Robust Optimization for Decision-Making Under Endogenous Uncertainty

Robust Optimization for Decision-making under Endogenous Uncertainty

Nikolaos H. Lappas, Chrysanthos E. Gounaris*

Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213

Abstract

This paper contemplates the use of robust optimization as a framework for addressing problems that involve endogenous uncertainty, i.e., uncertainty that is affected by the decision maker’s strategy. To that end, we extend generic polyhedral uncertainty sets typically considered in robust optimization into sets that depend on the actual decisions. We present the derivation of robust counterpart models in this setting, and we discuss relevant algorithmic con- siderations for solving these models to guaranteed optimality. Besides capturing the functional changes in parameter correlations that may be induced by given decisions, we show how the use of our decision-dependent uncertainty sets allows us to also eradicate conservatism effects from parameters that become irrelevant in view of the optimal decisions. We quantify these benefits via a number of case studies, demonstrating our proposed framework’s versatility to be utilized in the context of various applications. Keywords: Robust Optimization, Endogenous Uncertainty, Decision-dependent Uncer- tainty Sets

1 Introduction

Uncertainty is inherent in virtually every system we wish to optimize. Parameters aﬀected by various external forces such as market prices and demand, unexpected disruptive events such as equipment malfunctions and natural disasters, or simply incomplete information about the system under study may render solutions of deterministic optimization models suboptimal or even infeasible

1 Author to whom all correspondence should be addressed ([email protected]).

1 when parameter realizations deviate from their nominal values. To that end, multiple approaches have been proposed so as to account for uncertainty during the decision-making process.

The various alternative methodologies that have been developed serve diﬀerent purposes, and selecting which one to adopt should be based on careful examination of the characteristics of the application in question. For example, when the number of uncertain parameters is relatively small, multi-parametric programming [1] can map the parameter space in order to oﬀer closed- form solutions of optimization problems in terms of the former and to provide robustness estimates of any given solution. When detailed probabilistic information about the system parameters is available, often in the form of discrete scenario trees, stochastic programming [2] can be utilized to optimize system performance in expectation. Bounds on the variability of such performance can then be applied as explicit constraints on various statistically meaningful metrics [3]. In stochastic programming, the assumption of full recourse is commonly utilized and is often implemented by penalizing infeasibilities in the objective function. It should be mentioned, however, that this practice would not be suitable in applications where constraint violations cannot be tolerated (e.g., due to system safety concerns), are not meaningful (e.g., equipment physical limitations), or cannot be fairly “monetized.”

On the other hand, robust optimization (RO) [4–6] oﬀers an attractive option for applications where distributional information about the uncertainty is limited and/or where solution feasibility is top priority. For such settings, RO seeks solutions that remain feasible for any possible uncertainty realization from within a postulated uncertainty set, which captures applicable correlations among uncertain parameters. Although one can derive such sets based on probabilistic information [7,

8], precise knowledge of probability distributions is not typically required to construct such sets.

Multiple types of uncertainty sets (e.g., polyhedral, ellipsoidal, cardinality-constrained budgets) can be used in this context, exploiting correlations among uncertain parameters as a mechanism to control the trade-oﬀ between robustness and performance.

A common characteristic across traditional uncertainty sets utilized in RO is their constant nature; that is, the range of parameter realizations they admit does not depend on the values we choose for the decision variables. This fact leads to an important limitation, namely that constant sets do not suﬃce to model settings where one’s decisions directly aﬀect the underlying probability distribution from which parameter realizations draw (e.g., entering a market sooner provides access

2 to a larger demand). Furthermore, a subtle manifestation arises in cases where one’s decisions render a model parameter that is referenced in the uncertainty set physically meaningless (e.g., the yield of a process step that was not selected in the optimal ﬂowsheet). Not only would it be unrealistic to postulate a correlation involving such a parameter, doing so might lead to overly conservative solutions as a result of an eﬀective relaxation of the uncertainty set’s projection to the space of parameters that remain relevant in view of the optimal decisions.

In order to overcome the above challenges, we recently proposed the use of decision-dependent uncertainty sets (DDUS) in the context of robust optimization models for process scheduling applications [9]. In that work, the eﬀects of task processing times were removed from the uncertainty set whenever the decision was taken to not execute such tasks, oﬀering a more realistic representation of the uncertainty encountered in the actual application, as well as leading to less conservative solutions compared to robust solutions reported in prior literature. Expanding upon a previously published shorter version [10], the contributions of the current paper are four-fold:

• We classify decision-making contexts involving uncertain parameters of endogenous nature

for which the use of DDUS is warranted.

• We propose DDUS of generic polyhedral form that feature decision-dependency in both left-

and right-hand sides, and we discuss ways to instantiate such sets.

• We derive the robust counterpart model in the context of our proposed DDUS, and we discuss

applicable algorithmic approaches to solve the former to guaranteed optimality.

• We assess the performance beneﬁts as well as the computational burden associated with using

our proposed DDUS over a series of case studies that are addressed with RO for the ﬁrst time

in the open literature.

The remainder of this paper is structured as follows. We begin by providing a brief overview of the static RO methodology in Section 2. In Section 3, we discuss the possibility for uncertain parameters to exhibit an endogenous nature, and we introduce our novel DDUS that allow us to model this setting. We then derive the robust counterpart under DDUS in Section 4, while in Section 5, we present a comprehensive computational study involving a number of case studies

3 stemming from a diverse set of application contexts. We conclude the paper with some ﬁnal remarks in Section 6.

Notation. Lowercase letters with regular typeface denote scalars (e.g., a), lowercase letters with bold typeface denote column vectors (e.g., a), while uppercase letters in bold typeface denote

th matrices (e.g., A). We use |a| to indicate the size of vector a, and we use ai to denote its i element. We also use e to denote the vector of ones, 0 to denote the vector of zeros, E to denote the matrix of all ones, and O to denote the matrix of all zeros. Sizes for these are to be inferred from the expressions in which they participate. The ◦ operator corresponds to the element-wise product of two equally-sized vectors or matrices, while for a quantity a, we use a and a to represent this quantity’s applicable lower and upper bound, respectively. Finally, an equation involving vector or matrix terms (e.g., a ≤ b) is to be viewed as a set of equations referencing the terms element-wise

(e.g., ai ≤ bi ∀ i), while an implication or equivalence with a set of equations in its sides (e.g.,

{a = b} ⇔ {c ≤ d}) is also to be viewed element-wise (e.g., {ai = bi} ⇔ {ci ≤ di} ∀ i).

2 Static Robust Optimization

Let us consider the mixed-integer linear optimization problem (1),1 which features continuous decision variables x, discrete decision variables w, and inequality constraints indexed over the set M.

min x1 x,w |x| |w| s.t. x ∈ R , w ∈ {0, 1} (1) > > > > > cmx + q Cmx + rmw + q Rmw ≤ am + q bm ∀ m ∈ M (x, w) ∈ F

For ease of exposition, the various expressions in the constraints have been partitioned into terms that do and terms that do not involve parameters q, which we shall consider to be uncertain

|q| and about to realize from within an uncertainty set Q ⊆ R . The quantities cm, Cm, rm,

Rm, am, and bm are therefore constants known with certainty, for all m ∈ M, while the set

|x| |w| F ⊆ R × {0, 1} is introduced to represent the space that results from intersecting all linear constraints (both equalities and inequalities) that do not reference any of the uncertain parameters

1 We remark that, although in this paper we focus on problems that can be represented via mixed-integer linear formulations, the RO methodology can be extended to nonlinear models as well.

4 q. For clarity, we mention that we consider all constraints in the set M to reference at least one parameter from q; that is, at least one of the elements of Cm, Rm, or bm must be non-zero, for each constraint m ∈ M. Furthermore, the objective function has been brought into the set of constraints using a standard epigraph reformulation (where, without loss of generality, the continuous variable x1 is used to denote the epigraph variable). Consequently, any mixed-integer linear problem that does not reference uncertain parameters in equality constraints2 can be equivalently transformed to the form (1). We also highlight that this formulation can accommodate uncertain parameters in both left- and right-hand sides as well as in objective function coeﬃcients.

In this setting, all decisions are to be taken in a single stage, before the actual realizations of the uncertain parameters are observed. The fundamental idea behind single-stage (a.k.a., static)

RO is to guarantee the constraint satisfaction for any uncertain parameter realization from within a suitably chosen uncertainty set, Q, and to then seek the best feasible solution, as assessed against the worst possible such realization. Consequently, the RO formulation can be represented via problem (2).

min x1 x,w |x| |w| s.t. x ∈ R , w ∈ {0, 1} (2) > > > > > cmx + q Cmx + rmw + q Rmw ≤ am + q bm ∀ q ∈ Q ∀ m ∈ M (x, w) ∈ F

We remark that, under the reasonable assumption of a non-empty, non-singleton uncertainty set, formulation (2) typically involves inﬁnitely many constraints, and in order solve it, one shall typically apply standard reformulation techniques and numerical algorithms used in semi-inﬁnite programming.

2 This is due to the fact that a trivial or infeasible solution would arise if an equality is to be enforced for more that one distinct realization of the uncertain parameters [11]. In certain models, equalities that reference uncertain parameters can be eliminated via solving out of the model some suitably chosen state variable. If this is not possible, then the reader is referenced to the methodology of Adjustable Robust Optimization [5], where the introduction of recourse decision variables can facilitate this case via coeﬃcient matching, as has also been demonstrated in our previous work [9].

5 2.1 Solution Approaches

There are mainly two, methodologically-distinct approaches for dealing with the semi-infinite nature of the RO model (2). The first one is based on reformulating the problem into an equivalent, but finite-sized monolithic model using duality arguments (e.g., linear, conic, or Fenchel, depending on the type of the uncertainty set). This approach is advantageous inasmuch it provides an explicit closed form robust counterpart, which can then be solved via off-the-shelf optimization software without additional programming effort. A comprehensive list of robust counterparts for various uncertainty sets and types of constraints (both linear and non-linear) can be found in Ben-Tal et al. [12], or Gorissen et al. [6].

The alternative approach constitutes an adversarial methodology inspired by the cutting plane algorithm, which was originally proposed by Kelley [13], and was later formalized for RO by Mu- tapcic and Boyd [14]. The algorithm relies on the sequential identification of violated scenarios, namely uncertain parameter realizations q∗ ⊆ Q that cause infeasibility of one or more constraints in view of the current decision (x∗, w∗). These violations are subsequently enforced by adding in the master problem suitable deterministic-like constraints to guarantee the feasibility of the offend- ing constraints for these identified scenarios. The algorithm exits successfully when a certificate is obtained that the current master problem solution no longer violates any scenarios from within the uncertainty set. Conversely, when no feasible solution can satisfy the constraints already added in the master problem, then the problem can be announced robust infeasible. The main advantage of this approach lies on the fact that it can accommodate a much wider variety of settings, such as more complicated uncertainty sets (e.g., sets referencing discrete random variables or non-convex sets for which duality gaps arise). On the other hand, this approach can only provide certified robust feasible solutions only after successful termination of the iterative algorithm, meaning that a premature exit (due to time limit, for instance) will not lead to a practically useful outcome. The performance of the aforementioned approaches has been assessed in the literature (see, e.g., the extensive computational work by Bertsimas et al. [15]).

6 2.2 Traditional Uncertainty Sets

As mentioned earlier, one of the most critical elements of applying the RO methodology is the selection of the uncertainty set against which robust feasibility is sought. The shape and size of the set can have a direct impact on the conservatism of the robust optimal solution as well as the tractability of the robust counterpart. Hence, extensive literature effort has focused on defining uncertainty sets that manage the trade off among these two aspects while providing reasonably accurate descriptions of uncertainty in practical applications. Popular uncertainty sets that have been proposed in the literature include the box [16], ellipsoidal [17], cardinality-constrained [18], conic [19], and convex-constrained sets [20]. Furthermore, uncertainty sets describing the possible realizations of discrete uncertain parameters have also been proposed [21], enabling the modeling of scenario-based information.

In this work, we will focus on polyhedral uncertainty sets of the general form shown in (3), which generalize the cardinality-constrained sets. These sets are gaining increasing popularity in the literature due to their simplicity as well as their flexibility to accommodate all types of affine correlations, such as budgets or factor models [22]. Most importantly, polyhedral sets offer consid- erable numerical advantages by yielding robust formulations that belong to the same model class as the deterministic base model describing the problem of interest; for example, if the deterministic model is of mixed-integer linear form, then so will be the robust counterpart formulated in view of a polyhedral uncertainty set.   |q|  q ∈ R :      Q = Hq ≤ d (3)      q ≤ q ≤ q

Here, the vector d has size equal to the number of the parameter correlations (other than simple parameter bounds) one wishes to model in this set, storing their right-hand side constants.

Therefore, the matrix H, which stores the left-hand side coefficients multiplying the parameters in these correlations, has size |d| × |q|. The vectors q and q represent the applicable box bounds for the admissible realizations of the continuous uncertain parameters. Note that the constant nature of this set stems from the absence of decision variables in its definition. From an application point of view, this means that the uncertainty one seeks insurance against does not get affected in any

7 way from the decisions made. As we discuss later, this assumption can be quite conservative in many real life settings.

3 The Case of Endogenous Uncertainty

Uncertain parameters can be classiﬁed as exogenous, when they are not aﬀected by one’s decisions (e.g., weather conditions), or as endogenous, when the decision maker can manipulate their realization or ability to be observed. Most optimization problems studied in RO literature consider only the former type of uncertainty, which can be modeled using traditional, constant (i.e., not decision-dependent) uncertainty sets. In order to properly motivate the use of DDUS, we provide in the following some background on endogenous uncertainty.

Endogeneity arises due to various reasons. In certain cases, a decision may render a parameter referenced in a model physically meaningless (e.g., the price of a product under development that did not hit the market). In other cases, a decision may aﬀect the timing of a parameter realization

(e.g., the time at which we observe the true magnitude of a well production rate depends on when we decide to drill the well), essentially dictating whether the parameter reveals itself before the second, third, or some other later stage. Finally, a decision may affect an uncertain parameter’s stochastic support or distribution from which this parameter draws (e.g., the technology which we choose to invest on will affect the range of possible yields for the process). In the stochastic programming literature, endogenous uncertainty where the decisions can affect the underlying distribution is classified as Type-I endogenous uncertainty, while in the other cases it is classified as Type-II endogenous uncertainty [23, 24].

There are many important application settings where uncertainty is subject to the optimizer’s decisions (see, e.g., Jonsbr˚atenet al. [25], Goel and Grossmann [23]). In a monopolistic market, for instance, a decision to increase the production will have a negative eﬀect on product prices.

Other examples arise in oilﬁeld development planning [26], network capacity expansion [23], network interdiction problems [27], and the planning of clinical trials [28], to name but a few. These problems have traditionally been studied as two-stage or multi-stage stochastic programming problems, where the cartesian product between the set of decision variables aﬀecting the realization of uncertainty and the original set of discrete scenarios is considered, creating essentially a copy of

8 the scenario tree for each possible combination of decisions. As a result, the computational effort required to address the above problems can be prohibitive, especially in the case of multi-stage settings where the enforcement of non-anticipativity constraints dominates the model size. More recently, significant efforts have been made towards better modeling and solution approaches that are based on reducing the size of the scenario tree and efficiently enforcing the non-anticipativity restrictions [24]. Alternative approaches that model these problems as Markov decision processes have also been proposed [29].

At the same time, the above application settings constitute examples where the use of traditional

(constant) uncertainty sets in RO can prove limited. Since constant uncertainty set coeﬃcients do not allow functional changes in the applicable correlations they model (e.g., they cannot adapt to changes in the underlying distributions), one must make conservative approximations to be inclusive of all possibilities, admitting unrealistic worst-case realizations and leading to suboptimal solutions. To that end, we propose in this paper a more promising strategy that involves directly encoding in the uncertainty set itself the dependence of the parameters on the actual decisions.

3.1 Decision-dependent Uncertainty Sets

In order to address problems with endogenous uncertainty through RO in a generic enough fashion, we extend the constant polyhedral uncertainty set of the form (3) into a set that depends on the decision variables, as in (4).

  |q|  q ∈ R :      Q (x, w) = Hq + H0 (v(x, w) ◦ q) ≤ Gw + G0v(x, w) + d , (4)    n > o   H e 6= 0 ∨ v(x, w) = e ⇒ q ≤ q ≤ q 

|x| |w| |q| Here, v(x, w): R ×{0, 1} 7→ {0, 1} is a vector of problem-speciﬁc, binary-valued functions of our decision variables that indicate the materialization of each uncertain parameter in the vector q (see below for how to select these functions), the matrix H0 contributes left-hand side terms that must be removed when parameters do not materialize, while the matrices G and G0 contribute respectively direct and materialization-related decision dependency to the right-hand sides. The quantities q and q retain their deﬁnitions as bounds for the admissible realizations of parameters

9 q, but care should now be taken so that these bounds are wide enough to remain valid under all possible feasible decisions (x, w) for which the corresponding elements of v(x, w) attain the value of one. Conversely, if a parameter that does not participate in a constant left-hand side term also happens to not materialize (i.e., the corresponding materialization indicator attains the value of zero), leading to a situation where this parameter vanishes from the set in light of the specific decisions made, then there is no need to account for the parameter’s bounds at all. This is reflected by referencing the bound constraints inside the apodosis of the implication statement in the DDUS (4). We remark that our definition of a DDUS constitutes a generalization of the specially structured cardinality-constrained sets presented in the works by Poss [30] and Nohadani and Sharma [31] that only involve direct right-hand side decision dependency, covering only a subset of the modeling capabilities we introduce in this work. More specifically, our description encompasses their case by setting v(x, w) = e and G0 = O. Finally, it is noteworthy to mention that DDUS of the form (4) retain the properties of their constant precursors (3) with regards to the model class of the resulting robust counterpart formulation. More specifically, for a mixed- integer linear deterministic model as the basis, the robust counterpart under DDUS will also be mixed-integer linear programming representable.

Evaluation of Materialization Indicators: An uncertain parameter is said to materialize if and only if our decisions do not cause it to vanish from the model, which would happen if all occurrences of the parameter in the formulation are multiplied by variables (or expressions of variables) for which a value of zero has been chosen. There are many reasons why a parameter may not materialize under certain decisions. Often, parameters serve merely as big-M coefficients in terms that do not activate. Other parameters may be associated with tasks that are controlled by a decision of whether to execute them or not, in which latter case these parameters lose their physical meaning (they become unobservable). For example, the efficiency of a compressor will only materialize if the decision is made to turn on that compressor. If the decision is made to leave the compressor at its “off” state, then the efficiency that the compressor would have otherwise attained becomes irrelevant. Consequently, a formal description of the materialization indicator functions v(x, w) can be obtained by declaring a set of state variables v ∈ {0, 1}|q| and introducing

10 the following constraints in the RO counterpart model.

^ {v = 0} ⇔ {Cmx + Rmw = bm} (5) m∈M

We remark that these constraints often lead to an intuitive solution, usually of the form vi = wk, which relates a specific uncertain parameter qi to a specific binary decision wk that governs the materialization of the former. In such cases when the mapping from variables w to variables v is known to the modeler a priori, the materialization indicators can be directly replaced by the native binary variables (or the binary-valued expressions of those) governing their materialization. This allows for the numerical efficiency of not having to introduce new variables v, alleviating also any possible tractability burden associated with introducing the conditional conjunctive constraints (5) in the final RO model.

3.2 Illustrative Modeling Capabilities

In general, non-materialized parameters do not aﬀect the performance of our ﬁnal solution.

Furthermore, non-materialized parameters that lose their physical meaning in light of the optimal decisions should also not be referenced in the uncertainty set, giving rise to our need to generalize the left-hand sides of sets (4) into constant and decision-dependent terms for full modeling ﬂexibility.

We now brieﬂy illustrate a number of modeling conveniences that our proposed DDUS of the form (4) aﬀord us. Note that these are only a handful of possible examples that can be envisioned, given our sets’ quite general form.

1. Our DDUS allow for the introduction of decision-dependent distributional supports. For

example, Eq. (6) can model decision-dependent bounds for a parameter’s realization.

q ≤ q|{w=1} w + q|{w=0} (1 − w) (6)

2. Our DDUS allow for decision dependency in correlations among uncertain parameters. Among

other uses, this enables the modeling of uncertainty that arises from a set of possible discrete

scenarios. For example, Eq. (7) can model a decision-dependent cardinality budget, where

the maximum number of parameters that can attain their upper bound realization is limited

11 Figure 1: Cardinality budget DDUS for various decisions w1 + w2 + w3 = n, where n may vary between 0 (top left) and 3 (bottom right). The black dots signify the extremal admissible realizations in each case.

based on (e.g., investment type) decisions. Figure 1 illustrates this concept.

q1 + q2 + q3 ≤ w1 + w2 + w3 (7)

3. Our DDUS allow for eliminating the eﬀect of parameters that did not materialize as a result of

our decisions. For example, Eq. (8) removes the contribution of non-materialized parameters

by dynamically projecting the set into the lower-dimensional space of only the materialized

parameters. Figure 2 illustrates the diﬀerence in the shape of the applicable uncertainty set

when parameter q2 does and does not materialize, as governed by its corresponding materi-

0 alization indicator binary variable, v2. Here, the vector q holds the values for the nominal realizations.

0 0 0 v1q1 + v2q2 + v3q3 ≤ 1.1 q1v1 + q2v2 + q3v3 (8)

12 Figure 2: Polyhedral DDUS for the cases v = [1, 0, 1]> (shaded) and v = [1, 1, 1]> (transparent), corresponding to two and three materialized parameters, respectively.

4 Derivation of the Robust Counterpart

This section will focus on extending the reformulation based approach for solving mixed-integer linear RO problems of the generic form (2) under endogenous uncertainty modeled via DDUS of the form (4); that is, under the setting Q ← Q(x, w).

The first step towards deriving the finite-sized robust counterpart is to reformulate the semi- infinite problem (2) into its equivalent bi-level problem (9). Note how the materialization indicator variables v have become outer-level state variables, which are set to their appropriate values via constraints (5). However, as discussed earlier, these constraints often have an intuitive solution, allowing us to eliminate variables v (as well as the constraints in which these are referenced) from the formulation.

min x1 x,w,v |x| |w| |q| s.t. x ∈ R , w ∈ {0, 1} , v ∈ {0, 1}   >  max (Cmx + Rmw − bm) q   q     |q|   s.t. q ∈ R  > > ≤ am − cmx − rmw ∀ m ∈ M (9)  0 0   Hq + H (v ◦ q) ≤ Gw + G v + d     >   H e 6= 0 ∨ v = e ⇒ q ≤ q ≤ q  V {v = 0} ⇔ {Cmx + Rmw = bm} m∈M (x, w) ∈ F

13 We proceed by applying linear duality on the inner maximization problems, turning them into inner minimization problems. To that end, we introduce new sets of non-negative continuous variables pm, sm and tm to serve as the dual variables corresponding to the general correlations, upper bound facets, and lower bound facets of the uncertainty set, respectively.3 The ﬁnal step in the derivation calls for simply dropping the inner minimization operators, resulting in model (10).

min x x,w,v, 1 pm,sm,tm |x| |w| |q| s.t. x ∈ R , w ∈ {0, 1} , v ∈ {0, 1} |d| |q| |q| pm ∈ R+ , sm ∈ R+ , tm ∈ R+ ∀ m ∈ M 0 > > > > > (Gw + G v + d) pm + q sm − q tm ≤ am − cmx − rmw ∀ m ∈ M > 0> H pm + v ◦ H pm + sm − tm = Cmx + Rmw − bm ∀ m ∈ M (10)

> H e = 0 ∧ v = 0 ⇒ {sm ≤ 0} ∀ m ∈ M

> H e = 0 ∧ v = 0 ⇒ {tm ≤ 0} ∀ m ∈ M V {v = 0} ⇔ {Cmx + Rmw = bm} m∈M (x, w) ∈ F

We note that, unlike the traditional approach when one uses constant uncertainty sets, the robust counterpart corresponding to a DDUS involves bilinear terms between variables w and pm as well as between v and pm. Fortunately, since w and v are binary variables, these bilinear products can be linearized exactly using the standard McCormick (a.k.a., Glover) linearization

W technique. For this linearization, two sets of auxiliary non-negative continuous variables Pm and V > Pm is introduced, for each constraint m ∈ M, in order to replace respectively the products pmw > 4 and pmv in the formulation. The ﬁnal robust counterpart is presented in (11).

3 Note how the dual variables are subscripted over index m, since we have a separate such set of dual variables for each original constraint m ∈ M. 4 W We remark that variables in Pm need not be deﬁned (and the related linearization constraints need not be V included in the formulation) whenever the corresponding elements of matrix G are zero. Similarly, the set Pm can be restricted whenever the corresponding elements of matrix H0 and the corresponding elements of matrix G0 are both zero. Furthermore, in cases where the mapping from w to v is known, common variables can be V W utilized among various elements of variable sets Pm and Pm .

14 min x x,w,v, 1 pm,sm,tm, W V Pm ,Pm |x| |w| |q| s.t. x ∈ R , w ∈ {0, 1} , v ∈ {0, 1} |d| |q| |q| W |d|×|w| V |d|×|q| pm ∈ R+ , sm ∈ R+ , tm ∈ R+ , Pm ∈ R+ , Pm ∈ R+ ∀ m ∈ M > W 0 V > > > > > e G ◦ Pm + G ◦ Pm e + d pm + q sm − q tm ≤ am − cmx − rmw ∀ m ∈ M > > 0 V H pm + e H ◦ Pm e + sm − tm = Cmx + Rmw − bm ∀ m ∈ M > H e = 0 ∧ v = 0 ⇒ {sm ≤ 0} ∀ m ∈ M

> H e = 0 ∧ v = 0 ⇒ {tm ≤ 0} ∀ m ∈ M

W > Pm ≤ pme ∀ m ∈ M > W > ew = E ⇒ Pm ≥ pme ∀ m ∈ M > W ew = O ⇒ Pm ≤ O ∀ m ∈ M V > Pm ≤ pme ∀ m ∈ M > V > ev = E ⇒ Pm ≥ pme ∀ m ∈ M > V ev = O ⇒ Pm ≤ O ∀ m ∈ M V {v = 0} ⇔ {Cmx + Rmw = bm} m∈M (x, w) ∈ F (11)

Treatment of Implication Constraints: We should highlight that our final model (11) contains a number of implication constraints, which we have deliberately refrained from reformulating into a linearly representable form via, for example, the well-known big-M technique. The reason is that such reformulations require valid upper bounds for the expressions referenced in the apodoses of these implications, which we may not always know. More specifically, with the exception of the materialization indicator definitions of Eq. (5), our implication constraints reference dual variables for which we do not a priori possess safe upper bounds, in general.5 In fact, noting that an overly restricted dual LP results in an overly relaxed primal LP, the choice of a large but not rigorous dual variable upper bound values has the potential to admit solutions that do not insure against

5 In general, the dual variables must remain unbounded, in order to accommodate arbitrarily strong sensitivity of the optimal solution on the primal constraint right-hand sides.

15 the full uncertainty set we want to consider, i.e., are robust infeasible.

To that end, no implication constraint, except perhaps those of Eq. (5) (if present), should be enforced via an explicit reformulation. Instead, they should be enforced procedurally during the branch and bound search using, for example, the logical constraint facility of modern mixed-integer linear solvers [32–34]. In our implementation, we follow this numerically safe approach for all implication constraints, which in our experience exhibits also better numerical stability compared to solving reformulated models that are poorly scaled due to the presence of very large big-M coeﬃcients.

5 Computational Experiments

In this section we investigate the use of RO in the context of ﬁve problems that feature endogenous uncertainty and that have been previously studied in the literature using stochastic programming methodologies. The DDUS we introduced in this paper now facilitate our ability to cast and study these applications as RO problems as well. For reference, we provide in Appendix A the exact deterministic models that we used as the basis for deriving robust counterparts in each of the

ﬁve case studies we contemplate. More details about the problem statements, their notation, and exact data for the various benchmark instances can be found in the original literature source. For each case study, we deﬁne suitable polyhedral uncertainty sets that adhere to the general DDUS form of Eq. (4). The symbols ψ and φ in these sets refer to constants that we use to tune their size and shape, respectively. Exact values for these constants are provided later.

5.1 Case Studies

5.1.1 Case Study I: Capacity Expansion Problem

The ﬁrst case study is from Goel and Grossmann [23] and contemplates the capacity expansion of a process network, where intermediate product B has to be processed by process 3 so as to produce a high-value product A. Product B itself can be either purchased directly from the market, produced in-house by process 1 using raw material C, or produced in-house by process 2 using raw material D. The above options are not mutually exclusive and can be utilized in conjunction.

Moreover, if processes 1 or 2 are chosen, the cost for the installation of necessary equipment has to

16 be accounted for. The overall objective is to maximize the proﬁt within a 10-period horizon.

The endogenous uncertain parameters involved in this problem are the production yields of

6 processes 1 and 2, namely θ1,t and θ2,t, for each time period t ∈ T . Their endogeneity stems from the fact that, if a unit is not operated in a given time period, the corresponding production yield does not retain a physical meaning. To that end, the binary-valued indicator variable associated

θ with the materialization of each uncertain θi,t parameter would be vi,t = bi,t, where bi,t are the binary decisions of operating process i in time period t. Furthermore, the problem features one set of exogenous uncertain parameters, the total demands for product A, ξt, for each period t. Since ξ these are purely right-hand side parameters, their corresponding materialization indicator is vt = 1. We deﬁne a budget correlation among the materialized production yields for each of processes 1 and 2, which reﬂects the fact that these processes are expected to perform close to their nominal performance, on average across the horizon. In addition, each of the uncertain parameters is

0 0 0 bounded around their nominal realization values, θi,t and ξt . The values for θi,t were chosen as the 0 average among the low and high scenario levels reported in the source paper, while the value for ξt was set to 2.1 tons/hr, as reported in Apap and Grossmann [24]. The DDUS we use is as follows:

   θ ∈ , θ ∈ , ξ ∈ ∀ t ∈ T :   1,t R 2,t R t R     X X   b θ ≥ (1 − ψ φ) b θ0 ∀ i ∈ {1, 2}   i,t i,t i,t i,t  Q (b1,t, b2,t) = t∈T t∈T  0 0   {bi,t = 1} ⇒ (1 − ψ) θi,t ≤ θi,t ≤ (1 + ψ) θi,t ∀ t ∈ T ∀ i ∈ {1, 2}      0 0   (1 − ψ) ξt ≤ ξt ≤ (1 + ψ) ξt ∀ t ∈ T  (12)

5.1.2 Case Study II: Oﬀshore Oil Planning Problem

The second study originates from Goel and Grossmann [26] (specifically, example 4 in that paper), where an oil company has identified 5 offshore oil reserves and wants to plan its activities so as to develop them. In order to extract oil from each reserve, a well platform has to be installed with the necessary pipelines to transfer the oil to the production platform, and ultimately, to the shore. In addition, the company has at its disposal multiple extraction technologies, which come at

6 We remark that parameters θ3,t are not considered uncertain in this case study.

17 given costs and which can achieve diﬀerent initial production rates. The objective is to maximize the net present value of the oilﬁeld development project over a 15-year planning horizon.

The endogenous uncertain parameters involved in this problem are the initial deliverabilities,

InitDelivf , of each ﬁeld f ∈ F . These parameters are of endogenous nature because they do not retain a physical meaning if a well is not drilled. The materialization indicator variables in this

InitDeliv P case are vf = t∈T bf,t, where bf,t are binary variables in the model to decide if a well is InitDeliv A B C A/B/C drilled on ﬁeld f in time period t. Equivalently, vf = yf +yf +yf , where yf are binary variables that decide the type of extraction technology (A, B, or C) used in each case. Note that the model enforces that, for each ﬁeld f ∈ F , at most one of these variables can attain the value of

InitDeliv one, and hence, the indicators vf are properly restricted to binary evaluations. Finally, the total reserve sizes, Sizef , for each ﬁeld f ∈ F , are also uncertain. Since these are purely right-hand

Size side parameters, their materialization indicators are vf = 1. Motivated from the probability distributions assumed in the referenced paper, we impose budget correlations to restrict how far the sums (across all ﬁelds) of initial deliverabilities can deviate from their nominal values. In addition, each of the oilﬁeld sizes is bounded around their nominal

0 realization values, Sizef , while the initial deliverabilities are bounded depending on which technology was chosen in each case. More speciﬁcally, for technologies A, B, and C, respectively, we chose

0 0 0 0 values αf = 1.0 InitDelivf , βf = 1.1 InitDelivf , and γf = 1.2 InitDelivf , where InitDelivf is the nominal realization value for the initial deliverability of each ﬁeld f. All nominal values were chosen as the medium scenario levels reported in the paper. The DDUS we use is as follows:

   InitDelivf ∈ R, Sizef ∈ R ∀ f ∈ F :       X A B C X A B C   yf + yf + yf InitDelivf ≥ (1 − ψ φ) αf yf + βf yf + γf yf     f∈F f∈F       yA + yB + yC InitDeliv ≥ (1 − ψ) α yA + β yB + γ yC ∀ f ∈ F   f f f f f f f f f f  A B C   Q yf , yf , yf = A B C A B C  yf + yf + yf InitDelivf ≤ (1 + ψ) αf yf + βf yf + γf yf ∀ f ∈ F       yA + yB + yC = 1 ⇒ {InitDeliv ≥ (1 − ψ) min {α , β , γ }} ∀ f ∈ F   f f f f f f f       yA + yB + yC = 1 ⇒ {InitDeliv ≤ (1 + ψ) max {α , β , γ }} ∀ f ∈ F   f f f f f f f     0 0   (1 − ψ) Sizef ≤ Sizef ≤ (1 + ψ) Sizef ∀ f ∈ F  (13)

18 5.1.3 Case Study III: Clinical Trial Planning Problem

The third case study comes from Colvin and Maravelias [28] (speciﬁcally, example 2 in that paper) and constitutes an R&D portfolio optimization problem. Here, a pharmaceutical company has the opportunity to pursue the development of 5 potential drugs. Before any of them can generate revenue for the company, however, it has to undergo a series of 3 clinical trial phases of given durations. Only those drugs that succeed in all three phases are approved by the regulator and enter the market. There are costs associated with performing each trial, and the company has to plan the use of its limited R&D resources over the next 36-month horizon so as to maximize its portfolio’s net present value.

The endogenous uncertain parameters involved in this problem are αi,j, indicating the success or not of a clinical trial of drug i ∈ I in trial phase j ∈ J. Endogeneity arises due to the fact that such a parameter has no physical meaning if the corresponding trial is not attempted. The

α P materialization indicator variable in this case is vi,j = t∈T Xi,j,t, where Xi,j,t is the binary decision to pursue trial (i, j) in time period t ∈ T .

According to the data provided in the referenced paper, a trial has a higher probability to succeed in later phases than in earlier ones. To that end, we impose phase-speciﬁc correlations to restrict the average number of failures that may occur in the corresponding trials in a manner that is consistent with this observation. We remark that, unlike the previous case studies, the uncertain parameters in this example are of discrete (binary) nature, which in principle would prohibit us from formulating the robust counterpart via the duality-based derivation procedure described in Section 4. However, the special nature of the uncertainty set, namely the facts that (i) no continuous uncertain parameters are referenced in the correlations, and (ii) the coeﬃcients of the binary uncertain parameters constitute a totally unimodular matrix for all feasible decisions Xi,j,t, allow us to simply relax the integral domains of the uncertain parameters into their continuous relaxation without increasing the level of conservativeness. Therefore, the DDUS we use is as follows:

19    α ∈ ∀ (i, j) ∈ I × J :   i,j R     X X X X   Xi,j,tαi,j ≥ (1 − ψi,j) Xi,j,t ∀ j ∈ J  Q (Xi,j,t) = i∈I t∈T i∈I t∈T (14)  ( )   X   X = 1 ⇒ {0 ≤ α ≤ 1} ∀ (i, j) ∈ I × J  i,j,t i,j   t∈T 

5.1.4 Case Study IV: Pre-disaster Network Investment

The fourth case study is motivated from the work of Peeta et al. [27], where a highway network consists of 9 arcs that are subject to random failures. These failures can be prevented by investing on strengthening the corresponding arcs. The objective is to optimally select the arcs that have to be strengthened so as to guarantee the existence of a route from the source to the sink after the occurrence of the failures, while minimizing the necessary investment costs.

The endogenous uncertain parameters of this problem are the disaster outcomes, ξe, for each arc e ∈ E. More speciﬁcally, ξe attains the value of 1 if the arc e is not functional after the catastrophic event, and 0 if this arc remains functional. The endogeneity of these parameters arises from the fact that our decision to strengthen an arc makes it ineligible for failure, forcing the applicable upper bound realization for each parameter ξe to depend on ye, the binary decision to pursue the investment to strength arc e. However, despite their endogenous nature, these are right-hand side

ξ uncertain parameters, and hence, their materialization indicator variable would be ve = 1, i.e., constant and independent of our decisions. In other words, parameters ξe are an example of mere distributional support decision dependency, similar to the case demonstrated earlier via Eq. (6).

We shall now construct a DDUS for this application by introducing arc-speciﬁc facets to disallow the possibility of failure for those arcs that have indeed been strengthened. Furthermore, in order to reproduce the scenarios investigated in the referenced work, we postulate a budget correlation among failures of non-strengthened arcs, excluding from the set pessimistic scenarios where too many arcs fail simultaneously. We remark that, similarly to the previous case study, the constructed set satisﬁes the total unimodularity property, and thus, the binary uncertain parameters ξe can be relaxed into their continuous relaxation without any compromises in terms of conservativeness. The

DDUS we deﬁne for this case study is as follows:7

7 We remark that, in light of the variable upper bounds (ξe ≤ 1 − ye), the box upper bound (ξe ≤ 1) is redundant

20    ξ ∈ ∀e ∈ E :   e R       ξ ≤ 1 − y ∀ e ∈ E   e e  Q (ye) = X X (15)  ξe ≤ ψ (1 − ye)    e∈E e∈E       0 ≤ ξe ≤ 1 ∀e ∈ E 

5.1.5 Case Study V: Sizes Problem

The last case study originates in the work of Jonsbr˚atenet al. [25], where a product is available in 3 different sizes and the production takes place in 3 time periods with uncertain demands for each product size and period. An order for a specific size can be satisfied by products of larger size, an action that induces a fixed penalty cost. The objective is to decide the product sizes and the production levels for each time period in a way that minimizes the production costs while meeting the product demand.

The uncertain parameters in this case are the production costs, pi,t, and the demands, Di,t, for each product size i ∈ I and each period t ∈ T . The production costs are endogenous, since they are only materialized if it is decided to indeed run the production line. Consequently, the

p materialization indicator variable for a parameter pi,t shall be vi,t = yi,t, where yi,t is the binary variable in the model associated with the production decision. On the other hand, the demands

D are exogenous, right-hand side parameters; hence, vi,t = 1. For each product size i, we define a budget correlation among the associated–yet materialized– production costs, which reflects the fact that the latter are expected to realize close to their nominal value across the horizon, on average. We also define symmetric bounds around the nominal realization values for both the production costs and the demand. These nominal values, which are

0 0 respectively denoted as pi,t and Di,t, were chosen as the average among the low and high scenario levels reported in the paper. The DDUS we use is as follows:

and could have beeen omitted from the set’s deﬁnition in order to simplify the robust counterpart model.

21    p ∈ ,D ∈ ∀ t ∈ T ∀ i ∈ I :   i,t R i,t R     X X   y p ≤ (1 − ψ φ) y p0 ∀ i ∈ I   i,t i,t i,t i,t  Q (yi,t) = t∈T t∈T (16)  0 0   {yi,t = 1} ⇒ (1 − ψ) pi,t ≤ pi,t ≤ (1 + ψ) pi,t ∀ t ∈ T ∀ i ∈ I      0 0   (1 − ψ) Di,t ≤ Di,t ≤ (1 + ψ) Di,t ∀ t ∈ T ∀ i ∈ I 

5.2 Results

In order to investigate the trade-off between robustness and optimality, we instantiated and solved each case study under three different levels of uncertainty, namely low (L), medium (M), and high (H). More specifically, for case studies I, II, IV and V, these levels of uncertainty correspond to (ψ, φ) settings of (0.1, 0.25), (0.2, 0.50), and (0.3, 0.75), respectively. For case study III, we chose values ψi,j = 5 (1 − pi,j) ψ, where ψ refers to the three levels above for low/medium/high and constants pi,j correspond to the probability of drug i successfully undergoing trial in phase j (data from the referenced paper). All computational experiments reported in this paper were conducted using the commercial MILP solver CPLEX 12.7 [33], which ran on a single-thread

(limited via appropriate solver option) of an Intel Xeon E5-2687Wv3 @ 3.10GHz processor with

4GB of available RAM.

Table 1 reports normalized optimal objective values, elucidating the amount of risk premium

(diﬀerence between nominal deterministic and robust optimal solutions) that has to be paid so as to insure against the postulated level of uncertainty. The table also compares with robust optimal solutions obtained using standard, non-decision-dependent uncertainty sets (non-DDUS).8

We observe that DDUS-based RO solutions feature better objective values compared to their non-

DDUS counterparts, recovering much of the risk premium paid by the latter (yet without any compromises in the level of insurance against risk). This can be attributed to the fact that, in the case of non-DDUS, the non-materialized parameters (which are not dynamically removed from a non-DDUS) attain their best possible values (since they do not aﬀect the optimal solution), driving the materialized ones (which can negatively impact the optimal solution) to attain worse

8 The non-DDUS were derived by setting all materialization indicators to the value of one, v = e, and by replacing the right-hand side term Gw with its conservative envelope, max(x,w)∈F:{v(x,w)=e} Gw.

22 realizations.

Table 1: Robust optimal objective values for three levels of uncertainty (low, medium and high) when utilizing DDUS and non-DDUS. The entries are normalized against the optimal values of the corresponding nominal deterministic instances (= 100). Case studies I–III correspond to maximization problems (robustness leads to decrease in optimal value), while case studies IV–V correspond to minimization problems (robustness leads to increase in optimal value).

DDUS non-DDUS Case Study Det. LMHLMH

I 100 90 84 79 71 65 44

II 100 87 79 60 69 63 41

III 100 71 65 54 57 45 36

IV 100 118 123 125 135 147 152

V 100 128 132 139 141 149 154

Table 2 presents formulation sizes and associated solution statistics. We observe that, for the problems we investigated, DDUS-based RO did not require excessive sacrifices in terms of computational tractability. More specifically, the reformulation procedure did not introduce any new binary variables, since in all cases the materialization indicators could be mapped directly into binary variables that were native to (i.e., already present in) the deterministic model. On the other hand, the number of constraints and continuous variables increased significantly.

The computational times remained relatively small, and although larger instances would be better suited for investigating the impact on CPU times, the observed performance was encourag- ing. This opens up the prospect to apply DDUS to more challenging problems with more complex deterministic formulations and a larger number of uncertain parameters. In any case, we have to highlight that the number of binary variables–and hence the combinatorial complexity–in our robust counterparts increases much more mildly (if at all) with the number of uncertain parameters, as compared to the stochastic programming methodologies followed in the previous literature.

Furthermore, it does not increase with the number of correlations in which these parameters participate, nor with the number of scenarios considered. As a result, our approach can provide insights for many problems involving endogenous uncertainty that may not be currently addressable by

23 means of stochastic programming due to a prohibitive computational cost.

Table 2: Solution statistics (averaged across the three uncertainty levels) and model sizes for the deterministic (nominal) and robust (under DDUS and non-DDUS) formulations.

Case Study Formulation Avg. CPU Avg. Nodes Bin. Cont. Constraints time (in sec) Variables Variables

Deterministic 0.1 25 45 105 231 I Robust non-DDUS 0.4 33 45 872 489 Robust DDUS 1.4 57 45 1,120 657

Deterministic 2.0 108 125 632 410 II Robust non-DDUS 2.6 211 125 4,769 1,847 Robust DDUS 7.1 584 125 5,487 2,912

Deterministic 0.1 56 84 221 335 III Robust non-DDUS 0.5 103 84 1,477 1,781 Robust DDUS 1.4 182 84 1,865 2,612

Deterministic 0.1 32 18 25 81 IV Robust non-DDUS 0.1 35 18 121 477 Robust DDUS 0.1 55 18 159 782

Deterministic 0.1 22 22 36 103 V Robust non-DDUS 0.1 97 22 98 551 Robust DDUS 0.1 182 22 129 642

6 Conclusions

In this paper, we demonstrated that robust optimization can be an amenable framework for addressing decision-making problems under endogenous uncertainty, which are abundant in practice. In order to do this efficiently, we extended generic polyhedral uncertainty sets into their decision-dependent counterparts, and we showed how the latter offer additional modeling flexibility as well as reduce solution conservatism. Our proposed approach was illustrated through a number of applications mined from the stochastic programming literature, and our computational studies showcased that using decision-dependent uncertainty sets can provide considerably less conservative solutions while maintaining the general computational tractability advantages of robust

24 optimization. Finally, we remark that, although the discussion was focused on decision-dependent uncertainty sets of polyhedral form, the modeling paradigm exposed in this paper can be readily extended to decision-dependent uncertainty sets of more involved structure.

7 Acknowledgments

The authors gratefully acknowledge support from the National Science Foundation (grant No.

CBET-1510787). N.H.L. further acknowledges support from the University of Patras via an Andreas

Mentzelopoulos scholarship.

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27 Appendix A Deterministic Models

For reference, we present in this appendix the deterministic models we used as the basis for deriving robust counterparts in the ﬁve case studies we investigated. We have kept the notation as in the original models presented in the literature. Compared to the latter, however, we have solved out a few state variables in order to eliminate equality constrains that involved uncertain parameters. In certain cases, we have also chosen to expand the index list of some parameters in order to make the problem deﬁnition more general as well as more interesting from an application point of view. Any deviations of our models from the original sources are clearly documented after the presentation of each model.

28 Capacity Expansion Problem from Goel and Grossmann [23]

max ζ

X sales purch inv s.t. ζ ≤ + βtxt − αtxt − γtwt t∈T X X exp QE − FEi,tyi,t + VEi,tyi,t t∈T i∈I X X y rate − FOi,tbi,t + VOi,tyi,t t∈T i∈I X w rate rate rate − VOi,t (2 + θ3,t) θ1,ty1,t + (2 + θ3,t) θ2,ty2,t + (1 + θ3,t) y3,t t∈T inv rate rate rate purch sales wt ≥ θ1,tθ3,ty1,t + θ2,tθ3,ty2,t + θ3,ty3,t + xt − xt δt ∀ t ∈ T

sales xt ≥ ξt ∀ t ∈ T

inﬂow rate inﬂow L1 b1,t ≤ θ1,ty1,t ≤ U1 b1,t ∀ t ∈ T

inﬂow rate inﬂow L2 b2,t ≤ θ2,ty2,t ≤ U2 b2,t ∀ t ∈ T

inﬂow rate rate rate inﬂow L3 b3,t ≤ θ1,tθ3,ty1,t + θ2,tθ3,ty2,t + θ3,ty3,t ≤ U3 b3,t ∀ t ∈ T > X exp bi,t ≤ yi,t ∀ t ∈ T ∀ i ∈ I τ=1 exp yi,t ≤ bi,t ∀ t ∈ T ∀ i ∈ I Q Q QE wi,t = wi,t−1 + yi,t ∀ t ∈ T ∀ i ∈ I rate Q θ1,ty1,t ≤ w1,t ∀ t ∈ T rate Q θ2,ty2,t ≤ w2,t ∀ t ∈ T rate rate rate Q θ1,tθ3,ty1,t + θ2,tθ3,ty2,t + θ3,ty3,t ≤ w3,t ∀ t ∈ T

ζ ∈ R QE h QE i rate rate Q h Qi yi,t ∈ 0,Ui , yi,t ∈ 0,Ui , wi,t ∈ 0,Ui ∀ t ∈ T ∀ i ∈ I sales sales purch h purchi inv inv xt ∈ 0,Ut , xt ∈ 0,Ut , wt ∈ 0,Ut ∀ t ∈ T exp bi,t, yi,t ∈ {0, 1} ∀ t ∈ T ∀ i ∈ I

rate Compared to the model presented in the original source, the variables wi,t were solved out by eliminating equality constraints in which they participated. Furthermore, the parameters θi were augmented to θi,t in order to capture the possibility for time-dependent variations in the production yields. Data associated with the new variables were inherited from the original variables θi (i.e., same values across all periods t ∈ T ).

29 Oﬀshore Oil Planning Problem from Goel and Grossmann [26]

max ζ X X cum cum prod prod b s.t. ζ ≤ + cf,t qf,t + cf,t qf,t − cf,tbf,t t∈T f∈F X b exp out out − cwp,tbwp,t + cwp,tExpandwp,t − cwp,tqwp,t wp∈WP X b out out + cwp,wp0,tbwp,wp0,t − cwp,wp0,tqwp,wp0,t wp0∈WP " # X exp X b out out − cpp,tExpandpp,t + cwp,pp,tbwp,pp,t − cwp,pp,tqwp,pp,t p∈PP wp∈WP shr shr −ct qt

X A A B B C C − cf yf + cf yf + cf yf f∈F prod X cum qf,t ≤ InitDelivf bf,t − κf InitDelivf qf,t ∀ t ∈ T ∀ f ∈ F t∈T t cum X prod qf,t = qf,τ ∆τ ∀ t ∈ T ∀ f ∈ F τ=1 cum qf,t ≤ Sizef ∀ t ∈ T ∀ f ∈ F

out prod X out qwp,t = qwp,t + qwp0,wp,t ∀ t ∈ T ∀ wp ∈ WP wp0∈WP out X out X out qwp,t = qwp,pp,t + qwp,wp0,t ∀ t ∈ T ∀ wp ∈ WP pp∈PP wp0∈WP out X out qpp,t = qwp,pp,t ∀ t ∈ T ∀ pp ∈ PP wp∈WP shr X out qt = (1 − shrink) qpp,t ∀ t ∈ T pp∈PP out qwp,t ≤ Capwp,t ∀ t ∈ T ∀ wp ∈ WP out qpp,t ≤ Cappp,t ∀ t ∈ T ∀ pp ∈ PP

Capwp,t = Capwp,t−1 + Expandwp,t ∀ t ∈ T ∀ wp ∈ WP

Cappp,t = Cappp,t−1 + Expandpp,t ∀ t ∈ T ∀ pp ∈ PP A B C yf + yf + yf ≤ 1 ∀ f ∈ F X A B C bf,t = yf + yf + yf ∀ f ∈ F t∈T X bwp,t ≤ 1 ∀ wp ∈ WP t∈T X bpp,t ≤ 1 ∀ pp ∈ PP t∈T X X bwp,t = bwp,wp0,t + bwp,pp,t ∀ t ∈ T ∀ wp ∈ WP wp0∈WP pp∈PP t X 0 bwp,wp0,t ≤ bwp0,τ ∀ t ∈ T ∀ wp ∈ WP ∀ wp ∈ WP τ=1 t X bwp,pp,t ≤ bpp,τ ∀ t ∈ T ∀ pp ∈ PP ∀ wp ∈ WP τ=1 0 bwp,wp0,t + bwp0,wp,t ≤ 1 ∀ t ∈ T ∀ wp ∈ WP ∀ wp ∈ WP

(Model continued on next page) t out X qf,t ≤ M bf,τ ∀ t ∈ T ∀ f ∈ F τ=1 t out X qwp,t ≤ M bwp,τ ∀ t ∈ T ∀ wp ∈ WP τ=1 t out X qpp,t ≤ M bpp,τ ∀ t ∈ T ∀ pp ∈ PP τ=1 t out X 0 qwp,wp0,t ≤ M bwp,wp0,τ ∀ t ∈ T ∀ wp ∈ WP ∀ wp ∈ WP τ=1 t out X qwp,pp,t ≤ M bwp,pp,τ ∀ t ∈ T ∀ pp ∈ PP ∀ wp ∈ WP τ=1

Expandwp,t ≤ Mbwp,t ∀ t ∈ T ∀ wp ∈ WP

Expandpp,t ≤ Mbpp,t ∀ t ∈ T ∀ pp ∈ PP

ζ ∈ R out prod cum qf,t , qf,t , qf,t ∈ R+ ∀ t ∈ T ∀ f ∈ F out qpp,t, Cappp,t, Expandpp,t ∈ R+ ∀ t ∈ T ∀ pp ∈ PP out qwp,t, Capwp,t, Expandwp,t ∈ R+ ∀ t ∈ T ∀ wp ∈ WP out 0 qwp,wp0,t ∈ R+ ∀ t ∈ T ∀ wp ∈ WP ∀ wp ∈ WP out qwp,pp,t ∈ R+ ∀ t ∈ T ∀ pp ∈ PP ∀ wp ∈ WP shr qt ∈ R+ ∀ t ∈ T A B C bf,t, yf , yf , yf ∈ {0, 1} ∀ t ∈ T ∀ f ∈ F

bpp,t ∈ {0, 1} ∀ t ∈ T ∀ pp ∈ PP

bwp,t ∈ {0, 1} ∀ t ∈ T ∀ wp ∈ WP

0 bwp,wp0,t ∈ {0, 1} ∀ t ∈ T ∀ wp ∈ WP ∀ wp ∈ WP

bwp,pp,t ∈ {0, 1} ∀ t ∈ T ∀ pp ∈ PP ∀ wp ∈ WP

deliv Compared to the model presented in the original source, the variables qf,t were solved out by eliminating equality constraints in which they participated. Furthermore, the model was augmented to involve a set of available extraction technologies, namely A, B and C, only one of which may be

A B C used per oil ﬁeld, as dictated by an SOS1 set of decision variables yf , yf and yf . The absence of a technology investment on a given ﬁeld imposes a no-production constraint, while the utilization

A B C of a technology contributes a field-specific objective cost, denoted as cf , cf and cf for each of the A B three available options. In our case study, we chose these costs to be (in $ mil.) cf = 0.5, cf = 1, C and cf = 3, for all fields f ∈ F .

31 Clinical Trials Planning Problem from Colvin and Maravelias [28]

max ζ

X X h low i s.t. ζ ≤ + αi,1 revi Xi,1,t i∈I t∈T X X h med D L i + αi,2 revi Xi,2,t − γi (Zi,2,t) − γi (t + τi,2Xi,2,t) i∈I t∈T X X h max D L i + αi,3 revi Xi,3,t − γi (Zi,2,t + Zi,3,t) − γi (t + τi,3Xi,3,t) i∈I t∈T

Yi,j,t = Yi,j,t−1 + Xi,j,t−τi,j ∀ t ∈ T : t > 1 ∀ j ∈ J ∀ i ∈ I

Zi,j,t = Zi,j,t−1 + Xi,j−1,t−τi,j − Xi,j,t ∀ t ∈ T : t > 1 ∀ j ∈ {2, 3} ∀ i ∈ I

Zi,1,1 = 1 − Xi,1,1 ∀ i ∈ I

Zi,1,1 = Zi,1,t−1 − Xi,1,t ∀ t ∈ T : t > 1 ∀ i ∈ I X Xi,j,t ≤ 1 ∀ t ∈ T : t > 1 ∀ i ∈ I t∈T t X X X max ρi,j,rXi,j,t0 ≤ ρr ∀ t ∈ T ∀ r ∈ R 0 i∈I j∈J t =t−τi,j +1 t−1 X Xi,j,t ≤ αi,j−1 Xi,j−1,t0 ∀ t ∈ T : t > 1 ∀ j ∈ {2, 3} ∀ i ∈ I t0=1 ζ ∈ R

Yi,j,t,Zi,j,t ∈ [0, 1] ∀ t ∈ T ∀ j ∈ J ∀ i ∈ I

Xi,j,t ∈ {0, 1} ∀ t ∈ T ∀ j ∈ J ∀ i ∈ I

32 Pre-disaster Network Investment Problem from Peeta et al. [27]

min ζ X s.t. ζ ≥ (ceye + lexe) e∈E X ceye ≤ B e∈E X X xe − xe = 1 e=(0,j)∈E e=(j,0)∈E X X xe − xe = 0 ∀ i ∈ V \{0,D} e=(i,j)∈E e=(j,i)∈E X X xe − xe = −1 e=(D,j)∈E e=(j,D)∈E

xe ≤ 1 − ξe ∀ e ∈ E

ζ ∈ R

xe ∈ [0, 1] ∀ e ∈ E

ye ∈ {0, 1} ∀ e ∈ E

33 Sizes Problem from Jonsbr˚atenet al. [25]

min ζ   X X  X  s.t. ζ ≥ (σ yi,t + pi,tzi,t) + ρ xi,j,t t∈T i∈I j∈I: j

zi,t ≤ Myi,t ∀ i ∈ I X zi,t ≤ ct ∀ t ∈ T i∈I X xi,j,t ≥ Di,t ∀ t ∈ T ∀ i ∈ I j∈I: j≥i ζ ∈ R

zi,t ∈ [0, ct] ∀ t ∈ T ∀ i ∈ I

xi,j,t ∈ [0, ct] ∀ t ∈ T ∀ j ∈ I : j < i ∀ i ∈ I

yi,t ∈ {0, 1} ∀ t ∈ T ∀ i ∈ I

Compared to the model presented in the original source, the parameters pi were augmented to pi,t in order to allow for time-dependent variations in the costs of production. Data associated with the new variables were inherited from the original variables pi (i.e., same values across all periods t ∈ T ).