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doi: 10.3934/jimo.2022090
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A bi-objective integrated mathematical model for blood supply chain: Case of Turkish red crescent

1. 

Department of Industrial Engineering, Çankaya University, Ankara, Turkey

2. 

TRT General Directorate, Ankara, Turkey

*Corresponding author: Benhür Satır

Received  November 2021 Revised  April 2022 Early access June 2022

Various criteria feature in blood supply chain (BSC) designs, where cost-based and time-based are the most commonly found in the literature. In the current study, total annual cost is used together with a new time-based objective. The total time spent in the transportation of blood products is considered as time lost, and weight is given to that time according to the product amount and then normalized with respect to shelf life. In using cost and time objectives, we developed a bi-objective mixed-integer mathematical programming model for the BSC of Turkish Red Crescent (TRC, the singular authority controlling BSC throughout Turkey), including collection, production, and distribution echelons, and also considering bag-type decisions for whole-blood collection. The objective of the study was to propose a BSC design model and solution approach. With all real-life TRC instances resolved optimally, a linear programming relaxation-based heuristic was developed for large-scale problem sizes. Real-life data were obtained from the TRC and the remainder from open-to-public sources. The study's main finding is that cost and time objectives alone produce significantly different designs, whilst using them together to form efficient-frontier solutions for decision-makers adds practical value.

Citation: Benhür Satır, Vahdi Yolcu. A bi-objective integrated mathematical model for blood supply chain: Case of Turkish red crescent. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022090
References:
[1]

M. AraniY. ChanX. Liu and M. Momenitabar, A lateral resupply blood supply chain network design under uncertainties, Applied Mathematical Modelling, 93 (2021), 165-187.  doi: 10.1016/j.apm.2020.12.010.

[2]

A. M. AraújoD. SantosI. Marques and A. Barbosa-Povoa, Blood supply chain: A two-stage approach for tactical and operational planning, OR Spectrum, 42 (2020), 1023-1053.  doi: 10.1007/s00291-020-00600-1.

[3]

M. ArvanR. Tavakkoli-Moghaddam and M. Abdollahi, Designing a bi-objective and multi-product supply chain network for the supply of blood, Uncertain Supply Chain Management, 3 (2015), 57-68. 

[4]

M. Y. N. AttariS. H. R. Pasandideh and S. T. Akhavan Niaki, A hybrid robust stochastic programming for a bi-objective blood collection facilities problem (Case study: Iranian blood transfusion network), Journal of Industrial and Production Engineering, 36 (2019), 154-167. 

[5]

G. BrunoA. DiglioC. Piccolo and L. Cannavacciuolo, Territorial reorganization of regional blood management systems: Evidences from an Italian case study, Omega, 89 (2019), 54-70. 

[6]

H. M. Dilaver, A Mathematical Modeling Approach for Managing Regional Blood Bank Operations, M. Sc. thesis, Bilkent University in Ankara, 2018.

[7]

M. Eskandari-KhanghahiR. Tavakkoli-MoghaddamA. A. Taleizadeh and S. H. Amin, Designing and optimizing a sustainable supply chain network for a blood platelet bank under uncertainty, Engineering Applications of Artificial Intelligence, 71 (2018), 236-250. 

[8]

B. FahimniaA. JabbarzadehA. Ghavamifar and M. Bell, Supply chain design for efficient and effective blood supply in disasters, International Journal of Production Economics, 183 (2017), 700-709. 

[9]

M. Fazli-KhalafS. Khalilpourazari and M. Mohammadi, Mixed robust possibilistic flexible chance constraint optimization model for emergency blood supply chain network design, Annals of Operations Research, 283 (2019), 1079-1109.  doi: 10.1007/s10479-017-2729-3.

[10]

S. B. Ghorashi, M. Hamedi and R. Sadeghian, Modeling and optimization of a reliable blood supply chain network in crisis considering blood compatibility using MOGWO, Neural Computing and Applications, (2019), 1–28.

[11]

Habertürk, Corona and Ramadan Reduced Daily Donations from 9 Thousand Units to 2 Thousand, News of Habertürk, 2020. Available from: https://www.haberturk.com/korona-ve-ramazan-birlesince-gunluk-kan-bagisi-9-bin-uniteden-2-bin-e-dustu-haberler-2664729.

[12]

N. HaghjooR. Tavakkoli-MoghaddamH. Shahmoradi-Moghadam and Y. Rahimi, Reliable blood supply chain network design with facility disruption: A real-world application, Engineering Applications of Artificial Intelligence, 90 (2020), 103493. 

[13]

S. M. Hosseini-Motlagh, M. R. G. Samani and S. Cheraghi, Robust and stable flexible blood supply chain network design under motivational initiatives, Socio-Economic Planning Sciences, (2020), 70.

[14]

S. M. Hosseini-MotlaghM. R. G. Samani and S. Cheraghi, Robust and stable flexible blood supply chain network design under motivational initiatives, Socio-Economic Planning Sciences, 70 (2020), 100725. 

[15]

S. M. Hosseini-MotlaghM. R. G. Samani and S. Homaei, Blood supply chain management: Robust optimization, disruption risk, and blood group compatibility (a real-life case), Journal of Ambient Intelligence and Humanized Computing, 11 (2020), 1085-1104. 

[16]

İ KaradaǧM. E. Keskin and V. Yiǧ it, Re-design of a blood supply chain organization with mobile units, Soft Computing, 25 (2021), 6311-6327. 

[17]

S. Khalilpourazari and A. Arshadi Khamseh, Bi-objective emergency blood supply chain network design in earthquake considering earthquake magnitude: A comprehensive study with real world application, Annals of Operations Research, 283 (2019), 355-393.  doi: 10.1007/s10479-017-2588-y.

[18]

S. KhalilpourazariS. SoltanzadehG. W. Weber and S. K. Roy, Designing an efficient blood supply chain network in crisis: Neural learning, optimization and case study, Annals of Operations Research, 289 (2020), 123-152.  doi: 10.1007/s10479-019-03437-2.

[19]

R. LotfiY. Z. MehrjerdiM. S. PishvaeeA. Sadeghieh and G. W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization, 11 (2021), 221.  doi: 10.3934/naco.2020023.

[20]

R. Lotfi, S. Safavi, A. Gharehbaghi, S. Ghaboulian Zare, R. Hazrati and G. W. Weber, Viable supply chain network design by considering blockchain technology and cryptocurrency, Mathematical Problems in Engineering, (2021).

[21]

R. Lotfi, B. Kargar, A. Gharehbaghi and G. W. Weber, Viable medical waste chain network design by considering risk and robustness, Environmental Science and Pollution Research, (2021), 1–16.

[22]

R. LotfiN. Mardani and G. W. Weber, Robust bi-level programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534. 

[23]

R. LotfiB. KargarS. H. HoseiniS. NazariS. Safavi and G. W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, International Journal of Energy Research, 45 (2021), 1-18. 

[24]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G. W. Weber, A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project, Journal of Industrial & Management Optimization, (2020). doi: 10.3934/jimo. 2020158.

[25]

G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in multi-objective mathematical programming problems, Applied mathematics and computation, 213 (2009), 455-465.  doi: 10.1016/j.amc.2009.03.037.

[26]

A. NagurneyA. H. Masoumi and M. Yu, Supply chain network operations management of a blood banking system with cost and risk minimization, Computational Management Science, 9 (2012), 205-231.  doi: 10.1007/s10287-011-0133-z.

[27]

J. Nahofti Kohneh, H. Derikvand, M. Amirdadi and E. Teimoury, A blood supply chain network design with interconnected and motivational strategies: A case study, Journal of Ambient Intelligence and Humanized Computing, (2021), 1–21.

[28]

NTV, Reduced Blood Stocks Caused TRC to Offer VIP Service to Donors, News of NTV, 2020. Available from: https://www.ntv.com.tr/galeri/saglik/kan-stoklari-azaldi-kizilay-bagiscilara-vip-hizmet-baslatti,d-XQI2jsHk6VBnoVXd1xRQ/EMpFQyAyoUuOP_sDU5AjXQ.

[29]

A. F. OsorioS. C. Brailsford and H. K. Smith, A structured review of quantitative models in the blood supply chain: a taxonomic framework for decision making, International Journal of Production Research, 53 (2015), 7191-7212. 

[30]

A. F. OsorioS. C. BrailsfordH. K. Smith and J. Blake, Designing the blood supply chain: How much, how and where?, Vox sanguinis, 113 (2018), 760-769. 

[31]

D. Rahmani, Designing a robust and dynamic network for the emergency blood supply chain with the risk of disruptions, Annals of Operations Research, 283 (2019), 613-641.  doi: 10.1007/s10479-018-2960-6.

[32]

G. ȘahinH. Süral and S. Meral, Locational analysis for regionalization of Turkish Red Crescent blood services, Computers & Operations Research, 34 (2007), 692-704. 

[33]

M. R. G. SamaniS. M. Hosseini-Motlagh and S. F. Ghannadpour, A multilateral perspective towards blood network design in an uncertain environment: Methodology and implementation, Computers & Industrial Engineering, 130 (2019), 450-471. 

[34]

S. A. Seyfi-Shishavan, Y. Donyatalab, E. Farrokhizadeh and S. I. Satoglu, A fuzzy optimization model for designing an efficient blood supply chain network under uncertainty and disruption, Annals of Operations Research, (2021), 1–55.

[35]

P. N. ThanhO. Péton and N. Bostel, A linear relaxation-based heuristic approach for logistics network design, Computers & Industrial Engineering, 59 (2010), 964-975. 

[36]

A. S. Torrado and A. Barbosa-Póvoa, Towards an optimized and sustainable blood supply chain network under uncertainty: A literature review, Cleaner Logistics and Supply Chain, (2022). doi: 10.1007/s00291-020-00600-1.

[37]

Turkish Rec Crescent, Emergency Blood Call, News of Turkish Rec Crescent, 2020. Available from: https://www.kizilay.org.tr/Haber/KurumsalHaberDetay/5116.

[38]

M. Yegül, Simulation Analysis of the Blood Supply Chain and a Case Study, M. Sc. thesis, Middle East Technical University in Ankara, 2007.

[39]

M. Yegül, Blood Supply Network Design, Ph. D thesis, Middle East Technical University in Ankara, 2016.

[40]

V. Yolcu, Logistics Management for Blood Collection and Blood Products Distribution in Turkish Red Crescent, M. Sc. thesis, Çankaya University in Ankara, 2019.

[41]

B. ZahiriS. A. TorabiM. Mohammadi and M. Aghabegloo, A multi-stage stochastic programming approach for blood supply chain planning, Computers & Industrial Engineering, 122 (2018), 1-14. 

[42]

Y. Zhou, T. Zou, C. Liu, H. Yu, L. Chen and J. Su, Blood supply chain operation considering lifetime and transshipment under uncertain environment, Applied Soft Computing, (2021), 106.

show all references

References:
[1]

M. AraniY. ChanX. Liu and M. Momenitabar, A lateral resupply blood supply chain network design under uncertainties, Applied Mathematical Modelling, 93 (2021), 165-187.  doi: 10.1016/j.apm.2020.12.010.

[2]

A. M. AraújoD. SantosI. Marques and A. Barbosa-Povoa, Blood supply chain: A two-stage approach for tactical and operational planning, OR Spectrum, 42 (2020), 1023-1053.  doi: 10.1007/s00291-020-00600-1.

[3]

M. ArvanR. Tavakkoli-Moghaddam and M. Abdollahi, Designing a bi-objective and multi-product supply chain network for the supply of blood, Uncertain Supply Chain Management, 3 (2015), 57-68. 

[4]

M. Y. N. AttariS. H. R. Pasandideh and S. T. Akhavan Niaki, A hybrid robust stochastic programming for a bi-objective blood collection facilities problem (Case study: Iranian blood transfusion network), Journal of Industrial and Production Engineering, 36 (2019), 154-167. 

[5]

G. BrunoA. DiglioC. Piccolo and L. Cannavacciuolo, Territorial reorganization of regional blood management systems: Evidences from an Italian case study, Omega, 89 (2019), 54-70. 

[6]

H. M. Dilaver, A Mathematical Modeling Approach for Managing Regional Blood Bank Operations, M. Sc. thesis, Bilkent University in Ankara, 2018.

[7]

M. Eskandari-KhanghahiR. Tavakkoli-MoghaddamA. A. Taleizadeh and S. H. Amin, Designing and optimizing a sustainable supply chain network for a blood platelet bank under uncertainty, Engineering Applications of Artificial Intelligence, 71 (2018), 236-250. 

[8]

B. FahimniaA. JabbarzadehA. Ghavamifar and M. Bell, Supply chain design for efficient and effective blood supply in disasters, International Journal of Production Economics, 183 (2017), 700-709. 

[9]

M. Fazli-KhalafS. Khalilpourazari and M. Mohammadi, Mixed robust possibilistic flexible chance constraint optimization model for emergency blood supply chain network design, Annals of Operations Research, 283 (2019), 1079-1109.  doi: 10.1007/s10479-017-2729-3.

[10]

S. B. Ghorashi, M. Hamedi and R. Sadeghian, Modeling and optimization of a reliable blood supply chain network in crisis considering blood compatibility using MOGWO, Neural Computing and Applications, (2019), 1–28.

[11]

Habertürk, Corona and Ramadan Reduced Daily Donations from 9 Thousand Units to 2 Thousand, News of Habertürk, 2020. Available from: https://www.haberturk.com/korona-ve-ramazan-birlesince-gunluk-kan-bagisi-9-bin-uniteden-2-bin-e-dustu-haberler-2664729.

[12]

N. HaghjooR. Tavakkoli-MoghaddamH. Shahmoradi-Moghadam and Y. Rahimi, Reliable blood supply chain network design with facility disruption: A real-world application, Engineering Applications of Artificial Intelligence, 90 (2020), 103493. 

[13]

S. M. Hosseini-Motlagh, M. R. G. Samani and S. Cheraghi, Robust and stable flexible blood supply chain network design under motivational initiatives, Socio-Economic Planning Sciences, (2020), 70.

[14]

S. M. Hosseini-MotlaghM. R. G. Samani and S. Cheraghi, Robust and stable flexible blood supply chain network design under motivational initiatives, Socio-Economic Planning Sciences, 70 (2020), 100725. 

[15]

S. M. Hosseini-MotlaghM. R. G. Samani and S. Homaei, Blood supply chain management: Robust optimization, disruption risk, and blood group compatibility (a real-life case), Journal of Ambient Intelligence and Humanized Computing, 11 (2020), 1085-1104. 

[16]

İ KaradaǧM. E. Keskin and V. Yiǧ it, Re-design of a blood supply chain organization with mobile units, Soft Computing, 25 (2021), 6311-6327. 

[17]

S. Khalilpourazari and A. Arshadi Khamseh, Bi-objective emergency blood supply chain network design in earthquake considering earthquake magnitude: A comprehensive study with real world application, Annals of Operations Research, 283 (2019), 355-393.  doi: 10.1007/s10479-017-2588-y.

[18]

S. KhalilpourazariS. SoltanzadehG. W. Weber and S. K. Roy, Designing an efficient blood supply chain network in crisis: Neural learning, optimization and case study, Annals of Operations Research, 289 (2020), 123-152.  doi: 10.1007/s10479-019-03437-2.

[19]

R. LotfiY. Z. MehrjerdiM. S. PishvaeeA. Sadeghieh and G. W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization, 11 (2021), 221.  doi: 10.3934/naco.2020023.

[20]

R. Lotfi, S. Safavi, A. Gharehbaghi, S. Ghaboulian Zare, R. Hazrati and G. W. Weber, Viable supply chain network design by considering blockchain technology and cryptocurrency, Mathematical Problems in Engineering, (2021).

[21]

R. Lotfi, B. Kargar, A. Gharehbaghi and G. W. Weber, Viable medical waste chain network design by considering risk and robustness, Environmental Science and Pollution Research, (2021), 1–16.

[22]

R. LotfiN. Mardani and G. W. Weber, Robust bi-level programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534. 

[23]

R. LotfiB. KargarS. H. HoseiniS. NazariS. Safavi and G. W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, International Journal of Energy Research, 45 (2021), 1-18. 

[24]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G. W. Weber, A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project, Journal of Industrial & Management Optimization, (2020). doi: 10.3934/jimo. 2020158.

[25]

G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in multi-objective mathematical programming problems, Applied mathematics and computation, 213 (2009), 455-465.  doi: 10.1016/j.amc.2009.03.037.

[26]

A. NagurneyA. H. Masoumi and M. Yu, Supply chain network operations management of a blood banking system with cost and risk minimization, Computational Management Science, 9 (2012), 205-231.  doi: 10.1007/s10287-011-0133-z.

[27]

J. Nahofti Kohneh, H. Derikvand, M. Amirdadi and E. Teimoury, A blood supply chain network design with interconnected and motivational strategies: A case study, Journal of Ambient Intelligence and Humanized Computing, (2021), 1–21.

[28]

NTV, Reduced Blood Stocks Caused TRC to Offer VIP Service to Donors, News of NTV, 2020. Available from: https://www.ntv.com.tr/galeri/saglik/kan-stoklari-azaldi-kizilay-bagiscilara-vip-hizmet-baslatti,d-XQI2jsHk6VBnoVXd1xRQ/EMpFQyAyoUuOP_sDU5AjXQ.

[29]

A. F. OsorioS. C. Brailsford and H. K. Smith, A structured review of quantitative models in the blood supply chain: a taxonomic framework for decision making, International Journal of Production Research, 53 (2015), 7191-7212. 

[30]

A. F. OsorioS. C. BrailsfordH. K. Smith and J. Blake, Designing the blood supply chain: How much, how and where?, Vox sanguinis, 113 (2018), 760-769. 

[31]

D. Rahmani, Designing a robust and dynamic network for the emergency blood supply chain with the risk of disruptions, Annals of Operations Research, 283 (2019), 613-641.  doi: 10.1007/s10479-018-2960-6.

[32]

G. ȘahinH. Süral and S. Meral, Locational analysis for regionalization of Turkish Red Crescent blood services, Computers & Operations Research, 34 (2007), 692-704. 

[33]

M. R. G. SamaniS. M. Hosseini-Motlagh and S. F. Ghannadpour, A multilateral perspective towards blood network design in an uncertain environment: Methodology and implementation, Computers & Industrial Engineering, 130 (2019), 450-471. 

[34]

S. A. Seyfi-Shishavan, Y. Donyatalab, E. Farrokhizadeh and S. I. Satoglu, A fuzzy optimization model for designing an efficient blood supply chain network under uncertainty and disruption, Annals of Operations Research, (2021), 1–55.

[35]

P. N. ThanhO. Péton and N. Bostel, A linear relaxation-based heuristic approach for logistics network design, Computers & Industrial Engineering, 59 (2010), 964-975. 

[36]

A. S. Torrado and A. Barbosa-Póvoa, Towards an optimized and sustainable blood supply chain network under uncertainty: A literature review, Cleaner Logistics and Supply Chain, (2022). doi: 10.1007/s00291-020-00600-1.

[37]

Turkish Rec Crescent, Emergency Blood Call, News of Turkish Rec Crescent, 2020. Available from: https://www.kizilay.org.tr/Haber/KurumsalHaberDetay/5116.

[38]

M. Yegül, Simulation Analysis of the Blood Supply Chain and a Case Study, M. Sc. thesis, Middle East Technical University in Ankara, 2007.

[39]

M. Yegül, Blood Supply Network Design, Ph. D thesis, Middle East Technical University in Ankara, 2016.

[40]

V. Yolcu, Logistics Management for Blood Collection and Blood Products Distribution in Turkish Red Crescent, M. Sc. thesis, Çankaya University in Ankara, 2019.

[41]

B. ZahiriS. A. TorabiM. Mohammadi and M. Aghabegloo, A multi-stage stochastic programming approach for blood supply chain planning, Computers & Industrial Engineering, 122 (2018), 1-14. 

[42]

Y. Zhou, T. Zou, C. Liu, H. Yu, L. Chen and J. Su, Blood supply chain operation considering lifetime and transshipment under uncertain environment, Applied Soft Computing, (2021), 106.

Figure 1.  Production of Blood Products and Blood Supply Chain Structure of TRC
Figure 2.  Linear relaxation-based heuristic
Figure 3.  BDC and RBC locations of TRC in Turkey
Figure 4.  Optimal BDC - RBC assignments of COST Model and TIME Model
Figure 5.  CT and TC solutions
Figure 6.  CT and TC' solutions
Figure 7.  TC' solutions considered by Decision Maker
Figure 8.  Solutions under Current Assignments (CA)
Figure 9.  Total and Marginal Costs w.r.t. $ \pi _{\alpha =1} $
Figure 10.  Percentage of TS demand met by apheresis donations and average $ x_{k} $
Table 1.  Survey on selected related works, with a comparison to the current study
Reference BSC Echelons Objective(s) Decisions Solution
Collection Production Inventory Distribution Single/Multi Aggregated Bag type WB test Lateral transport. Real-life case Heuristic developed
[14] Y Y Y Y S —— Y N N Y Y
[32] N N N Y M Y N N N Y N
[5] Y N N N S —— N N N Y N
[39] Y Y Y Y S —— N N N Y Y
[33] Y Y Y Y M Y N Y N Y N
[10] Y Y Y Y M Y N Y N N Y
[4] Y Y N Y M Y N Y N Y N
[16] Y N N Y M Y N N N Y N
[41] Y Y Y Y M N N N Y N Y
[1] Y Y Y Y M Y N N Y N N
[2] Y Y Y Y S —— N N N Y Y
[3] Y Y N Y M N N Y Y N N
Our study Y Y N Y M N Y Y Y Y Y
Y: Yes, N: No, M: Multi, S: Single, ——: Not applicable.
Reference BSC Echelons Objective(s) Decisions Solution
Collection Production Inventory Distribution Single/Multi Aggregated Bag type WB test Lateral transport. Real-life case Heuristic developed
[14] Y Y Y Y S —— Y N N Y Y
[32] N N N Y M Y N N N Y N
[5] Y N N N S —— N N N Y N
[39] Y Y Y Y S —— N N N Y Y
[33] Y Y Y Y M Y N Y N Y N
[10] Y Y Y Y M Y N Y N N Y
[4] Y Y N Y M Y N Y N Y N
[16] Y N N Y M Y N N N Y N
[41] Y Y Y Y M N N N Y N Y
[1] Y Y Y Y M Y N N Y N N
[2] Y Y Y Y S —— N N N Y Y
[3] Y Y N Y M N N Y Y N N
Our study Y Y N Y M N Y Y Y Y Y
Y: Yes, N: No, M: Multi, S: Single, ——: Not applicable.
Table 2.  Usage and shelf lives of blood products
Product Treatment Case Shelf Life
ES Erythrocyte suspension Surgery with major blood loss, treatment of anemic patients, premature infants 42 days
TS Thrombocyte suspension Major blood loss, cancer treatment 5 days
FFP Fresh frozen plasma Blood loss and curbs in surgery, treatment of liver disease, treatment of burn injuries 2 years
Product Treatment Case Shelf Life
ES Erythrocyte suspension Surgery with major blood loss, treatment of anemic patients, premature infants 42 days
TS Thrombocyte suspension Major blood loss, cancer treatment 5 days
FFP Fresh frozen plasma Blood loss and curbs in surgery, treatment of liver disease, treatment of burn injuries 2 years
Table 3.  Notations
Notations for the COST Model:
        Superscripts:
AS Apheresis Set
BC Buffy Coat
ES Erythrocyte Suspension
FFP Fresh Frozen Plasma
JH Relation between an RBC and a TFC
JJ Relation between one RBC and another RBC
JK Relation between an RBC and a BDC
JL Relation between an RBC and a CL
LP Liquid Plasma
TB Top & Bottom (bag type)
TS Thrombocyte Suspension
TS-A Thrombocyte Suspension obtained by apheresis donation
TT Top & Top (bag type)
        Indices:
$ i $ Cities in Turkey $ i \in S^{I}=\lbrace 1,2,\ldots ,I\rbrace $
$ j $ Regional blood centers (RBCs) $ j \in S^{J}=\lbrace j:j=1,2,\ldots, J\rbrace $
$ k $ Blood donation centers (BDCs) $ k \in S^{K}=\lbrace k:k=1,2,\ldots, K\rbrace $
$ h $ Transfusion centers (TFCs) $ h \in S^{H}=\lbrace h:h=1,2,\ldots ,H\rbrace $
$ l $ Central laboratories (CLs) $ l \in S^{L}=\lbrace l:l=1,2,\ldots ,L\rbrace $
$ t $ Transportation modes $ t \in S^{T}=\lbrace t:t=1,2,\ldots ,T\rbrace $
$ S_{i}^{H} $ Set of TFCs located within a city i $ S_{i}^{H} \subset S^{H} $
        Parameters:
            Donation amounts:
$ B_{k} $ Annual donation amount in BDC k
$ A_{k} $ Annual apheresis donation amount in BDC k
            WB and BP bag amounts:
$ ABB $ Average amount of blood in a whole blood bag (liters/bag)
$ AB^{BC} $ Average amount of BC in one ready-for-sale bag (liters/bag)
$ AB^{TS} $ Average amount of TS in one ready-for-sale bag (liters/bag)
$ AB^{TS-A} $ Average amount of TS (obtained by apheresis donation) in one ready-for-sale bag (liters/bag)
$ AB^{LP} $ Average amount of LP in one ready-for-sale bag (liters/bag)
$ AB^{FFP} $ Average amount of FFP in one ready-for-sale bag (liters/bag)
$ AB^{ES} $ Average amount of ES in one ready-for-sale bag (liters/bag)
            BP production amounts:
$ TT^{ES} $ Average amount of ES produced with one T & T bag (liters)
$ TT……{LP} $ Average amount of LP produced with one T & T bag (liters)
$ TB^{ES} $ Average amount of ES produced with one T & B bag (liters)
$ TB^{LP} $ Average amount of LP produced with one T & B bag (liters)
$ TB^{BC} $ Average amount of BC produced with one T & B bag (liters)
$ AS^{TS} $ Average amount of TS produced with one apheresis set (liters)
            BP demand amounts:
$ D_{h}^{TS} $ Annual TS demand at TM h (liters/year)
$ D_{h}^{ES} $ Annual ES demand at TM h (liters/year)
$ D_{h}^{FFP} $ Annual FFP demand at TM h (liters/year)
            Transportation costs: (Note: TL refers to Turkish Lira; ISO code: TRY)
$ CT_{jkt}^{JK} $ Unit cost of transportation from RBC j to BDC k with mode t (TL/liter/kilometer)
$ CT_{j'jt}^{JJ} $ Unit cost of transportation from RBC j' to RBC j with mode t (TL/liter/kilometer)
$ CT_{jht}^{JH} $ Unit cost of transportation from RBC j to TFC h with mode t (TL/liter/kilometer
$ CT_{jlt}^{JL} $ Unit cost of transportation from RBC j to CL l with mode t (TL/liter/kilometer)
            Blood bag costs:
$ CB^{TT} $ T & T bag cost (TL/bag)
$ CB^{TB} $ T & B bag cost (TL/bag)
$ CB^{AS} $ Apheresis set cost (TL/set)
            Operation and production costs:
$ CO^{TT} $ Estimated operational cost of whole blood donation with T & T bag (TL/bag)
$ CO^{TB} $ Estimated operational cost of whole blood donation with T & B bag (TL/bag)
$ CP^{ES} $ Estimated production cost of ES (TL/bag)
$ CP^{LP} $ Estimated production cost of LP (TL/bag)
$ CP^{BC} $ Estimated production cost of BC (TL/bag)
$ CP^{FFP} $ Estimated production cost of FFP (TL/bag)
$ CP^{TS} $ Estimated production cost of TS (TL/bag)
            Destruction costs:
$ CD^{TT} $ Estimated destruction cost of whole blood in T & T bag (TL/bag)
$ CD^{TB} $ Estimated destruction cost of whole blood in T & B bag (TL/bag)
$ CD^{AS} $ Estimated destruction cost of apheresis (TL/set)
$ CD^{ES} $ Estimated destruction cost of ES (TL/bag)
$ CD^{LP} $ Estimated destruction cost of LP (TL/bag)
$ CD^{BC} $ Estimated destruction cost of BC (TL/bag)
$ CD^{FFP} $ Estimated destruction cost of FFP (TL/bag)
$ CD^{TS} $ Estimated destruction cost of TS (TL/bag)
            Lot size:
$ LS_{j} $ Average lot size of blood products in a vehicle per TFC shipment j dispatched from an RBC via ground transportation mode
            Distances:
$ CL_{kj} $ 1, if BDC k is close to RBC j; 0, otherwise
$ DS_{jkt}^{JK} $ Distance from BDC k to RBC j with mode t (kilometers)
$ DS_{j'jt}^{JJ} $ Distance from RBC j to RBC j' with mode t (kilometers)
$ DS_{jht}^{JH} $ Distance from TFC h to RBC j with mode t (kilometers)
$ DS_{jlt}^{JL} $ Distance from CL l to RBC j with mode t (kilometers)
            Decision variables:
$ x_{k} $ Rate of whole blood donations collected with T & B bag in BDC k
$ y_{kjt} $ 1, if BDC k is assigned to RBC j with mode t; 0, otherwise
$ z_{ijt} $ 1, if TFCs in city i are assigned to RBC j with mode t; 0, otherwise
$ u_{jlt} $ 1, if RBC j is assigned to CL l with mode t; 0, otherwise
$ p_{k}^{BC} $ Annual production of BC in BDC k (liters/year)
$ pd_{k}^{BC} $ Annual BC produced and destroyed in BDC k (liters/year)
$ p_{k}^{TS-BC} $ Annual production of TS (from BC) in BDC k (liters/year)
$ p_{k}^{TS} $ Annual production of TS in BDC k (liters/year)
$ pd_{k}^{TS} $ Annual TS produced and destroyed in BDC k (liters/year)
$ p_{k}^{TS-A} $ Annual production of TS (with apheresis) in BDC k (liters/year)
$ pd_{k}^{TS-A} $ Annual TS produced (with apheresis) and destroyed in BDC k (liters/year)
$ p_{k}^{LP} $ Annual production of LP in BDC k (liters/year)
$ pd_{k}^{LP} $ Annual LP produced and destroyed in BDC k (liters/year)
$ p_{k}^{FFP} $ Annual production of FFP in BDC k (liters/year)
$ pd_{k}^{FFP} $ Annual FFP produced and destroyed in BDC k (liters/year)
$ p_{k}^{ES} $ Annual production of ES in BDC k (liters/year)
$ pd_{k}^{ES} $ Annual ES produced and destroyed in BDC k (liters/year)
$ s_{jj't}^{TS} $ Annual lateral transshipment amount of TS from RBC j to RBC j' with mode t (liters/year)
$ s_{jj't}^{ES} $ Annual lateral transshipment amount of ES from RBC j to RBC j' with mode t (liters/year)
$ s_{jj't}^{FFP} $ Annual lateral transshipment amount of FFP from RBC j to RBC j' with mode t (liters/year)
$ ps_{kjt}^{ES} $ Annual ES produced from donated whole blood at BDC k and sent to RBC j with mode t (and then used to satisfy demand) (liters/year)
$ psd_{kjt}^{ES} $ Annual ES produced from donated whole blood at BDC k and sent to RBC j with mode t (but then destroyed at RBC j) (liters/year)
$ ps_{kjt}^{TS} $ Annual TS produced from donated whole blood at BDC k and sent to RBC j with mode t (and then used to satisfy demand) (liters/year)
$ psd_{kjt}^{TS} $ Annual TS produced from donated whole blood at BDC k and sent to RBC j with mode t (but then destroyed at RBC j) (liters/year)
$ ps_{kjt}^{FFP} $ Annual FFP produced from donated whole blood at BDC k and sent to RBC j with mode t (and then used to satisfy demand) (liters/year)
$ psd_{kjt}^{FFP} $ Annual FFP produced from donated whole blood at BDC k and sent to RBC j with mode t (but then destroyed at RBC j) (liters/year)
$ \pi $ Total annual BSC cost (TL)
Notations for the COST Model:
        Superscripts:
AS Apheresis Set
BC Buffy Coat
ES Erythrocyte Suspension
FFP Fresh Frozen Plasma
JH Relation between an RBC and a TFC
JJ Relation between one RBC and another RBC
JK Relation between an RBC and a BDC
JL Relation between an RBC and a CL
LP Liquid Plasma
TB Top & Bottom (bag type)
TS Thrombocyte Suspension
TS-A Thrombocyte Suspension obtained by apheresis donation
TT Top & Top (bag type)
        Indices:
$ i $ Cities in Turkey $ i \in S^{I}=\lbrace 1,2,\ldots ,I\rbrace $
$ j $ Regional blood centers (RBCs) $ j \in S^{J}=\lbrace j:j=1,2,\ldots, J\rbrace $
$ k $ Blood donation centers (BDCs) $ k \in S^{K}=\lbrace k:k=1,2,\ldots, K\rbrace $
$ h $ Transfusion centers (TFCs) $ h \in S^{H}=\lbrace h:h=1,2,\ldots ,H\rbrace $
$ l $ Central laboratories (CLs) $ l \in S^{L}=\lbrace l:l=1,2,\ldots ,L\rbrace $
$ t $ Transportation modes $ t \in S^{T}=\lbrace t:t=1,2,\ldots ,T\rbrace $
$ S_{i}^{H} $ Set of TFCs located within a city i $ S_{i}^{H} \subset S^{H} $
        Parameters:
            Donation amounts:
$ B_{k} $ Annual donation amount in BDC k
$ A_{k} $ Annual apheresis donation amount in BDC k
            WB and BP bag amounts:
$ ABB $ Average amount of blood in a whole blood bag (liters/bag)
$ AB^{BC} $ Average amount of BC in one ready-for-sale bag (liters/bag)
$ AB^{TS} $ Average amount of TS in one ready-for-sale bag (liters/bag)
$ AB^{TS-A} $ Average amount of TS (obtained by apheresis donation) in one ready-for-sale bag (liters/bag)
$ AB^{LP} $ Average amount of LP in one ready-for-sale bag (liters/bag)
$ AB^{FFP} $ Average amount of FFP in one ready-for-sale bag (liters/bag)
$ AB^{ES} $ Average amount of ES in one ready-for-sale bag (liters/bag)
            BP production amounts:
$ TT^{ES} $ Average amount of ES produced with one T & T bag (liters)
$ TT……{LP} $ Average amount of LP produced with one T & T bag (liters)
$ TB^{ES} $ Average amount of ES produced with one T & B bag (liters)
$ TB^{LP} $ Average amount of LP produced with one T & B bag (liters)
$ TB^{BC} $ Average amount of BC produced with one T & B bag (liters)
$ AS^{TS} $ Average amount of TS produced with one apheresis set (liters)
            BP demand amounts:
$ D_{h}^{TS} $ Annual TS demand at TM h (liters/year)
$ D_{h}^{ES} $ Annual ES demand at TM h (liters/year)
$ D_{h}^{FFP} $ Annual FFP demand at TM h (liters/year)
            Transportation costs: (Note: TL refers to Turkish Lira; ISO code: TRY)
$ CT_{jkt}^{JK} $ Unit cost of transportation from RBC j to BDC k with mode t (TL/liter/kilometer)
$ CT_{j'jt}^{JJ} $ Unit cost of transportation from RBC j' to RBC j with mode t (TL/liter/kilometer)
$ CT_{jht}^{JH} $ Unit cost of transportation from RBC j to TFC h with mode t (TL/liter/kilometer
$ CT_{jlt}^{JL} $ Unit cost of transportation from RBC j to CL l with mode t (TL/liter/kilometer)
            Blood bag costs:
$ CB^{TT} $ T & T bag cost (TL/bag)
$ CB^{TB} $ T & B bag cost (TL/bag)
$ CB^{AS} $ Apheresis set cost (TL/set)
            Operation and production costs:
$ CO^{TT} $ Estimated operational cost of whole blood donation with T & T bag (TL/bag)
$ CO^{TB} $ Estimated operational cost of whole blood donation with T & B bag (TL/bag)
$ CP^{ES} $ Estimated production cost of ES (TL/bag)
$ CP^{LP} $ Estimated production cost of LP (TL/bag)
$ CP^{BC} $ Estimated production cost of BC (TL/bag)
$ CP^{FFP} $ Estimated production cost of FFP (TL/bag)
$ CP^{TS} $ Estimated production cost of TS (TL/bag)
            Destruction costs:
$ CD^{TT} $ Estimated destruction cost of whole blood in T & T bag (TL/bag)
$ CD^{TB} $ Estimated destruction cost of whole blood in T & B bag (TL/bag)
$ CD^{AS} $ Estimated destruction cost of apheresis (TL/set)
$ CD^{ES} $ Estimated destruction cost of ES (TL/bag)
$ CD^{LP} $ Estimated destruction cost of LP (TL/bag)
$ CD^{BC} $ Estimated destruction cost of BC (TL/bag)
$ CD^{FFP} $ Estimated destruction cost of FFP (TL/bag)
$ CD^{TS} $ Estimated destruction cost of TS (TL/bag)
            Lot size:
$ LS_{j} $ Average lot size of blood products in a vehicle per TFC shipment j dispatched from an RBC via ground transportation mode
            Distances:
$ CL_{kj} $ 1, if BDC k is close to RBC j; 0, otherwise
$ DS_{jkt}^{JK} $ Distance from BDC k to RBC j with mode t (kilometers)
$ DS_{j'jt}^{JJ} $ Distance from RBC j to RBC j' with mode t (kilometers)
$ DS_{jht}^{JH} $ Distance from TFC h to RBC j with mode t (kilometers)
$ DS_{jlt}^{JL} $ Distance from CL l to RBC j with mode t (kilometers)
            Decision variables:
$ x_{k} $ Rate of whole blood donations collected with T & B bag in BDC k
$ y_{kjt} $ 1, if BDC k is assigned to RBC j with mode t; 0, otherwise
$ z_{ijt} $ 1, if TFCs in city i are assigned to RBC j with mode t; 0, otherwise
$ u_{jlt} $ 1, if RBC j is assigned to CL l with mode t; 0, otherwise
$ p_{k}^{BC} $ Annual production of BC in BDC k (liters/year)
$ pd_{k}^{BC} $ Annual BC produced and destroyed in BDC k (liters/year)
$ p_{k}^{TS-BC} $ Annual production of TS (from BC) in BDC k (liters/year)
$ p_{k}^{TS} $ Annual production of TS in BDC k (liters/year)
$ pd_{k}^{TS} $ Annual TS produced and destroyed in BDC k (liters/year)
$ p_{k}^{TS-A} $ Annual production of TS (with apheresis) in BDC k (liters/year)
$ pd_{k}^{TS-A} $ Annual TS produced (with apheresis) and destroyed in BDC k (liters/year)
$ p_{k}^{LP} $ Annual production of LP in BDC k (liters/year)
$ pd_{k}^{LP} $ Annual LP produced and destroyed in BDC k (liters/year)
$ p_{k}^{FFP} $ Annual production of FFP in BDC k (liters/year)
$ pd_{k}^{FFP} $ Annual FFP produced and destroyed in BDC k (liters/year)
$ p_{k}^{ES} $ Annual production of ES in BDC k (liters/year)
$ pd_{k}^{ES} $ Annual ES produced and destroyed in BDC k (liters/year)
$ s_{jj't}^{TS} $ Annual lateral transshipment amount of TS from RBC j to RBC j' with mode t (liters/year)
$ s_{jj't}^{ES} $ Annual lateral transshipment amount of ES from RBC j to RBC j' with mode t (liters/year)
$ s_{jj't}^{FFP} $ Annual lateral transshipment amount of FFP from RBC j to RBC j' with mode t (liters/year)
$ ps_{kjt}^{ES} $ Annual ES produced from donated whole blood at BDC k and sent to RBC j with mode t (and then used to satisfy demand) (liters/year)
$ psd_{kjt}^{ES} $ Annual ES produced from donated whole blood at BDC k and sent to RBC j with mode t (but then destroyed at RBC j) (liters/year)
$ ps_{kjt}^{TS} $ Annual TS produced from donated whole blood at BDC k and sent to RBC j with mode t (and then used to satisfy demand) (liters/year)
$ psd_{kjt}^{TS} $ Annual TS produced from donated whole blood at BDC k and sent to RBC j with mode t (but then destroyed at RBC j) (liters/year)
$ ps_{kjt}^{FFP} $ Annual FFP produced from donated whole blood at BDC k and sent to RBC j with mode t (and then used to satisfy demand) (liters/year)
$ psd_{kjt}^{FFP} $ Annual FFP produced from donated whole blood at BDC k and sent to RBC j with mode t (but then destroyed at RBC j) (liters/year)
$ \pi $ Total annual BSC cost (TL)
Table 4.  Additional Notations
Additional Notations for the TIME Model:
        Parameters:
            Shelf-life:
$ SL^{TS} $ Shelf-life of TS(minutes)
$ SL^{ES} $ Shelf-life of ES(minutes)
$ SL^{FFP} $ Shelf-life of FFP(minutes)
            Transportation Times:
$ TR_{jkt}^{JK} $ Average time of transportation from RBC j to BDC k with mode t (minutes)
$ TR_{j'jt}^{JJ} $ Average time of transportation from RBC j' to RBC j with mode t (minutes)
$ TR_{jht}^{JH} $ Average time of transportation from RBC j to TFC h with mode t (minutes)
        Decision variables
$ w^{TS-JK} $ Total annual weighted-time in transportation of TS from BDC k to RBC j (liters*minutes)
$ w^{TS-JJ} $ Total annual weighted-time in transportation of TS from RBC j' to RBC j (liters*minutes)
$ w^{TS-JH} $ Total annual weighted-time in transportation of TS from RBC j to TFC h (liters*minutes)
$ w^{ES-JK} $ Total annual weighted-time in transportation of ES from BDC k to RBC j (liters*minutes)
$ w^{ES-JJ} $ Total annual weighted-time in transportation of ES from RBC j' to RBC j (liters*minutes)
$ w^{ES-JH} $ Total annual weighted-time in transportation of ES from RBC j to TFC h (liters*minutes)
$ w^{FFP-JK} $ Total annual weighted-time in transportation of FFP from BDC k to RBC j (liters*minutes)
$ w^{FFP-JJ} $ Total annual weighted-time in transportation of FFP from RBC j' to RBC j (liters*minutes)
$ w^{FFP-JH} $ Total annual weighted-time in transportation of FFP from RBC j to TFC h (liters*minutes)
$ \omega^{TS} $ Total annual weighted-shelf-life in transportation of TS (liters*shelf-life)
$ \omega^{ES} $ Total annual weighted-shelf-life in transportation of ES (liters*shelf-life)
$ \omega^{FFP} $ Total annual weighted-shelf-life in transportation of FFP (liters*shelf-life)
$ \tau $ Total annual weighted-shelf-life in transportation (liters*shelf-life)
Additional Notations for the TIME Model:
        Parameters:
            Shelf-life:
$ SL^{TS} $ Shelf-life of TS(minutes)
$ SL^{ES} $ Shelf-life of ES(minutes)
$ SL^{FFP} $ Shelf-life of FFP(minutes)
            Transportation Times:
$ TR_{jkt}^{JK} $ Average time of transportation from RBC j to BDC k with mode t (minutes)
$ TR_{j'jt}^{JJ} $ Average time of transportation from RBC j' to RBC j with mode t (minutes)
$ TR_{jht}^{JH} $ Average time of transportation from RBC j to TFC h with mode t (minutes)
        Decision variables
$ w^{TS-JK} $ Total annual weighted-time in transportation of TS from BDC k to RBC j (liters*minutes)
$ w^{TS-JJ} $ Total annual weighted-time in transportation of TS from RBC j' to RBC j (liters*minutes)
$ w^{TS-JH} $ Total annual weighted-time in transportation of TS from RBC j to TFC h (liters*minutes)
$ w^{ES-JK} $ Total annual weighted-time in transportation of ES from BDC k to RBC j (liters*minutes)
$ w^{ES-JJ} $ Total annual weighted-time in transportation of ES from RBC j' to RBC j (liters*minutes)
$ w^{ES-JH} $ Total annual weighted-time in transportation of ES from RBC j to TFC h (liters*minutes)
$ w^{FFP-JK} $ Total annual weighted-time in transportation of FFP from BDC k to RBC j (liters*minutes)
$ w^{FFP-JJ} $ Total annual weighted-time in transportation of FFP from RBC j' to RBC j (liters*minutes)
$ w^{FFP-JH} $ Total annual weighted-time in transportation of FFP from RBC j to TFC h (liters*minutes)
$ \omega^{TS} $ Total annual weighted-shelf-life in transportation of TS (liters*shelf-life)
$ \omega^{ES} $ Total annual weighted-shelf-life in transportation of ES (liters*shelf-life)
$ \omega^{FFP} $ Total annual weighted-shelf-life in transportation of FFP (liters*shelf-life)
$ \tau $ Total annual weighted-shelf-life in transportation (liters*shelf-life)
Table 5.  Optimal objective function values for COST and TIME Models
MODEL OBJECTIVE FUNCTION OPTIMAL VALUE
COST $ \pi $ 3, 229.71
TIME $ \tau $ 1, 278.74
MODEL OBJECTIVE FUNCTION OPTIMAL VALUE
COST $ \pi $ 3, 229.71
TIME $ \tau $ 1, 278.74
Table 6.  Ranges for $ \tau $ and $ \pi $
OBJECTIVE FUNCTION VALUE
MODEL $ \pi $ $ \tau $
COST 3, 229.71 4, 064.39
TIME 3, 751.98 1, 278.74
OBJECTIVE FUNCTION VALUE
MODEL $ \pi $ $ \tau $
COST 3, 229.71 4, 064.39
TIME 3, 751.98 1, 278.74
Table 7.  CT and TC solutions
COST GIVEN TIME
(CT) RESULTS
TIME GIVEN COST
(TC) RESULTS
# $ \pi $ $ \tau $ # $ \pi $ $ \tau $
CT1 3, 751.98 1, 278.74 TC1 4, 064.40 3, 229.71
CT2 3, 263.53 1, 557.30 TC2 1, 284.55 3, 281.93
CT3 3, 260.05 1, 835.87 TC3 1, 278.84 3, 334.16
CT4 3, 258.72 2, 114.44 TC4 1, 278.86 3, 386.39
CT5 3, 257.55 2, 393.00 TC5 1, 279.15 3, 438.62
CT6 3, 252.06 2, 671.57 TC6 1, 278.74 3, 490.84
CT7 3, 249.53 2, 950.13 TC7 1, 278.74 3, 543.07
CT8 3, 243.35 3, 228.70 TC8 1, 278.74 3, 595.30
CT9 3, 242.91 3, 507.27 TC9 1, 278.74 3, 647.52
CT10 3, 235.74 3, 785.83 TC10 1, 278.74 3, 699.75
CT11 3, 229.71 4, 064.40 TC11 1, 278.74 3, 751.98
COST GIVEN TIME
(CT) RESULTS
TIME GIVEN COST
(TC) RESULTS
# $ \pi $ $ \tau $ # $ \pi $ $ \tau $
CT1 3, 751.98 1, 278.74 TC1 4, 064.40 3, 229.71
CT2 3, 263.53 1, 557.30 TC2 1, 284.55 3, 281.93
CT3 3, 260.05 1, 835.87 TC3 1, 278.84 3, 334.16
CT4 3, 258.72 2, 114.44 TC4 1, 278.86 3, 386.39
CT5 3, 257.55 2, 393.00 TC5 1, 279.15 3, 438.62
CT6 3, 252.06 2, 671.57 TC6 1, 278.74 3, 490.84
CT7 3, 249.53 2, 950.13 TC7 1, 278.74 3, 543.07
CT8 3, 243.35 3, 228.70 TC8 1, 278.74 3, 595.30
CT9 3, 242.91 3, 507.27 TC9 1, 278.74 3, 647.52
CT10 3, 235.74 3, 785.83 TC10 1, 278.74 3, 699.75
CT11 3, 229.71 4, 064.40 TC11 1, 278.74 3, 751.98
Table 8.  TC' solutions
TIME GIVEN COST
(TC') RESULTS
# $ \pi $ $ \tau $
TC'1 3, 229.71 4, 064.40
TC'2 3, 234.93 1, 518.23
TC'3 3, 240.15 1, 419.26
TC'4 3, 245.37 1, 340.95
TC'5 3, 250.60 1, 335.11
TC'6 3, 255.82 1, 329.18
TC'7 3, 261.04 1, 328.35
TC'8 3, 266.26 1, 309.58
TC'9 3, 271.49 1, 300.34
TC'10 3, 276.71 1, 299.13
TC'11 3, 281.93 1, 284.55
TIME GIVEN COST
(TC') RESULTS
# $ \pi $ $ \tau $
TC'1 3, 229.71 4, 064.40
TC'2 3, 234.93 1, 518.23
TC'3 3, 240.15 1, 419.26
TC'4 3, 245.37 1, 340.95
TC'5 3, 250.60 1, 335.11
TC'6 3, 255.82 1, 329.18
TC'7 3, 261.04 1, 328.35
TC'8 3, 266.26 1, 309.58
TC'9 3, 271.49 1, 300.34
TC'10 3, 276.71 1, 299.13
TC'11 3, 281.93 1, 284.55
Table 9.  Ranges for $ \tau $ and $ \pi $
OBJECTIVE FUNCTION VALUE
MODEL $ \pi $ $ \tau $
TIME-CA 3, 415.29 1, 361.12
COST-CA 3, 341.05 1, 409.78
OBJECTIVE FUNCTION VALUE
MODEL $ \pi $ $ \tau $
TIME-CA 3, 415.29 1, 361.12
COST-CA 3, 341.05 1, 409.78
Table 10.  Ranges for $ \tau $ and $ \pi $
OBJ. MODELS % deviation
TC'8 COST-CA TIME-CA w.r.t.
COST-CA
w.r.t.
TIME-CA
$ \pi $ 3, 266.26 3, 341.05 3, 415.29 2.29% 4.56%
$ \tau $ 1, 295.93 1, 409.78 1, 361.12 8.79% 5.03%
OBJ. MODELS % deviation
TC'8 COST-CA TIME-CA w.r.t.
COST-CA
w.r.t.
TIME-CA
$ \pi $ 3, 266.26 3, 341.05 3, 415.29 2.29% 4.56%
$ \tau $ 1, 295.93 1, 409.78 1, 361.12 8.79% 5.03%
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