---
title: \linespread{1}\Huge Green contracting problem in liner shipping
author:
- \small Lingyu Zhang^[Faculty of Business Administration, Hamburg University (UHAM), [e-mail]]
- \small Christopher Dirzka^[Department of Economics, Copenhagen Business School (CBS), [e-mail]]
- \small Michele Acciaro^[\textbf{Corresponding author:} Department of Strategy and Innovation, Copenhagen Business School (CBS), [e-mail]]
date: \small September 2022
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\begin{center}
\vspace*{2cm}
\title{\linespread{1}\Huge Green contracting problem in liner shipping}
\vspace*{2cm}
\end{center}

This paper solves the green contracting problem. The problem involves contracts that split ownership and operations. Under information asymmetry, this agency problem poses an (in-)direct burden to the public good. A novel non-parametric approach that links an additive slacks-based model and a network data envelopment analysis (ASBM-NDEA) within a game-theoretical setting is proposed. Solving this chain decision-making problem, the approach provides insights into cooperative and non-cooperative contractual dynamics with the objective to raise individual utility and mitigate system externalities. To illustrate the approach, a sample which includes 3,878 contracts between owners and operators in the liner shipping industry, covering the years 2015-2020, was gathered. In reference to the underlying case study, the results highlight that the \textit{modus operandi} in this industry, which implies operator dominance and scale economies give rise to less-than-optimal contracts. In contrast, cooperative contracting, that is, shared obligations to mitigate the ecological footprint and scale-to-operations matching enable more sustainable operations. Given the green contracting problem's presence in diverse settings, the paper contributes to broader sustainable development agenda and insofar beyond the outlined industry case. Specifically, the paper supports policy action which assists transparent cooperative negotiations and climate target setting within contracts.

\begin{quote}
Keywords: \textit{Green contracting problem, Network additive slacks-based model (ASBM-NDEA), Game theoretical approach, Carbon emissions, Liner operation}
\end{quote}

# 1. Introduction

Building upon the seminal work by Schaltegger and Sturm [-@schaltegger_okologische_1990], which introduced the guiding principle to balance private recourse usage and the impact on public goods, sustainable operations research gained significant importance within operation reserch [@hart_beyond_1997; @hart_global_1999; kleindorfer_sustainable_2005; @marcus_green_2009; @drake_om_2013; @assenhove_sustainable_2019]. Scholarly discourse has proposed diverse parametric and non-parametric approaches to optimize these desired- and undesired outputs in operations [@hailu_non-parametric_2001; @fernandez_multiple-output_2002; @seiford_modeling_2002; @fare_modeling_2004; @kuosmanen_measuring_2005; @huppes_framework_2005; @quariguasi_frota_neto_methodology_2009; @chen_measuring_2012; @kounetas_measurement_2021]. Critical to these approaches is the notion that individual decision-making units (DMUs) utilize input(s) to produce output(s) within single-stage processes. Such processes can yield insights in abstracted systems, but fail to provide substantive insights into real-world operations.

In the real-world, firm operations can be specified as chain decision-making processes, that are, interdependent processes linked to customers, suppliers, partners, or competitors- which influence sustainable operations [see coopetition theory @modi_how_2020]. As an illustration - firm $A$ intends to provide logistic services and chooses to lease the transport means through the firm $B$, establishing a contractual relationship between $A \rightarrow B$. This relation splits ownership and operations, i.e., $B$ transfers operations and the obligation to pay a leasing rate to $$A. Besides the leasing rate, $A$ has to cover the operational costs, while $B$ remains obligated to cover the capital costs. Interdependence is established between the operational costs (covered by $A$) and the pre-contractual decisions by $B$. Under a sustainable operation perspective, $B$ selected the transport means’ baseline performance, while $A$ bears the responsibility for such a decision, which results under information asymmetry in less-than-optimal contracts. The illustration showcases that while the operation by $A$ is a single-stage process (i.e., asset produces an output with a given efficiency), in the real world the decisions by $B$ are interlinked with the operation by $A$ (multi-stage process). This gives rise the to question: \textit{Which admenments to contractural negeotions, underpinned by self-interest, are imperative to ensure indivdual utility maximinition, while inducing a common strive to mitigate implication on public goods?}

The chain-decision making process outlined in this case describes an agency problem involving public goods, which is specified in this study as a \textit{green contracting problem} (GCP). The carbon footprint (undesired output) relates to the contractual partners interlinked decision-making process. By contrast, the contract's utility (desired output) is distinct and underpinned by the contractual negotiation process. Guided by the theoretical work by [@chen_additive_2020], this study solves the green contracting problem by joining an additive slacks-based model and a network data envelopment analysis (ASBM-NDEA). In light of varying firm interaction dynamics which impact contractual negations, this study implements and advances the work through a game-theoretical approach. Specifically, this study illustrates the GCG through a classic agency problem in transportation science, that is, the charter market [@rana_model_1988] in the liner shipping industry.

Liner operations refer to ocean containerized cargo services, which operate along fixed services and schedules [reviews by @christiansen_ship_2004; @christiansen_ship_2013; @christiansen_liner_2020]. As the tether between buyer and supplier relations, the industry is critical to the global merchandise trade [@fransoo_critical_2013]. Yet, in light of the most recent IPCC report [-@ipcc_climate_2021], the industry is under scrutiny given the significant carbon footprint [@imo_fourth_2020]. Similar to other industries, liner operators suffer under uncertain markets. Operators are required to adjust capacity to these disruptions, seasonal changes and cycles. Due to capital-intensive production units, agile asset management is key to survival [@agarwal_ship_2008; @agarwal_network_2010]. This involves the ability to \textit{charter-in}, and \textit{charter-out} [p. 473, @christiansen_ship_2013]. The contract or charterparty involves a shipowner (owner) and an operator (charterer). In liner shipping, the most common charterparty is the time-charter contract: An owner transfers the ship's operational control to the charterer for a specific period. The charterer pays operational costs, while the owner covers the capital costs. Under the owner's and charter's self-interest to maximize individual utility, i.e., negotiating a suitable charter rates and lowering emissions (associated with lower bunkering costs), this study applies the GCP to illustrate the approach. Joining the charter contract sample by Kumar et al. [@kumar_suppliers_2021], including 3,878 contract in the observation period from January 2015 until December 2020, with operational information gathered collaboration with the Clean Cargo Initiative (CCI) [-@cci_clean_2020], enables this study to provide unique industry insights. Guided by prior research concerning chartering operations in the shipping industry [@kohn_gam_2011; @agnolucci_energy_2014; @adland_does_2017; @dirzka_principal-agent_2021], the variables utilized in the case study were selected.

This study appeals the sustainable operations and transportation science research streams by introducing a green contracting problem.

=> REQUIRES INPUT BY SECTION 4.2. and 4.3.

Subsequently, the section [2](#sec:sec2) outlines the additive slacks-based model and network data envelopment model proposed by Chen and Zhu (2020) and our extension, which incorporates a game theoretical perspective. Such perspective incorporates a cooperative-, owner-dominance, and charter-dominance view. The data description is outlined in section [3](#sec:sec3), which includes serves to introduce the case study and information sourcing process. The results under respective games are described, as well as, the methodological and policy implications are summarised in section [4](#sec:sec4). In section [5](#sec:sec5), the limitation are discussed and future research avenues.

# 2. Game Theoretical ASBM-NDEA Methodology {#sec:sec2}

Standard data envelopment analysis (DEA) models include the assumption that DMUs are black boxes that consume input(s) to produce output(s) [@avkiran_opening_2009]. As a result, the essential internal structures in operational processes are neglected when a system is composed of several stages and evolves over a time period. An emerging stream within operations research considers network DEA models as a means to rectify this problem [@chen_second_2017]. Core to network DEA models is the engagement with possible conflicts among different stages that arises due to the intermediate measures which connect those stages [@liang2008dea].

Initial scholarly debate examined models, which assume that outputs in the first stage are just input in the second stage. Subsequent developments extended the network DEA models, i.e., each stage consists out of individual inputs and individual output(s) [@cook2010network]. While such formulation is closer to actual operations, its solution process requires converting the non-linear program to a linear program to locate optimal global results. In this case, second-order cone programming (SOCP) was proposed as a mature convex optimization to convert the non-linear models into solvable linear models [@chen2020conic].

In contrast to DEA models, such as the CCR or BCC model [@charnes1978measuring; @banker1984some], which assume a proportional change in inputs and output while disregarding the effect of non-radial slacks in the efficiency [@tone2010dynamic], non-radial network DEA models can address the slacks. These non-radial slack-based models split among ratio-form based model (SBM) and additive-form based model (ASBM) [@tone2001slacks]. As the first work to integrate input and output slacks into the additive DEA model, Charnes et al. [-@charnes1985foundations] accounted for the substitutional changes. Green et al. [-@green1997note] advanced the ASBM by incorporating a non-linear format, which allows to benchmark DMUs without the certainty of achieving a global optimization.

\begin{figure}
\begin{center}
\includegraphics[width=10cm,height=5cm]{payload/00 graphic/stage_model.png}
\caption{The Network-ASBM Model}
\label{fig:fig1}
\end{center}
\end{figure}

Despite the SBM popularity in the OR research stream [@tone2009network; @fukuyama2010slacks], the network ASBM model showcases some advantages: network ASBMs enable to outline efficiency more directly compared to SBMs. In addition, network ASBMs do not require changes in production possibility set and constraints when linked to the SOCP technology. Moreover, in network ASBMs the decision-makers can assign the weights flexibility (compared to the SBM-based network model by Kao [-@kao2009efficiency]) based on individual choice and are not bound to follow pre-designed weights based on a weight function. Lastly, such an approach can be divided into cooperative and uncooperative models in existing research [@cook2014data], which will be further discussing in the subsequent sections.

Our initial modelling approach considers a two-stage general network model as outlined in fig.\ref{fig:fig1}. Let $DMU_{j}(j = 1,. . .,n)$ be the set of $n$ given homogeneous Decision Making Units (DMUs), term $j$ in the subscript represents the $j_{th}$ $DMU (j = 1,. . .,n)$. Appendix [A.](#apA) lists the notations used in this section.

Due to this study's focus on eco-efficiency, which split among economic value and environmental damage, we pre-process the undesirable outputs. Omitting these outputs would lead to incomplete results in practice [@yang2009incorporating]. The network DEA model allows to include the undesirables as inputs and outputs [e.g., @hua2008performance]. This works implements the established approach by Seiford and Zhu [-@seiford2002modeling], which uses a monotone decreasing transformation to handle the undesirable outputs in DEA models. Initially, a negative term ($-1$) is multiplied with function, and a proper translation vector is added to change all terms to non-negative variables. The translated dataset results in $\bar{y}_{u j}^{1}=-y_{u j}^{1}+v_{1}>0$ and $\bar{y}_{l j}^{2}=-y_{l j}^{2}+v_{2}>0$, with $v_{1},v_{2}$ representing the proper translation vectors and $\bar{y}_{u j}^{1}>0$ and $\bar{y}_{l j}^{2}>0$ to handle the undesirable outputs in this model.

In order to deal with the non-discretionary inputs (exogenous variables) in the production process, we apply the monotone decreasing transformation by Hua [-@hua2007eco]:

\begin{equation}
\bar{x}_{j}=k-x_{j}
\end{equation}

## 2.1. Competitive Interaction

In this section, we extend the general cooperative network model adopted
by [@chen_additive_2020] to analyze this non-cooperative two-stage
network structure. Our network model is showed in \ref{fig1} . Assume we
have N homogeneous Decision Making Units (DMU), $DMU_{j}$ means the
$j_{th}$ $DMU (j = 1,. . .,n)$. At the beginning, we assume the first
stage is more important than the second stage , so the first stage is
the leader and the second stage is the follower.

\begin{figure} [h]
\centering
\includegraphics[height=13cm,width=10cm]{Figure1.eps}
\caption{The Structure of the ASBM network model}
\label{fig:fig1}
\end{figure}

In the first stage, each $DMU_j$ consumes M inputs denoted as $x_{mj}$
(m=1,2,···，M), $K$ non-discretionary inputs denoted as $x_{kj}$
(k=1,2,···，K) to produce P desirable outputs denoted as $y_{pj}$
(p=1,2,···，P), and $U$undesirable outputs $y_{uj}$ (m=1,2,···，U). In
addition, we denote there are D intermediate measures $z_{dj}$
(d=1,2,3···, D) linking two stages.

Before the estimation of the efficiency, we applied the approach by
[@seiford_modeling_2002] to transfer the non-discretionary variables: a
negative term (−1) is multiplied with the function, and a proper
translation vector is added to change all terms to non-negative
variables. Therefore, the translated data set for undesirable outputs in
the first stage are listed as follows:

\begin{equation}
\bar{y}_{u j}^{1}=-y_{u j}^{1}+v_{1}>0
\end{equation}

Moreover, the non-discretionary inputs cannot be improved and the higher
values means favorable environment, so here the non-discretionary input
is converted into a reverse non-discretionary input as [@hua2007]did in
their work: applying a linear monotone decreasing transformation and
then add a proper translation vector $w$, we have:

\begin{equation}
\bar{x}_{k j}^{1}=-x_{k j}^{1}+w>0
\end{equation}

A specific $DMU_o$ (subscript o denotes the DMU estimated，o
=1，2，3....n). The efficiency for the specific DMU is defined in
Equation (3), which adapted by[@chen_additive_2020]:

\begin{equation}
min \frac{1}{P+U+M+D}\left(\sum_{p=1}^{P} \frac{y_{p o}^{1}}{y_{p o}^{1}+s_{p}^{+}}+\sum_{u=1}^{U} \frac{\bar{y}_{u o}^{1}}{\bar{y}_{u o}^{1}+s_{u}^{+}}+\sum_{m=1}^{M} \frac{x_{m o}-s_{m}^{-}}{x_{m o}}+\sum_{d=1}^{D} \frac{z_{d o}}{z_{d o}+t_{d}^{+}}\right)
\end{equation}

Where $s_{m}^{-}, s_{p}^{+}, s_{u}^{+}, t_{d}^{+}$ are respectively the
excess inputs, good input deficits and bad output excesses (slack
vectors) in the first stage, $\lambda_{j}^{1}$ is the unknown intensity
vector in the first stage. The constraints are as follows:

\begin{align}&\sum_{j=1}^{n} x_{m j} \lambda_{j}^{1}+s_{m}^{-}=x_{m o}, \forall m, \\ &\sum_{j=1}^{n} \bar x_{k j} \lambda_{j}^{1}=\bar x_{k o}^{1}, \forall k, \\ &\sum_{j=1}^{n} y_{p j} \lambda_{j}^{1}-s_{p}^{+}=y_{p o}^{1}, \forall p, \\ &\sum_{j=1}^{n} y_{u j} \lambda_{j}^{1}-s_{u}^{+}=y_{u o}^{1}, \forall u, \\ &\sum_{j=1}^{n} z_{d j} \lambda_{j}^{1}-t_{d}^{+}=z_{d o}, \forall d \\ &\sum_{j=1}^{n} \lambda_{j}^{1}=1\end{align}

Without the last constraint, then the market shows CRS (constant return
of scale), which is common in a competitive environment. As the most
market is not a perfectly competitive market, we assume VRS (variable
returns to scale). Then we apply SOCP to solve the non-linear problem
and transfer the equation above:

\begin{equation}
\begin{array}{c}\frac{1}{P+U+M+D} \sum_{p=1}^{P} \frac{y_{p o}^{1}}{y_{p o}^{1}+s_{p}^{+}}+\frac{1}{P+U+M+D} \sum_{u=1}^{U} \frac{\bar{y}_{u o}^{1}}{\bar{y}_{u o}^{1}+s_{u}^{+}} \\+\frac{1}{P+U+M+D} \sum_{d=1}^{D} \frac{z_{d o}}{z_{d o}+t_{d}^{+}}+\frac{1}{P+U+M+D} \sum_{m=1}^{M} \frac{x_{m o}^{1}-s_{m}^{-}}{x_{m o}^{1}}\end{array}
\end{equation}

For each term here, we define its upper bound as
$\xi_{p}^{1}, \xi_{u}^{2}, \xi_{d}^{3}, \xi^{4}$ respectively. Thus, the
model is equivalent to the following one:

\begin{equation}\min \sum_{p=1}^{P} \xi_{p}^{1}+\sum_{p=1}^{U} \xi_{u}^{2}+\sum_{d=1}^{D} \xi_{d}^{3}+\xi^{4}\end{equation}

\begin{align}\text { s.t } &\frac{1}{P+U+M+D} \frac{y_{p o}^{1}}{y_{p o}+s_{p}^{+}} \leq \xi_{p}^{1}, \forall p, \\&\frac{1}{P+U+M+D} \frac{\bar{y}_{u o}^{1}}{\bar{y}_{u o}^{1}+s_{u}^{+}} \leq \xi_{u}^{2}, \forall u, \\&\frac{1}{P+U+M+D} \frac{z_{d o}}{y_{d o}+t_{d}^{+}} \leq \xi_{d}^{3}, \forall d, \\&\frac{1}{P+U+M+D} \sum_{m=1}^{M} \frac{x_{m o}-s_{m}^{-}}{x_{m o}} \leq \xi^{4}, \forall m\end{align}

The first constraint can be transformed as:

\begin{equation}
(P+U+M+D)\left(y_{p o}^{1}+s_{p}^{+}\right) \xi_{p}^{1} \geq\left(\sqrt{y_{p o}^{1}}\right)^{2}
\end{equation}
which is equivalent to:
\begin{equation}
\sqrt{\left(\sqrt{y_{p o}^{1}}\right)^{2}+\left(\frac{1}{2}(P+U+M+D)\left(y_{p o}^{1}+s_{p}^{+}\right)-\xi_{p}^{1}\right)^{2}} \leq \frac{1}{2}\left((P+U+M+D)\left(y_{p o}^{1}+s_{p}^{+}\right)+\xi_{p}^{1}\right)
\end{equation}

For the other constraints, we applied the same approaches used by
[@chen_additive_2020] to rewrite the structure. Then the problem can be
converted to the following optimization model:

\begin{equation}
\min \sum_{p=1}^{P} \xi_{s}^{1}+\sum_{u=1}^{U} \xi_{u}^{2}+\sum_{d=1}^{D} \xi_{d}^{3}+\xi^{4}
\end{equation}

\begin{align} \text { s.t }&\left\|\frac{\sqrt{y_{p o}^{1}}}{\frac{1}{2}(P+U+M+D)\left(y_{p o}+s_{p}^{+}\right)-\xi_{p}^{1}}\right\|_{2} \leq \frac{1}{2}(P+U+M+D)\left(y_{p o}^{1}+s_{p}^{+}\right)+\xi_{p}^{1}, \forall p \text { , }\\ &\left\|\frac{\sqrt{y_{u o}^{1}}}{2}(P+U+M+D)\left(\bar{y}_{u o}^{1}+s_{u}^{+}\right)-\xi_{u}^{2}\right\|_{2} \leq \frac{1}{2}(P+U+M+D)\left(\bar{y}_{u o}^{1}+s_{u}^{+}\right)+\xi_{u}^{2}, \forall u,\\ &\left\|\frac{\sqrt{z_{d o}}}{\frac{1}{2}(P+U+M+D)\left(z_{d o}+t_{d}^{+}\right)-\xi_{d}^{3}}\right\|_{2} \leq \frac{1}{2}(P+U+M+D)\left(z_{d o}+t_{d}^{+}\right)+\xi_{d}^{3}, \forall d\\ &\frac{1}{P+U+M+D} \sum_{m=1}^{M} \frac{x_{m o}-s_{m}^{-}}{x_{m o}} \leq \xi^{4}, \forall m\end{align}

At the same time, it still satisfies the constraints （4)-(9).

After solving this equation, we can get the optimal results
($\lambda_{j}^{1^{*}}, s_{m}^{-*}, s_{p}^{+*}, s_{u}^{+*}, t_{d}^{+*}$).
The projections are the recommendations for improvement for each
$DMU_{j}$ , the projections for the first stage are as follows:

\begin{align}&\sum_{j=1}^{n} x_{m j}^{1} \lambda_{j}^{1}=x_{m o}^{1}-s_{m}^{-*} \\&\sum_{j=1}^{n} y_{p j}^{1} \lambda_{j}^{1}=y_{p o}^{1}+s_{p}^{+*} \\&\sum_{j=1}^{n} y_{u j}^{1} \lambda_{j}^{1}=y_{u o}^{1}+s_{u}^{+*} \\&\sum_{j=1}^{n} z_{d j} \lambda_{j}^{1}=z_{d o}+t_{d*}^{+}\end{align}

Like the Figure(1) shows, we assume that in the second stage, H inputs
denoted by $x_{hj}$ (h=1,2,···, H) produces S desirable outputs $y_{sj}$
(m=1,2,···，S) and L undesirable outputs $y_{lj}$ (l=1,2,···，L). Before
the efficiency estimation, the process of the undesirable output in the
second stage is similar to what we did in the first stage.

\begin{equation}
\bar{y}_{l j}^{2}=-y_{l j}^{2}+v_{2}>0
\end{equation}

Since the second stage is the follower, it must accept the optimal
solutions from the first stage and thus keeps continuity between
different stages (therefore the followers must accept the projections of
the first stage regarding intermediates), therefore:

\begin{equation}
\sum_{j=1}^{n} z_{d j} \lambda_{j}^{1}=\sum_{j=1}^{n} z_{d j} \lambda_{j}^{2}, \forall d
\end{equation}
And this projection is already decided in the first stage and the
player in the second stage has to accept the target. To sum up, they
must satisfy the constraint:

\begin{equation}
\sum_{j=1}^{n} z_{d j} \lambda_{j}^{2}=z_{d o}+t_{d}^{+*}, \forall d
\end{equation}
Similarly, the efficiency of the second stage is as follows:

\begin{equation}\min \frac{1}{R+L+H}\left(\sum_{r=1}^{R} \frac{y_{r o}^{2}}{y_{r o}^{1}+s_{r}^{+}}+\sum_{l=1}^{L} \frac{\bar{y}_{l o}^{2}}{\bar{y}_{l o}^{2}+s_{l}^{+}}+\sum_{h=1}^{H} \frac{x_{h o}^{2}-s_{h}^{-}}{x_{h o}^{2}}\right)\end{equation}

constraint to:

\begin{align}&\sum_{j=1}^{n} x_{h o}^{2}+s_{h}^{-}=x_{h o}^{2}, \forall h, \\&\sum_{j=1}^{n} y_{r o} \lambda_{j}^{2}-s_{r}^{+}=y_{r o}, \forall r \\&\sum_{j=1}^{n} \bar{y}_{l o} \lambda_{j}^{2}-s_{l}^{+}=y_{l o}, \forall l \\&\sum_{j=1}^{n} \lambda_{j}^{2}=1\end{align}

Where $s_{h}^{-}, s_{r}^{+}, s_{l}^{+}, t_{d}^{-}$ are respectively the
excess inputs, good input deficit, and bad output excess (slack vectors)
in the second stage, $\lambda_{j}^{2}$ is the unknown intensity vector
in the second stage. And we use the same approach to convert it to a
SOCP problem as follows:

\begin{equation}
\min \sum_{r=1}^{R} \xi_{r}^{1}+\sum_{l=1}^{L} \xi_{l}^{2}+\xi^{3}
\end{equation}

\begin{align}
&\left\|\begin{array}{c}
\sqrt{y_{r o}^{2}} \\
\frac{1}{2}(R+L+H)\left(y_{r o}^{2}+s_{r}^{+}\right)-\xi_{s}^{1}
\end{array}\right\| \leq \frac{1}{2}\left((R+H+L)\left(y_{r o}^{2}+s_{r}^{+}\right)+\xi_{r}^{1}\right), \forall r,\\
&\left\|\begin{array}{c}
\sqrt{\bar{y}_{l o}^{2}} \\
\frac{1}{2}(R+L+H)\left(\bar{y}_{l o}^{2}+s_{l}^{+}\right)-\xi_{l}^{2}
\end{array}\right\| \leq \frac{1}{2}\left((R+L+H)\left(\bar{y}_{l o}^{2}+s_{l}^{+}\right)+\xi_{l}^{2}\right), \forall l,\\
&\frac{1}{R+L+H}\left(\sum_{h=1}^{H} \frac{x_{h o}^{2}-s_{h}^{-}}{x_{h o}^{2}}\right) \leq \xi^{3}, \forall h
\end{align}

it still satisfies the constraints （23)-(26)

When the second stage is the dominated stage, the approach is the same,
so we listed the part in the \ref{Appendix}.

## 2.2 Cooperative Two-stage ASBM Model

Under the cooperative supply chain model, those two stages cooperate to
maximize efficiency jointly. Compared with Chen and Zhu's model, we add
the undesirable outputs into the model by [@chen_additive_2020] and
estimated the overall efficiency of the whole supply chain:

Here we assume the weights on two stages are $w$ and $1-w$, the
processes of the undesirable outputs and the non-discretionary inputs
are the totally the same like what we did in the leader-follower models,
(so 2, 5,7). then the efficiency of the whole supply chain is as
follows:

\begin{equation}
\begin{aligned}
&w \frac{1}{P+U+M+D}\left(\sum_{p=1}^{P} \frac{y_{p o}^{1}}{y_{p o}+s_{p}^{+}}+\sum_{u=1}^{U}
\frac{\bar{y}_{u o}^{1}}{y_{u o}^{1}+s_{u}^{+}}+\sum_{m=1}^{M} \frac{x_{m o}^{1}-s_{m}^{-}}{x_{m o}^{1}}+\sum_{d=1}^{D} \frac{z_{d o}}{z_{d o}+t_{d}^{+}}\right) \\
&+(1-w) \frac{1}{R+L+H+D}\left(\sum_{r=1}^{R} \frac{y_{r o}^{2}}{y_{r o}^{2}+s_{r}^{+}}+\sum_{l=1}^{L} \frac{\bar{y}_{l o}^{2}}{y_{l o}^{2}+s_{l}^{+}}+\sum_{h=1}^{H} \frac{x_{h o}^{2}-s_{h}^{-}}{x_{h o}^{2}}+\sum_{d=1}^{D} \frac{z_{d o}-t_{d}^{-}}{z_{d o}}\right)
\end{aligned}
\end{equation}
which is constraint to
\begin{equation}
\text { s.t. }\begin{array}{l}
\sum_{j=1}^{n} x_{m j} \lambda_{j}^{1}+s_{m}^{-}=x_{m o}, \forall m \\
\sum_{j=1}^{n} x_{h j}^{2} \lambda_{j}^{2}+s_{h}^{-}=x_{h o}^{2}, \forall h \\
\sum_{j=1}^{n} \bar x_{k j} \lambda_{j}^{1}=\bar x_{k o}^{1}, \forall k, \\
\sum_{j=1}^{n} y_{i j} \lambda_{j}^{2}-s_{r}^{+}=y_{r o}, \forall r \\
\sum_{j=1}^{n} \bar{y}_{l j} \lambda_{j}^{2}-s_{l}^{+}=\bar{y}_{l o}, \forall l \\
\sum_{j=1}^{n} y_{p j} \lambda_{j}^{1}-s_{p}^{+}=y_{p o}, \forall p \\
\sum_{j=1}^{n} \bar{y}_{u j}^{1} \lambda_{j}^{1}-s_{u}^{+}=\bar{y}_{u o}^{1}, \forall u \\
\sum_{j=1}^{n} \lambda_{j}^{1}=1, \sum_{j=1}^{n} \lambda_{j}^{2}=1, \\
s_{m}^{-}, s_{h}^{-}, s_{r}^{+}, s_{l}^{+}, s_{p}^{+}, s_{u}^{+}, \lambda_{j}^{1}, \lambda_{j}^{2} \geq 0
\end{array}.
\end{equation}
Importantly,, the intermediates are fixed in our model, therefore:

\begin{equation}
\sum_{j=1}^{n} \lambda_{j}^{1} z_{d j}=\sum_{j=1}^{n} \lambda_{j}^{2} z_{d j}, \forall d
\end{equation}

The corresponding model can be converted into a SOCP problem (see
[@chen_additive_2020]).

\begin{equation}
\min \sum_{p=1}^{p} \xi_{p}^{1}+\sum_{u=1}^{U} \xi_{u}^{2}+\sum_{r=1}^{R} \xi_{r}^{3}+\sum_{l=1}^{L} \xi_{l}^{4}+\sum_{d=1}^{D} \xi_{d}^{5}+\xi^{6}
\end{equation}

\begin{align}
&\left\|\begin{array}{c}
\sqrt{y_{p o}^{1} w}\\
\frac{1}{2}\left((P+U+M+D)\left(y_{p o}^{1}+s_{p}^{+}\right)-\xi_{p}^{1}\right)\end{array}\right\|_{2} \leq \frac{1}{2}\left((P+U+M+D)\left(y_{p o}^{1}+s_{p}^{+}\right)+\xi_{p}^{1}\right). \forall p\\
&\left\|\begin{array}{c}
\sqrt{\bar{y}_{u o}^{1} w} \\
\frac{1}{2}\left((P+U+M+D)\left(\bar{y}_{u o}^{1}+s_{u}^{+}\right)-\xi_{u}^{2}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((P+U+M+D)\left(\bar{y}_{u o}^{1}+s_{u}^{+}\right)+\xi_{u}^{2}\right) . \forall u,\\
&\left\|\begin{array}{c}
\sqrt{y_{r o}^{2}(1-w)} \\
\frac{1}{2}\left((P+U+M+D)\left(\bar{y}_{u o}^{1}+s_{u}^{+}\right)-\xi_{u}^{2}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((R+L+H+D)\left(y_{r o}^{2}+s_{r}^{+}\right)+\xi_{r}^{3}\right) . \forall \mathrm{r},\\
&\left\|\begin{array}{c}
\sqrt{\bar{y}_{l o}^{2}(1-w)} \\
\frac{1}{2}\left((R+L+H+D)\left(\bar{y}_{l o}^{2}+s_{l}^{+}\right)-\xi_{l}^{4}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((R+L+H+D)\left(\bar{y}_{l o}^{2}+s_{l}^{+}\right)+\xi_{l}^{4}\right) . \forall l,\\
&\left\|\begin{array}{c}
\sqrt{z_{d o} w} \\
\frac{1}{2}\left((P+U+M+D)\left(z_{d o}+t_{d}^{+}\right)-\xi_{d}^{5}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((P+U+M+D)\left(z_{d o}+t_{d}^{+}\right)+\xi_{d}^{5}\right) . \forall d,\\
&\frac{w}{P+U+M+D} \sum_{m=1}^{M} \frac{x_{m o}-s_{m}^{-}}{x_{m o}}+\frac{1-w}{R+L+H+D}\left(\sum_{h=1}^{H} \frac{x_{h o}^{2}-s_{h}^{-}}{x_{h o}^{2}}+\sum_{d=1}^{D} \frac{z_{d o}-t_{d}^{-}}{z_{d o}}\right) \leq \xi^{6}
\end{align}

# 3. Data Description {#sec:sec3}

To illustrate the \textit{green contracting problem} and insofar the proposed game theortical ASBM-NDEA approch, a case study is introduced. The GCP is specifed as an agency problem with implications on public goods the GCP, involving an agent and a principal. In this problem the princiapl contracts the agent to provide an operation in their name, giving some authority to the agent [@ross_economic_1973]. Under information asymmetry (or self-interest, bounded rationality and risk aversion), the agency relations give rise to less-than optimal contractural agrranments, i.e., designated as the principal-agent problem [@eisenhardt_agency_1989; @stiglitz_principal_1989]. Building upon works in transportation science, which pointed to the principal-agent problem as a barrier to sustainbile operations [@blumstein_overcoming_1980; @weber_reflections_1997], this section introduces variable selction process and the sample in regards to the charter market in the liner shipping industry.

In the charter market, the agent is the party which knows more about the contract object's sustaible perfroannce. Given that the shipowner establishes the technological baseline, it can be postulated that the onwer is the agent, while the principal operates the ship and pays the charter rate and operational costs. The princiapl-agent problem arises when the agent can not communicate -underlying information asymmetries- the basline via the charter rate to the princiapl. In short, the problem can be descibed as a transcation in which the party responsible for the investment decisions is not the party, which pays the future costs that arise from the investment [@vernon_identification_2012]. These split incentives casuse to the priorly noted less-than optimal contractural aegrranments, which disinzentives eco-technologies [@acciaro_energy_2013; @jafarzadeh_framework_2014].

The variable selection is guided by reference works [see @kohn_gam_2011; @agnolucci_energy_2014; @adland_does_2017; @dirzka_principal-agent_2021] which identifed the princapal-agent problem and adverse consequences on sustainbile performance in the charter market. Aligned with the game theortical ASBM-NDEA approch, the selection process splits into an owner perspective on charter rate negotions and charterer perspective on resulting operations.

=> IS AEcoI INCLUDED IN THE FIRST OR SECOND STAGE? -- <span style="background-color: Yellow">I did not get it. What is AECoI</mark> THIS IS THE ABB for: Asset eco-investements (index)

In the first stage, the asset size (\textit{ACap}) and insofar scale economies impact on freight rates and environmental performance is allocated. Capital costs (\textit{CAPEX}) is another cirtical variable, allowing to assess the yield range by the owner and is estimated based on depreciated newbuild prices and transformed the value to daily cost via a standard 8-years financing based on LIBOR. The negoiated contract length (\textit{CLen}) serves to mirror the willingness of owners and charterers to engage in a contractual relation, besides indicating the market sentiment. Similary, the market freight rate (\textit{MRate}) provides insights into the overall supply-demand equlirbium. Subject to various ship sizes and designs (i.e., narrow beam, eco-design, or standard design), the rate is assigned to indidivaul contracts. Specifically, the eco-design and supplementary investments by (\textit{AEcoI}) the owner into the ship are covered by an index, which constructed based on the work by Bouman et al. [-@bouman_state---art_2017]. The undesired output is the Estimated Index Value (\textit{ATEff}), an index devised by Faber et al. [-@faber_estimated_2015] to capture the technical performance (i.e., engine performance). The link between the owner and charter perspective is the negotiated charter rate (\textit{CRate}), consituting the intermediate in ASBM-NDEA approach. In the second stage, bunkering costs (\textit{OPEX}) that are impacted by the descisions in the prior stage are included. These bunkering costs are equivalent to approximately 30–50% [@fransoo_critical_2013]in overall OPEX and covered by the charterer. The costs are estiamted using the bunker consumption throughout the contract length and respective prices tied to the contract starting date. To capture the operational behavior dynamics (\textit{AOps}) throughout the contract, the sample includes an operational intensity index, which is designed as sailed distance and contract lengths ratio. The undesired output refers to carbon emissions (\textit{CEm}) emitted during operations and is calculated by considering bunker consumption and respectives types, i.e., the carbon emissions in tons are divided by the asset size, sailed distances and contract length. The desired output is the ships productivty, which is specifed as the transported cargo units by the charterer per sailed distance and contract length. To set the variable selection into context, the tab.\ref{tab:tab1} provides futher desciptions and respective relations to the reference works.

\begin{table}[!htbp] \centering
\caption{Variable choice and literature base}
\label{tab:tab1}
\small
\begin{tabular}{@{\extracolsep{5pt}} p{0.5cm}p{0.5cm}p{0.6cm}p{0.7cm}p{6.9cm}p{1cm}p{1cm}p{1cm}p{1cm}}
\\[-2ex]\hline
Stage & Opr. & Units & Source & Description & \rotatebox{70}{Köhn and Helen (2011)} & \rotatebox{70}{Agnolucci et al. (2014)} & \rotatebox{70}{Aldand et al. (2017)} & \rotatebox{70}{Dirzka et al. (2021)} \\ \hline

\textbf{I.} & $x_{m1}$ & TEU & CCI & \textit{ACap}: Ship capacity in twenty-foot eqv. unit & \text{*} & \text{*} & \text{*} & \text{*} \\
& $x_{m2}$ & USD & CL & \textit{$\sfrac{CAPEX}{day}$}: Capital expense per day & & & & \\
& $x_{m3}$ & Days& K/CCI & \textit{CLen}: Contractual agreement length & \text{*} & \text{*} & & \text{*} \\
& $x_{m4}$ & diml. & CL & \textit{AEcoI}: Investment index in eco-systems & & & \text{*} \\
& $x_{k}$ & USD & CL & \textit{$\sfrac{MRate}{day}$}: Market freight rate per day & \text{*} & \text{*} & \text{*} & \\
& $y_{p} $ & diml. & CL & \textit{ATEff}: Estimated Index Value (EIV) & & \text{*} & \text{*} & \\
\textbf{Int.} & $Z_{d}$ & USD & K & \textit{$\sfrac{CRate}{day}$}: Contract freight rate per day & \text{*} & \text{*} & \text{*} & \\
\textbf{II.} & $x_{h1}$ & USD & CL/CCI & \textit{$\sfrac{OPEX}{day}$}: Operational expenses per day & & \text{*} & \text{*} & \\
& $x_{h2}$ & diml. & CCI & \textit{AOps}: Operational intensity in distance per day & & & & \text{*} \\
& $y_{l}$ & diml. & CCI & \textit{CEm}: Operational carbon emissions & \text{*} & & \text{*} & \text{*}\\
& $y_{r}$ & diml. & CCI & \textit{AProd}: Contract productivity in distance per day & & & & \\

\hline \\[-1.8ex]
\end{tabular}
\subcaption*{Note: Source abbriavtions refer to the Kumar et al. (K), Clean Cargo Initiative (CCI) and Clarksons Shipping Intelligence (CL) database.
\end{table}

The sample used to illustrate the GCP was sourced via the database by Kumar et al. [-@kumar_suppliers_2021] and includes 12.628 contracts, which were issued throughout the years 2015-2020. Each contract in the database is linked to the asset (i.e., IMO identifiacation number) used in the transaction, charterer, owner and charter rate in USD per day. Besides, the Clean Cargo Initiative's (CCI) [-@cci_clean_2020] database was used to map actual operations (i.e., bunker consumption and insofar carbon emissions). The CCI is a business-to-business initiative, which provides a green shipping forum to ocean carriers, freight forwarders, and cargo owners. An annual survey these information on an asset-level, allowing granualrity above the any other sample gathered by regulators (i.e., Data Collection System by the International Maritime Organization (DCS) and Monitoring, Reporting, and Verification (MRV) system by the European Union). At last, supplementary infromation were acquired via an ocean analytics provider, Clarkson Shipping Intelligence (CL). The asset identficantion number was used to join each database, which resulted in 3881 individual contracts issued by 25 charter firms with no fragmentary information sets.

As shown in the tab.\ref{tab:tab2}, the descriptive statistics and correlation analysis indicates that: [i] The average ship in the sample can carry 2974 twenty-foot containers (TEU), which is below the average across the ocean liner shipping sector, indicating that rather smaller tonnage is contracted in the charter market. Naturally the asset size is positively correlated the charter rate, costs and productivity. [ii] The contract length which ranges between 4 and 1860 days suggests a positive relation between the length the owner and charter would do business with each other and the market sentiment. A possible interpretation is that higher freight market incentivce charterers to \textit{lock-in} rates. Yet, the contract length displays no signficant realtion in reagrdes to sustainble performance. [iii] Design and operational performance is signifcantly negativly related, i.e., higher the design performance less emissions is the asset emitting during the operation. It shall be noted that design performance is interrelated with scale economies and insofar assets with less usage --in liner shipping larger ships are usually newer once. [iv] The eco-systems investment index relates postivley to charter rates and carbon emssions, while the first observation can be resasoned by some frieght premium attributed to such investments, the latter one is ambigous. [v] Operataional intensity is most signficantly related to carbon emissions (higher than productivity), which is in line with common undersatnding behind pollution in transportion.

\begin{table}[!htbp] \centering
\caption{Descriptive statistics and correlation}
\label{tab:tab3}
\small
\begin{tabular}{@{\extracolsep{2pt}} lllllll}
\\[-2ex]\hline
& Opr.& N & Mean & Std & Min & Max \\ \hline
Contract rate (USD/day) & CRate & $3,881$ & $10,026.650$ & $5,069.280$ & $3,200$ & $41,000$ \\
Contract length (days) & CLen & $3,881$ & $308.490$ & $146.030$ & $4$ & $1,860$ \\
Market rate (USD/day) & MRate & $3,881$ & $11,377.280$ & $6,577.700$ & $3,700$ & $39,875$ \\
Asset operations (nm/day) & AOps & $3,881$ & $203.220$ & $67.760$ & $25.120$ & $1,180.220$ \\
Asset productivity (tranported-TEU/nm/day) & AProd & $3,881$ & $497.980$ & $478.390$ & $14.630$ & $3,169.390$ \\
Carbon emissions (ton/day) & CEm & $3,881$ & $89.960$ & $58.650$ & $2.970$ & $489.240$ \\
Asset capacity (nominal TEU) & ACap & $3,881$ & $2,974.690$ & $2,173.820$ & $342$ & $13,566$ \\
Asset technical efficency (index) & ATEff & $3,881$ & $21.570$ & $5.360$ & $8.480$ & $49.410$ \\
Asset eco-investment (index) & AEcoI & $3,881$ & $110.100$ & $8.690$ & $100$ & $142.860$ \\
Asset OPEX (USD/day) & OPEX & $3,881$ & $9,424.450$ & $6,491.830$ & $814.890$ & $47,190.310$ \\
Asset CAPEX (USD/day) & CAPEX & $3,881$ & $6,557.290$ & $2,707.290$ & $1,995.260$ & $19,689.760$ \\
\hline \\[-1.8ex]
\end{tabular}
\begin{tabular}{@{\extracolsep{-4pt}} llllllllllll}
\\[-2ex]\hline

& CRate & CLen & MRate & AOps & AProd & CEm & ACap & ATEff & AEcoI & OPEX & CAPEX \\ \hline
CRate & & & & & & & & & & & \\
CLen & 0.28 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & & & & & & & \\
MRate & 0.83 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.22 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & & & & & & \\
AOps & 0.33 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.03 \textasteriskcentered \textasteriskcentered & 0.37 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & & & & & \\
AProd & 0.69 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.14 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.79 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.65 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & & & & \\
CEm & 0.53 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.08 & 0.65 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.78 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.9 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & & & \\
ACap & 0.74 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.18 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.84 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.5 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.95 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.82 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & & \\
ATEff & -0.44 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.13 & -0.42 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.37 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.49 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.42 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.52 \textasteriskcentered \textasteriskcentered \textasteriskcentered & & & & \\
AEcoI & 0.24 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.1 & 0.3 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.14 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.21 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.15 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.2 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.12 & & & \\
OPEX & 0.5 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.07 & 0.61 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.64 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.79 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.86 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.73 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.37 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.13 & & \\
CAPEX & 0.48 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.12 & 0.62 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.39 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.72 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.68 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.75 \textasteriskcentered \textasteriskcentered \textasteriskcentered & -0.35 \textasteriskcentered \textasteriskcentered \textasteriskcentered & 0.08 & 0.6 \textasteriskcentered \textasteriskcentered \textasteriskcentered & \\

\hline \\[-1.8ex]
\end{tabular}
\end{table}

# 4. Results and Discussion {#sec:sec4}

Applying the ASBM-NDEA in a game-theoretical setting results in three distinct outputs. The Inter output addresses the cooperative contracting, while the Owner and Charterer outputs relate to contracting with one side dominating the contractual relation. Individual outputs are displayed in tab. 2. The outputs’ average displayed in the columns: 𝑦𝑖, 𝑦𝑜 and 𝑦𝑐.

To summarize, cooperative contracting appears to be the most suitable to enhance sustainable operations. Yet, it shall also be noted that with owner-dominated contracts it is possible to approach the best practice frontier. In contrast, charterer dominance is associated with less-than-optimal contracts. Given that the underlying industry structure is skewed, i.e., major charterer dominance and fragmented supply (see Davies 1986; Shashikumar 1995; Besides, the projections indicate a mismatch between scale economies and scale-to-operations. This implies that owners supply the market with unsuitable assets and operators charter assets which cannot be utilized productivity. Across the distinct outputs and years, the results show that owners and charterer should settle contracts with -7.5% to -17.5% less capacity to achieve more optimal contracts. Specifically, in the year 2020, which was marked by a major disruption, this result is apparent.
On the whole, the results show that asset productivity (utilization) should be strengthened, while operational intensity (voyage speed) is close to the optimum. In addition, the results point to possible improvements in regards to the technical baseline, which was capture by the Estimated Index Value (EIV) and investments into eco-systems. At last, it shall be pointed out that under owner dominance the charter rate should rise vice versa under chartered dominance the rate should diminish. Under cooperative contracts, the charter rate should also be higher to reach shared objectives, but less than under owner dominance.

\begin{table}[!htbp] \centering
\caption{Game Theoretical ASBM-NDEA Efficiency}
\label{tab:tab4}
\small
\begin{tabular}{@{\extracolsep{2pt}} lcccccccccc}
\\[-2ex]\hline
& \multicolumn{3}{c}{\textbf{Cooperative}} & \multicolumn{6}{c}{\textbf{Non-cooperative}} \\
Year & \multicolumn{3}{c}{} & \multicolumn{3}{c}{Shipowner-dominated} & \multicolumn{3}{c}{Charterer-dominated} \\ \cmidrule(l{3pt}r{3pt}){5-7} \cmidrule(l{3pt}r{3pt}){8-10}
& Overall & Shipowner & Charterer & Overall & Shipowner & Charterer & Overall & Shipowner & Charterer \\ \hline
$2015$ & $0.823$ & $0.884$ & $0.763$ & $0.766$ & $0.824$ & $0.709$ & $0.772$ & $0.863$ & $0.682$ \\
$2016$ & $0.850$ & $0.914$ & $0.786$ & $0.807$ & $0.839$ & $0.775$ & $0.802$ & $0.867$ & $0.736$ \\
$2017$ & $0.831$ & $0.900$ & $0.762$ & $0.782$ & $0.856$ & $0.708$ & $0.786$ & $0.862$ & $0.710$ \\
$2018$ & $0.852$ & $0.905$ & $0.799$ & $0.801$ & $0.845$ & $0.758$ & $0.792$ & $0.863$ & $0.721$ \\
$2019$ & $0.844$ & $0.898$ & $0.789$ & $0.819$ & $0.839$ & $0.799$ & $0.799$ & $0.841$ & $0.758$ \\
$2020$ & $0.834$ & $0.879$ & $0.789$ & $0.790$ & $0.826$ & $0.755$ & $0.793$ & $0.840$ & $0.745$ \\
\hline \\[-1.8ex]
\end{tabular}
\end{table}

For the whole chain (the average efficiency of the shipowner and charterer), the efficiency of the cooperative model is always higher than the uncooperative model. It could be the motivation to develop the cooperative model in the time charter market. The values in Column 1 are always higher than the values in Column 4 and Column 7.
The efficiency of the cooperative model is always bigger or equal to the efficiencies in uncooperative model: this result is consistent with the conclusion of Li et al (2012), therefore, our results help to verify that the solutions of the cooperative are the global optimal for the players in the network.
Under three scenarios, Shipowner’s efficiencies are always higher than the Charterer's. For those two actors in this supply chain, the shipowner is easier to get higher efficiency. (It may indicate that it is easier to make the economic-related stage efficient, but harder for the eco-related efficient).

=> INPUT REQUIRED: IS RATE THE MARKET OR CONTRACT RATE? --<span style="background-color: Yellow">the Rates are the contract rates as the marke is the environmental variable, the managers cannot change it</mark>

\begin{table}[!htbp] \centering
\caption{Game Theoretical ASBM-NDEA Projections}
\label{tab:tab5}
\small
\begin{tabular}{@{\extracolsep{22pt}} lcccccc}
\\[-2ex]\hline
& \multicolumn{6}{c}{Cooperative} \\ \cmidrule(l{3pt}r{3pt}){2-7}
& 2015 & 2016 & 2017 & 2018 & 2019 & 2020 \\ \hline

$ \Delta CLen$ & $0.563$ & $0.719$ & $0.446$ & $0.337$ & $0.261$ & $0.534$ \\
$ \Delta CRate$ & $$-$0.106$ & $$-$0.428$ & $$-$0.132$ & $$-$0.087$ & $$-$0.315$ & $$-$0.158$ \\
$ \Delta CEm$ & $$-$0.155$ & $$-$0.015$ & $$-$0.192$ & $$-$0.124$ & $$-$0.069$ & $$-$0.103$ \\
$ \Delta AOps$ & $$-$0.183$ & $$-$0.136$ & $$-$0.235$ & $$-$0.226$ & $$-$0.341$ & $$-$0.179$ \\
$ \Delta AProd$ & $0.559$ & $0.588$ & $0.706$ & $0.393$ & $0.242$ & $0.361$ \\
$ \Delta ACap$ & $$-$0.215$ & $$-$0.194$ & $$-$0.189$ & $$-$0.187$ & $$-$0.219$ & $$-$0.289$ \\
$ \Delta AEcoI$ & $$-$0.047$ & $$-$0.044$ & $$-$0.030$ & $$-$0.029$ & $$-$0.036$ & $$-$0.019$ \\
$ \Delta ATEff$ & $$-$0.129$ & $$-$0.070$ & $$-$0.117$ & $$-$0.082$ & $$-$0.157$ & $$-$0.121$ \\
$ \Delta CAPEX$ & $$-$0.355$ & $$-$0.349$ & $$-$0.333$ & $$-$0.327$ & $$-$0.414$ & $$-$0.385$ \\
$ \Delta OPEX$ & $$-$0.497$ & $$-$0.365$ & $$-$0.307$ & $$-$0.358$ & $$-$0.194$ & $$-$0.518$ \\

\hline \\[-1.8ex]
\end{tabular}
\begin{tabular}{@{\extracolsep{22pt}} lcccccc}
\\[-2ex]\hline
& \multicolumn{6}{c}{Shipowner-dominated} \\ \cmidrule(l{3pt}r{3pt}){2-7}
& 2015 & 2016 & 2017 & 2018 & 2019 & 2020 \\ \hline

$ \Delta CLen$ & $0.455$ & $1.410$ & $0.449$ & $0.430$ & $0.328$ & $0.717$ \\
$ \Delta CRate$ & $1.478$ & $0.921$ & $0.713$ & $0.886$ & $0.604$ & $1.172$ \\
$ \Delta CEm$ & $$-$0.174$ & $$-$0.008$ & $$-$0.195$ & $$-$0.115$ & $$-$0.030$ & $$-$0.106$ \\
$ \Delta AOps$ & $$-$0.183$ & $$-$0.132$ & $$-$0.235$ & $$-$0.214$ & $$-$0.293$ & $$-$0.176$ \\
$ \Delta AProd$ & $0.566$ & $0.583$ & $0.729$ & $0.409$ & $0.268$ & $0.359$ \\
$ \Delta ACap$ & $$-$0.169$ & $$-$0.166$ & $$-$0.180$ & $$-$0.184$ & $$-$0.187$ & $$-$0.231$ \\
$ \Delta AEcoI$ & $$-$0.013$ & $$-$0.016$ & $$-$0.045$ & $$-$0.040$ & $$-$0.036$ & $$-$0.043$ \\
$ \Delta ATEff$ & $$-$0.121$ & $$-$0.108$ & $$-$0.119$ & $$-$0.095$ & $$-$0.173$ & $$-$0.113$ \\
$ \Delta CAPEX$ & $$-$0.382$ & $$-$0.421$ & $$-$0.314$ & $$-$0.343$ & $$-$0.397$ & $$-$0.362$ \\
$ \Delta OPEX$ & $$-$0.484$ & $$-$0.347$ & $$-$0.301$ & $$-$0.367$ & $$-$0.142$ & $$-$0.487$ \\

\hline \\[-1.8ex]
\end{tabular}
\begin{tabular}{@{\extracolsep{22pt}} lcccccc}
\\[-2ex]\hline
& \multicolumn{6}{c}{Charterer-dominated} \\ \cmidrule(l{3pt}r{3pt}){2-7}
& 2015 & 2016 & 2017 & 2018 & 2019 & 2020 \\ \hline

$ \Delta CLen$ & $0.625$ & $0.544$ & $0.521$ & $0.406$ & $0.310$ & $0.399$ \\
$ \Delta CRate$ & $$-$1.315$ & $$-$1.337$ & $$-$1.065$ & $$-$1.034$ & $$-$0.696$ & $$-$1.134$ \\
$ \Delta CEm$ & $$-$0.285$ & $$-$0.117$ & $$-$0.424$ & $$-$0.367$ & $$-$0.223$ & $$-$0.191$ \\
$ \Delta AOps$ & $$-$0.220$ & $$-$0.134$ & $$-$0.272$ & $$-$0.291$ & $$-$0.373$ & $$-$0.212$ \\
$ \Delta AProd$ & $0.211$ & $0.268$ & $0.271$ & $0.213$ & $0.142$ & $0.105$ \\
$ \Delta ACap$ & $$-$0.255$ & $$-$0.217$ & $$-$0.227$ & $$-$0.223$ & $$-$0.227$ & $$-$0.301$ \\
$ \Delta AEcoI$ & $$-$0.071$ & $$-$0.064$ & $$-$0.031$ & $$-$0.041$ & $$-$0.038$ & $$-$0.049$ \\
$ \Delta ATEff$ & $$-$0.084$ & $$-$0.064$ & $$-$0.100$ & $$-$0.067$ & $$-$0.159$ & $$-$0.088$ \\
$ \Delta CAPEX$ & $$-$0.266$ & $$-$0.309$ & $$-$0.345$ & $$-$0.353$ & $$-$0.414$ & $$-$0.376$ \\
$ \Delta OPEX$ & $$-$0.382$ & $$-$0.389$ & $$-$0.180$ & $$-$0.201$ & $$-$0.102$ & $$-$0.279$ \\

\hline \\[-1.8ex]
\end{tabular}
\end{table}

The definition of the projection: the suggestion of how the inefficient DMUs could change to become efficient. The values in tables above are all percentages. In other words, the projections are the modification recommendation for the inputs and outputs.

1. from 2015-2020, TEUs should be reduced in three scenarios (regardless of who is dominating the market, it is suggested that ships should become smaller) and "ship size should be reduced to balance low capacity utilization". And ship owners should not invest too much money in ship size.
2. EC0_investment is quite reasonable and getting better over time. Shipowners are doing a good job in this ec0_clean_investment.
3. Regarding the time charter rates, the direction of change of time charter rates is different in three cases. When the shipowner dominates the market, the time charter rate should go up; when the charterer dominates the market, the time charter rate should go down. However, from the perspective of cooperation, the rates are still too high.
4. Capital expenditure and operating costs should always be reduced: common sense in cost saving. Regardless of who dominates the market, inefficient actors should reduce the value of these two variables.
5. Speed variation: Speed should be reduced, which is not perfect, but not a major issue either.
6. EIV change. EIV (design fuel efficiency), at the beginning, the distance between the current situation and the frontier (the frontier is the efficiency that DMU wants to achieve) is quite long, but in the next years it becomes closer.
7. productivity can be increased to achieve higher efficiencies
8. In three cases, carbon emissions are quite good.

## 4.1. Post-Hoc Analysis

MARKET POWER MEASURE OWNER CHARTER POWER - owner charter differnce - EFFICENCY

Size of Charter (tonnage share) - high - low

\begin{figure}
\begin{center}
\includegraphics[width=10cm,height=5cm]{payload/00 graphic/stage_model.png}
\caption{Posthoc analysis: Market share and efficiency}
\label{fig:fig2}
\end{center}
\end{figure}

From the picture above, we can see the smaller charterers companies have higher average efficiencies. But the bigger companies have more efficient DMUs. When one organization have more efficient DMUs, they can gain higher benefit from the cooperation (Lozano, 2012). From the distribution of the average efficiencies for those charterer companies, we can see the bigger charterer have more efficient DMUs even though the average scores maybe lower. Hence, the smaller companies are more willing to cooperate with the bigger charterer companies for information-sharing.
At the same time, there are some obvious inefficient observations in bigger charterer companies, the wise player should learn from the efficient DMUs through information-sharing platform of the same organization.

## 4.2. Methodological contribution

Data envelopment analysis (DEA) models serve as a powerful decision support method to assess the performance of a set of peer decision making unit (DMU) units by identifying efficient units, while also providing frontier predictions for inefficient units [@charnes1978measuring]. However, most traditional DEA models such as the CCR or BCC models (Charnes, Cooper, and Rhodes [-@charnes1978measuring]; Banker, Charnes, and Cooper [@banker1986]) have many weaknesses. On the one hand, traditional DEA models follow the concept of radial efficiency measures, leading to another problem of assuming a change in the ratio of inputs to outputs, which is inappropriate in many practical situations [@Zhou2007]. Therefore, there is a large literature on non-radial DEA models to address this issue - in contrast to radial DEA models, non-radial DEA models deal directly with input excesses and output shortfalls, and therefore all these slack variables can have different values. In detail, these non-radial slack models are divided into models based on the ratio form (SBM) and models based on the additive form (ASBM) ([-@Tone2001], [@charnes1985foundations] ).

On the other hand, the traditional DEA models cannot address network supply chains with at least two participants because those models assume that DMUs are black boxes that consume inputs to produce outputs [@Avkiran2009]. Thus, when a system consists of multiple phases and evolves over time, the underlying internal structure of the operational process is ignored. An emerging school of thought in operations research considers network DEA models (NDEA models) as a means to correct this problem (K. Chen and Zhu 2017). The core of NDEA models deals with possible conflicts between different stages due to intermediate measures that connect these stages [@Liang2008]. Moreover, The initial academic debate examined models that assumed that the output of the first stage was only an input to the second stage. Subsequent developments have extended the network DEA model in which each stage consists of a single input and a single output (Cook et al., 2010).

Although such a formulation is closer to the actual operation, its solution process requires converting the nonlinear procedure into a linear one to find the best overall result. To solve this nonlinear problem, SBM-based NDEA requires that the unknown weights between different stages are calculated by the model in advance - the weights that the manager wants may be different from the weights obtained [@Kao2018]. In contrast to the methods applied in general SBM-NDEA models, the novel ASBM NDEA studied by Chen and Zhu [-@chen_additive_2020] provides the manager with the freedom to decide the weights.The ASBM NDEA model applies second-order conic programming (SOCP), a well-established convex optimization that allows nonlinear problems to be solved. By assuming that all inputs are minimized and all outputs are maximized, they deal with both phases of the network from a centralized control perspective.

Firstly, our proposed model incorporates the leader-follower structure of the non-radial ASBM model. In practice, not every participant in this supply chain is willing to cooperate and work together to obtain higher overall efficiency. They may seek to maximize their own payoff (efficiency). Especially when a subsystem in the network has a higher advantage to decide the intermediate[@liang2008]. There is a large literature on leader-follower structural DEA models, but this focuses almost exclusively on radial DEA models (e.g., [@liang2008], [@li2012dea]. In a leader-follower model, the efficiency of the leader is first assessed, and then the efficiency of the follower is measured using the parameters of the leader. However, to the best of our knowledge, there is no work so far that discusses the leader-follower structure in the non-radiative ASBM network model. In addition, investigating efficiency in cooperative and noncooperative situations in empirical cases could provide quantitative support for whether participants in that supply chain need to be motivated to cooperate. In addition, participants in the supply chain can apply our model to identify their weaknesses and find solutions to improve efficiency in different situations. To elaborate, followers in this supply chain can discover the most practical approach by considering the leader's metrics rather than waiting for the leader's actions. Thus, our work enriches the DEA approach to solve game theory problems in networked supply chains.

Furthermore, by combining undesirable outputs and non-discretionary inputs, we extend the ASBM network model to a generalized eco-efficiency network problem. In contrast to the top DEA application areas-banking and education-environmental and sustainability issues have attracted increasing attention [@Ju2022]. For example, [@An2017] and [@Ji2017] studied the issue of carbon emissions in DEA models. Our study also focuses on eco-efficiency, so assuming that all outputs should be maximized would not be close to the reality of the environment [@zhu2017], and omitting them would lead to incomplete results in practice (Yang and Pollitt 2009). To address this issue, this paper adopts the established approach of [@seiford_modeling_2002], which uses monotonically decreasing transformations in DEA models to deal with undesirable outputs.

In addition to undesirable outputs, our study discusses the effects of environmental variables. The assumption that all inputs should be minimized (so all inputs are discretionary inputs) is inappropriate under certain practical environmental problems because environmental variables (exogenous non-discretionary inputs) are fixed and beyond the control of each DMU manager. Non-discretionary inputs reflect the specificity of the environment and determine whether it is favorable or unfavorable, thus affecting the performance of the DMU ([@hua2007]; [@banker1986]). To the best of our knowledge, the model proposed in this study is the first attempt to incorporate both undesirable outputs and nondiscretionary inputs into the ASBM NDEA, making it a more general model for eco-efficiency evaluation that can even be applied to various other supply chain operations.

Nevertheless, our model has some limitations: first, the weights of the two sectors in the supply chain should be assigned before the overall efficiency is calculated. In our model, in this paper we assume that the overall efficiency value is the arithmetic mean of the efficiency values of the two actors. However, in future studies, the assigned weights between the two actors can be adjusted based on the results of actual questionnaires or expert inquiries.

In addition, the model can be extended from different perspectives. 1) Some of the outputs from the first stage can be used in the second stage for cases where there is partial sharing of outputs between the first and second stages. 2) Future studies can discuss the case when the intermediate variables generated in the first stage have other losses than 100% as inputs to the second stage.

## 4.3. Managerial and policy contribution

This paper contributes to the operations research literature within transportation science by introducing the green contracting problem. To solve this chain decision-making problem with externalities, this paper proposed to implement the theoretical approach outlined by Chen and Zhu (2020). By linking the approach to a game-theoretical setting, this paper was able to design a novel solution method.

Applying the ASBM-NDEA to a classic agency problem we showed that approach is suitable to address real-world operations, which involve interdependencies among market participants. The charter market in liner shipping was selected as a case study, showing that the status quo in the industry, that is, charter dominance in contractual relational results in less-than-optimal contracts. Yet, this case also pointed to cooperative contracting and scale- to-operations matching as a critical tool to achieve green contracts. While this outcome is case study-specific, some broader implications can be noted: The green contracting problem appears in diverse relations which produce externalities and that split ownership and operations. Under the assumption that the market can govern transactions as a whole system, rather than self-interested parties with no common objective, higher productivity and a lower ecological footprint can be achieved. Guided by the seminal works concerning agency relations (i.e., Ross 1973; Stiglitz 1989; Eisenhardt 1989), this paper argues that transparency in the marketplace can be a driver toward sustainable operations. Critical to overcoming information asymmetries and insofar allowing more foresight are policies that strive to internalize externalizes within transactions. Carbon taxation can be a tool to incentive the right technology and behavioral choices (Krass, Nedorezov, and Ovchinnikov 2013; Drake, Kleindorfer, and Wassenhove 2016).

Correspondingly, if the charter is unable to validate the information by the shipowner, the *market of lemons* problem emerges [@akerlof_market_1970], and the average contract quality deteriorates.

Suboptimal and inefficient markets are the results of misplaced incentives, incomplete or erroneous information, regulations, market structure, financing, and habits. Market failures refer to imperfections within the market that result in the inefficient allocation of resources. Market barriers refer to obstacles, which are not market failures and instead describe rational conditions, i.e., heterogeneous markets and external circumstances [@sanstad_normal_1994; @jaffe_energy_1994; @brown_market_2001; @sorrell_reducing_2000; @sorrell_economics_2004].

# 5. Limitations and Conclusion {#sec:sec5}

In this study, we examined the green contracting problem. This chain decision-making problem with externalities is novel within transportation science research. It captures the strive by contractual partners to maximize individual utilities, while interdependence concerning sustainable operation is present. Our approach to solve this problem is based on the theoretical contribution by Chen and Zhu (2020). We show that the ASBM-NDEA model, which joins an additive slack-based model and a network data envelopment analysis and is solved by second-order cone programming, is suitable for real-world multi-stage problems. Conversely to other non-linear DEA models, it requires no weight designation placed on the different stages, nor does it rely on changing the production possibility set and constraints. We advance this model by incorporating a game-theoretical perspective, enabling us to estimate (un-)cooperative scenarios.

To validate the approach, we outline the green contracting problem in liner shipping. We gathered a sample in the period over January 2017 until December 2020 covering the whole contractural process, i.e., transaction procedure and results of exercising a contract. The sample’s granularity allows us to construct a robust case and measure eco-efficiency, including economic value produced and environmental damage caused. Against this background, the game-theoretical approach includes three distinct scenarios to measure efficiencies under varying market dynamics. The uncooperative scenario is split among an owner or charterer contractural dominated process. While these scenarios point to higher individual utilities skewed towards the dominant actors, the overall efficiencies range below the one estimated in the cooperative scenario.

Our results show that there is a board mismatch in assigned capacity across the scenarios, leaving under- utilization as the main policy issue to be dealt with. In addition, we show that mutualism in contract relations is key -higher contract length enables more stable relationships and allowing owners to recoup investments in higher design efficiency. While operational efficiency, tied primarily to transport intensity, is close to the optimum, highlighting the limits charterers are confronted with to achieve higher environ- mental performance if the baseline is not well calibrated. Overall, the scenarios highlight that contrast relations can be improved, resulting in a shift closer to the frontier.
Our work provides a distinct methodological and theoretical contribution to transportation science re- search by engaging not only with the not yet well understood green contracting problem but proposing, as well, a suitable solution approach. As opposed to prior works on eco-efficiency in logistics (e.g., Quar- iguasi Frota Neto et al. 2009) or general production processes (e.g., Kounetas, Polemis, and Tzeremes 2021), we highlight that businesses operate not in isolation but rather within complex networks, which implies interdependence in decision-making and respective results. This interdependence implies that re- ciprocal responsibility in contractual agreements to counter environmental degradation should be placed in a key role. Consequently, market participants and regulators are encouraged to recognize that the status quo -in our case study- leads to less-than-optional outcomes concerning individual utility and pub- lic welfare. Therefore, we argue a strive towards more transparent market environments might foster a higher degree in sustainable operations.

While this work’s approach and results can be applied to other chain decision-making problems with ex- ternalities, some limitations shall be noted. We recognize that contractual relations are rather complex in itself and involve (in-)directly more than just an owner and charter. As the parties are dependent on vari- ous other market participants (i.e., owner and shipbuilder versus charter and freight forwarder or terminal operator), we propose to broaden the horizon. In light of this, a systems dynamics perspective builds upon the current approach might provide further insights. We selected to outline the green contracting problem some bais in the data curation process was unavoidable concerning the real-world case. The core information on environmental damage was gathered from ocean-shippers, primarily larger stakeholders in the liner shipping sector. In order to assure complete information, we omitted contracts with incomplete sets, leading to baizes in regards to market power dynamics and a sole focus on ocean shipping rather than the whole sector, which includes short-sea operations. This might pose a problem concerning the scenario results, which require further investigation into interactions among smaller market participants with more balanced dynamics and will be broadened to other industries.

## References

<div id="refs"></div>

\newpage

## Appendix

### Appendix A. Notation {#apA}

\begin{table}[!htbp] \centering
\caption{}
\label{tab:tab_apa}
\small
\begin{tabular}{@{\extracolsep{5pt}} p{3cm}p{2cm}p{10cm}}
\\[-2ex]\hline
Class & Notation & Description \\ \hline
\textbf{Stage I.} &$x_{mj}$ & Inputs $(m=1,2,\cdots,M)$\\
&$x_{kj}$ & Non-discretionary inputs $(k=1,2,3,\cdots,K)$\\
&$y_{pj}$ & Desirable outputs $(p=1,2,\cdots,P)$\\
&$y_{uj}$ & Undesirable outputs $(u=1,2, \cdots,U)$\\
&$s_{k}^{-}$ & Input excess $(k=1,2,3,\cdots,K)$\\
&$s_{p}^{+}$ & Good input deficit $( p=1,2,\cdots,P)$\\
&$s_{u}^{+}$ & Bad output excess $( u=1,2, \cdots,U)$\\
&$\lambda_{j}^{1}$ & Unknown intensity vector $( j=1,2, \cdots,n)$\\
&$E_{j}^{1}$ & Efficiency $( j=1,2, \cdots,n)$\\
\textbf{Intermediates} &$z_{dj}$ & Intermediates stage-linking $( d=1,2,3,\cdots, D)$\\
&$t_{d}^{+}$ & Output excess in stage I. $( d=1,2,3,\cdots, D)$\\
&$t_{d}^{-}$ & Input deficit in stage II. $( d=1,2,3,\cdots, D)$\\
\textbf{Stage II.} &$x_{hj}$ & Inputs $(h=1,2,\cdots,H)$\\
&$y_{rj}$ & Desirable outputs $(r=1,2,\cdots,R)$\\
&$y_{lj}$ & Undesirable outputs $(l=1,2,\cdots,L)$\\
&$s_{h}^{-}$ & Inputs excess $(h=1,2,\cdots,H)$\\
&$s_{r}^{+}$ & Good input deficit $(r=1,2,\cdots,R)$\\
&$s_{l}^{+}$ & Bad output excess $(l=1,2,\cdots,L)$\\
&$\lambda_{j}^{2}$ & Unknown intensity vector $( j=1,2 ,\cdots,n)$\\
&$E_{j}^{2}$ & Efficiency $(j=1,2, \cdots,n)$\\
\textbf{Data processing} &$v$ & Translation vector to handle undesirable outputs\\
&$\bar{y}_{uj}^{1}$ & Translated undesirable variables in the first stage\\
&$\bar{y}_{lj}^{2}$ & Translated undesirable variables in the second stage\\
&$k$ & Translation vector handling the non-discretionary inputs\\
&$\bar{x}_{j}$ & Translated non-discretionary inputs\\
\hline \\[-1.8ex]
\end{tabular}
\end{table}

### Appendix B. Constraints {#apB}

In case the second stage is dominants, the efficiency is specified:

\begin{equation}
\min \frac{1}{R+L+H+D}\left(\sum_{r=1}^{R} \frac{y_{r 0}^{2}}{y_{r 0}^{1}+s_{r}^{+}}+\sum_{l=1}^{L} \frac{\bar{y}_{l 0}^{2}}{\bar{y}_{l 0}^{2}+s_{l}^{+}}+\sum_{h=1}^{H} \frac{x_{h 0}^{2}-s_{h}^{-}}{x_{h 0}^{2}}+\sum_{d=1}^{D} \frac{z_{d 0}-t_{d}^{-}}{z_{d 0}}\right)
\end{equation}

we have the following constraints:

\begin{align}
&\sum_{j=1}^{n} x_{h j}^{2} \lambda_{j}^{2}+s_{h}^{-}=x_{h 0}^{2}, \forall h \\
&\sum_{j=1}^{n} y_{r j}^{2} \lambda_{j}^{2}-s_{r}^{+}=y_{r 0}^{2}, \forall r \\
&\sum_{j=1}^{n} \bar{y}_{l j}^{2} \lambda_{j}^{2}-s_{l}^{+}=\bar{y}_{l 0}^{2}, \forall l \\
&\sum_{j=1}^{n} z_{d j} \lambda_{j}^{2}+t_{d o}^{-}=z_{d 0}, \forall d \\
&\sum_{j=1}^{n} \lambda_{j}^{2}=1
\end{align}

\begin{equation}
\min \sum_{r=1}^{R} \xi_{r}^{1}+\xi_{l}^{2}+\xi^{3}
\end{equation}

\begin{align}
\text { s.t. }
&\left\|\begin{array}{c}
\sqrt{y_{r 0}^{2}} \\
\frac{1}{2}\left((R+L+H+D)\left(y_{r 0}^{2}+s_{r}^{+}\right)-\xi_{r}^{1}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((R+L+H+D)\left(y_{r 0}^{2}+s_{r}^{+}\right)+\xi_{r}^{1}\right), \forall r,\\
&\left\|\begin{array}{c}
\sqrt{\bar{y}_{l 0}^{2}} \\
\frac{1}{2}\left((R+L+H+D)\left(y_{r 0}^{2}+s_{r}^{+}\right)-\xi_{r}^{1}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((R+L+H+D)\left(\bar{y}_{l 0}^{2}+s_{r}^{+}\right)+\xi_{l}^{2}\right), \forall l,\\
&\frac{1}{R+L+H+D}\left(\sum_{h=1}^{H} \frac{x_{h o}^{2}-s_{h}^{-}}{x_{h o}^{2}}+\sum_{d=1}^{D} \frac{z_{d o}-t_{d}^{-}}{z_{d o}}\right) \leq \xi^{3}
\end{align}

Constraint Sets (66)-(70).

The first stage is forced to accept the optimal intermediate products from the second stage, leading to the effiency estimation in the first stage:

\begin{equation}
\min \frac{1}{P+U+M}\left(\sum_{p=1}^{P} \frac{y_{P 0}^{1}}{y_{p 0}^{1}+s_{p}^{+}}+\sum_{u=1}^{U} \frac{\bar{y}_{u 0}^{1}}{\bar{y}_{u 0}^{1}+s_{u}^{+}}+\sum_{m=1}^{M} \frac{x_{m 0}-s_{m}^{-}}{x_{m 0}}\right)
\end{equation}

Based on the variable return to scale following constrains apply:

\begin{align}
&\sum_{j=1}^{n} x_{m j} \lambda_{j}^{1}+s_{m}^{-}=x_{m 0}, \forall m \\
&\sum_{j=1}^{n} y_{p j} \lambda_{j}^{1}-s_{p}^{-}=y_{p 0}^{1}, \forall p \\
&\sum_{j=1}^{n} y_{u j} \lambda_{j}^{1}-s_{u}^{-}=y_{u 0}^{1}, \forall u \\
&\sum_{j=1}^{n} z_{d j} \lambda_{j}^{1}+t_{d}^{-*}=z_{d 0}, \forall d \\
&\sum_{j=1}^{n} \lambda_{j}^{1}=1
\end{align}

The intermediate product in the second stage is $z_{d o}-t_{d}^{-*}$, dervied by the dominated stage.
Applying the same conversion procedure the efficeicy in the first stage:

\begin{equation}
\min \sum_{p=1}^{P} \xi_{p}^{1}+\xi_{u}^{2}+\xi^{3}
\end{equation}

\begin{align}
\text { s.t. }
&\left\|\begin{array}{c}
\sqrt{y_{p 0}^{1}} \\
\frac{1}{2}\left((P+U+M)\left(y_{p 0}^{1}+s_{p}^{+}\right)-\xi_{p}^{1}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((P+U+M)\left(y_{p 0}^{1}+s_{p}^{+}\right)+\xi_{p}^{1}\right), \forall p\\
&\left\|\begin{array}{c}
\sqrt{y_{u 0}^{1}} \\
\frac{1}{2}\left((P+U+M)\left(y_{u 0}^{1}+s_{u}^{+}\right)-\xi_{u}^{1}\right)
\end{array}\right\|_{2} \leq \frac{1}{2}\left((P+U+M)\left(y_{u 0}^{1}+s_{u}^{+}\right)+\xi_{u}^{1}\right), \forall u,\\
&\frac{1}{\mathrm{P}+\mathrm{U}+\mathrm{M}} \sum_{m=1}^{M} \frac{x_{m 0}^{1}-S_{m}^{-}}{x_{m 0}^{1}} \leq \xi^{2}
\end{align}

Constraint Sets (76)-(80)