Single Stage Plant-Warehouse Location Problem (SSPWLP): A State of Art Formulation

Abstract

Abstract Kaufman et. al. (1977) first considered this problem (SSPWLP). Goods flow from plants to warehouses to markets. Here we need to locate plants and warehouses of appropriate capacities so that sum total of location cost of plants and warehouses and distribution cost of goods from plant to warehouses to markets is minimized. They considered normalized decision variables (see Sharma and Muralidhar (2009)). However they used the formulation style of Geoffrion and Graves (1974). We use normalized variables but use the variable style of Sharma (1991) and Sharma and Berry (2007) that reduces the number of variables. We give several strong linking constraints by drawing from the works of Sharma and Namdeo (2005) and Sharma and Berry (2007). Below we give the formulation in brief. Details can be seen in the body of the paper. Variables names are self explanatory and makes understanding the model easier. New Formulation of SSPWLP: Min sum(i,j), cpw(i,j)*xpw(i,j) + sum(j,k), cwm(j,k)*xwm(j,k) Sum(i), yp(i)*fp(i) + sum(j), yw(j)*fw(j) (0) Sum(i,j), xpw(i,j) = 1 (0a) Sum(j,k), xwm(j,k) = 1 (0b) Sum(j), xwm(j,k) >= d(k) for all k (0c) Sum(j), xpw(i,j) <= capp(i) for all i (1) Sum(j), xpw(i,j) <= capp(i)*yp(i) for all i (2) Sum(i), xpw(i,j) <= capw(j) for all j (3) Sum(i), xpw(i,j) <= capw(j)*yw(j) for all j (4) xpw(i,j) <= yp(i)*capw(j) for all i, j (5) xpw(i,j) <= yw(j)*capp(i) for all i,j (6) xwm(j,k) <= d(k)*yw(j) for all j,k (7) Sum(j), xpw(i,j) <= yp(i) for all i (8) Sum(i), xpw(i,j) <= yw(j) for all j (9) Flow balance constraint: Sum(i), xpw(i,j) = sum(k), xwm(j,k) for all j (10) Sum(i), capp(i)*yp(i) >= 1 (11) Sum(j), capw(j)*yw(j) >= 1 (12) xpw(i,j) >= 0 for all i,j; xwm(j,k) >= 0 for all j,k (13) yp(i) = (0,1) for all i and yw(j) = (0,1) for all j (14) This (the above formulation of SSPWLP) is the best formulation of SSPWLP that is amenable to solution by LP relaxation and attendant branch and bound and/or branch and cut solution procedure. In literature the distribution phase between plant and warehouse (where plant and warehouse are to be located) is solved by Sharma and Agarwal (2014) as MID_CPLP were Lagrangian Relaxation (LR) was deployed to get RHS_CPLP and LHS_CPLP (different classes of capacitated plant location problems) that were attempted by well known relaxations (LR and Linear Programming) already available in literature (see Priyanka Verma and Sharma (2011)). Computational investigation is underway to determine efficacy of different new linking constraints given in this paper.