Geometric Programming Approach to an Interactive Fuzzy Inventory Problem

Table 1

𝛼-level interval of coefficient parameters.

Br	𝜇^𝐿𝑀(𝑥)	Br	𝜇^𝑅𝑀(𝑥)	(𝑀𝛼)=[𝑀^𝐿𝛼,𝑀^𝑅𝛼]
L:	1−(𝑀2−𝑥)/(𝑀2−𝑀1)	L:	1−(𝑥−𝑀2)/(𝑀3−𝑀2)	[𝑀1+𝛼(𝑀2−𝑀1),𝑀3−𝛼(𝑀3−𝑀2)]
L:	1−(𝑀2−𝑥)/(𝑀2−𝑀1)	P:	1−((𝑥−𝑀2)/(𝑀3−𝑀2))^2	[𝑀1+𝛼(𝑀2−𝑀1),𝑀2+√1−𝛼(𝑀3−𝑀2)]
L:	1−(𝑀2−𝑥)/(𝑀2−𝑀1)	E:	𝛼2(1−𝑒^−𝛽2((𝑀3−𝑥)/(𝑀3−𝑀2)))	[𝑀1+𝛼(𝑀2−𝑀1),𝑀3+((𝑀3−𝑀2)/𝛽2)log(1−𝛼/𝛼2)]
P:	1−((𝑀2−𝑥)/(𝑀2−𝑀1))^2	L:	1−(𝑥−𝑀2)/(𝑀3−𝑀2)	[𝑀2−√1−𝛼(𝑀2−𝑀1),𝑀3−𝛼(𝑀3−𝑀2)]
P:	1−((𝑀2−𝑥)/(𝑀2−𝑀1))^2	P:	1−((𝑥−𝑀2)/(𝑀3−𝑀2))^2	[𝑀2−√1−𝛼(𝑀2−𝑀1),𝑀2+√1−𝛼(𝑀3−𝑀2)]
P:	1−((𝑀2−𝑥)/(𝑀2−𝑀1))^2	E:	𝛼2(1−𝑒^−𝛽2((𝑀3−𝑥)/(𝑀3−𝑀2)))	[𝑀2−√1−𝛼(𝑀2−𝑀1),𝑀3+((𝑀3−𝑀2)/𝛽2)log(1−𝛼/𝛼2)]
E:	𝛼1(1−𝑒^−𝛽1((𝑥−𝑀1)/((𝑀2−𝑀1)))	L:	1−(𝑥−𝑀2)/(𝑀3−𝑀2)	[𝑀1−((𝑀2−𝑀1)/𝛽1)log(1−𝛼/𝛼1),𝑀3−𝛼(𝑀3−𝑀2)]
E:	𝛼1(1−𝑒^−𝛽1((𝑥−𝑀1)/(𝑀2−𝑀1)))	P:	1−((𝑥−𝑀2)/(𝑀3−𝑀2))^2	[𝑀1−((𝑀2−𝑀1)/𝛽1)log(1−𝛼/𝛼1),𝑀2+√1−𝛼(𝑀3−𝑀2)]
E:	𝛼1(1−𝑒^−𝛽1((𝑥−𝑀1)/(𝑀2−𝑀1)))	E:	𝛼2(1−𝑒^−𝛽2((𝑀3−𝑥)/(𝑀3−𝑀2)))	[𝑀1−((𝑀2−𝑀1)/𝛽1)log(1−𝛼/𝛼1),𝑀3+((𝑀3−𝑀2)/𝛽2)log(1−𝛼/𝛼2)]

L, P, and E stand for linear, parabolic, and exponential membership functions, respectively, 𝛼₁,𝛼₂>1;𝛽₁,𝛽₂>0;0<𝛼<1.