An exact and a heuristic approach for the transportation-p-facility location problem

Abstract

The delineation of the transportation network is a strategic issue for all over the place. The problem of locating new facilities among several existing facilities and minimizing the total transportation cost are the main topics of the location network system. This paper addresses the transportation-p-facility location problem (T-p-FLP) which makes a connection between the facility location problem and the transportation problem, where p corresponds to the number of facilities. The T-p-FLP is a generalization of the classical transportation problem in which we have to seek where and how we impose the p-number of facilities such that the total transportation cost from existing facility sites to the potential facility sites will be minimized. The exact approach, based on the iterative procedure, and a heuristic approach as applied to the T-p-FLP are discussed and corresponding results are compared. An experimental example is incorporated to explore the efficiency and effectiveness of our proposed study in reality. Finally, a summary is given together with suggestions for future studies.

Appendix A

Here, we describe the iteration formulas used in the solution methodology section. Now, we refer to

$$\begin{aligned} Z(x,y) = \sum _{i=1}^{m}\sum _{j=1}^{p} \alpha _{i}u_{ij}^B\phi (c_{i},d_{i};x_{j},y_{j}), \end{aligned}$$

where $\phi (c_{i},d_{i};x_{j},y_{j}) =[{(c_{i}-x_{j})^{2}+(d_{i}-y_{j})^{2}}]^{1/2}$ and the terms $u_{ij}^B$ are constants. Differentiating Z with respect to $(x_j,y_j)$ and equating with 0, we get

$$\begin{aligned}&\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B(c_i-x_j)}{\phi (c_{i}, d_{i};x_{j},y_{j})}=0 \quad ( j=1,2,\ldots , p), \end{aligned}$$

(A.6)

$$\begin{aligned}&\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B(d_i-y_j)}{\phi (c_{i}, d_{i};x_{j},y_{j})}=0 \quad ( j=1,2,\ldots , p). \end{aligned}$$

(A.7)

Now, from Eqs. (A.6)–(A.7) we obtain

$$\begin{aligned}&\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^Bc_i}{\phi (c_{i},d_{i};x_{j},y_{j})} -x_j\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B}{\phi (c_{i},d_{i};x_{j},y_{j})} =0 \quad ( j=1,2,\ldots , p), \\&\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^Bd_i}{\phi (c_{i},d_{i};x_{j},y_{j})} -y_j\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B}{\phi (c_{i},d_{i};x_{j},y_{j})} =0 \quad ( j=1,2,\ldots , p). \end{aligned}$$

Then,

$$\begin{aligned}&x_j=\frac{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^Bc_i}{\phi (c_{i}, d_{i};x_{j},y_{j})}}{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B}{\phi (c_{i},d_{i};x_{j},y_{j})}} \quad ( j=1,2,\ldots , p), \\&y_j=\frac{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^Bd_i}{\phi (c_{i}, d_{i};x_{j},y_{j})}}{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B}{\phi (c_{i},d_{i};x_{j},y_{j})}} \quad ( j=1,2,\ldots , p). \end{aligned}$$

These equations are solved iteratively. The iteration equations for $(x_j,y_j)$ are as follows:

$$\begin{aligned}&x_j^{k+1}=\frac{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^Bc_i}{\phi (c_{i}, d_{i};x_{j}^k,y_{j}^k)}}{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^B}{\phi (c_{i},d_{i};x_{j}^k,y_{j}^k)}} \quad ( j=1,2, \ldots ,p;~k\in {\mathbb {N}}), \end{aligned}$$

(A.8)

$$\begin{aligned}&y_j^{k+1}=\frac{\sum _{i=1}^{m}\frac{\alpha _iu_{ij}^Bd_i}{\phi (c_{i},d_{i};x_{j}^k,y_{j}^k)}}{\sum _{i=1}^{m} \frac{\alpha _iu_{ij}^B}{\phi (c_{i},d_{i};x_{j}^k,y_{j}^k)}} \quad ( j=1,2,\ldots ,p;~k\in {\mathbb {N}}), \end{aligned}$$

(A.9)

where $\phi (c_{i},d_{i};x_{j}^k,y_{j}^k) =[{(c_{i}-x_{j}^k)^{2}+(d_{i}-y_{j}^k)^{2}}]^{1/2}$. The initial estimates of $(x_j,y_j)$ are simply chosen by the weighted mean coordinates:

$$\begin{aligned}&x_j^{0}=\frac{\sum _{i=1}^{m}\alpha _iu_{ij}^Bc_i}{\sum _{i=1}^{m} \alpha _iu_{ij}^B} \quad ( j=1,2,\ldots , p), \end{aligned}$$

(A.10)

$$\begin{aligned}&y_j^{0}=\frac{\sum _{i=1}^{m}\alpha _iu_{ij}^Bd_i}{\sum _{i=1}^{m} \alpha _iu_{ij}^B} \quad (j=1,2,\ldots , p). \end{aligned}$$

(A.11)