range(0, day)
        return sum(self._allocated(P, a, L) for a in As
                   for L in range(day-a+1, max_days+1))

    def _allocated(self, P, a, L):
        P_nom = self.price_ave
        e = self.price_elastic
        return math.ceil(self._demand(a, L)*(sum(P[a:a+L])/(P_nom*L))**e)

    def _demand(self, a,L): #XXX: fictional non-constant demand function
        return abs(1-a)/L + 2*(a**2)/L**2

这是解决它的一种方法mystic：

"""
This file is 'local.py'
"""
n_days = 7
n_rooms = 50
P_nom = 85
P_bounds = 0,None
P_elastic = 2

import hotel
h = hotel.hotel([n_rooms]*n_days, P_nom, P_elastic)

def objective(price, hotel):
    return -hotel.profit(price)

def constraint(price, hotel): # <= 0
    return -min(hotel.remaining(price))

bounds = [P_bounds]*n_days
guess = [P_nom]*n_days

import mystic as my

@my.penalty.quadratic_inequality(constraint, kwds=dict(hotel=h))
def penalty(x):
    return 0.0

# using a local optimizer, starting from the nominal price
solver = my.solvers.fmin
mon = my.monitors.VerboseMonitor(100)

kwds = dict(disp=True, full_output=True, itermon=mon,
            args=(h,),  xtol=1e-8, ftol=1e-8, maxfun=10000, maxiter=2000)
result = solver(objective, guess, bounds=bounds, penalty=penalty, **kwds)

print([round(i,2) for i in result[0]])

结果：

>$ python local.py 
Generation 0 has Chi-Squared: -4930.000000
Generation 100 has Chi-Squared: -22353.444547
Generation 200 has Chi-Squared: -22410.402420
Generation 300 has Chi-Squared: -22411.780268
Generation 400 has Chi-Squared: -22413.908944
Generation 500 has Chi-Squared: -22477.869093
Generation 600 has Chi-Squared: -22480.144244
Generation 700 has Chi-Squared: -22480.280379
Generation 800 has Chi-Squared: -22485.563188
Generation 900 has Chi-Squared: -22485.564265
Generation 1000 has Chi-Squared: -22489.341602
Generation 1100 has Chi-Squared: -22489.345912
Generation 1200 has Chi-Squared: -22489.351219
Generation 1300 has Chi-Squared: -22491.994305
Generation 1400 has Chi-Squared: -22491.994518
Generation 1500 has Chi-Squared: -22492.061127
Generation 1600 has Chi-Squared: -22492.573672
Generation 1700 has Chi-Squared: -22492.573690
Generation 1800 has Chi-Squared: -22492.622064
Generation 1900 has Chi-Squared: -22492.622230
Optimization terminated successfully.
         Current function value: -22492.622230
         Iterations: 1926
         Function evaluations: 3346
STOP("CandidateRelativeTolerance with {'xtol': 1e-08, 'ftol': 1e-08}")
[1.15, 20.42, 20.7, 248.1, 220.56, 41.4, 160.09]

再次使用全局优化器：

"""
This file is 'global.py'
"""
n_days = 7
n_rooms = 50
P_nom = 85
P_bounds = 0,None
P_elastic = 2

import hotel
h = hotel.hotel([n_rooms]*n_days, P_nom, P_elastic)

def objective(price, hotel):
    return -hotel.profit(price)

def constraint(price, hotel): # <= 0
    return -min(hotel.remaining(price))

bounds = [P_bounds]*n_days
guess = [P_nom]*n_days

import mystic as my

@my.penalty.quadratic_inequality(constraint, kwds=dict(hotel=h))
def penalty(x):
    return 0.0

# try again using a global optimizer
solver = my.solvers.diffev
mon = my.monitors.VerboseMonitor(100)

kwds = dict(disp=True, full_output=True, itermon=mon, npop=40,
            args=(h,),  gtol=250, ftol=1e-8, maxfun=30000, maxiter=2000)
result = solver(objective, bounds, bounds=bounds, penalty=penalty, **kwds)

print([round(i,2) for i in result[0]])

结果：

>$ python global.py 
Generation 0 has Chi-Squared: 3684702.124765
Generation 100 has Chi-Squared: -36493.709890
Generation 200 has Chi-Squared: -36650.498677
Generation 300 has Chi-Squared: -36651.722841
Generation 400 has Chi-Squared: -36651.733274
Generation 500 has Chi-Squared: -36651.733322
Generation 600 has Chi-Squared: -36651.733361
Generation 700 has Chi-Squared: -36651.733361
Generation 800 has Chi-Squared: -36651.733361
STOP("ChangeOverGeneration with {'tolerance': 1e-08, 'generations': 250}")
Optimization terminated successfully.
         Current function value: -36651.733361
         Iterations: 896
         Function evaluations: 24237
[861.07, 893.88, 398.68, 471.4, 9.44, 0.0, 244.67]

我认为为了产生更合理的定价，我会将P_bounds值更改为更合理的值。

于 2018-01-29T04:31:16.613 回答