Robust gasoline closed loop supply chain design with redistricting, service sharing and intra-district service transfer

Transportation Research Part E 123 (2019) 121–141

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Robust gasoline closed loop supply chain design with redistricting, service sharing and intra-district service transfer

T

R. Saedinia, Behnam Vahdani , F. Etebari, B. Afshar Nadjafi ⁎

Department of Industrial Engineering, Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

ARTICLE INFO

ABSTRACT

Keywords: CLSC network design Redistricting Resource sharing Service transfer Robust optimization

This paper presents a bi-objective model to design a closed loop supply chain with simultaneous consideration of districting regions, facility location–allocation, service sharing, and intra-district service transfer. The first objective function is to minimize the total costs, and the second one is to minimize the maximum volume of surplus demand from service facilities. In this model, costs, returns, and maximum service supply are uncertain; a robust optimization approach is used to take into account the uncertainty. Additionally, the validity of the proposed mathematical model on a real case study in the oil and gas industry sector has been investigated.

1. Introduction The rapid economic changes and increased competitive pressures in global markets have persuaded companies to emphasize the importance of supply chain and integrated logistics (Kalaitzidou et al., 2015). In addition, considering environmental concerns and social responsibilities, an increasing number of companies have applied reverse logistic approaches and forward supply chains (Zeballos et al., 2014), making reverse logistics and closed loop supply chains (CLSCs) as the two significant aspects of every industry. Currently, governmental rules alongside green supply chain management, active in recovering and exterminating the wastes and other dangerous products, forces top managers to pay attention to and reconsider the processes in supply chain networks (Vahdani and Mohammadi, 2015). In this context, the demand beyond the capacity of the service providers is one of the critical issues in deciding whether to provide service for customers or to facilitate the supply chain networks (SCNs), owing to their significant impact on offering high-quality services (Jasmand et al., 2012). Therefore, it can be argued that these service providers face a lack of capacity in responding to the demand. This deficiency is generally due to the fact that service providers typically consider the low capacity in providing service for their customers, who are in turn affected by unfavorable conditions and economic stagnation (Yang et al., 2003). Moreover, since it is impossible for service providers to increase capacity over the planning horizon, the need to use alternative and effective approaches in SCNs is essential. One of these effective approaches is the use of the spatial disparity model (Ko et al., 2015). Districting is defined as clustering together the regions of customers as service locations and demand units, in a way that can lead to better customer service quality and to increased customer satisfaction. The advantage of districting that balances the volume of demand in the regions is intra-district service transfer (Duque et al., 2012), which is used to determine how to share service capacity within each district. This sharing occurs through the transfer of surplus services at those facilities that have capacity surplus. Therefore, in addition to balancing demand among districts, district service transfer makes the workload of service facilities within each district well-adjusted. It should be noted that the process is essential for maximizing the efficient usage of service capacities and

⁎

Corresponding author. E-mail address: [email protected] (B. Vahdani).

https://doi.org/10.1016/j.tre.2019.01.015 Received 19 July 2018; Received in revised form 16 December 2018; Accepted 29 January 2019 1366-5545/ © 2019 Elsevier Ltd. All rights reserved.

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minimizing usage disparity among service facilities. Accordingly, through using this approach, management can be carried out effectively to share the capacity of service providers within regions, in such a way that the regional service providers with additional service capacity can create an appropriate balance in the demand volume as well as the level of providing service for customers through transferring their additional capacity to high demand service providers (Li et al., 2014). It should be indicated that the transfer of an integrated service is not easy to model, because the service transfer should not be made between facilities that have either additional services or high demand. It is even possible to waste parts of the transferred service during the act of transfer. Consequently, the task of modeling these conditions requires advance considerations of different conditions of highly-complex transfers (Ko et al., 2015). On the other hand, uncertainty is one of the new challenges related to the design and planning of SCNs that researchers face. A type of this uncertainty, environmental uncertainty, can affect the performance of supply chain members, including suppliers and manufacturers (Vahdani et al., 2013). Another type of uncertainty known as system uncertainty can also affect such supply chain processes as production and distribution. The existence of such uncertainties can particularly affect the quality of the necessary decisions for the supply chain at various levels. Considering the aforementioned cases, researchers have tried to tackle these types of uncertainties through the use of uncertainty planning methods (Dai and Zheng, 2015), and one of the effective methods is the use of a robust optimization approach that through predicting the occurrence of the worst possible conditions, associated with uncertainties, can examine the current state of the system and make appropriate decisions. Applying the above-mentioned considerations to a CLSC can create advantages, such as setting up the capacity of service providers, preventing the establishment of new surplus facilities, and proper utilization of the service provider’s capacity. Therefore, this study proposes a bi-objective mathematical model to investigate the gap identified in the research. The first objective function, which is a common goal for the SCN design, is to minimize the costs of CLSC networks, including fixed opening costs and transportation costs. The second objective function is to minimize the maximum volume of surplus demand from service providers to achieve an appropriate balance in the demand volume across all regions. The proposed model considers several decisions on location-allocation, volume of product flow, districting regions, and capacity level of service facility, to name a few. Moreover, a robust optimization approach is used to consider the uncertainties in the planning and designing of the SCN. In addition, as several factors should be considered in the proposed model, a real case study in the oil and gas sector is launched to demonstrate the applicability of the present study along with the implementation of the model under both definitive and uncertain conditions. In addition, since the proposed model is bi-objective, one of the most effective interactive methods provided by Torabi and Hassini (2008) is opted for solve the model. This method is based on fuzzy programming which, similar to other fuzzy programming methods, allows the decision maker to calculate the degree of satisfaction with each of the objective functions in achieving their optimal values. However, in this method, unlike other methods in the literature, the gaps between satisfaction degrees of the objective functions are not required to be considered. This leads to more effective solutions, making the results more acceptable to decision-makers. Also, due to its one-step approach, this method requires fewer computations compared to the other methods in the literature. Based on above-mentioned explanations, the novel aspects of this study compared to its counterparts are summarized as follows:

• developing a new framework for coordination of redistricting, service sharing and intra-district service transfer in a CLSC, and • proposing a bi-objective robust mathematical programming model to reduce the demand overload and its disparity in a CLSC. The rest of the paper is organized as follows. Section 2 presents a review of the literature. Both definition and formulation of the problem are described in detail in Section 3. The proposed solution approach is presented in Section 4, and the case study and the obtained results are examined in Section 5. Section 6 concludes the paper and presents various sensitivity analysis and management insights. 2. Literature review Since the subject of this research is to consider the two issues of redistricting and the design of a CLSC, here is an overview of the recent research on these issues. Considering the application of a fuzzy programming method to solve the proposed model, a review of the literature on the subject is also presented. 2.1. Redistricting problem One of the most important decisions in a CLSC planning is the distribution of products among customers. In this context, there are many concerns in real-world issues that have been less investigated in the literature. Among them, we can point out to the imbalance in customer demand, that becomes more critical when we know that, in the previous studies, each customer has been considered as an area. Therefore, it has been tried to meet the demands for each area by opening distribution centers. An approach that can partly solve the imbalances in customer demand is districting customers’ spatial units prior to launching the distribution process. On the other hand, in order to plan effectively, each region should have a server that responds to the demand of its covered customers, so that the customers receive the required products from the main distribution centers. However, it may not always be possible for all customers to receive services desirably, as the issue is affected by two reasons: (1) changes in customer demand; (2) limited capacity of the server facility. In order to minimize these problems, sharing of surplus capacity among server facilities can be used as an effective solution. 122

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One of the first studies on districting geographic regions was done by Duque et al. (2011). They stated that one of the most important components of decision making in the districting regions was the contiguousness of the area units. Therefore, they took advantage of the constraints of the traveling salesman problem to consider the concept and developed a mathematical programming model (MPM). However, in the proposed model, only the allocation of spatial units to regions was considered, without taking into account the decisions related to the providers (e.g., finding the appropriate place for their positioning) and the decisions to share surplus capacity. Moreover, since the network was intended to have a level, the way of allocating the areas to other facilities in multilevel networks was not investigated. It should be indicated that the objective function of the study was to minimize the sum of the dissimilarities between all pairs of spatial units within each region. In addition, in this research, it was assumed that there are a certain number of regions with their own spatial units which, unlike the previous models, were not known in advance, leading to the increases in the efficiency and applicability of the proposed model. Through using a compromise approach, we have attempted to establish a proper balance between the number of created regions and the dissimilarities between all pairs of the spatial unit. Other development in this research are to consider thresholds for critical criteria in making decisions. These criteria could include the following: the number of households per region, a spatial unit per region, and a population per region. In fact, considering these criteria is the major innovation of this study. An effective algorithm has also been developed to solve the model. However, similar to the previous investigation, location decisions, sharing of surplus capacity and allocation of these areas to other facilities are not considered. Li et al. (2014) also developed an innovative algorithm called memory-based randomize greedy and edge reassignment (MERGE) to solve the model proposed by Duque et al. (2011); however, instead of the objective function employed by Duque et al. (2011), they proposed a new objective function for the problem. To this end, they provided a measurement index to maximize compactness. Since the gaps noted in previous research were still observable, Kim et al. (2015) presented a new MPM for the problem investigated by Duque et al. (2011). The objective function was different as their aim was to maximize the relationship between the center and its hinterlands. They also considered the flow between the regions, and a heuristic method was developed to solve the model. However, the shortcomings of the previous studies were still present. Ko et al. (2015) proposed a location-allocation for redistricting and balancing the demand of different regions. Although the number of regions was not known in the model, a service facility was devised for each region. In addition, the balancing of demand in different regions was done through sharing the surplus capacity of the service facility. The proposed model is one of the comprehensive studies present in the context of the discussion. However, in this research, the network is considered to be one stage, making it impossible to integrate decisions with other considerations in multilevel networks. Accordingly, the present study attempts to examine the above considerations as well as investigating the decisions related to the design of a CLSC. 2.2. Designing closed loop supply chain As noted in the previous section, the districting of customers’ spatial units has not yet been considered in the design of such multilevel networks as CLSCs. In this section, we examine the literature on CLSCs from two other perspectives. In the first part, a number of studies on the design of the oil and gas CLSC networks are reviewed and then, supplemented by an overview of multi-objective models. Santoso et al. (2015) presented a MPM to design a CLSC for the distribution of liquefied petroleum gas in Indonesia. The network included filling stations, distribution centers, retailers and customers. In the proposed model, location decisions were discarded, but decisions regarding the allocation of facilities to each other and the flow between them were taken into account. Also, the distribution costs included the fixed and variable costs. The constraints of the proposed mathematical model were the classical constraints of flow balance and capacity of facilities. However, this study proposed a model for a real problem, in which, as with any other existing research, each customer is considered as an area. Santoso et al. (2016) developed their research by adding the considerations of the routing problem, simultaneous pickup, and delivery to its CLSC. The study considered the constraints on the time of arrival and departure of vehicles, and the developed model considered the decisions on allocation of facilities, routing of vehicles, and their scheduling, simultaneously. Similar to the previous research, location decisions were not taken into account, so the research did not offer a novel feature in the design of its network. Paydar et al. (2017) developed a MPM to design an oil CLSC in which minimizing the risk of waste collection was considered as an objective. The network included original oil suppliers, manufacturers, distribution/collection centers and vendors. Decisions were made to locate and allocate facilities and flows between them, and inventory costs were also included in the study. From the point of view of network designing, similar to the previous research in literature, the study only considered the classical constraints of flow balance and capacity of facilities. There were also constraints (e.g., other location-inventory models) related to the level of inventory of various facilities. In terms of modeling, it was complete in comparison to the previous two models. However, there was no new feature in the design of its CLSC. The only noticeable feature of their study was to consider the risk of engine oils returned by vendors met by the parameter, which was estimated by the failure mode and effect analysis method. Several studies have been conducted in the literature on the design of a CLSC network that formulated the considered issues as multi-objective mathematical models. Here are a few of them that have different objective functions with decisions taken regarding the location-allocation of facilities and the volume of flow between them. The constraints in the research include the balance of flow, the opening and capacity of facilities, and the allocation of facilities. However, each of these studies has used various methods, such as interval planning, robust optimization, and fuzzy programming, to consider the uncertainty in the parameters of the developed models. Also, the numbers of supply chain members were different. the innovation of these studies were in their various objective 123

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functions. Ramezani et al. (2013) considered three objective functions in the proposed model. The first objective function was to maximize the total revenue from its CLSC network. The second objective function was to maximize the level of service, forward and reverse chains of SCN, provided to customers. The third objective function was to minimize the total amount of defective raw materials received from the suppliers. Vahdani and Mohammadi (2015) considered a queuing system to serve customers in a CLSC. In that way, they assumed that products did not have the same priority, and provided a multi-priority queuing system for serving customers. The objective functions considered in their research included minimizing the total costs of the CLSC network and minimizing the maximum allowable waiting times in the queue of products. Jindal and Sangwan (2017) presented a two-objective mathematical model - the first objective function was to maximize the CLSC profit, coming from deducting network costs from its revenue, and the second objective function was to minimize the environmental impacts of transportation between facilities in the reverse chain. Fathollahi-Fard et al. (2018) presented a two-objective mathematical model. The objective functions in this research included minimizing the total costs of the CLSC network and minimizing the social aspects concerning the job opportunities. Moreover, several review articles have been presented related to studies conducted on the reverse network design and CLSCs, that researchers can refer to (Souza, 2013; Agrawal et al., 2015; Govindan et al., 2015; Govindan et al., 2017a, 2017b). 2.3. Fuzzy programming solution approach Various solution methodologies have been developed for solving multi-objective mathematical optimization models. These solution methodologies can be classified into three main categories: prior, posterior or generation, and interactive (Hwang and Masud, 2012). The most popular and applicable method among the approaches available in the prior category is the weighting method. For the generation category, the most famous and practical methods are AUGMECON and AUGMECON2. Introduced by Mavrotas (2009), AUGMECON is an enhancement of the original -constraint method, AUGMECON2, proposed by Mavrotas and Florios (2013), is an upgrading of AUGMECON. Among interactive methods, interactive fuzzy solution methodologies are of great importance and popularity, owing to their ability in calculating the degree of satisfaction of each of the objective functions in achieving the optimal solution. They are also more flexible in considering the importance of each objective function for decision makers. The primary fuzzy solution for multi-objective mathematical optimization models, established by Zimmermann (1978), is the min-max method, however, due to its inefficiency and ineffectiveness in offering solutions (Lai and Hwang, 1993), to overcome this defect, several methods have been proposed. Mahaptra et al. (2006) improved min-max Method. In the developed approach, the decision maker could achieve the optimal results due to his/her expectations. Also, Islam and Roys (2006) presented a new fuzzy multi-objective planning method called primal geometric programming. To develop the weakness of min-max approach, Lai and Hwang (1993) provided an interactive fuzzy approach called LH. Selim and Ozkarahan (2008) presented a new interactive fuzzy goal programming (IFGP) for solving multi-objective problems. In this method, they took advantage of a modified merged function based on Werners (1988) method. Li et al. (2006) expanded the fuzzy compromise method (called LZL) originally introduced by Guua and Wu (1999) through spontaneous computation of appropriate membership thresholds in place of selecting them. Finally, to solving multi-objective problems, Torabi & Hassini presented a novel fuzzy approach called TH to eliminate the disadvantages of previous methods. Their TH approach is a combination of LH and IFGP methods, and has been used in several studies (e.g., Wu et al., 2018; Pourjavad and Mayorga, 2018; Singh and Goh, 2018). Finally, in order to consider the research gap, a brief comparative analysis of the literature on the design of CLSC is presented in Table 1. 3. Problem definition In this research, a multi-echelon CLSC model is considered. The members of forward SCN include suppliers, manufacturers, distributors and regions of customers, while reverse SCN members include customers’ areas, collection, recovery and disposal centers. In the forward supply chain, the suppliers will provide the manufacturers’ needs, and the manufacturers in turn, after the production process, make the product distribution process based on their customers' demand. However, a number of products received by the customers are returned for various possible reasons. In this case, the collection and recovery centers will make decisions on recovery and disposal of the returned products after collecting them and conducting the related examinations. Also returned products are shipped to suppliers to observe whether they are recyclable. Otherwise, they will be transported to disposal centers. There are a number of spatial units in the customer areas: some are spatial units of demand and some are spatial regions of the service provider to provide services to the demand units. In the proposed model, various decisions can be taken on the locationallocation, the volume of flow between different facilities, districting regions, the level of capacity of each facility, the choice of service providers in each region, the determination of sink in each region, sharing of service providers, and balancing in the network. This districting leads to an increase in the level of customers’ access to the required service, since it creates specified units under responsibility of each service provider and accordingly determines the level of demand for each region. Another advantage of the proposed model is its intra-district service transfer. Through this, management can be carried out effectively to share the capacity of service providers within the regions in such a way that service providers of regions with additional servicing capacity can appropriately balance the demand volume and the level of service offering for their customers through the transfer of their additional capacity to high demand service providers. It should be noted that achieving an appropriate balance between the demand and servicing for each region is necessary to create an appropriate balance in the demand volume among 124

Keyvanshokooh et al. (2013) Ramezani et al. (2013) Özceylan et al. (2014) Khatami et al. (2015) Vahdani and Mohammadi (2015) Santoso et al. (2015) Santoso et al. (2016) Keyvanshokooh et al. (2016) Talaei et al. (2016) Özceylan et al. (2017) Masoudipour et al. (2017) Paydar et al. (2017) Jindal and Sangwan (2017) Jeihoonian et al. (2017) Ahmadzadeh and Vahdani (2017) Tosarkani and Amin (2018) Zeballos et al. (2018) Haddadsisakht and Ryan (2018) Sahebjamnia et al. (2018) Kim et al. (2018) Zhen et al. (2018) Fathollahi-Fard et al. (2018) The current research

Authors and year

Table 1 A brief literature review.

125

✓ ✓

✓ ✓

✓ ✓

✓

✓ ✓ ✓

✓ ✓

✓

✓

✓

✓

✓

✓ ✓ ✓

✓

✓

✓

✓ ✓ ✓ ✓

✓

✓ ✓

✓

✓

✓ ✓

✓

✓

✓

✓

✓

✓ ✓ ✓ ✓

✓ ✓

✓

Case study based

Theoretical

Single

Multiple

Type of research

Objective function

✓

✓ ✓

✓ ✓

✓ ✓

✓

Certain

✓

✓ ✓ ✓

✓ ✓

✓

✓ ✓

✓ ✓

✓

✓

✓ ✓

Uncertain

Conditions

✓

✓ ✓ ✓ ✓

✓ ✓

✓

✓ ✓

✓ ✓ ✓

✓

✓

✓ ✓ ✓ ✓

✓

Locationallocation

✓

✓ ✓ ✓ ✓

✓ ✓

✓

✓ ✓

✓ ✓ ✓ ✓ ✓

✓ ✓ ✓

✓ ✓ ✓ ✓

✓

Flow

✓

Districting/ Redistricting

Network design decisions

✓

Service sharing

✓

Intra-district Service transfer

✓

✓

✓

✓

✓ ✓ ✓ ✓ ✓

✓ ✓

✓ ✓

✓

Commercial solver

✓

✓

Exact

Solution approaches

✓ ✓

✓

✓

✓ ✓

✓

Heuristic/Metaheuristic

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Zone k Suppliers/(Oil field) (I)

Production facility /(Oil refinery) (J)

Customers zones (K)

Zone 1

Spatial region 18/ demand

demand demand demand Spatial region 15/service facility (storage tank)

Disposal center (Plain) (M)

Collection centers/ Recovery centers Petroleum sludge complex (L)

Fig. 1. The graphic schematic representation of the problem.

different regions. To achieve this objective, the service transfer should neither be made between facilities that have an additional service nor between facilities with a high demand. There is also the possibility of wasting parts of the transferred service during the transfers. Therefore, in order to model these conditions, different situations of the transfers that can be highly complex should be examined. Fig. 1 presents a graphic representation of the problem. Also, the main assumptions considered in the problem formulation are as follows:

• The number of customer zones is predetermined. • At most, one service facility (server) can exist in each spatial region. • The number of districts is predetermined. • Potential locations of servers are predetermined. • A server, with a positive net service balance over the demand, is not permissible to receive service from other servers. • A server, with a negative net service balance, is not permissible to send service. • For each server, “its individual service supply plus the sum of entire service transmitted to it,” is permissible to surpass the original extreme capacity of the server. • Some parameters of the proposed model are uncertain. These parameters include rate of return of used products, constant costs of

setting up by supplier, opening production facility/distribution center, opening collection center/recovery center, opening collection center/recovery center, transportation costs and maximum service supply at the facility.

3.1. Sets and indices I : Set of potential suppliers i I J : Set of potential production facilities/distribution centers j J K : Set of customer zones (districts) including spatial regions k K E : Set of entire spatial regions A : Set of spatial regions indicating a demand resource at districts a , a E S : Set of spatial regions with a potential service facility (server) at districts s, s L : Set of potential collection centers/recovery centersl L M : Set of potential disposal centers m M N : Set of available capacity level for facilitates

E

3.2. Parameters da : Demand of spatial region a rak : Rate of return of utilized products from spatial region a assigned to district k sa : Average disposal portion fin : Constant cost of setting up by supplier i with capacity level n

onj : Constant cost of opening production facility/distribution center j with capacity level n

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hln : Constant cost of opening collection center/recovery center l with capacity level n

n cam : Constant cost of opening disposal center m with capacity level n cxij : Cost of transportation per unit of products from supplier i to production facility/distribution center j

cujk : Cost of transportation per unit of products from production facility/distribution center j to district k

cqkl : Cost of transportation per unit of returned products from districtk to collection center/recovery center l cpli : Cost of transportation per unit of recovered products from collection center/recovery center l to supplieri ctlm : Cost of transportation per unit of scrapped products from collection center/recovery center l to disposal center m cesak : Cost of transportation per unit of products from service facility in spatial region s to customer in spatial region a in district k cvss'k : Transfer cost per unit of products between service facilities in spatial regions s, s in district k

cawin : Capacity with the level n for supplier i cay nj : Capacity with the level n for production facility/distribution center j cazln : Capacity with the level n for collection center/recovery center l

n cavm : Capacity with the level n for disposal center m carin : Capacity with the level n for recovery product for supplier i L1: A large number L1 |A| L2 : The total capacity of service facilities : A large number s k K rak da

gsmax : Extreme service supply at the facility in spatial region s

µ : Permissible surplus ratio for a district : Rate of service loss in a service transfer; 0

<1

3.3. Decision variables xij : Number of products transported from supplier i to production facility/distribution center j

ujk : Number of products transported from production facility/distribution center j to district k

Qkl : Number of returned products transported from districtk to collection center/recovery center l pli : Number of recovered products transported from collection center/recovery center l to supplier i Tlm : Number of scrapped products transported from collection center/recovery center l to disposal center m

k : Imaginary flow from spatial region a to a in customer zonek for contiguity assessment for each couple of spatial regions Haa

gsk : Supply of service of the facility located at spatial regions within customer zone k

bsk : Balance of net service of the service facility in spatial region s within customer zone k already transfer of service; demarcated as “a facility’s own supply” minus “the directly allocated demand” bsk + : Positive quota of net service balance bsk ; max{bsk , 0}

bsk : Negative quota of net service balance bsk ; min{bsk , 0}

c k ' : Service transfer from the service facility in spatial region s to that in spatial region s within customer zone k ; demarcated as positive and negative for outgoing ss

and incoming service, respectively

k k c k+ ' : Positive quota of transfer of service css ; max{css, 0} ss

c k ' : Negative quota of transfer of service cssk ; min{cssk, 0} ss

zmsk : Net surplus in the service facility in spatial region s within customer zone k after services transfer

Lsk : The sign of balance of net servicebsk ; 1 if and only ifbsk is positive and 0 otherwise Rak : 1 if spatial region a is allocated to customer zone k and 0 otherwise sinkak : 1 if spatial region a in customer zone k is the sink and 0 otherwise k Fas : 1 if the server in spatial region s is the service supplying facility for spatial region a within customer zone k and 0 otherwise win : 1 if supplier i is selected for setting up with capacity level n ; 0 otherwise yjn : 1 if a production facility/distribution center is opened at location j with capacity level n ; 0 otherwise

zln : 1 if a collection center/recovery center is opened at location l with capacity level n ; 0 otherwise

n vm : 1 if a disposal center is opened at location m with capacity level n ; 0 otherwise

3.4. Mathematical model

minw1 = i I n N

fin win +

+

j J n N

onj yjn +

ctlm Tlm + l L m M

l L i I

hln zln + l L n N

cpli pli +

camn vmn + m M n N

cesak da Fask + a A s S k K

cx ij x ij + i I j J

s S s' S k K s s'

minw2 = maxzmsk

s

S, k

cujk ujk + j J k k

k K l L

cqkl Qkl

cvss'k c k ' ss

(1) (2)

K

The objective function (1) minimizes the total CLSC costs. The first term shows the costs of setting up of suppliers to meet customers’ requirements. The second to forth terms calculate the total inauguration costs of production facility/distribution centers, 127

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collection, recovery and disposal centers. The fifth to eleventh terms calculate the total cost of transportation such that fifth to ninth terms consider the total transportation costs between facilities. The tenth term calculates the transportation costs between the facility of service providers and the requested locations for demand. The eleventh term calculates the costs of moving products between service providers. In addition, the objective function (2) minimizes the maximum additional demand facility of service providers. S.t.:

ujk = j J

s S

da Rak

gsk

k

(3)

gsk

(1 + µ )

a A

K

s S

k

K

(4)

The constraint (3) shows that the volume of product transferred from distribution centers to each customer region is equal to the level of supply of all service-offering facility in that region. Constraint (4) guarantees that the total demand of each district is satisfied by the sum of supplies from its servers at a permissible surplus ratio µ if overloaded.

xij = i I

ujk

j

J

(5)

k K

Tlm = sa

Qkl

m M

l

L

(6)

k K

pli = (1

sa)

i I

Qkl

l

L

(7)

k K

Qkl = l L

rak da Rak

k

K

(8)

a A

The constraints (5) to (8) balance the flows among suppliers, production facility/distribution centers, customers’ regions, collection/recovery, and disposal centers. It should be mentioned that the constraint (8) ensures that used products are collected from all components in each customer region.

win cawin

x ij j J

i

I

(9)

n N

win carin

pli l L

i

I

(10)

n N

yjn cayjn

xij i I

j

J

(11)

n N

yjn cayjn

ujk k K

j

J

(12)

n N

zln caz ln

Qkl k K

l

L

m

M

(13)

n N

vmn cavmn

Tlm l L

(14)

n N

Tlm +

z ln caz ln

pli

m M

i I

pli

x ij

l L

l

L

(15)

n N

i

I

(16)

j J

The constraints (9) to (16) represent the facility capacity constraints. The constraints (9) and (10) also ensure that if the supplier does not perform the setting up process, it will not be possible to meet the needs and demands of the production facility/distribution centers and receive recovered products from collection/recovery centers. The constraints (11) to (15) also ensure that flows between facilities are possible if the related facility is opened. In other words, production, distribution, collection, recovery, and disposal processes are performed only if the facility needed to perform these processes has been opened.

win

1

i

I

yjn

1

j

J

z ln

1

l

L

(17)

n N

(18)

n N

(19)

n N

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vmn

1

m

M

(20)

n N

The constraints (17) to (20) ensure that suppliers and production facility/distribution centers, collection/recovery and disposal centers can select at most one capacity level to carry out their own processes. k Haa

k Haa

(a | (a, a) E )

L1. sinkak

Rak

a

E, k

(a | (a, a) E )

k Haa

(L 1

1) Rak

a

E, k

K

k

(21) (22)

(a | (a, a) E )

sinkak = 1

K

K

(23)

a E

Constraint (21) ensures that spatial regions establishing a district are adjoining. The difference between inbound and outbound imaginary flows of spatial regiona is calculated in the left-hand side. If this quantity is positive, the spatial region is not considered an imaginary flow sink. Constraint (22) prohibits the imaginary flow among spatial regions of dissimilar districts. Constraint (23) ensures that each district contains only one sink.

Rak = 1

a

A

1

k

K

Rsk = 1

s

S

(24)

k K

Rak

(25)

a A

(26)

k K

1

Rsk

k

K

(27)

s S

Constraints (24) and (25) ensure that each spatial region is allocated to only one district and each district has at least one spatial region. Constraints (26) and (27) ensure that each server is allocated to only one district and that each district has at least one server.

Fask = Rak

a

A, k

K

(28)

s S

Fask gsk

a

Rsk

A, s

gsmax Rsk

gsk

s S k K

s

S, k

S, k

(29)

K

(30)

K

L2

(31)

Constraint (28) ensures that each spatial region is served by only one server in the district to which the spatial region belongs. Constraint (29) ensures that the facility is assigned to the same district as the spatial region. Constraint (30) ensures that the service supply of each server does not exceed the maximum service supply at the facility. Constraint (31) ensures that the total supplied service does not exceed the total supply capacity.

ck' + ck' = 0 ss

s, s

ss

cssk = 0

s

S, k

S, k S, k

ss

ck'

s ,s

ck'

ck'

s ,s '

ss

ss

s'

K, s

(32) (33)

K

'

c k +'

ss

S, k

K, s

s

'

(34)

K, s

s'

(35)

Constraints (32) and (33) ensure that the service transfers between two servers have accurate signs; they also prohibit transfer of service from a server to itself. Constraints (34) and (35) denote outbound and inbound transfers of service, respectively.

bsk = gsk zmsk =

da Fask

s

S, k

K

(36)

a A

bsk

(1

c k'

) s' S

ss

s

S, k

K (37)

s' s

Constraint (36) denotes the net balance of service supply in each server before service transfer. Constraint (37) represents the surplus in a server utilizing the net balance and the total transfer of service to this server, including service waste or expenditure 129

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through the transfer.

bsk +

bsk

s

S, k

K

(38)

bsk

bsk

s

S, k

K

(39)

bsk +

L2. Lsk

s

S, k

bsk

L2.

(Lsk

bsk + = bsk

1)

bsk

s

(40)

K

s

S, k

S, k

K

(41)

K

(42)

Constraints (38) and (39) ensure that bsk +andbsk have the accurate positive and negative signs correspondent to the value ofbsk . Constraints (40) and (41) ensure that only one of the bsk +and bsk becomes non-zero, depending on the sign ofbsk . Constraint (42) specifies the value of bsk + used in constraints (43) and (44) to impose an upper bound on the service transfer.

bsk +

c k +' s' s

ss

s

S, k

K (43)

s' s

c k +'

s, s

'

S, k

bsk

ck'

s, s'

S, k

c k +'

Rsk gsmax

bsk +

ss

bsk ,

ss

ss

c k ', ss

zmsk

s, s '

ss

s

(44)

K, s

s'

(45)

S, k

s'

K, s

(46)

R

(47)

k x ij , ujk , Qkl , Tlm, pli , Haa ,

bsk , c k '

K, s

'

bsk +,

c k +' , gsk ss

0

(48)

0

(49)

win, yjn , z ln, vmn, Rak , sinkak , Lsk , Fask

{0, 1}

(50)

Constraint (43) ensures that total service transfers creating from each server should not surpass its additional supply. Constraint (44) ensures that outbound transfer of service occurs only from servers with additional supply. Constraint (45) limits inbound transfer of service to servers with additional supply. Constraint (46) avoids transfer of service among districts; thus servers in each district serve only the spatial regions within the district. Constraints (47)–(50) are the limitations on the decision variables. 3.5. Robust counterpart mathematical model In order to create the robust counterpart model, rate of return of used products, Constant costs of setting up by supplier, opening production facility/distribution center, opening collection center/recovery center, opening collection center/recovery center, transportation costs, and maximum service supply at the facility are regarded as uncertain parameters. A case study from Iran is also embedded in this study. In addition, due to the international financial and technological constraints on the country, inflation and the devaluation of its national currency, cost parameters are also considered uncertain. Factors affecting the rate of return of the used products are parts of a product quality and the effectiveness of the service facilities. Considering the technological limitations, it is not possible to use new effective and efficient production processes. It is clear that this has a significant impact on product quality, and makes the exact estimation of this parameter impossible. On the other hand, most of the equipment of the service provider is outdated, and due to the cost of upgrading related to the equipment and the limited budget available for renovation and repair, it is not possible to increase the effectiveness and efficiency of the equipment. As a result, it is impossible to accurately estimate the maximum capacity of the equipment. With reference to the above mentioned issues, the planning of the proposed CLSC network under uncertainty conditions seems to be necessary. Moreover, the lack of proper service of this network can lead to customer dissatisfaction. Therefore, the above planning should be considered in the worst possible case. In the proposed model, we have changed each uncertainty factor in a closed bounded box. This set lets us create the most conservative possible solution so that the uncertainty related to several parameters can be considered at the same time in the worst possible conditions (Ben-Tal et al., 2009). A general view of the box is as follows:

uBox = {

Rn: |

t

¯| t

Gt ,

t = 1, 2,

(51)

, n}

where ¯t is the t normal value of the t th parameter of vector and the positive value of Gt denotes a “scale of uncertainty” and > 0 is the “level of uncertainty”. We suggest the following source (Ben-Tal et al., 2009) for further information on robust optimization approaches. By considering the robust optimization approach proposed by Ben-Tal and Nemirovsky (1999), the objective functions and the changed constraints are presented as follows. It should be noted that other constraints remain unchanged. Some of the features of this method are as follows: (1) this method does not require any information regarding the statistical distribution of 130

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parameters; (2) the obtained solution will never be infeasible — the maximum robustness of feasibility is guaranteed by this method; (3) it is highly risk-averse and conservative — it is therefore suitable for issues in which the cost of risk is really high, and (4) it provides the worst possible solution, thus providing a maximum degree of confidence for the decision maker (Ben-Tal et al., 2009; Vahdani et al., 2018). (52)

minw1 minw2 =

maxzmsk

s

S, k

(53)

K

S.t.: n

(f¯i win +

i I n N

+

f in )

+ j J n N cu jk )

(cu ¯ jk ujk + j J k k

(o¯nj yjn +

+ k K l L

da (ce ¯ sak Fask +

+ a A s S k K

ce sak )

o jn )

n

(h¯l zln +

+ l L n N cq kl )

(cq ¯ kl Qkl +

+ s S s' S k K

Qkl Qkl l L

gsk

cssk+

r¯ak (1

r ) da Rak

k

K

r¯ak (1 +

r ) da Rak

k

K

a A

a A

¯ (gsmax

g g max G s

¯ (gsmax

f n f Gin wi

g g max G s f in

f n f Gin wi

o n o G jn yj

h n h Gln z l

ca n ca Gmn vm

(cv ¯ ss'k c k ' + ss

cx cx Gij x ij cu cu G jk ujk

o jn

cp cp Gli pli ce k ceu G sak Fas

K

h ln

ca mn

S, k

K, s

s

cu jk

cu jk

cq kl

(60) (61) (62) (63)

l, n

(64)

m, n

(65) (66)

m, n

(67)

i, j

(68)

j, k

(69)

j, k

(70)

k, l

cq kl

ct lm

(71)

k, l

(72) (73)

l, m

ct lm

cp li

(74)

l, m

l, i

cp li ce sak

w1

(58)

i, j

cx ij

cp li )

(59)

ca mn cx ij

(cp ¯ li pli +

cx ij )

(57)

l, n

ct ct Glm Tlm

cp cp Gli pli

s, s

)

l L i I

(cx ¯ ij x ij + i I j J

(56)

j, n

h ln

cq cq G kl Qkl

ct ct Glm Tlm

S, k

ss k

+

+

(55)

j, n

cu cu G jk ujk

cq cq G kl Qkl

) Rsk

s

cv c '

ct lm )

ca mn )

(54)

i, n

o jn

ca n ca Gmn vm cx cx Gij x ij

) Rsk

max

m M n N

l L m M

i, n f in

o n o G jn yj

h n h Gln z l

max

(ca ¯ mn vmn +

+

(ct¯ lm Tlm +

+

s s'

l L

h ln )

(75)

l, i

(76) (77)

s, a, k 131

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R. Saedinia et al. ce k ceu G sak Fas

ce sak

cv k cv G ss'k css'

cv ss'k

cv k cv G ss'k css'

s , s, k

cv ss'k

bsk , c k ', zmsk

(78)

s, a, k s

s, s, k

s s

s

(80)

R

ss

(81)

k x ij , ujk , Qkl , Tlm, pli , Haa , bsk +, c k +' , gsk , ss

bsk , c k '

(79)

f in ,

o jn ,

h ln ,

ca mn ,

cx ij ,

cu jk ,

cq kl ,

ceu sak ,

cv ss 'k

0

0

ss

(82) (83)

win, yjn , z ln, vmn, Rak , sinkak , Lsk , Fask

{0, 1}

(84)

4. Solution approach Through combining the method provided in the previous section, a solution method is used along with a fuzzy solution approach derived from Torabi and Hassini (2008) to solve the proposed model. The steps of the presented hybrid solutions approach can be summarized as follows: Step 1: determining the parameters of uncertainty and considering the distribution functions required in the model. Step 2: formulating the proposed model with the parameters defined in the previous step. Step 3: converting the constraints of mixed-integer programming model to constraints of the certain counterpart by applying the approach outlined in the previous section. Step 4: converting the robust model to the equivalent certainty model by applying the approach outlined in the previous section. Step 5: specifying the positive ideal solution (PIS) and negative ideal solution (NIS) for each objective function. To calculate the positive and negative ideal solutions, i.e., (W1PIS , x1PIS ) and (W2PIS , x 2PIS ) , each certainty model is separately solved for each of the objective functions, and the PIS is obtained, and then the NIS is estimated as follows: W1NIS = W1 (x 2PIS ) ,W2NIS = W2 (x1PIS ) , Step 6: calculating a membership function using the formulation below.

1ifW1 < W1PIS µh (x ) =

W1NIS

W1NIS

W1 W1PIS

ifW1PIS

W1

W1NIS

0ifW1 > W1NIS In fact, µh (x ) represents the satisfaction degree of the h

(85) th

objective function.

Step 7: converting the certainty mixed integer programming model to a certainty single-objective mixed integer programming model using the integrated function which is calculated as follows:

max (x ) =

0

+ (1

)

h µh (x )

(86)

h

S.t.: 0

x

µh (x ),

(87)

h = 1, 2

F (x ),

0

and

(88)

[0, 1] th

where µh (x ) and 0 = min {µh (x )} show respectively the satisfaction degree of h objective function and the minimum degree of objective satisfaction. This formulation is determined as a convex combination from the lower bound of satisfaction degree of functions ( 0 ) and the total weighted of these degrees to achieve ( µh (x ) ), which guarantees to obtain a balanced solution. In addition, respectively indicate the relative importance of hth objective function and the coefficient of restitution. Parameter h is h and specified by the decision-makers based on h h = 1 and h > 0 . It should be noted that the range of belongs to [0,1]. Step 8: determining the parameters h , and and solving single-objective models created in the previous step. If the answer is satisfactory for decision makers, it stops; otherwise, in order to achieve new solutions, it changes the values of parameters and and, if needed, changes the value of h . In order to measure the decision makers’ satisfaction with the results, the range of satisfaction degrees (RSD), a dispersion index, is defined as follows (Torabi and Hassini, 2008):

RSD (x ) = max(µh (x )) h

min(µh (x ))

(89)

h

Actually, RSD evaluates the balancing amount of a compromise solution by computing the supreme diversity between the 132

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satisfaction degrees of objectives. It likewise specifies the level of consistency between desirable.

and . The greater value of RSD will be more

5. Case study In today's world, environmental pollution can cause a variety of diseases, including cancer. According to studies conducted by the Cancer Research Center, benzene and aromatic compounds are among the most dangerous compounds that can cause cancer and are also the most important air pollutants in metropolitan areas. Therefore, the Environmental Protection Agency has issued an order to distribute Euro 4 gasoline in order to reduce air pollution and protect human health (Heeb et al., 2008; Weiss et al., 2012). In the current study, both Shazand Arak Oil Refinery and Persian Gulf Star Oil Refinery, which are among the most important gasoline optimization projects in Iran, are considered as candidate refineries for opening. The distribution of Euro-4 gasoline in the central and southern parts of Iran has also been investigated: Tehran, Karaj, Qom, Semnan and Garmsar in the central part of Iran and Bandar Abbas, Ahvaz, Shiraz and Kerman in the southern part of Iran are among the cities that received Euro-4 gasoline at the early stages of optimization of gasoline. In order to provide service to these two parts of Iran, some cities with service facility are selected to perform the distribution of gasoline in these areas. Since refineries require crude oil to produce petroleum fuels, in this study, we have considered two oil fields of Masjed Soleyman and South Pars, as two the main oil fields. Since substantial amounts of sludge are produced annually in storage tanks for oil products, these materials are collected from different parts of the country to maintain a healthy environment and are usually maintained in a single location such that at the right time appropriate action on refining and recycling of consumables are taken, and waste products are disposed in the environment. In this case study, we have considered two ponds of sludge storage of petroleum products in Bandar Abbas and Ahwaz as the candidate centers for recycling and refining. Information on issues such as the production capacity and cost of refineries and suppliers is shown in Tables 2–4. Moreover, information required in relation to the gasoline demand for each city in each region is presented in Table 5. In this study, under deterministic and uncertain conditions, the problem was solved via the TH solution approach using GAMS software. In this solution, the importance of the objective functions is considered ( = 0.5, 0.5, respectively) and the coefficient of restitution is = 0.5. Fig. 2 illustrates problem solving in a deterministic state. As it is noticeable, the South Pars Oil Field is selected as the main provider of feed for the refineries, and both Shazand Arak Oil Refinery and Persian Gulf Star Oil Refinery are selected to produce and distribute Euro-4 gasoline. The central part of Iran is supplied through Shazand Arak and Persian Gulf Star refineries, and the southern part of the country is only supplied by Persian Gulf Star. It can be observed that the Oil Products Distribution Company of Tehran and Arak in the central part of Iran and Oil Products Distribution Company of Bandar Abbas in the southern part of the country have been selected as service provider facilities in these regions. The Oil Products Distribution Company in Tehran is responsible for transferring and supplying gasoline to Tehran, Karaj, Qom, Semnan and Garmsar, and the Oil Products Distribution Company of Arak is responsible to provide gasoline demand in Arak. The Oil Products Distribution Company in Bandar Abbas would also supply the demand for gasoline in southern cities. The sludge in tanks of providing service to regions is transferred to Bandar Abbas pond and after refining of raw materials and consumables, it is transferred to South Pars and the non-consumables are disposed in Assaluyeh plains. Given that the overload in the facility of the service providers should be minimized in this problem, the value of(12000 × 103lit) would be supplied to Tehran’s servicing facility, and is lower than the demand for the spatial regions allocated to this facility. The volume of additional demand (negative balance) in this facility is(2500 × 103lit) . As Arak’s servicing facility is provided with additional supply, and sharing is allowed only between service providers within each region; the value of (769.88 × 103lit) for additional supply in the servicing facility of the Oil Products Distribution Company of Arak in the central part of Iran would be transferred to Tehran’s servicing facility. Following this intra-district service transfer, and reducing the transfer value to 0.10, the additional volume of demand in Tehran’s servicing facility is (1807.1 × 103lit) . Thus, by this service transfer, the service shortage has significantly decreased in Tehran’s servicing facility. Finally, the value of the total cost obtained in a deterministic state is 467412942.79, the overload is (1807.1 × 103lit), and the time required to solve the problem is 374 s. Fig. 3 shows problem solving in an uncertain state with an uncertainty level of 0.3. In this case, the gasoline for the central part of Iran would only be provided by Shazand Arak Oil Refinery and for the southern part of the country through Gulf Persian Star Oil Refinery. In the central area of servicing facility, the Oil Products Distribution Company of Arak, in addition to the city of Arak, would also supply gasoline to Qom. Since in an uncertain state, the value of supply of gasoline to Tehran’s servicing facility is (7700 × 103lit) and the demand allocated to it is higher, the negative balance in this facility is(5800 × 103lit) , which the value (3400 × 103lit) of Arak’s servicing facility is transferred to servicing facility of the Oil Products Distribution Company of Tehran. The value of the total cost Table 2 Parameters related to potential suppliers. Potential suppliers (i )

Constant cost of setting up (Euro) (f in )

Capacity (Lit/day) (cawin )

South Pars

250,000× 103

18,000× 103

Masjed Soleyman

300,000× 103

20,000× 103

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Table 3 Parameters related to production facility/distribution center. Potential production facility/distribution center (j )

Constant cost of opening (Euro) (onj )

Capacity (Lit/day) (cay nj )

Persian Gulf Star refinery

100,000× 103

12,000× 103

Shazand Arak refinery

16,000× 103

90,000× 103

Table 4 Parameters related to collection /recovery center. Potential collection/recovery center (l )

Constant cost of opening (Euro) (hln)

Capacity (Lit/day) (caz ln)

Bandar Abbas pond

30,000× 103

20,000× 103

Ahwaz pond

25,000× 103

15,000× 103

Table 5 Demand of spatial region. Spatial region

Demand (da ) (Lit/day)

Tehran

8000× 103

Bandar Abbas

1000× 103

Shiraz

2500× 103

Garmsar

1000× 103

Qom

2000× 103

Bushehr

1000× 103

Karaj

2000× 103

Arak

1500× 103

Semnan

1500× 103

Ahvaz

1000× 103

obtained in an uncertain state is 614125719.74, the overload is (2740 × 103lit) , and the time required to solve the problem is 468 s. It can be observed that in an uncertain state, Tehran’s servicing facility is supplied the gasoline much less than the demand allocated to it, so the additional volume of demand for this facility is much higher than that of the deterministic state, and consequently the level of the service transfer to this facility in an uncertain state has also increase. It should be mentioned that the entire supply of additional service in Arak’s servicing facility has been transferred to Tehran’s servicing facility. With respect to the results obtained in the robust and deterministic states, it can be found robust model can yield worse solutions than deterministic model. This is a normal and expected phenomenon because the worst conditions are considered in robust models to reach the optimal solution and thus the resulting solutions are always worse than those of deterministic models. Therefore, the implementation risk of a robust model is much lower than that of a definitive model. The study models are coded in GAMS software, version 23.6.2, by using the CPLEX optimization solver in a PC with core i3, 2.1 GHz CPU and 4 GB RAM. 6. Sensitivity analysis and managerial insights Considering the objectives of the proposed model, in this section, the sensitivity analysis is used to examine the behavior of these objective functions with respect to the changes in the parameters of the proposed model. The parameters considered in the sensitivity analysis include the demand of each spatial region(da),the maximum supply of service for each facility(gsmax ) , and the rate of return of the utilized products(rak ) . Figs. 4–8 show the changing trend in the objective function values with respect to the change in these parameters. The excessive demand heterogeneity is an important issue, as a large volume of demand from service facilities severely affects the quality of a service, which is a critical issue in service planning. As shown in Figs. 4 and 5, a large volume of demand increases the costs and system overload. This increase could be due to various factors (e.g., the inadequate capacity of service facilities and distribution centers). In this case, with the increase of the demand, distribution centers and more service facilities should be created to meet the demand. As it is known, this approach leads to higher system costs and thus a reduction in profitability. On the other hand, service facilities may have sufficient capacity for all demands, but the volume of heterogeneous demand leads to unequal service by service facilities. For example, in the case study, there may be a high demand or a heterogeneous demand for cities assigned to some service facilities (oil distribution companies) compared to the other facilities. It should be indicated that this heterogeneity can have several factors. For example, trips between regions or in some cases immigration by residents of different regions can lead to an increase in demand and, as a result, will create these heterogeneities. Therefore, in order to compensate for this heterogeneity, an intra-district service transfer may occur, and the service sharing is performed between the over-served facilities and 134

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Fig. 2. An illustration of a solution representation under deterministic condition.

the over-demand facilities, leading to the balance in the volume of demand among the facilities within each region. However, this approach also imposes the cost of transporting the service between the service facilities and ultimately leads to an increase in the total cost of the system. It can be stated that an increase in the cost of this is far less than the creation of new distribution centers and facilities. These conditions occur when the volume of demand exceeds the total supply of service facilities in each region. Therefore, despite an intra-district service transfer, there would still be overload, so the higher the volume, the higher the demand, leading to the increase in the second objective, as shown in Fig. 5. Insight 1: Redistricting, service sharing, and intra-district service transfer can have significant effects on supply chain cost reductions if the demand is increased. Therefore, it is suggested that the above approaches be used, in respond to the increasing demand, instead of creating new service facilities and distribution centers. Insight 2: As indicated, the heterogeneity in the volume of the demand could lead to increased costs and system overload. It is therefore suggested that decision makers, before using the above model, carry out a comprehensive review of geographical areas, alongside their immigration and periodic travel, to achieve a more accurate estimation of the demand by regions. As shown in Fig. 6, with an increase in the maximum capacity of the service facilities, the amount of system overload decreases. The reason for this is that by increasing the capacity of service facilities, it is possible to meet more demand, so the overload of each service facility is reduced. It can be seen in this figure that an increase in capacity leads to a significant reduction in the value of the second objective function, to the point where the overload of the system can reach its optimal level, which is in fact the absence of overload. In this case, increasing the capacity of the service facilities could lead to unused capacity in the system, which can ultimately reduce the effectiveness of the system. Insight 3: Increasing the capacity of the service facility to a certain extent has a positive effect on the second objective function and reduces its value. That is, decision-makers can increase the efficiency and effectiveness of the system by adjusting the capacity of the service facility, as it could be prevented from unused capacities of service facilities. 135

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Fig. 3. An illustration of a solution representation under uncertain condition.

First objective function

1.2E+09 1.1E+09 1E+09 900000000 800000000 700000000 600000000 500000000 0

2000

4000

6000

8000

10000

Average of demand of spatial unit

Fig. 4. First objective function and average of demand.

Another parameter is the rate of return of used products. As shown in Figs. 7 and 8, an increase in this rate can increase both objective functions. The reason is that as this rate increases, shipments also increase in collection/recovery centers, suppliers, and disposal centers. There may be a need for the opening of collection/recovery or disposal facilities to service the used products. Therefore, the total cost of the system can increase significantly with an increase in the rate. By increasing the rate of return of the used products, service providers spend part of their capacity to transfer these products to collection/recovery centers. Therefore, less demand is provided by this facility, and the overhead of servicing facilities also increases. It can be observed that this rate has a significant effect on the values of the objective functions greater than 20%. Insight 4: The process of changes in this rate shows a significant impact on the performance of the system under investigation. 136

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Second objective function

3500 3000 2500 2000 1500 1000 500 0 0

2000

4000

6000

8000

10000

Average of demand of spatial unit

Fig. 5. Second objective function and average of demand.

Second objective function

1800 1500 1200 900 600 300 0 0

3000

6000

9000

12000

15000

Maximum service supply at the facility

Fig. 6. Second objective function and maximum service supply.

First objective function

758400000 758200000 758000000 757800000 757600000 757400000 757200000 0

0.2

0.4

0.6

0.8

Rate of return

Fig. 7. First objective function and rate of return.

Second objective function

3500 3200 2900 2600 2300 2000 0

0.2

0.4

0.6

0.8

Rate of return

Fig. 8. Second objective function and rate of return.

Therefore, decision-makers should adjust this rate by enhancing the quality of products and improving the performance of equipment, and even reducing it as much as possible. In addition, in Figs. 9 and 10, the effects of increasing the number of regions on the values of the objective functions are shown. As can be seen, with increasing number of spatial regions, the values of the two objective functions also increase so that the second 137

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780000000

First objective function

700000000 620000000 540000000 460000000 380000000 300000000 0

5

10

15

20

25

30

Number of regions (a)

Fig. 9. First objective function and number of regions.

Second objective function

5000 4000 3000 2000 1000 0 0

5

10

15

20

25

30

Number of regions (a)

Fig. 10. Second objective function and number of regions.

objective function is more sensitive to these changes. The reason for these changes is that increasing the number of spatial regions naturally increases the demand, so we need to open and use more facilities and increase the number of transports. On the other hand, more spatial regions are allocated to each service facility. While meeting the demand for these areas would be impossible due to insufficient supply capacity throughout the service facility, eventually it could lead to an increase in the system overload. Insight 5: The process of the above changes shows that increasing the number of spatial regions without increasing the total capacity of the service facilities could lead to inefficiencies in the system. Therefore, it is suggested that if decision makers settle on increasing the number of regions, they should also consider the requirements related to the increased service capacity. In addition, in order to show the appropriate behavior of the model under uncertainty, this problem has been investigated in an uncertain state at various levels of uncertainty. As shown in Figs. 11 and 12, both objective functions increase with an increase in level of uncertainty, and the solutions become worse. Moreover, in order to display the computational results in terms of CPU time, nine test problems are considered in large dimensions in Table 6. The results show that an increase in the number of spatial regions has a significant impact on the increase of CPU time. In order to show the robustness of the proposed model, for each uncertainty level, five realizations are randomly generated by uniform distribution in the corresponding uncertainty set to investigate the performance of the solutions acquired from the proposed model under certain and uncertain conditions. The computational results of these experiments are shown in Table 7. Alternatively, as the computational results display in Table 7, the robust model is obtained the solutions by both lower standard deviations and greater quality than the model under certain condition.

First objective function

1.2E+09 1.1E+09 1E+09 900000000 800000000 700000000 0

0.1

0.2

0.3

0.4

0.5

Level of uncertainty

Fig. 11. First objective function and level of uncertainty.

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0.6

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R. Saedinia et al. 3000 Second objective function

2500 2000 1500 1000 500 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Level of uncertainty

Fig. 12. Second objective function and level of uncertainty. Table 6 Computational results in in terms of CPU time. Test problems

Problem size I /J /K / A/S / L/ M /N

1 2 3 4 5 6 7 8 9

3/4/5/28/7/3/2/2 4/5/7/33/10/3/2/2 4/6/7/40/12/3/3/2 5/6/8/45/12/4/3/2 5/7/8/50/14/4/4/3 5/6/7/53/15/4/4/3 6/7/8/58/14/5/5/3 6/6/8/65/15/5/5/3 6/6/9/80/20/5/5/3

Time (second) First objective function

Second objective function

TH

10.230 29.014 62.512 134.025 178.86 195 268.185 387.926 5243.090

127.931 479.61 853.49 1000 1041 1108.043 1643.98 1001.303 13025.66

20.85 89.028 128.733 189.829 332.826 634.580 1287.54 1001.278 8477.88

Table 7 Summary of test results under realizations. Test problem

Problem size

1

3/4/5/28/7/ 3/2/2

2

4/5/7/33/ 10/3/2/2

Uncertainty level ( )

0.1 0.3 0.5 0.1 0.3 0.5

Objective functions values under realizations Mean of objective functions values under realizations

Standard deviation of objective function values under realizations

Deterministic (w1, w 2)

Robust (w1, w 2)

Deterministic (w1, w 2)

Robust (w1, w 2)

(43651391,120112.54) (43618841,121805.06) (43776246,120440.11) (522917629,178943.75) (523672561,177605.31) (523218234,175894.81)

(43581249,121136.74) (43570918,120714.16) (42623210,120828.09) (519984346,192854.22) (523368839,192335.4) (514134935,191652.47)

(10543,5771) (78106,10297) (86321,8718) (77811,20379) (96102,29218) (135162,49685)

(4741,2419) (12418,3504) (33096,13612) (141578,5612) (21617,14361) (55968,20456)

7. Conclusions The focus of the current research was to present an integrated framework to coordinate the decisions related to the design of a CLSC network, redistricting, service sharing, and intra-district service transfer within spatial regions. In many real-world problems, the distribution of products to customers is based on districting of the spatial regions. While researches in the design of SCNs have not taken into account districting or redistricting of the spatial regions decisions. Moreover, in all studies carried out in this area, with an increase demand from customers on the one hand and the limited capacity of distribution centers on the other hand, a designed network is required to create new facilities in responding to requests. Therefore, the SCN structure that has been created is fully affected by the volume of the demand and capacity of distribution centers, as sharing surplus capacity between existing distribution centers can respond to these conditions. Therefore, the contributions of this paper are twofold: (i) developing a new framework for coordination of redistricting, service sharing, and intra-district service transfer in a CLSC, and (ii) proposing a bi-objective robust mathematical programming model for reducing the demand overload and its disparity in a CLSC. Also, due to the uncertainty in estimating some of the parameters of the proposed model, affected by the circumstances of the case study, a robust optimization approach has been used to overcome these uncertainties. The results demonstrate that conducting a comprehensive survey of geographical areas, along with their immigration and travel times could play effective roles in the better estimation of the demand in different areas, leading to improvements in a system performance. It could also improve the performance of the system by reducing the rate of return of the utilized products, improving the quality of the manufactured products and improving the performance of the 139

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transmission equipment. There are several areas of this study that need further investigation. For example, one of these attractive topics is the vehicle routing problem in the spatial regions that is not considered in this research. Other studies can determine the route of vehicles and their number. It should also be explained that in this case, the concepts of split delivery can be considered so that each demand point may not be able to receive all of their needs from one vehicle, and they have to complete the remainder of their needs from other vehicles. Among other issues, it is possible to consider the priority between the new products and the returned ones from customers to receive services from the service facilities. Finally, Sustainable and green considerations can also be considered in relation to service providers and other supply chain members. References Agrawal, S., Singh, R.K., Murtaza, Q., 2015. A literature review and perspectives in reverse logistics. Resour. 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