Disciplina en el interior del centro anátomo-forense

Leave Office

18:00 Arrive Home

< 210 mins (SI) Arrive

Cinema Leave

Cinema Comedy Movie (SI)

Arrive Restaurant

Leave Restaurant Chinese Restaurant (SI,CR)

(movie m) ∧ (genre g) ∧ (hasGenre m g) ∧(surface g ’Comedy’)

(restaurant r) ∧ (cuisine c) ∧ (servesCuisine r c) ∧(surface c ’KOREAN’)

Figure 2-1: A TPN for Simon and Christian’s trip

two restaurant options. In the TPN graph, these grounded options for each activity share one common start event, which is represented by a double circle and indicates that the subsequent episodes are alternatives. In addition, each grounded activity is associated with a duration (highlighted in the figure). These durations encode the length of the movie or dinner. The constraints for traversals between locations (Table 2.1) are omitted from the graph to save space.

Leave Office

18:00 Arrive Home

< 210 mins 20:00 Joy

at AMC 16 (124 mins) 19:30 Norm of the

North at AMC 20 (90 mins)

Panda Express (30 mins)

Magic Wok (30 mins)

Figure 2-2: An expanded TPN with activity candidates

Next, Uhura passes the expanded TPN to BCDR to fill in the details of the plan, select the candidate for each activity, and determine their orders while meeting all the requirements. The result is a complete plan with grounded activities for both Simon and Christian. If no feasible plan can be found that meets all requirements, as in this example, BCDR will try relaxing the temporal bounds of some episodes in order to make room for completing all activities. This is the situation encountered by Simon in this example: due to the long travel times to and from the candidate

Traversal Durations

Office→ PE: [40,65] PE → Home: [25,30]

Office→ MW: [30,35] MW→ Home: [20,25]

Office→ AMC 16: [40,65] AMC 16→ Home: [25,30]

Office→ AMC 20: [30,35] AMC 20→ Home: [20,25]

PE →AMC 16: [25,45] AMC 16→ PE : [25,45]

PE →AMC 20: [35,55] AMC 20→ PE: [35,55]

MW→ AMC 16: [35,45] AMC 16→ MW: [35,45]

MW→ AMC 20: [40,55] AMC 20→ MW: [40,55]

Table 2.1: Travel times between locations (PE stands for Panda Express, and MW stands for Magic Wok)

Chinese restaurants, no solution can be found that meets all temporal requirements.

Hence Uhura engages Simon and Christian, and initiates a discussion about possible resolutions for his problem.

Uhura: Simon, you may have dinner at Magic Wok then watch the 8pm Joy at AMC 16. However, due to the length of the movie you have to remove the constraint on the time of arriving home. Is that OK?

Simon: No, I cannot delay my arrival time.

Uhura: OK, then Simon can you remove the constraint on departure time from Office? If so you may watch Norm of the North at 7:30pm, and arrive home on time.

Simon: No I cannot leave office before 6pm.

Uhura: Simon and Christian, How about not having dinner tonight?

Christian: That’s fine.

Simon: Sounds good. Thank you.

In this example, BCDR proposed different relaxations that resolves the conflicts in the Simon and Christian’s plan. Since they cannot suspend the constraints on the departure and arrival times, the first two proposals were rejected. BCDR incorporates their inputs on earlier solutions, and kept proposing new ones that respects all their feedback until an agreement is reached, which enables a feasible plan for both Simon and Christian (Figure 2-3).

18:00

Leave Office < 210 mins Arrive Home

19:30 Norm of the North at AMC 20 (90 mins) Drive to AMC 20

[30,35] Drive Home

[20,25]

(a) Simon’s trip with suspended dinner episode

17:45

Leave Office < 180 mins Arrive Home

Drive Home [40,55]

(b) Christian’s trip with suspended dinner episode Figure 2-3: A solution enabled by relaxed cuisine constraint

This example demonstrates the desired features of BCDR on resolving over-subscribed temporal plans using discrete relaxations: it allows Uhura to work col-laboratively with the users to resolve conflicts in over-subscribed plans, enumerating discrete relaxations in best-first order and providing rationale for the relaxations made.

2.2.2 Definitions

We extend the definition of TPN with an additional set of relaxation episodes, 𝑅𝐸, to support discrete relaxations. For its solutions, we also include a set of suspended episodes from 𝑅𝐸 that are necessary for making the solution consistent. Formally, we define a relaxable TPN as the following:

Definition 5. A relaxable TPN is a 9-tuple ⟨𝑃, 𝑄, 𝑉, 𝐸, 𝑅𝐸, 𝐿_𝑒, 𝐿_𝑝, 𝑓_𝑝, 𝑓_𝑒⟩, where:

∙ 𝑃 is a set of controllable finite domain discrete variables;

∙ 𝑄 is the collection of domain assignments to 𝑃;

∙ 𝑉 is a set of events representing designated time points;

∙ 𝐸 is a set of episodes between pairs of events 𝑣_𝑖, 𝑣_𝑗 ∈𝑉;

∙ 𝑅𝐸⊆𝐸 is a set of relaxable episodes that can be suspended;

∙ 𝐿𝑒 : 𝐸 → 2^𝑄 is a guard function that attaches conjunctions of assignments in 𝑄, 𝑞_𝑖 ∈𝑄, to some episodes 𝑒_𝑖 ∈𝐸;

∙ 𝐿𝑝 : 𝑃 → 2^𝑄 is a guard function that attaches conjunctions of assignments in 𝑄, 𝑞_𝑖 ∈𝑄, to some discrete variable 𝑝_𝑖 ∈𝑃;

∙ 𝑓𝑝 :𝑄→ ℛ⁺ is a function that maps each assignment to every controllable discrete variable, 𝑞_𝑖 ∈𝑄, to a positive reward value;

∙ 𝑓𝑒 : 𝑅𝐸 → ℛ⁺ is a function that maps the suspension to one relaxable temporal constraint𝑒_𝑖 ∈𝑅𝐸, to a positive cost value.

The solution to a relaxable TPN is a set of assignments to variables in𝑃 such that all activated episodes that are not suspended (not in 𝑅_𝑒) are temporally consistent.

Definition 6. The solution to a relaxable TPN is a 4-tuple ⟨𝐴, 𝑆, 𝑅𝑒, 𝐸^′⟩, where:

∙ 𝐴 is a complete set of assignments to variables in 𝑃;

∙ 𝑆is a set of additional assignments that defines ordering over activities in the TPN;

∙ 𝑅_𝑒 is a set of episodes in 𝑅𝐸 that are suspended.

∙ 𝐸^′ is a set of episodes that encodes the traversal activities between locations, gener-ated by the routing function. Each𝑒^′ ∈𝐸^′ encodes the traversal time associated with an agent’s movement between locations specified in 𝐴, following the order defined by𝑆.

In addition, given a solution, we may also define if its set of discrete relaxations are minimal. A set of discrete relaxations of a temporally consistent solution is minimal if and only if none of its strict subset still makes the solution temporally consistent:

Definition 7. A solution ⟨𝐴, 𝑆, 𝑅_𝑒, 𝐸^′⟩ has a minimal set of discrete relaxation if and only if:

∙ ⟨𝐴, 𝑆, 𝑅_𝑒, 𝐸^′⟩ is a temporally consistent solution;

∙ ⟨𝐴, 𝑆, 𝑅^′_𝑒, 𝐸^′⟩, where 𝑅^′_𝑒 ⊂𝑅𝑒 is not temporally consistent.