An Air Quality Management Model Based on an Interval Dual Stochastic-Mixed Integer Programming

Abstract

The issue of air pollution has become the focus of the world because of its significant influence to the economic development and public health. This paper proposes an interval dual stochastic-mixed integer programming (IDSIP) approach for regional air quality management. The IDSIP approach can be effectively communicated into the optimization processes and resulting solutions, which is formulated through integrating interval-parameter integer programming (IIP) within a two-stage stochastic programming (TSP) joint chance-constrained programming (CCP) and could deal with uncertainties expressed as not only probability distributions but also interval values. Moreover, the left-hand-side (LHS) constraints with stochastic variables could be handled at different risk levels with varied reliability scenarios. In the modeling formulation, penalties are imposed when expected policies are violated. The results indicate that reasonable solutions for air quality management system have been generated, which can help decision makers draw up productive strategies taking into account the trade-off between system economy and air quality under uncertainty.

Appendix

1.1 Left-Hand-Side Chance-Constrained Programming

In general, the CCP method can be used for dealing with uncertainties expressed as stochastic variables in the right hand (Loucks et al. 1981; Huang 1998; Li et al. 2007). However, parameters in the LHS of the constraints are stochastic variables frequently. The method of left-hand stochastic (LHS) will be illustrated as follows (Cao et al. 2011):

The left-hand stochastic of model can be formulated as follows:

$$ \min {f}^{\pm }={c}_1^{\pm }{x}_1^{\pm }+\cdots +{c}_n^{\pm }{x}_n^{\pm } $$

(8a)

subject to:

$$ {P}_{\mathrm{r}}\left\{{\xi}_{11}{x}_1^{\pm }+\cdots +{\xi}_{1n}{x}_n^{\pm}\tilde{\le}{b}_1^{\pm}\right\}\ge 1-{p}_i, $$

(8b)

…

$$ {P}_{\mathrm{r}}\left\{{\xi}_{l1}{x}_1^{\pm }+\cdots +{\xi}_{ln}{x}_n^{\pm}\tilde{\le}{b}_l^{\pm}\right\}\ge 1-{p}_i, $$

(8c)

…

$$ {a}_{\left(l+1\right)1}^{\pm }{x}_1^{\pm }+\cdots +{a}_{\left(l+1\right)n}^{\pm }{x}_n^{\pm}\le {b}_{\left(l+1\right)}^{\pm }, $$

(8d)

$$ {a}_{m1}^{\pm }{x}_1^{\pm }+\cdots +{a}_{mn}^{\pm }{x}_n^{\pm}\le {b}_m^{\pm }, $$

(8e)

$$ {x}_1^{\pm },\cdots {x}_n^{\pm}\ge 0, $$

(8f)

where ξ _ij (i = 1, 2, …, l and j = 1, 2, …, n) are stochastic variables, and c ^± _ij , a ^± _ij , and b ^± _i are interval numbers. Constraints of model (1) contain stochastic variables and can be violated with a low probability (p _i  ≤ 0.5). In other words, the probability of satisfying the constraints cannot be less than 1 − p _i .

Then, suppose that the constraints are independent with each other, and all the parameters (c ^± _ij ) in the objective function are not <0. Suppose that ξ = (ξ _i1, ξ _i2, …, ξ _in )^T obeys an n-dimensional normal distribution:

$$ \xi \sim N\left(\mu, V\right), $$

(9)

where μ = (a _i1, a _i2, …, a _in )^T is the expectation of ξ and V = (v _ij ) _n × n is the variance–covariance matrix of n. Parameters in the LHS of the constraints are positive in most of practical problems; thus, without loss of generality, we suppose that the expectation of ξ is positive, i.e., μ ≥ 0.

Since ξ obeys an n-dimensional normal distribution, the arbitrary linear combinations of its elements obey one-dimensional normal distribution (Tong 1990; Rose and Smith 1996, 2002). Thus, we have:

$$ {d}^T\xi \sim N\left({d}^T\mu, {d}^T Vd\right), $$

(10)

where d = (d ₁, d ₂, …, d _n )^T ∈ R ⁿ. Let

$$ \varOmega \sim {\xi}_{i1}{x}_1^{\pm }+\cdots +{\xi}_{in}{x}_n^{\pm }, $$

(11)

then Ω is a set of stochastic variables which obeys a one-dimensional normal distribution. Since each x ^± _ij (1 ≤ i ≤ l, 1 ≤ j ≤ n) is an interval containing a set of deterministic numbers, there are infinite linear combinations of ξ _i1, ξ _i2, …, ξ _in . In other words, there are infinite stochastic variables within Ω.

For any given y _j  ∈ x ^± _j (j = 1, 2, …, n), let Y = (y ₁, y ₂, …, y _n )^T.Then, we have:

$$ \begin{array}{c}\hfill \varOmega =\left\{{\xi}_{i1}{y}_1+{\xi}_{i2}{y}_2+\cdots +{\xi}_{in}{y}_n\left|{y}_j\in {x}_j^{\pm },j=1,2,\dots, n\right.\right\}\hfill \\ {}\hfill =\left\{{Y}^T\xi \right.\left|Y\in {X}^{\pm}\right\}\hfill \end{array} $$

(12)

According to (10), we have:

$$ {Y}^T\xi \sim N\left({Y}^T\mu, {Y}^T VY\right). $$

(13)

The stochastic variable (Y ^T ξ) obeys a one-dimensional normal distribution whose exception is Y ^T μ and variance is Y ^T VY. Then Y ^T ξ can be expressed as follows:

$$ {Y}^T\xi ={Y}^T\mu +Z\sqrt{Y^T VY}, $$

(14)

where Z ∼ N(0, 1).

Theorem 1

Constraint ${P}_r\left\{{\xi}_{i1}{x}_1^{\pm }+\cdots +{\xi}_{in}{x}_n^{\pm}\tilde{\le}{b}_i^{\pm}\right\}\ge 1-{p}_i$ can be converted to the following inequalities:

$$ \begin{array}{cc}\hfill \left(\mathrm{i}\right)\hfill & \hfill {\left({X}^{-}\right)}^T\left(\mu +{Z}^{\left(1-{p}_i\right)}\sqrt{D_i(V)}\right)\le {b}_i^{-}\hfill \end{array}, $$

(15a)

$$ {\left({X}^{+}\right)}^T\left(\mu +{Z}^{\left(1-{p}_i\right)}\sqrt{D_i(V)}\right)\le {b}_i^{-}, $$

(15b)

or

$$ \begin{array}{cc}\hfill \left(\mathrm{ii}\right)\hfill & \hfill {\left({X}^{-}\right)}^T\left(\mu +{Z}^{\left(1-{p}_i\right)}\sqrt{D_i(V)}\right)\le {b}_i^{+}\hfill \end{array}, $$

(16a)

$$ {\left({X}^{+}\right)}^T\left(\mu +{Z}^{\left(1-{p}_i\right)}\sqrt{D_i(V)}\right)\le {b}_i^{+}. $$

(16b)

From (9), we have $\mu +{Z}^{\left(1-{p}_i\right)}\sqrt{D_i(V)}={\left({a}_{ij}+{Z}^{\left(1-{p}_i\right)}\sqrt{v_{ij}}\right)}_{n\times 1}$.

Therefore, constraints (15a), (15b), (16a), and (16b) can be converted to:

$$ \begin{array}{cc}\hfill \left(\mathrm{i}\right)\hfill & \hfill {\displaystyle \sum_{j=1}^n{x}_j^{-}\left({a}_{ij}+{Z}^{\left(1-{p}_i\right)}\sqrt{v_{ij}}\right)}\le {b}_i^{-},\hfill \end{array} $$

(17a)

$$ {\displaystyle \sum_{j=1}^n{x}_j^{+}\left({a}_{ij}+{Z}^{\left(1-{p}_i\right)}\sqrt{v_{ij}}\right)}\le {b}_i^{-}, $$

(17b)

or

$$ \begin{array}{cc}\hfill \left(\mathrm{ii}\right)\hfill & \hfill {\displaystyle \sum_{j=1}^n{x}_j^{-}\left({a}_{ij}+{Z}^{\left(1-{p}_i\right)}\sqrt{v_{ij}}\right)}\le {b}_i^{+},\hfill \end{array} $$

(18a)

$$ {\displaystyle \sum_{j=1}^n{x}_j^{+}\left({a}_{ij}+{Z}^{\left(1-{p}_i\right)}\sqrt{v_{ij}}\right)}\le {b}_i^{+}. $$

(18b)