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A recourse-based nonlinear programming model for stream water quality management

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Abstract

A recourse-based nonlinear programming (RBNP) method is developed for stream water quality management under uncertainty. It can not only reflect uncertainties expressed as interval values and probability distributions but also address nonlinearity in the objective function. A 0-1 piecewise linearization approach and an interactive algorithm are advanced for solving the RBNP model. The RBNP is applied to a case of planning stream water quality management. The RBNP modeling system can provide an effective linkage between environmental regulations and economic implications expressed as penalties or opportunity losses caused by improper policies. The solutions can be used for generating a variety of alternatives under different combinations of pre-regulated targets, which are also associated with different water-quality-violation risk levels and varied potential economic penalty or loss values.

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Acknowledgments

This research was supported by the Natural Sciences Foundation of China (50979001) and Major Science and Technology Program for Water Pollution Control and Treatment (2009ZX07104-004). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.

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Authors and Affiliations

  1. MOE Key Laboratory of Regional Energy Systems Optimization, S-C Energy and Environmental Research Academy, North China Electric Power University, Beijing, 102206, China

    Y. P. Li & G. H. Huang

Authors
  1. Y. P. Li
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  2. G. H. Huang
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Correspondence to Y. P. Li.

Appendix: Solution method

Appendix: Solution method

A two-step method is proposed for solving the RBNP model (i.e. model 7a–7l). The submodel corresponding to f + can be formulated in the first step when the system objective is to be maximized; the second submodel (corresponding to f −) can then be formulated based on the solution of the first submodel. Thus, the first submodel can be formulated (assume that B ± > 0 and f ± > 0) as follows:

$$ \begin{aligned} {\text{Max }}f^{ + } = \sum\limits_{j = 1}^{{j_{1} }} {\left[ {c_{j}^{ + } x_{j}^{ + } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\gamma_{j}^{ + } x_{jk}^{\prime + } + \varphi_{j}^{ + } x_{jk}^{ + } } \right)} } \right]} + \sum\limits_{{j = j_{1} + 1}}^{{n_{1} }} {\left[ {c_{j}^{ + } x_{j}^{ - } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\gamma_{j}^{ + } x_{jk}^{\prime - } + \varphi_{j}^{ + } x_{jk}^{ - } } \right)} } \right]} \\ - \sum\limits_{j = 1}^{{j_{2} }} {\sum\limits_{h = 1}^{v} {p_{h} \left[ {e_{j}^{ - } y_{jh}^{ - } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\alpha_{j}^{ - } y_{jhk}^{\prime - } + \beta_{j}^{ - } y_{jhk}^{ - } } \right)} } \right]} } - \sum\limits_{{j = j_{2} + 1}}^{{n_{2} }} {\sum\limits_{h = 1}^{v} {p_{h} \left[ {e_{j}^{ - } y_{jh}^{ + } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\alpha_{j}^{ - } y_{jhk}^{\prime + } + \beta_{j}^{ - } y_{jhk}^{ + } } \right)} } \right]} } \\ \end{aligned} $$
(16)

subject to:

$$ \sum\limits_{j = 1}^{{j_{1} }} {\left[ {\left| {a_{rj} } \right|^{ - } {\text{Sign}}\left( {a_{rj}^{ - } } \right)x_{j}^{ + } + \left| {a_{rj}^{\prime } } \right|^{ - } {\text{Sign}}\left( {a_{rj}^{\prime - } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ + } } } \right]} + \sum\limits_{{j = j_{1} + 1}}^{{n_{1} }} {\left[ {\left| {a_{rj} } \right|^{ + } {\text{Sign}}\left( {a_{rj}^{ + } } \right)x_{j}^{ - } + \left| {a_{rj}^{\prime } } \right|^{ + } {\text{Sign}}\left( {a_{rj}^{\prime + } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ - } } } \right] \le b_{r}^{ + } , \, \forall r} $$
(17)
$$ \begin{aligned} \sum\limits_{j = 1}^{{j_{1} }} {\left[ {\left| {a_{tj} } \right|^{ - } {\text{Sign}}\left( {a_{tj}^{ - } } \right)x_{j}^{ + } + \left| {a_{tj}^{\prime } } \right|^{ - } {\text{Sign}}\left( {a_{tj}^{\prime - } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ + } } } \right]} + \sum\limits_{{j = k_{1} + 1}}^{{j_{1} }} {\left[ {\left| {a_{tj} } \right|^{ + } {\text{Sign}}\left( {a_{tj}^{ + } } \right)x_{j}^{ - } + \left| {a_{tj}^{\prime } } \right|^{ + } {\text{Sign}}\left( {a_{tj}^{\prime + } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ - } } } \right]} \\ + \sum\limits_{j = 1}^{{j_{2} }} {\left[ {\left| {a_{tjh} } \right|^{ + } {\text{Sign}}\left( {a_{tjh}^{ + } } \right)y_{jhk}^{ - } + \left| {a_{tjh}^{\prime } } \right|^{ + } {\text{Sign}}\left( {a_{tjh}^{\prime + } } \right)\sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{ - } } } \right]} \\ + \sum\limits_{{j = j_{2} + 1}}^{{n_{2} }} {\left[ {\left| {a_{tjh} } \right|^{ - } {\text{Sign}}\left( {a_{tjh}^{ - } } \right)y_{jhk}^{ + } + \left| {a_{tjh}^{\prime } } \right|^{ - } {\text{Sign}}\left( {a_{tjh}^{\prime - } } \right)\sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{ + } } } \right]} \, \ge \widetilde{{w_{h}^{ - } }}, \quad \forall t, \, h, \, k \\ \end{aligned} $$
(18)
$$ x_{jk}^{\prime + } = \left\{ \begin{array} {ll} 1, & {\text{if }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} < x_{jk}^{ + } \le \bar{M}_{k} \\ 0, & {\text{if otherwise}} \\ \end{array} \right., \quad j = 1,2, \ldots , \, j_{1} ; \, \forall k $$
(19)
$$ x_{jk}^{\prime - } = \left\{ \begin{array} {ll} 1,& {\text{if }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} < x_{jk}^{ - } \le \bar{M}_{k} \\ 0, & {\text{if otherwise}} \\ \end{array} \right., \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} ; \, \forall k $$
(20)
$$ \sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{\prime + } } = 1, \quad j = 1,2, \ldots , \, j_{1} ; \, \forall k $$
(21)
$$ \sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{\prime - } } = 1, \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} ; \, \forall k $$
(22)
$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} x_{jk}^{\prime + } \le x_{jk}^{ + } \le \bar{M}_{k} x_{jk}^{\prime + } , \quad j = 1,2, \ldots ,j_{1} ; \, \forall k $$
(23)
$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} x_{jk}^{ - } \le x_{jk}^{ - } \le \bar{M}_{k} x_{jk}^{\prime - } , \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} ; \, \forall k $$
(24)
$$ y_{jhk}^{ - } = \left\{ \begin{array}{ll} 1,& {\text{ if }}\underline{N}_{k} < y_{jhk}^{ - } \le \bar{N}_{k} \\ 0,& {\text{ if otherwise}} \\ \end{array} \right., \quad j = 1,2, \ldots ,j_{2} ; \, \forall h, \, k $$
(25)
$$ y_{jhk}^{\prime + } = \left\{ \begin{array}{ll} 1, & {\text{ if }}\underline{N}_{k} < y_{jhk}^{ + } \le \bar{N}_{k} \\ 0, & {\text{ if otherwise}} \\ \end{array} \right., \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h, \, k $$
(26)
$$ \sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{\prime - } } = 1, \quad j = 1, \, 2, \ldots ,j_{2} ; \, \forall h, \, k $$
(27)
$$ \sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{\prime + } } = 1, \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h, \, k $$
(28)
$$ \underline{N}_{k} y_{jhk}^{\prime - } \le y_{jhk}^{ - } \le \bar{N}_{k} y_{jhk}^{\prime - } , \quad j = 1,2, \ldots ,j_{2} ; \, \forall h, \, k $$
(29)
$$ \underline{N}_{k} y_{jhk}^{\prime + } \le y_{jhk}^{ + } \le \bar{N}_{k} y_{jhk}^{\prime + } , \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h, \, k $$
(30)
$$ x_{j}^{ + } , \, x_{jk}^{ + } \ge 0, \quad j = 1,2, \ldots ,j_{1} $$
(31)
$$ x_{j}^{ - } , \, x_{jk}^{ - } \ge 0, \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} $$
(32)
$$ y_{jh}^{ - } , \, y_{jhk}^{ - } \ge 0, \quad j = 1,2, \ldots ,j_{2} $$
(33)
$$ y_{jh}^{ + } , \, y_{jhk}^{ + } \ge 0, \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} $$
(34)

where x + j and x + jk (j = 1, 2, …, j 1) are upper bounds of the first-stage variables with positive coefficients in the objective function; x − j and x − jk (j = j 1 + 1, j 1 + 2, …, n 1) are lower bounds of the first-stage variables with negative coefficients; y − jh and y − jhk (j = 1, 2, …, j 2) are the second-stage variables with positive coefficients in the objective function; y + jh and y + jhk (j = j 2 + 1, j 2 + 2, …, n 2) are the second-stage variables with negative coefficients. Then, based on solutions of the first submodel (16–34), the second submodel (corresponding to f −) can be formulated as:

$$ \begin{aligned} {\text{Max }}f^{ - } & = \sum\limits_{j = 1}^{{j_{1} }} {\left[ {c_{j}^{ - } x_{j}^{ - } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\gamma_{j}^{ - } x_{jk}^{\prime - } + \varphi_{j}^{ - } x_{jk}^{ - } } \right)} } \right]} + \sum\limits_{{j = j_{1} + 1}}^{{n_{1} }} {\left[ {c_{j}^{ - } x_{j}^{ + } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\gamma_{j}^{ - } x_{jk}^{\prime + } + \varphi_{j}^{ - } x_{jk}^{ + } } \right)} } \right]} \\ - \sum\limits_{j = 1}^{{j_{2} }} {\sum\limits_{h = 1}^{v} {p_{h} \left[ {e_{j}^{ + } y_{jh}^{ + } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\alpha_{j}^{ + } y_{jhk}^{\prime + } + \beta_{j}^{ + } y_{jhk}^{ + } } \right)} } \right]} } - \sum\limits_{{j = j_{2} + 1}}^{{n_{2} }} {\sum\limits_{h = 1}^{v} {p_{h} \left[ {e_{j}^{ + } y_{jh}^{ - } + \sum\limits_{k = 1}^{{s_{j} }} {\left( {\alpha_{j}^{ + } y_{jhk}^{\prime - } + \beta_{j}^{ + } y_{jhk}^{ - } } \right)} } \right]} } \\ \end{aligned} $$
(35)

subject to:

$$ \begin{aligned} \sum\limits_{j = 1}^{{j_{1} }} {\left[ {\left| {a_{rj} } \right|^{ + } {\text{Sign}}\left( {a_{rj}^{ + } } \right)x_{j}^{ - } + \left| {a_{rj}^{\prime } } \right|^{ + } {\text{Sign}}\left( {a_{rj}^{\prime + } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ - } } } \right]} \\ + \sum\limits_{{j = j_{1} + 1}}^{{n_{1} }} {\left[ {\left| {a_{rj} } \right|^{ - } {\text{Sign}}\left( {a_{rj}^{ - } } \right)x_{j}^{ + } + \left| {a_{rj}^{\prime } } \right|^{ - } {\text{Sign}}\left( {a_{rj}^{\prime - } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ + } } } \right] \le b_{r}^{ - } , \, \forall r} \\ \end{aligned} $$
(36)
$$ \begin{aligned} \sum\limits_{j = 1}^{{j_{1} }} {\left[ {\left| {a_{tj} } \right|^{ + } {\text{Sign}}\left( {a_{tj}^{ + } } \right)x_{j}^{ - } + \left| {a_{tj}^{\prime } } \right|^{ + } {\text{Sign}}\left( {a_{tj}^{\prime + } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ - } } } \right]} + \sum\limits_{{j = k_{1} + 1}}^{{j_{1} }} {\left[ {\left| {a_{tj} } \right|^{ - } {\text{Sign}}\left( {a_{tj}^{ - } } \right)x_{j}^{ + } + \left| {a_{tj}^{\prime } } \right|^{ - } {\text{Sign}}\left( {a_{tj}^{\prime - } } \right)\sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ + } } } \right]} \\ + \sum\limits_{j = 1}^{{j_{2} }} {\left[ {\left| {a_{tjh} } \right|^{ - } {\text{Sign}}\left( {a_{tjh}^{ - } } \right)y_{jhk}^{ + } + \left| {a_{tjh}^{\prime } } \right|^{ - } {\text{Sign}}\left( {a_{tjh}^{\prime - } } \right)\sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{ + } } } \right]} \\ + \sum\limits_{{j = j_{2} + 1}}^{{n_{2} }} {\left[ {\left| {a_{tjh} } \right|^{ + } {\text{Sign}}\left( {a_{tjh}^{ + } } \right)y_{jhk}^{ - } + \left| {a_{tjh}^{\prime } } \right|^{ + } {\text{Sign}}\left( {a_{tjh}^{\prime + } } \right)\sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{ - } } } \right]} \ge \widetilde{{w_{h}^{ + } }}, \, \forall t,h,k \\ \end{aligned} $$
(37)
$$ x_{jk}^{\prime - } = \left\{ \begin{array}{ll} 1,& {\text{ if }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} < x_{jk}^{ - } \le \bar{M}_{k} \\ 0,& {\text{ if otherwise}} \\ \end{array} \right., \quad j = 1,2, \ldots ,j_{1} ; \, \forall k $$
(38)
$$ x_{jk}^{\prime + }= \left\{ \begin{array}{ll} 1, & {\text{ if }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} < x_{jk}^{ + } \le \bar{M}_{k}^{{}} \\ 0, & {\text{ if otherwise}} \\ \end{array} \right., \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} ; \, \forall k $$
(39)
$$ \sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{\prime - } } = 1, \quad j = 1,2, \ldots ,j_{1} ; \, \forall k $$
(40)
$$ \sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{\prime + } } = 1, \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} ; \, \forall k $$
(41)
$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} x_{jk}^{\prime - } \le x_{jk}^{ - } \le \bar{M}_{k} x_{jk}^{\prime - } , \quad j = 1,2, \ldots ,j_{1} ; \, \forall k $$
(42)
$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M}_{k} x_{jk}^{ + } \le x_{jk}^{ + } \le \bar{M}_{k} x_{jk}^{\prime + } , \quad j = j_{1} + 1,j_{1} + 2, \ldots , \, n_{1} ; \, \forall k $$
(43)
$$ y_{jhk}^{ + } = \left\{ \begin{array}{ll} 1,& {\text{if }}\underline{N}_{k} < y_{jhk}^{ + } \le \bar{N}_{k} \\ 0, & {\text{if otherwise}} \\ \end{array} \right., \quad j = 1,2, \ldots , \, j_{2} ; \, \forall h, \, k $$
(44)
$$ y_{jhk}^{\prime - } = \left\{ \begin{array}{ll} 1,& {\text{ if }}\underline{N}_{k} < y_{jhk}^{ - } \le \bar{N}_{k} \\ 0, & {\text{ if otherwise}} \\ \end{array} \right., \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h, \, k $$
(45)
$$ \sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{\prime + } } = 1, \quad j = 1,2, \ldots , \, j_{2} ; \, \forall h, \, k $$
(46)
$$ \sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{\prime - } } = 1, \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h, \, k $$
(47)
$$ \underline{N}_{k} y_{jhk}^{\prime + } \le y_{jhk}^{ + } \le \bar{N}_{k} y_{jhk}^{\prime + } , \quad j = 1,2, \ldots ,j_{2} ; \, \forall h, \, k $$
(48)
$$ \underline{N}_{k} y_{jhk}^{\prime - } \le y_{jhk}^{ - } \le \bar{N}_{k} y_{jhk}^{\prime - } , \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h, \, k $$
(49)
$$ 0 \le x_{j}^{ - } \le x_{{j\,{\text{opt}}}}^{ + } , \, 0 \le \sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ - } } \le \sum\limits_{k = 1}^{{s_{j} }} {x_{{jk\,{\text{opt}}}}^{ + } } , \quad j = 1,2, \ldots ,j_{1} $$
(50)
$$ x_{j}^{ + } \ge x_{{j{\text{opt}}}}^{ - } , \, \sum\limits_{k = 1}^{{s_{j} }} {x_{jk}^{ + } } \ge \sum\limits_{k = 1}^{{s_{j} }} {x_{{jk\,{\text{opt}}}}^{ - } } , \quad j = j_{1} + 1, \, j_{1} + 2, \ldots ,n_{1} $$
(51)
$$ y_{jh}^{ + } \ge y_{{jh\,{\text{opt}}}}^{ - } , \, \sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{ + } } \ge \sum\limits_{k = 1}^{{s_{j} }} {y_{{jhk\,{\text{opt}}}}^{ - } } , \quad j = 1,2, \ldots ,j_{2} ; \, \forall h $$
(52)
$$ 0 \le y_{jh}^{ - } \le y_{{jh\,{\text{opt}}}}^{ + } , \, 0 \le \sum\limits_{k = 1}^{{s_{j} }} {y_{jhk}^{ - } } \le \sum\limits_{k = 1}^{{s_{j} }} {y_{{jhk\,{\text{opt}}}}^{ + } } , \quad j = j_{2} + 1, \, j_{2} + 2, \ldots ,n_{2} ; \, \forall h $$
(53)

where x + j opt , x + jk opt (j = 1, 2, …, j 1), x − j opt , x − jk opt (j = j 1 + 1, j 1 + 2, …, n1), y − jh opt , y − jhk opt (j = 1, 2, …, j 2), y + jh opt and y + jhk opt (j = j 2 + 1, j 2 + 2, …, n2) are solutions of submodel (1). Solutions of x − j opt , x − jk opt (j = 1, 2, …, j 1), x + j opt , x + jk opt (j = j 1 + 1, j1 + 2, …, n1), y + jh opt , y + jhk opt (j = 1, 2, …, j 2), y − jh opt and y − jhk opt (j = j 2 + 1, j 2 + 2, …, n2) can be obtained through solving submodel (35–53). Then, through integrating solutions of submodels (16–34) and (35–53), final solutions for the RBNP model can be obtained as follows:

$$ x_{{j\,{\text{opt}}}}^{ \pm } = \left[ {x_{{j\,{\text{opt}}}}^{ - } , \, x_{{j\,{\text{opt}}}}^{ + } } \right]{\text{ or }}\left[ {\sum\limits_{k = 1}^{{s_{j} }} {x_{{jk\,{\text{opt}}}}^{ - } } , \, \sum\limits_{k = 1}^{{s_{j} }} {x_{{jk\,{\text{opt}}}}^{ + } } } \right], \quad j = 1,2, \ldots ,n_{1} $$
(54)
$$ y_{{jh\,{\text{opt}}}}^{ \pm } = \left[ {y_{{jh\,{\text{opt}}}}^{ - } ,y_{{jh\,{\text{opt}}}}^{ + } } \right]{\text{ or }}\left[ {\sum\limits_{k = 1}^{{s_{j} }} {y_{{jhk\,{\text{opt}}}}^{ - } } , \, \sum\limits_{k = 1}^{{s_{j} }} {y_{{jhk\,{\text{opt}}}}^{ + } } } \right], \, j = 1,2, \ldots ,n_{2} ; \, \forall h $$
(55)
$$ f_{\text{opt}}^{ \pm } = \left[ {f_{\text{opt}}^{ - } , \, f_{\text{opt}}^{ + } } \right] $$
(56)

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Li, Y.P., Huang, G.H. A recourse-based nonlinear programming model for stream water quality management. Stoch Environ Res Risk Assess 26, 207–223 (2012). https://doi.org/10.1007/s00477-011-0468-6

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