A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: I. Robust Linear Optimization and Robust Mixed Integer Linear Optimization

Abstract

Robust counterpart optimization techniques for linear optimization and mixed integer linear optimization problems are studied in this paper. Different uncertainty sets, including those studied in literature (i.e., interval set; combined interval and ellipsoidal set; combined interval and polyhedral set) and new ones (i.e., adjustable box; pure ellipsoidal; pure polyhedral; combined interval, ellipsoidal, and polyhedral set) are studied in this work and their geometric relationship is discussed. For uncertainty in the left hand side, right hand side, and objective function of the optimization problems, robust counterpart optimization formulations induced by those different uncertainty sets are derived. Numerical studies are performed to compare the solutions of the robust counterpart optimization models and applications in refinery production planning and batch process scheduling problem are presented.

1. Introduction

In many optimization applications, the problem data is assumed to be known with certainty. However, that is seldom the case in practice. Very often, the realistic data are subject to uncertainty due to their random nature, measurement errors or other reasons. Since the solution of an optimization problem often exhibits high sensitivity to the data perturbations as illustrated by Ben-Tal and Nemirovski¹, ignoring the data uncertainty could lead to solutions which are suboptimal or even infeasible for practical applications.

Robust optimization belongs to an important methodology for dealing with optimization problems with data uncertainty. In the first stage of this type of method, a deterministic data set is defined within the uncertain space, and in the second stage the best solution which is feasible for any realization of the data uncertainty in the given set is obtained. The corresponding second stage optimization problem is also called robust counterpart optimization problem. One major motivation for studying robust optimization is that in many applications the data set is an appropriate notion of parameter uncertainty, e.g., for applications in which infeasibility cannot be accepted at all (e.g., design of engineering structures like bridges considered in Ben-Tal and Nemirovski^2–3), and for those cases that the parameter uncertainty is not stochastic, or if no distributional information is available.

One of the earliest papers on robust counterpart optimization is related to the work of Soyster⁴, who considered simple perturbations in the data and aimed at finding a reformulation of the original linear programming problem such that the resulting solution would be feasible under all possible perturbations. This approach, however, is the most conservative one since it ensures feasibility against all potential realizations. Thus, it is highly desirable to provide a mechanism to allow tradeoff between robustness and performance. To address the issue of over-conservatism in worst-case models, Ben-Tal, Nemirovski and co-workers^{1, 5–7} and El-Ghaoui and co-workers^8–9 independently proposed the ellipsoidal set based robust counterpart formulation for dealing with parameter uncertainty within linear and quadratic programming problems. El-Ghaoui and Lebret⁸ studied the robust solutions to the uncertain least-squares problems, and El-Ghaoui et al.,⁹ studied uncertain semidefinite problems. Ben-Tal and Nemirovski^6–7 showed that when the uncertainty sets for a linear constraint are ellipsoids, the robust formulation turns out to be a conic quadratic problem. Ben-Tal et al.,⁵ considered LP problems where some of the decision variables must be determined before the realization of uncertain data, while the other decision variables can be set after the realization.

The robust optimization formulation introduced for linear programming problems with uncertain linear coefficients was extended by Lin et al.,¹⁰ and Janak et al.,¹¹ to mixed integer linear optimization (MILP) problems under uncertainty. They developed the theory of the robust optimization framework for general mixed-integer linear programming problems and considered both bounded and several known probability distributions. The robust optimization framework is later extended by Verderame and Floudas¹² who studied both continuous (general, bounded, uniform, normal) and discrete (general, binomial, Poisson) uncertainty distributions and applied the framework to operational planning problems. The work was further compared with the conditional-value risk based method in Verderame and Floudas¹³. For a recent review on planning and scheduling under uncertainty, the reader is directed to Verderame et al.,¹⁴, and for process scheduling under uncertainty to Li and Ierapetritou¹⁵.

Bertsimas and Sim¹⁶ considered robust linear programming with coefficient uncertainty using an uncertainty set with budgets. In this robust counterpart optimization formulation, a budget parameter is introduced to control the degree of conservatism of the solution. As it will be shown in this paper, this type of robust formulation is based on a combined interval and polyhedral uncertainty set. Bertsimas and coworkers¹⁷ extended and applied a robust optimization framework in the fields of linear and discrete programming. Bertsimas et al.,¹⁸ characterized the robust counterpart of a linear programming problem with uncertainty set described by an arbitrary norm. The ideas of the robust optimization approach in Bertsimas and Sim¹⁶ have also been extended to conic optimization problems in Bertsimas and Sim¹⁹, and also used by Bertsimas and Thiele²⁰ to address inventory control problems to minimize total costs.

Kouvelis and Yu²¹ proposed a framework for robust discrete optimization, which seeks to find a solution that minimizes the worst case performance under a set of scenarios for the data. Chen and Lin²² proposed an approximate algorithm to solve the robust design problem in a stochastic-flow network. Atamtürk and Zhang²³ described a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. They generalized the approach to multi-commodity network flow and design, and studied applications to lot-sizing and location-transportation problems. Atamtürk²⁴ introduced alternative formulations to robust mixed 0–1 programming with interval uncertainty objective coefficients. Averbakh²⁵ proposed a general approach for finding minmax regret solutions for a class of combinatorial problems with interval uncertain objective function coefficients, based on reducing the problem with uncertainty to a set of deterministic problems. Kasperski and Zielinski²⁶ considered a similar class of problems and presented a polynomial time approximation algorithm. Bertsimas and Sim¹⁷ proposed an approach to address data uncertainty for discrete optimization and network flow problems. They presented an algorithm for the special case of the robust network flow where only the objective uncertainty exists and the problem is a mixed 0–1 problem, and solved the problem by considering a polynomial number of nominal minimum cost flow problems in a modified network.

Chen et al.,²⁷ proposed an asymmetrical uncertainty set that generalizes the symmetric ones. Chen et al.,²⁸ studied the relationship between different Conditional Value-at-Risk (CVaR) bound based approximations to individual chance constraints and different set based robust optimization formulations and showed the equivalence between them. Fischetti and Monaci²⁹ developed a robustness framework denoted as “light robustness” approach to cope with the issue of overly conservative solutions in robust optimization. They placed a hard upper bound on the objective value and then minimize the degree of infeasibility with a fixed uncertainty set.

As pointed out by Goh and Sim³⁰, if the exact distribution of uncertainties is precisely known, optimal solutions to the robust problem would be overly and unnecessarily conservative. Conversely, if the assumed distribution of uncertainties is in fact different from the actual distribution, the optimal solution using a stochastic programming approach may perform poorly. So several recent works aim at bridging the gap between the conservatism of robust optimization and the specificity of stochastic programming, where optimal decisions are sought for the worst-case probability distributions within a family of possible distributions, defined by certain properties such as their support and moments. Specifically, El Ghaoui et al.,³¹ developed worst-case Value-at-Risk (VaR) bounds for a robust portfolio selection problem when only the bounds on the means and covariance matrix of the assets are known. Chen et al.,²⁷ introduced directional deviations as an additional means to characterize a family of distributions that were applied by Chen and Sim³² to a goal-driven optimization problem. Delage and Ye³³ studied distributionally robust stochastic programs where the mean and covariance of the primitive uncertainties are themselves subject to uncertainty. Ben-Tal et al.,³⁴ proposed a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set.

In this paper, we present a systematic study of the set induced robust counterpart optimization techniques for both linear optimization (LP) and mixed integer linear optimization (MILP) problems. The new contributions of the paper are as follows: we have proposed several novel uncertainty sets (i.e., adjustable box; pure ellipsoidal; pure polyhedral; combined interval, ellipsoidal, and polyhedral set) and derived their robust counterparts for both LP and MILP problems; for the first time in the literature, we have discussed the connection among six different uncertainty sets (including those studied in the literature, i.e., i.e., interval set introduced by Soyster⁴, combined interval and ellipsoidal set introduced by Ben-Tal and Nemirovski¹, combined interval and polyhedral set introduced by Bertsimas and Sim¹⁶) and the differences among their corresponding robust counterparts, from both the geometrical point of view and the computational studies.

The paper is organized as follows. In section 2, we introduce the set induced robust counterpart optimization for general linear and mixed integer linear optimization problems. In section 3, we introduce six different uncertainty sets and discuss their relationship from a geometric point of view. In section 4, we present the detailed robust counterpart formulations under different uncertainty sets for linear constraints and the derivation procedures. In section 5, a numerical example and a refinery production planning example are studied and the different robust counterpart optimization models are compared. In section 6, robust counterparts for mixed integer linear optimization problems are derived. In section 7, a numerical example and an application in process scheduling problem are presented. Finally, conclusions are presented in section 8.

2. Uncertainty Set Induced Robust Optimization

In set induced robust optimization, the uncertain data is assumed to be varying in a given uncertainty set, and the aim is to choose the best solution among those “immunized” against data uncertainty, that is, candidate solutions that remain feasible for all realizations of the data from the uncertainty set.

2.1 Robust linear optimization

Motivating Example 1 Consider the following linear optimization problem:

$$\begin{array}{cc}\hfill \text{max}& 8x_1+12x_2\hfill \\ \hfill s.t.& {\tilde{a}}_{11}{x}_1+{\tilde{a}}_{12}{x}_2\le 140\hfill \\ \hfill & {\tilde{a}}_{21}{x}_1+{\tilde{a}}_{22}{x}_2\le 72\hfill \\ \hfill & x_1,x_2\ge 0.\hfill \end{array}$$

Assume that the left hand side (LHS) constraint coefficients ${\tilde{a}}_{11}$, ${\tilde{a}}_{12}$, ${\tilde{a}}_{21}$, ${\tilde{a}}_{22}$ are subject to uncertainty and they are defined as follows:

$${\tilde{a}}_{11}=10+{\xi }_{11},{\tilde{a}}_{12}=20+2{\xi }_{12},{\tilde{a}}_{21}=6+0.6{\xi }_{21},{\tilde{a}}_{22}=8+0.8{\xi }_{22},$$

where ξ₁₁,ξ₁₂,ξ₂₁,ξ₂₂ are independent random variables. The random variables are distributed in the range [−1,1] (i.e., the constraint coefficients ${\tilde{a}}_{11}$, ${\tilde{a}}_{12}$, ${\tilde{a}}_{21}$, ${\tilde{a}}_{22}$ have maximum 10% perturbation around their nominal values 10, 20, 6, 8, respectively). Under the set induced robust optimization framework, finding a robust solution for the above example means to find the best possible candidate solution such that the feasibility of the constraints is maintained no matter what value the random variables realize within a certain set that belongs to the uncertain space defined by ξ_ijξ[−1,1].

In general, consider the following linear optimization problem

$$\begin{array}{cc}\hfill \text{max}& \mathit{cx}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _j{\tilde{a}}_{\mathit{ij}}{x}_j\le {\tilde{b}}_i{\forall }_i\hfill \end{array}$$ (2.1)

where ${\widehat{a}}_{\mathit{ij}}$ and ${\widehat{b}}_i$ represent the true value of the parameters which are subject to uncertainty. Assume that the uncertainty affecting each constraint is independent of each other and consider the i-th constraint of the above linear optimization problem where both the LHS constraint coefficients and RHS parameters are subject to uncertainty. Define the uncertainty as follows

$${\tilde{a}}_{\mathit{ij}}=a_{\mathit{ij}}+{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{\forall }_j\in J_i$$ (2.2a)

$${\tilde{b}}_i=b_i+{\xi }_{i0}{\widehat{b}}_i$$ (2.2b)

where a_ij and b_i represent the nominal value of the parameters; ${\widehat{a}}_{\mathit{ij}}$ and ${\widehat{b}}_i$ represent constant perturbation (which are positive); J_i represents the index subset that contains the variable indices whose corresponding coefficients are subject to uncertainty; and ξ_i0 and ξ_ij ∀_i, ∀_j ∈ are random variables which are subject to uncertainty. With the above definition, the original i-th constraint can be rewritten as:

$$\sum _{j\notin J_i}{a}_{\mathit{ij}}{x}_j+\sum _{j\in J_i}{\tilde{a}}_{\mathit{ij}}{x}_j\le {\tilde{b}}_i$$ (2.3)

which can be further reformulated as follows:

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[-{\xi }_{i0}{\widehat{b}}_i+\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j\right]\le b_i.$$ (2.4)

In the set induced robust optimization method, with a predefined uncertainty set U, the aim is to find solutions that remain feasible for any ξ in the given uncertainty set U so as to immunize against infeasibility, that is,

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\underset{\xi \in U}{\text{max}}\left\{-{\xi }_{i0}{\widehat{b}}_i+\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j\right\}\right]\le b_i.$$ (2.5)

Finally, replacing the original constraint in LP problem (2.1) with the corresponding robust counterpart constraints, the robust counterpart of the original LP problem is obtained:

$$\begin{array}{cc}\hfill \text{max}& \mathit{cx}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _j{a}_{\mathit{ij}}{x}_j+\left[\underset{\xi \in U}{\text{max}}\left\{-{\xi }_{i0}{\widehat{b}}_i+\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j\right\}\right]\le b_i{\forall }_i.\hfill \end{array}$$ (2.6)

Motivating Example 1 (Continued). Applying the robust counterpart formulation (2.6) to the two constraints of the motivating example 1, their corresponding robust counterpart constraints become

$$10x_1+20x_2+\underset{({\xi }_{11},{\xi }_{12})\in U_1}{\text{max}}\{{\xi }_{11}{x}_1+{\xi }_{12}{x}_2\}\le 140$$

$$6x_1+8x_2+\underset{({\xi }_{21},{\xi }_{22})\in U_2}{\text{max}}\{0.6{\xi }_{21}{x}_1+0.8{\xi }_{22}{x}_2\}\le 72$$

where U₁ and U₂ are predefined uncertainty sets for (ξ₁₁, ξ₁₂) and (ξ₂₁, ξ₂₂), respectively.

2.2 Robust mixed integer linear optimization

Motivating Example 2. Consider the following mixed integer linear optimization problem:

$$\begin{array}{cc}\hfill \text{max}& 3x_1+2x_2-10y_1-5y_2\hfill \\ \hfill s.t.& x_1+x_2\le 20\hfill \\ \hfill & x_1+2x_2\le 12\hfill \\ \hfill & a_{31}{x}_1+b_{31}{y}_1\le 0\hfill \\ \hfill & a_{42}{x}_2+b_{42}{y}_2\le 0\hfill \\ \hfill & x_1-x_2\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}.\hfill \end{array}$$

Assume that the left hand side (LHS) constraint coefficients of the third and the fourth constraints are subject to uncertainty and they are defined as follows:

$$a_{31}=1+0.1{\xi }_{31},b_{31}=20+2{\xi }_{33},a_{42}=1+0.1{\xi }_{42},b_{42}=-20+2{\xi }_{44},$$

where ξ₃₁,ξ₃₃,ξ₄₂,ξ₄₄ are independent uncertain parameters distributed in the range [−1,1]. The robust solution for the problem is among the candidate solutions that remain feasible for all realizations of the data from the uncertainty set. For example, if the uncertainty set is defined as the bounded box with range [−1,1] on each dimension, then the corresponding robust counterpart optimization solution should ensure the feasibility of all the constraints for any possible values of the uncertain parameters and maximize the objective at the same time.

Generally, consider the following mixed integer linear optimization (MILP) problem

$$\begin{array}{cc}\hfill \text{max}& \sum _m{c}_m{x}_m+\sum _k{d}_k{y}_k\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _m{\tilde{a}}_{\mathit{im}}{x}_m+\sum _k{\tilde{b}}_{\mathit{ik}}{y}_k\le {\tilde{p}}_i{\forall }_i\hfill \end{array}$$ (2.7)

where x and y represent the continuous and integer variables, respectively, and ${\tilde{a}}_{\mathit{im}}$, ${\tilde{b}}_{\mathit{ik}}$, ${\tilde{p}}_i$ represent the true value of the parameters which are possibly subject to uncertainty. Considering the i-th constraint of the above problem, we assume that the uncertain parameters in the i-th constraint are defined as follows:

$${\tilde{a}}_{\mathit{im}}=a_{\mathit{im}}+{\xi }_{\mathit{im}}{\widehat{a}}_{\mathit{im}},{\forall }_m\in M_i$$ (2.8a)

$${\tilde{b}}_{\mathit{ik}}=b_{\mathit{ik}}+{\xi }_{\mathit{ik}}{\widehat{b}}_{\mathit{ik}},{\forall }_k\in K_i$$ (2.8b)

$${\tilde{p}}_i=p_i+{\xi }_{i0}{\widehat{p}}_i$$ (2.8c)

where M_i and K_i represent the subsets that contain the continuous and discrete variable indices whose corresponding coefficients are subject to uncertainty, respectively; a_im, b_ik, p_i represent the nominal value of the parameters; ${\widehat{a}}_{\mathit{im}}$, ${\widehat{b}}_{\mathit{ik}}$, ${\widehat{p}}_i$ represent positive constant perturbation; and ξ_im, ξ_ik, ξ_i are random variables which are subject to uncertainty. With the above definitions, the original i-th constraint can be rewritten as follows:

$$\sum _{m\notin M_i}{a}_{\mathit{im}}{x}_m+\sum _{k\notin K_i}{b}_{\mathit{ik}}{b}_k+\sum _{m\in M_i}{\tilde{a}}_{\mathit{im}}{x}_m+\sum _{k\in K_i}{\tilde{b}}_{\mathit{ik}}{y}_k\le {\tilde{p}}_i$$ (2.9)

which after grouping the uncertain part can be further rewritten as:

$$\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\left\{-{\xi }_{i0}{\widehat{p}}_i+\sum _{m\in M_i}{\xi }_{\mathit{im}}{\widehat{a}}_{\mathit{im}}{x}_m+\sum _{k\in K_i}{\xi }_{\mathit{ik}}{\widehat{b}}_{\mathit{ik}}{y}_k\right\}\le p_i.$$ (2.10)

With a predefined uncertainty set U for the random variables ξ = {ξ_i0, ξ_im, ξ_ik}, the objective is to find solutions that remain feasible for any ξ in the set so as to immunize against infeasibility, that is:

$$\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\underset{\xi \in U}{\text{max}}\left\{-{\xi }_{i0}{\widehat{p}}_i+\sum _{m\in M_i}{\xi }_{\mathit{im}}{\widehat{a}}_{\mathit{im}}{x}_m+\sum _{k\in K_i}{\xi }_{\mathit{ik}}{\widehat{b}}_{\mathit{ik}}{y}_k\right\}\le p_i.$$ (2.11)

Correspondingly, the robust counterpart of the original MILP problem is obtained by replacing the original i-th constraint with its robust counterpart constraint (2.11):

$$\begin{array}{cc}\hfill \text{max}& \sum _m{c}_m{x}_m+\sum _k{d}_k{y}_k\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\underset{\xi \in U}{\text{max}}\left\{-{\xi }_{i0}{\widehat{p}}_i+\sum _{m\in M_i}{\xi }_{\mathit{im}}{\widehat{a}}_{\mathit{im}}{x}_m+\sum _{k\in K_i}{\xi }_{\mathit{ik}}{\widehat{b}}_{\mathit{ik}}{y}_k\right\}\le p_i{\forall }_i.\hfill \end{array}$$ (2.12)

Motivating Example 2 (Continued). Applying the robust counterpart formulation (2.12) to the two constraints of motivating example 2 and realizing that there is no RHS uncertainty (i.e., ${\widehat{p}}_i=0$), their corresponding robust counterpart constraints become:

$$x_1-20y_1+\underset{({\xi }_{31},{\xi }_{33})\in U_1}{\text{max}}\{0.1{\xi }_{31}{x}_1+2{\xi }_{33}{y}_1\}\le 0$$

$$x_2-20y_2+\underset{({\xi }_{42},{\xi }_{44})\in U_2}{\text{max}}\{0.1{\xi }_{42}{x}_2+2{\xi }_{44}{y}_2\}\le 0$$

where U₁ and U₂ are predefined uncertainty sets for (ξ₃₁, ξ₃₃) and (ξ₄₂,ξ₄₄), respectively.

The set induced robust counterpart formulations (2.6) and (2.12) depend on the selection of the uncertainty set U. In the subsequent sections, several different uncertainty sets are studied first and the corresponding robust counterpart optimization formulations are then derived.

3. Uncertainty Sets

As stated in the previous section, the formulation of robust counterpart optimization models is connected with the selection of the uncertainty set. In the sequel, several different uncertainty sets are introduced. For the sake of simplicity, we eliminate the constraint index i in the random vector ξ.

Definition 3.1 (Box Uncertainty Set) The box uncertainty set is described using the ∞ -norm of the uncertain data vector as follows

$$U_{\infty }=\{\xi \mid {\parallel \xi \parallel }_{\infty }\le \Psi \}=\{\xi \mid \mid {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\}$$ (3.1)

where Ψ is the adjustable parameter controlling the size of the uncertainty set.

Figure 3.1 illustrates the box uncertainty set for parameter ${\tilde{a}}_j$, j=1,2 defined by ${\tilde{a}}_j=a_j+{\xi }_j{\widehat{a}}_j,j=1,2$, where ${\tilde{a}}_j$ denotes the true value of the parameter, a_j denotes the nominal value of the parameter, ξ_j denotes the uncertainty and ${\widehat{a}}_j$ represents a constant perturbation.

Figure 3.1.

Illustration of box uncertainty set

If the uncertain parameters are known to be bounded in given intervals ${\tilde{a}}_{\mathit{ij}}\in [a_{\mathit{ij}}-{\widehat{a}}_{\mathit{ij}},a_{\mathit{ij}}+{\widehat{a}}_{\mathit{ij}}]{\forall }_j\in J_i$, then the uncertainty can be represented by ${\tilde{a}}_{\mathit{ij}}=a_{\mathit{ij}}+{\xi }_j{\widehat{a}}_{\mathit{ij}}$ and this results in the interval uncertainty set, which is a special case of box uncertainty set when Ψ = 1 (i.e., U_∞ = {ξ ||ξ_j ≤ 1, ∀_j ε J_i}). Note that in this paper, we specifically use the “interval uncertainty set” to denote the box set with Ψ = 1, and use the “box uncertainty set” to represent a general adjustable bounded set.

Definition 3.2 (Ellipsoidal Uncertainty Set) The ellipsoidal uncertainty set is described using the 2-norm of the uncertain data vector as shown in Figure 3.2,

$$U_2=\{\xi \mid {\parallel \xi \parallel }_2\le \Omega \}=\left\{\xi \mid \sqrt{\sum _{j\in J_i}{\xi }_j^2}\le \Omega \phantom{\mid }\right\}$$ (3.2)

where Ω is the adjustable parameter controlling the size of the uncertainty set.

Figure 3.2.

Illustration of ellipsoidal uncertainty set

Note that it is known from geometry that for bounded uncertainty ξ_j ε [−1, 1], when $\Omega \ge \sqrt{\mid J_i\mid }$ (where |J_i| is the cardinality of the set J_i), the entire uncertain space is covered by the ellipsoid uncertainty set.

Definition 3.3 (Polyhedral Uncertainty Set) The polyhedral uncertainty set is described using the 1-norm of the uncertain data vector as shown in Figure 3.3,

$$U_1=\{\xi \mid {\parallel \xi \parallel }_1\le \Gamma \}=\left\{\xi \mid \sum _{j\in J_i}\mid {\xi }_j\mid \le \Gamma \phantom{\mid }\right\}$$ (3.3)

where Γ is the adjustable parameter controlling the size of the uncertainty set.

Figure 3.3.

Illustration of polyhedral uncertainty set

Note that for bounded uncertainty ξ_j ε [−1, 1], when Γ ≤ |J_i|, the overall uncertain space is covered by the polyhedral uncertainty set.

The above three uncertainty sets can be further combined to generate new uncertainty sets. Bounded uncertainty is a type of important uncertainty characteristic which is widely studied in practice. We will further introduce several uncertainty sets which are generated by combining the ellipsoid, or polyhedron, or both ellipsoid and polyhedral uncertainty set with the interval uncertainty set.

Definition 3.4 (“Box+Ellipsoidal” Uncertainty Set)

This type of uncertainty set is the intersection between an ellipsoid and a box defined as follows,

$$U_{2\cap \infty }=\left\{\xi \mid \sum _{j\in J_i}{\xi }_j^2\le {\Omega }^2,\mid {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\phantom{\mid }\right\}$$ (3.4)

It is known from geometry that for an adjustable box defined by (3.1) and an adjustable ellipsoid defined by (3.2), in order to ensure that the intersection between them does not reduce to any one of them, the parameters should satisfy the following relationship

$$\Psi \le \Omega \le \Psi \sqrt{\mid J_i\mid }$$ (3.5)

Remark 3.1 As ε = 1, the above set (3.4) defines the intersection between interval and ellipsoid, which is referred as “interval+ellipsoidal” uncertainty set in this paper. This type of uncertainty set is important for bounded uncertainty since it makes no sense to construct an uncertainty set exceeding the bounded uncertain space. For this kind of uncertainty set, when Ω = 1, the ellipsoid is exactly inscribed by the box; when $\Omega =\sqrt{\mid J_i\mid }$, the ellipsoid is circumscribed by the box (i.e., the intersection between the box and ellipsoid is exactly the box). Figure 3.4 illustrates the geometry of this uncertainty set for the case that the dimension of the uncertain parameter space is 2 (i.e., |J_i|=2).

Figure 3.4.

Illustration of the “interval+ellipsoidal” uncertainty set

Definition 3.5 (“Box+Polyhedral” Uncertainty Set) This type of uncertainty set is the intersection between the polyhedral and the interval set defined with both 1-norm and infinite norm as follows,

$$U_{1\cap \infty }=\left\{\xi \mid \sum _{j\in J_i}\mid {\xi }_j\mid \le \Gamma ,\mid {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\phantom{\mid }\right\}$$ (3.6)

It is also known from geometry that for an adjustable box defined by (3.1) and an adjustable polyhedron defined by (3.3), the intersection between them does not reduce to any one of them if the parameters satisfy the following relationship

$$\Psi \le \Gamma \le \Psi \mid J_i\mid$$ (3.7)

Remark 3.2 As Ψ = 1, the above set defines the intersection between the interval and polyhedral set, which is referred as “interval+polyhedral” uncertainty set. For this uncertainty set, when Γ = 1, the polyhedron is exactly inscribed by the box and the intersection between the polyhedron and the box is exactly the polyhedron; when Γ = |J_i, the intersection between the polyhedron and the box is exactly the box, as shown in Figure 3.5.

Figure 3.5.

Illustration of combined interval and polyhedral uncertainty set

Definition 3.6 (“Box+Ellipsoidal+Polyhedral” Uncertainty Set) This type of uncertainty set is the intersection between the ellipsoidal, polyhedral and box set defined as follows,

$$U_{1\cap 2\cap \infty }=\left\{\xi \mid \sum _{j\in J_i}\mid {\xi }_j\mid \le \Gamma ,\sum _{j\in J_i}{\xi }_j^2\le {\Omega }^2,\mid {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\phantom{\mid }\right\}$$ (3.8)

For this type of uncertainty set, the intersection between polyhedron and ellipsoid is not reduced to any one of them if the adjustable parameters satisfy the following set of conditions:

$$\Psi \le \Omega \le \Psi \sqrt{\mid J_i\mid }$$ (3.9a)

$$\Omega \le \Gamma \le \Omega \sqrt{\mid J_i\mid }$$ (3.9b)

where the first equation is used to ensure that there is intersection between the ellipsoid and the box, the second equation is used to ensure that there is intersection between the ellipsoid and the polyhedron as shown in Figure 3.6.

Figure 3.6.

Illustration of combined interval, ellipsoidal and polyhedral uncertainty set

Illustration 3.1 Assume ${\tilde{a}}_1=20+2{\xi }_1$, ${\tilde{a}}_2=10+{\xi }_2$, ξ₁, ξ₂ ∈ [−1,1], then the corresponding ellipsoidal and polyhedral uncertainty sets for $\tilde{a}$ under different values of Ω and Γ can be illustrated as Figure 3.7 and Figure 3.8:

Figure 3.7.

Illustration of the relationship between ellipsoidal and polyhedral uncertainty set (Γ = Ω)

Figure 3.8.

Illustration of the relationship between ellipsoidal and polyhedral uncertainty set ($\Gamma =\Omega \sqrt{2}$)

From the above illustration in Figure 3.7, it can be observed that when Γ = Ω, the polyhedron is inscribed by the ellipsoid. On the other hand, it can be observed from Figure 3.8 that when $\Gamma =\Omega \sqrt{\mid J_i\mid }$, the ellipsoid is inscribed by the polyhedron, which verifies the analysis in the previous definitions.

The different uncertainty sets are summarized in Table 3.1. Considering different types of uncertainty characteristics (bounded or unbounded), we also list the suggested range for the adjustable parameter of different uncertainty sets. Based on these definitions of the uncertainty sets, the corresponding robust counterpart optimization formulations for linear optimization problems are derived in the next section.

Table 3.1.

Summary on the uncertainty set

Illustration	Type	Adjustable parameter	Suggested range for bounded uncertainty	Suggested range for unbounded uncertainty
	Box	Ψ	Ψ ≤ 1	Ψ < ∞
	Ellipsoidal	Ω	$\Omega \le \sqrt{\mid J_i\mid }$	Ω < ∞
	Polyhedral	Γ	$\Gamma \le \mid J_i\mid$	Γ < ∞
	Interval+Ellipsoidal	Ω	$\Omega \le \sqrt{\mid J_i\mid }$
	Box+Ellipsoidal	Ψ,Ω	$\Psi \le 1,\Psi \le \Omega \le \Psi \sqrt{\mid J_i\mid }$	$\Psi \le \Omega \le \Psi \sqrt{\mid J_i\mid }$
	Interval+Polyhedral	Γ	$\Gamma \le \mid J_i\mid$
	Box+Polyhedral	Ψ,Γ	$\Psi \le 1,\Psi \le \Gamma \le \Psi \mid J_i\mid$	$\Psi \le \Gamma \le \Psi \mid J_i\mid$
	Interval+Ellipsoidal+Polyhedral	Ω,Γ	$\Omega \le \sqrt{\mid J_i\mid },\Omega \le \Gamma \le \Omega \sqrt{\mid J_i\mid }$
	Box+Ellipsoidal+Polyhedral	Ψ Ω Γ	$\Psi \le 1,\Psi \le \Omega \le \Psi \sqrt{\mid J_i\mid }\Omega \le \Gamma \le \Omega \sqrt{\mid J_i\mid }$	$\Psi \le \Omega \le \Psi \sqrt{\mid J_i\mid }\Omega \le \Gamma \le \Omega \sqrt{\mid J_i\mid }$

Remark 3.3

• 1)All the parameter values should be non-negative.
• 2)The “interval+ellipsoidal”, “interval+polyhedral”, and “interval+ellipsoidal+polyhedral” uncertainty sets are not suggested for the unbounded uncertainty distribution since we don't want to restrict the set within a given interval.
• 3)The suggested parameter range for bounded uncertainty is based on the following: when the adjustable parameter's value is equal to the upper bound given in the table, the bounded uncertain space is entirely covered by the corresponding uncertainty set. Thus, further increase of the value of the parameter could lead to more conservative solution and will not improve the solution robustness.
• 4)The suggested range for unbounded uncertainty is based on that we want to avoid that the intersection between different uncertainty sets is reduced to any one of them.

4. Robust Counterpart Formulations for Linear Optimization Problems

To attain robust solutions, we look for solutions which are feasible for any realization of the uncertain data in a predefined uncertainty set. In the following subsections, we present the derivation procedure of the equivalent robust counterpart optimization models based on formulation (2.6). In order to eliminate the inner maximization problem in the i-th constraint of (2.6), we first transform the inner maximization problem into its conic dual, and then incorporate the dual problem into the original constraint.

We will first derive the robust counterpart formulation for LHS only uncertainty of a linear optimization problem, then we will extend it to the RHS only uncertainty, and finally to the case of LHS and RHS uncertainty appearing simultaneously.

4.1 Left Hand Side (LHS) Uncertainty

When only LHS uncertainty is considered in the i-th constraint of (2.1), the corresponding robust counterpart constraint (2.5) for the i-th constraint is reduced to

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\underset{\xi \in U}{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j\right\}\right]\le b_i.$$ (4.1)

The robust counterpart is derived for different uncertainty sets introduced in section 3 as follows.

Property 4.1 If the set U is the box uncertainty set (3.1), then the corresponding robust counterpart constraint (4.1) is equivalent to the following constraint:

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j\mid \right]\le b_i$$ (4.2)

Proof. For the box uncertainty set U_∞ = {ξ | |ξ_j| ≥ Ψ, ∀_j ∈ J_i}, we define P_∞ = [I_L×L;0_1×L], p_∞ = [0L×1;Ψ] and $K_{\infty }=\left\{[{\theta }_{L\times 1};t]\in R^{L+1}\mid {\parallel \theta \parallel }_{\infty }\le t\right\}$, where L is the cardinality of the uncertainty set (i.e., L = |J_i). Then the inner maximization problem in (4.1) can be rewritten as

$$\underset{\xi }{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j:P_{\infty }\xi +p_{\infty }\in K_{\infty }\right\}.$$

Defining dual variable y = [w_i; τ_i] ∈ R^L+1 and using the dual cone of $K_{\infty }:K_{\infty }^{\ast }\left\{[{\theta }_{L\times 1};t]\in R^{L+1}\mid {\parallel \theta \parallel }_1\le t\right\}$, the conic dual of the inner maximization problem can be formulated as

$$\underset{w,\tau }{\text{max}}\{\Psi {\tau }_i:w_{\mathit{ij}}={\widehat{a}}_{\mathit{ij}}{x}_j{\forall }_j,{\parallel w_i\parallel }_1\le {\tau }_i\}.$$

Since the above problem is a minimization problem, it can be further rewritten as the following equivalent formulation by replacing τ_i with ${\parallel w_i\parallel }_1={\sum }_{j\in J_i}\mid w_{\mathit{ij}}\mid$,

$$\underset{w}{\text{min}}\left\{\Psi \sum _{j\in J_i}\mid w_{\mathit{ij}}\mid :w_{\mathit{ij}}={\widehat{a}}_{\mathit{ij}}{x}_j{\forall }_j\right\}.$$

Realizing that ${\widehat{a}}_{\mathit{ij}}\ge 0$, we can reformulate the conic dual of the inner maximization problem as follows

$$\underset{w}{\text{min}}\left\{\Psi \sum _{j\in J_i}\mid w_{\mathit{ij}}\mid :w_{\mathit{ij}}={\widehat{a}}_{\mathit{ij}}{x}_j{\forall }_j\right\}=\Psi \sum _{j\in J_i}\mid {\widehat{a}}_{\mathit{ij}}{x}_j\mid =\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_{\mathit{j}}\mid .$$

Replacing the original inner maximization problem with the above conic dual, then the following constraint is obtained:

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j\mid \right]\le b_i.$$

Remark 4.1 Constraint (4.2) contains absolute value terms |x_j|. If the variable is positive, the absolute value operator can be directly removed. Otherwise, it can be further equivalently transformed to the following constraints because their corresponding feasible sets are identical:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}{u}_j\le b_i\hfill \\ \mid x_j\mid \le u_j,j\in J_i\hfill \end{array}\phantom{\}}$$

Thus, the absolute value term in (4.2) can be eliminated and the final equivalent robust formulation is obtained:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}{u}_j\le b_i\hfill \\ -u_j\le x_j\le u_j\hfill \end{array}\phantom{\}}.$$ (4.3)

Remark 4.2 When Ψ = 1 (i.e., the interval uncertainty set), the robust counterpart formulation is reduced to ${\sum }_j{a}_{\mathit{ij}}{x}_j+{\sum }_{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j\mid \le b_i$, which is exactly the robust counterpart formulation proposed by Soyster⁴, the so called “worst case scenario” robust model for bounded uncertainty.

Motivating Example 1 (Continued). Considering the first constraint of motivating example 1,

$$(10+{\xi }_{11})x_1+(20+2{\xi }_{12})x_2\le 140$$

and assuming that the uncertainty set related to ξ₁₁, ξ₁₂ is defined by (3.1), the corresponding robust counterpart for this constraint is:

$$10x_1+20x_2+\Psi (\mid x_1\mid +2\mid x_2\mid )\le 140.$$

The first robust counterpart constraint with different value of Ψ is illustrated in Figure 4.1(a). It can be observed that as the parameter value Ψ increases (i.e., the size the uncertainty set increases), the feasible set of the resulting robust counterpart optimization problem contracts.

Figure 4.1.

Illustration of the robust counterpart constraint (a) Box uncertainty set; (b) Ellipsoidal uncertainty set; (c) Polyhedral uncertainty set; (d) “Interval+Ellipsoidal” uncertainty set; (e) “Interval+Polyhedral” uncertainty set; (f) “Interval+Ellipsoidal+Polyhedral” uncertainty set

Similarly, for the second constraint of the example, the box uncertainty set induced robust counterpart is

$$6x_1+8x_2+\Psi (0.6\mid x_1\mid +0.8\mid x_2\mid )\le 72.$$

Notice that the robust counterpart formulation is constructed constraint by constraint and different parameter values can be applied for different constraints. The complete box uncertainty set induced robust counterpart formulation of this motivating example with different parameters Ψ₁ and Ψ₂ for the two constraints is

$$\begin{array}{cc}\hfill \text{max}& 8x_1+12x_2\hfill \\ \hfill s.t.& 10x_1+20x_2+{\Psi }_1(\mid x_1\mid +2\mid x_2\mid )\le 140\hfill \\ \hfill & 6x_1+8x_2+{\Psi }_2(0.6\mid x_1\mid +0.8\mid x_2\mid )\le 72\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

which is equivalent to the following problem since the variables are positive:

$$\begin{array}{cc}\hfill \text{max}& 8x_1+12x_2\hfill \\ \hfill s.t.& 10x_1+20x_2+{\Psi }_1(x_1+2x_2)\le 140\hfill \\ \hfill & 6x_1+8x_2+{\Psi }_2(0.6x_1+0.8x_2)\le 72\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

Property 4.2 If the set U is the ellipsoidal uncertainty set (3.2), then the corresponding robust counterpart constraint (4.1) is equivalent to the following constraint

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{x}_j^2}\right]\le b_i$$ (4.4)

Proof. Consider the ellipsoidal uncertainty set $U_2=\{\xi \mid \sqrt{{\sum }_{j\in J_i}{\xi }_j^2}\le \Omega \}$, we define P₂ = [I_L×L; 0_1×L], I diag{1,…1}, p₂ = [0_L×1;Ω] and $K_2=\left\{[{\theta }_{L\times 1};t]\in R^{L+1}\mid {\parallel \theta \parallel }_2\le t\right\}$, then the inner maximization problem in (4.1) can be denoted as

$$\underset{\xi }{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j:P_2\xi +p_2\in K_2\right\}.$$

Defining the dual variable y = [z_i; τ_i] ∈ R^L+1 and using the dual cone $K_2^{\ast }=K_2$, the conic dual of the inner maximization problem is

$$\text{min}\{\Omega {\tau }_i:z_i={\widehat{a}}_{i}x,{\parallel z_i\parallel }_2\le {\tau }_i\}.$$

Since it is a minimization problem, we can make equivalent transformation of above problem by replacing τ_i with ${\parallel z_i\parallel }_2=\sqrt{{\sum }_{z\in J_i}{z}_{\mathit{ij}}^2}$ and get

$$\underset{z}{\text{min}}\left\{\Omega \sqrt{\sum _{j\in J_i}{z}_{\mathit{ij}}^2}:z_i={\widehat{a}}_{i}x\right\}=\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{x}_j^2}.$$

After incorporating the above conic dual into the robust counterpart constraint, the following robust counterpart is obtained

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{x}_j^2}\right]\le b_i.$$

Motivating Example 1 (Continued). The corresponding robust constraint for the first constraint of the motivating example 1 is:

$$10x_1+20x_2+\Omega \sqrt{x_1^2+4x_2^2}\le 140,$$

and its robust counterpart constraint with different value of Ω is illustrated in Figure 4.1(b). It can be observed that as the parameter value Ω increases (i.e., the size the uncertainty set increases), the feasible set of the resulting robust counterpart optimization problem contracts.

Property 4.3 If the set U is the polyhedral uncertainty set (3.3), then the corresponding robust counterpart constraint (4.1) is equivalent to the following constraints

$$\{\begin{array}{c}\hfill \sum _j{a}_{\mathit{ij}}{x}_j+\Gamma p_i\le b_i\hfill \\ \hfill p_i\ge {\widehat{a}}_i\mid x_{\mathit{ij}}\mid ,{\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$ (4.5)

Proof. Consider the polyhedral uncertainty set $U_1=\left\{\xi \mid {\sum }_{j\in J_i}\mid {\xi }_j\mid \le \Gamma \phantom{\mid }\right\}$, define P₁ = [I_L×L;0_1×L], p₁ = [0_L×1;Γ], $K_1=\left\{[{\theta }_{L\times 1};t]\in R^{L+1}\mid {\parallel \theta \parallel }_1\le t\right\}$, then the set U₁ can be denoted as $U_1=\{\xi \mid P_1\xi +p_1\in K_1\}$ and the inner maximization problem in (4.1) can be denoted as

$$\underset{\xi }{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j:P_1\xi +p_1\in K_1\right\}.$$

Defining the dual variable y = [z_i; τ_i] Ψ R^L+1 and based on the fact that the dual cone of K₁ is

$$K_1^{\ast }=K_{\infty }=\left\{[{\theta }_{L\times 1};t]\in R^{L+1}\mid {\parallel \theta \parallel }_{\infty }\le t\right\}.$$

The conic dual of the inner optimization problem can be formulated as:

$$\underset{z,\tau }{\text{min}}\{\Gamma {\tau }_i:z_i={\widehat{a}}_{i}x,{\parallel z_i\parallel }_{\infty }\le {\tau }_i\}$$

which can be further rewritten as the following equivalent formulation by replacing τ_i with ${\parallel z_i\parallel }_{\infty }=\underset{j\in J_i}{\text{max}}\mid z_{\mathit{ij}}\mid$,

$${\parallel z_i\parallel }_{\infty }=\underset{j\in J_i}{\text{max}}\mid z_{\mathit{ij}}\mid ,$$

$$\underset{z}{\text{min}}\left\{\Gamma \underset{j\in J_i}{\text{max}}\mid z_{\mathit{ij}}\mid :z_i={\widehat{a}}_{i}x\right\}=\Gamma \underset{j\in J_i}{\text{max}}\mid {\widehat{a}}_{\mathit{ij}}{x}_j\mid .$$

Since the above problem is a minimization problem, we can introduce an auxiliary variable p_i to replace $\underset{j\in J_i}{\text{max}}\mid {\widehat{a}}_{\mathit{ij}}{x}_j\mid$ and obtain the following equivalent description:

$$\underset{z}{\text{min}}\left\{\Gamma \underset{j\in J_i}{\text{max}}\mid z_{\mathit{ij}}\mid :z_i={\widehat{a}}_{i}x\right\}=\Gamma p_i,p_i\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid ,{\forall }_j\in J_i.$$

Incorporating the above conic dual into the robust counterpart constraint, the following robust counterpart is obtained

$$\{\begin{array}{c}\hfill \sum _j{a}_{\mathit{ij}}{x}_j+\Gamma p_i\le b_i\hfill \\ \hfill p_i\ge {\widehat{a}}_i\mid x_j\mid ,{\forall }_j\in J_i\hfill \end{array}\phantom{\}}.$$

Remark 4.3 As shown in Remark 4.1, an equivalent robust formulation for (4.5) can be obtained by replacing the absolute value term |x_j| with auxiliary variable u_j and constraint −u_j ≤ x_j ≤ u_j as follows:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Gamma p_i\le b_i\hfill \\ p_i\ge {\widehat{a}}_i{u}_j,{\forall }_j\in J_i\hfill \\ -u_j\le x_j\le u_j,{\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$ (4.6)

Motivating Example 1 (Continued). The corresponding robust counterpart constraint for the first constraint of the motivating example 1 is

$$\{\begin{array}{c}10x_1+20x_2+\Gamma p\le 140\hfill \\ p\ge \mid x_1\mid ,p\ge 2\mid x_2\mid \hfill \end{array}\phantom{\}}$$

The above robust counterpart for the first constraint with different value of G is illustrated in Figure 4.1(c). It can be observed that as the parameter value G increases (i.e., the size the uncertainty set increases), the feasible set of the resulting robust counterpart optimization problem contracts.

Property 4.4 If the set U is the “box+ellipsoidal” uncertainty set (3.4), then the corresponding robust counterpart constraint (4.1) is equivalent to the following constraint

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j-z_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}\right]\le b_i$$ (4.7)

Proof. The “box+ellipsoidal” uncertainty set $U_{2\cap \infty }=\left\{\xi \mid {\sum }_{j\in J_i}{\xi }_j^2\le {\Omega }^2,\mid {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\phantom{\mid }\right\}$ can be denoted using conic representation as follows,

$$U_{2\cap \infty }=\{\xi \mid P_2\xi +p_2\in K_2,P_{\infty }\xi +p_{\infty }\in K_{\infty }\},$$

where K₂ and K_∞ have the same definition as in the previous proof. Thus the inner maximization problem of (4.1) becomes

$$\underset{\xi }{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j:P_2\xi +p_2\in K_2,P_{\infty }\xi +p_{\infty }\in K_{\infty }\right\}.$$

Defining the dual variable $y^1=[w_i,{\tau }_1]\in R^{L+1}$, $y^2=[z_i,{\tau }_2]\in R^{L+1}$ and using the dual cone $K_{\infty }^{\ast }=K_1$, $K_2^{\ast }=K_2$ the conic dual of the inner maximization problem can be formulated as follows:

$$\text{min}\{\Psi {\tau }_1+\Omega {\tau }_2:w_i+z_i={\widehat{a}}_{i}x,{\parallel w_i\parallel }_1\le {\tau }_1,{\parallel z_i\parallel }_2\le {\tau }_2\}.$$

After an equivalent transformation through replacing T₁ and T₂ with ${\parallel w_i\parallel }_1={\sum }_{j\in J_i}\mid w_{\mathit{ij}}\mid$ and ${\parallel z_i\parallel }_2=\sqrt{{\sum }_{j\in J_1}{z}_{\mathit{ij}}^2}$, respectively, we get

$$\underset{z,w}{\text{min}}\left\{\Psi \sum _{j\in J_i}\mid w_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{z}_{\mathit{ij}}^2}:w_i+z_i={\widehat{a}}_{i}x\right\},$$

which is further equivalent to

$$\underset{z}{\text{min}}\Psi \sum _{j\in J_i}\mid {\widehat{a}}_{\mathit{ij}}{x}_j-z_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{z}_{\mathit{ij}}^2}.$$

Since z_ij are decision variables, we can replace z_ij with ${\widehat{a}}_{\mathit{ij}}{z}_{\mathit{ij}}$ and get an equivalent problem:

$$\underset{z}{\text{min}}\Psi \sum _{j\in J_i}\mid {\widehat{a}}_{\mathit{ij}}{x}_j-{\widehat{a}}_{\mathit{ij}}{z}_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}.$$

Incorporating the above conic dual into the robust counterpart constraint and remove the minimization operator (it is a equivalent operation since the inner minimization is on the left hand side of a “less or equal to” constraint), the following robust counterpart is obtained

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j-z_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}\right]\le b_i.$$

Remark 4.4 As shown in Remark 4.1, an equivalent robust formulation for (4.7) can be obtained by replacing the absolute value term $\mid x_j-z_{\mathit{ij}}\mid$ with auxiliary variable u_ij and constraint $-u_{\mathit{ij}}\le x_j-z_{\mathit{ij}}\le u_{\mathit{ij}}$ as follows:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}{u}_{\mathit{ij}}+\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}\le b_i\hfill \\ -u_{\mathit{ij}}\le x_j-z_{\mathit{ij}}\le u_{\mathit{ij}}\hfill \end{array}\phantom{\}}$$ (4.8)

Remark 4.5 When ψ= 1 (i.e., the set U is defined as “interval+ellipsoidal” uncertainty set), the corresponding “interval+ellipsoidal” based robust counterpart optimization formulation takes the following form:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}{u}_{\mathit{ij}}+\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}\le b_i\hfill \\ -u_{\mathit{ij}}\le x_j-z_{\mathit{ij}}\le u_{\mathit{ij}}\hfill \end{array}\phantom{\}}$$ (9)

which is exactly the robust counterpart formulation proposed by Ben-Tal and Nemirovski¹ (i.e., a special case of the combined adjustable box and adjustable ellipsoidal based robust counterpart).

Motivating Example 1 (Continued). The “interval+ellipsoidal” based robust constraint for the first constraint of the motivating example 1 is:

$$\{\begin{array}{c}10x_1+20x_2+u_{11}+2u_{12}+\Omega \sqrt{z_{11}^{2}4z_{12}^2}\le 140\hfill \\ -u_{11}\le x_1-z_{11}\le u_{11},-u_{12}\le x_2-z_{12}\le 12\hfill \end{array}\phantom{\}}$$

The above constraint can be projected to the space spanned by the x₁,x₂ dimensions by fixing x₁ at different points and maximizing the corresponding x₂. The constraint can be illustrated as shown in Figure 4.1(d). Comparing the robust counterpart constraint illustration Figure 4.1(b) and Figure 4.1(d), it can be observed that for Ω =1, the two robust counterparts are the same, whereas for Ω =2, the “interval+ellipsoidal” based robust counterpart is less conservative because the resulting optimization feasible set is larger. This is consistent with the fact that as Ω ≤ 1, the intersection between interval and ellipsoid is exactly the ellipsoid, but as Ω > 1, the intersection between interval and ellipsoid is smaller than the ellipsoid itself.

Property 4.5 If the set U is the “box+polyhedral” uncertainty set (3.6), then the corresponding robust counterpart constraint (4.1) is equivalent to the following constraints

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{w}_{\mathit{ij}}+\Gamma z_i\le b_i\hfill \\ z_i+w_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i\hfill \\ -z_j\le 0,w_{\mathit{ij}}\ge 0\hfill \end{array}\phantom{\}}$$ (4.10)

Proof. The “box+polyhedral” uncertainty set $U_{1\cap \infty }=\left\{\xi \mid {\sum }_{j\in J_i}\mid {\xi }_j\mid \le \Gamma ,\mid {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\phantom{\mid }\right\}$ can be denoted as follows using conic representation:

$$U_{1\cap \infty }=\{\xi \mid P_1\xi +p_1\in K_1,P_{\infty }\xi +p_{\infty }\in K_{\infty }\}.$$

Defining the dual variable $y^1=[w_i,{\tau }_1]\in R^{L\times 1}$, $y^2=[v_i,{\tau }_2]\in R^{L\times 1}$ and using the dual cone $K_1^{\ast }=K_{\infty }$, $K_{\infty }^{\ast }=K_1$, the inner maximization problem is rewritten as

$$\underset{\xi }{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j:P_1\xi +p_1\in K_1,P_{\infty }\xi +p_{\infty }\in K_{\infty }\right\}.$$

The conic dual of the above problem can be formulated as follows

$$\text{min}\{\Psi {\tau }_i+\Gamma {\tau }_2:w_i+v_i={\widehat{a}}_{i}x,{\parallel w_i\parallel }_1\le {\tau }_1,{\parallel v_i\parallel }_{\infty }\le {\tau }_2\}.$$

We can further get the following equivalent transformation through replacing T₁ and T₂ with ${\widehat{a}}_{\mathit{ij}}{z}_{\mathit{ij}}$, respectively,

$$\underset{w,z}{\text{min}}\left\{\Psi \sum _{j\in J_i}\mid w_{\mathit{ij}}\mid +\Gamma \underset{j\in J_i}{\text{max}}\mid v_{\mathit{ij}}\mid :w_i+v_i={\widehat{a}}_{i}x\right\}.$$

Since the above problem is a minimization problem, it can be equivalently transformed to the following problem

$$\underset{w,z}{\text{min}}\left\{\Psi \sum _{j\in J_i}\mid w_{\mathit{ij}}\mid +\Gamma z_i:z_i\ge \mid {\widehat{a}}_{\mathit{ij}}{x}_j-w_{\mathit{ij}}\mid ,{\forall }_j\in J_i\right\}.$$

The above problem is further equivalent to the following problem since it is a minimization problem and optimal solution must be obtained with $\mid w_{\mathit{ij}}\mid \le \mid {\widehat{a}}_{\mathit{ij}}{x}_j\mid$

$$\underset{w,z}{\text{min}}\left\{\Psi \sum _{j\in J_i}\mid w_{\mathit{ij}}\mid +\Gamma z_i:z_i\ge \mid {\widehat{a}}_{\mathit{ij}}{x}_j\mid -\mid w_{\mathit{ij}}\mid ,{\forall }_j\in J,z_i\ge 0\right\},$$

which is further equivalent to the following problem by replacing $\mid w_{\mathit{ij}}\mid$ with w_ij and w_ij ≥ 0

$$\underset{w,z}{\text{min}}\left\{\Psi \sum _{j\in J_i}{w}_{\mathit{ij}}+\Gamma z_i:z_i\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid -w_{\mathit{ij}},{\forall }_j\in J,w_{\mathit{ij}}\ge 0,z_i\ge 0\right\}.$$

After incorporating the above conic dual into the robust counterpart constraint, the following robust counterpart is obtained

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{w}_{\mathit{ij}}+\Gamma z_i\le b_i\hfill \\ z_i+w_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i\hfill \\ -u_j\le 0,w_{\mathit{ij}}\ge 0\hfill \end{array}\phantom{\}}$$

Remark 4.6 As pointed out in Remark 4.1, an equivalent robust formulation for (4.10) can be obtained by replacing the term $\mid x_j\mid$ with auxiliary variable u_j and constraint $-u_j\le x_j\le u_j$ follows:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{w}_{\mathit{ij}}+\Gamma z_i\le b_i\hfill \\ z_i+w_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}{u}_j,{\forall }_j\in J_i\hfill \\ -u_j\le x_j\ge u_j,{\forall }_j\in J_i\hfill \\ z_i\ge 0,w_{\mathit{ij}}\ge 0\hfill \end{array}\phantom{\}}$$ (4.11)

Remark 4.7 When ψ = 1 (i.e., the set U is defined as the “interval+polyhedral” uncertainty set), the corresponding robust counterpart optimization formulation becomes:

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \sum _{j\in J_i}{w}_{\mathit{ij}}+\Gamma z_i\le b_i\hfill \\ z_i+w_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}{u}_j,{\forall }_j\in J_i\hfill \\ -u_j\le x_j\ge u_j,{\forall }_j\in J_i\hfill \\ z_i\ge 0,w_{\mathit{ij}}\ge 0\hfill \end{array}\phantom{\}}$$ (4.12)

which is exactly the robust counterpart proposed by Bertsimas and Sim¹⁶.

Motivating Example 1 (Continued). The corresponding “interval+polyhedral” based robust counterpart for the first constraint of the motivating example 1 is:

$$\{\begin{array}{c}10x_1+20x_2+w_1+w_2+{\Gamma }_z\le 140\hfill \\ z+w_1\ge \mid x_1\mid ,z+w_2\ge 2\mid x_2\mid \hfill \\ z,w_1,w_2\ge 0\hfill \end{array}\phantom{\}}$$

Figure 4.1(e) illustrates the projection of the above constraints to the x₁,x₂ dimensions. dimensions. Comparing the robust counterpart constraint illustration Figure 4.1(c) and Figure 4.1(e), it can be observed that for Γ =1, the robust counterpart constraint is the same, whereas for Γ =3, the “interval+polyhedral” based robust counterpart is less conservative. This is consistent with the fact that as Γ ≤ 1, the intersection between interval and polyhedron is exactly the polyhedron, but as Γ > 1, the intersection between interval and polyhedron is smaller than the polyhedron itself.

Property 4.6 If the set U is the “interval+ellipsoidal+polyhedral” uncertainty set (3.8), then the corresponding robust counterpart constraint (4.1) is equivalent to the following constraints

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}\mid p_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{\mathit{ij}}^2}+\Gamma z_i\right]\le b_i\hfill \\ z_i\ge \mid {\widehat{a}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\mid {\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$ (4.13)

Proof. Consider the “interval+ellipsoidal+polyhedral” uncertainty set

$$U_{1\cap 2\cap \infty }=\left\{\xi \mid \sum _{j\in J_i}\mid {\xi }_j\mid \le \Gamma ,\sum _{j\in J_i}{\xi }_j^2\le {\Omega }^2,\mid {\xi }_j\mid \le 1,{\forall }_j\in J_i\phantom{\mid }\right\}.$$

It can be denoted using conic representation as follows,

$$U_{1\cap 2\cap \infty }=\{\xi \mid P_1\xi +p_1,\in K_1,P_2\xi +p_2\in K_2,P_{\infty }\xi +p_{\infty }\in K_{\infty }\}.$$

Defining the dual variable y¹ = [p_i, τ₁], y² = [w_i, τ₂], y³ = [v_i, τ₃] and using the dual cone $K_1^{\ast }=K_{\infty }$, $K_2^{\ast }=K_2$, $K_{\infty }^{\ast }=K_1$, the inner maximization problem can be written as

$$\underset{\xi }{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j:\xi \mid P_1\xi +p_1,\in K_1,P_2\xi +p_2\in K_2,P_{\infty }\xi +p_{\infty }\in K_{\infty }.\right\}$$

The conic dual of the above problem can be formulated as follows

$$\min \{{\tau }_1+\Omega {\tau }_2+\Gamma {\tau }_3:p_i+w_i+v_i={\widehat{a}}_{i}x,{\Vert p_i\Vert }_1\le {\tau }_1,{\Vert w_i\Vert }_2\le {\tau }_2,{\Vert v_i\Vert }_{\infty }\le {\tau }_3\}.$$

.

After equivalent transformation through replacing τ₁, τ₂, τ₃ with ∥p_i∥₁, ∥w_i∥₂, ∥v_i∥_∞ respectively, we get

$$\underset{p,w,v}{\min }\left\{\sum _{j\in J_i}\mid p_{ij}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{ij}^2}+\Gamma \underset{j\in J_i}{\max }\mid v_{ij}\mid :p_i+w_i+v_i={\widehat{a}}_{i}x\right\}.$$

Replacing $\underset{j\in J_i}{\text{max}}\mid v_{\mathit{ij}}\mid$ with auxiliary variable z_i, get

$$\underset{p,w,z}{\min }\left\{\sum _{j\in J_i}\mid p_{ij}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{ij}^2}+\Gamma z_i:z_i\ge \mid {\widehat{a}}_i{x}_{ij}-p_{ij}-w_{ij}\mid ,\forall j\in J_i\right\}.$$

Incorporate the above conic dual and removing the minimization operator, then the following robust counterpart is obtained

$$\{\begin{array}{c}\sum _j{a}_{ij}{x}_j+\left[\sum _{j\in J_i}\mid p_{ij}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{ij}^2}+\Gamma z_i\right]\le b_i\hfill \\ z_i\ge \mid {\widehat{a}}_{ij}{x}_j-p_{ij}-w_{ij}\mid \forall j\in J_i\hfill \end{array}\phantom{\}}.$$

Remark 4.8 An equivalent robust formulation for (4.13) can be obtained by replacing the term |p_ij| with auxiliary variable v_ij and constraint −v_ij ≤ p_ij ≤ v_ij, replacing $\mid {\widehat{a}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\mid$ with auxiliary variable u_ij and constraint $-u_{\mathit{ij}}\le {\widehat{a}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\le u_{\mathit{ij}}$ as follows:

$$\{\begin{array}{c}\sum _j{a}_{ij}{x}_j+\sum _{j\in J_i}{v}_{ij}+\Omega \sqrt{\sum _{j\in J_i}{w}_{ij}^2}+\Gamma z_i\le b_i\hfill \\ -v_{ij}\le p_{ij}\le v_{ij},\forall j\in J_i\hfill \\ -z_i\le {\widehat{a}}_{ij}{x}_j-p_{ij}-w_{ij}\le z_i,\forall j\in J_i\hfill \end{array}\phantom{\}}$$ (4.14)

Motivating Example 1 (Continued). The corresponding “interval+ellipsoidal+polyhedral” uncertainty set induced robust constraint for the first constraint of the motivating example 1 is:

$$\{\begin{array}{c}10x_1+20x_2+\mid p_1\mid +\mid p_2\mid +\Omega \sqrt{w_1^2+w_2^2}+\Gamma z\le 140\hfill \\ z\ge \mid x_1-p_1-w_1\mid ,z\ge \mid 2x_2-p_2-w_2\mid \hfill \end{array}\phantom{\}}$$

The above constraints are also plotted on the x₁, x₂ dimensions as shown in Figure 4.1(f). Comparing the robust counterpart constraint illustration Figure 4.1(d) and Figure 4.1(f), it can be seen that for both Ω =1 and Ω =2, the “interval+ellipsoidal” set induced robust counterpart is more conservative than the combined interval, ellipsoidal and polyhedral set induced model because the resulting optimization feasible set is in general larger. This is consistent with the fact that when we further incorporate the polyhedral set to construct the uncertainty set, the size of the resulting uncertainty set is actually decreased.

Finally, as a summary to the above derivation, we list the robust counterpart formulations for linear optimization problems with LHS uncertainty as shown in Table 4.1. Note that in Table 4.1, we list the “interval+ellipsoidal” based robust counterpart formulation but not “box+ellipsoidal” based model by realizing that the “interval+ellipsoidal” set is important for the bounded uncertainty distribution. Similarly, the “interval+polyhedral” and “interval+ellipsoidal+polyhedral” set induced robust counterpart optimization formulations are listed in the table.

Table 4.1.

Robust counterpart formulation for the i-th linear constraint with LHS uncertainty

Uncertainty set	Robust counterpart formulation
Box	${\sum }_j{a}_{ij}{x}_j+\Psi \left[{\sum }_{j\in J_i}{\widehat{a}}_{ij}\mid x_j\mid \right]\le b_i$
Ellipsoidal	${\sum }_j{a}_{ij}{x}_j+\left[\Omega \sqrt{{\sum }_{j\in J_i}{\widehat{a}}_{ij}^2{x}_j^2}\right]\le b_i$
Polyhedral	$\{\begin{array}{c}\hfill {\sum }_j{a}_{ij}{x}_j+z_i\Gamma \le b_i\hfill \\ \hfill z_i\ge {\widehat{a}}_{ij}\mid x_j\mid \forall j\in J_i\hfill \end{array}\phantom{\}}$
Interval+ Ellipsoidal	${\sum }_j{a}_{ij}{x}_j+{\sum }_{j\in J_i}{\widehat{a}}_{ij}\mid x_j-z_{ij}\mid +\Omega \sqrt{{\sum }_{j\in J_i}{\widehat{a}}_{ij}^2{z}_{ij}^2}\le b_i$
Interval+ Polyhedral	$\{\begin{array}{c}{\sum }_j{a}_{ij}{x}_j+\left[z_i\Gamma +{\sum }_{j\in J_i}{p}_{ij}\right]\le b_i\hfill \\ z_i+p_{ij}\ge {\widehat{a}}_{ij}\mid x_j\mid \forall j\in J_i\hfill \\ z_i\ge 0,p_{ij}\ge 0\hfill \end{array}\phantom{\}}$
Interval+Ellipsoidal+Polyhedral	$\{\begin{array}{c}{\sum }_j{a}_{ij}{x}_j+\left[z_i\Gamma +{\sum }_{j\in J_i}\mid p_{ij}\mid +\Omega \sqrt{{\sum }_{j\in J_i}{w}_{ij}^2}\right]\le b_i\hfill \\ z_i\ge \mid {\widehat{a}}_{ij}{x}_j-p_{ij}-w_{ij}\mid \forall j\in J_i\hfill \end{array}\phantom{\}}$

Remark 4.9 For the sake of simplicity, only robust formulations with absolute value terms are listed in Table 4.1, and equivalent robust formulations after eliminating the absolute value terms can be found via equations 4.3, 4.5, 4.6, 4.8 and 4.11. In the rest part of the paper, the absolute value term in the other robust counterpart formulations can be eliminated in a similar way.

4.2 Right Hand Side (RHS) uncertainty

Consider the case that only RHS uncertainty exists in the i-th constraint of (2.1) as follows

$$\sum _j{a}_{ij}{x}_j\le {\tilde{b}}_i$$ (4.15)

where ${\tilde{b}}_i=b_i+{\xi }_i{\widehat{b}}_i$ and ξ_i is the random variable. Then the robust counterpart for the i-th constraint (2.5) is reduced to

$$\sum _j{a}_{ij}{x}_j+\left[\underset{\xi \in U}{\max }\left\{-{\xi }_i\widehat{b}\right\}\right]\le b_i$$ (4.16)

Property 4.7 For RHS only uncertainty of the i-th constraint (4.15), the uncertainty set induced robust counterpart constraint (4.16) is equivalent to the following constraint

$$\sum _j{a}_{ij}{x}_j+\Delta {\widehat{b}}_i\le b_i$$ (4.17)

where Δ is defined as Ψ, Ω, Γ, min(Ω, 1), min(Ω, 1), min(Ω, Γ, 1) for the box, ellipsoidal, polyhedral, “interval+ellipsoidal”, “interval+polyhedral”, and “interval+polyhedral+ellipsoidal” uncertainty sets, respectively.

Proof. Since the dimension of the uncertain space for RHS only uncertainty is one (i.e., |J_i = 1), all the previously discussed different uncertainty sets are reduced to 1-dimenstional interval sets which can be described as

$$U=\{{\xi }_i\mid \mid {\xi }_i\mid \le \Delta \}$$ (4.18)

where Δ is defined as Ψ,Ω, Γ, min(Ω, 1), min(Γ, 1), min(Ω, Γ, 1) for the box, ellipsoidal, polyhedral, “interval+ellipsoidal”, “interval+polyhedral”, and “interval+polyhedral+ellipsoidal” uncertainty sets, respectively.

Incorporating auxiliary variables x₀ and a constraint x₀ = −1, the constraint (4.16) can be rewritten as

$$\sum _j{a}_{ij}{x}_j+\left[\underset{\xi \in U}{\max }\left\{{\xi }_i\widehat{b}{x}_0\right\}\right]\le b_i.$$

With the above reformulation and following the derivation process for box uncertainty set of LHS uncertainty, the corresponding robust counterpart formulation is obtained

$$\sum _j{a}_{ij}{x}_j+\Delta {\widehat{b}}_i\mid x_0\mid \le b_i.$$

Notice that x₀ = −1, so the above constraint is reduced to:

$$\sum _j{a}_{ij}{x}_j+\Delta {\widehat{b}}_i\le b_i$$

which is the robust counterpart for RHS only uncertainty for linear optimization problem.

Finally, the robust counterpart formulations for different uncertainty sets are summarized in Table 4.2.

Table 4.2.

Robust counterpart for the i-th linear constraint with RHS uncertainty

Uncertainty set	Robust counterpart formulation
Box	${\sum }_j{a}_{ij}{x}_j+\Psi {\widehat{b}}_i\le b_i$
Ellipsoidal	${\sum }_j{a}_{ij}{x}_j+\Omega {\widehat{b}}_i\le b_i$
Polyhedral	${\sum }_j{a}_{ij}{x}_j+\Gamma {\widehat{b}}_i\le b_i$
Interval+Ellipsoidal	${\sum }_j{a}_{ij}{x}_j+\min (\Omega ,1){\widehat{b}}_i\le b_i$
Interval+Polyhedral	${\sum }_j{a}_{ij}{x}_j+\min (\Gamma ,1){\widehat{b}}_i\le b_i$
Interval+Ellipsoidal+Polyhedral	${\sum }_j{a}_{ij}{x}_j+\min (\Omega ,\Gamma ,1){\widehat{b}}_i\le b_i$

From the above analysis, it is observed that for RHS only uncertainty of a linear constraint, there is no difference in defining different uncertainty sets since all of them reduce to a simple interval.

4.3 Simultaneous LHS and RHS uncertainty

Let us consider the more general case where uncertainty appears on both the LHS and the RHS of the i-th constraint:

$$\sum _{j\notin J_i}{a}_{ij}{x}_j+\sum _{j\in J_i}{\tilde{a}}_{ij}{x}_j\le {\tilde{b}}_i.$$ (4.19)

Similarly, through incorporating auxiliary variable x₀ and a constraint x₀ = −1, moving the RHS to the LHS, the above constraint can be rewritten as

$$\sum _{j\notin J_i}{a}_{ij}{x}_i+\sum _{j\in J_i}{\tilde{a}}_{ij}{x}_j+{\tilde{b}}_i{x}_0\le 0$$ (4.20)

Thus, the robust counterpart formulations for simultaneous LHS and RHS uncertainty can be derived using the same procedure as shown in section 4.1 and they are summarized in Table 4.3. For a detailed derivation procedure, the reader is directed to Appendix A.

Table 4.3.

Robust counterpart formulation for the i-th linear constraint with LHS and RHS uncertainty

Uncertainty set	Robust counterpart formulation
Box	${\sum }_j{a}_{ij}{x}_j+\Psi \left[{\sum }_{j\in J_i}{\widehat{a}}_{ij}\mid x_j\mid +{\widehat{b}}_i\right]\le b_i$
Ellipsoidal	${\sum }_j{a}_{ij}{x}_j+\left[\Omega \sqrt{{\sum }_{j\in J_i}{\widehat{a}}_{ij}^2{x}_j^2+{\widehat{b}}_j^2}\right]\le b_i$
Polyhedral	$\{\begin{array}{c}{\sum }_j{a}_{ij}{x}_j+z_i\Gamma \le b_i\hfill \\ z_i\ge {\widehat{a}}_{ij}\mid x_j\mid \forall j\in J_i,z_i\ge {\widehat{b}}_i\hfill \end{array}\phantom{\}}$
Interval+ Ellipsoidal	${\sum }_j{a}_{ij}{x}_j+\left[{\sum }_{j\in J_i}{\widehat{a}}_{ij}\mid x_j-z_{ij}\mid +{\widehat{b}}_i\mid 1+z_{i0}\mid +\Omega \sqrt{{\sum }_{j\in J_i}{\widehat{a}}_{ij}^2{z}_{ij}^2+{\widehat{b}}_i^2{z}_{i0}^2}\right]\le b_i$
Interval+Polyhedral	$\{\begin{array}{c}{\sum }_j{a}_{ij}{x}_j+\left[z_i\Gamma +{\sum }_{j\in J_i}{p}_{ij}+p_{i0}\right]\le b_i\hfill \\ z_i+p_{ij}\ge {\widehat{a}}_{ij}\mid x_j\mid \forall j\in J_i,z_i+p_{i0}\ge {\widehat{b}}_i\hfill \\ z_i\ge 0,p_{ij}\ge 0,p_{i0}\ge 0\hfill \end{array}\phantom{\}}$
Interval+Ellipsoidal+Polyhedral	$\{\begin{array}{c}{\sum }_j{a}_{ij}{x}_j+\left[z_i\Gamma +{\sum }_{j\in J_i}\mid p_{ij}\mid +\mid p_{i0}\mid +\Omega \sqrt{{\sum }_{j\in J_i}{w}_{ij}^2+w_{i0}^2}\right]\le b_i\hfill \\ z_i\ge \mid {\widehat{a}}_{ij}{x}_j-p_{ij}-w_{ij}\mid \forall j\in J_i,z_i\ge \mid {\widehat{b}}_i+p_{i0}+w_{i0}\mid \hfill \end{array}\phantom{\}}$

Remark 4.10 The objective max cx can be equivalently transformed as follows:

$$\begin{array}{cc}\hfill \max & z\hfill \\ \hfill s.t.& z-cx\le 0\hfill \end{array}$$ (4.21)

Thus, the uncertainty in the objective coefficient $\tilde{c}$ can be treated as uncertainty in the following type of constraints

$$z-\tilde{c}x\le 0$$ (4.22)

Hence, the complete robust counterpart formulations for uncertainty in LHS, RHS and objective function are obtained.

5. Computational Studies for Robust Linear Optimization

Example 5.1 Consider the following linear optimization problem

$$\begin{array}{cc}\hfill \max & c_1{x}_1+c_2{x}_2\hfill \\ \hfill s.t.& a_{11}{x}_1+a_{12}{x}_2\le b_1\hfill \\ \hfill & a_{21}{x}_1+a_{22}{x}_2\le b_2\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

where [c₁ c₂] = [8 12] $\left[\begin{array}{cc}\hfill a_{11}\hfill & \hfill a_{12}\hfill \\ \hfill a_{21}\hfill & \hfill a_{22}\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 10\hfill & \hfill 20\hfill \\ \hfill 6\hfill & \hfill 8\hfill \end{array}\right]$, $\left[\begin{array}{c}\hfill b_1\hfill \\ \hfill b_2\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 140\hfill \\ \hfill 72\hfill \end{array}\right]$. The uncertain version of the above LP problem can be described as the following problem:

$$\begin{array}{cc}\hfill \max & {\tilde{c}}_1{x}_1+{\tilde{c}}_2{x}_2\hfill \\ \hfill s.t.& {\tilde{a}}_{11}{x}_1+{\tilde{a}}_{12}{x}_2\le {\tilde{b}}_1\hfill \\ \hfill & {\tilde{a}}_{21}{x}_1+{\tilde{a}}_{22}{x}_2\le {\tilde{b}}_2\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

where the possible uncertainty is related to the left hand side (LHS) constraint coefficients ${\tilde{a}}_{11}$, ${\tilde{a}}_{12}$, ${\tilde{a}}_{21}$, ${\tilde{a}}_{22}$, the right hand side (RHS) parameter ${\tilde{b}}_1$, ${\tilde{b}}_2$ and the objective (OBJ) coefficients ${\tilde{c}}_1$, ${\tilde{c}}_2$. Here we define the uncertainty as follows:

$${\tilde{c}}_j=c_j+{\widehat{c}}_j{\xi }_{i0},j=1,2;$$

$${\tilde{a}}_{ij}=1_{ij}+{\widehat{a}}_{ij}{\xi }_{ij},i=1,2,j=1,2;$$

$${\tilde{b}}_i=b_i+{\widehat{b}}_i{\xi }_i,i=1,2;$$

where ${\widehat{a}}_{\mathit{ij}}=0.1a_{\mathit{ij}}$, ${\widehat{b}}_i=0.1b_i$, ${\widehat{c}}_j=0.1c_j$ represent constant perturbation around their nominal values ξ₁₀,ξ₂₀,ξ₁₁,ξ₁₂,ξ₂₁,ξ₂₂,ξ₁ξ₂ are independent random variables.

When we only consider the LHS uncertainty, the different uncertainty set induced robust counterparts can be formulated as shown in section 4.1. For example, the ellipsoidal uncertainty set based robust counterpart is:

$$\begin{array}{cc}\hfill \max & 8x_1+12x_2\hfill \\ \hfill s.t.& 10x_1+20x_2+\Omega \sqrt{x_1^2+4x_2^2}\le 140\hfill \\ \hfill & 6x_1+8x_2+\Omega \sqrt{0.36x_1^2+0.64x_2^2}\le 72\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

Note that the same uncertainty set parameter value Ω is applied for both constraints here. In the sequel, this will be similarly applied for the rest cases without further explanation. The solution of the different uncertainty set induced robust counterparts is shown in Figure 5.1(a). Figure 5.1(b) illustrates the relationship between the “interval+ellipsoidal”, “interval+polyhedral”, “interval+ellipsoidal+polyhedral” models (based on LHS+RHS uncertainty for both constraints).

Figure 5.1.

Only LHS uncertainty for both constraints (|J_i| = 2)

Considering only the RHS uncertainty, the different uncertainty set induced robust counterparts can be formulated as shown in section 4.2. For example, the ellipsoidal uncertainty set based robust counterpart is as follows and the solution of the different robust counterparts are shown in Figure 5.2.

$$\begin{array}{cc}\hfill \max & 8x_1+12x_2\hfill \\ \hfill s.t.& 10x_1+20x_2+14\Omega \le 140\hfill \\ \hfill & 6x_1+8x_2+7.2\Omega \le 72\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

Figure 5.2.

Only RHS uncertainty for both constraints (|J_i| = 1)

Considering LHS and RHS uncertainty simultaneously, the ellipsoidal uncertainty set based robust counterpart is as follows and the solution is shown in Figure 5.3.

$$\begin{array}{cc}\hfill \max & 8x_1+12x_2\hfill \\ \hfill s.t.& 10x_1+20x_2+\Omega \sqrt{x_1^2+4x_2^2+196}\le 140\hfill \\ \hfill & 6x_1+8x_2+\Omega \sqrt{0.36x_1^2+0.64x_2^2+51.84}\le 72\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

Figure 5.3.

Simultaneous LHS and RHS uncertainty for both constraints (|J_i| = 3)

Considering LHS, RHS and OBJ uncertainty simultaneously, we first equivalently transform the objective uncertainty into constraint uncertainty as (4.22) and then the different uncertainty set induced robust counterparts can be derived based on the simultaneous LHS and RHS uncertainty. For example, the ellipsoidal uncertainty set based robust counterpart is as follows and the solution for simultaneous LHS, RHS and OBJ uncertainty is shown in Figure 5.4.

$$\begin{array}{cc}\hfill \max & z\hfill \\ \hfill s.t.& z-8x_1-12x_2+\Omega \sqrt{0.64x_1^2+1.44x_2^2}\le 0\hfill \\ \hfill & 10x_1+20x_2+\Omega \sqrt{x_1^2+4x_2^2+196}\le 140\hfill \\ \hfill & 6x_1+8x_2+\Omega \sqrt{0.36x_1^2+0.64x_2^2+51.84}\le 72\hfill \\ \hfill & x_1,x_2\ge 0\hfill \end{array}$$

Figure 5.4.

Simultaneous LHS, RHS and OBJ uncertainty (|J_i| = 3)

Based on the solution of the different cases of uncertainties, the following remarks can be made:

• (1)It can be observed from Figures 5.1(a), 5.3(a) and 5.4(a) that for the ellipsoidal set based robust counterpart, when Ω ≤ 1, the ellipsoidal and “interval+ellipsoidal” has the same solution because the corresponding uncertainty sets are the same; when $\Omega \le \sqrt{\mid J_i\mid }$, the “interval+ellipsoidal” solution reaches the worst case and does not decrease anymore because the “interval+ellipsoidal” uncertainty set is exactly the interval and does not change. For the polyhedral set induced robust counterpart, when Γ ≤ 1, the polyhedral and “interval +polyhedral” set induced models have the same solution; when Γ ≥ |J_i|, the “interval+polyhedral” solution reaches the worst case and does not decrease anymore. It can be concluded from those results that for bounded uncertainty, the uncertainty set should be combined with interval to avoid conservative solutions.
• (2)Comparing the “interval+ellipsoidal” and the “interval+polyhedral” set based model from Figures 5.1(a), 5.3(a) and 5.4(a), when $\Gamma =\Omega \sqrt{\mid J_i\mid }$, the “interval+polyhedral” set based solution is always worse than the “interval+ellipsoidal” based solution, which is verified by the fact that the “interval+polyhedral” uncertainty set is larger and completely covers the “interval+ellipsoidal” set; when Γ = Ω, the “interval+polyhedral” set based solution is always better than the “interval+ellipsoidal” based solution because the “interval+polyhedral” uncertainty set is smaller and completely covered by the “interval+ellipsoidal” set.
• (3)Comparing the “interval+ellipsoidal+polyhedral” set based model with others from Figures 5.1(b), 5.3(b) and 5.4(b), for every Ω value, we adjust the value of Γ between Ω and $\Omega \sqrt{\mid J_i\mid }$ and test three different values of Γ (as explained in section 3, only when $\Omega \le \Gamma \le \Omega \sqrt{\mid J_i\mid }$, the intersection between the ellipsoidal and polyhedral set does not reduce to any one of them). It can be observed that as the value of Γ increases from Ω to $\Omega \sqrt{\mid J_i\mid }$, the “interval+ellipsoidal+polyhedral” set based solution switches from the “interval+polyhedral” set based solution with Γ = Ω to the “interval+ellipsoidal” based solution with $\Gamma =\Omega \sqrt{\mid J_i\mid }$, because the intersection between the ellipsoid and polyhedron is exactly changing from the polyhedron with Γ = Ω to the ellipsoid with parameter $\Gamma =\Omega \sqrt{\mid J_i\mid }$.
• (4)For RHS only uncertainty, which is a special case where the number of uncertain parameters for every constraint is 1, the solution is identical for ellipsoidal and polyhedral set induced models, and also for the “interval+ellipsoidal” and “interval+polyhedral” uncertainty sets as shown in Figure 5.2. Furthermore, as Ω ≤ 1 and Γ ≤ 1, all the solutions are identical. This is consistent with the definition of the corresponding uncertainty set: as Ω = Γ ≤ 1, the four types of uncertainty sets are actually the same interval set.

Figure 5.5.

Refinery flowchart

Example 5.2 Refinery production planning problem

Petroleum refinery production planning involves several types of uncertainty, such as prices and product demands. The refinery topology shown in Figure 5.5 and the operational planning model originally proposed by Alen³⁵ are used. Leiras et al.,³⁶ illustrated the application of robust optimization framework which is based on the “interval+polyhedral” uncertainty set induced robust optimization methodology proposed by Bertsimas and Sim¹⁶.

In this example, the refinery includes three units: primary distillation unit (PDU), cracking and blending. It processes crude oil (x₁) to produce gasoline (x₂), naphtha (x₃), jet fuel (x₄), heating oil (x₅), and fuel oil (x₆), where x₇ ~ x₂₀ are intermediary streams. The objective function maximizes the profit, which considers the crude oil cost and operating cost of the distillation and cracker units. Constraints include the production yield, fixed proportion blending, production balances and production requirements. The deterministic model and the definitions of variables and parameters are shown below.

$$\max \sum _{t\in T}\sum _{j\in J^{\mathrm{prod}}}{p}_{jt}{x}_{jt}-\sum _{t\in T}\sum _{j\in J^{\mathrm{feed}}}{c}_{jt}{x}_{jt}$$ (5.1a)

$$s.t.x_{jt}\le {\mathit{cap}}_{jt}\forall j\in J^{\mathrm{feed}},t\in T$$ (5.1b)

$$x_{jt}\le {\Sigma }_i{\eta }_{ij}{x}_{it}\forall i\in I,t\in T$$ (5.1c)

$$x_{it}\le {\Sigma }_j{\sigma }_{ij}{x}_{jt}\forall i\in I,t\in T$$ (5.1d)

$$x_{it}\le {\Sigma }_j{\alpha }_{ij}{x}_{jt}\forall i\in I,t\in T$$ (5.1e)

$$x_{jt}\le {\mathit{prod}}_{jt}\forall i\in I,t\in T$$ (5.1f)

$$x_{it}\ge 0\forall i\in I,t\in T$$ (5.1g)

where (5.1a) represents the profit objective, (5.1b) is plant capacity constraint, (5.1c) is production yield constraint, (5.1d) is fixed proportion blending constraints, (5.1e) is production balance constraint, and (5.1f) is the production demand constraint.

The uncertain parameters we focus on are the cost c_jt, the prices of products p_jt, the yields η_ij and the demands prod_jt. We assume that those parameters are subject to bounded uncertainty and that there exists a maximum of 10% deviation of cost and price coefficients, 5% of demand coefficient and 1% of yield coefficient from their nominal values. It is also assumed that only the yields of products from the distillation unit are controlled. The cost c_jt and the prices of products p_jt appear in the same constraint and they are considered simultaneously. We applied the six different robust counterpart formulations for the three kinds of uncertainty separately, and all of them together subsequently. The solution of the nominal deterministic model is US$23,387.50/day. Considering the different types of uncertainty separately, the worst cases scenario results computed from box set induced robust counterpart optimization model with Ψ = 1 are listed in Table 5.1.

Table 5.1.

Objective function values for the worst-case scenario case

Uncertain parameter	Yield	Demand	Cost and price	Cost, price, yield and demand
Objective value	22665.00	23134.97	7113.92	6569.14

From the above results, it can be observed that the cost and price uncertainty has the most significant effect on the overall profit since the objective value is much less than the objective value of pure yield uncertainty or pure demand uncertainty. In the sequel, we first analyze the different types of uncertainty separately, and then consider them simultaneously.

• (1)Yield uncertainty. The set of yield constraints (5.1c) contain the uncertain parameters. In each constraint, the number of the uncertain parameters (i.e., |J_i|) is 1. It belongs to the LHS case uncertainty. Applying the six different kinds of robust optimization formulations, the results are shown in Figure 5.6. It can be observed that the results of different formulations are the same as the adjustable parameter is less than 1 because there is only 1 uncertain parameter in each individual constraint. When Ψ, Ω, Γ = 0, the solutions are the same as in the deterministic model. When Ψ = 1, $\Omega =\sqrt{\mid J_i\mid }$ and Γ = |J_i|, the results reach the worst case. When $\Gamma \rangle \sqrt{\mid J_i\mid }$ and Γ > |J_i|, the results of “interval+ellipsoidal”, “interval+polyhedral” and “interval+ellipsoidal+polyhedral” set induced models do not decrease anymore.
• (2)Cost and price uncertainty Since the uncertain parameters appear in the objective, we convert it into a constraint. The resulting problem has only LHS uncertainty and the number of the uncertain parameters (i.e., J_i) is 7. The results are shown in Figure 5.7. When Ψ,Ω,Γ = 0, the solutions are the same as in the deterministic model. When Ψ,Ω,Γ increase, the results of the box, ellipsoidal and polyhedral uncertainty set induced models will decrease or even become infeasible. If the uncertainty set is combined with an “interval” set, the solution will finally reach the worst case value and will not decrease anymore. In this study, the following parameters are applied, for “interval+ellipsoidal” model, $\Omega =\sqrt{\mid J_i\mid }=\sqrt{7}$; for “interval+polyhedral” set induced model, Γ = |J_i| = 7; for “interval+ellipsoidal+polyhedral” set induced model, we take $\Omega =\Gamma ∕\sqrt{\mid J_i\mid }$.
• (3)Demand uncertainty This belongs to the RHS uncertainty case and there is only one uncertain parameter in each constraint. The results are shown on Figure 5.8. From this figure, we can see that the results of different formulations are the same because there is only 1 uncertain parameter in each individual constraint. When Ψ,Ω,Ψ = 0, the solutions are the same as in the deterministic model. When Ψ,Ω,Γ = 1, the results reach the worst case solution. When $\Omega \rangle \sqrt{\mid J_i\mid }=1$, and Γ > |J_i| = 1, the results of the “interval+ellipsoidal”, “interval+polyhedral” and interval+ellipsoidal+polyhedral” induced models do not decrease anymore.
• (4)
  Simultaneous yield, price, cost, and demand uncertainty Here we consider all uncertainties together. The x axis is Γ_price and we set ${\Gamma }_{\mathit{yield}}={\Gamma }_{\mathit{demand}}=\frac{1}{7}{\Gamma }_{\mathit{price}}$ to plot the result using the same axis. The parameters are as follows:

  $$J_{\mathit{yield}}=1,J_{\mathit{price}}=7,J_{\mathit{demand}}=1$$

  $${\Psi }_{\mathit{yield}}={\Gamma }_{\mathit{yield}}∕\mid J_{\mathit{yield}}\mid ,{\Psi }_{\mathit{price}}={\Gamma }_{\mathit{price}}∕\mid J_{\mathit{price}}\mid ,{\Psi }_{\mathit{demand}}={\Gamma }_{\mathit{demand}}∕\mid J_{\mathit{demand}}\mid ,$$

  $${\Omega }_{\mathit{yield}}={\Gamma }_{\mathit{yield}}∕\sqrt{\mid J_{\mathit{yield}}\mid },{\Omega }_{\mathit{price}}={\Gamma }_{\mathit{price}}∕\sqrt{\mid J_{\mathit{price}}\mid },{\Omega }_{\mathit{demand}}={\Gamma }_{\mathit{demand}}∕\sqrt{\mid J_{\mathit{demand}}\mid }$$

  The results are shown as Figure 5.9. From this figure we can observe that when all the Ψ,Ω,Γ for yield, price, cost and demand are 0, the results are equal to those of the deterministic model. When Γ_price = 7, Γ_yield = Γ_demand = 1, and $\Omega =\sqrt{\mid J_i\mid }$, the results of the “interval+ellipsoidal”, “interval+polyhedral”, and “interval+ellipsoidal+polyhedral” set induced models reach the worst case and do not decrease anymore. At the same point, the “box” reaches the worst case also.

Figure 5.6.

Solution for yield uncertainty

Figure 5.7.

Solution for price and cost uncertainty

Figure 5.8.

Solution for demand uncertainty

Figure 5.9.

Solution for simultaneous yield, price, cost and demand uncertainty

Finally, from the above analysis, it can be concluded that for bounded uncertainty in the yield, demand and price/cost data, the uncertainty set should be combined with the interval set so as to avoid too conservative or even infeasible solutions. On the other hand, all the different models have the flexibility to adjust the solution between the worst-case scenario and the deterministic solution, depending on the selection of the adjustable parameters for their corresponding uncertainty set. To perform a more rigorous comparison of the different models' conservatism, the evaluation of the probabilistic guarantees of constraint violation is necessary, and this will be the subject of a forthcoming publication.

6. Robust Counterpart Formulations for Mixed Integer Linear Optimization Problems

In this section, different uncertainty set induced robust counterpart formulations are derived for a general mixed integer linear constraint. We first present the results for simultaneous constraint LHS and RHS uncertainty, and then extend the results to the case of objective function coefficients' uncertainty.

6.1 Uncertainty in LHS and RHS

For problem (2.7), introducing auxiliary variable x₀ and an additional constraint x₀ = −1, the original i-th constraint's robust counterpart (2.11) can be rewritten as

$$p_i{x}_0+\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\underset{\xi \in U}{\max }\left\{{\xi }_{i0}{\widehat{p}}_i{x}_0+\sum _{m\in M_i}{\xi }_{im}{\widehat{a}}_{im}{x}_m+\sum _{k\in K_i}{\xi }_{ik}{\widehat{b}}_{ik}{y}_k\right\}\le 0$$ (6.1)

With the following definition

$${\xi }_i=[{\xi }_{i0};\left\{{\xi }_{im}\right\};\left\{{\xi }_{ik}\right\}]$$ (6.2a)

$$A_i=[p_i,\left\{a_{im}\right\},\left\{b_{ik}\right\}]$$ (6.2b)

$${\widehat{A}}_i=[{\widehat{p}}_i,\left\{{\widehat{a}}_{im}\right\},\left\{{\widehat{b}}_{ik}\right\}]$$ (6.2c)

$$X=[x_0;\left\{x_m\right\};\left\{y_k\right\}]$$ (6.2d)

$$j\in J_i=\left\{0\right\}>M_i>K_i$$ (6.2e)

the robust counterpart (6.1) can be rewritten as:

$$\sum _j{A}_{ij}{X}_j+\underset{{\xi }_i\in U}{\max }\left\{\sum _{j\in J_i}{\xi }_{ij}{\widehat{A}}_{ij}{X}_j\right\}\le 0.$$ (6.3)

In order to eliminate the inner maximization problem in (6.3), we first transform the inner maximization problem into its conic dual, and then incorporate the dual problem into the original constraint. In the sequel, the robust counterpart formulations for the i-th mixed integer linear constraint in (2.7) with simultaneous LHS and RHS uncertainty will be directly given. Detailed proofs of all properties can be found in the Appendix B.

Property 6.1 If the set U is the box uncertainty set (3.1), then the corresponding robust counterpart constraint (6.3) becomes:

$$\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\Psi \left[\sum _{m\in M_i}{\widehat{a}}_{im}\mid x_m\mid +\sum _{k\in K_i}{\widehat{b}}_{ik}\mid y_k\mid +{\widehat{p}}_i\right]\le p_i$$ (6.4)

Proof. (see Appendix B).

Remark 6.1 Notice that the absolute value operators in constraint (6.4) can be directly removed while the corresponding variable is positive. The robust formulation can be further equivalently transformed to the following constraints:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\Psi \left[\sum _{m\in M_i}{\widehat{a}}_{im}{u}_m+\sum _{k\in K_i}{\widehat{b}}_{ik}{v}_k+{\widehat{p}}_i\right]\le p_i\hfill \\ \mid x_m\mid \le u_m\forall m\in M_i\hfill \\ \mid y_k\mid \le v_k\forall k\in K_i\hfill \end{array}\phantom{\}}$$ (6.5)

The above constraint set can be further rewritten as the following form:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\Psi \left[\sum _{m\in M_i}{\widehat{a}}_{im}{u}_m+\sum _{k\in K_i}{\widehat{b}}_{ik}{v}_k+{\widehat{p}}_i\right]\le p_i\hfill \\ -u_m\le x_m\le u_m\forall m\in M_i\hfill \\ -v_k\le y_k\le v_k\forall k\in K_i\hfill \end{array}\phantom{\}}$$ (6.6)

Motivating Example 2 (Continued). The robust counterpart for the original third constraint is as follows:

$$x_i-20y_1+\Psi (0.1x_1+2y_1)\le 0.$$

The final complete robust counterpart optimization model is:

$$\begin{array}{cc}\hfill \max & 3x_1+2x_2-10y_1-5y_2\hfill \\ \hfill s.t.& x_1+x_2\le 20\hfill \\ \hfill & x_1+2x_2\le 12\hfill \\ \hfill & x_1-20y_1+{\Psi }_1(0.1x_1+2y_1)\le 0\hfill \\ \hfill & x_2-20y_2+{\Psi }_2(0.1x_2+2y_2)\le 0\hfill \\ \hfill & x_1-x_2\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$

In the above formulation, different parameters Ψ₁, Ψ₂ are assigned to the two constraints. Note that the absolute value operator has been eliminated since the variables are all positive.

Property 6.2 If the set U is the ellipsoidal uncertainty set (3.2), then the corresponding robust counterpart constraint (6.3) is becomes:

$$\sum _m{a}_{im}{x}_m+\sum _m{b}_{ik}{y}_k+\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{im}^2{x}_m^2+\sum _{k\in K_i}{\widehat{b}}_{ik}^2{y}_k^2+{\widehat{p}}_i^2}\le p_i$$ (6.7)

Proof. (see Appendix B).

Motivating Example 2 (Continued). The robust counterpart constraint for the third constraint of motivating example 2 is

$$x_1-20y_1+{\Omega }_1\sqrt{0.01x_1^2+4y_1^2}\le 0.$$

Property 6.3 If the set U is defined as the polyhedral uncertainty set (3.3), then the corresponding robust counterpart constraint (6.3) becomes:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+z_i\Gamma \le p_i\hfill \\ z_i\ge {\widehat{a}}_{im}\mid x_m\mid \forall m\in M_i\hfill \\ z_i\ge {\widehat{b}}_{ik}\mid y_k\mid \forall k\in K_i\hfill \\ z_i\ge {\widehat{p}}_i\hfill \end{array}\phantom{\}}$$ (6.8)

Proof. (see Appendix B).

Remark 6.2 Similarly, as in Remark 6.1, the above robust formulation can be further transformed into the following equivalent constraint set after eliminating the absolute value operators:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+z_i\Gamma \le p_i\hfill \\ z_i\ge {\widehat{a}}_{im}{u}_m\forall m\in M_i\hfill \\ z_i\ge {\widehat{b}}_{ik}{v}_k\forall k\in K_i\hfill \\ z_i\ge {\widehat{p}}_i\hfill \\ -u_m\le x_m\le u_m\forall m\in M_i\hfill \\ -v_k\le y_k\le v_k\forall k\in K_i\hfill \end{array}\phantom{\}}$$ (6.9)

Motivating Example 2 (Continued). The corresponding robust formulation for the third constraint of the motivating example is:

$$x_1-20y_1+z_1{\Gamma }_1\le 0$$

$$z_1\ge 0.1x_1,z_1\ge 2y_1$$

Property 6.4 If the set U is the “interval+ellipsoidal” uncertainty set (3.4) with Ψ = 1, then the corresponding robust counterpart constraint (6.3) becomes:

$$\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\sum _{m\in M_i}{\widehat{a}}_{im}\mid x_m-z_{im}\mid +\sum _{m\in K_i}{\widehat{b}}_{ik}\mid y_k-z_{ik}\mid +{\widehat{p}}_i\mid 1+z_{i0}\mid +\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{im}^2{z}_{im}^2+\sum _{k\in K_i}{\widehat{b}}_{ik}^2{z}_{ik}^2+{\widehat{p}}_i^2{z}_{i0}^2}\le p_i$$ (6.10)

Proof. (see Appendix B).

Remark 6.3 Constraint (6.10) can be rewritten as

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\sum _{m\in M_i}{\widehat{a}}_{im}{u}_{im}+\sum _{m\in K_i}{\widehat{b}}_{ik}{u}_{ik}+{\widehat{p}}_i{u}_{i0}+\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{im}^2{z}_{im}^2+\sum _{k\in K_i}{\widehat{b}}_{ik}^2{z}_{ik}^2+{\widehat{p}}_i^2{z}_{i0}^2}\le p_i\hfill \\ u_{im}=\mid x_m-z_{im}\mid \forall m\in M_i\hfill \\ u_{ik}=\mid y_k-z_{ik}\mid \forall k\in K_i\hfill \\ u_{i0}=\mid 1+z_{i0}\mid \hfill \end{array}\phantom{\}}$$

which can be further equivalently transformed to the following constraint sets as shown in Remark 6.1:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\sum _{m\in M_i}{\widehat{a}}_{im}{u}_{im}+\sum _{m\in K_i}{\widehat{b}}_{ik}{u}_{ik}+{\widehat{p}}_i{u}_{i0}+\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{im}^2{z}_{im}^2+\sum _{k\in K_i}{\widehat{b}}_{ik}^2{z}_{ik}^2+{\widehat{p}}_i^2{z}_{i0}^2}\le p_i\hfill \\ -u_{im}\le x_m-z_{im}\le u_{im}\forall m\in M_i\hfill \\ -u_{ik}\le y_k-z_{ik}\le u_{ik}\forall k\in K_i\hfill \\ -u_{i0}\le 1+z_{i0}\le u_{i0}\hfill \end{array}\phantom{\}}$$ (6.11)

Motivating Example 2 (Continued). The robust counterpart formulation for the third constraint is

$$x_1-20y_1+{\Omega }_1\sqrt{0.01z_{31}^2+4z_{33}^2}\le 0$$

$$-y_{31}\le x_1-z_{31}\le u_{31}$$

$$-u_{33}\le y_1-z_{33}\le u_{33}$$

Property 6.5 If the set U is defined as the “interval+polyhedral” uncertainty set (3.6) with Ψ = 1, then the corresponding robust counterpart constraint (6.3) is equivalent to the following constraint sets:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\left[z_i{\Gamma }_i+\sum _{m\in M_i}{w}_{im}+\sum _{k\in K_i}{w}_{ik}+w_{i0}\right]\le p_i\hfill \\ z_i+w_{im}\ge {\widehat{a}}_{im}\mid x_m\mid \forall m\in M_i\hfill \\ z_i+w_{ik}\ge {\widehat{b}}_{ik}\mid y_k\mid \forall k\in K_i\hfill \\ z_i+w_{i0}\ge {\widehat{p}}_i\hfill \end{array}\phantom{\}}$$ (6.12)

Proof. (see Appendix B).

Remark 6.4 While the variables are positive, the absolute value operator can be directly removed. Otherwise, the robust formulation (6.12) can be rewritten as follows as shown in Remark 6.1:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\left[z_i{\Gamma }_i+\sum _{m\in M_i}{w}_{im}+\sum _{k\in K_i}{w}_{ik}+w_{i0}\right]\le p_i\hfill \\ z_i+w_{im}\ge {\widehat{a}}_{im}{u}_m\forall m\in M_i\hfill \\ z_i+w_{ik}\ge {\widehat{b}}_{ik}{v}_k\forall k\in K_i\hfill \\ z_i+w_{i0}\ge {\widehat{p}}_i\hfill \\ -u_m\le x_m\le u_m\forall m\in M_i\hfill \\ -v_k\le y_k\le v_k\forall k\in K_i\hfill \end{array}\phantom{\}}$$ (6.13)

Motivating Example 2 (Continued). Since all variables are positive, the robust counterpart for the third constraint becomes:

$$x_1-20y_1+z_1{\Gamma }_1+w_{31}+w_{33}\le 0$$

$$z_1+w_{31}\ge 0.1x_1,z_1+w_{33}\ge 2y_1$$

Property 6.6 If the set U is the “interval+ellipsoidal+polyhedral” uncertainty set (3.8) with Ψ = 1, then the corresponding robust counterpart constraint (6.3) becomes:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+z_i\Gamma +\sum _{m\in M_i}\mid q_{im}\mid +\sum _{k\in K_i}\mid q_{ik}\mid +\mid q_{i0}\mid +\Omega \sqrt{\sum _{m\in M_i}{w}_{im}^2+\sum _{k\in K_i}{w}_{ik}^2+w_{i0}^2}\le p_i\hfill \\ z_i\ge \mid {\widehat{a}}_{im}{x}_m-q_{im}-w_{im}\mid \forall m\in M_i\hfill \\ z_i\ge \mid {\widehat{b}}_{ik}{y}_k-q_{ik}-w_{ik}\mid \forall k\in K_i\hfill \\ z_i\ge \mid {\widehat{p}}_i+q_{i0}+w_{i0}\mid \hfill \end{array}\phantom{\}}$$ (6.14)

Proof. (see Appendix B).

Remark 6.5 As in Remark 6.1, the robust counterpart can be equivalently rewritten as follows by introducing auxiliary variables and eliminating the absolute value operators:

$$\{\begin{array}{c}\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+z_i\Gamma +\sum _{m\in M_i}{u}_{im}+\sum _{k\in K_i}{u}_{ik}+u_{i0}+\Omega \sqrt{\sum _{m\in M_i}{w}_{im}^2+\sum _{k\in K_i}{w}_{ik}^2+w_{i0}^2}\le p_i\hfill \\ -u_{im}\le q_{im}\le u_{im}\forall m\in M_i\hfill \\ -u_{ik}\le q_{ik}\le u_{ik}\forall k\in K_i\hfill \\ -u_{i0}\le q_{i0}\le u_{i0}\hfill \\ -z_i\le {\widehat{a}}_{im}{x}_m-q_{im}-w_{im}\le z_i\forall m\in M_i\hfill \\ -z_i\le {\widehat{b}}_{ik}{y}_k-q_{ik}-w_{ik}\le z_i\forall k\in K_i\hfill \\ -z_i\le {\widehat{p}}_i+q_{i0}+w_{i0}\le z_i\hfill \end{array}\phantom{\}}$$ (6.15)

Motivating Example 2 (Continued). The robust counterpart formulation for the third constraint is

$$x_1-20y_1+z_1{\Gamma }_1+u_{31}+u_{33}+{\Omega }_1\sqrt{w_{31}^2+w_{33}^2}\le 0$$

$$-u_{31}\le q_{31}\le u_{31},-u_{33}\le q_{33}\le u_{33}$$

$$-z_1\le 0.1x_1-q_{31}-w_{31}\le z_1,-z_1\le 2y_1-q_{33}-w_{33}\le z_1$$

The different uncertainty set induced robust counterpart formulations are summarized in Table 3.1. Finally, we point out that for the case of LHS only or RHS only uncertainty, the corresponding robust counterpart optimization formulations can be derived based on the above results of simultaneous LHS and RHS uncertainty.

For example, for LHS only uncertainty, we have ${\widehat{p}}_i=0$, then the box set induced robust counterpart (6.4) is reduced to

$$\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\Psi \left[\sum _{m\in M_i}{\widehat{a}}_{im}\mid x_m\mid +\sum _{k\in K_i}{\widehat{b}}_{ik}\mid y_k\mid \right]\le p_i$$ (6.16)

Similarly, for RHS only uncertainty, ${\widehat{a}}_{\mathit{im}}=0$, ${\widehat{b}}_{\mathit{ik}}=0$, then the box set induced robust counterpart (6.4) is reduced to

$$\sum _m{a}_{im}{x}_m+\sum _k{b}_{ik}{y}_k+\Psi {\widehat{p}}_i\le p_i$$ (6.17)

6.2 Objective Function Coefficients' Uncertainty

Considering the objective coefficients uncertainty in the mixed integer linear optimization problem (2.7):

$$\max \sum _m{\tilde{c}}_m{x}_m+\sum _k{\tilde{d}}_k{y}_k$$ (6.18)

To derive the corresponding robust counterpart formulation, the objective uncertainty is equivalently transformed into constraint LHS uncertainty as follows

$$\begin{array}{cc}\hfill & \max z\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.z-\sum _m{\tilde{c}}_m{x}_m+\sum _k{\tilde{b}}_k{y}_k\le 0\hfill \end{array}$$ (6.19)

Then the robust counterpart formulation can be applied on the resulting constraints which contain LHS only uncertainty.

Motivating Example 2 (continued). To derive the robust counterpart for the objective function coefficients' uncertainty in motivating example 2, the original objective function is transformed into the following constraint first:

$$z-(3x_1+2x_2-10y_1-5y_2)\le 0$$

Then, the set induced robust counterpart constraint for the resulting new constraint can be formulated. For example, the box set induced robust formulation is:

$$z-(3x_1+2x_2-10y_1-5y_2)+{\Psi }_0(0.3x_1+0.2x_2+y_1+0.5y_2)\le 0$$

The ellipsoidal set induced robust counterpart constraint is:

$$z-(3x_1+2x_2-10y_1-5y_2)+{\Omega }_0\sqrt{0.09x_1^2+0.04x_2^2+y_1^2+0.25y_2^2}\le 0$$

The polyhedral set induced robust counterpart constraint is:

$$\{\begin{array}{c}z-(3x_1+2x_2-10y_1-5y_2)+v_0{\Gamma }_0\le 0\hfill \\ v_0\ge 0.3x_1,v_0\ge 0.2x_2,v_0\ge y_1,v_0\ge 0.5y_2\hfill \end{array}\phantom{\}}$$

The “interval+ellipsoidal” set induced robust counterpart constraint is:

$$\{\begin{array}{c}z-(3x_1+2x_2-10y_1-5y_2)+(0.3u_{01}+0.2u_{02}+u_{03}+0.5u_{04})+{\Omega }_0\sqrt{0.09z_{01}^2+0.04z_{02}^2+z_{03}^2+0.25z_{04}^2}\le 0\hfill \\ -u_{01}\le x_1-z_{01}\le u_{01},-u_{02}\le x_2-z_{02}\le u_{02}\hfill \\ -u_{03}\le y_1-z_{03},\le u_{03},-u_{04}\le y_2-z_{04}\le u_{04}\hfill \end{array}\phantom{\}}$$

7. Computational Studies for Robust Mixed Integer Linear Optimization

Example 7.1 Consider the following mixed 0–1 programming problem

$$\begin{array}{cc}\hfill \max & 3x_1+2x_2-10y_1-5y_2\hfill \\ \hfill s.t.& x_1+x_2\le 20\hfill \\ \hfill & x_1+2x_2\le 12\hfill \\ \hfill & x_1-20y_1\le 0\hfill \\ \hfill & x_2-20y_2\le 0\hfill \\ \hfill & x_1-x_2\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$

Let us assume that all the objective function coefficients, the LHS and RHS of the constraints parameter are possibly subject to uncertainty. To find robust solutions of this problem, we first convert the objective uncertainty into LHS uncertainty as shown in Section 6.2:

$$\begin{array}{cc}\hfill \max & z\hfill \\ \hfill s.t.& z-(3x_1+2x_2-10y_1-5y_2)\le 0\hfill \\ \hfill & x_1+x_2\le 20\hfill \\ \hfill & x_1+2x_2\le 12\hfill \\ \hfill & x_1-20y_1\le 0\hfill \\ \hfill & x_2-20y_2\le 0\hfill \\ \hfill & x_1-x_2\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$

The corresponding uncertain version of the above problem can be represented using the general form as follows

$$\begin{array}{cc}\hfill \max & z\hfill \\ \hfill s.t.& A_{0}z+\tilde{A}x+\tilde{B}y\le \tilde{p}\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$

where $\tilde{A}=\{a_{\mathit{im}}+{\widehat{a}}_{\mathit{im}}{\xi }_{\mathit{im}}\}$, $\tilde{B}=\{b_{\mathit{ik}}+{\widehat{b}}_{\mathit{ik}}{\xi }_{\mathit{ik}}\}$, $\tilde{p}=\{p_i+{\widehat{p}}_i{\xi }_{i0}\}$, ξ_im, ξ_ik, ξ_i0 are independent uncertain parameters, a_im, b_ik and p_i are nominal data defined as follows

$$A_0=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right],\left\{a_{im}\right\}=\left[\begin{array}{cc}\hfill -3\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right],\left\{b_{ik}\right\}=\left[\begin{array}{cc}\hfill 10\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill -20\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -20\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left\{p_i\right\}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 20\hfill \\ \hfill 12\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill -4\hfill \end{array}\right]$$

Assuming 10% uncertainty level for the possible uncertainty (i.e., ${\widehat{a}}_{\mathit{im}}=0.1\mid a_{\mathit{im}}\mid$, ${\widehat{b}}_{\mathit{ik}}=0.1\mid b_{\mathit{ik}}\mid$, ${\widehat{p}}_i=0.1\mid p_i\mid$), the robust counterpart model under different uncertainty sets can be formulated as shown in section 6. Note that for the constraints containing only continuous variables, their corresponding robust counterpart constraints can be formulated using the method presented in section 4.

In this example, several different uncertainty cases are studied, which include LHS only uncertainty, RHS only uncertainty, OBJ only uncertainty, simultaneous LHS, RHS and OBJ uncertainty. Without giving a complete description of all the robust counterpart optimization models, we list several robust counterpart models using the box set induced robust counterpart formulation as follows:

• (1)
  Considering LHS only uncertainty for all the constraints, the box set induced robust counterpart model is

  $$\begin{array}{cc}\hfill \max & 3x_1+2x_2-10y_1-5y_2\hfill \\ \hfill s.t.& x_1+x_2+\Psi (0.1x_1+0.1x_2)\le 20\hfill \\ \hfill & x_1+2x_2+\Psi (0.1x_1+0.2x_2)\le 12\hfill \\ \hfill & x_1-20y_1+\Psi (0.1x_1+2y_1)\le 0\hfill \\ \hfill & x_2-20y_2+\Psi (0.1x_2+2y_2)\le 0\hfill \\ \hfill & x_1-x_2+\Psi (0.1x_1+0.1x_2)\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$

  Note that the same uncertainty set parameter Ψ is applied for all the constraints here. A similar setting will be applied for the rest of the models.
• (2)
  Considering simultaneous LHS and RHS uncertainty, the box set induced robust counterpart model is

  $$\begin{array}{cc}\hfill \max & 3x_1+2x_2-10y_1-5y_2\hfill \\ \hfill s.t.& x_1+x_2+\Psi (0.1x_1+0.1x_2+2)\le 20\hfill \\ \hfill & x_1+2x_2+\Psi (0.1x_1+0.2x_2+1.2)\le 12\hfill \\ \hfill & x_1-20y_1+\Psi (0.1x_1+2y_1)\le 0\hfill \\ \hfill & x_2-20y_2+\Psi (0.1x_2+2y_2)\le 0\hfill \\ \hfill & x_1-x_2+\Psi (0.1x_1+0.1x_2+0.4)\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$
• (3)
  Considering simultaneous LHS, RHS and OBJ uncertainty, the robust counterpart model is:

  $$\begin{array}{cc}\hfill \max & z\hfill \\ \hfill s.t.& z-(3x_1+2x_2-10y_1-5y_2)+\Psi (0.3x_1+0.2x_2+y_1+0.5y_2)\le 0\hfill \\ \hfill & x_1+x_2+\Psi (0.1x_1+0.1x_2+2)\le 20\hfill \\ \hfill & x_1+2x_2+\Psi (0.1x_1+0.2x_2+1.2)\le 12\hfill \\ \hfill & x_1-20y_1+\Psi (0.1x_1+2y_1)\le 0\hfill \\ \hfill & x_2-20y_2+\Psi (0.1x_2+2y_2)\le 0\hfill \\ \hfill & x_1-x_2+\Psi (0.1x_1+0.1x_2+0.4)\le -4\hfill \\ \hfill & 0\le x_1,x_2\le 10,y_1,y_2\in \{0,1\}\hfill \end{array}$$

Based on the solution of the robust formulations under different cases of uncertainties, the following remarks can be made:

• (1)For RHS only uncertainty, which is a special case where the number of uncertain parameters for every constraint is 1, the solution is identical for ellipsoidal and polyhedral set induced models, and also for the “interval+ellipsoidal”, “interval+polyhedral” and “interval+ellipsoidal+polyhedral” uncertainty set induced models as shown in Figure 7.3. Furthermore, as Ω ≤ 1 and Γ ≤ 1, all the solutions are identical because as Ω = Γ ≤ 1, the different uncertainty sets are actually the same interval set.
• (2)It can be observed from Figures 7.1(a), 7.2(a), 7.3(a), 7.4(a) and 7.5(a) that the ellipsoidal set based robust counterpart solution is equal or worse (even becomes infeasible with large Ω value) than the “interval+ellipsoidal” set based solution. Similarly, the polyhedral set based solution is equal or worse than the “interval +polyhedral” set based solution. This is because for the ellipsoidal set or polyhedral set, its combination with the interval set makes the resulting uncertainty set smaller, and thus less conservative. This suggests that for bounded uncertainty, the uncertainty set should be combined with interval to avoid conservative solutions.
• (3)Comparing the “interval+ellipsoidal” and the “interval+polyhedral” induced model from Figures 7.1(b), 7.2(b), 7.4(b) and 7.5(b), when $\Gamma =\Omega \sqrt{\mid J_i\mid }$, the “interval+polyhedral” based solution is always worse than the “interval+ellipsoidal” based solution, which is because the “interval+polyhedral” uncertainty set is larger and completely covers the “interval+ellipsoidal” set; when Γ = Ω, the “interval+polyhedral” based solution is always better than the “interval+ellipsoidal” based solution because the “interval+polyhedral” uncertainty set is smaller and completely covered by the “interval+ellipsoidal” set.
• (4)Comparing the “interval+ellipsoidal+polyhedral” set based model with others from Figures 7.1(b), 7.2(b), 7.4(b) and 7.5(b), it can be observed that as the Γ value increases from Ω to $\Omega \sqrt{\mid J_i\mid }$, the “interval+ellipsoidal+polyhedral” based solution switches from the “interval+polyhedral” based solution with Γ = Ω to the “interval+ellipsoidal” based solution with $\Gamma =\Omega \sqrt{\mid J_i\mid }$, because the intersection between ellipsoid and polyhedron is exactly changing from the polyhedral with Γ = Ω to the ellipsoid with parameter $\Gamma =\Omega \sqrt{\mid J_i\mid }$.

Figure 7.3.

Only RHS uncertainty for all constraints (|J_i| = 1)

Figure 7.1.

Only LHS uncertainty for all constraints (|J_i| = 2) Note: for polyhedral model, as $\Gamma \rangle \sqrt{2}$, model infeasible

Figure 7.2.

LHS+LHS uncertainty for all constraints (|J_i| = 3) Note: for polyhedral model and “box” model, infeasible for large Γ, Ψ,

Figure 7.4.

Only OBJ uncertainty (|J_i| = 4) Note: for polyhedral model, infeasible for large Γ

Figure 7.5.

Objective uncertainty (|J_i| = 4) and LHS uncertainty for the third and fourth constraints (|J_i| = 2)

Finally, from the above analysis, it can be concluded all the different models have the flexibility to adjust the solution between the worst-case scenario and the deterministic solution, depending on the selection of the adjustable parameters for their corresponding uncertainty set. On the other hand, the degree of conservatism of the models differs, and some models even become infeasible with relatively large uncertainty set parameter values.

Example 7.2 Process scheduling problem

This example involves the scheduling of a batch chemical process related to the production of two chemical products using three raw materials. The state-task-network (STN) representation of this example is shown in Figure 7.6. The deterministic MILP formulation (7.1) for the scheduling of this batch process is based on³⁷ and detailed problem data can be found in³⁷.

Figure 7.6.

State Task Network (STN) representation of the batch chemical process

Through this example, we study the different robust counterpart optimization formulations introduced in section 6 considering different types of uncertainty cases. The scheduling problem's MILP formulation is as follows:

$$\begin{array}{cc}\hfill & \max profit\hfill \\ \hfill s.t.& profit-\sum _{s\in S_p,n}\tilde{p}ric{e}_s{d}_{s,n}+\sum _{s\in S_r}\tilde{p}ric{e}_s(ST{I}_s-ST{F}_s)\le 0\hfill \end{array}$$ (7.1a)

$$\sum _{i\in I_j}w{v}_{i,j,n}\le 1\forall i\in I$$ (7.1b)

$$s{t}_{s,n}=s{t}_{s,n-1}-d_{s,n}-\sum _{i\in I_s}{\rho }_{s,i}^C\sum _{j\in J_i}{b}_{i,j,n}+\sum _{i\in I_s}{\rho }_{s,i}^P\sum _{j\in J_i}{b}_{i,j,n-1}\forall s\in S,\forall n\in N$$ (7.1c)

$$s{t}_{s,n}\le s{t}_s^{\max }\forall s\in S,\forall n\in N$$ (7.1d)

$$v_{i,j}^{\min }w{v}_{i,j,n}\le b_{i,j,n}\le v_{i,j}^{\max }w{v}_{i,j,n}\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1e)

$$\sum _n{d}_{s,n}\ge {\tilde{r}}_s\forall s\in S$$ (7.1f)

$$T{f}_{i,j,n}\ge T{s}_{i,j,n}+{\tilde{\alpha }}_{i,j}w{v}_{i,j,n}+{\tilde{\beta }}_{i,j}{b}_{i,j,n}\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1g)

$$T{s}_{i,j,n+1}\ge T{f}_{i,j,n}-H(1-w{v}_{i,j,n})\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1h)

$$T{s}_{i,j,n+1}\ge T{f}_{i\prime ,j,n}-H(1-w{v}_{i,\prime j,n})\forall i,i\prime \in I_j,\forall j\in J,\forall n\in N$$ (7.1i)

$$T{s}_{i,j,n+1}\ge T{f}_{i\prime ,j\prime ,n}-H(1-w{v}_{i,\prime j\prime ,n})\forall i,i\prime \in I_j,i\ne i\prime ,\forall j,j\prime \in J,\forall n\in N$$ (7.1j)

$$T{s}_{i,j,n+1}\ge T{s}_{i,j,n}\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1k)

$$T{f}_{i,j,n+1}\ge T{f}_{i,j,n}\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1l)

$$T{s}_{i,j,n}\le H\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1m)

$$T{f}_{i.j.n}\le H\forall i\in I,\forall j\in J_i,\forall n\in N$$ (7.1n)

Nomenclature for the process scheduling model (7.1)

i ∈	I tasks
Is	tasks which produce or consume state (s)
Ij	tasks which can be performed in unit (j)
j ∈ J	units
Ji	units which are suitable for performing task (i)
n ∈ N	event points representing the beginning of a task
s ∈ S	states
Sp	states belong to products
Sr	states belong to raw materials
prices	price of state (s)
STIs	initial amount of state (s)
STFs	final amount of state (s)
ds,n	amount of state (s) delivered to the market at event point (n)
wvi,j,n	binary, whether or not task (i) in unit (j) start at event point (n)
sts,n	continuous, amount of state (s) at event point (n)
${\rho }_{s,i}^P$, ${\rho }_{s,i}^C$	proportion of state (s) produced, consumed by task(i), respectively
bi,j,n	amount of material undertaking task (i) in unit (j) at event point (n)
stmaxs	available maximum storage capacity for state (s)
${\nu }_{i,j}^{\text{min}}$, ${\nu }_{i,j}^{\text{max}}$	minimum amount, maximum capacity of unit (j) when processing task (i)
rs	market demand for state (s) at the end of the time horizon
Tfi,j,n	time at which task (i) finishes in unit (j) while it starts at event point (n)
Tsi,j,n	time at which task (i) starts in unit (j) at event point (n)
αi,j, βi,j	variable term of processing time of task (i) in unit (j)
H	time horizon

In the above formulation, the objective function (7.1a) maximizes the profit; allocation constraints (7.1b) state that only one of the tasks can be performed in each unit at an event point (n); constraints (7.1c) represent the material balances for each state (s) expressing that at each event point (n) the amount is equal to that at event point (n-1), adjusted by any amounts produced and consumed between event points (n-1) and (n), and delivered to the market at event point (n); the storage and capacity limitations of production units are expressed by constraints (7.1d) and (7.1e); constraints (7.1f) are written to satisfy the demands of final products; and constraints (7.1g) to (7.1n) represent time limitations due to task duration and sequence requirements in the same or different production units.

In this example, uncertainties in material and product prices, processing times of tasks in different units, and product demands are studied. We assume bounded uncertainty and assign a maximum of 5% deviation of price data, 5% of processing times and 20% of demand data from their nominal values.

(1) Price uncertainty Considering only price uncertainty, then only constraint (7.1a) is affected, where $\tilde{p}{\mathit{rice}}_s$ are the uncertain parameters. For the process network in this example, there are three raw materials and two products, so the total number of uncertain parameters in the constraint is 5 (i.e., |J_i| = 5 ). We first study the ellipsoidal and polyhedral sets related robust formulations presented in section 6 and apply them on this constraint. The results are shown in Figure 7.7. From the results shown in Figure 7.7, it is seen that when Ω≤ 1 and Γ≤ 1, (a) the ellipsoidal and the “interval+ellipsoidal” set based solutions are identical, and (b) the polyhedral and the “interval+polyhedral” set based solutions are identical. This is because the corresponding uncertainty sets are also identical. As Ω > 1 and Γ > 1, the combined uncertainty sets based solutions are better because their uncertainty sets are smaller with the restriction of the bounded box comparing to the pure ellipsoidal and pure polyhedral set, whose corresponding solutions quickly deteriorate. The above analysis further verifies the earlier observation that for bounded uncertainty, a combination set is preferred to obtain less conservative solution. Finally, considering the “interval+ellispsoidal+polyhedral” set will only lead to solutions between the “interval+ellispoidal” and the “interval+polyhderal” cases and require a more complex model. Hence it is not suggested for the solution of robust scheduling problems.

Figure 7.7.

Price uncertainty (|J_i| = 5)

(2) Processing time uncertainty Here we consider only processing time uncertainty in constraints (7.1g), where ${\tilde{\alpha }}_{i,j}$ and ${\tilde{\beta }}_{i,j}$ are uncertain parameters. Thus, every such constraint has two uncertain parameters (i.e., |J_i| = 2 ). We study the ellipsoidal and polyhedral sets related robust formulations presented in section 6 and apply them on this constraint. The results are shown in Figure 7.8. From the solution, same conclusions can be made as in the analysis for price uncertainty.

Figure 7.8.

Processing time uncertainty (|J_i| = 2)

(3) Demand uncertainty Considering only demand uncertainty, then constraints (7.1f) are affected. For each one of these constraints, there is only uncertain parameter on the RHS of the constraint, and the uncertain parameter is the demand data ${\tilde{r}}_s$. Considering that the uncertainty is bounded, we only need to study the box set and those combined sets. Since the number of uncertain parameters is 1, for each constraint, the different uncertainty sets are reduced to 1-dimenstion interval set which can be described as

$$U=\{{\xi }_i\mid \mid {\xi }_i\mid \le \Delta \}$$ (7.2)

where Δ is defined as Ψ, min(Ω,1) , min(Γ,1) , min(Ω, Γ, 1) for the box, “interval+ellipsoidal”, “interval+polyhedral”, “interval+polyhedral+ellipsoidal” uncertainty set, respectively. Thus, the different uncertainty set induced robust counterpart formulations will be identical with same uncertainty set parameter value Δ. Here, we plot the result of their robust counterpart solution as shown in Figure 7.9.

Figure 7.9.

Demand uncertainty (|J_i| = 1)

Finally, we studied the worst-case scenario solution for the different uncertainty cases. The worst-case scenario solution means that the uncertainty set covers the whole uncertainty space. Among the different uncertainty sets to cover the whole bounded uncertain space, box uncertainty set takes the smallest size, and here the box set with Ψ = 1 (i.e., interval set) is applied for the three types on uncertainty individually and the results are shown in Table 7.1. Comparing the result, we can conclude that with the given uncertainty characteristics, the price uncertainty has the largest effect on the final profit, whereas the demand uncertainty has the least effect on the final profit.

Table 7.1.

Worst-case scenario solution

	Deterministic	Price uncertainty	Processing Time uncertainty	Demand uncertainty
Objective value	1088.75	959.56	974.95	1032.71

8. Conclusions

Set induced robust counterpart optimization techniques are systematically studied in this paper. Several important uncertainty sets are studied, including those studied in the literature and also several new ones proposed in this work. New uncertainty sets such as the adjustable box, ellipsoidal, polyhedral and “interval+ellipsoidal+polyhedral” set are introduced and their relationship with some well known uncertainty sets presented in the literature is discussed. The relationships between those different uncertainty sets are extensively discussed, and useful insights are gained for their corresponding robust counterpart models. For uncertainty in the left hand side, right hand side and objective function, the robust counterpart formulations induced by those different uncertainty sets for linear optimization problems and mixed integer linear optimization problems are derived. The different uncertainty set based robust counterpart formulations are also compared through numerical studies, a production planning and a process scheduling problem.

Table 6.1.

Summary on robust counterpart formulation for the i-th mixed integer linear constraint

Uncertainty Set	Robust Counterpart Formulation
Box	${\sum }_m{a}_{im}{x}_m+{\sum }_k{b}_{ik}{y}_k+\Psi \left[{\sum }_{m\in M_i}{\widehat{a}}_{im}\mid x_m\mid +{\sum }_{k\in K_i}{\widehat{b}}_{ik}\mid y_k\mid +{\widehat{p}}_i\right]\le p_i$
Ellipsoidal	${\sum }_m{a}_{im}{x}_m+{\sum }_k{b}_{ik}{y}_k+\Omega \sqrt{{\sum }_{m\in M_i}{\widehat{a}}_{im}^2{x}_m^2+{\sum }_{k\in K_i}{\widehat{b}}_{ik}^2{y}_k^2+{\widehat{p}}_i^2}\le p_i$
Polyhedral	$\{\begin{array}{c}{\sum }_m{a}_{im}{x}_m+{\sum }_k{b}_{ik}{y}_k+z_i\Gamma \le p_i\hfill \\ z_i\ge {\widehat{a}}_{im}\mid x_m\mid \forall m\in M_i\hfill \\ z_i\ge {\widehat{b}}_{ik}\mid y_k\mid \forall k\in K_i\hfill \\ z_i\ge {\widehat{p}}_i\hfill \end{array}\phantom{\}}$
Interval+ Ellipsoidal	${\sum }_m{a}_{im}{x}_m+{\sum }_k{b}_{ik}{y}_k+{\sum }_{m\in M_i}{\widehat{a}}_{im}\mid x_m-z_{im}\mid +{\sum }_{m\in K_i}{\widehat{b}}_{ik}\mid y_k-z_{ik}\mid +{\widehat{p}}_i\mid 1+z_{i0}\mid +\Omega \sqrt{{\sum }_{m\in M_i}{\widehat{a}}_{im}^2{z}_{im}^2+{\sum }_{k\in K_i}{\widehat{b}}_{ik}^2{z}_{ik}^2+{\widehat{p}}_i^2{z}_{i0}^2}\le p_i$
Interval+ Polyhedral	$\{\begin{array}{c}{\sum }_m{a}_{im}{x}_m+{\sum }_k{b}_{ik}{y}_k+z_i{\Gamma }_i+{\sum }_{m\in M_i}{w}_{im}+{\sum }_{k\in K_i}{w}_{ik}+w_{i0}\le p_i\hfill \\ z_i+w_{im}\ge {\widehat{a}}_{im}\mid x_m\mid \forall m\in M_i\hfill \\ z_i+w_{ik}\ge {\widehat{b}}_{ik}\mid y_k\mid \forall k\in K_i\hfill \\ z_i+w_{i0}\ge {\widehat{p}}_i\hfill \end{array}\phantom{\}}$
Interval+ Ellipsoidal+ Polyhedral	$\{\begin{array}{c}{\sum }_m{a}_{im}{x}_m+{\sum }_k{b}_{ik}{y}_k+z_i\Gamma +{\sum }_{m\in M_i}\mid q_{im}\mid +{\sum }_{k\in K_i}\mid q_{ik}\mid +\mid q_{i0}\mid +\Omega \sqrt{{\sum }_{m\in M_i}{w}_{im}^2+{\sum }_{k\in K_i}{w}_{ik}^2+w_{i0}^2}\le p_i\hfill \\ z_i\ge \mid {\widehat{a}}_{im}{x}_m-q_{im}-w_{im}\mid \forall m\in M_i\hfill \\ z_i\ge \mid {\widehat{b}}_{ik}{y}_k-q_{ik}-w_{ik}\mid \forall k\in K_i\hfill \\ z_i\ge \mid {\widehat{p}}_i+q_{i0}+q_{i0}\mid \hfill \end{array}\phantom{\}}$

Appendix A

Derivation of the robust counterpart for a linear constraint under simultaneous LHS and RHS uncertainty

Consider the i-th linear constraint of problem (2.1) with simultaneous LHS and the RHS uncertainty:

$$\sum _{j\notin J_i}{a}_{\mathit{ij}}{x}_j+\sum _{j\in J_i}{\tilde{a}}_{\mathit{ij}}{x}_j\le {\tilde{b}}_j$$ (A.1)

where ${\tilde{a}}_{\mathit{ij}}=a_{\mathit{ij}}+{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{\forall }_j\in J_i$, ${\tilde{b}}_i=b_i+{\xi }_{0i}{\widehat{b}}_i$. Incorporating auxiliary variable x₀ and an additional constraint x₀ = −1, the constraint can be rewritten as

$$b_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\left[{\xi }_{i0}{\widehat{b}}_i{x}_0+\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j\right]\le 0.$$ (A.2)

With a given uncertainty set U for ξ_i0 and ξ_ij, the corresponding set induced robust counterpart is

$$b_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\left[\underset{\xi \in U}{\text{max}}\left\{{\xi }_{i0}{\widehat{b}}_i{x}_0+\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{a}}_{\mathit{ij}}{x}_j\right\}\right]\le 0.$$ (A.3)

With the following definition

$${\xi }_i=[{\xi }_{i0};\left\{{\xi }_{\mathit{ij}}\right\}],$$ (A.4a)

$$A_i=[b_i,\left\{a_{\mathit{ij}}\right\}],$$ (A.4b)

$${\widehat{A}}_i=[{\widehat{b}}_i,\left\{{\widehat{a}}_{\mathit{ij}}\right\}],$$ (A.4c)

$$X=[x_0;\left\{x_j\right\}],$$ (A.4d)

$$J_i^{\prime }=J_i>\left\{0\right\},$$ (A.4e)

constraint (A.3) can be rewritten as

$$\sum _j{A}_{\mathit{ij}}{X}_j+\underset{\xi \in U}{\text{max}}\left\{{\xi }_i{\widehat{A}}_{i}X\right\}\le 0.$$ (A.5)

Property A.1 The box uncertainty set (3.1) induced robust counterpart formulation (A.5) is equivalent to

$$\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \left[\sum _{j\in J_i^{\prime }}{\widehat{a}}_{\mathit{ij}}\mid x_j\mid +{\widehat{b}}_i\right]\le b_i$$ (A.6)

Proof. Applying Property 3.1 on (A.5), we obtain the following equivalent problem

$$\sum _j{A}_{\mathit{ij}}{X}_j+\left[\Psi \sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}\mid X_j\mid \right]\le 0.$$

Expanding the above constraints using the previously defined variables, the resulting robust counterpart formulation is

$$b_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \left[\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j\mid +{\widehat{b}}_i\mid x_0\mid \right]\le 0.$$

Notice that x₀ = −1, so the absolute value operation on them is automatically eliminated. The final robust counterpart formulation is

$$\sum _j{a}_{\mathit{ij}}{x}_j+\Psi \left[\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j\mid +{\widehat{b}}_i\right]\le b_i.$$

Property A.2 The ellipsoidal uncertainty set induced robust counterpart formulation (A.5) is equivalent to

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{x}_j^2+{\widehat{b}}_i^2}\right]\le b_i$$ (A.7)

Proof. Applying Property 3.2 on (A.5), the ellipsoidal based uncertainty set induced robust counterpart is

$$\sum _j{A}_{\mathit{ij}}{X}_j+\left[\Omega \sqrt{\sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}^2{X}_j^2}\right]\le 0.$$

Expanding the above constraints, the resulting robust counterpart formulation is

$$b_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{x}_j^2+{\widehat{b}}_i^2{x}_0^2}\le 0.$$

Notice that x₀ = −1, so the final robust counterpart formulation is

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{x}_j^2+{\widehat{b}}_i^2}\right]\le b_i.$$

Property A.3 The polyhedral uncertainty set induced robust counterpart formulation (A.5) is equivalent to

$$\{\begin{array}{cc}\sum _j{a}_{\mathit{ij}}{x}_j+z_i\Gamma \le b_i\hfill & \hfill \\ z_i\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i,\hfill & z_i\ge {\widehat{b}}_i\hfill \end{array}\phantom{\}}$$ (A.8)

Proof. Applying Property 3.3 on (A.5), the ellipsoidal based uncertainty set induced robust counterpart is

$$\{\begin{array}{c}\sum _j{A}_{\mathit{ij}}{X}_j+\Gamma z_i\le 0\hfill \\ z_i\ge {\widehat{A}}_i\mid X_j\mid ,{\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$

Expanding the above constraints, the resulting robust counterpart formulation is

$$\{\begin{array}{cc}{b}_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+z_i\Gamma \le 0\hfill & \hfill \\ z_i\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i;\hfill & z_i\ge {\widehat{b}}_i{x}_0\hfill \end{array}\phantom{\}}$$

Notice that x₀ = −1, so the final robust counterpart formulation is

$$\{\begin{array}{cc}\sum _j{a}_{\mathit{ij}}{x}_j+z_i\Gamma \le b_i\hfill & \hfill \\ z_i\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i,\hfill & z_i\ge {\widehat{b}}_i\hfill \end{array}\phantom{\}}$$

Property A.4 The “Interval+ellipsoidal” uncertainty set induced robust counterpart formulation (A.5) is equivalent to

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j-z_{\mathit{ij}}\mid +{\widehat{b}}_i\mid 1+z_{i0}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2+{\widehat{b}}_i^2{z}_{i0}^2}\right]\le b_i$$ (A.9)

Proof. Applying Property 3.4 on (A.5), the “interval+ellipsoidal” based uncertainty set induced robust counterpart is

$$\sum _j{A}_{\mathit{ij}}{X}_j+\left[\sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}\mid X_j-z_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}\right]\le 0.$$

Expanding the above constraints, the resulting robust counterpart formulation is

$$b_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j-z_{\mathit{ij}}\mid +{\widehat{b}}_i\mid x_0-z_{i0}\mid +\Omega \sqrt{\sum _{j\in M_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2+{\widehat{b}}_i^2{z}_{i0}^2}\right]\le 0.$$

Notice that x₀ = −1, the final robust counterpart formulation is

$$\sum _j{a}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}\mid x_j-z_{\mathit{ij}}\mid +{\widehat{b}}_i\mid 1+z_{i0}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{a}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2+{\widehat{b}}_i^2{z}_{i0}^2}\right]\le b_i.$$

Property A.5 The “Interval+polyhedral” uncertainty set induced robust counterpart formulation (A.5) is equivalent to

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\left[z_i\Gamma +\sum _{j\in J_i}{p}_{\mathit{ij}}+p_{i0}\right]\le b_i\hfill \\ z_i+p_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i,z_i+p_{i0}\ge {\widehat{b}}_i\hfill \\ z_i\ge 0,p_{\mathit{ij}}\ge 0,p_{i0}\ge 0\hfill \end{array}\phantom{\}}$$ (A.10)

Proof. Applying Property 3.5 on (A.5), the robust counterpart is

$$\{\begin{array}{c}\sum _j{A}_{\mathit{ij}}{x}_j+\sum _{j\in J_i}{p}_{\mathit{ij}}+\Gamma z_i\le 0\hfill \\ z_i+p_{\mathit{ij}}\ge {\widehat{A}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i\hfill \\ z_i\ge 0,p_{\mathit{ij}}\ge 0\hfill \end{array}\phantom{\}}$$

Expanding the above constraints, the resulting robust counterpart formulation is

$$\{\begin{array}{c}{b}_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}{p}_{\mathit{ij}}+p_{i0}+z_i\Gamma \right]\le 0\hfill \\ z_i+p_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid ,{\forall }_j\in J_i;z_i+p_{i0}\ge {\widehat{b}}_i\mid x_0\mid \hfill \\ z_i\ge 0,p_{\mathit{ij}}\ge 0,p_{i0}\ge 0\hfill \end{array}\phantom{\}}$$

Notice that x₀ = −1, so the final robust counterpart formulation is

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\left[z_i\Gamma +\sum _{j\in J_i}{p}_{\mathit{ij}}+p_{i0}\right]\le b_i\hfill \\ z_i+p_{\mathit{ij}}\ge {\widehat{a}}_{\mathit{ij}}\mid x_j\mid {\forall }_j\in J_i,z_i+p_{i0}\ge {\widehat{b}}_i\hfill \\ z_i\ge 0,p_{\mathit{ij}}\ge 0,p_{i0}\ge 0\hfill \end{array}\phantom{\}}$$

Property A.6 The “Interval+polyhedral+ellipsoidal” uncertainty set induced robust counterpart formulation (A.5) is equivalent to

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\left[z_i\Gamma +\sum _{j\in J_i}\mid p_{\mathit{ij}}\mid +\mid p_{i0}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{\mathit{ij}}^2+w_{i0}^2}\right]\le b_i\hfill \\ z_i\ge \mid {\widehat{a}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\mid {\forall }_j\in J_i,z_i\ge \mid -{\widehat{b}}_i-p_{i0}-w_{i0}\mid \hfill \end{array}\phantom{\}}$$ (A.11)

Proof. Applying Property 3.6 on (A.5), the robust counterpart is

$$\{\begin{array}{c}\sum _j{A}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}\mid p_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}}+\Gamma z_i\right]\le 0\hfill \\ z_i\ge \mid {\widehat{A}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\mid {\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$

Expanding the above constraints, the resulting robust counterpart formulation is

$$\{\begin{array}{c}{b}_i{x}_0+\sum _j{a}_{\mathit{ij}}{x}_j+\left[\sum _{j\in J_i}\mid p_{\mathit{ij}}\mid +\mid p_{i0}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{\mathit{ij}}^2+w_{i0}^2}+z_i\Gamma \right]\le 0\hfill \\ z_i\ge \mid {\widehat{a}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\mid {\forall }_j\in J_i;z_i\ge \mid {\widehat{b}}_i{x}_0-p_{i0}-w_{i0}\mid \hfill \end{array}\phantom{\}}$$

Notice that x₀ = −1, so the final robust counterpart formulation is

$$\{\begin{array}{c}\sum _j{a}_{\mathit{ij}}{x}_j+\left[z_i\Gamma +\sum _{j\in J_i}\mid p_{\mathit{ij}}\mid +\mid p_{i0}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{\mathit{ij}}^2+w_{i0}^2}\right]\le b_i\hfill \\ z_i\ge \mid {\widehat{a}}_{\mathit{ij}}{x}_j-p_{\mathit{ij}}-w_{\mathit{ij}}\mid {\forall }_j\in J_i,z_i\ge \mid {\widehat{b}}_i+p_{i0}+w_{i0}\mid \hfill \end{array}\phantom{\}}$$

Appendix B

Derivation of the robust counterpart for a mixed integer linear constraint under simultaneous LHS and RHS uncertainty

As presented in section 6.1, the robust counterpart formulation for the i-th mixed integer linear constraint in problem (2.7) with simultaneous LHS and RHS uncertainty can be rewritten as (6.3), i.e.,

$$\sum _j{A}_{\mathit{ij}}{X}_j+\underset{{\xi }_i\in U}{\text{max}}\left\{\sum _{j\in J_i}{\xi }_{\mathit{ij}}{\widehat{A}}_{\mathit{ij}}{X}_j\right\}\le 0$$

where A_i,${\tilde{A}}_i$,X,ξ_i,J_i are defined in (6.2). In the follows, proofs for Properties 6.1–6.6 are presented.

B.1 Proof of property 6.1: Notice that the derivation procedure in Section 4 for the robust linear counterpart constraint also applies for the mixed integer linear constraint since it applies for both continuous and integer variable. So, applying Property 4.1 on constraint (6.3), we can obtain the following equivalent problem

$$\sum _j{A}_{\mathit{ij}}{X}_j+\left[\Psi \sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}\mid X_j\mid \right]\le 0.$$

Expand the above constraints using the definition in equation (6.2), the resulting robust counterpart formulation is:

$$p_i{x}_0+\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\Psi \left[\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}\mid x_m\mid +\sum _{k\in K_i}{\widehat{b}}_{\mathit{ik}}\mid y_k\mid +{\widehat{p}}_i\mid x_0\mid \right]\le 0.$$

Notice that x₀ = −1, so the final robust counterpart formulation (6.4) is obtained:

$$\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\Psi \left[\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}\mid x_m\mid +\sum _{k\in K_i}{\widehat{b}}_{\mathit{ik}}\mid y_k\mid +{\widehat{p}}_i\right]\le p_i.$$

B.2 Proof of property 6.2: Applying Property 4.2 on (6.3), we obtain the following equivalent problem

$$\mathrm{f}\sum _j{A}_{\mathit{ij}}{X}_j+\left[\Omega \sqrt{\sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}^2{X}_j^2}\right]\le 0.$$

Expanding the above constraints using the definition in equation (6.2), then the resulting robust counterpart formulation is

$$p_i{x}_0+\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}^2{x}_m^2+\sum _{k\in K_i}{\widehat{b}}_{\mathit{ik}}^2{y}_k^2+{\widehat{p}}_i^2{x}_0^2}\le 0.$$

Notice that x₀ = −1, so the final robust counterpart formulation (6.7) is obtained:

$$\sum _m{a}_{\mathit{im}}{x}_m+\sum _m{b}_{\mathit{ik}}{y}_k+\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}^2{x}_m^2+\sum _{k\in K_i}{\widehat{b}}_{\mathit{ik}}^2{y}_k^2+{\widehat{p}}_i^2}\le p_i.$$

B.3 Proof of property 6.3: Applying Property 4.3 on (6.3), we obtain the following equivalent problem

$$\{\begin{array}{c}\sum _j{A}_{\mathit{ij}}{X}_j+\Gamma z_i\le 0\hfill \\ z_i\ge {\widehat{A}}_{\mathit{ij}}\mid X_j\mid ,{\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$

Expanding the above constraints using the definition in equation (6.2), the resulting robust counterpart formulation is

$$\{\begin{array}{c}{p}_i{x}_0+\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+z_i\Gamma \le 0\hfill \\ z_i\ge {\widehat{a}}_{\mathit{im}}\mid x_m\mid {\forall }_m\in M_i\hfill \\ z_i\ge {\widehat{b}}_{\mathit{ik}}\mid y_k\mid {\forall }_k\in K_i\hfill \\ z_i\ge {\widehat{p}}_i\mid x_0\mid \hfill \end{array}\phantom{\}}$$

Notice that x₀ = −1, so the final robust counterpart formulation (6.8) is obtained:

$$\{\begin{array}{c}\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+z_i\Gamma \le p_i\hfill \\ z_i\ge {\widehat{a}}_{\mathit{im}}\mid x_m\mid {\forall }_m\in M_i\hfill \\ z_i\ge {\widehat{b}}_{\mathit{ik}}\mid y_k\mid {\forall }_k\in K_i\hfill \\ z_i\ge {\widehat{p}}_i\hfill \end{array}\phantom{\}}$$

B.4 Proof of property 6.4: Applying Property 4.4 on (6.3), we obtain the following equivalent problem

$$\sum _j{A}_{\mathit{ij}}{X}_j+\left[\sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}\mid X_j-z_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{\widehat{A}}_{\mathit{ij}}^2{z}_{\mathit{ij}}^2}\right]\le 0$$

Expanding the above constraints using the definition in equation (6.2), the resulting robust counterpart formulation is obtained:

$$p_i{x}_0+\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}\mid x_m-z_{\mathit{im}}\mid +\sum _{m\in K_i}{\widehat{b}}_{\mathit{ik}}\mid y_k-z_{\mathit{ik}}\mid +{\widehat{p}}_i\mid x_0-z_{i0}\mid +\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}^2{z}_{\mathit{im}}^2+\sum _{k\in K_i}{\widehat{b}}_{\mathit{ik}}^2{z}_{\mathit{ik}}^2+{\widehat{p}}_i^2{z}_{i0}^2}\le 0$$

Notice that x₀ = −1, so the final robust counterpart formulation (6.10) is obtained:

$$\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}\mid x_m-z_{\mathit{im}}\mid +\sum _{m\in K_i}{\widehat{b}}_{\mathit{ik}}\mid y_k-z_{\mathit{ik}}\mid +{\widehat{p}}_i\mid 1+z_{i0}\mid +\Omega \sqrt{\sum _{m\in M_i}{\widehat{a}}_{\mathit{im}}^2{z}_{\mathit{im}}^2+\sum _{k\in K_i}{\widehat{b}}_{\mathit{ik}}^2{z}_{\mathit{ik}}^2+{\widehat{p}}_i^2{z}_{i0}^2}\le p_i$$

B.5 Proof of property 6.5: Applying Property 4.5 on (6.3), we obtain the following equivalent problem

$$\{\begin{array}{c}\sum _j{A}_{\mathit{ij}}{X}_j+\sum _{j\in J_i}{w}_{\mathit{ij}}+\Gamma z_i\le 0\hfill \\ z_i+w_{\mathit{ij}}\ge {\widehat{A}}_{\mathit{ij}}\mid X_j\mid {\forall }_j\in J_i\hfill \\ z_i\ge 0,w_{\mathit{ij}}\ge 0\hfill \end{array}\phantom{\}}$$

Expanding the above constraints using the definition in equation (6.2), the resulting robust counterpart formulation is

$$\{\begin{array}{c}{p}_i{x}_0+\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\left[\sum _{m\in M_i}{w}_{\mathit{im}}+\sum _{k\in K_i}{w}_{\mathit{ik}}+w_{i0}+z_i\Gamma \right]\le 0\hfill \\ z_i+w_{\mathit{im}}\ge {\widehat{a}}_{\mathit{im}}\mid x_m\mid {\forall }_m\in M_i\hfill \\ z_i+w_{\mathit{ik}}\ge {\widehat{b}}_{\mathit{ik}}\mid y_k\mid {\forall }_k\in K_i\hfill \\ z_i+w_{i0}\ge {\widehat{p}}_i\mid x_0\mid \hfill \end{array}\phantom{\}}$$

Notice that x₀ = −1, so the final robust counterpart formulation (6.12) is obtained:

$$\{\begin{array}{c}\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\left[z_i{\Gamma }_i+\sum _{m\in M_i}{w}_{\mathit{im}}+\sum _{k\in K_i}{w}_{\mathit{ik}}+w_{i0}\right]\le p_i\hfill \\ z_i+w_{\mathit{im}}\ge {\widehat{a}}_{\mathit{im}}\mid x_m\mid {\forall }_m\in M_i\hfill \\ z_i+w_{\mathit{ik}}\ge {\widehat{b}}_{\mathit{ik}}\mid y_k\mid {\forall }_k\in K_i\hfill \\ z_i+w_{i0}\ge {\widehat{p}}_i\hfill \end{array}\phantom{\}}$$

B.6 Proof of property 6.6: Applying Property 4.6 on (6.3), we obtain the following equivalent problem

$$\{\begin{array}{c}\sum _j{A}_{\mathit{ij}}{X}_j+\left[\sum _{j\in J_i}\mid q_{\mathit{ij}}\mid +\Omega \sqrt{\sum _{j\in J_i}{w}_{\mathit{ij}}^2}+\Gamma z_i\right]\le 0\hfill \\ z_i\ge \mid {\widehat{A}}_{\mathit{ij}}{X}_j-q_{\mathit{ij}}-w_{\mathit{ij}}\mid {\forall }_j\in J_i\hfill \end{array}\phantom{\}}$$

Expanding the above constraints using the definition in equation (6.2), the resulting robust counterpart formulation is

$$\{\begin{array}{c}{p}_i{x}_0+\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+\sum _{m\in M_i}\mid q_{\mathit{im}}\mid +\sum _{m\in K_i}\mid q_{\mathit{ik}}\mid +\mid q_{i0}\mid +\Omega \sqrt{\sum _{m\in M_i}{w}_{\mathit{im}}^2+\sum _{k\in K_i}{w}_{\mathit{ik}}^2+w_{i0}^2}+z_i\Gamma \le 0\hfill \\ z_i\ge \mid {\widehat{a}}_{\mathit{im}}{x}_m-q_{\mathit{im}}-w_{\mathit{im}}\mid {\forall }_m\in M_i\hfill \\ z_i\ge \mid {\widehat{b}}_{\mathit{ik}}{y}_k-q_{\mathit{ik}}-v_{\mathit{ik}}\mid {\forall }_k\in K_i\hfill \\ z_i\ge \mid {\widehat{p}}_i{x}_0-q_{i0}-w_{i0}\mid \hfill \end{array}\phantom{\}}$$

Notice that x₀ = −1, so the final robust counterpart formulation (6.14) is obtained:

$$\{\begin{array}{c}\sum _m{a}_{\mathit{im}}{x}_m+\sum _k{b}_{\mathit{ik}}{y}_k+z_i\Gamma +\sum _{m\in M_i}\mid q_{\mathit{im}}\mid +\sum _{m\in K_i}\mid q_{\mathit{ik}}\mid +\mid q_{i0}\mid +\Omega \sqrt{\sum _{m\in M_i}{w}_{\mathit{im}}^2+\sum _{k\in K_i}{w}_{\mathit{ik}}^2+w_{i0}^2}\le p_i\hfill \\ z_i\ge \mid {\widehat{a}}_{\mathit{im}}{x}_m-q_{\mathit{im}}-w_{\mathit{im}}\mid {\forall }_m\in M_i\hfill \\ z_i\ge \mid {\widehat{b}}_{\mathit{ik}}{y}_k-q_{\mathit{ik}}-w_{\mathit{ik}}\mid {\forall }_k\in K_i\hfill \\ z_i\ge \mid {\widehat{p}}_i{x}_0-q_{i0}-w_{i0}\mid \hfill \end{array}\phantom{\}}$$