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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.20 No.4 pp.509-521
DOI : https://doi.org/10.7232/iems.2021.20.4.509

# Production-Distribution Problem Optimization in a Green Closed-Loop Supply Chain

Anastasia A. Kurilova*, Hafis Ahmed Oglu Hajiyev, Elvira Irekovna Abdullina, Alla Borisovna Plisova, Antonina Alexandrovna Arkhipenko
Togliatti State University, Russia
Azerbaijan State University of Economics (UNEC), Republic of Azerbaijan
Kazan Federal University, Naberezhnye Chelny Institute KFU, Russia
MIREA - Russian Technological University (RTU MIREA), Russia
Kuban State Agrarian University named after I.T. Trubilin, Russia
*Corresponding Author, E-mail: aakuriloval@yandex.ru
August 10, 2021 September 10, 2021 September 27, 2021

## ABSTRACT

Integrated production-distribution planning in traditional forward supply chain has attracted a lot of attention in recent years and its economic advantages are particularly noticeable. In this paper, the issue of integrated productiondistribution planning of a multi-level green closed loop, including recycling centers, manufacturing / remanufacturing factories and decentralized distribution centers, is investigated. A three-level bi-objective model with the aim of increasing profit and reducing the adverse environmental effects is presented. A revised multi-choice goal programming approach and a hierarchical iteration algorithm is used to solve it.

## 1. INTRODUCTION

Production-distribution planning is the main activity in supply chain optimization. In production planning, it is decided to employ and send down workers, manufacture in regular and extra times, sign lateral contracts, and examine machine capacity for a planning horizon. On the other hand, distribution planning decisions are made to determine which facilitates will supply market demands. In the past, suppliers, manufacturers, distributors, and retailers were recognized as independent businesses attempting to maximize their profits (Barbarosoğlu and Özgür, 1999). The process of maximizing the profit of independent companies does not mean raising the maximum profits of the entire supply chain. Therefore, it is crucial to know how to integrate production-distribution planning (PDP) decisions to ensure the overall importance and operational performance of the supply chain (Liang and Cheng, 2009).

Green supply chain management (GSCM) aims at taking into account environmental considerations and requirements in supply chain management (SCM), which covers the supply of raw materials, production and delivery process, satisfying customer requirements, and managing the life of the products utilized (Srivastava, 2008). Lin et al. (2014) classified the various approaches utilized in the Green Closed-Loop Supply Chain Network Design (CLGSCND) into three categories, i.e., optimization of fuel consumption, GHG emissions reductions, and waste management (Dekker et al., 2012;Roozitalab et al., 2017).

This study aims at optimizing an integrated production- distribution planning in the green closed-loop supply chain (GCLSC) and proposing a three-level bi-objective mathematical model. The remainder of this manuscript is structured as follows. The literature review is presented in section II. Section III covers hypotheses and mathematical models. Solutions are presented in section IV and numerical models in section V. Ultimately, the manuscript will be concluded in section VI.

## 2. LITERATURE REVIEW

The design and analysis of production-distribution systems have been a topic of interest for many years. In 2016, Ma et al. (2016) proposed a productiondistribution planning model using bi-level planning for supply chain management. Rafiei et al. (2018) proposed an integrated production and distribution planning problem in a four-level supply chain with two main objectives of minimizing the total cost of the supply chain and the target function and maximizing the level of services provided. Jing and Li (2018) investigated a multilevel decentralized closed-loop supply chain system planning covering a joint recycling center, several productions and reproduction factories, and numerous decentralized distributions to different regions. Recently, a solution approach has been designed by a hierarchical iterative strategy based on the self-adaptive genetic algorithm (SAGA). Manno et al. (2019) introduced a mathematical model for optimizing the production and distribution of marine equipment in Italy. They performed a full analysis of the effect of each cost on production and distribution decisions. Golsefidi and Jokar (2020) employed robust optimization tools to simultaneously optimize production, distribution, and inventory. They worked with genetic algorithms and refrigeration simulations for problem optimization.

This study is a continuum of works of Jing and Li (2018) and Golsefidi and Jokar (2020) in which the criteria of the bi-objective green closed-loop supply chain are considered aiming to increase profits and reduce adverse environmental effects and variable capacity of vehicles in the model.

## 3. MATHEMATICAL MODEL

This study focuses on a multi-level decentralized closed-loop supply chain system comprising several recycling centers, many productions and reproduction factories, and several decentralized distributors to different regions. The task of recycling centers is to decompose the returned product into its constructive parts. If these parts were standard, they can be stored in the inventory as high-quality parts, otherwise, will be on the shelf. Highquality parts are transferred to production/reproduction factories according to manufacturing demands.

If required, parts will be processed in production/ reproduction factories. High-quality parts are stored in the inventory of quality parts and reproduced parts manufactured by reprocessing high-quality parts are stored in the inventory of reproduced parts. Factories also manufacture new parts by processing raw materials. Products are manufactured by assembling reproduced and high-quality parts and are stored in the inventory of reproduced products. New products are manufactured by assembling new parts and are stored in new product inventory. According to downstream orders, new or reproduced products are delivered to various distributors. Each distributor can choose one or more factories for service and deliver new and reproduced products to retailers or customers for sale. At the same time, distributors collect expired products from retailers or customers and return them to recycling centers as returned products. The hypothesis of the model include the following:

• • Demand for reproduced and new products is definite and the sale price of reproduced products is lower than that of new products.

• • The cost of production varies from one factory to another.

• • The capacity differs for various vehicles.

• • Shipping costs vary between nodes.

• • Transportation of vehicles at each level and each period is allowed to emit a limited rate of greenhouse gases.

• •The processing of parts will emit greenhouse gases into the environment.

Indices

• t : The set of indices of periods {1,…,T}

• p: The set of indices of the product type {1,…P}

• c : The set of indices of the part type {1,…C}

• v : The set of indices of the type of vehicle {1,…V}

• i : The set of indices of production/reproductions factories {1,…I}

• j : The set of indices of distributors {1,…J}

• k : The set of indices of recycling centers {1,…K}

Parameters of the first-level model

• PPCkict The cost of purchasing each part (case) with the quality of c by the production / reproduction factory of i to the recycling center at the period of t

• URCCjkpt The cost of purchasing each returned product P of the recycling center of k to the distributor j at the period t

• SDTkpt The cost of implementation if the returned product p is separated and tested in the recycling center k at the period t

• UDTCkpt The cost of separating and testing each returned product P in the recycling center k at the period t

• UDCkct The cost of disposing each unit of part c in the recycling center k at the period t

• $ICQC kct R$ The cost of each part with the quality ofc in the recycling center of k at the period t

• $ICRP kpt R$ The cost of each returned part P in the recycling center k at the period t

• $UTC kicvt RF$ The cost of transporting each part with the quality of c from the recycling centerk to production/reproduction factories i atthe period t

• BOCpc A list of part c in the product p

• θct The rate of reproduction of part c at the period t

• $α ¯ k p R$ Maximum inventory of the returned product p in the recycling center k

• $β ¯ k c R$ Maximum inventory of the part with the quality of c in the recycling center

• $M D T k p R$ Maximum volume of the returned product p that can be separated and tested in each period in the recycling center k

• $c a p v R F$ The capacity of a vehicle v floating between the recycling center and the production / reproduction factory

• $d i s k i R F$ The distance between the recycling center k and production/reproduction factories i

• $E m i s k i c v t R F$ The rate of GHGs emission for each part with the quality of c by the vehicle v per each kilometer between the recycling center k and production/reproduction factories i

• $TE t max R F$ Maximum allowable emission of GHGs by the transportation system floating between the recycling center and factories at the period t

Parameters in the second-level model

• $M P N i j p t$ Mean price of each unit of product p by the distributor j to the production/reproduction factory i at the period t

• MPRijpt Mean price of each unit of reproduced product p by the distributor j to the production/ reproduction factory i at the period t

• SAipt The cost of implementation if the new product p is assembled in the factory i at the period t.

• SRAipt The cost of implementation if the reproduced product p is assembled in the factory i at the period t.

• UACipt The cost of assembling each unit of product p in the factory i at the period t

• URACipt The cost of assembling each unit of reproduced product p in the factory i at the period t

• SPict The cost of implementation if part c is newly processed in the factory i at the period t

• RSPict The cost of implementation if part c is newly reprocessed in the factory i at the period t

• UPCict The cost of processing each part c in the factory i at the period t

• URPCict The cost of reprocessing each part c in the factory i at the period t

• $I C N P i p t F$ The cost of inventory of each unit of new product p in the factory i at the period t

• $I C R M P i p t F$ The cost of inventory of each unit of reproduced product p in the factory i at the period t

• $I C Q C i c t F$ The cost of inventory of each part with the quality of c in the factory i at the period t

• $I C N C i c t F$ The cost of inventory of each unit of new part c in the factory i at the period t

• $I C R C i c t F$ The cost of inventory of each unit of reproduced part c in the factory i at the period t

• $U T C i j p v t F D$ The transportation cost of each unit of product p by the vehicle v from the factory i to the distributor j at the period t

• $β ¯ i c F$ Maximum inventory of each part with the quality of c in the factory i at the period t

• Maximum inventory of new part c in the factory i at the period t

• $ξ ¯ i c F$ Maximum inventory of reproduced part c in the factory i at the period t

• $λ ¯ i p F$ Maximum inventory of new product p in the factory i at the period t

• $x ¯ i p F$ Maximum inventory of reproduced product p in the factory i at the period t

• MAip Maximum volume of new product p that can be assembled in the factory i

• MRAip Maximum volume of reproduced product p that can be assembled in the factory i

• MPic Maximum volume of part c that can be processed in the factory i

• MRPic Maximum volume of part c that can be reprocessed in the factory i

• $c a p v F D$ The capacity of vehicle v floating between the production/reproduction factory and the distribution center

• $d i s i j F D$ The distance between the production / reproduction factory i and the distributor j

• $E m i s i j c v t F D$ The rate of emitting GHGs by the vehicle v to manufacture each unit of product p per kilometer between the production / reproduction factory i and the distributor j

• $E m i s p n c i c t F$ The rate of emitting GHGs during the process of each new part c in the production/ reproduction factory i and the distributor j

• $E m i s p r c i c t F$ The rate of emitting GHGs during the reprocessing of each new part c in the production/ reproduction factory i and the distributor j

• $O P E t max F$ Maximum allowable emission of GHGs at the period t during production/reproduction operations in factories

• $T E t max F D$ Maximum allowable emission of GHGs by the transportation system floating between the factory and the distributor j

• PPCkicvt These parameters have occurred at the first level of the model.

• BOCpc These parameters have occurred at the first level of the model.

The third-level model

• SPNjpt The sale price of each unit of new product p in the distributor j at the period t

• SPRjpt The sale price of each unit of reproduced product p in the distributor j at the period t

• DNMjpt Demand for a new product p in the market of distributor j at the period t

• DRMjpt Demand for a reproduced product p in the market of distributor j at the period t

• URCDjpt The cost of purchasing each returned product p paid to retailers or customers by the distributor j at the period t

• USNPjpt The cost of the lake of each new product p by the distributor at the period t

• USRPjpt The cost of the lake of each reproduced product p by the distributor at the period t

• $I C N P j p t D$ The inventory cost of each new product p in the distributor j at the period t

• $I C R M j p t D$ The inventory cost of reproduced product p in the distributor j at the period t

• $I C R P j p t D$ The inventory cost of each returned product p in the distributor j at the period t

• $U T C j k p t D R$ The cost of transporting each unit of returned product p by the distributor j to the recycling center k at the period t

• EPAjpt Maximum volume of the returned product p available in the market of distributor j at the period t

• $λ ¯ i p D$ Maximum inventory of new products p in the market of distributor j

• $x ¯ i p d$ Maximum inventory of reproduced products p in the market of distributor j

• $α ¯ i p d$ Maximum inventory of returned products p in the market of distributor j

• $c a p v D R$ The capacity of vehicle v floating between the distributor and the recycling center

• $d i s j k D R$ The distance between the distributor j and the recycling center k

• $E m i s j k p v t D R$ The rate of emitting GHGs by the vehicle v for each unit of returned product p per kilometer between the distributor j and the recycling center k

• $T E t max D R$ Maximum allowable emission of GHGs by the transportation system between the distributor and recycling centers at the period t

• URCCjkpvt This parameter has occurred in the first level of the model

• MPNijpvt This parameter has occurred in the second level of the model

• MRPijpvt This parameter has occurred in the third level of the model

Decision variables at the first level of the model

• afkicvt The volume of a part with the quality of c that is transported by the vehicle v from the recycling center k to the factory i at the period t

• dajkpvt The volume of the returned product p that is transported by the vehicle v from the distributor j to the recycling center k at the period t

• dtkpt The volume of returned product p that is separated and tested in the recycling center k at the period t

• dkct The volume of product c that is disposed in the recycling center k at the period t

• $α k p t R$ The inventory of returned product p in the recycling center k at the end of the period t

• $β k p t R$ The inventory of product with the quality of c in the recycling center k at the end of the period t

• σkpt Zero and one. If the returned product p is separated and tested in the recycling center k at the period t, it will gain the value of one, otherwise zero.

Decision variables at the second level of the model

• fdnijpvt The volume of new product p that is transported by the vehicle v from the factory i to the distributor at the period t

• fdrijpvt The volume of reproduced product p that is transported by the vehicle v from the factory i to the distributor at the period t

• xipt The volume of new product p that is assembled in the factory i at the period t

• yipt The volume of reproduced product p that is assembled in the factory i at the period t

• wict The volume of new part c that is processed in the factory i at the period t

• zict The volume of part c that is reprocessed in the factory i at the period t

• $λ i p t F$ The inventory of new product p in the factory i at the end of the period t

• $x i p t F$ The inventory of reproduced product p in the factory i at the end of the period t

• $β i c t F$ The inventory of part with the quality of c in the factory i at the end of the period t

• $ζ i c t F$ The inventory of reproduced part c in the factory i at the end of the period t

• $ξ i c t F$ Zero and one. If the new product p is assembled in the factory i at the period t, it will assigned the value of one, otherwise zero.

• ηipt Zero and one. If the reproduced product p is assembled in the factory i at the period t, it will assigned the value of one, otherwise zero.

• δipt Zero and one. If the new part c is assembled in factory i at the period t, it will gain the value of one, otherwise zero.

• πict Zero and one. If the new part c is processed in factory i at the period t, it will gain the value of one, otherwise zero.

• τict Zero and one. If the new part c is reprocessed in factory i at the period t, it will gain the value of one, otherwise zero.

• subcict The value for alternating part c in the factory i at the period t

• afkicvt This variable has occurred at the first level of the model

Decision variables at the third level of the model

• nssjpt The volume of the lake of new product p in the distributor j at the period t

• rss jpt The volume of the lake of reproduced product p in the distributor j at the period t

• γjpt The volume of returned product p that is collected by the distributor j from downstream markets at the period t

• $λ j p t D$ The inventory of new product p in the distributor j at the end of the period t

• $x j p t D$ The inventory of reproduced product p in the distributor j at the end of the period t

• $α j p t D$ The inventory of returned product p in the distributor j at the end of the period t

• dajkpvt This variable has occurred in the first level of the model.

• fdnijpvt These variables have occurred in the second level of the model.

• fdrijpvt These variables have occurred in the second level of the model.

### 3.1 The First Level: Recycling Centers

In recycling centers, it is attempted to increase total profit and diminish harmful environmental impacts, as follows:

$max F 1 R = ∑ k = 1 K ∑ i = 1 I ∑ c = 1 C ∑ v = 1 V ∑ t = 1 T P P C k i c t − ( ∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T U R C C j k p t d a i j p v t + ∑ k = 1 K ∑ p = 1 P ∑ t = 1 T S D T k p t σ k p t + ∑ k = 1 K ∑ p = 1 P ∑ t = 1 T U D T C T k p t d t k p t + ∑ k = 1 K ∑ c = 1 C ∑ t = 1 T U D C k c t d k c t + ∑ k = 1 K ∑ p = 1 P ∑ t = 1 T I C R P k p t R α k p t R + ∑ k = 1 K ∑ c = 1 C ∑ t = 1 T I C R P k c t R β k c t R + ∑ k = 1 K ∑ i = 1 I ∑ c = 1 C ∑ v = 1 V ∑ t = 1 T P P U T C k i c v t R F . a f k i c v t )$
(1)

Target function (1): Describing the total profit of recycling centers, revenue minus total expenses of recycling, total costs of separation and testing, total costs of disposal, total inventory costs, and total transportation costs.

Target function (2): The rate of emitting GHGs by each transportation vehicle for each high-quality part per kilometer between recycling centers and production / reproduction factories.

#### Limitations of the first level

Equations to retain inventory of returned products and high-quality parts are as follows:

$α k p t R = α k p , t − 1 R + ∑ j = 1 J ∑ v = 1 V d a j k p v t − d t k p t ∀ k , p , t$
(2)

$β k p t R = β k p , t − 1 R + ∑ p = 1 P B O C p c . d t k p t − d k p t ∀ k , c , t$
(3)

The value assigned for low-quality parts that must be disposed as follows (the capacity of disposing parts):

$∑ p = 1 P B O C p c . d t k p t ( 1 − θ k c t ) ≤ d k p t ∀ k , c , t$
(4)

Limits for inventory of returned products and highquality parts are as follows:

$α k p t R ≤ α ¯ k p R ∀ k , p , t$
(5)

$β k c t R ≤ β ¯ k c R ∀ k , c , t$
(6)

A limit for the capacity of separating and testing is as follows:

$d t k p t ≤ M D T k p R . σ k p t ∀ k , c , t$
(7)

The limitation for capacity of transportation from the recycling center to production/reproduction factories is as the following equation:

$∑ k = 1 K ∑ i = 1 I ∑ c = 1 C ∑ t = 1 T a f k i c v t ≤ c a p v R F ∀ v$
(8)

A limitation on the maximum allowable emission of GHGs by transportation vehicles is as follows:

$∑ k = 1 K ∑ i = 1 I ∑ v = 1 V ∑ t = 1 T d i s k i R F . E m i s k i c v t R F ≤ T E t max R F ∀ t$
(9)

The limitation of a negative integer for variables is as follows (the set of integers Z):

$a f k i c v t , d a j k p t , d t k p t , d k c t , α k p t R , β k c t R ≥ 0 ∈ Z , ∀ k , i , p , c , v , t$
(10)

The limitation of binary numbers for variables is according to equation (11), as follows:

$σ k p t ∈ ( 0 , 1 ) ∀ k , p , t$
(11)

### 3.2 The Second Level: Production/Reproduction Factories

Target function (1) – The second level: Describing how to maximize the total profit of all factories, including sale revenue minus total expenses of production, total costs of reproduction, total inventory costs, and total transportation costs.

$max F 1 R = ∑ i = 1 i ∑ j = 1 J ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T M P N i j p t . f d n i j p v t + ∑ i = 1 I ∑ j = 1 J ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T M P R i j p t . f d r i j p v t − ( ∑ i = 1 I ∑ p = 1 P ∑ t = 1 T ( S A i p t . η i p t + U A C i p t x i p t ) + ∑ i = 1 I ∑ c = 1 C ∑ t = 1 T ( S p c i t . π c i t + U p C c i t . w c i t ) + ∑ i = 1 I ∑ p = 1 P ∑ t = 1 T ( S R A i p t . δ i p t + U R A C i p t y i p t ) + ∑ c = 1 C ( S R P c i t . τ c i t + U R P C c i t . z c i t ) + ∑ i = 1 I ∑ p = 1 P ∑ t = 1 T ( I C N P i p t F . λ p i t F + I C R M P p i t F . x i p t F ) + ∑ i = 1 I ∑ c = 1 C ∑ t = 1 T ( I C Q C i c t F β i c t F + I C N C i c t F ζ i c t F + I C R C i c t F ξ i c t F ) + ∑ i = 1 I ∑ j = 1 J ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T ( U T C i j p v t F D f d n i j p v t + U T C i j p v t F D . f d r i j p v t ) + ∑ k = 1 K ∑ i = 1 I ∑ c = 1 C ∑ v = 1 V ∑ t = 1 T P P C k i c t . a f k i c v t )$
(12)

The target function of the second level minimizes the emission of GHGs by each transportation vehicle for each new and reproduced product per kilometer between recycling centers and production/reproduction factories, and the rate of emitting GHGs during the processing of new products and reprocessing of parts.

$m i n F 2 R = ∑ i = 1 I ∑ j = 1 J ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T d i s i j F D E m m i s i j p v t F D . ( f d n i j p v t + f d r i j p v t ) + ∑ i = 1 I ∑ c = 1 C ∑ t = 1 T E m i s p n v i c t F . w i c t + ∑ i = 1 I ∑ c = 1 C ∑ t = 1 T E m i s p n v i c t F . z i c t$
(13)

#### Limitations of the second level

Equations for retaining the inventory of high-quality parts, reproduced parts, new products, and reproduced produces are as follows:

$β i c t F = β i c , t − 1 F + ∑ k = 1 K ∑ v = 1 V a f k i c v t − z i c t ∀ i , c , t$
(14)

$ζ i c t F = ζ i c , t − 1 F + w i c t − ∑ p = 1 P B O C p c . x i p t − s u b c i c t ∀ i , c , t$
(15)

$ξ i c t F = ξ i c , t − 1 F + z i c t − ∑ p = 1 P B O C p c . y i p t − s u b c i c t ∀ i , c , t$
(16)

$λ i p t F = λ i p , t − 1 F + x i p t − ∑ j = 1 J ∑ v = 1 V f d n i j p v t ∀ i , p , t$
(17)

$x i p t F = x i p , t − 1 F + y i p t − ∑ j = 1 J ∑ v = 1 V f d r i j p v t ∀ i , p , t$
(18)

Limits for the inventory of high-quality parts, new parts, reproduced parts, new products, and reproduced parts are as follows:

$β i c t F ≤ β ¯ i c F ∀ i , c , t$
(19)

$ζ i c t F ≤ ζ ¯ i c F ∀ i , c , t$
(20)

$ξ i c t F ≤ ξ ¯ i c F ∀ i , c , t$
(21)

$λ i p t F ≤ λ ¯ i p F ∀ i , p , t$
(22)

$x i p t F ≤ x ¯ i p F ∀ i , p , t$
(23)

Limitations for the capacity of processing, reprocessing, and assembling new and reproduced products are applied as follows:

$w i c t ≤ M P i c . π i c t ∀ i , c , t$
(24)

$z i c t ≤ M R P i c . τ i c t ∀ i , c , t$
(25)

$x i p t ≤ M A i p η i p t ∀ i , p , t$
(26)

$y i p t ≤ M R A i p δ i p t ∀ i , p , t$
(27)

The limitation for the capacity of transportation from production/reproduction factories to distribution centers are as follows:

$∑ i = 1 I ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T ( f d n i j p v t + f d r i j p v t ) ≤ c a p v F D ∀ v$
(28)

The limitation for maximum allowable emission of GHGs by transportation vehicles are as follows:

$∑ i = 1 I ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T d i s i j F D . E m i s i j p v t F D . ( f d n i j p v t + f d r i j p v t ) ≤ T E t max F D ∀ t$
(29)

The limitation for maximum allowable emission of GHGs for the processing and reprocessing operations in factories is applied as follows:

$∑ i = 1 I ∑ c = 1 C E m i s p n c i c t F . w i c t + ∑ i = 1 I ∑ c = 1 C E m i s p n c i c t F . z i c t ≤ O P E t max F ∀ t$
(30)

The limitation of negative integer for variables is applied as follows:

(31)

The limitation for binary variables is applied as follows:

$π , τ , η , δ ∈ ( 0 , 1 ) ∀ i , j , p , c , t$
(32)

#### The third level: Distributors

Target function (1) – The third level: Describing how to maximize the total profit of all distributors, including sale revenue minus total expenses of purchases, total costs of recycling, total costs of deficiencies, total inventory costs, and total transportation costs

$max F 1 D = ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T S P N i j p t . ( D N M i p t − n s s j p t ) + ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T S P R i j p t . ( D R M i p t − r s s j p t ) + ∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T U R C C j k p t d a i j p v t − ( ∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T ( M P N i j p t . f d n i j p v t + M R N i j p t . f d r i j p v t ) + ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T U R C D j p t . γ j p t + ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T S ( U S N P j p t . n s s j p t + U S R P j p t . r s s j p t ) + ∑ j = 1 J ∑ p = 1 P ∑ t = 1 T ( I C N P j p t D . λ j j p t D + I C R M P j p t D . x j p t D + I C R P j p t d . α j p t D ) + ∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T U T C j k p v t D R . d a j k p v t )$
(33)

Target function (2) of the third level minimizes the emission of GHGs by each transportation vehicle for each returned products per kilometer between distributors and recycling centers.

$m i n F 2 R = ∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ v = 1 V ∑ t = 1 T d i s i k D R E m i s j k p v t D R . d a j k p v t$
(34)

#### Limitations of the third level of the model

Equations of retaining the inventory of new products, reproduced products, and returned products are as follows:

$λ j p t D = λ j p , t − 1 D + ∑ i = 1 I ∑ v = 1 V f d n i j p v t − ( D N M j p t − n s s j p t )$
(35)

$x j p t D = x j p , t − 1 D + ∑ i = 1 I ∑ v = 1 V f d r i j p v t − ( D R M j p t − r s s j p t )$
(36)

$α j p t D = α j p , t − 1 D + γ j p t + ∑ k = 1 K ∑ v = 1 V d a k j p v t ∀ j , p , t$
(37)

A limitation for the volume of collected returned products that cannot be greater than the volume of returned products available at the market is applied as follows:

(38)

Limitations for the lake of new and reproduced products are as the following:

(39)

(40)

Limits of the level of inventory of new and reproduced products are applied as follows:

$n s s j p t ≤ D N M j p t ∀ j , p , t$
(41)

(42)

Limits of the level of inventory of new, reproduced, and returned products are as follows:

$λ i p t D ≤ λ ¯ i p D ∀ i , p , t$
(43)

$x i p t D ≤ x i p D ∀ i , p , t$
(44)

$α i p t D ≤ α ¯ i p D ∀ i , p , t$
(45)

The limitation for the capacity of transportation vehicles is applied as follows:

$∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ t = 1 T d d a j k p v t ≤ c a p v D R ∀ v$
(46)

The limitation for maximum allowable emission of GHGs by transportation vehicles used by distributors is as the following equation:

$∑ j = 1 J ∑ k = 1 K ∑ p = 1 P ∑ v = 1 V d i s j k D R . E m i s j k p v t D R . ( d a j k p v t ) ≤ T E t max D R ∀ t$
(47)

The limitation of negative integer for variables is applied as follows:

$s j p t , r s s j p t , d a j k p v t , f d n i j p v t , f d r i j p v t , γ i p t , λ j p t D , x j p t D , α j p t D ≥ 0 ∈ Z , ∀ i , j , p , v , t$
(48)

## 4. SOLUTION

### 4.1 Revised Multi-Objective Goal Programming

Numerous solutions have been recently proposed, which offer a scientific framework for problems of deciding on the modeling of opposite objectives, including meta-goal programming (Tirkolaee et al., 2020), fuzzy meta-programming (Montazeri and Mahmoodi-k, 2015), multi-choice goal programming (Selim et al., 208), and revised multi-choice goal programming (Chang, 2008). This study was carried out based on the revised multichoice goal programming (Chang, 2007).

$min ∑ i = 1 2 [ w i ( d i + − d i − ) + α i ( e i + − e i − ) ]$
(49)

$F 1 ( x ) − d 1 + + d 1 − = y 1$
(50)

(51)

(52)

$F 2 ( x ) − d 2 + + d 2 − = y 2$
(53)

(54)

(55)

$e 1 + , e 1 − , e 2 + , e 2 − , d 1 + , d 1 − , d 2 + , d 2 − ≥ 0$
(56)

### 4.2 Hierarchical Iteration Algorithm

According to a 3-D structure of the model, the hierarchical iteration strategy was employed to solve problems as follows:

The first step: Initialization: The variable dajkpvt that is at the third level of the model is generated randomly according to its range of values.

The second step: Solving the first level of the model: The bi-objective model is solved in the first level using the goal programming approach. The value of a goal objective function $( F 1 k )$ and decision variables are obtained, where afkicvt is considered for the second level.

The third step: Solving the second level of the model considering values of afkicvt from the first level. The biobjective model of the second level is solved by the goal programming approach and the values of the goal objective function $( F 2 k )$ and decision variables of the second level are obtained, where values of fdrijpvt between fdnijpvt are considered for the third level.

The fourth step: Solving the third level of the model: By including decision variables of fdrijpvt and fdnijpvt from solving of the model in the second level, values of the goal objective function $( F 3 k )$ and decision variables are obtained for the third level, where values of dajkpvt are considered for the first level with a renewable iteration.

The fifth step: Stop criteria: The solving process is stopped if

$max ( w ¯ 1 , w ¯ 2 , w ¯ 3 ) ≤ w * w ¯ 1 = | F 1 K + 1 − F 1 K F 1 K + 1 | , w ¯ 2 = | F 2 K + 1 − F 2 K F 2 K + 1 | , w ¯ 3 = | F 3 K + 1 − F 3 K F 3 K + 1 |$

where w* is tolerance with a value of to stop the algorithm. If it is true, stop the algorithm, otherwise go back to the second level.

## 5. NUMERICAL RESULTS

By taking into account random values of parameters based on ranges defined in (Jing and Li, 2009), the proposed model was solved at a small size (i = j = k = p = t = c = v = 2) with a weight of both objective functions of 0.5 in GAMS. The values of objective functions for each level are presented in Table 1.

Results of this section show that the least value for the first objective function has been obtained in the first level. By moving from the first to the second level, the number of units will be increased by 54,000. This increase by moving from the second to the third level is by 167,000. The same is true for the second objective function. An increase in the value of the second objective function by moving from the second to the third level is higher than the corresponding value by moving from the first to the second level. Therefore, it is concluded that moving from the first to the second level will enhance the complexity of the problem. Similarly, the problem will be more complicated by moving from the second to the third level, with higher complexity than when moving from the first to the second level.

### 5.1 Analysis of Results

In this section, we compare the proposed model in two definite and robust states per nominal data and evaluated the sensitivity analysis related to the model (Table 2).

We solved both definite and robust states of the model using GAMS optimization software (version 25) and a computer with process Core Ram 6, 17. The results are summarized in Figure 1.

As observed in Figure 1, the highest effect was related to environmental costs, and factors such as supply chain risk had extremely limited impacts.

### 5.2 Experiment Design and Analysis Objective Functions’ Interactions

Experiment design is one of the strongest quality improvement and productivity enhancement techniques, where some experiments are carried out to make conscious changes in the process or system to evaluate their effect on the performance characteristics using the response received from the process or system. In fact, experiment design is the systematic manipulation of some of the variables, in which the effect of these alternations is assessed and the results obtained are implemented. In this analysis, the goal is to evaluate the effects of each objective function on the final results (the ultimate goal after balancing and single-targeting) and the impact of interactions of objective functions with each other.

• Step 1: Making the problem single-objective by weighting or other techniques (the problem becomes single-objective by the weighting method).

• Step 2: Designing the table of experiments. Given the fact that the model has four objectives, there is be 2k tests, one of which is impossible and thereby eliminated. The experiment is related to the negative levels of the four impossible objectives.

• Step 3: Analyzing the mathematical model variance and assessing model’s efficiency

• Step 4: Forming a reduced analysis of variance table

• Step 5: Drawing Pareto charts and ranking effects

In the experiment table, values of +1 and -1 show the level of factors, meaning that they show which factors (objective functions) are used in the set of objectives during the experiment and optimization process and can affect Z effect size. Therefore, there are 1-32 experiments, and the tests in which all factors are at level -1 are eliminated due to lack of defining the problem. Table 3 shows a number of designed tests and values of the response variable based on 2k factorial design:

### 5.3 Analysis of Variance

Following the implementation of the experiments and running the model based on the designed tests, its analysis of variance was performed and its reduced model plot was analyzed. Table 4 shows the analysis of variance of the reduced model:

The F value of model 15 showed 54, which demonstrated the significance of the model. P values below 0.05 indicated the significance of the model conditions. In this state, BD, AC, AB, D, B, A, which are the factors of the model, are significant models. Table 5 shows the appropriate statistics of the model:

The predicted R2 with a value of 0.6618 had a rational distance (0.7505) with the modified R2. In other words, the difference is below 0.2, which confirmed the accuracy of the model. Therefore, the main problem of the model was solved by reducing the model. Adeq Precision measures the signal to noise ratio, where a ratio above four is favorable. The ratio of the designed model was 14.653, which showed the sufficiency of the signal. The model can be used to move in the space of the design and regression model.

Analyzing the variability in observations through the analysis of variance is a completely mathematical method. Nevertheless, using the analysis method to test the lack of difference between treatment means needs the establishment of specific hypotheses.

### 5.5 Assuming the Normal Distribution of the Mathematical Model’s Output

The hypothesis of data normal distribution can be evaluated by drawing the residual histogram. If the hypothesis of $N ( 0 , σ 2 )$ is established for the residuals, the residual histogram should look like a normal distribution and be centered around zero. Unfortunately, small samples often show considerable changes. Therefore, lack of normal distribution does not necessarily mean that the hypotheses are serious flawed. Excessive distance from the formal distribution should be taken seriously and more assessed. There was no irrational evidence in the desired model, which confirmed the accuracy of the model according to the normal diagram and residuals. This is demonstrated in Figure 2:

### 5.6 Diagram of Residuals Based on Fitted Values

If the model and assumptions are correct, then the residuals should not show a specific structure or have a specific relationship with other variables, including the predicted response. This can be easily checked by drawing the residuals in terms of predicted values. In addition, non-constant variance can be expected in abnormal data with skewed distributions, in which the variance is usually a function of the mean. According to Figure 3, noncompliance was not observed in this regard and the adequacy of the model was confirmed.

### 5.6 Effect Ranking by Pareto Chart

One of the basic charts is to prioritize effects or values. In the test design method, one of the analytical reports is a Pareto report to rank effects. The priority of it is, as follows: A>B>D> AB > AC > BD

According to the results, the objective functions of fixed and variable costs and gas emission costs and impact chain risk were significant and the coefficient of these effects was negative, which means that they reduced the objective function Z. Meanwhile, the costs of the interaction of factors A (fixed and variable costs) and B (gas emission costs) and the interaction of factors D (supply chain risk) and B (gas emission costs) were positive and increased the value of the objective function Z. In addition, the interactions of factors A (fixed and variable costs) and C (costs of customer dissatisfaction) reduced (Z) model costs.

## 6. CONCLUSION

This study was carried out to investigate a threelevel supply chain including recycling centers in the first level, production/reproduction factories in the second level, and decentralized distributors in the third level. A bi-objective three-level model was proposed in order to maximize profit and diminish adverse environmental impacts. The proposed model determines the volume of products with parts processed in the recycling center and production/reproduction factories, as well as the number of products or parts delivered in each level. This model was solved using the revised multi-objective goal programming and a hierarchical iterative algorithm. Larger models can be solved considering uncertainty for some parameters such as demand, as well as employing metaheuristic algorithms such as NSGA-II.

## Figure

Comparison of various models.

Normality of the optimal results of the mathematical model.

Residuals trends.

## Table

Values of the objective function at different levels

Different solution models and their comparison

A number of tests designed based on 2k factorial design

Analysis of variance of the reduced model

Statistical calculations of reduced model parameters

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