Conflicting Bundle Allocation with Preferences in Weighted Directed Acyclic Graphs: Application to Orbit Slot Allocation Problems †
- We define a generic modeling framework for the path allocation problem with conflict (directed path allocation problem, or DPAP) and consider two optimization criteria (global utility and leximin).
- We instantiate this framework with two compact representations of conflicts, one based on a vertex conflict (vertex-constrained directed path allocation problem, or V-DPAP) and one based on a resource consumption conflict (resource-constrained directed path allocation problem, or R-DPAP)—note that V-DPAP comes from the path allocation in the directed acyclic graph (PADAG) problem defined in .
- We show that the decision problems associated with V-DPAP and R-DPAP are NP-complete, whatever the optimization criteria.
- We define several complete and incomplete allocation schemes for solving V-DPAP and R-DPAP.
- We evaluate all of the algorithmic approaches on dozens of orbit slot allocation benchmarks and discuss the obtained results.
2. Related Works
3. Directed Path Allocation Problems
- is a set of nodes; in our case, each node corresponds to an item that can be allocated to an agent, except for two specific nodes referred to as the source and the sink ;
- is the set of arcs of the acyclic graph, with the assumption that and are, respectively, the unique source and sink of the graph; an arc indicates that items and can be selected sequentially;
- is a utility function that associates a weight to each arc of the graph to represent a preference over the combinations of item selections; we assume that contains an arc from to labeled by utility 0, to deal with cases where no bundle of items can be selected in g.
- is a set of agents;
- is a set of single-source single-sink edge-weighted DAGs, as introduced in Definition 1;
- maps each graph g in to its owner a in ; we also denote by the set of graphs owned by agent a;
- is a path compatibility function that indicates whether a combination of paths (one path per graph) is feasible (value 1) or not (value 0).
4. V-DPAP: Vertex-Constrained Directed Path Allocation Problems
4.1. Framework Definition
4.2. Theoretical Complexity
- the set of nodes is ,
- the set of paths from to in g corresponds to the set of truth values for that satisfy the clause (decision diagram representation),
- the weight of every edge is set to 0, except for edges where , that have weight 1.
5. R-DPAP: Resource-Constrained Directed Path Allocation Problems
5.1. Framework Definition
- a set of disjunctive resources ;
- for each graph , a triple such that:
- and associate a start date and an end date, respectively, that together define a time window for each item;
- returns the resource required for each item. For any vertex , indicates that v does not require any resource in . In particular, the source and sink nodes do not consume any resource. Moreover, we assume that for two items v and belonging to the same graph and requiring the same resource in , the time windows of v and do not overlap;
- associates a duration with each item; resource must be used during time units within time window without any interruption (non-preemptive consumption).
- for all graphs , for all nodes , and ;
- there is no conflict for nodes in with respect to resource consumption. Formally, for each pair of distinct graphs g and , for each node and each node such that and (i.e., v and consume the same resource in ), either or holds.
5.2. Theoretical Complexity
- is a set of activities;
- is a function that assigns a processing time to each activity of A;
- is a function that assigns a release date to each activity of A;
- is a function that assigns a due date to each activity of A.
- the release dates are satisfied, i.e., , ;
- the machine performs at most one activity at each time step, i.e., with , either or holds;
- the maximum lateness is minimized, where .
- we consider a unique resource r;
- we consider an agent for each activity a in A;
- for each activity a in A, we consider the graph (illustrated in Figure 7a) that belongs to agent and that has the following features:
- its set of vertices is composed of three nodes: , , and ;
- its set of edges is composed of with a utility equal to 1, and , that both have a null utility;
- as illustrated in Figure 7b, node requires resource r during time units within time window ;
- the obtained R-DPAP is , with a function that assigns, for each activity a in A, agent to graph .
5.3. Relationship between R-DPAP and V-DPAP
- We consider the non-empty subsets S of one by one, following an increasing cardinality order. For a given set S, if there exists a subset of size such that is a conflict, S is marked as being a conflict but is not added to the set of minimal conflicts. Otherwise, we test whether there exists a schedule containing all the tasks in S. If not, S is marked as a conflict and added to the set of minimal conflicts.
- To determine whether there exists a schedule containing all the items in a set S, we use a dynamic programming algorithm. More precisely, we consider the subsets of S following an increasing cardinality order and we determine, for each subset , the minimum time at which all items in can be served in a feasible schedule. To do this, we start from and apply recursive formulas. If item belongs to graph g and is the last item visited, the minimum time at which the visit of i can end is given by , and visiting i at the latest position among the items in is feasible if and only if . From this, the minimum time at which all items in can be served in a feasible schedule is given by . It can be shown that at the end of the process, all the items in S can be scheduled if and only if . The dynamic programming algorithm described before has a time complexity that is exponential in the size of S; however, the number of requests is low for the practical application we are targeting.
6. V-DPAP Solution Methods
6.1. Utilitarian Allocation ()
6.2. Leximin Allocation ()
|Algorithm 1: Leximin algorithm.|
6.3. Approximate Leximin Allocation ()
|Algorithm 2: Approximate leximin algorithm.|
6.4. Greedy Allocation ()
6.5. Round-Robin Allocations ( and )
7. Experimental Evaluation
7.1.1. Constellation and Requests Features
7.1.2. From Requests to DAGs
- In the full satisfaction variant, each path goes through one orbit slot for each RT, except for a specific direct source-to-sink path that allows us to guarantee that there exists at least one feasible path for each request. In other words, it is not possible to skip one RT for an observation request, unless this request is not served at all.
- In the partial satisfaction variant, it is possible to skip some RTs for a request. In terms of generated graphs, it simply consists in adding edges with a null utility between successive virtual nodes, between the source and the first virtual node and between the last virtual node and the sink.
7.1.3. V-DPAP and R-DPAP Generation
- For generating the set of conflicts associated with a V-DPAP, we define a conflict for each pair of nodes corresponding to orbit slots that: 1. belong to the same satellite; 2. temporally overlap; and 3. are from different users. For the last assumption, we consider that it is possible to allocate to some user two orbit slots from the same satellite that overlap. In fact, as the allocation of an orbit slot consists in allowing an agent to dispose of the satellite during the associated time interval, two overlapping orbit slots and can be seen as a unique orbit slot that is the union of and . With this conflict generation scheme, all the conflicts obtained are binary. Note that it would be possible to compute these conflicts more finely, for instance, by following the approach proposed in .
- In the case of the function in R-DPAP, we follow the same process as in Example 4. More precisely, we create a resource for each satellite s of the constellation. Then, for each graph associated with an observation request r, for each vertex v in that corresponds to an orbit slot o (i.e., all vertices except source, sink, and the ones added between successive RTs), we define , (i.e., the temporal window associated with vertex v is exactly the temporal window associated with orbit slot o), (i.e., the duration associated with v is the minimum duration required) and where s is the satellite associated with orbit slot o. For each vertex v that is a virtual node, we consider that , , and . Note that R-DPAPs are next transformed into V-DPAPs as explained in Section 5.3, and in this case the conflicts obtained are not necessarily binary ones.
7.1.4. Instance Generation Parameters and Properties
7.1.5. Experimental Conditions
7.2. Results for the Full Request Satisfaction Mode
7.2.1. V-DPAP Results Analysis for
7.2.2. Sensitivity to Constellation Size
7.2.3. R-DPAP Results
7.3. Results for the Partial Request Satisfaction Mode
7.3.1. V-DPAP Results
7.3.2. R-DPAP Results
Data Availability Statement
Conflicts of Interest
|DAG||Directed acyclic graph|
|DPAP||Directed path allocation problem|
|PADAG||Path allocation in directed acyclic graph|
|ILP||Integer linear programming|
|MILP||Mixed integer linear programming|
|POI||Point of interest|
|Approximate leximin solver|
|Utilitarian MILP solver|
|Path round-robin solver|
|Node round-robin solver|
A Nash equilibrium is an allocation in which the modification of a path for a single agent does not improve its associated utility.
We have removed the worst 5% of values and the best 5% of values for the indicated range.
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|Number of orbital planes||2, 4, 8, 16|
|Number of satellites/plane||2|
|Scheduling horizon||Start||1 January 2020|
|Problems||Number of users||4|
|Requests||Number of requests/user||2|
|Requested observation Times||3 RTs/request|
|Maximum random time shift||1 h|
|Minimum slot duration||120 s|
|Satisfaction mode||full, partial|
|Algorithms||Type||, , , , ,|
|CPLEX time limit||120 s|
|Slots per RT||1.94||3.81||7.54||15.01|
|Slot duration (s)||618.10||616.44||616.91||616.66|
|Slots per RT||1.94||3.81||7.54||15.01|
|Slot duration (s)||618.10||616.44||616.91||616.66|
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Roussel, S.; Picard, G.; Pralet, C.; Maqrot, S. Conflicting Bundle Allocation with Preferences in Weighted Directed Acyclic Graphs: Application to Orbit Slot Allocation Problems. Systems 2023, 11, 297. https://doi.org/10.3390/systems11060297
Roussel S, Picard G, Pralet C, Maqrot S. Conflicting Bundle Allocation with Preferences in Weighted Directed Acyclic Graphs: Application to Orbit Slot Allocation Problems. Systems. 2023; 11(6):297. https://doi.org/10.3390/systems11060297Chicago/Turabian Style
Roussel, Stéphanie, Gauthier Picard, Cédric Pralet, and Sara Maqrot. 2023. "Conflicting Bundle Allocation with Preferences in Weighted Directed Acyclic Graphs: Application to Orbit Slot Allocation Problems" Systems 11, no. 6: 297. https://doi.org/10.3390/systems11060297