Interval Oriented Entropy Based Multi-item Solid Transportation Problem with Budget and Breakability

Abstract

To maximization the total profit and minimization the total cost we investigate two transportation models with interval transportation parameters and we present a solution of our newly developed models using appropriate methods. The solid transportation problem has been formulated with and without interval entropy function defined by Shannon's measure of entropy. Also an additional constraint on the total budget at each destination is imposed. When we transport the goods from plants to destination then few items should be damage due to bad condition of route and vehicles, for this reason we introduce the breakability in our respective models. We employ two different approaches (Hu and Wang's Approach and Mahato and Bhunia's Approach) based on interval ranking and made a comparison on it. To convert the multi-objective programming problem into single-objective problem we apply weighted sum method. The models are illustrated with numerical examples and solved using the LINGO.13 optimization software. The results of the models with and without entropy are also obtained and compared.

Entropy Based Uncertain Multi Objective Multi-Item Fixed Charge 4D Transportation Problem Involving Breakable and Substitute Items

“…Giri et al (2014) investigated an entropy based FCSTP under the fuzzy-random environment with all unit discount costs. Baidya et al (2015) formulated and solved a profitable in-terval value STP with interval entropy function. Dalman (2018a) investigated an entropy-based multi-item STP in uncertain environment.…”

Parametric Functional Representation of Interval Number with Arithmetic Operations

“…Several researches [1,2,9,11,15,16,32,34,35] used interval arithmetic for optimization by different techniques; interval arithmetic provides an efficient method of global optimization [10]. Interval arithmetic operations [3,[6][7][8]12,29,30] studied by several researches in various approaches and form for investigation their problem with interval in real or complex domain.…”

“…Let A =[4,5] and B =[2,3], then find the parametric form as well as symmetrical parametric form of the intervals and their corresponding arithmetic operations A + B, A − B, AB, A/B.Solution:The parametric interval-valued functional form of A and B can be written as a ( p) = 4 1− p 5 p and b ( p) = 2 1− p 3 p . The symmetrical form of interval valued function of the same intervals can be written as a ( p)…”

“…Let A =[4,5] and B =[2,3], then find the arithmetic operation using convex combination form of A + B, A − B, AB, A/B.Solution:The interval-valued function in convex combination form of A and B can be written as a ( p)= 5 p + 4 (1 − p) , b ( p) = 3 p + 2 (1 − p) .The addition A + B =[6,8] and the interval-valued function in convex combination form of the sum is s ( p) = 8 p + 6 (1 − p) .The subtraction A − B =[1,3] and convex combination form of the interval-valued function is d (p) = 5 p + 4 (1 − p) − {2 p + 3 (1 − p)} = 2 p + 1.The multiplication A × B =[8,15] and convex combination form of the interval-valued function for multiplication is m ( p) = (5 p + 4 (1 − p))(3 p …”