A generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variables

Abstract

In this paper, we propose a generalized Benders decomposition-based branch and cut algorithm for solving two stage stochastic mixed-integer nonlinear programs (SMINLPs) with mixed binary first and second stage variables. At a high level, the proposed decomposition algorithm performs spatial branch and bound search on the first stage variables. Each node in the branch and bound search is solved with a Benders-like decomposition algorithm where both Lagrangean cuts and Benders cuts are included in the Benders master problem. The Lagrangean cuts are derived from Lagrangean decomposition. The Benders cuts are derived from the Benders subproblems, which are convexified by cutting planes, such as rank-one lift-and-project cuts. We prove that the proposed algorithm converges in the limit. We apply the proposed algorithm to a stochastic pooling problem, a crude selection problem, and a storage design problem. The performance of the proposed algorithm is compared with a Lagrangean decomposition-based branch and bound algorithm and solving the corresponding deterministic equivalent with the solvers including BARON, ANTIGONE, and SCIP.

Appendices

Appendix 1: Mathematical formulation of stochastic pooling problem with contract selection

Sets

$i=\hbox {feeds}$

$j=\hbox {products}$

$l=\hbox {pools}$

$p=\hbox {qualities}$

$c=\hbox {contracts}$

$\omega =\hbox {scenarios}$

$T_X=(i, l)$ pairs for which input to pool connection allowed

$T_Y= (l, j)$ pairs for which pool to output connection allowed

$T_Z=(i, j)$ pairs for which input to output connection allowed

Parameters

$A^U_i=\hbox {maximum}$ available flow

$A^L_i=\hbox {minimum}$ available flow

$S^{U}_{l}=\hbox {maximum}$ pool size

$S^{L}_{l}=\hbox {minimum}$ pool size

$D^U_j=\hbox {maximum}$ product demand

$C_{ik}=\hbox {feed}$ concentration

$P^U_{jk}=\hbox {maximum}$ allowable product concentration

$P^L_{jk}=\hbox {minimum}$ allowable product concentration

$\alpha _l=\hbox {fixed}$ cost parameter for pools

$\beta _l=\hbox {variable}$ cost parameter for pools

$\delta _{i}=\hbox {fixed}$ cost for feed storage

$\xi _{i}=\hbox {variable}$ cost for feed storage

$\tau _{\omega }=\hbox {probability}$ of scenario $\omega$

$\psi _{i\omega }^{c}=\hbox {unit}$ price for feed i under contract c in scenario $\omega$

$d_{j\omega }=\hbox {product}$ unit price in scenario $\omega$

$\sigma _{i}^{c}=\hbox {minimum}$ purchasable amount of feed i under contract c

First stage variables

Binary variables

$\lambda _i=\hbox {whether}$ feed i exists

$\theta _l=\hbox {whether}$ pool l exists

Continuous variables

$S_{l}=\hbox {capacity}$ of pool l

$A_{i}=\hbox {capacity}$ of pool i

Second stage variables

Binary variables

$u_{i\omega }^{c}=\hbox {whether}$ contract c is selected in purchasing feed i in scenario $\omega$

Continuous variables

$y_{lj\omega }=\hbox {flow}$ from pool l to product j in scenario $\omega$

$z_{ij\omega }=\hbox {flow}$ of feed i to product j in scenario $\omega$

$q_{il\omega }=\hbox {proportion}$ of flow from input i to pool l in scenario $\omega$

$CT_{i\omega }=\hbox {cost}$ of purchasing feed i in scenario $\omega$

$CT_{i\omega }^{c}=\hbox {cost}$ of purchasing feed i under contract c in scenario $\omega$

$B_{i\omega }^{c}=\hbox {the}$ amount of feed i purchased under contract c in scenario $\omega$

We study pooling problem with purchase contracts under price uncertainty. The first stage decisions include whether to select the feeds and pools, the capacity of the feeds and pools, whether the feeds exist.

$$\begin{aligned}&A^L_{i}\lambda _{i} \le A_{i}\le A^U_{i}\lambda _{i}\quad \forall \, i \end{aligned}$$

(10)

$$\begin{aligned}&S^{L}_{l}\theta _{l} \le S_{l}\le S^U_{l}\theta _{l}\quad \forall \, l \end{aligned}$$

(11)

After the pools are selected, the uncertainties in the prices of the feeds and products are realized. We need to decide the material flows just as in a standard pooling problem. Equations (12)–(20) are the equations in the pq-formulation of the pooling problem.

$$\begin{aligned} \sum _{l:(i,l)\in T_{X}}\sum _{j:(l,j)\in T_{Y}} q_{il\omega }y_{lj\omega } + \sum _{j:(i,j)\in T_{Z}}z_{ij\omega }\le A_{i} \quad \forall \, i,\omega \end{aligned}$$

(12)

$$\begin{aligned} \sum _{j:(l,j)\in T_{Y}}y_{lj\omega }\le S_{l} \quad \forall \, l,\omega \end{aligned}$$

(13)

$$\begin{aligned} \sum _{l:(l,j)\in T_{Y}}y_{lj\omega }+\sum _{i:(i,j)\in T_{Z}}z_{ij\omega }\le D_{j\omega }^{U} \quad \forall \, j,\omega \end{aligned}$$

(14)

$$\begin{aligned} \sum _{i:(i,l)\in T_{X}}q_{il\omega }=\theta _{l} \quad \forall \, i,\omega \end{aligned}$$

(15)

$$\begin{aligned}&P^{L}_{jk}\left( \sum _{l:(l,j)\in T_Y}y_{lj\omega } + \sum _{i:(i,j)\in T_Z}z_{ij\omega }\right) \le \sum _{l:(l,j)\in T_{Y}}\sum _{i:(i,l)\in T_{X}}C_{ik}q_{il\omega }y_{lj\omega } + \sum _{i:(i,j)\in T_{Z}}C_{ik}z_{ij\omega }\nonumber \\&\quad \le P^{U}_{jk}\left( \sum _{l:(l,j)\in T_Y}y_{lj\omega } + \sum _{i:(i,j)\in T_Z}z_{ij\omega }\right) \quad \forall \, j,k,\omega \end{aligned}$$

(16)

$$\begin{aligned} 0\le q_{il\omega }\le \lambda _{i} \quad \forall (i,l)\in T_{X},\omega \end{aligned}$$

(17)

$$\begin{aligned} 0\le y_{lj\omega }\le \min \left\{ S_{l},D_{j}^{U}, \sum _{i:(i,l)\in T_{X}}A_{i}^{U}\right\} \quad \forall \, (l,j)\in T_Y \end{aligned}$$

(18)

$$\begin{aligned} 0\le z_{ij\omega }\le \min \big \{A_{i}^{U},D_{j}^{U}\big \} \quad \forall \, (i,j)\in T_Z,\omega \end{aligned}$$

(19)

$$\begin{aligned} \sum _{i:(i,l)\in T_{X}}q_{il\omega }y_{lj\omega }=y_{lj\omega }\quad \forall \, l,j,\omega \end{aligned}$$

(20)

In this case study, we also consider different types of purchase contracts when the prices are realized. The formulation of contracts is based Park et al. [32]. (21)–(24) decides that the total amount of feed i purchased can come from exactly one contract.

$$\begin{aligned} \sum _{l:(i,l)\in T_{X}}\sum _{j:(l,j)\in T_{Y}} q_{il\omega }y_{lj\omega } + \sum _{j:(i,j)\in T_{Z}}z_{ij\omega }=\sum _{c}B_{i\omega }^{c}\quad \forall \, i,\omega \end{aligned}$$

(21)

$$\begin{aligned} A_{i}^{L}u_{i\omega }^{c}\le B_{i\omega }^{c}\le A_{i}^{U}u_{i\omega }^{c} \quad \forall \, i,\omega ,c \end{aligned}$$

(22)

$$\begin{aligned} \sum _{c}u_{i\omega }^c\le \lambda _{i} \quad \forall \, i,\omega \end{aligned}$$

(23)

$$\begin{aligned} CT_{i\omega }=\sum _{c}CT_{i\omega }^{c} \quad \forall \, i,\omega \end{aligned}$$

(24)

(25) describes fixed price contract.

$$\begin{aligned} CT_{i\omega }^{f}=\psi _{i\omega }^{f}B_{i\omega }^{f} \quad \forall \, i,\omega \end{aligned}$$

(25)

(26)–(31) describe discount after a certain amount contract.

$$\begin{aligned} CT_{i\omega }^{d}=\psi _{i\omega }^{d1}B_{i\omega }^{d1}+\psi _{i\omega }^{d2}B_{i\omega }^{d2} \quad \forall \, i,\omega \end{aligned}$$

(26)

$$\begin{aligned} B_{i\omega }^{d}=B_{i\omega }^{d1}+B_{i\omega }^{d2} \quad \forall \, i,\omega \end{aligned}$$

(27)

$$\begin{aligned} B_{i\omega }^{d1} = B_{i\omega }^{d11}+ B_{i\omega }^{d12} \quad \forall \, i,\omega \end{aligned}$$

(28)

$$\begin{aligned} 0\le B_{i\omega }^{d11}\le \sigma _{i\omega }^{d} u_{i\omega }^{d1} \quad \forall \, i,\omega \end{aligned}$$

(29)

$$\begin{aligned} B_{i\omega }^{d12}=\sigma _{i\omega }^{d}u_{i\omega }^{d2} \quad \forall i,\omega \end{aligned}$$

(30)

$$\begin{aligned} 0\le B_{i\omega }^{d2}\le A^{U}_{i}u_{i\omega }^{d2} \quad \forall \, i,\omega \end{aligned}$$

(31)

(32)–(37) describe bulk discount contract.

$$\begin{aligned} CT_{i\omega }^b=\psi _{i\omega }^{b1}B_{i\omega }^{b1}+\psi _{i\omega }^{b2}B_{i\omega }^{b2} \quad \forall \, i,\omega \end{aligned}$$

(32)

$$\begin{aligned} B_{i\omega }^{b}=B_{i\omega }^{b1}+B_{i\omega }^{b2} \quad \forall \, i,\omega \end{aligned}$$

(33)

$$\begin{aligned} 0\le B_{i\omega }^{b1}\le \sigma _{i\omega }^{b}u_{i\omega }^{b1} \quad \forall \, i,\omega \end{aligned}$$

(34)

$$\begin{aligned} \sigma _{i\omega }^{b}u_{i\omega }^{b2}\le B_{i\omega }^{b2}\le A_{i}^{U}u_{i\omega }^{b2} \quad \forall \, i,\omega \end{aligned}$$

(35)

$$\begin{aligned} u_{i\omega }^{b1}+u_{i\omega }^{b2}=u_{i\omega }^{b} \quad \forall \, i,\omega \end{aligned}$$

(36)

Because of the decisions on purchase contracts, there are binary variables in the second stage decisions.

$$\begin{aligned} u_{i\omega }^{c}, u_{i\omega }^{d1}, u_{i\omega }^{d2}, u_{i\omega }^{b1}, u_{i\omega }^{b2}\in \{0,1\} \end{aligned}$$

(37)

The objective includes the fixed and variable cost of installing the pools and fixed cost of the feeds, purchase of feeds, and sales of final products.

$$\begin{aligned}&\min \sum _{l}\big (\alpha _{l}\theta _{l} + \beta _{l}S_{l}\big )+\sum _{i}\big (\delta _{i}\lambda _{i}+\xi _{i}A_{i}\big )\nonumber \\&\quad +\sum _{\omega }\tau _{\omega }\left( \sum _{i}CT_{i\omega }-\sum _{j}d_{j}\left( \sum _{l:(l,j)\in T_{Y}}y_{lj\omega }+\sum _{i:(i,j)\in T_{Z}}z_{ij\omega }\right) \right) \end{aligned}$$

(38)

Appendix 2: Convex hull reformulation of piecewise McCormick envelopes in the crude selection problem

In the crude selection problem described in Sect. 7.2, the bilinear terms come from the product of the split fractions and some material flows. The bilinear terms are only in the stage 2 problems. In order to obtain tight convex relaxations in the Benders subproblems, we use the hull relaxation of the piesewise McCormick envelopes.

We represent the split fractions by variable $q_i$, $i\in I$ where I is the set of indices for the $q_{i}$ variables. The material flows that occur in the bilinear terms are represented by $x_{j}$, $j\in J$ where J is the set of indices for the $x_{j}$ variables. Variables $x_{j}$, $j\in J_{i}$ are the material flows that occur in the bilinear terms with variable $q_i$. The bilinear terms are represented by $z_{ij}$, $i\in I, j\in J_i$. All the other variables in stage 2 are represented by $y_{2}$. The stage 1 variables are represented by $y_{1}$. Without loss of generality, we represent the constraints in a given subproblem by

$$\begin{aligned}&A_{1}y_{1}+A_{2}y_{2}+Bz\le d \end{aligned}$$

(39a)

$$\begin{aligned}&z_{ij} = q_{i}x_{j}\quad \forall \, i\in I, j\in J_i \end{aligned}$$

(39b)

where both the constraints and variable bounds in each subproblem are included. Since the McCormick envelopes of $z_{ij}$ can yield weak relaxations, we uniformly discretize each variable $q_i$ into m intervals where m is an integer that is greater than one. The piecewise McCormick envelopes can be represented using the following disjunctions.

$$\begin{aligned} \vee _{n\in \{1,2,\ldots ,m\}} \begin{bmatrix} A_{1}y_{1}+A_{2}y_{2}+Bz\le d\\ z_{ij}\ge q_{i}\cdot x_{j}^{L} + \frac{n-1}{m}(x_{j}-x_{j}^{L})\quad \forall \, j\in J_{i}\\ z_{ij}\ge q_{i}\cdot x_{j}^{U} + \frac{n}{m}(x_{j}-x_{j}^{U})\quad \forall \, j\in J_{i}\\ z_{ij}\le q_{i}\cdot x_{j}^{L} + \frac{n}{m}(x_{j}-x_{j}^{L})\quad \forall \, j\in J_{i}\\ z_{ij}\le q_{i}\cdot x_{j}^{U} + \frac{n-1}{m}(x_{j}-x_{j}^{U})\quad \forall \, j\in J_{i}\\ x_{j}^{L}\le x_{j}\le x_{j}^{U}\quad \forall \, j\in J_{i}\\ \frac{n-1}{m}\le q_{i}\le \frac{n}{m} \end{bmatrix} \quad \forall \, i\in I \end{aligned}$$

(40)

where $x_{j}^{U}$ and $x_{j}^{L}$ are the upper and lower bounds of variable $x_{j}$, respectively. In each disjunct n, variable $q_{i}$ is bounded between $\frac{n-1}{m}$ and $\frac{n}{m}$. The McCormick envelopes in disjunct n are changed accordingly with the bounds of $q_{i}$. The McCormick envelopes of the variables $x_{j}$ that occur in the same bilinear term of $q_{i}$ are all included in each disjunct. Note that constraint (39a) is included in all the disjuncts. If (39a) were not included in the disjuncts, the hull relaxation [11] of (40) would be the same as the normal McCormick envelope relaxation. We use the hull relaxation of (40) to have a tight relaxation for the Benders master problem in the crude selection problem. $m=5$ is used in the computational experiments.