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Deployment and redeployment of ambulances using a heuristic method and an Ant Colony Optimization

Ambulance Mobile App

Deployment and redeployment of ambulances using a heuristic method and an Ant Colony Optimization

Ambulance Mobile App

Coverage problems aim to cover a territory. This field of literature helps decision makers to implant a facility, a depot or an enterprise. Covering location problem is classified in a strategic level decision. However the covering allocation problem occupies the tactical or operational level. To meet the customers demand, authors of [1] considered the stochastic process of client arrival. This randomness depends to the clients demands. They extended the deterministic ‘Maximal Covering Location Problem’ to a queuing model which includes more real constraints such as the waiting time. Researchers use heuristic approaches to face such models complexity. Covering models are also applied to solve public facility problems, especially industrial waste disposal. Reference [2] minimized the cost and the distance configuring the disposal nodes. When the number of nodes is given, authors of [3] maximized the demand under limited resources. Among the applications of the coverage models, we find the healthcare system. To meet the incoming emergency calls, researchers made a planning of ambulance route; deployment of ambulance from waiting site to emergency location, then to the hospital and its redeployment from the unit care to another waiting site. Coverage models are classified under static, probabilistic and dynamic classes. The first class concerns deterministic static to minimize the number of ambulances required to cover all sectors demand [4]. The Maximal Covering Location Problem ‘MCLP’ ensured a distribution of vehicles to meet the maximum of demands [5]. The second class of coverage models sheds new light on probabilistic models. These models introduced the constraint of ambulance unavailability. Thus, in [6] researchers affected to each ambulance a probability of unavailability in their Maximum Expected Covering Location Problem. The third class introduces dynamic models. The Dynamic Double Standard Model is developed by [7] with the objective of maximizing the coverage and minimizing the relocation cost at time ݐ. Researchers use deterministic methods, simulation and heuristics to solve these problems. Code Shoppy The problem of coverage is known as a NP-hard one in the literature. Therefore, exact methods give the optimal results but may be too heavy. However, metaheuristic methods allow faster feasible solutions [8]. This study aims to minimize the lateness during an emergency intervention in order to improve the hospital management system of the city. We resort to the ACO algorithm to solve this problem because it is especially used for deployment and redeployment problems. Generally social insects and in particular ant colonies, solve relatively different complex problems. Following this finding, the ACOs appeared [9]. The remainder of paper is organized as follows. Section 2 deals with a description of the problem and a definition of the objective function. Section 3 presents a heuristic method and an ACO algorithm. Section 4 is devoted to the computational results about a real case of Casablanca region. The last section concludes our work.

This section contains the methods used for the resolution: a heuristic method and an ACO hybridized by a GLS ‘guided local search’. The local search serves to find an improved fitness around the current solution. Furthermore, the GLS has to guide the movement of the local search. We consider firstly that demands are expressed from an intervention sector. Then, the duration between two emergency calls is distributed according to the Poisson law of a periodicity equals to two (day and night). Our model deploys the nearest available ambulance in order to minimize the total lateness of the emergency intervention. Once located in hospital after a patient transportation, the available ambulance will be redeployed to another emergency intervention which it has not yet received a response, or will be allocated to a waiting site. A.HeuristicTo initialize the ACO algorithm, we develop a heuristic. It takes into account the fleet size of ambulances ܰand the number of ambulances at each waiting site ߨ௜. This relationship is done by the application (2). That to say: :݅The potential waiting site, :ܰThe total number of ambulances, ܲ: The set of fire stations, ܪ:The set of hospitals, :ܵThe set of intervention sectors ߨ:ܲ∪ܪ ⟶ℕ ܿݑݏℎ ݏܽ ∑ߨ ௜௜∈ ௉∪ு ܰ= (2)݀௜௝: The distance between the nodes i and j,ߚ௦௝: The average of calls inter-arrival duration such as s∈S and j∈ {1,…,݌} with ݌ is the number of periods. Whether the set: ܵ௜=ቄs∈S,such as dୗ౟,୧=min୧∈୔∪ୌdୱ୧ቅ⊂ S,ܵ ௜ contains the sectors closest to the waiting site )ܪ∪ܲ∈(݅. We can characterize ܵ ௜by ܨ ௜ the frequency sum of interventions related to ܵ ∈ݏ௜ (3). ܨ௜=∑݂௦௦∈ௌ೔ & ݂௦ =భഁೞ=௣∑ఉೞೕ೛ೕసభ(3) ܵ ௜ is also characterized by ܦ௜, the sum of the distances between the waiting site ݅ and the sectors ܵ∈ݏ௜ , plus the distances between these sectors and the nearest hospital (4). ܦ௜=∑݀௜௦௦∈ௌ೔+∑݊݅݉௛∈ு݀௦௛௦∈ௌ೔(4)Thus, we determine the number of vehicles ߨ௜according to the calculated distances and frequencies: ߨ௜=ி೔∗஽೔∑ி೔∗஽೔೔∈(ು∪ಹ)∗ܰ (5)B.ACO The ACO is initialized by the heuristic defined above and based on three components: the pheromone, the path visibility and the pheromone update. • Pheromone Ants have the possibility to communicate via substances called pheromones. They can secrete these substances in soil and build paths that can be followed by other ants. A colony is able to choose the shortest path under constraints. Thus, the pheromone quantity in the shortest path is greater than secreted pheromone in the other paths. So, the way within highest pheromone concentration is chosen by the majority of ants and presents the solution of the problem. In our work, each waiting site ݅is represented by a pheromone ℎ݌௜that is the ratio between the number of ambulances in waiting site ߨ௜ and the total number of ambulancesܰ . ℎ݌௜=గ೔ே (6) The ACO is based on the phenomenon of pheromone evaporation. A path that is not used absolutely or used for few times, its pheromone evaporates. We call ߙthe effectiveness of a path and 1−ߙ the pheromone evaporation rate. •Path visibility In this work, the visibility ݒof a path represents the inverse of the distance between a waiting site ݅and an intervention sector ݆. Since the distance decreases, the visibility increases. We have to define a set of sectors that can be covered by a waiting site ݅. The set of sectors ܵ௜ is composed by the minimal distance ܦ ݅between different sectors and the waiting site covering them (the nearest one) and, on the other hand, by the distance between this set of sectors and the nearest hospital (equation 4). Thus, we determine the visibility of a waiting site ݅according to equations (3) and (4) by: ݒ௜=ி೔∗஽೔∑ி೔∗஽೔೔∈(ು∪ಹ)(7) The solution of our problem is formulated using the two concepts; the pheromone and the visibility (equations 6 and 7). We add the effectiveness of the path ߙ and the rate of pheromone evaporation ߙ1− as: ݒ ∗ߙ =݊݋݅ݐݑ݈݋ݏ ௜+(1− )ߙ∗݌ℎ௜ (8) •Pheromone update The ACO will not be complete without the evaporation process. Thus, in order to avoid the non-optimality of solutions, it seems necessary to forget the bad ones. The pheromone update is done as follow: ℎ݌௜: The pheromone representing the waiting site ݅, ߨ௜: The number of ambulances in waiting site ݅, ݎݑ݈݈݁݅݁݉௜: The number of ambulances in waiting site ݅giving the best solution (fitness), ܰ: The total number of ambulances, ܯ: A fixed constant, At iteration ݐ, ℎ݌௜(ݐ)= ݌ℎ௜∗(1−ߙ),At iteration (ݐ+1), the formula becomes: ݌ℎ௜(1+ݐ)=ℎ݌ ௜(ݐ)−ܯ+|గ೔ି௠௘௜௟௟௘௨௥೔|ே(9) The following algorithm defines the interesting steps of the ACO to distribute ambulances in hospitals and fire stations as potential waiting sites: View More

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