A constraint-based procedure for scheduling and allocation with unrelated machines

This paper addresses the resolution of mixed Task Scheduling and Resource Allocation problems in an integrated way. Time and resource constraint propagation is specified on top of a solving procedure directed by a backtracking algorithm. To evaluate the approach, experiments are reported on randomly generated instances. The main features of these instances are general precedence constraints, unrelated and versatile machines. 1. Problem statement Introduction. This paper addresses an integrated constraint-based approach to solve mixed Task Scheduling and Resource Allocation (in short TSRA) problems. A TSRA problem consists of a set of tasks to be performed by a set of renewable resources. Resources are disjunctive, i.e., they process only one task at a time, and are called machines. To each task is associated a set of machines able to process it knowing that every task is processed without interruption within a time window on only one machine. Moreover, certain tasks may be linked together by general precedence constraints. Finally, the task duration depends on the machine on which the task is performed, i.e., the machines are unrelated. The objective is to derive necessary conditions on task scheduling and resource allocation for the existence of a schedule for which the makespan is lower than or equal to an upper bound. According to the notation scheme given in [2], the problem is equivalent to − i i d , r , prec RMPM (MPM stands for multi-purpose machines) which is NPcomplete considering NP-hardness of max i L outtree , 1 p P = . Some efficient algorithms are known for each independent problem, scheduling and allocation, but they are unable to reach optimality for the mixed problem. For an integrated solving of some TSRA problems, heuristic approaches are investigated in [1, 3, 6]. Since constraint-based approaches have been proved to be an efficient and flexible way for tackling scheduling problems, an extension taking into account allocation constraints seems a promising way of research. To our knowledge, the only works that investigated a constraint-based approach for TSRA problems have been proposed so far in [5, 9, 11]. In these papers task durations are resource-dependent. In [9], a time-window is associated with every task for each possible allocation. Constraint propagation leads to narrowing time-windows; this updating may suppress some alternatives for the allocation. Moreover, specific developments involve the aggregation in a cumulative resource of the set of resources that may be allocated to several tasks. This paper follows the principles presented in [5] and proposes a backtracking algorithm to solve TSRA problems in an integrated way. Typology. In [11], TSRA problems are cast into four categories. The distinctions between categories are made according to two underlying features of the resources which are the versatility (i.e., does a resource may execute various kinds of tasks?) and the similarity (i.e., does the duration of a task depend on the resource used to achieve it?). Hybrid Flow Shop problems [8] fall into the category of TSRA problems with nonversatile and similar (identical) resources . Job Shop problems with alternative sets of resources [9] are included in the category of TSRA problems with nonversatile and nonsimilar (unrelated) resources . Multi-Purpose Machines Problems considered in [6] and Multi-Processor Job Shop problems [3] are linked to TSRA problems with versatile and identical resources . Finally the most general type of problems refers to TSRA problems with versatile and unrelated resources; to the best of our knowledge, they are only studied in [5] for which the authors propose time and resource constraint propagation mechanisms. In the sequel, we are concerned by solving this last category TSRA problems. Modelling. A set T of tasks has to be achieved by a set M of machines. To a task T i ∈ are associated its start time i st , its finish time i ft , and the set of allowed machines ? ? i ⊆ ; its duration, denoted by μ , i p , depends on the machine i ? μ∈ it is allocated to. It follows that the duration of task i belongs to the interval ] i,μ M μ , i,μ M μ p max p min [ i i ∈ ∈ . Let μ T be the set of tasks to be performed on machine μ . For a task μ T i ∈ , one has i i r st ≥ and i μ , i i d p st ≤ + where i r stands for the release date and i d for the due date. A precedence relation between two tasks i ( μ T i ∈ ) and j is written μ , i i j p st st + ≥ . Resource capacity constraints state that two tasks assigned to a common machine must be sequenced: μ T j , i ∈ ∀ , ) 0 ft st ( i j ≥ − ∨ ) 0 ft st ( j i ≥ − . 2. Constraint propagation Precedences, limit times, and durations form a set of time constraints on which graph algorithms (e.g., Floyd-Warshall or Bellman-Ford procedures) can deduce the tightest adjustments of start and finish times in a polynomial complexity. For the processing of resource constraints, one way is to model these constraints like disjunctions of time constraints and to extend graph algorithms to handle them [5]. This extension consists in removing the disjunctions if an inconsistency with the initial time constraints appears. These deletions, realised following a method strongly related to the Upper-Lower Tightening algorithm proposed in the area of Temporal Constraint Satisfaction Problems [10], may resolve some conflicts in resource sharing or suppress some inconsistent resource allocations. The main drawback of this approach is to forget the semantics of the resource constraints and thus to do not exploit the specificity of joint scheduling and allocation. Another way is to separately consider time and resource constraints. The goal is to specify constraint propagation mechanisms for resource allocation and analyse their interaction with the propagations obtained on time constraints. Task scheduling. We retain two classical rules to detect precedences between each pair of conflicting tasks. Rule (1) is based on temporal reasoning [4]: for , T j , i μ ∈ if μ , j μ , i i j p p r d + < − then i j p . Rule (2) is based on energetic reasoning [7]. In the disjunctive case, the energy required by i on μ over an interval ∆ , termed μ , i w , is given by the intersection of ∆ with the processing of i. This energy is minimal, termed μ , i w , when the processing of i is realised for positions that overlap ∆ as less as possible. Rule (2) is expressed as follows: for , T j , i μ ∈ if