Heuristic approaches for solid transportation-p-facility location problem

Abstract

Determining optimum places for the facilities and optimum transportation from existing sites to the facilities belongs to the main problems in supply chain management. The solid transportation-p-facility location problem (ST-p-FLP) is an integration between the facility location problem and the solid transportation problem (STP). This paper delineates the ST-p-FLP, a generalization of the classical STP in which location of p-potential facility sites are sought so that the total transportation cost by means of conveyances from existing facility sites to potential facility sites will be minimized. This is one of the most important problems in the transportation systems and the location research areas. Two heuristic approaches are developed to solve such type of problem: a locate-allocate heuristic and an approximate heuristic. Thereafter, the performance of the proposed model and the heuristics are evaluated by an application example, and the obtained results are compared. Moreover, a sensitivity analysis is introduced to investigate the resiliency of the proposed model. Finally, conclusions and an outlook to future research works are provided.

Appendix A

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

$$\begin{aligned}&Z(x,y) = \sum _{i=1}^{m}\sum _{j=1}^{p}\sum _{k=1}^{l}\gamma _{i}w_{ijk}^B\phi _{k}(u_{i},v_{i};x_{j},y_{j}), \end{aligned}$$

where $\phi _k=\epsilon _k\varphi _k(u_{i},v_{i};x_{j},y_{j})$ and the terms $w_{ijk}^B$ are constants. Differentiating Z with respect to $x_j$ and $y_j$, respectively, and then equating to 0 gives:

$$\begin{aligned}&\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B(u_i-x_j)}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}=0\quad ( j=1,2,\ldots , p), \end{aligned}$$

(6.1)

$$\begin{aligned}&\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B(v_i-y_j)}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}=0\quad ( j=1,2,\ldots , p). \end{aligned}$$

(6.2)

Now, from Eqs. (6.1)–(6.2) we obtain:

$$\begin{aligned}&\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^Bu_i}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}-x_j\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}=0\quad ( j=1,2,\ldots , p), \\&\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^Bv_i}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}-y_j\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}=0\quad ( j=1,2,\ldots , p). \end{aligned}$$

Then,

$$\begin{aligned}&x_j=\frac{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^Bu_i}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}}{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}}\quad ( j=1,2,\ldots , p), \\&y_j=\frac{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^Bv_i}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}}{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B}{\varphi _k(u_{i},v_{i};x_{j},y_{j})}}\quad ( j=1,2,\ldots , p). \end{aligned}$$

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

$$\begin{aligned}&x_j^{r+1}=\frac{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^Bu_i}{\varphi _k(u_{i},v_{i};x_{j}^r,y_{j}^r)}}{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B}{\varphi _k(u_{i},v_{i};x_{j}^r,y_{j}^r)}}\quad ( j=1,2,\ldots ,p;~r\in \mathbb {N}), \\&y_j^{r+1}=\frac{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^Bv_i}{\varphi _k(u_{i},v_{i};x_{j}^r,y_{j}^r)}}{\sum _{i=1}^{m}\sum _{k=1}^{l}\frac{\epsilon _k\gamma _{i}w_{ijk}^B}{\varphi _k(u_{i},v_{i};x_{j}^r,y_{j}^r)}}\quad ( j=1,2,\ldots ,p;~r\in \mathbb {N}), \end{aligned}$$

where $\varphi _k(u_{i},v_{i};x_{j}^r,y_{j}^r)=[{(u_{i}-x_{j}^r)^{2}+(v_{i}-y_{j}^r)^{2}}+\delta _k]^{1/2}$. The initial estimates of $(x_j,y_j)$ are illustrated by the weighted mean coordinates as follows:

$$\begin{aligned}&x_j^{0}=\frac{\sum _{i=1}^{m}\sum _{k=1}^{l}\gamma _{i}w_{ijk}^Bu_i}{\sum _{i=1}^{m}\sum _{k=1}^{l}\gamma _{i}w_{ijk}^B}\quad ( j=1,2,\ldots , p), \\&y_j^{0}=\frac{\sum _{i=1}^{m}\sum _{k=1}^{l}\gamma _{i}w_{ijk}^Bv_i}{\sum _{i=1}^{m}\sum _{k=1}^{l}\gamma _{i}w_{ijk}^B}\quad ( j=1,2,\ldots , p). \end{aligned}$$