Managing Customer Arrivals in Service Systems with Multiple Servers

Transcription

1 Managing Cutomer Arrival in Service Sytem with Multiple Server Chrito Zacharia Department of Management Science, School of Buine Adminitration, Univerity of Miami, Coral Gable, FL Michael Pinedo Department of Information, Operation & Management Science, Stern School of Buine, New York Univerity, New York, NY. We analyze a dicrete multi-erver model for cheduling cutomer arrival under no-how. Cutomer may have different waiting cot coefficient and different no-how rate, reflecting their type and their hitory in attending cheduled appointment repectively. The challenge i to aign cutomer to time lot o that the ervice ytem utilize it reource efficiently, and cutomer experience hort waiting time. Theoretical and heuritic guideline are provided for the effective practice of appointment overbooking to offet no-how. For the cae of heterogeneou cutomer, we provide tructural propertie of an optimal chedule and we introduce a new equencing rule. When cutomer come from a homogeneou pool, recurive expreion for the performance meaure of interet are derived and we provide an upper bound for the optimal overbooking level. Extenive numerical experiment reveal further propertie and pattern that appear in the optimal olution, and motivate the development of two very well performing and computationally inexpenive heuritic olution. Our analyi demontrate the benefit of reource-pooling in containing operational cot and increaing cutomer throughput. Key word : ervice ytem; cheduling; no-how; overbooking; dicrete queue; parallel erver. Hitory : working paper, lat edited on Augut Introduction Appointment cheduling ytem are widely ued a a tool for managing cutomer arrival and matching upply and demand for ervice. It i common for cutomer to not how up for their cheduled ervice. Mied appointment reult in under-utilization of a ervice ytem valuable reource and limit the acce for other cutomer who could have filled the empty lot. Appointment overbooking i one operational trategy employed by ervice provider to addre the iue of no-how and at the ame time increae cutomer acce to ervice. On the other hand, overbooking potentially reult in an overcrowded facility, with increaed cutomer wait and ytem overtime. In thi tudy we demontrate that a enible practice of appointment overbooking can ignificantly improve the operational performance of a ervice ytem, while cutomer experience hort waiting time and better acce ervice. We addre the problem of cheduling cutomer arrival at a parallel-erver ytem under nohow. Cutomer have different waiting cot coefficient and different no-how rate, reflecting

2 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server their type and their hitory in attending cheduled appointment repectively. An optimal chedule balance the trade-off between the benefit of efficient reource utilization and the cot of cutomer waiting time. Thi tudy demontrate that an informed trategy for appointment overbooking can ignificantly improve the operational performance of a ervice ytem, while cutomer experience hort waiting time and better acce to care. A dicrete queueing model capture the random evolution of the ytem workload, baed on which we derive recurive expreion for the performance meaure of interet. The tak of finding an optimal chedule i modeled a an integer tochatic program which i analytically intractable and computationally expenive. A tight upper-bound (a olution to a convex program) retrict our earch for an optimal chedule on a contained olution pace. Our theoretical and experimental analyi reveal propertie and pattern that appear in the optimal cheduling trategy, and inform the development of two highly efficient heuritic olution. While motivated by the preing need of the healthcare ector, our model i applicable for a wide range of ervice ytem with appointment driven arrival. We avoid a reference to a particular application domain throughout thi tudy, by uing generic term a erver and ervice ytem. In outpatient care, our ervice ytem can be ued to model a diagnotic facility where it i crucial to utilize reource (e.g., CT can, X-ray generator, MRI) efficiently. Doctor are modeled a parallel erver in etting where continuity of care (ee Balaubramanian et al. ()) i not a concern. Nure are modeled a parallel erver when they are the bottleneck reource, and/or the preence of a doctor i not required (e.g., vaccination and immunization, routine lab teting, etc.). Other application domain include in-office conultation (e.g., financial, legal), on-ite cutomer upport (e.g., Apple Geniu Bar), entertainment, cometic ervice.. Related Literature Many paper have appeared in the literature on appointment cheduling, motly motivated by healthcare application. Cayirli and Veral (3), Gupta and Denton (8) provide overview of the literature, the reearch challenge and opportunitie. Hall () provide a comprehenive review of model and method ued for cheduling the delivery of patient care for all part of the healthcare ytem. The analyi may be baed on anyone of a variety of approache, including tochatic programming (e.g., Mancilla and Storer (), Mak et al. (4)), queueing theory (e.g., Green and Savin (8), Liu and Ziya (3), Kuiper et al. (4)), and tylized cheduling model (e.g., Robinon and Chen (), LaGanga and Lawrence (), Zacharia and Pinedo (4)). In the cae of homogeneou cutomer it i of interet to determine how many cutomer to chedule any given day and how to allocate thee cutomer to lot. The equencing of the cutomer i alo of interet when cutomer have different characteritic. In mot cae, finding an

3 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 3 optimal chedule i analytically intractable, and thu, the majority the literature ue enumeration, earch algorithm, imulation-baed technique and/or heuritic. A ervice ytem typically tart empty at the beginning of a working day, operate for a finite amount of time, and hut down until the next period. Therefore, it i important to perform tranient analyi for the random evolution of uch ytem. A pointed out in Bandi and Bertima (), tranient queue are difficult to analyze via claical queueing technique. Typically the analyi of rich queueing ytem over finite time horizon i addreed either by computer imulation (e.g., Millhier and Veral (4), Klaen and Yoogalingam (9)) or approximation (e.g., Araman and Glynn (), Honnappa et al. (4), Zacharia and Armony (5)). Mot of the literature focue on ingle-erver model. Kaandorp and Koole (7), Hain and Mendel (8), Klaen and Yoogalingam (9), Robinon and Chen (), Millhier and Veral (4) are ome recent work that conider the appointment cheduling problem with homogeneou cutomer who arrive on time for their cheduled appointment, if they do how up. Begen and Queyranne (), Cayirli et al. (), LaGanga and Lawrence (), Zacharia and Pinedo (4) account for cutomer heterogeneity a well. Even though the literature for the ingle erver ytem i quite extenive, the multi-erver cae ha received limited attention. A pointed out by Gupta and Wang () a well, appointment cheduling model become intractable if multiple feature are conidered imultaneouly. Very few tudie analyze ervice ytem with more than one erver, and, to the bet of our knowledge, in that cae only imulation tudie have been conducted. For example, Sickinger and Kolich (9) propoe and evaluate heuritic cheduling policie for a medical center with two computer tomography (CT) canner, Liu and Liu (998) tudy a block appointment ytem for clinic operation with multiple random arriving doctor, Zhu et al. (9) contruct a dicrete event imulation model to tudy pecialit outpatient clinic. A tylized dicrete queueing model with erver ha not been tudied analytically in the literature and bear unique challenge. The goal of thi tudy i to develop and analyze a dicrete multi-erver queueing model for deigning appointment ytem. We addre the following quetion: (a) how to effectively overbook capacity in order to offet cutomer no-how, (b) how to allocate the available ervice lot throughout the working day and furthermore, (c) how to account for patient heterogeneity. The objective function i the weighted um of expected ytem idle time, ytem overtime and cutomer waiting time. 3. Homogeneou Cutomer In thi ection we introduce a tylized dicrete queueing model that capture the random evolution of the ytem workload over time, baed on which we derive recurive expreion for the perfor-

4 4 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server mance meaure of interet in tranient tate. Conequently we preent the optimization problem under tudy and characterize the optimal overbooking trategy. 3.. Dicrete Multi-Server Scheduling Model Conider identical ervice provider working in parallel. Each one ha in her regular chedule n time lot available to erve cutomer in a working day. Beyond thee n regular lot, each one can erve cutomer in overtime lot a well. The ytem regular capacity i therefore n cutomer per day. Cutomer arrival are driven by cheduled appointment and the cheduler tak i to aign a number of cutomer to each time lot. We aume that cutomer how up with probability p = q (, ) at the beginning of their aigned lot and require determinitic ervice of one time lot. The number of cutomer to be cheduled throughout the working day (a deciion variable) i denoted by m, and we let y = m n denote the overbooking level. We conider only chedule that aign all m cutomer to the n regular time lot. That i, no cutomer i aigned a priori at the outet to an overtime lot. However, if the ervice provider have not erved all cutomer by the end of the n-th lot, then ome or all of them have to continue working overtime until the queue emptie out, while overtime cot are being incurred. Since cutomer are homogeneou in thi ection, their equencing i irrelevant, and a chedule can be completely characterized by the vector x = (x,..., x n ) Z n, where x t i the number of cutomer aigned to lot t, with m = n t= x t. In 5 we addre the optimal equencing of heterogeneou cutomer. Before we introduce the optimization problem, we characterize the random evolution of the ytem workload under any given chedule x. The number of new arrival at the beginning of each lot i a binomial random variable, not necearily identically ditributed, ince we allow to aign a different number of cutomer to different lot. The backlog of cutomer at the beginning of lot t, denoted by Z t, i captured by the recurion Z t = max{z t + A t, }, for t, () and Z =, where A t Binomial (x t, p) denote the number of new arrival at lot t. Our ervice ytem evolve randomly over time a a dicrete multi-erver queue with group arrival, i.e., a D A t/d/ queue. Note that a imilar recurion with () decribe the waiting time in a D/G/ queue, ee Janen and van Leeuwaarden (5). The probability ditribution of the queue length in tranient tate i derived a follow. Let f(k; n, p) be the probability that a Binomial (n, p) random variable take a value equal to k, i.e.,

5 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 5 ( ) n f(k; n, p) = p k ( p) n k, and let π j t (x) = Pr(Z t = j) denote the probability of a backlog of k j cutomer at the the beginning of lot t under chedule x. Let l t = t (x τ= τ ) denote the maximum poible backlog at the beginning of lot t. Auming that the ytem i empty at the beginning of the working day, then π (x) = and π j t (x) can be expreed recurively for t =, 3,..., n + a πt(x) i = min(,l t ) j= min(+i,l t ) j πt (x) j f(k; x t, p) for i = k= π j t (x)f( + i j; x t, p) for i l t j=max(,+i x t ) otherwie. Let I(x), O(x) and W (x) denote the expected erver total idle time, overtime, and cutomer aggregate waiting time, repectively, aociated with chedule x. Note that #(idle lot) + #(cutomer who how up) = n + (#overtime lot), () and therefore I(x) = O(x) + n mp. (3) The performance meaure of interet can be expreed repectively baed on () and (3) a l n+ O(x) = E(Z n+ ) = jπn+(x), j (4) j= j= l n+ I(x) = jπn+(x) j + n pm, (5) W (x) = x n t 3.. Optimization Problem l t t= i= j=max(, i+) π j t (x) i k=max(, j) rf(k; i, p) j+k +. (6) There are three cot (penaltie) aociated with an appointment chedule: cutomer waiting cot, erver idle time and overtime cot. If there are le than cutomer preent at the beginning of any one of the regular n time lot, then one or more provider remain idle and for each provider being idle an idle time cot c I i incurred. The cheduler may overbook certain time lot and aign more than cutomer in order to compenate for the no-how behavior. If more than cutomer are preent at the beginning of a time lot due to overbooking, then all but of thee cutomer have to wait. A waiting cot w i incurred for each time lot that a cutomer ha to wait before tarting ervice. Finally, an overtime cot c O i incurred for each overtime lot. We normalize the

6 6 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server objective function with repect to c I, i.e. c I =, and we conider the following nonlinear integer program: min (m,x) V (x) = I(x) + c O O(x) + ww (x).t. x t nonnegatve integer for all t =,,..., n, n x t = m. t= We denote the optimal olution with (m, x ) and the optimal overbooking level with y = m n. Lemma. x t for all t =,,..., n. Lemma i a direct conequence of the recurion in () and our model aumption. The optimal chedule ha at leat cutomer aigned to each lot, and the optimization problem now become to identify which lot (if any) to overbook and by how much. Let X y n = {x : n x t = n + y, x t, for t =,,..., n} t= denote the et of all feaible chedule that allocate y overbooked cutomer to the n lot, with every lot having at leat cutomer aigned to it. (P ) 3.3. Upper-Bound on the Optimal Overbooking Level In thi ection we demontrate that the optimal olution y olution of a dicrete convex optimization problem. of (P ) i bounded above by the Let x y = ( + y,,..., ) for ome nonnegative integer y be a chedule where all the overbooking (if any) occur during the firt time lot and exactly cutomer are aigned to lot, 3,..., n, and let A = {x y : y nonnegative integer}. Such chedule turn out to be optimal when we focu on optimizing ytem efficiency by ignoring cutomer waiting cot: min (m,x) I(x) + c O O(x).t. x t nonnegative integer for all t =,,..., n, x t for all t =,,..., n, n x t = m. t= The optimization problem (P ) balance the trade-off between maintaining high reource utilization during the regular length of the workday and incurring low overtime cot. Lemma. There i a ȳ uch that xȳ i a olution to the optimization problem (P ). When we look pat cutomer waiting cot, it i optimal to overbook a number cutomer at the very beginning of the working day. Thi policy guarantee that the ytem i a buy a poible, (P )

7 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 7 while overtime cot are kept at a moderate level (the overbooked cutomer are aborbed by potential no-how throughout the working day). A chedule within cla A i a olution to (P ) (minimizing the weighted um of all three cot) if the erver cot coefficient, c I and c O, are an order of magnitude larger than the cutomer weight. In practice, uch ytem could correpond to one where provider availability and ytem reource are ufficiently more cotly than having cutomer waiting. Further, the optimal overbooking level ȳ for (P ) turn out to provide an analytically tractable upper bound to the optimal overbooking level y for (P ). Let M y Binomial (n + y, p) denote the total number of cutomer who will how up for their appointment under chedule x y = ( + y,,..., ). Then and from (5), O(x y ) = E[max(, M y n)] = E[M y n] + E[max(, n M y )] n = pm n + (n k)f(k; m, p), (7) k= I(x y ) = O(x y ) + n pm n = (n k)f(k; m, p). (8) k= Recall that all appointment chedule within cla A minimize the idle time and overtime cot, and therefore (P ) can be written a min [I(x) + c y, x Xn y O O(x)] = min y, x X y n A [I(x) + c O O(x)] = min [I(x y) + c y O O(x y )]. Theorem. (i) I(x y ) i decreaing and dicretely convex in y on {,,...}. (ii) O(x y ) i increaing and dicretely convex in y on {,,...}. (iii) y ȳ. Since O(x y ) and I(x y ) are dicretely convex in y, efficient computational procedure can provide ȳ, which i an upper bound to y. A demontrated in 4, thi upper bound i tight and contain ueful information regarding the optimal olution to (P ). For the ret of 3.3 we provide an analytic characterization for the upper bound ȳ baed on the continuou relaxation of (P ). Conider the differentiable extenion of the binomial coefficient to non-integer v u defined a ( ) u v = Γ(u+), where Γ(t) = x t e x dx i the Gamma Γ(v+)Γ(u v+) function. Note note that d ( ) u v du = Γ(u + ) [Ψ(u + ) Ψ(u v + )], Γ(v + )Γ(u v + )

8 8 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server where Ψ(t) = Γ (t) i the Digamma function, the logarithmic derivative of the Gamma function. Γ(t) When v i integer, then d(u v) = ( ) u v du v i=. Therefore, either ȳ = (boundary olution), or ȳ = ŷ, v i or ȳ = ŷ, where ŷ atifie a firt order condition n ( [ ] n + ŷ k ( + c O ) (n k) )p k ( p) n+ŷ k ln( p) + + pc k n + ŷ i O =. (9) k= The optimal overbooking level y i bounded above by ȳ = m n. i= 3.4. Periodic Overbooking Heuritic The optimization problem (P ) i computationally very intene; the ize of the olution pace i Xn y = (y+n )!, exponential both in y and n, with y being ubject to optimization a well. In 3.3 y!(n )! an upper bound for the overbooking level wa developed, by conidering the problem where w =. It i demontrated in 4 that, a the waiting cot coefficient w increae, the optimal chedule become more uniform, without necearily oberving a decreae in the optimal overbooking level. Furthermore, periodic pattern appear in the middle egment of the chedule for large value of n. It appear that one can think of an optimal chedule to conit of three egment: a beginning tart-up egment, a middle tationary egment, and a final emptying-out egment. In the tart-up egment, the chedule tend to be ubject to more overbooking than in the other two. The middle egment appear to be more uniform and regular. The overbooked cutomer in the lat egment of the chedule tend to taper off (in order to avoid high overtime cot). If n i large then the middle egment of the chedule i quite ubtantial. However, when n i mall, the middle egment of the chedule may tend to diappear. The uniformity and regularity of the middle egment of the chedule i a motivation for the heuritic decribed in what follow. We propoe a computationally inexpenive heuritic olution baed on the evolution of a dicrete queue. Let v n,y = ( + y,,,..., ) be a ub-chedule of length n n. Let m be the number of cutomer allocated to egment v n,y, i.e., m = n + y, and let N = n n be the number of conecutive uch egment. We conider periodic chedule of the form x n,y = (v n,y ; v n,y ;...; v n,y } {{ ). } N time In other word, we conider chedule that demontrate a periodic pike of ize y every n lot, ee Figure. Periodic chedule give a renewal flavor to our arrival proce, rendering our tranient queue tractable. If we focu only on the queue length at the beginning of time lot t, t,..., t N +, where t i = (i )n +, then we can provide a cloed-form characterization of the ytem idle time

9 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 9 Figure A chedule with periodic overbooking. xt (cutomer per lot) +y t t t 3... t (lot index) and overtime. Conider the dicrete-time bulk queueing ytem that evolve according to (), with A i being independent and identically ditributed a A Binomial (m, p), i =,,..., N. In the queueing literature uch a ytem i referred to a a dicrete D A /D/n queue. A in Janen and van Leeuwaarden (5), the expected queue length right before the beginning of egment t i i given explicitly, not recurively, by Spitzer identity (ee Spitzer (956)) a i E(Z ti ) = τ= i τ= τ E[max(, Y τ)] = τ [E[Y τ] + E[max(, Y τ )]] τ= [ i τn ] = pτm τn τ + (τn k)f(k; τm, p) = (i )(pm n ) + k= i τn τ= τ (τn k)f(k; τm, p), () where Y τ = τ i= (A i ) denote the τ-fold convolution of (A ). The expected ytem overtime and idle time can now be expreed a k= N τn O(x n,y ) = E(Z ) = pm N n N + tn (τn + τ k)f(k; τm, p), () N τn and I(x n,y ) = O(x n,y ) + n N pm N = (τn τ k)f(k; τm, p). () Note that (7) and (8) are pecial cae of () and () repectively when n = n. Theorem. (i) I(x n,y ) i decreaing and dicretely convex in y on {,,...}. (ii) O(x n,y ) i increaing and dicretely convex in y on {,,...}. Periodic chedule, beide being analytically tractable, turn out to yield computationally inexpenive and very well performing heuritic olution. Algorithm decribe the Periodic Overbooking Heuritic (POH) in term of olving n convex program. A demontrated in 4, the middle egment of the optimal chedule often ha the pattern of one of the following three pecial cae: (a) y = for ome n > correponding to moderate overbooking, (b) n = correponding to frequent overbooking when no-how rate are high, (c) n = n when it i optimal to overbook only at the beginning of the chedule. τ= τ= k= k=

10 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server Algorithm Periodic Overbooking Heuritic (POH) : procedure POH(n,, w, q, c O ) : x POH e n initiation tate - no overbooking, e n i the vector of n one, 3: cot POH ( p)n only idle time cot 4: for n =,,..., n do 5: y arg min y [I(x n,y ) + c O(x n,y )] dicrete convex program 6: l n mod n 7: x (x n,y ; e l) e l i the vector of l one 8: cot [I(x) + c O(x) + ww (x)] 9: if cot < cot POH then : x POH x : cot POH cot : end if 3: end for 4: return x POH and cot POH 5: end procedure 3.5. Front-Loading Heuritic The overbooking level y contain much of the information regarding the optimal chedule. A demontrated in the following ection, the overbooked cutomer are allocated according to a frontloaded pattern: more cutomer are cheduled toward the beginning of the working day (in order to get an empty ytem running), and toward the end of the working day the a chedule become le dene (in order to avoid high overtime cot). We propoe a econd heuritic, the Front-Loading Heuritic (FLH), which predict the optimal overbooking level baed on our numerical analyi in 4., and allocate the overbooked cutomer in a front-loaded manner. Algorithm (ee Appendix) decribe in detail the FLH procedure. In 4.3 we compare the two heuritic olution and evaluate their performance. 4. Numerical Experiment In thi ection we diplay and dicu the reult of our numerical experiment. Our analyi reveal further propertie and pattern that appear in the optimal chedule, and provide u with additional inight into their overall tructure. Throughout our numerical analyi, following the literature (ee for example Robinon and Chen (, ), Zacharia and Pinedo (4)), we conider an overtime cot coefficient c O =.5 and value of the waiting cot coefficient w between and.6. Provider idlene cot more than cutomer wait, but it i le cotly than provider overtime. We interpret that value of w between and. correpond to an efficiency regime, value of w above. correpond to a quality regime, and between. and. to a hybrid quality and

11 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server Figure Optimal chedule. n =, {,, 3, 4, 5}, w {.,.5,.}, q {.5,.,.5,.,.5,.3}. = erver, w=., n= = erver, w=.5, n= = erver, w=., n= # of cutomer per lot # of cutomer per lot # of cutomer per lot q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 4 = erver, w=., n= 4 = erver, w=.5, n= 4 = erver, w=., n= # of cutomer per lot 3 # of cutomer per lot 3 # of cutomer per lot 3 q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 6 =3 erver, w=., n= 6 =3 erver, w=.5, n= 6 =3 erver, w=., n= # of cutomer per lot # of cutomer per lot # of cutomer per lot q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 7 =4 erver, w=., n= 7 =4 erver, w=.5, n= 7 =4 erver, w=., n= # of cutomer per lot # of cutomer per lot # of cutomer per lot q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 q=.5 q=. q=.5 q=. q=.5 q=.3 # of cutomer per lot =5 erver, w=., n= q=.5 q=. q=.5 q=. q=.5 q=.3 # of cutomer per lot =5 erver, w=.5, n= q=.5 q=. q=.5 q=. q=.5 q=.3 # of cutomer per lot =5 erver, w=., n= q=.5 q=. q=.5 q=. q=.5 q=.3 efficiency regime. Whenever we find it neceary, all the quantitie of interet are ubcripted by, the number of erver. 4.. Optimal Schedule Optimal chedule are diplayed in Figure for different no-how rate q, different waiting cot coefficient w, and for up to 5 erver. It i evident, and intuitive, that the optimal overbooking

12 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server level y = m n i increaing in q. A w increae, the optimal chedule become le front-loaded, without necearily oberving a decreae in the overbooking level. Further, many optimal chedule exhibit a pike at the firt appointment lot, in agreement with the eminal reult of Bailey (95) and Welch and Bailey (95) for the ingle erver cae. The overbooking level increae ignificantly with the number of parallel erver, and that increae i more prevalent for higher no-how rate. Recall that in 3.3 we etablihed that the olution to a dicretely convex optimization problem (the cheduling problem where w = ) provide an upper bound for the optimal overbooking level y. In Table we demontrate that thi upper bound i often tight. Conider for example the cae where q = % and = erver. The optimal olution to (P ) i (6,,,,,,,,, ) with an overbooking level of 4 cutomer. The optimal olution to (P ) for w =.5 ha the ame overbooking level, but the actual chedule (4,, 3,, 3,,,,, ) look quite different. The overbooked cutomer are pread out more uniformly over the working day with a moderate pike at the firt lot. Table Optimal chedule and overbooking level. n =, {,, 3}, w {.,.5,.,.5}, q {.5,.,.5,.,.5,.3}. = = = 3 w =. x = y = x = 3 y = x = y = q =.5 w =.5 x = y = x = y = x = y = w =. x = y = x = y = x = y = w =.5 x = y = x = y = x = y = w =. x = y = x = 4 y = x = y = 3 q =. w =.5 x = y = x = 3 y = x = y = w =. x = y = x = 3 y = x = y = w =.5 x = y = x = 3 y = x = y = w =. x = y = x = 5 y = 3 x = y = 4 q =.5 w =.5 x = y = x = 3 3 y = x = y = 4 w =. x = y = x = 3 3 y = x = y = 3 w =.5 x = y = x = 3 3 y = x = y = 3 w =. x = 3 y = x = 6 y = 4 x = y = 6 q =. w =.5 x = y = x = y = 4 x = y = 6 w =. x = y = x = y = 3 x = y = 5 w =.5 x = y = x = y = 3 x = y = 5 w =. x = 4 y = 3 x = 8 y = 6 x = y = 9 q =.5 w =.5 x = y = x = y = 5 x = y = 8 w =. x = y = x = y = 4 x = y = 7 w =.5 x = y = x = y = 4 x = y = 7 w =. x = 4 y = 3 x = 9 y = 7 x = y = q =.3 w =.5 x = y = 3 x = y = 7 x = y = w =. x = y = x = y = 6 x = y = w =.5 x = y = x = y = 5 x = y = 9 In Table we demontrate the effect of the number of lot n on the optimal chedule. There i evidence for a linear relationhip between the overbooking level and n, by oberving that the cae where n =, 5,, 5, 3, 35 and 4 lot have correponding overbooking level y =, 3, 4, 5, 6, 7 and 8 repectively. Moreover, a n increae, a periodic pattern appear in the middle egment of the chedule.

13 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 3 Table Optimal chedule a a function of n. =, q =.5, w =.. n x Next, we focu on the ytem performance a a function of the number of erver and the no-how rate q. Let ρ = pm n be the ytem utilization under the optimal chedule. The benefit of reource pooling are apparent in two dimenion: (a) decreaing the total cot ( V in ), (b) increaing the ytem utilization (ρ cutomer acce to ervice ( y decreaing increaing in ), and conequently increaing the increaing in ). Thee gain are illutrated in Figure 3. We alo oberve that higher no-how rate come along with a more cotly ytem, ince both waiting time and ytem idlene increae with the variability in the arrival proce. Figure 3 Benefit of reource pooling. n =, w =., {,, 3, 4, 5}, q {.,.,.3}. (a) Cot per erver (b) Sytem utilization (c) Overbooking level q =. q =. q =.3.98 q =. q =. q =.3 4 q =. q =. q =.3 V ρ y Predictive Model for the Optimal Overbooking Level The overbooking level y contain much of the information regarding the optimal chedule. In thi ection we employ predictive model to predict y a a function of the independent variable n,, w, and q. An equivalent way to capture y i through the proportion y p = y, where ȳ i the ȳ upper bound derived in 3.3. We ue the Logit model to predict the proportion y p, and Quadratic regreion to predict y, ee Greene (8).

14 4 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server In the Logit model the expected value for the proportion y p given the ytem parameter z = (n,, w, q) i E(y p z) = ( + e (β n+β +β 3 w+β 4 q+β 5 ) ), (3) for ome coefficient vector β = (β,..., β 5 ). The more elaborate Quadratic regreion i a form of linear regreion in which the relationhip between y and z i modeled a a econd-order polynomial E(y z) =Quad(β; z) β n + β n + β 3 nw + β 4 nq + β 5 n + β 6 + β 7 w + β 8 q (4) + β 9 + β w + β wq + β w + β 3 q + β 4 q + β 5. Our data et wa generated by olving 45 intance of the optimization problem (P ). For the olved intance the length of the working day n take value in {5, 6,..., 3} and we conider problem with up to = erver. The waiting cot coefficient w take value in {,.5,...,.6}, and the no-how rate q in {.5,.,...,.4}. The idle time cot coefficient c I i normalized to and the overtime cot coefficient i fixed at c O =.5. The maximum oberved overbooking level i 3 cutomer. Randomly choen, two third of the intance are ued a our training et, and the ret a our tet et. Though we could have conidered higher-order polynomial, a quadratic regreion fit our training data well, yielding a.98 R-quared. We evaluate the goodne-of-fit of our Logit model with the McFadden peudo R-quared, a propoed by McFadden (974). A.68 peudo R-quared indicate a very good explanation of our training data. Table 3 Predictive Model. (a) Quadradic Regreion n n nw nq n w q w wq w q q Coefficient (β) p-value R =.977 (b) Logit regreion n w q Coefficient (α) p-value McFadden peudo R =.678 (c) Predictive Power model [ prediction ] MAE Quadratic [ max(, Quad(β; z)).387 Logit ( + e (α n+α +α 3 w+α 4 q+α 5) ) ȳ ].564 [.] denote the cloet integer The reult of the two regreion are ummarized in Table 3 (a) and (b). The only tatitically non-ignificant variable (at level.5) in the Quadratic model for predicting y i the quare of the number of lot n, confirming our obervation of a linear relationhip in 4.. Regarding the Logit model for the proportion y p, the number of lot n i the only non-ignificant variable. Interetingly

15 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 5 though, it coefficient i negative, indicating that the larger the length of the working day, the le likely it i that the overbooking level achieve it upper bound. Finally, we evaluate the predictive power of the two model via the Mean Abolute Error (MAE) over the tet et. The prediction are made a in Table 3 (c). The proportion y p, etimated by the Logit model, i multiplied with the upper bound ȳ and rounded to the cloet integer. The Quadratic model i further corrected for negative value, which are mapped to zero-overbooking level. Both model perform very well, with MAE le than.6. The Quadratic model make the mot accurate prediction yielding the mallet MAE at.387. Building upon our prediction for the optimal overbooking level, we propoe the Front Loading Heuritic (FLH). In particular, given the ytem parameter n,, w and q, we overbook a number of cutomer a uggeted by the quadratic model, in agreement with the front-loaded pattern oberved in 4.. Algorithm (appearing in the Appendix) decribe in detail the FLH procedure Evaluation of POH and FLH Finally, we evaluate the performance of POH and FLH via their cot difference relative to the optimal chedule. The comparion i baed on all 45 intance decribed in 4.. Both POH and FLH yield very well performing chedule, with the average cot difference being 5.9% and 3.49% repectively. We diplay in Table 4 a ample comparion for a working day coniting of n = 6 lot (for example an 8-hour working day with lot of 3 min length) and for different value of, w, q. A olution to a convex program provide the overbooking level in the POH. On the other hand, FLH overbook a number of cutomer a uggeted by our predictive model in 4.. Periodic chedule, beide being analytically tractable, turn out to provide very well performing and computationally inexpenive heuritic olution. In mot of the cae POH provide the optimal overbooking level, but, due to it periodic nature, the allocation of the overbooked cutomer i not the optimal one. Only trivial cae, where all the overbooking occur at the firt lot, are olved to optimality. FLH, on the other hand, often provide chedule identical to the optimal one for non-trivial cae a well (ee Table 4). Even though on average FLH perform better than POH, it i not clear under what circumtance in general which heuritic i better than the other. 5. Heterogeneou Cutomer Conider now the more general model where cutomer are heterogeneou in two dimenion: they have different waiting cot coefficient and different no-how rate, reflecting their type and their hitory in attending cheduled appointment repectively. The cheduler would like to aign each one of the cutomer to arrive at the beginning of one of the time lot. Cutomer j, j =,..., m,

16 6 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server q =. q =.5 q =. Table 4 Performance of POH and FLH compared to the optimal chedule. n = 6, {,, 3}, w =.5 w =. w =.5 w =.5 w =. w =.5 w =.5 w =. w =.5 w {.5,.,.5}, q {.,.5,.}. = = = 3 ( ) ( 3 3 ) ( ) [ ] [ 3 3 ] [ ] { } { 3 3 } { }.%.%.3%.7% 4.7% 6.4% ( ) ( 3 3 ) ( ) [ ] [ 3 3 ] [ ] { } { 3 3 } { }.%.%.3%.83%.63% 6.94% ( ) ( 3 ) ( ) [ ] [ 3 ] [ ] { } { 3 } { }.%.%.%.%.%.% ( ) ( ) ( ) [ ] [ 4 4 ] [ ] { } { } { }.% 4.5% 3.87% 6.9% 6.4%.7% ( ) ( ) ( ) [ ] [ ] [ ] { } { } { }.%.%.55% 4.73% 4.85%.7% ( ) ( ) ( ) [ ] [ ] [ ] { } { } { }.%.%.8%.7%.6%.% ( ) ( ) ( ) [ ] [ 5 5 ] [ ] { } { } { }.7% 5.59% 7.9%.48%.5%.6% ( ) ( ) ( ) [ ] [ ] [ ] { } { } { }.%.7% 4.39%.5%.9%.93% ( ) ( ) ( ) [ ] [ ] [ ] { } { } { }.5%.74%.6%.% 6.6%.3% (.) denote x, [.] denote x POH, {.} denote x FLH, denote the POH cot difference, denote the FLH cot difference will how up with probability p j = q j (, ) at the beginning of the time lot he wa aigned, ha a waiting cot coefficient w j, and require one time lot of ervice. Let C = {,,..., m} denote the et of all cutomer to be cheduled. A chedule i defined a a pair of vector (σ, x), where σ = (σ, σ,..., σ m ) i a permutation of C, and x = (x, x,..., x n ) i uch that x t cutomer are aigned to lot t, with n t= x t = m, and cutomer (σ, σ,..., σ x ) are aigned to lot, cutomer (σ x +, σ x +,..., σ x +x ) are aigned to lot, and o on. For better preentation of our analyi, let t = t τ= x τ be the um of all cutomer cheduled up to lot t, t n. A a convention, =. Let I(σ, x) and O(σ, x) denote the expected total erver idle time and overtime aociated with chedule (σ, x) repectively. Note that #(erver idle lot) + #(cutomer who how up) = n + #(overtime lot),

17 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 7 and therefore I(σ, x) = O(σ, x) + n m p j. (5) Alo let B j (σ, x) be the random variable denoting the backlog of cutomer that have priority over cutomer j upon her arrival (if cutomer j how up). Then the expected waiting time of cutomer j, meaured in time lot, i W j (σ, x) = p j k j= Pr(B j (σ, x) = k) k. (6) The objective function i the weighted um of erver expected idle time, overtime, and cutomer waiting time V (σ, x) = c I I(σ, x) + c O O(σ, x) + and we conider the optimization problem (P 4 ) min (σ,x) V (σ, x).t. σ i a permutation of C, m w j W j (σ, x), (7) j= x t i a nonnegative integer for t =,,..., n, n x t = m. t= Theorem 3 characterize an optimal olution to (P 4 ) in term of a priority rule and a equencing rule. In order to diplay the latter, let B t be the random variable denoting the number of cutomer among (σ, σ,..., σ t ) till preent at the beginning of lot t, i.e., the backlog of cutomer that have priority over cutomer σ t, t n. Theorem 3. An optimal chedule (σ, x ) atifie the following propertie: (i) The cutomer aigned to each one of the lot,..., n are prioritized in decreaing order of their weight w j. (ii) For t { τ : τ n, x τ+ = }, for j = σ t hold: (P 4 ) and k = σ t +, the following equencing rule p j w j p k w k Pr( B t ) Pr( B t = ( ) mod, B t )p j Pr( B t ) Pr( B t = ( ) mod, B. (8) t )p k The following corollary i a direct conequence of Theorem 3. It demontrate implified expreion of the equencing rule in (8) with intuitive interpretation for four pecial cae. Corollary. (i) If m n <, then the equencing rule in Theorem 3 become p j w j p k w k. (ii) If p j = p for all j C, then the equencing rule in Theorem 3 become w j w k. (iii) If w j = w for all j C, then the equencing rule in Theorem 3 become p j p k.

18 8 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server (iv) If =, then the equencing rule in Theorem 3 become p jw j p j p k w k p k. A uggeted by (i) and (ii) of Corollary, cutomer with larger weight tend to be cheduled at lot where they are expected to encounter a horter queue and therefore incur maller waiting cot. From (iii) of Corollary, cutomer between conecutive overbooked lot are equenced in decreaing order of their no-how probability: the further away a cutomer i aigned from an overbooked lot, the more likely he i to how up and to encounter a horter queue. A a remark, the equencing rule in (iv) of Corollary i the Smallet Weighted Probability of Showing Up firt rule that appear in Zacharia and Pinedo (4), where they analyze a ingle erver model. 6. Concluion To the bet of our knowledge, thi i the firt tudy to develop and analyze a tylized appointment cheduling model for managing cutomer arrival in a ervice ytem with more than one erver. Theoretical and heuritic guideline are provided for the effective practice of appointment overbooking to offet no-how. When cutomer come from a homogeneou pool, recurive expreion for the performance meaure of interet are derived and we provide an upper bound for the optimal overbooking level. Our extenive numerical experiment reveal further propertie and pattern that appear in the optimal chedule, and motivate the development of two very well performing and computationally inexpenive heuritic olution. Periodic chedule have a imple form, are analytically tractable, and yield olution on average 5.% more cotly than the optimal one. Our front loading heuritic predict quite accurately the optimal overbooking level, often provide a olution identical to the optimal one, and ha an average cot difference of 3.5%. For the cae of heterogeneou cutomer, we provide tructural propertie of an optimal chedule and we introduce a new equencing rule. For the ake of analytical tractability, and in order to focu pecifically on the uncertainty reulting from no-how, we did not model the variability in ervice time. Determinitic ervice time, however, have been conidered in the literature (e.g. Green and Savin (8), LaGanga and Lawrence (7), Zacharia and Pinedo (4)) and have hown to yield ueful approximation and inight. Cutomer non-punctuality i not captured by our model. A dicued in, appointment cheduling model become intractable if multiple feature are conidered imultaneouly, and typically tranient analyi of rich queueing ytem i addreed either via computer imulation or approximation (for example diffuion procee). An intereting future reearch direction i to account for non-perfectly pooled ervice provider. In practice, ituation often arie where a tream of cutomer i dedicated to a particular provider (for example in a group phyician practice), wherea ome other cutomer do not have a preference and go to the next available provider.

19 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 9 Acknowledgment The author gratefully acknowledge the upport of the Alexander S. Onai Foundation. Appendix Proof of Lemma Aume x = (x, x,..., x n ) i a chedule with at leat one lot with le than cutomer aigned to it, and tet t be the firt uch lot, i.e., t = min{t : x t < }. Let m = n t= x t. We will how via contradiction that x cannot be optimal. Cae : Aume x t for all t t. Then Pr(A t ) = for all t =,,..., n +, and therefore, from (), Pr(Z t = ) = for all t =,,..., n +. From (4), (5) and (6) we get O(x) = I(x) = n mp > (9) W (x) =. Conider now the chedule (ee Figure 4) x = (x, x,..., x t, x t +, x t +,..., x t, x t, x t +,..., x n ) and let A t and Z t be the aociated new arrival and backlog at the beginning of lot t repectively. Then O(x ) = I(x ) = n (m + )p () W (x ) =. From (9) and (), and by the aumption that p >, chedule x i le cotly than x. Cae : Aume x t > for for ome t t, i.e., there i at leat one lot with more than cutomer aigned to it. Cae a: Suppoe that the firt lot with more than cutomer aigned to it appear after lot t, and let t be that lot, i.e., t = min{t : x t >, t > t } (ee Figure 4). Thi implie that lot,,..., t have at mot cutomer aigned to each one of them. Conider now the chedule ˆx = (x, x,..., x t, x t +, x t +,..., x t, x t, x t +,..., x n ). Then Pr(Ẑt = ) = Pr(Z t = ) = for all t =,,..., t,

20 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server and Pr(Ât > a) < Pr(A t > a) for all a, i.e., Â t < t A t. Therefore, from (), Ẑ t < t Z t for all t = t +, t +,..., n, concluding that O(ˆx) < O(x), I(ˆx) < I(x), and W (ˆx) < W (x). Cae b: Suppoe that there i a lot prior to t with more than cutomer aigned to it. Let t be the lat lot before t that ha more than cutomer aigned to it, i.e., t = max{t : x t >, t < t } (ee Figure 4). Thi implie that lot t +,..., t have exactly cutomer aigned to each one of them. Conider now the chedule x = (x, x,..., x t, x t, x t +,..., x t, x t +, x t +,..., x n ). Then x t = x t and therefore Z t d = Z t for all t =,,..., t. () Note that x t = x t and hence Pr(Ãt > a) < Pr(A t > a) for all a, i.e., Ât < t A t. Therefore, from (), Z t < t Z t for all t = t +, t +,..., t. () Next, we will how that Z t + < t Z t for all t = t +, t +,..., n. Since x t = x t = for all t = t +, t +,..., t, we get that Since Z t + = max{z t + Bino ((t t ) + x t + x t, p) (t t + ), } Z t + = max{ Z t + Bino ((t t ) + x t + x t, p) (t t + ), }. Z t d = Z t and x t + x t = x t + x t, we conclude that Z t d = Z t for all t = t +, t +,..., n. (3) From (), () and (3) we conclude that Proof of Lemma O( x) < O(x), I( x) < I(x), and W ( x) < W (x). Firt, recall that I(x) = O(x) + n mp, and therefore a chedule minimize the expected idle if and only if it minimize the expected overtime. It uffice to how that a chedule from cla A minimize the total expected overtime. Suppoe that for ome realization of the arrival proce d out of m cutomer will actually how up. Then the total number of overtime lot mut be at leat max(d n, ), for any chedule mathbfx uch that n t= = m.

21 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server Figure 4 Proof of Lemma. (a) Cae cutomer per lot x x t t lot index (b) Cae a cutomer per lot x ˆx t t t t lot index (c) Cae b cutomer per lot x x t t t t lot index Suppoe a chedule belong to cla A and that for ome realization of the arrival proce a out of m cutomer will actually how up. Then exactly min(a, n) will be erved during regular lot. The ret a min(a, n) will be erved during overtime lot, and thi i true for every a and for every realization of the arrival proce with a arrival. Note that a min(a, n) = max(, a n), the minimum poible number of overtime lot. Therefore, a chedule from cla A minimize the total expected overtime and idle time. Proof of Theorem (i) It i equivalent to how that I(x y ) i decreaing and dicretely convex in m = n + y on {n, n +,...}. Firtly note that n ( ) m + I(x y+ ) = (n k) p k ( p) m+ k k k= n ( ) m n ( ) m = (n k) p k ( p) m+ k + (n k) p k ( p) m+ k k k k= k= n ( ) m n ( ) m = ( p) (n k) p k ( p) m k + p (n k) p k ( p) m (k ) k k k= k= n ( ) j=k m = ( p)i(x y ) + p [n (j + )] p j ( p) m j j j= n ( ) n m ( ) m = ( p)i(x y ) + p (n j) p j ( p) m j p p j ( p) m j j j j= j= n ( ) n m ( ) m = ( p)i(x y ) + p (n j) p j ( p) m j p p j ( p) m j j j j= j=

22 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server concluding that = ( p)i(x y ) + pi(x y ) p = I(x y ) p n j= n j= ( ) m p j ( p) m j, j p < I(x y+ ) I(x y ) = p Therefore, I(x y ) i decreaing in m = n + y. ( ) m p j ( p) m j j n k= ( ) m p k ( p) m k <. (4) k For the proof of the dicrete convexity it uffice to how (from Theorem of Yüceer ()) that I(x y ) ha increaing difference in m on {n, n +,...}. Let P m = n ) k= pk ( p) m k and note that P m+ = From (4) and (5) n k= n = j= ( m + ( m k = ( p) ) p k ( p) m+ k k ) p k ( p) m+ k + n j= n k= ( ) m p k ( p) m k + p k n ( m k= ( m k ( ) m p k ( p) m+ k k n ( ) m p k ( p) m (k ) k ) j=k = ( p)p m + p p j ( p) m j j j= ( ) m = ( p)p m + pp m p p n ( p) m (n ) n < P m. [I(x y+ ) I(x y+ )] [I(x y+ ) I(x y )] = p[p m+ P m ] >, concluding that I(x y ) ha increaing difference in m on {n, n +,...}. (ii) Recall that O(x y ) = pm n + I(x y ), and therefore O(x y+ ) O(x y ) = p[(m + ) m] + [I(x y+ ) I(x y )] = p pp m (5) = p( P m ) (6) >. It i traightforward to verify that O(x y ) ha increaing difference in m = n+y on {n, n+,...} a well, ince O(x y ) = pm n + I(x y ).

23 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server 3 (iii) Let ȳ be an optimal olution to (P ) and let m = n + ȳ. Then c I I(xȳ+ ) + c O O(xȳ+ ) c I I(xȳ) + c O O(xȳ), implying (from (4) and (6)) that p (( + c O )P m c O ), (7) where P m = n ( m ) k= pk ( p) m k i the probability that the waiting time of the lat cutomer k of chedule xȳ+ i equal to zero. Let (y, x ) and be an optimal olution to (P ) and aume, for contradiction, that y > ȳ. Let t be the lat lot under chedule x that ha more than cutomer aigned to it, i.e., t = max{t : x t > }. Thi implie that lot t +,..., n have exactly cutomer aigned to each one of them. Conider now the chedule x = (x, x,..., x t, x t, x t +,..., x n), refer to Figure 5. We will how that I( x) + c O O( x) + ww ( x) < I(x ) + c O O(x ) + ww (x ), under the aumption that y > ȳ. Figure 5 Proof of Theorem (iii). cutomer per lot x x xȳ t t t lot index Clearly W (x ) > W ( x). It uffice to how that I( x) + c O O( x) < I(x ) + c O O(x ). Let u = n t=t x t be the number of cutomer aigned to lot t,..., t n under chedule x. Let alo B be the random variable denoting the backlog of cutomer at the beginning of lot t, which i tochatically the ame under both chedule x and x, and let b be realization of B. Then, uing imilar argument a in (7), the expected overtime and idle time cot under chedule x and x, ubcripted by b, are (n t +) b O b (x ) = b + p(u + ) (n t + ) + [(n t + ) b k]f(k; u +, p) (n t +) b O b ( x) = b + pu (n t + ) + [(n t + ) b k]f(k; u, p) I b (x ) = O b (x ) + n pm I b ( x) = O b ( x) + n p(m ). k= k= Therefore I( x) + c O O( x) I(x ) c O O(x ) = p (( + c O ) P ) c O, (8)

24 4 Zacharia and Pinedo: Managing Cutomer Arrival in Service Sytem with Multiple Server where P = (n t +) b Pr(B = b) f(k; u, p) y k= i the probability that the waiting time of the lat cutomer of chedule x i equal to zero. Uing imilar argument a in the proof of Lemma, and by the aumption that n t= x t = m m, we conclude that P < P m. (9) From (7), (8) and (9) it follow that I( x) + c O O( x) < I(x ) + c O O(x ), which i a contradiction. Therefore y ȳ. Proof of Theorem (i) For fixed τ, and for z τn, let H τ (z) = τn k= (τn k)( z k) pk ( p) z k. A in the proof of Theorem, H τ (z) i decreaing and dicretely convex in z on {τn, τn +, τn +...}. Therefore, H τ (z) i decreaing and dicretely convex in z on the ubet {τn, τn + τ, τn + τ,...}, concluding that H τ ( τ(n + y ) ) i decreaing and dicretely convex in y on {,,,...}. Then, note that I(x n,y ) = N τ= τ H τ( τ(n + y ) ), a linear combination of decreaing and dicretely convex function in y convex in y on {,,,...}. on {,,,...}. Therefore, I(x n,y ) i decreaing and dicretely (ii) Recall that O(x n,y ) = I(x n,y ) + p(n + y )N n N, and therefore O(x n,y ) i dicretely convex in y on {,,,...}. For the monotonicity property we need a little more work. Firt note that N O(x n,y ) = τ H ( τ τ(n + y ) ) + p(n + y )N n N. (3) τ= Uing imilar argument a in the proof of Theorem equation (4), we get that for z τn p < H τ (z + ) H τ (z) < p < H τ (z + ) H τ (z + ) < By adding all the inequalitie above we get that Therefore, from (3) and (3) O(x n,y +) O(x n,y ) =. p < H τ (z + τ) H τ (z + τ ) <. pτ < H τ (z + τ) H τ (z) < for z τn. (3) N τ= N > =, [ ( Hτ τ(n + y τ + ) ) ( H τ τ(n + y ) )] + pn τ= τ pτ + pn