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A Note on Appointment Scheduling with Piecewise LinearCost FunctionsDongdong Ge, Guohua Wan, Zizhuo Wang, Jiawei Zhang

To cite this article:Dongdong Ge, Guohua Wan, Zizhuo Wang, Jiawei Zhang (2013) A Note on Appointment Scheduling with Piecewise Linear CostFunctions. Mathematics of Operations Research

Published online in Articles in Advance 13 Nov 2013

. http://dx.doi.org/10.1287/moor.2013.0631

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MATHEMATICS OF OPERATIONS RESEARCH

Articles in Advance, pp. 1–8ISSN 0364-765X (print) � ISSN 1526-5471 (online)

http://dx.doi.org/10.1287/moor.2013.0631© 2013 INFORMS

A Note on Appointment Scheduling with Piecewise Linear Cost Functions

Dongdong GeResearch Center for Management Science and Information Analytics, School of Information Management and Engineering,

Shanghai University of Finance and Economics, Shanghai 200433, China, ge.dongdong@shufe.edu.cn

Guohua WanAntai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China, ghwan@sjtu.edu.cn

Zizhuo WangDepartment of Industrial and Systems Engineering, University of Minnesota, Minneapolis, Minnesota 55455, zwang@umn.edu

Jiawei ZhangStern School of Business, New York University, New York, New York 10012, jzhang@stern.nyu.edu

We consider the problem of determining the optimal schedules for a given sequence of jobs on a single processor. Theobjective is to minimize the expected total cost incurred by job waiting and processor idling, where the job processing timesare random variables. It is known in the prior literature that if the processing times are integers and the costs are linearfunctions satisfying a mild condition, then the problem can be solved in a polynomial number of expected cost evaluations. Inthis work, we extend the result to piecewise linear cost functions, which include many useful objective functions in practice.Our analysis explores the (hidden) dual network flow structure of the appointment scheduling problem and thus greatlysimplifies that of prior work. We also find the number of samples needed to compute a near optimal solution when only someindependent samples of processing times are known.

Key words : scheduling; stochastic programming; convex relaxation; sampling; dualityMSC2000 subject classification : Primary: 90B36, 90C27; secondary: 90C15, 90C10OR/MS subject classification : Primary: production/scheduling, healthcare; secondary: stochastic, mathematics, integerHistory : Received January 14, 2012; revised February 9, 2013. Published online in Articles in Advance.

1. Introduction. We consider the appointment scheduling problem for a given sequence of jobs. The deci-sion maker has to determine the starting time of each job, called the appointment time. A job cannot be startedbefore its appointment time or before the previous job is finished. Because of the randomness of processingtimes, a job may be finished earlier or later than the appointment time of the next job. If it is finished earlier, an“underage” cost will be incurred because of the idling of the processor. On the other hand, if a job is finishedlater, the system will experience an “overage cost” because of the waiting of the next job. The objective of theappointment scheduling problem is to determine the appointment time for each job such that the total expectedcost is minimized.

The appointment scheduling problem has many emerging applications in service management, particularlyin the healthcare industry. Although there has been extensive research on this problem,1 its computationalcomplexity was unknown until the recent work by Begen and Queyranne [2]. Begen and Queyranne provedthe existence of an optimal integer schedule for the linear cost functions given that the processing times areintegers. Under a mild technical condition (so-called �-monotonicity condition; see Definition 6.3 in Begenand Queyranne [2]) on the cost coefficients, they showed that the objective function is L-convex and proposedan algorithm for this problem with a polynomial number of expected cost evaluations. Moreover, they showedthat the expected cost evaluation can be computed in polynomial time when the random processing times areindependent and their supports are bounded. In Begen et al. [3], the authors developed a sampling approach whenthe distribution of the processing times is not known and only a set of independent samples is available. Theyanalyzed the subdifferential of the cost function and established a bound on the number of samples required toobtain a near-optimal solution (with multiplicative error) with high probability.

In this short note, we extend the study of Begen and Queyranne [2] and Begen et al. [3] to piecewise linearcost functions with integer break points. If appointment times and processing times are assumed to be integers,any cost function can be equivalently formulated as a piecewise linear function. As in Begen and Queyranne [2],we first prove the existence of an integer optimal solution by exploring the hidden network flow structure of theproblem. We then present a condition under which the problem can be solved in a polynomial number of costfunction evaluations. Our condition takes �-monotonicity as a special case and includes the case in which the

1 There is a rich literature on this subject. However, as we position our paper as a short note, we do not intend to provide a thorough reviewof the literature. Instead, we refer the readers to Cayirli and Veral [5], Gupta and Denton [6], and Cardoen et al. [4] for a detailed review.

1

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost Functions2 Mathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS

underage cost rate is uniform and the overage cost is convex. We then extend the sampling approach by Begenet al. [3] to a general class of functions and provide an alternative bound on the number of samples required toobtain a near-optimal solution (with additive error).

The remainder of the paper is organized as follows. In §2, we describe the model and prove the existenceof an integer optimal solution. In §3, we consider a convex relaxation to the problem and prove that undermild conditions the problem can be solved in polynomial time. In §4, we discuss the situation in which thedistribution of the processing times is unknown and study a sampling approach. We conclude and make the finalremarks in §5.

2. The model and the existence of integer optimal solutions. We closely follow Begen and Queyranne [2]to set up the model. Let S denote the set of jobs to be scheduled in a given session and let n = �S� be thetotal number of jobs. Each job i ∈ S has a random processing time denoted by pi4�5 where uncertainty isrepresented by �, taking values in a sample space ì. Let p4�5 = 4p14�51 : : : 1 pn4�55. The processing timesare assumed to be integer valued and bounded as in Begen and Queyranne [2]. More precisely, we make thefollowing assumption about the random processing times.

Assumption 1. For each i ∈ S, there exist a pair of nonnegative numbers, pi and pi, such that pi4�5 ∈

6pi1 pi7∩� for all � ∈ì, where � is the set of integers.

The n jobs must be scheduled sequentially in a predetermined order 1121 : : : 1 n. Let Ai denote the appointmenttime of job i ∈ S. We also define an 4n+15th job, a dummy job with processing time 0, to compute the overageand underage cost of the nth job (In the case that An+1 = T is predetermined, one can substitute An+1 by T inthe formulation and add a constraint for each Ai. The analysis and result in this paper are mostly the same inthat case. We will remark on the difference later on.) The appointment time of the 4n+ 15th job An+1 can beviewed as the scheduled makespan of the whole session. We denote the decision variable in this problem by� = 8A11A21 : : : 1An1An+19. A job cannot start before its appointment time nor before its predecessors havebeen completed, and the start and completion times of a job depend on the appointment times as well as therandom processing times p4�5. They can be computed recursively as follows. Let Si4�5 and Ci4�5 denote thestart time and completion time, respectively, of job i in scenario � ∈ì. Then

Si4�5= max4Ci−14�51Ai5

Ci4�5= Si4�5+pi4�5

for i = 11 : : : 1 n, and C04�5 = 0. If job i is completed before (after, respectively) the appointment time of jobi+1, it incurs an underage (overage, respectively) cost captured by function ui4 · 5 (oi4 · 5, respectively): �+ →�.More precisely, the underage cost and overage cost of job i are ui44Ai+1 −Ci4�55

+5 and oi44Ci4�5−Ai+15+5,

respectively. Here x+ = max4x105 is the positive part of real number x. We make the following assumptionabout the underage and overage cost functions.

Assumption 2. For each i ∈ S, ui4 · 5 and oi4 · 5 are nondecreasing, continuous, and satisfy ui405 = 0 andoi405= 0.

In the appointment scheduling problem, the objective is to minimize the expected total overage and underagecosts. We can formulate the problem as follows:

min�

E�

n∑

i=1

8oi44Ci4�5−Ai+15+5+ ui44Ai+1 −Ci4�55

+59

s.t. Ci4�5= max4Ai1Ci−14�55+pi4�5 ∀ i ∈ S1 ∀� ∈ì

0 (1)

It is easy to see that (1) can be equivalently written as the following form.

4P15 min�1C

E�

[ n∑

i=1

8oi44Ci4�5−Ai+15+5+ ui44Ai+1 −Ci4�55

+59

]

(2)

s.t. Ci4�5= max4Ai1Ci−14�55+pi4�5 ∀ i ∈ S1 ∀� ∈ì0 (3)

Let P1 = P1 = 0, Pi =∑

j<i pj and Pi =∑

j<i pj for i = 21 : : : 1 n+ 1. Define P=∏n+1

i=1 6Pi1 Pi7. Following thesame argument in the proof of Lemma 4.3 in Begen and Queyranne [2], we can conclude the existence of anoptimal solution �∗ ∈P to problem 4P15 as long as Assumptions 1 and 2 hold.

The next theorem establishes the conditions under which 4P15 has an integral optimal solution.

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost FunctionsMathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS 3

Theorem 1. In addition to Assumptions 1 and 2, if ui4 · 5 and oi4 · 5 are piecewise linear functions withinteger break points, then 4P15 has an integral optimal solution.

Proof. By the assumption that oi4 · 5 and ui4 · 5 are piecewise linear with integer break points, for each i ∈ Sand each nonnegative integer l, there exist ol

i 1 rli 1 u

li1 q

li such that

oi4x5= olix+ r li if x ∈ 6l1 l+ 15

andui4x5= ul

ix+ qli if x ∈ 6l1 l+ 150

Let �∗ be any optimal solution to 4P15 and C∗ be the corresponding job completion times. For each � ∈ì, let

I4�5= 8i ∈ S2 A∗

i+1 ≥C∗

i 4�590

It follows that, for i ∈ S\I4�5,C∗

i 4�5 >A∗

i+10

Therefore, the optimal value of 4P15 can be written as

E�

[

∑

i∈S\I4�5

4oloi 4�5

i 4C∗

i 4�5−A∗

i+15+ rloi 4�5

i 5+∑

i∈I4�5

4ului 4�5

i 4A∗

i+1 −C∗

i 4�55+ qlui 4�5

i 5

]

1

where loi 4�5= �C∗i 4�5−A∗

i+1� and lui 4�5= �A∗i+1 −C∗

i 4�5� (�x� is the largest integer that is less than or equalto x). Now we define a linear program as follows (we define pn+14�5= 0):

4LP5 min�1�

E�

[

∑

i∈S\I4�5

4oloi 4�5

i 4Yi4�5−Xi+15+ rloi 4�5

i 5+∑

i∈I4�5

4ului 4�5

i 4Xi+1 − Yi4�55+ qlui 4�5

i 5

]

s.t. Yi+14�5=Xi+1 +pi+14�5 ∀� ∈ì1 i ∈ I4�5

Yi+14�5≥ Yi4�5+pi+14�5 ∀� ∈ì1 i ∈ I4�5

Yi+14�5= Yi4�5+pi+14�5 ∀� ∈ì1 i ∈ S\I4�5

Yi+14�5≥Xi+1 +pi+14�5 ∀� ∈ì1 i ∈ S\I4�5

loi 4�5≤ Yi4�5−Xi+1 ≤ loi 4�5+ 1 ∀� ∈ì1 i ∈ S\I4�5

lui 4�5≤Xi+1 − Yi4�5≤ lui 4�5+ 1 ∀� ∈ì1 i ∈ I4�50

By Assumption 1, the set ì is finite, therefore 4LP5 is a finite-dimensional linear program. Note that any feasiblesolution 4�1�5 to 4LP5 is a feasible solution to 4P15 by setting 4�1C5 = 4�1�5. Furthermore, 4�1�5 =

4�∗1C∗5 is a feasible solution to 4LP5. Now notice that each constraint of 4LP5 contains exactly one 1 and one−1, thus the constraint matrix of 4LP5 is totally unimodular. Therefore, problem 4LP5 has an integral optimalsolution (see, e.g., Heller and Tompkins [7]). By the construction of 4LP5, this integral solution has the sameobjective value as 4�∗1C∗5 and thus is optimal to 4P15. This completes the proof.

3. Convex relaxation and polynomial time algorithm. In this section, we study how we can solve 4P15efficiently. We first rewrite the objective function based on the following observation. From constraint (3), wehave

Ci+14�5= max4Ci4�51Ai+15+pi+14�5= 4Ci4�5−Ai+15+

+Ai+1 +pi+14�50

It follows that4Ci4�5−Ai+15

+=Ci+14�5−Ai+1 −pi+14�50 (4)

Similarly, we have4Ai+1 −Ci4�55

+=Ci+14�5−Ci4�5−pi+14�50 (5)

Using (4) and (5), we can replace the objective function of 4P15 with

min�1C

E�

[ n∑

i=1

8oi4Ci+14�5−Ai+1 −pi+14�55+ ui4Ci+14�5−Ci4�5−pi+14�559

]

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost Functions4 Mathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS

For any �, �, and �, we define

h�4�1�5=

n∑

i=1

8oi4Ci+14�5−Ai+1 −pi+14�55+ ui4Ci+14�5−Ci4�5−pi+14�5590 (6)

Now we present a two-stage stochastic programming formulation of 4P15 as follows:

4P25 min�

g4�5 2= E�6f�4�571 (7)

where for each � ∈ì and �,

f�4�5 2= min�

h�4�1�5 (8)

s.t. Ci+14�5= max4Ai+11Ci4�55+pi+14�5 ∀ i ∈ S (9)

C14�5= p14�50 (10)

Here pn+14�5 = 0 and � = 4Ci4�52 i = 1121 : : : 1 n+ 15 are the second stage (recourse) variables. Notice thatwith constraint (9) and (10), there is only one feasible solution to each second stage problem so that theminimization is actually not necessary. We keep using this two-stage stochastic programming formulation sinceit will become useful when we relax constraint (9).

In the following, we assume that the cost functions oi4 · 5 and ui4 · 5 are convex. Although the objectivefunction of the second stage problem h�4�1�5 is convex in � for any given � and �, the feasible set is ingeneral not. This motivates us to relax constraint (9) by linear inequalities. By doing this, we get the followingtwo-stage stochastic programming problem as a convex relaxation to (7)–(10):

min�

G4�5 2= E�6F�4�571 (11)

where for each � ∈ì and �,

F�4�5 2= min�

h�4�1�5 (12)

s.t. Ci+14�5≥Ai+1 +pi+14�5 ∀ i ∈ S (13)

Ci+14�5≥Ci4�5+pi+14�5 ∀ i ∈ S (14)

C14�5= p14�50 (15)

It is clear that constraints (13)–(15) are relaxations of (9)–(10), thus problem (11)–(15) is convex relaxation ofproblem (7)–(10). Therefore, F�4�5 ≤ f�4�5 for each � ∈ ì and �, and G4�5 ≤ g4�5. In the following, wepresent a condition under which the convex relaxation is exact. We have the following theorem.

Theorem 2. If h�4�1�5 is nondecreasing in 4C24�51C34�51 : : : 1Cn4�55 for each � ∈ì and �, then theconvex relaxation (11)–(15) is exact, i.e., F�4�5= f�4�5 for each � ∈ì and �, and G4�5= g4�5.

Proof. Since problem (11)–(15) is a relaxation of problem (7)–(10) with the same objective function, itsuffices to show that there exists an optimal solution to problem (11)–(15) that is feasible to problem (7)–(10).

For any fixed � and �, suppose C∗4�5 is an optimal solution to problem (12)–(15). Now we define anothersolution C4�5 according to (9)–(10). Note that Ci4�5≤C∗

i 4�5 holds for all i. By the assumption that h�4�1�5is nondecreasing in C4�5, C must also be an optimal solution to (12)–(15). Therefore the theorem holds.

We now specify a condition on u4 · 5 and o4 · 5 such that the condition in Theorem 2 is satisfied.

Proposition 1. Assume u4 · 5 and o4 · 5 are both convex, nondecreasing, piecewise linear functions. Defineui and ui to be the smallest and largest slopes for ui4 · 5; oi and oi to be the smallest and largest slopes foroi4 · 5, respectively. If ui, oi and ui satisfies

ui +oi ≥ ui+1 for i = 11 : : : 1 n− 11 (16)

then the convex relaxation (11)–(15) is exact.

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost FunctionsMathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS 5

Before proving Proposition 1, we remark that the condition specified in Proposition 1 is quite general anda broad class of cost functions fit into this frame. In particular, we would like to specify two classes of costfunctions that satisfy condition (16):

P-uniform. In this case, there exists u ≥ 0, such that ui4x5 = u · x for all i ∈ S and oi4x5 can be any convexnondecreasing piecewise linear function. This case captures some practical models in which the idle cost rate isuniform.

Generalized �-monotonicity. In this case, it is assumed that both the underage and overage cost functionsare linear, i.e., oi4x5 = �i · x and ui4x5 = �i · x and �i + �i ≥ �i+1 for i = 11 : : : 1 n− 1. We call this case thegeneralized �-monotonicity because the �-monotonicity defined by Begen and Queyranne [2] is a special caseof the generalized �-monotonicity. In Begen and Queyranne [2], the cost coefficients are �-monotone if thereexist real numbers �i, 1 ≤ i ≤ n such that 0 ≤ �i ≤ �i and �i +�i ≥�i+1 +�i+1 for i = 11 : : : 1 n− 1.

Proof of Proposition 1. By Theorem 2, it suffices to verify that for each i, i = 11 : : : 1 n− 1, h�4�1�5 isnondecreasing in Ci+14�5 when (16) holds.

To show this, note that in (6), increasing each unit of Ci+1 will at least increase h�4�1�5 by oi +ui − ui+1,for i = 11 : : : 1 n − 1. Therefore, when (16) holds, h�4�1�5 must be nondecreasing and therefore the convexrelaxation is exact. �

Next, we study the characteristics of the objective function of the scheduling problem based on the two-stagestochastic program formulation. First, we show the convexity and submodularity of the objective function G4�5.The proof uses the following well-known result on submodularity.

Lemma 1 (A Direct Corollary of Theorem 2.7.6. in Topkis [13]). If X and T are lattices, S is a sub-lattice of X × T , f 4x1 t5 is submodular in 4x1 t5 on S, St is the section of S at t in T , and g4t5= infx∈St f 4x1 t5is finite on the projection çT S of S on T , then g4t5 is submodular on çT S.

We are now ready to prove the following theorem.

Theorem 3. The function G4�5 is convex and submodular in � on �n+1.

Proof. It suffices to show that F�4�5 is convex and submodular in � for every � ∈ ì. By assumption,oi4 · 5 is nondecreasing and convex for each i ∈ S, and Ci+14�5−Ai+1 −pi+1 is a linear function. Thus it followsthat oi4Ci+14�5−Ai+1 − pi+14�55 is jointly convex and submodular in 4Ci+14�51Ai+15. Here the convexity iseasy to see and the submodularity follows from Corollary 2.6.1 in Topkis [13].

Therefore, F�4�5 is convex in � because the feasible set defined by constraints (11)–(15) is a convex.It remains to show F�4�5 is submodular in �. We notice that inequalities (14)–(15) are linear and each con-

tains at most two entries with opposite signs. Therefore, the feasible region is a sublattice (see Example 2.2.7(b)in Topkis [13]). By Lemma 1, F�4�5= min� h�4�1�5 is submodular in �. This completes the proof.

Furthermore, one can prove that G4�5 also satisfies L-convexity, an important discrete convexity property.

Definition 1 (Murota [9]). f 2 �q →�∪� is L-convex if and only if f 4z5+ f 4y5≥ f 4z∨ y5+ f 4z∧ y5,∀ z1 y ∈�q and ∃ r ∈�2 f 4z+ 15= f 4z5+ r , ∀ z ∈�q .

Theorem 4. The function G4�5 is L-convex in � on �n+1.

Proof. Because of the submodularity by Theorem 3, we only need to prove that there exists r ∈ �, suchthat G4�+ 15=G4�5+ r , for all � ∈�n+1.

We claim G4�+15=G4�5. Given � ∈�n+1 and a realization � ∈ì, for any feasible solution � to problemF�4�5, � + 1 is also a feasible solution to problem F�4� + 15 and h�4�1�5 = h�4� + 11� + 15. ThereforeF�4�+ 15 ≤ F�4�5. Similarly, we can prove that F�4�5 ≤ F�4�+ 15. Therefore F�4�5 = F�4�+ 15 for any� ∈ì. Thus we conclude that G4�+ 15=G4�5.

By Theorem 1, there exists an integral optimal solution �∗ to minimize G4�5. By Theorem 4 and the Iwata’ssteepest descent scaling algorithm (see, e.g., Murota [10]), G4�5 can be minimized in O4�4n5EOn2 log p5 timewhere �4n5=O4n55 is the number of function evaluations required to minimize a submodular set function overan n-element ground set (Orlin [11]), EO is the time needed for an expected cost evaluation for G4�5, andp = maxi∈S pi. This extends the result of Begen and Queyranne [2] for linear cost functions.

Furthermore, when the processing times are stochastically independent, as shown by Begen and Queyranne [2],the expected cost can be computed in O4n2p25 time, and thus 4P15 can be solved in O4n9p2 log p5 time.

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost Functions6 Mathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS

Remark 1. When An+1 is predetermined, the objective function G4�5 may no longer be L-convex. However,one can show that G4�5 is L\-convex in this case. The function G4�5 is L\-convex if and only if H4�1�5 =

G4�−�15=E�6F�4�−�157 is submodular in 4�1�5. To see this, we define Di4�5=Ci4�5+� and it followsthat

F�4�1�5= minD

n∑

i=1

8oi4Di+14�5−Ai+1 −pi+14�55+ ui4Di+14�5−Di4�5−pi+14�559

s.t. Di+14�5≥Ai+1 +pi+14�51 ∀ i

Dn+14�5≥ T +�

Di+14�5≥Di4�5+pi+14�51 ∀ i

D14�5= p14�5+�0

Similar to the proof of Theorem 3, we conclude that F�4�1�5 is submodular in 4�1�5. Then by Murota [10],G4�5 can be minimized in O4EOn7 log p5 time, where EO is the time needed for an expected cost evaluationfor G4�5. Using a different approach, Begen and Queyranne [2] proved the same result when the cost functionsare linear.

4. A sampling approach. In this section we consider the sample average approximation (SAA) approachto this problem, i.e., to minimize the empirical cost function of N independent samples drawn from the truedistribution, and compare the sample optimal solution to the optimal solution of the original problem. We developa bound on the required number of independent samples to guarantee that the solution obtained by SAA is nearoptimal with high probability.

Following the previous discussion, we make the following assumption on the cost functions in this section:

Assumption 3. For every i ∈ S, the cost functions ui4 · 5 and oi4 · 5 are convex, nondecreasing, and piecewiselinear with integer break points. Furthermore, they all satisfy condition (16).

Under Assumption 3, the convex relaxation in (12)–(15) is exact. Suppose we have N samples denoted byp4�k5 = 4p14�k51p24�k51 : : : 1 pn4�k55 and let � = �4N 5 be the empirical distribution of �, i.e., Prob4� =

�k5 = 1/N for k = 1121 : : : 1N . We denote the objective function obtained from the empirical distribution byG4�5 = E�6F�4�57 and denote by � an optimal solution that minimizes G4�5. We also define �∗ to be thetrue optimal solution, i.e., �∗ is a minimizer of G4�5.

First, we show that an optimal solution to the sample problem can be found efficiently. When there are Nsamples, the cost function G4�5 for any � can be evaluated within Nn arithmetic operations. Therefore, by thesame argument as in the previous section, we have the following:

Proposition 2. Assume the cost functions satisfy Assumption 3. Then G4�5 can be minimized inO4n8N log p5 time when the number of samples is N .

For the case in which the piecewise linear cost functions do not satisfy condition (16), we remark that thesampling problem with the convex relaxation can still be solved efficiently. By applying the network program-ming algorithms in Ahuja et al. [1], for general piecewise linear cost functions with integer break points, eachcost evaluation of G4�5 can be completed in O4n2 logn log 4n2p55 time. Therefore G4 · 5 can be minimized inO4n9N log p logn log 4n2p55. Thus the sampling approach can still provide a useful approximation to the originalproblem in this case.

Next we specify the bound on the required number of samples such that the SAA solution � will be nearoptimal. We introduce the following form of the Hoeffding inequality:

Lemma 2 (van der Vaart and Wellner [14]). Let x11 : : : 1 xr be samples drawn from a random vari-able c. Then

P

(

∣

∣

∣

∣

1r

r∑

i=1

xi − c

∣

∣

∣

∣

≥ �

)

≤ 2 exp(

−r�2

C2

)

1

where C = sup c, c is the expectation of c.

Define omax 2= max8oti 2 i = 1121 : : : 1 n3 t ∈ Ii9 and umax 2= max8us

i 2 i = 1121 : : : 1 n3 s ∈ Ji9. We have thefollowing theorem regarding the bound in our sampling approach.

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost FunctionsMathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS 7

Theorem 5. Let � > 0 (accuracy level) and 0 <�< 1 (confidence level) be given. If

N ≥ 441/�52n54omax + umax52p2 log 4np/�51

then G4A5≤G4A∗5+ � with probability at least 1 − �.

The bound stated in Theorem 5 serves as a complement to the main result by Begen et al. [3]. For thelinear cost functions satisfying �-monotonicity, they obtained a bound on the samples, 440541/�524n24n+ 15 ·

49omax + 4umax5/�52 log 4245n2 + 55/�55, to guarantee that G4A5≤ 41 + �5G4A∗5 with probability at least 1 −�,

where � = min8ui1oi9. The bound in Theorem 5 is in terms of an additive error whereas the bounds by Begenet al. [3] is in terms of a multiplicative error. This difference makes it hard to directly compare the two boundsin general.

Proof of Theorem 5. The proof follows the idea introduced in Kleywegt et al. [8]. First note that for anyset of samples ì, the optimal solution A must satisfy Ai ≤

∑

j<i pj (see Lemma 4.3 in Begen and Queyranne [2]).Define C = n24umax +omax5p ≥

∑ni=14umax +omax5An. Note that C is an upper bound of the objective value for any

solution satisfying Ai ≤∑

j<i pj and any sample. We claim that, for any cost functions satisfying Assumption 3,when the number of samples N ≥ 441/�52n log 4np/�5C2, with probability at least 1 − �, we have both

�G4A∗5−G4A∗5� ≤�

2

and�G4A5−G4A5� ≤

�

20

To prove this claim, we note that there always exists an integral optimal solution according to Theorem 1.Thus, given the upper bound on Ai as above, there are at most 4np5n possible integral optimal solutions (eitherfor a sample problem or the stochastic problem itself). And by Lemma 2, for each fixed A, the probability that�G4A5−G4A5� ≥ � is less than 2 exp44−N�25/C25. Then we take a union bound over all the integral solutionsand the claim holds.

Therefore, given N = 441/�52n54omax + umax52p2 log 4np/�5 samples, with probability at least 1 − �,

G4A∗5+�

2≥ G4A∗5≥ G4A5≥G4A5−

�

21

i.e., Theorem 5 holds. �

5. Final remarks. In this note, we study the appointment scheduling problem with discrete random pro-cessing times. We extend the study by Begen and Queyranne [2] to piecewise linear cost functions. We showthat under mild conditions, we can solve this problem in polynomial time. We also discuss a sample averageapproximation approach to the problem. We prove a bound on the number of required samples to achieve anear-optimal solution with an additive error, which complements to the result by Begen et al. [3].

To conclude the paper, we briefly discuss the subdifferential of the expected total cost function G4�5. Begenet al. [3] give a closed-form formula for the subdifferential. However, we hope the discussion below will helpfurther understand the structure of the appointment scheduling problem. For the sake of simplicity, we discussthe linear cost function case where oi4x5= �i · x and ui4x5=�i · x. The case for piecewise linear cost functionscan be similarly derived with more complex notations. For any given � ∈ì and �, the dual of the second stageproblem (12)–(15) is

F�4�5= max�4�51�4�5

n∑

i=1

84Ai+1 +pi+14�55�i4�5+pi+14�5�i4�5−�iAi+1 −4�i+�i5pi+14�59+p14�5�04�5

s.t. �i4�5+�i4�5−�i+14�5=�i+�i−�i+1 ∀ i=1121: : : 1n−1

�n4�5+�n4�5=�n+�n1

�04�5−�14�5=−�11

�i4�51 �i4�5≥01 ∀ i=1121: : : 1n0

(17)

Let O∗4�5 be the set of optimal solutions of (17) (restricted to variable �4�5). Then, by Propositions 2.2 and2.3 in Shapiro et al. [12], the subdifferential of G4�5=E�6F�4�57 is given by

¡G4�5= 8ïG4�52 ïiG4�5= E�6�∗

i−14�57− �i−11 i = 1121 : : : 1 n+ 11�∗4�5 ∈O∗4�591 (18)

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Ge et al.: Appointment Scheduling with Piecewise Linear Cost Functions8 Mathematics of Operations Research, Articles in Advance, pp. 1–8, © 2013 INFORMS

0�2 �n – 1 �n

�1�n – 1

�n�0

�1

1 nn–1. . . . . .

Figure 1. Network flow model for (17).

where �0 = 0. One may interpret problem (17) as a production problem as shown in Figure 1. In particular,variable �i is the production quantity in time period i, �i is the inventory at the beginning of period i, anddemand in period i is �i +�i −�i+1. (17) can then be viewed as a problem of choosing a production plan tosatisfy the demand, while minimizing the total linear costs given in the objective function of (17). This problemcan be solved in linear time (see Whitin [15]). Therefore, for any �, �∗4�5 can be computed very efficiently.This fact might be helpful to compute the subdifferential of G4�5 defined in (18).

Acknowledgements. The authors thank the associate editor and the referees for their helpful comments thatled to the improvements of this paper. The research of the first author is supported by NSF of China under[Grant 71001062] and Shanghai Pujiang Program. The research of the second author is supported by NSF ofChina under [Grants 71125003 and 70872079]. The research of the fourth author is supported by NSF of Chinaunder [Grant 11371001].

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