Resource allocation for branch network system with considering heterogeneity based on DEA method

Abstract

In this paper, we proposed a new DEA approach to allocate the resource in branch network system which is not covered by the existing resource allocation works under a centralized decision-making environment. The branch network system is typically appears in multi-national or multi-regional corporations, which has many branches across multiple locations. Given the spatial distribution of the production, we imposed additional restrictions on resource allocation and divided the resource inputs into three groups: fixed inputs, regional inputs that allocated to the branches in the same area and common resource that an additional resource allocated to all the branches. Then, we generalize the model further to accommodate technological heterogeneity due to the difference in the geographical locations of the branches. And the objective of the proposed models is to maximize the gross profits of the entire organization, which is a natural assumption for a for-profit organization. Finally, an example was presented to illustrate the proposed approach with heterogeneous technology is more practically feasible and superior than the prior approach with homogeneous technology.

Appendix

The method to transform the model (3) into a liner programming.

Let $\lambda_{jk} \partial_{{i_{3} j}} = \mu_{{i_{3} \,jk}}$, $\lambda_{jk} = \mu_{{i_{3} \,jk}} + \eta_{{i_{3} \,jk}}$, $\eta_{{i_{3} \,jk}} = (1 - \partial_{{i_{3} }} )\lambda_{jk}$, $\forall i_{3} ,\forall j,k$, where $i_{3} = 1,2, \ldots ,m_{3}$, $j,k = 1, \ldots ,n$. Since $\lambda_{jk} = \mu_{{i_{3} \,jk}} + \eta_{{i_{3} \,jk}}$, $\forall i_{3} ,\forall j,k$, this paper can arbitrary select each of $i^{\prime}_{3} = 1,2, \ldots ,m_{3}$, and impose $\mu_{{i_{3} \,jk}} + \eta_{{i_{3} \,jk}} = \mu_{{i_{3}^{{\prime }} \,jk}} + \eta_{{i_{3}^{{\prime }} \,jk}} \left( {i_{3} = 1, \ldots m_{3} ,j,k = 1, \ldots ,n} \right)$.

Then, in this paper we impose $\lambda_{jk} = \mu_{{i_{3} \,jk}} + \eta_{{i_{3} \,jk}} = \mu_{jk} + \eta_{jk} ,\forall i_{3} ,j,k$ to preserve the linearity and convexity in the DEA model. Then model (3) can be transformed into the following linear programming model.

$$ \begin{aligned} & \quad \hbox{max} \quad \varPi = \sum\limits_{k = 1}^{n} {\left( {\sum\limits_{r = 1}^{s} {p_{r} \hat{y}_{rk} } - \sum\limits_{i = 1}^{m} {c_{i} (x_{{i_{1} k}} + \hat{x}_{{i_{2} k}} + \hat{x}_{{i_{3} k}} )} } \right)} \quad (A.1) \\ & s.t\quad \sum\limits_{j = 1}^{n} {(\mu_{jk} + \eta_{jk} } )x_{{i_{1} j}} \le x_{{i_{1} k}} ,\quad k = 1, \ldots ,n,\quad i_{1} \in U,\quad (A.2) \\ & \quad \quad \quad \sum\limits_{j = 1}^{n} {(\mu_{jk} + \eta_{jk} } )x_{{i_{2} j}} \le \hat{x}_{{i_{2} k}} ,\quad k = 1, \ldots ,n,\quad i_{2} \in A,\quad (A.3) \\ & \quad \quad \quad \sum\limits_{{k \in J_{t} }} {\hat{x}_{{i_{2} k}} } = \sum\limits_{{j \in J_{t} }} {x_{{i_{2} j}} } ,\quad t = 1, \ldots ,T,\quad i_{2} \in A,\quad (A.4) \\ & \quad \quad \quad \sum\limits_{j = 1}^{n} {\mu_{{i_{3} jk}} } x_{{i_{3} }} \le \partial_{{i_{3} k}} x_{{i_{3} }} ,\quad k = 1, \ldots ,n,\quad i_{3} \in C,\quad (A.5) \\ & \quad \quad \quad \sum\limits_{j = 1}^{n} {(\mu_{jk} + \eta_{jk} } )y_{rj} \ge \hat{y}_{rk} ,\quad k = 1, \ldots ,n,\quad r = 1, \ldots ,s,\quad (A.6) \\ & \quad \quad \quad \sum\limits_{{k \in J_{t} }} {\hat{y}_{rk} } \ge \sum\limits_{{j \in J_{t} }} {y_{rj} } ,\quad t = 1, \ldots ,T,\quad r = 1, \ldots ,s,\quad (A.7) \\ & \quad \quad \quad \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\partial }_{{i_{3} k}} \le \partial_{{i_{3} k}} \le \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\partial }_{{i_{3} k}} ,\quad k = 1, \ldots ,n,\quad i_{3} \in C,\quad (A.8) \\ & \quad \quad \quad \alpha \le \sum\limits_{{k \in J_{t} }} {\partial_{{i_{3} k}} } \le \beta ,\quad t = 1, \ldots ,T,\;i_{3} \in C,\quad (A.9) \\ & \quad \quad \quad \sum\limits_{k = 1}^{n} {\partial_{{i_{3} k}} } = 1,\quad i_{3} \in C,\quad (A.10) \\ & \quad \quad \quad \sum\limits_{j = 1}^{n} {(\mu_{jk} + \eta_{jk} } ) = 1,\quad k = 1, \ldots ,n,\quad (A.11) \\ & \quad \quad \quad \hat{x}_{{i_{2} k}} ,\hat{y}_{rk} \ge 0,\quad k = 1, \ldots ,n,\quad i_{2} \in A,\quad r = 1, \ldots ,s,\quad (A.12) \\ & \quad \quad \quad \mu_{jk} ,\eta_{jk} ,\partial_{{i_{3} k}} \ge 0,\quad j,\quad k = 1, \ldots ,n,\quad i_{3} \in C.\quad (A.13) \\ \end{aligned} $$

Since $(\mu_{jk} + \eta_{jk} )\partial_{{i_{3} \,jk}} = \mu_{{i_{3} \,jk}}$, multiplying $(\mu_{jk} + \eta_{jk} )$ on constraints (A.8) and (A.9) to ensure the range of $\mu_{{i_{3} \,jk}}$ is related to $\partial_{{i_{3} \,jk}}$, this paper thus has

$$ \begin{array}{ll} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\partial }_{{i_{3} k}} \left( {\mu_{jk} + \eta_{jk} } \right) \le \mu_{{i_{3} \,jk}} \le \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\partial }_{{i_{3} k}} \left( {\mu_{jk} + \eta_{jk} } \right),\quad j,\quad k = 1, \ldots ,n,\quad i_{3} \in C,\quad (A.14), \hfill \\ \alpha \sum\limits_{{j \in J_{t} }} {\left( {\mu_{jk} + \eta_{jk} } \right)} \le \sum\limits_{{j \in J_{t} }} {\mu_{{i_{3} \,jk}} } \le \beta \sum\limits_{{j \in J_{t} }} {\left( {\mu_{jk} + \eta_{jk} } \right)} ,\quad k = 1, \ldots ,n,\quad t = 1, \ldots ,T,\quad i_{3} \in C,\quad (A.15). \hfill \\ \end{array} $$

Moreover, this paper imposes constraints (A.14) and (A.15) on Model (A). The methods that transform the model (6) into a linear programming are the same with the above description, so we won’t repeat it here again.