Integer Programming Approaches for Appointment Scheduling with Random No-shows and Service Durations

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1 Vol. 00, No. 0, Xxxxx 0000, pp issn X eissn Integer Programming Approaches for Appointment Scheduling with Random No-shows and Service Durations Ruiwei Jiang Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, Siian Shen Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, Yiling Zhang Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, We consider a single-server scheduling problem given a fixed seuence of job arrivals with random noshows and service durations. The joint probability distribution of the uncertain parameters is assumed to be ambiguous and only the support and first moments are known. We formulate a class of distributionally robust optimization models that incorporate the worst-case expected cost and the worst-case conditional Value-at-Risk (CVaR) of appointment waiting, server idleness, and overtime as the objective or constraints. Our models flexibly adapt to different prior beliefs of no-show probabilities. We obtain exact mixed-integer nonlinear programming (MINLP) reformulations that facilitate decomposition algorithms, and derive valid ineualities to strengthen the reformulations. In particular, we derive the convex hulls for special cases of no-show beliefs, yielding polynomial-size linear programming reformulations for the least and the most conservative supports of no shows. We test various instances to demonstrate the computational efficacy of our approaches and provide insights for appointment scheduling under distributional ambiguity of multiple uncertainties. Key words : appointment scheduling; no-show uncertainty; mixed-integer programming; valid ineualities; totally unimodularity; convex hulls 1. Introduction We consider an appointment scheduling problem with random no-shows and service durations. The problem involves a single server and a set of appointments following a fixed order of arrivals at the server. A system operator needs to schedule an arrival time for each appointment. A common goal is to minimize the expected waiting time of all appointments, idle time and overtime of the server if the distributional information is fully accessible. The problem is fundamental for establishing service uality and operational efficiency in many service systems, including outpatient care and surgery planning in hospitals (see, e.g., Denton and Gupta 2003), call-center staffing 1

2 2 Article submitted; manuscript no. (Please, provide the manuscript number!) (see, e.g., Gurvich et al. 2010), and server operations in data centers (see, e.g., Shen and Wang 2014). In Section 1.1, we provide an extensive review of the literature on variants of the stochastic appointment scheduling problem under specific objectives, metrics, and applications. In reality, due to lack of data, it can be challenging to accurately estimate the (joint) probability distribution of no-shows and service durations. The data of no-shows could be limited because of low probability of occurrence and the heterogeneity of appointments. In view of a wide range of plausible substitutes (e.g., Log-Normal, Normal, and Uniform), we could also misspecify the servicetime distribution. Then with ambiguous estimates of no-show and service-duration distributions, we can schedule unnecessarily long (respectively, short) time in between appointments, resulting in significant server idleness (respectively, appointment waiting or server overtime). To address the issue of distributional ambiguity, recent papers have started investigating distributionally robust (DR) variants of the appointment scheduling problem. For example, Kong et al. (2013) provide the first study on the DR model using a cross-moment ambiguity set that consists of all distributions with common mean and covariance of the random service durations. They obtain a copositive cone programming reformulation and solve a semidefinite program approximation. The most relevant to this paper, Mak et al. (2015) study a DR model using a marginal-moment ambiguity set of the random service durations. They obtain exact and tractable reformulations by successfully solving a nonconvex optimization problem based on a binary encoding of its feasible region (see Section 3.2 for details). In this paper, we generalize the DR optimization model for appointment scheduling in Mak et al. (2015) by incorporating the uncertainty of no-shows and its distributional ambiguity. Moreover, we consider both risk-neutral and risk-averse models based on the system operator s risk preferences. In particular, for risk-averse operators, we consider the Conditional Value-at-Risk (CVaR) to guarantee the uality of service and reduce long waiting, idleness and overtime. To the best of our knowledge, this paper is among the first to consider both discrete (e.g., no-shows) and continuous (e.g., service durations) randomness in the DR scheduling problem. This generalization results in a challenging mixed-integer nonlinear program (MINLP), to which the approach by Mak et al. (2015) does not apply anymore. The main focus of this paper is to derive effective integer programming approaches for solving the generalized DR model. By deriving linearization and valid ineualities, we develop a decomposition algorithm to solve the DR model. In particular, the valid ineualities effectively accelerate the algorithm in computational experiments (see Section 5). Furthermore, we show that these ineualities recover the convex hulls in important special cases, leading to polynomial-size linear programming (LP) reformulations.

3 Article submitted; manuscript no. (Please, provide the manuscript number!) Literature Review There are often two phases associated with scheduling a set of appointments. In the first phase, system operators consider a server allocation problem (see, e.g., Denton et al. 2010, Gurvich et al. 2010, Shylo et al. 2012) to decide which servers to operate and assign appointments to open servers. In the second phase, which we focus on in this paper, the operators determine the arrival time of each appointment on their assigned servers. We assume a given and fixed seuence of appointment arrivals. In practice, system operators can follow a designated rule to seuence the appointments, e.g., the first-call-first-serve rule in outpatient clinics. For studies that also involve seuencing decisions, we refer to Denton et al. (2007), Gupta and Denton (2008), Mak et al. (2015), Mancilla (2009), Mak et al. (2014), He et al. (2015). For the studies of combining the two phases and considering integrated server allocation, appointment seuencing, and scheduling, we refer to Batun et al. (2011) and Deng et al. (2015), both assuming random service durations but without the consideration of no-shows. The studies of stochastic appointment scheduling (e.g., Gupta and Denton 2008, Erdogan and Denton 2013, Berg et al. 2014) generally assume uncertain service durations following known distributions. Denton and Gupta (2003) formulate a two-stage stochastic LP model for appointment scheduling and demonstrate that the optimal time intervals allocated in between appointments form a dome shape if the unit idleness costs are high relative to the unit waiting costs. Klassen and Yoogalingam (2009) use simulation optimization and demonstrate the robustness of the dome-shaped scheduling rule, but also show that one could benefit from considering a flatter plateau-dome rule in many scenario patterns of the random service durations. Mittal et al. (2014) consider a robust appointment scheduling problem, and derive a closed-form optimal solution for appointment time and a constant-factor approximation algorithm for optimal appointment seuencing. Begen and Queyranne (2011) consider stochastic appointment scheduling with discrete service durations and derive polynomial-time algorithms by exploiting the submodularity and L-convexity of the objective function. Begen et al. (2012) extends the results in Begen and Queyranne (2011) and consider a sample average approximation of the stochastic appointment scheduling problem. They derive an upper bound on the sample size reuired to achieve a near-optimal solution (with multiplicative error) to the original problem with high confidence. Ge et al. (2013) extend the results in Begen and Queyranne (2011), and complement the results in Begen et al. (2012) by considering piecewise linear cost functions and bounding the sample size for obtaining a near-optimal solution (with additive error) with high confidence. For a comprehensive survey of various scheduling problems including their models, theories, and applications, we refer to Pinedo (2012). The consideration of no-shows in scheduling problems dates back to the work by Ho and Lau (1992), who implement a heuristic approach to double book the first two arrivals and subseuently

4 4 Article submitted; manuscript no. (Please, provide the manuscript number!) schedule the rest of the appointments that arrive in a fixed order. Cayirli and Veral (2003) review the literature of outpatient appointment scheduling and point out the significant impact of patient no-shows in the related systems. Hassin and Mendel (2008) further analyze the impact of no-shows on a single-server ueuing model. Recently, Liu et al. (2010) find that an open access policy (also proposed in Robinson and Chen 2010) that calls for meeting today s demand today, performs well under low-volume patient arrivals in dynamic appointment scheduling with no-shows. Erdogan and Denton (2013) incorporate no-shows into a stochastic LP model proposed by Denton and Gupta (2003). A number of heuristic policies and approximation algorithms have also been proposed to schedule appointments under the no-show uncertainty (see, e.g., Muthuraman and Lawley 2008, Zeng et al. 2010, Cayirli et al. 2012, Lin et al. 2011, Luo et al. 2012, LaGanga and Lawrence 2012, Zacharias and Pinedo 2014, Parizi and Ghate 2015, Kong et al. 2015). As mentioned before, the issue of distributional ambiguity has been recently considered for appointment scheduling under random service durations (see Kong et al. 2013, Mak et al. 2015, Kong et al. 2015). We refer to Scarf et al. (1958), Dupačová (1987), Bertsimas and Popescu (2005), Bertsimas et al. (2010), Delage and Ye (2010) for generic DR modeling and computation with moment-based ambiguity sets of probability distributions of uncertain parameters Contributions of the Paper We summarize the main contributions of this paper as follows. 1. Depending on the system operator s risk preferences towards each performance metric, we formulate a class of DR models that incorporate the worst-case expected cost and the worst-case CVaR of waiting, idleness, and overtime as objective or constraints. Meanwhile, the DR models can flexibly adapt to different prior beliefs of the maximum number of consecutive no-shows, covering from the least conservative case (i.e., no consecutive no-shows) to the most conservative case (i.e., arbitrary no-shows). 2. We develop effective solution approaches for each DR model considered in this paper. The exact reformulations of the DR models result in mixed-integer trilinear programs. To address the computational challenge, we linearize and derive valid ineualities to strengthen the reformulations that facilitate decomposition algorithms. In particular, for the least conservative and the most conservative cases, our derivation leads to polynomial-size LP reformulations that can readily be implemented in LP solvers (e.g., Microsoft Excel). This can particularly benefit small business (e.g., community hospitals) with limited budget for scheduling. 3. We test diverse instances to show the computational efficacy of our approaches, and also to demonstrate the performance of DR models under multiple uncertainties, risk preferences, and levels of conservativeness for balancing tradeoffs between uality of service and operational cost.

5 Article submitted; manuscript no. (Please, provide the manuscript number!) Structure of the Paper The remainder of the paper is organized as follows. Section 2 formulates the DR expectation and DR CVaR models, as well as their variants based on different risk preferences and levels of conservativeness. In Sections 3 and 4, we derive MINLP reformulations and cutting-plane approaches for optimizing the proposed DR expectation and CVaR models, respectively. We also derive polynomial-size LP reformulations of the DR models for special cases of no-show supports. Section 5 tests a diverse set of scheduling instances and demonstrates the insights of DR schedules under various parameter settings. We summarize the paper and discuss future research directions in Section 6. Notation and Proofs: The convex hull of a set X is denoted by conv(x). The abbreviation w.l.o.g. represents without loss of generality. We follow the convention that j k=i a k = 0 if i > j. For presentation brevity, we relegate all detailed proofs to the appendices. 2. Formulations of DR Appointment Scheduling Consider n appointments arriving at a single server following a fixed order of arrivals given as 1,..., n. Each appointment i has a random service duration s i. We interpret the possibility of random no-show for appointment i by a 0-1 Bernoulli random variable i such that i = 1 if appointee i shows up, and i = 0 otherwise. We schedule an arrival time for each appointment, or euivalently, assign time intervals between appointments i and i + 1 for all i = 1,..., n Modeling Waiting, Idleness, and Overtime under Uncertainty Let variable x i represent the scheduled time interval between appointments i and i + 1, i = 1,..., n 1. Under the two types of uncertainties, one or multiple of the following three scenarios can happen: (i) an appointment cannot start on time due to overtime operations of previous appointments, (ii) the server is idle and waiting for the next appointment due to an early finish of previous appointments or no-shows, and (iii) the server cannot finish serving all appointments within a given time limit, denoted by T. For all i = 1,..., n, let variable w i represent the waiting time of appointment i, and variable u i represent the server idle time after finishing appointment i. Also, let variable W represent the server s overtime beyond the fixed time limit T to finish all n appointments. The feasible region of decision x is denoted as { } X = x : x i 0, i = 1,..., n, x i = T, (1) which ensures that we assign nonnegative time in between all consecutive appointments, and appointment n is scheduled to arrive before the end of the service horizon, i.e., time T. Note that variable x n 0 is a dummy variable to represent T n 1 x i, i.e., the time from the scheduled start of the last appointment to the server time limit.

6 6 Article submitted; manuscript no. (Please, provide the manuscript number!) Given decision vector x X and a joint realization of uncertain parameters (, s), the total cost of scheduling all n appointments based on appointment waiting time (i.e., w = [w 1,..., w n ] T ), server idleness (i.e., u = [u 1,..., u n ] T ), and server overtime (i.e., W ) is calculated by solving a linear program: Q(x,, s) := min w,u,w (c w i w i + c u i u i ) + c o W (2a) s.t. w i u i 1 = i 1 s i 1 + w i 1 x i 1 i = 2,..., n (2b) W u n = n s n + w n x n (2c) w i 0, w 1 = 0, u i 0, W 0, i = 1,..., n. (2d) The objective function (2a) minimizes a linear cost function of the waiting, idleness, and overtime, with nonnegative parameters c w i, c u i, and c o being the respective unit penalty costs. In this paper, we assume that these cost parameters satisfy c u i+1 c u i c w i+1 for all i = 1,..., n 1, i.e., the neighboring unit server idleness costs are relatively similar. This assumption is standard in the existing literature (see, e.g., Denton and Gupta (2003), Ge et al. (2013), Kong et al. (2013), Mak et al. (2015)). In fact, if this assumption does not hold and c u i+1 > c u i + c w i+1, then the system operator would rather enforce idleness even if appointment i + 1 has arrived and keep it waiting, than start appointment i + 1 right away and have idleness afterwards. This will not be realistic due to practical concerns. Under this assumption, constraints (2b) yield either the waiting time of appointment i or the server s idle time between finishing appointment i 1 and the arrival of appointment i (see Proposition 1 in Ge et al. (2013) for a proof), i.e., w i = max{0, i 1 s i 1 + w i 1 x i 1 }, and u i 1 = max{0, x i 1 i 1 s i 1 w i 1 }, i = 2,..., n. (3) Similarly, constraint (2c) computes either the overtime W or the idle time u n after finishing appointment n. Since appointment 1 always arrives at time 0 and there is no waiting of appointment 1, we have w 1 = 0 and reuire all waiting, idleness, and overtime to be nonnegative in constraints (2d). In formulation (2), we also note that the waiting time costs c w i w i are modeled from the perspective of servers (e.g., operating rooms). In particular, we assume that appointment no-shows take place after the server has been set up for serving the appointments. Hence, the waiting time costs stem from euipment and personnel idleness, as well as the losses of opportunities of serving other appointments, and are incurred regardless whether the appointments show up. From the perspective of appointments, the waiting time costs should be modeled as c w i w i i, i.e., the waiting time costs are waived if an appointment does not show up. In this paper, we focus on the DR models and solution methods for the former case, i.e., server-based waiting time costs. In Appendix A, we elaborate how our DR approaches can adapt for a more general setting that incorporates both server-based and appointment-based waiting time costs.

7 Article submitted; manuscript no. (Please, provide the manuscript number!) DR Models with Various Risk Measures and Supports of No-Shows The classical stochastic appointment scheduling approaches seek an optimal x X to minimize the expectation of random cost Q(x,, s) subject to uncertainty (, s) with a known joint probability distribution denoted as P,s. In this paper, we assume that P,s is only known belonging to an ambiguity set F(D, µ, ν) that is determined by the support D of (, s) and the mean values µ = [µ 1,..., µ n ] T and ν, where µ i represents the mean E[s i ] of appointment i for each i = 1,..., n, and ν represents E[ n i], i.e., the expected total number of appointment show-ups given n appointments. We consider support D = D D s where D models the support of random no-show parameter and D s models the support of random service duration parameter s. For random service durations, we assume upper and lower bounds of the duration of each appointment i and accordingly D s := {s 0 : s L i s i s U i, i = 1,..., n}. For random no-shows, we parameterize the support D that D = D (K) by an integer K {2,..., n + 1} such rules out consecutive no-shows in any K consecutive appointments. Accordingly, D (K) := { {0, 1} n : i+k 1 j=i j 1, i = 1,..., n K + 1 We note that (i) when K = 2, D (2) rules out all consecutive no-shows, and (ii) when K = n + 1, D (n+1) allows arbitrary no-shows and so D (n+1) = {0, 1} n. Also, note that D (k) D (k ) }. for all 2 k < k n + 1. Hence, the seuence of parameterized supports D (2) D (3) D (n+1) form a spectrum of conservativeness levels, with D (2) on the least conservative end and D (n+1) on the most conservative end. In practice, the system operator has the flexibility to select parameter K according to her targeted conservativeness. The probability of K consecutive no-shows out of the n appointments uickly decays as K increases, and so K can be expected to be much lower than n + 1 to keep D (K) guideline for the selection of K. reasonably conservative. In Section 2.3, we provide a practical and rigorous The ambiguity set F(D, µ, ν) is specified as F(D, µ, ν) := P,s 0 : D D s dp,s = 1 s D D i dp,s = µ i i = 1,..., n s D D s ( n i) dp,s = ν, (4) where P,s matches the mean values of service durations and the total number of no-shows, and respect the supports of random variables and s. Note that ambiguity set F(D, µ, ν) does not incorporate higher moments (e.g., variance and correlations) of service time and no-shows for the following reasons. First, with a small amount of data, it is often unclear whether/how the service

8 8 Article submitted; manuscript no. (Please, provide the manuscript number!) time and no-shows are correlated. Second, the introduction of higher moments undermines the computational tractability of the DR models, which can be achieved by using F(D, µ, ν) and the solution algorithm derived in Sections 3 and 4. Finally, as we find in the computational study (see Section 5), the DR models based on F(D, µ, ν) can already provide near-optimal solutions. In this case, the benefit of incorporating higher moments is not significant. We consider DR appointment scheduling models that impose a min-max DR objective and/or DR constraints. Specifically, given x X, we consider a risk measure ϱ(q(x,, s)) of Q(x,, s) where (i) a risk-neutral system operator sets ϱ(q(x,, s)) = E P,s [Q(x,, s)], i.e., the expected total cost of waiting, idleness, and overtime; (ii) a risk-averse system operator sets ϱ(q(x,, s)) = CVaR 1 ɛ (Q(x,, s)), i.e., the CVaR of the total cost with 1 ɛ (0, 1) confidence. CVaR is freuently applied in optimization models under uncertainty (see, e.g., Rockafellar and Uryasev (2000, 2002) for theories of CVaR and Sarin et al. (2014) for its application in a stochastic parallel machine scheduling problem). Then, the DR models impose a generic min-max DR objective in the form min x X sup P,s F(D,µ,ν) ϱ(q(x,, s)), (5a) and/or generic DR constraints in the form sup P,s F(D,µ,ν) ϱ(q(x,, s)) Q. (5b) where Q R represents a bounding threshold for the risk measure from above. Note that both DR objective (5a) and constraints (5b) protect the risk measure by hedging against all probability distributions in F(D, µ, ν). Depending on the system operator s risk preferences, the DR models can impose either or both of DR objective (5a) and constraints (5b), and use either expectation or CVaR as risk measures in (5a) (5b), i.e., ϱ(q(x,, s)) = E P,s [Q(x,, s)] or ϱ(q(x,, s)) = CVaR 1 ɛ (Q(x,, s)). Furthermore, the system operator can tune the cost parameters c w i, c u i, and c o to let Q(x,, s) represent different conseuences (e.g., performance metric, uality of service, resource opportunity cost, etc.) associated with waiting, idleness, and overtime in (5a) (5b). For example, by setting c o = 1 and c u i = c w i = 0 for all i, we propose sup P,s F(D,µ,ν) CVaR 1 ɛ (Q(x,, s)) Q (6) to constrain the CVaR of overtime W below threshold Q. For this particular cost parameter setting, constraint (6) guarantees that inf P,s F(D,µ,ν) P,s { W Q } 1 ɛ, and hence the overtime W is controlled under threshold Q with the smallest possible probability being no less than 1 ɛ.

9 Article submitted; manuscript no. (Please, provide the manuscript number!) Guideline of Selecting Parameter K In practice, a system operator may evaluate the probability of the random variables = ( 1,..., n ) belonging to D (K), i.e., P( D (K) ). Then, she can select parameter K such that P( D (K) ) exceeds a given threshold such as 90%. To this end, she can gradually increase the value of K from 2 until P( D (K) ) exceeds the threshold for the first time. For example, we derive P( D (K) ) under the assumption of independent no-shows. Observation 1. If the components of are jointly independent, then P( D (K) ) = 1 [Q n ] 1(K+1), where K = 2,..., n, p 0 = (n ν)/n represents the no-show probability of each appointment, and Q represents a (K + 1) (K + 1) matrix such that (i) Q i1 = 1 p 0 Q i(i+1) = p 0 for all i = 1,..., K, and (iii) Q (K+1)(K+1) = 1. for all i = 1,..., K + 1, (ii) Proof of Observation 1 We construct a Markov chain with K + 1 states, where state 1 represents a show-up and state i + 1 represents i consecutive no-show(s) for all i = 1,..., K. By construction, matrix Q is the one-step transition matrix of this Markov chain, where state K + 1 (i.e., K consecutive no-shows) is absorbing. Thus, [Q n ] 1(K+1), representing component (1, K + 1) of matrix Q n, euals to the probability of having K consecutive no-shows out of the n appointments. Based on Observation 1, the selection of K can be conveniently done in a spreadsheet. In Figure 1, we display an example of P( D (K) ) with n = 10, K = 1,..., 11, and (n ν)/n = 0.1,..., 0.9. From this figure, we observe that K = 2 is sufficient for P( D (K) ) 90% when (n ν)/n 0.1, i.e., when the no-show probability for each appointment is no greater than 0.1. This observation motivates us to select D = D (2) for scheduling appointments with low no-show probabilities. P( D (K) ) P( D (K) ) Figure K = # of Consecutive No Shows Ruled Out An example of P( D (K) ) for n = 10 appointments K = # of Consecutive No Shows Ruled Out Next, we develop solution methods for the DR models. For presentation brevity, we analyze DR expectation models in Section 3 and DR CVaR models in Section 4, and all other risk-neutral/riskaverse/hybrid DR models above can be similarly solved.

10 10 Article submitted; manuscript no. (Please, provide the manuscript number!) 3. Integer Programming Approaches for DR Expectation Models We analyze the DR expectation models via a generic objective form (5a) formulated as min x X sup P,s F(D,µ,ν) E P,s [Q(x,, s)], (7) which minimizes the worst-case expected total cost of waiting, idleness, and overtime. We first consider the inner maximization problem sup P,s F(D,µ,ν) E P,s [Q(x,, s)] for a fixed x X, where P,s is the decision variable. It can be detailed as a linear functional optimization problem where D = D (K) max P,s 0 s.t. Q(x,, s)dp,s D D s (8a) s i dp,s = µ i i = 1,..., n (8b) D D s ( ) i dp,s = ν (8c) D D s D D s dp,s = 1, (8d) for K {2,..., n + 1}. Letting ρ i, γ, and θ be dual variables associated with constraints (8b), (8c), and (8d), respectively, we present problem (8) in its dual form as µ i ρ i + νγ + θ (9a) min ρ R n,γ R,θ R s.t. s i ρ i + γ i + θ Q(x,, s) (, s) D D s. (9b) Here ρ = [ρ 1,..., ρ n ] T, γ, and θ are unrestricted in sign, and constraints (9b) are associated with primal variables P,s, (, s) D D s. Under the standard assumptions that µ i belongs to the interior of set { D D s s i dq,s : Q,s is a probability distribution over D D s } for each appointment i, and ν belongs to the interior of set { D D s ( n i)dq,s : Q,s is a probability distribution over D D s }, strong duality holds between (8) and (9). Furthermore, for a fixed (ρ, γ, θ), constraints (9b) are euivalent to θ max (,s) D Ds {Q(x,, s) n (ρ is i + γ i )}. Thus, due to the objective of minimizing θ, the dual formulation (9) is euivalent to { { }} Q(x,, s) (ρ i s i + γ i ). (10) min ρ R n,γ R µ i ρ i + νγ + max (,s) D D s 3.1. MINLP Reformulation and a Generic Cutting-Plane Approach Note that Q(x,, s) is a minimization problem and thus in (10) we have an inner max-min problem. Our next step is to analyze the structure of Q(x,, s) for given solution x and realized (, s). Consider Q(x,, s) in (2) in its dual form as Q(x,, s) = max y ( i s i x i )y i (11a)

11 Article submitted; manuscript no. (Please, provide the manuscript number!) 11 s.t. y i 1 y i c w i i = 2,..., n (11b) y i c u i i = 1,..., n (11c) y n c o, (11d) where variable y i 1 represents the dual associated with each constraint i in (2b) for all i = 2,..., n, variable y n represents the dual of constraint (2c), and constraints (11b), (11c), (11d) are associated with primal variables w i for all i = 2,..., n, variables u i for all i = 1,..., n, and variable W in (2), respectively. Therefore, formulation (10) is euivalent to { { min µ i ρ i + νγ + max Q(x,, s) ρ,γ (,s) D D s { } = min ρ,γ µ i ρ i + νγ + max y Y h(x, y, ρ, γ) }} (ρ i s i + γ i ) (12a), (12b) where Y represents the feasible region of variable y in (11) given by (11b) (11d), and { } h(x, y, ρ, γ) := max ( i s i x i )y i (ρ i s i + γ i ). (12c) (,s) D D s The derivation of h(x, y, ρ, γ) follows that we can interchange the order of max (,s) D D s max y Y in (12a). Combining the inner problem in the form of (12b) with the outer minimization problem in (7), we obtain a reformulation of the DR expectation model (7): min x X,ρ,γ,δ µ i ρ i + νγ + δ (13a) s.t. δ max y Y h(x, y, ρ, γ) max y Y,(,s) D D s { ( i s i x i )y i and } (ρ i s i + γ i ). (13b) We now derive structural properties of max y Y h(x, y, ρ, γ) as a function of variables x, ρ, and γ. Lemma 1. For any fixed variables x, ρ, and γ, max y Y h(x, y, ρ, γ) < +. Furthermore, function max y Y h(x, y, ρ, γ) is convex and piecewise linear in x, ρ, and γ with a finite number of pieces. We refer to Appendix B.1 for a proof. Lemma 1 indicates that constraint (13b) essentially describes the epigraph of a convex and piecewise linear function of decision variables in model (13). This observation facilitates us applying a separation-based decomposition algorithm to solve formulation (13) (or euivalently, the DR expectation model (7)), presented in Algorithm 1. This algorithm is finite because we identify a new piece of the function max y Y h(x, y, ρ, γ) each time when the set {L(x, ρ, γ, δ) 0} is augmented in Step 7, and this function has a finite number of pieces according to Lemma 1. The main difficulty of the above decomposition algorithm lies in solving the separation problem (14). In general, this problem is a mixed-integer trilinear program because of the integrality

12 12 Article submitted; manuscript no. (Please, provide the manuscript number!) Algorithm 1 A decomposition algorithm for solving DR expectation model (7). 1: Input: feasible regions X, Y, and D D s ; set of cuts {L(x, ρ, γ, δ) 0} =. 2: Solve the master problem min x X,ρ,γ,δ µ i ρ i + νγ + δ s.t. L(x, ρ, γ, δ) 0 and record an optimal solution (x, ρ, γ, δ ). 3: With (x, ρ, γ) fixed to be (x, ρ, γ ), solve the separation problem { } max h(x, y, ρ, γ) max ( i s i x i )y i (ρ i s i + γ i ) y Y y Y,(,s) D D s (14) and record an optimal solution (y,, s ). 4: if δ n ( i s i x i )y i n (ρ i s i + γ i ) then 5: stop and return x as an optimal solution to formulation (7). 6: else 7: add the cut δ n ( i s i x i )y i n (s i ρ i + i γ) to the set of cuts {L(x, ρ, γ, δ) 0} and go to Step 2. 8: end if restrictions forced by D and the trilinear terms i s i y i in the objective function. This creates obstacles for optimally solving the separation problem if presented in its current form. In Section 3.2, we linearize and reformulate the separation problem as a mixed-integer linear program (MILP) that can readily be solved by off-the-shelf software. Meanwhile, we derive valid ineualities to strengthen the mixed-integer feasible region of this MILP, which further accelerates the solution of the separation problem MILP Reformulation of the Separation Problem and Valid Ineualities Our approach is inspired by Mak et al. (2015), where the authors point out that an optimal solution y to a similar separation problem that does not involve no-show uncertainty, exists at an extreme point of polyhedron Y. They then successfully decompose the separation problem by appointment for each i = 1,..., n and reformulate it via the extreme points of Y. On the contrary, for fixed x, ρ, and γ in this paper, our separation problem is a mixed-integer trilinear program involving binary variables i, i = 1,..., n. Moreover, except for the case D = D (n+1) = {0, 1} n, h(x, y, ρ, γ) is not decomposable by appointment in view of the cross-appointment nature of D. As a result, max y Y h(x, y, ρ, γ) becomes much more challenging to tackle.

13 Article submitted; manuscript no. (Please, provide the manuscript number!) 13 Our analysis consists of the following steps. We start by showing the convexity of h(x, y, ρ, γ) in variable y. Then, it follows from fundamental convex analysis that maximizing convex function h(x, y, ρ, γ) on polyhedron Y will yield an optimal solution at one of the extreme points of Y. Also considering the cost of idleness, we extend extreme-point representation result in Mak et al. (2015) and reformulate the separation problem (14) using a polynomial number of binary variables to replace the continuous variables y i, i = 1,..., n. Lemma 2. For fixed x, ρ, and γ, function h(x, y, ρ, γ) is convex in variable y. We refer to Appendix B.2 for a proof. By Lemma 2, an optimal solution y to the separation problem (14) exists at one of the extreme points of Y consisting of linear constraints (11b) (11d). Consider { } Y = c o y n c u n, y n + c w n y n 1 c u n 1,, y 2 + c w 2 y 1 c u 1. (15) It can be observed (see, e.g., Mak et al. (2015)) that any extreme point ŷ of Y satisfy (i) either ŷ n = c u n or ŷ n = c o, and (ii) for all i = 1,..., n 1, dual constraint ŷ i+1 + c w i+1 ŷ i c u i at either the lower bound or the upper bound. is binding This observation motivates us to establish an alternative formulation of (14) using new binary variables. For notation convenience, we define a dummy variable y n+1, which always takes the lower-bound value c u n+1 := 0. There is a one-to-one correspondence between an extreme point of Y and a partition of the integers 1,..., n+1 into intervals. For each interval {k,..., j} {1,..., n+1} in the partition, y j takes on the lower bound value c u j and other y i eual to their upper bounds, i.e., y i = y i+1 + c w i+1, i = k,..., j 1. As a result, for each interval {k,..., j} in the partition and i {k,..., j}, the value of y i is given by: { c u y i = π ij := j + j cw l 1 i j n, c o + n cw l 1 i n, j = n + 1, and y n+1 = π n+1,n+1 := 0. Define binary variables t kj for all 1 k j n + 1, such that t kj = 1 if interval {k,..., j} belongs to the partition (i.e., t kj = 1 if y i = π ij ) and t kj = 0 otherwise. For a valid partition, we reuire each index i belonging to exactly one interval, and thus i n+1 t j=i kj = 1, i = 1,..., n+1. For notation convenience, we define x n+1 = n+1 = s n+1 := 0. Using binary variables t kj, we reformulate the separation problem (14) as ( ) max max ( i s i x i )π ij t kj t (,s) D D s s.t. j=k i=k (16) (ρ i s i + γ i ) (17a) t kj = 1 i = 1,..., n + 1 (17b) j=i t kj {0, 1}, 1 k j n + 1. (17c)

14 14 Article submitted; manuscript no. (Please, provide the manuscript number!) Note that the objective function (17a) contains trilinear terms i s i t kj with binary variables i and t kj, and continuous variables s i. To linearize formulation (17), we define p ikj i t kj and o ikj i s i t kj for all 1 k j n + 1 and k i j. Also, we introduce the following McCormick ineualities (18a) (18b) and (18c) (18d) for variables p ikj and o ikj, respectively. p ikj t kj 0, p ikj i 0, p ikj i t kj 1, p ikj 0, o ikj s L i p ikj 0, o ikj s U i p ikj 0, o ikj s i + s L i (1 p ikj ) 0, o ikj s i + s U i (1 p ikj ) 0. (18a) (18b) (18c) (18d) Thus, the separation problem (14) is euivalent to the following mixed-integer linear program. max t,,s,p,o (π ij o ikj x i π ij t kj ) j=k i=k s.t. (17b) (17c), (18a) (18d), (ρ i s i + γ i ) (19a) (19b) s i [s L i, s U i ], D {0, 1} n. (19c) Therefore, we can replace Steps 3 8 of Algorithm 1 proposed in Section 3 based on this MILP reformulation as follows: 3: With (x, ρ, γ) fixed to be (x, ρ, γ ), solve formulation (19) and record an optimal solution (t,, s, p, o ). 4: if δ n+1 n+1 j=k j i=k (π ijo ikj x i π ij t kj) n (ρ i s i + γ i ) then 5: stop and return x as an optimal solution to formulation (7). 6: else 7: add the cut δ n+1 n+1 j=k j i=k {L(x, ρ, γ, δ) 0} and go to Step 2. 8: end if ( πij o ikj π ij t kjx i ) n (s i ρ i + i γ) to the set of cuts Remark 1. We note that Algorithm 1 applies to various types of no-show support D. For example, we can specify D = { {0, 1} n : n (1 i) Q max }, where Q max represents the maximum number of no-shows. In this case, we only need to replace the definition of D in (19c) when applying Algorithm 1. In fact, Algorithm 1 is general in D selections, depending upon the operator s beliefs and/or preferences, and the computational tractability. In this paper, we specify D = D (K) due to its flexibility (see Section 2.3) and computational tractability (see, e.g., Proposition 1 and Theorem 2). We further identify a set of valid ineualities to strengthen the mixed-integer feasible region of formulation (19). We summarize the valid ineualities in the following proposition and delegate its proof in Appendix B.3.

15 Article submitted; manuscript no. (Please, provide the manuscript number!) 15 Proposition 1. The following ineualities are valid for set F = {(t,, s, p, o) : (19b) (19c)}: i K+2 s i s i l=i K+2 p ikj = i i = 1,..., n + 1, (20a) j=i (o ikj s L i p ikj ) s L i 1 i n + 1, (20b) j=i (o ikj s U i p ikj ) s U i 1 i n + 1, (20c) j=i p lki + i+k 1 p iki + j=i+1 i+k 1 l=i j=i+k 1 p (i+1)(i+1)j p lkj t kj 1 k < j n + 1, k i j K + 1, (20d) p l(i+1)j i K+2 j=i+k LP Reformulations of the DR Expectation Model t ki i = K 1,..., n, (20e) t (i+1)j i = 1,..., n K + 2. (20f) In this section, we present the main results of the paper as the derivation of convex hulls of the separation problem (14) for D = D (2) (i.e., no conservative no-shows) and D = D (n+1) arbitrary no-shows). This leads to polynomial-size LP reformulations of the DR expectation model (7). Case 1. (No Consecutive No-Shows) Recall that F represents the mixed-integer feasible region of formulation (19), i.e., F = {(t,, s, p, o) : (19b) (19c)}. We show that the valid ineualities identified in Proposition 1 are sufficient to describe conv(f ). We first notice that when K = 2: (i) ineualities (20d) are euivalent to p ikj + p (i+1)kj t kj for all 1 k < j n + 1 and k i j 1, and (ii) ineualities (20e) and (20f) are identical and euivalent to (i.e., p iki + j=i+1 p (i+1)(i+1)j t ki i = 1,..., n. (21) This leads to the convex-hull result in the following theorem, of which a detailed proof is relegated to Appendix B.4. Theorem 1. Polyhedron CF := {(t,, s, p, o) : (17b), (18a), (18c), (20a) (20d), (21)} is the convex hull of set F, i.e., CF = conv(f ). Therefore, the separation problem (14) is euivalent to the following LP reformulation: max t,,s,p,o (π ij o ikj x i π ij t kj ) j=k i=k (ρ i s i + γ i )

16 16 Article submitted; manuscript no. (Please, provide the manuscript number!) s.t. (t,, s, p, o) CF. To combine the separation problem with the outer minimization problem in (13), we present the above reformulation in its dual form: min s.t. (α i + s U i τ U i s L i τ L i ) (22a) j 1 min{j,n} (α i σ ikj ) + λ ikj + φ i i=k i=k i=j π ij x i 1 k j n + 1, (22b) i=k ζ i γ 1 i n, (22c) τ L i τ U i ρ i 1 i n, (22d) σ ikj + s L i ϕ L ikj s U i ϕ U ikj + ζ i s L i τ L i + s U i τ U i max{2i j,i 1} n l=min{2i k 1,i} 1 min{j 1,i} l=max{k,i 1} λ lkj φ l 0 1 k j n + 1, k i j, (22e) ϕ L ikj + ϕ U ikj + τ L i τ U i π ij 1 k j n + 1, k i j, (22f) ϕ L ikj, ϕ U ikj, τ L i, τ U i, λ ikj, φ i, σ ikj 0 1 k j n + 1, k i j, (22g) where we denote a b := max{a, b} and a b := min{a, b} for a, b R for notation convenience. Here the dual variables α i, σ ikj, ϕ L/U ikj, ζ i, τ L/U i, λ ikj, and φ i are associated with constraints (17b), (18a), (18c), (20a), (20b) (20c), (20d), and (21) respectively (after transforming all ineualities into the form), and constraints (22b) (22f) are associated with primal variables t kj, i, s i, p ikj, and o ikj respectively. In (22b), the term min{j,n} i=j φ i becomes φ j for all 1 j n, and will disappear for j = n + 1. In (22e), when k i < j, the term min{j 1,i} λ l=max{k,i 1} lkj becomes λ ikj λ (i 1)kj ; when k < i = j, it becomes a singleton λ (i 1)kj ; and when k = i = j, it does not appear. Similarly, when 2 k = i = j n, the term max{2i j,i 1} n l=min{2i k 1,i} 1 φ l becomes φ i φ i 1 ; when j > i = k or k = i = j = n + 1, the term only contains φ i 1 ; when k < i = j or 1 = k = i = j, the term only contains φ i ; and in all other cases, i.e., when 1 k < i < j n + 1, the term does not appear. We can then reformulate the DR expectation model in a LP form as follows. Theorem 2. Under no-consecutive no-show assumption, i.e., D = D (2), the DR expectation model (7) is euivalent to the following linear program: min µ i ρ i + νγ + (α i + s U i τ U i s L i τ L i ) s.t. (22b) (22g), x i = T, x n+1 = 0, x i 0 i = 1,..., n.

17 Article submitted; manuscript no. (Please, provide the manuscript number!) 17 Case 2. (Arbitrary No-Shows): Given D = {0, 1} n and D s = n [sl i, s U i ], the optimization problem defining function h(x, y, ρ, γ) (see (12c)) is separable by each appointment, i.e., { } h(x, y, ρ, γ) = max ( i s i x i )y i (ρ i s i + γ i ) (,s) D D s = max {( i s i x i )y i (ρ i s i + γ i )}. i {0,1}, s i [s L i,su i ] To reformulate separation problem (14), recall the observations on polyhedron Y in Section 3.2 and again we represent the extreme points of Y based on variables t kj. It follows that ( ) max h(x, y, ρ, γ) = max max {( i s i x i )π ij (ρ i s i + γ i )} y Y t 0 i {0,1}, s i [s L i,su i ] s.t. j=k i=k t kj (23a) t kj = 1 i = 1,..., n + 1 (23b) j=i t kj {0, 1}, 1 k j n + 1. (23c) Because the constraint matrix formed by constraints (23b) (23c) is totally unimodular (TU, see Mak et al. (2015)), we can relax the integrality constraints (23c) without loss of optimality. Hence, formulation (23a) (23c) is a LP model in variables t kj and we can take its dual as: min α,β s.t. n+1 α i α i i=k i=k (24a) β ij, 1 k j n + 1 (24b) β ij max {( i s i x i )π ij (ρ i s i + γ i )}, i {0,1}, s i [s L i,su i ] i = 1,..., n, j = i,..., n + 1 (24c) β n+1,n+1 = 0, (24d) where dual variables α i, i = 1,..., n + 1 are associated with primal constraints (23b), dual constraints (24b) are associated with primal variables y kj, each variable β ij represents the value of max i {0,1},s i [s L i,su i ] {( is i x i )π ij (ρ i s i + γ i )}, and β n+1,n+1 = 0 because n+1 = s n+1 = π n+1,n+1 = 0. Finally, note that for each i = 1,..., n, the related objective function ( i s i x i )π ij (ρ i s i + γ i ) is linear in variables i and s i, and thus each of constraints (24c) is euivalent to β ij π ij x i s L i ρ i (25a) β ij π ij x i s U i ρ i (25b) β ij π ij x i s L i ρ i γ + s L i π ij (25c) β ij π ij x i s U i ρ i γ + s U i π ij (25d)

18 18 Article submitted; manuscript no. (Please, provide the manuscript number!) because i {0, 1} and s i {s L i, s U i } at optimality. It follows that formulation (13), as well as the DR expectation model (7), is euivalent to the following LP model when D = D (n+1) : min x,ρ,γ,α,β µ i ρ i + νγ + α i s.t. (24b), (24d), (25a) (25d), 4. Solution Approaches for DR CVaR Models x i = T, x n+1 = 0, x i 0, i = 1,..., n. In this section, we reformulate DR CVaR constraints in (6) with general cost parameters c o, c u i, and c w i. We begin by representing the CVaR by an alternative definition (see, e.g., Rockafellar and Uryasev (2000, 2002)): CVaR 1 ɛ (Q(x,, s)) = inf {z + 1ɛ } E P,s [Q(x,, s) z] +, z R where [a] + := max{a, 0} for a R. It follows that sup P,s F(D,µ,ν) CVaR 1 ɛ (Q(x,, s)) = sup P z R,s F(D,µ,ν) = inf sup z R P,s F(D,µ,ν) = inf z R { z + 1 ɛ inf {z + 1ɛ } E P,s [Q(x,, s) z] + {z + 1ɛ E P,s [Q(x,, s) z] + } (26a) } sup E P,s [Q(x,, s) z] +, (26b) P,s F(D,µ,ν) where euality (26a) follows from the Sion s minimax theorem (Sion 1958) because z + 1 E ɛ P,s [Q(x,, s) z] + is convex in variable z, concave (in particular, linear) in variables P,s, and F(D, µ, ν) is weakly compact MILP Reformulation and Decomposition Algorithm Based on a similar dualization process in Section 3 (see the primal and dual formulations (8) and (9)), we reformulate the inner maximization problem in (26b) as a minimization problem, and combine it with the outer minimization problem to obtain ( ) inf z + 1 µ i ρ i + νγ + θ z,ρ,γ,θ ɛ s.t. s i ρ i + γ i + θ [Q(x,, s) z] + = inf z,ρ,γ,θ s.t. (, s) D D s ( ) z + 1 µ i ρ i + νγ + θ ɛ s i ρ i + γ i + θ 0 (, s) D D s (27a)

19 Article submitted; manuscript no. (Please, provide the manuscript number!) 19 s i ρ i + γ i + θ Q(x,, s) z (, s) D D s, (27b) where constraints (27a) and (27b) are derived based on the definition of [ ] +. Thus, the DR CVaR constraint (6) is euivalent to ( ) Q z + 1 µ i ρ i + νγ + θ (28a) ɛ { } min ρ i s i + γ i + θ 0 (28b) (,s) D D s { } θ + z max Q(x,, s) (ρ i s i + γ i ), (28c) (,s) D D s where constraint (28a) is linear, but (28b) and (28c) need further analysis. First, we replace constraint (28b) by euivalent linear constraints in the following proposition, whose proof is relegated to Appendix C.1. Proposition 2. For fixed ρ i and γ, and D = D (K) euivalent to linear constraints: n K+1 θ + β i + with K {2,..., n + 1}, constraint (28b) is (s L i χ L i s U i χ U i η i ) 0, (29a) η i + min{i,n K+1} j=max{i K+1,1} β j γ 1 i n, (29b) χ L i χ U i ρ i 1 i n, (29c) β i, χ L i, χ U i, η i 0 1 i n. (29d) Second, note that the right-hand side of constraint (28c) is euivalent to that of constraint (13b) in the reformulated DR expectation model, and so the reformulated separation problem (19) and Algorithm 1 described in Section 3 can be easily adapted to handle constraint (28c). Furthermore, the valid ineualities (20a) (20f) can be incorporated to accelerate solving the adapted separation problem and implementing the decomposition algorithm LP Reformulations of the DR CVaR Model We derive LP reformulations for the DR CVaR constraint (6) when D = D (2) no-shows) and when D = D (n+1) (i.e., arbitrary no-shows). (i.e., no consecutive Case 1. (No Consecutive No-Shows) Recall that DR CVaR constraint (6) is euivalent to constraints (28a), (29a) (29d) with K = 2, and (28c). When D = D (2), we apply Theorem 1 to further reformulate (28c) as linear constraints θ + z n+1 (α i + s U i τi U s L i τi L ) and (22b) (22g), resulting in the following proposition.

20 20 Article submitted; manuscript no. (Please, provide the manuscript number!) Proposition 3. When D = D (2), the DR CVaR constraint (6) is euivalent to linear constraints (28a), (29a) (29d) with K = 2, n+1 (α i + s U i τ U i s L i τ L i ) θ + z, and (22b) (22g). We remark that the LP reformulation in Proposition 3 is of the size O(n 3 ) because constraints (22b) (22g) incorporate O(n 3 ) decision variables and linear constraints. In this section, we focus on a specific DR CVaR constraint (6) that restricts overtime only. That is, c u i = c w i = 0 for all 1 i n and c o = 1, and Q(x,, s) = Q W (x,, s) := min w,u,w W subject to constraints (2b) (2d). Next, we derive a more compact LP reformulation of this DR CVaR constraint with O(n 2 ) variables and constraints. To that end, we derive an O(n 2 ) LP reformulation for constraint (28c). We begin by specializing the extreme point representation of polyhedron Y for Q(x,, s) = Q W (x,, s). Lemma 3. When c u i = c w i = 0 for all 1 i n and c o = 1, the set of extreme points of polyhedron Y defined in (15) is { n l=k e l : k = 1,..., n} {0 n }, where e l represents an n-dimensional unit vector with component l eualing to one and any other component eualing to zero; 0 n is an n-dimensional zero vector. Recall the observation in Section 3.2 that each extreme point (y 1,..., y n+1 ) of Y is associated with a partition of set {1,..., n + 1} into intervals. The result in Lemma 3 follows from (16) when the cost parameters take the above specified values. Define binary variables t k for all 1 k n to represent the set of extreme points of Y, such that t k = 1 if the extreme point is n l=k e l and t k = 0 otherwise. Note that extreme point 0 n is represented by t k = 0 for all 1 k n. For a valid representation, we reuire n t k 1. It follows that the right-hand side of (28c) (with Q(x,, s) = Q W (x,, s)) is euivalent to max t,,s s.t. ( ) ( i s i x i ) t k i=k (ρ i s i + γ i ) t k 1, D, s D s, t {0, 1} n as a mixed-integer bilinear program with binary vectors and t, and continuous vector s. We linearize the bilinear terms by defining p ki t k i and o ki t k i s i for all 1 k i n. Also, we introduce McCormick ineualities (30b) (30c) and (30d) (30e) for variables p ki and o ki, respectively to further reformulate the separation problem as a mixed-integer linear program: max t,,s,p,o i=k (o ki x i t k ) (ρ i s i + γ i ) (30a) s.t. p ki t k 0 1 k i n, (30b) p ki i 0, p ki i t k 1, p ki 0 1 k i n, (30c) o ki s L i p ki 0, o ki s U i p ki 0 1 k i n, (30d)

21 Article submitted; manuscript no. (Please, provide the manuscript number!) 21 o ki s i + s L i (1 p ki ) 0, o ki s i + s U i (1 p ki ) 0 1 k i n, (30e) t k 1, (30f) D, s D s, t {0, 1} n. (30g) Similar as before, we aim to derive the convex hull of the feasible region of problem (30), i.e., the mixed-integer feasible region described by constraints (30b) (30g). We denote the feasible region as set G and derive conv(g) in the following theorem, whose proof is relegated to Appendix C.2. Theorem 3. When D = D (2), the following ineualities are valid for set G = {(t,, s, p, o) : (30b) (30g)}: s i s i p kn n, (31a) p ki + p k(i+1) t k 1 k i n 1, (31b) (p ki t k ) i 1 1 i n, (31c) ( ) pki + p k(i+1) t k + i + i i n 1, (31d) (o ki s L i p ki ) s L i 1 i n, (31e) (o ki s U i p ki ) s U i 1 i n. (31f) Furthermore, polyhedron CG := {(t,, s, p, o) : (30b), (30d), (31a) (31f)} is the convex hull of set G, i.e., CG = conv(g). Theorem 3 provides us a compact LP reformulation of the right-hand side of constraint (28c) with O(n 2 ) variables and constraints: max t,,s,p,o i=k (o ki x i t k ) s.t. (t,, s, p, o) CG. (ρ i s i + γ i ) (32a) Finally, by resorting to the dual formulation of (32), we represent constraint (28c) as (32b) n 1 (α i s L i τ L i + s U i τ U i ) φ i θ + z (33a) α i n 1 (α i σ ki ) + (λ ki φ i ) i=k min{i,n 1} l=max{i 1,1} i=k x i 1 k n, (33b) i=k max{i,n 1} φ l ζ γ 1 i n, (33c) l=n