# A robust optimization model for green supplier selection and order
allocation in a closed-loop supply chain considering cap-and-trade mechanism

Hossein Mirzaee1,2
<EMAIL_ADDRESS>Hamed Samarghandi2
<EMAIL_ADDRESS>Keith Willoughby2
<EMAIL_ADDRESS>Corresponding author2 Edwards School of Business,
University of Saskatchewan, Saskatoon, SK, Canada

###### Abstract

Due to increasing air pollution, which is a consequence of the environmental
effects of production in various industries, green supply chain management
(GSCM) has attracted the attention of both scholars and practitioners. Green
supplier selection is one of the most important problems in GSCM, which
satisfies a firm’s environmental goals as well as its economic targets. In
this paper, for the first time, a green supplier selection problem considering
both green and non-green evaluation criteria in a closed-loop supply chain is
studied, and a cap-and-trade mechanism as a way of controlling the air
pollution caused by manufacturers is proposed. To solve the described problem,
a multi-objective robust optimization (RO) model, as an effective approach to
handle uncertainty, is proposed. A numerical example using randomly generated
data, accompanied by the analysis based on the proposed approach is elaborated
to validate the presented model. The results prove that the developed model
for green supplier selection is able to effectively enhance the decision-
making process of the experts. By illustrating the trade-off in robustness
between the model and proposed solutions, as well as the effect of the
deviation penalty on the closeness of results to the achieved solution, we
show how firms can make optimal decisions when assigning the parameters.
Furthermore, analyses show that allowance amount (cap) and allowance prices in
the cap-and-trade system impact the firms’ costs and the amount of carbon
released. Finally, we show that the cap-and-trade mechanism results in a
better solution in terms of the total utility of the supply chain compared to
the penalty-based system.

Keywords: green supplier selection; cap-and-trade; robust optimization;
closed-loop supply chain

##  1 Introduction

Due to the increasing environmental issues that have arisen recently, GSCM has
become an important topic for both practitioners and researchers (Sang Liu,
2016). According to Srivastava (2007), GSCM brings environmental
considerations to supply chain management; this includes product design, raw
material and parts sourcing, production processes, and transportation of
finished products to customers. One of the most critical topics in GSCM is
supplier selection, since about 70% of the cost of the final product arise
from component parts and raw material (Ghodsypour O’brien, 2001). Thus,
procuring the raw materials from proper suppliers significantly impacts the
characteristics of the final product. Accordingly, companies bestow privilege
to the partners that score the highest on environmental criteria (Rao Holt,
2005).

Supplier selection is the process of evaluating various suppliers to select
the best one(s) according to the high-priority criteria for the manufacturers
(Ding ., 2015). In case of multiple sourcing, where more than one partner can
be selected for outsourcing, a solution must encompass the selected suppliers
and the size of the order to be placed with each of them. To evaluate the
suppliers, the manufacturers should first specify their appraisal criteria.
According to the literature of GSCM, there are two general types of measures
in supplier selection: non-green and green.

Non-green criteria ascertain a company’s competitiveness compared to other
firms in the market. Among the non-green measures used in the selection
process in the literature, the most common ones are: cost, quality, and lead
time (Mirzaee ., 2018). In contrast, green criteria contemplate those of a
company’s activities with an adverse impact on the quality of water, air, and
soil. The most utilized measures in evaluating the green criteria are the
toxicity level of the materials used in products, recyclability, green
production, and environmental management systems (Rezaei ., 2016), as well as
pollution production (i.e., the amount of carbon emission) (F. Yu ., 2018).

Considering green criteria mentioned above by manufacturers is caused by some
incentives that government makes for them to reduce the pollution level. One
of these incentives can be restricting the emission level in a way that going
beyond that level is costly for polluters. Carbon emission is usually measured
by production level as well as equipment used in the supply chain (F. Yu .,
2018). Companies usually produce products to maximize their profit; they do
not supervise the amount of carbon emission during the production process.
Recently, a mechanism called cap-and-trade has been proposed by various
governments to keep the level of $CO_{2}$ emissions at a reasonable level. In
this mechanism, a cap is defined as the maximum carbon emission allowance for
manufacturers, and is treated as the emission quota when producing the
products and responding to the customer demands. Furthermore, companies can
sell or buy carbon emission allowances from each other in a trading market (Xu
., 2022). In this situation, companies alter their objective to profit
maximization not only by manufacturing the proper amount of products, but also
by buying or selling carbon emission allowances. Generally, in the cap-and-
trade mechanism, the assigned cap is reduced each year by the government to
reach an acceptable level of carbon pollution.

In a supply chain, the amount of production, and consequently, the decisions
related to raw material outsourcing are highly impacted by market demand and
procurement cost. Typically, these factors are not under the companies’
control. Although firms can impact these parameters by marketing and product
pricing, they cannot predict their exact values because of unpredictable
factors such as rivals’ strategies. This uncertainty prevents the manufacturer
from forecasting precise values for demand and procurement cost. Also, in a
closed-loop supply chain, the quantity of returns consisting of used or
rejected products is not easy to predict and should be considered uncertain
(Pishvaee ., 2011). Consequently, a significant challenge for decision-makers
is handling the uncertainty of the parameters such that their proposed
solutions represent the real-life scenarios. In the context of suply chain,
there are three common ways to control the uncertainty of the input data:
stochastic optimization, fuzzy set theory, and robust optimization (RO)
(Tordecilla ., 2021).

The majority of studies available in the literature of green supplier
selection has considered the uncertain parameters as stochastic or fuzzy. One
of the challenges that the decision-makers face with stochastic programming is
estimating the correct probability distribution of the uncertain parameters
(Gorissen ., 2015). However, in many cases, historical data related to the
uncertain parameters is not available. Hence, the estimated probability
distributions might not be reliable (Vahdani ., 2012). Furthermore, despite
the high probability of correct prediction of the uncertain parameters, in
very rare cases, it may be wrong and cause solution infeasibility. Even though
this issue may occur with a very small probability, it can lead to a serious
rise in the costs (Pishvaee ., 2011). What about fuzzy set theory? When using
fuzzy set theory, exhaustive knowledge and comprehension about the parameters
is required to generate an accurate membership function. Undoubtedly,
obtaining such comprehensive consciousness about the market and its dynamics
poses an enormous challenge for decision-makers, and is considered one of the
obstacles of effectively employing the fuzzy set theory for real-world problem
optimization. (Memon ., 2015).

Robust optimization is a functional technique to handle uncertainties, and is
mostly free from the flaws mentioned above (Jabbarzadeh ., 2019). Both fuzzy
optimization and stochastic optimization rely on probability distribution of
random variables (L. Chen ., 2022), which, once correctly identified, can
accurately reflect the uncertain characteristics of random variables. However,
obtaining the precise probability distribution information is closely related
to the number of data samples as well as the accuracy of the prediction
methods. This increases the complexity of calculations and reduces confidence
(Firouzmakan ., 2019). Contrary to fuzzy and stochastic programming, RO
defines the uncertain parameters in a problem as an uncertain set with bounds
or scenarios, and makes decisions based on the average, or the worst case.
Accordingly, after realizing any uncertain parameters, the optimal solution is
achieved with good generalization, and does not have the mentioned complexity
(Lu ., 2020).

RO considers a number of scenarios as possible outcomes, and assigns a
probability of occurrence to each outcome. The objective of RO is to obtain a
solution which is nearly optimal and feasible under each scenario (Arabzad .,
2015). In this approach, one must still use historical data to find the
probability of occurrence of each scenario, but this problem is solved by
considering the possibility of solution infeasibility.

Stochastic programming optimizes the expected value of objective function over
the set of all possibilities. On the contrary, robust optimization generates a
solution which is feasible for all scenarios, and considers the infeasibility
as a cost in the model (Shabani, 2014; Bertsimas Sim, 2003). In other words,
although finding the correct probability of occurrence of the scenarios is
still a challenge for robust optimization, it is dealt with by assuming the
cost of infeasibility in the model. In stochastic models, failing to predict
the right value of parameters leads to solution infeasibility, elevated costs,
and futility of the decision support system. RO, however, intrinsically
considers infeasibility with an assigned cost. In other words, RO facilitates
the probability prediction challenge by delivering a single optimal solution
that is feasible under all scenarios (Ahmadvand Sowlati, 2022)

Although there exist numerous studies in the area of supplier selection, only
a few papers have considered both green and non-green criteria at the same
time. Also, to the best of our knowledge, this paper is the first study that
employs robust optimization to handle uncertainties of the green supplier
selection (GSS) problem and cap-and-trade mechanism for carbon emission. Based
on the literature, RO is one of the best options to solve various problems in
supply chain. In other words, as confirmed by the literature, RO is an
efficient approach to deal with uncertainty. However, this approach has not
been applied to the green supplier selection problem.

In other words, in this study, we present a GSS model which is embedded in a
closed-loop supply chain framework. The model regards both green and non-green
measurements in the presence of distinct quantitative and qualitative factors.
Moreover, it is assumed that the market operates under the cap-and-trade
regulations imposed by the government. To deal with the described problem, an
RO approach is employed. It is shown that the proposed RO is able to generate
solutions that are close to real cases.

This study aims to address the following research questions:

  * •

the impact of cap-and-trade, as an environmental mandate, on the supply chain;

  * •

finding an effective approach for controlling the intrinsic uncertainty of the
model’s parameters;

  * •

selecting the best supplier among all candidates in a supply chain considering
economic and environmental criteria; and,

  * •

determining clear cause-and-effect relationships between various factors to
help government entities with their decision making efforts.

The structure of the remainder of the paper is as follows. Section 2
summarizes the relevant research. Section 3 outlines the RO model of the
considered GSS problem. Section 4 explains the computational experiments of
the proposed model and discusses the results. Conclusions and future study
directions are presented in section 5.

##  2 Literature review

The goal of green supply chain management is reduction of the harmful effects
of the supply chain’s activities on the environment. In this regard, firms
have to identify the most effective measures for evaluating the environmental
performance of their suppliers. Research in green supplier selection, in
comparison with traditional supplier selection (which generally deals with the
firm’s profit and quality of the products) is limited. Hence, the common
environmental criteria used in the literature of green supplier selection can
be identified. Some of the most used environmental criteria in the literature
include:

  * •

Environmental management system: this criteria defines the suppliers’ policies
for making the production process environmentally friendly; one example for
such policies is the ISO 14001 certificate. (Handfield ., 2002; Hsu Hu, 2009;
Lee ., 2009; Awasthi ., 2010; Bai Sarkis, 2010; Kuo ., 2010; Mafakheri .,
2011; Yeh Chuang, 2011; Amin Zhang, 2012; Govindan ., 2013; Tseng Chiu, 2013;
Hu ., 2015; Hashemi ., 2015; Rezaei ., 2016; Qin ., 2017; Kumar ., 2017; Gupta
Barua, 2017);

  * •

Pollution production: this criteria measures the amount of pollution that a
manufacturer creates (Amin Zhang, 2012; Govindan ., 2013; Hashemi ., 2015; Hu
., 2015; Kannan ., 2015; Huang ., 2016; Rezaei ., 2016; Qin ., 2017; Luthra .,
2017; Kumar ., 2017);

  * •

Recyclability: refers to the capability of suppliers in using recycled
material in their manufacturing process (Yeh Chuang, 2011; Amin Zhang, 2012;
Hu ., 2015; Kannan ., 2015; Govindan Sivakumar, 2016);

  * •

Green product: this measure evaluates the ability of suppliers in using green
technology as well as environmentally friendly material (Handfield ., 2002;
Lee ., 2009; Amin Zhang, 2012; Tseng Chiu, 2013);

  * •

Product toxicity: assesses the suppliers regarding the level of toxic
substance used in their products (Hu ., 2015; Kannan ., 2015; Rezaei ., 2016;
Gupta Barua, 2017).

Table LABEL:tab:literature_review_on_GSS summarizes the findings of the
mentioned papers. One of the challenges of the supplier selection problem is
that firms tend to maximize their profit while trying to have a great
performance from the environmental point of view. Therefore, companies are
required to identify the non-green criteria affecting the firm’s profit as
well as green criteria. Different studies have been completed to identify the
most important criteria in traditional supplier selection problem. Dickson
(1966), Lehmann O’shaughnessy (1974), Weber . (1991), and Cheraghi . (2004)
conducted research to identify non-green supplier evaluation measures. Those
studies showed that cost, delivery performance, and quality are the three most
used and important criteria in this particular problem area.

The next step of supplier selection and order allocation process is assessing
the candidates regarding the criteria mentioned above. Researchers have
applied different techniques to solve this problem. Chai . (2013) classified
these techniques in three main groups: multi-criteria decision-making (MCDM),
artificial intelligence (AI), and mathematical programming (MP). A list of the
papers that apply these techniques to solve the supplier selection problem is
presented in table LABEL:tab:literature_review_on_GSS. MCDM is a framework
that helps the decision makers find the best alternative between multiple
options based on various criteria. To select the best option, MCDM sorts the
alternatives based on their scores. AI refers to the science of making the
computers able to work, learn, and think intelligently, like humans. MP is a
useful approach to address the supply chain management problems in a clear
way; constraints or equations can implement assumptions that make the problem
realistic. In this paper, an MP is presented to model the problem.

Applied research invariably involves the development of methods to more
closely align decisions to real-life situations. One such approach to augment
applicability involves considering the supply chain as a closed-loop system.
Firms in the supply chain can collect used or rejected products and deploy
them again in the production system, thus contributing to reduced,
environmentally harmful waste (Cao ., 2020). Moreover, pollution control
systems can also provide enhanced applicability for solutions. Cap-and-trade
is an interesting mechanism that is becoming more popular in today’s consumer
market. This mechanism is proved to be one of most effective approaches of GHG
emission control (Xu ., 2021). Furthermore, based on Y. Yu . (2021), cap-and-
trade is more effective than the carbon tax system on reducing GHG emissions
because it produces a higher social welfare and generates more profit for the
manufacturers. B. Zhang Xu (2013) presented a multi-product production
planning model under the C&T system and proposed a profit-maximization model
to achieve the firm’s optimal policy. In their analysis, it was indicated that
C&T curbs the emission level better than carbon tax system. Another research
supporting the effectiveness of C&T is conducted by Yh. Chen . (2020), which
further compared emission reduction effects of carbon tax and C&T schemes.
They showed that both mechanisms stimulate clean innovation, but C&T is more
efficient. They stated that government can control air pollution by assigning
the proper carbon cap for manufacturers in a trade-off between environmental
and economical objectives.

The emission of greenhouse gas was restricted by the Kyoto Protocol (Oberthür
Ott, 1999) for the first time. The Kyoto Protocol is an international treaty
developed within the United Nations Framework Convention on Climate Change
(UNFCCC). It was adopted in Kyoto, Japan in 1997 and came into effect in 2005.
The Protocol proposes cap-and-trade, which is a flexible framework for
reducing air pollution.

The existing literature regards cap-and-trade as one of the most effective
ways to control the carbon emission (Golpîra Javanmardan, 2022). B. Zhang Xu
(2013) considered cap-and-trade mechanism in their production planning and
presented a profit maximization model to find the firm’s best policy. Gong
Zhou (2013) designed a dynamic production model to scrutinize the impacts of
creating a carbon trading market on production policy and found an emission
trading policy as well as the optimal production plan under the cap-and-trade
regulation. Shen . (2014) studied California’s cap-and-trade scheme with the
goal of implementing it as China’s carbon reduction program. Li . (2018)
studied the impact of cap-and-trade system on manufacturers’ optimal
operational decision and showed that customers’ green preferences act as an
incentive for greening the production technology. S. Zhang . (2019) considered
two scenarios for carbon allowance prices: dynamic and static. They
investigated the effects of cap-and-trade market on manufacturers’ decisions
under both scenarios and showed that upgrading the production technology is
positively correlated with penalties imposed for extra emissions.

In the context of supply chain management, cap-and-trade is a relatively new
topic. Thus, the need for more research in this area is deeply felt. To the
best of the authors’ knowledge cap-and-trade regulations have not been
addressed in the research body related to green supplier selection.

###  2.1 Research contributions

Although there are numerous studies on the supplier selection problem, we know
of only a small group of papers addressing both green and non-green factors in
the process of supplier evaluation while considering a closed-loop structure
with uncertain parameters. Furthermore, as the literature shows, there is no
paper that considers RO for green supplier evaluation as well as the cap-and-
trade mechanism. This paper is an attempt to fill these gaps by fostering the
contributions presented below.

  * •

A multi-objective mathematical model for a closed-loop supplier selection and
order allocation evaluating the candidates in terms of both environmental and
economical criteria is presented. The developed model helps firms achieve a
more environmentally friendly manufacturing system. Model realism is enhanced
by developing an approach that considers two groups of conflicting criteria
(green and non-green).

  * •

The cap-and-trade mechanism, as a method to manage air pollution, is employed
in the model. Analyses on the cap and market prices of carbon are performed to
help firms and governments determine the values of the parameters to achieve
better results. These analyses can subsequently translate to lower cost and
carbon emission, as well as more environmentally friendly products. Moreover,
based on the analysis conducted on the cap-and-trade approach, this mechanism
is demonstrated as a proper approach for carbon emission reduction.

  * •

The generalized model is solved by the RO approach to handle the uncertainty
embedded in the problem. Sensitivity analysis has been conducted on two
parameters to illustrate the trade-off between model robustness and solution
robustness, and solution deviation. This can inform decision-makers of the
best parameter values.

Table 1: Literature Review Summary

References Criteria Solving Approach Uncertainty Approach Closed-loop Findings
Green Non-green MCDM MP AI Fuzzy SO RO Handfield . (2002) - AHP as a decision
support model is used to find out the trade-offs between environmental
dimensions. Hsu Hu (2009) - Hazardous substance management (HSM) is
incorporated into supplier selection using ANP approach. Lee . (2009) The
Delfi method and fuzzy AHP are used to support supplier selection decision.
Awasthi . (2010) - A fuzzy multi-criteria approach is presented that consists
three steps: finding the best criteria, scoring the suppliers using fuzzy
TOPSIS, and sensitivity analysis of criteria weights on supliers evaluation.
Bai Sarkis (2010) A new supplier selection technique using rough set and grey
system theory is presented. Kuo . (2010) A green supplier selection model is
generalized using a hybrid method called ANN-MADA. Mafakheri . (2011) A two-
stage multi-criteria dynamic programming approach for suppier selection and
order allocation is proposed. Yeh Chuang (2011) A green partner selection
model is solved by genetic algorithm to find the set of pareto optimal
solutions. Amin Zhang (2012) A framework for supplier evaluation in a closed-
loop supply chain is generalized. Govindan . (2013) Triple Bottom Line
approach for supplier selection using a fuzzy multi-criteria model is
developed. Tseng Chiu (2013) Environmental and non-environmental criteria for
selecting the best partners are identified by evauating the weight of criteria
and using grey relational analysis. Hu . (2015) A novel evaluation system of
green supplier selection under the mode of low carbon economy is investigated.
Hashemi . (2015) A comprehensive green supplier selection model using ANP and
Grey relational analysis is proposed. Kannan . (2015) Fuzzy Axiomatic Design
is proposed to to select the best green supplier for a plastic manufacturing
company. Huang . (2016) A game-theoretic model is presented in order to
investigate the impacts of supplier selection, transportation mode selection,
the product line design, and pricing strategies on profits and greenhouse
gases emissions. Rezaei . (2016) Best worst method is used to find the best
suppliers among the qualified suppliers. Luthra . (2017) An integrated
approach of AHP, VIKOR, and multi-criteria optimization is developed to solve
evaluate the sustainable supplier selection. Qin . (2017) - TOMID approach to
solve a green supplier selection problem considering interval type-2 fuzzy
sets is extended. Kumar . (2017) Suppliers’ performance is evaluated based on
Green Practices using the fuzzy-extended Elimination and Choice Expressing
Reality approach. Gupta Barua (2017) - Supplier evaluation is done usnig a
three phase methodology including three phases: identifying green criteria,
ranking determined criteria using a novel best worst method, and ranking
suppliers using fuzzy TOPSIS. Moheb-Alizadeh Handfield (2018) A sustainable
and efficient supply chain is designed using a multi-objective MINLP model for
supplier selection and order allocation with stochastic demand. This paper

##  3 Mathematical model formulation

###  3.1 Robust optimization

An interesting contribution that makes the results of the supplier selection
problem valuable is considering the intrinsic uncertainty of the parameters;
wrong estimation of parameter values may lead to colossal losses in an
uncertain environment (Wang ., 2021). Obviously, this requires an effective
approach for handling such uncertainty. To the best of our knowledge, RO has
not been applied to solve the problems in the literature of green supplier
evaluation. In this paper, RO is employed because it handles infeasibility,
and is confirmed to be an efficient approach to deal with data uncertainty by
treating uncertainty as a deterministic trait, while not limiting uncertain
parameter values to point estimates (Jeyakumar ., 2014). Accordingly, to
decrease its possibility, RO assigns a cost to infeasibility. Other popular
approaches are capable of controlling uncertainty, but in case of solution
infeasibility, which may tremendously increase the costs, other approaches are
required. In other words, the expected cost of infeasibility can be
significant and must be taken into consideration.

Unlike deterministic methods that feature a specific value for the parameters,
RO considers various parameter values and obtains solutions under all possible
scenarios. Mulvey . (1995) introduced RO as a new approach for uncertain
problems. RO performs a trade-off between two kinds of robustness; namely,
model robustness (i.e., the obtained solution yields the least infeasibility
in all scenarios), and solution robustness(i.e., the solution is as close to
the optimal value as possible in all scenarios). Simply put, the goal of RO is
to achieve a robust solution which ensures all the real scenarios are nearly
optimal and feasible.

Mulvey . (1995) assumed $x$ and $y$ as design variables (in which their
optimal value is not dependent upon the real value of the uncertain
parameters), and control variables (in which their optimal value depends on
the value of the uncertain parameters). Correspondingly, there are two groups
of constraints: control constraints, and structural constraints. The former
are those that include noisy parameters, while the latter feature the
constraints with no uncertain parameters. The basic structure of the RO model
is presented below.

$\displaystyle Min\;\;c^{T}x+d^{T}y$ (1) $\displaystyle Ax=b$ (2)
$\displaystyle Bx+Cy=e$ (3) $\displaystyle x,y\geq 0$ (4)

where $A$ and $b$ are deterministic parameters, and $B$ and $C$ are uncertain
parameters. In other word, equations (2) and (3) convey the structural and
control constraints, respectively. There are a finite set of scenarios $s$
defined for the RO problem with the probability $Pr_{s}$ for each scenario.
Then, the model formulation transforms to the following:

$\displaystyle
Min\;\;\sigma(x,y_{1},...,y_{s})+\omega\rho(\delta_{1},...,\delta_{s})$ (5)
$\displaystyle Ax=b$ (6) $\displaystyle B_{s}x+C_{s}y+\delta_{s}=e_{s}$ (7)
$\displaystyle x,y_{s}\geq 0$ (8)

in which $\delta_{s}$ indicates the permitted infeasibility in constraint (7)
under scenario $s$. The objective function includes two parts. The first part
represents the solution robustness, and the second part is a measure of model
robustness. As the value of the penalty of infeasibility, $\omega$ helps to
make a trade-off between model robustness and solution robustness. The first
part of the presented objective function can be reformulated as follows:

$\sigma(x,y_{1},...,y_{s})=\sum_{s}Pr_{s}\xi_{s}+\lambda\sum_{s}Pr_{s}\left(\xi_{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{s^{\prime}}\right)^{2}$
(9)

where $\lambda$ is the penalty for solution variance and takes positive
values. $\xi_{s}$ is the value of the objective function in each scenario. In
this equation, the first part shows the average value of the objective in
different scenarios, and the second part indicates the objective variance.
Furthermore, equation (9) can be written in the following form:

$\sigma(x,y_{1},...,y_{s})=\sum_{s}Pr_{s}\xi_{s}+\lambda\sum_{s}Pr_{s}\mid\xi_{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{s^{\prime}}\mid$
(10)

CS. Yu Li (2000) state that, due to non-linearity, this equation is very
complicated and time-consuming to solve. Instead, they propose the following
formulation to linearize the equation.

$\displaystyle\sigma(x,y_{1},...,y_{s})=\sum_{s}Pr_{s}\xi_{s}+\lambda\sum_{s}Pr_{s}\left[\left(\xi_{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{s^{\prime}}\right)+2\theta_{s}\right]$
(11)
$\displaystyle\left(\xi_{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{s^{\prime}}\right)+\theta_{s}\geq
0\qquad\qquad\forall s$ (12) $\displaystyle\theta_{s}\geq 0\qquad\qquad\forall
s$ (13)

The variable $\theta_{s}$ is used to prevent
$\xi_{s}-\sum_{s^{\prime}}P_{s^{\prime}}\xi_{s^{\prime}}$ from being negative.
According to the above new formulation, the objective function can be written
as follows.

$Min\;\;\sum_{s}Pr_{s}\xi_{s}+\lambda\sum_{s}Pr_{s}\left[\left(\xi_{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{s^{\prime}}\right)+2\theta_{s}\right]+\omega\sum_{s}Pr_{s}\delta_{s}$
(14)

where $\omega$ is defined as the weight of trade-off between model robustness
and solution robustness. $\omega$ is a number without unit, which indicates
the penalty assigned to infeasibility; if $\omega$ increases, the probability
of obtaining an infeasible solution decreases.

###  3.2 Model description

In this section, a multi-objective, multi-product, multi-supplier, and multi-
period robust model for green supplier selection and order allocation in a
closed-loop supply chain is presented. Two groups of criteria are presented
for supplier evaluation: green and non-green. The non-green criteria are cost,
quality, and delivery time, while green criteria are divided into two
categories: quantitative (carbon emission amount), and qualitative
(environmental management system level, recyclability level, green product
level, product toxicity level).

In this closed-loop supply chain, the manufacturer sells its products to the
customer, but some products may be rejected due to their low quality or other
problems. Moreover, the manufacturer tends to use second-hand products in the
production process to benefit from their low costs. Second-hand products must
be collected from the customers. A percentage of the collected and rejected
products are usable. The reminder are disposed. Therefore, the input material
and component parts include product purchased from suppliers, reusable parts
or material collected from the final consumer market, and usable parts from
rejected products (see figure 1). Other assumptions of the model are presented
below.

Assumptions

  * •

Shortage is allowed and will be back-ordered. A penalty will be charged for
each unit of shortage.

  * •

The supply chain operates under cap-and-trade regulations. According to this
mechanism, each manufacturer is allowed to emit a predetermined amount of
carbon. Any extra quantity of carbon emission must be purchased from other
manufacturers in a trade market. Conversely, manufacturers can sell their
unused carbon emission allowance to generate profit. Note that a carbon tax
system allocates a price to carbon emission and puts it on the market to find
the emission reduction mechanism. On the other hand, cap-and-trade assigns an
emission level to the manufacturers and allows the market to find the
allowance price. So, in cap-and-trade mechanism. In other words, buyers and
sellers offer their desired prices, which can be different based on their
profit level, and the manufacturer considers the maximum offer for selling its
allowance and the minimum offer for buying extra allowance.

  * •

Uncertain parameters include demand, cost of products purchased from
suppliers, percentage of the returned products, percentage of the collected
second-hand products, percentage of reusable collected products, percentage of
usable rejected products, and delivery delay.

  * •

Carbon emission is a function of $CO_{2}$ released in the production process
as well as $CO_{2}$ released during material and product transportation.
Purchased products from a supplier can be transported by either the
manufacturer’s trucks (buyer) or the supplier’s trucks. The manufacturer is
responsible for the carbon emission during the transportation process only if
the manufacturer’s trucks are employed. Using the manufacturer’s truck for
transportation is cheaper but increases the amount of carbon released by the
manufacturer.

  * •

Different kinds of trucks exist for transporting the procured materials and
products according to the weight or volume of the loads.

  * •

A constant interest rate is applied to all of the costs through time.

Figure 1: Closed-loop supply chain structure

Indices

$i$: products $(i=1,\cdots,I)$

$j$: suppliers $(j=1,\cdots,J)$

$t$: periods $(t=1,\cdots,T)$

$k$: vehicles $(k=1,\cdots,K)$

$n$: echelons in supply chain $(n=1,2)$ ($n=1$ and $n=2$ show the supplier and
buyer, respectively).

$m$: set of prices offered by buyers or sellers at the carbon emission trading
market

$s$: scenarios $(s=1,\cdots,S)$

Parameters

$Pr_{s}$: probability of scenario $s$

$C_{ijt}^{s}$: cost of product $i$ purchased from supplier $j$ in period $t$
under scenario $s$

$ir$: interest rate

$h_{it}$: holding cost of product $i$ in period $t$

$f_{it}$: backorder cost of product $i$ in period $t$

$L_{jt}^{s}$: number of days in which products purchased from supplier $j$ in
period $t$ under scenario $s$ received after the specified time

$G_{ijt}$: penalty for each day of late delivery of product $i$ purchased from
supplier $j$ in period $t$

$e_{it}^{s}$: percentage of product $i$ rejected in period $t$ under scenario
$s$ from the customer due to low quality

$u_{it}^{s}$: percentage of usable parts after disassembly of product $i$
rejected by customer in period $t$ under scenario $s$

$p_{it}^{s}$: percentage of used product $i$ collected in period $t$ under
scenario $s$ from the customer

$v_{it}^{s}$: percentage of reusable parts after disassembly of product $i$
collected as used product in period $t$ under scenario $s$

$O_{ijt}$: manufacturer’s loss caused by rejected product $i$ purchased from
supplier $j$ in period $t$

$SP_{t}^{m}$: price offered by seller $m$ in the carbon emission allowance
market in period $t$

$BP_{t}^{m}$: price offered by buyer $m$ in the carbon emission allowance
market in period $t$

$DC_{it}$: disassembly cost for product $i$ in period $t$

$RC_{it}$: remanufacturing cost of product $i$ in period $t$

$DP_{it}$: disposal cost of product $i$ in period $t$

$TC_{jtkn}$: transportation cost of products purchased from supplier $j$ in
period $t$ by truck type $k$ belonging to the $n$th echelon of supply chain

$DE_{it}^{s}$: customers’ demand for product $i$ in period $t$ under scenario
$s$

$d_{j}$: distance of supplier $j$ from the producer

$CET_{jtkn}$: amount of carbon emitted per kilometer of transportation from
supplier $j$ in period $t$ by truck type $k$ belonging to $n$th echelon of
supply chain

$CEP_{ijt}$: amount of carbon emitted during production using product $i$
purchased from supplier $j$ in period $t$

$CER_{it}$: amount of carbon emitted during remanufacturing or recycling
product $i$ in period $t$

$EM_{ijt}$: environmental management system score assigned to supplier $j$
from which product $i$ in period $t$ is purchased. This score is determined by
experts at the manufacturer’s firm evaluating the performance of the suppliers
regarding environmental policies(e.g. possession of an ISO 14001 certificate).

$GP_{ijt}$: green product score of supplier $j$ from which product $i$ in
period $t$ is purchased. This is the manufacturer’s strategy to purchase
products with minimum environmental effect during its life cycle.

$RE_{ijt}$: recyclability score of product $i$ purchased from supplier $j$ in
period $t$

$PT_{ijt}$: toxicity score of product $i$ purchased from supplier $j$ in
period $t$

$CAP_{t}$: amount of carbon emission determined by the government as an upper
bound for the manufacturer in period $t$

$M_{k}$: breaking point of loads for each kind of truck used for
transportation

Decision variables

$x_{ijt}$: number of product $i$ purchased from supplier $j$ in period $t$

$r_{it}$: number of remaining product $i$ at the end of period $t$

$b_{it}$: number of back-ordered product $i$ at the end of period $t$

$\alpha_{t}$: allowance of carbon emission the manufacturer needs to buy from
the market in period $t$

$\beta_{t}$: allowance of carbon emission the manufacturer wants to sell to
the other manufacturers in period $t$

$\tau_{t}$: the best price for the manufacturer to buy carbon allowance in
period $t$

$\phi_{t}$: the best price for the manufacturer to sell carbon allowance in
period $t$

$q_{jtkn}$: binary variable, equals 1 if the buyer places an order to supplier
$j$ in period $t$ transported by truck $k$ belonging to $n$th echelon; 0
otherwise

$W_{jtkn}$: auxiliary continuous variable to determine the amount of load from
supplier $j$ in period $t$ transported by truck type $k$ belonging to $n$th
echelon

$\delta_{ijt}^{s+}$: under-fulfillment (shortage) of product $i$ purchased
from supplier $j$ in period $t$

$\delta_{ijt}^{s-}$: over-fulfillment (storage) of product $i$ purchased from
supplier $j$ in period $t$

According to the criteria introduced, three objectives are defined. The first
objective, $\xi_{1}^{s}$, is cost, which is a measure of non-green criteria.
this cost includes uncertain parameters and is formulated as:

$\displaystyle\underset{s\in\\{1,2,\ldots,S\\}}{\xi_{1}^{s}}$
$\displaystyle=\sum_{j}\sum_{t}L_{jt}^{s}\sum_{i}G_{ijt}x_{ijt}+\sum_{i}\sum_{t}e_{it}^{s}\sum_{j}O_{ijt}x_{ijt}+\sum_{i}\sum_{j}\sum_{t}x_{ijt}C_{ijt}^{s}$
(15)
$\displaystyle+\sum_{i}\sum_{t}r_{it}h_{it}+\sum_{i}\sum_{t}b_{it}f_{it}+\sum_{i}\sum_{t}e_{it}^{s}\sum_{j}x_{ijt}DC_{it}$
$\displaystyle+\sum_{i}\sum_{t}p_{it}^{s}\sum_{j}x_{ijt}DC_{it}+\sum_{i}\sum_{t}e_{it}^{s}u_{it}^{s}RC_{it}\sum_{j}x_{ijt}$
$\displaystyle+\sum_{i}\sum_{t}p_{it}^{s}v_{it}^{s}RC_{it}\sum_{j}x_{ijt}+\sum_{i}\sum_{t}e_{it}^{s}(1-u_{it}^{s})DP_{it}\sum_{j}x_{ijt}$
$\displaystyle+\sum_{i}\sum_{t}p_{it}^{s}(1-v_{it}^{s})DP_{it}\sum_{j}x_{ijt}+\sum_{j}\sum_{t}\sum_{k}\sum_{n}TC_{jtkn}q_{jtkn}$
$\displaystyle-\sum_{t}\alpha_{t}\tau_{t}+\sum_{t}\beta_{t}\phi_{t}$

In this equation, the first term calculates the penalty for rejected items. It
is an indication of the quality of the products. The second term is the cost
of delay. The remaining terms calculate other features including purchasing
cost, holding cost, back-order cost, disassembly cost of rejected and
collected products, remanufacturing cost for the usable parts obtained from
collected and rejected products, disposal cost of the unusable parts, and
profit/cost of selling/buying carbon emission allowance. Of note, three
different kinds of criteria (cost, quality, and delivery performance) in this
objective, all of which are combined in the cost function $\xi_{1}^{s}$.

The second objective is the amount of carbon emission, which is a measure of
quantitative green criteria. Due to parameter uncertainty $\xi_{2}^{s}$ is
written as:

$\displaystyle\underset{s\in\\{1,2,\ldots,S\\}}{\xi_{2}^{s}}$
$\displaystyle=\sum_{j}d_{j}\sum_{k}\sum_{t}CET_{jtk2}q_{jtk2}+\sum_{i}\sum_{j}\sum_{t}x_{ijt}CEP_{ijt}$
(16)
$\displaystyle+\sum_{i}\sum_{t}(e_{it}^{s}u_{it}^{s}+p_{it}^{s}v_{it}^{s})CER_{it}\sum_{j}x_{ijt}$

The first term of equation (16) calculates the amount of carbon released
during transportation of purchased products from supplier $j$ by trucks
belonging to the manufacturer. The second term measures the carbon emission
during the production phase, and the third term calculates the carbon released
during the remanufacturing or recycling process fed by components and parts
obtained from disassembling the collected and rejected products.

The third objective is a measure of qualitative green criteria. This objective
function does not include uncertain parameters, and is defined as:

$\displaystyle z_{3}$
$\displaystyle=\sum_{i}\sum_{j}\sum_{t}LC_{ijt}x_{ijt}+\sum_{i}\sum_{j}\sum_{t}ER_{ijt}x_{ijt}+\sum_{i}\sum_{j}\sum_{t}RE_{ijt}x_{ijt}$
(17) $\displaystyle+\sum_{i}\sum_{j}\sum_{t}TS_{ijt}x_{ijt}$

The robust model of the described problem is therefore formulated as:

$\displaystyle\begin{split}&Min\;z_{1}=\sum_{s}Pr_{s}\xi_{1}^{s}+\lambda_{1}\sum_{s}Pr_{s}\left[\left(\xi_{1}^{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{1}^{s^{\prime}}\right)+2\theta_{1}^{s}\right]\\\
&\qquad\qquad\qquad+\omega\sum_{s}Pr_{s}\left(\sum_{i}\sum_{j}\sum_{t}\left(\delta_{ijt}^{s^{-}}+\delta_{ijt}^{s^{+}}\right)\right)\end{split}$
(18)
$\displaystyle\begin{split}&Min\;z_{2}=\sum_{s}Pr_{s}\xi_{2}^{s}+\lambda_{2}\sum_{s}Pr_{s}\left[\left(\xi_{2}^{s}-\sum_{s}Pr_{s^{\prime}}\xi_{2}^{s^{\prime}}\right)+2\theta_{2}^{s}\right]\\\
&\qquad\qquad\qquad+\omega\sum_{s}Pr_{s}\left(\sum_{i}\sum_{j}\sum_{t}\left(\delta_{ijt}^{s^{-}}+\delta_{ijt}^{s^{+}}\right)\right)\end{split}$
(19) $\displaystyle Max\;z_{3}$ (20)
$\displaystyle\left(\xi_{1}^{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{1}^{s^{\prime}}\right)+\theta_{1}^{s}\geq
0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall s$ (21)
$\displaystyle\left(\xi_{2}^{s}-\sum_{s^{\prime}}Pr_{s^{\prime}}\xi_{2}^{s^{\prime}}\right)+\theta_{2}^{s}\geq
0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall s$ (22)
$\displaystyle\begin{split}&\sum_{j}d_{j}\sum_{k}CET_{jtk2}q_{jtk2}+\sum_{i}\sum_{j}x_{ijt}CEP_{ijt}\\\
&\qquad\qquad+\sum_{i}(e_{it}^{s}u_{it}^{s}+p_{it}^{s}v_{it}^{s})CER_{it}\sum_{j}x_{ijt}\leq
CAP_{t}-\alpha_{t}+\beta_{t}\qquad\qquad\forall t,s\end{split}$ (23)
$\displaystyle\tau_{t}=\max_{\forall
m}BP_{t}^{m}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
t$ (24) $\displaystyle\phi_{t}=\min_{\forall
m}SP_{t}^{m}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
t$ (25)
$\displaystyle\begin{split}&u_{it}^{s}e_{it}^{s}\sum_{j}x_{ijt}+v_{it}^{s}p_{it}^{s}\sum_{j}x_{ijt}+\sum_{j}x_{ijt}+b_{it}+r_{it-1}+\sum_{j}\delta_{ijt}^{s-}\\\
&\qquad\qquad=DE_{it}^{s}+r_{it}+b_{it-1}+\sum_{j}\delta_{ijt}^{s+}\qquad\qquad\qquad\qquad\qquad\forall
i,t,s\end{split}$ (26)
$\displaystyle\sum_{i}x_{ijt}=\sum_{k}M_{k}\sum_{n}W_{jtkn}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,t$ (27) $\displaystyle W_{jt1n}\leq
q_{jt1n}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,t,n$ (28) $\displaystyle W_{jt(k+1)n}\leq
q_{jt(k+1)n}+q_{jtkn}\qquad\qquad\qquad\qquad\qquad\forall
j,t,n;(k=1,\cdots,(K-1))$ (29) $\displaystyle W_{jtKn}\leq
q_{jt(K-1)n}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,t,n$ (30)
$\displaystyle\sum_{k}q_{jtkn}=1\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,t,n$ (31)
$\displaystyle\sum_{k}W_{jtkn}=1\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,t,n$ (32) $\displaystyle 0\leq W_{jtkn}\leq
1\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,k,t,n$ (33) $\displaystyle
C_{ij(t+1)}^{s}=C_{ijt}^{s}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,j,t,s$ (34) $\displaystyle
h_{i(t+1)}=h_{it}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,t$ (35) $\displaystyle
f_{i(t+1)}=f_{it}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,t$ (36) $\displaystyle
G_{ij(t+1)}=G_{ijt}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,j,t$ (37) $\displaystyle
O_{ij(t+1)}=O_{ijt}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,j,t$ (38) $\displaystyle
DC_{i(t+1)}=DC_{it}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,t$ (39) $\displaystyle
RC_{i(t+1)}=RC_{it}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,t$ (40) $\displaystyle
DP_{i(t+1)}=DP_{it}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
i,t$ (41) $\displaystyle
TC_{j(t+1)kn}=TC_{jtkn}(1+ir)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
j,t,k$ (42)

Equation (18) aims to minimize the costs while trying to obtain the highest
profit in the carbon emission trading market. Due to defining different
scenarios, this objective is written as a robust objective to obtain the
closest solution to all scenarios while keeping infeasibility at the lowest
level. Equation (19) is another robust objective aiming to minimize the carbon
emission. Note that equations (18) and (19) go in opposite directions. In
other words, the first and second objectives restrict each other. Therefore,
the optimal solution is found through a trade-off between these two
objectives. The third objective maximizes quality by choosing the suppliers
with the best qualitative performance. Suppliers’ qualitative scores are
defined by experts on a scale of $0-10$.

Equations (21) and (22) transform the first and second objectives to linear
functions. These equations ensure that the deviation of each scenario from the
average objective value is a positive amount. Equation
(LABEL:eq:23_cap_constraint) adheres to the manufacturer’s carbon emission
allowance, yet makes it possible to buy or sell the allowance. Buying and
selling price is defined in equations (24) and (25), where the maximum and
minimum offered price is selected to sell and buy the allowance, respectively.
Equation (26) balances the inventories.

Equations (27) to (33) confirm that the purchased products from each supplier
are assigned to a specific truck category based on their size. For instance,
if three different kinds of trucks are classified by their load size, and the
order from a specific supplier fits on the first category, the variable
$x_{ijt}$ will be a linear combination of first and second breaking points.
Hence, $q_{jt1n}$ will be equal to 1 and variables $W_{jt1n}$ and $W_{jt2n}$
will be positive. Equations (34)-(42) refer to the interest rate applied to
the prices.

###  3.3 Solution procedure

To solve the proposed RO model a two-step procedure was followed. In the first
step, each of the three objectives are independently solved , i.e., the model
is solved three times; each time, one of the objectives is considered and the
other two objectives are removed from the model. This way, the optimum value
of each objective is found in the absence of the other objectives. Obviously,
in multi-objective models the value of each objective (in the presence of the
other objectives) is worse than or equal to their global optimal value. Thus,
there is always a deviation between objective function values in multi-
objective models and their optimal values; the goal is to minimize this
deviation, which will be done in the second step. In the second step, the
model is reformulated as a single objective mathematical model whose objective
function is minimizing the normalized deviation of the objectives from their
optimal values. Assume that the values of the first, second, and third
objective are demonstrated as $z_{1}$, $z_{2}$, and $z_{3}$, respectively; and
the optimal values are illustrated as $z_{1}^{*}$, $z_{2}^{*}$, and
$z_{3}^{*}$. Then, the objective function is as follows.

$Min\;z_{total}=\left[\frac{z_{1}-z_{1}^{*}}{z_{1}^{*}}+\frac{z_{2}-z_{2}^{*}}{z_{2}^{*}}+\frac{z_{3}^{*}-z_{3}}{z_{3}^{*}}\right]$
(43)

In equation (43), $z_{1}$ and $z_{2}$ are minimization objectives and their
values will be greater than or equal to their optimal values, $z_{1}^{*}$ and
$z_{2}^{*}$, respectively. On the other hand, $z_{3}$ is a maximization
objective and its value will be smaller than or equal to its optimal value,
$z_{3}^{*}$. As the deviation of an objective from its optimal value cannot be
negative, equation (43) is written such that it remains positive
$\left(z_{1}-z_{1}^{*}\geq 0;z_{2}-z_{2}^{*}\geq 0;z_{3}^{*}-z_{3}\geq
0\right)$.

##  4 Computational results

In order to validate the described RO model for green supplier evaluation, we
present results of various instances of the model. The experiments were
conducted based on the data set of table 2, which was generated randomly using
uniform distribution. Note that the parameters related to price were generated
for the first period, and the next periods were calculated according to model
constraints. Also, three different scenarios were considered, namely
pessimistic, most realistic, and optimistic.

Table 2: Data for Numerical Example

Sets and | Scenarios | Amounts
---|---|---
parameters |  |
$i$ |  | $4$
$j$ |  | $5$
$t$ |  | $4$
$k$ |  | $3$
$s$ |  | $3$
$m$ |  | $3$
$ir$ |  | $0.04$
$\lambda_{1}$ |  | $15$
$\lambda_{2}$ |  | $15$
$\omega$ |  | $50$
$Pr^{s}$ |  $1$, 2, 3 |  $0.2$, 0.6, 0.2
$C_{ij1}^{s}$ |  $1$, 2, 3 |  $U(10,23)$, $U(11.5,26)$, $U(13,30)$
$h_{i1}$ |  | $U(28,35)$
$f_{i1}$ |  | $U(33,41)$
$L_{jt}^{s}$ | $1,2,3$ | $U(0,5)$
$G_{ijl}$ |  | $U(6,12)$
$e_{it}^{s}$ |  $1$, 2, 3 |  $U(0.03,0.092)$, $U(0.035,0.126)$, $U(0.04,0.145)$
$p_{it}^{s}$ |  $1$, 2, 3 |  $U(0.02,0.08)$, $U(0.023,0.092)$, $U(0.027,0.105)$
$u_{it}^{s}$ |  $1$, 2, 3 |  $U(0.6,0.9),U(0.62,0.93)$, $U(0.63,0.94)$
$v_{it}^{s}$ |  $1$, 2, 3 | $U(0.6,0.9),U(0.72,0.93),U(0.73,0.94)$
$O_{ij1}$ |  | $U(5,11)$
$SP_{t}^{m}$ |  | $U(4000,4020)$
$BP_{t}^{m}$ |  | U(3980,4000)
$DC_{i1}$ |  | U(4,7)
$RC_{i1}$ |  | U(10,17)
$DP_{i1}$ |  | U(3,5)
$TC_{j1kn}$ |  | $TC_{j111}=U(28,37),\;TC_{j112}=U(29,38),\;TC_{j121}=U(35,40)$
|  |  $TC_{j122}=U(36,41)$, $TC_{j131}=U(39,52)$, $TC_{j132}=U(40,53)$
$DE_{it}^{s}$ | $1,2,3$ |  $U(2500,4600)$, $U(2930,4760)$, $U(3070,4990)$
$d_{j}$ |  | $U(3,7)$
$CET_{j}tkn$ |  |  $CET_{jt1n}=U(0.29,0.37)$, $CET_{jt2n}=U(0.33,0.46)$
|  | $CET_{jt3n}=U(0.41,0.49)$
$CEP_{ijt}$ |  | $U(0.006,0.012)$
$CER_{ijt}$ |  | $U(0.006,0.012)$
$EM_{ijt}$ |  | $U(1,10)$
$GP_{ijt}$ |  | $U(1,10)$
$RE_{ijt}$ |  | $U(1,10)$
$PT_{ijt}$ |  | $U(1,10)$
$CAP_{t}$ |  | $U(170,200)$
$M_{k}$ |  | ${3000,6000,14000}$

The results of the solved model are shown in table 3. The results determine
the best suppliers and the optimal order allocations. For instance,
$X_{112}=2617$ indicates that the manufacturer should place an order for 2617
of product 1 to the first supplier in the second period. Moreover, the optimal
shortage and storage amounts, which are calculated based on the optimal orders
are provided. The over-fulfillment and under-fulfillment values express the
extra orders and the ignored demand, respectively. These amounts are
calculated based on the relative importance the manufacturer assigns to
solution optimality, i.e., achieving the best possible answer, compared to
customer satisfaction. Furthermore, the optimum carbon trade quantity in each
period is given. The results demonstrate that the optimal quantities for
purchasing carbon allowance are 96.939, 64.684, 88.764, and 134.056 in periods
1, 2, 3, and 4, respectively. By using this data, managers will be able to
find the right amount to order from each supplier considering the defined
criteria, and how much carbon allowance they need to buy or sell in each
period.

Table 3: Results of Numerical Example Variables | Results
---|---
$X_{ijt}$ | $X_{112}=2,617\;\;\;X_{141}=1,825\;\;\;X_{142}=2,506\;\;\;X_{152}=1,627\;$
|
$X_{153}=4,091\;\;\;\;X_{211}=2,558\;\;\;\;X_{212}=381\;\;\;\;\;\;X_{214}=3,121$
| $X_{231}=126\;\;\;\;\;\;\;X_{243}=810\;\;\;X_{252}=2,340\;\;\;X_{253}=1,908$
| $X_{311}=2,962\;\;\;X_{312}=2,756\;\;\;X_{343}=2,813\;\;\;X_{412}=2,861$
| $X_{414}=2,690\;\;\;X_{421}=3,382$
$r_{it}$ | $0$
$b_{it}$ | $b_{14}=4,243\;\;\;b_{34}=3,056\;\;\;b_{43}=3,657\;\;\;b_{44}=3,937$
$\alpha_{t}$ | $\alpha_{1}=96.939\;\;\;\alpha_{2}=64.684\;\;\;\alpha_{3}=88.764\;\;\;\alpha_{4}=134.056$
$\beta_{t}$ | $0$
$\delta_{ijt}^{s+}$ | $\delta_{111}^{s+}=178\;\;\;\delta_{112}^{s+}=159\;\;\;\delta_{124}^{s+}=202\;\;\;\delta_{153}^{s+}=147$
|
$\delta_{214}^{s+}=123\;\;\;\delta_{222}^{s+}=101\;\;\;\delta_{243}^{s+}=106\;\;\;\delta_{251}^{s+}=92$
|
$\delta_{312}^{s+}=97\;\;\;\delta_{313}^{s+}=113\;\;\;\delta_{341}^{s+}=126\;\;\;\delta_{354}^{s+}=145$
|
$\delta_{432}^{s+}=103\;\;\;\delta_{441}^{s+}=143\;\;\;\delta_{443}^{s+}=174\;\;\;\delta_{454}^{s+}=105$
$\delta_{ijt}^{s-}$ | $\delta_{111}^{s-}=184\;\;\;\delta_{114}^{s-}=212\;\;\;\delta_{123}^{s-}=150\;\;\;\delta_{142}^{s-}=164$
|
$\delta_{221}^{s-}=94\;\;\;\delta_{232}^{s-}=104\;\;\;\delta_{253}^{s-}=109\;\;\;\delta_{254}^{s-}=127$
|
$\delta_{314}^{s-}=153\;\;\;\delta_{321}^{s-}=131\;\;\;\delta_{322}^{s-}=99\;\;\;\delta_{343}^{s-}=117$
|
$\delta_{411}^{s-}=148\;\;\;\delta_{412}^{s-}=105\;\;\;\delta_{443}^{s-}=183\;\;\;\delta_{444}^{s-}=107$
$z_{1}$ | $640,743$
$z_{2}$ | $43,627$
$z_{3}$ | $1,133,472$
$z_{Total}$ | $1.699$

To analyze the model and check the sensitivity of its objectives to
parameters, several analyses were performed. The results will be discussed in
the following sections.

###  4.1 Sensitivity analysis on $\omega$

One of the important parameters in robust models is $\omega$, the value that
weighs the trade-off between optimality and feasibility. In this model,
infeasibility is defined as over-fulfillment or under-fulfillment. Over-
fulfillment elevates the warehousing costs, while under-fulfillment results in
lower customer satisfaction; customer’s order is affected by the possibility
of under-fulfillment. To decrease the risk of infeasibility, one should
increase the penalty of receiving an infeasible solution, which, in turn,
leads to reduced model optimality. Therefore, there is always a trade-off
between model infeasibility and optimality. Understandably, it is not easy for
decision-makers to determine the exact value of $\omega$. Consequently, a
suitable way to investigate this issue is to measure different values of the
objectives and the amount of model infeasibility when distinct values for
$\omega$ are considered. Afterward, an analysis of the results enables the
decision-makers to manipulate the preferred value of $\omega$.

By increasing the value of $\omega$, due to the imposed penalty, the model
tries to decrease the infeasibility. Therefore, as shown in figure 2(a), when
$\omega$ increases, infeasibility, which is defined as the amount of over-
fulfillment and under-fulfillment, decreases. On the other hand, by increasing
the value of $\omega$, firms will be penalized more for infeasibility; in this
situation, first and second objective functions, which are formulated as
robust equations, will deteriorate, and consequently $z_{Total}$ will become
worse. This is shown in figure 2(b).

By scrutinizing these results, decision-makers can assign a precise value to
$\omega$ based on the goals defined by the firm. For instance, if the
objective values are more important, a lower weight for model robustness
should be set. On the other hand, if customer satisfaction is more critical,
more penalty must be in place for infeasibility (higher values for $\omega$),
which leads to worse objective value.

Furthermore, in figure 2(b), a comparison is performed between the values of
robust and deterministic models. The figure depicts that for a large domain of
$\omega$ values ($\omega>25$) the objective function of the deterministic
model is better than the robust model. This occurs due to the penalty imposed
to the robust model for deviation and infeasibility, i.e., there is more
restriction on the objective function. Regardless, by handling uncertainty,
robust models are able to achieve solutions that are closer to real world
scenarios. This enhances their applicability.

Figure 2: Sensitivity analysis on $\omega$

###  4.2 Sensitivity analysis on $\lambda$

Another crucial aspect is the analysis of the impact of the penalty on the
deviation of scenarios from the expected value of the objective. Note that the
model considers various scenarios, and one cannot be certain which one will
happen. Hence, the solution method seeks to achieve a solution which is the
closest possible value to the optimum of all scenarios. In this sense,
$\lambda$ is the weight parameter that alludes to the importance of solution
closeness to the real-world application. A larger value for $\lambda$ implies
a higher level of importance for deviation from the average; plus, the
objective value deteriorates due to the increased penalty. This is illustrated
in figure 3. As displayed in figure 3(a), by increasing the value of
$\lambda_{1}$, the deviation between the objective value in each scenario and
the average objective value decreases. On the other hand, according to figure
3(b), the value of the first objective function increases. In other words, the
first objective function value exacerbates as it is a minimization objective.
Consequently, total objective value deteriorates.

The same occurs for $\lambda_{2}$, but since in this paper the values of the
second objective are significantly smaller than the first objective,
$\lambda_{2}$ has negligible effect on the deviation of the objective function
(see table 4). It does not mean that the second objective is not important,
because the total objective is normalized using equation (43) to avoid
overestimating the importance of objectives with large values. However, the
deviation is fixed on a small amount (11.148) compared to the first objective
(486,736), and adding the value of $\lambda_{2}$ does not have an impact on
the deviation, although the second objective function worsens (figure 3(c)).
Therefore, it is recommended to set the value of $\lambda_{2}$ to a small
number to avoid deteriorating the second objective, which represents the
amount of emission.

In other words, putting more penalty on the deviation of objective values in
different scenarios reduces the deviation, but worsens the objective values.
This provides a road map for the decision-makers to choose values of
$\lambda_{1}$ and $\lambda_{2}$ according to their goals.

Figure 3: Sensitivity analysis on $\lambda$ Table 4: Effect of $\lambda_{2}$ on Deviation of Second Objective Value $\lambda_{2}$ | $z_{Total}$ | $z_{1}$ | $z_{2}$ | $z_{3}$ | $\delta$ | Deviation from Second Objective
---|---|---|---|---|---|---
0 | 12.557 | 486,736 | 43,182 | 1,170,111 | 4,283 | 11.148
2 | 12.56 | 486,736 | 43,193 | 1,170,111 | 4,283 | 11.148
5 | 12.572 | 486,736 | 43,237 | 1,170,111 | 4,283 | 11.148
10 | 12.587 | 486,736 | 43,291 | 1,170,111 | 4,283 | 11.148
15 | 12.602 | 486,736 | 43,346 | 1,170,111 | 4,283 | 11.148
22 | 12.623 | 486,736 | 43,422 | 1,170,111 | 4,283 | 11.148
23 | 12.626 | 486,736 | 43,433 | 1,170,111 | 4,283 | 11.148
24 | 12.629 | 486,736 | 43,444 | 1,170,111 | 4,283 | 11.148

###  4.3 Sensitivity analysis on CAP

As mentioned in section 1, governments usually reduce the assigned cap in cap-
and-trade mechanism each year to decrease air pollution111As an example, see
the cap-and-trade mechanism of Ontario, Canada (accessed March 13, 2020):
https://www.ontario.ca/page/cap-and-trade.. Therefore, an analysis of cap
reduction is required to predict the probable effects of this restriction and
make the right actions toward the change.

Figure 4(a) shows a reverse relationship between objective values and carbon
cap. It indicates that by decreasing the carbon allowance cap, the cost
objective function ($\times 10^{5}$) increases, thus worsening the total
objective value. This matter can be investigated from different perspectives.
From government’s point of view, the total utility decreases by decreasing the
cap, but it results in less carbon emission, which can be deemed as a positive
result for government due to health- and environmental-related costs. From the
manufacturer’s point of view, decreasing the cap is not an ideal option,
because they are forced to purchase more allowance, or they will make less
money for not being able to sell as much allowance, or they will have to
upgrade to more expensive green technologies.

In addition, this restriction affects the amount of allowance to buy or sell.
As displayed in figure 4(b), while the cap decreases, the amount of total
carbon allowance the manufacturer needs to buy (sell), increases (decreases).
This is the main driver for the increased cost.

Furthermore, according to figure 4(c), changing the cap does not have a
discernible impact on the optimal amount of carbon emission ($z_{2}$); no
trend can be verified in this figure. For the parameter values featured in
this study, a cap reduction does not significantly affect the carbon released
by manufacturers since they consider the carbon allowance market as a source
of profit. In other words, manufacturers focus on the cost function while
trying to satisfy the restrictions related to carbon emission.

Using these results help the government assign the proper carbon emission cap
and minimize the impact on manufacturers in terms of emission costs.

Figure 4: Sensitivity analysis on CAP†
†Values on the x-axis show cap reduction. For example, -0.1 means cap is
decreased by 10%.

###  4.4 Sensitivity analysis on cap-and-trade market prices

Buying and selling prices are critical parameters in the cap-and-trade system
with a direct impact on the amount of carbon allowance being traded by firms.
According to table 5, by increasing the selling price, the values of the first
and total objectives decrease. This can be traced to the growth in the optimal
value of the amount of sold allowance.

As shown in table 5, when $BP>SP$ the manufacturers turn into brokerages. In
this case, firms can buy allowance in the market and sell it at a higher price
for a profit. Therefore, the optimal values of variables that control buying
and selling allowance in the market will be positive in the same period. To
avoid this situation, government can restrict the carbon market price such
that $BP$ is always less than $SP$ in carbon trading market.

Table 5: Results of sensitivity analysis on carbon market prices Parameter | Price change (%) | $z_{Total}$ | $z_{1}$ | $z_{2}$ | $z_{3}$ | $\alpha$ | $\beta$
---|---|---|---|---|---|---|---
BP | -20 | 13.24 | 847,973 | 40,670 | 1,365,767 | 296 | 0
| -10 | 12.783 | 728,194 | 40,670 | 1,365,767 | 296 | 0
| -5 | 12.9 | 555,835 | 43,466 | 1,147,220 | 377 | 0
| 0 | 12.602 | 486,736 | 43,346 | 1,170,111 | 387 | 0
| 5 | 10.108 | 39,832 | 40,497 | 1,396,833 | 5,343 | 5,085
| 10 | -0.306 | 1,363 | 3,684 | 5,571,751 | 3.9×108 | 3.9×108
SP | -10 | -0.307 | 1,363 | 3,681 | 5,568,767 | 9.9×107 | 9.9×107
| -5 | -0.307 | 1,363 | 3,688 | 5,605,797 | 135,493 | 136,376
| 0 | 12.602 | 486,736 | 43,346 | 1,170,111 | 387 | 0
| 5 | 12.602 | 486,736 | 43,346 | 1,170,111 | 385 | 0
| 10 | 12.602 | 486,736 | 43,346 | 1,170,111 | 387 | 0
| 20 | 12.602 | 486,736 | 43,346 | 1,170,111 | 387 | 0

###  4.5 Cap-and-trade versus paying penalty to the government

Another regulation possibility to investigate is forcing the manufacturers to
pay a penalty to the government for carbon emission once a certain cap is
exceeded. This policy does not allow formation of a carbon emission trading
market. As shown in figures 5(a)-5(c), for the cap changes below 40 percent,
cost of pollution is less in cap-and-trade compared to the penalty-based
mechanism. However, the amount of carbon released under cap-and-trade policy
is more than the emissions in the government penalty regulation. Therefore, in
case there is a high priority on the carbon reduction objective, government
must decrease the cap in the C&T mechanism to a value which is less than 40%
of the original amount. Also, $z_{Total}$ in cap-and-trade is lower in the
given range. In other words, cap-and-trade performs better than the penalty-
based system for manufacturers. Namely, cap-and-trade derives a better total
utility; thus, it is preferred to the penalty-based approach.

If the cap is lowered more than 40 percent, no superiority exists between the
two mechanisms. This occurs since the cap is decreased below the 40 percent
mark, the amount of allowance the manufacturer needs to buy (or the amount of
penalty the manufacturer should pay to the government) will be positive; and
the quota to sell will gradually tend to zero. In other words, the
manufacturer will not be able to generate profit by selling emission
allowance. Hence, the gap between the cap-and-trade mechanism and the penalty
system will vanish. The above scrutiny is an important result for the
governments to assign effective values for the cap according to the defined
emission targets, while not hurting the businesses.

Based on the presented results, cap-and-trade culminates in better outcomes in
terms of total model utility, although with higher carbon emission which is
the result of assigning equal weights to economic selection criteria, green
selection criteria, and carbon emission level. Obviously, assigning more
weight to carbon emission level translates to solutions with less carbon
emission.

Another inquiry on the two carbon control systems is based on the fact that
cap-and-trade mechanism allocates a constant carbon emission permission to the
entire market. Therefore, the manufacturers buy the allowance of the other
players in the carbon trading market, while the total allowed carbon emission
is a fixed value. AS a result, if the firms choose not to sell their allowance
in the market, or if they consume all of their own allowance, other firms will
not be able to place orders anymore.

Conversely, in penalty-based systems, although there is a cap determined by
the government, the total amount of carbon released can exceed the cap because
manufacturers can emit as much carbon as they need by paying the penalty to
the government. Therefore, the total carbon released by manufacturers may be
more in this system. Note that this argument cannot be confirmed by the
results of this paper because data from all of the manufacturers in the market
is required to audit the total amount of carbon emission. Consequently, this
topic is left for study in future research efforts.

Figure 5: Cap-and-trade vs. penalty-based

##  5 Conclusion

To support decisions related to outsourcing in a green supply chain and to
help with procuring the best materials from the best suppliers in an uncertain
environment encompassing both green and traditional criteria, a multi-
objective robust optimization model is generalized in this paper. The cap-and-
trade mechanism is considered to evaluate the role of this pollution control
system. A numerical example is presented to analyze different aspects of the
generalized model and the solution approach.

The optimal activity levels of a firm to achieve the maximum profit while
meeting the environmental goals are obtained. The proposed model enables the
decision makers to work with a more flexible decision support system due to
the analysis done on $\omega$ and $\lambda$. The results show the effect of
cap amount and the trade prices on the firm’s objective. Therefore,
manufacturers will be well-informed about selecting the best suppliers, the
amount of orders to place with each of the selected suppliers, and trading in
the allowance market to meet their objectives. Additionally, the superiority
of cap-and-trade mechanism compared to penalty-based system regarding the
total utility of supply chain from a micro-economic perspective is shown in
this paper.

Since this paper is one of the first to consider cap-and-trade mechanisms in
the green supply chain, more research in this area can be pursed. For
instance, this paper analyzes the cap changes and its effects on the
objectives from the manufacturer’s point of view. Accordingly, a motivating
topic for future research is finding the optimal value of the cap assigned by
the government to reduce the adverse environmental effects of production
industries, while remaining business friendly. One could explore a pricing
study to find a reasonable range for the allowance prices according to
different factors. Specifically, decreasing the cap will leave all of the
manufacturers with less allowance for carbon emission. Therefore, the
manufacturers will have less allowance to offer in the trade market, resulting
in elevated trading prices. In this case, some manufacturers may decide to
simply sell their allowance to generate profit. This presents a challenge from
the existing problem’s perspective.

In this paper, we studied the difference between cap-and-trade system and
penalty-based mechanism to control the air pollution from the manufacturer’s
point of view. If one has access to data about the total amount of carbon
released by all manufacturers in the market, one could study the differences
between the two particular pollution control regimes from the government’s
perspective.

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