Operator Assignment and Routing Problems in Home Health Care Services

Transcription

1 8th IEEE International Conference on Automation Science and Engineering August 20-24, 2012, Seoul, Korea Operator Assignment and Routing Problems in Home Health Care Services Semih Yalçındağ 1, Andrea Matta 2 and Evren Şahin 3 Abstract Human resource planning in Home Health Care (HHC) services is a critical activity on which the quality of the delivered care depends. From the admission of the patient, the service organization has to decide the home operators who will follow the patients during their stay as well as the detailed care delivery plan. The robust resource planning is crucial for operations in the HHC structure, to avoid process inefficiencies, treatment delays and low quality of service. Many of HHC providers pursue the objective of continuity of care which is maintaining the same (principal) operator throughout the time the patient stays in the system. This paper proposes a two stage approach for assignment and routing decisions in HHC organizations. The main goal is to analyze the interaction between the assignment and routing processes in a single district. All the proposed cases are applied on a instance generated from real data obtained by an Italian HHC provider. Keywords: home health care, continuity of care, assignment, routing, traveling salesman problem I. INTRODUCTION Home Health Care (HHC) service is an alternative to the conventional hospitalization and consists of delivering medical, paramedical and social services to patients at their homes rather than in a hospital. The development of the HHC concept can be attributed to the ageing of populations, social changes in families, the increase in the number of people with chronical diseases, the improvements in medical technologies, the advent of new drugs and the governmental pressures to contain health care costs. The goal of the service is to help patients to improve or keep their best clinical, social and psychological conditions. Many and heterogeneous resources are involved in the HHC service delivery, including operators (e.g., nurses, physicians, physiotherapists, social workers, psychologists, home support workers, etc.) and material resources (consumable and non-consumable) that are used to provide care at home. The coordination of these resources is a difficult activity that has to deal with several sources of variability. Hence, robust resource planning is crucial to increase the operational efficiency, the responsiveness of the organization and the quality of the provided service. In particular, random events such as variations in patients conditions, resource unavailabilities and longer durations of transport in the surrounding territory, affect the service delivery process. The most critical source of randomness is the sudden variation of the service amount 1 S. Yalçındağ is with Dipartimento di Meccanica, Politecnico di Milano and Laboratoire Génie Industriel, Ecole Centrale Paris 2 A. Matta is with Dipartimento di Meccanica, Politecnico di Milano 3 E.Şahin is with Laboratoire Génie Industriel, Ecole Centrale Paris required by patients, which makes the short term resource planning more and more complex. In this study, we focus on the assignment of operators to patients, and the routing process of operators. The assignment problem refers to the decision of which operators will take care of which patients. The routing problem specifies the sequence in which the patients are visited. In the literature there are many works that address the operator scheduling problem and there are also many different application of this problem [10]. In particular, the operator assignment problem of the HHC services and the machine scheduling of the manufacturing problem has some similarities [17]. The manufacturing problem deals with preparing a schedule for a number of job on a set of machines and is largely addressed with numerical approaches like linear programming or heuristics [11]. The connection between these problems can be stated as the comparisons of the operators in the HHC domain with the machines and the newly admitted patients with the jobs. However, the operator assignment problem differs from the machine scheduling problem because of the operator characteristics and the patient demand variability. Moreover, the scheduling problem in the HHC context is also differs with the continuity of care issue. The continuity of care [8], [16] is pursued by several HHC providers to assign a patient to only one operator who is responsible for the care during his/her stay in the HHC service. Since loss of information between operators is avoided and the patient does not need to develop new relations with new operators, the continuity of care is considered as a crucial indicator of the service quality [8]. Despite the importance of both assignment and routing problems, there are not enough works in the literature dealing with these problems in HHC, specifically in the continuity of care context. This paper proposes a two stage approach for assignment and routing decisions in HHC organizations. The purpose of this paper is to assess how the different approaches available for the assignment and routing problems behave when applied in sequence with the single district assumption. We focus on the interaction between assignment and routing, where the output of the assignment problem is incorporated as an input into the routing problem. Specifically, we analyze if different operator assignment models have any impact on the routing process of the HHC services. At assignment level we use a mathematical programming model and two different policies; all approaches aim at balancing the workloads among the operators. After solving the assignment problem, a TSP (Travel Salesman Problem) /12/$ IEEE 329

2 model is formulated for solving the routing problem. The adoption of the care continuity context facilitates the routing process of operators by considering independent routes of the operators. Such an approach, i.e., one assignment model and many (as the number of operators) routing models, has not been investigated in literature yet and it is object of study in this work. The rest of the paper is organized as follows. In Section II, we provide a literature review related to the assignment and routing problems. In Section III, we describe the assignment and routing models. Computational experiments are reported in Section IV. Finally, some concluding remarks and future research directions are presented in Section V. II. LITERATURE REVIEW Existing literature on human resource planning of HHC services is devoted basically to five main aspects: the resource dimensioning, partitioning of a territory into districts, allocation of resources to districts, assignment of care providers to patients or the visits and the resource scheduling and/or routing [15]. The first step is the resource dimensioning issue. Here, the number of operators is determined to meet the predetermined care demand with the minimum cost and the adequate service quality. The second step is partitioning of a territory into districts. This consists of grouping small geographic areas into larger clusters, which are named districts, according to relevant criteria where each district is under the responsibility of a multidisciplinary team. Once districts are determined, resources are assigned to districts and then to patients equitably. After that, the successive step is the routing process. Among these aspects, we focus on the literature of the assignment and routing issues and we only present some of the existing works, for more details please refer to the work of Yalcindag et al. [15]. Only a few papers take into account the continuity of care. Borsani et al. [16] propose an assignment model and scheduling model where the output of the assignment model is used as the input to the scheduling model. The scheduling model provides the visit details without explicitly considering the TSP problem. The objective of the assignment process is to ensure workload balance among operators while respecting continuity of care, qualification requirements and geographical coherence constraints. Hertz and Lahrichi [1] propose two mixed integer programming models for assigning operators to patients. One model includes linear constraints and a quadratic objective function, while the other includes nonlinear constraints. The objective is to balance the operators workloads while respecting constraints related to maximum acceptable loads and assigning each patient to exactly one nurse of each type. Ben Bachouch et al. [12] develop a VRP model with time windows as a mixed linear programming model with the objective of minimizing the total distance traveled by the operators. The model is subject to visits and operators time windows, nurses meal breaks, care continuity and the restriction on the nurses maximum distance travel limit constraints. Lanzarone and Matta [4] evaluate the advantages of modeling the patient demand uncertainty in the problem of workload balancing among HHC operators. For this purpose, an integer programming model for balancing the workload amount among the operators of a specific category is formulated while preserving the continuity of care. Elbanani et al. [3] develop a model for determining routes for operators that incorporates constraints of the VRP with the medical and continuity of care constraints with the objective of minimizing the total traveling cost of operators. More recently, Trautsamwieser et al. [2] develop a model for the daily planning of the HHC services. The goal of the work is securing the HHC services in times of natural disasters. They develop the daily scheduling model as a VRP with state-dependent breaks. The objective of the model is minimizing the sum of travel times and waiting times, and also the dissatisfaction levels of the patients and health care operators subject to the assignment constraints, working time restrictions, time windows and mandatory breaks. Lanzarone et al. [5] develop a stochastic patient model to estimate the patient requests along his care pathway, and to use these estimates for a workload balancing model among the operators. Historical data of one of the largest Italian public HHC providers is adopted to develop and validate the model. In the literature, only a few existing studies focus on both the routing or the assignment aspects of the HHC planning processes [13], [16]. Thus, focusing on these problems are still important to analyze the interactions between the assignment and routing processes. In particular, it is also crucial to investigate under which circumstances the routing process is more useful and efficient.(e.g. fully independent or partially independent districts etc.). To this end, in this paper we work on these aspects since it seems to be an important step towards the development of a complete framework for resource planning in HHC. III. ASSIGNMENT AND ROUTING MODELS A. Problem Definition The assignment and routing problems in the HHC services are used to determine which operator will provide the service to which patients (visit) in which sequence with a predetermined objective such as balancing the workloads of the operators, minimizing the total distance traveled etc.. In this work, we assume that the assignment is held within a single category of operators (nurse or doctor). In the real setting, operators are divided into districts (as groups) based on their main skills and geographical areas to serve. Within each district, all operators are homogeneous with same professional capabilities. In this paper, it is assumed that districts are known before the assignment process is done; also, assignment and routing are executed within a single district. A new patient is admitted into the HHC service after an initial assessment of his/her clinical and social needs. It is assumed that the duration of visits and traveling times does not depend on the specific pair of the patient and operator. Thus, we express the demand of the newly admitted patient 330

3 as the total amount of time requested for the visits including the travel time. The assignment process is held to balance the workloads of home operators while satisfying a set of specific constraints of HHC operations. In this study, operators are assigned to the patients at the beginning of a considered week, t = 1,...,T where T is the total number of weeks. Then, for the successive periods the process is repeated to assign newly admitted patients. Patients might be admitted to the system during any week, but they are assigned to one of the operators in the beginning of the following week. Thus, each week single assignment problem is solved to obtain the lists of the operators. Models are proposed under continuity of care where the newly admitted patient has to be assigned to only one principal operator in the set Ω of all operators. This assignment is preserved throughout the length of stay of the patient in the HHC services. Each operator k, with k Ω = {1,...,K}, has one main skill that is used to handle a set of patients. The main skill refers to the patients for which the operator is best suited to care. Each operator has a deterministic capacity a k (in terms of minutes) which is the maximum amount of time that the operator can accomplish according to his/her working contract. In particular, in the case of the excess patient demand, it is possible to handle the excess amount with the additional workload capacity. As far as the assignment level, three approaches are analyzed: two structural policies and one mixed integer programming model (MILP). The first policy considers randomness of the patient demand when assignments are decided for minimizing the work in excess of operators. The second policy captures the practicalities of HHC providers by assigning the new patient to the operator with the smallest workload. The MILP model provides assignments under the assumption of deterministic patient demand. These three schemes are used to analyze either incorporating randomness into the problem has any benefit in terms of workload balancing. The main difference between the mathematical programming model and the structural policies is that the first model is able to assign all of the new patients of a week together, while the policy assignments can only assign one patient at a time after having ranked newly admitted patients with some criteria. Since one assignment at each time can be held, all patients are sorted according to their expected demand and each time the patient with the highest expected demand is selected for the assignment. At routing decision level, a TSP model is proposed to create the routes for HHC operators in each week t. The routing problem takes place just after the assignment problem. B. Assignment Models under Continuity of Care This section describes the assignment models that are considered in this work. Structural policies are taken from the study of Lanzarone and Matta [7] while the MILP model is taken from the work of Lanzarone et al. [6]. 1) Mathematical Programming Model with Deterministic Demand: In the continuity of care case, patients are assigned to the principle operator. Once they enter the HHC structure, they keep this principle operator during their length of stay. Each newly admitted patient i (with i = 1,...,n new ) has a deterministic demand λ i (expressed in minutes) which corresponds to the weekly care volume needed by the patient. More specifically, this value is composed of the visiting time plus the travelling time to reach patient home. For simplicity of exposition we assume that visiting times of all patients are equal to each other. Since we do not know the visiting sequence of patients, we have to use an average travel time. This is calculated by using the travel times to reach the patient i form any other patient and then by finding the average value. Since the continuity of care is preserved, w 0 k is used to denote the initial workload for the operator k at the beginning of each week where the assignment problem is solved. The objective of the model is to balance the operators utilization rates, u k. This can be reached by assigning the new patients to the operators with the lowest utilization rates. Thus, the corresponding objective function is maximizing the function h, where h is the minimum utilization rate of all operators. The corresponding model is as follows: max h (1) s.t. K k=1 x ik = 1 i (2) w k = w 0 n new k + λ i x ik k (3) i=1 h w k a k k (4) x ik {0,1} i,k (5) w k 0 k (6) The decision variable x ik takes the value 1 if the newly admitted patient i is assigned to the operator k and 0 otherwise. The other decision variable w k is a continuous variable and is used to calculate the total workload of operator k for both the newly and previously admitted patients. Equation (2) implies that all newly admitted patients must be assigned to only one operator. Equation (3) defines the total workload of each operator k. Inequality (4) expresses the minimum utilization rate h, which is maximized in the objective function. The considered model is a variant of the classical Bin Packing Problem (BPP) [9], [14]. The basic BPP deals with finding the smallest number of identical fixed sized bins so that all of the objects are assigned to these bins. Since the assignment model in this part aims at balancing the workload of the operators and also ensuring the continuity of care, the presented model turns out to be different than the classical BPP problem. 2) Structural Policies: In this part, the randomness of the patient demand and correspondingly the random workload of the operator is considered. Each operator is assumed to have an initial workload randomly distributed with probability 331

4 density function Φ k (w k ). Distribution Φ k is assumed triangular with parameters α k, β k and γ k (with 0 α k β k γ k ). The parameter α k is the minimum amount that the workload can assume, β k is the mode of the distribution and γ k is the maximum workload amount. The operator capacity, a k, is assumed to be in the second part of the triangular distribution (0 β k a k γ k, k Ω). The policy considers a penalty or cost function that is a power function of the amount of care that the operator k provides in surplus to his/her capacity a k. This function is a random variable because it depends on the operator initial workload and the demand of the newly admitted patient (if assigned to the operator). For more details about this cost function and related issues refer to the work of Lanzarone and Matta [7]. The policy minimizes the maximum value of the cost increase and named as the Maximum Cost (MC) Policy. In a certain sense, the policy minimizes the probability of incurring in largely oversaturated operators. The best assignment decision is determined by considering all of the possible comparisons between all of the possible pairs of operators that are in the set compatible operators Ω. The maximum cost policy can be shown as follows. Given two operators i and j (i, j Ω) with parameters α k, β k, γ k (with k = i, j), the newly admitted patient has to be assigned to the operator i if γ i a i γ j a j and to operator j if γ j a j γ i a i. The second considered policy is the Expected Available Capacity (EAC) Policy which captures the way of planning in HHC organizations. The variability of the patient demand is generally neglected by planners. The only information used for assigning each new patient to the principal operator is the expected workload of each operator (in the practice planners use standard values present in the therapeutic project of the patients in charge). According to this procedure, the new patient is assigned with the operator k with the highest expected available capacity AC k. Assuming triangular initial workloads, the expected available capacity can be calculated by the difference between the capacity a k and the expected workload: AC k = a k α k + β k + γ k 3 C. Routing Model In this part, we formulate a TSP model with a flow based formulation with patterns to find the routes of each operator. The routes are obtained according to the assignments that are provided by one of the models presented in the previous part. Since we have K available operators in the system, the TSP model should be solved for each operator independently to gather the individual routes. In other words, after obtaining the assignment solution, K independent static TSP models with deterministic demand should be solved to obtain individual routes. In real practice, a patient may require more than one visit per week. Thus, the service can be provided according to one of the predetermined schedules. Since there can be (7) many combination for the visiting days, patterns are used to identify all of the possible combinations. For example, if a patient requires two visits per week, he/she can be visited by one of the following predetermined patterns: Monday- Wednesday or Tuesday-Thursday etc. Thus, incorporating these patterns into the model is very important to cope with real life needs. It is important to note that, the assignment lists of operators are obtained for a fixed time period (week), t, whereas the routes are obtained in a daily base (for each d). Thus, routes are obtained for each day of each week. Here we provide the flow based formulation with the single commodity flow in the complete directed network G = (N,Ar). Each node in the network G corresponds to a patient except node 1 that is selected as the common health care center where all operators depart for service and come back after they finish their tour in a given day. A pattern in the model is denoted by p P and a day is denoted by d D. We also define a subset of P as P to identify patterns where there is a request from patient on day d. To make it clear, subset P contains the patterns in which the A p,d value of the matrix A takes value 1 for any day d. Note that the A matrix contains all the pattern list, with their daily information. Since the patients are assigned to one of the operators, the big patient set is partitioned into smaller subsets. The patient set is defined as i = j = 1,...,N where N is the subset of the all patient set. The binary variable y d i j indicates whether or not the operator goes directly form patient i to patient j on day d when i j and c i j is the corresponding travel time. fi d j is the flow amount in an arc (i, j) when i j and z jp is a binary variable to decide either pattern p is assigned to patient j or not. We also define the parameter γ j to identify the frequency visits requested by each patient j. The TSP model with patterns is as follows: min y d i j.c i j i N j N d D (8) s.t. y d i j 1 i N j, d, j 1 (9) y d i j = y d ji i N i N j, d, j 1 (10) y d i j z jp i N p P j, d, j 1 (11) y d i j z jp i N p P j, d, j 1 (12) z jp = 1 j, j 1 (13) p P fi d j f ji d = z jp i N i N p P j, d, j 1 (14) f1 d j = z jp d, j 1 (15) j N j N p P f d p P i j (N 1)y d i j i, j, d (16) z jp A pd = γ j j (17) d D 332

5 y d i jc i j + V z jp a i N j N j N p P d (18) The objective function (8) is minimizing the total distance traveled by the operator. The constraints (9) ensures that the operator can visit patient j at most one time on day d. The constraints (10) are the flow conservation constraints. The constraints (11) guarantees that, according to the assigned pattern p, if patient j requires any service on day d the operator must visit the patient on that day. The constraints (12) ensures that there will be no visit to patient j if there is no request on the associated patten p on day d. The constraints (13) guarantees that each patient is assigned to only one pattern. The constraints (14) and (15) are the flow conservation constraint on f variables to eliminate sub-tours. Constraints (16) ensure that a flow can only take place in an existing arc. Constraints (17) ensure that the number of visits required by each patient will be provided. Constraints (18) guarantees that the workload of the operator in terms of the travel time and service time on each day does not exceed the maximum working time limit a where the term V is denoting the constant visit time. After solving the assignment model and K independent TSP models, we can obtain the routes for each operator. Having the complete list of patients including the previously admitted and newly admitted ones, the new routes are constructed for each operator. And these processes are repeated for each week, t. Thus, to obtain results for more patients and more periods, we should solve both models in a rolling base. IV. COMPUTATIONAL STUDY In this part, we analyze and compare the developed models with a instance generated from real data obtained by an Italian provider. The analysis is based on the interactions between the operator assignment and routing problem of the HHC structure. More specifically, our goal is to analyze if there is any impact of different assignment approaches on the routing problem for a single district. A. Performance Indicators Average total travel time of all periods, C, traversed by all operators is used to identify how long does it take to finish all of tours on average. This indicator can be calculated as follows: C = T t=1 C t T (19) C t is the total travel time in week t obtained for all operators and T is the total number of weeks. C 1 is used to show the average total travel times just after the assignment process and before the routing process with the average travel distances. In particular, C 2 is used to provide the average total travel time after the routing process with the real travel distances. The calculation of the workload level indicator for each operator k is considered to be mean utilization measuring the workload performance and it can be calculated as: u k = T t=1 w kt T a k k (20) where w kt is the workload level of operator k in week t. The range of the u k value among all operators, denoted as Z, is expressing the workload balancing performance of the assignment and routing processes. A smaller Z value corresponds to better balancing level. In the following part, Z 1 is used to show the workload performance of the assignment problem before the routing process and Z 2 is used for after the routing process. u is used to show the average utilization level of all operators which is calculated by summing all u k values and dividing this sum with the total number of operators. B. Results In this part, independent interactions of the assignment and routing problems are tested on a real instance by using the policy approach and the mathematical programming model. We obtain results for the assignment problem with seven and eight operators and for the routing problem with seven operators. It is assumed that the operators has same operation skills. Each operator is characterized by his/her capacity, defined as the weekly time according to his/her contract and these capacities are defined as 50, 50, 50, 40, 40, 40, 35 and 30 hours. For the assignment problem, the visiting and traveling times are assumed to be fixed for all visits and is considered as 1 hour per visit. Patients, their locations, their visiting requirements and operator capacities are obtained by using real data provided by an Italian HHC provider. In our set up, we consider each week as 6 working days (Monday to Saturday) and we solve the presented models for consecutive 4 weeks. An initial assignment for each operator is performed at the initial week (week 0) with all of patients that are already assigned and the successive assignment are executed on each of the following weeks on a rolling base. We also consider the case where patients leave the system so in such cases we delete these patients from the assignments lists. These assignments are then executed on a set of sample paths, generated with the Monte Carlo approach from the estimated demand distributions. These sample paths are set of scenarios that are used to evaluate the performance of the models under different requirements of patients. Thus, a sample path includes the values of the randomly generated patient demands. Since we need to respond different amount of patient s needs during each week, we define 14 different visiting patterns (schedules) and these patterns are incorporated in the routing level. As described in the previous parts, assignment lists are obtained for the Maximum Cost Policy (MC), Expected Available Capacity Approach (EAC) and Mixed Integer Linear Programming (MILP) model. A total of 256 patients are considered with execution of 10 sample paths and the corresponding results are presented in Tables I and II. It can be observed from Table I that for the assignment process, using 8 operators instead of 7 leads decreases in the 333

6 TABLE I PERFORMANCE INDICATORS FOR THE ASSIGNMENT PROBLEM (MEAN VALUE ± HALF-WIDTH 95 CONFIDENCE INTERVAL) Operator Variable EAC MC MILP Z ± ± ± Op. u Z ± ± ± Op. u TABLE II PERFORMANCE INDICATORS FOR THE ROUTING PROBLEM (MEAN VALUE ± HALF-WIDTH 95 CONFIDENCE INTERVAL) Operator Variable EAC MC MILP Z ± ± ± Op. C ± ± ±125.4 average utilization level, u. This shows that having 7 operators is not sufficient and all operators are almost saturated. Thus, working with 8 operators provides better solutions. Since the operators are saturated with the 7 operator case, Z 1 turns to be very similar for all 3 experiments. Incorporating one more operator increases relative differences between the workload levels obtain by 3 models. It can also be observed that on average the MCP policy provides better workload balancing levels for both models with 7 and 8 operators. So we can conclude that uncertainty seems to be crucial to balance the workload levels in the assignment level. In addition to this, deterministic case, MILP, also provides better result than the real case, EAC. Thus, HHC providers would gain more if they use the MILP model instead of considering the expected available capacities. For the routing process, all three models provide almost the same results in terms of both workload balancing, Z 2 and average total travel time, C 2. In addition to the average total travel time obtained with the real distances (after the routing process), C 2, we also calculate the same value with the average travel distances (after the assignment process), C 1. As a result, we obtain C 1 as 2219,5. Comparing this value with the C 2 values of the Table II, we observe that routing process does not able to decrease the travel times very much. The main reason for this observation is the size of the considered district. The relative distances within this district are small so going from one patient to another is not very time consuming. Thus, in such a case where all the patients are close to each other, we do not need to perform any routing operation. The importance and effectiveness of the routing operation can usually be observed in a bigger or integrated districts. V. CONCLUSIONS In this work, we propose a sequential assignment and routing approaches under continuity of care and we analyze their interactions. We conclude that incorporating uncertainty into the assignment model provides better workload balancing levels. Moreover, since the selected district is small and independent, we conclude that it is not necessary to perform any routing process. Results obtained with the average travel distances is almost as good as the one obtained from the routing process. An on going activity is the development of a partially integrated districting model. With this model, we will able to work with relatively larger district and we will also able to show the affect of the routing process. 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