It is assumed there are *m* alternative establishment points and *n* demand points, and then the question to be considered: to satisfy demand points, a supplier need to establish production facilities to manufacture certain commodities. Each facility provides products for several demand points, and each demand point can be divided to several facilities. It is assumed that products can be subdivided infinitely. The studied issues of this paper include: how to select positions of production facilities and ensure their supply (capacity), how to distribute products to demand points and minimize sum of fixed cost of facility construction, variable cost of supply setting, and the cost of distributing products to demand points.

To facilitate expression, the author firstly defines used parameters and decision variables in this paper as follows:

Sets and parameters

*M* refers to set of alternative production facilities, *M* = {1, 2, ..., *m*};

*N* refers to set of demand points, *N* = {1, 2, ...,*n*};

*f*_{i} refers to fixed cost for constructing production facilities in *i* position, *i* ∈ *M*;

*q*_{i} refers to variable cost for setting unit supply in *i* position, *i* ∈ *M*;

*c*_{ij} refers to unit distribution cost of distributing demand point *j* to production facility *i*, *i* ∈ *M*, *j* ∈ *N*;

*B*_{i} refers to maximal supply of production facility which can be set in *I* position, *i* ∈ *M*;

*d*_{j} refers to demand in demand point *j*, *j* ∈ *N*;

Decision variables
$$\begin{array}{}{y}_{i}=\left\{\begin{array}{ll}1,& \text{Construct facility in}i\text{position}\\ \\ 0,& \text{Or}\end{array}\right.,i\in M,\end{array}$$

*u*_{i} refers to supply of production facility in *I* position, *i* ∈ *M*;

*x*_{ij} refers to demand ratio of demand point *j* distributed to facility *i*, *i* ∈ *M*, *j* ∈ *N*.

When parameters in model are confirmed, location of production facility can
be shown in following mathematical model:
$$(\mathrm{D}\mathrm{F})\phantom{\rule{1em}{0ex}}min\phantom{\rule{1em}{0ex}}{\displaystyle \sum _{i=1}^{m}\left({f}_{i}{y}_{i}+{q}_{i}{u}_{i}\right)+\sum _{i=1}^{m}\sum _{j=1}^{n}{c}_{ij}{d}_{j}{x}_{ij}}$$(1)
$$s.t.{u}_{i}\le {B}_{i}{y}_{i}\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in M$$(2)
$$\sum _{i=1}^{m}{x}_{ij}\text{=1\hspace{0.17em}}\mathrm{\forall}j\in N$$(3)
$${x}_{ij}\le {y}_{i}\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in M,\mathrm{\forall}j\in N$$(4)
$$\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in M$$(5)
$${y}_{i}\in \{0,1\}\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in M$$(6)
$${x}_{ij}\ge 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in M,\mathrm{\forall}j\in N$$(7)
$${u}_{i}\ge \text{0\hspace{0.17em}}\mathrm{\forall}i\in M$$(8)

Objective function (1) refers to cost of minimized system, including fixed cost of facility construction, variable cost of supply setting, and the cost of distributing products to demand points. Constraint function (2) means capacity of production facility cannot exceed maximal supply which can be provided by the facility, and supply can only be set in production facilities with station. Constraint function (3) means the total ratio of demand of demand point *j*(*j* ∈ *N*) being distributed to all facilities is 1. Constraint function (4) means only production facilities with station can provide products for demand points. Constraint function (5) means total demand distributed to production facility should not exceed supply of this facility. Constraint function (6), (7), and (8) restrict decision variables.

There are many deficiencies in selected facility location model because many uncertain factors may happen during the running of facilities. Normal running of facilities will be influenced and great inconvenience will be brought to social life once being influenced by these uncertain factors. Uncertain facility supply is one of major factors which affect normal running of facilities. Therefore, the uncertainty of facility supply must be considered when selecting address for facilities. This paper assumes facility supply not equal to initially set supply, which means there is fluctuation in facility supply. Fluctuation of facility supply is assumed to be a random variable. Some scholars consider uncertainty of supply when designing logistics transportation network. For example, Thapalia et al. (2012) [8] mentioned design problem of single commodity flow transportation network in random edge capacity with target functions of network constructing and running cost and maximal expectation for not satisfying demands. Thapalia *et al*. (2012) [8] described the possible appearing of uncertain parameters by scenario analysis. However, it is difficult to set scenes or happening ratio of scenes. When there are many scenes, the problem scale will increase, and solving of model will cost more time. Then the model cannot be applied in practical problem. Aiming at deficiencies in above research, in the following text, this paper assumes the fluctuation of facility supply as random variables with incomplete probability distribution information, and comes up with robust location model based on uncertain facility supply. The advantage of this model is that there is no need to know specific probability distribution of fluctuation of facility supply, but only needs to estimate its interval.

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