Multi-Item Newsvendor Problem with Wasserstein Ambiguity Sets

In this example, we consider the multi-item newsvendor problem discussed in the paper Chen et al. (2020). This newsvendor problem determines the order quantity \(w_i\) of each of the \(I\) products under a total budget \(d\). The unit selling price and ordering cost for each product item are denoted by \(p_i\) and \(c_i\), respectively. The uncertain demand of each product item is denoted by the random variable \(\tilde{z}_i\). Once the demand realizes, the selling quantity \(y_i\) is expressed as \(\min{x_i, z_i}\), and the newsvendor problem can be written as the following distributionally robust optimization model,

with \(s\) the scenario index, and \(u\) the auxiliary random variable. The recourse decision \(y_i\) follows the approximated adaptation \(\overline{\mathcal{A}}(\{\{1\}, \{2\}, \dots, \{S\}\}, [I+1])\) indicating that \(y_i\) adapts to different scenarios \(s\) and is affinely adaptive to the random variables \(\pmb{z}\) and the auxiliary variable \(u\). The model maximizes the worst-case expectation over a Wasserstein ambiguity set \(\mathcal{F}\), expressed as follows.

with \(\theta \geq 0\) the parameter capturing the distance between the distribution \(\mathbb{P}\) and the empirical distribution \(\hat{\pmb{z}}\). The support \(\mathcal{Z}_s\) for each sample \(s\) is defined as

In this numerical experiment, parameters of the model and the ambiguity set are specified as follows:

• The number of products: \(I=2\);
• Sample size: \(S=50\);
• The unit cost of each product: \(c_i=1\)
• The unit price of each product: \(p_i\) is randomly generatd from a uniform distribution on \([1, 5]\).
• The upper bound of demand: the array \(\overline{\pmb{z}}\) is randomly generated from the uniform distribution on \([0, 100]^I\);
• The sample data of demands: the array \(\hat{\pmb{z}}\in\mathbb{R}^{S\times I}\) is randomly generated from the uniform distribution on \([\pmb{0}, \overline{\pmb{u}}]\);
• The budget \(d=50 I\);
• The Wasserstein metric parameter \(\theta=0.01 \min\{\overline{\pmb{z}}\}\).

The RSOME code for implementing the model above is given as follows.

from rsome import dro
from rsome import norm
from rsome import E
from rsome import grb_solver as grb
import numpy as np
import numpy.random as rd

# model and ambiguity set parameters
I = 2
S = 50
c = np.ones(I)
d = 50 * I
p = 1 + 4*rd.rand(I)
zbar = 100 * rd.rand(I)
zhat = zbar * rd.rand(S, I)
theta = 0.01 * zbar.min()

# modeling with RSOME
model = dro.Model(S)                        # create a DRO model with S scenarios
z = model.rvar(I)                           # random demand z
u = model.rvar()                            # auxiliary random variable

fset = model.ambiguity()                    # create an ambiguity set
for s in range(S):
    fset[s].suppset(0 <= z, z <= zbar,
                    norm(z - zhat[s]) <= u) # define the support for each scenario
fset.exptset(E(u) <= theta)                 # the Wasserstein metric constraint
pr = model.p                                # an array of scenario probabilities
fset.probset(pr == 1/S)                     # support of scenario probabilities

x = model.dvar(I)                           # define first-stage decisions
y = model.dvar(I)                           # define decision rule variables
y.adapt(z)                                  # y affinely adapts to z
y.adapt(u)                                  # y affinely adapts to u
for s in range(S):
    y.adapt(s)                              # y adapts to each scenario s

model.minsup(-p@x + E(p@y), fset)           # worst-case expectation over fset
model.st(y >= 0)                            # constraints
model.st(y >= x - z)                        # constraints
model.st(x >= 0)                            # constraints
model.st(c@x == d)                          # constraints

model.solve(grb)                            # solve the model by Gurobi

Being solved by Gurobi...
Solution status: 2
Running time: 0.0271s

Reference

Chen, Zhi, Melvyn Sim, Peng Xiong. 2020. Robust stochastic optimization made easy with RSOME. Management Science 66(8) 3329–3339.