Weight Restriction Approach in a Two-stage Network Structure: A DEA-based Approach

Abstract

The standard data envelopment analysis model aimed to maximize the efficiency scores using the most favourable input/output weights. In two-stage network structures, the presence of intermediate undesirable measures leads to a distinct domain of weight restriction study. In the presence of undesirable intermediate measures and equipped with non-cooperative game theory, this study developes a joint weight restriction approach to derive a common set of weights. Concurrently, the overall efficiency of the system can be evaluated. Numerical instances of 34 OECD countries in 2012 have been presented to elucidate the details and applicability of the proposed method.

Executive Summary

The conventional data envelopment analysis (DEA) is designed to assist each decision-making unit in selecting its most favourable input/output weights, enabling the units to optimize their performance. Additionally, the performance of different units is attained through the use of different sets of weights. As a result, comparison and ranking of units on a common basis seem such a considerable challenge. The object of common weight restriction is more highlighted in the literature when the subject toys around a network structure process. The two-stage production process normally considers the desirable intermediate measure. In many real situations, the intermediate measures consist of desirable and undesirable outputs. On these occasions, the challenge of a common set of weights has attracted the research attention. Among major approaches for analysing two-stage network assisting DEA, non-cooperative game theory is in our scope. Equipped with this perspective for analysing the two-stage network structure, the motivation of this article is to derive a common set of weights in two-stage systems with undesirable intermediate measures. Despite numerous advancements in this area, the present study introduces a model based on a joint weight restriction approach, addressing two critical issues. First, the proposed weight restriction approach aims to generate the possible minimum amount of undesirable intermediate output in the first stage. Second, the second stage optimizes its performance while keeping the first-stage efficiency at its current quantity. Concurrently, the individual and the overall efficiencies can be at hand in implementing the proposed model. Further, the proposed model generates positive weights and prevents weight dissimilarity. To elucidate the details and applicability of the proposed model, a comparison with existing models has been demonstrated through an experimental data set. The comparison can facilitate the enhancement of even more robust methods for addressing the challenges.

Keywords

Data envelopment analysis (DEA)Non-cooperative approach Joint weight restriction Two-stage network structure

According to the World Commission on Environment and Development, pollution levels have increased while resource scarcity remains. Consequently, this challenge and pressure have led to a growing interest in the use of efficiency and productivity management with regard to undesirable and pollutant outputs. To achieve this goal, parametric and non-parametric techniques have been applied in production theory. DEA, pioneered by Charnes et al. (1978) and extended by Banker et al. (1984), has recently made a substantial contribution in analysing undesirable outputs. DEA is a non-parametric technique for evaluating the relative efficiency of a set of homogeneous decision-making units (DMUs) by using a ratio of the weighted sum of outputs to the weighted sum of inputs, subject to the condition that this ratio does not exceed one for any DMU. Also, conventional DEA models determine a set of weights such that the efficiency of a target DMU relative to the other DMUs is maximized. This flexibility in the selection of input and output weights often causes more than one DMU being evaluated as efficient, leading to the inability to be fully discriminated. Furthermore, the set of weights will typically be different for the various DMUs. Also, the input and output weights may not be most favourable to themselves. As a possible answer to reduce this flexibility, a common set of weights has been suggested for efficiency evaluation in the background of applying DEA for efficiency evaluation. The idea was first developed by Ganely and Cubbin (1992) and Roll and Golany (1993). To derive a common set of weights, several approaches have been proposed in the DEA literature. For example, Hashimoto and Wu (2004) proposed a DEA-CP (compromise programming) model which aims at seeking a common set of weights across the DMUs by combining the DEA and the compromise programming of Kao and Hung (2005), also proposed a similar compromise solution approach for generating common weights under the DEA framework. Wang et al. (2009) suggested ranking DMUs by imposing a minimum weight restriction, which also produces a common weight for the DMUs to be compared. Wu et al. (2010) proposed a procedure for deriving a common set of weights based on satisfaction degree. Pourhabib et al. (2018) proposed a weight restriction approach to generate a common set of weights. The approach was conducted from two viewpoints. Their first contribution was proposing a single lower bound to restrict all weights. Although the proposed model is always feasible and optimal, it is always positive. In the second perspective, the weights are restricted separately. The proposed model was feasible and always obtained dissimilar positive weights. In the above-mentioned studies, a DMU is commonly modelled as a single process that transforms inputs into outputs. In real-world occasions, apart from consuming inputs and producing desirable outputs, the DMUs also generate undesirable outputs. In real occasions, joint production of desirable and undesirable outputs creates difficulties in the measurement of overall performance in two-stage network structures. Cook et al. (2010) have presented four categories in two-stage structure studies. Liang et al. (2008) employed the concept of the leader–follower approach to characterize a two-stage network structure. As far as we are aware, there are different DEA-based works that consider undesirable variables in network-structured production systems. It is worth noting that different proposed approaches can handle this situation through the assumption of an appropriate technology. For example, Lozano et al. (2013) presented a directional distance function network DEA approach that takes undesirable outputs into account and applies it to evaluate the efficiency of 39 Spanish airports. Maghbouli et al. (2014) employed cooperative and non-cooperative game theory to assess the relative performance of two-stage network structures. It should be pointed out that the existing papers in the DEA literature have widely employed envelopment structures to analyse two-stage structure systems in the presence of intermediate or final undesirable factors. Kiani Mavi et al. (2019) proposed an alternative approach to find the common set of weights in a two-stage network DEA based on goal programming to detect the effects of both eco-efficiency and eco-innovation in the presence of undesirable inputs, intermediate products and the outputs by the context of big data.

Afsharian et al. (2021) reviewed some DEA approaches employing a common set of weights. Interestingly enough, the proposed DEA models in the literature employ centralized management scenarios. In the context of the two-stage production system, Li et al. (2022) proposed a novel DEA approach for measuring the eco-efficiency of two-stage structures with undesirable intermediate measures. In another effort, Chu and Zhu (2021) developed a novel method that first projects the intermediate measures and then applies the multiplier DEA models to evaluate the overall and divisional efficiencies allowing weight flexibility for intermediate measures. With a glimpse into CO₂ reduction and energy consumption, Madadi et al. (2022) suggested moving a DMU from the current frontier to the frontier with better efficiency conditions, then employing the weak disposability axioms, the combined DEA and goal programming technique found the appropriate reallocation model. Despotis et al. (2022) considered a dual role for intermediate measures, as both consumer and producer. The proposed approach is expressed on multiplier and envelopment forms under constant and variable return-to-scale assumptions. For further research and different perspectives, we may refer to Namin and Khamseh (2022), Thi Le Hang Nguyen et al. (2021), Koronakos et al. (2021), Mozaffari et al. (2021), Omrani et al. (2022), Song et al. (2022) Kiaei and Kazemi (2019) and Liu et al. (2015). Modelling two-stage production systems with intermediate undesirable outputs is an important and interesting subject in the context of DEA. The motivation of this study is the application of the joint weight restriction method for searching a common set of weights. Compared with various DEA-based approaches in the past that served centralized management scenarios, this article speaks about the multiplier format of network DEA. The structure of the proposed method emphasizes undesirable intermediate measures. Since the before-mentioned approaches apply the envelopment models, the main contribution of the study is employing the issue of the multiplier format of DEA in two-stage network structures with undesirable intermediate measures. Another attribute of the proposed approach with the multiplier DEA-based format includes the restriction of all weights with a single lower bound. Also, compared with previous studies which follow the centralized perspective in analysing network structures, this article proposes the leader–follower game theory to assess the relative overall and divisional performance of the DMUs.

PRELIMINARIES

Assume that there are n DMUs and each $D M U_{j} (j = 1, ..., N)$ feed $m$ inputs $x_{i j} (i = 1, ..., m)$ to generate $s$ outputs $y_{r j} (r = 1, ... s)$ . The following DEA model known as CCR¹ measures the efficiency score of $D M U_{o}$ as follows:

\begin{array}{l} M a x E f f_{o} = \frac{\sum_{r = 1}^{s} u_{r} y_{r o}}{\sum_{i = 1}^{m} v_{i} x_{i o}} \\ s . t . \sum_{r = 1}^{s} u_{r} y_{r j} - \sum_{i = 1}^{m} v_{i} x_{i j} \leq 0, j = 1, ..., n, \\ u_{r}, v_{i} \geq 0, f o r a l l r, i . \end{array}

(1)

In evaluation with CCR model (1), $D M U_{o}$ is called efficient if the optimal solution of the model (1) is equal to unity, that is, $E f f_{o}^{*} = 1$ . Otherwise, it is called an inefficient unit. Standard CCR model (1) has the unit invariant property that is organized by Lovell and Pastor (1995). Applying this property, each DMU is allowed to select the most advantageous weights for maximizing its efficiency score also normalize the weights. Hence, scaling data cause model (1) to its equivalent format. The equivalent model (2) is as follows:

\begin{array}{l} M a x E f f_{o} = \frac{\sum_{r = 1}^{s} u_{r} y_{r o}}{\sum_{i = 1}^{m} v_{i} x_{i o}} \\ s . t . \\ \sum_{r = 1}^{s} u_{r} y_{r j} - \sum_{i = 1}^{m} v_{i} x_{i j} \leq 0, j = 1, ..., n, \\ 0 \leq u_{r} \leq 1, r = 1, ..., s \\ 0 \leq v_{i} \leq 1, i = 1, ..., m . \end{array}

(2)

Theorem 1. There exists a scale of data that causes model (1) to be equivalent to model (2).

Proof: Assume that $u_{r}^{*}$ and $v_{i}^{*} (i = 1, ..., m, r = 1, ..., s)$ are the optimal solution to model (1) and suppose that $ϕ$ is maximum of $u_{r}^{*}$ and $v_{i}^{*} (i = 1, ..., m, r = 1, ..., s)$ , that is, $ϕ = \max \{u_{r}^{*}, v_{i}^{*}\} (i = 1, ..., m, r = 1, ..., s)$ . If the input/output weights are divided by $ϕ$ , then the result gives a feasible solution for model (2).

As Theorem 1 claims, the resulting score is the best attainable efficiency level for each DMU. However, there are some shortcomings. First, the efficiency score of different DMUs obtained by different sets of weights may not be comparable. Second, with reference to the flexibility in the selection of weights, there is the possibility of the existence of more than one efficient unit. Hence, all DMUs cannot be discriminated. Pourhabib et al. (2018) developed a nonlinear and feasible model for joint weight restriction. Their model generates a set of common weights by setting the bounded weight variables. The model has the following format:

\begin{array}{l} M i n \frac{\sum_{j = 1}^{N} s_{j}}{ϕ} \\ s . t \\ \sum_{r = 1}^{s} u_{r} y_{r j} - \sum_{i = 1}^{m} v_{i} x_{i j} + S_{j} = 0, j = 1, \dots, N \\ ϕ \leq v_{i} \leq 1, i = 1, \dots, m \\ ϕ \leq u_{r} \leq 1, r = 1, \dots, s \\ u_{r}, v_{i}, ϕ \geq 0 f o r a l l i a n d r . \end{array}

(3)

The objective function minimizes the ratio $\frac{\sum_{j = 1}^{n} s_{j}}{ϕ}$ for all units employing a joint single lower bound $ϕ$ and an upper bound of unity. Applying this modification, the generated common set of weights is not similar and strictly positive. It can be easily proved that model (3) is always feasible.

A NON-COOPERATIVE WEIGHT RESTRICTION APPROACH IN THE TWO-STAGE PROCESS

In this section, a two-stage decision process is addressed. In this two-stage structure, the intermediate measures consist of desirable and undesirable outputs. Consider a two-stage production process as shown in Figure 1.

Figure 1.

Two-stage Process of $D M U_{j}$ .

Suppose again that there are $N$ DMUs and for the first stage of $D M U_{j} (j = 1, ..., N)$ the observed data on the vectors of inputs, desirable outputs and undesirable outputs are $x_{j} = (x_{i j}, x_{2 j}, ..., x_{m j}) \geq 0$ , $v_{j} = (v_{1 j}, v_{2 j}, ..., v_{K j}) \geq 0$ and $w_{j} = (w_{1 j}, w_{2 j}, ..., w_{F j}) \geq 0$ , respectively. The first-stage outputs $(v_{j}, w_{j})$ are employed as the inputs for the second stage. The second stage applies the intermediate measure $(v_{j}, w_{j})$ and external input vector $z_{j} = (z_{1 j}, z_{2 j}, ..., z_{T j})$ . The output of the second stage or the final product of $D M U_{j}$ is recorded as $y_{j} = (y_{1 j}, x_{2 j}, ..., y_{s j}) \geq 0$ .

In what follows, a weight restriction approach introduced in the previous section is executed in this two-stage decision process. An interesting area where a common set of weights DEA approaches can be applied is the efficiency measurement of multi-stage organizations which generally employs the centralized approach. This article grasps the idea of measuring two-stage network structures in the presence of intermediate undesirable outputs, with respect to leader–follower game theory. Because the common set of weights is applied to each observation under consideration, this article employs a non-cooperative management scenario. The major advantage of the approach is summarized as follows. The first stage is conducted in a format that generates the smallest allowable amount of undesirable intermediate measure, and, therefore, provides a common set of weights applying the multiplier format of the joint weight restriction approach. Having fixed the efficiency of the first stage represented by the optimality of the first stage, the second stage is solved with a multiplier DEA-based weight restriction approach. Besides the case outlined above, equipped with non-cooperative game theory, the leader and follower preferences raise a question. In this case, the leader determines the most efficient status and based on the optimal solution of the leader, the follower identifies its optimality. Without loss of generality, assume that the first stage is the leader and the second stage is the follower. In the first step, one needs to recall the smallest allowable amount of undesirable intermediate outputs. According to Kao and Hwang (2020), fixing the amount of desirable outputs at the current level, the minimum amount of undesirable factors which can be allowed to generate is $w_{f o}^{*} = w_{f o} - s_{f} (f = 1, ..., F)$ . The amount of slack variable $s_{f}$ can be mathematically formulated as

\begin{array}{l} M a x \sum_{f = 1}^{F} s_{f} \\ s . t . \\ \sum_{j = 1}^{N} λ_{j} v_{k j} = v_{k o} (k = 1, ..., K) \\ \sum_{j = 1}^{N} λ_{j} w_{f j} + s_{f} = w_{f o} (f = 1, ..., F) \\ λ_{j}, s_{f} \geq 0 j = 1, ..., N, f = 1, ..., F . \end{array}

(4)

By plugging the first-stage outputs $(v_{j}, w_{j})$ into this additive model (4), the allowable minimum amount of intermediate undesirable outputs is calculated. Determining the optimal solution of model (4), the triple $(x_{j}, v_{j}, w_{j}^{*})$ is applied in the joint restriction model. According to the weight restriction approach model (3), the model for achieving the common set of weights in the first stage has the following algebraic representation:

\begin{array}{l} M i n \frac{\sum_{j = 1}^{N} d_{j}}{φ} \\ s . t . \\ (\sum_{k = 1}^{K} u_{k} v_{k j} + \sum_{f = 1}^{F} γ_{f} w_{f j}^{*}) - \sum_{i = 1}^{m} v_{i} x_{i j} + d_{j} = 0, j = 1, \dots, N \\ φ \leq u_{k} \leq 1, = 1, \dots . K \\ φ \leq γ_{f} \leq 1, f = 1, \dots, F \\ φ \leq v_{i} \leq 1, i = 1, \dots, m \end{array}

(5)

Employing model (5) provides strictly positive common set of weights for the first stage. Looking closely to model (5), using slack variable $d_{j} (j = 1, ..., N)$ in the first constraint of the model, the objective function minimizes the deviation of all units. What is more, the single lower bound, which was introduced for all weights, is maximized at the same time. Another feature of model (5) is its nonlinearity and applying Charnes and Cooper (1962) transformation can be converted to linear programming. Efficiency score $(E_{1}^{*})$ for the first stage can be calculated with the presented common set of weights as $E_{1}^{*} = \frac{\sum_{k = 1}^{K} u_{k}^{*} v_{k j} + \sum_{f = 1}^{F} γ_{f}^{*} w_{f j}^{*}}{\sum_{i = 1}^{m} v_{i}^{*} x_{i j}}$ . Based upon the leader–follower non-cooperative game theory, the second stage is considered as the follower. Therefore, having obtained the efficiency of the first stage, the second stage has been evaluated by preserving the efficiency status of the first stage. Hence, applying the optimal solutions of the first stage, the second stage uses the optimal triple vector $(v^{*}, w^{* *}, z)$ as its input to produce the final output $y$ . Applying the optimal solution of the first stage guarantees that the first stage’s efficiency remains unchanged. Since the undesirable factor in the first stage has been allowed to achieve its minimum level, $w^{* *}$ can feed up to the second stage with that amount. Also, the optimal value for the desirable intermediate output of the first stage gets its optimal value $v_{k j}^{*} = v_{k j} - s_{k}$ where the slack variable $s_{k}$ calculated from the model is similar to model (4) with the difference that the undesirable intermediate outputs satisfy in this constraint. $\sum_{j = 1}^{N} λ_{j} w_{f j}^{*} = w_{f o}^{*} (f = 1, ..., F)$ . Also, the slack variable can be used to calculate the optimal value of the intermediate output of the first stage. To improve the divisional efficiency scores obtained by a common set of weights, the improved leader in the first stage can employ the reduced desirable and undesirable intermediate factors in the second stage. The reduced measures are determined as $v_{k j}^{*} = (1 - E_{1}^{*}) v_{k j} (k = 1, ..., K), w_{f j}^{* *} = (1 - E_{1}^{*}) w_{f j}^{*} (f = 1, ..., F)$ . Note that these improvements affect the divisional efficiency score obtained by a common set of weights. On the basis of the joint weight restriction approach, the possible common set of weights for the second stage can be formulated as follows:

\begin{array}{l} M i n \frac{\sum_{j = 1}^{n} d_{j}^{'}}{φ} \\ s . t . \\ \sum_{r = 1}^{s} η_{r} y_{r j} - (\sum_{f = 1}^{F} γ_{f}^{'} w_{f j}^{* *} + \sum_{k = 1}^{K} u_{k}^{'} v_{k j}^{*} + \sum_{t = 1}^{T} μ_{t} z_{t j}) \\ + d_{j}^{'} = 0, j = 1, ..., N \\ φ \leq μ_{f}^{'} \leq 1, k = 1, ..., K \\ φ \leq γ_{f}^{'} \leq 1, f = 1, ..., F \\ φ \leq η_{r} \leq 1, r = 1, ..., s \\ φ \leq μ_{t} \leq 1, t = 1, ..., T \end{array}

(6)

In the above model (6), the optimal adjusted solution of the first stage, that is, $v_{k j}^{*}$ and $w_{f j}^{* *}$ , are the inputs of the second stage. These improved measures affect the efficiency score of the second stage. The term $\sum_{f = 1}^{F} {γ^{'}}_{f} w_{f j}^{* *} + \sum_{k = 1}^{K} {u^{'}}_{k} v_{k j}^{*}$ employs different weight coefficients toys around the optimal results of the first stage. That is, the improved intermediate measures are denoted as $v_{k j}^{*}$ and $w_{f j}^{* *}$ seek to increase the efficiency of the second stage based on a common set of weights. These adjusted values by plugging the external input $z$ and final output $y$ are recorded in the model (6) with their related weight coefficient. These weights are restricted with the same lower bound $φ$ and the upper bound unity. The second stage efficiency score is calculated with the common set of weights as $E_{2}^{*} = \frac{\sum_{r = 1}^{s} η_{r}^{*} y_{r j}}{\sum_{f = 1}^{F} {γ^{'}}_{f}^{*} w_{f j}^{* *} + \sum_{k = 1}^{K} {u^{'}}_{k}^{*} v_{k j}^{*} + \sum_{t = 1}^{T} μ_{t}^{*} z_{t j}}$ . Note that the system is efficient if and only if the two component processes are efficient, overall efficiency is calculated as the weighted sum of outputs compared to the weighted sum of inputs. However, the intermediate measure takes both roles; outputs for the first stage and input for the second stage. To gain further insight into the practical implication of the proposed approaches, a real case consisting of 34 OECD countries in 2012 taken by Kiani Mavi et al. (2019) is analysed.

NUMERICAL EXAMPLE

The practicability of the proposed weight restriction methodology in a two-stage process is highlighted by a real data set consisting of 34 OECD countries in 2012. The data set is taken from Kiani Mavi et al. (2019). Table 1 reports the data set.

Table 1:

Data Set for OECD Countries.

DMU	Inputs			Intermediate Input/Output		Outputs
DMU	x ₁	x ₂	x ₃	z ₁	z ₂	y ₁	y ₂	y ₃	y ₄
Australia	12,241.40	130.38	7,682,300	156,395.10	531,325.6	3,408.385	4,565.211	2.10	13,090
Austria	4,429.89	88.97	82,531	42,824.84	80,150.24	4,763.259	18,412.394	2.80	8,368
Belgium	4,955.98	124.63	30,280	52,009.18	119,380.7	4,156.244	41,673.751	1.70	11,187
Canada	19,516.48	170.55	9,093,510	183,744.35	729,206.9	4,518.514	29,025.962	1.20	17,347
Chile	8,603.14	101.21	743,532	27,707.87	109,908.8	335.284	507.844	2.10	3,623
Czech Rep.	5,337.98	141.75	77,230	20,940.24	129,749.8	3,249.889	21,044.499	15.60	6,575
Denmark	2,901.68	72.54	42,256	33,892.71	56,087.33	7,088.553	9,226.707	4.80	15,976
Estonia	689.70	179.37	42,390	2,508.12	21,856.28	3,338.510	1,307.609	13.10	1,194.00
Finland	2,721.26	155.72	303,890	26,998.01	63,138.29	7,187.926	3,724.745	6	12,722
France	30,052.0	103.08	547,557	280,851.12	487,262.7	4,169.848	113,250.612	3.5	26,040
Germany	42,755.65	91.51	384,880	375,251.35	945,186	4,399.672	193,799.441	2	129,368
Greece	4,979.89	88.19	128,900	23,986.20	102,436.8	2,643.854	855.057	1.8	8,003
Hungary	4,388.16	99.92	90,530	13,468.05	57,400.82	2,522.846	14,470.677	8.1	2,575
Iceland	190.09	443.62	100,250	1,537.66	4,461.415	5,679.519	93.401	0.80	5,248
Ireland	2,184.31	61.50	68,890	23,927.12	57,922.48	3,605.941	21,914.723	3.5	5,027
Israel	3,696.04	94.86	21,640	29,331.48	79,338.09	8,255.404	9,634.614	2	579
Italy	25,474.18	76.40	294,140	213,049.13	440,470.2	1,943.470	29,711.963	11.4	59,238
Japan	65,559.50	100.46	364,560	490,886.28	1,406,855	5,201.3171	105,075.614	6.8	57,269
Korea Rep	26,119.54	160.73	97,466	130,560.50	696,523	6,456.626	130,460.428	7	4,494
Luxemburg	260.06	81.33	2,590	6,179.45	11,213.64	4,594.529	714.838	0.7	251
Mexico	54,475.98	95.69	1,943,950	126,198.17	623,880.1	241.802	45,418.667	0.5	12,098
Netherland	8,998.33	101.51	33,690	86,668	195,406.6	4,561.231	69,039.552	2.4	9,076
New Zealand	2,397.65	130.26	263,310	19,069.09	79,397.17	4,008.711	723.215	1.50	2,270
Norway	2,695.09	101.42	365,245	52,274.62	53,527.82	5,569.446	4,818.883	3	14,628
Poland	18,294.71	110.70	306,210	52,421.48	393,515.6	1,850.717	12,220.495	2.50	15,741
Portugal	5,397.24	80.73	91,600	22,607.35	64,325.35	3,615.146	1,694.368	3.4	1,505
Slovak Rep.	2,736.15	122.98	48,088	9,847.83	42,885.65	2,717.589	7,574.416	9.10	481
Slovenia	1,017.24	121.51	20,140	4,768.86	18,340.85	4,216.835	1,466.834	7.10	71,887
Spain	23,419.92	80.17	500,210	136,926.17	322,873.5	26,52.551	16,346.454	11.6	21,327
Sweden	5,118.45	118.16	407,340	57,874.20	55,537.4	6,670.028	17,096.659	10.5	2,264
Switzerland	4,700.91	60.17	39,516	68,483.50	52,523.47	4,481.074	53,294.078	6.8	2,264
Turkey	27,797.25	83.67	769,630	82,325.66	442,170.5	1,168.599	2,176.908	1.2	9,800
United Kingdom	32,772.20	79.81	241,930	271,950.95	568,763.8	4,185.689	69,223.897	6.7	48,971
United States	159,815.82	135.93	9147,420	1,669,151.7	6680,052	4,117.674	148,530.552	0.3	270,735

Source: Kiani Mavi et al. (2019).

Employing the proposed weight restriction approach for airport performance, the process is considered a two-stage structure. The first stage is called eco-efficiency, and the second stage is related to the eco-innovation process. The inputs and outputs are recorded as follows:

For the first stage, there are three inputs, which are characterized by the total labour force (x₁), energy consumption (x₂) and land area (x₃).The outputs of the first stage consist of both desirable and undesirable outputs. The only desirable intermediate measure is GDP (v₁). The undesirable intermediate factors are total green gas emissions (CO₂ equivalent emissions) (w₁).The second stage employs no external inputs. Since the undesirable output of the first stage did not leave the first stage, the second stage employs the whole outputs of the first stage without using any external inputs. The output of the second stage or the final outputs are characterized as the number of researchers in research and development (y₁), high technology export (y₂), ISO 14,001 certificates (y₃) and electricity production from renewable sources (y₄). Figure 2 depicts the process as a two-stage network structure.

Figure 2:

Two-stage Process of Eco-efficiency and Eco-innovation.

To assess the common set of weights, the first stage, or the eco-efficiency stage, is assumed as the leader. For handling intermediate undesirable outputs of the first stage, employing model (4), the smallest allowable amount of undesirable intermediate outputs is calculated. Applying the triple $(x_{j}, v_{j}, w_{j}^{*})$ in model (5), for the first stage (leader stage), efficiency scores with the optimal values for the target points are reported in Table 2. The common set of weights for stage 1 obtained with model (5) is recorded as $u_{1}^{*} = 0.01$ (the assigned weight for desirable output), $v_{1}^{*} = 0.17, v_{2}^{*} = 0.49, v_{3}^{*} = 0.01$ (the weights for inputs) and $γ_{1}^{*} = 0.01$ (the weight for undesirable output). The second column of Table 2 reports the efficiency score of the first stage labelled as $E_{1}^{*}$ . As Figure 2 shows, the intermediate undesirable factor has not left the first stage. Hence, model (6) takes the optimal values of both desirable intermediate measure $(v_{1}^{*})$ and undesirable intermediate measure $(w_{1}^{* *})$ into consideration. The results of model (6) for the second stage are summarized in the third column of Table 2. Employing model (6) with the optimal solution of leader stage one, the common set of weights is recorded as $η_{1}^{*} = η_{2}^{*} = η_{3}^{*} = η_{4}^{*} = 0.01$ (the weights assigned for final outputs). The assigned weights for intermediate desirable and undesirable inputs are drawn as ${u^{'}}_{1}^{*} = 0.01$ (the common weight assigned for desirable intermediate output) and ${γ^{'}}_{1}^{*} = 0.23$ (the common assigned to undesirable input). Employing the common set of weights obtained from model (6) for the second stage (follower stage), the efficiency score of the second stage (follower stage) is titled as $E_{2}^{*}$ .

Table 2:

Efficiency Score for the First Stage and Second Stage with Proposed Approach.

DMU	First Stage $(E_{1}^{*})$	Second Stage $(E_{2}^{*})$	Overall Efficiency
1	0.08	0.01	0.05
2	0.47	0.05	0.26
3	0.77	0.07	0.42
4	0.04	0.02	0.03
5	0.06	0.02	0.04
6	0.22	1	0.61
7	0.63	0.06	0.35
8	0.08	0.07	0.08
9	0.14	0.05	0.1
10	0.47	0.03	0.25
11	0.62	0.05	0.34
12	0.2	0.03	0.12
13	0.14	0.08	0.11
14	0.03	0.4	0.22
15	0.39	0.07	0.23
16	0.59	0.04	0.32
17	0.52	0.03	0.28
18	0.59	0.02	0.31
19	0.42	0.06	0.24
20	1	0.05	0.53
21	0.08	0.03	0.06
22	0.8	0.06	0.43
23	0.11	0.03	0.07
24	0.23	0.03	0.13
25	0.15	0.04	0.1
26	0.22	0.02	0.12
27	0.18	0.06	0.12
28	0.2	0.9	0.55
29	0.27	0.02	0.15
30	0.21	0.03	0.12
31	0.99	0.05	0.52
32	0.12	0.01	0.07
33	0.6	0.03	0.32
34	0.25	0.02	0.14
Average	0.35	0.11	0.23
Variance	0.08	0.06	0.03

As Table 2 shows, when the first process is leader, there is only one efficient unit, that is, unit #20. As model (5) admits, the common set of weights is positive in terms of inputs, intermediate desirable and adjusted undesirable output. Intermediate undesirable output has not left the process. Therefore, the second stage fed up with optimal solutions of the first stage (both intermediate desirable and undesirable measures). The second stage employs no external input and by using the optimal solutions of the first stage can produce final products. As the third column of Table 2 reports, keeping the efficiency of the first stage unchanged, in the second stage, there is one efficient unit, unit#6, in the second stage. The last column of Table 2 shows the overall efficiency of the whole system, while the intermediate measure has transformed. The overall efficiency can be calculated by employing the formula $\frac{E_{1}^{*} + E_{2}^{*}}{2}$ . Applying the average of the two stages in the case study, there is no overall efficient unit among these 34 countries. In the first stage, unit#20 was efficient; employing the optimal solution of the first stage, the second stage achieves the one efficient unit, unit#6. In a nutshell, there is no unit which is efficient in both stages. Referring to the last row of Table 2, the average of efficiencies in the first stage is greater than the average of efficiencies in the second stage and overall efficiency. Generally speaking, the first stage has the highest amount of variance and average, 0.35 and 0.08, compared with the second stage and overall efficiencies. To further investigate the proposed weight restriction method, the results are compared with the proposed method of Kiani Mavi et al. (2019). In the goal programming technique employed by the authors, the intermediate measures remain unchanged. Generally, the multi-objective linear programming with a centralized perspective is deemed as the common set of weights problem. The common set of weights achieved for the data set of Table 1 in Kiani Mavi et al. (2019) goal programming method is reported in Table 3.

Table 3:

The Generated Weights of Kiani Mavi et al. (2019) and the Proposed Approach.

	Items	Common WeightsKiani Mavi et al. (2019)	Common WeightsProposed approach
Inputs	Labour force (x₁)	0.002999	0.17
	Energy use (x₂)	0.5455	0.49
	Land area (x₃)	0.0001	0.01
Intermediate outputs (stage 1)	GDP (v₁)	0.0008194	0.01
Intermediate outputs (stage 1)	Green gas emission (w₁)	0.0001	0.01
Intermediate inputs (stage 2)	GDP (v₁)	0.003461	0.01
Intermediate inputs (stage 2)	Green gas emission (w₁)	0.0001	0.23
Outputs	Researchers in R&D (y₁)	0.0001	0.01
	High technology expert (y₂)	0.0001	0.01
	ISO 14000 certificates (y₃)	0.4511	0.01
	Electricity production from renewable sources (y₄).	0.0001	0.01

For more comparison, the last column of Table 3 represents the weights generated by the proposed weight restriction approach. Employing the third column of a common set of weights, the results of the goal programming method proposed by Kiani Mavi et al. (2019) for this case study are summarized in Table 4.

Table 4:

Efficiency Score for the First Stage and the Second Stage with Kiani Mavi et al. (2019).

DMU	First Stage $(E_{1}^{*})$	Second Stage $(E_{2}^{*})$	Overall Efficiency
1	0.1379	0.1038	0.1041
2	0.4494	0.0839	0.3451
3	0.4357	0.1023	0.3391
4	0.1328	0.1235	0.1006
5	0.1356	0.1320	0.1024
6	0.1505	0.3188	0.0955
7	0.4780	0.0938	0.3565
8	0.0193	1	0.0003
9	0.1704	0.1219	0.1193
10	0.9213	0.0665	0.8030
11	0.9999	0.0987	1
12	0.2281	0.1470	0.1799
13	0.1338	0.2436	0.0869
14	0.00050	0.3588	0.0023
15	0.3715	0.1259	0.2772
16	0.3296	0.1052	0.2557
17	0.9116	0.0790	0.8274
18	0.9388	0.0945	0.9868
19	0.4360	0.1925	0.4146
20	0.1088	0.0932	0.0757
21	0.2185	0.1587	0.1776
22	0.6747	0.0963	0.5698
23	0.1389	0.1412	0.1024
24	0.4069	0.0509	0.2988
25	0.2319	0.2396	0.2049
26	0.2444	0.1109	0.1843
27	0.0956	0.2779	0.0578
28	0.0534	0.7753	0.0051
29	0.5717	0.0877	0.4781
30	0.3761	0.0644	0.2685
31	1	0.0604	0.7531
32	0.2697	0.1617	0.2329
33	0.9998	0.0767	0.9422
34	0.6403	0.1230	0.6616
Average	0.39	0.18	0.34

As Table 4 shows, the Kiani Mavi et al. (2019) method illustrates one overall efficient unit, #unit11. In addition, in the first and second stages, unit#31 and #8 are recorded as efficient units respectively. To compare the results of these two different approaches, the number of efficient units in the first stage and second stage is equal. In both approaches, it was recorded as one efficient unit. But our proposed method with a weight restriction perspective reflects no overall efficient unit. Statistically speaking, the average of overall and divisional efficiencies in the latter approach is greater than the former approach. This is due to the difference in using various perspectives for a two-stage network structure. The employed non-cooperative approach preserves the first-stage efficiency unchanged. In the latter approach, the cooperative point of view is employed. Taking a look at the common weights generated by the approaches in Table 3, Kiani Mavi et al. (2019) generate the lowest weights in comparison with the proposed approach. Furthermore, for the second input and the third final output the weights with Kiani Mavi et al. (2019) are recorded as 0.5455 and 0.4511. In the proposed approach, the generated weights are depicted as 0.49 and 0.01. From the common weight results of Table 3, it seems that the proposed approach can be regarded as a more suitable one for this application. Since the generated weights are not close to zero as the counterpart approach acts. The proposed approach determines three positive weights with a high deviation of zero assigned to the first, and second input for the first stage and the undesirable input for the second stage, that is, 0.49, 0.17 and 0.23, respectively. The counterpart approach achieves two weights with a similar deviation of zero. The second input for the first stage with a weight assigned of 0.5455 and the third final output with a weight assigned of 0.4511. From the efficiency evaluation from the results of Tables 2 and 4, our approach generally does not generate higher overall and individual efficiencies for each unit. It can be seen that the proposed approach adopts the overall efficiency as the average of the two stages’ efficiency. Based on the above comparison, none of the units are efficient in employing the proposed approach of non-cooperative theory. That is, all the countries need to adjust for better performance. Moreover, our approach generates a pair of divisional efficiencies for each unit. Utilizing the proposed method for finding a common set of weights with a leader–follower perspective, this method can detect the pseudo-inefficiency in a two-stage process. That is, the inefficiency is spread over the system, and the existing methods in the literature ignore it and may overestimate the efficiency score. Hence, the proposed common set of weights may detect the pseudo-inefficiency and ensure that the existing envelopment models may overestimate the efficiency score

CONCLUSION

Aiming at determining a common set of weights, considerable attention has been paid to two-stage network analysis. The existing studies on a two-stage network structure just consider envelopment models of DEA. Furthermore, when the intermediate measure consists of desirable and undesirable, the existing approach did not employ multiplier models. The proposed approach generates a common set of weights which is more acceptable to all DMUs and more convincing to those making decisions in practical applications. This article introduced a joint weight restriction reduction for generating a common set of weights in a two-stage network structure. The contribution of this article is applying a non-cooperative game theory in determining a common set of weights in the presence of both desirable and undesirable intermediate measures. An illustrative example of 34 OECD countries revealed the applicability of the proposed method.

Footnotes

ACKNOWLEDGEMENT

The authors would like to thank Professor Alireza Amirteimoori and Professor Dimitrrios Sotrios for their motivation and support.

DECLARATION OF CONFLICTING INTERESTS

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

FUNDING

The authors received no financial support for the research, authorship and/or publication of this article.

NOTES

Azam Pourhabib Yekta is an Assistant Professor of applied mathematics and operation research at Islamic Azad University (Sowmesara Branch) in Iran. Her areas of research interest include performance measurement and management, efficiency and productivity analysis and multi-criteria decision-making, with special emphasis on the quantitative methods of performance measurement. She holds an MSc in applied mathematics and received her PhD in applied mathematics and operational research from the Islamic Azad University Lahijan Branch. Her research projects are focused on efficiency analysis and productivity evolution and measurement in manufacturing industries.

E-mail: apourhabibyekta@yahoo.com

Mahnaz Maghbouli is an Assistant Professor of applied mathematics and operation research at Islamic Azad University (Aras Branch) in Iran. The realm of her interest includes the board area of performance measurement emphasizing quantitative models and Math Education, especially those based on the broad set of methods known as data envelopment analysis (DEA). She holds an MSc in applied mathematics and received her PhD in applied mathematics and operational research from Islamic Azad University, Guilan Science and Research Branch. Her research projects are focused on the analysis of efficiencies and productivity evolution in manufacturing industries.

E-mail: mmaghbouli@gmail.com

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Weight Restriction Approach in a Two-stage Network Structure: A DEA-based Approach

Abstract

Executive Summary

Keywords

PRELIMINARIES

A NON-COOPERATIVE WEIGHT RESTRICTION APPROACH IN THE TWO-STAGE PROCESS

Two-stage Process of D M U j .

NUMERICAL EXAMPLE

Data Set for OECD Countries.

Two-stage Process of Eco-efficiency and Eco-innovation.

Efficiency Score for the First Stage and Second Stage with Proposed Approach.

The Generated Weights of Kiani Mavi et al. (2019) and the Proposed Approach.

Efficiency Score for the First Stage and the Second Stage with Kiani Mavi et al. (2019).

CONCLUSION

Footnotes

ACKNOWLEDGEMENT

DECLARATION OF CONFLICTING INTERESTS

FUNDING

NOTES

References

Two-stage Process of $D M U_{j}$ .