Efficiency measurement for mixed two-stage nonhomogeneous network processes with shared extra intermediate resources

Abstract

Unreasonable allocation of shared resources reduces the system efficiency and is a considerable operational risk. Sub-processes with insufficient portion of shared resources could not help accomplish complicated tasks, and overstaffing and idle resources will occur in the sub-processes assigned with redundant shared resources. This unfair portion distribution may cause internal contradictions among sub-processes and even lead to the collapsing of the entire system. This study proposes a data-driven, mixed two-stage network data envelopment analysis model. This method aims to reasonably define the allocation portion of shared extra intermediate resources among several nonhomogeneous subsystems and measure the overall system performance. A data set of 58 international hotels is used to test the features of the proposed model.

Keywords

shared resource allocation / mixed two-stage system / data envelopment analysis / efficiency

Introduction

Data envelopment analysis (DEA) is a data-driven approach that measures the relative efficiency or performance of a set of decision-making units (DMUs) with multiple inputs and outputs. This method was first proposed by ^{Charnes et al. (1978)}. Numerous extension studies based on this approach have been conducted (^{Amin and Toloo, 2004}; ^{Chen et al., 2014}; ^{An et al., 2019}; Song et al., 2020).

Traditional DEA studies regard each DMU as a black box, but the internal structure is ignored. Many studies investigated a simple two-stage network structure that uses intermediate measures as connection products between two stages to avoid the bias caused by ignoring the structure. ^{Chen et al. (2006)} evaluated the effect of IT on the two-stage bank’s operation process. The first stage uses fixed assets, number of employees, and IT investment to generate deposit dollars. As an intermediate measure, deposit dollars are utilized to output securities and loans in the second stage. Several studies have also applied DEA to this simple two-stage process (^{Seiford and Zhu, 1999}; ^{Kao and Hwang, 2008}; ^{An et al., 2018}).

The studies mentioned above are based on a simple two-stage structure and involve inputs, outputs, and intermediate measures but without any additional intermediate measure. However, many real-life cases must be represented by highly complicated two-stage structures. ^{Yin et al. (2019)} discussed the fundamental structure of hotel operations and management model by considering additional inputs and comparing it with the simple two-stage structure. Other types of two-stage process have been proposed to avoid misleading due to the ignored operations of internal processes (^{Liang et al., 2006}; ^{Cook et al., 2010}; ^{Halkos et al., 2014}; ^{Kao, 2014}; ^{Li et al., 2014}).

Mixed two-stage network structure is a special structure that mixes series and parallel to a two-stage network process. This structure is common in real life, such as in airports, hospital operations, military systems, and hotel chains. In this study, we propose a mixed two-stage nonhomogeneous network process with shared extra intermediate resources. In the two serial stages, the second stage has several parallel nonhomogeneous sub decision-making units (SDMUs). These SDMUs consume the respective inputs from the first stage and have shared extra intermediate resources. This mixed structure can be applied to the real life. For example, in a complete supply chain, the first stage uses initial investment to produce various semi-finished products and distributes them to the second stage. The nonhomogeneous SDMUs of the second stage use respective intermediate measures to produce the final productions. These nonhomogeneous SDMUs also require shared extra intermediate inputs, such as advertising and management costs. These shared extra intermediate inputs cannot be conveniently divided but directly affect the performance evaluation.

The rest of this paper is organized as follows. Section 2 contains the literature review in the field of nonhomogeneous DEA and network systems. Section 3 proposes a mixed two-stage DEA model. Section 4 presents the empirical data of 58 Taiwan international hotels that are applied in the proposed model. Section 5 comprises the drawn conclusions.

Literature review

Nonhomogeneous DEA

Traditional DEA measures the performance of a set of homogeneous DMUs. However, this method cannot reasonably handle actual nonhomogeneous issues that remain to be measured. ^{Cook et al. (2012)} presented a case in which some DMUs did not produce certain outputs and therefore could be categorized into different nonhomogeneous groups. A DEA-type model was developed to measure this nonhomogeneous situation. ^{Cook et al. (2013)} also proposed a DEA-based methodology to solve this nonhomogeneous problem. Each DMU was divided into a set of business sub-units, and the weighted average of sub-group efficiency values was assumed as the overall efficiency for the DMU. Different from the above works that investigated the nonhomogeneous issue on the output side, ^{Li et al. (2016)} developed DEA models to handle nonhomogeneous DMUs with input configurations. Another nonhomogeneous problem is partial input-to-output effects. ^{Imanirad et al. (2013)} extended the traditional DEA methodology to estimate the technical efficiency score of nonhomogeneous DMUs with partial input-to-output effect. ^{Imanirad et al. (2015)} also extended their earlier work for efficiency measurement, in which the nonhomogeneous DMUs had different input/output profiles. ^{Li et al. (2017)} presented a model to measure the efficiency for nonhomogeneous DMU with given output bundles produced by multiple procedures or processes. ^{Zhu et al. (2018)} established a cross-like efficiency model to evaluate efficiency for homogenous DMUs. The proposed model can also be used to range the DMUs with missing outputs or inputs. For parallel systems with nonhomogeneous subunits, ^{Du et al. (2015)} proposed DEA models to measure the overall performance of entire system and the efficiency decomposition for each nonhomogeneous subunit.

Network systems

From the perspective of system structure, existing research on shared resources/inputs can mainly be divided into three categories, namely, two-stage series, parallel, and mixed structures. Numerous studies on shared resources are based on two-stage series process. In two-stage bank operation process, ^{Chen et al. (2006)} developed a nonlinear DEA programming model to measure the effect of IT considering three types of shared input in two stages. ^{Chen et al. (2010)} emphasized that in many real-life situations, some inputs cannot be split up, which would lead to inaccurate results in efficiency evaluation. They developed a set of DEA models to measure the efficiency for two-stage systems with shared input that cannot be split. ^{Zha and Liang (2010)} proposed a product-form cooperative efficiency model to measure the performance of a two-stage production process in series. ^{Amirteimoori (2013)} divided shared resources into two interdependent stages arranged in series. ^{Wu et al. (2016b)} indicated a situation in which the undesirable intermediate measure from the first stage could be used or processed by the second stage. New resources would be fed back to the first stage from the second stage. In this situation, shared resources were used by two stages, because the inputs cannot be conveniently split up.

Another part of research on shared resources is based on parallel structure systems. ^{Cook and Tuenter (2000)} and ^{Cook and Hababou (2001)} proposed DEA models to evaluate the sales and service efficiency for Canadian Bank branches with parallel structure. ^{Jahanshahloo et al. (2004a}; ^2004b) measured the efficiency of a multicomponent bank branch parallel system. ^{Kao (2012)} discussed the distribution of the same resources for a parallel system in his example, which was used by ^{Beasley (1995)}. ^{Rogge and De Jaeger (2012)} presented a shared input DEA model to estimate the cost efficiency for municipality and each fraction. In their research, waste cost was shared among six parallel waste fractions. ^{Bian et al. (2015)} measured the efficiency for a general parallel system with shared inputs and outputs. In the general parallel system, each subsystem had a set of common, dedicated, and shared inputs/outputs. A general DEA model was proposed to handle the efficiency evaluation issue. ^{Wu et al. (2016a)} treated the transportation as a parallel system with shared resources. They proposed a DEA model to measure the performance of transportation systems in China.

The third type of research on shared resources is based on mixed structure. ^{Yu and Lin (2008)} measured the efficiency and effectiveness of railways’ operations with a mixed structure by a multiactivity network DEA model. ^{Yu and Fan (2009)} presented a DEA model named MSNDEA to estimate the efficiency of multimode transit firm. The structure of transit firm was a mixed structure, in which the first stage was a parallel and was connected in series with the second stage. ^{Huang et al. (2014)} proposed a modified two-stage DEA approach based on the envelope form to evaluate the efficiency and effectiveness of Taiwan’s international tourist hotels with a mixed structure framework. The first stage is a production process, and the second stage is a parallel system. As an intermediate input, marketing expense helps each parallel subsystem to produce their own output.

As reviewed in this section, the literature of serial system focuses on the internal structure. However, this type of system still ignores the internal details of each stage. As another type of network systems, parallel system usually limits that each parallel sub-process must be the same as others, i.e., the parallel system has insufficient ability to address the cases with parallel nonhomogeneous sub-processes. Mixed system is the integration version of series and parallel systems. Mixed system is suitable for complex cases and ensures that the parallel sub-processes are homogeneous. In the present study, a mixed system suitable for nonhomogeneous DMUs is considered. This structure compensates for the short board that the ordinary mixed structure cannot solve the nonhomogeneous problem. To the best of our knowledge, the sharing of extra intermediate resources among nonhomogeneous parallel sub-processes in the second stage of mixed two-stage systems has not been discussed yet. This study aims to propose a multiplier-based network DEA model to estimate the overall efficiency of mixed two-stage nonhomogeneous network processes with shared extra intermediate resources. We adopt the weighted additive efficiency decomposition method proposed by ^{Chen et al. (2014)} to calculate the efficiency for two stages. The optimal portion of shared resources for each sub-process is determined by the proposed model, which varies in accordance with intervals.

Mixed two-stage DEA models for shared extra intermediate resources

As shown in Fig. 1, a mixed two-stage nonhomogeneous network process has two stages that are connected in series. The second stage is composed of multiple parallel nonhomogeneous SDMUs. Among the nonhomogeneous SDMUs, their inputs and outputs are not exactly the same. Nevertheless, all SDMUs have shared extra intermediate resources.

DMUs with mixed two-stage network structure are supposed. The second stage of each DMU_j (j = 1,…, n) has h parallel nonhomogeneous SDMUs, and each SDMU_k (k = 1,…, h) produces its own productsEMBED Equation.DSMT4 (r_kÎR_k, k = 1,…, h) by consuming p shared extra intermediate resources e_t (t = 1,…, p). Q_k refers to the intermediate measures z_{q_k} (q_kÎQ_k, k = 1,…, h) that come from the first stage by using m initial inputs x_i (i = 1,…, m).

Intermediate resources e_t (t = 1,…, p) are shared by SDMUs; hence, we denote a_tk (t = 1,…, p; k = 1,…, h) as the portion of shared extra intermediate resources used by SDMU_k (k = 1,…, h). Thus, we have {{\displaystyle \sum }}_{k=\mathrm{1}}^h{a}_{tk}=\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p). Similar to ^{Cook and Hababou (2001}), the intervals of a_tk (t = 1,…, p; k = 1,…, h) are specified by users, namely, L_tk£ a_tk£ U_tk (t = 1,…, p; k = 1,…, h).

The DMU being evaluated is denoted as DMU_o. We will discuss the additive efficiency of the parallel nonhomogeneous SDMUs in the second stage. The weighted average of the two-stage efficiency is used to calculate the total efficiency for DMU_o. The efficiency of the first stage can be measured under the assumption of constant returns to scale.

E_o^{(\mathrm{1})}=\max {\displaystyle \frac{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}}{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{io}}},

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{\displaystyle \frac{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}j}}{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{ij}}}\le \mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(j=\mathrm{1},\dots ,n),

(1)w_{q_k},v_i\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(q_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}i=\mathrm{1},\dots ,m),

where w_{q_k} and v_i are unknown nonnegative weights associated with the intermediate measures z_{q_k} and initial inputs x_i, respectively.

In two-stage DEA study, ^{Kao and Hwang (2008)} and ^{Liang et al. (2008)} argued that the multipliers associated with the intermediate measures are the same in two stages. In parallel systems, ^{Kao and Hwang (2010)} and ^{Kao (2012)} emphasized that the same inputs (or outputs) should have the same multiplier, regardless of what process assumes (or produces). Hence, we can measure the efficiency of each parallel nonhomogeneous SDMU_k (k = 1,…, h) in the second stage via model (2):

E_o^{(\mathrm{2}k)}=\max {\displaystyle \frac{{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}o}}{{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}},

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{\displaystyle \frac{{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}j}}{{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}j}+{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{tj}}}\le \mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{a}_{tk}=\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p),

L_{tk}\le a_{tk}\le U_{tk}(t=\mathrm{1},\dots ,p;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h),

(2)u_{r_k},w_{q_k},g_t,a_{tk}\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(r_k\in R_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{q}_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}t=\mathrm{1},\dots ,p),

where u_{r_k} indicates the unknown weights associated with the final outputs of SDMU_k, w_{q_k} is the same as the multipliers of the intermediate measures in model (1), g_t denotes the unknown weights of the shared extra intermediate resources e_t, and a_tk represents the portions of shared extra intermediate resources e_t used by SDMU_k.

We take the approach of weighted average of efficiency for SDMU_k to derive the aggregate measure of efficiency for the second stage, which was used to estimate the overall efficiency of nonhomogeneous parallel network processes by ^{Du et al. (2015)}. Weight d_k represents the relative importance of SDMU_k, where {{\displaystyle \sum }}_{k=\mathrm{1}}^h{\delta }_k=\mathrm{1}. The aggregate efficiency of the second stage can then be estimated by model (3):

E_o^{(\mathrm{2})}=\max {{\displaystyle \sum }}_{k=\mathrm{1}}^h{\delta }_k\times {\displaystyle \frac{{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}o}}{{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}},

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{\displaystyle \frac{{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}j}}{{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}j}+{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{tj}}}\le \mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{a}_{tk}=\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p),

L_{tk}\le a_{tk}\le U_{tk}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h),

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{\delta }_k=\mathrm{1},

(3)u_{r_k},w_{q_k},a_{tk},g_t,{\delta }_k\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(r_k\in R_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{q}_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}t=\mathrm{1},\dots ,p).

Next, we will find a feasible scheme for weight d_k (k = 1,…, h). As mentioned above, weight d_k (k = 1,…, h) represents the relative importance of SDMU_k. This scheme is reasonable, considering it lets the proportion of the resources used by SDMU_k to the total resources in the second stage represent the weight d_k, which reflects the relative importance or relative operating scale for SDMU_k (^{Chen et al., 2014}; ^{Wang and Chin, 2010}). d_k can be obtained by model (4):

(4){\delta }_k={\displaystyle \frac{{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h),

where the denominator EMBED Equation.DSMT4 represents the total resources for the second stage, including intermediate measures and shared extra intermediate resources, and the numerator EMBED Equation.DSMT4 represents the size of SDMU_k. The objective function in model (3) can be rewritten as follows:

(5)E_o^{(\mathrm{2})}=\max {\displaystyle \frac{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}o}}{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}}.

The overall efficiency and stage efficiency for the mixed two-stage systems can be estimated by the additive efficiency decomposition approach proposed by ^{Chen et al. (2014)}. f⁽¹⁾ and f⁽²⁾ represent the relative weights for the first and second stages, respectively. Similar to weight d_k, f⁽¹⁾ and f⁽²⁾ can also be determined by the proportion of the resources used by the second stage to the total resources in the two-stage system. The weights f⁽¹⁾ and f⁽²⁾ are accordingly displayed as follows:

(6a){\varphi }^{(\mathrm{1})}={\displaystyle \frac{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{io}}{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{io}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}},

(6b){\varphi }^{(\mathrm{2})}={\displaystyle \frac{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{io}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}},

where the denominators in (6a) and (6b) represent all of the resources consumed by DMU_o, and the numerators in (6a) and (6b) represent the resources used by the first and second stages, respectively.

The overall efficiency of DMU_o is the weighted average of the two stages’ efficiencies, which is solved by the following problem:

\begin{array}{l}{E}_o=\max ({\varphi }^{(\mathrm{1})}\times E_o^{(\mathrm{1})}+{\varphi }^{(\mathrm{2})}\times E_o^{(\mathrm{2})})\\ \mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}=\max {\displaystyle \frac{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}o}}{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{io}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{to}}},\end{array}

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{\displaystyle \frac{{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}j}}{{{\displaystyle \sum }}_{i=\mathrm{1}}^m{v}_i{x}_{ij}}}\le \mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(j=\mathrm{1},\dots ,n),

{\displaystyle \frac{{{\displaystyle \sum }}_{r_k\in R_k}{u}_{r_k}{y}_{r_{k}j}}{{{\displaystyle \sum }}_{q_k\in Q_k}{w}_{q_k}{z}_{q_{k}j}+{{\displaystyle \sum }}_{t=\mathrm{1}}^p{g}_t{a}_{tk}{e}_{tj}}}\le \mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{a}_{tk}=\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p),

L_{tk}\le a_{tk}\le U_{tk}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h),

(7)v_i,u_{r_k},w_{q_k},a_{tk},g_t\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(i=\mathrm{1},\dots ,m;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{r}_k\in R_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{q}_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}t=\mathrm{1},\dots ,p).

Under the Charnes–Cooper transformation (^{Charnes and Cooper, 1962}), the fractional (7) can be converted into linear program model (8):

E_o=\max {{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}o},

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}j}-{{\displaystyle \sum }}_{i=\mathrm{1}}^m{\nu }_i{x}_{ij}\le \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}j}-{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}j}-{{\displaystyle \sum }}_{t=\mathrm{1}}^p{\gamma }_t{a}_{tk}{e}_{tj}\le \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{i=\mathrm{1}}^m{\nu }_i{x}_{io}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{\gamma }_t{a}_{tk}{e}_{to}=\mathrm{1},

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{a}_{tk}=\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p),

L_{tk}\le a_{tk}\le U_{tk}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h),

(8){\nu }_i,{\mu }_{r_k},{\omega }_{q_k},a_{tk},{\gamma }_t\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(i=\mathrm{1},\dots ,m;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{r}_k\in R_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{q}_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}t=\mathrm{1},\dots ,p).

Obviously, model (8) is a nonlinear programming. We set g_ta_tk = l_tk to linearize the nonlinear programming model (8). We have:

E_o=\max {{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}o},

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}j}-{{\displaystyle \sum }}_{i=\mathrm{1}}^m{\nu }_i{x}_{ij}\le \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}j}-{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}j}-{{\displaystyle \sum }}_{t=\mathrm{1}}^p{\lambda }_{tk}{e}_{tj}\le \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{i=\mathrm{1}}^m{\nu }_i{x}_{io}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{\lambda }_{tk}{e}_{to}=\mathrm{1},

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{\lambda }_{tk}-{\gamma }_t=\mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p),

{\gamma }_t{L}_{tk}\le {\lambda }_{tk}\le {\gamma }_t{U}_{tk}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h),

(9){\nu }_i,{\mu }_{r_k},{\omega }_{q_k},{\gamma }_t,{\lambda }_{tk}\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(i=\mathrm{1},\dots ,m;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{r}_k\in R_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{q}_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}t=\mathrm{1},\dots ,p).

({\mu }_{r_k}^{*},{\omega }_{q_k}^{*},{\gamma }_t^{*},{\lambda }_{tk}^{*}) represents the optimal solution for model (9), then a_{tk}^{*}={\lambda }_{tk}^{*}/{\gamma }_t^{*}. The optimal overall efficiency is E_o^{*}={{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}^{*}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}^{*}{y}_{r_{k}o}.

At last, the decomposition of stage efficiency can be obtained, when the second stage is given pre-emptive priority. Under the overall efficiency constant, the decomposition efficiency of the second stage is as follows:

E_o^{(\mathrm{2})}{}^{*}={{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}o},

\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}j}-{{\displaystyle \sum }}_{i=\mathrm{1}}^m{\nu }_i{x}_{ij}\le \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}j}-{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}j}-{{\displaystyle \sum }}_{t=\mathrm{1}}^p{\lambda }_{tk}{e}_{tj}\le \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}j=\mathrm{1},\dots ,n),

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{r_k\in R_k}{\mu }_{r_k}{y}_{r_{k}o}-E_o^{*}{{\displaystyle \sum }}_{i=\mathrm{1}}^m{\nu }_i{x}_{io}=E_o^{*},

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{q_k\in Q_k}{\omega }_{q_k}{z}_{q_{k}o}+{{\displaystyle \sum }}_{k=\mathrm{1}}^h{{\displaystyle \sum }}_{t=\mathrm{1}}^p{\lambda }_{tk}{e}_{to}=\mathrm{1},

{{\displaystyle \sum }}_{k=\mathrm{1}}^h{\lambda }_{tk}-{\gamma }_t=\mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p),

{\gamma }_t{L}_{tk}\le {\lambda }_{tk}\le {\gamma }_t{U}_{tk}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(t=\mathrm{1},\dots ,p;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h),

(10){\nu }_i,{\mu }_{r_k},{\omega }_{q_k},{\gamma }_t,{\lambda }_{tk}\ge \mathrm{0}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}(i=\mathrm{1},\dots ,m;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{r}_k\in R_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{q}_k\in Q_k,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}k=\mathrm{1},\dots ,h;\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}t=\mathrm{1},\dots ,p).

E_o^{(\mathrm{1})}{}^{**} represents the decomposition efficiency of the first stage with second-emptive priority. The first stage’s efficiency is then calculated as:

(11)E_o^{(\mathrm{1})}{}^{**}={\displaystyle \frac{E_o^{*}-{\varphi }^{(\mathrm{2})}{}^{*}\times E_o^{(\mathrm{2})}{}^{*}}{{\varphi }^{(\mathrm{1})}{}^{*}}},

where f^(1)* and f^(2)* are the optimal weights obtained from models (6a) and (6b), respectively. When the first stage is given the pre-emptive priority, the model for decomposition efficiency of the second stage is similar to model (10); hence, we omit it.

All the steps of our method are summarized as follows to present our approach in this study clearly.

Algorithm of our approach:

Step 1: Measure the efficiencyE_o^{(\mathrm{1})} for the first stage using model (1).

Step 2: Measure the efficiency E_o^{(\mathrm{2})} for the second stage using models (2–4).

Step 2.1: Measure efficiency E_o^{(\mathrm{2}k)} for each parallel nonhomogeneous SDMU_k in the second stage using model (2).

Step 2.2: Calculate the relative weight d_k for SDMU_k using model (4).

Step 2.3: Measure the efficiency E_o^{(\mathrm{2})} for the second stage using model (3).

Step 3: Calculate relative weights f⁽¹⁾ and f⁽²⁾ for the first and second stages using models (6a) and (6b), respectively.

Step 4: Measure efficiency E_o for mixed structure system using model (7).

Step 5: Calculate the optimal solution a_{tk}^{*} for allocation portion of shared resources and the optimal stage efficiency E_o^{(\mathrm{1})\mathrm{**}} (or E_o^{(\mathrm{1})\mathrm{*}}) and E_o^{(\mathrm{2})\mathrm{*}} (or E_o^{(\mathrm{2})\mathrm{**}}).

In summary, our method is an integrated study of three types of problems, namely, serial system, parallel system, and nonhomogeneous DMU efficiency evaluation. Our method is based on the principle of maximizing system efficiency, evaluating the overall efficiency of the system, rationally allocating extra intermediate resources, and obtaining appropriate stage efficiency (Fig. 2).

Illustrative application

Data description

In this section, we collate and use the empirical data provided by ^{Huang et al. (2014)} to test our models. The data consist of 58 Taiwan international tourist hotels. The authors considered mixed two-stage nonhomogeneous network processes, as depicted in Fig. 3. All factors are clarified as follows:

• Operating expenses: The cost of hotel operation, including cost of food and beverage (F&B), maintenance expenses, and expenditures for water and electricity, which are all measured in millions New Taiwan Dollar (NT$).

• Rooms: The number of rooms supplied for renting.

• Catering space: The square of floor space for F&B, which is measured in square feet (ft²).

• Employees: The number of employees, including workers in guest rooms and catering and management staff.

• Occupancy service capacity: The number of room nights available, which is the quotient of the number of rooms sold and the occupancy rate of the room.

• Catering service capacity: The product of catering space and the number of employees of F&B department.

• Marketing expense: The cost for marketing activity, such as advertisement and website operation, which is measured in millions NT$.

• Occupancy revenue: The operational revenue gained from room renting, which is measured in millions NT$.

• Catering revenue: The operational revenue gained from F&B sales, which is measured in millions NT$.

The first stage is the production process with four original inputs and two intermediate outputs. The two intermediate outputs are transported to two parallel and nonhomogeneous sub-processes, occupancy service process, and catering service process, which belong to the second stage. The two sub-processes use shared extra intermediate resources for marketing expense. The shard marketing expense affects the two sub-processes but cannot be conveniently split up. Finally, the two sub-processes produce their own output as the final outputs of the second stage. The data of the number of lodging guests, which is one factor of the final output of the occupancy division in the study of ^{Huang et al. (2014)}, are missing; consequently, we have to omit the factor in our study. Table 1 shows the descriptive statistics of all factors.

Efficiency analysis

Table 2 reports the efficiencies estimated from the proposed model when the intervals of a_tk are set as [0.2, 0.8]. The number and the name of hotels are shown in the first and second columns, respectively. The overall efficiency and the ranking-based overall efficiency for DMUs are listed in the fourth and third columns, respectively. The fifth column reports the efficiency of the first stage when the first stage is given pre-emptive priority and the overall efficiency keeps constant. The data in the sixth column are the efficiency of the second stage with secondary priority. The seventh and eighth columns are the reverse of the fifth and sixth columns, which show the efficiency of the first stage with secondary priority and the efficiency of the second stage with pre-emptive priority, respectively. The optimal weights of two stages are listed in the last two columns.

The overall efficiency differs, which illustrates that the proposed models have strong recognition capability. The maximum efficiency is 0.9443 (DMU₁₃), and the minimum is 0.3235 (DMU₅₁). All DMUs can be sorted in accordance with the obtained overall efficiency value. This finding indicates that the proposed method can be used to evaluate the performance of DMUs accurately, that is, it can clearly distinguish the small overall efficiency differences among DMUs.

On the premise that the overall efficiency remains unchanged, the efficiency of the two stages can also be obtained. Figures 4 and 5 are the scatter diagrams of the decomposition efficiency of two stages, where x-coordinate and y-coordinate are the overall efficiency and stage efficiency, respectively. Figure 4 illustrates the efficiency of the distribution for two stages when the first stage is given pre-emptive priority. On the contrary, Fig. 5 is based on the situation that the second stage has pre-emptive priority. Comparison of the two figures shows that the dots in Fig. 5 are more concentrated, and fewer extreme dots exist. For example, six dots are clearly far from the population dots in Fig. 4, whereas five lonely dots are scattered in Fig. 5. These extreme dots are all marked out in Figs. 4 and 5. From the perspective of the number of extreme dots, the efficiency decomposition method that gives the second stage pre-emptive priority is more suitable for these hotels with mixed two-stage network structure. Nonetheless, the number of DMUs with extreme dots in Fig. 5 is the same as the number in Fig. 4. In other words, the effect of the two efficiency decomposition principles on DMU is similar. Most of the extreme dots in Fig. 5 are about the first stage, such as DMUs 51, 44, 43, and 55. The ordinates of these dots are significantly smaller in Fig. 5 than in Fig. 4. This condition may seem unreasonable, given the blue dots are generally above the orange dots from both figures. However, this result is caused by the principle of efficiency decomposition. As shown in Fig. 5, the second stage has higher priority than the first stage. The efficiency of the second stage will thus be maximized as much as possible when the efficiency is decomposed, and the efficiency of the first stage will naturally decline. In fact, both efficiency decomposition principles are valid. As mentioned above, from the overall trend, the results of the two methods are similar without very serious and large changes, that is, the blue dots on both pictures are above the orange dots on the whole. Therefore, when using this method, the decision maker should carefully choose which stage should be given greater priority.

Shared resource allocation

Next, we discuss the effect of the interval of a_tk on overall efficiency. In Table 3, when the interval of a_tk is L_tk = 0.2, U_tk = 0.8, the overall efficiency is listed in the third column, and the fourth and fifth columns report the portions of shared extra intermediate resources obtained by occupancy and catering service process, respectively. Similarly, the center-right and last sets of data are based on the a_tk interval of [0.1, 0.9] and [0, 1], respectively.

The absolute optimal efficiency depends on whether the optimal portion is in the interval of a_tk. For example, in three different interval settings, the efficiency of DMU₆ is always 0.6242, because the optimal portion of it is (a^(1)*, a^(2)*) = (0.6568, 0.3432) Î [0.2, 0.8] Ì [0.1, 0.9] Ì [0, 1], which is the optimal portion locating among the three intervals at the same time. For DMU₁, however, its optimal overall efficiency varies with the intervals. When L_tk = 0, U_tk = 1, the maximum optimal of DMU₁ is 0.8550, and the optimal portion is (a^(1)*, a^(2)*) = (0.9283, 0.0717) Î [0, 1]. Nevertheless, (0.9283, 0.0717) Ï [0.2, 0.8]. As a result, [L_tk, U_tk] = [0.2, 0.8], the efficiency of DMU₁ is only 0.6824, and (a^(1)*, a^(2)*) = (0.8, 0.2), which indicates that the efficiency is a locally optimal solution.

For DMU₅, the efficiency increases with the expansion of the interval. When [L_tk, U_tk] = [0.2, 0.8], the efficiency EMBED Equation.DSMT4= 0.7561, and (a^(1)*, a^(2)*) = (0.8, 0.2). When [L_tk, U_tk] = [0, 1], the efficiency of DMU₅ becomes 0.8550, and (a^(1)*, a^(2)*) = (1, 0). This result suggests that assigning all marketing expense to occupancy service process is the most beneficial way to maximize the efficiency for Emperor Hotel. The results of DMUs 13, 22, 24, 43, and 58 imply similar meaning. In turn, the results of DMUs 44, 49, 55, and 57 indicate that marketing expense mainly used for catering service process contributes considerably to efficiency. For DMUs 19, 27, 47, and 54, the marketing expense shares for both sub-processes is 50%, which means that the two sub-processes have the same effect on overall efficiency. Setting interval as [0, 1] gives the most accurate and reasonable result for all DMUs. Nonetheless, in some real-life cases, decision makers have to set an interval with nonzero limit to ensure the basic benefits of all sub-processes. Therefore, decision makers need to accord to the different results to specify the interval.

Summary of test results

The key factor affecting the accuracy and robustness of results is the interval setting of a_tk. From the perspective of accuracy, the definition of accuracy should be combined with the actual situation. If all shared resources can be given to a certain department, then the result is the most accurate when the interval of a_tk is [0, 1]. At this time, if the result obtained in the other interval of a_tk is different from the result obtained in [0, 1], it may be regarded as inadequately accurate. However, in practice, shared resources are often divided into various departments. The results are hence obtained within a reasonable interval of a_tk. Although they are not the most ideal results, they are the most objective and accurate results.

For robustness, we can conclude that the results obtained by our model are relatively robust. To illustrate the robustness clearly, we first make the following three definitions:

• Efficiency change rate b₁ = (efficiency value of interval [0.1, 0.9] / efficiency value of interval [0.2, 0.8] - 1) ´ 100%;

• Efficiency change rate b₂ = (efficiency value of interval [0, 1] / efficiency value of interval [0.1, 0.9] - 1) ´ 100%;

• Average efficiency change rate b₀ = (b₁ + b₂) / 2

Among the 58 DMUs, the average efficiency change rate b₀ of 48 DMUs is less than 1% (see Table 4). The average efficiency change values b₀ of 12 DMUs among the 48 DMUs are zero. Most DMUs thus have good robustness under different values of a_tk. Among the other 10 DMUs, only 3 DMUs have an average efficiency change rate b₀ higher than 0.05, and the maximum value of b₀ is 0.06375. DMU₆ is taken as an example, and the efficiency change rates b₁ and b₂ are both 0, indicating that the set three intervals have no effect on the actual efficiency value of DMU₆. For DMU₁₀, the efficiency change rate b₁ = 0.00162, and the change rate b₂ = 0, indicating that the efficiency of DMU₁₀ is affected in the range of [0.2, 0.8]. However, the efficiency performance area is robust when the interval is expanded to [0.1, 0.9] or more. For DMU₅, the efficiency change rate b₂ = 0.09126 is the maximum of all the change rates, indicating that the efficiency volatility is larger when the interval is expanded from [0.1, 0.9] to [0, 1] but still does not exceed 10%. In general, the obtained results have the accuracy to adapt to different actual situations and high robustness.

Conclusions

In this study, we propose a multiplier-based network DEA model to estimate the overall efficiency of mixed two-stage nonhomogeneous network processes with shared extra intermediate resources. Mixed two-stage nonhomogeneous network processes commonly occur in real life. We use the data set of 58 Taiwan international hotels to test the features of the proposed model. The results show that the proposed model can accurately determine the efficiency of each DMU and rank all DMUs based on their efficiencies. Different efficiency decomposition principles cause different stage efficiencies for the same DMU. The interval of the shared resource portion may affect the optimal efficiency values for certain DMUs. In future research, we will consider general system structures, such as multistage mixed structures. Panel data are also worthy of future study and analysis. Shared extra intermediate resources can be applied to other research fields, such as transportation (^{Zhen et al., 2019}), environment (^{Yang et al., 2019}), and supply chain (^{Brunaud et al., 2018}).

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