In real life while dealing with any problem the information consists of data, which is not easily comprehensible due to the presence of many impurities like ambiguity, inconsistency, impreciseness, inconstant data/information. So, a new concept of fuzzy set (FS) theory was proposed and later neutrosophic set (NS) theory has emerged that deals with ambiguity on a broader level. Quadripartitioned neutrosophic set (QNS) is an extension of NS that consists of four independent variables as truth, contradiction, ignorance, and falsity degree of membership denoting uncertainty in data, where contradiction and ignorance together make indeterminacy of NS and have more scope to deal with impurities then NS. In actual life, transportation problems (TPs) have existed since ages and have been a matter of concern because it directly has an effect on cost and time value of TPs. In the existing literature, many methodologies/algorithms have been proposed in various fuzzy and neutrosophic environments, but till yet no TPs have not been presented in the QNS environment. In this paper, a unique solution to quadripartitioned neutrosophic transportation problem (QNTP) has been proposed where cost is fuzzy but demand and supply are crisp. To solve the QNTP, a novel algorithm is proposed to find a basic feasible solution and to check its optimality two novel algorithms are proposed. To express the validity of the proposed method, a real-life problem has been solved effectively. In future, the work will be extended to the entire fuzzy transportation problem (TP) in QNS environment, i.e., entire demand, supply, and cost are fuzzy in nature, and to other fuzzy environments.