1. Solve the following problems.

a) Define fuzzy set and give the examples.

b) Let A and B be two fuzzy set, 𝐴 = {(2,1), (3,0.5)}; 𝐡 = {(3,1), (4,0.5)}, Fine A + B

c) Discuss the procedure for solving fuzzy linear programming problem with right hand size as fuzzy numbers.

 

 

2. Consider the fuzzy sets A, B and C defined on 𝑋 = [0,10] of real numbers by the membership grade function

π‘₯ 1

𝐴 = π‘₯ + 2 , 𝐡 = 2, 𝐢 = 1 + 10(π‘₯ βˆ’ 2)2

Calculate 𝐴 βˆͺ 𝐡, 𝐴 βˆͺ 𝐢 π‘Žπ‘›π‘‘ 𝐴 ∩ 𝐢̅.

 

3. Solve the following fuzzy linear programming problem

 

π‘€π‘Žπ‘₯ 𝑧 = 0.5π‘₯1 + 0.2π‘₯2

𝑆𝑒𝑏𝑗𝑒𝑐𝑑 π‘‘π‘œ

π‘₯1 + π‘₯2 ≀ 𝐡1 2π‘₯1 + π‘₯2 ≀ 𝐡2

 

 

π‘€β„Žπ‘’π‘Ÿπ‘’

 

 

1 π‘“π‘œπ‘Ÿ π‘₯ ≀ 300.

 

 

1 π‘“π‘œπ‘Ÿ π‘₯ ≀ 400.

 

𝐡1(π‘₯) = {

 

400βˆ’π‘₯

 

 

100

 

π‘“π‘œπ‘Ÿ 300 < π‘₯ ≀ 400

 

𝐡2

 

(π‘₯) = {

 

500βˆ’π‘₯

 

 

100

 

π‘“π‘œπ‘Ÿ 400 < π‘₯ ≀ 500

 

0 π‘“π‘œπ‘Ÿ π‘₯ > 400. 0 π‘“π‘œπ‘Ÿ π‘₯ > 500.

 

 

4. Solve the following fuzzy linear programming problem

 

π‘€π‘Žπ‘₯ 𝑧 = 5π‘₯1 + 4π‘₯2

𝑆𝑒𝑏𝑗𝑒𝑐𝑑 π‘‘π‘œ

(4,2,1)π‘₯1 + (5,3,1)π‘₯2 ≀ (24,5,8)

(4,1,2)π‘₯1 + (1,0.5,1)π‘₯2 ≀ (12,6,3)

π‘₯1, π‘₯2 β‰₯ 0