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Tactical Planning in Healthcare with Approximate Dynamic Programming Martijn Mes & Peter Hulshof Department of Industrial Engineering and Business Information Systems University of Twente The Netherlands Sunday, October 6, 2013 INFORMS Annual Meeting 2013, Minneapolis, MN OUTLINE 1. Introduction 2. Problem formulation 3. Solution approaches Integer Linear Programming Dynamic Programming Approximate Dynamic Programming 4. Our approach 5. Numerical results 6. Managerial implications 7. What to remember INFORMS Annual Meeting 2013 2/30 INTRODUCTION Healthcare providers face the challenging task to organize their processes more effectively and efficiently Growing healthcare costs (12% of GDP in the Netherlands) Competition in healthcare Increasing power from health insures Our focus: integrated decision making on the tactical planning level: Patient care processes connect multiple departments and resources, which require an integrated approach. Operational decisions often depend on a tactical plan, e.g., tactical allocation of blocks of resource time to specialties and/or patient categories (master schedule / block plan). Care process: a chain of care stages for a patient, e.g., consultation, surgery, or a visit to the outpatient clinic INFORMS Annual Meeting 2013 3/30 CONTROLLED ACCESS TIMES Tactical planning objectives: 1. Achieve equitable access and treatment duration. 2. Serve the strategically agreed target number of patients. 3. Maximize resource utilization and balance the workload. We focus on access times, which are incurred at each care stage in a patient’s treatment at the hospital. Controlled access times: To ensure quality of care for the patient and to prevent patients from seeking treatment elsewhere. Payments might come only after patients have completed their health care process. INFORMS Annual Meeting 2013 4/30 TACTICAL PLANNING AT HOSPITALS IN OUR STUDY Typical setting: 8 care processes, 8 weeks as a planning horizon, and 4 resource types. Current way of creating/adjusting tactical plans: In biweekly meeting with decision makers. Using spreadsheet solutions. Our model provides an optimization step that supports rational decision making in tactical planning. Care pathway Knee Knee Hip Hip Shoulder Shoulder Stage 1. Consultation 2. Surgery 1. Consultation 2. Surgery 1. Consultation 2. Surgery Week 1 5 6 2 10 4 2 Week 2 10 5 7 0 9 8 Patient admission plan Week 3 Week 4 Week 5 12 11 9 10 12 11 4 6 8 7 0 1 7 8 8 5 5 7 Week 6 5 9 3 4 9 10 INFORMS Annual Meeting 2013 Week 7 4 5 2 4 3 8 5/30 PROBLEM FORMULATION [1/2] Discretized finite planning horizon 𝑡 ∈ 1,2, … , 𝑇 Patients: Set of patient care processes 𝑔 ∈ {1,2, … , 𝐺} Each care process consists of a set of stages 1,2, … , 𝑒𝑔 A patient following care process 𝑔 follows the stages 𝐾𝑔 = 𝑔, 1 , 𝑔, 2 , … , (𝑔, 𝑒𝑔 ) Resources: Set of resource types 𝑟 ∈ {1,2, … , 𝑅} Resource capacities 𝜂𝑟,𝑡 per resource type and time period To service a patient in stage 𝑗 = (𝑔, 𝑎) of care process 𝑔 requires 𝑠𝑗,𝑟 of resource 𝑟 From now on, we denote each stage in a care process by a queue 𝑗. INFORMS Annual Meeting 2013 6/30 PROBLEM FORMULATION [2/2] After service in queue i, we have a probability 𝑞𝑖,𝑗 that the patient is transferred to queue j. Probability to leave the system: qi,0 = 1 − 𝑗∈𝐾𝑔 𝑞i,j Newly arriving patients joining queue i: 𝜆𝑖,𝑡 For each time period, we determine a patient admission plan: 𝑥𝑡,𝑗 = 𝑥𝑡,𝑗,0 , 𝑥𝑡,𝑗,1 , … , where 𝑥𝑡,𝑗,𝑢 indicates the number of patients to serve in time period t that have been waiting precisely u time periods at queue j. Waiting list: 𝑆𝑡,𝑗 = 𝑆𝑡,𝑗,0 , 𝑆𝑡,𝑗,1 , … Time lag 𝑑𝑖,𝑗 between service in i and entrance to j (might be medically required to recover from a procedure). Total patients entering queue j: 𝑆𝑡,𝑗,0 = 𝜆𝑗,𝑡 + 𝑞𝑖,𝑗 𝑥𝑡−𝑑𝑖,𝑗 ,𝑖,𝑢 ∀𝑖 ∀𝑢 INFORMS Annual Meeting 2013 7/30 ASSUMPTIONS 1. All patients arriving at a queue remain in the queue until service completion. 2. Unused resource capacity is not transferable to other time periods. 3. Every patient planned according to the decision 𝑥𝑡,𝑗 will be served in queue j in period t, i.e., no deferral to other time periods. 4. We use time lags 𝑑𝑖,𝑗 = 1. 5. We use a bound U on u. 6. We temporarily assume: patient arrivals, patient transfers, resource requirements, and resource capacities are deterministic and known. INFORMS Annual Meeting 2013 8/30 MIXED INTEGER LINEAR PROGRAM Number of patients in queue j at time t with waiting time u Number of patients to treat in queue j at time t with a waiting time u [1] Updating waiting list & bound on u Limit on the decision space [1] Hulshof PJ, Boucherie RJ, Hans EW, Hurink JL. (2013) Tactical resource allocation and elective patient admission planning in care processes. Health Care Manag Sci. 16(2):152-66. 9/30 PROS & CONS OF THE MILP Pros: Suitable to support integrated decision making for multiple resources, multiple time periods, and multiple patient groups. Flexible formulation (other objective functions can easily be incorporated). Cons: Quite limited in the state space. Model does not include any form of randomness. Rounding problems with fraction of patients moving from one queue to another after service. INFORMS Annual Meeting 2013 10/30 MODELLING STOCHASTICITY [1/2] We introduce 𝑊𝑡 : vector of random variables representing all the new information that becomes available between time t−1 and t. We distinguish between exogenous and endogenous information: Patient arrivals from outside the system Patient transitions as a function of the decision vector 𝑥𝑡−1 , the number of patients we decided to treat in the previous time period. INFORMS Annual Meeting 2013 11/30 MODELLING STOCHASTICITY [2/2] Transition function to capture the evolution of the system over time as a result of the decisions and the random information: Where Stochastic counterparts of the first three constraints in the ILP formulation. INFORMS Annual Meeting 2013 12/30 OBJECTIVE [1/2] Find a policy (a decision function) to make decisions about the number of patients to serve at each queue. Decision function 𝑋𝑡𝜋 𝑆𝑡 : function that returns a decision 𝑥𝑡 ∈ 𝑋𝑡 (𝑆𝑡 ) under the policy 𝜋 ∈ 𝛱. The set 𝛱 refers to the set of potential policies. The set 𝑋𝑡 (𝑆𝑡 ) refers to the set of feasible decisions at time t, which is given by: Equal to the last three constraints in the ILP formulation. INFORMS Annual Meeting 2013 13/30 OBJECTIVE [2/2] Our goal is to find a policy π, among the set of policies 𝛱, that minimizes the expected costs over all time periods given initial state 𝑆0 : Where 𝑆𝑡+1 = 𝑆 𝑀 (𝑆𝑡 , 𝑥𝑡 , 𝑊𝑡+1 ) and 𝑥𝑡 ∈ 𝑋𝑡 (𝑆𝑡 ). By Bellman's principal of optimality, we can find the optimal policy by solving: Compute expectation evaluating all possible outcomes 𝑤𝑖,𝑗 representing a realization for the number of patients transferred from i to j, with 𝑤0,𝑗 representing external arrivals and 𝑤𝑗,0 patients leaving the system. 14/30 DYNAMIC PROGRAMMING FORMULATION Solve: Where Solved by backward induction INFORMS Annual Meeting 2013 15/30 THREE CURSUS OF DIMENSIONALITY State space 𝑆𝑡 too large to evaluate 𝑉𝑡 (𝑆𝑡 ) for all states: 1. Suppose we have a maximum 𝑆 for the number of patients per queue and per number of time periods waiting. Then, the number of states per time period is 𝑆 𝐽 ×|𝑈| . Suppose we have 40 queues (e.g., 8 care processes with an average of 5 stages), and a maximum of 4 time periods waiting. Then we have 𝑆 160 states, which is intractable for any 𝑆 > 1. 2. Decision space 𝑋𝑡 (𝑆𝑡 ) (combination of patients to treat) is too large to evaluate the impact of every decision. 3. Outcome space (possible states for the next time period) is too large to computing the expectation of ‘future’ costs). Outcome space is large because state space and decision space is large. INFORMS Annual Meeting 2013 16/30 APPROXIMATE DYNAMIC PROGRAMMING (ADP) How ADP is able to handle realistic-sized problems: Large state space: generate sample paths, stepping forward through time. Large outcome space: use post-decision state Large decision space: problem remains (although evaluation of each decision becomes easier). Post-decision state: Used as a single representation for all the different states at t+1, based on 𝑆𝑡 and the decision 𝑥𝑡 . State 𝑆𝑡𝑥 that is reached, directly after a decision has been made in the current pre-decision state 𝑆𝑡 , but before any new information 𝑊𝑡+1 has arrived. Simplifies the calculation of the ‘future’ costs. INFORMS Annual Meeting 2013 17/30 TRANSITION TO POST-DECISION STATE Besides the earlier transition function, we now define a transition function from pre 𝑆𝑡 to post 𝑆𝑡𝑥 . With Expected transitions of the treated patients Deterministic function of the current state and decision. Expected results of our decision are included, not the new arrivals. INFORMS Annual Meeting 2013 18/30 ADP FORMULATION We rewrite the DP formulation as where the value function 𝑉𝑡𝑥 (𝑆𝑡𝑥 ) for the ‘future costs’ of the postdecision state 𝑆𝑡𝑥 is given by We replace this function with an approximation 𝑉𝑡𝑛 (𝑆𝑡𝑥 ). We now have to solve With 𝑣𝑡𝑛 representing the value of decision 𝑥𝑡𝑛 . INFORMS Annual Meeting 2013 19/30 ADP ALGORITHM 1. Initialization: Initial approximation 𝑉𝑡0 𝑆𝑡 , ∀𝑡, initial state 𝑆1 and n=1. 2. Do for t=1,…,T Solve: 𝑛 𝑥 If t>1 update approximation 𝑉𝑡−1 (𝑆𝑡−1 ) for the previous post decision state Sxt-1 using the value 𝑣𝑡𝑛 resulting from decision 𝑥𝑡𝑛 . Find the post decision state 𝑆𝑡𝑥 . Obtain a sample realization 𝑊𝑡+1 and compute new pre-decision state 𝑆𝑡+1. 3. Increment n. If 𝑛 ≤ 𝑁 go to 2. 4. Return 𝑉𝑡𝑁 𝑆𝑡𝑥 , ∀𝑡. INFORMS Annual Meeting 2013 20/30 VALUE FUNCTION APPROXIMATION [1/3] What we have so far: ADP formulation that uses all of the constraints from the ILP formulation and uses a similar objective function (although formulated in a recursive manner). ADP differs from the other approaches by using sample paths. These sample paths visit one state per time period. For our problem, we are able to visit only a fraction of the states per time unit (≪ 1%). Remaining challenge: To design a proper approximation for the ‘future’ costs 𝑉𝑡𝑛 𝑆𝑡𝑥 … That is computationally tractable. provides a good approximation of the actual value. Is able to generalize across the state space. INFORMS Annual Meeting 2013 21/30 VALUE FUNCTION APPROXIMATION [2/3] Basis functions: Particular features of the state vector have a significant impact on the value function. Create basis functions for each individual feature. Examples: “total number of patients waiting in a queue” or “longest waiting patient in a queue”. We now define the value function approximations as: Where 𝜃𝑓𝑛 is a weight for each feature 𝑓 ∈ 𝐹, and 𝜙𝑓 (𝑆𝑡𝑥 ) is the value of the particular feature given the post-decision state 𝑆𝑡𝑥 . INFORMS Annual Meeting 2013 22/30 VALUE FUNCTION APPROXIMATION [3/3] The basis functions can be observed as independent variables in the regression literature → we use regression analysis to find the features that have a significant impact on the value function. We use the features “number of patients in queue j that are u periods waiting” in combination with a constant. This choice of basis functions explain a large part of the variance in the computed values with the exact DP approach (R2 = 0.954). We use the recursive least squares method for non-stationary data to update the weights 𝜃𝑓𝑛 . INFORMS Annual Meeting 2013 23/30 DECISION PROBLEM WITHIN ONE STATE Our ADP algorithm is able to handle… a large state space through generalization (VFA) a large outcome space using the post-decision state Still, the decision space is large. Again, we use a MILP to solve the decision problem: Subject to the original constraints: Constraints given by the transition function 𝑆𝑥𝑀 (𝑆𝑡 , 𝑥𝑡 ). Constraints on the decision space 𝑋𝑡 (𝑆𝑡 ). INFORMS Annual Meeting 2013 24/30 EXPERIMENTS Small instances: To study convergence behavior. 8 time units, 1 resource types, 1 care process, 3 stages in the care process (3 queues), U=1 (zero or 1 time unit waiting), for DP max 8 patients per queue. 8 × 83×2 = 2,097,152 states in total (already large for DP given that decision space and outcome space are also huge). Large instances: To study the practical relevance of our approach on real-life instances inspired by the hospitals we cooperate with. 8 time units, 4 resource types, 8 care processes, 3-7 stages per care process, U=3. INFORMS Annual Meeting 2013 25/30 CONVERGENCE RESULTS ON SMALL INSTANCES Tested on 5000 random initial states. DP requires 120 hours, ADP 0.439 seconds for N=500. ADP overestimates the value functions (+2.5%) caused by the truncated state space. 120 100 80 60 40 20 0 0 50 100 150 DP State 1 200 ADP State 1 250 300 DP State 2 350 400 450 500 ADP State 2 INFORMS Annual Meeting 2013 26/30 PERFORMANCE ON SMALL AND LARGE INSTANCES Compare with greedy policy: fist serve the queue with the highest costs until another queue has the highest costs, or until resource capacity is insufficient. We train ADP using 100 replication after which we fix our value functions. We simulate the performance of using (i) the greedy policy and (ii) the policy determined by the value functions. We generate 5000 initial states, simulating each policy with 5000 sample paths. Results: Small instances: ADP 2% away from optimum and greedy 52% away from optimum. Large instances: ADP results 29% savings compared to greedy. INFORMS Annual Meeting 2013 27/30 MANAGERIAL IMPLICATIONS The ADP approach can be used to establish long-term tactical plans (e.g., three month periods) in two steps: Run N iterations of the ADP algorithm to find the value functions given by the feature weights for all time periods. These value functions can be used to determine the tactical planning decision for each state and time period by generating the most likely sample path. Implementation in a rolling horizon approach: Finite horizon approach may cause unwanted and short-term focused behavior in the last time periods. Recalculation of tactical plans ensures that the most recent information is used. Recalculation can be done using the existing value function approximations and the actual state of the system. INFORMS Annual Meeting 2013 28/30 WHAT TO REMEMBER Stochastic model for tactical resource capacity and patient admission planning to… achieve equitable access and treatment duration for patient groups; serve the strategically agreed number of patients; maximize resource utilization and balance workload; support integrated and coordinated decision making in care chains. Our ADP approach with basis functions… allows for time dependent parameters to be set for patient arrivals and resource capacities to cope with anticipated fluctuations; provides value functions that can be used to create robust tactical plans and periodic readjustments of these plans; is fast, capable of solving real-life sized instances; is generic: object function and constraints can easily be adapted to suit the hospital situation at hand. INFORMS Annual Meeting 2013 29/30 QUESTIONS? Martijn Mes Assistant professor University of Twente School of Management and Governance Dept. Industrial Engineering and Business Information Systems Contact Phone: +31-534894062 Email: m.r.k.mes@utwente.nl Web: http://www.utwente.nl/mb/iebis/staff/Mes/