### Title: Multi-item multi-stage transportation problem with breakability

Authors: Abhijit Baidya; Uttam Kumar Bera; Manoranjan Maiti

Addresses: Department of Mathematics, National Institute of Technology, Agartala, Jirania-799055, West Tripura, India ' Department of Mathematics, National Institute of Technology, Agartala, Jirania-799055, West Tripura, India ' Department of Applied Mathematics, Vidyasagar University, Midnapore-721102, WB, India

Abstract: In this manuscript, we formulate and solve four newly developed multi-stage multi item transportation models with breakability and restriction on transportation. Normally, due to bad condition of the road and vehicle, the breakable transported units got damaged. Here, the units are first transported from plants to distribution centres (DCs) and then from DC to customers. Four multi-item multi-stage transportation problems (MIMSTPs) are formulated to minimise the total transportation cost where the unit transportation costs, capacities of plants and DCs and demands of customers for the four models are crisp, fuzzy, stochastic and fuzzy-stochastic in nature respectively. For all models, breakability coefficient is taken deterministic. For the fuzzy model, generalised triangular fuzzy number (GTFN) and mean of cut (MC) method are considered. The chance constraint programming technique is used to find the deterministic form of the stochastic and fuzzy-stochastic models. The models are illustrated with numerical examples and solved using the LINGO-13 solver. In particular, with restricted number of DCs, the optimum results are also obtained. Some sensitivity analysis is also presented for model 1 with respect to breakability and deficit in demand.

Keywords: transportation problem; TP; multi-stage TP; breakability; space constraint; mean of α-cut; MC; method; chance constraint programming.

International Journal of Operational Research, 2018 Vol.31 No.4, pp.510 - 544