An approach to optimize block surgical schedules

European Journal of Operational Research 235 (2014) 138–148

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Discrete Optimization

An approach to optimize block surgical schedules Sangdo Choi a,1, Wilbert E. Wilhelm b,⇑ a b

Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL 32611-6565, USA Department of Industrial and Systems Engineering, Texas A&M University, 3131 TAMU, College Station, TX 77843-3131, USA

a r t i c l e

i n f o

Article history: Received 5 October 2012 Accepted 15 October 2013 Available online 30 October 2013 Keywords: Operations research in health service Block surgical schedule Sequential newsvendor Normal distribution No-shows

a b s t r a c t We provide an approach to optimize a block surgical schedule (BSS) that adheres to the block scheduling policy, using a new type of newsvendor-based model. We assume that strategic decisions assign a specialty to each Operating Room (OR) day and deal with BSS decisions that assign sub-specialties to time blocks, determining block duration as well as sequence in each OR each day with the objective of minimizing the sum of expected lateness and earliness costs. Our newsvendor approach prescribes the optimal duration of each block and the best permutation, obtained by solving the sequential newsvendor problem, determines the optimal block sequence. We obtain closed-form solutions for the case in which surgery durations follow the normal distribution. Furthermore, we give a closed-form solution for optimal block duration with no-shows. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction A block surgical schedule (BSS) prescribes the duration of the time block reserved for each specified surgery sub-specialty and sequences time blocks within each operating room (OR) day to achieve the objective of minimizing the sum of expected earliness and tardiness costs. Tactical-level decisions compose a BSS for use over the intermediate-term (e.g., month or quarter), allowing flexibility within a long-term, strategic plan, for example, to accommodate seasonal demand changes. With the goal of synthesizing a methodology to prescribe a BSS, specific research objectives of this paper are 1 a method to optimize the planned duration of each block, minimizing the sum of expected earliness and lateness costs; 2 a method to optimize the sequence (i.e., permutation) of blocks in each OR day; and 3 an extension of our method to prescribe an optimal planned block duration when no-shows are considered. Each hospital provides a unique capacity for performing surgery through the numbers of ORs and surgical skills it offers. A surgical suite typically comprises several ORs, each of which is equipped to support one (e.g., cardiology, neurological, or orthopedic) or several (e.g., general surgery, ENT) specialties. The typical surgical specialty comprises a number of subspecialties. For instance, orthopedics includes hip replacement, knee replacement, femur fixation, and shoulder repair sub-specialties. Surgeries that require the same sub-specialty are medically

⇑ Corresponding author. Tel.: +1 979 458 2348, fax: +1 979 458 4299. 1

E-mail addresses: s.choi@ufl.edu (S. Choi), [email protected] (W.E. Wilhelm). Tel.: +1 352 392 1464; fax: +1 352 392 3537.

0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.10.040

homogeneous and require the same medical expertise of the surgeon or surgeons involved (van Oostrum et al., 2008). Allocation (or assignment) decisions are made for the longer term (e.g., six or 12 months); we assume that they assign one specialty to each OR day. Based on specialty-to-OR-day-assignment decisions, the current paper prescribes time blocks for subspecialties within the specialty for the intermediate term (e.g., month or quarter). To the best of our knowledge, little research had dealt with determining block duration and sequence within an OR day. We deal with the block scheduling policy in this study. A block is the amount of time during which a specific sub-specialty is assigned to an OR. A block may be planned with the duration of two hours, half of a day, or a day, for example, to permit a surgeon to perform a series of surgeries. An alternative, the open scheduling policy, under which each surgeon can schedule his/her surgeries at any time, was common in the 1960s and 1970s but is rarely used in practice today, because it does not utilize surgeons’ time as efficiently as block scheduling (Blake, Dexter, & Donald, 2002). Once BSS determines a schedule of time blocks, including the duration and sequence of each, the day-by-day schedule for a week may be used cyclically, that is, for each week over the intermediate-term planning horizon. A cyclic schedule avoids the need to prescribe a new schedule every week and promotes coordination among surgeons, staff and other departments (e.g., post-anesthesia care unit (PACU), intensive care unit (ICU)), affording each surgeon the opportunity to promote his/her efficiency by performing surgeries consecutively and by establishing routine office hours that are compatible with the BSS. A BSS, which is analogous to a master production schedule in a manufacturing environment, has a number of important uses. A

S. Choi, W.E. Wilhelm / European Journal of Operational Research 235 (2014) 138–148

BSS defines aggregate resource requirements of peri-operative activities and ancillary departments (e.g., PACU, ICU, nursing), not only of ORs and surgeons. Nurse managers must ensure that the set of ORs and PACUs run compatibly each day of the week (Blake & Donald, 2002) so that actual decisions adhere to the BSS as strictly as possible. Like Dexter and Hopwood (1999), Rohleder, Sabapathy, and Shorn (2005), and Samanlioglu et al. (2010), this paper focuses on ORs and does not deal with other departments. An appropriate BSS allows hospital managers to accommodate random events (e.g., a short-term shortage of surgeons or anesthetists), seasonal fluctuations in demand (e.g., summer or Christmas time), or strategic decisions that alter program emphasis (e.g., to respond to an increasing popularity of cosmetic surgery) (Blake & Donald, 2002). In particular, the operational-level uses the BSS to schedule individual patients; if actual demand levels were to deviate significantly from the those upon which BSS was based, a hospital manager should update the BSS to better accommodate them. This research contributes from several perspectives. It provides a closed form for optimal block durations, which are given by newsvendor solutions, for the case in which surgery durations are independent and normally distributed. Hospital managers can make use of this closed form to balance the risk of a planned duration that is too short, which could force a late start of the next block (i.e., delay), should actual time exceed it; and the risk of a planned duration that is too long, which could result in expediting the next block, should actual time be less. Furthermore, we deal with a new type of newsvendor problem, which is, in fact, a series of time-based newsvendor problems that we call the sequential newsvendor (SNV) problem. The classic newsvendor model deals with a single time period with random demand, which is a known distribution. It prescribes the optimal order quantity to minimize the sum of costs related to expected demand over and under the order quantity. Our model prescribes the duration of each surgery time block to minimize the sum of costs related to expected early and late completion (i.e., before and after the end of the time block, respectively). Optimal block durations can be obtained via a newsvendor problem that prescribes the optimal planned ending time. We prove that the smallest-variance-first-rule (SV) optimally sequences blocks if each surgery follows a normal distribution. This research also suggests an approach to find the optimal block duration when subject to no-shows, a new and emerging topic in the healthcare setting. However, because surgery typically deals with serious health issues, no-shows are not likely to occur with the high frequency they do, for example, at primary care clinics. If no-shows occur frequently, the risk of idleness increases and is often hedged by overbooking. This research analyzes the effect of no-shows using both the ratio of earliness cost to lateness cost and the probability that a patient will be a no-show to hedge by managing planned block duration. The remainder of this paper is organized as follows. Section 2 reviews the intermediate-term surgical scheduling literature. Section 3 presents preliminaries and Section 4 describes our solution approach. Section 5 describes the optimal block duration with no-shows. Section 6 provides insights for hospital management. Finally, Section 7 concludes and offers suggestions for future research.

2. Literature review Few studies have addressed the tactical level of decision making that prescribes a block surgical schedule for the intermediate term. Complicating matters, there is no commonly accepted definition of intermediate-term surgical scheduling (Testi, Tanfani, & Torre, 2007; van Oostrum et al., 2008). Blake and Donald (2002), Blake

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and Donald (2002),Fei, Chu, Meskens, and Artiba (2008), and Fei et al. (2008) described the intermediate-term surgical scheduling process in detail, comparing it with master production scheduling in manufacturing. van Oostrum et al. (2008) discussed the pros and cons of intermediate-term surgical schedule, compared centralized and decentralized processes for planning such schedules, addressed various implementation issues and discussed suitability for hospitals with different organizational foci and culture. Due to the absence of a standard definition, various studies have assigned surgeries to ORs as part of strategic, tactical, or operational decisions. The strategic problem of assigning specialties to ORs essentially assumes that each OR day comprises a single time block and determines the number of OR-days for each specialty. One line of research on intermediate-term decisions has investigated assigning the expected number of surgeries associated with each specialty to a specific OR day. In contrast, we regard this assignment problem as a strategic-level decision and assume that the assignment of specialties to OR days is given. Santibanez, Begen, and Atkins (2007) assigned specialties to time blocks at the intermediate term, assuming that both the total amount of OR time and the number of patients are predetermined for each specialty over the planning horizon. Following Santibanez et al. (2007), our approach invokes the assumption that the number of patients is forecast for each sub-specialty. Guinet and Chaabane (2003) and Jebali, Hajalouane, and Ladet (2006) combined the assignment of specialties to OR days, typically a strategic-level problem, and the sequencing of surgeries in each OR, considered an operational-level issue, in one model. A number of operations research methodologies have been used to assign surgeries to ORs or blocks. Both deterministic integer programs (Kharraja, Albert, & Chaabane, 2006; Blake & Donald, 2002; Zhang, Dessouky, & Belson, 2009; Fei, Chu, & Meskens, 2009) and stochastic programs (Denton, Miller, Balasubramanian, & Huschka, 2010; Beliën, Demuelemeester, & Cardoen, 2009) have been used to prescribe intermediate-term surgical schedules. Kharraja et al. (2006) modeled the assignment of specialties to days of pre-specified duration as a cutting stock problem with the objective of minimizing penalties for under- and over-use of ORs. Blake and Donald (2002) and Zhang et al. (2009) developed an analytical solution and incorporated it in a simulation model that captures randomness (e.g., random arrivals, no-shows) and non-linearities (e.g., non-proportional allocation of demand). Fei et al. (2008) studied surgery assignment using a set-partitioning formulation and branch-and-price. Denton et al. (2010) and Beliën et al. (2009) used stochastic optimization at the operational level to assign surgeries to ORs on a day. A number of studies have used newsvendor models to prescribe block duration. Several studies (Strum, May, & Vargas, 2000; Olivares, Terwiesch, & Cassorla, 2008; Wachtel & Dexter, 2010) have employed the newsvendor model to optimize the duration of a single block; they do not deal with sequencing blocks. This approach is more closely related to ours than is the assignment problem used, for example, by Guinet and Chaabane (2003) and Jebali et al. (2006). Guerriero and Guido (2010) and May, Spangler, Strum, and Vargas (2011) employed a newsvendor model at the strategic level to determine OR time for a specialty. Strum et al. (2000) developed a newsvendor model to find the optimal block duration based on historical workloads (e.g., numbers of surgeries performed, numbers of staff hours). Olivares et al. (2008) applied a newsvendor model to determine how much OR time to reserve for a specific cardiac surgery to balance the costs of reserving too much vs too little OR time. Wachtel and Dexter (2010) gave a systematic review of the behavioral and experimental literature associated with newsvendor problems relevant to OR management and commented on the potential significance of these studies.

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In contrast to earlier studies, we employ a newsvendor model to prescribe planned end-time (accordingly, block durations as well) and the sequential newsvendor model to specify block sequence. No prior research has addressed the block-sequence problem. We are the first to provide a closed-form solution for the case in which surgery durations are independent and normally distributed. Using the closed form that we obtain, we are able to derive the optimal rule to sequence blocks. 3. Preliminaries This section introduces notation and assumptions used in the subsequent presentation. We also discuss both decision and associated random variables. Finally, we formulate the objective function, which minimizes the sum of expected earliness and lateness costs. 3.1. Assumptions and notation We deal with a general number of blocks to cast our results in the most generic form possible, even though there may only be one, two, or (at most) four blocks for each OR day. We focus on a single OR because, once each surgical specialty has been assigned to an each OR day, a problem involving multiple ORs can be decomposed into a set of independent problems, each involving a single OR. Other researchers (Blake & Donald, 2002; Zhang et al., 2009; Santibanez et al., 2007) have considered related departments such as nurse capacity, bed capacity, and PACU. Alternative assignments may occur in practice in order to coordinate the capacities of surgery and other departments. For example, a subspecialty may be assigned blocks in several ORs during a day or to blocks in different ORs on different days. Given the assignment of sub-specialties to OR days, our models can be applied to each OR day. The incidence of such alternatives may depend upon the size of the hospital; our analysis follows the recommendations of our healthcare collaborator. We emphasize on-time performance rather than nurse, bed, PACU capacities. We assume that the forecast employed to support the strategic decisions that assign specialties to OR days is compatible with the forecast used at the tactical level to prescribe the BSS, which partitions each OR day into time blocks for sub-specialties associated with the relevant specialty. We assume an environment in which strategic decisions assign specialties to OR days based on a longterm (e.g., annual) forecast. Tactical decisions assign each sub-specialty to a time block within the appropriate OR day, based on an intermediate-term forecast, which can be expected to be more accurate because it deals with a shorter time horizon and may include a mix of actual and forecast needs. Operational-level decisions, which are made on a daily basis, assign actual patients to specific times within time blocks, matching sub-specialty need with the BSS (e.g. Dexter, Ledolter, & Watchel, 2005). We assume that one surgery begins as soon as the previous one ends. Most prior studies for intermediate-term surgical scheduling (Blake & Donald, 2002; Dexter et al., 2005; Denton et al., 2010; Fei et al., 2009; Fei et al., 2008; Kharraja et al., 2006; Rohleder et al., 2005; Samanlioglu et al., 2010; van Oostrum et al., 2008) have dealt with numbers of surgeries assigned to blocks, or did not consider waiting- and idle-times between consecutive surgeries (Blake et al., 2002; Santibanez et al., 2007). The assumption implicit in these studies is the same as ours. Most prior studies for operational-level scheduling have assumed that each surgery is scheduled to begin at the expected completion time of the previous surgery (Choi, 2012; Gupta, 2007; Pinedo, 2009), although some incorporate a multiple of the standard deviation of the surgery duration as a safety time to manage risk (Gul, Denton, Fowler, & Huschka, 2011). If the previous surgery completes before the scheduled start

of the next surgery, OR idleness is incurred; if it finishes after the scheduled start time, the next surgery (i.e., both patient and surgeon who are ready at the scheduled start time) must wait. In contrast, we assume that each surgery begins when the previous surgery ends. This assumption appears to be reasonable in our study because patients are typically prepared well in advance of their scheduled start time and successive surgeries within each block are likely to be performed by the same surgeon so that s/he would be available as well. If successive surgeries are performed by different surgeons, our assumption would require schedulers to communicate with the surgeon who will perform the next surgery and facilitate her/his readiness ahead of the scheduled start time. This is done currently to the extent possible but may entail establishing different procedures to effect routinely. If the scheduled start time were enforced when the previous surgery is completed early, our assumption would lead to lower bounds on optimal block durations. We feel that it is appropriate for the intermediate term (e.g., month of quarter) that block time duration be considered rather than the waitingand idle-times of individual surgeries. A companion paper (Choi & Wilhelm, 2012) studied the problem of sequencing individual surgeries at the operational (i.e., daily) level, considering the waitingand idle-times between surgeries. Each medical procedure is designated by one of many thousands of current procedure terminology (CPT) codes. Each surgery specialty (e.g., orthopedic) may deal with hundreds of CPT codes and each sub-specialty (e.g., joint replacement; bone fractures; knee, spine or shoulder repair) within the index set I of sub-specialties that constitute the specialty may deal with dozens of CPT codes. Further, a given surgery may involve a combination of CPT codes. For example, shoulder repair deals with a large number of CPT codes, of which about 15 procedures are performed commonly. Examples of these five-digit codes are 29805 (diagnostic shoulder arthroscopy), 29826 (shoulder arthroscopy with subacromial decompression), 29807 (labral repair), 29827 (rotator cuff repair), 23430 (bicep tenodesis), and 23120 (acronioclavicular joint resection). One of the authors recently had shoulder-repair surgery that involved the combination of the first four of these CPT codes. We use b S i to denote the index set of surgery types associated with sub-specialty i, each an individual CPT code or a combination that is common. We envision a BSS process that forecasts ni , the expected number of surgeries to be performed within sub-specialty i 2 I; and qis , Si . the portion of sub-specialty i surgeries that will be of type s 2 b ^ is and r ^ is , the mean and Historical data can be used to estimate l Si . variance, respectively, of the duration of surgeries of type s 2 b With this information, the planning process can determine a representative surgery duration for each sub-specialty, which can be interpreted as the duration of a randomly selected surgery to be performed by this sub-specialty. The random duration of the representative surgery, Di , can be expressed as the convex combination of individual, mutually independent, surgery-type durations P b is ; s 2 b b is . This representative surgery of sub-speD qis D S i : Di ¼ s2b Si P ^ is and cialty i has a mixture distribution with mean li ¼ qis l s2b Si P 2 2 ^2 variance ri ¼ q r and, by the Central Limit Theorem (CLT) s2b S i is is (Casella & Berger, 2001), is normally distributed because jb S i j, for each i 2 I, is large, as described in the paragraph above. Based on this analysis, we treat the duration of surgeries of sub-specialty i as i.i.d. normal random values. We determine block duration for each sub-specialty, expressed in terms of representative surgeries. We assume that subspecialty i involves ni representative surgeries. While a sub-specialty that performs surgeries requiring long durations (e.g., orthopedic) may have ni ¼ 1, indicating a single surgery in a block, other sub-specialties with surgeries that require short durations

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(e.g., eye surgery) may perform several surgeries in a time block. A single surgeon or a group of surgeons may perform a series of surgeries in a time block. We now define the notation we use in the subsequent presentation. If sub-specialty i is assigned to one block and the duration of each surgery is Di hours with mean li and variance r2i , the block must accommodate the total surgery time, the ni -fold convolution  i :¼ ni li and variance r  2i :¼ ni r2i . of Di , which has mean l Index sets and indices I Sub-specialties K Sequence positions for time blocks D Permutations of time blocks Parameters ce Earliness penalty cost Lateness penalty cost cl b Ratio of earliness cost to lateness cost,

i2I k2K d2D

b ¼ ce =cl .

We use map D : I ! K to represent a set of sequences (or permutations), each of which assigns each sub-specialty to one and only one sequence position. We use subscripts ½k for kth block sequence position and i for sub-specialty to avoid potential confusion. We assume that each sub-specialty is assigned to a block of time, so the numbers of assigned sub-specialties to the OR, time blocks, and sequence positions are equal. Each sub-specialty should be assigned to one block and each block should be assigned to one position in a sequence, hence jKj ¼ jIj. The total number of sequences possible is jKj!, where jKj is the number of blocks or sub-specialties. Decision variables prescribe planned block durations (i.e., xd½k gives the planned duration of the kth block) and block sequence (i.e., permutation d) for one day in the OR. The planned end time of the block in the kth position, given d, is prescribed by yd½k , where

blocks. If the last surgery of the sub-specialty assigned to the block completes earlier than the planned completion time of the block, earliness results; otherwise, lateness is incurred. Because no surgery will be started after the last block, its earliness corresponds to surgeon and OR idleness; and its lateness, to surgeon and staff overtime. Our objective function penalizes the expected earliness,   þ  þ  or E yd½k  T d½k and the E xd½1 þ    þ xd½k  Bd½1      Bd½k  þ  or expected lateness, E Bd½1 þ    þ Bd½k  xd½1      xd½k  þ  , of each block k 2 K. The former represents the cost E T d½k  yd½k of expediting the start time of the next surgery; and the latter, the cost of delaying the start time of the next surgery. In the case of the last surgery in the sequence, the cost of earliness represents idleness, and the cost of lateness represents overtime premium. We use the same lateness penalty, cl , for all blocks k 2 K because analytical results depend mainly upon parameters of surgery durations (i.e., mean and standard deviation). On-time performance is important in health-care delivery systems, because a BSS coordinates surgeons, nurses, and anesthesiologists and influences other departments like PACU. We build a schedule that balances the expected costs of earliness and lateness associated with each block, defining objective function f½k ðxd½1 þ    þ xd½k Þ; k 2 K; d 2 D

f½k ðxd½1 þ   þ xd½k Þ : ¼ ce E

by day because it depends on forecast workloads and sub-specialties assigned to each day. The random duration of the kth block in the sequence, Bd½k is the  ½k ¼ n½k l½k and variance n½k -fold convolution of D½k and has mean l

r 2½k ¼ n½k r2½k . Random duration Bd½k must be compared with decision variable xd½k , which prescribes planned block duration. We define T d½k :¼ Bd½1 þ    þ Bd½k as the random end time to complete all surgeries assigned to blocks ½1 through ½k and compare it with decision variable yd½k , the planned end time of block ½k. Fig. 1 shows the relationship between decision variables and related random variables for four time blocks: the former are indicated below the time line; and the latter, above. In Fig. 1, lateness is incurred in association with the third block; earliness, with other

xd½1 þ   þ xd½k  Bd½1     Bd½k

þ 

 þ  ; þ cl E Bd½1 þ   þ Bd½k  xd½1     xd½k

ð1Þ

and equivalently, f½k ðyd½k Þ; k 2 K; d 2 D

yd½k ¼ xd½1 þ    þ xd½k . The planned end time of the last block yd½jKj corresponds to the end of the OR day and is important in deciding the number of hours that the staff will be required to work and the amount of overtime that is required. The utilization of an OR, as determined by yd½jKj , relative to the length of the work day may vary



f½k ðyd½k Þ :¼ ce E

  þ  þ  þ cl E T d½k  yd½k : yd½k  T d½k

ð2Þ

In this context, the classical newsvendor problem optimizes the duration of one block, balancing the penalties for expected early and late completion relative to the prescribed duration. We deal with a series of blocks during an OR day, optimizing the duration and sequence of each. d specifies a particular sequence (i.e., permutation) of blocks. The objective function for each sequence is the sum of costs associated with each sequence position. Surgeries in each time block can be completed earlier or later than the prescribed block duration. Because one block follows another, the effects of early and late completion cascade and must be considered in prescribing the completion time for the next block. The penalty for late completion of a block before the last one of the day reflects unplanned waiting time, equivalently, delay. The penalty for late completion of the last block of the day reflects overtime costs that must be borne by the hospital.

Fig. 1. Relationship between decision and random variables.

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  is a newsvendor problem. f½k ðyd½k Þ, equivalently f½k xd½1 þ    þ xd½k , is

3.2. Mathematical model This subsection describes two optimization models, one in terms of xd½k and another, which transforms xd½k to decision variable yd½k . We focus on the latter model to prescribe optimal durations and sequence, because it reduces a complicated problem to a series of newsvendor problems, the sequential newsvendor model. In this subsection, we describe the two mathematical models and show how to exploit the sequential newsvendor model. Lastly, we depict the sequential newsvendor model graphically. In minimizing the sum of expected earliness and lateness costs, the sequential newsvendor problem (SNV), in terms of decision variable xd½k ; k 2 K, is:

ðSNV d ðxÞÞ min min

d2D xd :k2K ½k

X



f½k xd½1 þ    þ xd½k



ð3Þ

k2K

s:t: xd½k P 0 k 2 K; d 2 D:

ð4Þ

^ We seek to determine the optimal planned duration ^ xd½k of the kth block, k 2 K and the optimal block sequence (i.e., permutation) ^ To solve SNV d ðxÞ, we need to show that both the first order necd.

essary condition (FONC) and the second order necessary condition (SONC) are satisfied; i.e., the Hessian matrix of the objective func^ xd½k (Bazaraa, tion should be semi-positive at the optimal point ^ Sherali, & Shetty, 2006). Constraint (4) requires decision variable xd½k ; k 2 K to be non-negative. Solving the problem with decision variables xd½k ; k 2 K is challenging because it requires a complex Hessian matrix to be evaluated. Instead, we employ a linear transformation to use alternative decision variables y½k ; k 2 K and do not use a Hessian matrix, as described in the following subsection. With planned end-time decision variable y½k ; k 2 K, problem SNV d ðxÞ can be transformed to SNV d ðyÞ:

ðSNV d ðyÞÞ min min

d2D yd :k2K ½k

X f½k ðyd½k Þ

ð5Þ

Hence, the objective functions of both SNV d ðxÞ and SNV d ðyÞ are con^½k , is vex. The optimal solution to newsvendor problem NV ½k ðy½k Þ; y the value at which the distribution function F T ½k ðy½k Þ is equal to the critical ratio:

^½k Þ ¼ F T ½k ðy

ce

cl 1 ¼ ; þ cl 1 þ b

ð8Þ

which is the same for each k 2 K because we assume that cost parameters are the same for all blocks. We now suppress superscript d to streamline presentation. We discuss constraint (6), which requires yd½k to increase with k, in the following section. We now introduce a new function, gðdÞ, to explain the sequential newsvendor problem:

( Z  ¼ min gðdÞ : ð6Þ; ð7Þ; and gðdÞ ¼ min d2D

) X f½k ðyd½k Þ :

ð9Þ

k2K

Fig. 3 depicts the sequential newsvendor problem, representing relationships among gðdÞ and NV d½k ðyd½k Þ. There are jDj ¼ jKj! possible sequences, and each d 2 D has an associated gðdÞ. Z  of (9) ^ achieves its minimum at sequence d. Given sequence d 2 D; NV d½k ðyd½k Þ defines the objective function of a newsvendor problem that prescribes the optimal, planned ^d½k ; k 2 K. Given d; gðdÞ is the sum of optimal end time of block ½k; y values to NV d½k ðyd½k Þ, for all k 2 K. Z  in (9) is minimized by prescribing the best sequence (i.e., permutation) of blocks. We first determine gðdÞ by summing the solutions of the jKj newsvendor problems, giving the optimal block durations for a given sequence d, then find the best sequence as described in following section.

k2K

s:t: yd½k1 6 yd½k yd½k

k ¼ 2; . . . ; jKj; d 2 D

P 0 k 2 K; d 2 D:

ð6Þ ð7Þ

Fig. 2 depicts the variable transformation in two dimensional space. While the feasible area is the first quadrant of x-space, owing to constraints (4), the feasible area in y-space is half of the first quadrant as shown in Fig. 2(b) owing to constraints (6) and (7). Variable transformation from x- to y-space recasts SNV d ðxÞ as SNV d ðyÞ, which is able to utilize well-known properties of the newsvendor problem. Given d, each ½k term in (5),

   NV d½k yd½k minf½k ðyd½k Þ; yd½k

a convex objective function (Bazaraa et al., 2006) and its solution satisfies FONC and SONC at the optimal point (Porteus, 2002).

4. Solution approach In this section, we describe solution approaches to prescribe optimal block durations and optimal sequence. We show that the   newsvendor solution, NV d½k yd½k ; k 2 K, gives the optimal planned end times for a given sequence d 2 D. We derive the closed form of the objective value and prove that the SV rule is optimal to sequence blocks for the case in which surgeries are independent and normally distributed. Section 4.1 devises optimal block durations for the unconstrained version of SNVðyÞ and Section 4.2 devises the closed-form solution to the constrained version. Section 4.3 determines the optimal sequence, ^ d.

Fig. 2. Variable transformation from x-space to y-space.

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Fig. 3. Sequential newsvendor problem.

4.1. Unconstrained optimal block durations We first seek the unconstrained (i.e., without constraints (6) and (7)) optimal block durations for a given sequence d. In this subsection, we assume that sequence d is fixed, so we suppress this superscript. We may solve problem SNVðxÞ using a dynamic programming approach; however, it is hard to prove optimality for a general number of blocks, because x½k appears in f½k ðx½1 þ    þx½k Þ; . . ., and f½jKj ðx½1 þ    þ x½jKj Þ so that these functions are not separable. However, Proposition 1 establishes separability of the transformed problem, SNVðyÞ. Subsequently, we are able to solve independent newsvendor problems NV ½k ðy½k Þ; k 2 K, owing to separability. Proposition 1. Define the unconstrained sequential newsvendor problem USNVðyÞ by relaxing constraints (6) and (7) to obtain:

ðUSNVðyÞÞ min

X f½k ðy½k Þ:

ð10Þ

k2K

Problem USNVðyÞ is separable with respect to y½k :

n o min f½1 ðy½1 Þ þ f½2 ðy½2 Þ þ    þ f½jKj ðy½jKj Þ

 min f½1 ðy½1 Þ þ min f½2 ðy½2 Þ þ    þ min f½jKj ðy½jKj Þ:

0

distributed random durations of surgeries associated with subspecialties, each assigned to a block [1] through ½k (i.e., T ½k :¼ B½1 þ    þ B½k ) and let F T ½k be the normal distribution function of T ½k ,  ½k :¼ l  2 :¼ r  ½1 þ    þ l  ½k and variance r 2 which has mean l ½k

½1

 2½k . þ þ r (i) Problem NV ½k ðyÞ, which prescribes the optimal planned end ^½k for each k 2 K time of the kth block, has optimal solution, y such that:

^½k Þ ¼ F T ½k ðy

cl 1 ¼ ¼ UðzÞ; ce þ cl 1 þ b

ð12Þ

 ½k ; r  2 Þ; y  ½k þ zr  ½k ; k 2 K; UðzÞ is the ^½k ¼ l where T ½k  Nðl ½k standard normal distribution function; and z is the normal score. x½k for each (ii) The corresponding, optimal block duration, ^ ^½1 , and k 2 K, can be obtained by definition: ^ x½1 ¼ y ^ ^½k  y ^½k1 ; k ¼ 2; . . . ; jKj. x½k ¼ y

ð11Þ

Proof. After relaxing (6) and (7), problem USNVðyÞ is separable with respect to y½k ; k 2 K because NV ½k ðy½k Þ; k 2 K is independent of other variables y½k0  ; k ð – kÞ 2 K.

 k and Proposition 2. Given random block duration B½k with mean l  2½k ; k 2 K, let T ½k be the sum of the independent, normally variance r

h

Based on Proposition 1, we can solve individual NV ½k ðy½k Þ; k 2 K ^½k ; k 2 K, that problems independently to optimize USNVðyÞ; any y satisfies FONC and SONC optimizes USNVðyÞ. NV ½k ðy½k Þ is a newsvendor-type problem that prescribes the planned end time of the ½kth block to minimize the sum of expected earliness and lateness costs. Newsvendor problem NV ½k ðy½k Þ, which is associated with random variable T ½k ; k 2 K, can be solved independently according to Proposition 2, which follows. Even though T ½k ’s are not independent random variables, we can solve problems NV ½k ðy½k Þ; k 2 K independently after relaxing constraints (6) because E½ðy½k  T ½k Þþ  and E½ðT ½k  y½k Þþ  are functions of y½k and T ½k is essentially a parameter that gives information about all surgeries through the kth block.

Proof.  k ¼ E½T ½k  ¼ E½B½1  þ   þ E½B½k  ¼ l  1 þ  þ (i) T ½k ¼ B½1 þ  þ B½k ; l l k and r 2½k ¼ V½T ½k  ¼ V½B½1  þ   þ V½B½k  ¼ r 21 þ  þ r 2k ; k 2 K. Because individual B½k are normally distributed, T ½k is also. ^½k , as defined by (12), is the optimal solution to newsvendor y problem NV ½k ðy½k Þ (Nahmias, 2008; Porteus, 2002). (ii) follows from the definition of the variable transformation from x to y. h z is the value of the standard unit normal for which its cdf UðzÞ is equal to the critical ratio associated with the newsvendor solution as well as the solution to our problem, which prescribes the duration of a block. In fact, because (we assume that) cost parameters ce and cl are the same for each block the same z and UðzÞ prel

1 are scribe the duration of each block. Thus z and UðzÞ ¼ cecþcl ¼ 1þb

independent of block sequence; they are the same each block over all possible block sequences.

144

S. Choi, W.E. Wilhelm / European Journal of Operational Research 235 (2014) 138–148

4.2. Constrained optimal block durations Now, we solve constrained optimization problem SNVðyÞ, focusing on constraints (6) and (7), which require 0 6 y½k1 6 y½k ; k ¼ 2; . . . ; jKj, correspondingly, that 0 6 x½k ; k 2 K. According to the Karush Kuhn Tucker (KKT) conditions, if the optimal solution to unconstrained problem USNVðyÞ satisfies constraints (6) and (7), it is the global optimal solution to constrained problem SNVðyÞ as shown Fig. 4(a). Otherwise, the optimal solution is on the ^d½k1 ¼ y ^d½k for one or more k 2 K as shown boundary so that y Fig. 4(b). We show the KKT conditions analytically. At the constrained ^½k to SNVðyÞ, KKT conditions must hold (Bazaraa optimal solution y et al., 2006):

0 B B B B @

@ ^ Þ f ðy @y½1 ½1 ½1

0

...

...

0

0

@ ^ Þ f ðy @y½2 ½2 ½2

0

...

0

0

...

@ ^ Þ f ðy @y½k ½k ½k

...

0

0 0

...

0

@ ^ Þ f ðy @y½jKj ½jKj ½jKj

1 0 ... ... B 0 1 0 ... þ u½1 B @0 ... ... ... 0 ... ... 0 0 0 0 ... ... B0 1 0 ... þ u½2 B @ 0 0 1 . . . 0 ... ... 0 0 0 0 ... ... B0 0 0 ... þ u½jKj B @0 ... ... 0 0 ... ... 0 1 0 0 0 ... 0 B0 0 ... 0C C ¼B @0 ... 0 0A 0 ... 0 0

... 1

1 C C C C A

F T ½1 ðy½1 Þ ¼ F T ½2 ðy½2 Þ ¼

1 ¼ 0:038; 1 þ 25

ð15Þ

^½1 ¼ l½1  1:768r½1 ¼ 1:823 and so that z ¼ 1:768. Then y ^½2 ¼ l½2  1:768r½2 ¼ 1:749, so that y½1 > y½2 and these values y are not feasible with respect to (6). In this case, the planned end time of the second block is less than the planned end time of the first block, which would mean that the planned duration of the second block were negative; i.e., ^ x½1 ¼ 1:823 and ^x½2 ¼ 0:074. This example is an extreme case because T ½2 has a much larger variance and a smaller mean than T ½1 . Furthermore, the ratio of the two costs, b, is huge because the cost of earliness is 25 times of the cost of lateness. Considering two consecutive blocks, Proposition 3 establishes restrictions on parameters to assure that mathematically feasible, optimal solutions are also practically feasible. Based on the defini ½k þ zr  ½k , we impose condition l  k P jzjr  k ; k 2 K, tion of y½k ¼ l which, in turn, assures that y½k P 0 holds. Proposition 3 establishes that this condition also implies that y½1 6 y½2 relative to two independent and normally distributed distributions and this result is extended to all k 2 K by Corollary 4.

0 0C C 0A 0 1 0 0C C þ  0A 0 1 0 0 C C 0 A 1 ð13Þ

^½k1  y ^½k Þ ¼ 0 k ¼ 2; . . . ; jKj; u½k1 ðy

Next, we derive a condition to assure that a solution to unconstrained problem USNVðyÞ satisfies (6) and (7). Even though random variable T ½k has a larger mean and variance than T ½k1 , it is numerically possible, but not practically feasible, for an optimal ^½k < y ^½k1 for some k). We first give solution to violate (6) (i.e., y an example to demonstrate the relevant issues. Consider two normal distributions representing block surgery durations with l1 ¼ 2; r1 ¼ 0:1 and l2 ¼ 1; r2 ¼ 0:7. Then, for sequence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ! 2; l½1 ¼ 2; l½2 ¼ 3; r½1 ¼ 0:1 and r½2 ¼ :12 þ :72 . Assume ^½1 and y ^½2 are such that that b ¼ 25. Optimal solutions y

ð14Þ

where u½k is a Lagrangian multiplier associated with constraint (6) of k 2 K. If the optimal solution to USNVðyÞ is feasible with respect ^½k Þ ¼ 0 and u½k ¼ 0 hold to (6) and (7) as shown in Fig. 4(a), @y@ f½k ðy ½k

^½k is not necessarily equal to y ^½kþ1 . If for all k 2 K. Because u½k ¼ 0; y the optimal solution to unconstrained problem USNVðyÞ violates ^½k Þ – 0 and either constraints (6) or (7) as shown Fig. 4(b), @y@ f½k ðy ½k

^½k1 ¼ y ^½k (i.e., the optimal solution lies on the boundary, so that y ^x½k ¼ 0) and the Lagrangian multiplier u½k is non-zero for some k 2 K (Bazaraa et al., 2006). In our analysis, we concentrate on the former case, which is depicted by Fig. 4(a).

Proposition 3. Consider two block durations Bi , which are indepen i; r  2i Þ; i ¼ 1; 2. We require that dent and normally distributed: Nðl  1 þ zr  1 and F B1 ðy½1 Þ ¼ F B1 þB2 ðy½2 Þ ¼ 1=ð1 þ bÞ ¼ UðzÞ so that y½1 ¼ l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2       y½2 ¼ l1 þ l2 þ z r1 þ r2 . If li P jzjri ; i ¼ 1; 2, then y½1 6 y½2 ; in other words,

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l 1 þ zr 1 6 l 1 þ l 2 þ z r 21 þ r 22 :

ð16Þ

Proof. Case (i): z P 0. This case occurs if b 6 1. Using þjzj to denote z P 0, inequality (16) becomes

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l 1 þ jzjr 1 6 l 1 þ l 2 þ jzj r 21 þ r 22 :

ð17Þ

In this case, Eq.(17) is trivially true. Case (ii): z < 0. This case occurs if b > 1. Using jzj to denote z < 0, the equivalent of inequality (16) is

Fig. 4. Graphical depiction of KKT conditions.

S. Choi, W.E. Wilhelm / European Journal of Operational Research 235 (2014) 138–148

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 1  jzjr 1 6 l 1 þ l 2  jzj r 21 þ r 22 :

ð18Þ

We must now show that (16) holds in the form of (18) in case (ii). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Combining the fundamental relationship r 21 þ r 22 6 r 1 þ r 2  i P jzjr  i ; i ¼ 1; 2, the following inequality holds: with conditions, l

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  21 þ r  22  r  1 Þ 6 jzjr 2 6 l  2: jzjð r  1 to the left- and right-most terms, Adding l

l 1 þ jzj

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r 21 þ r 22  r 1 6 l 1 þ l 2 ;

which can be rearranged to establish (18) when z < 0, showing that inequality (16) holds for both positive and negative z values. h We now generalize Proposition 2 to prescribe planned end times for all blocks, relying on the Proposition 3, which deals with a two-block case.    ½k ; r  2½k such that l  ½k P jzjr  ½k ; k 2 K, Corollary 4. Given B½k  N l  ½k ¼ l  ½1 þ    þ l  ½k and varlet T ½k :¼ B½1 þ    þ B½k have mean l 2 ¼ r  2½1 þ    þ r  2½k . Optimal block durations ^ x½k ; k 2 K can be iance r ½k ^½k ; k 2 K as follows: obtained from optimal planned end times y

^x½1 ¼ y ^½1 ¼ l  ½1 þ zr  ½1  ½1 þ zr  ½1 ¼ l  ½k þ zr  ½k  l  ½k1  zr  ½k1 ^x½k ¼ y ^½k  y ^½k1 ¼ l "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# Xk Xk1  ½k þ z ¼l r 2½l  r 2½l ; k P 2: l¼1

l¼1

ð19Þ

ð20Þ

^½k ¼ y

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xk l ½l þ z r 2½l :

k X l¼1

l¼1

Optimal block duration ^x½k can be obtained by definition: ^x1 ¼ y ^1 P 0 and ^ ^½k  y ^½k1 P 0; k P 2. h x½k ¼ y  ½k P jzjr  ½k ; k 2 K is  ½k P jzjr  ½k ; k 2 K is satisfied, l If condition l  ½1 þ    þ l  ½k P jzjðr  ½1 þ    þ r  ½k Þ P jzj also satisfied because l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 r ½1 þ    þ r ½k . If condition l ½k P jzjr ½k ; k 2 K is satisfied, both ^½k ; k 2 K and optimal block durations optimal planned end-times y ^ ^½k1 6 y ^½k ; x½k ; k 2 K will be non-negative; accordingly, 0 6 y k ¼ 2; . . . ; jKj. If we use different cost parameter values for the last block to reflect the fact that lateness for this block is actually overtime and earliness is idleness, optimal solutions, ^ x½jKj ; k 2 K are given as follows:

^x½jKj ¼ y ^½jKj  y ^½jKj1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ½jKj þ ^z r  2½jKj  z r  2½jKj1 ;  2½1 þ    þ r  2½1 þ    þ r ¼l

solutions depend mainly upon parameters such as mean and vari^ are not significantly different. ance, if two cost ratios (i.e., b and b) 4.3. Optimal block sequence NV ½k ðy½k Þ defines a newsvendor problem that prescribes the planned end time of block ½k; SNVðyÞ seeks the sum of optimal solutions to all NV ½k ðy½k Þ; k 2 K and defines the gðdÞ value for each permutation d. Z  in (9) is the objective function value associated with the optimal sequence of newsvendor solutions; i.e., the minimum of gðdÞ over sequences d 2 D. Hence, the next problem we d, which we address solve is to determine the optimal sequence, ^ as the sequential newsvendor problem. In the previous subsection, we prescribe optimal block durations for a fixed sequence. We want to find the minimum gðdÞ over all d 2 D. We show that each gðdÞ can be expressed in a closed form when surgery durations assigned to each block are independent and normally distributed and use this form to derive the optimal rule to sequence blocks. Consider the duration of the kth block, y½k . We suppress superscripts and subscripts for clarity, defining f ðyÞ and ; r  2 Þ, so that the objective function of NV ½k ðy Þ becomes T  Nðl ½k

f ðyÞ ¼ ce E½ðy  TÞþ  þ cl E½ðT  yÞþ : We use Lemmas 5–7 to derive a closed form expression for the optimal value of min f ðyÞ. ; r  2 Þ, Lemma 5. For T  Nðl

  r ðyl Þ2  ÞU y  l : E½ðy  TÞþ  ¼ pffiffiffiffiffiffiffi e r 2 þ ðy  l r 2p Proof. See the Appendix.

 ½k P jzjr  ½k ; k 2 K of Proposition 3 is satisfied, Proof. If condition l ^½k1 6 y ^½k ; k ¼ 2; . . . ; jKj, because Proposition 3 can be applied to y each pair of successive blocks (e.g., [1] and [2], [2] and [3], and so on). The proof relies on the fact that each T ½k1 is normally distributed (i.e., equivalent to B1 in Proposition 3) and each B½k is independent and normally distributed so that T ½k ¼ T ½k1 þ B½k (i.e., 1 equivalent to B1 þ B2 in Proposition 3), where B1  Nðl  ½k1 ; r  2½1 þ    þ r  2½k1 Þ and B2  Nðl  ½k ; r  2½k Þ. The optimal þ þ l ^½k of the kth block is given by planned end time y

ð21Þ

^cl 1 such that Uð^zÞ ¼ l ¼ , where ^cl corresponds to lateness ^c þ ^ce 1 þ b ^ e (i.e, overtime) cost; ^c , to earliness (i.e., idleness) cost. The optimal

145

h

; r  2 Þ, Lemma 6. For T  Nðl

    r ðyl Þ2  yl : E½ðT  yÞþ  ¼ pffiffiffiffiffiffiffi e r 2 þ ðl  yÞ 1  U r 2p Proof. See the Appendix.

h

Now, we simplify Z ¼ miny f ðyÞ, expressing Z as an increasing  . We invoke Lemma 7 for a single block. function of r b of the problem ; r  2 Þ, the optimal value Z Lemma 7. If T  Nðl miny f ðyÞ is defined as:

 2 b ¼ ðce þ cl Þ pr ffiffiffiffiffiffiffi ez ; Z 2p l 1 where UðzÞ ¼ 1þb ¼ cecþcl . Proof. See the Appendix.

h

We next apply Lemma 7 to a particular sequence to obtain a closed-form for gðdÞ for a general number of blocks. Proposition 8. Objective function gðd1 Þ, evaluated for a particular sequence, w.l.o.g. d1 : 1 ! 2 !  ! jKj (i.e., ½k ¼ k; k 2 K), can be expressed as

ðce þ cl Þ 2    gðd1 Þ ¼ pffiffiffiffiffiffiffi ez fr 1 þ r2 þ    þ rjKj g; 2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðce þ cl Þ 2  21 þ r 1 þ r  22 þ    ¼ pffiffiffiffiffiffiffi ez fr 2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2jKj g;  21 þ    þ r þ r 1 where UðzÞ ¼ 1þb .

ð22Þ

146

Proof. For

S. Choi, W.E. Wilhelm / European Journal of Operational Research 235 (2014) 138–148

sequence

1 ! 2 !  ! jKj; T ½k

r 21 þ    þ r 2½k . Apply Lemma 7 to each block k 2 K.

has h

variance

Proposition 9 analyzes (22) to prescribe the optimal block sequence.    ½k ; r  2½k for each k 2 K. The optimal Proposition 9. Let B½k  N l sequence with the optimal planned end-times that minimize the sum of expected earliness and lateness (idleness and overtime associated with the last block, respectively) is the SV. Proof. Without loss of generality, sequence B’s according to smallest variance first and renumber so that r½k1 6 r½k ; k ¼ 2; . . . ; jKj.  ½k ¼ l  ½1 þ    þ l  ½k and Define T ½k :¼ B½1 þ    þ B½k with mean l 2 2 2 ¼ r   variance r þ    þ r . Swapping the first two blocks in ½1 ½k ½k the sequence without changing the sequence of other blocks, we obtain

5. Extensions: no-shows Patient no-shows play a major role in deteriorating schedule performance (Lin, Muthuraman, & Lawley, 2011) because the no-show rate can be significant; for example, they have been reported to be from 22% to more than 50% (Guse, Richardson, Carle, & Schmidt, 2003) in health-care clinics. Surgery-patient no-shows may result from immediate cancellations before scheduled surgery, due, for example, to failure of patients to prepare for surgery as instructed. Hospital managers can overbook patients to minimize the expected idle time caused by no-shows or employ the following analysis to manage planned block durations appropriately. Let a denote the probability of a no-show, a discrete event, and hðdi Þ denote the probability distribution function (p.d.f.) for Di , the duration of a representative surgery of sub-specialty i. Define a new 0 p.d.f., h ðdi Þ with discrete mass representing a no-show and continuous random duration as follows: 0

For sequence 1 ! 2 !    ! N;  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e cl Þ 2 b ¼ ðcpþ  21 þ r 1 þ r  22 þ    þ r  22 þ    þ r  2jKj ;  21 þ r ffiffiffiffiffiffiffi ez r Z 2p and; for sequence 2 ! 1 !    ! N;  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e cl Þ 2 b ¼ ðcpþ  22 þ r 2 þ r  21 þ    þ r  21 þ    þ r  2jKj :  22 þ r ffiffiffiffiffiffiffi ez r Z 2p Corresponding terms in the two square brackets are the same, except for the first ones. Thus, it can be seen that the SV rule optimally sequences the first two blocks. So, fix the first block in position. In a similar manner, switching the blocks in the second and third positions shows the SV rule optimally sequences these two blocks as well. By comparing successive pairwise switches, the SV rule can be seen to give the optimal permutation of all blocks. h Proposition 9 shows that the SV rule gives the optimal sequence of blocks when surgery durations are independent and normally distributed. Based on our preliminary analysis, it does not appear possible to obtain a closed from of gðdÞ for block durations that follow a distribution other than the normal.

h ðdi Þ ¼



a

if di ¼ 0; ð1  aÞhðdi Þ if di > 0:

The associated distribution function of surgery duration, considering the possibility of a no-show, H0 ðxÞ, is defined as H0ðxÞ :¼ a þ ð1  aÞHðxÞ, where HðxÞ is the distribution function of hðxÞ. We have to use Lebesgue integration rather than Riemann integration to form H0 ðxÞ, the distribution function of surgery duration with the possibility of a no-show (Folland, 1999), because Riemann integration for a no-show event is 0. For a single block, we can find the optimal duration ^ x as the value at which distribution function H0 ðxÞ is equal to the critical ratio:

H0 ð^xÞ ¼ a þ ð1  aÞHð^xÞ ¼

1 : 1þb

xO and ^ xN denote the optimal solutions for the original case Let ^ without no-shows and the new case with no-shows, respectively. The corresponding distribution functions are given by:

Hð^xO Þ ¼

1 ; 1þb

Fig. 5. Optimal block durations with no-show and without no-show.

ð23Þ

147

S. Choi, W.E. Wilhelm / European Journal of Operational Research 235 (2014) 138–148

and

Hð^xN Þ ¼

1  a  ab : 1  a  ab þ b

ð24Þ

xÞ with ranges of 0 6 a 6 :3 and Fig. 5 gives the values of Hð^ 0:5 6 b 6 1:5 to show how optimal block duration changes as a function of a and b. Hð^ xÞ is a decreasing function of both a and b. When there are no-shows, we may increase the number of patients scheduled in a given block or decrease the optimal block duration for a given number of patients. 6. Managerial insights This paper provides managerial insights into BSS, based on the assumptions that forecasts provide the expect number of surgeries to be performed by each surgical sub-specialty, that a representative surgery-duration distribution that is normally distributed can be derived for each sub-specialty based on historical data, that all surgery durations are mutually independent, and that each surgery begins when the previous one ends. Our analysis results in an easy way to compute the optimal planned duration (equivalently, planned end time) of each time block and shows that time blocks can be optimally sequenced using the easy-to-implement SV rule. If surgeons in each sub-specialty were responsible for setting the planned duration of its block, they might neglect surgery-duration uncertainty, resulting in a naive planned block duration equal to the sum of the expected durations of its surgeries. This would parallel the current practice of scheduling the starting time of each surgery to be the sum of the expected durations of surgeries that precede it. Alternatively, each sub-specialty might take a myopic approach, neglecting the impact of other sub-specialties on the schedule because they do not exchange information, but considering uncertainty by applying a newsvendor model to set planned block duration, say x0½k ; k 2 K according to

 ½k þ zr  ½k x0½k ¼ l 1 such that UðzÞ ¼ 1þb . In contrast, the planned block durations (equivalently, planned end times) that our method prescribes deals optimally with uncertainty and depends upon b, the ratio of earliness-to-lateness cost penalties. If b ¼ 1 (i.e., ce ¼ cl ), the optimal block durations for a gi ½1 ; ^  ½2 ; . . . ; ^ x½1 ¼ l x½2 ¼ l x½jKj ven permutation can be specialized to ^  ½jKj . This case actually corresponds to the naive approach and ¼l shows that it is actually optimal if b ¼ 1. If b < 1 (i.e., ce < cl ), z is positive. In other words, if the penalty cost of lateness is greater than that of earliness, the block duration is longer than in the case  ½k ; k 2 K) to minimize the risk of delaying the next of b ¼ 1 (i.e., l block. In this case, the planned block duration that our method x½k , would be less than the duration that the myowould prescribe, ^ pic method would prescribe, x0½k (i.e., ^ x½k < x0½k ), indicating that our

method is better able to manage the risk of delaying the next block. If b > 1 (i.e., ce > cl ), z is negative and the block duration is shorter than in the case of b ¼ 1 to minimize the risk of idleness. In this case, our method prescribes planned block durations that are longer than the myopic approach (i.e., ^ x½k > x0½k ), indicating that our method is better able to deal with the snowball effect created by variances accumulating for successive blocks. We now formalize the relationship between the planned block durations that our method and the myopic method prescribe. Proposition 10. Consider the planned block duration for kth block as prescribed by our method, x½k , and the myopic method, x0½k :

^x½k 6 x0½k if b 6 1 ^x½k > x0½k otherwise:

Proof. We use the following fundamental relationship for both cases (i) and (ii):

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2½1 þ    þ r 2½k 6 r 2½1 þ    þ r 2½k1 þ r ½k : Case (i) b 6 1. In this case, ce 6 cl ; i.e., z P 0.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2½1 þ    þ r 2½k  r 2½1 þ    þ r 2½k1

l ½k þ z ^x½k

 ½k 6r

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2½1 þ    þ r 2½k  r 2½1 þ    þ r 2½k1 6 l ½k þ zr ½k

6 x0½k :

Case (ii) b > 1. In this case ce > cl ; i.e., z < 0. The proof parallels that of case (i). h The mean and variance of T ½k , which determine the expected earliness and lateness of the last block, would be the same no matter which sub-specialty is put in that sequence position; i.e.,     ^^d½jKj ¼ f½jKj y ^d½jKj ; k 2 K; d 2 D. Thus, the planned end time f½jKj y ^½jKj , which is the planned number of OR hours for of block ½jKj; y the day, does not depend on the sequence. In other words, the sub-specialty with largest variance comes for free in the last sequence position but would add to total cost if it displaced another sub-specialty with a lower variance in an earlier sequence position. To hedge no shows, a primary question is whether planned block durations should be lengthened or reduced. Our approach is different from an overbooking policy that defines the optimal number of surgeries in a given block time, because we seek the optimal block duration, given the forecast number of surgeries including no-shows. Considering no-shows reduces optimal block duration in comparison with the case without no-shows. A hospital manager can apply criterion (24) to prescribe optimal planned block durations to hedge no-shows. 7. Conclusion This paper presents new methods to prescribe optimal planned duration and sequence of time blocks, each of which reserves OR resources for a particular surgical sub-specialty. Further, rather than using an overbooking policy, it gives a closed form to prescribe optimal planned block duration to hedge no shows. Results lend considerable insights for managing OR resources. The methods we propose can be implemented easily and, we expect, would result in improved performance through managing the BSS process and optimizing the sum of expected earliness and lateness costs. Effectively planned block durations can also be expected to facilitate scheduling actual patients at the operational level. Our findings suggest several avenues for future research. For example, a BSS may affect staff scheduling, PACU, and other relevant departments. Incorporating such ancillary departments in BSS planning is fertile opportunity for the future research. Finally, our model of the sequential newsvendor problem can be applied in time-sensitive environments other than health care (e.g., JIT delivery) in which both earliness and lateness must be minimized. Acknowledgements This material is based in part on work supported by the National Science Foundation on Grant No. CMMI 1129693. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Appendix A. Proofs A.1. Proof of Lemma 5 Proof.

Z

y

 Þ2 ðtl 1  ðy  tÞ pffiffiffiffiffiffiffi e 2r 2 dt  2pr 1 Z y Z y  Þ2 ðtl 1   tÞ Þ pffiffiffiffiffiffiffi e 2r 2 dt þ ðl ¼ ðy  l  2pr 1 1  Þ2 ðtl 1   pffiffiffiffiffiffiffi e 2r 2 dt  2pr   ðyl Þ2  ÞUðy  lÞ þ pr ffiffiffiffiffiffiffi e 2r 2 ¼ ðy  l  r 2p

E½ðy  TÞþ  ¼

A.2. Proof of Lemma 6 Proof.

Z

1

 Þ2 ðtl 1  ðt  yÞ pffiffiffiffiffiffiffi e 2r 2  2pr y Z 1 Z 1  Þ2 ðtl 1   yÞ  Þ pffiffiffiffiffiffiffi e 2r 2 dt þ ðl ¼ ðt  l  2pr y y

E½ðT  yÞþ  ¼

 Þ2 ðtl 1   pffiffiffiffiffiffiffi e 2r 2 dt  2pr   Z 1  Þ2 ðtl r ðtl Þ2 1  1 pffiffiffiffiffiffiffi e 2r 2 dt þ ðl  yÞ ¼  pffiffiffiffiffiffiffi e 2r 2  2pr 2p y y     r ðy2rl2Þ2  yl :  ¼ pffiffiffiffiffiffiffi e þ ðl  yÞ 1  U r 2p

A.3. Proof of Proposition 7 ^ be the solution that minimizes f ðyÞ, and z be the Proof. Let y ^  standard normal score z ¼ yrl at the optimal solution,  l ^  1  l þ zr . ^¼l UðzÞ ¼ U yrl ¼ 1þb ¼ cecþcl , and y

   

^l y r ð^yl Þ2 ^  ^Þ ¼ ce pffiffiffiffiffiffiffi e r 2 þ ðy  lÞU f ðy r 2p     Þ2  

ð^ l y  ^l r  y  y ^Þ 1  U þ cl pffiffiffiffiffiffiffi e r 2 þ ðl r 2p 



r 2 2  zUðzÞ þ cl pr  zð1  UðzÞÞ ffiffiffiffiffiffiffi ez  r ¼ ce pffiffiffiffiffiffiffi ez þ r 2p 2p  r 2  zUðzÞðce þ cl Þ  r  zcl ¼ ðce þ cl Þ pffiffiffiffiffiffiffi ez þ r 2p r 2 ¼ ðce þ cl Þ pffiffiffiffiffiffiffi ez ; 2p which is an increasing function of

r.

h

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