4-Dimensional Transportation Problem for Substitute and Complementary Items Under Rough Environment Sharmistha Halder (Jana) 1(B ) , Debasis Giri 2 , Barun Das 3 , Goutam Panigrahi 4 , and Manoranjan Maiti 5 1 Department of Mathematics, Midnapore College, Midnapore, India sharmistha792010@gmail.com 2 Department of Computer Science and Engineering, MAKAUT, Kolkata, India debasis giri@hotmail.com 3 Department of Mathematics, SKBU, Purulia, India bdasskbu@gmail.com 4 Department of Mathematics, NITD, Durgapur, India panigrahi goutam@rediffmail.com 5 Department of Applied Mathematics with Oceanology and Computer Programming, Midnapore, India mmaiti2004@yahoo.co.in Abstract. In this paper, an innovative 4-dimensional multifarious breakable items transportation problem (4DMBITP) has been proposed. Here, per unit selling expenses, per unit purchasing prices, per unit trans- portation expenditures, fixed charge, availability of the sources, demands of the destination, conveyances capacities and total available budget are expressed by rough intervals. The transported items are substitutable and complementary in nature. The demand of the items at the desti- nation are directly related to the substitutability and complementary nature of the products and own selling price. The suggested model is converted into a deterministic one using lower and upper approximation intervals following Hamzehee et al. [1] as well as Expected Value Tech- nique. The converted model is optimized through Generalized Reduced Gradient (GRG) techniques using LINGO 14 software. Finally, numeri- cal examples are presented to illustrate the preciseness of the proposed model. Keywords: 4-dimensional TP · Rough interval · Substitutable and complimentary items · Fixed charge · Budget constraint 1 Introduction Transportation problems (TP) are one of the most common special type lin- ear programming problems involving constraints (source constraint and demand constraint). TP is one of the optimization problems most widely used by pri- vate and public sectors. In genuine circumstance, we handle with constraints c  Springer Nature Switzerland AG 2020 O. Castillo et al. (Eds.): ICITAM 2019, SCI 863, pp. 855–872, 2020. https://doi.org/10.1007/978-3-030-34152-7_65