24.1 INTRODUCTION

To measure the level of “corporate sustainability” of organizations in the private sector, this chapter ¹ discusses how to unify natural and managerial disposability concepts under radial and non‐radial measurements. See Chapter 15 for a description of corporate sustainability. The proposed approach provides us four unified efficiency measures related to natural disposability and/or managerial disposability. These unified efficiency measures serve as a computational basis for identifying a new type of scale efficiency measures, none of which has been discussed in previous chapters, including Chapter 18.

After theoretically discussing the proposed scale efficiency measures, this chapter applies the proposed approach to examine the level of corporate sustainability among petroleum firms in the United States. The petroleum firms are generally classified into two categories. One of the two categories includes a group of “integrated companies,” which have a large supply chain for spanning both upstream and downstream functions in the US petroleum industry. The other category includes “independent companies” that focus on the upstream function in their business operations.

As an application of the proposed approach, this chapter examines whether the integrated companies outperform the independent companies, because a large supply chain incorporated into the former group provides them with a scale merit in their operations and gives a business opportunity to directly contact consumers² and identify their opinions on the operation of petroleum firms. Thus, it is easily imagined that a large supply chain can enhance the level of corporate sustainability in the US petroleum industry. Moreover, vertical integration between upstream and downstream may be a promising business trend for better performance in the petroleum industry. This chapter on US industry and Chapter 16 on world industry explore different aspects and structures regarding the petroleum business. See Chapter 13 for a description of the current trend on oil and gas businesses in the world.

The remainder of this chapter is organized as follows. Section 24.2 discusses underlying concepts for disposability unification that are incorporated in the proposed radial and non‐radial approaches. Section 24.3 describes non‐radial measurement under natural, managerial and natural and managerial disposability concepts. Section 24.4 describes radial measurement under the three types of disposability concepts. Section 24.5 discusses a computational flow for disposability unification. Section 24.6 applies the proposed non‐radial and radial approaches to evaluate the unified (environmental and operational) performance of US petroleum companies. Section 24.7 summarizes this chapter.

24.2 UNIFICATION BETWEEN DISPOSABILITY CONCEPTS

Disposability Unification: Returning to Stage III of Figure 15.6, Figure 24.1 visually describes two functional forms in the horizontal axis (x) and in the vertical axis (g and b). One of the two efficiency frontiers, the upper line, expresses the convex relationship between the input (x) and the desirable output (g). The other efficiency frontier indicates a similar (convex) relationship between the input (x) and an undesirable output (b).

The importance of Figure 15.6 is that it visually describes the three stages (Stages I, II and III) of disposability unification. Figure 24.1 duplicates only the last stage (Stage III) for the unification of natural and managerial disposability concepts. This chapter unifies these disposability concepts based upon Stage III of Figure 15.6.

As identified from the left‐hand side of Figure 24.1, the two efficiency frontiers have an increasing trend of the desirable output (g) and the undesirable output (b) along with an enhanced input (x). The rationale for the shape similarity is because the undesirable output is the “byproduct” of the desirable output. However, as shown on the right‐hand side of Figure 24.1, the two frontiers have different structures. That is, the efficiency frontier for an undesirable output increases and then decreases with input enhancement. As discussed in Chapter 21, such a case occurs under DC, or eco‐technology innovation, for pollution reduction. Thus, the final stage of disposability unification must attain the status of Stage III for the proposed unification, as depicted in both Figure 15.6 and Figure 24.1 because the unification between natural and managerial disposability concepts serves as part of the important process of DEA environmental assessment.

In discussing the proposed disposability unification, it is necessary for us to reorganize Figure 24.1 in the x and g & b space. For this purpose, Figure 24.2 visually specifies the necessary condition for the proposed unification. Figure 24.2 visually assumes that the two types of outputs (g and b) increase with increased input (x). The input change, not listed on the figure, increases the desirable output (g) but decreases the undesirable output (b) from a certain point after eco‐technology is introduced into the operation of an organization. To describe the situation more clearly, Figure 24.2 shows a supporting line that we can identify on the efficiency frontier between g and b. For example, the line (a–c) indicates no occurrence of DC. The line (d–e) shows weak DC that indicates the starting situation on DC. The line (f–h) indicates strong DC, corresponding to the occurrence of eco‐technology innovation on an undesirable output. See Chapter 23 for a detailed description of the three types of DC. In the case of multiple components of the three production factors, the occurrence of strong DC, or the negative slope of the supporting hyperplane, implies that an increase of the input vector leads to an increase in some components of the desirable output vector, but simultaneously decreases some components of an undesirable output vector without worsening the other components. In the disposability unification, it is necessary for us, hereafter, to incorporate such a possible occurrence of strong DC in the proposed formulations.

24.3 NON‐RADIAL APPROACH FOR DISPOSABILITY UNIFICATION

UENM under Variable RTS and DTS: Returning to Chapter 16 (non‐radial measurement), this chapter first utilizes a non‐radial model that incorporates the unification of natural and managerial disposability concepts (e.g., Goto et al., 2014). The proposed model is designed to measure the level of unified efficiency under natural and managerial disposability (UENM) of the k‐th DMU.

As mentioned in Chapter 16, the k‐th DMU needs to decrease some input(s) and increase some desirable outputs to improve its operational performance under natural disposability. In contrast, the DMU may increase the inputs to improve its environmental performance under managerial disposability. To satisfy the two conflicting requirements, the proposed non‐radial model separates inputs into two categories according to the two disposability concepts. Then, unification of the two disposability concepts uses the following non‐radial (NR) model:

(24.1)

The number of original m inputs are newly separated into (under natural disposability) and (under managerial disposability) in the model in Equation (24.1) in the manner that it maintains m = + . One of the two input categories uses inputs whose slacks ( for i = 1,…, ) are formulated under natural disposability. For example, the number of employees belongs to the input category. The minus in the number of inputs in the first category produces a better performance under natural disposability. The other category contains inputs whose slacks ( for q = 1,…, ) are formulated under managerial disposability. For example, the amount of capital investment for eco‐technology innovation belongs to the input category. Another example is R&D expenditure. The amount of capital investment is used to attain technology innovation that is important for enhancing the level of production activity and that of environmental protection. The more inputs in the second category produce the better performance under managerial disposability because of green technology development.

The level of UENM regarding the k‐th DMU under variable RTS and DTS is determined by

(24.2)

where all slack variables are determined on the optimality of Model (24.1). The equation within the parenthesis, obtained Model (24.1), indicates the level of unified inefficiency under the two disposability concepts. The degree of UENM is obtained by subtracting the level of inefficiency from unity as specified in Equation (24.2).

Model (24.1) has the following dual formulation:

(24.3)

Here, (i = 1,…, ), (q = 1,…, ), u_r (r = 1,…, s) and w_f (f = 1,…, h) are all dual variables related to the first, second, third and fourth groups of constraints in Model (24.1), respectively. The dual variable (σ), which is unrestricted (URS), is obtained from the last equation of Model (24.1).

An important feature of Model (24.1) is that it incorporates natural and managerial disposability in such a manner that inputs and two types of outputs are classified by the two disposability concepts. In the previous chapters, these disposability concepts are equally treated in the evaluation of UENM.

Problem of Model (24.3): The supporting hyperplane, obtained by Model (24.3), becomes in a simple case where three production factors have a single component. The simple case is for our descriptive convenience. This chapter considers that stands for an amount for eco‐technology investment, for example. The supporting hyperplane indicates where all the dual variables, except σ, are positive. The dual variable (σ) is unrestricted in sign. It is easily thought that the supporting hyperplane does not fit with our expectation. For instance, an increase in the input () decreases the undesirable output because of . The sign should be opposite. The following three conditions are necessary for us to make the supporting hyperplane fit with our expectation to apply DEA to performance analysis for energy and environment. First, if increases or decreases, then b increases or decreases because is the input under natural disposability. Second, if increases or decreases, then b decreases or increases because indicates the input under managerial disposability (e.g., eco‐technology investment). Thus, the two expectations clearly indicate that the unknown parameter () should be positive and the other () should be negative in their signs, respectively. Finally, both u and σ should be unrestricted in these signs to incorporate the possible occurrence of DC, as depicted in Figure 24.2, where the slope of the supporting hyperplane is determined by u and the intercept is determined by σ.  Consequently, the supporting hyperplane should be in the simple case.

UENM(DC): To reorganize Model (24.3) in such a manner that it can fit with Figure 24.2, Sueyoshi and Goto (2014b) proposed the following non‐radial model in which the k‐th DMU can simultaneously attain the status of natural and managerial disposability along with the possible occurrence of desirable congestion (DC):

(24.4)

where v_i (i = 1,…, ), v_q (q = 1,…, ), u_r (r = 1,…, s) and w_f (f = 1,…, h) are parameters to express the direction of the supporting hyperplane measured by Model (24.4). The dual variable (σ), which is unrestricted (URS), indicates the intercept of the expected supporting hyperplane.

Model (24.4), which is a dual model, has the following primal (dual’s dual) formulation that unifies the two disposability concepts:

(24.5)

Model (24.5) has two unique features in the structure. One of the two features is that all the inputs are classified into two categories, as mentioned previously. The first category includes a group of inputs whose slacks ( for i = 1,…, ) are formulated under natural disposability. The other category contains a group of inputs whose slacks ( for q = 1,…, ) are formulated under managerial disposability. The other unique feature is that the constraints regarding a desirable output vector are expressed by equality because they do not have any slacks. In contrast, the other groups of constraints are expressed by inequality because they have slacks in Model (24.5). The existence of slacks indicates the status of inequality.

The unified efficiency under natural and managerial disposability, or , of the k‐th DMU is measured by

(24.6)

where all slack variables are determined on the optimality of Model (24.5). The equation within the parenthesis, obtained from the optimality of Model (24.5), indicates the level of unified inefficiency under natural and managerial disposability. The level of unified efficiency is obtained by subtracting the level of inefficiency from unity under the occurrence of DC, or eco‐technology innovation, on undesirable outputs. It is important to note that Models (24.4) and (24.5) under non‐radial measurement, corresponding to Stage III of Figure 15.6.

UENM(DC) under Constant RTS and DTS: Model (24.5) may drop from the formulation as follows:

(24.7)

Model (24.7) drops from the formulation. Then, Equation (24.6) measures the level of under constant RTS and DTS as follows:

(24.8)

where all slacks are determined on the optimality of Model (24.7). The subscript c stands for constant RTS and DTS, so not the subscript v in Equation (24.8).

Scale Efficiency under Natural and Managerial Disposability: After obtaining the two types of UENM(DC) for the k‐th DMU, this chapter uses

(24.9)

to determine the level of scale efficiency in the case of non‐radial models. The scale efficiency indicates the level at which each DMU effectively operates by considering the operational size. Thus, the degree of scale efficiency can be considered as a level of efficiency for size utilization under the possible occurrence of eco‐technology innovation.

At the end of this section, it is important to note that the magnitude control factor (MCF: ε) is usually incorporated into the proposed non‐radial models. See Chapter 16 which discusses the use of MCF in the non‐radial approach. Since Chapter 16 has not used the MCF in its empirical study on petroleum firms, this chapter does not incorporate it in the proposed formulations, in order to maintain empirical consistency between Chapter 16 and this chapter.

24.4 RADIAL APPROACH FOR DISPOSABILITY UNIFICATION

UENM under variable RTS and DTS: Since Chapter 17 has already discussed Stage I of Figure 15.6, this chapter starts with Stage II for radial measurement. First, shifting our description from a non‐radial (Section 24.3) to a radial approach, this chapter follows the previous research of Sueyoshi and Goto (e.g., Sueyoshi and Goto 2012b, 2014a) that proposed the following radial (R) model that corresponds to Model (24.1):

(24.10)

Model (24.10) combines Models (17.5) and (17.10) by a straightforward unification. The level of regarding the k‐th DMU under variable RTS and DTS is determined by 

 (24.11)

where all slacks are determined on the optimality of Model (24.10). The equation within the parentheses, obtained from the optimality of Model (24.10), indicates the level of unified inefficiency. The UENM is obtained by subtracting the level of inefficiency from unity.

The dual formulation of Model (24.10) becomes as follows:

(24.12)

As discussed for UENM measured by the non‐radial approach, Models (24.10) and (24.12) have a difficulty in measuring technology innovation, as well. A description on the rationale in Section 24.3 is applicable to Models (24.10) and (24.12).

UENM(DC): To discuss the structure of the supporting hyperplane, as depicted in Figure 24.2, this chapter reorganizes Model (24.12) whose hyperplane becomes in the case where three production factors have a single component. The corresponding formulation of Model (24.12), satisfying the requirement on the case of multiple components, becomes as follows:

(24.13)

Along with the slight modification, Model (24.13) drops from the second constraint of Model (24.12). The rationale is that may be infeasible because u_r is unrestricted in the sign so that the dual variable cannot become negative, or the equation becomes infeasible in the worst case. To avoid such a computational difficulty, this chapter reorganizes Model (24.12) with a structure like Model (24.13).

The primal (dual’s dual) formulation of Model (24.13) becomes as follows:

(24.14)

Model (24.14) corresponds to Model (24.5) in the non‐radial measurement. The level of regarding the k‐th DMU under variable RTS and DTS is determined by

(24.15)

where the inefficiency score and all slack variables are determined by the optimality of Model (24.14). The equation within the parentheses, obtained from Model (24.14), indicates the level of unified inefficiency. The degree of is obtained by subtracting the level of inefficiency from unity. Here, the magnitude of the inefficiency score (ξ*) is determined by desirable outputs in Model (24.14) because these outputs are byproducts of desirable outputs. The influence of undesirable outputs exists as slacks on undesirable outputs , as specified in Equation (24.15). It is important to note that Models (24.13) and (24.14) under radial measurement correspond to Stage III of Figure 15.6.

Meanwhile, under constant RTS and DTS, the level of regarding the k‐th DMU is determined by

(24.16)

The degree of on the k‐th DMU under constant RTS and DTS is determined by

(24.17)

All the slack variables used in Equation (24.17) are determined by the optimality of Model (24.16).

Scale Efficiency under Natural and Managerial Disposability: After obtaining the two types of UENM(DC), this chapter measures

(24.18)

as the level of scale efficiency in the case of radial measurement.

24.5 COMPUTATIONAL FLOW FOR DISPOSABILITY UNIFICATION

Figure 24.3 visually describes the computational flow for disposability unification and efficiency measurement. Although this chapter uses non‐radial models for our empirical study, this description can be applied to the radial framework.

As depicted in Figure 24.3, the proposed approach measures the level of unified efficiency (UE). See Chapters 16 and 17 on UE measurement. As discussed in the two chapters, we measure UE under natural disposability (UEN) and UE under managerial disposability (UEM). After measuring UE, UEN and UEM, the proposed approach combines two disposability concepts for a unified efficiency measure (UENM). By unifying them, this chapter considers two conditions related to the occurrence of congestion. One of the two conditions is that it is possible for us to measure UENM under no occurrence of undesirable congestion (UC). The other condition is that we can consider the possible occurrence of desirable congestion (DC), or eco‐technology innovation on undesirable outputs, to measure the level of UENM(DC). The rationale on the first condition is that this chapter is interested in measuring the level of scale efficiency (SENM) on US petroleum companies. In this case, it is necessary for us to assume that no UC occurs in the transportation system from upstream to downstream. As a result, this chapter can determine the degree of SENM(DC). Meanwhile, the rationale on the second condition is because this chapter is interested in the performance assessment of US petroleum companies under a possible occurrence of eco‐technology innovation and its related managerial challenge.

Figure 24.4 visually describes the computation of SENM(DC) in the horizontal axis (x: input) and the vertical axis (g: desirable output; b: undesirable output). The efficiency frontier for the desirable output is a piece‐wise linear contour line ({A}–{B}–{C}–{D}–{E}) and that of the undesirable output is the contour line ({F}–{G}–{H}–{I}–{J}–{K}). The upper contour line has an increasing shape because it is structured by no occurrence of UC, while the other (lower) one has an increasing and decreasing shape because it is structured by the possible occurrence of DC, or eco‐technology innovation. Based upon the type of RTS and DTS, this chapter classifies the efficiency frontier for the desirable output into two cases. One of the two cases is the contour line ({A}–{B}–{C}–{D}–{E}) under variable RTS. The other case is the straight line from the origin (O) to L that is structured by constant RTS.

Assuming that the undesirable output is a byproduct of the desirable output, this chapter measures the magnitude of UENM(DC)_v on the observed DMU{P} by dividing the distance between {P} and Q by the distance between N and Q under variable RTS. Meanwhile, the magnitude of UENM(DC)_c is measured by the distance between {P} and Q divided by that between L and Q under constant RTS. Consequently, the degree of SENM(DC) is measured by

The degree indicates the ratio between distance N to Q divided by that between L to Q. The assumption on “byproduct” makes it possible to measure the degree of SENM(DC) based upon the magnitude of desirable outputs. See Models (24.14) and (24.16).

24.6 US PETROLEUM INDUSTRY

24.7 SUMMARY

To measure the level of corporate sustainability regarding energy companies, this chapter discussed how to unify natural and managerial disposability by radial and non‐radial measurements. The unified efficiency measures included , , and as well as two scale efficiency measures: SENM(DC)^NR and SENM(DC)^R.

As an application, this chapter used the proposed non‐radial approach to examine the corporate sustainability of petroleum firms in the United States. Integrated companies have large supply chains for spanning both upstream and downstream functions. In contrast, independent companies have only the upstream function in their business operations. This empirical study found that the integrated companies outperformed the independent companies, because the large supply chain incorporated into the former group provided them with a scale merit in their operations and gave them a business opportunity to directly contact consumers. Consequently, this large supply chain can enhance the level of corporate sustainability in the US petroleum industry. Moreover, vertical integration in upstream and downstream may be a promising business trend toward better environmental performance in the petroleum industry.