key: cord-0942944-uzsqyyzt
authors: Tlili, Takwa; Masri, Hela; Krichen, Saoussen
title: Towards an efficient collection and transport of COVID-19 diagnostic specimens using genetic-based algorithms
date: 2021-12-09
journal: Appl Soft Comput
DOI: 10.1016/j.asoc.2021.108264
sha: 47ec9ba4bcaeea1d54c5f2e83853b0de59bbea53
doc_id: 942944
cord_uid: uzsqyyzt

The speed by which the COVID-19 pandemic spread throughout the world makes the emergency services unprepared to answer all the patients’ requests. The Tunisian ministry of health established a protocol planning the sample collection from the patients at their location. A triage score is first assigned to each patient according to the symptoms he is showing, and his health conditions. Then, given the limited number of the available ambulances in each area, the location of the patients and the capacity of the nearby hospitals for receiving the testing samples, an ambulance scheduling and routing plan needs to be established so that specimens can be transferred to hospitals in short time. In this paper, we propose to model this problem as a Multi-Origin-Destination Team Orienteering Problem (MODTOP). The objective is to find the optimal one day tour plan for the available ambulances that maximizes the collected scores of visited patients while respecting duration and capacity constraints. To solve this NP-hard problem, two highly effective approaches are proposed which are Hybrid Genetic Algorithm (HGA) and Memetic Algorithm (MA). The HGA combines (i) a k-means construction method for initial population generation and (ii) a one point crossover operator for solution recombination. The MA is an improvement of HGA that integrates an effective local search based on three different neighborhood structures. Computational experiments, supported by a statistical analysis on benchmark data sets, illustrate the efficiency of the proposed approaches. HGA and MA reached the best known solutions in 54.7% and 73.5% of instances, respectively. Likewise, MA reached a relative error of 0.0675% and performed better than four existing approaches. Real-case instances derived from the city of Tunis were also solved and compared with the results of an exact solver Cplex to validate the effectiveness of our algorithm.

With the emergence of the COVID-19 pandemic, many healthcare facilities found themselves overwhelmed by the number of patients. The hospitals

were not an exception as they were struggling to deal with the outbreak of the fast-moving pandemic spread. Several logistic management problems were raised due to limited resources and unusual time pressure. We herein cite some recent researches carried to tackle routing problems related to the pandemic. For instance, Pacheco et al. [1] studied the problem of vehicle routing for the urgent delivery of face shields during the COVID-19 pandemic in Spain. The problem was modeled as a pick up and delivery Vehicle Routing Problem (VRP) variant, where dierent drivers volunteers to pick up face shields from makers, deliver material to face shield makers and deliver face shields to demand points. A heuristic based on a multi-start insertion algorithm was implemented. Recently, Singgih [2] considered the problem of deployment of mobile laboratories that are equipped with the testing capability to handle the over-demand situation in Indonesia during COVID-19 pandemic. The author presented a heuristic method to dene the optimal location of a single mobile laboratory. Zhang et al. [3] studied the problem of transport of high-risk individuals being transferred for medical isolation in epidemic areas in China where the number of available quarantine vehicles is limited. The problem was solved using a water wave optimization metaheuristic. Chen et al. [4] designed a multi-vehicle multi-trip routing problem to model the contactless food distribution for closed gated communities.

In this paper, we consider another major issue encountered during the pandemic: how to manage the logistics associated with the collection of patients' specimens at their places. We should rst highlight the fact that at-home testing of suspect COVID-19 cases could ease pressure on hospitals and emer-J o u r n a l P r e -p r o o f Journal Pre-proof gency services and prevent the spread of infection. Hence, the transportation of specimens in a reliable and ecient manner is essential for eective patient care, allowing faster diagnosis and patient treatment.

Exploring the Tunisian case, at the early stage of the pandemic, a special hotline was deployed to answer the requests of patients. The rst step of the protocol consists in receiving the patient calls notifying the facility that they are seeking care due to COVID-19 symptoms (e.g. fever, cough, fatigue, headache, shortness of breath, loss of smell or taste, sore throat or other). The severity of COVID-19 symptoms can range from very mild to severe. For instance, people who are older have a higher risk of enduring severe conditions from COVID-19, and the risk increases with age. People who have existing chronic medical conditions also may have a higher risk of serious illness. Another important information, is about the social circle of people the patient met lately, if they had a close contact with someone who has COVID-19 or if they have travelled to/from hot-spots areas. According to the dierent information, a rst triage of the demands in done to assign scores to the dierent requests. Given the limited size of the ambulances eet available to carry the specimens collection task, the logistic management was a challenging problem.

In this paper, we propose a solution approach to nd an ecient routing plan for the set of ambulances collecting the patients' specimens at their locations.

The objective is to respond to the maximum number of requests by maximizing the collected patients' scores. Each ambulance route starts from a given depot in the target testing area, and nishes its one day tour at a hospital having a COVID-19 laboratory. As the testing capacities for each hospital are limited by the number of testing kit supplies and working hours of the health ocers, the number of collected specimens at one hospital should not exceed this capacity. The problem is modeled as an integer linear problem, derived from team orienteering vehicle routing problem (TOP) [5, 6] .

The TOP was rst proposed by Chao et al. in 1996 [5] . It can be represented by a directed complete graph where start and end points are specied along with other customers. Each costumer has an associated score. The goal is to determine a xed number of routes, limited in length, that visit some locations and maximize the sum of the collected scores. Furthermore, each customer can be served at most once. The TOP was used to model dierent routing problems in rescue and emergency situations [7, 8, 9, 10] . Given the computational challenge of TOP (NP-hard), heuristic and metaheuristic algorithms are very suitable for nding near-optimal solutions for large sized J o u r n a l P r e -p r o o f Journal Pre-proof instances that cannot be solved exactly in an acceptable computation time [5] .

Our problem diers from the well-known TOP through the assumptions that the nal depot has a determined capacity and also by the fact that the ambulances are located in dierent starting points. The considered problem can be also dened as an open VRP [11] given that the itinerary of an ambulance is not a closed circuit. We also assume that the generated route ends at one of the hospitals having capacity restrictions. An illustrative example of ambulance routing in the city of Tunis is presented in Figure 1 .

As a solution approach, we propose two variants of genetic algorithms.

A Hybrid Genetic Algorithm (HGA) that combines k-means method with regular evolutionary operators. The initial population is generated based on a cluster-rst route-second approach which starts by grouping the patients into a set of clusters based on their locations. The number of clusters is equal to the number of available ambulances. Then, in order to get the routes, a scheduling of the patients is performed for each cluster without violating the route duration and hospital capacity constraints. A one point crossover operator is used for solution recombination, and a inversion and swap operators are used for the mutation task.

A Memetic Algorithm (MA) that integrates a Local Search (LS) procedure into the HGA to strengthen the exploitation process and enhance its performance. The main concept of LS technique is to improve interactively the solution using local modications. The proposed version of MA is based on three eective neighborhood structures which are designed to expand the search space and accelerate the convergence of HGA. In summary, the key point is that both genetic operators and LS neighborhood structures are carefully selected in order to run jointly when producing the solution.

In the literature, GAs have for long been recognized as powerful optimization tools for complex routing problems [12] . Also, numerous local search heuristics [13] showed very promising results on solving dierent TOP extensions [14, 15] . MA oers a framework to combine the exploration power of GA and the exploitation eectiveness of local search. Such combination outcomes a robust metaheuristic that has demonstrated a signicant success to handle J o u r n a l P r e -p r o o f Journal Pre-proof several NP-hard problems [16] . Motivated by these facts, we propose to implement an HGA and an MA to tackle the TOP.

In order to validate our proposed algorithms, we experiment it using a TSPbased benchmark proposed by Fischetti et al. [17] . Computational experiments include a comparison of our algorithms with ve existing methods from the literature as well as a real-word case study. HGA and MA reached the best known solutions in 54.7% and 73.5% of instances, respectively. In terms of the number of average relative percentage deviation, MA produced challenging results by achieving 0.0675% and performed better than four state-of-the-art approaches.

To make more rigorous comparisons, statistical tests have been conducted and proved the competitiveness of our approaches versus the state-of-the-art methods.

The experiments on a real case data set show that the MA improves consistently the exploration of the search space as it produces high quality solutions compared to the HGA. We should note that this enhanced algorithm performance in solving the MODTOP is to the detriment of the required running time.

Hospital Patient Figure 1 : Illustrative example of ambulance routing problem in Tunis city

The remainder of this paper is structured as follows. Section 2 reviews the related work. Section 3 presents the problem description and the proposed mathematical model. Section 4 details the proposed approach and its imple- In all the previous cited references the considered objective functions are related to either time or cost of the generated routes with the assumptions that all the requests (i.e. of pick up/service or delivery) are satised. In the other side, some routing models applied to emergency and rescue assume that only a subset of requests will be covered and the objective is to maximize the number of served requests or the number of collected rewards of the visited points. These assumptions t with TOP model.

On its broad context, the TOP proposed by Chao et al. [5] is a routing problem where the goal is to determine the path for each team member, in order to maximize the total collected score by the team given a limited time span.

The TOP is NP-hard [5] . Only few researches focused on using exact methods [28, 29] to solve the TOP. Such methods become highly time-consuming as the problem instances increase in size. Therefore, the main body of literature dealing with TOP is dominated by heuristic and metaheuristics methods [7, 9, 10] .

Dang et al. [7] proposed an ecient heuristic approach for TOP based on an interval graph model and an inspired particle swarm optimization. Chen et al. [8] used the TOP to model the problem of optimal team deployment 

The Tunisian ministry of health established a protocol to be followed in order to schedule the visits for suspected COVID-19 patients requesting an at-home test. The Figure 2 describes the adopted process. As a rst step, the call center answers the requests of patients suspected to be COVID-19

positive. An electronic information form is lled for each patient to get the following information:

The symptoms (e.g. fever, cough, fatigue, headache, shortness of breath, loss of smell or taste, sore throat or other)

The age and health condition of the patient.

Whether he was in a close contact of someone who is diagnosed with

Whether he travelled lately to a hot-spot area During the triage phase, a score is assigned automatically to each patient.

For those having a score greater than a certain threshold, they will be scheduled in a waiting list, to be visited by an ambulance at their place to take a We should point out that by the end of the tours, the collected tests will be delivered to a set of hospitals having limited capacities (e.g. the capacity depends on the number of testing kit supplies and working hours of the health ocers). Hence, another decision along the routing is to assign each vehicle to one of the hospitals while respecting the capacity constraints. We where each hospital has a limited capacity c h . Furthermore, a non-negative travel time t ij is associated with each arc (i, j) ∈ A. The total time taken to visit the points on each of the paths cannot exceed the specied limit T max . The set of patients is denoted by N c . Each patient i ∈ N c has a predened prot p i (i.e. score) and a service time s i . to serve all patients. The objective is to nd the subset of served patients along with the corresponding visiting sequences such that the total collected prot is maximized. We should note that each customer is visited at most once by only one ambulance.

Used parameters as well as the decision variables are described as follows.

Parameters:

travel time from i to j s i service time for patient i T m ax maximum total travel time for an ambulance Decision variables:

Route construction constraints (2) k∈K j∈Nc

i∈N k∈K

Time constraints (7) i∈N j∈N

Decision variables type constraints

The objective function (1) Constraints (4) ensure that each ambulance visits exactly one hospital.

Constraints (5) ensure that the hospitals are the end-points of the ambulances' routes.

Constraints (6) state that for each hospital h, the total number of collected specimens is lower than the capacity of the hospital c h . Constraints (7) checks that the route duration of each ambulance does not exceed the T max . Constraints (8) and (9) are related to patient visiting assumptions.

Constraints (8) ensure that each patient is visited at most once by one of the ambulances.

Constraints (9) set values of the decision variable y ik . A patient i is considered to be visited by the ambulance k if this vehicle crossed one of the arcs (i, j) where j ∈ N .

Constraints (10) impose binary restrictions to x and y decision variables.

Due to its NP-hardness [31] , the MODTOP cannot be solved optimally within a reasonable computational time particularly for large-scale instances.

Therefore, we designed two genetic based algorithms which are (1) Hybrid

Genetic Algorithm (HGA) and (2) Memetic Algorithm (MA). The general structure of the proposed approaches is based on the combination of k-means clustering and genetic operators. The k-means clustering process was proved to be ecient in simplifying logistic networks and improving the routing solution ( [32] , [33] ). Furthermore, in order to prevent the HGA from being trapped into local optimum, we developed the MA that incorporates a local search (LS) procedure to improve the solution in local search scope.

For both of HGA and MA, the rst step is to generate an initial population based on a cluster-rst route-second method that integrates K-means algorithm and route construction procedure (section 4.3). Common GA operators are proposed as follows.

Survivor selection mechanism adapting Steady-state" technique (section 4.4).

Ospring generation using Random one point crossover" (section 4.5).

Mutation operator applying Inversion and swap" methods (section 4.6).

Three The aforementioned framework is detailed in Figure 3 . In the proposed encoding, K − 1 zeros are employed to separate the set of routes. Given V P the number of visited patients and U P the number of unvisited patients. The total length of the chromosome is equal to V P + (K − 1) + (K × 2), where (K × 2) is the number of depots and hospitals in the chromosome.

The quality of the solution s is evaluated using a tness function dened as f (s) = T S(s), where T S(s) is the total score of s.

An example of the solution for COVID-19 context is shown in Figure 4 .

Given 10 patients to be visited by 2 ambulances based on 2 medical depots. The 10 diagnostic specimens are transported to 2 dierent hospitals. The visiting sequence of route 1 from o1 to h1 is 1 − 2 − 3 − 7, and the visiting sequence of route 2 from o2 to h2 is 10 − 6 − 9. 

The initial population can be viewed as a cluster-rst route-second"

heuristic.

Cluster-rst step: K-means algorithm is used to assign a given set of sub-patients P ∈ N c to clusters ( [34] ). It is about generating a set of N clusters, named C, where C = n 1 , n 2 , . . . , N . Each cluster (n i ) involves a set of patients and the number of clusters is equal to the number of routes (i.e. ambulances). At the end of this step, k-means generates a set of k clusters containing the patients to visit. Each feasible cluster should satisfy the constraints described in the problem formulation of section 3. K-means method is detailed in algorithm 1.

Route-second step: In the resulting clusters from step 1, the patients in each cluster are sorted randomly. Each obtained route contains a number of nearby patients with random order. To generate an initial population, a set of generated routes needs to be feasible. Based on the encoding previously described in section 4.1, each route R is adjusted as follows.

Add a depot number o as a rst element of R. Add the patients sequentially in a random order to R. Add a random hospital number h at the end of R while checking its capacity constraint.

If the route duration exceeds T max then remove the patients with the lowest scores until the duration constraint is satised.

The aforementioned steps construct a set of feasible routes to be in-

This procedure is iterated multiple times by varying the subset of patients P , given as input to the k-means, until the population size is reached. The initial subset P ∈ N c is randomly generated containing from 80% up to 100% of all patients. for all c ∈ C do 5: for all j ∈ P do 6:

Calculate the Euclidean distance to each centroid;

Assign the closest node to centroid to create the cluster ; e + +; 13: until No change in the centroid

The survivor selection of the candidate solutions is an important step in genetic algorithms. The most promising chromosomes are included in the next generation and will be used as parents in the crossover operations.

There are dierent selection techniques, e.g roulette-wheel, rank selection, tournament, elitism and steady-state ( [35] ).

In the proposed algorithm, a chromosome is more likely to be selected if its tness function value f is high and the steady-state mechanism is adopted.

The main idea of steady-state is that the candidate solutions are allowed in the current population to become a part of the new population. The steadystate selection process used in HGA and MA is detailed as follows.

Step 1: Identify the best solutions of the population.

Step 2: Remove nb bad chromosomes.

Step 3: The rest of the current population survives to the new generation without going through selection process.

J o u r n a l P r e -p r o o f for all x ∈ Sorted_list_of _unvisited_patient(s) do 5: s ← best_insertion(s, x); 6: end for 7: end for As an improve to the produced osprings, a new insertion heuristic is introduced. Insertion heuristics have proven to be popular methods for solving a variety of vehicle routing and scheduling problems [36] . The best_insertion(s, x) contains three steps Compute the geographical center of all routes (i.e. genes) Assign x to the nearest route that hasn't exceeded the capacity constraint If the generated route exceed T M ax , mark x as unvisited.

In order to maintain the population diversity, a mutation operator is performed after the crossover previously described. The mutation process applied in the proposed algorithm integrates two dierent types which are 

In the proposed LS, three neighborhood structures have been developed to explore the solution space. An illustrative example is illustrated in Figure   ( 6) to better explain the developed operators. In the illustrative example, the initial routes are: 

(c) Insert node: Inserts a new patient in a route where the location consumes the least travel time. A detailed explanation of the insert operator is introduced in [6] .

As shown in case (c), the unvisited patient x 11 is supposed (1) to be the least time consuming compared to patient x 10 and (2) its insertion does not exceed the maximum travel time. A single arc is deleted (x 6 , H 1 ) and two other arcs are created (x 6 , x 11 ) and (x 11 , H 1 ). After the insert node operator, the new routes are: 

For performance assessment, we carried out 135 benchmark instances grouped into three classes, called`generations'. Each generation contains 45 problems with up to 400 vertices. The corresponding T max value for each instance is computed as 50% of the shortest Hamiltonian cycle length ( [39] ).

The tested benchmark is a TSP-based data set adapted by Fischetti et al. Where V is the set of vertices, d 1i is the distance from vertex 1 to vertex i and d max is the distance from vertex 1 to its farthest vertex.

The performance of designed metaheuristics depends inherently of its parameter settings. To tune the dierent parameters, we apply an automatic procedure called F-Race to determine the best conguration. The F-Race is an oine automatic statistical procedure proposed by [40] to enable conguration of parameterized algorithms. The tuning was performed on a random selection of 10 large instances from the benchmark and the real case instances, with the total time limit set to 600 seconds. To evaluate the performance of each parameter, we test each considered value from Table 1 while xing the other parameters to their nal values. For each algorithm, the total number of congurations in this preliminary experiment is equal to 3 × 3 × 2 × 2 = 36.

We report in Table 1 , the set of candidate congurations and the nal retained values. with state-of-the-art methods. For the sake of completeness, we also report the processor details of the computing environments used to test the ve solution methods ( Table 2 ).

The CPU time depends on a variety of factors, such as hardware, compiler and programming language used to develop the dierent approaches.

Therefore, we did not directly compare the executing time eciency of the algorithms. The bold values mean the best-so-far results found by the algorithms.

* means BKS achieved.

-means not available value. In Table 5 In Table 6 , we provide the number of times the BKS is found (#Best) and the average relative percentage deviation (ARPD) over the total number of benchmark instances for the generations. Since there are unaddressed instances for GRASP-SR and ALNS algorithms, we excluded 18 instances (Table 2) . Therefore, we did not deeply compare the run time eciency of the algorithms. The statistical results reported in Table 7 shows that MA outperforms, in all generations, 2-PIA, GRASP-PR and EA4OP with a general p-value equals to 0.00, 0.00 and 0.03, respectively. Compared to HGA, the test is signicant on generation 1 (p-value = 0.04) and generation 3 (p-value = 0.02) but it is not the case for generation 2 where p-value > 0.05. Regarding GRASP-SR and ALNS, the proposed MA shows a competitive results with p-value > 0.05. A real dataset of 30 instances has been generated assuming that there are 6 hospitals and 6 depots. Each hospital has a maximum capacity of specimen tests (Table 8 ) and in each depot, there are a prexed number of ambulances (Table 9) . Given the aforementioned dataset, the problem instances were solved using Cplex 12.6.2 with default settings. We got to solve small instances with a time limit of 10800 seconds. 

In the small-scaled instances in which the number of patients is less than 25, Cplex was able to nd optimal solutions within three hours. For the rest of instances, Cplex either terminates with an out of memory error or it is stopped as we have set the maximum resolution time, while the HGA and MA continue to generate solutions within a very limited CPU time. This fact underlines the usefulness of metaheuristics in solving such NP-hard problems.

The computational diculty increases signicantly with problem size. For instance, Cplex takes more than 2 hours to solve the problem instance Cov 10 . To handle the same instance, HGA nds a near optimal solution in only 7.8 seconds, with a %Gap that amounts to only 0.5%. Although, MA succeeds to reach the best solution in 10.9 seconds. Among all instances solved by Cplex, HGA nds the optimal solutions in 9 instances out of 10. In fact, the average gap among all instances is equal to 0.5% which is considerably interesting. MA attains all the optimal values generated by Cplex with a general AVG time equals to 8.85 seconds which is a worst result compared to HGA that solved the instances in 7.02 seconds. In terms of solution values, MA outperforms HGA in all the tested data set. 6 . Conclusions and future works COVID-19 related issues become a top priority for researchers worldwide, notably in combinatorial optimization eld. In this paper, we studied the collection and transport of COVID-19 specimens. The problem can be described as follows. Given (1) a set of suspected patients requesting COVID-19 tests at home, each one of them is associated with a priority and (2) a set of ambulances located in dierent hospitals. The objective is to select the subset of urgent patients to be visited in priority as well as to determine the order and the optimal itinerary to collect the COVID-19 specimens test. Some restrictions are imposed in our model, such as the hospital capacity and the daily working time of the ambulance driver. We modeled the collection and transportation of COVID-19 specimens as a new variant of the team orienteering problem, named multi-origin-destination team orienteering problem.

Given the problem complexity (NP-hard) a hybrid genetic algorithm combining the k-means along with the evolutionary operators is proposed. A memetic algorithm considering a local search is also implemented to improve the convergence speed and fully exploit the solution space. Compared to the current state-of-the-art, the two algorithms are proved to be ecient as they matched in many cases the best reported results on dierent TOP benchmarks. Experiments on real-case benchmark data sets indicated that the both HGA and MA produce high quality solutions with reasonable computational requirements for small sized instances. The numerical results for large sized instances, supported by statistical tests, prove the eciency of HGA in generating better approximation of the global optimum. However, MA is computationally less ecient than the HGA, this may be explained by the constructive nature of the considered neighborhoods.

J o u r n a l P r e -p r o o f

As future works, we can consider the dynamic case where new requests are coming while serving the actual patient, also we can solve dierent objective functions in a multi-objective problem. Finally, we suggest to integrate our proposed approaches into a decision support system in order to assist dynamically the ambulance drivers of COVID-19 specimens to accomplish their works optimally.

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