# Viable Supply Chain Network Design by considering Blockchain Technology and Cryptocurrency

Authors: Reza Lotfi; Soroush Safavi; Alireza Gharehbaghi; Sara Ghaboulian Zare; Reza Hazrati; Gerhard-Wilhelm Weber
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Nowadays, using Blockchain Technology (BCT) is growing faster in each country. It is essential to apply BCT in Supply Chain Network Design (SCND) and is considered by the designer and manager of SC. This research indicates Viable Supply Chain Network Design (VSCND) by applying BCT. A new form of two-stage robust optimization is suggested. Facility locations and activation BCT for VSCND is the first stage of decisions; finally, we determine flow transshipment between components in the next stage. The GAMS-CPLEX is used for solving the model. The results show that running BCT will decrease 0.99% in costs. There is an economic justification for using BCT when demand is high. A fix-and-optimize and Lagrange relaxation (LR) generate lower and upper bound to estimate large scale in minimum time. The gap between the main model and fix-and-optimize is better than the LR algorithm. Finally, this research suggests equipping VSCND by BCT that becomes more resilient against demand fluctuation, sustainable, and agile.


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## Body

## 1. Introduction

One of the new discussions in the present age is the Viability of Supply Chain (VSC) in postpandemic adaptation. Ivanov and Dolgui [1] proposed the viability of SC that includes resiliency, sustainability, and agility. They suggested that organizational, informational, technological, financial, and process-functional structure should improve and change as follows:(1)
Organizational (subcontractor and backup supplier, workplace resilience, facility fortification)(2)
Informational (data analytics, visibility, digital twins, supplier portals, blockchain)(3)
Technological (robotics, additive manufacturing, smart manufacturing and warehousing, industry 4.0)(4)
Financial (business-government, revenue management, liquidity reserves)(5)
Process-functional (flexible capacity and sourcing, inventory and capacity buffers, Omnichannel, product diversification, and substitution) [2]Therefore, using BCT is clearly toward VSC. The BCT can help in the clarity and agility of SC [3]. Smart contract executes contract automatically between layers of SC. After every event happens, such as transactions by customers in retailers, instant settlement and checkout processes are done in all SC layers. This technology runs information sharing, and all components of SC are aware of demands until the end of chain.Smart contract in BCT runs contracts well between nodes, and financial transaction runs without human intervention. However, establishing BCT in SC needs programming and receiving correct input until giving the correct output. Therefore, it can mitigate risks and transaction costs [4]. Smart contract reduces costs compared to traditional contract and when it runs, tractable and irreversible transactions are established [5]. In the blockchain industry, Ethereum environments can implement smart contracts for SC as well. When a smart contract starts, there is no way to stop it. Thus, this procedure decreases demand fluctuation, raises awareness of demand variation in all SC layers, and decreases bullwhip effects [3]. This subject increases SC’s resiliency and sustainability through SC’s flexibility by sharing information on transactions and demands, decreasing flow quantity in SC and CO2 emission, and increasing satisfaction and welfare. Walmart for food supply ecosystem, Ford for supplying cobalt, De Beers for tracking diamond, UPS for tracking logistics, FedEx for tracking shipments are companies that used BCT to add transparency in SC. We should use this technology to manage SC clearly, be resilient against demand, and be sustainable for the future [6, 7]. In Figure 1, we can see the trend of using BCT and the market size of BCT in USA from 2016 to 2028 and the market share of Europe in 2020; it can show that the growth rate is 82.4% from 2021 to 2028. Therefore, we should consider this trend, and every SC that uses this technology can be successful in the future.Figure 1
The market size of BCT in the USA and the market share of Europe [8].The main contribution and motivation of this study are as follows:(1)
Applying BCT to increase viability and agility of SCND(2)
Using a new mathematical model for showing the effects of BCT on SCND(3)
Adding resiliency, sustainability, and robustness to VSCND by considering BCTWe organized this paper as follows. In Section2, we study on related work and show gap research in scope SCND. In Section 3, we determine VSCND mathematical model. In Section 4, the findings and results of the proposed model with sensitivity analysis are explained. In Sections 5 and 6, the managerial insights and conclusion and outlook are determined.


## 2. Related Work about SCND

Increasing the presence of investors in the cryptocurrency industry has caused other people in the community to be attracted to this industry. This subject has led many researchers to focus on this area, which is described below. Nayak and Dhaigude [9] reviewed and evaluated 178 articles that employed BCT in SCs. They presented that the benefits of BCT include data management, improving transparency, improving response time smart contract management, operational efficiency, disintermediation, immutability, and intellectual property management. Moreover, one of the most important advantages of BCT is increasing SC resiliency, reducing disruptions, helping risk management, and establishing multilayer protection [10].We surveyed SC by applying BCT with considering the mathematical model and other tools presented below.Choi [11] proposed global SC operations with air logistics with a mean-variance (MV) approach for risk analysis by applying the BCT era. They utilized their model by helping with the newsvendor problem. Choi and Luo [12] surveyed data quality problems for emerging markets in sustainable fashion SC. Their model includes decentralized SC and implementing BCT to enhance the profit and transparency of SC and increase welfare. They used newsvendor to the model problem and compared the decentralized SCM model with BCT and centralized model. In other works of Choi [11], he proposed BCT for diamond SC. He used BCT for exploring diamonds and compared the traditional model with the BCT platform. He found that using BCT platform can reduce cost and is beneficial to all parties in the luxury SC.Nayak and Dhaigude [9] suggested a conceptual model for sustainable SC management (SCM) using BCT. They used Interpretive Structural Modeling (ISM) and Matrice d’Impacts Croise’s Multiplication Appliquée a UN Classement (MICMAC) to draw a conceptual model, identify factors, and show effects of BCT on SCM. Rahmanzadeh et al. [13] designed a tactical SC planning model with open innovation consideration within a BCT. They proposed using BCT for registering, collecting, and refining ideas in open innovation. They used a fuzzy mathematical model to tackle uncertainty in the home appliances domain. They found that, by spending 1% of the total cost of SC, they can receive good ideas and decrease 41% of the total cost. Dolgui et al. [3] developed multiple logistics service providers by considering a smart contract for flexible flow shop scheduling. They used dynamic control theory for a running model.Manupati et al. [14] proposed a multiechelon green SC for the production allocation problem by implementing a blockchain approach. They suggested a Mixed Integer Nonlinear Programming (MINLP) for a carbon taxation policy for the greenness of the model. They showed that the distributed ledger-based blockchain approach enables minimizing total cost and carbon emissions. Liu et al. [15] surveyed a green agrifood SC by applying information service based on blockchain and big data (ISBD). They applied Stackelberg’s game theory to draw the model and solve investment decision problems. They compared four models, surveyed the freshness of agrifood by BCT, and used BCT for information sharing between producer and retailer. De Giovanni [4] explained that BCT could manage risks and transaction costs. He drew a situation that suppliers and retailers can activate by a traditional online platform or BCT. They removed transaction costs by BCT. Finally, they determined a smart wholesale price and revenue sharing contract to increase coordination between layers. They utilized game theory to solve the model.Alkahtani et al. [16] considered an e-agricultural SCM with BCT equipment in a cooperative situation. They applied the BCT into the agricultural SCM. Further, a Fuzzy Inference System (FIS) and Analytical Hierarchy Process (AHP) are embedded to show uncertainties in the model. They wanted to maximize profit with advertisement costs constraints and space constraints. As this model was NLP, they employed Karush–Kuhn–Tucker (KKT) to solve the model. They found that the effect of BCT can reduce the costs of SCM. Zhong et al. [17] implemented a Stackelberg game for container shipping lines and surveyed it using blockchain. Their model uses two stages to define the freight rate. They found that entering BCT can regulate the freight rate and improve the price in the container shipping market. The first stage is entering the blockchain or not, and the next step is assigning container shipping lines. They used the Stackelberg game and Cournot game for the second stage to show BCT effects.We classified the literature review in Table1 and the application of BCT in SCND. It can be seen that we want to design VSCND by BCT platform, which are not studied yet. We present VSCND through (flexible capacity (resilience strategy), sustainability constraints, and agility (by considering BCT)) for SCND. Finally, we add uncertainty through the robust scenario by defining a new form of the objective function.Table 1
Survey of SCND with considering BCT.
Ref.ProblemPlatformObjectivesUncertaintyMethodIndustryEconomicEnvironmentalEnergySocialOthers[11]SC operationsBCT✓————ProbablyScenario newsvendor + risk—[12]Data quality problemsBCT✓————ProbablyNewsvendorFashion[11]Diamond SCBCT✓————ProbablyStackelberg gameDiamond[9]Sustainable SCBCT————✓—ISM + MICMACNumerical example (NE)[13]Tactical with open innovationBCT✓————Fuzzy∗MILPHome appliances[3]Flexible flow shop schedulingBCT————✓—Dynamic optimal controlNE[14]Production allocation problemBCT✓—————MINLP + NSGA-IINE[15]ISBD green in agrifood SCBig data + BCT✓————ProbablyStackelberg gameNE[4]SCMBCT✓————StochasticStackelberg gameNE[1]VSCDigital concept✓—————Dynamics control approach—[2]VSCDigital concept✓—————Dynamic game-theoretic—[18]Sustainable SC—✓✓—✓——MILP,∗MOGWO, MORDA, AECAluminum[19]Open-CLSC—✓——✓✓RobustMILP + AWTNE[16]E-agricultural SCMBCT✓————FIS + AHPNLP + KKTE-agriculture[17]Container shipping linesBCT✓—————Stackelberg and Cournot gameNE[20]Waste VSC—✓————RobustScenario + riskIranThis researchVSCNDBCT✓————New robust stochasticMILPComputer∗ MILP: Mixed Integer Linear Programming; MOGW: Multiobjective Gray Wolf Optimizer; MORDA: Multiobjective Red Deer Algorithm; AEC: Augmented Epsilon Constraint; AWT: Augmented Weighted Tchebycheff.


## 3. Problem Description

In this research, we try to design VSCND with considering BCT. The previous section shows a lack of research in resilience and sustainable SC considering BCT. In the present study, we have customers, retailers, manufacturers, suppliers that transact in the BCT environment. The current SC uses BCT and smart contracts and executes contracts automatically between layers of SC. After customers transact in retailers, all payments automatically expand in SC layers, each component withdraws its share at once, and no charge remains in the retailers’ accounts. Eventually, we present VSCND through the flexible capacity facility (resilience strategy), sustainability constraints, and agility (by considering BCT) for SCND.In Figure2, we used Ethereum smart contract technology that pays at once after payment of customers. We want to consider BCT instead of a traditional transaction. Therefore, using BCT enables reducing variable costs. We draw a model that can select applying traditional transactions with higher variable costs or select BCT with lower variable costs. Still, there are fix  and maintenance costs for establishing BCT.Figure 2
VSCND with considering BCT.Hence, we need this assumption is as follows.Assumption 1.
(i)
All demands should be prepared, and the shortage is not permitted (agility)(ii)
Flow and capacity constraints are active in forwarding SC (agility)(iii)
A BCT is used for payment in layers of SC decreasing variable cost (agility)(iv)
Sustainability constraints include emission and energy consumption are defined (sustainability)(v)
Flexible scenario-based capacity for the facility is as resilience strategy (resiliency)(vi)
Using robust stochastic programming to cope with fluctuation of demand (resiliency) [2, 21](vii)
Variable cost after activating BCT is less than that without BCT


### 3.1. Model 1: VSCNDBCT

(1)minZ=1−λ∑spsΓs+zα∑psΓs−∑spsΓs+λmaxΓs,(2)Γs=FC+VCs,(3)FC=FCF+FCBT,(4)FCF=∑s′fss′xss′+∑mfmmxmm+∑dfcdxdd+∑rfrrxrr,(5)FCBT=fbt⋅xbt+∑t∑smbttsxbt,(6)VCs=VC1s1−xbt+VC2sxbt=VC1s−VC1s−VC2sxbt,∀s,(7)VC1s=∑p∑t∑s′∑mvsms′mptsqsms′mpts+∑m∑dvmcmdptsqmdmdpts+∑d∑rvdrdrptsqdrdrpts+∑r∑cvrcrcptsqrcrcpts,∀s,(8)VC2s=∑p∑t∑s′∑mvsms′mpts′qsms′mpts+∑m∑dvmcmdpts′qmdmdpts+∑d∑rvdrdrpts′qdrdrpts+∑r∑cvrcrcpts′qrcrcpts,∀s,subject to the following.Flow quantity constraints:(9)∑rqrcrcpts≥dcpts,∀c,p,t,s,(10)∑dqdrdrpts=∑cqrcrcpts,∀r,p,t,s,(11)∑mqmdmdpts=∑rqdrdrpts,∀d,p,t,s,(12)∑s′qsms′mpts=∑dqmdmdpts,∀m,p,t,s.Resiliency strategy (flexible capacity based on scenarios):(13)∑cqrcrcpts≤ρrCaprrptsxrr,∀r,p,t,s,(14)∑rqdrdrpts≤ρdCapddptsxdd,∀d,p,t,s,(15)∑dqmdmdpts≤ρmCapmmptsxmm,∀m,p,t,s,(16)∑mqsms′mpts≤ρs′Capss′ptsxss′,∀s′,p,t,s.Sustainability strategy (allowed emission and energy consumption):(17)∑p∑s′∑memsms′mptsqsms′mpts+∑m∑demmcmdptsqmdmdpts+∑d∑remdrdrptsqdrdrpts+∑r∑cemrcrcptsqrcrcpts≤EMts,∀t,s,(18)∑p∑s′∑mensms′mptsqsms′mpts+∑m∑denmcmdptsqmdmdpts+∑d∑rendrdrptsqdrdrpts+∑r∑cenrcrcptsqrcrcpts≤ENts,∀t,s.Decision variables:(19)xss′,xmm,xdd,xrr,xbt∈0,1,∀s′,m,d,r,(20)qsms′mpts,qmdmdpts,qdrdrpts,qrcrcpts≥0,∀s′,m,d,r,p,t,s.The objective function (1) minimizes the weighted expected and maximum cost function in each scenario. We proposed this form to increase robustness against demand disruption and consider the worst case. Constraint (2) includes the summation of fix and variable costs. Constraints (3) to (5) include the fix cost of establishing facilities and BCT network in a central server and BCT maintenance. Constraints (6) to (8) indicate the variable cost after establishing facilities without running BCT and considering BCT. Constraints (9) show satisfaction of demand. Constraints (10) to (12) show flow quantity between facilities. Constraints (13) to (16) show resiliency strategy and facility and capacity constraints dependent of scenario. Constraints (17) and (18) show sustainability strategy and total emissions and energy are less than maximum emission and energy. Constraints (19) and (20) are decision variables, and constraints (19) are locations and binary variables and activating BCT. Constraints (20) are positive flow variables.


### 3.2. Linearization of Max and Absolute Function

We need to change objective function (1) and equation (6) from Mix Integer Nonlinear Programming (MINLP (to Mixed Integer Programming (MIP))) by operational research method. This process decreases the time solution [22, 23].Linearization of max function and absolute function is as follows.Ifk=maxΩs, then we can replace these constraints with the model k≥Ωs,∀s.Ifk=Ωs, then we can replace these constraints with the model k=αs+βs,Ωs=αs−βs,αs,βs≥0,∀s.We can change and linearize a binary and a nonnegative variable that is produced.Supposez=Ax, if A is a nonnegative and positive variable and x is a binary variable. Therefore, we can replace these constraints with the model [24](21)z≥0,(22)z≤Mx,(23)z≤A,(24)z≥A−1−xM.It means that ifxis zero, z is zero based on equations (21) and (22). If x is 1, then z is A based on equations (23) and (24).


### 3.3. Linearization of VSCNDBCT

We can write linearization of VSCNDBCT as follows.Linearization of VSCNDBCT:(25)minZ=1−λ∑spsΓs+zα∑psvas+vbs+λδ,subject to(26)δ≥Γs,∀s,(27)Γs−∑spsΓs=vas−vbs,∀s,(28)vas,vbs≥0,∀s,(29)VCs=VC1s−ηs,∀s,(30)ηs≤VC1s−VC2s,∀s,(31)ηs≥VC1s−VC2s−M1−xbt,∀s,(32)ηs≥0,∀s,(33)ηs≤Mxbt,∀s,constraints (2)–(5) and (7)–(20).We linearize model (1) by changing MINLP to MIP. When we linearize it, speed solving and the complexity of the model are more straightforward than MINLP in all solvers. We show the complexity of VSCND; the numbers of binary, free, and nonnegative variables and constraints are calculated as follows:(34)binary variables=s′+m+d+r+1,(35)positive variables=p⋅t⋅ss′⋅m+m⋅d+d⋅r+r⋅c+3s,(36)free variables=11+2s,(37)constraints=10+10s+p⋅t⋅sc+2r+2d+2m+s′.It can be seen that scenario sets positively affect the number of constraints and positive and free variables and make the model scale grow. Therefore, applying methods that reduce scenarios such as Lagrangian relaxation (LR) and fix-and-optimize helps solve fast in minimum time.


### 3.4. Lagrangian Relaxation (Lower Bound for Large Scale)

Lagrangian relaxation (LR) is an exact method that relaxes complex constraints in a difficult problem, and because of removing constraints, solution time decreases. This method removes complex constraints and adds a penalty to the objective with the Lagrange multiplier. In practice, this relaxed problem can give us a lower bound for minimization and can often be solved more quickly than the main problem [25–27].Formulation of a MIP and changing to LR form are as follows:(38)ZIP≔mincTx,subject to(39)Ax≥b,(40)Dx≥d,(41)xinteger.In this step, we remove constraint (39) and add it to objective function (38) with the form of penalty cost and used the Lagrange coefficient for this situation:(42)Zλ≔mincTx+λb−Ax,subject to constraints (40) and (41).We need to find the maximum ofλ that minimizes Zλ:(43)ZD≔maxλ≥0Zλ,Zλ≔mini=1,...,mcTxi+λTb−Axi,subject to constraints (40) and (41).In each iteration, we updateλ based on these methods:(44)λt+1=λt+γt∇Zλt,λ0=0,t=0,λt+1=λt+γtb−Axi,λt+1=max0,λt+γtb−Axi,∑t=0∞γt=∞,limt⟶∞γt=0,γt=μtz∗−zλt∑i=1mbi−∑j=1naijxt2,γt+1=αγt,0≤α≤1,T>1.Therefore, based on these steps, the objective function (45) is LR of cost based on objective (25) that relax constraint (9). Figure 3 shows the steps of the LR algorithm that we applied to this research:(45)minLR obj=Z+∑c∑p∑t∑sudcpts∑rqrcrcpts−dcpts,Subject to constraints (1)–(8), (10)–(20) and (26)–(33).Figure 3
Solution approach for lower bound LR algorithm.As can be seen, the complexity of the LR algorithm is as follows: binary, positive, and free variables are the same as equations (34)–(36). But constraints change into equation (46). This subject decreases the time solution:(46)constraints=11+10s+p⋅t⋅s2r+2d+2m+s′.


### 3.5. Fix-and-Optimize (Upper Bound for Large Scale)

Because this problem is Np-hard, we need an algorithm to decrease the scale of this problem, produce an appropriate upper bound, and solve the model in minimum time. We suggest a new fix-and-optimize algorithm that creates an upper bound based on Figure4. Relaxation of each constraint makes the upper bound for the objective function and equal to or greater than the amount of the primary objective function [21]. Fixing the binary variables by this method produce an appropriate upper bound. Finally, the objective function of the proposed method is always equal to or bigger than the main model.Figure 4
Fix-and-optimize solution approach (upper bound).This technique produces a close upper bound for the primary model explained in Helber and Sahling [28].These steps of the new fix-and-optimize are as follows:(1)
Relax binary variables by changing them between zero and one and solve the model (relax constraint (19)).(2)
After solving step 1, we receive a lower bound, and our model is LP form.(3)
Loop:(a)
Summarize binary variables for each set and round them up(b)
Solve the model with a new fix binary variable(c)
If the model obtains optimum value, it is an upper bound, and we save it in a list(d)
Loop until the difference between two steps is less than 0.001(4)
Sort ascending objective function and report less upper bound objective function.The complexity of the fix-and-optimize algorithm is as follows: binary variables are removed, and free, nonnegative variables and constraints are the same as equations (35)–(37). This subject decreases the time solution.


## 3.1. Model 1: VSCNDBCT

(1)minZ=1−λ∑spsΓs+zα∑psΓs−∑spsΓs+λmaxΓs,(2)Γs=FC+VCs,(3)FC=FCF+FCBT,(4)FCF=∑s′fss′xss′+∑mfmmxmm+∑dfcdxdd+∑rfrrxrr,(5)FCBT=fbt⋅xbt+∑t∑smbttsxbt,(6)VCs=VC1s1−xbt+VC2sxbt=VC1s−VC1s−VC2sxbt,∀s,(7)VC1s=∑p∑t∑s′∑mvsms′mptsqsms′mpts+∑m∑dvmcmdptsqmdmdpts+∑d∑rvdrdrptsqdrdrpts+∑r∑cvrcrcptsqrcrcpts,∀s,(8)VC2s=∑p∑t∑s′∑mvsms′mpts′qsms′mpts+∑m∑dvmcmdpts′qmdmdpts+∑d∑rvdrdrpts′qdrdrpts+∑r∑cvrcrcpts′qrcrcpts,∀s,subject to the following.Flow quantity constraints:(9)∑rqrcrcpts≥dcpts,∀c,p,t,s,(10)∑dqdrdrpts=∑cqrcrcpts,∀r,p,t,s,(11)∑mqmdmdpts=∑rqdrdrpts,∀d,p,t,s,(12)∑s′qsms′mpts=∑dqmdmdpts,∀m,p,t,s.Resiliency strategy (flexible capacity based on scenarios):(13)∑cqrcrcpts≤ρrCaprrptsxrr,∀r,p,t,s,(14)∑rqdrdrpts≤ρdCapddptsxdd,∀d,p,t,s,(15)∑dqmdmdpts≤ρmCapmmptsxmm,∀m,p,t,s,(16)∑mqsms′mpts≤ρs′Capss′ptsxss′,∀s′,p,t,s.Sustainability strategy (allowed emission and energy consumption):(17)∑p∑s′∑memsms′mptsqsms′mpts+∑m∑demmcmdptsqmdmdpts+∑d∑remdrdrptsqdrdrpts+∑r∑cemrcrcptsqrcrcpts≤EMts,∀t,s,(18)∑p∑s′∑mensms′mptsqsms′mpts+∑m∑denmcmdptsqmdmdpts+∑d∑rendrdrptsqdrdrpts+∑r∑cenrcrcptsqrcrcpts≤ENts,∀t,s.Decision variables:(19)xss′,xmm,xdd,xrr,xbt∈0,1,∀s′,m,d,r,(20)qsms′mpts,qmdmdpts,qdrdrpts,qrcrcpts≥0,∀s′,m,d,r,p,t,s.The objective function (1) minimizes the weighted expected and maximum cost function in each scenario. We proposed this form to increase robustness against demand disruption and consider the worst case. Constraint (2) includes the summation of fix and variable costs. Constraints (3) to (5) include the fix cost of establishing facilities and BCT network in a central server and BCT maintenance. Constraints (6) to (8) indicate the variable cost after establishing facilities without running BCT and considering BCT. Constraints (9) show satisfaction of demand. Constraints (10) to (12) show flow quantity between facilities. Constraints (13) to (16) show resiliency strategy and facility and capacity constraints dependent of scenario. Constraints (17) and (18) show sustainability strategy and total emissions and energy are less than maximum emission and energy. Constraints (19) and (20) are decision variables, and constraints (19) are locations and binary variables and activating BCT. Constraints (20) are positive flow variables.


## 3.2. Linearization of Max and Absolute Function

We need to change objective function (1) and equation (6) from Mix Integer Nonlinear Programming (MINLP (to Mixed Integer Programming (MIP))) by operational research method. This process decreases the time solution [22, 23].Linearization of max function and absolute function is as follows.Ifk=maxΩs, then we can replace these constraints with the model k≥Ωs,∀s.Ifk=Ωs, then we can replace these constraints with the model k=αs+βs,Ωs=αs−βs,αs,βs≥0,∀s.We can change and linearize a binary and a nonnegative variable that is produced.Supposez=Ax, if A is a nonnegative and positive variable and x is a binary variable. Therefore, we can replace these constraints with the model [24](21)z≥0,(22)z≤Mx,(23)z≤A,(24)z≥A−1−xM.It means that ifxis zero, z is zero based on equations (21) and (22). If x is 1, then z is A based on equations (23) and (24).


## 3.3. Linearization of VSCNDBCT

We can write linearization of VSCNDBCT as follows.Linearization of VSCNDBCT:(25)minZ=1−λ∑spsΓs+zα∑psvas+vbs+λδ,subject to(26)δ≥Γs,∀s,(27)Γs−∑spsΓs=vas−vbs,∀s,(28)vas,vbs≥0,∀s,(29)VCs=VC1s−ηs,∀s,(30)ηs≤VC1s−VC2s,∀s,(31)ηs≥VC1s−VC2s−M1−xbt,∀s,(32)ηs≥0,∀s,(33)ηs≤Mxbt,∀s,constraints (2)–(5) and (7)–(20).We linearize model (1) by changing MINLP to MIP. When we linearize it, speed solving and the complexity of the model are more straightforward than MINLP in all solvers. We show the complexity of VSCND; the numbers of binary, free, and nonnegative variables and constraints are calculated as follows:(34)binary variables=s′+m+d+r+1,(35)positive variables=p⋅t⋅ss′⋅m+m⋅d+d⋅r+r⋅c+3s,(36)free variables=11+2s,(37)constraints=10+10s+p⋅t⋅sc+2r+2d+2m+s′.It can be seen that scenario sets positively affect the number of constraints and positive and free variables and make the model scale grow. Therefore, applying methods that reduce scenarios such as Lagrangian relaxation (LR) and fix-and-optimize helps solve fast in minimum time.


## 3.4. Lagrangian Relaxation (Lower Bound for Large Scale)

Lagrangian relaxation (LR) is an exact method that relaxes complex constraints in a difficult problem, and because of removing constraints, solution time decreases. This method removes complex constraints and adds a penalty to the objective with the Lagrange multiplier. In practice, this relaxed problem can give us a lower bound for minimization and can often be solved more quickly than the main problem [25–27].Formulation of a MIP and changing to LR form are as follows:(38)ZIP≔mincTx,subject to(39)Ax≥b,(40)Dx≥d,(41)xinteger.In this step, we remove constraint (39) and add it to objective function (38) with the form of penalty cost and used the Lagrange coefficient for this situation:(42)Zλ≔mincTx+λb−Ax,subject to constraints (40) and (41).We need to find the maximum ofλ that minimizes Zλ:(43)ZD≔maxλ≥0Zλ,Zλ≔mini=1,...,mcTxi+λTb−Axi,subject to constraints (40) and (41).In each iteration, we updateλ based on these methods:(44)λt+1=λt+γt∇Zλt,λ0=0,t=0,λt+1=λt+γtb−Axi,λt+1=max0,λt+γtb−Axi,∑t=0∞γt=∞,limt⟶∞γt=0,γt=μtz∗−zλt∑i=1mbi−∑j=1naijxt2,γt+1=αγt,0≤α≤1,T>1.Therefore, based on these steps, the objective function (45) is LR of cost based on objective (25) that relax constraint (9). Figure 3 shows the steps of the LR algorithm that we applied to this research:(45)minLR obj=Z+∑c∑p∑t∑sudcpts∑rqrcrcpts−dcpts,Subject to constraints (1)–(8), (10)–(20) and (26)–(33).Figure 3
Solution approach for lower bound LR algorithm.As can be seen, the complexity of the LR algorithm is as follows: binary, positive, and free variables are the same as equations (34)–(36). But constraints change into equation (46). This subject decreases the time solution:(46)constraints=11+10s+p⋅t⋅s2r+2d+2m+s′.


## 3.5. Fix-and-Optimize (Upper Bound for Large Scale)

Because this problem is Np-hard, we need an algorithm to decrease the scale of this problem, produce an appropriate upper bound, and solve the model in minimum time. We suggest a new fix-and-optimize algorithm that creates an upper bound based on Figure4. Relaxation of each constraint makes the upper bound for the objective function and equal to or greater than the amount of the primary objective function [21]. Fixing the binary variables by this method produce an appropriate upper bound. Finally, the objective function of the proposed method is always equal to or bigger than the main model.Figure 4
Fix-and-optimize solution approach (upper bound).This technique produces a close upper bound for the primary model explained in Helber and Sahling [28].These steps of the new fix-and-optimize are as follows:(1)
Relax binary variables by changing them between zero and one and solve the model (relax constraint (19)).(2)
After solving step 1, we receive a lower bound, and our model is LP form.(3)
Loop:(a)
Summarize binary variables for each set and round them up(b)
Solve the model with a new fix binary variable(c)
If the model obtains optimum value, it is an upper bound, and we save it in a list(d)
Loop until the difference between two steps is less than 0.001(4)
Sort ascending objective function and report less upper bound objective function.The complexity of the fix-and-optimize algorithm is as follows: binary variables are removed, and free, nonnegative variables and constraints are the same as equations (35)–(37). This subject decreases the time solution.


## 4. Results and Discussion

This section had a case study and estimated parameters defined in the notation list by the manager’s information from the computer supply chain. We tried to show the performance of the proposed model. The number of sets is presented in Table2, and amount of parameters are assigned in Table 3. The probability of occurrence is the same, and optimistic, pessimistic, and possible scenarios happen.Table 2
The number of sets, variables, and constraints.
Problems′⋅m⋅d⋅r⋅c⋅p⋅t⋅sBinary variablePositive variableFree variableConstraintCost functionTime (seconds)P1-main3.3.3.3.3.3.3.313981176882839008.4131.012Table 3
Parameters of case study.
ParametersValueUnitParametersValueUnitddcpts[U (2000, 2100)].1000. (s-1)/(s−1-).0.2 + 0.9)Num.fdd[U (80,85)].1000$vsms′mptsU (0.002,0.003)$/Num.frd[U(200,210)].1500$vmdmdptsU (0.004,0.005)$/Num.frr[U(200,210)].200;$vdrdrptsU (0.002,0.003)$/Num.fbt[U(200,210)].200$vrcrcptsU (0.001,0.002)$/Num.pps1/s%vsms′mpts′0.9.U (0.002,0.003)$λ50%vmdmdpts′0.9.U (0.004,0.005)$M10500-vdrdrpts′0.9.U (0.002,0.003)$emsms′mptsU (4,5).10−4Ton/Num.vrcrcpts′0.9.U (0.001,0.002)$emmdmdptsU (4,5).10−4Ton/Num.mbtts[U(3,4)].100$emdrdrptsU (4,5).10−4Ton/Num.esms′mptsU (4,5).10−3$/Num.emrcrcptsU (4,5).10−4Ton/Num.emdmdptsU (4,5).10−3$/Num.ensms′mptsU (1,2).10−4MJ/Num.edrdrptsU (4,5).10−3$/Num.enmdmdptsU (1,2).10−4MJ/Num.ercrcptsU (4,5).10−3$/Num.endrdrptsU (1,2).10−4MJ/Num.Capss′pts[U (5500,6600)].1000Num.enrcrcptsU (1,2).10−4MJ/Num.Capmmpts[U (55000,66000)].1000Num.EMts50000⋅s′⋅m⋅d⋅r⋅sTonCapddpts[U (3300,4400)].1000Num.ENts70000⋅s′⋅m⋅d⋅r⋅sMJCaprrpts[U (3300,4400)].1000Num.ρs′=ρm=ρd=ρr=90%fss′[U(200,210)].1000$zα0.5-fmm[U(200,210)].4000$[]: signWe applied a computer with this configuration: CPU 3.2 GHz, Processor Core i3-3210, 6.00 GB RAM, 64-bit operating system. Finally, we solve the mathematical models by GAMS-CPLEX solver.We drew the potential location for assigning components of VSCND in Iran (c.f. Table4 and Figure 5). After solving the model, it suggests that we activate BCT and determine the location and flow of SC components. The objective function is 2839008.413, shown in Table 2, and the final location-allocation is drawn in Figure 6. Eventually, we compare the model with BCT and without BCT in Table 5. We can see that without BCT, costs are almost 0.99% greater than with BCT.Table 4
Assigning location for the facility of SC and BCT.
Problem: P1VariablesCitySupplierxss′YazdChabaharBehbahan011ManufacturerxmmKhorramdarehZahedanKashan001DCxddKashanKashmarSanandaj111RetailerxrrTehranBorojerdKhoy111Figure 5
Potential location for the facilities.Figure 6
Final location of the SC facility with BCT platform.Table 5
Compare P1 with BCT and without BCT.
ModelP1-with BCTP1-without BCTGap (%)Profit2839008.4132867283.5460.99


### 4.1. Variation on the Scale of the Main Model

We determine several large-scale problems in Table6. When the scale of problems increases, the time and cost increase as shown in Figures 7 and 8. As shown in Figure 8, the time solution is exponential and NP-hard on a large scale. Therefore, we need to solve the model by heuristic, metaheuristic [29], and new exact solution in minimum time on a large scale.Table 6
Cost and time solution for different problems.
Prob.s′⋅m⋅d⋅r⋅c⋅p⋅t⋅sBinary var.Positive var.Free var.ConstraintCost functionTime (seconds)P13.3.3.3.3.3.3.313981176882839008.4131.012P23.3.3.3.3.3.3.51316352111402868628.5382.857P34.4.4.4.4.4.4.31730811715763931302.81515.437P44.4.4.6.4.4.4.41951321923543813256.04826.237P55.5.5.6.5.5.5.522137652153105525665.563538.483P68.8.8.6.8.8.8.531716952119260——P712.12.12.12.12.12.8.5492764952146140——P820.8.8.12.15.15.20.749105002125191180——P930.8.8.12.20.20.30.1059384003031636110——P1040.8.8.12.30.35.40.156917640045412646160——Figure 7
Cost function for problems.Figure 8
Time solution for problems.


### 4.2. Variation on the Conservative Coefficient

The conservative coefficient (λ) is the number of conservative decision-makers. We change it by varying between 0 and 1 that the conservation of decision-maker has been changed. If the conservative coefficient increases to 1, the cost function grows in Table 7 and Figures 9 and 10. If λ increases by 25%, the cost will increase by 0.3%, but the time solution does not change significantly.Table 7
Variation on conservative coefficient.
ProblemConservative coefficient (λ)Cost functionTime (seconds)Cost variation (%)P10.002822175.2060.911−0.59P10.253031098.1871.3736.77P1-main model0.52839008.4131.0120.00P10.752847425.0161.2670.30P11.002855003.8610.8960.56Figure 9
The cost function for variation on conservative coefficient.Figure 10
Time solution for variation on conservative coefficient.


### 4.3. Variation on Demand

As shown in Table8, the application of BCT decreases and is not economical when decreasing demand happens definitely. When demand changes and increases, the cost will grow, and establishing BCT is required to decrease cost (c.f. Figure 11). By reducing 40% for demand, the cost function decreases by 22%, and BCT is not applicable.Table 8
Variation of demand.
ProblemVariation of demand (%)Cost functionTime (seconds)BCTCost variation (%)P1−502208505.1821.230No need−22.21P1−402065840.5011.009No need−22.21P1−202381482.6101.678Needed−27.23P1-main model02839008.4131.012Needed0.00P1+202954586.6911.119Needed4.07P1+403072017.1060.763Needed8.21Figure 11
Effects of variation on demand.


### 4.4. Producing Bounds for the Main Model

We generate lower and upper bounds and compare the main model with the LR algorithm and a new fix-and-optimize strategy (c.f. Table9). In addition, by relaxing constraint (19) that is LP form, we generate a lower bound for validating the LR algorithm. The comparison of the three methods is determined in Figures 12 and 13. We see that the fix-and-optimize strategy produces a suitable solution with a gap minimum of 10% and the LR algorithm produces a lower bound with a gap of less than 32% for the main model. Also, it can be seen that the cost function of P5 shows a difference between methods for three methods (c.f. Figure 14). We suggest using both approaches, but the fix-and-optimize strategy is better than the LR algorithm for large-scale problems.Table 9
Comparing three algorithms with the main model.
ProblemLP model lower bound (A)LR model lower bound (B)Main model (B)Fix-and-opt. upper bound (C)GAP1(%)GAP2(%)CostTime (seconds)CostTime (seconds)CostTime (seconds)CostTime (seconds)P11603085.1310.2561917317.4561.5442839008.4131.0122841080.440.832−32.470.07P21621637.9560.3421938656.3333.3322868628.5382.8572869908.321.108−30.170.04P32696941.8481.2202794259.922.4683931302.81515.4373941819.722.501−28.920.27P42726058.8080.6742748952.71349.7483813256.04826.2373826966.7430.755−27.910.36P54186057.71114.9214479146.692534.8245525665.563538.4836119280.3716.347−18.9410.74P61.259045 E + 7456.623————1.250087 E + 7978.651——P72.5671 E + 72464.913————2.376011 E + 740152.25——Figure 12
The cost function for models.Figure 13
Time solution for models.Figure 14
Attaining bounds for problem P5.


## 4.1. Variation on the Scale of the Main Model

We determine several large-scale problems in Table6. When the scale of problems increases, the time and cost increase as shown in Figures 7 and 8. As shown in Figure 8, the time solution is exponential and NP-hard on a large scale. Therefore, we need to solve the model by heuristic, metaheuristic [29], and new exact solution in minimum time on a large scale.Table 6
Cost and time solution for different problems.
Prob.s′⋅m⋅d⋅r⋅c⋅p⋅t⋅sBinary var.Positive var.Free var.ConstraintCost functionTime (seconds)P13.3.3.3.3.3.3.313981176882839008.4131.012P23.3.3.3.3.3.3.51316352111402868628.5382.857P34.4.4.4.4.4.4.31730811715763931302.81515.437P44.4.4.6.4.4.4.41951321923543813256.04826.237P55.5.5.6.5.5.5.522137652153105525665.563538.483P68.8.8.6.8.8.8.531716952119260——P712.12.12.12.12.12.8.5492764952146140——P820.8.8.12.15.15.20.749105002125191180——P930.8.8.12.20.20.30.1059384003031636110——P1040.8.8.12.30.35.40.156917640045412646160——Figure 7
Cost function for problems.Figure 8
Time solution for problems.


## 4.2. Variation on the Conservative Coefficient

The conservative coefficient (λ) is the number of conservative decision-makers. We change it by varying between 0 and 1 that the conservation of decision-maker has been changed. If the conservative coefficient increases to 1, the cost function grows in Table 7 and Figures 9 and 10. If λ increases by 25%, the cost will increase by 0.3%, but the time solution does not change significantly.Table 7
Variation on conservative coefficient.
ProblemConservative coefficient (λ)Cost functionTime (seconds)Cost variation (%)P10.002822175.2060.911−0.59P10.253031098.1871.3736.77P1-main model0.52839008.4131.0120.00P10.752847425.0161.2670.30P11.002855003.8610.8960.56Figure 9
The cost function for variation on conservative coefficient.Figure 10
Time solution for variation on conservative coefficient.


## 4.3. Variation on Demand

As shown in Table8, the application of BCT decreases and is not economical when decreasing demand happens definitely. When demand changes and increases, the cost will grow, and establishing BCT is required to decrease cost (c.f. Figure 11). By reducing 40% for demand, the cost function decreases by 22%, and BCT is not applicable.Table 8
Variation of demand.
ProblemVariation of demand (%)Cost functionTime (seconds)BCTCost variation (%)P1−502208505.1821.230No need−22.21P1−402065840.5011.009No need−22.21P1−202381482.6101.678Needed−27.23P1-main model02839008.4131.012Needed0.00P1+202954586.6911.119Needed4.07P1+403072017.1060.763Needed8.21Figure 11
Effects of variation on demand.


## 4.4. Producing Bounds for the Main Model

We generate lower and upper bounds and compare the main model with the LR algorithm and a new fix-and-optimize strategy (c.f. Table9). In addition, by relaxing constraint (19) that is LP form, we generate a lower bound for validating the LR algorithm. The comparison of the three methods is determined in Figures 12 and 13. We see that the fix-and-optimize strategy produces a suitable solution with a gap minimum of 10% and the LR algorithm produces a lower bound with a gap of less than 32% for the main model. Also, it can be seen that the cost function of P5 shows a difference between methods for three methods (c.f. Figure 14). We suggest using both approaches, but the fix-and-optimize strategy is better than the LR algorithm for large-scale problems.Table 9
Comparing three algorithms with the main model.
ProblemLP model lower bound (A)LR model lower bound (B)Main model (B)Fix-and-opt. upper bound (C)GAP1(%)GAP2(%)CostTime (seconds)CostTime (seconds)CostTime (seconds)CostTime (seconds)P11603085.1310.2561917317.4561.5442839008.4131.0122841080.440.832−32.470.07P21621637.9560.3421938656.3333.3322868628.5382.8572869908.321.108−30.170.04P32696941.8481.2202794259.922.4683931302.81515.4373941819.722.501−28.920.27P42726058.8080.6742748952.71349.7483813256.04826.2373826966.7430.755−27.910.36P54186057.71114.9214479146.692534.8245525665.563538.4836119280.3716.347−18.9410.74P61.259045 E + 7456.623————1.250087 E + 7978.651——P72.5671 E + 72464.913————2.376011 E + 740152.25——Figure 12
The cost function for models.Figure 13
Time solution for models.Figure 14
Attaining bounds for problem P5.


## 5. Managerial Insights and Practical Implications

As managers of the SC, we should move forward to using novel technology in SC to decrease cost and increase resiliency and agility. BCT and cryptocurrency, renewable energy, and Internet of Things are new technologies that all of us need to use to improve the performance of SC. BCT and cryptocurrency are some of the best technologies that can facilitate financial transactions between customers and suppliers. As a result, operational and extra costs have been removed and make SC lean and agile. This research designed a SCND by considering and establishing BCT and cryptocurrency. We apply SC run traditional or consider BCT. Therefore, using BCT enables reducing variable costs. We draw a model that can select BCT with higher variable costs or select BCT with lower variable costs, fix cost, and maintenance costs for establishing BCT. We found that applying cryptocurrency can help SC to attain sustainability by decreasing cost and being agile. When demand is high, using BCT and cryptocurrency is an economic justification. Finally, we design a model for all risk-neutral decision-makers until risk-averse and cover all spectrum types. We surveyed this model on a large scale and contributed fix-and-optimize and Lagrange relaxation to generate upper and lower bound when the solver cannot solve in logical time.


## 6. Conclusions and Outlook

Using a novel technology by SC is grown every day. Technologies, such as cryptocurrency and blockchain, make SC agile and lean and decrease operational costs. VSCND applied resiliency, sustainability, and agility by embedding cryptocurrency and BCT. We used a new robust two-stage stochastic MIP for modeling this problem. We utilized GAMS-CPLEX to solve the problem. We found that establishing BCT and cryptocurrency for transaction reduce costs for SC with high demand.The findings are as follows:(1)
The cost and time increase when the scale of problems increases in Figures7 and 8.(2)
If the conservative coefficient increases to 1, the cost function grows in Table7 and Figures 9 and 10. When the conservative coefficient increases by 25%, the cost increases by 0.3%, but the time solution does not change significantly.(3)
The application of BCT decreases and is not economical when decreasing demand happens. When demand increases, it increases the cost function, and establishing BCT is needed for reducing cost (c.f. Figure11). By reducing 40% of demand, the cost function decreases by 22%, and BCT is not applicable.(4)
We utilize fix-and-optimize and LR to generate lower and upper bound to estimate large scale in minimum time. The results show that the gap between the main model and fix-and-optimize is better than the LR algorithm.One of the important research limitations is solving the large-scale main model. We propose applying exact algorithms such as benders decomposition, branch and price, branch-and-cut, and metaheuristic algorithms [30, 31]. We can add other resilience, sustainable strategy, and multiobjectives [18, 19] to the model and increase resiliency and sustainability such as backup suppliers and improve node complexity.Further, we suggest considering robust risk criteria such as Robust Conditional and Entropic Value at Risk (RCVaR and EVaR) [32, 33]. Researchers intend to investigate uncertainty methods such as robust convex and stochastic programming to close to the real world [34]. Using fuzzy and novel robust optimization as a data-driven approach is advantageous for a risk-averse decision-maker in the recent decade. Eventually, we suggest equipping all SC with novel technology such as Internet of Things (IoT) and RFID [35] to increase SC’s viability.



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*Source: 7347389-2021-11-08.xml*