A COMPARATIVE STUDY OF MULTI- OBJECTIVE SHORTEST PATH PROBLEMS

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A COMPARATIVE STUDY OF

MULTI-OBJECTIVE SHORTEST PATH

PROBLEMS

K.Karthikeyan,

School of Advanced Sciences, Mathematics Division, VIT University, Vellore – 632 014

Tamil Nadu, India. E-mail: [email protected]

Abstract :

For a multi-objective shortest path problem (MOSPP) of a network, there may be several Pareto optimal solutions. The decision maker can select the best one or the most satisfactory solution depending on the priority and nature of the problem. In this paper, we use the concept of Properly Pareto optimal solution for MOSSP and proved that it is stronger than Pareto optimal solution using Geoffrion’s approach by means of an example.

Key words: Properly Pareto optimal solution; Pareto optimal solution; Geoffrion’s technique; Martin’s label setting algorithm.

1. INTRODUCTION

. Multi-objectives such as optimization cost, time, distance, risk, delay, reliability, quality of service and environment impact etc., may arise in network optimization problems. Several researchers have studied multi-objective network optimization problems because of their large number of applications in science, engineering and management The main purpose of the network optimization problems is to optimize the performance with respect to predefined objectives. While dealing with the above said networks, we require the computation of shortest path from one node to another called shortest path problem (SPP).

When only one objective is considered in the network, the shortest path problem is called single objective shortest path problem (SOSPP). When more than one objective is considered in the network, the shortest path problem is called multiple objective shortest path problems (MOSPP). In MOSPP, our aim is to find Pareto optimal paths (Hwardg and Masud (1979)) from one node to other nodes usually. Here, in this paper, we change the MOSSP problem in to a SOSPP by Geoffrion’s technique and then find the solution called Properly Pareto Optimal Solution from start node to other.

In most of the cases, the consideration of one objective function for a SPP does not lead to any realistic solution. A Multi objective shortest path problem (MOSPP) of a network consists of more than one objective of either maximization type or minimization type or of mixed type. In the case of single objective, the best path indicates the optimal path. The concept of optimality is replaced by Pareto optimality in multi objective case due to conflicting nature of the objectives. A Pareto optimal solution [1] or non dominated solution for a multi objective problem is one for which no objective function can be improved without a simultaneous detriment to atleast one of the other objectives. Hence for a MOSPP, there may be several Pareto optimal solutions. Among them, the decision maker can select the best one or the most satisfactory solution depending upon the priority and nature of the problem.

Xavier Gandibleux et. al. [2] presents a straight extension of the label setting algorithm proposed by Martins in 1985 for a shortest path problem with multiple objectives. This extended version computes all the e_cient paths from a given source vertex, to all the other vertices of the network. The algorithm can cope with problems in which the \cost values" associated with the network arcs are all positive, and it \optimizes" multiple objectives simultaneously. Karthikeyan et. al. [6 ] introduced the concept of Properly Pareto optimal solution for MOSPP

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use the concept of Properly Pareto optimal solution for MOSPP, to the network given in [3] and establish that there is only one Properly Pareto Optimal solution for the network between the source and end nodes which is better than the (non dominated) Pareto optimal solutions by means of examples.

2. PRELIMINARIES

In the paper[3], two set of objectives, one with triangular fuzzy number as edge weights and another with a three objective shortest path problem are considered for MOSPP. When these two sets are solved using Martin’s label setting algorithm, to find the non dominated solutions, they concluded that the number of non dominated solutions is less for the multi objective shortest path problem with objectives as triangular fuzzy number, when compared to three objective shortest path problems. In this paper, we used the concept of Properly Pareto optimal solutions for the given MOSPP and established that there is a unique solution between the source node and end node of the MOSPP for the network with the three objectives.

Throughout this paper, we follow Mangasarian’s conventions for vectors in Rn will be followed. Suppose that x, y are in Rn . Then

x = y if and only if xi = yi for all i x





y if and only if xi  yi for all i

x  y if and only if xi  yi for all i and xr < yr for some r x < y if and only if xi < yi for all i

Before we could proceed, let us define the following terms

2.1. Definition

A feasible point

x

₀ is said to be an efficient solution of (P) if

x

₀  X and there exists no other feasible

point x such that

f

(

x

)



f

(

x

₀

)

and

f

(

x

)



f

(

x

₀

)

.

2.2. Definition

A feasible point

x

₀ is said to be Properly efficient if it is an efficient solution of (P) and if there

exists a real number

M



0

such that for each i,

f x

_i

( )

₀



f x

_i

( )



M f x

(

_j

( )



f x

_j

( ))

₀ for some j

such that

f x

_j

( )



f x

_j

( )

₀ whenever x is feasible to (P) and

f x

_i

( )



f x

_i

( )

₀ .

2.3. Definition

The value of a path is the sum of the weights of the edges constituting the path. Let S be the set of all paths between any two nodes. The value of a path pS of a single objective shortest path problem (SOSPP) is a

function

W S

:



R

such that

( , )

( )

_ij

i j

W p





d

where

d

_ij is the weight of the edge (i, j). The value of a path

pS a multi objective shortest path problem (MOSPP) is a vector function

W S

:



R

m such that

1 2

( )

(

( ),

( ),....

m

( ))

W P



W P W

P

W

P

and each component

W

r, r  { 1,2,… m } is defined as a function

:

r

W

S



R

. If P is a path (v1, v2 , … vn ) then

1 , 1 1

( )

n

r r

i i i

W

P

d

  





2.4. Definition

For a multi-objective shortest path problem (MOSPP) with S as the set of all paths between two specific nodes of a network, a path

p





S

is said to be Pareto minimum if there does not exist any other path

S

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2.5. Definition

For a multi-objective shortest path problem, where

S

is the set of all paths between two specified nodes

of a network, a path



p

_in

_S

_{is said to be Properly efficient minimum path if}

p

_{is a Pareto minimum path}

and there exists a scalar M > 0 such that for each i, for each path

p

in

S

for which

*

( )

(

)

i i

W p



W p

we

have

* *

(

)

( )

(

( )

(

)

i i r r

W p



W p



M W p



W p

for some r such that

*

( )

(

)

r r

W p



W p

Result: Every properly non-dominated path in a MOSPP is a non-dominated path but the converse is not true which is dominated by the following example

3. GEOFFRION’S APPROACH TO SOLVE THE MULTI-OBJECTIVE SHORTEST PATH PROBLEM

3.1 GEOFFRION’S LEMMA

Let >0 in Rk with 1

1

k

i i







be fixed. If

x

₀ is an optimal solution for the following scalar minimization problem ( P ). Minimize



t

f

(

x

)

subject to

x



X

, then

x

0 is a properly efficient solution for (P). Consider the following mathematical programming problem:

P: Minimize

(

f

₁

(

x

),

f

₂

(

x

),...

f

_k

(

x

))

subject to

g

(

x

)



0 ,

x



R

n

where

f

_i

:

R

n



R

and

g

:

R

n



R

m

Result : Let  > 0 with i = 1,

x

₀ is an optimal solution of P ,  Rk,

P : Minimize

(



t

f

(

x

))

such that

g

(

x

)



0 ,

x



R

n

Then

x

₀ is a properly efficient solution of P.

The following steps are adopted for the new approach

Step1: Convert the multi-objective function of the network in to a single objective function of the network using Geoffrion technique.

Step2: Solve the SOSPP by using any one of the known algorithm.

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Example 3.1: Consider the following network with three objectives

.

4, 3, 6

8, 9, 15

16, 1, 10 4, 5, 4

1,1,1

7, 5, 18

16, 9, 20

Solution: Change the given MOSPP to SOSPP by using Geoffrion’s technique by taking  = 1/3, we have

. 4.333

10.666

1 9 4.333

10

15

Solve this SOSPP by using Dijkstra’s algorithm we get the Properly efficient solution. The solution for the above SOSPP is followed.

<1, 2> with the value of the path as <8, 9,15> <1, 3> with the value of the path as <7, 5, 8> <1, 3, 4> with the value of the path as <23, 6, 28> <1, 3, 4, 5> with the value of the path as <27, 11, 32>

3.2. COMPARISON TABLE

Sl No. From source node s =1 to

end node t = 5 (Pareto optimal solutions)

Value of path by Pareto optimal solution

Path and the value of path by Properly Pareto Optimal solution

1 1-3-5 (23, 14, 38) 1-3-4-5 with value of

path (27, 11, 32)

2 1-3-4-5 (27, 11, 32)

3 1-2-4-5 (16, 17, 25)

4 1-2-3-4-5 (29, 16, 30)

4. CONCLUSION

It is clear from the above example that there is only one path from source node 1 to final node 5, which is <1,3, 4, 5> with the path value as <27, 11, 32> called as Properly Pareto optimal path. But in the paper[3], they

1

3

2 ₄

5

1

3

2 ₄

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REFERENCES

[1] Sastry, V.N., Janakiraman, T.N. and Ismail Mohideen, S.(2005): New Polynomial Time Algorithm to Compute a Set of Pareto Optimal Paths for Multi Objective Shortest Path Problem, International Journal of Computer Mathematics, 82(3), pp. 289 –300. [2] Xavier Gandibleux, Fredric Beugnies, sabine Randriamasy (2004): Martins Algorithm revisited for Multi objective shortest path

problem with a maxmin cost function,Quarterly journal of the Belgian, French & Italian Operations Research Societies.

[3] S.ISMAIL MOHIDEEN and B.RAJESH (2010): A COMPARATIVE ANALYSIS OF MULTI OBJECTIVE SHORTEST PATH PROBLEMS, International Journal of Engineering Science and Technology, 2(7), pp. 3241-3243

[4] Sastry.V.N., Ismail Mohideen.S.(1999): A Modified Algorithm to Compute Pareto Optimal Vectors, Journal of Optimization Theory and Applications, 103, pp. 241 – 244.

[5] L.R.Foulds, (1992): Graph Theory Applications, Springer-Verlag New York, Inc, 234-236.