A Two-Stage Chance-Constrained Mean-Risk Stochastic Programming Model for Single-Machine Scheduling. Manuscript Draft Not to be shared

Transcription

1 A Two-Stage Chance-Constrained Mean-Risk Stochastic Programming Model for Single-Machine Scheduling Manuscript Draft Not to be shared Kerem Bülbül Sabancı University, Industrial Engineering Program, Orhanlı-Tuzla, Istanbul, Turkey. Simge Küçükyavuz Department of Integrated Systems Engineering, Ohio State University, Columbus, Ohio, USA. Nilay Noyan, Halil Şen Sabancı University, Industrial Engineering Program, Orhanlı-Tuzla, Istanbul, Turkey. June 3, 26 Abstract: The traditional single-machine literature assumes that parameters/data including the processing times are known with certainty at the time of decision making and most of the classical stochastic scheduling studies focus on analyzing the expected performance by assuming that the uncertain parameters follow specific distributions. In this study, we consider single-machine scheduling problems in the presence of uncertain parameters, and introduce a new class of risk-averse stochastic programming models. In particular, we develop a chance-constrained two-stage stochastic programming model; it features a mean-risk objective, where the conditional value-at-risk (CVaR) is specified as the risk measure, and a joint probabilistic constraint is enforced on the feasibility of the second-stage problem corresponding to the optimal timing problem. To solve this computationally challenging scenario-based optimization model, we devise an exact Benders decomposition-based branch-and-cut algorithm. We present an extensive computational study to demonstrate the value of the proposed model and the computational effectiveness of the solution algorithm. Keywords: single-machine scheduling; stochastic scheduling; stochastic programming; risk-averse two-stage; conditional value-at-risk; probabilistic constraints; Benders decomposition; branch-and-cut. Introduction The machine scheduling literature has mostly ignored the risk that may arise from the uncertainty inherent in the problem parameters. For instance, processing times may be uncertain due to the possible machine breakdowns, variable setup times, and inconsistency of the worker performance and tool quality. Research has either adopted a deterministic point of view or a risk-neutral approach where in most cases distributional assumptions are put in place for uncertain parameters and the focus is on optimization featuring the expected values of the random performance measures. In practice, the performance of either approach is closely related to the degree of uncertainty (random variability). Clearly, with increased uncertainty the solution of a deterministic or a risk-neutral scheduling model may produce undesirable performance for some realizations of the random data and be considered as risky. Building upon this perspective, in this study we focus on developing scheduling optimization models that explicitly factor in the uncertainty in the parameters and hedge against the risk resulting from the inherent randomness. In this regard, we employ scenario-based stochastic programming one of the fundamental approaches to model uncertainty, and propose a new two-stage stochastic programming model featuring a joint probabilistic (chance) constraint and a risk measure. We review the most closely related studies and refer to Pinedo (28) and Atakan et al. (26) for an excellent overview of conventional stochastic scheduling and a detailed review on recent developments in machine

2 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 2 scheduling under uncertainty. There are only a few papers studying risk-averse machine scheduling problems involving a risk measure (Beck and Wilson, 27; Sarin et al., 24; Atakan et al., 26). Sarin et al. (24) and Atakan et al. (26) focus on minimizing the very popular and widely applied risk measures conditional value-at-risk (CVaR) and value-at-risk (VaR) in single-machine scheduling, respectively, and both studies propose a scenario-based optimization model independent of the specific performance measure of interest and devise a scenario decomposition-based algorithm. The decomposable structure of CVaR allows Sarin et al. (24) to adapt a classical Benders decomposition-based method for minimizing CVaR similar to those in Ahmed (26) and Noyan (22). On the other hand, the VaR minimization problem does not exhibit such a direct decomposable structure due to the presence of the scenario-linking constraint that reflects the fundamental structure of VaR. To deal with the VaR-related linking constraint, Atakan et al. (26) adapt the Lagrangian relaxationbased scenario decomposition method proposed by Carøe and Schultz (999), and employ this method to obtain lower bounds on the optimal VaR. In addition, the authors provide a stabilized cut generation algorithm to solve the Lagrangian dual problem, and obtain promising schedules for the original problem by using a primal heuristic. Beck and Wilson (27) also minimize VaR; however, they focus on a specific performance measure the makespan in job shops. These authors use the term probabilistic minimum makespan or α-minimum makespan instead of VaR at the confidence level of α. They assume that the processing times are independent random variables with known probability distributions and develop several heuristic search algorithms which integrate Monte Carlo sampling with deterministic scheduling algorithms. Thus, instead of solving a scenario-based optimization model, Beck and Wilson (27) base their solution method on solving a particular deterministic counterpart problem and evaluating the given solutions using Monte Carlo simulation. A machine scheduling problem is commonly tackled in two-phases. The output of the first phase is a feasible job processing sequence for each machine present in the problem. In second phase, the optimal start and completion times of the jobs are calculated given the fixed processing sequences. The sequencing component gives rise to the notoriously complex combinatorial structure of machine scheduling problems; however, the optimal timing problem solved in the second phase is simple in many well-known settings. In particular, on a single machine the optimal timing problem is straightforward for regular performance measures, such as the total tardiness and total completion time, which are non-decreasing in the completion times. The optimal timing solution for this case is obtained trivially by scheduling jobs consecutively without any idle time in between. Consequently, the completion times may be expressed in closed form for any feasible sequence, and we can model a single-machine scheduling problem with uncertain parameters as a single-stage stochastic program as in Atakan et al. (26). For non-regular performance measures, on the other hand, the optimal timing problem is not trivial any more, but it can often be solved by a low-order polynomial time algorithm or as a linear programming problem (Kanet and Sridharan, 2). In this case, it is possible to set the job start and completion time decisions optimally by inserting idle times in the schedule upon revealed uncertainty. This rationale is is naturally captured in a two-stage stochastic programming model, where in the second stage an optimal timing problem is solved. In our study, we focus on the latter non-trivial situations and introduce a class of risk-averse models for machine scheduling problems in a two-stage stochastic programming setting. There has been increasing interest in the stochastic programming literature to develop risk-averse two-stage optimization models. A line of research focuses on incorporating risk measures to model decision makers risk preferences (Ahmed, 26; Schultz and Tiedemann, 26; Fábián, 28; Miller and Ruszczyński, 2; Noyan, 22). As discussed above, incorporating a risk measure is crucial to capture the effect of the inherent random variability. In such two-stage modeling approaches, it is common to guarantee that the second-stage problem is feasible under each joint realization of the random parameters, which may lead to overly conservative solutions. To avoid such overly conservative solutions one can relax some constraints in the second-stage problem and

3 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 3 penalize the violation in the relaxed constraints (see, e.g., Noyan, 22). Alternative to this quantitative approach, the violation of the relaxed constraints can also be kept under control by qualitative means. The related research stream studies chance-constrained two-stage programs (see,e.g., Luedtke, 24; Hong et al., 25; Liu et al., 26; Zeng et al., 24; Prékopa and Unuvar, 25) and introduces a probabilistic constraint on the infeasibility of the second-stage problem. Chance (or probabilistic) constraints are widely used in many engineering applications in a variety of fields to restrict the probability of certain undesirable events, and they are closely related to the concept of reliability and service-level constraints. The interested reader is referred to Prékopa (995) and Dentcheva (26) for reviews on chance-constrained optimization models. From a modeling perspective, Liu et al. (26) and Sarin et al. (24) are the studies most closely related to ours. Liu et al.(26) study a class of risk-neutral chance-constrained two-stage stochastic programming models, which involve additional costs associated with the infeasibility of the second-stage subproblems. By considering the cost of the recovery actions taken when a second-stage subproblem is infeasible, they extend the model proposed by Luedtke (24) which only focuses on guaranteeing the feasibility of the second-stage problem with a specified high probability. We extend the study of Liu et al. (26) and Sarin et al. (24) (and also Noyan (22)) by incorporating a risk measure into the objective function and enforcing a joint chance constraint on the feasibility of the second-stage problem, respectively. Thus, we propose a hybrid modeling approach which takes into account both the quantitative and qualitative aspects of risk. This risk-averse approach also provides a flexible way of modeling preferences in stochastic multi-criteria decision making. The existing risk-averse studies mentioned previously consider a single performance measure, such as the total weighted tardiness. On the other hand, our model can incorporate several conflicting criteria, and varying the specified confidence level of CVaR and the probability level of the joint chance constraint allows us to balance the trade-off between the multiple conflicting criteria. Stochastic programming models are generally known to be computationally challenging partially attributed to the potentially large number of scenario-dependent variables and constraints. Introducing integer variables and a joint chance-constraint, resulting in non-convex programs in the finite probability case, further complicate the solution of these models. In the context of machine scheduling, the inherent combinatorial complexity of these problems is an additional confounding factor. The computational challenges of scenario-based stochastic programming models for single-machine scheduling problems have already been illustrated in Sarin et al. (24) and Atakan et al. (26). Various decomposition-based solution methods have been proposed to deal with stochastic programming models with an emphasis on the risk-neutral two-stage stochastic programs. Along these lines, to solve our proposed model, we develop a Benders decomposition-based branch-and-cut algorithm, which relies on the recent developments proposed in Luedtke (24) and Liu et al. (26). In particular, we adapt the feasibility and optimality cuts presented in Luedtke (24) and Liu et al. (26), respectively, to guarantee the satisfaction of the joint chance-constraint and the exact calculation of the optimal second-stage objective function value. In addition, we use the basic linear representation of CVaR to approximate the CVaR measure in the objective function (similar to Noyan, 22; Sarin et al., 24). Our proposed models are general enough to reflect the randomness in any type of parameter, and the solution methods are designed to be independent from the structure of particular scheduling performance measure of interest. We also note that our proposed modeling framework and the solution method are not limited to singlemachine scheduling but could also be applied to a wide variety of different scheduling problems. Moreover, the proposed framework is not limited to CVaR. We may also consider other risk measures, for which the proposed algorithmic framework would be valid. Ahmed (26) investigates several risk measures for which the mean-risk function preserves the convexity, and thus, slight variants of the existing Benders decomposition based methods are applicable to solve the mean-risk two-stage stochastic linear programs. To the best of our knowledge, the

4 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 4 proposed flexible hybrid risk-averse modeling approach is a first in the scheduling literature. The rest of the paper is organized as follows. In Section 2, we describe the two-stage chance-constrained mean-risk optimization model and our scenario decomposition-based solution method in a general form. In Section 3, we introduce the new risk-averse optimization model for a single-machine scheduling problem and present the corresponding mathematical programming formulations. Section 4 is dedicated to the computational study, while Section 5 contains our concluding remarks. 2. Stochastic Optimization Models In this section, we present the proposed stochastic optimization model and a Benders decomposition method for its solution. To emphasize the versatility of our approach, we present the model and the method in a general form. In Section 3, we describe a particular scheduling problem, where we specify the performance and risk criteria explicitly. However, the model and method we present are more generally applicable to other performance criteria in scheduling, as well as other operations research problems outside of the scheduling domain. 2. General Form of the Model Weconsiderafiniteprobability space(ω,2 Ω,P). LetΩ = {ω,...,ω m } with corresponding probabilities ψ,...,ψ m, and S = {,...,m}. We denote the first-stage decision variables by x X and assume that X R n + is a non-empty polyhedron defined by the deterministic constraints of the problem. The vector ξ(ω) denotes the input data of the second-stage problem, some of which are random. The random performance measure associated with the second-stage problem designated by the functional Q(x, ξ(ω), β(ω)) depends on the first-stage decision vector x and the state of the second-stage problem β(ω) {,}. We assume that there are two possible states of the system: ideal and non-ideal, which are denoted by β(ω) = and β(ω) =, respectively. Then, f(x,ω) = c x+q(x,ξ(ω),β(ω)) is the random total cost associated with any first-stage decision x. In addition to the second-stage performance measure Q(x, ξ(ω), β(ω)) of primary interest, we assume that there is a secondary performance measure (goal). It is highly desirable that the decisions satisfy the goal constraints described by T(ω)x + W(ω)y h(ω) for some second-stage decision y R n2 +. Note that these are not hard constraints that prescribe a feasible set of operations, but they depict a desirable set of operations. For a given x X and an elementary event ω s Ω, we define the ideal second-stage problem (ISSP) as follows: Q(x,ξ(ω s ),β s = ) = minimize y {q s y : T s x+w s y h s, y R n2 + }. () Here, ξ(ω s ) = (q(ω s ),T(ω s ),W(ω s ),h(ω s )) := (q s,t s,w s,h s ) denotes the realization of the random data ξ(ω) under the elementary event ω s, and β s = β(ω s ). This ideal problem characterizes the set of operations that are desirable with respect to our goal; however, satisfying this goal typically comes at a high cost Q(x, ξ(ω), ). In fact, if we adopt the convention that Q(x,ξ(ω),) = whenever () is infeasible and aim at minimizing f(x,ω) with Q(x,ξ(ω),β(ω) = ), then the goal constraints are required to hold for every scenario, and we may end up with an overly conservative solution. Therefore, we propose a chance-constrained optimization model that enforces the goal constraints with a specified high probability by taking the path suggested by Liu et al. (26). In particular, we strike a middle ground by not enforcing the goal constraints for every scenario and not ignoring the undesirable outcomes in the objective. Instead, when the first-stage decisions lead to infeasible or highly costly decisions in the ideal second-stage problem we allow a subset of the goal constraints to be violated at the expense of incurring additional penalties in the objective function by designing a non-ideal second-stage problem (NISSP): Q(x,ξ(ω s ),β s = ) = minimize y, u {q s y+ q s u : T s x+w s y+u h s, y R n2 +, u R r +}. (2) This is referred to as the simple recovery problem in Liu et al. (26). We assume that there exists a feasible first-stage decision vector x such that Q(x,ξ(ω),), is finite for all ω Ω. Note that we can represent the

5 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 5 second-stage cost as Q(x,ξ(ω s ),β s ) = minimize y, u {q s y+ q s u : T s x+w s y+u h s,u M s β s y R n2 +, u R r +}, where M s is an r-dimensional vector of large enough constants that represent componentwise upper bounds on the vector u. The general form of the chance-constrained two-stage mean-risk stochastic linear programming problem (CCMRP) is then given by minimize ( λ)e ( c x+q(x,ξ(ω),β(ω)) ) +λσ ( c x+q(x,ξ(ω),β(ω)) ) (3) subject to P{x P(ω)} ǫ, (4) β(ω) = = x P(ω), ω Ω, (5) x X, β(ω) {,}, ω Ω, (6) wherep(ω) = {x X : y R n2 +, W(ω)y h(ω) T(ω)x}foranyω Ω, andσ( )isafunctional representing the risk associated with the random outcome f(x,ω) by a scalar value. In fact, P(ω) is the set of first-stage decisions for which the ideal second-stage problem under the specified scenario is feasible. In this mean-risk approach, λ is a non-negative trade-off coefficient referred to as the risk coefficient representing the exchange rate of mean cost for risk. This is a novel model and combines quantitative and qualitative measures of risk in the objective (3) and constraints (4), respectively. Observe that the joint probabilistic constraint (4) ensures that the ideal second-stage problem () is feasible for a subset of the scenarios, denoted by S, which satisfies the condition s S ψ s ǫ. In other words, the joint probabilistic constraint (4) guarantees that u s = for s S, and for the remaining (infeasible) scenarios, denoted by S, we penalize the violation amounts by introducing the term q s u s into the objective function. In a scheduling context, the most common parameters of interest include the processing times, release dates, due dates, and weights. The central assumption in deterministic machine scheduling theory and practice is that the exact values of these parameters (or their point estimates) are available to the dispatcher at the time the job processing sequence or the full schedule is constructed. Clearly, this assumption may not be warranted in many practical settings, and the model proposed in this research allows for randomness in any one of these parameters. Ultimately, our objective in the scheduling application presented in Section 3 is to account for the risk associated with the inherent uncertainty using the quantitative and qualitative risk management constructs embedded into CCMRP. In our scheduling application, the first-stage decisions x refer to the non-preemptive static job sequencing decisions, which will be determined by the dispatcher before any uncertainty is revealed. This fixed job processing sequence is referred to as a non-preemptive static list policy in the stochastic scheduling terminology(see, e.g., Pinedo, 28). A central theme in scheduling is the service level in a make-to-order system, in which jobs have associated due dates or deadlines. The joint chance constraint in CCMRP will help attain a high service level. In line with this rationale, in the ideal second-stage problem the due dates will have to be met, while in the non-ideal case they can be violated by incurring additional costs. As the random performance criterion, we will take both the earliness commonly regarded as inventory holding cost and a cost component related to tardiness into account. However, it is important to recognize that the model and the associated solution techniques in this section are more general and would be able to handle other scheduling performance criteria, environments, and restrictions as well. In this work, we let the risk measure σ( ) be given by CVaR α ( ). CVaR is a widely applied risk measure with several useful properties. In particular, it satisfies the axiomatic properties monotonicity, subadditivity, positive homogeneity, and translation equivariance required for coherence (Artzner et al., 999). Even if coherence is a very well received concept, we acknowledge that some recent studies (see, e.g., Kou et al., 23; Atakan et al., 26) question the necessity of the subadditivity axiom. In our study, we follow the common school of thought

6 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 6 in the literature, which views coherence as a desirable concept from a risk management perspective, and benefit from the computationally tractable representation of CVaR in our formulations. Moreover, note that CVaR can be used to express a wide range of risk preferences, including risk-neutral (for α = ) and pessimistic worst-case (for sufficiently large values of α) preferences. In Appendix A, we present some basic definitions and results related to CVaR for the case of finite probability spaces and the convention that smaller values of random variables are preferred. Remark 2. Note that our proposed model subsumes some well-known two-stage models as special cases: for ǫ =, we obtain the traditional risk-neutral two-stage model and its counterpart minimizing CVaR (as in Noyan (22) and Sarin et al. (24)) for λ = and λ =, respectively. Setting λ = yields the model proposed by Liu et al. (26), and omitting the terms related to risk and the second stage from the objective function provides us with the model proposed by Luedtke (24). By using the translation invariance property of CVaR and the linear representation of CVaR α provided in (47) in Appendix A, where η denotes VaR α the value-at-risk at the confidence level α and w s stands for the realization of the random performance measure in excess of VaR α under scenario s, the deterministic equivalent formulation of the two-stage problem CCMRP is obtained as follows: minimize c x+( λ) ( ) ( ψ s q s y s + q ) s u s +λ η + ψ s w s α s S s S subject to x X, u s M s β s, s S (9) T s x+w s y s +u s h s, s S () ψ s β s ǫ, () s S w s q s y s + q s u s η, s S (2) y s R n2 +, β s {,}, u s R r +, s S w R m +, η R. The constraints (2) establish the relationship between VaR α and w s,s S, and these values are then used to calculate CVaR through the final term in the objective (7). The knapsack constraint () is the linear reformulation of the joint chance constraint (4). In the equal probability case, i.e., when ψ s = m,s S, these constraints can be replaced by β s mǫ. Zhang et al. (24) give valid inequalities for a related s S deterministic equivalent formulation of a multi-stage risk-neutral chance-constrained program without recovery. (7) (8) 2.2 Benders Decomposition-Based Branch-and-Cut Algorithm As discussed in Section, the challenging nature of the scheduling problems stems from the sequencing aspect. A sequence can only be represented by employing a set of binary variables and related constraints. The interested reader is referred to Keha et al. (29) and Atakan et al. (26), respectively, for discussions and comparisons of alternate integer programming formulations of machine scheduling problems in deterministic and stochastic settings. In the formulation of CCMRP, the integrality restrictions and constraints enforcing feasible job processing sequences are embedded into X. In contrast, for a majority of the well-studied machine environments and performance criteria including makespan, total (weighted) completion time, total (weighted) tardiness, total (weighted) earliness/tardiness the optimal timing problem only features continuous variables. Section 3 will illustrate these concepts with a concrete machine scheduling example.

7 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 7 Motivated by the common property of machine scheduling problems mentioned above, we assume that the second-stage problem in CCMRP only contains continuous decision variables. This is already reflected in our definitions of the second-stage problems ISSP and NISSP specified in () and (2), respectively. Consequently, CCMRP lends itself to reformulation via the Benders decomposition principle. The first-stage decisions x, β, and the related constraints (8), (), and (2) are kept in the master problem, while the constraints (9)-() are relegated to the Benders subproblem. The variable θ s is defined to represent a lower bound on the optimal objective function value of the second-stage problem under scenario s improved iteratively by adding Benders optimality cuts and replaces the relevant terms in the objective (7) and constraints (2). Ultimately, the Benders relaxed master problem takes the following form: (RMP) minimize c x+( λ) s Sψ s θ s +λ subject to (8),(), ( η + α ) ψ s w s feasibility cuts (x, β) F, (4) (multi) optimality cuts (x, β, θ) O, (5) s S (3) w s θ s η, s S (6) w R m +, η R, β {,} m. (7) Here, F and O denote the set of feasibility and optimality cuts generated so far, respectively. At some node of the branch-and-cut tree, some of the β variables are typically fixed, and the formulation (3)-(7) can be augmented with the additional restrictions β s =, s S, and β s =, s S. Finally, before proceeding with the details of the feasibility and optimality cuts, we point out that the representation of CVaR α in (RMP) trivially follows from its definition and is very closely related to the sub-gradient based algorithm see Algorithm 2 in Noyan (22). The same representation was employed recently by (Sarin et al., 24) as well. However, note that there are other options to represent and approximate the CVaR α term in the objective function (see, e.g., Philpott and de Matos, 22; Zhang et al., 26). Feasibility cuts: In the setup of CCMRP, the constraints (5) mandate that given a first-stage solution ( x, β), the second-stage problem associated with scenario b must be feasible if the corresponding state variable β b is set to zero in (RMP). However, if β b = at the current solution of (RMP), then there is no restriction on the feasibility of the second stage problem. Thus, if β b = for some b S, we check whether ISSP associated with scenario b specified in () is feasible. If ISSP is infeasible, then we cannot have β b = and need to add a feasibility cut following the procedure described in Luedtke (24). We summarize this procedure for completeness. First, we obtain the extreme ray π associated with the dual of () for scenario b that yields the inconsistent solution. Next, wesolvethefollowingsinglescenariooptimization problemforallscenarioss S andα = π T b : h s (α) = minimize αx subject to T s x+w s y h s, x X, y R n2 +. Having obtained the values h s (α) for s S where α = π T b, we sort them to obtain a permutation σ such that: h σ (α) h σ2 (α) h σm (α). Let a be the largest integer such that a j= ψ σ j(α) ǫ. Under equal probabilities, we directly obtain a = mǫ. Then, consider a subset Υ = {υ,υ 2,...,υ l } {σ,σ 2,...,σ a } such that h υi (α) h υi+ (α) for i =,...,l,

8 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 8 where υ = σ and υ l+ = σ a+. It turns out that the inequality αx+ l ( hυi (α) h (α)) υi+ β υi h υ (α) (8) i= is a feasibility cut that excludes the current first-stage solution ( x, β) with β b = from further consideration (Luedtke, 24, Theorem 2). Note that σ is always in the subset Υ (due to a facet condition), hence υ = σ. Then, the separation problem to find the most violated inequality (8) at the current solution ( x, β) with integral β aims to find Υ that minimizes the left-hand side of (8). It turns out that because β {,} m, the separation problem can be solved trivially. If β σ =, then Υ = {σ }. To see this, note that for a given ( x, β), the first term in the left-hand side of (8) is a constant, and the second term involving β is non-negative. As a result, this choice of Υ provides the minimum possible second term (zero). On the other hand, if β σ =, then because β s {,} for s S it is enough for us to find the next scenario σ i with i {2,...,a} such that β σi = and add this element, if it exists, to Υ, i.e., Υ = {σ,σ i } for this choice of i. Then the second term in the left-hand side of (8) is equal to h σ(α) h σi(α). Note that adding σ j to Υ = {σ,σ i } for j < i, where β j = by the definition of i, does not change the left-hand side value h σ(α) h σi(α) = h σ(α) h σj(α) +h σj(α) h σi(α). Furthermore, adding σ j to Υ = {σ,σ i } for j > i cannot decrease the left-hand side. Finally, any other choice Υ = {σ,σ j } for j i, increases the left-hand side to h σ(α) h σj(α)+h σj(α) h σa+(α) if j < i, and h σ(α) h σj(α) for j > i. Note that inequalities (8) can also be added at fractional β values during the branch-and-bound process to solve (RMP). In this case, the separation problem can be solved as a shortest path problem (Atamtürk et al., 2) or with sorting (Günlük and Pochet, 2). One may also use the stronger mixing inequalities proposed in Luedtke et al. (2); Küçükyavuz (22); Abdi and Fukasawa (26), although at the expense of harder separation problems. If ISSP associated with scenario s is feasible for all s S such that β s =, then there is no inconsistency in the current solution; hence, we do not need any feasibility cuts. However, we may need to add optimality cuts, which we describe next. Optimality cuts: Given a first-stage solution (x,β), let π s be the dual vector associated with the optimal basis of the ideal second-stage problem () for ω s at this iteration. We have two types of optimality cuts: If β s = at the current solution of the master problem and the associated original second-stage problem () is feasible, then we solve the original second-stage problem () and obtain the optimal dual vector π s. It is easy to see that θ s π s (h s T s x) is an optimality cut, and it is valid for all x X (more precisely x P(ω s )) if β s =. It needs to be lifted to be valid also for all x X if β s =. To do this, we solve the following problem with α = π s T s : Then we add the optimality cut of the form v s (α) = min q s y+ q s u+αx subject to W s y+u h s T s x x X, y R n2 +, u R r +. θ s π s (h s T s x) ( π s h s v s (α) ) β s. (9) If β s = at the current solution of the master problem, then we solve the non-ideal second-stage problem (2) for scenario s. Let π s be the corresponding optimal dual vector. It is easy to see that θ s π s (h s T s x) is an optimality cut, and it is valid for all x X (see Remark 2.2). We can obtain

9 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 9 a stronger version of this valid cut by considering the case β s =. To do this, we solve the following lifting problem with α = π s T s : v s (α) = min q s y+αx In this case, we add the optimality cut of the form subject to W s y h s T s x x X, y R n2 +. θ s ( π s ) (h s T s x) ( ( π s ) h s v s (α) ) ( β s ). (2) Remark 2.2 Suppose that for a given x X the second-stage problem () for scenario s is feasible. Then, it is easy to show that π s (h s T s x) π s (h s T s x). Note that the optimality cuts (9) and(2) are valid for general probability case. For details on the optimality cuts we refer to Liu et al. (26). Wesolveoptimizationproblemstoobtainthecoefficientsv s ( π s T s )and v s (π s T s ). Theseoptimizationproblems are formulated as mixed-integer LPs, when we have integer/binary variables in the first stage. Instead of solving these additional mixed-integer linear programs we could solve the associated LP relaxations to obtain the lifted cuts; however, note that there is a trade-off between the computational efficiency and the effectiveness of the generated cuts. Note that throughout this section, we assume that the primary performance measure of interest, and the goal constraint, are linearly representable. This is definitely the case when we consider optimal timing problems that are representable by linear programs, and goal constraints that limit the tardiness. However, if we have goals, such as limiting number of tardy jobs, that require discrete decisions, then the second-stage problems involve binary decisions, and the duals of the second stage problem are not readily available. However, in this case, we can utilize the developments from two-stage stochastic integer programming that convexify the second-stage problems to enable a Benders decomposition algorithm (see, Gade et al., 24; Zhang and Küçükyavuz, 24; Qi and Sen, 26, and references therein). 3. Proposed Stochastic Scheduling Model Machine scheduling problems featuring both tardiness and earliness costs have been popular since the advent of just-in-time manufacturing in the 98 s. The general rationale in these problems is that earliness costs represent inventory holding costs for job completions prior to the due dates, and tardiness costs are born out of the cost of the loss of customer goodwill or a contractual obligation if jobs are delivered after their due dates. In this paper, we consider a similar setting. Our primary concern is to minimize the total earliness costs while guaranteeing a certain service level for the customers. To thisend, weallowthatthedeliverytimesmayexceedtheduedatesatmostbyasmallpre-definedpercentageand only with a small probability, and such violations incur an (excess) tardiness cost in the objective function. From the viewpoint of the earliness/tardiness scheduling literature, this characteristic of our problem reminds of the deterministic earliness/tardiness scheduling problems with due-windows, in which no penalties are accumulated as long as jobs are finished in their respective due windows. Our model is general enough to subsume meaningful practical settings as special cases. For instance, by adjusting the values of the parameters as appropriate, it is possible to define a problem such that all due dates are required to be satisfied and the objective is to minimize the total earliness cost. These concepts will be made concrete in the context of the scheduling model introduced in the sequel. In particular, we discuss the random performance measure of interest and detail the stochastic goal constraints relevant to the joint probabilistic constraint. The general two-stage stochastic programming

10 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling model introduced in the previous section allows for randomness in all parameters, but in this section we only focus on the uncertainty in the processing times for notational convenience and due to the practical relevance of this particular setting. Parameters: N : set of jobs; N := {,...,n}, d j : due date of job j, j : maximum tardiness level allowed for job j N expressed as a fraction of d j, S : set of scenarios; S := {,...,m}, ψ s : probability associated with scenario s S, p s j : processing time of job j N under scenario s S, ǫ : desired probability level of the chance-constraint, λ : trade-off coefficient for the mean-risk objective (the exchange rate of mean for risk). α : confidence level of CVaR, M s : the big-m parameter; it is set as j ps j min j {d j }. Decision Variables: x jk : the binary assignment variable which takes on value if job j N is scheduled at position k N, I s k E s k T s k u s k and is zero otherwise, : idle time inserted between the jobs at positions k and k in the sequence under scenario s S, : earliness of the kth job in the sequence under scenario s S, : tardiness of the kth job in the sequence under scenario s S, is the excess tardiness that exceeds the maximum tardiness level allowed in the chance-constraint of the kth job in the sequence under scenario s S, β s : takes the value if the second-stage problem is infeasible under scenario s, η : variable representing VaR at the confidence level of α, w s : variable representing the excess amount with respect to VaR α under scenario s. In a stochastic scheduling setting, when processing times are uncertain, it is inevitable that some jobs are tardy with respect to their due dates. Therefore, to attain desirable service levels, we consider a chance constraint that limits tardiness to a given percentage, j, of the due date of job j N. Suppose that we are given a sequence of jobs κ() κ(2) κ(n). The chance constraint representing the service level restriction is given by P(T k (ω)/d κ(k) κ(k), k N) ǫ, (2) where T k (ω) is a random variable representing the tardiness of the kth job in the sequence, whose realization for scenario s S is given by Tk s,k N. Note, however that the sequence of the jobs is not given and is decided in the first stage. The stochastic goal constraint, written with respect to the first-stage variables, takes the following form: Let the first-stage feasible set be X := {x {,} n : T k (ω) n k= x jkd j x jk j k N. (22) k= x jk =, j N, k= x jk =, k N}. (23) j=

11 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling Given a sequencing decision vector x X, we have Q( x,ξ(ω s ),) = n k= Es k, and the ideal second-stage problem, which satisfies the goal constraint (22), is given by Q( x,ξ(ω s ),) = minimize E k, (24) k= subject tot k x jk d j j, k N, (25) k= k I i +E k T k = i= x jk d j j= k i= j= x ji p s j, k N, (26) I k,e k,t k, k N. (27) Thus, the feasible region of the second-stage problem {y R n2 + : W(ω s )y h(ω s ) T(ω s )x} featured in P(ω s ) is specified by (25)-(27). Instead of satisfying the goal constraint (22) under each elementary event ω, we enforce a joint constraint on satisfying it with a specified high probability of ǫ in the first-stage problem. Accordingly, for a small fraction of scenarios (as determined by ǫ), we relax (22) in the modified (non-ideal) second-stage problem and penalize the total excess tardiness in the objective function. Thus, we enforce a reliability-based approach to ensure that the tardiness does not exceed a percentage of the due date length for each job under a selected set of scenarios, and penalize the excess tardiness for the remaining undesirable scenarios. This leads to the non-ideal second stage problem: Q( x,ξ(ω s ),) = minimize (E k +u k ), (28) k= subject tot k u k x jk d j j, k N, (29) k= k I i +E k T k = i= x jk d j j= k i= j= x ji p s j, k N, (3) I k,e k,t k,u k, k N. (3) Note that this problem is feasible for all x X (for this part we have a relatively complete recourse and we do not need feasibility cuts). In the model of our interest, which is referred to as TSRA-S, we let the risk associated with the first-stage sequencing decisions be given by the random variable Q(x, ξ(ω), β(ω)), which denotes the random total earliness and excess tardiness, where excess tardiness of job j N is defined to be tardiness beyond an acceptable limit, a given percentage j of the due date of job j N. Let E k (ω) be a random variable representing the earliness of the kth job in the sequence, whose realization for scenario s S is given by Ek s,k N, and U k (ω) represent the random excess tardiness of the kth job in the sequence in scenario s. In other words, U k (ω) = (T k (ω) d κ(k) κ(k) ) + (the realization of U k (ω) for scenario s S is denoted by u s k ). Therefore Q(x,ξ(ω),β(ω)) = k N (E k(ω)+u k (ω)). In this setup, VaR α (Q(x,ξ(ω),β(ω))) provides an upper bound on the total earliness and excess tardiness that is exceeded only with a small probability of α. On the other hand, CVaR α (Q(x,ξ(ω),β(ω))) is a measure of severity of the total earliness and excess tardiness if it is larger than VaR α (Q(x,ξ(ω),β(ω))). In summary, the deterministic equivalent formulation of the stochastic scheduling problem is m n ( ) (APDF) minimize( λ) ψ s (Ek s +u s k)+λ η + m ψ s w s, (32) α s= k= s=

12 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 2 subject to x jk =, j N, (33) k= x jk =, k N, (34) j= m ψ s β s ǫ, (35) s= u s k M s β s, s S,k N, (36) Tk s u s k x jk d j j, s S,k N, j= k Ii s +Ek s Tk s = i= w s x jk d j j= k i= j= x ji p s j, s S,k N, (Ek s +u s k) η, s S, k= I s k,e s k,t s k,u s k, s S,k N, (37) x jk {,}, j,k N, (38) β s {,}, s S, (39) w s, s S, (4) η. 3. Benders Decomposition for our Scheduling Model We solve the relaxed master problem(rmp), given by (3)-(7), where X is as defined in (23), to obtain a sequence given by x X. Denoting the dual variablesassociatedwith(29)and(3)byπ k andπ k, respectively, thedualofthenon-idealsecondstageproblem (given in (28)-(3)) is as follows: D s (x) maximize π k k= j= x jk d j j + k= π k d j x jk j= k i= j= x ji p s j, (T k ) subject toπ k π k, k N, (E k ) π k, k N, (I k ) π i k N, i=k (u k ) π k k N, π k, k N, π k R, k N. (4) Let (ˆπ,ˆπ ) is an optimal solution for D s (x) then θ s is a valid optimality cut for all x X. k= j= x jk ( ˆπ kd j j + ˆπ kd j p s j i=k ˆπ i ) (42) Denoting the dual variables associated with (25) and (26) by π k second stage problem (given in (24)-(27)) is as follows: D s (x) maximize x jk d j j + d j x jk π k k= j= k= π k j= and π k, respectively, the dual of the ideal k i= j= x ji p s j,

13 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 3 (T k ) subject toπ k π k, k N, (E k ) π k, k N, (I k ) π i k N, i=k π k, k N, π k R, k N. If ISSP is infeasible (and consequently D s (x) is unbounded), then we need to add a feasibility cut to the master problem. For this we will follow (Luedtke, 24, Section 2.3) as described in Section Suppose ISSP is feasible and ˆπ = (ˆπ,ˆπ ) is an optimal solution for D s (x) then k θ s x jk d j j + d j x jk x ji p s j θ s ˆπ k k= j= k= j= ( x jk k= ˆπ k j= ˆπ kd j j + ˆπ kd j p s j i=k ˆπ i ) i= j= is an optimality cut which is valid for all x X only if β s = i.e., valid only for the ideal second stage. Therefore, it needs to be lifted to be valid for all x X when β s = too. 3.. Lifting To this end, we solve the following lifting problem v s (ˆπ ) n ( T s = minimize (E k +u k ) x jk ˆπ kd j j + ˆπ kd j p s j k= subject tot k u k where ˆπ T s is the coefficient of x. k= j= x jk d j j, k N, k= k I i +E k T k = i= x jk d j j= k i= j= i=k ˆπ i x ji p s j, k N, x jk =, j N, k= x jk =, k N, j= I k,e k,t k,u k, k N, x jk [,], j,k N, Note that the integrality of x is relaxed. In our computational study, we observed that solving the continuous version gives rise to better results in terms of solution time and optimality gaps. The lifted optimality cut is as follows θ s which is valid for all x X. k= j= ( x jk ˆπ kd j j + ˆπ kd j p s j i=k ˆπ i ) ), +v s (ˆπT s )β s (43) 3..2 Feasibility Cuts If ISSP is infeasible (D s (x) is unbounded), then we add a feasibility using the method described in Section 2.2 (see, also, Luedtke, 24).

14 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 4 With the extreme ray π = ( π, π ) of f b (x) where b is the scenario under which the the ideal second stage problem is infeasible is at our hand, we solve the following single scenario optimization problem for all scenarios s S: h s ( π T b ) = minimize n k= j= subject tot k x jk (p b j i=k π i π kd j j π kd j ), x jk d j j, k N, k= k I i +E k T k = i= x jk d j j= k i= j= x ji p b j, k N, x jk =, j N, k= x jk =, k N, j= I k,e k,t k, k N, x jk [,], j,k N, where π T b is the coefficient of x i.e., α of the single scenario optimization subproblems (5) in (Luedtke, 24). As explained in Section 2.3 of (Luedtke, 24), the selection of the domain of first stage variables is important and there is a trade-off between stronger inequalities and shorter computation time. However, in our computational studies, we have observed that for this particular problem, choosing x {,} n increases the computational time substantially but does not yield significantly stronger (better) cuts. Having obtained the values h s (α) for s S where α = π T b, we then sort them to obtain a permutation σ such that: h σ (α) h σ2 (α) h σm (α). Then we obtain a subset Υ = {υ,υ 2,...,υ l } {σ,σ 2,...,σ a } such that h υi (α) h υi+ (α) for i =,...,l, where υ = σ and υ l+ = σ a+. Then the inequality is valid. k= j= x jk (p b j i=k π i π kd j j π kd j )+ l ( hυi (α) h (α)) υi+ β υi h υ (α) (44) i= The pseudo-code of the proposed algorithm is given in Algorithm. 4. Computational Study In the first part of our computational study, we aim to demonstrate the value of the proposed risk-averse stochastic programming model for decision making purposes. To this end, we evaluate the solutions provided by our model under different input parameter settings and contrast them against those produced by their risk-neutral counterparts. In the second part of the study, we investigate the computational effectiveness of the proposed algorithm. The computational study is conducted on a workstation with two 2.3GHz Intel(R) Xeon(R) E5-263 CPUs with Hyper-Threading enabled and 64 GB of memory running on Windows 8.. The proposed algorithm which is referred to as BDS in this section is implemented in C++ using the Concert Technology component library

15 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 5 Algorithm : Solving TSRA-S by Benders decomposition F, O // Initialization 2 repeat // Main Loop 3 Solve (RMP) given by (3) (7); 4 if infeasible then 5 TSRA-S is infeasible; 6 break; 7 else 8 CutFound False; 9 Let ( x,β,θ,ϑ,η ) be an optimal solution of (RMP); for s S do if β s = then 2 Solve the non-ideal second stage problem (given in (28)-(3)); 3 Let (ˆπ,ˆπ ) be the optimal dual solution and ẑ be the corresponding OFV; 4 if θ s < ẑ then 5 Create and add an optimality cut of the form (42) to the description of O; 6 CutFound True; 7 else 8 Solve the ideal second stage problem (given in (24)-(27)); 9 if infeasible then 2 Let ( π, π ) be the extreme ray associated with D s (x); 2 Create and add a feasibility cut of the form (44) to the description of F; 22 CutFound True; 23 else 24 Let (ˆπ,ˆπ ) be the optimal dual solution and ẑ be the corresponding OFV; 25 if θ s < ẑ then 26 Create and add an optimality cut of the form (43) to the description of O; 27 CutFound True; 28 until CutFound = False; of IBM CPLEX Note that in the presence of a control callback such as the lazy constraint callback used in BDS CPLEX switches off dynamic search feature, sets the number of threads to one and operates under deterministic parallel search mode. To be able to increase the active thread number, we manually turn on the opportunistic parallel search mode. Furthermore, in order to urge CPLEX to place greater emphasis on proving optimality through moving the best bound value, the MIPEmphasis switch is set to 3 (IBM ILOG CPLEX, 23). 4. Generation of the Problem Instances In our computational study, we have used Data Set of Atakan et al.(26). The authors focus on the uncertainty in the processing times and assume that each scenario is equally likely. In particular, they generate each scenario representing a joint realization of the processing times of all jobs as follows: first, for each job j {,...,n}, an estimated processing time is drawn from an integer uniform distribution U[, 9], and then this value is perturbed randomly up or down for each scenario. However, the key element in their data generation scheme is that the distributions of the perturbations (specified

16 Bülbül, Küçükyavuz, Noyan,Şen: Risk-averse Machine Scheduling 6 as the mixtures of the uniform distributions) depend on the corresponding estimated expected processing times. For details about the parameters of the mixture distributions and the expectation and the standard deviation of perturbations we refer to Atakan et al. (26). Their setup ensures that the processing times fluctuate around their estimated expected values with a high probability and take significantly larger values with a small probability; this seems to be consistent with the common observations in many manufacturing environments as highlighted by the authors. On the other hand, they generate the due dates by taking into consideration only the tardiness issue; therefore, we follow an alternative approach and generate the due dates from a discrete uniform distribution U [ a l P, a u P ], where a l and a u are respectively the lower and upper bounds of the due date tightness factor, and P is the sum of the average processing times, i.e., P = j N s S ps j /m. We consider two intervals of the form [a l,a u ]: [.7,.9] and [,.3]. In our analysis, we let j = for all j N. Hence the goal constraint (22) is equivalent to max k N { Tk (ω) n k= x jkd j }. (45) We focus on investigating the impact of the following parameters which are essential components of the proposed modeling approach: the probability level of the chance-constraint (ǫ), the relative tardiness level allowed in the chance-constraint ( ), the trade-off coefficient of the mean-risk objective (λ), the confidence level of CVaR (α). In the first part of our study, we consider the following values of these parameters when n =, m = : ǫ {,.,,,,,,}, {.,}, λ {,.,}, and α {,5}. In the second part, we mainly focus on varying the size of the problem instances and allow n and m take values from {,5,2,25} and {,2,3}. In addition, parameter settings with ǫ =.5, λ =, and α = are also tested. 4.2 Model Analysis Table indicates the relative percentage decreases in CVaR and increases in expectation for the solutions of the risk-averse models in comparison to those of the risk-neutral model (λ = ) for an illustrative problem instance. The risk-averse solutions exhibit improvements in CVaR over their (chanceconstrained) risk-neutral counterparts, albeit at times at the expense of the increase in the expected total earliness and excess tardiness (TE and ET) in order to hedge against the uncertainty. We also observe that the trade-off between the expectation and the CVaR criteria becomes more pronounced when the α and λ values are larger. On the other hand, we cannot observe such patterns with respect to the changes in the ǫ parameter. This can be partially attributed to the hybrid structure of proposed modeling approach and the presence of conflicting criteria in the objective function. We control the tardiness values using a maximum-based qualitative measure (45) (via a joint chance constraint) and total sum-based quantitative measures (expectation and CVaR in the objective function). Moreover, the non-additive CVaR measure is introduced on the summation of two conflicting criteria (TE and ET), which also complicates the dynamics of the model. In Table 2, for the solutions of the model under several parameter settings we calculate the probability of satisfying the stochastic goal constraint (45); recall that the goal constraint introduces an upper bound ( ) on the ratio of the tardiness and due date for each job (equivalently, on the maximum ratio of the tardiness and due date over all jobs). The corresponding numerical results are reported in Table 2. Recall also that, ǫ is the specified lower bound on the probability of meeting the goal constraint enforced via the joint-chance constraint in our model, the attained probability levels for a given sequence may be higher than this lower bound. When we drop the joint-chance constraint, i.e., set ǫ =, the attained probability level can be even ; see the instances with λ =,α = (risk-neutral case) and λ =,α =. In other words, a penalty term in the objective for excess tardiness (both in expectation and in CVaR) alone does not ensure that desirable service levels are met, and chance constraints are indeed necessary to guarantee a reliability-based target performance. In general, the attained probability levels appear to be equal to their lower bounds ( ǫ). For the largest λ and α values considered (λ = and α = 5), we also observe slight improvements in the attained probability