This paper presents a green closedloop tire supply chain model. The relevant model is twoobjective fuzzy model, the first objective is to minimize costs and maximize profits and the second objective is to minimize environmental issues in the chain. The components of this model include suppliers, manufacturers, distributors, product collectors, reproduction centers and recycling centers. In the considered green supply chain network, recycling and product collection centers are responsible for product collection referring them to recycling or reproduction centers or to destroy them if not repairable. In addition, it is possible to develop production and distribution centers under the name of new distribution centers and new production centers. One of the most important decisions obtained from the current model is to determine the optimal location of each component based on the potential locations and also to determine the optimal amount of production, distribution, collection, recycling and reproduction of the product. The following schematic figure is provided for further specification of the model.
In the above figure, the manufacturer receives the raw material from the supplier and sells it to distribution centers. There is a connection between distribution centers, collection centers, reproduction and recycling centers, while recycling centers have the possibility to yield revenue.
2.1. Assumptions
 Potential locations for production, distribution, reproduction and recycling centers are determined in advance.
 The capacity of supply, distribution, reproduction, recycling and disposal is limited.
 The rates of return, recycling, and reproduction are uncertain.
 The costs of raw material supply and production are uncertain.
 Market demand (customers) is uncertain.
 Only one transportation model takes place in this network (for example, only road transportation).
 There is a fee considered for collecting used tires from each market. This cost encourages the end consumer to deliver the used tire to the retailer, and also encourages the retailer to receive and deliver used tires to collection centers.
 The market demand for the new tire must be satisfied, otherwise the shortage fee will be applied, but the satisfaction of the product demand in Retread is optional.
 Transport companies buy the tires they need directly from distribution centers and transport them to reproduction centers for reproduction.
Subscripts
\(i\)

Set of potential locations

\(j\)

Set of supplier centers

\(k\)

Production centers

\(l\)

New manufacturing centers

\(m\)

Distribution Centers

\(n\)

New distribution centers

\(o\)

Product collection centers

\(p\)

Reproduction centers

\(q\)

Recycling centers

\(r\)

Set of various products

Parameters
\(\tilde{{RSC}_{j}}\)

j Cost of raw materials from the supplier

\({FROC}_{r}\)

p Fixed cost of ordering for the product

\({VROC}_{r}\)

r Variable cost of product purchase

\(\tilde{{PSC}_{kr}}\)

r Production cost product, k production center

\({FCM}_{k}\)

k Cost of construction of production center

\({FCMM}_{m}\)

m Cost of construction of distribution centers

\({DCM}_{l}\)

l Cost of developing a new production center

\({DCMN}_{n}\)

n Cost of developing a new distribution center

\({FCO}_{O}\)

o Cost of construction of collection centers

\({CCO}_{O}\)

O Collection fee of collection center

\({FCL}_{l}\)

l Cost of construction of reproduction centers

\({FCQ}_{q}\)

Cost of construction of recycling centers

\(TC\)

Total network flow costs

\({RI}_{r}\)

r Revenue from resale of the product

\({RCI}_{r}\)

r Revenue from product recycling

\({RUI}_{r}\)

r Income from sales to the market for reuse of the product

\({MI}_{r}\)

r Revenue from the sale of product to the market for energy recovery

\({POPK}_{k}\)

K The amount of pollution produced by the factory

\({POPQ}_{q}\)

Q The amount of pollution produced by the recycling center

\({POPP}_{p}\)

P The amount of pollution produced by the reproduction center

\({CONP}_{mp}\)

m The amount of fuel consumption transported from the distribution center to the reproduction centers p

\({POPO}_{o}\)

o The amount of pollution caused by the destruction or burning of tires used by the product collection center

\({CAPJ}_{j}\)

J Supplier capacity

\({CAPK}_{k}\)

K Manufacturer capacity

\({CAPM}_{m}\)

m Distributor capacity

\({CAPO}_{o}\)

O Capacity of the product collection center

\({CAPP}_{p}\)

P Capacity of the reproduction center

\({CAPQ}_{q}\)

Q Recycling center capacity

\({CAPL}_{l}\)

L Capacity of new production center

\({CAPN}_{n}\)

n Capacity of the new distribution center

\({\tilde{DEM}}_{kr}\)

r The amount of customer demand for a new product from the manufacturer K

\({\tilde{DEMP}}_{pr}\)

Customer demand for the reproduced product r from the manufacturer p

\({DefC}_{kr}\)

r Shortage cost for new product from manufacturer k

\({TCMP}_{MP}\)

Cost of transporting materials from M distribution centers to P reproduction centers

Decision variables
\({XJ}_{j}\)

If supplier j is selected to supply the raw materials required for the tire 1 and otherwise zero

\({XK}_{ki}\)

If the production center k is constructed in place i 1 and otherwise zero

\({YK}_{kir}\)

Production rate r by production center k in place i

\({XM}_{mi}\)

If the distribution center m is located at location i 1 and otherwise zero

\({YM}_{mir}\)

The distribution rate of the distribution center m at location i for product r

\({XO}_{Oi}\)

If the product collection center o is located in place i 1 and otherwise zero

\(\tilde{{YO}_{oir}}\)

Collection rate of product r in product collection center O in place I

\({XP}_{pi}\)

If the product reproduction center p is located in place i 1 and otherwise zero

\(\tilde{{YP}_{pir}}\)

Product r reproduction rate in reproduction center p in place I

\({XQ}_{qi}\)

If the recycling center q is constructed in place i 1 and otherwise zero

\(\tilde{{YQ}_{qir}}\)

Product r recycling rate in recycling center q in place I

\(X\)

The amount of total network flow between different levels of the supply chain

\({XL}_{li}\)

If a new production center L is constructed at location i 1 and otherwise zero

\({XN}_{ni}\)

If a new distribution center n is constructed at location I 1 and otherwise zero

\({YL}_{lir}\)

r Production rate in new production center l in place i

\({YN}_{nir}\)

Product r distribution rate in distribution center n in place i

\({XDEF}_{kr}\)

The shortage of commodity K for the product r

\({XTCMP}_{MPr}\)

The flow of r product from distribution centers M to reproduction centers P

\({XRI}_{r}\)

Product r resale price

\({XRCI}_{r}\)

Product r recycling value

\({XRUI}_{r}\)

Market selling price for product r reuse

\({XMI}_{r}\)

The selling price of the product r to the market for energy recovery

Furthermore, we address the objective functions of the problem, which include economic and environmental objective functions. The economic objective is to minimize the costs of the entire network, which are as follows:
1) The cost of supplying raw materials, including the fixed cost of ordering and the variable cost of purchase. 2) The cost of construction and development of production centers. 3) The cost of construction and development of the distribution center. 4) The cost of constructing collection centers. 5) The cost of constructing new production centers. 6) The cost of constructing recycling centers. 7) Total network flow costs including the transportation of materials and products between two different levels of the supply chain.
The economic objective can also be presented as maximizing revenue:
1) Revenue from resale 2) Revenue from recycling 3) Revenue from resale to market for reuse 4) Revenue from resale to market for energy recovery
The objective functions and problem constraints are modeled in the following.
\(\text{max}z1={RI}_{r}.{XRI}_{r}+{RCL}_{r}.{XRCL}_{r}+{RUI}_{r}{XRUI}_{r}+{MI}_{r}{XMI}_{r}[\sum _{j}^{}{\tilde{RSC}}_{j}.{XJ}_{j}+\sum _{k}^{}{\tilde{PSC}}_{k}.{YK}_{ki}\) \(+\sum _{k}^{}{FCM}_{k}{XK}_{ki}.+\sum _{m}^{}{FCMM}_{m}.{XM}_{mi}+\sum _{l}^{}{DCM}_{L}.{XL}_{li}\) \(+\sum _{n}^{}{DCMN}_{n}.{XN}_{ni}\) \(+\sum _{o}^{}{FCO}_{o}.{XO}_{oi}+\sum _{o}^{}{CCO}_{o}\tilde{{YO}_{oi}}\) \(+\sum _{l}^{}{FCL}_{l}{XL}_{lI}+\sum _{Q}^{}{FCQ}_{q}{.XQ}_{qi}+\sum _{k}^{}{DefC}_{k}.{XDEF}_{k}\) \(+\sum _{m}^{}\sum _{p}^{}{TCMP}_{mp}]+TC+FROC+VCOC\) (1)
\(\text{min}z2=\sum _{k}^{}{POPK}_{k}.{XK}_{ki}\) \(+\sum _{q}^{}{POPQ}_{q}.{XQ}_{qi}\) \(+\sum _{p}^{}{POPP}_{p}.{XP}_{pi}\) \(+\sum _{m}^{}\sum _{p}^{}{CONP}_{mp}.{XM}_{mi}+\sum _{o}^{}{POPo}_{o}.{XO}_{Oi}\) (2)
\({XJ}_{j}\le 1 \forall j\) (3)
\({XK}_{ki}\le 1 \forall k,i\) (4)
\(\sum _{i}^{}{XK}_{ki}=1 \forall k\) (5)
\({YK}_{kir}\ge {\tilde{DEM}}_{kr} \forall k,i,r\) (6)
\({YK}_{kir}\le {CAPK}_{k} \forall k,i,r\) (7)
\({XM}_{mi}\le 1 \forall m,i\) (8)
\(\sum _{i}^{}{XM}_{mi}=1 \forall m\) (9)
\({YM}_{mi}\le {CAPM}_{m} \forall m,i\) (10)
\({XO}_{Oi}\le 1 \forall o,i\) (11)
\(\sum _{i}^{}{XO}_{Oi}=1 \forall o\) (12)
\(\tilde{{YO}_{oi}}\le {CAPM}_{m} \forall o,i,m\) (13)
\({XP}_{pi}\le 1 \forall p,i\) (14)
\(\sum _{i}^{}{XP}_{pi}=1 \forall p\) (15)
\(\tilde{{YP}_{pi}}\le {CAPP}_{p} \forall p,i\) (16)
\({XQ}_{qi}\le 1 \forall q,i\) (17)
\(\sum _{i}^{}{XQ}_{qi}=1 \forall q\) (18)
\(\tilde{{YQ}_{qir}}\le {CAPQ}_{q} \forall q,i,r\) (19)
\(X=\sum _{r}^{}\sum _{m}^{}\sum _{p}^{}{XTCMP}_{mpr}\) (20)
\({XL}_{li}\le 1 \forall l,i\) (21)
\(\sum _{i}^{}{XL}_{li}=1 \forall l\) (22)
\({YL}_{lir}\le {CAPL}_{l} \forall l,i,r\) (23)
\({XN}_{ni}\le 1 \forall n,i\) (24)
\(\sum _{i}^{}{XN}_{ni}=1 \forall n\) (25)
\({YN}_{nir}\le {CAPN}_{n} \forall n,i,r\) (26)
\({XDEF}_{k}={\tilde{DEM}}_{kr}+{\tilde{DEMP}}_{pr}{YK}_{kir} \forall k,i,r,p\) (27)
\(\sum _{k}^{}\sum _{i}^{}{YK}_{kir}=\sum _{m}^{}\sum _{i}^{}{YM}_{mir} \forall r\) (28)
\(\sum _{m}^{}\sum _{i}^{}{YM}_{mir}=\sum _{o}^{}\sum _{i}^{}{\tilde{YO}}_{oir}+\sum _{p}^{}\sum _{i}^{}{\tilde{YP}}_{pir}+\sum _{q}^{}\sum _{i}^{}{\tilde{YQ}}_{qir}+\sum _{l}^{}\sum _{i}^{}{YL}_{li}+\sum _{n}^{}\sum _{i}^{}{YN}_{ni} \forall r\) (29)
Equation 1 seeks to maximize profits or minimize costs. Revenues as well as expenses are separately presented in the previous section. Equation 2 seeks to minimize environmental issues. Equation 3 states that it is possible to supply from each supplier up to one time. Equation 4 indicates that each plant is established at most once. Equation 5 shows that the total location of a factory equals one. Equation 6 shows that the amount of production for each factory should be more than the amount of demand for that factory. Equation 7 shows that the output of each plant should not exceed the capacity of that plant. Equation 8 shows that each distribution center is established at most once. Equation 9 states that a distribution center can only be established in one potential location. Equation 10 shows that the total amount of distribution for each distribution center cannot exceed the capacity of that distribution center. Equation 11 shows that each product collection center can only be constructed once.
Equation 12 shows that there is only one place for each product collection center. Equation 13 states that the total amount of product collected by each collection center cannot exceed its capacity. Equation 14 shows that each reproduction center can only be constructed once. Equation 15 shows that each reproduction center is constructed in only one place. Equation 16 shows that the total amount produced by each reproduction center could not exceed its capacity. Equation 17 shows that each recycling center can be established only once. Equation 18 shows that only one site is allocated for the construction of each recycling center. Equation 19 shows that the capacity of each recycling center is limited. Equation 20 states that the total network flow is equal to the total amount of transport from distribution centers to reproduction centers. Equation 21 shows that each new production center can be constructed only once. Equation 22 shows that there is only one location for each production center. Equation 23 indicates that the total amount of production for the reproduction centers should be less than its capacity. Equation 24 shows that each new distribution center can only be constructed in one location. Equation 25 shows that the total potential locations for each new distribution center are just one location. Equation 26 indicates the capacity limit of each new distribution center. Equation 27 indicates the degree of shortage. Equation 28 shows that the total amount of production should be the same as the amount of collection. Equation 29 indicates the constraint of closedloop supply chain.