Maxz = ∑ ∑ ∑ ∑ rvct . Zklct + ∑ ∑ ∑ ∑ RVRnct . RRjnct − ∑ ∑ ∑ φlct . UDlct

t k l c t k l c t c l

− ∑ ∑ ∑ ∑ FOrit . SOijrt − ∑ ∑ ∑ ∑ BCirt . Xijrt − ∑ ∑ ∑ HCPrjt . SPrjt

t r i j t r i j t j r

− ∑ ∑ ∑ ∑ PCjct . Yjkct − ∑ ∑ ∑ ∑ HCkct . Skct′t

t c k j t c k j

− ∑ ∑ ∑ ∑ TCSPijrt . Xijrt − ∑ ∑ ∑ ∑ TCSP1nitc . X1nitc

i j r t i j r t

− ∑ ∑ ∑ ∑ TCPWjkct . (RYTkjct + RYkjct + Yjkct)

j k c t

− ∑ ∑ ∑ ∑ TCWRklct . (Zklct + RZlkct) − ∑ ∑ ∑ ∑ TCPDjmct . RDjmct

k l c t j m c t

− ∑ ∑ ∑ ∑ rvct−3 . RZlkct − ∑ ∑ ∑ ∑ ICjct . (RYTkjct + RYkjct)

t>3 k l c j k c t

− ∑ ∑ ∑ ∑ DCmct . RDjmct.

j m c t

Eq. (1) expresses the difference between the total revenues and total costs of the supply chain, there by maximizing the overall profit.

∑ Xijrt + X1nict ≤ CAPSir . SSi, ∀i, n, c, ∀r, ∀t. (2)

Constraint (2) represents the capacity limitation of each supplier for providing raw materials in each period.

∑ Yjkct . UCAPPjc ≤ CAP. Pjc, ∀c, ∀t, ∀j. (3) Constraint (3) ensures that the quantity of each product produced by a manufacturer in a given period does not exceed the manufacturer's available production capacity.

∑ ∑ UCAPDmc. RDjmct ≤ CAPDm .SDm, ∀m, ∀t. (4)

Constraint (4) ensures that the quantity of each product sent to a disposal center in a given period does not exceed the center's disposal capacity.

∑ ∑ UCAPRnc. RRjnct ≤ CAPRn .SRn, ∀n, ∀t. (5)

Constraint (5) ensures that the quantity of products processed at each recycling center does not exceed its available recycling capacity.

∑ ∑ Vc. Yjkt ≤ CAPWk. SWk, ∀k, t = 1. (6)

Constraint (6) ensures that the quantity of products stored in each warehouse does not exceed its available storage capacity.

∑ Vc. (Skx,t−1,t−1 + ∑ Yjktt) ≤ CAPWk ⋅ SWk , ∀k, t = 2. (7)

∑ Vc ⋅ (Sk,t,−2,t−1 + Skk,t−1,t,−1 + ∑ Yjkct) ≤ CAPWk ⋅ SWk, ∀k, t = 3. (8)

∑ Vc ⋅ (Skx,t−3,t−1 + Skx,t−2,t−1 + Skk,t−1,t−1 + ∑ Yjkkt) ≤ CAPWk SWk, ∀k, t ≥ 4. (9)

Constraints (7)-(9) impose the warehouse capacity restrictions by ensuring that the aggregate storage space occupied by incoming products and inventories remaining from previous periods remains within the warehouse capacity. These constraints also enforce a maximum product holding time of three periods.

∑ RYTkjct = 0, ∀k, ∀c, t = 1. (10)

Skctt = max (∑ Yjkct − ∑ Zklct , 0), ∀k, ∀c, t = 1. (11)

∑ RYTkjct = 0, ∀k, ∀c, t = 2. (12)

Sk,t,−1,t = max (Skc,t−1,t−1 − ∑ Z , 0), ∀k, ∀c, t = 2. (13)

Skctt = max (∑ Yjkct + min(Skc,t−1,t−1 − ∑ Zklct, 0) , 0), ∀k, ∀c, t = 2 . (14)

∑ RYT = 0, ∀k, ∀c, t = 3. (15)

Skc,t−2,t = max (Skc,t−2,t−1 − ∑ Zklct, 0) , ∀k, ∀c, t = 3. (16) Skctt = max(∑j Yjkct + min(Skc,t−1,t−1 + min(Skc,t−2,t−1 − ∑t Zklct, 0), 0), 0),

(17)

∀k, ∀c, t = 3.

∑ RYTkjct = max (Skc,t−3,t−1 − ∑ Zklct,, 0), ∀k, ∀c, t ≥ 4. (18)

Skc,t−2,t = max(Skc,t−2,t−1 + min(Skc,t−3,t−1 − ∑ Zklct, 0) , 0), ∀k, ∀c, t ≥ 4. (19)

Skc,t−1,t = max(Skct−1,t−1

+ min(Skc,t−2,t−1 (20)

+ min(Skc,t−2,t−1 + Skc,t−3,t−1 − ∑ Zkict, 0) , 0) , 0) , ∀k, ∀c, t ≥ 4.

Skct = max (∑ Yjkt

+ min(Skc,t−1,t−1 (21)

+ min(Skc,t−2,t−1

+ min(Skc,t−3,s−1 − ∑ Zkct, 0) , 0) , 0) , 0), ∀k, ∀c, t ≥ 4.

Constraints (10)-(21) track the inventory of products stored in the warehouse and identify the quantities that must be returned upon reaching the end of their allowable storage life. Given that the maximum product lifetime is limited to three periods, products that entered the warehouse three periods earlier and remain unsold due to the absence of demand are returned to the manufacturer.

∑ Xijrt − ∑ ∑ Yjkct .UCAPSrc = SPrjt, ∀j, ∀r, t = 1. (22)

i k c

∑ (Xijrt + ∑ r3c RZlkct ) − ∑ ∑ Yjkct .UCAPSrc + SPrj,t−1 = SPrjt , ∀j, ∀r, ∀t ≥ 2 . (23)

i C,K,L k c

Constraints (22) and (23) establish the balance relationship between suppliers and manufacturers while accounting for the inventory of raw materials and components at the manufacturing facilities.

M ⋅ SOijrt ≥ Xijtr, ∀i, ∀r, ∀r, ∀t. (24)

Constraint (24) links the continuous flow variable X to the binary ordering decision variable SO. Whenever a positive quantity is ordered, SO is activated and takes the value 1; otherwise, SO remains 0. This formulation ensures that the fixed ordering cost is incurred only when an order is placed, as reflected in the objective function.

∑ Zklct ≤ ∑ Yjkct, ∀c, ∀k, t = 1. (25)

∑ Zklct ≤ ∑ Yjkct + Skc,t−1,t−1, ∀c, ∀k, t = 2. (26)

∑ Zklct ≤ ∑ Yjkct + Skc,t−1,t−1 + Skc,t−2,t−1, ∀c, ∀k, t = 3. (27) ∑ Zklct ≤ ∑ Yjkct + Skc,t−1,t−1 + Skk,t−2,t−1 + Skc,t−3,t−1, ∀c, ∀k, t ≥ 4. (28)

Constraints (26)-(28) ensure that the total quantity of products shipped from each warehouse to all retailers does not exceed the total amount of products received at that warehouse from all manufacturers. This balance accounts for both the products arriving in the current period and the inventory carried over from previous periods.

∑ Zklct ≤ ∑ Yjkct + Skc,t−1,t−1, ∀c, ∀k, t = 2. (29)

SCklt . TWRkl ≤ EXFt, ∀k, ∀l, ∀t. (30)

EXFt ⋅ (1 − SCklt) ≤ TWRkl, ∀k, ∀l, ∀t. (31)

Constraints (29)-(31) enforce the retailer coverage requirements. Retailers whose service time exceeds the maximum allowable threshold are regarded as uncovered, and their demand is therefore not considered in the model. In contrast, retailers located within the specified service coverage radius are considered covered, and their demand is included in the optimization model.

∑ SCklt ≤ 1, ∀l, ∀t. (32)

Constraint (32) guarantees that each retailer is covered by at most one integrated warehouse–collection center.

∑ Zklct + UDlct ≥ ∑ SCklt . Dlct, ∀l, ∀c, ∀t. (33)

Constraint (33) ensures that the demand of each covered retailer is satisfied, as much as possible, by products available in warehouses or produced in the current period. Any unmet demand is accounted for as lost sales in the model.

Zkct ≤ SCklt ⋅ Dlct, ∀k, ∀l, ∀c, ∀t. (34)

Constraint (34) specifies that a retailer located within the warehouse coverage radius cannot receive more than its required demand in each period, while retailers outside the coverage area receive no shipments.

RZlkct = r1c . Zklc ,t−3, ∀l, ∀k, ∀c, t ≥ 4. (35)

Constraint (35) defines the quantity of returned products from each retailer to each warehouse in each period. These returned products are proportional to the products shipped to retailers three periods earlier, according to a specified return rate.

RZlkct = 0, ∀l, ∀k, ∀c, t = 1,2,3. (36)

Constraint (36) indicates that no returns from retailers to warehouses occur during the first three periods.

∑ RZlkct = ∑ RYkjct, ∀k, ∀c, ∀t. (37)

Constraint (37) states that products returned from retailers to integrated warehouse–collection centers are subsequently transferred directly from warehouses to manufacturing centers.

∑ RDjmct = (1 − r2c) (∑ RYTkjct + ∑ RYkjct ) + r3c ∑ RZlkct , ∀j, ∀c, ∀t. (38)

m k k L,K

Eq. (38) shows that the quantity of products sent from manufacturing centers to disposal centers equals (1 − r2_c) of the total returned products received from warehouses and retailers.

∑ RRjnct = (r2c) (∑ RYTkjct + ∑ RYkjct ) + (1 − r3c) ∑ RZlkct , ∀j, ∀c, ∀t. (39)

n k k L,K

∑ RRjnct = (r2c) (∑ RYTkjct + ∑ RYkjct ) + (1 − r3c) ∑ RZlkct , ∀j, ∀c, ∀t. (40)

n k k L,K

Eqs. (39) and (40) represent the allocation of returned products to recycling centers versus manufacturing centers. A portion of returned products is directed to recycling centers for reuse, while the remaining portion is sent to disposal centers.

Xijrt, Yjkct, Zklct, RZlkct, RYTkjct, RYkjct, RRjnct, RDjmct, SPrjt, UDlct, Skcc′t

(41)

≥ 0, ∀i, j, r, t, k, c, l, n, m.

Yjkct, Zklct, RZlkct, RYTkjct , RYkjct , RRjnct, RDjmct, UDlct, Skct′t ∈ int, ∀j, t, k, c, l, n, m. (42)

SCklt, SOijtt ∈ {0,1}, (43)

The comparison of objective function values between the base case (without considering product reuse) and the proposed model (with consideration of product reuse) shows that higher revenue is achieved in the base case. In small-scale instances, the difference between the two cases is negligible; however, as the problem size increases, the gap between their objective function values becomes more significant. The widening gap is due to the fact that product reuse is not always cost-effective, although in some industries it is necessary due to raw material shortages and environmental concerns. Another important result concerns the output of different scenario-based solution methods. The relationship indicating that the WS objective value is greater than the HN objective value has been examined. In addition, the Value of Stochastic Solution (VSS) and Expected Value of Perfect Information (EVPI) indicators were computed to evaluate stochastic solutions. These indicators reflect the value of considering uncertainty in the model. The comparison shows that only in problems 4, 8, and 12 do these measures not provide significant savings. Another key result is the comparison of two uncertainty modeling approaches used in this study: the scenario-based stochastic programming approach and the Malloy approach. Since in the Malloy approach all scenarios are solved simultaneously, the here-and-now approach was used for comparison purposes. The results indicate that for both small- and large-scale instances, there is a relatively small difference between the objective function values of the two methods [16], [17].

Based on the obtained results, the following managerial implications can be proposed:

Management should make every effort to reduce product returns, since although returned products can be partially recovered and reused as input for suppliers, they reduce the overall profit of the supply chain. Reuse of recovered products is applicable in industries such as automotive batteries; however, it is not feasible for all products. Therefore, managers should carefully evaluate the applicability of product reuse in their specific systems. Uncertainty modeling approaches significantly help in understanding different market conditions, enabling managers to better estimate the potential profit of the supply chain under various scenarios. Sensitivity analysis shows that as the problem size increases, the changes tend to become more linear, allowing managers to obtain more predictable profit estimations for larger supply chain systems. The reuse rate of recovered products can also be considered a key factor in supply planning decisions [18].

As demonstrated in this chapter, the following key conclusions were obtained:

I. The comparison between the base case (without product reuse) and the proposed model shows that the base case is more profitable across different problem sizes; however, due to resource scarcity in the production of certain products, product reuse becomes inevitable. Therefore, this necessity should be considered in future mathematical models.

II. The use of scenario-based stochastic programming and Malloy’s approach leads to cost savings. Out of 12 test problems, only 3 instances did not show significant savings. Therefore, with approximately 70% confidence, these approaches can be considered beneficial [19].