Approximate Dynamic Programming for Dynamic Capacity Allocation with Multiple Priority Levels

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1 Aroximate Dynamic Programming for Dynamic Caacity Allocation with Multile Priority Levels Alexander Erdelyi School of Oerations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA Huseyin Toaloglu School of Oerations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA May 21, 2010 Abstract In this aer, we consider a quite general dynamic caacity allocation roblem. There is a fixed amount of daily rocessing caacity. On each day, jobs of different riorities arrive randomly and we need to decide which jobs should be scheduled for which days. Waiting jobs incur a holding cost deending on their riority levels. The objective is to minimize the total exected cost over a finite lanning horizon. We begin by formulating the roblem as a dynamic rogram, but this formulation is comutationally difficult as it involves a high dimensional state vector. To address this difficulty, we use an aroximate dynamic rogramming aroach that decomoses the dynamic rogramming formulation by the different days in the lanning horizon to construct searable aroximations to the value functions. We use the value function aroximations for two uroses. First, we show that the value function aroximations can be used to obtain a lower bound on the otimal total exected cost. Second, the value function aroximations can be used to make the job scheduling decisions over time. Comutational exeriments indicate that the job scheduling decisions made by our aroach erform significantly better than a variety of benchmark strategies.

2 Many comanies face the roblem of allocating limited rocessing caacity to jobs that arrive randomly over time. This is a common roblem in roduction environments, where a comany utilizes a certain roduction resource to serve the demands from customers with different rofit margins. Similarly, clinics and owners of medical diagnostics resources need to allocate aointment slots to atients with different needs and urgency levels. The essential tradeoff in these roblems is between using the available rocessing caacity for a low riority job that is currently available and delaying the rocessing of the low riority job with the hoe that a high riority job can use the available caacity later. This tradeoff is further comlicated by the fact that the rocessing caacity is erishable. If the rocessing caacity that is available on a day is not used by that day, then it ends u being wasted. In this aer, we consider a generic caacity allocation roblem that catures the fundamental tradeoffs mentioned above. We consider a setting where we have a fixed amount of daily rocessing caacity. Jobs of different riorities arrive randomly over time and we need to decide which jobs should be scheduled for which days. The jobs that are waiting to be rocessed incur a holding cost deending on their riority levels and we are interested in finding a job scheduling olicy that minimizes the total exected cost over a finite lanning horizon. It is ossible to formulate such a caacity allocation roblem as a dynamic rogram, but this dynamic rogramming formulation ends u being comutationally difficult to solve as it involves a high dimensional state vector. In this aer, we roose an aroximate dynamic rogramming strategy that is based on decomosing the dynamic rogramming formulation by the different days in the lanning horizon. Our dynamic rogramming decomosition aroach starts with a deterministic linear rogramming formulation for the caacity allocation roblem. This deterministic linear rogram is constructed under the assumtion that the job arrivals take on their exected values and we can schedule fractional numbers of jobs. In the linear rogram, there is one caacity constraint for each day in the lanning horizon, ensuring that the jobs scheduled on a day cannot exceed the daily available caacity. We use Lagrangian relaxation to relax these caacity constraints for every day in the lanning horizon excet for one articular day, so that we obtain a deterministic linear rogram for a caacity allocation roblem with limited caacity only on one day. This latter deterministic linear rogram, in turn, indicates how we can decomose the original dynamic rogramming formulation of the roblem by the different days in the lanning horizon. Ultimately, our dynamic rogramming decomosition strategy rovides searable aroximations to the value functions and these value function aroximations can be used to make the job scheduling decisions. There are several interesting roerties of our dynamic rogramming decomosition method. To begin with, it is ossible to show that our dynamic rogramming decomosition method rovides a lower bound on the otimal total exected cost. Such a lower bound becomes useful when assessing the otimality ga of a subotimal control olicy. It turns out that the deterministic linear rogramming formulation mentioned above also rovides such a lower bound, but the lower bound obtained by our dynamic rogramming decomosition method is rovably tighter than the one obtained by the deterministic linear rogram. Furthermore, the job scheduling decisions made by our aroach can be comuted by solving a min cost network flow roblem. This is a significant result since it allows us to 2

3 obtain integer solutions without imosing integrality requirements on the decision variables. Finally, our comutational exeriments indicate that the job scheduling decisions that are made by our dynamic rogramming decomosition method can rovide significant imrovements over numerous benchmarks based on the deterministic linear rogramming formulation. Two aers in the literature are articularly relevant to our work. First, Patrick et al. (2008) consider the dynamic allocation of aointment slots in a clinic and their roblem is quite similar to ours. They rovide a dynamic rogramming formulation, but motivated by the high dimensionality of the state vector, they use linear aroximations to the value functions. Linear value function aroximations imly that the marginal value of an additional caacity does not deend on how much caacity is left on a articular day. However, it is intuitive to exect that if we have fewer units of caacity left on a articular day, then the marginal value of an additional caacity should increase. The value function aroximations generated by our dynamic rogramming decomosition method are nonlinear and convex, indeed caturing the intuitive exectation that the marginal value of an additional caacity increases as we have fewer units of caacity left on a articular day. Second, Erdelyi and Toaloglu (2009a) use rotection level olicies for caacity allocation roblems. A rotection level olicy is similar to a rioritized first come first serve olicy. As the jobs arrive over time, it commits the caacity starting from the highest riority jobs, but the rotection levels dictate how much caacity should be left untouched for the future job arrivals. The authors visualize the total exected cost over the lanning horizon as a function of the rotection levels and use samled gradients of the total exected cost to search for a good set of rotection levels. The imlicit assumtion behind using a set of rotection levels is that the job arrivals are stationary so that using the same rotection levels every day is exected to erform reasonably well. In contrast, our dynamic rogramming decomosition aroach does not assume or require stationary job arrivals. Dynamic rogramming decomosition ideas first aeared in the network revenue management literature. In network revenue management roblems, an airline oerates a set of flight legs to serve the requests for itineraries that arrive randomly over time. Whenever an itinerary request arrives, the airline has to decide whether to accet or reject the itinerary request. The fundamental tradeoff is between acceting an inexensive itinerary request right now and reserving the caacity for an exensive itinerary request that may arrive in the future. It is ossible to formulate network revenue management roblems as dynamic rograms, but this aroach requires keeing track of the remaining caacity on all of the flight legs and the resulting dynamic rograms end u having high dimensional state vectors. Therefore, it is customary to try to decomose the roblem by the flight legs. Williamson (1992) is one of the first to decomose the network revenue management roblem by the flight legs to generalize the exected marginal seat revenue heuristic of Belobaba (1987). A discussion of early decomosition aroaches is given in Talluri and van Ryzin (2005). Liu and van Ryzin (2008) and Kunnumkal and Toaloglu (2009) use decomosition ideas while incororating customer choice behavior into network revenue management roblems. Zhang and Adelman (2009) are the first to establish that one can use dynamic rogramming decomosition methods to obtain uer bounds on the otimal total exected revenue. Toaloglu (2009) shows that decomosition methods can be visualized as an alication of Lagrangian relaxation to the dynamic rogramming formulation of the network revenue management roblem. Erdelyi and Toaloglu 3

4 (2009b) use decomosition ideas to make overbooking decisions in network revenue management. It is worthwhile to note that our caacity allocation roblem can be visualized as a revenue management roblem as it involves making the most use out of limited and erishable caacity. However, traditional revenue management roblems involve only the decision of whether to accet or reject a job, whereas our caacity allocation roblem involves the decision of when to schedule a job, ossibly along with whether to accet or reject a job. Zhang (2010) develos a dynamic rogramming decomosition method that imroves the uer bounds on the otimal total exected revenue rovided by the methods in Liu and van Ryzin (2008), Kunnumkal and Toaloglu (2009) and Zhang and Adelman (2009). The develoment in Zhang (2010) is for the network revenue management setting and it does not immediately extend to our caacity allocation setting for two reasons. First, it is common in the network revenue management literature to assume that there is at most one customer arrival at each time eriod. Therefore, the random variable that is of interest at each time eriod corresonds to the itinerary that the arriving customer urchases. Such a random variable has a reasonably small number of ossible realizations and one can comute exectations simly by enumerating over all of its ossible realizations. In our caacity allocation setting, the random variable that is of interest on each day corresonds to the numbers of job arrivals at different riority levels. The number of ossible realizations for such a random variable grows exonentially with the number of riority levels. For examle, if we have four riority levels and the number of job arrivals at each riority level can take 100 different values, then number of ossible realizations of job arrivals is on the order of one hundred million. Second, the aroach roosed by Zhang (2010) rovides nonsearable aroximations to the value functions. Nonsearable value function aroximations do not ose a difficulty in the network revenue management setting when finding a set of itineraries to make available for urchase. In contrast, our caacity allocation setting requires jointly deciding which arriving jobs with different riority levels should be scheduled for which days and the searable structure of the value function aroximations that we obtain lays a crucial role in making the job scheduling decisions in a tractable fashion. The idea of decomosing large scale dynamic rograms by using duality arguments has a history in the literature. The links between dynamic rograms and duality are studied in the book chater by Ruszczynski (2003). Adelman and Mersereau (2008) use the term weakly couled to refer to dynamic rograms that would decomose if a few constraints did not link the different comonents of the state vector. They develo a duality framework for such dynamic rograms. Under the broad umbrella of aroximate dynamic rogramming, there is rising interest in aroximate solutions of large scale dynamic rograms. The books by Bertsekas and Tsitsiklis (1996) and Powell (2007) rovide excellent coverage of this work. Although dynamic rogramming decomosition ideas are not covered in these books, our work in this aer and the success of these ideas in the network revenue management literature indicate their caability in dealing with large scale dynamic rograms. With rising interest in health care alications, there are a several dynamic caacity allocation roblems that originate within this roblem domain. The aer by Patrick et al. (2008) that we describe above is one such work. Gerchak et al. (1996) focus on a setting where limited oerating room 4

5 caacity has to be allocated between emergency and elective surgery. They characterize the structure of the otimal olicy and show that rotection level olicies may not be otimal. Guta and Wang (2008) resent a model that considers customer choice behavior when the customers choose aointment slots. Similar to Gerchak et al. (1996), these authors also indicate that the form of the otimal olicy may have comlicated structure and they roose aroximation strategies. In this aer, we make the following research contributions. 1) We develo a method to make job scheduling decisions in a caacity allocation model with jobs of different riorities. Our aroach decomoses the roblem by the different days in the lanning horizon and solves a sequence of dynamic rograms with scalar state variables. 2) We show that our aroach rovides a lower bound on the otimal total exected cost and this lower bound is at least as tight as the one rovided by a deterministic linear rogramming formulation. 3) Comutational exeriments demonstrate that our method erforms significantly better than benchmark strategies based on linear rogramming formulations. The rest of the aer is organized as follows. In Section 1, we formulate the caacity allocation roblem as a dynamic rogram. This formulation is comutationally difficult due to the size of the state vector. In Section 2, we give a deterministic linear rogram, which serves as a starting oint for our decomosition strategy. In Section 3, we decomose the dynamic rogramming formulation of the roblem by building on the deterministic linear rogram. In Section 4, we demonstrate how to overcome some comutational hurdles in our dynamic rogramming decomosition method. In Section 5, we dwell on how we can aly the olicy obtained by our dynamic rogramming decomosition method. In Section 6, we rovide comutational exeriments. In Section 7, we conclude. 1 Problem Formulation We have a fixed amount of daily rocessing caacity. At the beginning of each day, we observe the arrivals of jobs of different riorities and we need to decide which jobs should be scheduled for which days. The jobs that are waiting to be rocessed incur a holding cost deending on their riority levels, but we also have the otion of rejecting a job immediately by incurring a enalty cost. The objective is to minimize the total exected cost over a finite lanning horizon. The roblem takes lace over the set of days T = {1,..., τ}. The set of ossible riority levels for the jobs is P. The number of riority jobs that arrive on day t is given by the random variable D t so that D t = {D t : P} catures the job arrivals on day t. We assume that the job arrivals on different days are indeendent. While such an assumtion is reasonable in many situations, there can be some settings where it would be roblematic to assume that the job arrivals on different days are indeendent. For examle, if the jobs that are rejected come back to the system, then there can be correlations among the job arrivals on different days. Even in such cases, if there are few rejected jobs that come back to the system when comared with the fresh job arrivals or if the rejected jobs get service elsewhere, then it can be reasonable to assume that the job arrivals on different days are indeendent. Furthermore, the quality of service that we rovide on earlier days may influence the job arrivals in the future, creating correlations among the job arrivals on different days, but such an involved stochastic rocess for the job arrivals is not within the scoe of our model. If we schedule a 5

6 riority job that arrives on day t for day j, then we incur a holding cost of h jt. The enalty cost of rejecting a riority job that arrives on day t is r t. We use the convention that h jt = whenever it is infeasible to schedule a riority job arriving on day t for day j. For examle, since it is infeasible to schedule a job for a day in the ast, we naturally have h jt = whenever j < t. Similarly, if it is not ossible to schedule a job more than S days into the future, then we can cature this constraint by letting h jt = whenever j t S. Once we schedule a job for a articular day, this decision is fixed and cannot be changed. For notational brevity, we assume that all jobs consume one unit of rocessing caacity, but it is ossible to extend our aroach to multile units of caacity consumtion. Given that we are on day t, we let x jt be the remaining caacity on day j so that x t = {x jt : j T } catures the state of the remaining caacities as observed at the beginning of day t. Naturally, given that we are on day t, we do not need to kee track of the remaining caacities on the days in the ast and we can use x t = {x jt : j = t, t + 1,..., τ} as our state variable. However, we refer to use x t = {x jt : j T } for notational uniformity so that the state variable on different days ends u having the same number of dimensions. We let u jt be the number of riority jobs that we schedule for day j on day t so that u t = {u jt : P, j T } catures the decisions that we make on day t. In this case, the set of feasible decisions on day t is given by { U(x t, D t ) = u t Z P T + : u jt x } jt j T, u jt D t P, (1) P where the first set of constraints ensure that the decisions that we make do not violate the remaining caacities and the second set of constraints ensure that the decisions that we make do not violate the job arrivals. On the other hand, the total cost that we incur on day t is given by h jt u jt + [ r t D t ] u jt, j T P P j T where the first term corresonds to the holding cost for the jobs that we schedule for different days in the lanning horizon and the second term corresonds to the enalty cost for the jobs that we reject. We note that without loss of generality, it is ossible to assume that r t = 0 for all P, t T. To see this, we write the cost function above as [ j T P h jt ] r t u jt + P r t D t. Since P r t D t is a constant that is indeendent of the decisions, assuming that r t = 0 for all P, t T is equivalent to letting c jt = h jt r t and working with the holding cost c jt instead of h jt. By working with the holding cost c jt = h jt r t, we a riori assume that all of the job arrivals on a day are rejected and immediately charge the enalty cost P r t D t, but for each riority job that we schedule for day j, we reimburse the enalty cost r t back and charge the holding cost h jt instead. Throughout the aer, we indeed assume that r t = 0 for all P, t T and work with the holding costs {c jt : P, j, t T }. We use V t (x t ) to denote the minimum ossible total exected cost over days {t,..., τ} given that the state of the remaining caacities on day t is x t. Letting e j be the T dimensional unit vector with a one in the element corresonding to j T, we can comute the value functions {V t ( ) : t T } by solving the otimality equation { V t (x t ) = E min u t U(x t,d t) j T { c jt u jt + V t+1(x t }} u jt e j) j T P j T P (2) 6

7 with the boundary condition that V τ+1 ( ) = 0. The otimal total exected cost over the whole lanning horizon is V 1 (x 1 ), where x 1 catures the initial state of the caacities and it is a art of the roblem data. The knowledge of {V t ( ) : t T } can be used to find the otimal decisions on each day. In articular, if the state of the remaining caacities and the job arrivals on day t are resectively given by x t and D t, then we can solve the otimization roblem inside the exectation in (2) to find the otimal decisions on this day. Unfortunately, the number of dimensions of the state vector x t is equal to the number of days, which can easily be on the order of hundreds for ractical alications. Furthermore, solving the otimality equation in (2) requires taking an exectation over the job arrivals on each day and this exectation can be difficult to comute. These considerations render finding the exact solution to the otimality equation in (2) intractable. In the next two sections, we give two ossible aroaches for finding aroximate solutions to the otimality equation in (2). The first aroach involves solving a deterministic linear rogram, whereas the second aroach builds on this deterministic linear rogram to construct searable aroximations to the value functions. 2 Deterministic Linear Programming Formulation One aroach for finding aroximate solutions to the otimality equation in (2) involves solving a deterministic linear rogram that is formulated under the assumtion that the job arrivals take on their exected values and it is ossible to schedule fractional numbers of jobs for different days. In articular, using the decision variables {u jt : P, j T, t T } with the same interretation as in the revious section, we can solve the roblem min subject to c jt u jt (3) t T j T P u jt x j1 j T (4) t T P j T u jt u jt E{D t } P, t T (5) 0 P, j T, t T (6) to aroximate the otimal total exected cost over the lanning horizon. Constraints (4) in the roblem above ensure that the decisions that we make over the lanning horizon do not violate the remaining caacity on each day, whereas constraints (5) ensure that the decisions that we make on each day do not violate the exected numbers of job arrivals. One imortant use of roblem (3)-(6) occurs when we want to obtain lower bounds on the otimal total exected cost. In articular, letting zlp be the otimal objective value of roblem (3)-(6), the next roosition shows that we have zlp V 1(x 1 ) so that the otimal objective value of roblem (3)-(6) rovides a lower bound on the otimal total exected cost. Such lower bounds become useful when we want to gain insight into the otimality ga of any subotimal control olicy. 7

8 Proosition 1 We have z LP V 1(x 1 ). Proof Our roof is standard, but it allows us to demonstrate a useful inequality. Given that the job arrivals over the lanning horizon are D = {D t : P, t T }, we let π (D) be the total cost incurred by the otimal olicy and zlp (D) be the otimal objective value of roblem (3)-(6) that is obtained after relacing the right side of constraints (5) with {D t : P, t T }. Relacing the right side of constraints (5) with {D t : P, t T } and solving roblem (3)-(6) corresonds to making the decisions after observing all of the job arrivals over the whole lanning horizon. Furthermore, roblem (3)-(6) allows scheduling fractional numbers of jobs. On the other hand, the otimal olicy makes the decisions for a articular day after observing the job arrivals only on that day. Furthermore, the otimal olicy schedules integer numbers of jobs. Therefore, it holds that z LP (D) π (D). Taking exectations and noting that zlp (D) is a convex function of D, Jensen s inequality imlies that E{π (D)} E{z LP (D)} z LP (E{D}) = z LP, (7) where the equality follows from the definition of z LP. The result follows from the fact that E{π (D)} gives the total exected cost incurred by the otimal olicy and is equal to V 1 (x 1 ). Therefore, (7) imlies that both z LP and E{z LP (D)} rovide lower bounds on the otimal total is looser, but this lower bound can be comuted simly exected cost. The lower bound rovided by zlp by solving roblem (3)-(6). On the other hand, the lower bound rovided by E{zLP (D)} is tighter, but there is no closed form exression for the exectation in E{zLP (D)} and comuting this exectation requires estimation through Monte Carlo samles. Nevertheless, our comutational exeriments indicate that this extra comutational burden usually ays off and the lower bound rovided by E{zLP (D)} can be significantly tighter than the lower bound rovided by zlp. Another imortant use of roblem (3)-(6) occurs when we want to construct an aroximate olicy to make the decisions on each day. In articular, letting {µ j : j T } be the otimal values of the dual variables associated with constraints (4), we can use µ j to estimate the value of a unit of remaining caacity on day j. This idea suggests using the linear function j T µ j x jt as an aroximation to the value function V t (x t ). In this case, if the state of the remaining caacities and the job arrivals on day t are resectively given by x t and D t, then we can relace V t+1 (x t j T P u jt e j) on the right side of (2) with j T j[ µ xjt P jt] u and solve the roblem { min c jt u t U(x t,d t ) u jt + j T P j T µ j [ x jt P u jt] } (8) to make the decisions on day t. We refer to this decision rule as the LVF decision rule, standing for linear value functions. We use the LVF decision rule as a benchmark strategy later in our comutational exeriments. An immediate shortcoming of the LVF decision rule is that it constructs the value function aroximations by only using exected values of the job arrivals. In the next section, we build on roblem (3)-(6) to construct a value function aroximation strategy that addresses the random nature of the job arrivals more carefully. 8

9 3 Dynamic Programming Decomosition In this section, we develo a method that constructs searable aroximations to the value functions by decomosing the otimality equation in (2) into a sequence of otimality equations with scalar state variables. We begin with a duality argument on roblem (3)-(6). We let {µ j : j T } be the otimal values of the dual variables associated with constraints (4) in roblem (3)-(6). We choose an arbitrary day i in the lanning horizon and relax constraints (4) for all of the other days by associating the dual multiliers {µ j : j T \{i}} with them. In this case, letting 1( ) be the indicator function, the objective function of roblem (3)-(6) can be written as c jt u jt + [ µ j x j1 ] u jt t T j T P j T \{i} t T P = [ c jt 1(j i) ] µ j u jt + µ j x j1. t T j T P j T \{i} Therefore, by linear rogramming duality, the otimal objective value of roblem (3)-(6) is the same as the otimal objective value of the roblem z LP = min subject to t T j T P [ c jt 1(j i) ] µ j u jt + j T \{i} µ j x j1 (9) u it x i1 (10) t T P j T u jt u jt E{D t } P, t T (11) 0 P, j T, t T. (12) Comaring roblem (9)-(12) with roblem (3)-(6) and ignoring the constant term j T \{i} µ j x j1 in the objective function, we observe that roblem (9)-(12) is the deterministic linear rogramming formulation for a caacity allocation roblem where there is limited caacity only on day i and the caacities on all of the other days are infinite. In this caacity allocation roblem, if we schedule a riority job that arrives on day t for day j, then we incur a holding cost of c jt 1(j i) µ j. We refer to this roblem as the caacity allocation roblem focused on day i. Thus, the discussion in this aragrah and Proosition 1 imly that z LP j T \{i} µ j x j1 rovides a lower bound on the otimal total exected cost in the caacity allocation roblem focused on day i. On the other hand, if there is limited caacity only on day i and the caacities on all of the other days are infinite, then the set of feasible decisions can be written as { U i (x it, D t ) = u t Z P T + : u it x it, u jt D t P j T P }. (13) The definition above is similar to the one in (1), but it only ays attention to the remaining caacity on day i. In this case, if there is limited caacity only on day i and the holding cost of scheduling a riority job that arrives on day t for day j is c jt 1(j i) µ j, then we can obtain the otimal total 9

10 exected cost by comuting the value functions {v it ( ) : t T } through the otimality equation { { [ v it (x it ) = E min c jt 1(j i) ] u t U i (x it,d µ j u jt + v i,t+1(x it }} u it ) t) j T P P (14) with the boundary condition that v i,τ+1 ( ) = 0. The otimality equation in (14) is similar to the one in (2), but it kees track of the remaining caacity only on day i. This is emhasized by the subscrit i in the value functions. The discussion in the revious aragrah imlies that zlp j T \{i} µ j x j1 v i1 (x i1 ). Furthermore, the next roosition shows that v i1 (x i1 ) + j T \{i} µ j x j1 V 1 (x 1 ) and we obtain the chain of inequalities zlp v i1 (x i1 ) + µ j x j1 V 1 (x 1 ). j T \{i} Therefore, we can solve the otimality equation in (14) to obtain a lower bound on the otimal total exected cost in the original caacity allocation roblem and this lower bound is at least as tight as the one rovided by the otimal objective value of roblem (3)-(6). Proosition 2 We have v it (x it ) + j T \{i} µ j x jt V t (x t ) for all i T, t T. Proof Since we have v i,τ+1 ( ) = V τ+1 ( ) = 0 for all i T and µ j 0 for all j T by dual feasibility to roblem (3)-(6), the result holds for day τ + 1. We assume that the result holds for day t + 1 and let {û jt : P, j T } be an otimal solution to the otimization roblem inside the exectation in (2), ϑ t (x t, D t ) be the otimal objective value of this roblem and ϑ it (x it, D t ) be the otimal objective value of the otimization roblem inside the exectation in (14). We have ϑ t (x t, D t ) = c jt û jt + V t+1(x t û jt e j) j T P j T P c jt û jt + v i,t+1(x it û it ) + [ µ j x jt ] û jt j T P P j T \{i} P = [ c jt 1(j i) ]û µ j jt + v i,t+1(x it û it ) + µ j x jt ϑ it (x it, D t ) + j T P P j T \{i} j T \{i} µ j x jt, where the first inequality follows from the induction assumtion and the second inequality follows from the fact that {û jt : P, j T } is a feasible but not necessarily an otimal solution to the otimization roblem inside the exectation in (14). The result follows by taking exectations in the chain of inequalities above and noting that V t (x t ) = E{ϑ t (x t, D t )} and v it (x it ) = E{ϑ it (x it, D t )}. Proosition 2 suggests using the lower bound v it (x it ) + j T \{i} µ j x jt as an aroximation to V t (x t ). In this aroximation, given that we are on day t, the comonent v it (x it ) catures the value of the remaining caacity on day i, whereas the comonent j T \{i} µ j x jt catures the values of the remaining caacities on days other than day i. We note that the comonent j T \{i} µ j x jt is somewhat trivial in the sense that it exactly corresonds to how the LVF decision rule assesses the values of the remaining caacities on days other than day i. To obtain more sohisticated estimates for the values of 10

11 the remaining caacities on days other than day i, we roose comuting {v it ( ) : t T } for all i T. In this case, given that we are on day t, the comonent v it (x it ) catures the value of the remaining caacity on day i and we can use i T v it(x it ) as an aroximation to V t (x t ). In the next section, we resolve the comutational issues that are related to solving the otimality equation in (14). 4 Solving the Otimality Equation The value functions {v it ( ) : t T } comuted through the otimality equation in (14) involve a scalar state variable. Therefore, it is tractable to store these value functions. However, solving the otimality equation in (14) still requires dealing with an otimization roblem with P T decision variables and comuting an exectation that involves the high dimensional random variable D t, which are nontrivial tasks. In this section, we resolve the comutational issues that are related to solving the otimality equation in (14). We begin with the next roosition, which shows that the otimization roblem inside the exectation in (14) can be solved efficiently. Proosition 3 We can solve the otimization roblem inside the exectation in (14) by a sort oeration. Proof As mentioned above, the otimality equation in (14) assumes that there is limited caacity only on day i. Therefore, there are three things we can do with a job. We can schedule it for day i, schedule it for a day other than day i or reject it. If we decide to schedule the job for a day other than day i or reject it, then we simly follow the otion with the smallest cost since there is no limit on the number of jobs we can schedule for a day other than day i or reject. To make the idea recise, we let K = max j T {x j1 } so that the remaining caacities never exceed K. By using induction over the days of the lanning horizon, we can show that {v it ( ) : t T } are convex in the sense that 2 v it (k) v it (k + 1) + v it (k 1) for all k = 1,..., K 1, t T. Thus, letting K = {0,..., K 1} and k i,t+1 = v i,t+1(k +1) v i,t+1 (k) for all k K, we associate the decision variables {w k i,t+1 : k K} with the first differences { k i,t+1 : k K} and write the otimization roblem inside the exectation in (14) as min subject to j T P [ c jt 1(j i) ] µ j u jt + k i,t+1 wi,t+1 k + v i,t+1 (0) k K u it + P k K u jt + y t = D t j T w k i,t+1 1 w k i,t+1 = x it P k K u jt, y t, wk i,t+1 Z + P, j T, k K, where we use the slack variables {y t : P} in the second set of constraints above. For a articular riority level, we consider the decision variables {u jt : j T \ {i}} and y t in the roblem above. As far as the constraints are concerned, these decision variables aear only in the second set of constraints. Therefore, if any one of the decision variables {u jt : j T \ {i}} 11

12 and y t takes a nonzero value in the otimal solution, then this decision variable has to be the one with the smallest objective function coefficient. This imlies that we can relace all of these decision variables with a single decision variable, say z t, and the objective function coefficient of z t would be equal to the smallest of the objective function coefficients of the decision variables {u jt : j T \ {i}} and y t. Since the objective function coefficients of the decision variables {u jt : j T \ {i}} are {c jt µ j : j T \ {i}} and the objective function coefficient of the decision variable y t is zero, letting ĉ t = min{min{c jt µ j : j T \ {i}}, 0}, the last roblem above becomes min c it u it + ĉ t z t + k i,t+1 wi,t+1 k + v i,t+1 (0) (15) P P k K subject to wi,t+1 k = x it (16) P u it + k K u it + z t = D t P (17) w k i,t+1 1 k K (18) u it, z t, wk i,t+1 Z + P, k K. (19) We note that the decision variable z t corresonds to the least cost otion for a riority job among the otions of scheduling this job for a day other than day i and rejecting it. Using constraints (17), we write z t = D t u it for all P and lug them into the objective function to obtain c it u it + ĉ [ t D t ] u it + k i,t+1 wi,t+1 k + v i,t+1 (0) P P k K = [ c it ] ĉ t u it + ĉ t D t + k i,t+1 wi,t+1 k + v i,t+1 (0). P P k K Therefore, droing the constant term P ĉ t D t + v i,t+1(0) from the objective function, roblem (15)-(19) can be written as min subject to P [ c it ] ĉ t u it + k i,t+1 wi,t+1 k (20) k K P u it + k K w k i,t+1 = x it (21) u it D t P (22) w k i,t+1 1 k K (23) u it, wk i,t+1 Z + P, k K. (24) Problem (20)-(24) is a knasack roblem with each item consuming one unit of caacity. The items are indexed by P and k K. The (dis)utilities of items P and k K are resectively c it ĉ t and k i,t+1. The caacity of the knasack is x it. We can ut at most D t units of item P and one unit of item k K into the knasack. The result follows by the fact that a knasack roblem with each item consuming one unit of caacity can be solved by sorting the objective function coefficients and filling the knasack starting from the item with the smallest objective function coefficient. Another difficulty with the otimality equation in (14) is that we need to comute the exectation of the otimal objective value of the otimization roblem inside the first set of curly brackets. The roof 12

13 of Proosition 3 shows that this otimization roblem is equivalent to roblem (20)-(24), which is, in turn, a knasack roblem where each item consumes one unit of caacity and there is a random uer bound on how many units of a articular item we can ut into the knasack. Powell and Cheung (1994) derive a closed form exression for the exectation of the otimal objective value of such a knasack roblem. We briefly describe how their result relates to our setting. To reduce the notational clutter, we note that for aroriate choices of the set of items N = {1,..., N}, (dis)utilities {β n : n N }, integer valued knasack caacity Q and integer valued random uer bounds α = {α n : n N }, roblem (20)-(24) is a knasack roblem of the form min β n z n (25) subject to n N z n = Q (26) n N z n α n n N (27) z n Z + n N. (28) Using ζ(α) to denote the otimal objective value of the roblem above as a function of α, we are interested in comuting E{ζ(α)}. For a given realization of α, we can solve roblem (25)-(28) by sorting the objective function coefficients of the decision variables and filling the knasack starting from the item with the smallest objective function coefficient. We assume that β 1 β 2... β N, in which case it is otimal to start from the item with the smallest index. We let φ(n, q) be the robability that the nth item uses the qth unit of the available knasack caacity in the otimal solution. If we know φ(n, q) for all n N, q = 1,..., Q, then we can comute the exectation of the otimal objective value of roblem (25)-(28) as Q E{ζ(α)} = β n φ(n, q). q=1 n N Comuting φ(n, q) turns out to be not too difficult. Since the otimal solution fills the knasack starting from the item with the smallest index, for the nth item to use the qth unit of caacity in the knasack, the total caacity consumed by the first n 1 items should be strictly less than q and the total caacity consumed by the first n items should be greater than or equal to q. Therefore, we have φ(n, q) = P{α α n 1 < q α α n } and we can comute φ(n, q) as long as we can comute the convolutions of the distributions of {α n : n N }. The discussion in the revious two aragrahs rovides a method to comute the exectation of the otimal objective value of the otimization roblem inside the first set of curly brackets in (14). In the next section, we describe how we can use the value functions comuted through the otimality equation in (14) to make the job scheduling decisions on each day. 5 Alying the Greedy Policies At the end of Section 3, we roose using j T v jt(x jt ) as an aroximation to V t (x t ). In this case, if the state of the remaining caacities and the job arrivals on day t are resectively given by x t and D t, then 13

14 we can relace V t+1 (x t j T P u jt e j) on the right side of (2) with j T v j,t+1(x jt P u jt ) and solve the roblem { min c jt u u jt + v j,t+1 (x jt } u jt ) (29) t U(x t,d t) j T P j T P to make the decisions on day t. We refer to this decision rule as the DPD decision rule, standing for dynamic rogramming decomosition. The roblem above involves P T decision variables. The next roosition shows that we can efficiently solve this roblem as a min cost network flow roblem. Proosition 4 We can solve roblem (29) as a min cost network flow roblem. Proof As mentioned in the roof of Proosition 3, it is ossible to show that {v jt ( ) : j T, t T } are convex functions. In this case, letting K and k j,t+1 be defined as in the roof of Proosition 3, we associate the decision variables {wj,t+1 k : k K, j T } with the first differences { k j,t+1 : k K, j T } and write roblem (29) as min c jt u jt + k j,t+1 wj,t+1 k + v j,t+1 (0) j T P j T k K j T subject to u jt + wj,t+1 k = x jt j T P k K u jt D t P j T wj,t+1 k 1 k K, j T u jt, wk j,t+1 Z + P, k K, j T. We define the new decision variables {y t : P} as y t = j T u jt for all P, in which case the roblem above becomes min c jt u jt + k j,t+1 wj,t+1 k + v j,t+1 (0) (30) j T P j T k K j T subject to u jt + wj,t+1 k = x jt j T (31) P k K u jt y t = 0 P (32) j T y t D t P (33) wj,t+1 k 1 k K, j T (34) u jt, y t, wk j,t+1 Z + P, k K, j T. (35) Ignoring the constant term j T v j,t+1(0) in the objective function, the roblem above is a min cost network flow roblem. To see this, we consider a network comosed of two sets of nodes N 1 = {j T } and N 2 = { P}, along with a sink node. The decision variable u jt corresonds to an arc connecting node j N 1 to node N 2, the decision variable y t corresonds to an arc connecting node N 2 to 14

15 the sink node and the decision variable wj,t+1 k corresonds to an arc connecting node j N 1 to the sink node. The suly of node j N 1 is x jt and constraints (31) are the flow balance constraints for the nodes in N 1. The suly of node N 2 is zero and constraints (32) are the flow balance constraints for the nodes in N 2. The flow balance constraint for the sink node is redundant and omitted. Constraints (33) and (34) act as simle uer bounds on the arcs. Figure 1 shows the structure of the network for the case where T = {1, 2}, P = {a, b} and K = {l, m}. Since roblem (29) can be solved as a min cost network flow roblem, its continuous relaxation naturally rovides integer solutions. Therefore, we can solve roblem (29) as a linear rogram. 6 Comutational Exeriments In this section, we numerically comare the erformance of the searable value function aroximations constructed through the method in Section 3 with a number of benchmark strategies. 6.1 Exerimental Setu We resent two sets of comutational exeriments. In the first set of comutational exeriments, we generate a base roblem and modify its certain attributes to obtain test roblems with different characteristics. In the base roblem, the number of days in the lanning horizon is T = 100 and the set of riority levels is P = {1, 2, 3}. We assume that it is not ossible to schedule a job more than S days into the future so that h jt = whenever we have j t S. We use the holding costs h jt = φ 1.25 j t and the enalty costs r t = β φ 1.25 S 1 for all P, t T and t j t + S 1. We note that the riority level one has the lowest riority since it incurs the lowest holding and enalty costs. In the base roblem, we use S = 7, φ = 2 and β = 5. The exected numbers of daily job arrivals for the first, second and third riority levels are resectively 40, 20 and 10 and the coefficient of variation for the numbers of daily job arrivals is CV = 0.3. The daily rocessing caacity is 70 so that the daily exected job arrivals match the daily rocessing caacity. In the second set of comutational exeriments, we work with test roblems whose cost structures are insired by the health care alication in Patrick et al. (2008). In articular, we assume that there is a target duration for each riority level. If we can schedule a job within the target duration corresonding to its riority level, then we do not incur a holding cost. Otherwise, we incur a holding cost of f for each day that we run over the target duration for a riority job. Similar to the first set of comutational exeriments, we assume that it is not ossible to schedule a job more than S days into the future. Therefore, using b to denote the target duration for a riority job, the holding costs are of the form if j t 1 h jt = 0 if t j t + b 1 (j t b + 1) f if t + b j t + S 1 if t + S j for all P, j, t T. In the exression above, the first case corresonds to scheduling a job for a day in the ast and the fourth case corresonds to scheduling a job more than S days into the future, which 15

16 are both infeasible. The second case corresonds to scheduling a job within its target duration and the third case corresonds to scheduling a job beyond its target duration. The enalty costs are given by r t = (S b + 1) f for all P, t T. Throughout the second set of comutational exeriments, the set of riority levels is P = {1, 2, 3} and we use T = 100 and S = 11. Similar to the first set of comutational exeriments, the exected numbers of daily job arrivals for the first, second and third riority levels are resectively 40, 20 and 10 and the coefficient of variation for the numbers of daily job arrivals is CV = 0.3. We generate six test roblems by using different combinations for f = (f 1, f 2, f 3 ) and b = (b 1, b 2, b 3 ). In Table 1, we show the values that we use for f and b to generate the six test roblems in the second set of comutational exeriments. 6.2 Benchmark Strategies We comare the DPD decision rule introduced in Section 5 with three benchmark strategies. Our first benchmark strategy is the LVF decision rule that we describe at the end of Section 2. In contrast to the basic version of this benchmark strategy, we test the erformance of a dynamic version that udates the values of dual variables {µ j : j T } eriodically over time. In articular, we divide the lanning horizon into M equal segments and resolve roblem (3)-(6) on days t m = 1 + (m 1)τ/M for m = 1,..., M. On day t m, we relace the set of days T in roblem (3)-(6) with T m = {t m,..., τ} and the right side of constraints (4) with the current remaining caacities {x j,tm : j T m }. In this way, we only focus on the remaining ortion of the lanning horizon with the current values for the remaining caacities. Letting {µ j : j T m} be the otimal values of the dual variables associated with constraints (4), we use these dual variables in roblem (8) until we reach the beginning of the next segment and solve roblem (3)-(6) again. We use M = 5 in all of our test roblems. To motivate our second benchmark strategy, we note that the LVF decision rule uses the dual solution of roblem (3)-(6). Naturally, it is ossible to use the rimal solution to roblem (3)-(6) to make the job scheduling decisions. In articular, given that we are on day t, we relace the set of days T in roblem (3)-(6) with {t,..., τ} and the right side of constraints (4) with the current remaining caacities {x jt : j = t,..., τ}. Furthermore, we relace the right side of constraints (5) for day t with the current realizations of the job arrivals. For the other days, we continue using the exected values of the job arrivals. In this way, we only focus on the remaining ortion of the lanning horizon with the current values for the remaining caacities and the current realization of the job arrivals for day t. After these modifications, we solve roblem (3)-(6) and letting {û jt : P, j, t = t,..., τ} be an otimal solution, we aly the decisions given by {û jt : P, j = t,..., τ} on day t. We refer to this benchmark strategy as the PLP decision rule, standing for rimal linear rogram. The third benchmark strategy is a simle first come first serve olicy. In articular, this benchmark strategy schedules the job arrivals on a articular day starting from the jobs with the highest riority, without reserving caacity for the future job arrivals. We do not exect this benchmark strategy to erform well, but we use it to demonstrate the benefit from carefully reserving caacity for the future job arrivals. We refer to this benchmark strategy as the FC decision rule, standing for first come. 16

17 6.3 Comutational Results We begin by resenting our results for the first set of comutational exeriments. In articular, Table 2 shows the results for the base roblem in the first set of comutational exeriments. The second column in this table gives the lower bound on the otimal total exected cost rovided by the otimal objective value of roblem (3)-(6). The third column gives the lower bound on the otimal total exected cost rovided by the exectation of the erfect hindsight total cost. Using the notation in the roof of Proosition 1, the two lower bounds in the second and third columns resectively corresond to zlp (E{D}) and E{z LP (D)}. Proosition 2 shows that the DPD decision rule also rovides a lower bound on the otimal total exected cost, but this lower bound turns out to be only marginally better than the one rovided by the otimal objective value of roblem (3)-(6) and we do not reort this bound. The fourth, fifth, sixth and seventh columns in Table 2 resectively give the total exected costs incurred by the DPD, LVF, PLP and FC decision rules. We estimate these total exected costs by simulating the erformances of the different benchmark strategies under 100 job arrival trajectories with common random numbers. The eighth, ninth, tenth and eleventh columns give the ercentage of jobs rejected by the four benchmark strategies. The twelfth column gives the ercent ga between the total exected costs incurred by the DPD and LVF decision rules. Similarly, the thirteenth and fourteenth columns give the ercent erformance gas between the DPD decision rule and the remaining two benchmark strategies. The last column gives the ercent ga between the total exected cost incurred by the DPD decision rule and the tightest of the lower bounds reorted in the second and third columns. Thus, the last column gives an uer bound on the otimality ga of the DPD decision rule. Table 2 suggests that the DPD decision rule erforms significantly better than all of the benchmark strategies for the base roblem. The erformance gas between the DPD decision rule and the two linear rogramming based decision rules is 4 to 8%. It is interesting that although the lower bound rovided by the DPD decision rule does not imrove the one rovided by roblem (3)-(6) by much, the olicy rovided by the DPD decision rule erforms significantly better than the olicies that are based on roblem (3)-(6). The FC decision rule erforms substantially worse, with a erformance ga of about 18%. In the roof of Proosition 1, we show that zlp (E{D}) E{z LP (D)} and the second and third columns indicate that the ga between these two lower bounds can be significant. Therefore, estimating the lower bound E{zLP (D)} by using Monte Carlo samles can be worth the effort. In Tables 3 through 8, we vary different attributes of the base roblem to generate test roblems with different characteristics. The first columns in these tables indicate the arameter that we modify and its value. The interretations of other columns remain the same as in Table 2. In Table 3, we begin by varying the number of riority levels. The DPD decision rule erforms consistently better than all of the benchmark strategies. The erformance ga between the DPD and PLP decision rules tend to decrease with the increasing number of riority levels, but the erformance ga is still about 5% even when we have five riority levels. The FC decision rule rejects the smallest number of jobs, but this does not translate into better erformance. In Table 4, we vary how far we are allowed to look into the future when scheduling the jobs. The DPD decision rule continues to rovide imrovements over the other benchmark strategies. For the 17

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