# A data envelopment analysis approach for resource allocation with undesirable outputs: an application to home appliance production companies

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## Abstract

Traditional data envelopment analysis (DEA) models use all multiple inputs and outputs to estimate efficiency scores of decision-making units (DMUs). Each unit may consist of several subunits in cases such as manufacturing systems, and each subunit may produce both desirable and undesirable outputs. Providing information about the proportion of resources for each subunit could assist managers in making better decisions for increasing the efficiency of production systems. The current study proposes a new approach for resource allocation and efficiency estimation of production units by considering partial impacts among inputs and outputs in the DEA framework. A weak disposable technology is used in these evaluations, and an empirical application of the proposed approach for obtaining performance of home appliances production companies is provided for illustration purposes.

## Keywords

Data envelopment analysis (DEA) partial impacts undesirable outputs weak disposability resource allocation## List of Symbols

- \( j \in J, j = 1, \ldots ,n \)
collection of DMUs

- \( m = 1, \ldots ,M \)
set of good outputs

- \( s = 1, \ldots ,S \)
set of bad outputs

- \( i = 1, \ldots ,I \)
set of inputs

- \( k = 1, \ldots ,K \)
number of bundles

- \( {\text{DMU}}_{o} \)
DMU under evaluation

- \( v_{mj} \)
*m*th good output of*j*th DMU- \( w_{sj} \)
*s*th bad output of*j*th DMU- \( x_{ij} \)
*i*th input of*j*th DMU- \( v_{mo} \)
*m*th good output of DMU_{o}- \( w_{so} \)
*s*th bad output of DMU_{o}- \( x_{io} \)
*i*th input of DMU_{o}- \( \mu_{m} \)
weight for the

*r*th good output- \( \theta_{s} \)
weight for the

*s*th bad output- \( \rho_{i} \)
weight for the

*i*th input- \( \beta_{ik} \)
split variable for \( x_{ij} \) in

*k*th bundle- \( I_{k} \)
set of inputs in

*k*th bundle- \( T_{i} \)
set of all

*k*bundles that have*i*as a member- \( R_{k} \left( {O_{G} , O_{B} } \right) \)
*k*th bundle that has good or bad outputs- \( \varepsilon \)
non-Archimedean and very small number

- \( H_{kj} \)
convex combination of

*k*bundles’ efficiencies- \( e_{agg} \)
aggregate efficiency

- \( e_{ove} \)
overall efficiency

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