Ev Evolutionary Computation plus Dynamic Pr Programming for the - - PowerPoint PPT Presentation

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Ev Evolutionary Computation plus Dynamic Pr Programming for the - - PowerPoint PPT Presentation

Oct 19, 2023 30 likes •225 views

Ev Evolutionary Computation plus Dynamic Pr Programming for the Bi-Ob Objec ective e Travel elling Thie Th ief f Proble lem Junhua Wu, Sergey Polyakovskiy, Markus Wagner, Frank Neumann Frank.Neumann@Adelaide.edu.au Project page:

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Ev Evolutionary Computation plus Dynamic Pr Programming for the Bi-Ob Objec ective e Travel elling Th Thie ief f Proble lem

Junhua Wu, Sergey Polyakovskiy, Markus Wagner, Frank Neumann Frank.Neumann@Adelaide.edu.au Project page: https://cs.adelaide.edu.au/~optlog/research/ttp.php Or google “travelling thief Adelaide” Tuesday, July 17, 10:40-12:20, Conference Room D (3F)

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Composed of the merging of the Traveling Salesman Problem and the Knapsack Problem

1 3 4 2 5 8 7 6

The Travelling Thief Problem (TTP)

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Composed of the merging of the Traveling Salesman Problem and the Knapsack Problem

1 3 4 2 5 8 7 6

The Travelling Thief Problem (TTP)

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The Travelling Thief Problem (TTP)

Composed of the merging of the Traveling Salesman Problem and the Knapsack Problem

1 3 4 2 5 8 7 6

1 2 5

17

8 10 11 13 14 15 7 9 3 4 6 12

Knapsack

16

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The Travelling Thief Problem (TTP)

Composed of the merging of the Traveling Salesman Problem and the Knapsack Problem

1 3 4 2 5 8 7 6

2 5

17

6 8 10 11 12 13 14 15 7 9 3 4 1 2 5

14

7

Knapsack

1 16

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THE TRAVELING THIEF PROBLEM (TTP)

Goal: Visit each city exactly once, maximising the total profit 𝑄 such that the total weight does not exceed the knapsack capacity 𝑋, where 𝑄 is defined as: 𝑄 = $

%&' (

𝑞% 𝑦% − 𝑆 $

%&'

  • 𝑢%,%0'

where 𝑦% = 1 0 depending on whether the item 𝑗 is picked 1 or not 0 , and 𝑢%,4 is defined as: 𝑢%,4 = 𝑒(Π%, Π4) 𝑤(:; − 𝑋

<=

𝑤(:; − 𝑤(%- 𝑋 where Π% is the city at tour position 𝑗 in tour Π, and 𝑋

<= is the

current weight of the knapsack at city Π%.

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The Bi-Objective TTP

a natural extension: maximise the reward for a given weight of collected items, or determine the least weight subject to bounds imposed on the reward

  • Objective one: profit P as defined before
  • Objective two: total accumulated weight
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Packing-While-Travelling (PWT)

  • …
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Weight Total Reward

ρ1->(z1, w1, ) ρ1->(z2, w2) ρ1->(z3, w3) ρ1->(z4, w4) ρ1->(z5, w5) π1

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Weight Total Reward

(z1, w1) (z2, w2) (z3, w3) (z4, w4) (z5, w5) (π1, ρ1) (π2, ρ2) (π3, ρ3) (π4, ρ4) (π5, ρ5)

(the “natural” approach would be the following)

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Solving the Bi-Obj. TTP

  • Many single-objective TTP heuristics take a good

TSP tour as a starting point. What does this mean here?

  • TSP solvers; CONCORDE (CON), ACO, LKH and LKH2

1000 2000 3000 4000

Weight

  • 5000

5000

Total Reward Max reward:4791.466 Corresponding tour length:586

eil76_n75_uncorr_01.ttp, inver over

  • 2000

2000 4000 6000 8000 10000

Total Reward

ACO_Bounded01 CON_Bounded01 INV_Bounded01 LKH_Bounded01 LKH2_Bounded01 ACO_Bounded06 CON_Bounded06 INV_Bounded06 LKH_Bounded06 LKH2_Bounded06 ACO_SimilarWeights01 CON_SimilarWeights01 INV_SimilarWeights01 LKH_SimilarWeights01 LKH2_SimilarWeights01 ACO_SimilarWeights06 CON_SimilarWeights06 INV_SimilarWeights06 LKH_SimilarWeights06 LKH2_SimilarWeights06 ACO_Uncorrelated01 CON_Uncorrelated01 INV_Uncorrelated01 LKH_Uncorrelated01 LKH2_Uncorrelated01 ACO_Uncorrelated06 CON_Uncorrelated06 INV_Uncorrelated06 LKH_Uncorrelated06 LKH2_Uncorrelated06

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Indicators

Def 3.2: Given q different DP fronts, let 𝜚 denote a set of possible unique solution points derived by 𝜐1.. 𝜐q. Then 𝜕 is a Pareto front formed by the points of 𝜚 and 𝜕 is named as the surface of 𝜚. Given a tour 𝜐𝜌, and its corresponding solution set T𝜌:

  • Surface Contribution: number of objective vectors

contributed by T𝜌

  • Hypervolume: volume covered by T𝜌 w.r.t (0,C)
  • Loss of Contribution:
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Parent Selection Mechanisms

  • Rank-Based Selection (RBS), Fitness-Proportionate

Selection (FPS), Tournament Selection (TS), Arbitrary Selection (AS), Uniformly-at-Random Selection (UAR)

Crossover and Mutation Operators

  • TSP-only: multi-point crossover, 2-opt mutation,

jump

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Experimental Study

  • 2 indicators X 8 parent selection strategies
  • TTP instances from the classes eil51, eil76, eil101;

three knapsack types Assessment

  • 30 repetitions, Welch’s t-test with UAR as a

baseline (like the Student's t-test, but more reliable when the two samples have unequal variances and unequal sample sizes)

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5 10 15 20 25 AS-BST AS-EXT FPS RBS-EXP RBS-HAR RBS-IQ TS Bounded SimilarWeights Uncorrelated 5 10 15 20 25 30 AS-BST AS-EXT FPS RBS-EXP RBS-HAR RBS-IQ TS Bounded SimilarWeights Uncorrelated 5 10 15 20 25 30 AS-BST AS-EXT FPS RBS-EXP RBS-HAR RBS-IQ TS Bounded SimilarWeights Uncorrelated 5 10 15 20 25 AS-BST AS-EXT FPS RBS-EXP RBS-HAR RBS-IQ TS Bounded SimilarWeights Uncorrelated

animation with “appear”

Loss Surface Contribution Loss Hypervolume

hypervolume hypervolume total reward total reward Note: bars are sums of log-scaled p-values

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Comparison of bi-obj. approaches with single-

  • bjective MA2B

MA2B by El Yafrani and Ahiod [GECCO’16] Fitness-Proportionate Selection Loss of Hypervolume Loss of Surface Contribution

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Summary

  • Bi-Objective TTP: profit vs. weight
  • Dynamic programming provides provably optimal

trade-off fronts for a given tour

  • Indicator-based EA with a population of tours: with

”loss of surface contribution” and “loss of hypervolume”

  • Best bi-objective approaches beat single-objective

state-of-the-art