1 Scheduling Plans of Tasks Internship report. Supervisors: Amal El Fallah Seghrouchni, Safia-Kedad Sidhoum Davide Andrea Guastella Abstract—We present a heuristic algorithm for solving the is accomplished correctly and by satisfying all the constraints. problem of scheduling plans of tasks. The plans are ordered Moreover, due to both critical contexts and dynamic envi- vectors of tasks, and tasks are basic operations carried out by ronment, it is important to orchestrate the operations of the resources. Plans are tied by temporal, precedence and resource constraints that makes the scheduling problem hard to solve in sensors within a relatively short time. polynomial time. As further case study, consider the in-flight airplane The proposed heuristic, that has a polynomial worst-case time safety [5]. In this context there is a need for detecting and complexity, searches for a feasible schedule that maximize the resolving data errors which could come from faulty sensor number of plans scheduled, along a fixed time window, with measurements, inaccurate data processing, or poor information respect to temporal, precedence and resource constraints. transmission, that can lead to catastrophic accidents as in the case of the Air France 447’s accident. Our scheduling I. INTRODUCTION technique can be employed as a fault recovery technique: for example, once the fault has been detected by the on- Scheduling is a decision-making process that is used on board sensors, a system scheduler could discards the remaining a regular basis in many manufacturing and services indus- flight plans, and schedules a set of emergency plans into the tries [1]. It deals with the allocation of resources to tasks over current flight schedule. These emergency plans are strictly time given time periods and its goal is to optimize one or more constrained, and their orchestration necessarily needs to lead objectives. to a schedule which can guarantee the safety of passengers. From a theoretical point of view, a scheduling problem is a constrained combinatorial optimization problem where a set of task must be ordered in such a way that all these are arranged, III. DEFINITIONS according to one or more constraints, to constitute a schedule Definition 1 (Plan of Tasks). A plan of tasks Π is a partial that minimize or maximize a given objective function. k ordering of tasks J k to address a specific goal. Each plan One of the most popular scheduling problem is the resource- i Π is characterized by a priority value α . The structure of constrained project scheduling problem (RCPSP) [2]: the k k the plan is depicted by an activity-on-node (AON) network objective of resource-constrained scheduling consists in de- where the nodes and the arcs represent the tasks and the veloping a schedule such that a set of tasks is completed as precedence relations respectively [6]. The precedence relations early as possible considering both the precedence relationships are described by the notation ≺. For example, given two plans and the restricted availabilities of resources. Πk and Πj, Πk ≺ Πj indicates that the plan Πk must be The scheduling problem we are facing can be thought as scheduled before Π . a particular case of the RCPSP problem: in our case we are j A plan is defined by the following notation: facing the problem of scheduling plans of tasks subject to both precedence, resource and temporal constraints. Also, rather k k Πk = (J ; :::; J ) than scheduling tasks as in the RCPSP, our problem aims at 1 nk selecting a maximum number of plans. where k is an arbitrary index for the plan, nk is the number k of tasks in the plan Πk and Ji , with 0 ≤ i ≤ nk is the i-th II. CASE STUDIES task of the plan Πk. The set of tasks is topologically sorted Nowadays, airborne platforms such as Remote Piloted Air (see3). Vehicles (RPAS) are employed in different scenario including A plan Πk could not be scheduled if at least one task arXiv:2102.03555v1 [cs.AI] 6 Feb 2021 k conflicts, surveillance and rescue [3]. In these scenario, air- Ji 2 Πk could not be scheduled due to unsatisfied constraints. k borne platforms operate in highly dynamic environments with A plan Πk is characterized by a set of resources R ⊆ R. a low predictability. In this context, onboard instruments (i.e. Definition 2. A graph G = (V; E) consists of a set of vertices sensors) allow the platform, hence the mission manager, to V , and a set of edges E ⊆ V ×V . In a directed graph the edges collect knowledge from the field. Sensors carried by RPAS are directed from one vertex to another. A directed acyclic are now able to perform a large panel of functions such as graph (or DAG) is a directed graph with no directed cycles: a image acquisition, spectrum analysis, and object tracking [4]. directed cycle is a path that starts from any vertex u and ends All these sensors play a major role in operation and their in u. optimization has become essential. Because of the criticality of the context and the mission’s Definition 3 (Topological Sort). A topological sort of a objectives, it is important to develop a method that orchestrates directed acyclic graph (DAG) G = (V; E) is a linear ordering the operations conducted by the sensors, such that the mission of all its vertices such that if the graph G contains an edge 2 (u; v) then u appears before v in the ordering [7,8]. If the interrupted once started. Moreover, we assume that each task graph contains a cycle, then no linear ordering is possible. could consume more than one resource during its execution. For example, in Figure1 a directed acyclic graph is showed, Definition 7 (Resource). A resource ρ is any hardware or for which a topological sorting can be found, since it has software tool that tasks can use to handle information. A no cycles. Figure2 shows a possible topological sorting for resource ρ has a limited availability value Bρ. Resources the graph in Figure1, where the different colors depict the are typically distinct in renewable and nonrenewable [6]: different frontiers in the corresponding DFS graph. renewable resources have a fixed value of availability in each time period, while nonrenewable resources have a fixed value Definition 4 . frontier f G = (V; E) (Frontier) A for a graph is of availability along the entire project’s time horizon. In the ad- f ⊆ V a set of nodes such that the maximum distance between dressed scheduling problem, we deal with renewable resources each node u 2 f and the root node in the corresponding DFS with a fixed availability value of Bρ = 1; 8ρ 2 R, in each graph is the same. time period. In particular, resources with a fixed availability value of Bρ = 1 are also called unary or disjunctive resources. Definition 8 (Resource availability). The availability Bρ 2 N0 7 5 of a resource ρ represents the maximum value of availability of the resource ρ in each time period. The abstract amount of usage of the resource ρ by the task 11 8 3 k k Ji , in each time period, is represented by biρ = f0; 1g. In this work we are concerned with solving the problem of scheduling plans of tasks by using a heuristic that maximizes the number of plans scheduled in a fixed time window, taking 2 9 10 into account precedence, time, and resource constraints. Fig. 1: A Directed Acyclic Graph. IV. PROBLEM STATEMENT A. Constraints 1) Temporal constraints: When a plan Πk is scheduled, for k k each task Ji 2 Πk both the starting time si and the comple- 3 5 7 8 11 2 9 10 k k k tion time Ci = si + pi must be inside the temporal window k k [Ws;We] and also inside the temporal window [ri ; di ], where k k k Fig. 2: A possible topological sorting for the DAG in Figure1. (ri ≥ Ws) ^ (di ≤ We); 8Ji 2 Πk: k k The value of si for a task Ji 2 P ik is calculated as the Definition 5 (Task). A task J k is an atomic operation that k k i maximum value between Ws, the release date ri of the task Ji a subset of resources must execute. Each task J k belongs k i augmented by the time lag δij, and the maximum completion uniquely to one plan Πk. Tasks are characterized by the k k time Cj for each predecessor Jj 2 Πk. Formally, following elements: k k • a processing time p 2 , p ≥ 1; i N0 i k k k k k si = max(Ws; ri ; maxj2predk (Cj + δij)); (1) • a release date ri 2 N; i k • a due date di 2 N0; k k k • a time lag δij 2 N, if Ji ≺ Jj ; k k k k k where pred is the set of predecessors of the task J . • a set of resources R = fρ ; :::; ρ g, R ⊆ R , which i i i 1 n i Given a set P of plans, the plan Π 2 P satisfies the the task J k is assigned to. k i temporal constraints if The tasks within a plan can be dependent by a precedence graph that provides the precedence constraints between the 9sk 8J k 2 Π :(sk ≥ rk) ^ (sk + pk ≤ dk): tasks. As for the plans, the precedence relations between tasks i i k i i i i i are described by the notation ≺. For example, given two tasks 2) Precedence constraints: In order to maintain the prece- k k k k J1 and J2 belonging to the same plan Πk, J1 ≺ J2 indicates dence constraints between plans, a set P of plans to be sched- k k that the task J1 precedes the execution of the task J2 .