Vertical decomposition approach to solve single stage capacitated warehouse location problems

In this paper vertical decomposition approach is used for the first time to solve single stage capacitated warehouse location problems (SSCWLP). Stages of warehouse location problems are decomposed to get smaller sized problems, which are relatively easier to solve. New formulation of SSCWLP is developed using the style of (R.R.K. Sharma, 1991); and worked out by applying vertical decomposition approach. Relaxation of flow balance constraints leads to two different versions of capacitated plant location problem (CPLP), referred to as RHS_CPLP and LHS_CPLP. It is found that RHS_CPLP is exactly identical to CPLP (R. Sridharan, 1986); but LHS_CPLP is different from RHS_CPLP. Reference (R. Sridharan, 1986) gives 13 relaxations of CPLP (which is found identical to RHSCPLP for this paper). We extend these relaxations for the problem LHSCPLP also. 13 different relaxations of SSCWLP (with the proofs comparing their strengths) are developed. We found that the strong relaxation of SSCWLP given in (R.R.K. Sharma and V. Berry, 2007), is same as relaxation R3_O given in this paper. Also, there are other relaxations of SSCWLP that are better than strong relaxation proposed in (R.R.K. Sharma and V. Berry, 2007).