Planning for potential increases in disbursements and risk of managed funds conditional on desired short-term performance levels

ABSTRACT

As increasing disbursements (such as costs and fees) and risk (such as portfolio risk and leverage risk) may affect managed fund performance, analysts go beyond risk-adjusted return measures for performance appraisal. A methodology that assesses performance in a multidimensional framework is data envelopment analysis (DEA). A variant of DEA is inverse DEA. In this paper, inverse DEA is applied to determine output (investment income and benefit payments) targets for a given fund to perform at a desired efficiency level when increase in disbursements and risk at known levels is envisaged. An output-oriented inverse DEA model assuming variable returns to scale is formulated with theoretical underpinning. The proposed modelling framework ensures that, when an input augmented fund with estimated output targets is included in the observed fund set, the frontier of best performance established with the observed fund set does not change. The inverse DEA model is applied to a sample of Australian superannuation funds to demonstrate how a given fund may obtain pathways to improve its performance under different input-augmentation scenarios. As different pathways suggest feasible output targets, fund managers may find them valuable in forward planning.

KEYWORDS:

• Performance appraisal
• two-stage production process
• managed funds
• inverse data envelopment analysis
• scenario analysis

JEL CLASSIFICATION:

• C44
• C67
• G23

I. Introduction

A managed fund is an investment scheme where money of investors is pooled and invested across a range of asset classes such as shares, bonds, property and infrastructure assets with the aim of providing them a return sometime in the future. Managed funds are popular due to easiness in investing in them and accessibility to professionally managed portfolios. Investors, especially those with limited funds, view managed funds as an opportunity to invest in different asset classes and market sectors that they may not be able to access on their own. On the other hand, fund management is not an easy undertaking. When a managed fund performs poorly, its investors may withdraw all or part of their shares. Such action may trigger adverse outcomes. Excessive redemption could even lead to closure of managed funds.¹ Increase in redemption may lead to increase in costs as well. That in turn may affect returns negatively. Poor fund management practices may make it worse. Managing investment risk is another challenge. Ma, Tang, and Gomez (2019) report that high percentage of fund managers have their pay directly linked to fund performance. Hence, impact of potential increase in disbursements (costs and expenses) and portfolio risk on fund performance are issues that may concern fund managers. We investigate these issues by considering fund management as a production process where multiple inputs are transformed into multiple outputs. A methodology that assesses performance of managed entities (in our case managed funds) conceptualized as a production process is data envelopment analysis (DEA).

The research question we address here is how much increase in outputs of a given fund is required to maintain its performance at the current level or at a level higher than that when its inputs increase to known levels. DEA-type models that answer such questions are referred to as inverse DEA models. Numerous variants of inverse DEA models are developed for the single-stage production process. Recent DEA studies conceptualize overall fund management process as a network of sub-processes. See Basso and Funari (2016) for a survey of studies that conceptualize fund management as a single-stage process and Fukuyama and Galagedera (2021) for a list of studies that conceptualize fund management as a network process.

Kazemi and Galagedera (2022) address the aforementioned research question assuming fund management as a serially linked two-stage production process operating under constant returns to scale (CRS) technology. In serially linked production processes the subprocesses are linked through intermediate measures. Assuming a two-stage network structure, Kazemi and Galagedera (2022) develop an inverse DEA model to determine intermediate and output targets for an input augmented entity to maintain its performance at a pre-specified level. Empirical studies of managed fund performance appraisal using DEA generally assume variable returns to scale (VRS). In this paper, we contribute to inverse DEA literature in general and managed fund performance appraisal literature in particular by extending Kazemi and Galagedera (2022) CRS formulation to the VRS case with theoretical support.

We make several contributions. First, we formulate an inverse DEA model to analyse input pressure on output conditional on pre-specified relative performance levels for a general two-stage production process operating under VRS technology. We demonstrate application of the inverse DEA model using a sample of 69 Australian superannuation funds (SFs). SFs are a type of managed funds. We outline a procedure to obtain a set of output and intermediate targets for an input augmented SF when a desired performance level is pre-specified. This way we obtain input, intermediate and output measures for a new (hypothetical) fund that performs at a pre-specified relative efficiency level. Second, under our modelling framework, when the hypothetical fund is added to the observed fund set, it does not change the frontier of best performance established by the observed fund set. This we highlight as a methodological advancement as the hypothetical fund is feasible with respect to known levels of attainment. Third, our procedure enables fund managers to avoid risk of under preparedness. For example, when portfolio management expenses (a typical input) increases suddenly (a possibility when portfolio management is outsourced), the fund manager may not be in a position to react quickly if there is no prior knowledge of implication of such a change on fund performance. By adopting the proposed procedure, a fund manager may prepare for such eventuality through scenario analysis akin to sensitivity analysis in business analytics.

The rest of the paper is organized as follows. Section II includes a brief review of managed fund performance appraisal and inverse DEA literature. Section III details formulation of an inverse DEA model with theoretical underpinning considering the associated conventional DEA model as its foundation. The results obtained in the empirical application are discussed in section IV. After robustness check of results in section V, section VI concludes the paper with some remarks.

II. Literature review

Managed fund performance appraisal

Empirical evidence suggests that capital flows to managed funds are associated with performance. Therefore, managed fund performance appraisal receives considerable attention in the literature. Investors generally rely on simple measures of fund performance. Cremers, Fulkerson, and Riley (2022) cite numerous empirical studies to support this view. Fund managers on the other hand tend to compare their fund performance with a benchmark stated in the prospectus. This is not reasonable always as there may be a discrepancy between the prospectus-stated benchmark and the fund’s actual investment strategy. Cremers, Fulkerson, and Riley (2022) reveal that benchmark discrepancies are common in managed fund industry and when discrepancy exists, prospectus benchmark typically understates risk. Traditional measures of fund management performance consider risk and return as factors of performance. Empirical studies however highlight that, for comprehensive assessment of managed fund performance factors other than return and risk may be required. Two other factors deemed important in fund management performance appraisal are size and cost. For example, in a comparative study of common mutual funds and Islamic mutual funds in Malaysia, Mansor, Bhatti, and Ariff (2015) find that there is no difference in performance before fees and fees reduces market-reported gross returns substantially. They report reduction in returns after fees is more pronounced in the case of Islamic equity mutual funds. Mansor et al. (2020) report a similar finding. Al Rahahleh and Bhatti (2022) concur in the case of Shariah-compliant mutual funds in Saudi Arabia. Recognising that many factors may affect managed fund performance, researchers opt for methods that assess performance in a multi-dimensional framework. A method that fits into this category is DEA.

In DEA, performance of a given fund is assessed compared to that of the other funds with respect to a frontier of best performance established by known levels of attainment. Hence, DEA efficiency scores are relative. When performance is assessed on a scale between 0 and 1 inclusive, efficient funds receive the score 1 and inefficient funds receive scores less than 1. Because DEA assesses performance without reference to a theoretically established benchmark, the concern of comparing performance to an inaccurate benchmark highlighted in Cremers, Fulkerson, and Riley (2022) does not arise in DEA application.

Murthi, Choi, and Desai (1997) are one of the earliest applications of DEA for managed fund performance appraisal. They assess performance of 731 mutual funds considering expense ratio, standard deviation, total load and turnover as inputs and annual return as output. Studies that use DEA for managed fund performance appraisal can be categorized broadly under two types based on how they conceptualize fund management operation in production economics sense. Some studies consider fund management operation as a single-stage production process. Others consider fund management operation as a network process. In the single-stage case, fund management is viewed as a process where inputs are transformed into outputs with no knowledge of the transformation process. Later studies conceptualize fund management process as a network process comprising of multiple stages. As a result, they are able to decompose DEA-assessed overall performance into individual stage-level performance with each stage representing a different management perspective. For example, Premachandra et al. (2012) consider mutual fund management process as a serially linked two-stage process with operational management as the first stage and portfolio management as the second stage. Recent DEA studies that consider fund management function as a network process include Sánchez-González, Sarto, and Vicente (2017), Galagedera et al. (2018) and Galagedera (2018, 2020). See Kao (2017) for a comprehensive review of development of network DEA and its empirical applications.

All DEA studies cited in this sub-section use conventional models. In conventional DEA, the aim is to assess relative performance with input and output measures of a given set of entities, referred to as decision making units (DMUs). The objective is to determine for a DMU a set of output targets conditional on inputs or determine a set of input targets conditional on outputs to achieve relative efficiency.

Inverse DEA

Another strand of literature is on sensitivity of DEA results to variation in input and/or output. Zhang and Cui (1999) develop a model to determine input targets required to achieve a prespecified performance level when outputs are fixed. They present their model as a reverse version of a conventional DEA model. Wei, Zhang, and Zhang (2000) develop two DEA-type models linking change in input/output measures to current level of performance. Wei, Zhang, and Zhang (2000) formulate these problems as multi-objective optimization problems (MOOPs) and refer to them as inverse DEA models. Yan, Wei, and Hao (2002) incorporate input/output preferences in inverse DEA. Jahanshahloo et al. (2002) adopt an inverse DEA model to estimate output when inputs change. Jahanshahloo et al. (2004) estimate inputs using an inverse DEA model when outputs and performance level increase. Hadi-Vencheh and Forougi (2006) accommodate simultaneous increase in some inputs (outputs) and decrease in some other inputs (outputs) under an inverse DEA framework. Ghobadi (2017) proposes an inverse DEA model to estimate input and output levels simultaneously to maintain a given level of performance. More variants of inverse DEA models are discussed in Amin and Al-Muharrami (2018) and Emrouznejad, Yang, and Amin (2019). For reviews of inverse DEA literature, see Ghobadi and Jahangiri (2015), Zhang and Cui (2016) and Emrouznejad, Yang, and Amin (2019).

Recent applications of inverse DEA include An et al. (2019) and Amin and Boamah (2021) in the banking sector, Wegener and Amin (2019) in the oil and gas industry,Emrouznejad, Yang, and Amin (2019) in the manufacturing sector and Kamyab, Mozaffari, and Gerami (2020) in the education sector. Lim (2020) proposes an inverse optimization model accounting for frontier changes in conjunction with environmental factors. We apply inverse DEA to find answers to the research question; by how much should the outputs of a managed fund increase to perform at the current efficiency level or at a level higher than that when inputs increase. In DEA-based fund performance appraisal studies, disbursements and risk are typical inputs. Fund managers have to deal with such situations when economic environment deteriorates and financial markets become unstable. Fund flows tend to affect disbursements and market instability may affect fund portfolio risk. In our empirical investigation, we discuss how inverse DEA may facilitate fund managers in making informed decisions when disbursements and risk changes in the short-term.

The next section discusses development of an output-oriented inverse DEA model for the serially linked two-stage production process depicted in Figure 1.

Figure 1. Two-stage production process.

III. Model development

Modelling two-stage production process for performance appraisal

Consider the two-stage production process shown in Figure 1.

Suppose the number of DMUs assessed is n and at stage A, each DMU transforms $i_A$ inputs, ${\mathit{X}}^A\in {\mathrm{\Re }}_{+}^{i_A}$ to generate $r_A$ independent outputs, ${\mathit{Y}}^A\in {\mathrm{\Re }}_{+}^{r_A}$ and D outputs, ${\mathit{Z}}^{\mathit{AB}}\in {\mathrm{\Re }}_{+}^D$ that in turn serve as inputs at stage B. Because ${\mathit{Z}}^{\mathit{AB}}$ is inherent to the two-stage system serving a dual purpose (as output at stage A and as input at stage B), $Z^{\mathit{AB}}$ are referred to as intermediate measures. At stage B, each DMU transforms $i_B$ independent inputs, ${\mathit{X}}^B\in {\mathrm{\Re }}_{+}^{i_B}$ and ${\mathit{Z}}^{\mathit{AB}}\in {\mathrm{\Re }}_{+}^D$ to generate $r_B$ outputs, ${\mathit{Y}}^B\in {\mathrm{\Re }}_{+}^{r_B}.$Let ${\mathit{X}}_j^A=\left(x_{1j}^A,x_{2j}^A,\dots ,x_{i_A\mathit{j}}^{\mathit{A}}\right)$ denote observed inputs of $\mathrm{DM}{\mathrm{U}}_j$ at stage A, ${\mathit{X}}_j^B=\left(x_{1j}^B,x_{2j}^B,\dots ,x_{i_B\mathit{j}}^{\mathit{B}}\right)$ denote observed inputs of $\mathrm{DM}{\mathrm{U}}_j$ at stage B, ${\mathit{Z}}_j^{AB}=\left(z_{1j}^{AB},z_{2j}^{AB},\dots ,z_{Dj}^{AB}\right)$ denote intermediate measures that links stage A and stage B of $\mathrm{DM}{\mathrm{U}}_j$, ${\mathit{Y}}_j^A=\left(y_{1j}^A,y_{2j}^A,\dots ,y_{r_A\mathit{j}}^{\mathit{A}}\right)$ denote observed outputs of $\mathrm{DM}{\mathrm{U}}_j$ at stage A and ${\mathit{Y}}_j^B=\left(y_{1j}^B,y_{2j}^B,\dots ,y_{r_B\mathit{j}}^{\mathit{B}}\right)$ denote observed outputs of $\mathrm{DM}{\mathrm{U}}_j$ at stage B, and ${\lambda }^A=\left({\lambda }_1^A,{\lambda }_2^A,\dots ,{\lambda }_n^A\right)$ and ${\mathit{\lambda }}^B=\left({\lambda }_1^B,{\lambda }_2^B,\dots ,{\lambda }_n^B\right)$ denote two vectors of intensity variables.

A non-parametric production technology of the two-stage process under the variable returns to scale assumption is:

(1) $\mathcal{T}=\left\{\left({\mathit{X}}^A,{\mathit{X}}^B,{\mathit{Z}}^{\mathit{AB}},{\mathit{Y}}^A,{\mathit{Y}}^B\right)\left|\begin{array}{c}{\sum }_{j=1}^n{\lambda }_j^A{\mathit{X}}_j^A\le {\mathit{X}}^A;{\sum }_{j=1}^n{\lambda }_j^B{\mathit{X}}_j^B\le {\mathit{X}}^B;\\ {\sum }_{j=1}^n{\lambda }_j^A{\mathit{Y}}_j^A\ge {\mathit{Y}}^A;{\sum }_{j=1}^n{\lambda }_j^B{\mathit{Y}}_j^B\ge {\mathit{Y}}^B;\\ {\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge {\mathit{Z}}^{\mathit{AB}};{\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\le {\mathit{Z}}^{\mathit{AB}};\\ {\sum }_{j=1}^n{\lambda }_j^A=1;{\sum }_{j=1}^n{\lambda }_j^B=1;{\mathit{\lambda }}^A\ge \mathbf{0};{\mathit{\lambda }}^B\ge \mathbf{0}\end{array}\right.\right\}$(1)

Our aim is to formulate a DEA-type model to determine a set of intermediate and output targets for a given DMU so that it maintains its overall efficiency at the current level or at a level higher than that after its inputs increase at known levels. Since we seek to determine outputs with given levels of inputs under the VRS assumption, the model we formulate is VRS output-oriented. Kazemi and Galagedera (2022) address a research question similar to ours under the CRS assumption. They formulate a CRS output-oriented multiplier model first assuming overall efficiency as a weighted sum of stage-level efficiencies with value of output at each stage as a proportion of total value of output as weights. Thereafter, they formulate a model that solves the research question structured on the dual of their multiplier model. Following Kazemi and Galagedera (2022), we formulate a DEA-type model to solve the research question under the VRS case considering the dual model of the VRS multiplier version as the foundation.² The dual model of the VRS multiplier version that assesses output-oriented relative efficiency of DMU₀ for the case depicted in Figure 1 is

(2) ${\theta }_0^{\ast }=\mathit{max}\left\{{\theta }_0\left|\begin{array}{c}{\sum }_{j=1}^n{\lambda }_j^A{\mathit{X}}_j^A\le {\mathit{X}}_0^A;{\sum }_{j=1}^n{\lambda }_j^B{\mathit{X}}_j^B\le {\mathit{X}}_0^B;\\ {\sum }_{j=1}^n{\lambda }_j^A{\mathit{Y}}_j^A\ge {\theta }_0{\mathit{Y}}_0^A;{\sum }_{j=1}^n{\lambda }_j^B{\mathit{Y}}_j^B\ge {\theta }_0{\mathit{Y}}_0^B;\\ {\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}-{\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\ge \left({\theta }_0-1\right){\mathit{Z}}_0^{AB};\\ {\sum }_{j=1}^n{\lambda }_j^A=1;{\sum }_{j=1}^n{\mathit{\lambda }}_j^B=1;{\mathit{\lambda }}^A\ge \mathbf{0};{\mathit{\lambda }}^B\ge \mathbf{0};{\theta }_0\mathrm{is}\mathrm{free}\end{array}\right.\right\}$(2)

Optimal value of model (2), ${\theta }_0^{\ast }$ is $\ge 1$ and by assumption, ${\mathit{Z}}_0^{AB}>0$. Therefore, $\left({\theta }_0^{\ast }-1\right){\mathit{Z}}_0^{AB}\ge \mathbf{0}$. Then, from the third constraint of model (2) we have ${\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge {\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}$ implying intermediate inputs consumed at stage B may not exceed intermediate outputs produced at stage A. This is the common assumption of studies that formulate envelopment models for production processes with network representation. Output-oriented VRS relative efficiency of DMU₀ is given by 1/${\theta }_0^{\ast }$. When ${\theta }_0^{\ast }$ is $>1$, DMU₀ is deemed inefficient.

Studies that assess overall relative efficiency of DMU₀ with the two-stage process shown in Figure 1, generally do so with model (2) without its third constraint and including${\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge {\mathit{Z}}_0^{AB}$ and ${\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\le {\mathit{Z}}_0^{AB}$ instead. That is consistent with production technology defined in (1). However, Chen, Cook, and Zhu (2010) point out that, when DMU₀ is inefficient, information for its frontier projection may not be readily obtained that way. They propose replacing ${\mathit{Z}}_0^{AB}$ with ${\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ which is unrestricted. We take a different approach. We add ${\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ and ${\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\le {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ to model (2) while retaining ${\displaystyle {\sum }_{j=1}^n{\lambda }_j^A}{Z}_j^{AB}-{\displaystyle {\sum }_{j=1}^n{\lambda }_j^B}\mathit{Z}\mathit{jAB}\ge \left({\theta }_0-1\right)Z_0^{AB}$. We shall see later that including ${\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ and ${\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\le {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ along with $\sum _{\mathit{j}=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}-{\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge \left({\theta }_0-1\right){\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ in model (2) has practical advantages. Besides, when model (2) is solved replacing ${\mathit{Z}}_0^{AB}$ with unrestricted ${\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ without including ${\displaystyle {\sum }_{j=1}^n{\lambda }_j^A}{Z}_j^{AB}\ge \mathit{Z}0\mathit{AB}$ and ${\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\le {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$, optimal value of ${\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ may be zero.

Modelling input increase and desired performance under a unified framework

Suppose stage A inputs of inefficient $\mathrm{DM}{\mathrm{U}}_0$ increase from ${\mathit{X}}_0^A$ to ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$ where $\mathbf{\Delta }{\mathit{X}}_0^A\ge 0$ and its stage B inputs increase from ${\mathit{X}}_0^B$ to ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$ where $\mathbf{\Delta }{\mathit{X}}_0^B\ge 0$ and at least one element of $\mathbf{\Delta }{\mathit{X}}_0^A$ and $\mathbf{\Delta }{\mathit{X}}_0^B$ is $\ne 0$. Let ${\mathit{\beta }}_0^A={\beta }_{10}^A,{\beta }_{20}^A,\dots ,{\beta }_{r_{A}0}^{\mathit{A}}$ and ${\mathit{\beta }}_0^B={\beta }_{10}^B,{\beta }_{20}^B,\dots ,{\beta }_{r_{B}0}^{\mathit{B}}$ denote a set of output targets and ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}}={\tilde{\mathrm{z}}}_{10}^{\mathit{AB}},{\tilde{z}}_{20}^{\mathit{AB}},\dots ,{\tilde{\mathrm{z}}}_{\mathit{D}0}^{\mathit{AB}}$ denote a set of intermediate targets of $\mathrm{DM}{\mathrm{U}}_0$. A DEA-type model that estimates ${\mathit{\beta }}_0^A$, ${\mathit{\beta }}_0^B$ and ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$to maintain a pre-specified relative efficiency level, $1/{\theta }_0^{\prime }\left(\ge 1/{\theta }_0^{\ast }\right)$ with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$ and ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$ is

(3) ${\beta }_0^{A*},{\beta }_0^{B*}=\mathit{max}{\beta }_0^A,{\beta }_0^B{\displaystyle {\sum }_{j=1}^n{\delta }_j^A}{X}_j^A\le \left(X_0^A+\mathit{\Delta }{X}_0^A\right);{\displaystyle {\sum }_{j=1}^n{\delta }_j^B}{X}_j^B\le \left(X_0^B+\mathit{\Delta }{X}_0^B\right);{\displaystyle {\sum }_{j=1}^n{\delta }_j^A}{Y}_j^A\ge \mathit{\theta }0{\beta }_0^A;{\displaystyle {\sum }_{j=1}^n{\delta }_j^B}{Y}_j^B\ge \mathit{\theta }0{\beta }_0^B;{\displaystyle {\sum }_{j=1}^n{\delta }_j^A}{Z}_j^{AB}-{\displaystyle {\sum }_{j=1}^n{\delta }_j^B}{Z}_j^{AB}\ge \mathit{\theta }0-1{\tilde{Z}}_0^{\mathit{AB}};{\displaystyle {\sum }_{j=1}^n{\delta }_j^A}{Z}_j^{AB}\ge {\tilde{Z}}_0^{\mathit{AB}};{\displaystyle {\sum }_{j=1}^n{\delta }_j^B}{Z}_j^{AB}\le {\tilde{Z}}_0^{\mathit{AB}};{\beta }_0^A\ge Y_0^A;{\beta }_0^B\ge Y_0^B{\displaystyle {\sum }_{j=1}^n{\delta }_j^A}=1;{\displaystyle {\sum }_{j=1}^n{\delta }_j^B}=1;{\delta }^A\ge 0;{\delta }^B\ge 0;{\beta }_0^A,{\beta }_0^B,{\tilde{Z}}_0^{\mathit{AB}}\text{\hspace{0.17em}are\hspace{0.17em}free}$(3)

Model (3) is built with model (2) constraint set as the foundation. Modifications include (i) replacing ${\mathit{Z}}_0^{AB}$ with ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$ which is unrestricted (ii) adding ${\mathit{\beta }}_0^A\ge {\mathit{Y}}_0^A;{\mathit{\beta }}_0^B\ge {\mathit{Y}}_0^B$ to ensure that the required outputs ${\mathit{\beta }}_0^A$ and ${\mathit{\beta }}_0^B$ are at least as much as the observed outputs of $\mathrm{DM}{\mathrm{U}}_0$ so that no output increases at the expense of another output and (iii) adding ${\sum }_{j=1}^n{\delta }_j^A{\mathit{Z}}_j^{AB}\ge {\tilde{\mathbf{Z}}}_0^{\mathit{AB}};{\sum }_{j=1}^n{\delta }_j^B{\mathit{Z}}_j^{AB}\le {\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$ to eliminate the possibility of obtaining zero intermediate measures. We shall see later that including ${\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}-{\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge \left({\theta }_0-1\right){\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ together with ${\sum }_{j=1}^n{\lambda }_j^A{\mathit{Z}}_j^{AB}\ge {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ and ${\sum }_{j=1}^n{\lambda }_j^B{\mathit{Z}}_j^{AB}\le {\tilde{\mathit{Z}}}_0^{\mathit{AB}}$ has another advantage. The objective function of model (3) $\mathit{max}\left\{{\mathit{\beta }}_0^A,{\mathit{\beta }}_0^B\right\}$ is equivalent to $\mathit{Max}\left\{{\beta }_{10}^A,{\beta }_{20}^A,..,{\beta }_{r_{A}0}^{\mathit{A}},{\beta }_{10}^B,{\beta }_{20}^B,..,{\beta }_{r_{B}0}^{\mathit{B}}\right\}$. Hence, model (3) is a multi-objective optimization problem (MOOP).

In MOOPs, objectives are in conflict. Therefore, MOOP solutions generally offer trade-off among objectives. A solution set that comprises best possible trade-offs among the objectives is referred to as a Pareto optimal set. A feasible solution is Pareto optimal when no other feasible solution improves one objective without worsening at least one other objective. Numerous algorithms that produce Pareto optimal solutions to MOOPs are suggested in the literature. Suppose ${\mathit{\delta }}^{\mathit{A}\ast },{\mathit{\delta }}^{\mathit{B}\ast },{\mathit{\beta }}_0^{A\ast },{\mathit{\beta }}_0^{B\ast }$,${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$ is a Pareto optimal solution of model (3) obtained through a MOOP solution method.

Theoretical underpinning

Theorem 1.

Suppose ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast },{\mathit{\beta }}_0^{A\ast },{\mathit{\beta }}_0^{B\ast }$ is obtained in a Pareto optimal solution of model (3). When a new DMU with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, outputs ${\mathit{\beta }}_0^{A\ast },{\mathit{\beta }}_0^{B\ast }$ and intermediate measures ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$ is added to the observed DMU set, relative efficiency of the new DMU is $1/{\theta }_0^{\prime }$.

Proof:

Follows from Lemma 1 and Lemma 2.

Lemma 1.

Optimal value of model (4), ${\varphi }_0^{\ast }={\theta }_0^{\prime }$.

(4) ${\varphi }_0^{\ast }=\mathit{max}\left\{{\varphi }_0\left|\begin{array}{c}{\sum }_{j=1}^n{\omega }_j^A{X}_j^A\le \left(X_0^A+\mathbf{\Delta }{X}_0^A\right);{\sum }_{j=1}^n{\omega }_j^B{X}_j^B\le \left(X_0^B+\mathbf{\Delta }{X}_0^B\right);\\ {\sum }_{j=1}^n{\omega }_j^A{Y}_j^A\ge {\varphi }_0{\beta }_0^{A\ast };{\sum }_{j=1}^n{\omega }_j^B{Y}_j^B\ge {\varphi }_0{\beta }_0^{B\ast };\\ {\sum }_{j=1}^n{\omega }_j^A{Z}_j^{AB}-{\sum }_{j=1}^n{\omega }_j^B{Z}_j^{AB}\ge \left({\varphi }_0-1\right){\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast };\\ {\sum }_{j=1}^n{\omega }_j^A=1;{\sum }_{j=1}^n{\omega }_j^B=1;{\omega }^A\ge 0;{\omega }^B\ge 0;,\mathit{\hspace{0.17em}}{\varphi }_0\mathrm{is}\mathrm{free}\end{array}\right.\right\}$(4)

Proof.

Since constraints of model (4) are a subset of model (3) constraint set, ${\mathit{\omega }}^A={\mathit{\delta }}^{\mathit{A}\ast }$, ${\mathit{\omega }}^B={\mathit{\delta }}^{\mathit{B}\ast }$, ${\varphi }_0={\theta }_0^{\prime }$ is a feasible solution of model (4). This implies that the optimal value of model (4), ${\varphi }_0^{\ast }$ should be at least as much as ${\theta }_0^{\prime }$. Suppose ${\varphi }_0^{\ast }$ is strictly greater than ${\theta }_0^{\prime }$. Then, there exists $\mathit{\kappa }>1$ such that ${\varphi }_0^{\ast }=\mathit{\kappa }{\theta }_0^{\prime }$. From model (4) constraints, we have that ${\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{X}}_j^A\le {\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\sum }_{j=1}^n{\delta }_j^{B\ast }{\mathit{X}}_j^B\le {\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, ${\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Y}}_j^A\ge {\theta }_0^{\prime }\left(\mathit{\kappa }{\mathit{\beta }}_0^{A\ast }\right)$, ${\sum }_{j=1}^n{\delta }_j^{B\ast }{\mathit{Y}}_j^B\ge {\theta }_0^{\prime }\left(\mathit{\kappa }{\mathit{\beta }}_0^{B\ast }\right)$ and ${\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Z}}_j^{AB}-{\sum }_{j=1}^n{\delta }_j^{B\ast }{\mathit{Z}}_j^{AB}\ge \left(\mathit{\kappa }{\theta }_0^{\prime }-1\right){\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$. Further, as ${\mathit{\delta }}^{\mathit{A}\ast },{\mathit{\delta }}^{\mathit{B}\ast },{\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast },{\mathit{\beta }}_0^{A\ast },{\mathit{\beta }}_0^{B\ast }$ is a Pareto optimal solution of model (3), we have $\mathit{\kappa }{\mathit{\beta }}_0^{A\ast }\ge {\mathit{Y}}_0^A$ and $\mathit{\kappa }{\mathit{\beta }}_0^{B\ast }\ge {\mathit{Y}}_0^B$ and ${\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Z}}_j^{AB}\ge {\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$and ${\sum }_{j=1}^n{\delta }_j^{B\ast }{\mathit{Z}}_j^{AB}\le {\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$. Hence, ${\mathit{\delta }}^{\mathit{A}\ast },{\mathit{\delta }}^{\mathit{B}\ast },{\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast },\mathit{\kappa }{\mathit{\beta }}_0^{A\ast },\mathit{\kappa }{\mathit{\beta }}_0^{B\ast }$ should be a feasible solution of model (3). In that case, ${\mathit{\delta }}^{\mathit{A}\ast },{\mathit{\delta }}^{\mathit{B}\ast },{\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast },{\mathit{\beta }}_0^{A\ast },{\mathit{\beta }}_0^{B\ast }$ cannot be a Pareto optimal solution of model (3) because $\mathit{\kappa }{\mathit{\beta }}_0^{A\ast },\mathit{\kappa }{\mathit{\beta }}_0^{B\ast }$ are greater than ${\mathit{\beta }}_0^{A\ast }$, ${\mathit{\beta }}_0^{B\ast }$. Therefore, $\mathit{\kappa }>1$ is a contradiction leading to ${\varphi }_0^{\ast }={\theta }_0^{\prime }$.

Lemma 2.

Suppose a new DMU with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, intermediate measures ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$ and outputs ${\mathit{\beta }}_0^{A\ast }$, ${\mathit{\beta }}_0^{B\ast }$ is added to the observed DMU set. The optimal value of model (5), ${\psi }_0^{\ast }$ is ${\theta }_0^{\prime }$.

(5) ${\psi }_0^{\ast }=\mathit{max}\left\{{\psi }_0\left|\begin{array}{c}{\sum }_{j=1}^n{\xi }_j^A{X}_j^A+{\xi }_{n+1}^A\left(X_0^A+\mathbf{\Delta }{X}_0^A\right)\le \left(X_0^A+\mathbf{\Delta }{X}_0^A\right);\\ {\sum }_{j=1}^n{\xi }_j^B{X}_j^B+{\xi }_{n+1}^B\left(X_0^B+\mathbf{\Delta }{X}_0^B\right)\le \left(X_0^B+\mathbf{\Delta }{X}_0^B\right);\\ {\sum }_{j=1}^n{\xi }_j^A{Y}_j^A+{\xi }_{n+1}^A{\beta }_0^{A\ast }\ge {\psi }_0{\beta }_0^{A\ast };{\sum }_{j=1}^n{\xi }_j^B{Y}_j^B+{\xi }_{n+1}^B{\beta }_0^{B\ast }\ge {\psi }_0{\beta }_0^{B\ast };\\ {\sum }_{j=1}^n{\xi }_j^A{Z}_j^{AB}+{\xi }_{n+1}^A{\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }-{\sum }_{j=1}^n{\xi }_j^B{Z}_j^{AB}+{\xi }_{n+1}^B{\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }\ge \left({\psi }_0-1\right){\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast };\\ {\sum }_{j=1}^{n+1}{\xi }_j^A=1;{\sum }_{j=1}^{n+1}{\xi }_j^B=1;{\xi }^A\ge 0;{\xi }^B\ge 0;,\mathit{\hspace{0.17em}}{\psi }_0\mathrm{is}\mathrm{free}\end{array}\right.\right\}$(5)

Proof:

We show here that, in an optimal solution of model (5), ${\xi }_{n+1}^{A\ast }={\xi }_{n+1}^{B\ast }=0$ and therefore model (5) reduces to model (4). See Appendix A for the proof.

Remark 1. When a new DMU with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, intermediate measures ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$ and outputs ${\mathit{\beta }}_0^{A\ast }$, ${\mathit{\beta }}_0^{B\ast }$ replaces $\mathrm{DM}{\mathrm{U}}_0$ in the observed DMU set, relative efficiency of the new DMU is $1/{\theta }_0^{\prime }$. We verified this empirically.

Remark 2. When a new DMU with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, intermediate measures ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$ and outputs ${\mathit{\beta }}_0^{A\ast },$ ${\mathit{\beta }}_0^{B\ast }$ is added to the observed DMU set, the frontier of best performance established by the observed DMU set does not change. We verified this empirically.

Now, consider the case where $\mathrm{DM}{\mathrm{U}}_0$ is efficient (${\theta }_0^{\ast }$ estimated in model (2) is 1) and its inputs are augmented to ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$.

Remark 3. Suppose an optimal solution of model (6) is ${\mathit{\zeta }}^{\mathit{A}\ast },{\mathit{\zeta }}^{\mathit{B}\ast },{\eta }_0^{AB\ast },{\phi }_0^{\ast }$.

(6) ${\phi }_0^{\ast }=\mathit{max}\left\{{\phi }_0\left|\begin{array}{c}{\sum }_{j=1}^n{\zeta }_j^A{X}_j^A\le \left(X_0^A+\mathbf{\Delta }{X}_0^A\right);\\ {\sum }_{j=1}^n{\zeta }_j^B{X}_j^B\le \left(X_0^B+\mathbf{\Delta }{X}_0^B\right);\\ {\sum }_{j=1}^n{\xi }_j^A{Y}_j^A\ge {\phi }_0{Y}_0^A;{\sum }_{j=1}^n{\xi }_j^B{Y}_j^B\ge {\phi }_0{Y}_0^B;\\ {\sum }_{j=1}^n{\xi }_j^A{Z}_j^{AB}\le {\eta }_0^{AB};{\sum }_{j=1}^n{\xi }_j^B{Z}_j^{AB}\ge {\eta }_0^{AB};\\ {\sum }_{j=1}^n{\zeta }_j^A=1;{\sum }_{j=1}^n{\zeta }_j^B=1;{\zeta }^A\ge 0;{\zeta }^B\ge 0;{\eta }_0^{AB},\mathit{\hspace{0.17em}}{\phi }_0\mathrm{are}\mathrm{free}\end{array}\right.\right\}$(6)

When a new DMU with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, intermediate measures ${\eta }_0^{AB\ast }$ and outputs ${\phi }_0^{\ast }{\mathit{Y}}_0^A$, ${\phi }_0^{\ast }{\mathit{Y}}_0^B$ is added to the observed DMU set, optimal value of model (7), is 1.

(7) ${\tau }_0^{*}=\mathit{max}{\tau }_0{\displaystyle {\sum }_{j=1}^n{\varsigma }_j^A}{X}_j^A+{\varsigma }_{n+1}^A\left(X_0^A+\mathit{\Delta }{X}_0^A\right)\le \left(X_0^A+\mathit{\Delta }{X}_0^A\right);{\displaystyle {\sum }_{j=1}^n{\varsigma }_j^B}{X}_j^B+{\varsigma }_{n+1}^B\left(X_0^B+\mathit{\Delta }{X}_0^B\right)\le \left(X_0^B+\mathit{\Delta }{X}_0^B\right);{\displaystyle {\sum }_{j=1}^n{\varsigma }_j^A}{Y}_j^A+{\varsigma }_{n+1}^A\left({\phi }_0^{*}{Y}_0^A\right)\ge {\tau }_0\left({\phi }_0^{*}{Y}_0^A\right);{\displaystyle {\sum }_{j=1}^n{\varsigma }_j^B}{Y}_j^B+{\varsigma }_{n+1}^B\left({\phi }_0^{*}{Y}_0^B\right)\ge {\tau }_0\left({\phi }_0^{*}{Y}_0^B\right);{\displaystyle {\sum }_{j=1}^n{\varsigma }_j^A}{Z}_j^{AB}+{\varsigma }_{n+1}^A{\eta }_0^{AB*}\le \mathit{Z}=\mathit{AB};{\displaystyle \sum _{\mathit{j}=1}^n{\varsigma }_j^B}{Z}_j^{AB}+{\varsigma }_{n+1}^B{\eta }_0^{AB*}\ge \mathit{Z}=\mathit{AB};{\displaystyle {\sum }_{j=1}^{n+1}{\varsigma }_j^A}=1;{\displaystyle {\sum }_{j=1}^{n+1}{\varsigma }_j^B}=1;{\varsigma }^A\ge 0;{\varsigma }^B\ge 0;\mathit{Z}=\mathit{AB},{\tau }_0\text{\hspace{0.17em}are\hspace{0.17em}free}$(7)

In this case (where $\mathrm{DM}{\mathrm{U}}_0$ is efficient), when a new DMU with inputs ${\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, ${\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$, intermediate measures $\mathit{Z}=\mathit{AB}*$ and outputs $\left({\phi }_0^{\ast }{\mathit{Y}}_0^A\right)$, $\left({\phi }_0^{\ast }{\mathit{Y}}_0^B\right)$ is added to the observed DMU set, new DMU relative efficiency remains 1. We verified this empirically.

Solving model (3)

A single point that maximizes all objective functions of a MOOP simultaneously may not exist. This difficulty prompts obtaining solutions that satisfy certain conditions. Pareto optimal solution point is one type and weak Pareto optimal solution point is another. A solution point is weak Pareto optimal if it is not possible to move to another solution point that improves all objective functions simultaneously. There are several methods to obtain Pareto optimal and weak Pareto optimal solutions. A commonly used approach of finding approximate solutions to a MOOP is assigning preferences (weights) to the objectives - ‘weighted sum method’. For insights on this method, see Marler and Arora (2010).

In the weighted sum method, each individual objective function (in our case ${\beta }_{10}^A,{\beta }_{20}^A,..,{\beta }_{r_{A}0}^{\mathit{A}},{\beta }_{10}^B,{\beta }_{20}^B,..,{\beta }_{r_{B}0}^{\mathit{B}}$) is assigned a nonnegative weight and the weighted sum is maximized subject to the constraints of the problem. Then MOOP reduces to a single-objective problem. Suppose ${\mathit{w}}^A=w_1^A,w_2^A,\dots ,w_{rA}^A$ $>0$ and ${\mathit{w}}^B=w_1^B,w_2^B,\dots ,w_{rB}^B$ $>0$ are a set of weights attached to the outputs.³ Suppose for given ${\mathit{w}}^{A^{\prime }}$ and ${\mathit{w}}^{B^{\prime }}$ an optimal solution to model (8) is ${\mathit{\beta }}_0^{A\ast },{\mathit{\beta }}_0^{B\ast }$ and the corresponding objective function value is ${\mathit{w}}^{A^{\prime }}{\mathit{\beta }}_0^{A\ast }+{\mathit{w}}^{B^{\prime }}{\mathit{\beta }}_0^{B\ast }$. Now, when any one of ${\beta }_{r0}^{A\ast }$,${\beta }_{r0}^{B\ast }$ changes, the optimal value of the objective function remains unchanged only when at least one other ${\beta }_{r0}^{A\ast }$,${\beta }_{r0}^{B\ast }$ value changes. Therefore, an optimal solution obtained in model (8) which is a linear programme is Pareto optimal. For details on the necessary conditions for Pareto optimality of MOOP solutions, see Goicoechea, Hansen, and Duckstein (1982).

(8) $\mathit{max}\left\{w^{A^{\prime }}{\beta }_0^A+w^{B^{\prime }}{\beta }_0^B\left|\begin{array}{c}{\sum }_{j=1}^n{\delta }_j^A{X}_j^A\le \left(X_0^A+\mathbf{\Delta }{X}_0^A\right);{\sum }_{j=1}^n{\delta }_j^B{X}_j^B\le \left(X_0^B+\mathbf{\Delta }{X}_0^B\right);\\ {\sum }_{j=1}^n{\delta }_j^A{Y}_j^A\ge {\theta }_0^{\prime }{\beta }_0^A;{\sum }_{j=1}^n{\delta }_j^B{Y}_j^B\ge {\theta }_0^{\prime }{\beta }_0^B;\\ {\sum }_{j=1}^n{\delta }_j^A{Z}_j^{AB}-{\sum }_{j=1}^n{\delta }_j^B{Z}_j^{AB}\ge \left({\theta }_0^{\prime }-1\right){\tilde{\mathbf{Z}}}_0^{\mathit{AB}};\\ {\sum }_{j=1}^n{\delta }_j^A{Z}_j^{AB}\ge {\tilde{\mathbf{Z}}}_0^{\mathit{AB}};{\sum }_{j=1}^n{\delta }_j^B{Z}_j^{AB}\le {\tilde{\mathbf{Z}}}_0^{\mathit{AB}};\\ {\beta }_0^A\ge Y_0^A;{\beta }_0^B\ge Y_0^B\\ {\sum }_{j=1}^n{\delta }_j^A=1;{\sum }_{j=1}^n{\delta }_j^B=1;\\ {\delta }^A\ge 0;{\delta }^B\ge 0;{\beta }_0^A,{\beta }_0^B,{\tilde{\mathbf{Z}}}_0^{\mathit{AB}}\mathrm{arefree}\end{array}\right.\right\}$(8)

Remark 4 presents conditions under which model (8) may be infeasible. Chen and Wang (2021) point out that inverse DEA models in general may not always produce feasible solutions. They discuss a procedure that may avoid VRS inverse DEA model infeasibility.

Remark 4. Model (8) is infeasible when ${\mathit{Y}}_0^A>{\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Y}}_j^A/{\theta }_0^{\ast }$ for at least one $Y_{r0}^A:\mathit{r}=1,2,\dots ,r_A$ and/or when ${\mathit{Y}}_0^B>{\sum }_{j=1}^n{\delta }_j^{B\ast }{\mathit{Y}}_j^B/{\theta }_0^{\ast }$ for at least one $Y_{r0}^B:\mathit{r}=1,2,\dots ,r_B$.

Suppose an optimal solution of model (2) is ${\mathit{\lambda }}^{\mathit{A}\ast },{\mathit{\lambda }}^{\mathit{B}\ast }$ and ${\theta }_0^{\ast }$. Now, since ${\mathit{\lambda }}^{\mathit{A}\ast },{\mathit{\lambda }}^{\mathit{B}\ast }\ge 0$ and ${\sum }_{j=1}^n{\lambda }_j^{A\ast }=1;\mathit{\hspace{0.17em}}{\sum }_{j=1}^n{\lambda }_j^{B\ast }=1$, at least one of ${\lambda }_j^{A\ast }$ and ${\lambda }_j^{B\ast }$>0. Let ${\mathit{\delta }}^A={\mathit{\lambda }}^{\mathit{A}\ast }$ and ${\mathit{\delta }}^B={\mathit{\lambda }}^{\mathit{B}\ast }$ in model (8). Then, we have that ${\mathit{\lambda }}^{\mathit{A}\ast },{\mathit{\lambda }}^{\mathit{B}\ast }$ satisfy the first and the sixth sets of constraints in model (8). From the fourth set of constraints we have $-{\sum }_{j=1}^n{\delta }_j^B{\mathit{Z}}_j^{AB}\ge -{\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$ and by adding ${\sum }_{j=1}^n{\delta }_j^A{\mathit{Z}}_j^{AB}$ to both sides we obtain ${\sum }_{j=1}^n{\delta }_j^A{\mathit{Z}}_j^{AB}-{\sum }_{j=1}^n{\delta }_j^B{\mathit{Z}}_j^{AB}\ge {\sum }_{j=1}^n{\delta }_j^A{\mathit{Z}}_j^{AB}-{\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$. The L.H.S. of this inequality is the same as the L.H.S. of the third constraint. Now choose ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$ such that ${\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Z}}_j^{AB}-{\tilde{\mathbf{Z}}}_0^{\mathit{AB}}=\left({\theta }_0^{\ast }-1\right){\tilde{\mathbf{Z}}}_0^{\mathit{AB}}$. Then, ${\mathit{\lambda }}^{\mathit{A}\ast },{\mathit{\lambda }}^{\mathit{B}\ast },{\theta }_0^{\ast }$ and ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }={\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Z}}_j^{AB}/{\theta }_0^{\ast }$ satisfies the first, third, fourth and sixth sets of constraints of model (8). From the second and the fifth sets of constraints we have ${\sum }_{j=1}^n{\delta }_j^A{\mathit{Y}}_j^A\ge {\theta }_0^{\prime }{\mathit{\beta }}_0^A\ge {\theta }_0^{\prime }{\mathit{Y}}_0^A$ and ${\sum }_{j=1}^n{\delta }_j^B{\mathit{Y}}_j^B\ge {\theta }_0^{\prime }{\mathit{\beta }}_0^B\ge {\theta }_0^{\prime }{\mathit{Y}}_0^B$. If ${\mathit{\lambda }}^{\mathit{A}\ast },{\mathit{\lambda }}^{\mathit{B}\ast },{\theta }_0^{\prime }={\theta }_0^{\ast }$ satisfies these conditions, then ${\mathit{\lambda }}^{\mathit{A}\ast },{\mathit{\lambda }}^{\mathit{B}\ast },{\theta }_0^{\ast }$ and ${\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }={\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Z}}_j^{AB}/{\theta }_0^{\ast }$ becomes a feasible solution of model (8). Model (8) becomes infeasible when ${\mathit{Y}}_0^A>{\sum }_{j=1}^n{\delta }_j^{A\ast }{\mathit{Y}}_j^A/{\theta }_0^{\ast }$ for at least one $Y_{r0}^A:\mathit{r}=1,2,\dots ,r_A$ and/or when ${\mathit{Y}}_0^B>{\sum }_{j=1}^n{\delta }_j^{B\ast }{\mathit{Y}}_j^B/{\theta }_0^{\ast }$ for at least one $Y_{r0}^B:\mathit{r}=1,2,\dots ,r_B$.

IV. Application to superannuation funds

We demonstrate application of our method using a sample of funds in the Australian superannuation fund industry which is highly regulated. Superannuation funds (SFs) provide investment opportunities for workers to save money during their working life to access on retirement and hence they are like retirement pension benefit schemes. More than half the funds of Australian managed fund industry is with SFs.

There are five major types of SFs: corporate, industry, public sector, retail and self-managed. A self-managed superannuation fund (SMSF) is do-it-yourself type fund where usually the member/s (no more than six) are the trustees. Association of Superannuation Funds of Australia (2022) reports 604,087 SMSFs, 12 corporate, 31 industry, 32 public sector and 83 retail funds. We do not consider SMSFs because their management structure is different from other fund types. Approximately half of the other 158 funds belongs to the retail type (52,5%) followed by industry (20.3%), public sector (19.6%) and corporate (7.6%) type. Assets managed by these funds exceed $2 trillion.

Empirical framework

We conceptualize fund management process as a two-stage production process depicted in Figure 2. At stage A, net assets (beginning-period net assets) are administered incurring costs (administration and operating expenses) to make payments to SF members (benefit payments) and secure funds for investment at stage B (total investments). Total investment is an intermediate measure. At stage B, investments are managed incurring expenses and fees (investment expenses) at risk (total risk, portfolio risk and leverage risk) to generate income (investment income).⁴

Figure 2. Superannuation fund assets (financial resource) management process.

We classify administration and operating expenses and investment expenses as disbursements. Total risk is measured as SD of past five-year annual rates of return. Leverage risk is computed as total liability relative to total assets. Portfolio risk is computed as a ratio of weighted sum of funds allocated to different asset classes to total funds available for investment.⁵ To be included in the sample, we require an SF to have positive values for all input, intermediate and output measures over the five-year period 2017–2021. Sixty-nine funds in the annual fund-level superannuation statistics back series (APRA 2021) satisfy this requirement. Out of the 69 funds, 6 are corporate, 28 are industry, 10 are public sector and 25 are retail type. Some summary statistics of the input, intermediate and output measures of the sample considered in the analysis are reported in Table 1. Investment expenses has the largest and portfolio risk has the smallest coefficient of variation.

Table 1. Summary statistics of input, intermediate and output variables.

Statistic (n=69)	Stage A process				Stage B process
	Input variables		Output variable	Intermediate variable	Input variables				Output
	Administration and operating expenses (000’ $)	Beginning period net assets (000’ $)	Benefit payments (000’ $)	Total investments (000’ $)	Investment expenses (000’ $)	Portfolio risk	Total risk (SD)	Leverage risk	Investment income (000’ $)
Mean	54,726	17,037,037	712,695	20,859,262	44,063	4.2983	0.0623	0.0296	559,140
Median	20,270	5,344,720	250,764	6,204,255	9,065	4.3393	0.0623	0.0252	182,958
SD	94,179	32,169,094	1,280,710	39,612,499	95,916	0.5004	0.0100	0.0274	992,675
Maximum	548,846	182,378,081	7,166,105	241,601,963	522,990	5.1830	0.0863	0.2034	5,091,052
Minimum	240	79,926	4,842	86,533	37	1.0046	0.0354	0.0025	40
Range	548,606	182,298,155	7,161,263	241,515,430	522,953	4.1783	0.0509	0.2008	5,091,012
CV	1.72	1.89	1.80	1.90	2.18	0.12	0.16	0.93	1.78

Overall performance

Table 2 presents overall efficiency of SFs estimated in model (2) by fund type. Twenty-one funds are overall efficient. Average and median overall efficiency scores are 0.7811 and 0.7523 respectively suggesting that the distribution of overall efficiency scores is right skewed. Corporate funds have the smallest average efficiency closely followed by public sector funds. Retail funds have the largest average efficiency. This dominance in average performance is reflected in the composition of the efficient frontier as well. Our findings with 2021 cross-sectional data are consistent with previous studies of SFs. See, for example Sun and Galagedera (2021).

Table 2. Overall efficiency by fund type.

Fund type	Number of funds	Average efficiency	Standard deviation	Minimum	Number of efficient funds (%)
Corporate	6	0.6751	0.1824	0.4599	1 (4.76)
Industry	28	0.7991	0.1815	0.4739	9 (42.86)
Public sector	10	0.6864	0.1683	0.4255	1 (4.76)
Retail	25	0.8243	0.1769	0.4583	10 (47.62)
All funds	69	0.7811	0.1829	0.4255	21 (100.00)

Forward planning

Here we discuss how a SF may obtain information useful for decision making in the short-term when increase in disbursements and portfolio risk is predicted conditional on a desired level of short-term performance. For the demonstration we select two funds with relative efficiency 0.8588 (ranked 28) and 0.7112 (ranked 42). We refer to them as SF1 and SF2 to maintain anonymity. Table 3 gives their observed input, intermediate and output measures. The reason we select these funds is to show that intermediate and output targets estimated in model (8) may have different patterns.

Table 3. Observed input, intermediate and output variable values of selected funds.

Fund label	Administration and operating expenses (000’ $)	Beginning period net assets (000’ $)	Benefit payments (000’ $)	Total investments (000’ $)	Investment expenses (000’ $)	Portfolio risk	Total risk (SD)	Leverage risk	Investment income (000’ $)
SF1	209,214	111,998,948	3,917,904	112,071,653	125,417	3.7531	0.0528	0.0237	1,638,330
SF2	35,066	12,668,020	505,006	14,626,599	38,454	4.321	0.0451	0.0316	400,865

Intermediate and output targets for SF1 to reach different efficiency levels with observed input levels

First, we discuss how to obtain pathways to improve relative performance of inefficient SF1 while maintaining its inputs as observed. Relative efficiency of SF1 is 0.8588. For the discussion, we consider 0.9, 0.95, 0.975 and 0.99 as desired efficiency levels. A pathway is defined by a set of intermediate and output targets that enable inefficient SF1 to achieve a desired efficiency level with its observed inputs. Intermediate and output targets are obtained by solving model (8) with ${\theta }_0^{\prime }$ set as the inverse of the desired efficiency level, substituting $\mathbf{\Delta }{\mathit{X}}_0^A=0$ and $\mathbf{\Delta }{\mathit{X}}_0^B=0$ (no change in inputs) and setting the weights of the two objectives ${\mathit{\beta }}_0^A$ (benefit payments) and ${\mathit{\beta }}_0^B$ (investment income) at 1 (${\mathit{w}}^{A^{\prime }}={\mathit{w}}^{B^{\prime }}=1$). Table 4 reports the results. Each row shows a pathway that SF1 may follow to reach the desired relative efficiency level with current levels of inputs. We discuss sensitivity of the results to variation in the weights in section 5.1.

Table 4. Intermediate and output targets that attains desired performance levels of SF1 with observed inputs.

	Stage A output target	Intermediate measure target	Stage B output target
	Benefit payments (000’ $)	Total investments (000’ $)	Investment income (000’ $)
Observed value	3,917,909	112,071,653	1,638,330
Desired relative performance
0.9	4,105,982 (4.80)	152,749,403 (36.30)	2,055,550 (25.47)
0.95	4,334,092 (10.62)	152,749,403 (36.30)	2,169,747 (32.44)
0.975	4,448,147 (13.53)	152,749,403 (36.30)	2,226,846 (35.92)
0.99	4,516,580 (15.28)	152,749,403 (36.30)	2,261,105 (38.01)

For example, SF1 may achieve 0.95 relative efficiency with current levels of inputs by increasing its benefit payments from $3,917,909,000 to $4,334,092,000 (10.62% increase), investment income from $1,638,330,000 to $2,169,747,000 (32.44% increase) and total investments from $112,071,653,000 to $152,749,403,000 (36.3% increase). It is possible for SF1 to reach desired efficiency levels within the range 0.9 and 0.99 with total investments unchanged at $152,749,403,000. Further, when SF1 achieves 0.9 relative efficiency, SF1 may reach higher levels of efficiency with relatively small increases in benefit payments and investment income.

Model (8) may not always produce an optimal solution. See remark 4. Another possibility is that model (8) may have alternate optimal solutions. Existence of alternative optimal solutions is an advantage in a business setting as that would increase available choices.

Each row of Table 4 represents a point on the Pareto efficient frontier under the relative efficiency scenario given in column 1. In the MOOP solution procedure ‘constraint method’, multiple Pareto optimal solutions can be obtained for a given scenario. A detailed description of application of ‘constraint method’ is available in Cohon (1978).

Intermediate and output targets for SF1 to reach different efficiency levels when disbursements increase

Here, we discuss intermediate and output targets required for SF1 when its disbursements increase by 2%, 5%, 7.5% and 10% with no change in risk to maintain its relative efficiency as observed at 0.8588, and increased to 0.9 and 0.95. Table 5 reports the results.

Table 5. Intermediate and output targets for SF1 to reach different performance levels when its administration and operating expenses and investment expenses increase.

Increase in disbursements	Stage A output target	Intermediate measure target	Stage B output target
	Benefit payments (000’ $)	Total investments (000’ $)	Investment income (000’ $)
Observed value	3,917,909	112,071,653	1,638,330
Desired relative performance: 0.8588 (observed efficiency level)
2%	3,958,844 (1.04)	153,048,762 (36.56)	1,968,987 (20.18)
5%	4,020,247 (2.61)	153,497,800 (36.96)	1,980,373 (26.68)
7.5%	4,071,416 (3.92)	153,871,998 (37.30)	1,989,861 (27.29)
10%	4,122,585 (5.22)	154,246,196 (37.63)	1,999,349 (27.89)
(b) Desired relative performance: 0.9
2%	4,148,882 (5.90)	153,048,762 (36.56)	2,063,505 (25.95)
5%	4,213,232 (7.54)	153,497,800 (36.96)	2,075,437 (33.72)
7.5%	4,266,858 (8.91)	153,871,998 (37.30)	2,085,381 (34.36)
10%	4,320,483 (10.28)	154,246,196 (37.63)	2,095,324 (35.00)
(c) Desired relative performance: 0.95
2%	4,379,375 (11.78)	153,048,762 (36.56)	2,178,144 (32.95)
5%	4,447,301 (13.51)	153,497,800 (36.96)	2,190,739 (33.72)
7.5%	4,503,905 (14.96)	153,871,998 (37.30)	2,201,235 (34.36)
10%	4,560,510 (16.40)	154,246,196 (37.63)	2,211,731 (35.00)

Intermediate and output targets are estimated in model (8). Parentheses give the percentage increase from the observed value.

The results reveal that in all three cases, benefit payments and investment income targets increase with increasing disbursements. For example, when disbursements increase by 5%, SF1 should target 26.68% increase in investment income and 2.61% increase in benefit payments to maintain its performance at the current level. If SF1 is to improve its performance to 0.9 efficiency level when disbursements increase by 5%, second row of panel (b) reveal that SF1 has to increase benefit payments and investment income further. Here, we increase both types of disbursements (administration and operating expenses and investment expenses) simultaneously. An SF manager may consider increasing one disbursement type at a time to determine which type may require relatively lower output targets.

Intermediate and output targets for SF2 to reach different efficiency levels when disbursements and risk increase

Here we discuss intermediate and output targets required for SF2 to maintain its relative efficiency as observed at 0.7112 and at 0.75 and 0.8 when administration and operating expenses and portfolio risk increases simultaneously. The levels of increase in administration and operating expenses are the same as in the previous case. Portfolio risk is measured as a ratio. Hence, we specify its potential increase in terms of number of SDs of portfolio risk distribution. The levels of increase in portfolio risk considered are 0.25SD, 0.5SD, 0.75SD and 1SD. Table 6 reports the results.

Table 6. Intermediate and output targets for SF2 to reach different performance levels when its administration and operating expenses and portfolio risk increase.

Increase in administration and operating expenses and portfolio risk	Stage A output target	Intermediate measure target	Stage B output target
	Benefit payments (000’ $)	Total investments (000’ $)	Investment income (000’ $)
Observed value	505,006	14,626,599	400,865
Desired relative performance: 0.7112 (observed efficiency level)
2% and 0.25SD	505,006 (0.00)	11,678,779 (−20.15)	516,731 (28.90)
5% and 0.5 SD	505,006 (0.00)	11,967,688 (−18.18)	529,249 (32.03)
7.5% and 0.75SD	505,006 (0.00)	12,208,446 (−16.53)	539,681 (34.63)
10% and 1SD	505,006 (0.00)	12,449,203 (−14.89)	550,113 (37.23)
(b) Desired relative performance: 0.75
2% and 0.25SD	505,006 (0.00)	14,246,568 (−2.60)	662,460 (65.26)
5% and 0.5 SD	505,006 (0.00)	14,551,334 (−0.51)	676,390 (68.73)
7.5% and 0.75SD	505,006 (0.00)	14,805,305 (1.22)	687,999 (71.63)
10% and 1SD	505,006 (0.00)	15,059,277 (2.96)	699,607 (74.52)
(c) Desired relative performance: 0.8
2% and 0.25SD	505,006 (0.00)	16,662,207 (13.92)	824,399 (105.65)
5% and 0.5 SD	505,006 (0.00)	16,995,729 (16.20)	840,660 (109.71)
7.5% and 0.75SD	505,006 (0.00)	17,273,664 (18.10)	854,210 (113.09)
10% and 1SD	505,006 (0.00)	17,551,599 (20.00)	867,761 (116.47)

Intermediate and output targets are estimated in model (8). Parentheses give the percentage increase from the observed value. SD = standard deviation of the distribution of portfolio risk.

Table 6 reveals that when administration and operating expenses increase within 2% and 10% and portfolio risk increases within 0.25SD and 1SD, no increase in benefit payments is required to maintain SF2 relative efficiency at the observed level, 0.75 and at 0.8 as long as the corresponding total investment targets and investment income targets are met. An interesting observation here is that for some scenarios of simultaneous increase in administration and operating expenses and portfolio risk, total investments target is less than the observed. For example, when administration and operating expenses increase by 2% and portfolio risk increases by 0.25SD, SF2 may improve its performance from 0.7112 to 0.75 by decreasing total investments by 2.6% and increasing investment income by 65.26%. Under our modelling framework, total investments (an intermediate measure) is left unbounded in model (8) as it is internal to the two-stage production process. Therefore, in model (8) solutions it is possible to obtain lower than observed targets for total investments. Opportunity to achieve desired performance levels with reduced total investments may be viewed as a favourable situation as excess funds may be utilized elsewhere.

Panel (c) of Table 6 reveals that when the desired relative performance is 0.8, SF2 may have to increase investment income excessively. This may not be achievable in the short-term. Amin, Al-Muharrami, and Toloo (2019) add to the discussion on practicality of such targets by suggesting that targets should be flexible enough to cope with external and internal constraints.

V. Robustness of the results

Variation in output preference scheme

The results we discussed thus far is based on a priori assignment of weights $w_r^A=1$ for $\mathit{r}=1,2,\dots ,r_A$ and $w_r^B=1$ for $\mathit{r}=1,2,\dots ,r_B$ in the objective function of model (8). Any set of positive weights may be used in the weighting method to obtain a Pareto optimal solution. Cohon (1978) suggests an orderly procedure in weight selection. The decision maker may then select a preferred intermediate and output target set from among the available Pareto optimal solutions. As an illustration, we present in Table 7 the targets for SF2 estimated in model (8) under two different weighting schemes when its administration and operating expenses increases by 10% and portfolio risk increases by 1SD. The weighting schemes investigate sensitivity of intermediate and output targets to variation in benefit payments weight given two desired performance levels (0.75 and 0.8).

Table 7. Intermediate and output targets under different weighting schemes when SF2 administration and operating expenses increases by 10% and portfolio risk increases by 1SD.

Weighting scheme	Stage A output target	Intermediate measure target	Stage B output target
	Benefit payments (000’ $)	Total investments (000’ $)	Investment income (000’ $)
Maintains efficiency at 0.75
(1, 1)	505,006	15,059,277	699,607
(5, 1)	548,194	12,057,035	562,381
Maintains efficiency at 0.8
(1, 1)	505,006	17,551,599	867,761
(5, 1)	584,740	12,860,838	639,063

The parentheses in the first column give the weights assigned to benefit payments and investment income. The reported values of total benefit payments, total investments and investment income are obtained in model (8).

The results reveal that when the weight on benefit payments increases from 1 to 5 leaving the weight attached to investment returns unchanged at 1, there is a trade-off between benefit payments and investment income. That is when benefit payment objective is preferred to investment income generation objective, the results in panels (a) and (b) of Table 8 reveal higher targets for benefit payments and reduced targets for investment income. This may not happen always. For example, when the weights are changed to (10,1) in the scenario shown in panel (a), the targets obtained under the weighting scheme (5,1) do not change. In other words, change in preference schemes may not always result in changes in the targets. No change in output targets under different preference schemes imply no opportunity for further trade-off.

VI. Concluding remarks

In this contribution we propose an inverse DEA model for a two-stage production process that enables investigation of the association between inputs, intermediates, outputs and relative performance analytically. Such an analysis is akin to sensitivity analysis encountered in business analytics. We demonstrate application of the inverse DEA model using a sample of Australian superannuation funds (SFs). This is new to SF literature. Application highlights how a fund manager may ascertain information to plan for potential changes in costs, expenses and fund portfolio risk to achieve desired short-term performance levels. Under our empirical set up that would mean planning to achieve the required output targets estimated in the inverse DEA model.

In some SFs, increase in certain input types may require higher increases in the outputs to reach a desired performance level than certain other input types. Managers of such SFs can clearly identify increase in which input measures requires manageable increases in the outputs. By narrowing down the possibility set, SF managers will be in a better position to select a realistic action plan. When output targets are known in advance, SF manager can consider them as benchmarks when allocating resources. It is important that fund managers are better prepared to deal with potential changes in factors of performance especially under different environmental conditions. When fund managers are well informed, they will be in a better position to face up to challenges. For example, Islamic equity fund managers operate under shariah screening and monitoring. Such restrictions may have implications on their performance compared to managers of conventional funds. Extending this study to managed funds in different jurisdictions may shed more light on this issue.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/00036846.2023.2206618

Notes

¹ Macquarie Property Securities Trust, Macquarie Balanced Fund and Macquarie Small Companies Growth Trust offered in Australia by Macquarie Group are among the several managed funds that succumbed to redemption pressure. https://www.investinfo.com.au/content/dam/wealth/public/documents/investment_news/Macquarie_Small_Companies_Growth_Trust_Termination.pdf (Accessed 10 May 2021).

In May 2020, Mercer Investments (Australia) Limited issued notice of termination of Mercer Diversified Alternatives Fund giving decreased fund size as the reason. Notice of Fund Termination – Mercer Portfolio Service, https://www.mercerportfolioservice.com.au (Accessed 10 May 2021)

MLC Investment Trust MLC Platinum Global Fund is another. MLC Investments Limited issued termination notices effective January 2020.) https://www.mlcam.com.au/content/dam/mlcam/pdf/mlc/MLC-platinum-global-fund-termination-notice-letter.pdf (Accessed 10 May 2021)

² The linear programming counterpart of multiplier model that estimates relative efficiency of DMU₀ with two-stage production process given in Figure 1 is $\mathit{Min}{\sum }_{\mathit{i}=1}^{i_A}{\omega }_i^A{x}_{i0}^A-{\xi }^A+{\sum }_{\mathit{i}=1}^{i_B}{\omega }_i^B{x}_{i0}^B+{\sum }_{d=1}^D{\pi }_d{z}_{d0}^{AB}-{\xi }^B$ subject to ${\sum }_{\mathit{i}=1}^{i_A}{\omega }_i^A{x}_{ij}^A-{\xi }^A-{\sum }_{d=1}^D{\pi }_d{z}_{dj}^{AB}-{\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^A{y}_{rj}^A\ge 0$ for $\mathit{j}=1,2,\dots ,\mathit{n}$;${\sum }_{\mathit{i}=1}^{i_B}{\omega }_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\pi }_d{z}_{dj}^{AB}-{\xi }^B-{\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^B{y}_{rj}^B\ge 0$ for $\mathit{j}=1,2,\dots ,\mathit{n}$; ${\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^B{y}_{r0}^B+{\sum }_{d=1}^D{\pi }_d{z}_{d0}^{AB}+{\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^A{y}_{r0}^A=1$ such that ${\omega }_i^A,{\omega }_i^B,{\mu }_r^A,{\mu }_r^B,{\pi }_d\ge 0$; ${\xi }^A,{\xi }^B$ unrestricted. Model (2) gives the dual of this model.

³ Through the weights, the user indicates the relative importance of outputs. If a priori articulation of preferences is not essential, may use equal weights. Yan, Wei, and Hao (2002) highlight that introduction of input/output preferences close to reality in inverse DEA analysis would be beneficial to management in short-term production planning.

⁴ The outputs are chosen to have the same unit of measure because the objective function of model (8) is expressed as a weighted sum of the outputs (benefit payments and investment income). Otherwise, marginal rates of substitution of outputs which more often than not are subjective is needed.

⁵ Portfolio risk is computed as [6(equity and commodities) + 4(property and infrastructure) + 2(cash and fixed deposits) + (other)]/total available for investment.

⁶ This subtraction is valid as from the fourth constraint of model (A), we have that ${\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^{B\ast }{\alpha }_{i0}^B+{\vartheta }^{\mathit{B}\ast }+{\sum }_{d=1}^D{\pi }_d^{\ast }{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }\ge {\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^{B\ast }{\beta }_{r0}^{B\ast }$.

Appendix A

Proof of Lemma 2

Consider the dual of model (5) given by (A) where $\left\{{\alpha }_{io}^A:\mathit{i}=1,2,\dots ,i_A\right\}={\mathit{X}}_0^A+\mathbf{\Delta }{\mathit{X}}_0^A$, $\left\{{\alpha }_{io}^B:\mathit{i}=1,2,\dots ,i_B\right\}={\mathit{X}}_0^B+\mathbf{\Delta }{\mathit{X}}_0^B$ and $\left\{{\tilde{\mathrm{z}}}_{10}^{\mathit{AB}\ast },{\tilde{z}}_{20}^{\mathit{AB}\ast },\dots ,{\tilde{\mathrm{z}}}_{\mathit{D}0}^{\mathit{AB}\ast }\right\}={\tilde{\mathbf{Z}}}_0^{\mathit{AB}\ast }$.

(A) $\mathit{Min}{\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^A{\alpha }_{i0}^A+{\vartheta }^A+{\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^B{\alpha }_{i0}^B+{\sum }_{d=1}^D{\pi }_d{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }+{\vartheta }^B$(A)

Subject to ${\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^A{x}_{ij}^A+{\vartheta }^A-{\sum }_{d=1}^D{\pi }_d{z}_{dj}^{AB}-{\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^A{y}_{rj}^A\ge 0$ $\mathit{j}=1,2,\dots ,\mathit{n}$

${\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^A{\alpha }_{i0}^A+{\vartheta }^A-{\sum }_{d=1}^D{\pi }_d{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }-{\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^A{\beta }_{r0}^{A\ast }\ge 0$

${\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\pi }_d{z}_{dj}^{AB}+{\vartheta }^B-{\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^B{y}_{rj}^B\ge 0$ $\mathit{j}=1,2,\dots ,\mathit{n}$

${\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^B{\alpha }_{i0}^B+{\sum }_{d=1}^D{\pi }_d{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }+{\vartheta }^B-{\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^B{\beta }_{r0}^{B\ast }\ge 0$

${\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^B{\beta }_{r0}^{B\ast }+{\sum }_{d=1}^D{\pi }_d{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }+{\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^A{\beta }_{r0}^{A\ast }=1$

${\rho }_i^A,{\rho }_i^B,{\mu }_r^A,{\mu }_r^B,{\pi }_d\ge 0$; ${\vartheta }^A,{\vartheta }^B$ are free

Let ${\rho }_i^{A\ast },{\rho }_i^{B\ast },{\mu }_r^{A\ast },{\mu }_r^{B\ast },{\pi }_d^{\ast },{\vartheta }^{\mathit{A}\ast },{\vartheta }^{\mathit{B}\ast }$ denote an optimal solution of model (A). Suppose the optimal value of model (A) is greater than 1 and the second and the fourth constraints of model (A) are satisfied at equality. Then, by adding the second and the fourth constraints of model (A), we have

(A.1) ${\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^{A\ast }{\alpha }_{i0}^A+{\vartheta }^{\mathit{A}\ast }+{\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^{B\ast }{\alpha }_{i0}^B+{\vartheta }^{\mathit{B}\ast }-\left({\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^{A\ast }{\beta }_{r0}^{A\ast }+{\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^{B\ast }{\beta }_{r0}^{B\ast }\right)$(A.1)

From the last constraint of model (A), (A.1) reduces to

(A.2) ${\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^{A\ast }{\alpha }_{i0}^A+{\vartheta }^{\mathit{A}\ast }+{\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^{B\ast }{\alpha }_{i0}^B+{\vartheta }^{\mathit{B}\ast }+{\sum }_{d=1}^D{\pi }_d{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }=1$(A.2)

L.H.S. of (A.2) is the objective function of model (A). This is a contradiction, suggesting that if the optimal value of model (A) is greater than 1, the L.H.S. of the second and the fourth constraints of model (A) cannot be equal to zero. Now assume that the optimal value of model (A) is greater than 1 and one of second and fourth constraints of model (A) is satisfied at strict inequality at optimality. Without loss of generality, consider that the second constraint of model (A) so that

(A.3) ${\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^{A\ast }{\alpha }_{i0}^A+{\vartheta }^{\mathit{A}\ast }-{\sum }_{d=1}^D{\pi }_d^{\ast }{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }-{\sum }_{\mathit{r}=1}^{r_A}{\mu }_r^{A\ast }{\beta }_{r0}^{A\ast }>0$(A.3)

Since the optimal value of model (A) is greater than 1, we have that

(A.4) ${\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^{B\ast }{\alpha }_{i0}^B+{\vartheta }^{\mathit{B}\ast }+{\sum }_{d=1}^D{\pi }_d^{\ast }{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }+{\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^{A\ast }{\alpha }_{i0}^A+{\vartheta }^{\mathit{A}\ast }>1$(A.4)

Now from the last constraint of model (A) and (A.3) we have that

(A.5) ${\sum }_{\mathit{i}=1}^{i_A}{\rho }_i^{A\ast }{\alpha }_{i0}^A+{\vartheta }^{\mathit{A}\ast }+{\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^{B\ast }{\beta }_{r0}^{B\ast }>1$(A.5)

Subtracting (A.5) from (A.4), we obtain that ${\sum }_{\mathit{i}=1}^{i_B}{\rho }_i^{B\ast }{\alpha }_{i0}^B+{\vartheta }^{\mathit{B}\ast }+{\sum }_{d=1}^D{\pi }_d^{\ast }{\tilde{z}}_{\mathit{d}0}^{\mathit{AB}\ast }-{\sum }_{\mathit{r}=1}^{r_B}{\mu }_r^{B\ast }{\beta }_{r0}^{B\ast }$> 0.⁶ This result reveals that, if the optimal value of model (A) is greater than one and the second constraint is satisfied at strict inequality the fourth constraint should also satisfy at strict inequality. Hence, from the complementary slackness property of primal-dual linear programming problems, the decision variables ${\xi }_{n+1}^A,{\xi }_{n+1}^B$ in model (5) that corresponds to the second and the fourth constraints of model (A) should be zero. Then model (5) reduces to model (4).