Abstract
This paper proposes a novel Inventory-Location problem that integrates supplier selection decisions to design a three-echelon supply chain network, under a continuous (s,Q) inventory control policy at the warehouses. In this problem, a set of warehouses must be selected within a set of potential locations to serve several customers or demand zones, additionally involving the selection of the suppliers for fulfilling incoming orders from the located warehouses. The optimal solution must be determined while minimizing total system costs including supplier selection, transportation (i.e., suppliers-warehouses and warehouses-customers), inventory (i.e., cycle and safety stock), and warehouse location costs. A key element of the problem is the consideration of variable lead-times for the warehouses, which are dependent on the selection of the supplier that serve them, thus increasing model complexity. Accordingly, an efficient algorithm based on the Generalized Benders Decomposition is developed and implemented to solve the proposed Mixed Integer, Nonlinear, Nonconvex, Programming Model. The proposed solution approach relies on a convenient model formulation and decomposition that yields a Mixed Integer Linear master problem and a continuous, convex subproblem. A wide set of medium-sized synthetic instances are optimally solved in affordable times, denoting the efficiency and effectiveness of the proposed model along with the proposed solution approach. Significant scientific and managerial insights are provided and discussed.
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References
Aǧrali S, Geunes J, Taşkin ZC (2012) A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements. J Glob Optim 54(3):551–581
Amid A, Ghodsypour SH, O’Brien C (2006) Fuzzy multiobjective linear model for supplier selection in a supply chain. Int J Prod Econ 104(2):394–407
Amid A, Ghodsypour SH, O’Brien C (2011) A weighted maxmin model for fuzzy multi-objective supplier selection in a supply chain. Int J Prod Econ 131(1):139–145
Amiri-Aref M, Klibi W, Babai MZ (2018) The multi-sourcing location inventory problem with stochastic demand. Eur J Oper Res 266(1):72–87
Araya-Sassi C, Miranda PA, Paredes-Belmar G (2018) Lagrangian relaxation for an inventory location problem with periodic inventory control and stochastic capacity constraints. Math Probl Eng. https://doi.org/10.1155/2018/8237925
Axsäter S (2015) Inventory control, vol 225. Springer, Cham
Baron O, Milner J, Naseraldin H (2011) Facility location: a robust optimization approach. Prod Oper Manag 20(5):772–785
Bashiri M, Tabrizi MM (2010) Supply chain design: a holistic approach. Expert Syst Appl 37(1):688–693
Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer Math 4(1):238–252
Benyoucef L, Xie X, Tanonkou GA (2013) Supply chain network design with unreliable suppliers: a Lagrangian relaxation-based approach. Int J Prod Res 51(21):6435–6454
Bhutta KS, Huq F, Frazier G, Mohamed Z (2003) An integrated location, production, distribution and investment model for a multinational corporation. Int J Prod Econ 86:201–216. https://doi.org/10.1016/S0925-5273(03)00046-X
Birge JR, Louveaux F (2011) Introduction to stochastic programming. Springer series in operations research and financial engineering, vol 49. Springer, New York, NY
Bradley JR, Arntzen BC (1999) Inventory in seasonal demand environments. Oper Res 47(6):795–806
Burke GJ, Carrillo JE, Vakharia AJ (2009) Sourcing decisions with stochastic supplier reliability and stochastic demand. Prod Oper Manag 18(4):475–484
Cabrera G, Miranda PA, Cabrera E, Soto R, Crawford B, Rubio JM, Paredes F (2013) Solving a novel inventory location model with stochastic constraints and (R, s, S) inventory control policy. Math Probl Eng 2013(Dc):1–12
Cárdenas-Barrón LE, Melo RA, Santos MC (2021) Extended formulation and valid inequalities for the multi-item inventory lot-sizing problem with supplier selection. Comput Oper Res 130:105234
Chen Q, Li X, Ouyang Y (2011) Joint inventory-location problem under the risk of probabilistic facility disruptions. Transp Res Part B Methodol 45(7):991–1003
Christopher M (2005) Logistics and supply chain management: creating value-added networks. Prentice Hall, London, p 305
Coyle JJ, John J, Bardi EJ, Langley CJ (2003) The management of business logistics: a supply chain perspective. South-Western/Thomson Learning, Mason
Current J, Min H, Schilling D (1990) Multiobjective analysis of facility location decisions. Eur J Oper Res 49(3):295–307
Darmawan A, Wong H, Thorstenson A (2021) Supply chain network design with coordinated inventory control. Transp Res Part E Logist Transp Rev 145(2):102168
Daskin MS, Coullard CR, Shen ZJM (2002) An inventory-location model: formulation, solution algorithm and computational results. Ann Oper Res 110:83–106. https://doi.org/10.1023/A:1020763400324
de Treville S, Schürhoff N, Trigeorgis L, Avanzi B (2014) Optimal sourcing and lead-time reduction under evolutionary demand risk. Prod Oper Manag 23(12):2103–2117
de Vries H, van de Klundert J, Wagelmans APM (2020) The roadside healthcare facility location problem a managerial network design challenge. Prod Oper Manag 29(5):1165–1187
Diabat A, Battaïa O, Nazzal D (2015) An improved Lagrangian relaxation-based heuristic for a joint location-inventory problem. Comput Oper Res 61:170–178
Dönmez Z, Kara BY, Karsu Ö, Saldanha-da-Gama F (2021) Humanitarian facility location under uncertainty: critical review and future prospects. Omega 102:102393
Drezner T (2014) A review of competitive facility location in the plane. Logist Res 7(1):114
Eiselt HA, Marianov V (2011) Foundations of location analysis, vol 155. Springer, New York, NY
Eiselt HA, Marianov V (2015) Applications of location analysis, vol 232. Springer, Berlin
Emirhüseyinoğlu G, Ekici A (2019) Dynamic facility location with supplier selection under quantity discount. Comput Ind Eng 134(March):64–74
Erlebacher SJ, Meller RD (2000) The interaction of location and inventory in designing distribution systems. IIE Trans 32(2):155–166
Escalona P, Ordóñez F, Marianov V (2015) Joint location-inventory problem with differentiated service levels using critical level policy. Transp Res Part E Logist Transp Rev 83:141–157
Farahani RZ, Rashidi Bajgan H, Fahimnia B, Kaviani M (2015) Location-inventory problem in supply chains: a modelling review. Int J Prod Res 53(12):3769–3788
Farahani RZ, Fallah S, Ruiz R, Hosseini S, Asgari N (2019) OR models in urban service facility location: a critical review of applications and future developments. Eur J Oper Res 276(1):1–27
Feng Q, Shi R (2012) Sourcing from multiple suppliers for price-dependent demands. Prod Oper Manag 21(3):547–563
Gebennini E, Gamberini R, Manzini R (2009) An integrated production-distribution model for the dynamic location and allocation problem with safety stock optimization. Int J Prod Econ 122(1):286–304
Gen M, Syarif A (2005) Hybrid genetic algorithm for multi-time period production/distribution planning. Comput Ind Eng 48(4):799–809
Geoffrion AM (1972) Generalized benders decomposition. J Optim Theory Appl 10(4):237–260
Geoffrion A, McBride R (1978) Lagrangean relaxation applied to capacitated facility location problems. AIIE Trans 10(1):40–47
Georgiadis G, Rajaram K (2013) The retail planning problem under demand uncertainty. Prod Oper Manag 22(5):1200–1213
Ghassemi A, Asl-Najafi J, Yaghoubi S (2018) A dynamic bi-objective closed-loop supply chain network design considering supplier selection and remanufacturer subcontractors. Uncertain Supply Chain Manag 6:117–134
Guerrero WJ, Prodhon C, Velasco N, Amaya CA (2013) Hybrid heuristic for the inventory location-routing problem with deterministic demand. Int J Prod Econ 146(1):359–370
Guerrero Campanur A, Olivares-Benitez E, Miranda PA, Perez-Loaiza RE, Ablanedo-Rosas JH (2018) Design of a Logistics Nonlinear System for a Complex, Multiechelon, Supply Chain Network with Uncertain Demands. Complexity 2018:1–16
Hajipour V, Niaki STA, Akhgar M, Ansari M (2021) The healthcare supply chain network design with traceability: A novel algorithm. Comput Ind Eng 161(August):107661
Haq AN, Kannan G (2006) Design of an integrated supplier selection and multi-echelon distribution inventory model in a built-to-order supply chain environment. Int J Prod Res 44(10):1963–1985
Harris FW (1913) How many parts to make at once. Factory Mag Manag 10(2):135–136
Jayaraman V (1998) Transportation, facility location and inventory issues in distribution network design. Int J Oper Prod Manag 18(5):471–494
Khalilzadeh M, Derikvand H (2018) A multi-objective supplier selection model for green supply chain network under uncertainty. J Model Manag 13(3):605–625
Kirschstein T, Meisel F (2019) A multi-period multi-commodity lot-sizing problem with supplier selection, storage selection and discounts for the process industry. Eur J Oper Res 279(2):393–406
Kuderinova N, Rebezov M, Zhenzhebir V, Belousova M, Odinokova E, Burlankov P, Antonova V, Kushnir K, Rotanov E (2021) Optimizing the combined problem of facility location and multi-objective supplier selection using a comprehensive benchmarking method. Ind Eng Manag Syst 20(2):248–257
Kumar SK, Tiwari MK (2013) Supply chain system design integrated with risk pooling. Comput Ind Eng 64(2):580–588
Lee K, Ozsen L (2020) Tabu search heuristic for the network design model with lead time and safety stock considerations. Comput Ind Eng 148(August):106717
Liao Z, Rittscher J (2007) A multi-objective supplier selection model under stochastic demand conditions. Int J Prod Econ 105(1):150–159
Lin YH, Tian Q, Zhao Y (2022) Locating facilities under competition and market expansion: formulation, optimization, and implications. Prod Oper Manag 31(7):3021–3042
List GF, Mirchandani PB, Turnquist MA, Zografos KG (1991) Modeling and analysis for hazardous materials transportation. Risk analysis, routing/scheduling and facility location. Transp Sci 25(2):100–114
Liu Y, Dehghani E, Jabalameli MS, Diabat A, Lu CC (2020) A coordinated location-inventory problem with supply disruptions: a two-phase queuing theory–optimization model approach. Comput Ind Eng 142(December 2019):106326
Melo MT, Nickel S, Saldanha-da-Gama F (2009) Facility location and supply chain management—a review. Eur J Oper Res 196(2):401–412
Miranda PA, Garrido RA (2004) Incorporating inventory control decisions into a strategic distribution network design model with stochastic demand. Transp Res Part E Logist Transp Rev 40(3):183–207
Miranda PA, Garrido RA (2006) A simultaneous inventory control and facility location model with stochastic capacity constraints. Networks Spat Econ 6(1):39–53
Miranda PA, Garrido RA (2008) Valid inequalities for Lagrangian relaxation in an inventory location problem with stochastic capacity. Transp Res Part E Logist Transp Rev 44(1):47–65
Miranda PA, Garrido RA (2009) Inventory service-level optimization within distribution network design problem. Int J Prod Econ 122(1):276–285
Miranda PA, Tapia-Ubeda FJ, Hernandez V, Cardenas H, Lopez-Campos M (2019) A simulation based modelling approach to jointly support and evaluate spare parts supply chain network and maintenance system. IFAC-PapersOnLine 52(13):2231–2236
Miranda PA, Cabrera G (2010) Inventory location problem with stochastic capacity constraints under periodic review ( R , s , S ). In: International conference on industrial logistics - logistics and sustainability, pp 289–296.
Mota B, Gomes MI, Carvalho A, Barbosa-Povoa AP (2018) Sustainable supply chains: an integrated modeling approach under uncertainty. Omega 77:32–57
Mourits M, Evers JJM (1995) Distribution network design an integrated planning support framework. Int J Phys Distrib Logist Manag 25(5):43–57
Nasiri GR, Kalantari M, Karimi B (2021) Fast-moving consumer goods network design with pricing policy in an uncertain environment with correlated demands. Comput Ind Eng 153(November 2020):106997
Ortiz-Astorquiza C, Contreras I, Laporte G (2017) Formulations and approximation algorithms for multilevel uncapacitated facility location. INFORMS J Comput 29(4):767–779
Ortiz-Astorquiza C, Contreras I, Laporte G (2018) Multi-level facility location problems. Eur J Oper Res 267(3):791–805
Ortiz-Astorquiza C, Contreras I, Laporte G (2019) An exact algorithm for multilevel uncapacitated facility location. Transp Sci 53(4):1085–1106
Owen SH, Daskin MS (1998) Strategic facility location: a review. Eur J Oper Res 111(3):423–447
Ozsen L, Daskin MS, Coullard CR (2009) Facility location modeling and inventory management with multisourcing. Transp Sci 43(4):455–472
Perez Loaiza RE, Olivares-Benitez E, Miranda Gonzalez PA, Guerrero Campanur A, Martinez Flores JL (2017) Supply chain network design with efficiency, location, and inventory policy using a multiobjective evolutionary algorithm. Int Trans Oper Res 24(1–2):251–275
Porteus E (2002) Foundations of stochastic inventory theory, vol 26. Standford University Press, Redwood City
Pourhejazy P, Kwon O (2016) The new generation of operations research methods in supply chain optimization: a review. Sustainability 8(10):1033
ReVelle CS, Swain RW (1970) Central facilities location. Geogr Anal 2(1):30–42
Rienkhemaniyom K (2015) A multi-criteria model for supplier selection and supply chain network design. Chiang Mai Univ J Nat Sci 14(4):389–413
Ross A, Khajehnezhad M, Otieno W, Aydas O (2017) Integrated location-inventory modelling under forward and reverse product flows in the used merchandise retail sector: a multi-echelon formulation. Eur J Oper Res 259(2):664–676
Şahin G, Süral H (2007) A review of hierarchical facility location models. Comput Oper Res 34(8):2310–2331
Sajedinejad A, Chaharsooghi SK (2018) Multi-criteria supplier selection decisions in supply chain networks : a multi-objective optimization approach. Ind Eng Manag Syst 17(3):392–406
Shahabi M, Tafreshian A, Unnikrishnan A, Boyles SD (2018) Joint production–inventory–location problem with multi-variate normal demand. Transp Res Part B Methodol 110:60–78
Shavandi H, Bozorgi B. (n.d.) Developing a location-inventory model under fuzzy environment.
Shen Z-JM, Coullard C, Daskin MS (2003) A joint location-inventory model. Transp Sci 37(1):40–55
Simchi-Levi D, Kaminsky P, Simchi-Levi E (2003) Designing and managing the supply chain: concepts, strategies, and case studies. McGraw-Hill/Irwin, Irvine
Simchi-Levi D, Chen X, Bramel J (1997) The logic of logistics—simchi, Levi, Bramel.pdf.
Snyder LV (2006) Facility location under uncertainty: a review. IIE Trans Institute Ind Eng 38(7):547–564
Snyder LV, Daskin MS, Teo CP (2007) The stochastic location model with risk pooling. Eur J Oper Res 179(3):1221–1238
Tanonkou GA, Benyoucef L, Xie X (2006) Integrated facility location and supplier selection decisions in a distribution network design. In: 2006 IEEE international conference on service operations and logistics, and informatics, vol 44. IEEE, pp 399–404
Tapia-Ubeda FJ, Miranda PA, Macchi M (2018a) A Generalized benders decomposition based algorithm for an inventory location problem with stochastic inventory capacity constraints. Eur J Oper Res 267(3):806–817
Tapia-Ubeda FJ, Miranda PA, Roda I, Macchi M, Orlando D (2018b) An inventory-location modeling structure for spare parts supply chain network design problems in industrial end-user sites. IFAC-PapersOnLine 51(11):968–973
Tapia-Ubeda FJ, Miranda PA, Gutiérrez-Jarpa G, Durán O (2019) Supplier selection for spare parts supply chain networks. IFAC-PapersOnLine 52(13):2237–2242
Tapia-Ubeda FJ, Miranda PA, Roda I, Macchi M, Durán O (2020) Modelling and solving spare parts supply chain network design problems. Int J Prod Res 58(17):5299–5319
Tcha DW, Lee BI (1984) A branch-and-bound algorithm for the multi-level uncapacitated facility location problem. Eur J Oper Res 18(1):35–43
Thanh PN, Bostel N, Péton O (2008) A dynamic model for facility location in the design of complex supply chains. Int J Prod Econ 113(2):678–693
Tracey M, Lim JS, Vonderembse MA (2005) The impact of supply-chain management capabilities on business performance. Supply Chain Manag 10(3):179–191
Truong TH, Azadivar F (2005) Optimal design methodologies for configuration of supply chains. Int J Prod Res 43(11):2217–2236
Vanteddu G, Chinnam RB, Gushikin O (2011) Supply chain focus dependent supplier selection problem. Int J Prod Econ 129(1):204–216
Wang H, Alidaee B (2019) The multi-floor cross-dock door assignment problem: rising challenges for the new trend in logistics industry. Transp Res Part E Logist Transp Rev 132:30–47
White JA, Case KE (1974) On covering problems and the central facilities location problem. Geogr Anal 6(3):281–294
Wilson RH (1934) A scientific routine for stock control. Harv Bus Rev 13(1):116–128
Yao Z, Lee LH, Jaruphongsa W, Tan V, Hui CF (2010) Multi-source facility location–allocation and inventory problem. Eur J Oper Res 207(2):750–762
Yu H, Solvang WD, Li S (2014) An integrated optimization model for single-product supply chain network design considering supplier selection. In: 2014 5th IEEE conference on cognitive infocommunications (CogInfoCom). IEEE, pp 443–448
Zhang X, Li Z, Wang Y (2020) A review of the criteria and methods of reverse logistics supplier selection. Processes 8(6):705
Acknowledgements
This research work was performed by ANID FONDECYT grant number 1123046. In addition, this research was also supported by “Proyecto Iniciación a la Investigación 2022” of Universidad Católica del Norte grant number VRIDT No. 055/2022. Furthermore, this research work was supported partially by “Postdoctorado DI 2019” of Pontificia Universidad Católica de Valparaíso, Grant Agreement Number 37.0/2019.
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Appendices
Appendix A
This appendix provides a simple analysis to demonstrate the convenience of selecting a single supplier instead of two suppliers for serving a warehouse with known demand mean and variance. For this demonstration, we assume that, in cases of employing two suppliers with different lead times, the perceived lead time at the warehouse would be the average value, considering the proportion of demand served by each supplier as weighing parameter. Any other assumption may potently increase the perceived lead time at the warehouse, for example integrating the lead time variance, or considering the maximum lead time of the two values, yielding more advantages of selecting a single supplier.
Following the notation in Sect. 3, omitting the index i of the warehouse, and considering that θ is the proportion of demand served by supplier 2 respect to the demand served by supplier 1, the total costs perceived by the warehouse can be written as follows (all remaining costs does not depend on supplier selection):
This analysis aims at demonstrating that selecting one supplier (θ = 0 or θ = 1) always yield a lower cost compared to operating the two of them (0 < θ < 1). Without loss of generality, we assume that total cost of employing suppler 1 \((\theta = 0)\) is lower than selecting supplier 2 \((\theta = 1)\), i.e., the following expression holds.
The second derivative of G(θ) the expression is:
Which is allays negative when LT1 ≠ LT2 and the costs function is strictly concave. Accordingly, any value of 0 < θ < 1 will always provide a more expensive cost than θ = 0 and selecting suppler 1 is always better than employing the two suppliers.
Notice that:
-
When the lead times are the same, the selection would be driven by the transportation costs, and again one supplier will be better than two suppliers (except when the transportation costs are the same, where choosing one suppler is also optimum).
-
If fixed cost of selecting suppliers are added in to the analysis, the costs function when 0 < θ < 1 is a vertical shift up, which does not affect the obtained result.
Appendix B
In this appendix, a numerical example is presented considering 2 suppliers, 5 potential warehouses, and 5 customers, for which the proposed formulation is solved by using the proposed GBD-based algorithm. In this case, after integrating the five initial cuts (P = 1, 2, 3, 4, 5), the algorithm employed 8 iterations for obtaining the final optimal solution. The MP solutions for all iterations are presented in Fig.
30 (except for the iteration 4). In addition, Table 9 and Fig. 31 show information about solutions obtained at each iteration.
The next is the set of cuts integrated into the MP, including the first five cuts obtained with the artificial or modified MP (i.e., P = 1, 2, 3, 4, 5), and the seven cuts obtained after solving the respective SP at each iteration.
Appendix C
Base Instance 1 | Base Instance 2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
#Customers (nC) | #Customers (nC) | |||||||||
5 | 10 | 15 | 20 | 5 | 10 | 15 | 20 | |||
#Suppliers (nS) = 1 | #Warehouses (nW) = 5 | OF | 1,653,895.92 | 2,860,993.11 | 4,572,433.18 | 6,162,400.77 | 1,341,407.58 | 2,548,128.58 | 3,857,099.02 | 5,184,189.95 |
Time | 3.67 | 7.56 | 6.73 | 7.31 | 0.42 | 0.95 | 1.58 | 2.34 | ||
IT | 21 | 19 | 20 | 20 | 18 | 21 | 20 | 21 | ||
NSS | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
NLW | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ||
#Warehouses (nW) = 10 | OF | 1,653,895.92 | 2,860,993.11 | 4,572,433.18 | 6,162,400.77 | 1,334,676.43 | 2,523,856.89 | 3,833,785.20 | 5,173,655.55 | |
Time | 154.73 | 640.28 | 2,444.92 | 4,336.02 | 15.47 | 364.33 | 951.34 | 2,163.33 | ||
IT | 137 | 237 | 438 | 476 | 90 | 194 | 264 | 382 | ||
NSS | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
NLW | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 5 | ||
#Suppliers (nS) = 2 | #Warehouses (nW) = 5 | OF | 902,217.35 | 1,771,093.71 | 3,003,583.48 | 3,928,232.99 | 1,341,407.58 | 2,519,762.03 | 3,685,399.79 | 4,885,462.65 |
Time | 0.09 | 0.06 | 1.55 | 0.59 | 0.84 | 1.02 | 1.54 | 1.39 | ||
IT | 9 | 9 | 13 | 13 | 22 | 22 | 22 | 19 | ||
NSS | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | ||
NLW | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | ||
#Warehouses (nW) = 10 | OF | 711,512.07 | 1,331,319.76 | 2,339,769.96 | 2,995,884.98 | 1,264,357.39 | 2,237,598.46 | 3,246,475.83 | 4,154,951.58 | |
Time | 2.16 | 4.38 | 138.55 | 323.36 | 37.36 | 347.52 | 837.80 | 704.27 | ||
IT | 23 | 44 | 119 | 144 | 110 | 160 | 246 | 245 | ||
NSS | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ||
NLW | 2 | 2 | 2 | 2 | 3 | 5 | 4 | 6 | ||
#Suppliers (nS) = 3 | #Warehouses (nW) = 5 | OF | 902,217.35 | 1,771,093.71 | 3,003,583.48 | 3,928,232.99 | 1,127,656.11 | 2,116,536.19 | 3,269,642.37 | 4,373,393.58 |
Time | 0.02 | 0.08 | 0.72 | 0.34 | 0.30 | 0.28 | 0.61 | 1.73 | ||
IT | 9 | 9 | 13 | 13 | 16 | 15 | 18 | 17 | ||
NSS | 2 | 2 | 2 | 2 | 1 | 3 | 3 | 3 | ||
NLW | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ||
#Warehouses (nW) = 10 | OF | 711,512.07 | 1,331,319.76 | 2,339,769.96 | 2,995,884.98 | 1,064,691.21 | 1,884,622.88 | 2,888,657.25 | 3,687,404.66 | |
Time | 1.72 | 4.61 | 161.25 | 312.48 | 11.09 | 87.61 | 598.34 | 393.86 | ||
IT | 24 | 43 | 116 | 146 | 70 | 122 | 203 | 160 | ||
NSS | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ||
NLW | 2 | 2 | 2 | 2 | 3 | 5 | 4 | 5 | ||
#Suppliers (nS) = 4 | #Warehouses (nW) = 5 | OF | 902,217.35 | 1,771,093.71 | 3,003,583.48 | 3,928,232.99 | 1,114,799.06 | 2,080,184.58 | 3,233,290.76 | 4,329,027.10 |
Time | 0.05 | 0.11 | 0.56 | 0.55 | 0.28 | 0.16 | 0.89 | 1.22 | ||
IT | 9 | 10 | 13 | 13 | 14 | 15 | 18 | 17 | ||
NSS | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ||
NLW | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | ||
#Warehouses (nW) = 10 | OF | 711,512.07 | 1,331,319.76 | 2,339,769.96 | 2,995,884.98 | 1,051,834.15 | 1,849,241.84 | 2,844,935.85 | 3,635,912.71 | |
Time | 2.64 | 6.67 | 171.83 | 394.81 | 8.92 | 60.72 | 467.98 | 319.95 | ||
IT | 27 | 42 | 130 | 162 | 63 | 110 | 175 | 144 | ||
NSS | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ||
NLW | 2 | 2 | 2 | 2 | 3 | 5 | 4 | 5 | ||
#Suppliers (nS) = 5 | #Warehouses (nW) = 5 | OF | 796,223.69 | 1,588,750.60 | 2,715,683.21 | 3,534,363.11 | 1,114,799.06 | 2,064,311.57 | 3,182,573.28 | 4,260,916.25 |
Time | 0.16 | 0.44 | 0.41 | 0.89 | 0.19 | 0.31 | 0.64 | 0.77 | ||
IT | 8 | 9 | 11 | 13 | 14 | 15 | 16 | 15 | ||
NSS | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ||
NLW | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | ||
#Warehouses (nW) = 10 | OF | 686,625.76 | 1,295,891.76 | 2,246,120.38 | 2,883,470.01 | 1,051,834.15 | 1,847,216.18 | 2,837,172.63 | 3,635,912.71 | |
Time | 2.38 | 4.25 | 105.48 | 239.78 | 12.80 | 64.20 | 498.11 | 435.16 | ||
IT | 27 | 41 | 108 | 141 | 70 | 113 | 178 | 157 | ||
NSS | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | ||
NLW | 3 | 3 | 4 | 4 | 3 | 5 | 5 | 5 |
Base Instance 3 | Base Instance 4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
#Customers (nC) | #Customers (nC) | |||||||||
5 | 10 | 15 | 20 | 5 | 10 | 15 | 20 | |||
#Suppliers (nS) = 1 | #Warehouses (nW) = 5 | OF | 995,112.73 | 2,515,167.29 | 4,200,934.27 | 5,258,949.57 | 1,238,599.35 | 2,832,975.75 | 4,325,107.64 | 5,424,879.07 |
Time | 0.16 | 0.44 | 0.75 | 2.03 | 0.30 | 0.14 | 1.20 | 1.75 | ||
IT | 12 | 16 | 19 | 19 | 13 | 13 | 17 | 17 | ||
NSS | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
NLW | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ||
#Warehouses (nW) = 10 | OF | 947,693.14 | 2,429,290.47 | 4,037,254.73 | 5,065,233.24 | 1,057,054.89 | 2,046,949.19 | 3,262,833.68 | 4,198,252.61 | |
Time | 3.92 | 367.42 | 1,088.72 | 1,667.41 | 10.47 | 38.22 | 254.28 | 352.20 | ||
IT | 63 | 208 | 296 | 320 | 73 | 84 | 137 | 160 | ||
NSS | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
NLW | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | ||
#Suppliers (nS) = 2 | #Warehouses (nW) = 5 | OF | 911,715.73 | 2,339,536.84 | 3,759,968.83 | 4,775,702.42 | 1,104,478.66 | 2,715,635.28 | 4,066,848.29 | 4,977,718.31 |
Time | 0.11 | 0.95 | 0.59 | 1.14 | 0.72 | 0.98 | 3.28 | 1.34 | ||
IT | 13 | 19 | 17 | 18 | 22 | 22 | 24 | 22 | ||
NSS | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ||
NLW | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | ||
#Warehouses (nW) = 10 | OF | 872,213.99 | 2,250,428.74 | 3,618,518.00 | 4,606,189.29 | 966,004.74 | 1,982,957.17 | 3,146,368.06 | 3,959,410.78 | |
Time | 4.48 | 415.91 | 567.47 | 782.55 | 15.53 | 46.00 | 305.33 | 254.34 | ||
IT | 59 | 189 | 184 | 204 | 88 | 97 | 173 | 163 | ||
NSS | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ||
NLW | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ||
#Suppliers (nS) = 3 | #Warehouses (nW) = 5 | OF | 712,827.76 | 1,802,342.72 | 2,797,581.10 | 3,669,264.02 | 1,104,478.66 | 2,715,635.28 | 4,066,848.29 | 4,977,718.31 |
Time | 0.09 | 0.14 | 0.30 | 0.44 | 0.77 | 1.05 | 2.41 | 1.80 | ||
IT | 9 | 12 | 12 | 12 | 22 | 21 | 22 | 21 | ||
NSS | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | ||
NLW | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ||
#Warehouses (nW) = 10 | OF | 646,117.64 | 1,611,471.85 | 2,551,847.98 | 3,369,148.32 | 923,324.60 | 1,939,914.68 | 2,914,593.43 | 3,715,791.39 | |
Time | 2.97 | 27.83 | 43.02 | 139.97 | 14.38 | 36.25 | 35.56 | 44.16 | ||
IT | 32 | 70 | 82 | 103 | 76 | 85 | 77 | 71 | ||
NSS | 2 | 2 | 2 | 2 | 3 | 2 | 3 | 3 | ||
NLW | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | ||
#Suppliers (nS) = 4 | #Warehouses (nW) = 5 | OF | 712,827.76 | 1,802,342.72 | 2,797,581.10 | 3,669,264.02 | 1,028,530.32 | 2,464,786.74 | 3,704,009.89 | 4,617,788.52 |
Time | 0.05 | 0.19 | 0.33 | 0.16 | 0.48 | 1.31 | 1.50 | 2.38 | ||
IT | 9 | 12 | 12 | 12 | 19 | 17 | 19 | 20 | ||
NSS | 2 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | ||
NLW | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ||
#Warehouses (nW) = 10 | OF | 646,417.64 | 1,611,471.85 | 2,551,847.98 | 3,369,148.32 | 691,545.15 | 1,318,757.45 | 2,018,605.18 | 2,610,180.59 | |
Time | 2.75 | 26.53 | 44.48 | 143.33 | 2.81 | 4.58 | 8.33 | 15.55 | ||
IT | 32 | 70 | 82 | 103 | 40 | 51 | 47 | 44 | ||
NSS | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | ||
NLW | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | ||
#Suppliers (nS) = 5 | #Warehouses (nW) = 5 | OF | 712,827.76 | 1,802,342.72 | 2,797,581.10 | 3,669,264.02 | 997,555.19 | 2,359,739.83 | 3,574,912.38 | 4,482,864.95 |
Time | 0.05 | 0.33 | 0.34 | 0.27 | 0.28 | 0.70 | 1.91 | 1.69 | ||
IT | 9 | 12 | 12 | 12 | 17 | 18 | 20 | 18 | ||
NSS | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ||
NLW | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ||
#Warehouses (nW) = 10 | OF | 641,973.51 | 1,610,049.60 | 2,538,434.70 | 3,357,335.96 | 691,545.15 | 1,318,757.45 | 2,018,605.18 | 2,610,180.59 | |
Time | 2.91 | 29.30 | 52.64 | 130.23 | 2.72 | 5.88 | 7.22 | 16.08 | ||
IT | 32 | 72 | 77 | 104 | 43 | 49 | 40 | 43 | ||
NSS | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ||
NLW | 4 | 5 | 5 | 5 | 3 | 3 | 3 | 3 |
Base Instance 5 | ||||||
---|---|---|---|---|---|---|
#Customers (nC) | ||||||
5 | 10 | 15 | 20 | |||
#Suppliers (nS) = 1 | #Warehouses (nW) = 5 | OF | 2,178,244.28 | 4,328,128.83 | 6,116,851.14 | 8,124,073.96 |
Time | 0.53 | 1.77 | 7.83 | 6.84 | ||
IT | 15 | 18 | 19 | 19 | ||
NSS | 1 | 1 | 1 | 1 | ||
NLW | 1 | 2 | 2 | 2 | ||
#Warehouses (nW) = 10 | OF | 1,663,819.94 | 3,331,834.97 | 4,812,300.25 | 6,098,933.05 | |
Time | 29.98 | 675.22 | 1,665.34 | 1,786.20 | ||
IT | 94 | 260 | 348 | 311 | ||
NSS | 1 | 1 | 1 | 1 | ||
NLW | 2 | 3 | 4 | 4 | ||
#Suppliers (nS) = 2 | #Warehouses (nW) = 5 | OF | 1,935,285.14 | 3,788,038.48 | 5,311,398.25 | 7,204,257.02 |
Time | 0.66 | 3.77 | 7.09 | 4.11 | ||
IT | 19 | 23 | 23 | 22 | ||
NSS | 1 | 1 | 1 | 1 | ||
NLW | 2 | 2 | 2 | 2 | ||
#Warehouses (nW) = 10 | OF | 1,293,913.57 | 2,617,644.85 | 3,838,034.86 | 4,996,407.90 | |
Time | 9.78 | 159.97 | 502.75 | 457.14 | ||
IT | 52 | 95 | 172 | 146 | ||
NSS | 2 | 2 | 2 | 2 | ||
NLW | 2 | 3 | 4 | 4 | ||
#Suppliers (nS) = 3 | #Warehouses (nW) = 5 | OF | 1,140,038.99 | 2,283,616.46 | 3,058,355.76 | 4,034,370.07 |
Time | 0.20 | 0.50 | 0.48 | 0.58 | ||
IT | 10 | 15 | 15 | 15 | ||
NSS | 1 | 1 | 1 | 1 | ||
NLW | 1 | 2 | 2 | 2 | ||
#Warehouses (nW) = 10 | OF | 1,076,136.77 | 2,137,499.81 | 2,860,586.43 | 3,696,629.23 | |
Time | 3.38 | 65.48 | 194.55 | 60.67 | ||
IT | 32 | 98 | 115 | 102 | ||
NSS | 2 | 3 | 3 | 3 | ||
NLW | 2 | 3 | 3 | 3 | ||
#Suppliers (nS) = 4 | #Warehouses (nW) = 5 | OF | 1,140,038.99 | 2,255,318.32 | 3,015,394.18 | 3,991,818.42 |
Time | 0.08 | 0.41 | 0.92 | 1.66 | ||
IT | 10 | 15 | 17 | 17 | ||
NSS | 1 | 2 | 2 | 2 | ||
NLW | 1 | 2 | 2 | 2 | ||
#Warehouses (nW) = 10 | OF | 964,500.00 | 1,925,217.76 | 2,593,771.64 | 3,430,290.72 | |
Time | 1.41 | 9.94 | 24.81 | 9.19 | ||
IT | 20 | 53 | 57 | 51 | ||
NSS | 2 | 3 | 3 | 3 | ||
NLW | 2 | 3 | 3 | 3 | ||
#Suppliers (nS) = 5 | #Warehouses (nW) = 5 | OF | 1,140,038.99 | 2,255,318.32 | 3,015,394.18 | 3,991,818.42 |
Time | 0.06 | 0.56 | 1.69 | 1.41 | ||
IT | 10 | 15 | 17 | 17 | ||
NSS | 1 | 2 | 2 | 2 | ||
NLW | 1 | 2 | 2 | 2 | ||
#Warehouses (nW) = 10 | OF | 964,500.00 | 1,925,217.76 | 2,593,771.64 | 3,430,290.72 | |
Time | 1.78 | 10.31 | 26.23 | 9.16 | ||
IT | 20 | 53 | 57 | 51 | ||
NSS | 2 | 3 | 3 | 3 | ||
NLW | 2 | 3 | 3 | 3 |
Appendix D
AVERAGE | ||||||
---|---|---|---|---|---|---|
#Customers (nC) | ||||||
5 | 10 | 15 | 20 | |||
#Suppliers (nS) = 1 | #Warehouses (nW) = 5 | OF | 1.481.451,97 | 3.017.078,71 | 4.614.485,05 | 6.030.898,67 |
Time | 1,02 | 2,17 | 3,62 | 4,06 | ||
IT | 15,80 | 17,40 | 19,00 | 19,20 | ||
NSS | 1,00 | 1,00 | 1,00 | 1,00 | ||
NLW | 1,40 | 2,00 | 2,20 | 2,40 | ||
#Warehouses (nW) = 10 | OF | 1.331.428,07 | 2.638.584,93 | 4.103.721,41 | 5.339.695,04 | |
Time | 42,92 | 417,09 | 1.280,92 | 2.061,03 | ||
IT | 91,40 | 196,60 | 296,60 | 329,80 | ||
NSS | 1,00 | 1,00 | 1,00 | 1,00 | ||
NLW | 2,00 | 2,20 | 2,60 | 3,20 | ||
#Suppliers (nS) = 2 | #Warehouses (nW) = 5 | OF | 1.239.020,89 | 2.626.813,27 | 3.965.439,73 | 5.154.274,68 |
Time | 0,48 | 1,36 | 2,81 | 1,72 | ||
IT | 17,00 | 19,00 | 19,80 | 18,80 | ||
NSS | 1,60 | 1,80 | 1,80 | 1,80 | ||
NLW | 2,20 | 3,00 | 3,00 | 3,20 | ||
#Warehouses (nW) = 10 | OF | 1.021.600,35 | 2.083.989,80 | 3.237.833,34 | 4.142.568,91 | |
Time | 13,86 | 194,75 | 470,38 | 504,33 | ||
IT | 66,40 | 117,00 | 178,80 | 180,40 | ||
NSS | 2,00 | 2,00 | 2,00 | 2,00 | ||
NLW | 2,60 | 3,20 | 3,20 | 3,60 | ||
#Suppliers (nS) = 3 | #Warehouses (nW) = 5 | OF | 997.443,77 | 2.137.844,87 | 3.239.202,20 | 4.196.595,79 |
Time | 0,28 | 0,41 | 0,90 | 0,98 | ||
IT | 13,20 | 14,40 | 13,80 | 15,60 | ||
NSS | 1,60 | 2,00 | 2,00 | 2,20 | ||
NLW | 2,00 | 2,80 | 3,00 | 3,00 | ||
#Warehouses (nW) = 10 | OF | 884.356,46 | 1.780.965,80 | 2.711.091,01 | 3.492.971,72 | |
Time | 6,71 | 44,36 | 206,54 | 190,23 | ||
IT | 46,80 | 83,60 | 118,60 | 116,40 | ||
NSS | 2,20 | 2,40 | 2,60 | 2,60 | ||
NLW | 2,80 | 3,40 | 3,20 | 3,40 | ||
#Suppliers (nS) = 4 | #Warehouses (nW) = 5 | OF | 979.682,70 | 2.074.745,21 | 3.150.771,88 | 4.107.226,21 |
Time | 0,19 | 0,43 | 0,84 | 1,19 | ||
IT | 12,20 | 13,80 | 15,80 | 15,80 | ||
NSS | 1,80 | 2,60 | 2,60 | 2,80 | ||
NLW | 2,00 | 2,80 | 2,80 | 3,20 | ||
#Warehouses (nW) = 10 | OF | 813.161,80 | 1.607.201,73 | 2.469.786,12 | 3.208.283,47 | |
Time | 3,71 | 21,69 | 143,49 | 176,57 | ||
IT | 36,40 | 65,20 | 98,20 | 100,80 | ||
NSS | 2,40 | 2,80 | 2,80 | 2,80 | ||
NLW | 2,80 | 3,40 | 3,20 | 3,40 | ||
#Suppliers (nS) = 5 | #Warehouses (nW) = 5 | OF | 952.288,94 | 2.014.092,61 | 3.057.228,83 | 3.987.845,35 |
Time | 0,15 | 0,47 | 1,00 | 1,00 | ||
IT | 11,60 | 13,80 | 15,20 | 15,00 | ||
NSS | 2,00 | 2,60 | 2,60 | 2,80 | ||
NLW | 2,00 | 2,60 | 2,60 | 3,20 | ||
#Warehouses (nW) = 10 | OF | 807.295,71 | 1.599.426,55 | 2.446.820,91 | 3.183.438,00 | |
Time | 4,52 | 22,79 | 137,94 | 166,08 | ||
IT | 38,40 | 65,60 | 92,00 | 99,20 | ||
NSS | 2,80 | 3,20 | 3,20 | 3,20 | ||
NLW | 3,00 | 3,80 | 4,00 | 4,00 |
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Tapia-Ubeda, F.J., Miranda-Gonzalez, P.A. & Gutiérrez-Jarpa, G. Integrating supplier selection decisions into an inventory location problem for designing the supply chain network. J Comb Optim 47, 2 (2024). https://doi.org/10.1007/s10878-023-01100-y
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DOI: https://doi.org/10.1007/s10878-023-01100-y