doi: 10.3934/jimo.2022078
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A vendor-managed inventory model based on optimal retailers selection and reliability of supply chain

1. 

Ph.D. Candidate - Operations Research, Department of Management, Faculty of Economics and Administrative Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

2. 

Associate Professor, Department of Management, Faculty of Economics and Administrative Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

*Corresponding author: Nasser Motahari Farimani

Received  November 2021 Revised  March 2022 Early access May 2022

Fund Project: This work was supported in part by: Research Deputy of Ferdowsi University of Mashhad, under Grant No. 57130

One of the most common and successful approaches for the integrated supply chain management (SCM) is the vendor-managed inventory (VMI). In VMI, a vendor takes control of the inventory decisions for retailers. To establish a long-run relationship between the vendor and the retailer, it is necessary to consider two impactful factors: the vendors and retailers' reliability, and the optimal selection of retailers. For this purpose, the redundancy allocation problem (RAP), as an effective technique for increasing the reliability of vendors, is used in this paper. Also, the reliability of retailers as well as the reliability of the relationship between retailers and vendors are considered. In the retailer selection, process decisions on impactful criteria are simultaneously considered. For the retailers' selection, one analysis hierarchical process (AHP) is performed for each vendor, and the weights of retailers are obtained. Then, the obtained weights are plugged into the model as the inputs of the designed model. Since the developed model is a non-differentiable, non-convex, and mixed-integer function, genetic algorithm (GA) and particle swarm optimization (PSO) are leveraged to solve the formulated model. Finally, the efficiency of the presented method is verified through a case study with data collected from the electronic supply chain.

Citation: Azam Modares, Nasser Motahari Farimani, Vahideh Bafandegan Emroozi. A vendor-managed inventory model based on optimal retailers selection and reliability of supply chain. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022078
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show all references

References:
[1]

M. Akbari KaasgariD. M. Imani and M. Mahmoodjan, Optimizing a vendor managed inventory (VMI) supply chain for perishable products by considering discount: Two calibrated meta-heuristic algorithms, Comput. Ind. Eng., 103 (2017), 227-241. 

[2]

M. Ardakan and M. Rezvan, Multi-objective optimization of reliability-redundancy allocation problem with cold-standby strategy using NSGA-Ⅱ, Reliability Engineering & System Safety, 172 (2017).

[3]

Z. AshrafD. MalhotraP. Muhuri and Q. M. Danish Lohani, Interval tType-2 fuzzy vendor managed inventory system and its solution with particle swarm optimization, Int. J. Fuzzy Syst., 23 (2021), 2080-2105. 

[4]

R. D. AstantiY. Daryanto and P. K. Dewa, Low-carbon supply chain model under a vendor-managed inventory partnership and carbon cap-and-trade policy, J. Open Innov. Technol. Mark. Complex, 8 (2022), 30. 

[5]

M. BaghelS. Agrawal and S. Silakari, Survey of meta-heuristic algorithms for combinatorial optimization, International Journal of Computer Applications, 58 (2012), 21-31. 

[6]

C. Bai and J. Sarkis, Determining and applying sustainable supplier key performance indicators, Supply Chain Management: An International Journal, 19 (2014), 275-291. 

[7]

Q. BaiM. Jin and X. Xu, Effects of carbon emission reduction on supply chain coordination with vendor-managed deteriorating product inventory, Int. J. Prod. Econ., 208 (2019), 83-99. 

[8]

M. BaranyB. BertokZ. Kovacs and L. T. Fan, Solving vehicle assignment problems by process-network synthesis to minimize cost and environmental impact of transportation, Clean Technol Environ Policy, 13 (2011), 637-642. 

[9]

E. Bazan, Vendor managed inventory (VMI) with consignment stock (CS) agreement for a two-level supply chain with an imperfect production process with/without restoration interruptions, International J. Production Economics, 157 (2014), 289-301.  doi: 10.1016/j.ijpe.2014.02.010.

[10]

M. Bieniek, The ubiquitous nature of inventory: Vendor managed consignment inventory in adverse market conditions, European J. Oper. Res., 291 (2021), 411-420.  doi: 10.1016/j.ejor.2019.07.070.

[11]

S. BenjaafarY. Li and M. Daskin, Carbon footprint and the management of supply chains: Insights from simple models, IEEE Trans Autom. Sci. Eng., 10 (2013), 99-116. 

[12]

Z. Chen, Optimization of production inventory with pricing and promotion effort for a single-vendor multi-buyer system of perishable products, International J. Production Economics, 203 (2018), 333-349. 

[13]

H.-K. Chiou and G.-H. Tzeng, An extended approach of multicriteria optimization for MODM problems, Adv. Soft Comput., 21 (2003), 111-116. 

[14]

D. Choudhary and R. Shankar, The value of VMI beyond information sharing in a single supplier multiple retailers supply chain under a nonstationary (Rn, Sn) policy, Omega, 51 (2015), 59-70. 

[15]

A. DarkoA. P. C. ChanE. E. Ameyaw and P. Owusu EK, Review of application of analytic hierarchy process (AHP) in construction, International J. Construction Management, 19 (2019), 436-452.  doi: 10.1080/15623599.2018.1452098.

[16]

M. A. Darwish and O. M. Odah, Vendor managed inventory model for single-vendor multi-retailer supply chains, European J. Operational Research, 204 (2010), 473-484. 

[17]

A. Diabat, Hybrid algorithm for a vendor managed inventory system in a two-echelon supply chain, European J. Oper. Res., 238 (2014), 114-121.  doi: 10.1016/j.ejor.2014.02.061.

[18]

Y. DuanJ. Luo and J. Huo, Buyer–vendor inventory coordination with quantity discount incentive for fixed lifetime product, International J. Production Economics, 128 (2010), 351-357. 

[19]

A. M. Fakoor Saghih and A. Modares, A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components, J. Industrial & Management Optimization.

[20]

H. Garg and S. Sharma, Multi-objective reliability-redundancy allocation problem using particle swarm optimization, Computers & Industrial Engineering, 64 (2013), 247-255. 

[21]

A. GharaeiM. Karimi and S. Shekarabia, An integrated multiproduct, multi-buyer supply chain under penalty, green, and quality control polices and a vendor managed inventory with consignment stock agreement: The outer approximation with equality rel, Appl. Math. Model., 69 (2019), 223-254.  doi: 10.1016/j.apm.2018.11.035.

[22]

M. GibsonJ. J. BernardoC. Chung and R. Badinelli, A comparison of interactive multiple-objective decision making procedures, Computers & Operations Research, 14 (1987), 97-105. 

[23]

P. Giovanni, Smart Supply Chains with vendor managed inventory, coordination, and environmental performance, European J. Oper. Res., 292 (2021), 515-531.  doi: 10.1016/j.ejor.2020.10.049.

[24]

H. Golpîra, Optimal integration of the facility location problem into the multi-project multi-supplier multi-resource Construction Supply Chain network design under the vendor managed inventory strategy, Expert Syst. Appl., 139 (2020), 112841. 

[25]

S. Goyal, A one-vendor multi-buyer integrated inventory model: A comment, European J. Operational Research, 82 (1995), 209-210. 

[26]

S. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241. 

[27]

K. Halat and A. Hafezalkotob, Modeling carbon regulation policies in inventory decisions of a multi-stage green supply chain: A game theory approach, Computers and Industrial Engineering, 128 (2019), 807-830. 

[28]

J. HanJ. Lu and G. Zhang, Tri-level decision-making for decentralized vendormanaged inventory, Inf. Sci., 421 (2017), 85-103.  doi: 10.1016/j.ins.2017.08.089.

[29]

X. HongW. Chunyuan and L. Xu, Multiple-vendor, multiple-retailer based vendor-managed inventory, Ann. Oper. Res., 238 (2016), 277-297.  doi: 10.1007/s10479-015-2040-0.

[30]

B. HuC. MengD. Xu and Y. J. Son, Supply chain coordination under vendor managed inventory-consignment stocking contracts with wholesale price constraint and fairness, International J. Production Economics, 202 (2018), 21-31. 

[31]

M. A. KaasgariD. M. Imani and M. Mahmoodjanloo, Optimizing a vendor managed inventory (VMI) supply chain for perishable products by considering discount: Two calibrated meta-heuristic algorithms, Comput. Ind. Eng., 103 (2017), 227-241. 

[32]

V. Karbasi-bonabM. Yousefi Nejad Attari and E. Neishabouri, Presenting a bi-objective vendor managed inventory model with fuzzy demand for multiple vendor, Journal of Decisions and Operations Research, 2 (2) (2018), 147-168. 

[33]

M. Karimi and A. H. Niknamfar, A vendor-managed inventory system considering the redundancy allocation problem and carbon emissions, International J. Management Science and Engineering Management, 12 (2017), 269-279. 

[34]

J. Kennedy and R. Eberhart, Particle Swarm Optimization, In Proceedings of ICNN'95 - International Conference on Neural Networks, 1995. doi: 10.1109/ICNN. 1995.488968.

[35]

H. Kim and P. Kim, Reliability–redundancy allocation problem considering optimal redundancy strategy using parallel genetic algorithm, Reliability Engineering & System Safety, 159 (2017), 153-160. 

[36]

D. Konur and B. Schaefer, Integrated inventory control and transportation decisions under carbon emissions regulations: LTL vs TL carriers, Transport Res E-Log, 68 (2014), 14-38. 

[37]

S. K. KonakA. Smith and D. W. Coit, Efficiently solving the redundancy allocation problem using tabu search, IIE Transaction, 35 (2017), 515-526. 

[38]

S. Krichanchai and B. L. Maccarthy, The Adoption of vendor managed inventory for hospital pharmaceutical, Int. J. Logist. Manag., 3 (2018), 755-780. 

[39]

P. Kusuma and M. Kallista, Collaborative vender managed inventory model by using multi agent system and continuous review (r, Q)) replenishment policy, J. Applied Engineering Science, (2022).

[40]

G. D. LiD. Yamaguchi and M. Nagai, A grey-based decision-making approach to the supplier selection problem, Mathematical and Computer Modelling, 46 (2007), 573-581. 

[41]

J. Li and J. Wang, An extended QUALIFLEX method under probability hesitant fuzzy environment for selecting green suppliers, Int. J. Fuzzy Syst., 19 (2017), 1866-1879.  doi: 10.1007/s40815-017-0310-5.

[42]

J. Li and X. Zou, Dynamics of an epidemic model with none-local infections for diseases with latency over a patchy environment, J. Math. Biol., 60 (2010), 645-686.  doi: 10.1007/s00285-009-0280-9.

[43]

X. LiW. WangX. ZhaoJ. ZaiQ. ZhaoY. Li and A. Cha, Transmission dynamics and evolutionary history of 2019-nCoV., Med Virol, 92 (2020), 501-511. 

[44]

W. LiuG. Y. KeJ. Chen and L. Zhang, Scheduling the distribution of blood products: A vendor-managed inventory routing approach, Transportation Research Part E: Logistics and Transportation Review, 140 (2020), 101964.  doi: 10.1016/j.tre.2020.101964.

[45]

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Figure 1.  General steps of research
Figure 2.  The framework of obtaining the weight of the retailers
Figure 3.  The weight of retailers
Figure 4.  The structure of the inventory management system
Figure 5.  Inventory levels for vendors
Figure 6.  Series-parallel system
Figure 7.  Mean $ S\big/N $ ratios of the parameters of GA (model by objective function 2 and constraints)
Figure 8.  Mean $ S\big/N $ ratios of the parameters of GA
Figure 9.  Mean $ S\big/N $ ratios of the parameters of PSO
Figure 10.  Chromosome representation with objective function 2 (solution encoding)
Figure 11.  Chromosome representation with the GA approach (solution encoding)
Figure 12.  Convergence diagram of the best implementation of the PSO
Figure 13.  Convergence diagram of the best implementation of the GA
Table 1.  Some studies on VMI problems
Study Retailer Vendor Distributor Objectives Constraints Main contribution
$ \begin{array}{c} \rm{{Astantiet.}}, \rm{{ al. (2022)}} \end{array} $ Single Single No Cost No Designing a new model by considering the effect of defect rate, deterioration, and carbon emissions in terms of CO2.
$ \begin{array}{c} \rm{{Lotfi, et. al.}} \rm{{(2022)}} \end{array} $ Multiple Multiple NO Cost Yes Appling data-driven robust optimization in health care SC to cope uncertainty and disruption
$ \begin{array}{c} \rm{{Kusuma et.}} \rm{{al.( 2022)}} \end{array} $ Multiple Single Single Cost No Eliminating the exclusive relationship between vendor and customer.
$ \begin{array}{c} \rm{{Shang et.}} \rm{{al. (2022)}} \end{array} $ Multiple Single Multiple Cost Yes Designing a location-inventory routing problem with both deterministic demand and uncertain demand.
$ \begin{array}{c} \rm{{Gharaei et.}} \rm{{al. (2018)}} \end{array} $ Multiple Single NO Cost Yes Integrating QC and environmental policies with SCs and inventory systems
$ \begin{array}{c} \rm{{Liu et. al.}} \rm{{(2020) }} \end{array} $ Multiple Multiple NO Cost Yes Applying VMI to blood products distribution, and designing decomposition algorithm to solve problems
$ \begin{array}{c} \rm{{Wettasinghe}} \rm{{et. al.}} \rm{{(2020)}} \end{array} $ Single Single NO Cost Yes Incorporating the inventory positioning decision into the VMI model to determine the optimal system base stock level, delivery quantity to retailer and cycle length.
$ \begin{array}{c} \rm{{Wei et. al.}} \rm{{(2020)}} \end{array} $ Multiple Single NO Cost No Considering stochastic learning effect in VMI and RMI
$ \begin{array}{c} \rm{{Poursoltan}} \rm{{et. al. (2020)}} \end{array} $ Single Single NO Cost Yes Considering the manufacturing and remanufacturing process and percentage of defective products randomly
$ \begin{array}{c} \rm{{Najafnejhad }} \rm{{et. al. (2021)}} \end{array} $ Multiple Single NO Cost Yes Considering upper inventory limits as decision variables
$ \begin{array}{c} \rm{{Keshavarz-}} \rm{{Ghorbani}} \rm{{et. al. (2021)}} \end{array} $ Multiple Single NO Cost Yes Investigating and analyzing the system behavior under the learning curve and Optimizing a two-level closed-loop VMI.
$ \begin{array}{c} \rm{{Karbasi-}} \rm{{bonab, et al.}} \rm{{(2018)}} \end{array} $ Multiple Multiple NO $ \begin{array}{c} 1. \rm{Cost}\\2. \rm{warehouse\;space} \end{array} $ Minimizing the total supply chain cost and optimizing the warehouse space
$ \begin{array}{c} \rm{{Sadeghi,}} \rm{{et al. (2013)}} \end{array} $ Multiple Multiple NO Cost Yes Finding the order quantities along with the number of shipments received by retailers and vendors such that the total inventory cost of the chain is minimized.
$ \begin{array}{c} \rm{{Sadeghi,}} \rm{{et al. (2014)}} \end{array} $ Multiple Single Yes $ \begin{array}{c} 1. \rm{Cost}\\2. \rm{RAP} \end{array} $ Yes Maximizing the system reliability of producing items within constraints using the RAP approach.
Karimi & Niknamfar (2017) Multiple Single No $ \begin{array}{c} \rm{Carbon}\;\rm{emission}\;\rm{cost}\;\rm{RAP} \end{array} $ Yes Maximizing the system reliability of using the RAP approach and minimizing the carbon emission cost
$ \begin{array}{c} \rm{{Present}}\;\rm{{study}} \end{array} $ Multiple Multiple No $ \begin{array}{c} 1. \rm{Cost}\\2. \rm{{ RAP}}\\3. \rm{Selection}\;\rm{of\;retailers} \end{array} $ Yes Considering the cost, reliability of vendors and retailers, and retailers' selection, simultaneously.
Study Retailer Vendor Distributor Objectives Constraints Main contribution
$ \begin{array}{c} \rm{{Astantiet.}}, \rm{{ al. (2022)}} \end{array} $ Single Single No Cost No Designing a new model by considering the effect of defect rate, deterioration, and carbon emissions in terms of CO2.
$ \begin{array}{c} \rm{{Lotfi, et. al.}} \rm{{(2022)}} \end{array} $ Multiple Multiple NO Cost Yes Appling data-driven robust optimization in health care SC to cope uncertainty and disruption
$ \begin{array}{c} \rm{{Kusuma et.}} \rm{{al.( 2022)}} \end{array} $ Multiple Single Single Cost No Eliminating the exclusive relationship between vendor and customer.
$ \begin{array}{c} \rm{{Shang et.}} \rm{{al. (2022)}} \end{array} $ Multiple Single Multiple Cost Yes Designing a location-inventory routing problem with both deterministic demand and uncertain demand.
$ \begin{array}{c} \rm{{Gharaei et.}} \rm{{al. (2018)}} \end{array} $ Multiple Single NO Cost Yes Integrating QC and environmental policies with SCs and inventory systems
$ \begin{array}{c} \rm{{Liu et. al.}} \rm{{(2020) }} \end{array} $ Multiple Multiple NO Cost Yes Applying VMI to blood products distribution, and designing decomposition algorithm to solve problems
$ \begin{array}{c} \rm{{Wettasinghe}} \rm{{et. al.}} \rm{{(2020)}} \end{array} $ Single Single NO Cost Yes Incorporating the inventory positioning decision into the VMI model to determine the optimal system base stock level, delivery quantity to retailer and cycle length.
$ \begin{array}{c} \rm{{Wei et. al.}} \rm{{(2020)}} \end{array} $ Multiple Single NO Cost No Considering stochastic learning effect in VMI and RMI
$ \begin{array}{c} \rm{{Poursoltan}} \rm{{et. al. (2020)}} \end{array} $ Single Single NO Cost Yes Considering the manufacturing and remanufacturing process and percentage of defective products randomly
$ \begin{array}{c} \rm{{Najafnejhad }} \rm{{et. al. (2021)}} \end{array} $ Multiple Single NO Cost Yes Considering upper inventory limits as decision variables
$ \begin{array}{c} \rm{{Keshavarz-}} \rm{{Ghorbani}} \rm{{et. al. (2021)}} \end{array} $ Multiple Single NO Cost Yes Investigating and analyzing the system behavior under the learning curve and Optimizing a two-level closed-loop VMI.
$ \begin{array}{c} \rm{{Karbasi-}} \rm{{bonab, et al.}} \rm{{(2018)}} \end{array} $ Multiple Multiple NO $ \begin{array}{c} 1. \rm{Cost}\\2. \rm{warehouse\;space} \end{array} $ Minimizing the total supply chain cost and optimizing the warehouse space
$ \begin{array}{c} \rm{{Sadeghi,}} \rm{{et al. (2013)}} \end{array} $ Multiple Multiple NO Cost Yes Finding the order quantities along with the number of shipments received by retailers and vendors such that the total inventory cost of the chain is minimized.
$ \begin{array}{c} \rm{{Sadeghi,}} \rm{{et al. (2014)}} \end{array} $ Multiple Single Yes $ \begin{array}{c} 1. \rm{Cost}\\2. \rm{RAP} \end{array} $ Yes Maximizing the system reliability of producing items within constraints using the RAP approach.
Karimi & Niknamfar (2017) Multiple Single No $ \begin{array}{c} \rm{Carbon}\;\rm{emission}\;\rm{cost}\;\rm{RAP} \end{array} $ Yes Maximizing the system reliability of using the RAP approach and minimizing the carbon emission cost
$ \begin{array}{c} \rm{{Present}}\;\rm{{study}} \end{array} $ Multiple Multiple No $ \begin{array}{c} 1. \rm{Cost}\\2. \rm{{ RAP}}\\3. \rm{Selection}\;\rm{of\;retailers} \end{array} $ Yes Considering the cost, reliability of vendors and retailers, and retailers' selection, simultaneously.
Table 2.  The weight of retailers to vendors using AHP
Retailer/Vendor $ j=1 $ $ j=2 $ $ j=3 $
$ k=1 $ 0.113 0.134 0.144
$ k=2 $ 0.041 0.052 0.056
$ k=3 $ 0.105 0.089 0.089
$ k=4 $ 0.086 0.067 0.069
$ k=5 $ 0.061 0.084 0.084
$ k=6 $ 0.102 0.094 0.088
$ k=7 $ 0.082 0.091 0.062
$ k=8 $ 0.093 0.085 0.085
$ k=9 $ 0.073 0.058 0.078
$ k=10 $ 0.077 0.065 0.076
Retailer/Vendor $ j=1 $ $ j=2 $ $ j=3 $
$ k=1 $ 0.113 0.134 0.144
$ k=2 $ 0.041 0.052 0.056
$ k=3 $ 0.105 0.089 0.089
$ k=4 $ 0.086 0.067 0.069
$ k=5 $ 0.061 0.084 0.084
$ k=6 $ 0.102 0.094 0.088
$ k=7 $ 0.082 0.091 0.062
$ k=8 $ 0.093 0.085 0.085
$ k=9 $ 0.073 0.058 0.078
$ k=10 $ 0.077 0.065 0.076
Table 3.  Notations used in designed model
Sets
$ J $ The set of vendors indexed by$ j $;
$ K $ The set of retailers indexed by$ k $;
$ S $ The set of subsystems indexed by$ s $;
Decision variables
$ q_{jk} $ Amount of product ordered to $ j^{th} $ vendor by $ k^{th} $ retailer.
$ x_{jk} $ Binary variable taking the value of 1 when $ k^{th} $ retailer allocated to $ j^{th} $ vendor; otherwise, it is equal to 0.
$ M_{sj} $ The number redundant machine of $ s^{th} $ subsystem to $ j^{th} $ vendor.
$ q_{j} $ Amount of product ordered by $ j^{th} $ vendor $ (q_{j} =\sum _{k=1}^{K}q_{jk} ) $.
$ b_{j} $ The maximum shortage for $ j^{th} $ vendor
Parameters
$ A_{jk} $ Ordering cost of $ k^{th} $ retailer supplied by $ j^{th} $ vendor.
$ a_{j} $ Setup cost of $ j^{th} $the vendor per unit time.
$ H_{jk} $ The unit holding cost of $ k^{th} $ retailer's product supplied by $ j^{th} $ vendor per unit time.
$ h_{j} $ Holding cost of $ j^{th} $ vendor's products per unit time.
$ d_{jk} $ The expected demand of $ k^{th} $retailer supplied by $ j^{th} $ vendor per unit time.
$ d_{j} $ The expected demand of $ j^{th} $ vendor per unit time.
$ R_{sj} $ Reliability of $ j^{th} $ vendor production in $ s^{th} $ subsystem.
$ R_{k} $ Reliability of $ k^{th} $ retailer.
$ R_{jk} $ Reliability of relationship between $ k^{th} $ retailer and $ j^{th} $ vendor.
$ \lambda _{k} $ The failure rate of $ k^{th} $ retailer activity (parameter of exponential distribution).
$ \lambda _{jk} $ The failure rate of relationship between $ k^{th} $ retailer and $ j^{th} $vendor (parameter of exponential distribution)
$ v_{k} $ Final order quantity $ k^{th} $retailer from all vendors for the product.
$ p_{j} $ Production rate of $ j^{th} $ vendor.
$ \pi_{j} $ Linear backorder cost per unit of $ j^{th} $ vendor.
$ \hat{\pi }_{j} $ Fixed backorder cost per unit of $ j^{th} $ vendor (time-independent).
$ F_{jk} $ Product transportation cost from $ j^{th} $ vendor to $ k^{th} $ retailer.
$ S_{jk} $ Safety stock of $ j^{th} $ vendor for $ k^{th} $ retailer.
$ f $ Space required storing one unit of the product.
$ \rm{Cap}_{j} $ Maximum of space capacity for $ j^{th} $ vendor.
$ \rm{Sho}_{j} $ Maximum of permissible shortage for $ j^{th} $ vendor.
$ C_{sj} $ The number of $ s^{th} $ the subsystem for $ j^{th} $ vendor.
$ \rm{bud}_{j} $ Available budget to purchasing and installation machines for $ j^{th} $ vendor.
$ u_{jk} $ Minimum of total order quantity for $ j^{th} $ vendor by $ k^{th} $ retailer.
$ w_{jk} $ Weight of $ k^{th} $retailer to $ j^{th} $ vendor.
Sets
$ J $ The set of vendors indexed by$ j $;
$ K $ The set of retailers indexed by$ k $;
$ S $ The set of subsystems indexed by$ s $;
Decision variables
$ q_{jk} $ Amount of product ordered to $ j^{th} $ vendor by $ k^{th} $ retailer.
$ x_{jk} $ Binary variable taking the value of 1 when $ k^{th} $ retailer allocated to $ j^{th} $ vendor; otherwise, it is equal to 0.
$ M_{sj} $ The number redundant machine of $ s^{th} $ subsystem to $ j^{th} $ vendor.
$ q_{j} $ Amount of product ordered by $ j^{th} $ vendor $ (q_{j} =\sum _{k=1}^{K}q_{jk} ) $.
$ b_{j} $ The maximum shortage for $ j^{th} $ vendor
Parameters
$ A_{jk} $ Ordering cost of $ k^{th} $ retailer supplied by $ j^{th} $ vendor.
$ a_{j} $ Setup cost of $ j^{th} $the vendor per unit time.
$ H_{jk} $ The unit holding cost of $ k^{th} $ retailer's product supplied by $ j^{th} $ vendor per unit time.
$ h_{j} $ Holding cost of $ j^{th} $ vendor's products per unit time.
$ d_{jk} $ The expected demand of $ k^{th} $retailer supplied by $ j^{th} $ vendor per unit time.
$ d_{j} $ The expected demand of $ j^{th} $ vendor per unit time.
$ R_{sj} $ Reliability of $ j^{th} $ vendor production in $ s^{th} $ subsystem.
$ R_{k} $ Reliability of $ k^{th} $ retailer.
$ R_{jk} $ Reliability of relationship between $ k^{th} $ retailer and $ j^{th} $ vendor.
$ \lambda _{k} $ The failure rate of $ k^{th} $ retailer activity (parameter of exponential distribution).
$ \lambda _{jk} $ The failure rate of relationship between $ k^{th} $ retailer and $ j^{th} $vendor (parameter of exponential distribution)
$ v_{k} $ Final order quantity $ k^{th} $retailer from all vendors for the product.
$ p_{j} $ Production rate of $ j^{th} $ vendor.
$ \pi_{j} $ Linear backorder cost per unit of $ j^{th} $ vendor.
$ \hat{\pi }_{j} $ Fixed backorder cost per unit of $ j^{th} $ vendor (time-independent).
$ F_{jk} $ Product transportation cost from $ j^{th} $ vendor to $ k^{th} $ retailer.
$ S_{jk} $ Safety stock of $ j^{th} $ vendor for $ k^{th} $ retailer.
$ f $ Space required storing one unit of the product.
$ \rm{Cap}_{j} $ Maximum of space capacity for $ j^{th} $ vendor.
$ \rm{Sho}_{j} $ Maximum of permissible shortage for $ j^{th} $ vendor.
$ C_{sj} $ The number of $ s^{th} $ the subsystem for $ j^{th} $ vendor.
$ \rm{bud}_{j} $ Available budget to purchasing and installation machines for $ j^{th} $ vendor.
$ u_{jk} $ Minimum of total order quantity for $ j^{th} $ vendor by $ k^{th} $ retailer.
$ w_{jk} $ Weight of $ k^{th} $retailer to $ j^{th} $ vendor.
Table 4.  Controllable factors and their levels (model by objective function 2 and constraints)
Parameter Notation Level Optimal level
Level 1 Level 2 fLevel 3
GA Popsize $ A $ 50 80 100 100
$ p_{c} $ B 0.4 0.5 0.6 0.6
$ p_m $ C 0.05 0.1 0.2 0.2
Parameter Notation Level Optimal level
Level 1 Level 2 fLevel 3
GA Popsize $ A $ 50 80 100 100
$ p_{c} $ B 0.4 0.5 0.6 0.6
$ p_m $ C 0.05 0.1 0.2 0.2
Table 5.  Controllable factors and their levels
Parameter Notation Level Optimal level
Level 1 Level 2 Level 3
GA Popsize $ A $ 50 100 150 100
$ p_{c} $ B 0.4 0.5 0.8 0.4
$ p_{m} $ C 0.1 0.2 0.3 0.3
PSO $ C_{1} $ A 1 1.5 2 2
$ C_{2} $ B 1 1.5 2 2
$ W $ C 0.7 0.8 0.9 0.9
Parameter Notation Level Optimal level
Level 1 Level 2 Level 3
GA Popsize $ A $ 50 100 150 100
$ p_{c} $ B 0.4 0.5 0.8 0.4
$ p_{m} $ C 0.1 0.2 0.3 0.3
PSO $ C_{1} $ A 1 1.5 2 2
$ C_{2} $ B 1 1.5 2 2
$ W $ C 0.7 0.8 0.9 0.9
Table 6.  The results of the objective functions using GA and PSO
Solution of model $ Z_{1} ^{*} $ $ Z_{2} ^{*} $ $ Z_{3} ^{*} $
GA 300988.2 2.72 4559
PSO 298305.4 2.72 4540
Solution of model $ Z_{1} ^{*} $ $ Z_{2} ^{*} $ $ Z_{3} ^{*} $
GA 300988.2 2.72 4559
PSO 298305.4 2.72 4540
Table 7.  The results of the suggested model using the GA and PSO method
GA PSO
Vendor Decision variables Retailer Vendor Decision variables Retailer
$ b_{j} $ $ q_{j} $ $ q_{jk} $ $ x_{jk} $ $ b_{j} $ $ q_{j} $ $ q_{jk} $ $ x_{jk} $
0 0 1 0 0 1
0 0 2 0 0 2
0 0 3 0 0 3
0 0 4 0 0 4
Vendor1 1530 8934 2300 1 5 Vendor1 1480 8702 2300 1 5
4000 1 6 4000 1 6
2634 1 7 2402 1 7
0 0 8 0 0 8
0 0 9 0 0 9
0 0 10 0 0 10
0 0 1 0 0 1
0 0 2 0 0 2
2498 1 3 2498 1 3
2200 1 4 2200 1 4
Vendor2 1620 9098 0 0 5 Vendor2 1620 9098 0 0 5
0 0 6 0 0 6
0 0 7 0 0 7
2300 1 8 2300 1 8
0 0 9 0 0 9
2100 0 10 2100 1 10
2340 1 1 2340 1 1
2200 1 2 2200 1 2
0 0 3 0 0 3
0 0 4 0 0 4
Vendor3 1480 7915 0 0 5 Vendor3 1480 7915 0 0 5
0 0 6 0 0 6
0 0 7 0 0 7
0 0 8 0 0 8
3375 1 9 3375 1 9
0 0 10 0 0 10
GA PSO
Vendor Decision variables Retailer Vendor Decision variables Retailer
$ b_{j} $ $ q_{j} $ $ q_{jk} $ $ x_{jk} $ $ b_{j} $ $ q_{j} $ $ q_{jk} $ $ x_{jk} $
0 0 1 0 0 1
0 0 2 0 0 2
0 0 3 0 0 3
0 0 4 0 0 4
Vendor1 1530 8934 2300 1 5 Vendor1 1480 8702 2300 1 5
4000 1 6 4000 1 6
2634 1 7 2402 1 7
0 0 8 0 0 8
0 0 9 0 0 9
0 0 10 0 0 10
0 0 1 0 0 1
0 0 2 0 0 2
2498 1 3 2498 1 3
2200 1 4 2200 1 4
Vendor2 1620 9098 0 0 5 Vendor2 1620 9098 0 0 5
0 0 6 0 0 6
0 0 7 0 0 7
2300 1 8 2300 1 8
0 0 9 0 0 9
2100 0 10 2100 1 10
2340 1 1 2340 1 1
2200 1 2 2200 1 2
0 0 3 0 0 3
0 0 4 0 0 4
Vendor3 1480 7915 0 0 5 Vendor3 1480 7915 0 0 5
0 0 6 0 0 6
0 0 7 0 0 7
0 0 8 0 0 8
3375 1 9 3375 1 9
0 0 10 0 0 10
Table 8.  Optimal number of machine using GA and PSO
$ M_{Sj} $ Machine 1 Machine 2 Machine 3 Machine 4 Machine 5
Vendor 1 1 2 1 2 1
Vendor 2 1 3 1 1 2
Vendor 3 2 3 1 2 1
$ M_{Sj} $ Machine 1 Machine 2 Machine 3 Machine 4 Machine 5
Vendor 1 1 2 1 2 1
Vendor 2 1 3 1 1 2
Vendor 3 2 3 1 2 1
Table 9.  Data related to obtain weighted of the AHP method, holding cost each of the retailer, ordering cost, demand of the retailers, reliability, the failure rate of retailers
$ {{ H}}_{{ jk}} $
1 15 13 14 12 18 17 19 14 12 17
2 17 18 15 20 20 18 19 17 16 19
3 18 13 15 10 13 12 10 15 17 13
$ {{ w}}_{{ jk}} $
1 0.113 0.041 0.105 0.086 0.061 0.102 0.082 0.093 0.073 0.077
2 0.134 0.052 0.089 0.067 0.084 0.094 0.091 0.085 0.058 0.065
3 0.144 0.056 0.089 0.069 0.084 0.088 0.062 0.085 0.078 0.076
$ {{ F}}_{{ jk}} $
1 10 10 12 14 13 14 10 11 12 13
2 10 11 10 13 12 11 11 12 13 11
3 13 12 11 12 13 13 10 11 12 10
$ {{ d}}_{{ jk}} $
1 2530 2768 4325 3021 3452 2876 3041 2870 3056 2690
2 3456 3405 2345 2900 3425 3409 4072 3023 3456 1025
3 3400 2313 3645 3965 3956 4357 3456 2047 3056 4076
$ {{ \alpha }}_{{ jk}} $
f1 0.85 0.86 0.78 0.7 0.78 0.75 065 074 0.83 0.78
2 0.84 0.82 0.75 0.68 0.70 0.8 0.84 0.76 0.83 0.78
3 0.76 0.74 0.82 0.75 0.67 0.78 0.76 0.83 0.80 0.76
$ {{ A}}_{{ jk}} $
1 25 16 85 84 39 60 45 32 66 56
2 42 60 91 54 65 97 94 78 65 76
3 74 65 64 90 80 67 72 64 55 60
$ {{ u}}_{{ jk}} $
1 2500 2500 2300 2300 2300 2200 2300 2500 2300 2200
2 2300 2300 2200 2200 2300 2300 2400 2300 2300 2100
3 2340 2200 2350 2300 2350 2400 2300 2200 2300 2400
$ {{ v}}_{{ k}} $
Vendors 3500 3000 3000 6000 4000 3500 4000 3000 3500 4500
$ {{\lambda }}_{{ k}} $
Vendors 0.2 0.2 0.3 0.2 0.4 0.2 0.3 0.2 0.4 0.3
$ {{ H}}_{{ jk}} $
1 15 13 14 12 18 17 19 14 12 17
2 17 18 15 20 20 18 19 17 16 19
3 18 13 15 10 13 12 10 15 17 13
$ {{ w}}_{{ jk}} $
1 0.113 0.041 0.105 0.086 0.061 0.102 0.082 0.093 0.073 0.077
2 0.134 0.052 0.089 0.067 0.084 0.094 0.091 0.085 0.058 0.065
3 0.144 0.056 0.089 0.069 0.084 0.088 0.062 0.085 0.078 0.076
$ {{ F}}_{{ jk}} $
1 10 10 12 14 13 14 10 11 12 13
2 10 11 10 13 12 11 11 12 13 11
3 13 12 11 12 13 13 10 11 12 10
$ {{ d}}_{{ jk}} $
1 2530 2768 4325 3021 3452 2876 3041 2870 3056 2690
2 3456 3405 2345 2900 3425 3409 4072 3023 3456 1025
3 3400 2313 3645 3965 3956 4357 3456 2047 3056 4076
$ {{ \alpha }}_{{ jk}} $
f1 0.85 0.86 0.78 0.7 0.78 0.75 065 074 0.83 0.78
2 0.84 0.82 0.75 0.68 0.70 0.8 0.84 0.76 0.83 0.78
3 0.76 0.74 0.82 0.75 0.67 0.78 0.76 0.83 0.80 0.76
$ {{ A}}_{{ jk}} $
1 25 16 85 84 39 60 45 32 66 56
2 42 60 91 54 65 97 94 78 65 76
3 74 65 64 90 80 67 72 64 55 60
$ {{ u}}_{{ jk}} $
1 2500 2500 2300 2300 2300 2200 2300 2500 2300 2200
2 2300 2300 2200 2200 2300 2300 2400 2300 2300 2100
3 2340 2200 2350 2300 2350 2400 2300 2200 2300 2400
$ {{ v}}_{{ k}} $
Vendors 3500 3000 3000 6000 4000 3500 4000 3000 3500 4500
$ {{\lambda }}_{{ k}} $
Vendors 0.2 0.2 0.3 0.2 0.4 0.2 0.3 0.2 0.4 0.3
Table 10.  Data related to holding cost, ordering, shortage, maximum production capacity and shortage capacity, maximum available budget
Parameter Vendor 1 Vendor 1 Vendor 1
$ a_{j} $ 1000 1100 1050
$ h_{j} $ 9 10 7
$ p_{j} $ 30000 28000 25000
$ \pi _{j} $ 5 7 6
$ \hat{\pi }_{j} $ 2 3 2
$ \text{Cap}_{j} $ 64000 58000 78000
$ \text{sho}_{j} $ 750 750 750
$ \text{bud}_{j} $ 300000 400000 430000
Parameter Vendor 1 Vendor 1 Vendor 1
$ a_{j} $ 1000 1100 1050
$ h_{j} $ 9 10 7
$ p_{j} $ 30000 28000 25000
$ \pi _{j} $ 5 7 6
$ \hat{\pi }_{j} $ 2 3 2
$ \text{Cap}_{j} $ 64000 58000 78000
$ \text{sho}_{j} $ 750 750 750
$ \text{bud}_{j} $ 300000 400000 430000
Table 11.  Data related to reliability of vendor in subsystems, The purchasing cost of the machine for each of the retailer
$ C_{sj} $ Vendor 1 Vendor 2 Vendor 3
Subsystem 1 15000 17000 11000
Subsystem 2 12000 9000 18000
Subsystem 3 16000 18000 12000
Subsystem 4 15000 9000 10000
Subsystem 5 13000 14000 12000
$ C_{sj} $ Vendor 1 Vendor 2 Vendor 3
Subsystem 1 15000 17000 11000
Subsystem 2 12000 9000 18000
Subsystem 3 16000 18000 12000
Subsystem 4 15000 9000 10000
Subsystem 5 13000 14000 12000
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