Fuzzy multi objective linear programming for traveling

salesman problem

Guided by: Presented by: Prof. T.R. Gulati Vishwadeep Gautam Dept. of Mathematics, (12312032), MSM I.I.T. Roorkee

Content

IntroductionMembership function and Linear Programming Multi objective linear programming Fuzzy multi objective linear programming(FMOLP)FMOLP approach for Traveling Salesman Problem Case study of Traveling Salesman Problem Conclusion References

Introduction

Traveling Salesman Problem (TSP) is an important real life problem in artificial intelligence and operations research domain.

TSP is well-known among NP-hard combinatorial optimization problems. It represents a class of problems which are analogous to finding the least-cost sequence for visiting a set of cities, starting and ending at the same city in such a way that each city is visited exactly once

TSP in order to simultaneously minimize the cost, distance and time.

Why we use FMOLP

Information about real life systems is often available in the form of vague descriptions. Hence, fuzzy methods are designed to handle vague terms, and are most suited to finding optimal solutions to problems with vague parameter.

Fuzzy multi-objective linear programming (FMOLP), an amalgamation of fuzzy logic and multi objective linear programming, deals with flexible aspiration levels or goals and fuzzy constraints with acceptable deviations.

Membership Function and LPA membership function (x) is characterized by : X → [0,1], x

X, where x is a real number describing on object or it’s attribute, X is the universal of discourse and A is subset of X.

A Fuzzy Membership function is mapped on interval [0,1]which is an arbitrary grade of truth.

The general linear programming Problem model, for maximization, proposed by Dantzig is:

Max Z = (1) Subject to ≤ ; (j=1,2…,m) and 0 Where Z is the objective function, are the decision variables,

m is the number of constraints, n is the number of decision variables, and are the given resources.

Multi Objective Linear ProgrammingLinear Programming is limited by the fact that it can

deal only with single objective function and does not incorporate soft constraints. So we use multi objective linear programming .

A general Linear Multiple Criteria Decision Making model can be represented as follows:

Find a vector such that = [] which maximize k objective functions with n variables and m constraints as: = ; i=1,2,…,k., (2) Subject to ; j=1,2,…m., Where are given crisp values

Cont.

In precise form, multiple objective problems can be represented by following Multi-Objective Linear Programming model:

optimize Z = CX (3) Subject to AX b.Where, Z = [] is vector of objectives, C is K N matrix of

constants and X is N 1 vector of decision variables, A is MN matrix of constant and b is M 1vector of constants.

Fuzzy multi-objective linear programmingConsidering the following Multi-Objective Linear

Programming model max Z = CX (4)

Subject to AX ≤ b Adopted Fuzzy model by Zimmerman is given by, max CX (5)subjected to AX bWhere = [,] are goals. and are fuzzy inequalities that are fuzzifications of and respectively.

Coun.

For measurement of satisfaction levels of objectives and constraints Zimmerman suggested simplest type of Membership function given by,

()=, k = 1,2,…,n. (6)

Where Represent tolerance for objective which is decided by decision maker.In case of minimizing objective function, Fuzzy Membership function is,

()=, k = 1,2,…,n. (7)

Another class of Fuzzy Membership functions:()=, I = 1,2,…,m. (8)

is tolerance for fuzzy resource for constraint.

Cont.As a result objective function becomes (X), (X),…, (X), (X), (X),…, (X)} (9)According to Fuzzy Sets, membership function of intersection of

any two or more sets is minimum Membership function of these sets, so objective function become

(X), (X),…, (X), (X), (X),…, (X)} (10) From this representation we haveMax CX (11)subjected to 1 - (-X) / , K = 1,2,…,n. 1 - ()/, I = 1,2,…,m and 1 ,X 0 , where is overall satisfaction level achieved with respect to solution.

FMLOP Approach For TSP

Considering the situation when decision maker has to determine optimal solution of TSP with min(cost, time, distance). Let be the link from city i to j and = (12)

Let be the cost of traveling from city(i) to city(j), also let be the total estimated cost of entire route for TSP and is tolerance for estimated cost, then objective function for minimization of cost is given as

min. (13)

Cont.

Let be the distance from city(I) to city(j). , also let be the total estimated distance of route for TSP and is tolerance for distance, then objective function for minimization of distance is given as

min. (14)Similarly let be the time spent in traveling from city(i)

to city(j) and be the corresponding aspiration level for objective function for minimization of total time and be tolerance. The objective function is given as

min. (15)

Cont.We have the restriction in TSP that every city should be visited from exactly one its neighboring city, and vice versa. i.e. = 1, j . (16)And

= 1, i . (17)A route can not be selected more than once, that is +1, ,j and non-negativity constraints is 0. These constraints collectively are expressed in vector form and fuzzy membership functions are defined for all objective functions.

Case Study for TSP

Assuming that salesman starts from his home city 0; has to visit the three cities exactly once. A map of the cities to be visited is shown in Figure and the cities listed along with

their cost, time and distance matrix in table-1. Let triplet (c,d,t) represents; cost in rupees, distance in kilometers, and time in hours Respectively for the corresponding couple of cites.

Table:1 The matrix for time, cost and distance for each pair of cities

City 0 ( C , d , t )

1 ( C , d , t )

2 ( C , d , t )

3 ( C , d , t )

0 0 , 0 , 0 20 , 5 , 4 15 , 5 , 5 11 , 3 , 2

1 20 , 5 , 4 0 , 0 , 0 30 , 5 , 3 10 , 3 , 3

2 15 , 5 , 5 30 , 5 , 3 0 , 0 , 0 20 , 10 , 2

3 11 , 3 , 2 10 , 3 , 3 20 , 10 , 2 0 , 0 , 0

Cont.Let links be the decision variable of selection of link (i, j) from city(i) to city(j). The three objective function are formulated for cost, distance and time respectively. Their Aspiration levels are set as 65, 16, 11 by solving each objective function subject to the given constraints in the TSP and their corresponding tolerances are decided as 5, 2, 1.Objective functions:Min = 20+15+11+20+30+10+15+30+20 +11+10+2065 (18)Tolerance = 5.

Cont.

Min = ++++++5++10 +++1016 (19)=2.Min = ++++++5++ +++11 (20) =1.The fuzzy membership function for cost, distance and time objective function are given as under based on equation (18),(19) and (20).

Cont.

)=-65)/5)=-16)/2)=-11)/1

The FMOLP with max-min approachMax CX (24) Subject to 1 – ( - 65 )/51 – ( - 16 )/21 – ( - 11 )/1

+ + = 1 (25) + + = 2 (26)

Cont.

+ + = 1 (27) + + = 1 (28) + + = 1 (29) + + = 1 (30) + + = 1 (31) + + = 1 (32) + = 1 (33) + = 1 (34)

Cont.

+ = 1 (35) + = 1 (36) + = 1 (37) + = 1 (38) 0 ; 0 ;

Table 2:Solution Solution Route

1 65 , 5 16 , 2 _ 0.80 ( , , )

2 65 , 5 16 , 2 11 , 1 _ No feasible Solution

2 65 , 5 16 , 2 11 , 4 0.55 ( , , , )

3 65 , 5

16 , 2 11 , 5 0.62 ( , , , )

Cont.

As given in table 2, only and are considered and is omitted; an optimal route with = 0.8 is obtained.

When is also considered , solution becomes infeasible on these tolerances. Again by relaxing tolerance in to 4, solution becomes feasible. In this case, the optimal path is achieved with = 0.55 .

By increasing tolerance in from 4 to 5, an optimal solution with

= 0.62 is obtained.These results show that by adjusting tolerance an

optimal solution to Multi-Criteria TSP can be determined.

Conclusion

In this work, the analysis of the symmetric TSP as a Fuzzy problem with vague decision parameters .

general lesson can be taken from this study is: Multi objective TSP exists in uncertain or vague environment

where route selection is done by exploiting these parameters. The tolerances are introduced by the decision maker to

accommodate this vagueness. By adjusting these tolerances, a range of solutions with

different aspiration level are found from which decision maker can choose the one that best meets his satisfactory level within the given domain of tolerances.

References

Angel E, Bampis E, Gourvès L, Approximating the Pareto curve with Local Search for Bi-Criteria TSP (1,2) Problem, Theor. Comp. Sci., 35-146, (2004).

Bellman RE, Zadeh LA, Decision-Making in a Fuzzy environment, Management Science, p. 17, (1970).

Borges PC, Hansen PH, A study of Global Convexity for a Multiple Objective Traveling Salesman Problem. In: Ribeiro C. C. and Hansen P. (Eds.), Essays and Surveys in Metaheuristics, Kluwer Academic, pp. 129-150, (2000).

Gupta, A., Warburton, A.: Approximation Methods for Multiple Criteria Traveling Salesman Problems. In: Sawaragi, Y. (ed.): Towards Interactive and Intelligent Decision Support Systems: Proceedings of the 7th International Conference on Multiple Criteria Decision Making. Springer- Verlag Berlin 211-217, (1986).

Cont.Tung, C. T.: A Multi criteria Pareto-optimal Algorithm for the Traveling

Salesman Problem. Asia-Pacific Journal of Operational Research 11, 103-115, (1994)

Yan, Z., Zhang, L., Kang, L., Lin, G.: A New MOEA for Multi-objective TSP and Its Convergence Property Analysis. In: Fonseca, C. M., Fleming, P. J., Zitzler, E., Deb, K., Thiele, L. (eds.): Evolutionary Multi-Criterion Optimization,

Proceedings of Second International Conference, EMO2003. Springer Berlin 342-354, (2003)

Zadeh, L., Fuzzy Logic and its Applications, Academic Press, New York. (1965).

Zeleny M. Linear Multiple-Objective Programming: Springer: Berlin. (1974).Zimmerman, H.J., Fuzzy programming and linear programming with

several objective functions. Fuzzy sets and syst 1:45-55, (1978).

Thank you