An Approximation-Based Approach for Chance-Constrained Vehicle Routing and Air Traffic Control Problems

Abstract

We proposed a polynomial approximation-based approach to solve a specific type of chance-constrained optimization problem that can be equivalently transformed into a convex program. This type of chance-constrained optimization is in great needs of many applications, and most solution techniques are problem-specific. Our essential contribution is to provide an all-purpose solution approach through Monte Carlo and establish the linkage between our obtained optimal solution with the true optimal solution. Thanks to fast-advancing computer hardware, our method would be increasingly appealing to businesses, including small businesses. We present the numerical results including the air traffic flow management (ATFM) and the capacitated routing problem (CVRP) with stochastic demand to show that our approach with Monte Carlo will yield high-quality, timely, and stable solutions. We apply the approach to the ATFM problem to efficiently solve the weather-affected traffic flow management problem. Since there are massive independent approximation processes in the polynomial approximation-based approach, a distributed computing framework is designed to carry out the computation. For the CVRP problem, we conclude that our chance-constrained method has some strategic advantages to serve a logistics company well when resource costs and service guarantees are of concern.

Appendix

We place all mathematical and theoretical proofs in this electronic companion appendix.

1.1 Proof of Proposition 1

Both F _λ(x) and $\bar F_\lambda (x)$ are convex functions with respect to x.

Proof

We only need to show that F _λ(x) is convex and $\bar F_\lambda (x)$ will use the same argument. Since g(x) is a convex function, $\max \{g(x),0\}$ is a convex function. Likewise, max_i=1,…,ℓ{A _i x − b _i} and max_i=1,…,n{−x _i, 0} are also convex functions. Since the “max” operator preserves the convexity with respect to x, F _λ(x) is a convex function with respect to x.

1.2 Proof of Theorem 2

For a qualified approximation $\bar g(x)$ of g(x) with a given 𝜖 > 0, let ν ^∗ and x ^∗ be the optimal value and optimal solution of model (4), respectively. Let $\bar \nu ^*$ and $\bar x^*$ be the optimal value and optimal solution of model (5). We have

$$\displaystyle \begin{aligned} ||x^*-\bar x^*||<\epsilon,\ |\nu^*-\bar \nu^*|<||c||\epsilon. \end{aligned} $$

(66)

Proof

We only need to show $||x^*-\bar x^*||< \epsilon$ because x ^∗ is the optimal solution of Model (4), it will satisfy x ^∗≥ 0, A _i x ^∗≤ b _i, i = 1, …, ℓ, and g(x ^∗) ≤ 0 and so does $\bar x^*$ of Model (5) such that $\bar x^*\geq 0$, $A_i\bar x^*\leq b_i, i=1,\ldots ,\ell$ and $\bar g(\bar x^*)\leq 0$. We have

$$\displaystyle \begin{aligned} \nu^*=c'x^*, \bar\nu^*=c'\bar x^* \end{aligned}$$

if $||x^*-\bar x^*||< \epsilon$ is true. Since F _λ(x) and $\bar F_\lambda (x)$ are unconstrained convex functions, we have

$$\displaystyle \begin{aligned} 0\in\partial F_\lambda (x^*)=c+\lambda\text{convex hull of }\{\{0\},\partial g(x^*)\} \end{aligned}$$

$$\displaystyle \begin{aligned} \mathrm{and} 0\in\partial \bar F_\lambda (\bar x^*)=c+\lambda\text{convex hull of }\{0,\nabla \bar g(\bar x^*)\} \end{aligned}$$

Thus, there exists z ₁ ∈convex hull of {{0}, ∂g(x ^∗)} and z ₁ ∈convex hull of $\{0,\nabla \bar g(\bar x^*)\}$. When z ₁ ≠ 0, we have $z_1=\nabla \bar g(\bar x^*)$ and z ₁ ∈ ∂g(x ^∗). Thus, by condition B, we have

$$\displaystyle \begin{aligned} ||\nabla \bar g(x^*)-\nabla \bar g(\bar x^*)||<\epsilon. \end{aligned}$$

Also, by condition D, a continuous function $\bar g(x)$ will imply $||x^*-\bar x^*||<\epsilon$.

1.3 Simulation Results of $\mathbb {E}(|x-X|), x\in [-1,1]$ at x = 0.5

1.4 Simulation Results of $\mathbb {E}(|x-X|), x\in [-1,1]$ at x = 0.1

1.5 Proof of

Proof

First, we assume that ϕ(y) is twice differentiable on [0, 1] because if otherwise, we can apply Theorem 3 to construct a (convex) Bernstein polynomial, which approximates ϕ(y) to within $\dfrac {\epsilon }{2}$ on [0, 1] using a degree of k > 2. We then use the obtained Bernstein polynomial to replace ϕ(y). We use ϕ′(y) and ϕ″(y) to denote the first- and second-order derivatives of ϕ(y), respectively. Let

$$\displaystyle \begin{aligned} B_k(\phi'';y)=\sum_{j=0}^k{k\choose j}y^j(1-y)^{k-j}\phi''(j/k) \end{aligned} $$

(67)

represent the Bernstein polynomial of degree k for ϕ″(y). Let us observe that y ^j(1 − y)^k−j ≥ 0 on [0, 1] and that in (67) are being approximated by the sum of non-negative multiples of the polynomials y ^j(1 − y)^k−j. For k ≥ 2, define p _k(y) by

$$\displaystyle \begin{aligned} p_k^{\prime\prime}(y)=B_{k-2}(\phi'';y),\ p_k^{\prime}(0)=\phi'(0),\ p_k(0)=\phi(0). \end{aligned} $$

(68)

We see that p _k(y) is a polynomial of degree at most k. We also define β _j,k(y) for 2 ≤ j ≤ k, by

$$\displaystyle \begin{aligned} \beta_{j,k}^{\prime\prime}(y)=y^{j-2}(1-y)^{k-j},\ \beta_{j,k}^{\prime}(0)=\beta_{j,k}(0)=0. \end{aligned} $$

(69)

To complete the definition of polynomials β _j,k(y), we define

$$\displaystyle \begin{aligned} \beta_{0,k}(y)=\mathrm{sign}[\phi(0)],\ \beta_{1,k}(y)=x\mathrm{sign}[\phi'(0)]. \end{aligned} $$

(70)

The relevance of the choice of functions (70) will be seen later. We then have

$$\displaystyle \begin{aligned} p_k(y)=\sum_{j=0}^kc_j\beta_{j,k}(y), \end{aligned} $$

(71)

where c _j ≥ 0 and $\beta _{j,k}^{\prime \prime }(y)\geq 0$ on [0, 1]. Now, given any 𝜖 > 0, applying Theorem 3, we have

$$\displaystyle \begin{aligned} |B_{k-2}(\phi'';y)-\phi''(y)|\leq \epsilon \end{aligned} $$

(72)

on [0, 1]. That is

$$\displaystyle \begin{aligned} |p_k^{\prime\prime}(y)-\phi''(y)|\leq \epsilon \end{aligned} $$

(73)

on [0, 1] and therefore, for y ∈ [0, 1],

$$\displaystyle \begin{aligned} \bigg|\int_0^y(p_k^{\prime\prime}(t)-\phi''(t))dt\bigg|\leq \int_0^y|p_k^{\prime\prime}(t)-\phi''(t)|dt\leq \epsilon y\leq \epsilon. \end{aligned} $$

(74)

Using (68), the inequality (74) gives

$$\displaystyle \begin{aligned} |p_k^{\prime}(y)-\phi'(y)|\leq \epsilon \end{aligned} $$

(75)

for y ∈ [0, 1]. Similarly, another integration shows that

$$\displaystyle \begin{aligned} |p_k(y)-\phi(y)|\leq \epsilon \end{aligned} $$

(76)

for y ∈ [0, 1]. Note that the polynomial β _j,k(y) may be ψ ₀(y), ψ ₁(y), ψ ₂(y), …. We set

$$\displaystyle \begin{aligned} \psi_j(y)=\beta_{j,k}(y),\ 2\leq j\leq k,\ \psi_0(y)=\mathrm{sign}[\phi(0)],\ \psi_1(y)=x\mathrm{sign}[\phi'(0)] \end{aligned} $$

(77)

where for j ≥ 2,

$$\displaystyle \begin{aligned} \beta_{j,k}^{\prime\prime}(y)=y^{j-2}(1-y)^{k-j}=x^{j-2}\sum_{i=1}^{k-j}(-1)^i{k-j\choose i}y^i \end{aligned} $$

(78)

and we have

$$\displaystyle \begin{aligned} \beta_{j,k}(y)=y^j\sum_{i=0}^{k-j}(-1)^i\dfrac{{k-j\choose i}y^i}{[(i+j)(i+j-1)]}. \end{aligned} $$

(79)

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