A bi-objective integrated mathematical model for blood supply chain: Case of Turkish red crescent

Benhür Satır; Vahdi Yolcu

doi:10.3934/jimo.2022090

Article Contents

2023, Volume 19, Issue 5: 3383-3418. Doi: 10.3934/jimo.2022090

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A bi-objective integrated mathematical model for blood supply chain: Case of Turkish red crescent

Benhür Satır^1, , and
Vahdi Yolcu^2,

1.
Department of Industrial Engineering, Çankaya University, Ankara, Turkey
2.
TRT General Directorate, Ankara, Turkey

^*Corresponding author: Benhür Satır
^*Corresponding author: Benhür Satır

Received: November 2021

Revised: April 2022

Early access: June 2022

Published: May 2023

Abstract / Introduction Full Text(HTML) Figure(10) / Table(10) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. Production of Blood Products and Blood Supply Chain Structure of TRC

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Figure 2. Linear relaxation-based heuristic

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Figure 3. BDC and RBC locations of TRC in Turkey

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Figure 4. Optimal BDC - RBC assignments of COST Model and TIME Model

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Figure 5. CT and TC solutions

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Figure 6. CT and TC' solutions

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Figure 7. TC' solutions considered by Decision Maker

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Figure 8. Solutions under Current Assignments (CA)

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Figure 9. Total and Marginal Costs w.r.t. $ \pi _{\alpha =1} $

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Figure 10. Percentage of TS demand met by apheresis donations and average $ x_{k} $

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Table 1. Survey on selected related works, with a comparison to the current study

Reference	BSC Echelons				Objective(s)		Decisions			Solution
Reference	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.

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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

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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)

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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)

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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

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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

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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

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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

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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

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Table 10. Ranges for $ \tau $ and $ \pi $

OBJ.	MODELS			% deviation
OBJ.	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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