Fuzzy multi-objective linear programming for traveling salesman problem

1. 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

2. Content
Introduction
Membership 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

3. 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.

4. 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.

5. Membership Function and LP
A membership function µ 𝐴(x) is characterized by µ 𝐴 : 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 = 𝑖=1
𝑛
𝐶𝑖 𝑋𝑖 (1)
Subject to 𝑖=1
𝑛
𝑎𝑖𝑗 𝑥𝑖 ≤ 𝑏𝑖 ; (j=1,2…,m) and 𝑥𝑖 ≥ 0
 Where Z is the objective function, 𝒙𝒊 are the decision variables, m is
the number of constraints, n is the number of decision variables, and 𝒃𝒊 are
the given resources.

6. Multi Objective Linear Programming
Linear 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 𝑋 𝑇= [𝑥1, 𝑥2, … , 𝑥 𝑛] which maximize k objective
functions with n variables and m constraints as:
𝑍𝑖 = 𝑗=1
𝑛
𝐶𝑖𝑗 𝑥𝑗 ; i=1,2,…,k., (2)
Subject to 𝑖=1
𝑛
𝑎𝑖𝑗 𝑥𝑖 ≤ 𝑏𝑗 ; j=1,2,…m., Where 𝑐𝑖𝑗 , 𝑎𝑖𝑗 𝑎𝑛𝑑 𝑏𝑗 are given
crisp values

7. 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 = [𝑧1, 𝑧2, … , 𝑧 𝑛] is vector of objectives, C is K ×N matrix of
constants and X is N ×1 vector of decision variables, A is M×N matrix
of constant and b is M × 1vector of constants.

8. Fuzzy multi-objective linear programming
Considering the following Multi-Objective Linear Programming model
max Z = CX (4)
Subject to AX ≤ b
Adopted Fuzzy model by Zimmerman is given by,
max 𝑍0 ≤ CX (5)
subjected to AX ≤ b
Where 𝑍0 = [𝑧1
0
,𝑧2
0
, … , 𝑧 𝑛
0] are goals. ≥ and ≤ are fuzzy inequalities that
are fuzzifications of ≥ and ≤ respectively.

9. Coun.
For measurement of satisfaction levels of objectives and constraints
Zimmerman suggested simplest type of Membership function given by,
𝜇1𝑘(𝐶 𝑘X)=
0, if 𝐶 𝑘X ≤ 𝑍 𝑘
0
−𝑡 𝑘
1 − (𝐶 𝑘X − 𝑍 𝑘
0
)/𝑡 𝑘 , if 𝑍 𝑘
0
− 𝑡 𝑘 ≤ 𝐶 𝑘X ≤ 𝑍 𝑘
0
1, if 𝐶 𝑘X ≥ 𝑍 𝑘
0
, k = 1,2,…,n.
(6)
Where 𝑡 𝑘 Represent tolerance for objective 𝑍 𝑘 which is decided by
decision maker.
In case of minimizing objective function, Fuzzy Membership function is,

10. 𝜇1𝑘(𝐶 𝑘X)=
0, if 𝐶 𝑘X ≥ 𝑍 𝑘
0
+ 𝑡 𝑘
1 − (𝑍 𝑘
0
−𝐶 𝑘X)/𝑡 𝑘 , if 𝑍 𝑘
0
≤ 𝐶 𝑘X ≤ 𝑍 𝑘
0
+𝑡 𝑘
1, if 𝐶 𝑘X ≤ 𝑍 𝑘
0
,
k = 1,2,…,n. (7)
Another class of Fuzzy Membership functions:
𝜇2𝑖(𝑎𝑖X)=
0, if 𝑎𝑖X ≥ 𝑏𝑖 + 𝑑𝑖
1 − (𝑎𝑖X−𝑏𝑖)/𝑑𝑖 , if 𝑏𝑖 ≤ 𝑎𝑖X ≤ 𝑏𝑖 + 𝑑𝑖
1, if 𝑎𝑖X ≤ 𝑏𝑖
,
I = 1,2,…,m. (8)
𝑑𝑖 is tolerance for fuzzy resource 𝑏𝑖 for 𝑖 𝑡ℎ
constraint.

11. Cont.
As a result objective function becomes
𝑚𝑎𝑥 𝑋 {𝜇11(𝐶1X), 𝜇12(𝐶2X),…, 𝜇1𝐾(𝐶 𝐾X), 𝜇21(𝑎1X), 𝜇22(𝑎2X),…,
𝜇2𝑚(𝑎 𝑚X)} (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
𝑚𝑎𝑥 𝑋min{𝜇11(𝐶1X), 𝜇12(𝐶2X),…, 𝜇1𝐾(𝐶 𝐾X), 𝜇21(𝑎1X), 𝜇22(𝑎2X),…,
𝜇2𝑚(𝑎 𝑚X)} (10)
From this representation we have
Max CX ≥ 𝑍0 (11)
subjected to 𝛼 ≤ 1 - (𝑍0-𝐶 𝑘X) / 𝑡 𝑘, K = 1,2,…,n.
𝛼 ≤ 1 - (𝑎𝑖X − 𝑏𝑖)/𝑑𝑖 , I = 1,2,…,m and 𝛼 ≥ 1 ,X ≥0 ,
where 𝛼 is overall satisfaction level achieved with respect to solution.

12. FMLOP Approach For TSP
Considering the situation when decision maker has to determine
optimal solution of TSP with min(cost, time, distance). Let 𝑋𝑖𝑗be the link
from city i to j and
𝑋𝑖𝑗 =
1, 𝑐𝑖𝑡𝑦 𝑖 → 𝑐𝑖𝑡𝑦(𝑗)
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(12)
Let 𝐶𝑖𝑗 be the cost of traveling from city(i) to city(j), also let 𝑍1
0
be the
total estimated cost of entire route for TSP and 𝑡1is tolerance for
estimated cost, then objective function for minimization of cost is
given as
𝑍1: min 𝑖=1
𝑛
𝑗=1
𝑛
𝐶𝑖𝑗 𝑋𝑖𝑗 ≤ 𝑍1
0
. (13)

13. Cont.
Let 𝑑𝑖𝑗 be the distance from city(I) to city(j). , also let 𝑍2
0
be the total
estimated distance of route for TSP and 𝑡2 is tolerance for distance,
then objective function for minimization of distance is given as
𝑍2: min 𝑖=1
𝑛
𝑗=1
𝑛
𝑑𝑖𝑗 𝑋𝑖𝑗 ≤ 𝑍2
0
. (14)
Similarly let 𝑡𝑖𝑗 be the time spent in traveling from city(i) to city(j) and
𝑍3
0
be the corresponding aspiration level for objective function for
minimization of total time and 𝑡3 be tolerance. The objective function
is given as
𝑍3: min 𝑖=1
𝑛
𝑗=1
𝑛
𝑡𝑖𝑗 𝑋𝑖𝑗 ≤ 𝑍3
0
. (15)

14. Cont.
We have the restriction in TSP that every city should be visited from
exactly one its neighboring city, and vice versa. i.e.
𝑖=1
𝑛
𝑋𝑖𝑗 = 1, ∀ j . (16)
And
𝑗=1
𝑛
𝑋𝑖𝑗 = 1, ∀ i . (17)
A route can not be selected more than once, that is
𝑋𝑖𝑗+𝑋𝑗𝑖 ≤1, ∀ i,j
and non-negativity constraints is
𝑋𝑖𝑗 ≥ 0.
These constraints collectively are expressed in vector form and fuzzy
membership functions are defined for all objective functions.

15. 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.

16. 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

17. Cont.
Let links 𝑋𝑖𝑗 be the decision variable of selection of link (i, j) from city(i)
to city(j). The three objective function 𝑍1, 𝑍2 , 𝑍3 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 𝑍1=
20𝑥01+15𝑥02+11𝑥03+20𝑥10+30𝑥12+10𝑥13+15𝑥20+30𝑥21+20𝑥23
+11𝑥30+10𝑥31+20𝑥32 ≤65 (18)
Tolerance 𝑡1= 5.

18. Cont.
Min 𝑍2 = 5𝑥01+5𝑥02+3𝑥03+5𝑥10+5𝑥12+3𝑥13+5𝑥20+5𝑥21+10𝑥23
+3𝑥30+3𝑥31+10𝑥32 ≤16 (19)
𝑡2=2.
Min 𝑍3 = 4𝑥01+5𝑥02+2𝑥03+4𝑥10+3𝑥12+3𝑥13+5𝑥20+3𝑥21+2𝑥23
+2𝑥30+3𝑥31+2𝑥32 ≤11 (20)
𝑡3=1.
The fuzzy membership function for cost, distance and time objective
function are given as under based on equation (18),(19) and (20).

19. Cont.
µ(𝑍1)=
0
1 − (
1
𝑍1-65)/5
µ(𝑍2)=
0
1 − (
1
𝑍2-16)/2
µ(𝑍3)=
0
1 − (
1
𝑍3-11)/1

20. The FMOLP with max-min approach
Max CX ≥ 𝑍0 (24)
Subject to
𝛼 ≤1 – ( 𝑍1 - 65 )/5
𝛼 ≤1 – ( 𝑍1 - 16 )/2
𝛼 ≤1 – ( 𝑍1 - 11 )/1
𝑥01+ 𝑥02 + 𝑥03 = 1 (25)
𝑥10 + 𝑥12 + 𝑥13 = 2 (26)

21. Cont.
𝑥20 + 𝑥21 + 𝑥23 = 1 (27)
𝑥30 + 𝑥31 + 𝑥32 = 1 (28)
𝑥10 + 𝑥20 + 𝑥30 = 1 (29)
𝑥01 + 𝑥21 + 𝑥31 = 1 (30)
𝑥02 + 𝑥12 + 𝑥32 = 1 (31)
𝑥03 + 𝑥13 + 𝑥23 = 1 (32)
𝑥01 + 𝑥10 = 1 (33)
𝑥02 + 𝑥20 = 1 (34)

22. Cont.
𝑥03 + 𝑥30 = 1 (35)
𝑥12 + 𝑥21 = 1 (36)
𝑥13 + 𝑥31 = 1 (37)
𝑥23 + 𝑥32 = 1 (38)
𝛼 ≥ 0 ;
𝑥𝑖𝑗 ≥ 0 ;

23. Table 2:
Solution
Solution 𝒁 𝟏 𝒕 𝟏 𝒁 𝟐 𝒕 𝟐 𝒁 𝟑 𝒕 𝟑 𝜶 Route
1 65 , 5 16 , 2 _ 0.80 ( 𝑥03 , 𝑥31 , 𝑥20 )
2 65 , 5 16 , 2 11 , 1 _ No feasible Solution
2 65 , 5 16 , 2 11 , 4 0.55 ( 𝑥03 , 𝑥31 , 𝑥12 , 𝑥20 )
3 65 , 5 16 , 2 11 , 5 0.62 ( 𝑥03 , 𝑥31 , 𝑥12 , 𝑥20 )

24. Cont.
As given in table 2, only 𝑍1 and 𝑍2 are considered and 𝑍3 is omitted;
an optimal route with 𝜶 = 0.8 is obtained.
When 𝑍3 is also considered , solution becomes infeasible on these
tolerances. Again by relaxing tolerance in 𝑍3 to 4, solution becomes
feasible. In this case, the optimal path is achieved with 𝜶 = 0.55 .
By increasing tolerance in 𝑍3from 4 to 5, an optimal solution with
𝜶 = 0.62 is obtained.
These results show that by adjusting tolerance an optimal solution to
Multi-Criteria TSP can be determined.

25. 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.

26. References
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Management Science, p. 17, (1970).
Borges PC, Hansen PH, A study of Global Convexity for a Multiple Objective
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and Surveys in Metaheuristics, Kluwer Academic, pp. 129-150, (2000).
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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).

27. 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).

28. Thank you