An accelerated benders decomposition algorithm for sustainable supply chain network design under uncertainty: A case study of medical needle and syringe supply chain
Mir Saman Pishvaee, Jafar Razmi, Seyed Ali Torabi
Electronic supplementary material
Part I. GSCALP categories and subcategories vs. concerned network design decisions
Concerned network design decisions / Impactsubcategories / Stakeholder categories /
Facility location / Capacity / Technology selection / Aggregate material flow /
PC / CC / PC / CC / PC / Amount of Production / Amount of delivered
products to customers / Amount of collected
products / Amount of EOL products processed by each EOL option /
Freedom of Association and Collective Bargaining / Worker
Child Labour
Fair Salary
Working Hours
Forced Labour
Equal opportunities/Discrimination
ü / Health and Safety
Social Benefits/Social Security
ü / ü / ü / Health and Safety / Consumer
Feedback Mechanism
Consumer Privacy
Transparency
ü / ü / End of life responsibility
Fair competition / Value chain actors (not including consumers)
Promoting social responsibility
Supplier relationships
Respect of intellectual property rights
Access to material resources
Access to immaterial resources / Local community
ü / ü / ü / ü / ü / Delocalization and Migration
Cultural Heritage
Safe & healthy living conditions
Respect of indigenous rights
Community engagement
ü / ü / ü / ü / ü / Local employment
Secure living conditions
Public commitments to sustainability issues / Society
ü / ü / ü / ü / ü / Contribution to economic development
Prevention & mitigation of armed conflicts
Technology development
Corruption
PC: production center CC: collection center
Part II. The crisp equivalent model
Min E[W1]= iptfi(1)pt+fi(2)pt+fi(3)pt+fi(4)pt4xipt+kqgk(1)q+gk(2)q+gk(3)q+gk(4)q4ykq+ijtρi(1)t+ρi(2)t+ρi(3)t+ρi(4)t4+cij1+cij2+cij3+cij44uijt+jkφk1+φk2+φk3+φk44+ajk1+ajk2+ajk3+ajk44sjk+klβl1+βl2+βl3+βl44+bkl1+bkl2+bkl3+bkl44vkl+kmτm1+τm2+τm3+τm44+hkm1+hkm2+hkm3+hkm44wkm
+knθn1+θn2+θn3+θn44+rkn1+rkn2+rkn3+rkn44zkn
+koυo1+υo2+υo3+υo44+eko1+eko2+eko3+eko44γko
-eiptekqsvcei,ekpqtfei(1)pt+fei(2)pt+fei(3)pt+fei(4)pt4+gek(1)q+gek(2)q+gek(3)q+gek(4)q4Ψei,ekpqt
Min E[W2]= iptexi(1)pt+exi(2)pt+exi(3)pt+exi(4)pt4xipt+kqeyk(1)q+eyk(2)q+eyk(3)q+eyk(4)q4ykq+ijtep(1)t+ep(2)t+ep(3)t+ep(4)t4+etij1+etij2+etij3+etij44uijt+jkeo(1)+eo(2)+eo(3)+eo(4)4+ecjk1+ecjk2+ecjk3+ecjk44sjk+klee(1)+ee(2)+ee(3)+ee(4)4+eskl1+eskl2+eskl3+eskl44vkl+kmev(1)+ev(2)+ev(3)+ev(4)4+elkm1+elkm2+elkm3+elkm44wkm+kner(1)+er(2)+er(3)+er(4)4+enkn1+enkn2+enkn3+enkn44zkn+koea(1)+ea(2)+ea(3)+ea(4)4+edko1+edko2+edko3+edko44γko
-eiptekqsveei,ekpqtexei(1)pt+exei(2)pt+exei(3)pt+exei(4)pt4+eyek(1)q+eyek(2)q+eyek(3)q+eyek(4)q4Ψei,ekpqt
Max EW3=
wc.ip txiptjcpi1pt+jcpi2pt+jcpi3pt+jcpi4pt4upi+kqykqjcck1q+jcck2q+jcck3q+jcck4q4ucksimax1jc+simax2jc+simax3jc+simax4jc4- simin1jc+simin2jc+simin3jc+simin4jc4-eip tekqsvjei,ekpqtjcpie1pt+jcpie2pt+jcpie3pt+jcpie4pt4upei+jccek1q+jccek2q+jccek3q+jccek4q4ucekΨei,ekpqtsimax1jc+simax2jc+simax3jc+simax4jc4- simin1jc+simin2jc+simin3jc+simin4jc4-simin1jc+simin2jc+simin3jc+simin4jc4simax1jc+simax2jc+simax3jc+simax4jc4- simin1jc+simin2jc+simin3jc+simin4jc4 + wt.ip txiptvpi(1)p+vpi(2)p+vpi(3)p+vpi(4)p4(1-edpi)simax(1)pt+simax(2)pt+simax(3)pt+simax(4)pt4- simin(1)pt+simin(2)pt+simin(3)pt+simin(4)pt4+kqykqvck(1)q+vck(2)q+vck(3)q+vck(4)q4(1-edck)simax(1)pt+simax(2)pt+simax(3)pt+simax(4)pt4- simin(1)pt+simin(2)pt+simin(3)pt+simin(4)pt4-simin(1)pt+simin(2)pt+simin(3)pt+simin(4)pt4simax(1)pt+simax(2)pt+simax(3)pt+simax(4)pt4- simin(1)pt+simin(2)pt+simin(3)pt+simin(4)pt4
+ ws. simax1ws+simax2ws+simax3ws+simax4ws4-ip tld1t+ld2t+ld3t+ld4t4xiptsimax1ws+simax2ws+simax3ws+simax4ws4- simin1ws+simin2ws+simin3ws+simin4ws4
+ wp.simax1pr+simax2pr+simax3pr+simax4pr4-ijtpr1t+pr2t+pr3t+pr4t4uijtsimax1pr+simax2pr+simax3pr+simax4pr4- simin1pr+simin2pr+simin3pr+simin4pr4
s.t.
ituijt≥2-2αj1dj3+2αj1-1dj(4), ∀j,
ksjk≥ωj2, ∀j,
ksjk≤ωj3, ∀j,
jsjk= oγko+nzkn+lvkl, ∀k,
lvkl= mwkm, ∀k,
juijt≤pxipt2αi2-1πi1p+2-2αi2πi2p , ∀i, t,
jsjk≤qykq2αk3-1ηk1q+2-2αk3ηk2q , ∀k,
kγko≤2αo4-1χo(1)+2-2αo4χ0(2), ∀o,
kvkl≤2αl5-1δl(1)+2-2αl5δl(2), ∀l,
kwkm≤2αm6-1ζm(1)+2-2αm6ζm(2), ∀m,
kzkn≤2αn7-1ξn(1)+2-2αn7ξn(2), ∀n,
ptxipt≤1, ∀i,
qykq≤1, ∀k,
2Ψei,ekpqt≤xeipt+ yekq, ∀ei, ek, p, q, t,
Ψei,ekpqt≥xeipt+ yekq-1, ∀ei, ek, p, q, t,
xipt, ykq, xeipt, yekq,Ψei,ekpqt∈ 0, 1 ∀i, k, ei,ek,p, q, t,
uijt, sjk, vkl, wkm, γko, zkn≥0, ∀ i, j, k, l, m, n,o, t.
Part III. Complexity of the concerned model
To analyze this issue analytically, we first present the compact form of the developed model (i.e., the aggregated single-objective model which is formed in step 4 and solved in step 5) as follows.
Max OBT= iptftiptxipt+kqgtkqykq+ijtctijtuijt+jkatjksjk+klbtklvkl+kmhtkmwkm+knttknzkn+koqtkoγko-eiptekqstei,ekpqtΨei,ekpqt+constant / (E.1)s.t. ituijt≥dtj, ∀j, / (E.2)
ksjk≤rtj, ∀j, / (E.3)
k sjk≥ntj, ∀j, / (E.4)
jsjk= oγko+nzkn+lvkl, ∀k, / (E.5)
lvkl= mwkm, ∀k, / (E.6)
juijt≤pxiptπtip , ∀i, t, / (E.7)
jsjk≤qykqηtkq , ∀k, / (E.8)
kγko≤χto, ∀o, / (E.9)
kvkl≤δtl, ∀l, / (E.10)
kwkm≤ζtm, ∀m, / (E.11)
kzkn≤ξtn, ∀n, / (E.12)
ptxipt≤1, ∀i, / (E.13)
qykq≤1, ∀k, / (E.14)
2Ψei,ekpqt≤xeipt+ yekq, ∀ei, ek, p, q, t, / (E.15)
Ψei,ekpqt≥xeipt+ yekq-1, ∀ei, ek, p, q, t, / (E.16)
xipt, ykq, xeipt, yekq,Ψei,ekpqt ∈ 0, 1 ∀i, k, ei,ek,p, q, t, / (E.17)
uijt, sjk, vkl, wkm, γko, zkn≥0, ∀ i, j, k, l, m, n,o, t. / (E.18)
In the above-mentioned model, parameters are presented in their closed form. For example ftipt (the first coefficient in the objective function) and dtj (the right hand side of the first constraint) are equal to the following terms.
ftipt=ϱ3.wc.jcpi1pt+jcpi2pt+jcpi3pt+jcpi4pt4.upisimax1jc+simax2jc+simax3jc+simax4jc4- simin1jc+simin2jc+simin3jc+simin4jc4w3α-PIS-w3α-NIS+ϱ3.wt.vpi(1)p+vpi(2)p+vpi(3)p+vpi(4)p4.(1-edpi)simax(1)pt+simax(2)pt+simax(3)pt+simax(4)pt4- simin(1)pt+simin(2)pt+simin(3)pt+simin(4)pt4w3α-PIS-w3α-NIS
-ϱ3.ws.ld1t+ld2t+ld3t+ld4t4simax1ws+simax2ws+simax3ws+simax4ws4- simin1ws+simin2ws+simin3ws+simin4ws4w3α-PIS-w3α-NIS
-ϱ1fi(1)pt+fi(2)pt+fi(3)pt+fi(4)pt4w1α-NIS-w1α-PIS-ϱ2exi(1)pt+exi(2)pt+exi(3)pt+exi(4)pt4w2α-NIS-w2α-PIS / (E.19)
dtj=2-2αj2dj3+2αj2-1dj(4) / (E.20)
Now, let ignore the capability of the proposed model in considering multiple capacity levels and production technologies for facilities and the saving resulted from opening joint production-collection centers. If these assumptions are to be relaxed, then model (E.1) to (E.18) can be divided into two sub-models as follows:
Sub-model (1):
Max SO1= iftixi+ijctijuij / (E.21)s.t. iuij≥dtj, ∀j, / (E.22)
juij≤xiπti , ∀i, / (E.23)
xi ∈ 0, 1 ∀i, / (E.24)
uij≥0, ∀ i, j. / (E.25)
Sub-model (2):
Max SO2= kgtkyk+jkatjksjk+klbtklvkl+kmhtkmwkm+knttknzkn+koqtkoγko / (E.26)
s.t. ksjk≤rtj, ∀j, / (E.27)
ksjk≥ntj, ∀j, / (E.28)
jsjk= oγko+nzkn+lvkl, ∀k, / (E.29)
lvkl= mwkm, ∀k, / (E.30)
jsjk≤ykηtk , ∀k, / (E.31)
kγko≤χto, ∀o, / (E.32)
kvkl≤δtl, ∀l, / (E.33)
kwkm≤ζtm, ∀m, / (E.34)
kzkn≤ξtn, ∀n, / (E.35)
yk ∈ 0, 1 ∀ k, / (E.36)
sjk, vkl, wkm, γko, zkn≥0, ∀ j, k, l, m, n,o. / (E.37)
It is obvious that sub-model (1) is a capacitated facility location problem (CFLP). Heretofore, Davis and Ray (1969) showed that CFLP is a NP-hard problem. Additionally, the investigations of Mirchandi and Francis (1990) indicate that this problem is strongly NP-hard. Therefore, since the original model (E.1) to (E.18) includes sub-model (1) which is similar to a CFLP, the complexity of the original model is at least equal to CFLP (i.e., it is NP-hard). However, it should be noted that the original model also includes sub-model (2) and the structure of this sub-model is somehow similar to CFLP. According to this issue and recalling the abovementioned two relaxed assumptions, it can be concluded that the complexity of the original model is significantly higher than CFLP.
Part IV. Case study data
Table E.1. The demand and returned products of each customer zone
Returned products (millions) / Demand (millions) / Customer zone /297) / 278, / 244, / (229, / 390) / 365, / 320, / (300, / (1) Mashhad
313) / 290, / 259, / (229, / 410) / 380, / 340, / (300, / (2) Yazd
175) / 160, / 149, / (141, / 230) / 210, / 195, / (185, / (3) Shiraz
117) / 104, / 91, / (88, / 180) / 160, / 140, / (135, / (4) Urmia
114) / 101, / 81, / (72, / 175) / 155, / 125, / (110, / (5) Ardebil
217) / 202, / 189, / (168, / 285) / 265, / 248, / (220, / (6) Rasht
156) / 130, / 114, / (91, / 240) / 200, / 175, / (140, / (7) Kermanshah
132) / 116, / 92, / (84, / 165) / 145, / 115, / (105, / (8) Karaj
229) / 198, / 168, / (137, / 300) / 260, / 220, / (180, / (9) Zanjan
206) / 168, / 145, / (114, / 270) / 220, / 190, / (150, / (10) Hamadan
225) / 206, / 183, / (168, / 295) / 270, / 240, / (220, / (11) Ghazvin
408) / 376, / 352, / (304, / 510) / 470, / 440, / (380, / (12) Tehran
296) / 276, / 256, / (224, / 370) / 345, / 320, / (280, / (13) Esfahan
244) / 214, / 202, / (183, / 320) / 280, / 265, / (240, / (14) Kerman
221) / 202, / 191, / (175, / 290) / 265, / 250, / (230, / (15) Sari
0) / 0, / 0, / (0, / 375) / 368, / 360, / (350, / (16) Iraq
0) / 0, / 0, / (0, / 355) / 340, / 330, / 320, / (17) Turkey
0) / 0, / 0, / (0, / 250) / 235, / 220, / (210, / (18) Afghanistan
85) / 84, / 80, / (72, / 112) / 110, / 105, / (95, / (19) Bandar Abbas
169) / 165, / 158, / (150, / 215) / 210, / 200, / (190, / (20) Semnan
104) / 100, / 92, / (88, / 130) / 125, / 115, / (110, / (21) Khoramabad
69) / 65, / 61, / (57, / 90) / 85, / 80, / (75, / (22) Birjand
63) / 59, / 56, / (52, / 82) / 78, / 74, / (68, / (23) Boshehr
Table E.2. The fixed cost and capacity data for production centers
Fixed cost (million Rials)() / Capacity (millions)
() / Capacity level (p) / Production technology (t) / Potential production centers (i) /
152000) / 146000, / 140000, / (133000, / 1620) / 1580, / 1540, / (1500, / (1) / (1) / (1) Varamin
157000) / 152000, / 148000, / (140000, / 1780) / 1740, / 1700, / (1690, / (2)
154000) / 152500, / 152000, / (151500, / 1620) / 1580, / 1540, / (1500, / (1) / (2)
167000) / 166000, / 164500, / (163000, / 1780) / 1740, / 1700, / (1690, / (2)
152000) / 150000, / 145000, / (140500, / 1620) / 1580, / 1540, / (1500, / (1) / (3)
155000) / 152000, / 151000, / (148000, / 1780) / 1740, / 1700, / (1690, / (2)
180000) / 173500, / 172500, / (171000, / 1620) / 1580, / 1540, / (1500, / (1) / (4)
182000) / 178000, / 177200, / (176000, / 1780) / 1740, / 1700, / (1690, / (2)
160000) / 158000, / 153000, / (143000, / 1720) / 1690, / 1630, / (1580, / (1) / (1) / (2) Saveh
163000) / 160000, / 156000, / (149000, / 2070) / 2030, / 1950, / (1896, / (2)
173000) / 168000, / 164000, / (158000, / 1720) / 1690, / 1630, / (1580, / (1) / (2)
178000) / 172000, / 169000, / (163000, / 2070) / 2030, / 1950, / (1896, / (2)
163000) / 159000, / 155000, / (152000, / 1720) / 1690, / 1630, / (1580, / (1) / (3)
169000) / 164000, / 161000, / (158000, / 2070) / 2030, / 1950, / (1896, / (2)
190000) / 187000, / 183000, / (177000, / 1720) / 1690, / 1630, / (1580, / (1) / (4)
192000) / 190000, / 188000, / (180000, / 2070) / 2030, / 1950, / (1896, / (2)
147000) / 140000, / 135000, / (129000, / 1360) / 1300, / 1250, / (1200, / (1) / (1) / (3) Semnan
152000) / 148000, / 143000, / (136000, / 1630) / 1560, / 1500, / (1440, / (2)
150500) / 150000, / 149000, / (148000, / 1360) / 1300, / 1250, / (1200, / (1) / (2)
165000) / 161000, / 160000, / (158000, / 1630) / 1560, / 1500, / (1440, / (2)
150000) / 145000, / 139000, / (135500, / 1360) / 1300, / 1250, / (1200, / (1) / (3)
151000) / 149000, / 146000, / (144000, / 1630) / 1560, / 1500, / (1440, / (2)
174000) / 170000, / 169000, / (167000, / 1360) / 1300, / 1250, / (1200, / (1) / (4)
178000) / 174000, / 173200, / (171000, / 1630) / 1560, / 1500, / (1440, / (2)
147000) / 140000, / 135000, / (129000, / 1360) / 1300, / 1250, / (1200, / (1) / (1) / (4) Ghom
152000) / 148000, / 143000, / (136000, / 1630) / 1560, / 1500, / (1440, / (2)
150500) / 150000, / 149000, / (148000, / 1360) / 1300, / 1250, / (1200, / (1) / (2)
165000) / 161000, / 160000, / (158000, / 1630) / 1560, / 1500, / (1440, / (2)
150000) / 145000, / 139000, / (135500, / 1360) / 1300, / 1250, / (1200, / (1) / (3)
151000) / 149000, / 146000, / (144000, / 1630) / 1560, / 1500, / (1440, / (2)
174000) / 170000, / 169000, / (167000, / 1360) / 1300, / 1250, / (1200, / (1) / (4)
178000) / 174000, / 173200, / (171000, / 1630) / 1560, / 1500, / (1440, / (2)
161000) / 159000, / 153000, / (144000, / 1760) / 1720, / 1700, / (1650, / (1) / (1) / (5) Arak
164000) / 161000, / 157000, / (150000, / 2110) / 2070, / 2040, / (1980, / (2)
174000) / 169000, / 165000, / (159500, / 1760) / 1720, / 1700, / (1650, / (1) / (2)
179000) / 173000, / 169000, / (164000, / 2110) / 2070, / 2040, / (1980, / (2)
163000) / 160000, / 156000, / (153000, / 1760) / 1720, / 1700, / (1650, / (1) / (3)
170000) / 165000, / 162000, / (159000, / 2110) / 2070, / 2040, / (1980, / (2)
191000) / 188000, / 184000, / (178000, / 1760) / 1720, / 1700, / (1650, / (1) / (4)
193000) / 191000, / 189000, / (181000, / 2110) / 2070, / 2040, / (1980, / (2)
152000) / 146000, / 140000, / (133000, / 1620) / 1580, / 1540, / (1500, / (1) / (1) / (6) Zanjan
157000) / 152000, / 148000, / (140000, / 1940) / 1900, / 1850, / (1800, / (2)
154000) / 152500, / 152000, / (151500, / 1620) / 1580, / 1540, / (1500, / (1) / (2)
167000) / 166000, / 164500, / (163000, / 1940) / 1900, / 1850, / (1800, / (2)
152000) / 150000, / 145000, / (140500, / 1620) / 1580, / 1540, / (1500, / (1) / (3)
155000) / 152000, / 151000, / (148000, / 1940) / 1900, / 1850, / (1800, / (2)
180000) / 173500, / 172500, / (171000, / 1620) / 1580, / 1540, / (1500, / (1) / (4)
182000) / 178000, / 177200, / (176000, / 1940) / 1900, / 1850, / (1800, / (2)
149000) / 142000, / 138000, / (131000, / 1360) / 1300, / 1250, / (1200, / (1) / (1) / (7) Ghazvin
154000) / 150000, / 145000, / (138000, / 1630) / 1560, / 1500, / (1440, / (2)
152500) / 151500, / 151000, / (150000, / 1360) / 1300, / 1250, / (1200, / (1) / (2)
167000) / 163000, / 161500, / (161000, / 1630) / 1560, / 1500, / (1440, / (2)
150000) / 147000, / 140000, / (138500, / 1360) / 1300, / 1250, / (1200, / (1) / (3)
152000) / 150000, / 148000, / (146000, / 1630) / 1560, / 1500, / (1440, / (2)
176000) / 171500, / 170500, / (169000, / 1360) / 1300, / 1250, / (1200, / (1) / (4)
180000) / 176000, / 175200, / (173000, / 1630) / 1560, / 1500, / (1440, / (2)
0) / 0, / 0, / (0, / 1760) / 1720, / 1700, / (1650, / (1) / (1) / (8) Ashtian
[Already Active]
4200) / 3800 / 3100, / (2500, / 2110) / 2070, / 2040, / (1980, / (2)
15000) / 14000, / 13000, / (12000, / 1760) / 1720, / 1700, / (1650, / (1) / (2)
18000) / 17000, / 16000, / (15000, / 2110) / 2070, / 2040, / (1980, / (2)
13000) / 11000, / 10100, / (9500, / 1760) / 1720, / 1700, / (1650, / (1) / (3)
24000) / 23000, / 21000, / (18000, / 2110) / 2070, / 2040, / (1980, / (2)
21500) / 20500, / 19500, / (18000, / 1760) / 1720, / 1700, / (1650, / (1) / (4)
27000) / 25000, / 23000, / (21000, / 2110) / 2070, / 2040, / (1980, / (2)
Table E.3. The fixed cost and capacity data for collection centers