Moment-based distributionally robust joint chance constrained optimization for service network design under demand uncertainty

Abstract

This paper proposes a distributionally robust joint chance constrained (DRJCC) programming approach to optimize the service network design (SND) problem under demand uncertainty. The distributionally robust method does not need complete distribution information and utilizes restricted historical data knowledge, which is significant in scarce data situations. The joint consideration of chance constraints enables more effective control of event probability, by which network managers can realize the purpose of controlling the overall service level of multi-commodities in a service network. DRJCC optimization can also help decision-makers adjust the network’s conservativeness, robustness, and service rates by setting the probability parameters of the chance constraints. We reformulate the DRJCC model by addressing the corresponding distributionally robust joint chance constraints with the worst-case Conditional Value-at-Risk method and Lagrange duality theory. The model is approximately reformulated as a mixed-integer linear program, which is easier to solve than the mixed-integer semi-definite programming model in existing literature. We also develop two benchmark approaches for comparison: Bonferroni inequality approximation and scenario-based stochastic program. Comparative numerical studies demonstrate the robustness and the validation of the proposed formulations. A case study is conducted to demonstrate the industrial performance of the uncertain SND under the DRJCC formulation. We explore the impact of the confidence level parameter on operational cost and real service level, reveal the general correlation between them. We also extract several risk-averse managerial insights for logistics fleet managers.

Appendices

Appendices

1.1 Appendix A: An alternative proof of Lemma 1

In this part, we provide an alternative way to prove Lemma 1 with the aid of the indicator function, Lagrange duality theory, and the method of inequality transformation.

Proof

We rewrite the DRJCC (18) as:

$$\begin{aligned} \sup _{\mathbb {P} \in \mathscr {P}} \mathbb {P}\left\{ \begin{aligned}&\varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \ge 0 \quad \forall k \in K \end{aligned} \right\} \le \epsilon . \end{aligned}$$

(84)

Inspired by previous works (El Ghaoui et al. 2003; Calafiore et al. 2009; Zymler et al. 2013b), we can express the left part of DRJCC (84) as:

$$\begin{aligned}&\sup \limits _{\mathbb {P} \in \mathscr {P}} \mathbb {P}\Big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \ge 0 \quad \forall k \in K \Big \} \nonumber \\ {}&= \left\{ \begin{aligned} \sup \quad&\int _{\mathbb {R}^{|K|}} \mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi }) d\mathbb {P}(\varvec{\xi }) \\ s.t. \quad&\int _{\mathbb {R}^{|K|}} d\mathbb {P}(\varvec{\xi }) = {1} \\&\int _{\mathbb {R}^{|K|}} \varvec{\xi } d\mathbb {P}(\varvec{\xi }) =\varvec{\hat{\mu }} \\&\int _{\mathbb {R}^{|K|}} (\varvec{\xi }- \varvec{\hat{\mu }})(\varvec{\xi }- \varvec{\hat{\mu }})^\textsf{T} \textrm{d}\mathbb {P}(\varvec{\xi }) = \hat{\varvec{\Gamma }} \\&\mathbb {P} \ge 0 \end{aligned} \right. , \end{aligned}$$

(85)

where $\mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi })$ is an indicator function that equals 1 if $\varvec{\xi } \in \mathcal {A} = \Big \{ \varvec{\xi } {\tiny \quad } |{\tiny \quad } \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \ge 0, \forall k \in K \Big \}$ and 0 otherwise. It can be expressed as a matrix form as follows:

$$\begin{aligned}&\sup \quad \int _{\mathbb {R}^{|K|}} \mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi }) d\mathbb {P}(\varvec{\xi }) \end{aligned}$$

(86)

$$\begin{aligned} s.t. \quad&\int _{\mathbb {R}^{|K|}} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{d}\mathbb {P}(\varvec{\xi }) = \begin{bmatrix} \hat{\varvec{\Gamma }}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^\textsf{T} &{} \varvec{\hat{\mu }} \\ \varvec{\hat{\mu }}^\textsf{T} &{} {1} \end{bmatrix} ,\end{aligned}$$

(87)

$$\begin{aligned}&\mathbb {P} \ge 0 . \end{aligned}$$

(88)

Then, we consider the Lagrangian function:

$$\begin{aligned} \mathscr {L}(\mathbb {P}, M) = \int _{\mathbb {R}^{|K|}} \mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi }) d\mathbb {P}(\varvec{\xi }) + <M, \quad \begin{bmatrix} \hat{\varvec{\Gamma }}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^\textsf{T} &{} \varvec{\hat{\mu }} \\ \varvec{\hat{\mu }}^\textsf{T} &{} {1} \end{bmatrix} - \int _{\mathbb {R}^{|K|}} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{d}\mathbb {P}(\varvec{\xi })> , \end{aligned}$$

(89)

where $M \in \mathbb {S}^{(|K|+1) \times (|K|+1)}$ is a symmetric matrix, the matrix of the dual variables of the constraints. The notation $<.,.>$ denotes the Frobenius product of two matrices. The problem is transformed into the following one:

$$\begin{aligned} \sup \limits _{\mathbb {P} \ge 0} \inf \limits _{M} \mathscr {L}(\mathbb {P}, M) . \end{aligned}$$

(90)

Because $\begin{bmatrix} \hat{\varvec{\Gamma }}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^\textsf{T} &{} \varvec{\hat{\mu }} \\ \varvec{\hat{\mu }}^\textsf{T} &{} {1} \end{bmatrix} \succ 0$, the strong duality holds (Bonnans and Shapiro 2013). Now, we consider the Lagrangian dual of the above problem.

$$\begin{aligned} \inf \limits _{M } \sup \limits _{\mathbb {P} \ge 0} \mathscr {L}(\mathbb {P}, M) . \end{aligned}$$

(91)

For any fixed $M \in \mathbb {S}^{(|K|+1) \times (|K|+1)}$, the inner problem is:

$$\begin{aligned}&\sup \limits _{\mathbb {P} \ge 0} \mathscr {L}(\mathbb {P}, M) \nonumber \\ {}&= <M, \quad \begin{bmatrix} \hat{\varvec{\Gamma }}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^\textsf{T} &{} \varvec{\hat{\mu }} \\ \varvec{\hat{\mu }}^\textsf{T} &{} {1} \end{bmatrix}> + \sup \limits _{\mathbb {P} \ge 0} \int _{\mathbb {R}^{|K|}} \Bigg ( \mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi }) - \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \Bigg ) \textrm{d}\mathbb {P}(\varvec{\xi }) \end{aligned}$$

(92)

$$\begin{aligned}&= \left\{ \begin{aligned}&<M, \quad \begin{bmatrix} \hat{\varvec{\Gamma }}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^\textsf{T} &{} \varvec{\hat{\mu }} \\ \varvec{\hat{\mu }}^\textsf{T} &{} {1} \end{bmatrix}> \quad \text {if}\, \mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi }) - \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \le 0 \quad \forall \varvec{\xi } \in \mathbb {R}^{|\textrm{K}|} \\&+ \infty \quad \qquad \qquad \qquad \quad \qquad \quad \quad \text {otherwise} \end{aligned} \right. . \end{aligned}$$

(93)

Note that $\mathbb {I}_{\{\mathcal {A}\}}(\varvec{\xi }) \le \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \quad (\forall \xi )$ can be divided into two situations ($\mathbb {I}$ can only be 0 or 1). First, for $\forall \varvec{\xi } \notin \mathcal {A}$, we have $\begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \ge 0$, which means M is a positive semi-definite matrix, i.e., $M \in S^{(m+1) \times (m+1)}_+$; second, for $\varvec{\xi } \in \mathcal {A}$, we have $\begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \ge 1$. For the second situation, we formulate the following model and ensure that its objective is not smaller than one, even in the worst case.

$$\begin{aligned}&\inf \limits _{\varvec{\xi } \in \mathbb {R}^{|K|}} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \end{aligned}$$

(94)

$$\begin{aligned} s.t. \quad&\varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \ge 0 \quad \forall k \in K . \end{aligned}$$

(95)

Note that:

$$\begin{aligned}&\varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \ge 0 \quad (\forall k \in K ) \nonumber \\ \Leftrightarrow&\min \limits _{k \in K} \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \ge 0 . \end{aligned}$$

(96)

We define $f(\varvec{\xi }) = \min \limits _{\alpha _k \in \varvec{\alpha }} \alpha _k \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \ge 0,$ where $\alpha _k$ denotes the scaling parameter, $\alpha _k \in \varvec{\alpha } = \{ \alpha _k \in \mathbb {R} | \alpha _k > 0, k \in K \}$. The scaling parameters used to improve the approximation quality (Chen et al. 2010).

We still apply the Lagrange dual to handle the program (94). The Lagrangian function is:

$$\begin{aligned} \mathscr {L}(\varvec{\xi }, \Upsilon ) = \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} - \Upsilon \Big ( \min \limits _{\alpha _k \in \varvec{\alpha }} \alpha _k \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \Big ) . \end{aligned}$$

(97)

where $\Upsilon$ is the dual variable of constraint (96) and $\Upsilon \ge \varvec{0}$. Because constraint (96) can hold strictly, which means there exists a $\varvec{\xi }_0$ such that $f(\varvec{\xi }_0) > 0$, the strong duality holds for the above program based on the Slater condition. That is:

$$\begin{aligned} \inf \limits _{\varvec{\xi } \in \mathbb {R}^{|K|}} \sup \limits _{\Upsilon } \mathscr {L}(\varvec{\xi }, \Upsilon ) =\sup \limits _{\Upsilon } \inf \limits _{\varvec{\xi } \in \mathbb {R}^{|K|}} \mathscr {L}(\varvec{\xi }, \Upsilon ) . \end{aligned}$$

(98)

Now we have:

$$\begin{aligned} \sup \limits _{\Upsilon } \inf \limits _{\varvec{\xi } \in \mathbb {R}^{|K|}} \mathscr {L}(\varvec{\xi }, \Upsilon ) \ge 1 . \end{aligned}$$

(99)

Hencse, there exists a $\Upsilon \ge 0$ that satisfies $\inf \limits _{\varvec{\xi } \in \mathbb {R}^{|K|}} \mathscr {L}(\varvec{\xi }, \Upsilon ) \ge 1$; then, for all $\varvec{\xi }$:

$$\begin{aligned}&\begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} \ge 1 + \Upsilon \Big ( \min \limits _{\alpha _k \in \varvec{\alpha }} \alpha _k \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \Big ) \quad \forall \varvec{\xi } \in \mathscr {Z} \end{aligned}$$

(100)

$$\begin{aligned} \Leftrightarrow&\begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} - \Upsilon \Big ( \min \limits _{\alpha _k \in \varvec{\alpha }} \alpha _k \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \Big ) -1 \ge 0 \quad \forall \varvec{\xi } \in \mathscr {Z} \end{aligned}$$

(101)

$$\begin{aligned} \Leftrightarrow&\min \limits _{\varvec{\xi } \in \mathscr {Z}} \max \limits _{\alpha _k \in \varvec{\alpha }} \left\{ \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} - \Upsilon \Big ( \alpha _k \big \{ \varvec{\xi }^\textsf{T}{1}_k - \varvec{S}^\textsf{T}{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \Big ) -1 \right\} \ge 0 \end{aligned}$$

(102)

$$\begin{aligned} \Leftrightarrow&\max \limits _{\alpha _k \in \varvec{\alpha }} \min \limits _{\varvec{\xi } \in \mathscr {Z}} \left\{ \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} - \Upsilon \Big ( \alpha _k \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \Big ) -1 \right\} \ge 0 \end{aligned}$$

(103)

$$\begin{aligned} \Leftrightarrow&\min \limits _{\varvec{\xi } \in \mathscr {Z}} \left\{ \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix}^\textsf{T} \textrm{M} \begin{bmatrix} \varvec{\xi } \\ {1} \end{bmatrix} - \Upsilon \Big ( \alpha _k \big \{ \varvec{\xi }^\textsf{T}\varvec{1}_k - \varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1} \big \} \Big ) -1 \right\} \ge 0 \quad \forall k \in K \end{aligned}$$

(104)

$$\begin{aligned} \Leftrightarrow&\textrm{M} - \begin{bmatrix} \varvec{0} &{} \frac{1}{2}\Upsilon \alpha _k\varvec{1}_k \\ \frac{1}{2}\Upsilon \alpha _k\varvec{1}_k ^\textsf{T} &{} 1 + \Upsilon \alpha _k( -\varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1}) \end{bmatrix} \succeq 0 \quad \forall k \in K . \end{aligned}$$

(105)

The equal transfer between (102) and (103) is supported by the classical saddle point theorem.

Consequently, the Lagrangian dual function of the chance constraints follows that:

$$\begin{aligned}&\inf \limits _{M \in \mathbb {S}^{|K|+1}} \sup \limits _{\mathbb {P} \ge 0} \mathscr {L}(\mathbb {P}, M) = \inf \limits _{M, \Upsilon } <M, \quad \begin{bmatrix} \hat{\varvec{\Gamma }}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^\textsf{T} &{} \varvec{\hat{\mu }} \\ \varvec{\hat{\mu }}^\textsf{T} &{} {1} \end{bmatrix}> \end{aligned}$$

(106)

$$\begin{aligned} s.t. \quad&M - \begin{bmatrix} \varvec{0} &{} \frac{1}{2}\Upsilon \alpha _k\varvec{1}_k \\ \frac{1}{2}\Upsilon \alpha _k\varvec{1}_k ^\textsf{T} &{} 1 + \Upsilon \alpha _k( -\varvec{S}^\textsf{T}\varvec{1}_k + \varvec{y_{No}^{k\tau }}^\textsf{T} \varvec{1} - \varvec{y_{oN}^{k\sigma }} ^\textsf{T} \varvec{1}) \end{bmatrix} \succeq 0 \quad \forall k \in K , \end{aligned}$$

(107)

$$\begin{aligned}&M \succeq 0 , \end{aligned}$$

(108)

$$\begin{aligned}&\Upsilon \ge 0, \end{aligned}$$

(109)

which is an SDP problem. The objective should not be larger than $\epsilon$ according to the original DRJCC (84).

Because $\Upsilon \ge 0$ and El Ghaoui et al. (2000) prove that its optimal value is uniformly bounded from below by a positive number, we can divide the matrix inequalities by $\Upsilon$. For simplicity, we denote $1/\Upsilon$ by $\Upsilon$ and $M/\Upsilon$ by M. We rewrite the constraints and obtain the results of Lemma 1. $\square$

1.2 Appendix B: The long tables of numerical stuties

The long tables of numerical studies are displayed here for ease of reading (Tables 9, 10, 11).

Table 9 Confidence level parameter analysis under scenario 3

Table 10 Performance under different scenarios with different confidence level for instance 6$\times$12$\times$5

Table 11 Operational cost analysis of the case study