Intuitionistic Fuzzy Transportation Problem by Zero Point Method

Abstract

The transportation problems (TPs) support the optimal management of the transport deliveries. In classical TPs the decision maker has information about the crisp values of the transportation costs, availability and demand of the products. Sometimes in the parameters of TPs in real life there is ambiguity and vagueness caused by uncontrollable market factors.Uncertain values can be represented by fuzzy sets (FSs) of Zadeh. The FSs have the degrees of membership and nonmembership. The concept of intuitionistic fuzzy sets (IFSs) originated in 1983 as an extension of FSs. Atanasov's IFSs also have a degree of hesitansy to representing the obscure environment.In this paper we formulate the TP, in which the transportation costs, supply and demand values are intuitionistic fuzzy pairs (IFPs), depending on the diesel prices, road condition, weather and other factors. Additional constraints are included in the problem: limits for the transportation costs. Its main objective is to determine the quantities of delivery from producers to buyers to maintain the supply and demand requirements at the cheapest transportation costs. The aim of the paper is to extend the fuzzy zero point method (FZPM [35]) to the intuitionistic FZPM (IFZPM) to find an optimal solution of the intuitionistic fuzzy TP (IFTP) using the IFSs and index matrix (IM) concepts, proposed by Atanassov. The solution algorithm is demonstrated by a numerical example. Its optimal solution is compared with that obtained by the intuitionistic fuzzy zero suffix method (IFZSM). This work on Sect. I and Sect. II is supported by the project of Asen Zlatarov University under Ref. No. NIX-423/2019 "Innovative methods for extracting knowledge management" The work on Sect. III and Sect. IV is supported by the Ministry of Education and Science under the Programme "Young scientists and postdoctoral students", approved by DCM # 577/17.08.2018.

Two-Stage Intuitionistic Fuzzy Transportation Problem through the Prism of Index Matrices

“…Step 2. For solving the first stage on the 2-S IFTP we can apply one of the algorithms, outlined in our papers [47], [48], [52]. In the program code of the developed algorithms was used a part of Microsoft Visual Studio.NET 2010 C project's.…”

“…Step 9. Determining the optimal plan at second stage of the 2-S IFTP -X * [L J ,U, {x * l * j * ,u g }] after execution of one of the algorithms, presented in [47], [48], [52] with the obtained cost IFIM C * . The optimal intuitionistic fuzzy transportation cost at the second stage is calculated by:…”

“…In this section we will define 2-S IFTP extending the IFTP from [48]: A trader supplies a product to 4 different companies {l 1 , l 2 , l 3 , l 4 }. Let a product be produced at the manufacturers {k 1 , k 2 , k 3 } in quantities c k i ,R (for 1 ≤ i ≤ 3).…”

“…Step 2. The conditions for limiting the transportation costs are checked according to proposed approach in [48]. The problem is also balanced.…”

“…In our previous works [46], [47], [48], [52], we have proposed for the first time an intuitionistic fuzzy modified distribution algorithm, a zero-suffix and a zero-point method to determine an optimal solution of the IFTP, interpreted by the IFSs and IMs [18], [19] concepts. The concept of index matrices was introduced to enable two matrices with different dimensions to be summed.…”