A Method for Assessing the Efficiency in Two-Stage Production Systems in the Presence of Dual-Role Factors

Chongyu MA; Jianwei REN; Lijun TANG; Chunhua CHEN; Chenxi FENG

doi:10.21078/JSSI-2024-0012

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Journal of Systems Science and Information ›› 2024, Vol. 12 ›› Issue (5) : 624-646. DOI: 10.21078/JSSI-2024-0012

A Method for Assessing the Efficiency in Two-Stage Production Systems in the Presence of Dual-Role Factors

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Abstract

Due to the existence of dual-role factors, it is difficult to evaluate the production efficiency of two-stage systems. Unlike single-stage systems, two-stage systems involve intermediate products that serve as both inputs and outputs. Hence, to overcome existing obstacles, we propose a novel approach called the two-stage enhanced Russell model with dual-role factors (T-ERM-D) to assess the overall efficiency of two-stage production systems. Furthermore, divisional models are developed to evaluate the efficiency of each individual stage. The 0-1 programming is applied to deal with dual-role factors. To handle the non-linearity of these models, the Charnes-Cooper transformation is employed to convert them into linear ones. Using the proposed models, we evaluate efficiency scores of 10 supply chains involving suppliers and producers. By comparing the results obtained from new models with those obtained from models that do not consider dual-role factors, we validate the advantages of the proposed approach.

Key words

data envelopment analysis / two stages / enhanced Russell model / dual-role factors / efficiency assessment

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Chongyu MA , Jianwei REN , Lijun TANG , Chunhua CHEN , Chenxi FENG. A Method for Assessing the Efficiency in Two-Stage Production Systems in the Presence of Dual-Role Factors. Journal of Systems Science and Information, 2024, 12(5): 624-646 https://doi.org/10.21078/JSSI-2024-0012

1 Introduction

In recent years, significant attention has been directed towards the study of two-stage production systems. Evaluating the production efficiency of such systems is crucial for identifying deficiencies within actual production activities. Data Envelopment Analysis (DEA) has emerged as a widely employed method for assessing the efficiency of production departments or enterprises. Its key advantage lies in its capability to handle multiple inputs and outputs simultaneously. The pioneering work of Charnes and Cooper^[1] in 1978 introduced the first model under constant returns to scale (CRS), laying the foundation for the application of DEA. Since then, DEA has been extended to tackle complex problems across various domains. Banker, et al.^[2] introduced the variable returns to scale (VRS) technique by introducing a new variable that determines whether a business operates in regions where returns to scale increase, remain constant, or decrease. Notably, DEA finds wide-ranging application in fields such as supply chain production, cold-chain transportation, urban construction, and bank management.

Two-stage systems typically involve dual-role factors, which increases the difficulty of efficiency evaluation. In actual production, a dual-role factor is not only an input but also an output. There are several typical dual-role variables (e.g., service quality, service capability, capital assets, corporate reputation and research funding). Beasley^[3] proposed that these variables serve as both inputs and outputs, playing two distinct roles simultaneously. Cook, et al.^[4] regarded the co-product in the production process as a dual-role factor, considering it as an expected output under certain conditions, but an undesirable output when it goes beyond a certain range. Pouralizadeh^[5] considered the cost of polluting gases and internal consumption within power plants as dual-role factors when studying the issue of production in the electricity supply chain.

Nevertheless, there is a scarcity of academic literature focusing on dual-role factors within the framework of DEA. Specifically, limited research explores the role of dual-role factors in two-stage or multi-stage production processes despite the extensive development of DEA methods. The majority of studies related to two-stage systems have disregarded the significance of dual-role factors. For example, Kao and Hwang^[6] decomposed the overall efficiency of a two-stage production process without considering dual-role factors, by multiplying the efficiencies of the two phases. Similarly, Chen, et al.^[7] constructed an input-oriented DEA model for the two sub-stages and expressed the overall efficiency formula as the weighted sum of the efficiency of the two stages. To the best of our knowledge, no significant research has been undertaken thus far to address the incorporation of dual-role factors in these contexts.

Two approaches are employed to handle dual-role factors. Firstly, a 0-1 programming method is developed to determine whether dual-role factors function as inputs or outputs. Izadikhah, et al.^[8] introduced a super-efficiency Enhanced Russell Model (ERM) that utilizes 0-1 programming to measure efficiency considering dual-role factors. Similarly, Ghiyasi and Cook^[9] proposed a classification model using 0-1 programming to ascertain whether a dual-role factor should be classified as an input or an output.

Secondly, in two-stage or multi-stage systems, dual-role factors are simultaneously considered as inputs and outputs. One prominent example is the intermediate product between two consecutive stages, which acts as an output for the previous stage and an input for the subsequent stage. For instance, Maleki and Kazemi^[10] developed a novel model to address the two-stage insurance company problem where intermediate products, including direct written premiums and reinsurance premiums, were regarded as dual-role factors. Additionally, Li, et al.^[11] extended the model proposed by Chen, et al.^[7] and presented an output-oriented two-stage model that considers intermediate variables as both inputs and outputs.

The objective of this study is to propose a DEA method incorporating dual-role factors for evaluating the production efficiency of two-stage systems. Maleki and Kazemi^[10] first investiaged the use of the ERM model to evaluate the efficiency of two-stage production. Izadikhah, et al.^[8] remodeled the ERM model in a single-stage production process, introducing dual-role factors into the program. However, none of these studies addressed the problem of two-stage efficiency assessment involving dual-role factors. Based on the above two concepts, the present paper applies the ERM model to a two-stage production process and incorporates dual-role factors into the production process. This novel method can identify weak links in the production. Our research results prove that the new method combines the advantages of the above two methods and compensate for the omission of dual-role factors in the previous literature.

The present paper is organized as follows. Section 2 discusses the previous research on two-stage and dual-role factors. In Section 3, a novel two-stage enhanced Russell model with dual-role factors is proposed. Moreover, the novel nonlinear model is transformed into a linear program by a Charnes-Cooper transformation. In Section 4 the efficiency evaluation models of two sub-stages are developed and transformed into linear programs. We then use a case study to verify the established model and analyze the results in Section 5. Section 6 concludes this paper and provides suggestions and directions for future research.

2 Literature Review

Considerable progress has been made in the field of DEA methodology for assessing the efficiency of two-stage production systems, with numerous creative models and equations having been examined and developed. However, to include dual-role factors, previous research has usually considered a single production process and rarely two or more stages. This section reviews the literature on dual-role factors and two-stage systems respectively.

2.1 Dual-Role Factors

There are few studies on the production processes of two or more stages considering dual-role factors. In the existing literature, the investigation and discussion of dual-role factors predominantly pertain to single-stage production processes. For example, Eydi and Rastgar^[12] proposed a decision model that considers dual-role data and fuzzy state by using the a-cut method. Toloo, et al.^[13] first constructed an interval DEA model based on pessimistic and optimistic viewpoints to measure interval efficiency. To determine the state of dual-role factors, they also proposed a model that combined the pessimistic and optimistic models to identify the respective states of each imprecise dual-role factor. A mixed binary linear model based on a modified production possibility set and the effect dual-role factor was proposed by Toloo, et al.^[14] When addressing the issue of dual-role factors, these factors were optimally specified to indicate that they are non-discretionary inputs or outputs. Ebrahimi, et al.^[15] developed a linear epsilon-based DEA model by calculating the upper and lower bounds of the efficiency scores in the presence of interval factors of inputs, outputs, and dual-role factors by taking into account the fixed and unified production boundary of all DMUs, in the situation of imprecise data and imprecise dual-role factors. Ding, et al.^[16] proposed a two-step DEA method for selecting suppliers involving dual-role factors. Lee and Saen^[17] introduced a new DEA model to measure corporate sustainability management with dual-role factors by applying a combined approach. Noveiri, et al.^[18] developed an alternative DEA method considering dual-role factors. Weighting methods were used by Ghazi and Lotfi^[19] to develop some weighted DEA models in the presence of dual-role factors. Chen^[20] discussed the problem about how to handle dual-role factors in DEA. Keshavarz and Toloo^[21] utilized the DEA method to assess the performance of several third-party reverse logistics providers in the presence of dual-role factors.

Coping with classification problem on dual-role factors is significant. Chen^[22] discussed the classification of dual-role factors in two opposite methods. Cook, et al.^[23] proposed a DEA allocation model based on the most basic dual-role factor model. The key step is to allocate a portion of the production output to the factor that plays an input role, with the aim of maximizing overall efficiency through redistribution.

The application of the DEA model in the presence of dual-role factors is mainly in supply chain management, transportation, bank management, and other areas. In a single stage, dealing with vehicle selection problem in cold chain management, Shabani, et al.^[24] constructed input-oriented and output-oriented free disposal hull (FDH) models and FDH super-efficiency models to offer a comprehensive ranking of DMUs in the presence of dual-role factors. Azadi and Saen^[25] proposed a new chance-constrained DEA method in the presence of dual-role factors and random data. Azizi, et al.^[26] added a virtual DMU to account for dual-role factors and undesirable outputs and to distinguish between fully efficient DMUs. Izadikhah, et al.^[8] proposed a super-efficiency ERM method to assess supplier sustainability in the presence of dual-role factors and quantity discounts, solving the problem that previous ERM models were unable to rank DMUs. In a two-stage or multi-stage system, Pouralizadeh^[5] developed a network DEA model with dual-role factors to assess the efficiency of the electricity supply chain. Su and Sun^[27] re-established a multi-stage DEA model under CRS conditions that considered both dual-role influencing factors and undesirable outputs. Mirhedayatian et al.^[28] proposed a network Slack Based Measure (SBM) model for dealing with dual-role factors and undesirable outputs.

Table 1 presents discussions on DEA methods in the presence of dual role factors. In general, existing literature on studying dual-role factors mostly focuses on single stages, with only a few references addressing two-stage or multi-stage processes.

Table 1 DEA methods in the presence of dual role factors

Discussions	References
Single-stage	Izadikhah, et al.^[8]; Eydi and Rastgar^[12]; Toloo, et al.^[13]; Toloo, et al.^[14]; Ebrahimi, et al.^[15]; Ding, et al.^[16]; Lee and Saen^[17]; Noveiri, et al.^[18]; Ghazi and Lotfi^[19]; Chen^[20]; Keshavarz and Toloo^[21]; Chen^[22]; Cook, et al.^[23] Shabani, et al.^[24]; Azadi and Saen^[25]; Azizi, et al.^[26]
Two-stage or multi-stage	Su and Sun^[27]; Mirhedayatian, et al.^[28]; Pouralizadeh^[5]

2.2 Two-Stages System

There is a specific relationship directly between the production efficiency of the two-stage system and each stage. In other words, the two-stage production efficiency can usually be expressed by the efficiency of each stage. In studying the two-stage production process, Kao and Hwang^[6] firstly represented the overall efficiency as the product of the efficiencies of the two respective sub-stages. Chen, et al.^[7] constructed an additive efficiency decomposition model by defining the overall efficiency as the sum of the efficiencies of the sub-stages under both CRS and VRS assumptions. Li, et al.^[11] formulated an output-oriented two-stage production model based on Chen, et al.^[7]'s model. They weighed the efficiency in each stage by its share of output. To address the limitation of the additive efficiency decomposition method proposed by Chen, et al.^[7], which was focused on by Ang and Chen^[29] and Guo, et al.^[30], Despotis, et al.^[31] developed a method to assess unbiased efficiency scores for the individual stages in a two-stage system. Then researchers began to apply different models and methods to the two-stage system. Maleki and Kazemi^[10] suggested a revised model under VRS, based on the ERM model to measure the efficiency of a two-stage process. Zhao, et al.^[32] focused on the coordination efficiency problem between suppliers and manufacturers in a two-stage production system, where coordinated efficiency is defined as the ratio of the efficiency of coordinated systems to the efficiency of uncoordinated systems.

Considering the influence of psychological factors of decision-makers on the evaluation results, Chen, et al.^[33] introduced regret theory to describe the psychological preference of decision-makers, and constructed a two-stage cross-efficiency model based on regret in centralized and decentralized decision-making environments, respectively. Atta, et al.^[34] introduced the two-stage DEA methodology to measure energy efficiency among nine West African countries. The first stage used the dynamic SBM model and the Malmquist Productivity Index (MPI) to evaluate the relative efficiency. The second stage adopted the Tobit regression to analyze the influencing factors. Abbasi, et al.^[35] proposed an alpha-cut-based model for evaluating decision unit performance in fuzzy two-stage DEA. He and Zhu^[36] developed a dynamic hybrid two-stage data DEA model for assessing the ecological efficiency of industrial systems. Ma and Zhao^[37] developed a hybrid two-stage DEA considering the series-parallel internal structure and the cooperative-Stackelberg relationship between the stages. Taking into account the impact of fairness concerns, Wu, et al.^[38] proposed a novel utility-based two-stage DEA methodology. To measure cost-effectiveness, Raheleh, et al.^[39] proposed a cooperative and a non-cooperative model of a two-stage game to measure cost-efficiency using centralized and Stackelberg game models. Li, et al.^[40] developed a conditional two-stage DEA model with dual-role factors for assessing multiple processes. Li, et al.^[41] proposed a modified DEA for monitoring the green supplier performance in the presence of dual-role factors. Wanke, et al.^[42] proposed a combinatorial model based on the directional distance function. Izadikhah, et al.^[43] developed a linear model to determine the upper and lower efficiency bounds of the two sub-phases in a non-cooperative environment, and evaluated the overall efficiency of the DMU in a cooperative environment. Esfandiari, et al.^[44] proposed two robust DEA models for measuring the performance of two-stage processes. Ma, et al.^[45] proposed a two-stage undesirable fixed-sum output model to assess the efficiency of industrial water treatment. Li, et al.^[46] applied a two-stage DEA model to measure the productivity change of China's banks.

Table 2 demonstrates the discussions on DEA methods in two-stage systems. In summary, existing literature often overlooks dual-role factors when studying DEA methods in two-stage systems, which is a limitation of existing DEA methods.

Table 2 DEA methods in two-stage systems

Discussions	References
Ignoring dual-role factors	Kao and Hwang^[6]; Chen, et al.^[7]; Li, et al.^[11]; Despotis, et al.^[31]; Maleki and Kazemi^[10]; Zhao, et al.^[32]; Chen, et al.^[33]; Atta, et al.^[34]; Abbasi, et al.^[35]; He and Zhu^[36]; Ma and Zhao^[37]; Wu, et al.^[38]; Raheleh, et al.^[39]; Wanke, et al.^[42]; Izadikhah, et al.^[43]; Esfandiari, et al.^[44]; Ma, et al.^[45]; Li, et al.^[46]
Considering dual-role factors	Li, et al.^[40]; Li, et al.^[41]

3 The Proposed DEA Model

Many researchers explore production issues in different fields. For example, Shen and Zhang^[47] studied the investment in green technology in the production of manufacturer-retailer supply chains. Hu and Qi^[48] explored the production recovery systems. Figure 1 shows the process of a two-stage system considering dual-role factors. Assume that there is a set of n DMUs.

x_{i j}

(i = 1, \dots, M)

is the ith input consumed in the first phase and

y_{r j}

(r = 1, \dots, S)

is the rth output produced in the second phase, where j represents the jth DMU.

z_{d j}

(d = 1, \dots, D)

represents the dth intermediate product which denotes the output for stage 1 and input for stage 2.

h_{f j}

(f = 1, \dots, F)

is defined as the fth dual-role factor that plays both the role of inputs and outputs in the first stage. In the above production process,

z_{d j}

(d = 1, \dots, D)

is also classified as the dual-role factor because it is the output of the first stage and the input of the second stage. Therefore,

h_{f j}

and

z_{d j}

are both dual-role factors. In the aforementioned production process diagram, all the inputs of the second stage are derived entirely from the outputs of the first stage, and no additional inputs are introduced. Dual-role factors, whether serving as inputs or outputs, solely participate in the production of the first stage.

Figure 1 Two-stage production system with dual-role factors

Full size|PPT slide

Define

X = (x_{1}, x_{2}, \dots, x_{M}), Z = (z_{1}, z_{2}, \dots, z_{D}), Y = (y_{1}, y_{2}, \dots, y_{S}), H = (h_{1}, h_{2}, \dots, h_{F})

. In the first stage of production, without taking dual-role factors into account, Maleki and Kazemi^[10] defined

T_{1}

as the production possibility set (PPS) to describe the production relations of the first stage. They proved that the set of technologies satisfies the convexity axiom and the strong disposability of inputs. The technology set is expressed as follows:

T_{1} = {(X, Z) : \sum_{j}^{} λ_{j} X_{j} \leq X, \sum_{j}^{} λ_{j} Z_{j} = Z, \sum_{j}^{} λ_{j} = 1, λ_{j} \geq 0, j = 1, \dots, n},

(1)

where

λ_{j}

denotes the intensity of stage 1. The constraint

\sum_{j} λ_{j} X_{j} \leq X

is the strong disposability of inputs. The constraint

\sum_{j} λ_{j} Z_{j} = Z

is the weak disposability of intermediate products. The constraint

\sum_{j} λ_{j} = 1

implies that the scale of the production process is under the assumption of VRS for stage 1. Considering the influence of dual-role factors in the first stage, we add the constraints of dual-role factors based on the above theory and propose a new PPS. Define

T_{1}^{^{'}}

as the new technology set. If

H_{j}

are considered as the outputs in the first stage, then the production technology set consists of the following parts:

T_{1}^{^{'}} = {(X, Z, H) : \sum_{j}^{} λ_{j} X_{j} \leq X, \sum_{j}^{} λ_{j} Z_{j} = Z, \sum_{j}^{} λ_{j} H_{j} \geq H, \sum_{j}^{} λ_{j} = 1, λ_{j} \geq 0} .

(2)

If dual-role factors play the role of input, the technology set

T_{1}^{^{″}}

can replace

T_{1}^{^{'}}

. Then we will conclude that:

T_{1}^{^{″}} = {(X, Z, H) : \sum_{j}^{} λ_{j} X_{j} \leq X, \sum_{j}^{} λ_{j} Z_{j} = Z, \sum_{j}^{} λ_{j} H_{j} \leq H, \sum_{j}^{} λ_{j} = 1, λ_{j} \geq 0} .

(3)

Similarly, define

T_{2}^{^{'}}

as the PPS of stage 2. To describe the second stage of production relations, Maleki and Kazemi^[10] proposed that the second-stage technology set satisfies the convexity axiom and strong disposability of the output. The following axiom for the associated production set is:

T_{2} = {(Z, Y) : \sum_{j}^{} μ_{j} Z_{j} = Z, \sum_{j}^{} μ_{j} Y_{j} \geq Y, \sum_{j}^{} μ_{j} = 1, μ_{j} \geq 0, j = 1, \dots, n},

(4)

where

μ_{j}

denotes the intensity of stage 2. The constraint

\sum_{j}^{} μ_{j} Y_{j} \geq Y

is the strong disposability of outputs and the constrain

\sum_{j}^{} μ_{j} = 1

implies that the scale of the production process is under the assumption of VRS for stage 2. Taking into account the variable remuneration for scale, when dual-role factors work as outputs, the overall associated set of production

T^{^{'}}

can be expressed in the following form based on the proposed axioms of the first and second stages:

\begin{aligned} T^{^{'}} = {(X, H, Z, Y) : \sum_{j} λ_{j} X_{j} \leq X, \sum_{j} λ_{j} H_{j} \geq H, \sum_{j} λ_{j} Z_{j} = Z, \sum_{j} μ_{j} Z_{j} = Z, \\ \sum_{j} μ_{j} Y_{j} \geq Y, \sum_{j} λ = 1, \sum_{j} μ = 1, λ_{j} \geq 0, μ_{j} \geq 0, j = 1, \dots, n} . \end{aligned}

(5)

When dual-role factors act as inputs, the overall PPS is constructed by:

\begin{aligned} T^{^{'}} = {(X, H, Z, Y) : \sum_{j} λ_{j} X_{j} \leq X, \sum_{j} λ_{j} H_{j} \leq H, \sum_{j} λ_{j} Z_{j} = Z, \sum_{j} μ_{j} Z_{j} = Z, \\ \sum_{j} μ_{j} Y_{j} \geq Y, \sum_{j} λ = 1, \sum_{j} μ = 1, λ_{j} \geq 0, μ_{j} \geq 0, j = 1, \dots, n} . \end{aligned}

(6)

Based on the above theory, we propose a novel two-stage enhanced Russell model with dual-role factors (T-ERM-D) to assess efficiency scores of DMUs in

T^{^{'}}

and

T^{^{″}}

. It can be expressed as:

\begin{aligned} E = min ω_{1} E_{1} + ω_{2} E_{2} = ω_{1} \frac{\frac{1}{M + \hat{F}} (\sum_{i = 1}^{M} θ_{i} + \sum_{f = 1}^{F} D_{f} θ_{f})}{\frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f = 1}^{F} (1 - D_{f}) ϕ_{f})} + ω_{2} \frac{\frac{1}{D} {\sum_{d = 1}^{D} \bar{ϕ}}_{d}}{\frac{1}{S} \sum_{r = 1}^{S} β_{r}} \\ s.t. \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq θ_{i} x_{i o}, i = 1, 2, \dots, M, \\ \sum_{j = 1}^{n} λ_{j} z_{d j} = ϕ_{d} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} λ_{j} D_{f} h_{f j} \leq θ_{f} D_{f} h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} λ_{j} (1 - D_{f}) h_{f j} \geq ϕ_{f} (1 - D_{f}) h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} μ_{j} z_{d j} = {\bar{ϕ}}_{d} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} μ_{j} y_{r j} \geq β_{r} y_{r o}, r = 1, 2, \dots, S, \\ \hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), \\ \sum_{j = 1}^{n} λ_{j} = 1, \sum_{j = 1}^{n} μ_{j} = 1, j = 1, 2, \dots, n, \\ λ_{j} \geq 0, μ_{j} \geq 0, θ_{i} \leq 1, θ_{f} \leq 1, ϕ_{d} \geq 1, ϕ_{f} \geq 1, {\bar{ϕ}}_{d} \leq 1, β_{r} \geq 1, D_{f} \in {0, 1} . \end{aligned}

(7)

In the above model, the overall efficiency is defined as a convex combination of the individual stage efficiencies, where

E_{1}, E_{2}

represent the production efficiency of the first and second phases, respectively.

ω_{1}

is the weight of the efficiency of stage 1 and

ω_{2}

is the weight of the efficiency of stage 2. The two variables should be selected to reflect the relative significance of the two phases and there exist a relationship between them:

ω_{1} + ω_{2} = 1

. Assuming that

E^{*}, E_{1}^{*}, E_{2}^{*}

represent optimal values of overall, stage 1, stage 2 efficiency, respectively, there exists

E^{*} = ω_{1} E_{1}^{*} + ω_{2} E_{2}^{*}

θ_{i}

denotes the input contraction of stage 1.

β_{r}

denotes the output extension of stage 2.

θ_{f}

denotes the input contraction of dual-role factor in the first stage, and

ϕ_{f}

denotes the output contraction of dual-role factors in the first stage.

ϕ_{d}

denotes the output extension of stage 1.

{\bar{ϕ}}_{d}

denotes the input contraction of the second stage.

\hat{F}

\bar{F}

are the number of dual-role factors acting as inputs and outputs, respectively. What needs to be pointed out is that, since we treat intermediate products as weak disposability, the second and fifth constraints concerning intermediate products in the model are represented as equations based on the PPS.

The 0-1 programming is applied to deal with dual-role factors. The value of the variable

D_{f}

is either 0 or 1. If

D_{f} = 0

, dual-role factors are treated as outputs. At this point, the constraint

\sum_{j = 1}^{n} λ_{j} (1 - D_{f}) h_{f j} \geq ϕ_{f} (1 - D_{f}) h_{f o}

holds and the constraint

\sum_{j = 1}^{n} λ_{j} D_{f} h_{f j} \leq θ_{f} D_{f} h_{f o}

is excluded. The constraint

\sum_{j = 1}^{n} λ_{j} (1 - D_{f}) h_{f j} \geq ϕ_{f} (1 - D_{f}) h_{f o}

can be written as

\sum_{j = 1}^{n} λ_{j} h_{f j} \geq ϕ_{f} h_{f o}

. If

D_{f} = 1

, it implies that dual-role factors play the role of inputs in the first phase. In this case, the constraint

\sum_{j = 1}^{n} λ_{j} D_{f} h_{f j} \leq θ_{f} D_{f} h_{f o}

is retained. It can be converted to

\sum_{j = 1}^{n} λ_{j} h_{f j} \leq θ_{f} h_{f o}

The additive efficiency decomposition method applied by Chen, et al.^[7] biases the efficiency assessments in favor of the second stage^[31]. The weight of the second stage is never greater than that of the first stage. This raises an issue because the weight reflects the importance of a stage, and when the production significance of the second stage is greater, its weight should also be greater. However, the fact is that the weight of the first stage is greater, which is unreasonable. In Model (7),

ω_{1}

and

ω_{2}

are not depended on the DMU being assessed. Their values are determined by decision-makers. If a stage is more important for the enterprise, decision-makers can appropriately increase the weight of that stage. In this case, for different DMUs at the same stage, their weight magnitudes are all equal. In other words, the size of a stage is the same from each DMU. So, there is no bias towards any stage as the values of

ω_{1}

and

ω_{2}

are arbitrarily assigned.

Model (7) should be converted from a nonlinear model to a linear one using the Charnes-Cooper transformation.

Let's redefine three new

δ_{1 f} = D_{f} θ_{f}, δ_{2 f} = D_{f} ϕ_{f}, δ_{3 j} = D_{f} λ_{j}

. For these three variables, the following constraints exist:

\begin{aligned} 0 \leq δ_{1 f} \leq D_{f} M, δ_{1 f} \leq θ_{f} \leq δ_{1 f} + M (1 - D_{f}), \\ 0 \leq δ_{2 f} \leq D_{f} M, δ_{2 f} \leq ϕ_{f} \leq δ_{2 f} + M (1 - D_{f}), \\ 0 \leq δ_{2 j} \leq D_{f} M, δ_{3 j} \leq λ_{j} \leq δ_{3 j} + M (1 - D_{f}) . \end{aligned}

(8)

In the above constraint, M denotes a positive number with a large numeric value. It must be noted that if

D_{f} = 0

, then

δ_{1 f} = δ_{2 f} = δ_{3 j} = 0

. If

D_{f} = 1

, then

δ_{1 f} = θ_{f}, δ_{2 f} = ϕ_{f}, δ_{3 j} = λ_{j}

Based on the above analysis, we acquire the following new model:

\begin{aligned} E = min ω_{1} E_{1} + ω_{2} E_{2} = ω_{1} (\frac{\frac{1}{M + \hat{F}} (\sum_{i = 1}^{M} θ_{i} + \sum_{f}^{F} δ_{1 f})}{\frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f}^{F} (ϕ_{f} - δ_{2 f}))}) + ω_{2} (\frac{\frac{1}{D} (\sum_{d = 1}^{D} {\bar{ϕ}}_{d})}{\frac{1}{S} (\sum_{r = 1}^{S} β_{r})}) \\ s.t. \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq θ_{i} x_{i o}, i = 1, \dots, M, \\ \sum_{j = 1}^{n} λ_{j} z_{d j} = ϕ_{d} z_{d o}, d = 1, \dots, D, \\ \sum_{j = 1}^{n} δ_{3 j} h_{f j} \leq δ_{1 f} h_{f o}, f = 1, \dots, F, \\ \sum_{j = 1}^{n} (λ_{j} h_{f j} - δ_{3 j} h_{f j}) \geq ϕ_{f} h_{f o} - δ_{2 f} h_{f o}, f = 1, \dots, F, \\ \sum_{j = 1}^{n} μ_{j} z_{d j} = {\bar{ϕ}}_{d} z_{d o}, d = 1, \dots, D, \\ \sum_{j = 1}^{n} μ_{j} y_{r j} \geq β_{r} y_{r o}, r = 1, \dots, S, \\ \hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), D_{f} \in {0, 1}, \\ \sum_{j = 1}^{n} λ_{j} = 1, \sum_{j = 1}^{n} μ_{j} = 1, \\ λ_{j}, μ_{j} \geq 0, θ_{i} \leq 1, θ_{f} \geq 1, ϕ_{d} \geq 1, ϕ_{f} \geq 1, {\bar{ϕ}}_{d} \leq 1, β_{r} \geq 1, \\ 0 \leq δ_{1 f} \leq D_{f} M, δ_{1 f} \leq θ_{f} \leq δ_{1 f} + M (1 - D_{f}), \\ 0 \leq δ_{2 f} \leq D_{f} M, δ_{2 f} \leq ϕ_{f} \leq δ_{2 f} + M (1 - D_{f}), \\ 0 \leq δ_{2 j} \leq D_{f} M, δ_{3 j} \leq λ_{j} \leq δ_{3 j} + M (1 - D_{f}) . \end{aligned}

(9)

Obviously, the deformed model above is still nonlinear. According to the Charnes-Cooper transformation, we define that:

\frac{1}{\frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f}^{F} (ϕ_{f} - δ_{2 f})} = t_{1}, \frac{1}{\frac{1}{S} (\sum_{r = 1}^{S} β_{r})} = t_{2} .

Let

t_{1}

and

t_{2}

be multiplied by all the constraints in Model (8), except for constraints

\hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), D_{f} \in {0, 1}

. This step is necessary because

\hat{F}

and

\bar{F}

in the objective function are not affected by

t_{1}

and

t_{2}

in in the subsequent transformations. Therefore, to limit values of

\hat{F}

and

\bar{F}

, constraints

\hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), D_{f} \in {0, 1}

are needed in Model (9). And the following variables can be acquired:

\begin{array}{rcl} θ_{i}^{1} = t_{1} θ_{i}, θ_{i}^{2} = t_{2} θ_{i}, ϕ_{f}^{1} = t_{1} ϕ_{f}, ϕ_{f}^{2} = t_{2} ϕ_{f}, ϕ_{d}^{1} = t_{1} ϕ_{d}, ϕ_{d}^{2} = t_{2} ϕ_{d}, {\bar{ϕ}}_{d}^{1} = t_{1} {\bar{ϕ}}_{d}, {\bar{ϕ}}_{d}^{2} = t_{2} {\bar{ϕ}}_{d}, \\ θ_{f}^{1} = t_{1} θ_{f}, θ_{f}^{2} = t_{2} θ_{f}, D_{i}^{1} = t_{1} D_{i}, D_{i}^{2} = t_{2} D_{i}, β_{r}^{1} = t_{1} β_{r}, β_{r}^{2} = t_{2} β_{r}, λ_{j}^{1} = t_{1} λ_{j}, λ_{j}^{2} = t_{2} λ_{j}, \\ μ_{j}^{1} = t_{1} μ_{j}, μ_{j}^{2} = t_{2} μ_{j}, δ_{1 f}^{1} = t_{1} δ_{1 f}, δ_{1 f}^{2} = t_{2} δ_{1 f}, δ_{2 f}^{1} = t_{1} δ_{2 f}, δ_{2 f}^{2} = t_{2} δ_{2 f}, δ_{3 j}^{1} = t_{1} δ_{3 j}, δ_{3 j}^{2} = t_{2} δ_{3 j} . \end{array}

(10)

Then Model (9) can be converted to the program below:

\begin{aligned} E = min ω_{1} E_{1} + ω_{2} E_{2} = ω_{1} (\frac{1}{M + \hat{F}} (\sum_{i = 1}^{M} θ_{_{i}}^{1} + \sum_{f = 1}^{F} δ_{_{1 f}}^{1})) + ω_{2} (\frac{1}{D} \sum_{d = 1}^{D} {\bar{ϕ}}_{_{d}}^{2}) \\ s.t. \\ \sum_{j = 1}^{n} λ_{_{j}}^{1} x_{i j} \leq θ_{_{i}}^{1} x_{i o}, \sum_{j = 1}^{n} λ_{_{j}}^{2} x_{i j} \leq θ_{_{i}}^{2} x_{i o}, i = 1, 2, \dots, M, \\ \sum_{j = 1}^{n} λ_{_{j}}^{1} z_{d j} = ϕ_{_{d}}^{1} z_{d o}, \sum_{j = 1}^{n} λ_{_{j}}^{2} z_{d j} = ϕ_{_{d}}^{2} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} δ_{_{3 j}}^{1} h_{f j} \leq δ_{_{1 f}}^{1} h_{f o}, \sum_{j = 1}^{n} δ_{_{3 j}}^{2} h_{f j} \leq δ_{_{1 f}}^{2} h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} (λ_{_{j}}^{1} h_{f j} - δ_{_{3 j}}^{1} h_{f j}) \geq ϕ_{_{f}}^{1} h_{f o} - δ_{2 f}^{1} h_{f o}, \sum_{j = 1}^{n} (λ_{_{j}}^{2} h_{f j} - δ_{_{3 j}}^{2} h_{f j}) \geq ϕ_{_{f}}^{2} h_{f o} - δ_{2 f}^{2} h_{f o}, \\ \frac{1}{D + \bar{F}} \sum_{d = 1}^{D} ϕ_{_{d}}^{1} + \sum_{f = 1}^{F} (ϕ_{_{f}}^{1} - δ_{2 f}^{1}) = 1, \frac{1}{S} \sum_{r = 1}^{S} β_{_{r}}^{2} = 1, \\ \hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), \\ \sum_{j = 1}^{n} μ_{_{j}}^{1} z_{d j} = {\bar{ϕ}}_{_{d}}^{1} z_{d o}, \sum_{j = 1}^{n} μ_{_{j}}^{2} z_{d j} = {\bar{ϕ}}_{_{d}}^{2} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} μ_{j} y_{r j} \geq β_{r} y_{r o}, r = 1, 2, \dots, S, \\ \sum_{j = 1}^{n} λ_{_{j}}^{1} = t_{1}, \sum_{j = 1}^{n} μ_{_{j}}^{1} = t_{2}, \sum_{j = 1}^{n} λ_{_{j}}^{2} = t_{1}, \sum_{j = 1}^{n} μ_{_{j}}^{2} = t_{2}, j = 1, 2, \dots, n, \\ λ_{_{j}}^{1} \geq 0, μ_{_{j}}^{1} \geq 0, λ_{_{j}}^{2} \geq 0, μ_{_{j}}^{2} \geq 0, \\ θ_{_{i}}^{1} \leq t_{1}, θ_{_{f}}^{1} \leq t_{1}, ϕ_{_{d}}^{1} \geq t_{1}, ϕ_{_{f}}^{1} \geq t_{1}, {\bar{ϕ}}_{_{d}}^{1} \leq t_{1}, β_{_{r}}^{1} \geq t_{1}, \\ θ_{_{i}}^{2} \leq t_{2}, θ_{_{f}}^{2} \leq t_{2}, ϕ_{_{d}}^{2} \geq t_{2}, ϕ_{_{f}}^{2} \geq t_{2}, {\bar{ϕ}}_{_{d}}^{2} \leq t_{2}, β_{_{r}}^{2} \geq t_{2}, \\ 0 \leq δ_{_{1 f}}^{1} \leq D_{_{f}}^{1} M, δ_{_{1 f}}^{1} \leq θ_{_{f}}^{1} \leq δ_{_{1 f}}^{1} + M (t_{1} - D_{_{f}}^{1}), \\ 0 \leq δ_{_{1 f}}^{2} \leq D_{_{f}}^{2} M, δ_{_{1 f}}^{2} \leq θ_{_{f}}^{2} \leq δ_{_{1 f}}^{2} + M (t_{2} - D_{_{f}}^{2}), \\ 0 \leq δ_{_{2 f}}^{1} \leq D_{_{f}}^{1} M, δ_{_{2 f}}^{1} \leq ϕ_{_{f}}^{1} \leq δ_{_{2 f}}^{1} + M (t_{1} - D_{_{f}}^{1}), \\ 0 \leq δ_{_{2 f}}^{2} \leq D_{_{f}}^{2} M, δ_{_{2 f}}^{2} \leq ϕ_{_{f}}^{2} \leq δ_{_{2 f}}^{2} + M (t_{2} - D_{_{f}}^{2}), \\ 0 \leq δ_{_{3 j}}^{1} \leq D_{_{f}}^{1} M, δ_{_{3 j}}^{1} \leq λ_{_{j}}^{1} \leq δ_{_{3 j}}^{1} + M (t_{1} - D_{_{f}}^{1}), \\ 0 \leq δ_{_{3 j}}^{2} \leq D_{_{f}}^{2} M, δ_{_{3 j}}^{2} \leq λ_{_{j}}^{2} \leq δ_{_{3 j}}^{2} + M (t_{2} - D_{_{f}}^{2}), \\ D_{_{f}}^{1} \in {0, t_{1}}, D_{_{f}}^{2} \in {0, t_{2}}, D_{f} \in {0, 1} . \end{aligned}

(11)

Now we set

\frac{1}{M + \hat{F}} = t

, and constraints in Model (11) are multiplied by t respectively. The following linear model can be obtained.

\begin{aligned} E = min ω_{1} E_{1} + ω_{2} E_{2} = ω_{1} (\frac{1}{M + \hat{F}} (\sum_{i = 1}^{M} θ_{i}^{1} + \sum_{f}^{F} δ_{1 f}^{1})) + ω_{2} (\frac{1}{D} (\sum_{d = 1}^{D} {\bar{ϕ}}_{d}^{2})) \\ s.t. \\ \sum_{j = 1}^{n} λ_{j}^{1} x_{i j} \leq θ_{i}^{1} x_{i o}, \sum_{j = 1}^{n} λ_{j}^{2} x_{i j} \leq θ_{i}^{2} x_{i o}, i = 1, \dots, M, \\ \sum_{j = 1}^{n} λ_{j}^{1} z_{d j} = ϕ_{d}^{1} z_{d o}, \sum_{j = 1}^{n} λ_{j}^{2} z_{d j} = ϕ_{d}^{2} z_{d o}, d = 1, \dots, D, \\ \sum_{j = 1}^{n} δ_{3 j}^{1} h_{f j} \leq δ_{1 f}^{1} h_{f o}, \sum_{j = 1}^{n} δ_{3 j}^{2} h_{f j} \leq δ_{1 f}^{2} h_{f o}, f = 1, \dots, F, \\ \sum_{j = 1}^{n} (λ_{j}^{1} h_{f j} - δ_{3 j}^{1} h_{f j}) \geq ϕ_{f}^{1} h_{f o} - δ_{2 f}^{1} h_{f o}, \sum_{j = 1}^{n} (λ_{j}^{2} h_{f j} - δ_{3 j}^{2} h_{f j}) \geq ϕ_{f}^{2} h_{f o} - δ_{2 f}^{2} h_{f o}, \\ \sum_{j = 1}^{n} μ_{j}^{1} z_{d j} = {\bar{ϕ}}_{d}^{1} z_{d o}, \sum_{j = 1}^{n} μ_{j}^{2} z_{d j} = {\bar{ϕ}}_{d}^{2} z_{d o}, d = 1, \dots, D, \\ \sum_{j = 1}^{n} μ_{j}^{1} y_{r j} \geq β_{r}^{1} y_{r o}, \sum_{j = 1}^{n} μ_{j}^{2} y_{r j} \geq β_{r}^{2} y_{r o}, r = 1, \dots, S, \\ \hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), D_{f} \in {0, 1}, M t + \hat{F} = 1, \\ \sum_{j = 1}^{n} λ_{j}^{1} = t_{1}, \sum_{j = 1}^{n} λ_{j}^{2} = t_{2}, \sum_{j = 1}^{n} μ_{j}^{1} = t_{1}, \sum_{j = 1}^{n} μ_{j}^{2} = t_{2}, \\ λ_{j}^{1}, λ_{j}^{2}, μ_{j}^{1}, μ_{j}^{2} \geq 0, \\ θ_{i}^{1} \leq t_{1}, θ_{i}^{2} \leq t_{2}, θ_{f}^{1} \geq t_{1}, θ_{f}^{2} \geq t_{2}, ϕ_{d}^{1} \geq t_{1}, ϕ_{d}^{2} \geq t_{2}, \\ ϕ_{f}^{1} \geq t_{1}, ϕ_{f}^{2} \geq t_{2}, {\bar{ϕ}}_{d}^{1} \leq t_{1}, {\bar{ϕ}}_{d}^{2} \leq t_{2}, β_{r}^{1} \geq t_{1}, β_{r}^{2} \geq t_{2}, \\ 0 \leq δ_{1 f}^{1} \leq D_{f}^{1} M, δ_{1 f}^{1} \leq θ_{f}^{1} \leq δ_{1 f}^{1} + M (t_{1} - D_{f}^{1})), \\ 0 \leq δ_{1 f}^{2} \leq D_{f}^{2} M, δ_{1 f}^{2} \leq θ_{f}^{2} \leq δ_{1 f}^{2} + M (t_{2} - D_{f}^{2})), \\ 0 \leq δ_{2 f}^{1} \leq D_{f}^{1} M, δ_{2 f}^{1} \leq ϕ_{f}^{1} \leq δ_{2 f}^{1} + M (t_{1} - D_{f}^{1})), \\ 0 \leq δ_{2 f}^{2} \leq D_{f}^{2} M, δ_{2 f}^{2} \leq ϕ_{f}^{2} \leq δ_{2 f}^{2} + M (t_{2} - D_{f}^{2})), \\ 0 \leq δ_{2 j}^{1} \leq D_{f}^{1} M, δ_{3 j}^{1} \leq λ_{j}^{1} \leq δ_{3 j}^{1} + M (t_{1} - D_{f}^{1})), \\ 0 \leq δ_{2 j}^{2} \leq D_{f}^{2} M, δ_{3 j}^{2} \leq λ_{j}^{2} \leq δ_{3 j}^{2} + M (t_{2} - D_{f}^{2})), \\ D_{f}^{1} \in {0, t_{1}}, D_{f}^{2} \in {0, t_{2}}, D_{f} \in {0, t}, \\ \frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d}^{1} + \sum_{f}^{F} (ϕ_{f}^{1} - δ_{2 f}^{1})) = 1, \frac{1}{S} (\sum_{r = 1}^{S} β_{r}^{2}) = 1. \end{aligned}

(12)

4 The Divisional Models

By analyzing the PPS and the overall efficiency model established above, divisional models of two-stage systems can be expressed as follows:

\begin{aligned} E_{1} = min \frac{\frac{1}{M + \hat{F}} (\sum_{i = 1}^{M} θ_{i} + \sum_{f}^{F} D_{f} θ_{f})}{\frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f}^{F} (1 - D_{f}) ϕ_{f})} \\ s.t. \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq θ_{i} x_{i o}, i = 1, \dots, M, \\ \sum_{j = 1}^{n} λ_{j} z_{d j} = ϕ_{d} z_{d o}, d = 1, \dots, D, \\ \sum_{j = 1}^{n} λ_{j} D_{f} h_{f j} \leq θ_{f} D_{f} h_{f o}, f = 1, \dots, F, \\ \sum_{j = 1}^{n} λ_{j} (1 - D_{f}) h_{f j} \geq ϕ_{f} (1 - D_{f}) h_{f o}, f = 1, \dots, F, \\ \hat{F} = \sum_{f = 1}^{F} D_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - D_{f}), \\ \sum_{j = 1}^{n} λ_{j} = 1, j = 1, \dots, n, \\ λ_{j} \geq 0, θ_{i} \leq 1, θ_{f} \leq 1, ϕ_{d} \geq 1, ϕ_{f} \geq 1, D_{f} \in {0, 1}, \end{aligned}

(13)

\begin{aligned} E_{2} = min \frac{\frac{1}{D} (\sum_{d = 1}^{D} {\bar{ϕ}}_{d})}{\frac{1}{S} (\sum_{r = 1}^{S} β_{r})} \\ s.t. \\ \sum_{j = 1}^{n} μ_{j} z_{d j} = {\bar{ϕ}}_{d} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} μ_{j} y_{r j} \geq β_{r} y_{r o}, r = 1, 2, \dots, S, \\ \sum_{j = 1}^{n} μ_{j} = 1, j = 1, 2, \dots, n, \\ μ_{j} \geq 0, {\bar{ϕ}}_{d} \leq 1, β_{r} \geq 1. \end{aligned}

(14)

Obviously, the above two models are nonlinear. For the efficiency model of the first stage, define

δ_{1 f} = D_{f} θ_{f}

δ_{2 f} = D_{f} ϕ_{f}

δ_{3 j} = D_{f} λ_{j}

. According to the Charnes-Cooper transformation, we transform the formula as follows:

\frac{1}{\frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f = 1}^{F} (ϕ_{f} + δ_{2 f}))} = t

Then we can obtain the model below:

\begin{aligned} E_{1} = min \frac{1}{M + \hat{F}} \sum_{i = 1}^{M} θ_{i} + \sum_{f = 1}^{F} δ_{1 f} \\ s.t. \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq θ_{i} x_{i o}, i = 1, 2, \dots, M, \\ \sum_{j = 1}^{n} λ_{j} z_{d j} = ϕ_{d} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} δ_{3 j} h_{f j} \leq δ_{1 f} h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} (λ_{j} - δ_{3 j}) h_{f j} \geq (ϕ_{f} - δ_{2 f}) h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} λ_{j} = t, j = 1, 2, \dots, n, \\ \frac{1}{D + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f = 1}^{F} (ϕ_{f} - δ_{2 f})) = 1, \\ \hat{F} = \sum_{f = 1}^{F} {\tilde{D}}_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - {\tilde{D}}_{f}), \\ λ_{j} \geq 0, θ_{i} \leq t, θ_{f} \leq t, ϕ_{d} \geq t, ϕ_{f} \geq t, \\ 0 \leq δ_{1 f} \leq D_{f} M, δ_{1 f} \leq θ_{f} \leq δ_{1 f} + M (t - D_{f}), \\ 0 \leq δ_{2 f} \leq D_{f} M, δ_{2 f} \leq ϕ_{f} \leq δ_{2 f} + M (t - D_{f}), \\ 0 \leq δ_{3 j} \leq D_{f} M, δ_{3 f} \leq λ_{j} \leq δ_{3 f} + M (t - D_{f}), \\ D_{f} \in {0, t}, {\tilde{D}}_{f} \in {0, 1} . \end{aligned}

(15)

Define

\frac{1}{M + \hat{F}} = t_{1}

, and let it multiply all of constraints in Model (15). Then the linear program of first stage is presented below:

\begin{aligned} E_{1} = min \sum_{i = 1}^{M} θ_{i} + \sum_{f = 1}^{F} δ_{1 f} \\ s.t. \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq θ_{i} x_{i o}, i = 1, 2, \dots, M, \\ \sum_{j = 1}^{n} λ_{j} z_{d j} = ϕ_{d} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} δ_{3 j} h_{f j} \leq δ_{1 f} h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} (λ_{j} - δ_{3 j}) h_{f j} \geq (ϕ_{f} - δ_{2 f}) h_{f o}, f = 1, 2, \dots, F, \\ \sum_{j = 1}^{n} λ_{j} = t, j = 1, 2, \dots, n, \\ \frac{1}{D t_{1} + \bar{F}} (\sum_{d = 1}^{D} ϕ_{d} + \sum_{f = 1}^{F} (ϕ_{f} - δ_{2 f})) = 1, \\ \hat{F} = \sum_{f = 1}^{F} {\tilde{D}}_{f}, \bar{F} = \sum_{f = 1}^{F} (1 - {\tilde{D}}_{f}), M t_{1} + \hat{F} = 1, \\ λ_{j} \geq 0, θ_{i} \leq t, θ_{f} \leq t, ϕ_{d} \geq t, ϕ_{f} \geq t, t_{1} \geq 0, \\ 0 \leq δ_{1 f} \leq D_{f} M, δ_{1 f} \leq θ_{f} \leq δ_{1 f} + M (t - D_{f}), \\ 0 \leq δ_{2 f} \leq D_{f} M, δ_{2 f} \leq ϕ_{f} \leq δ_{2 f} + M (t - D_{f}), \\ 0 \leq δ_{3 j} \leq D_{f} M, δ_{3 f} \leq λ_{j} \leq δ_{3 f} + M (t - D_{f}), \\ D_{f} \in {0, t}, {\tilde{D}}_{f} \in {0, t_{1}} . \end{aligned}

(16)

For the efficiency model of the second stage, we also apply the Charnes-Cooper transformation.

Define

\frac{1}{\frac{1}{S} (\sum_{r = 1}^{S} β_{r})} = t_{}

, then the linear pattern can be expressed as:

\begin{aligned} E_{2} = min \frac{1}{D} \sum_{d = 1}^{D} {\bar{ϕ}}_{d} \\ s.t. \\ \sum_{j = 1}^{n} μ_{j} z_{d j} = {\bar{ϕ}}_{d} z_{d o}, d = 1, 2, \dots, D, \\ \sum_{j = 1}^{n} μ_{j} y_{r j} \geq β_{r} y_{r o}, r = 1, 2, \dots, S, \\ \sum_{j = 1}^{n} μ_{j} = t, j = 1, 2, \dots, n, \\ \frac{1}{S} (\sum_{r = 1}^{S} β_{r}) = 1, \\ μ_{j} \geq 0, β_{r} \geq t, {\bar{ϕ}}_{d} \leq t . \end{aligned}

(17)

For our proposed Models (12), (16) and (17), there is an equation

E = ω_{1} E_{1} + ω_{2} E_{2}

5 A Case Study

5.1 Data

Zhao, et al.^[32] investigated the efficiency of supplier-manufacturer in a two-stage system. We select supplier-producer in supply chains as the research object. The data of supplier-producer in supply chains was collected by Mirhedayatian, et al.^[28].

Figure 2 illustrates a visual representation of the supplier-producer structure including dual-role factors. Within the diagram, one type of dual-role factor is present in the supplier, along with two types of inputs provided by the supplier, one type of output transported from the supplier to the producer, and one type of output that flows from the producer to the subsequent stage. The specific factors pertaining to each category are depicted in Figure 3. Table 3 lists two inputs of suppliers including material purchase cost and transportation cost.

Figure 2 Supplier-producer structure

Full size|PPT slide

Figure 3 Category of the factors

Full size|PPT slide

Table 3 The data of supplier

DMU	Material purchase cost	Transportation cost
1 Behnoush	290	220
2 Abali	300	345
3 Kafir	288	350
4 Zam	320	330
5 Khazar	290	275
6 Damdaran	340	210
7 Sara	325	370
8 Ramak	330	250
9 Pegah	349	320
10 Varna	295	355

Number of parts from supplier to producer is an intermediate product. R&D cost is a dual-role factor, as outlined in Table 4. Mirhedayatian, et al.^[28] viewed R&D cost as a factor with a dual role. Clearly, R&D cost serves as an input. In the eyes of decision-makers, R&D cost acts as a proxy for “supplier innovation”, which leads to high-quality work, successful marketing and sales, better goods and services, and more efficient processes. Therefore, it can also be regarded as an output. The output data of producers in Table 5 includes one item: number of green products.

Table 4 The data of intermediate products and dual-role factors

DMU	R&D cost $h_{1 j}$	Number of parts from supplier to producer $Z_{1 j}$
1 Behnoush	1450	236
2 Abali	1345	279
3 Kafir	1570	247
4 Zam	1620	289
5 Khazar	1565	275
6 Damdaran	1394	298
7 Sara	1444	320
8 Ramak	1530	327
9 Pegah	1629	297
10 Varna	1455	217

Table 5 The data of producers

DMU	Number of green products $y_{1 j}$
1 Behnoush	490
2 Abali	523
3 Kafir	539
4 Zam	597
5 Khazar	479
6 Damdaran	623
7 Sara	589
8 Ramak	532
9 Pegah	508
10 Varna	639

5.2 Results and Analysis

Assuming that the supplier and the producer have equal weights, which means

ω_{1} = ω_{2} = 0.5

, there exists

E^{*} = 0.5 E_{1}^{*} + 0.5 E_{2}^{*}

. The computation of the overall and divisional efficiency of 10 DMUs utilizing Models (12), (16), and (17) is displayed in Table 6. Among the 10 supply chain companies, Varna achieves the highest overall efficiency score of 0.8726, indicating that Varna is highly efficient and performs well in terms of overall efficiency. Table 6 also indicates that in the first stage, five DMUs achieved a score of 1, signifying their efficiency in the first stage. In the second stage, only Varna is efficient with a score of 1, while the remaining DMUs are inefficient. Notably, for Behnoush, Kafir, Khazar, Damdaran, and Sara, the variable

D_{f}

can be either 0 or 1, which indicates that dual-role factors have the same effect as both inputs and outputs. As for Abali, Ramak, and Varna, the variable

D_{f}

is always 0, suggesting that dual-role factors only serve as outputs in the first stage of production. Zam and Pegah have a value of 1 for the variable, which signifies that dual-role factors serve as inputs.

Table 6 The overall efficiency and divisional efficiency

DMU	Overall efficiency	Stage 1 efficiency	Stage 2 efficiency	$D_{f}$	Ranking
1 Behnoush	0.8526	1	0.7051	1 or 0	4
2 Abali	0.7246	0.8126	0.6366	0	7
3 Kafir	0.8706	1	0.7411	1 or 0	2
4 Zam	0.7690	0.8364	0.7015	1	6
5 Khazar	0.7958	1	0.5915	1 or 0	8
6 Damdaran	0.8550	1	0.7100	1 or 0	3
7 Sara	0.8126	1	0.6251	1 or 0	5
8 Ramak	0.6842	0.8159	0.5525	0	10
9 Pegah	0.6940	0.8071	0.5809	1	9
10 Varna	0.8726	0.7451	1	0	1

Table 6 is summarized in Figure 4. In Figures 5, 6, and 7, the efficiency scores for the first stage, second stage, and overall process are illustrated. Figure 5 reveals that Behnoush, Kafir, Khazar, Damdaran, and Sara all achieve a production efficiency score of 1 in the first stage. This indicates that these five DMUs are efficient in the first stage. Abali, Zam, Ramak, and Pegah all have production efficiency scores above 0.8, indicating a high level of production. Varna, however, ranks last with an efficiency score of 0.7451, suggesting a relatively lower production efficiency.

Based on the data presented in Figure 6, Varna emerges as the top performer in the second stage with a production efficiency score of 1. Ramak's production efficiency score is 0.5525, ranking last among the ten DMUs. This indicates that while Varna is fully efficient, Ramak encountered certain issues in its production process that need to be addressed for overall improvement. Additionally, Pegab, Khazar, Sara, and Abali exhibit second-stage efficiency scores of 0.5809, 0.5915, 0.6251, and 0.6366, respectively, ranking ninth, eighth, seventh, and sixth. These scores suggest that their production processes are inefficient and require the development of new production plans to enhance second-stage efficiency.

Figure 7 illustrates overall production efficiency scores for both stages. Notably, none of the enterprises achieve a perfect efficiency score of 1, which indicate that all companies face inefficiencies in the production process. Varna, Kafir, Damdaran, and Behnoush attain efficiency scores of 0.8726, 0.8706, 0.8550, and 0.8525, respectively, ranking second, third, fourth, and fifth. This implies that efficiency scores of these four enterprises are relatively similar. However, Pegah and Ramak have production efficiency scores of 0.6940 and 0.6842, respectively, indicating significantly lower efficiency levels that require prompt adjustments.

Table 7 The overall efficiency and divisional efficiency without dual-role factors

DMU	Overall efficiency	Stage 1 efficiency	Stage 2 efficiency	Ranking
1 Behnoush	0.7629	0.8207	0.7051	4
2 Abali	0.7075	0.7784	0.6366	9
3 Kafir	0.7218	0.7025	0.7411	8
4 Zam	0.7460	0.7905	0.7015	5
5 Khazar	0.7262	0.8608	0.5915	7
6 Damdaran	0.8550	1	0.7100	1
7 Sara	0.7263	0.8274	0.6251	6
8 Ramak	0.7763	1	0.5525	3
9 Pegah	0.6826	0.7842	0.5809	10
10 Varna	0.8024	0.6048	1	2

The aforementioned analysis results demonstrate the feasibility of the newly developed model in evaluating the efficiency of the two-stage system. Considering that five DMUs achieve a production efficiency score of 1 in the first stage when accounting for dual-role factors, it can be inferred that the overall performance of these ten DMUs in the first stage is commendable. However, in terms of overall efficiency, these five DMUs fall short of achieving a score of 1 due to factors pertaining to the second stage.

Behnoush obtains

E

=0.8526,

E_{1}

=1,

E_{2}

=0.7051, which illustrates that the poor performance of the second stage leads to the inefficiency of the whole process. Therefore, decision-makers should develop corresponding plans to enhance production efficiency and rectify shortcomings. For instance, since all inputs for the second stage are derived from the first stage, it may be feasible to introduce additional inputs for the second stage to improve efficiency. Alternatively, changing the type or proportion of inputs in the first stage could increase the number of intermediate products flowing into the second stage.

This case study shows that the methodology in this paper can effectively assess the overall efficiency of the two-stage system in the presence of dual-role factors and attain considerable outcomes.

To make a comparison, Maleki and Kazemi's model is utilized to calculate the overall and divisional efficiency scores of the two-stage production process without considering dual-role factors, as displayed in Table 7. It is evident that efficiency scores of the second stage remains the same in both cases since dual-role factors are only involved in the first stage.

Table 7 reveals that, when dual-role factors are ignored, Damdaran and Ramak achieve a production efficiency score of 1 in the first stage, indicating that these two DMUs are fully efficient in stage 1. However, when compared to the scenario where dual-role factors are taken into account, these DMUs display relatively lower efficiency scores in the first stage, ultimately affecting overall efficiency. Concerning the overall efficiency, only Damdaran and Varna attain scores exceeding 0.8, while the majority fall within the range of 0.7 to 0.8. In conclusion, efficiency scores when ignoring dual-role factors are lower than efficiency scores when considering these factors.

In this specific case's production process, R&D cost plays a crucial role in the first stage. Concrete data indicates that R&D cost holds considerable numerical significance. Failing to acknowledge the impact of this factor in the production process leads to an underestimation of production efficiency. Therefore, the newly developed model, which incorporates dual-role factors, surpasses the model that overlooks this factor when measuring production efficiency.

6 Conclusion

The application of two-stage DEA models has been the focus of growing research and is essential for evaluating production efficiency. However, the presence of dual-role factors poses obstacles to the evaluation of production efficiency in two-stage systems.

To address the issues and fill the gaps in the existing literature, this paper presented a T-ERM-D model based on the ERM model and divisional models, where the 0-1 programming was applied to deal with dual-role factors. To address the nonlinearity in these novel models, the Charnes-Cooper transformation was applied to convert them into linear ones. The practicality of the models was verified through the data of 10 supply chains. To showcase the advantages of the new approach, the calculation results were compared with the results ignoring dual-role factors. The findings indicated that the results excluding dual-role factors significantly underestimated the production efficiency. The models proposed can help enterprises identify any inefficient stages, improve production efficiency, and achieve sustainable development.

Building upon this research, further exploration into dual-role factors is possible. This study specifically focuses on two-stage production processes. Future research can extend the application of the ERM model to multi-stage systems. Additionally, factors such as undesirable outputs and inputs can also be integrated into the models.

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Funding

the National Natural Science Foundation of China(72262024)

the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(NJYT22095)

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Abstract

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1 Introduction

2 Literature Review

2.1 Dual-Role Factors

Table 1 DEA methods in the presence of dual role factors

2.2 Two-Stages System

Table 2 DEA methods in two-stage systems

3 The Proposed DEA Model

Figure 1 Two-stage production system with dual-role factors

4 The Divisional Models

5 A Case Study

5.1 Data

Figure 2 Supplier-producer structure

Figure 3 Category of the factors

Table 3 The data of supplier

Table 4 The data of intermediate products and dual-role factors

Table 5 The data of producers

5.2 Results and Analysis

Table 6 The overall efficiency and divisional efficiency

Figure 4 DEA scores

Figure 5 Stage 1 scores

Figure 6 Stage 2 scores

Figure 7 Overall efficiency scores

Table 7 The overall efficiency and divisional efficiency without dual-role factors

6 Conclusion

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References

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Abstract

Key words

Cite this article

1 Introduction

2 Literature Review

2.1 Dual-Role Factors

Table 1 DEA methods in the presence of dual role factors

2.2 Two-Stages System

Table 2 DEA methods in two-stage systems

3 The Proposed DEA Model

Figure 1 Two-stage production system with dual-role factors

4 The Divisional Models

5 A Case Study

5.1 Data

Figure 2 Supplier-producer structure

Figure 3 Category of the factors

Table 3 The data of supplier

Table 4 The data of intermediate products and dual-role factors

Table 5 The data of producers

5.2 Results and Analysis

Table 6 The overall efficiency and divisional efficiency

Figure 4 DEA scores

Figure 5 Stage 1 scores

Figure 6 Stage 2 scores

Figure 7 Overall efficiency scores

Table 7 The overall efficiency and divisional efficiency without dual-role factors

6 Conclusion

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References

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Funding