Solving The Shortest Path Tour Problem

27
Outline Introduction Applications Problem Description The state-of-the-art A modified graph method A dynamic programming based algorithm Computational results Conclusions Solving The Shortest Path Tour Problem P.Festa, F.Guerriero, D.Lagana, R.Musmanno by Shokirov Nozir Sabanci University Shokirov Nozir Solving The Shortest Path Tour Problem

Transcript of Solving The Shortest Path Tour Problem

Page 1: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Solving The Shortest Path Tour ProblemP.Festa, F.Guerriero, D.Lagana, R.Musmanno

by

Shokirov NozirSabanci University

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 2: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Outline1 Introduction2 Applications

Robot Motions ControlWarehouse ManagementScientific Contribution

3 Problem Description4 The state-of-the-art5 A modified graph method

On the reduction of the SPTP to the SPPSoving the SPTP as SPP

6 A dynamic programming based algorithm7 Computational results

Test Problems8 Conclusions

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 3: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Introduction

The shortest path tour problem (SPTP) is avariant of the shortest path problem (SPP) and

appeared for the first time in the scientificliterature in Bertsekas’s dynamic programming

and optimal control book in 2005.

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 4: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Robot Motions ControlWarehouse ManagementScientific Contribution

Robot Motions Control

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 5: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Robot Motions ControlWarehouse ManagementScientific Contribution

Applications of the SPTP arise for example in thecontext of the manufacture workpieces,where a robot has to perform at least one

operation selected from a set of S types ofoperations. In such case, the problem may bemodeled as a SPTP in which operations areassociated with nodes of a directed graphand the time needed for a tool change isrepresented by the distance between two

nodes.

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 6: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Robot Motions ControlWarehouse ManagementScientific Contribution

Warehouse Management

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 7: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Robot Motions ControlWarehouse ManagementScientific Contribution

Main Scientific Contribution

1 Analysing some basic theoretical properties of the SPTP2 Designing a dynamic programming-based algorithm

(DPA) for solving it3 Efficiently reducing SPTP to a classical SPP through

modified graph algorithm (MGA)

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 8: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Problem Description

Consider a directed graph G = (N ,A) defined by a set ofnodes N := 1, ..., n and a set of arcs A := (i , j) ∈ NxN : i 6= j ,where |A| = m. A non-negative length cij is assigned to eacharc (i , j) ∈ A. Moreover, lets S denote a certain number ofnode subsets T1, ...,TS such that Th ∩ Tk = , h, k = 1, ..., S ,h 6= k .Given two nodes i1, iπ ∈ N , i1 6= iπ, the path Pi1,iπ from i1 to iπ isdefined as a sequence of nodes Pi1,iπ = i1, ..., iπ such that(ij , ij+1) ∈ A, j = 1, ..., π − 1.Length of path Pi1,iπ is l(Pi1,iπ) =

∑π−1j=1 cj ,j+1

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 9: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Problem Description

The SPTP aims at finding a shortest path Ps, dfrom origin node s ∈ V to destination d ∈ Vin the directed graph G, such that it visits

successively and sequentially the followingsubsets Tk , k = 0, ..., S + 1, such that T0 = sand TS+1 = d . Note that sets Tk , k = 1, ..., S ,must be visited in exactly the same order in

which they are defined.

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 10: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Instance on a small directed graph G

N = {s = 1, 2, 3, 4, 5, 6, d = 7} ,S = 2,T0 = {s = 1} ,T1 = {3} ,T2 = {2, 4} ,T4 =

{d = 7} .

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 11: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

The state-of-the-art

The state of art consits of the expanded graphmethod (EGA) propesed by Festa.

The EGA relies on a polynomial-time reduction algorithm thattransforms any SPTP instance defined on a single-stage graphG into a single-source single-destination SPP instance definedon a multi-stage graph G’ = (V’, A’) with S+2 stages eachreplicating G, and such that V’ = 1, ... , (S+2)n and |A’| =(S+1)m.

The overal worst coplexity of EGA is O(Smlogn)

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 12: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

SPTP reduction to SPP

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 13: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

On the reduction of the SPTP to the SPPSoving the SPTP as SPP

On the reduction of the SPTP to the SPPGiven an istance of the SPTP on a directed graph G = (N, A) the followingdefinition is applied.

Definition

Let G (a) = (N (a),A(a), c (a)) be a weighted directed graph obtained from G insuch a way that;

1 N (a) =S+1⋃k=0

Tk

2 A(a) =S⋃

k=0A(a)k where A

(a)k

:={(i, j) ∈ Tk xTk+1 : i ∈ Tkandj ∈ Tk + 1

};

3 c (a) : A(a) → Z+ is a function that associates an integer non-negativenumber c (a) to each arc (i , j) ∈ A(a), where c

(a)ij := l(Pij) is the length of

a shortest path from node i ∈ Tk to node j ∈ Tk+1 on graph G

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 14: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

On the reduction of the SPTP to the SPPSoving the SPTP as SPP

Modefied Graph Algorithm (MGA)

TheoremThe worst case computational complexity of MGA is O(n3)

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 15: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

A dynamic programming based algorithm

We view the SPTP as an extension of the weight constrainedshortes path problem. To each path Ps,i from node s to node iwe associate a label yi , which stores information about thelength of the path and the resource consumption along thepath, that is, yi = (l(Ps,i , ri)).

TheoremThe computational complexity of DPA is O(S2n3cmax).

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 16: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Test Problems

(a) Table1: Grid random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 17: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Test Problems

(b) Table2: Fully random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 18: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on grid random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 19: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on grid random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 20: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on grid random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 21: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on fully random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 22: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on fully random networks

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 23: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on dense network-based test problems

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 24: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Test Problems

Results on dense network-based test problems

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 25: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Conclusions

1.This paper exhibits the results of the studyconcerning a variant of SPP, namely the SPPT

2. Two competitve solving algorithms are provided

1 Modefied graph-based algortihm (MGA)2 Dynamic programming-based algorithm

(DPA)

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 26: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Conclusions

3. The DPA outperforms MGA on the SPTP instances that aregenerated from grid random netwroks, while the latter ismore competitive than the former as long as fully randomand fully dense netwroks are involved.

4.The most efficient proposed algorithm outperforms clearlythe state-of-art solution strategy

5.Perfromance of proposed algorithms depend mainly on thestructure of the networks.

Shokirov Nozir Solving The Shortest Path Tour Problem

Page 27: Solving The Shortest Path Tour Problem

OutlineIntroductionApplications

Problem DescriptionThe state-of-the-art

A modified graph methodA dynamic programming based algorithm

Computational resultsConclusions

Questions and Comments

Shokirov Nozir Solving The Shortest Path Tour Problem