Resource allocation and target setting: a CSW–DEA based approach

Abstract

Resource allocation and target setting is part of the strategic management process of an organization. In this paper, we address the important issue of “optimally” allocating additional resources to the different operating units. Three different managerial interpretations of this question are presented, differing on the assumptions on the expected output increases. In each case, using multiplier data envelopment analysis (DEA) models and common set of weights (CSW), a new procedure for resource allocation and target setting is proposed. The proposed approach is innovative in its use of CSW and multi objective optimization, both of which are consistent with the centralized decision making character of the problem. The validity and usefulness of the proposed CSW–DEA models is shown using different datasets.

Appendices

Appendix 1

1.1 Proof of theorem 1

Suppose ${\mathbf{(u,v,\alpha }}_{{\mathbf{1}}} {\mathbf{,\alpha }}_{{\mathbf{2}}} {\mathbf{,}}...{{\varvec{\upalpha}}}_{{\mathbf{n}}} {\mathbf{)}}$ is a weak efficient solution of model (4). It is clear that this solution will be a feasible solution of model (5). By contradiction assume that this solution is not a weak efficient solution of model (5), then there is a feasible solution ${\mathbf{(}}\overline{{\mathbf{u}}} {\mathbf{,}}\overline{{\mathbf{v}}} {\mathbf{,}}\overline{{{\varvec{\upalpha}}}}_{{\mathbf{1}}} {\mathbf{,}}\overline{{{\varvec{\upalpha}}}}_{{\mathbf{2}}} {\mathbf{,}}...{\mathbf{,}}\overline{{{\varvec{\upalpha}}}}_{{\mathbf{n}}} {\mathbf{)}}$ of model (5) such that ${\mathbf{uy}}_{{\mathbf{j}}} < \overline{{\mathbf{u}}} {\mathbf{y}}_{{\mathbf{j}}} \quad \& \,\quad {\mathbf{vx}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {{{\varvec{\upalpha}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} > \,\overline{{\mathbf{v}}} {\mathbf{x}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {\overline{{{\varvec{\upalpha}}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} ,j = 1,2,...,n$. According to the constraints and variables of models (4) and (5), $\left( {{\mathbf{uy}}_{{\mathbf{j}}} } \right),\left( {\overline{{\mathbf{u}}} {\mathbf{y}}_{{\mathbf{j}}} } \right),\left( {{\mathbf{vx}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {{{\varvec{\upalpha}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} } \right),\,\left( {\overline{{\mathbf{v}}} {\mathbf{x}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {\overline{{{\varvec{\upalpha}}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} } \right) > 0,j = 1,2,...,n$ and we have ${\mathbf{uy}}_{{\mathbf{j}}} < \overline{{\mathbf{u}}} {\mathbf{y}}_{{\mathbf{j}}} \quad \& \,\quad \frac{1}{{{\mathbf{vx}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {{{\varvec{\upalpha}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} }} < \,\frac{1}{{\overline{{\mathbf{v}}} {\mathbf{x}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {\overline{{{\varvec{\upalpha}}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} }},j = 1,2,...,n$. This implies that $\frac{{{\mathbf{uy}}_{{\mathbf{j}}} }}{{{\mathbf{vx}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {{{\varvec{\upalpha}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} }} < \frac{{\overline{{\mathbf{u}}} {\mathbf{y}}_{{\mathbf{j}}} }}{{\overline{{\mathbf{v}}} {\mathbf{x}}_{{\mathbf{j}}} {\mathbf{ + }}\left\| {\overline{{{\varvec{\upalpha}}}}_{{\mathbf{j}}} } \right\|_{{\mathbf{1}}} }},\,\,j = 1,2,...,n$ which contradicts the weak efficiency of ${\mathbf{(u,v,\alpha }}_{{\mathbf{1}}} {\mathbf{,\alpha }}_{{\mathbf{2}}} {\mathbf{,}}...{{\varvec{\upalpha}}}_{{\mathbf{n}}} {\mathbf{)}}$ for model (4).

Appendix 2

2.1 Proof of theorem 2

First of all, note that the constraints of models (6) and (10) are equivalent, i.e. both models define the same feasible region. Moreover, their objective functions are also the same. Thus, the objective function in model (6) is the sum of the differences between the weighted sum of the outputs and the weighted sum of the inputs and this is maximized. In model (10), the slack variables $d_{j}$ is equal to the opposite difference, i.e. the weighted sum of the inputs and the weighted sum of the outputs, and the sum of these slacks variables is minimized. This shows that models (6) and (10) are equivalent.