applied integer programming: modeling and solution · applied integer programming: modeling and...

Traveling Salesman ProblemTransformation to the TSP

Applications of the TSP

Applied Integer Programming: Modeling andSolution

Chen, Batson, DangSection 6.1 - 6.3

Henrik Fredriksson

Blekinge Institute of Technology

April 15, 2015

Henrik Fredriksson Applied Integer Programming: Modeling and Solution



Modeling Combinatorical Optimization Problems II

1 Traveling Salesman ProblemImportance of the TSP

2 Transformation to the TSPShortest Hamiltonians PathTSP with Repeated City VisitsMultiple TSPClustered TSPMaximum TSP

3 Applications of the TSPMachine Sequencing Problems in Various Manufacturing SystemsSequencing Problems in Electronic IndustryVehicle Routing for Delivery and Dispatching



Applications of the TSPImportance of the TSP

Traveling Salesman Problem

A traveling salesman is to visit a number of cities and the distanceconnecting two cities are known; the problem is to find a shortest routethat starts from a home city, visits other cities exactly once, and returnsto the home city.

Became popular in () (Dantzig, Fulkerson and Johnson)

Perhaps the most well-studied COP

Thousands of publications. Over 1000 distinct papers cited

Representative of NP-hard COPs

Primary driving force in novel optimization and solution algorithms

Many AI algorithms, GA, SA, Tabu search, heuristics developed to(at least partly) solve the TSP




Milestones of TSP instances solved to optimality

Year No. of Cities Data Set Research Team

49 dantzig42 Dantzig, Fulkerson, Johnson 64 random points Held and Karp 67 random points Camerini, Fratta, Maffioli 120 grl20 Grotschel 318 lin318 Crowder and Padberg 532 att532 Padberg and Rinaldi 666 gr666 Grotschel and Holland 2392 pr2392 Padberg and Rinaldi 7397 pla7397 Applegate, Bixby, Chvatal, Cook 13, 509 usa13509 Applegate, Bixby, Chvatal, Cook 15, 112 dl5112 Applegate, Bixby, Chvatal, Cook 24, 978 sw24978 Applegate, Bixby, Chvatal, Cook 33, 810 pla33810 Applegate, Bixby, Chvatal, Cook 85, 900 pla85900 Applegate, Bixby, Chvatal, Cook




Slow progress ( to mid-s )

Lack of applications. Could only solve small instances of the problem

Rapid progression (mid-s - mid-s)

Increased computational capability and introduction ofbranch-and-cut technique

TSP can be used as a benchmark for IP algorithms




Some definitions

Definition

Let G = (V ,E ) be a graph where V is the set of vertices (nodes) and Eis the set of edges (arcs). If the TSP is defined over a directed graph,then we have a asymmetric TSP. If the the TSP is defined over aundirected graph, then we have a symmetric TSP.

Definition

A route is sequence of distinct nodes

(v1, v2, . . . , vn−1, vn)

such that (vi , vi+1) ∈ E ,∀i = 1, . . . n − 1. If the route contains all nodesof the graph, it is called a Hamiltonian route




Definition

A cycle is sequence of distinct nodes

(v1, v2, . . . , vn−1, vn, v1)

such that (vi , vi+1), (vn, v1) ∈ E ,∀i = 1 . . . , n − 1. If the cycle containsall nodes of the graph, it is called a Hamiltonian cycle

Finding the shortest Hamiltonian cycle gives is a solution to the TSP.




Example of Hamiltonian cycle

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Shortest Hamiltonians PathTSP with Repeated City VisitsMultiple TSPClustered TSPMaximum TSP

Shortest Hamiltonian Paths

Finding the shortest Hamiltonian path in graph G = (V ,E ) can betransformed into a TSP in the following way:

1 Construct a new graph G ′ by:1 adding a virtual node to G2 connect the new node with all the others with distance zero

2 Solve the TSP in the new graph G ′





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If a starting position for the salesman is given, say node 1. Then we dothe following transformation:

1 Add the arcs (vi , v1), i = 2, . . . n with distance zero to the graph2 Solve the TSP

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TSP with Repeated City Visits

Suppose we require that the salesman has to visit each city at least once,instead of exactly once. Consider the following transformation:

Whenever there exists a route from vn to vk in G . Construct a newgraph with arcs (vn, vk) where the distance equals the total cost ofthe shortest path between them.





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Multiple TSP

In the multiple TSP, m salesmen has to visit n customer cities. Letfp, p = 1, 2, . . . n be the fixed cost if salesman p is activated. Theproblem is determine how many of the salesmen should be utilized suchthat their total traveling distance is minimized and each city is visitedonce by one and only one salesman and then return to the home city(node 0). If the network is directed we can transform the multiple TSPto to a standard asymmetric TSP by the following steps:





1 Arrange the fixed cost in ascending order

f0 ≤ f1 ≤ . . . ≤ fm−1

2 Add dummy nodes labeled −1,−2, . . . ,−(m − 1) as a home city forsalesman 2, 3, . . .m, respectively.

3 Add the arcs (−i , j),∀i = 1, 2, . . . ,m − 1 and each (0, j) ∈ A withdistance

c ′−i,j = c0,j +1

2fi

4 Add the arcs (j ,−i) whenever (j , 0) ∈ A with distance

c ′j,−i = cj,0 +1

2fi

5 Add the arcs (−i ,−(i − 1)) for every pair of i = 1, 2, . . . ,m− 1 withdistance

c ′−i,−(i−1) =1

2fi−1 −

1

2fi .





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1 2(f0−

f 1)

1 2(f1−

f 2)





Suppose we obtain the solution

(0, 1), (1, 4), (4,−2), (−2,−1)(−1, 2), (2, 3), (3, 0).

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Salesman 0 visites cities 1 and 4; salesman 2 visits none and salesman 1visits cities 2 and 3.





Clustered TSP

Let G = (N,A) be a digraph where the nodes have been partioned into kdisjoint subsets Ni , i = 1, 2, . . . k . The problem of the clustered TSP is tofind the minimum cost Hamiltonian cycle with the constraint that nodeswithin the same cluster must be visited consecutively.The problem can be transformed to the standard TSP by adding an(large) additional cost to the arcs between nodes that do not belong tothe same cluster.





Maximum TSP

Let G = (N,A) be a graph with a positive or negative arc valuecij , (i , j) ∈ A. The problem is to find a Hamiltonian cycle(v1, v2, . . . vn, v1) such that

c1,2 + . . . + cn−1,n + cn,1

is maximal.The problem can be transformed into a standard (minimum) TSP bysetting c ′ij = −cij or c ′ij = −cij + M (M large constant) if c ′ij becomesnegative.




Machine Sequencing Problems in Various Manufacturing SystemsSequencing Problems in Electronic IndustryVehicle Routing for Delivery and Dispatching

Machine Sequencing Problems in Various ManufacturingSystems

Job scheduling

Suppose that n jobs with a given processing time has to be processed ona single machine. The jobs can be processed in any order but somemachine job setup times are dependent, e.g. job i must precede job j .Objective is to find a sequence of jobs so all jobs are processed such thatthe total process time is minimized.

Assembly line

In assembly line systems, jobs can be grouped together as clusters. Thejob within the same cluster must be completed before proceding to thenext cluster. This type of sequencing can be considered as a clusteredTSP.





Cellular manufacturing

In cellular manufacturing the aim of to group together similair parts to beprocessed in a machine cell together. Aneja and Kamoun (1999) showedthat the problem of sequencing jobs by a robot in a machine cell can beformulated as a TSP.

Flow shop sequencing

Suppose we have n jobs with given process time that are to be processedon m machines in the same order. Each machine can work on one job atthe time and must be completed without interruption. No waiting time isassumed, that is when job j is completed the j + 1 job in the queue isstarting immediatly. The objective is to finish the last job as soon aspossible. This problem can be considered as a n-city shortestHamiltonian path problem which in turn can be tranformed into an n + 1node TSP by adding a virtual node.





Sequencing Problems in Electronic Industry

Drilling holes on IC boards

A number of holes are needed on integrated circuit (IC) boards formounting chips and other hardware. The holes are typically produced bya programmed drilling machine. The TSP is to minimize the the totaltraveling time of the drill.





Vehicle Routing for Delivery and Dispatching

School bus routing

The problem to schedule school buses to pick up and transport childrento and from schools can be viewed as a multiple TSP if it possible toneglect the time windows and bus capacities constraints. Otherwise theproblem is a vehicle routing problem

Parcel/postal delivery/dispatching

This problem is a modified problem of the TSP where a vehicle is has tovisit a set of streets (arcs) instad of nodes. A path traversing all arcsexactly once is called an Eulerian path.





Meals/Clinic on wheels

This version of the TSP is about dispatching a fleet of vehicles in urbanor rural environments to satisfy the demand for some commodity orservice, e.g. deliver food to elderly on regular basis or medical services incommunities.





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