Fuzzy shortest path problem

Computers ind. Engng Vol. 27, Nos 1--4, pp. 465--468, 1994

Pergamon

Elsevier Science Ltd. Printed in Great Britain

0360-8352(94)00150-2

Fuzzy Shortest Path P r o b l e m SHINKOH O K A D A * a n d M I T S U O G E N t * A s h i k a g a J u n i o r C o l l e g e , A s h i k a g a 326, J a p a n t A s h i k a g a I n s t i t u t e o f T e c h n o l o g y , A s h i k a g a 326, J a p a n

As a preliminary, we define the order relation between intervals by using two parameters. It is clarified that some traditional definitions for ordering intervals [5] are special cases of this definition. On the other hand, when selecting an interval among many ones alternatively, a decision maker may have a variety of preferences, for instance, he may expect to avoid the selection of worst case, or to select an interval with a maximal average value and so on. In this definition, setting the values of two parameters call reflect decision maker's preference on the order relation. Some standard algorithms for shortest path problem have been proposed by Dijkstra [3], Belhnan and Ford [2] and Floyd and Warshall [4, 8]. In this paper, using above definition, we propose an algorithm based on Dijkstra's method which is solvable within O(n 2) in worst case. The some paths with minimal duration times can be obtained through the algorithm, and one of them may be selected according to the decision maker's preference.

abstract We discuss the problem of finding the shortest paths from a fixed origin to a specified nodes in a network with arcs represented as intervals on real line. As a preliminary, we define the order relation between intervals by using two parameters. The decision maker's preference can be reflected on the order relation according to setting values of two parameters. An algorithm for this problem is proposed on the basis of Dijkstra's method. The some paths with nfinimal duration times can be obtained through the algorithm. As a large scale example, the United States driving map with 70 cities and 211 routes between cities is exanrined K e y w o r d s : shortest path problem, interval analysis, order relation, network. 1

Introduction

The problem of finding a shortest path fl'om a specified origin node to a specified destination node is a fundamental problem that appears in mauy applications. It generates essential information in transportation, routing, and conmmnications applications. We assume that the network on which the shortest path problem is defined is a connected network G = (A;, A) where A; is a set of nodes with n element, and A is a set of arcs. Here, the lengths of the arcs are assumed to be transportation time or cost instead of the geographical distances. As the time and cost fluctuate depending on traffic conditions, payload and so on, it is not practical to represent each arc as a real nunrber. In this case, it is applicable to utilize fuzzy numbers or intervals instead of real numbers as the values of arcs. It is, however, not so easy to determine a membership function of a fuzzy number and to deal with arithmetic between fuzzy numbers. Therefore, throughout this paper, we deal with intervals, which are special casc of fuzzy numbers, for the sake of computational efficiency and ease of data acquisition. We discuss the probleur of finding the shortest paths from a fixed origin to a specified node in a network G with arcs represented as intervals.

2

Preliminaries

Before proposing tile subsequent problem and algorithm, we present several definitions. Throughout this paper, lower case letters a, b,... and upper case letters A, B , . . . denote real numbers and closed intervals respectively. An interval A is defined by the following ordered pairs in brackets [1]: A

=

[aL,a n]={t[a

=

< a ° , a W >-- {tl a c - a 'V < t < a C + a 'v}

L
~}

where a L, a n, a c and a w are the left limit, right limit, center and width of A respectively. Tile center and width of an interval are calculated as follows: an + a C

a L

--

2 an _ aL a w

~_

2

465

466

S e l e c t e d p a p e r s f r o m the 16th A n n u a l C o n f e r e n c e on C o m p u t e r s a n d I n d u s t r i a l E n g i n e e r i n g

If a ~ _> 0, A is called a non-negative interval. The addition of intervals and multiplication by a real number k are defined as follows: A + B

A + B

=

[a L , a R] -.[- [b L , b '~]

=

[aL + b L,a n+b"],

= =

< aC, aW > + < bC,bW > < a C + b C , a W + b W >,

km = k[a L,a n] = {

[kaL' kaa] [ka R,ka n]

[Property

i. For any interval A, A ~ , Z A (reflexivity). ii. For any intervals A and B and any parameters

~,/3(~ #/3), A

fork>0 fork<0,

Relation

between

Intervals

[Definition 1] Let a higher interval be preferred. For arbitrary a and/3 ( - 1 _ and B = < be ,b w > is defined as follows: ,~

w) and bc - a c > / 3 ( b W - a w)

iv. For any intervals A and B, A ~ , ~ B or B _--4~,~A. From the conditions i, ii and iii in Property 1, the relations ~ , Z are contented with the mxiom of order relation in ease of a ¢/3, and is not totally, but partially ordered. This sometimes leads to the indeeisive ease in which neither A __.*,~ B nor B ~ , ~ A holds. Hence, we eannot seleet a mininmm interval among lots of intervals by means of the order relation Definition 2. We can obtain some minimal intervals. The number of lninimal intervals becomes smaller, as the values of parameter a and/3 are closer. Therefore, we call control the number of minimal intervals by setting the parameters.

bc - a c >_ o,(b ~ - a

4

Let costs be represented by intervals in a nfinimization problem. A decision maker must prefer a lower cost among them. The order relation A __*,~ B is defined as follows: [ D e f i n i t i o n 2] Let a lower interval be preferred. For arbitrary a and/3 ( - 1 _
---~,Z B a n d B _ ~ , ~ C A -~,~ C (transitivity).

We consider the order relation between intervals. A number of methods for the ordering and ranking intervals have been proposed [5, 7]. We have presented the definition for ordering intervals reflecting decision maker's preference by using a parameter [7]. In this paper, expanding this definition and using two parameters, we introduce new definitions. Let profits be represented by intervals in a mmximization problem. A decision maker nmst prefer a higher profit among them. Hence, using two parameter a and/3 ( - 1 < / 3 _< a _< 1), The order relation A __.,,~ B is defined as follows:

A ~o,o B

BandB~,zA A = B (antisymmetry).

iii. For any intervals A, B and C, A

Order

-~,~ =~

k A = k < aC,a w > = < kaC,{ k [ a w > .

3

1]

¢~ bc - a c > o l ( a w - b w)

and bc - a c >_ fl(a w - b w) Now we will discuss the order relation __.~,~only for the subsequent minimization problem. [ A s s e r t i o n 1] The following relations hold: A _ ~ , _ l B ~ a L_
(1) (2)

Relation (1) seems to be the strict order relation that A is smaller than B. It is considered that the right limit and center of an interval correspond to the worst case and expectation of the interval respectively. Hence, relation (2) seems to be adequate order relation for a decision maker's preference. Definition 2 leads to the following properties:

Fuzzy Shortest

Path

Problem

Now we introduce the shortest path 1)rol)lem with arcs represented as intervals as follows: [ F u z z y - S P P ] Let G = (Af,A) be a network, where Af is a set of nodes with n elements, and A is a set of arcs. Let Dij be a length represented as a non-negative interval between adjacent nodes i and j. If nodes i aad j are not adjacent, Dij implies +oo. The problem is to find the paths with the minimal total length fi'om the origin to the destination on the network G. It is assumed that there exists the path fi'om the origin to each node in the network. On the basis of the algorithm proposed by Dijkstra [3], we propose an algoritlnn for solving the FuzzySPP. First, we need to prepare the following definitions: Uj : a set of minimal lengths fi'om node 1 to node j , minimalo,~{S} : an operator to get some minilnal elements of the set S by using the order relation ~,~, n : the number of nodes, P : the set of nodes with permanent labels, T : the set of nodes with temporary labels.

Selected papers from the 16th Annual Conference on Computers and Industrial Engineering

Figure 1: The d,'iving times map in United States

Table 1: Routes and duration times from Boston to Los Angels routes

total duration time 1

[2826, 4430] =< 3628, 8O2 >

,5

~',..ci --~

dm'atio,l times between cities

"~3 ---, 33 --~ 44 --~ [63,119] [245,367] [20~13] 58 [ 6 4 ~ 0 5 ] 70 [232,343] ---,

3

1 [68~28]

[750,1150] 3 ----+ 23 -----4 33 ~ 46 [502,702] [249,373] [2860, 44061 ,6 • [20-~ 13l 58 [645~.~05] 70 [232,343] =< 3633,773 > [277,415] [750,11501

I [15~57]~ 8 [27~35]'--49 [18~75]~ 12 [31~19]---47023 [232-"4,348] [2924,4384] =< 3654,730 > 33 46 56 58 [502,702] [249,3731 [277,415] [750,1150] 1 ~ ~; ---4 9 ~ 12 --4 40 ---+ [2938,4380] 5_~5 7] 55 [27_~3 5] 63 [18_~7 5] 70 [601,937] [130,194] =< 3659,721 > ,I [I [517,775] [722,1082] [277,415] -----4 .--4 ---4 23 1 [15~57] ~ [273~35] 9 [18~75 ] 12 [31~19] --"+ [3O53,4377] 33 46 56 58 62 [23~48] 70 =< 3715,662> [502,702] [249,373] [277,415] [485,627] [394,516] [3080,4372] =< 3726,646 >

1

~

3

1 [205__~,313] 3 63 [722,10821 3 I [20~13] [3209,4365] =< 3787, 578 > 16 56 [249,373] I [205_~13] 3 [3286,4360] 47 ----<3823,537 > 34 [126,1901 [3094, 4368] =< 3731,637 >

~

12

~

23

~

33

46 [20~.~ 13] 56 [5 5.~5 2] 58 [31_~1 9] 70 [232,348] [502,702] [249,373] [277,4151 [75~501 40 ----+ 41 55

[552__~52] 12 [601,937] [130,194] [517,775] 70 [277,415] ~ 12 23 ---4 33 ---4 [31~19] [23..~48] 70 [5o2,7o2] 58 62 [277,415] [485,627] [304,516] [87~i] 2 [134_~00] 5 [770_~30] 26 ---4 48 57 62 [19~96] 70 [308,462] [656,760] [408,562] [394,516l

467

Selected papers from the 16th Annual Conference on Computers and Industrial Engineering

468

The algorithm consists of the following 3 steps: [step 1] A DM gives the values of parameters c~ and according to his preference. The sets Uj, P and T are initialized as follows:

Uj = {Dlj} f o r j = 1,2 . . . . . n, P = {1} and T = { 2 , . . . , n } .

Go to step 2. [step 2] If T = 0, then terminate. Otherwise, find j* E T such that minimalo,o{Uj~TUj} C Uj., and set T = T \ {j*},P = Pt_J {j*}. Go to step 3. [step 3]

For all j E T ,

Uj = minimalo,a{UjU(Uj. +Dj.j)}. Retm'n to step 2. For illustrative nmnerical example, see Ol~ada and Gen [7].

5

Large Scale Example

Consider a computational large scale example composed of 70 cities (nodes) and 211 routes between adjacent cities (arcs) [6] shown in Figure 1. Each duration time is omitted for lack of space. Let a duration time (unit : minute) between adjacent cities be represented by an interval whose width takes about 20 % of the duration time. Here, we consider a problem obtaining some routes with minimal total duration times from Boston (node number 1) to Los Angeles (node number 70). First, It is assumed that the decision maker expects to select a route with the shortest average duration time and to avoid routes with the possibility of the longest driving time (the worst case). Hence, he will give the values of parameters as a = 1, /~ = 0. Applying the proposed algorithm to the example, 9 routes with minimal duration times are obtained as shown in Table 1. It is noted that these intervals are non-dominated with regard to the centers and right limits of intervals. However, the decision maker may be bothered by selecting a preferable route, as there exist so many non-dominated routes obtained out of this example. Next, if setting a = 0.7, ~ = 0.3, only 3 routes with duration times [2860, 4406], [2924, 4384] and [2938, 4380] are obtained. Assuming that the decision maker prefers the route with the possibility of the shortest driving time, the route with duration time [2826,4430] is obtained by setting

a = -1, ~ = -1. Therefore, the above results shows that the decision maker's preference can be reflected on the order relation through setting parameters. 6

Conclusion

We defined an order relation between intervals reflecting a decision maker's preference by setting values for two parameter, and proposed an algorithm for solving shortest path problem represented arcs as intervals on the basis of this order relation. A large scale example was examined through this algorithm, and we were able to obtain some routes with non-dominated duration times reflected a decision maker's preference. In future research, we will apply this algorithm to the other network problem such as minimum cost flow problem, scheduling problem and so on.

References [1] Alefeld, G. and Herzberger, J. : Introduction to Interval Computations (translated by Rokne,J.), Academic Press, New York, pp.1-9 (1983). [2] Bellman, R. : "On a Routing Problem," QAM, Vo1.16, pp.87-90 (1958). [3] Dijkstra, E. : "A note on Two Problems in Connexion with Graphs," Numerische Mathematik, Vol.1, pp.269-271 (1959). [4] Floyd, R. W. : "Algorithm 97, Shortest Path," Communications of ACM, Vol.5, pp.345 (1962). [5] Ishibuchi, H. and Tanaka, H. : "Multiobjective Programming in Optimization of the Interval Objective Function", European Journal of Operational Research, Vol.48, pp.219-225 (1990). [6] "United States Distances and Driving Times Map," Mobil Road Atlas 7t'ip Planning Guide, Mobil Travel Guide Publication, New York (1993). [7] Okada, S. and Gen, M. : "Order Relation between Intervals and Its Application to Shortest Path Problem," Computers and Industrial Engineering, Vol.25, Nos 1-4, pp.147-150 (1993). [8] Warshall, S. : "A theorem on Boolean Matrices," Journal of ACM, Vol.9, pp.ll-12 (1962).