Energy-Efficient Fast Delivery by Mobile Agents Andreas B¨ artschi joint work with Thomas Tschager ETH Department of Computer Science
Model of delivery Setting undirected graph G = ( V , E ) with edges e ∈ E having lengths ℓ e 400 km 300 km 400 km one package: source s and target t s t 400 km 600 km k agents, each starting at a node p i , energy consumption w i & velocity v i Assumptions Task agents cooperate by Find an efficient delivery in terms of global, centralized coordination energy consumption E � terms of form w i · ℓ e handovers possible at nodes V � terms of form v − 1 delivery time T · ℓ e as well as inside edges i ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 2 / 11
1 Introduction Model of delivery Outline 2 Previous work Agents with weights only Agents with velocities only 3 Combining the two measures Combining energy- and time-efficiency OPT characterization Algorithm ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 3 / 11
Agents with weights Each agent has its individual energy consumption (weight) w i . Can we optimize the total energy consumption E needed to deliver the package? w 4 = 7 ℓ/ 100km w 5 = 5 ℓ/ 100km 400 km 300 km 400 km s t 400 km 600 km w 1 = 12 ℓ/ 100km w 2 = 6 ℓ/ 100km w 3 = 5 ℓ/ 100km not on shortest path vertex handovers decreasing weights [1] B., Chalopin, Das, Disser, Graf, Hackfeld, Penna: Energy-Efficient Delivery by Heterogeneous Mobile Agents ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 4 / 11
Agents with velocities Each agent has its individual velocity v i . Can we optimize the total time T needed to deliver the package? v 4 = 40km /h v 5 = 60km /h 400 km 300 km 400 km s t 400 km 600 km v 1 = 20km /h v 2 = 20km /h v 3 = 76km /h not on shortest path (multiple) in-edge-handovers increasing velocities [2] B., Graf, Mihal´ ak: Collective fast delivery by energy-efficient agents ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 5 / 11
Combining energy- and time-efficiency Each agent has its individual weight w i and velocity v i . We want a delivery that is both efficient in its energy consumption and its delivery time: Fastest delivery among all energy-optimum ones. ⇒ this talk Task: lexicographically minimize ( E , T ). Energy-optimum delivery among all fastest ones. Task: lexicographically minimize ( T , E ). ⇒ NP-hard [2] Tradeoff between the two measures. Task: minimize ε · T + (1 − ε ) · E , ε ∈ (0 , 1). [2] B., Graf, Mihal´ ak: Collective fast delivery by energy-efficient agents ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 6 / 11
Agents with weights and velocities Each agent has its individual weight w i and velocity v i . Among all energy-optimum deliveries, can we find the fastest? ( E , T ) = (84 , 38) w 4 = 7 ℓ/ 100km w 5 = 5 ℓ/ 100km v 4 = 100km /h v 5 = 80km /h 400 km 300 km 400 km s ( E , T ) = (84 , 40) t 400 km 600 km w 1 = 12 ℓ/ 100km w 2 = 6 ℓ/ 100km w 3 = 5 ℓ/ 100km v 1 = 10km /h v 2 = 30km /h v 3 = 50km /h decreasing tuples ( w i , v − 1 not on shortest path 0 or 1 in-edge-handovers ) i ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 7 / 11
Characterization of an optimum delivery Theorem (OPT characterization) There is an optimum delivery, with the involved agents denoted by 1 , . . . , i , i + 1 , . . . , k, in which the following holds for each consecutive pair of agents: Decreasing weights: w i ≥ w i +1 . If w i = w i +1 , then v i < v i +1 . If w i = w i +1 , then agent i + 1 does not move without the package. v i < v i +1 < . . . w 1 = w 2 = . . . > w i = w i +1 = . . . > . . . = w k x y � �� � � �� � � �� � W 1 W 2 W 3 ⇒ First look at the problem for each weight class W j separately. ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 8 / 11
Uniform energy consumption weights 7 4 (3 , 5) 9 5 1 10 1 (3 , 1) 6 2 6 11 (3 , 4) 9 2 9 2 19 4 s t s t 4 4 1 4 2 9 6 14 5 2 8 (3 , 2) Example: 4 agents, weight 3, velocities v 1 = 1 , v 2 = 2 , v 3 = 4 , v 4 = 5. Approach: 1 Move closest agent to source s . Costs ( E , T )[ p 1 ] = (3 · 3 , 3 / 1) = (9 , 3). 2 Order agents by increasing velocity. Transform graph to DAG. Compute ( E , T )[ p i ] = the energy consumption E [ p i ] and delivery time T [ p i ] of an optimum delivery of the package from s to p i , using only agents 1 , . . . , i − 1. ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 9 / 11
Uniform energy consumption weights 7 4 (3 , 5) 9 5 1 10 1 (3 , 1) 6 2 6 11 (3 , 4) 9 2 9 2 19 4 s t s t 4 4 1 4 2 9 6 14 5 2 8 (3 , 2) Example: 4 agents, weight 3, velocities v 1 = 1 , v 2 = 2 , v 3 = 4 , v 4 = 5. Approach: 1 Move closest agent to source s . Costs ( E , T )[ p 1 ] = (3 · 3 , 3 / 1) = (9 , 3). 2 Order agents by increasing velocity. Transform graph to DAG. Compute ( E , T )[ p 2 ] = ( E , T )[ p 1 ] + (5 · 3 , 5 / 1) = (24 , 8) . ( E , T )[ p 3 ] = min { ( E , T )[ p 1 ] + (9 · 3 , 9 / 1) , ( E , T )[ p 2 ] + (4 · 3 , 4 / 2) } = (36 , 10) . ( E , T )[ p 4 ] = min { . . . , ( E , T )[ p 2 ] + (4 · 3 , 4 / 2) , . . . } = (36 , 10) . 3 Compute optimum delivery time among energy-optimum deliveries: ( E , T ) = (66 , 12). ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 9 / 11
Vertex handovers i i + 1 i i + 1 i i + 1 u v u v u v w i > 2 w i +1 w i < 2 w i +1 If w i � = 2 w i +1 , then agent i does not handover the package to i + 1 inside an edge. Assume we have w i � = 2 w j ∀ i , j . ⇒ We can use the precomputed weight class solutions. Define the subproblems ( E , T )[ j , y ] = the energy consumption E [ j , y ] and delivery time T [ j , y ] of an optimum delivery of the package from s up to node y , using only agents from the first j weight classes W 1 , . . . , W j . ⇒ We can compute ( E , T )[ j , y ] from all smaller subproblems ( E , T )[ j − 1 , x ]! ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 10 / 11
In-edge handovers NO i + 1 i + 2 ? ? i + 1 i velocity u v u v w i = 2 w i +1 = 4 w i +2 arrival time at u We can have at most one in-edge-handover per edge. But: Which agents are involved in an in-edge-handover? Pareto frontier! Adapt previous methods as follows: 1 Incorporate the Pareto frontier into the weight class computations. 2 Incorporate in-edge-handovers into the main dynamic program. Running time: Preprocessing O ( APSP + k + | V | ). O ( k | V | 2 + APSP ) Per weight class: O ( | V | · k 2 + | V | 2 · k ). Main dynamic program: O ( k · | V | · | V | ). ETH Department of Computer Science Andreas B¨ artschi FCT Bordeaux September 13, 2017 11 / 11
Recommend
More recommend
