solving the shortest path tour problem
TRANSCRIPT
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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