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

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.

Appendix A

1.1 Creating the input, good output and bad output bundle

Following the notations used in Imanirad et al [27], input, good output and bad output bundles may be generated for each k=1,…,K, where $I_{k}$ and $R_{k} \left( {O_{G} , O_{B} } \right)$ represent a set of inputs, and good output and bad output, respectively. These bundles are generated so that $R_{k}$is from a mutually exclusive set and for each k, $\left( {I_{k} , R_{k} \left( {O_{G} , O_{B} } \right)} \right)\quad k = 1, \ldots ,K$ is maximal based on the following definition.

Definition 1

An input, good output and bad output bundle $\left( {I_{k} , R_{k} \left( {O_{G} , O_{B} } \right)} \right)$ is considered to be maximal if it possesses the following two properties:

• 1. every good output v and every bad output w in $R_{k} \left( {O_{G} , O_{B} } \right)$ are influenced by every input i in $I_{k}$and no other input outside of $I_{k}$ may influence any good output v or any bad output w in $R_{k} \left( {O_{G} , O_{B} } \right)$;

• 2. there exists no output outside of $R_{k} \left( {O_{G} , O_{B} } \right)$ whose input bundle is identical to that of $R_{k} \left( {O_{G} , O_{B} } \right)$.

1.2 Algorithm 1 (creating maximal bundles)

• Step 1: Define S to be an empty set.

• Step 2: For each good output and bad output $\left( {m,s} \right)$, derive $I\left( {m,s} \right)$, the set of all inputs i that influence $\left( {m,s} \right)$; transfer $I\left( {m,s} \right)$ to S and set the bundle counter as$k = 1$.

• Step 3: For each $I\left( {m,s} \right)$ in S, compare it with every other $I\left( {m^{\prime},s^{\prime}} \right)$ in S, and identify all $I\left( {m^{\prime},s^{\prime}} \right)$ that have the same input elements as in$I\left( {m,s} \right)$. If no such $\left( {m^{\prime},s^{\prime}} \right)$ is identified, create bundle ($I_{k} , R_{k} \left( {O_{G} , O_{B} } \right))$ using $I\left( {m,s} \right)$ and $\left( {m,s} \right)$ so that $\left( {I_{k} , R_{k} \left( {O_{G} , O_{B} } \right)} \right) = \left( {I\left( {m,s} \right),\left( {m,s} \right)} \right)$. Remove $I\left( {m,s} \right)$from S. Go to step 4. Otherwise, group good output and bad output $\left( {m,s} \right)$ and all identified $\left( {m^{\prime},s^{\prime}} \right)$ (having the same input sets) together to derive $R_{k} \left( {O_{G} , O_{B} } \right)$ and create bundle $\left( {I_{k} , R_{k} \left( {O_{G} , O_{B} } \right)} \right)$ using $I\left( {m,s} \right)$ and $R_{k} \left( {O_{G} , O_{B} } \right)$ so that $\left( {I_{k} , R_{k} \left( {O_{G} , O_{B} } \right)} \right) = \left( {I\left( {m,s} \right),\left( {m,s} \right)} \right)$. Remove $I\left( {m,s} \right)$ and all identified $I\left( {m^{\prime},s^{\prime}} \right)$ from S. Go to step 4.

• Step 4: If S is nonempty, set $k = k + 1$ and go to step 3. Otherwise, terminate having formed the set of all bundles.