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Mathematics 2019, 7(3), 281; https://doi.org/10.3390/math7030281

Constrained FC 4D MITPs for Damageable Substitutable and Complementary Items in Rough Environments

1
Department of Mathematics, Midnapore College (Autonomous), Midnapore 721101, India
2
Department of Computer Science, Vidyasagar University, Midnapore 721102, India
3
Department of Mathematics, Sidho Kanho Birsha University, Purulia 723104, India
4
Department of Mathematics, National Institute of Technology, Durgapur 713209, India
5
Department of Mathematics, Vidyasagar University, Midnapore 721102, India
*
Author to whom correspondence should be addressed.
Received: 2 January 2019 / Revised: 25 February 2019 / Accepted: 27 February 2019 / Published: 19 March 2019
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Abstract

Very often items that are substitutable and complementary to each other are sent from suppliers to retailers for business. In this paper, for these types of items, fixed charge (FC) four-dimensional (4D) multi-item transportation problems (MITPs) are formulated with both space and budget constraints under crisp and rough environments. These items are damageable/breakable. The rates of damageability of the items depend on the quantity transported and the distance of travel i.e., path. A fixed charge is applied to each of the routes (independent of items). There are some depots/warehouses (origins) from which the items are transported to the sales counters (destinations) through different conveyances and routes. In proposed FC 4D-MITP models, per unit selling prices, per unit purchasing prices, per unit transportation expenditures, fixed charges, availabilities at the sources, demands at the destinations, conveyance capacities, total available space and budget are expressed by rough intervals, where the transported items are substitutable and complementary in nature. In this business, the demands for the items at the destinations are directly related to their substitutability and complementary natures and prices. The suggested rough model is converted into a deterministic one using lower and upper approximation intervals following Hamzehee et al. as well as Expected Value Techniques. The converted model is optimized through the Generalized Reduced Gradient (GRG) techniques using LINGO 14 software. Finally, numerical examples are presented to illustrate the preciseness of the proposed model. As particular cases, different models such as 2D, 3D FCMITPs for two substitute items, one item with its complement and two non substitute non complementary items are derived and results are presented. View Full-Text
Keywords: four-dimensional transportation problem; fixed charge; space constraint; budget constraint; substitute and complimentary items; rough interval four-dimensional transportation problem; fixed charge; space constraint; budget constraint; substitute and complimentary items; rough interval
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Halder Jana, S.; Jana, B.; Das, B.; Panigrahi, G.; Maiti, M. Constrained FC 4D MITPs for Damageable Substitutable and Complementary Items in Rough Environments. Mathematics 2019, 7, 281.

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