CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 61, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš 
Copyright © 2017, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-51-8; ISSN 2283-9216 

A Mixed Integer Nonlinear Model for Closed-Loop Supply 

Chains with Incentive and Proximity based Product Returns 

Antonio Yamzon, Veanney Ventura, Paolo Guico, Charlle Sy* 

Industrial Engineering Department, De La Salle University, 2401 Taft Avenue, Manila1004, Philippines, 

charlle.sy@dlsu.edu.ph 

Electronics firms in different locations are being required to collect used products due to environmental and 

health hazards. In order to meet environmental requirements, firms carry out collection activities and provide 

incentive offers to attract product returns. A mixed integer non-linear programming model for a Closed-Loop 

Supply Chain including decisions for collection activities, incentive offers and recovery options is formulated and 

validated. Quantity is modeled as a function of incentive offers and distance between the collection centers 

(distribution/retail centers) and consumers. The quality of product returns follows an arbitrary probability 

distribution, which is shifted by incentive level. Quality of product returns dictates the possible recovery options, 

which these can undergo. The model is subjected to scenario analysis. This is comprised of conditions wherein 

rebate, or discount incentives is preferred and when low or high incentive levels are favored. High stockout cost 

to secondary consumers encouraged the model to perform more cash rebate activities to stimulate more product 

returns to satisfy secondary consumer demand. In another situation, high cost of activities while having high 

stockout cost to secondary consumers induced the model to carryout discount activities as this would generate 

sales rather than the cash rebate which simply incentivizes the participation in the takeback program.  

1.  Introduction 

Electronics firms in different locations are being required to collect used products due to environmental and 

health hazards, in order to meet requirements, firms carry out collection activities and provide incentive offers 

to attract product returns. These product returns are then sorted to decide on the appropriate recovery options 

to use (Donmez and Turkay, 2013).  These could in the form of refurbishing, remanufacturing, cannibalizing and 

controlled disposal. Incentives have been shown as the most significant factor influencing consumer 

participation in product take-back programs (Balisado, 2013). The importance of this relationship was 

highlighted by Aras and Aksen (2007), stating that items collected without incentives are impractical for recovery. 

Firms therefore must decide on proper take-back programs and incentive levels to meet legislated collection 

requirements and obtain items of adequate quality for recovery while maximizing the profit potential of recovery. 

The type of incentive to offer: discount or cash rebate, is another relevant decision to consider. This is because 

of the difference in consumer reaction to the two types. The discount incentives increase demand for new 

products, but typically yields lower consumer participation than cash rebate (Lundin, 2012). On the other hand, 

previous literature on incentive-quality relationship includes that of Matter et al. (2015), who conducted a case 

study, which focused on the effect of institutionalizing tax and rebate incentives on solid waste recycling 

schemes. The findings of this research suggest that the quality of recycled products was unreliable without the 

implementation of these tax incentives. Govindan et al. (2015) further state that complexity increases in 

managing closed loop supply chains (CLSC) since demand and product returns are likewise inherently 

uncertain. Yamzon et al. (2016) demonstrated the effects of introducing different types of incentives to the 

quantity and quality of product returns in a CLSC. The distance between consumers and the product take-back 

program has been shown to be the second most significant factor in influencing consumer participation. Through 

a Design of Experiments (DOE) conducted by Balisado (2013), it was identified that the location of the collection 

centers was the second most significant factor in consumer participation. Therefore, the location of product take-

back programs is also a relevant decision, which should be incorporated in a CLSC model. This study considers 

these decisions simultaneously with the allocation of collected items to each recovery option. This is because 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1761006

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Yamzon A., Ventura V., Guico P., Sy C., 2017, A mixed integer nonlinear model for closed-loop supply chains with 
incentive and proximity based product returns, Chemical Engineering Transactions, 61, 49-54  DOI:10.3303/CET1761006  

49



recovery options are highly dependent on the quantity and quality of product returns available. To date, no other 

study has been made which incorporates incentive types, continuous incentive-quality relationship, and 

proximity-quantity relationship for product takeback in a closed-loop supply chain model. 

2.  Model Development 

The following section presents the model development for the CLSC understudy. A model for the network is 

formulated using Mixed Integer Non-Linear Programming (MINLP). The model aims to make investment, 

operational and marketing decisions that maximize the profit of the firm while meeting targets in product 

collection, investment budget and virgin raw material usage. The indices used along with the relevant 

parameters and decision variables are initially presented in Table 1 and Table 2. These are followed by a 

discussion on the network requirements for the CLSC. 

Table 1: Indices 

Notation Definition Notation Definition 

𝑖 Potential distributor or retailer facilities 𝑙 Available activities which offer discount 

incentives  

𝑗 Fixed primary consumers and 

secondary consumers 

𝑚 Available activities which offer cash rebate 

incentives 

𝑘 Type of recovery options to be 

performed 

𝑜 Available activities which do not offer 

incentives 

𝑡 Time periods in the planning horizon 𝑟 Raw Material or component types 

Table 2: Parameters and decision variables 

Notation Definition 

𝑑𝑗𝑡  Demand from customer j on time period t  

𝑓(𝑛𝑗𝑙𝑡
𝑎 ) Quantity of product returns from consumer j from activity l as a function of discounts and 

distance offered on time period t 

𝑓(𝑛𝑗𝑚𝑡
𝑏 ) Quantity of product returns from consumer j from activity m as a function of cash rebates 

and distance offered on time period t 
𝑓𝑜

𝑐 Quantity of product returns from activity o 

𝑞𝑘
𝑢 Upper limit of the acceptable quality level for recovery option k 

𝑞𝑘
𝑙  Lower limit of the acceptable quality level for recovery option k 

𝑄𝑢𝑎𝑙𝑙(𝑞/𝑑)  Probability distribution function of acquiring product returns of quality q using activity l 
offering a discount of d 

𝑄𝑢𝑎𝑙𝑚(𝑞/𝑖)  Probability distribution function of acquiring product returns of quality q using activity m 
offering a cash incentive of i 

𝑥𝑖𝑘𝑡 Quantity of products k transferred from supplier to distributor i in period t 

𝑦𝑖𝑗𝑘𝑡 Quantity of products k transferred from distributor i to customer j in period t 

𝑝𝑘𝑡 Quantity of products that have undergone recovery k in period t  

𝑐𝑘𝑡 Capacity for recovery option k in period t 

𝑢𝑖𝑗𝑘𝑡 Quantity of products that could be recovered under option k from consumer j returned to 
distributor i in period t 

𝑤𝑖𝑘𝑡 Quantity of products that could be recovered under option k returned from distributor i to the 
supplier/manufacturer in period t 

𝑛𝑗𝑙𝑡
𝑎  Discount offered for customer j in activity l on period t 

𝑛𝑗𝑚𝑡
𝑏  Cash Rebate offered for customer j in activity m on period t 

𝑏𝑖𝑡 1, if distributor i is operational on time-period t and 0, otherwise 

 

2.1.  Downstream flow of products 
The inventory of new and refurbished products stored in each distribution or retailer (d/r) facility for each period 

is defined by Eq(1). This amount is equal to the amount of products held from the previous period, plus the 

products that arrived in the current period less the products sent out in the current period. Eq(2) ensures that 

items moving out from the supplier/manufacturer (s/m) facility are limited by the amount of items manufactured 

or refurbished on the given period. Meanwhile, constraint Eq(3) provides the production and recovery capacity 

limits. The production, refurbish, remanufacture and cannibalize quantities in each period should be less than 

or equal to the corresponding capacity. Consequently, Eq(4) shows that the total assigned capacity for each 

activity is dependent on the capacity expansion that could be performed in each period.  

50



𝐼𝑖𝑘𝑡 = 𝐼𝑖𝑘𝑡−1 + 𝑥𝑖𝑘𝑡 − ∑ 𝑦𝑖𝑗𝑘𝑡              𝑗                    ∀   𝑖, 𝑘, 𝑡 (1) 

∑ 𝑥𝑖𝑘𝑡   ≤   ∑ 𝑝𝑘𝑡

𝑘

                                  

𝑖

 ∀   𝑘, 𝑡 (2) 

𝑝𝑘𝑡 ≤  𝐶𝑘𝑡                                                   ∀   𝑘, 𝑡 (3) 

𝑒𝑘𝑡 + 𝐶𝑘𝑡 =  𝐶𝑘𝑡+1                                   ∀   𝑘, 𝑡   (4) 

The succeeding constraints Eq(5) and Eq(6) apply the demand of primary and/or secondary consumers. The 

total deliveries plus the shortage must be greater than or equal to the demand, while the total deliveries must 

be less than or equal to the demand. Note that the variable 𝑞𝑗𝑙𝑡
𝑎  is added to the demand for new products, in 

Eq(5) and Eq(6), if discount activities are considered because it is assumed that consumers who participate in 

the aforementioned purchase new products upon returning their used items.  

∑ ∑ 𝑦𝑖𝑗𝑘𝑡

𝑘

+ 𝑠𝑗𝑡 ≥ 𝑑𝑗𝑡  +  𝑞𝑗𝑙𝑡
𝑎              

𝑖

 ∀   𝑗, 𝑡 (5) 

∑ ∑ 𝑦𝑖𝑗𝑘𝑡𝑘 ≤ 𝑑𝑗𝑡  +  𝑞𝑗𝑙𝑡
𝑎                       𝑖   ∀   𝑗, 𝑡 (6) 

            

2.2.  Upstream flow of products 
Constraints Eq(7) - Eq(9) define the quantity of product returns collected by selected activities. This is equal to 

the binary variable for selecting activities multiplied by the corresponding quantity function for product returns.  

𝑄𝑗𝑙𝑡
𝑎   = 𝑔𝑗𝑙𝑡

𝑎 𝑓(𝑛𝑗𝑙𝑡
𝑎 )  ∀ 𝑗, 𝑙, 𝑡                         (7) 

𝑄𝑗𝑚𝑡
𝑏   = 𝑔𝑗𝑚𝑡

𝑏 𝑓(𝑛𝑗𝑚𝑡
𝑏 )                           ∀ 𝑗, 𝑚, 𝑡      (8) 

𝑄𝑗𝑜𝑡
𝑐   = 𝑔𝑗𝑜𝑡

𝑐 𝑓𝑜
𝑐                                         ∀ 𝑗, 𝑜, 𝑡            (9) 

In Eq(7) and Eq(8), the binary variables 𝑔𝑗𝑙𝑡
𝑎  and 𝑔𝑗𝑚𝑡

𝑏  can be excluded if the corresponding quantity function, 

𝑓(𝑛𝑗𝑙𝑡
𝑎 ) or 𝑓(𝑛𝑗𝑚𝑡

𝑏 ), have values of 0 at incentive levels of 0. This is because incentive levels automatically have 

values of 0 if the corresponding activity is not chosen. However, the binary variables must be included if the 

functions do not have values of 0 at incentive levels of 0 in order to ensure that no product returns come from 

activities which are not selected. The constraint Eq(10) enforces product collection targets 𝜏𝑡. The total quantity 

of product returns collected by the different activities must be greater than or equal to the product collection 

target in each period. 

∑ (∑ 𝑄𝑗𝑙𝑡
𝑎

𝑙 + ∑ 𝑄𝑗𝑚𝑡
𝑏

𝑚 + ∑ 𝑄𝑗𝑜𝑡
𝑏

𝑜 )𝑗 ≥ 𝜏𝑡  ∀ 𝑡                     (10) 

Constraint Eq(11) computes for the quantity of product returns collected which can undergo each recovery 

option. The quantity of refurbishable product returns collected is the sum of the collection of each activity type 

multiplied by the probability of collecting refurbishable product returns from each activity. 

𝑅𝑗𝑘𝑡 = ∑ (𝑙  𝑄𝑗𝑙𝑡
𝑎 . 𝑝𝑟𝑜𝑏𝑗𝑘𝑙𝑡

𝑎 ) + ∑  ( 𝑄𝑗𝑚𝑡
𝑏 . 𝑝𝑟𝑜𝑏𝑗𝑘𝑚𝑡

𝑏
𝑚 ) + ∑ (𝑄𝑗𝑜𝑡

𝑐 . 𝑝𝑟𝑜𝑏𝑗𝑘𝑜𝑡
𝑐

𝑜 )     ∀  𝑗, 𝑘, 𝑡                     (11) 

Non-linearity arises from the above constraint because the variables for quantity, such as 𝑞𝑗𝑙𝑡
𝑎 , and the probability 

variables for quality are multiplied to each other. Both of these variables are functions of incentive levels. Eq(12) 

and Eq(13) compute the probability of acquiring recoverable product returns from consumer j with activities l or 

m on time t. This is the integral of the quality function for discount activities from the imposed quality limits. For 

instance, the limits are from the minimum quality level (𝑞𝑘
𝑙 ) to maximum quality level (𝑞𝑘

𝑢) given a corresponding 

discount level.  

𝑝𝑟𝑜𝑏𝑗𝑘𝑙𝑡
𝑎  = ∫ 𝑄𝑢𝑎𝑙𝑙(𝑞/𝑑 = 𝑛𝑗𝑙𝑡

𝑎 ) 𝑑𝑞
𝑞𝑘

𝑢

𝑞𝑘
𝑙                   ∀ 𝑗, 𝑙, 𝑡                   (12) 

𝑝𝑟𝑜𝑏𝑗𝑘𝑚𝑡
𝑏  = ∫ 𝑄𝑢𝑎𝑙𝑚(𝑞/𝑖 = 𝑛𝑗𝑚𝑡

𝑏 ) 𝑑𝑞
𝑞𝑘

𝑢

𝑞𝑘
𝑙              ∀ 𝑗, 𝑚, 𝑡                                             (13) 

Eq(14) and Eq(15) define the quantity of product returns moving from the facilities. In Eq(15), the total amount 

delivered to each d/r facility from a primary consumer is set to be equal to the quantity collected for each recovery 

option.  

∑ 𝑢𝑖𝑗𝑘𝑡 ≤ 𝑉𝑗𝑘𝑡𝑖                                                            ∀ 𝑗, 𝑘, 𝑡                (14) 

𝑅𝑖𝑘𝑡 =  ∑  𝑗 𝑢𝑖𝑗𝑘𝑡 + 𝑅𝑖𝑘(𝑡−1) − 𝑤𝑖𝑘𝑡                      ∀ 𝑖, 𝑘, 𝑡  (15) 

Eq(16) computes the product returns available for each recovery option. This constraint links the downstream 

and upstream flow of products through the variable 𝒑𝒌𝒕. Note that for each recovery option, the amount received 

𝒑𝒌𝒕 refers to product returns which are strictly in the quality level of the respective recovery options. For example, 

51



remanufacturing could utilize returned products that are remanufacturable and could likewise accommodate 

those to be refurbished. This is in line with the prescribed quality level discussed in the previous section. As a 

result, refurbishable product returns would be added to the quantity of remanufacturable product returns, while 

the quantity which has been chosen for refurbishing would be subtracted. Similarly, for other recovery options, 

the remaining product returns for higher order recovery options are added.  

𝑝𝑘𝑡 =  ∑ ∑ 𝑤𝑖𝑘𝑡𝑖
𝑘= 𝑘𝑚𝑎𝑥
𝑘= 𝑘 − ∑ 𝑝𝑘,𝑡

𝑘= 𝑘𝑚𝑎𝑥
𝑘= 𝑘+1             ∀ 𝑘, 𝑡                      (16) 

Eq(17) to Eq(19) define logical requirements for the d/r facilities and incentives. Eq(17) ensures product returns 

are not delivered to closed d/r facilities. While Eq(18) and Eq(19) ensure that incentive levels are 0 for activities 

which are not selected.  

∑ 𝑥𝑖𝑘𝑡𝑘 + ∑  𝑗 ∑ 𝑢𝑖𝑗𝑘𝑡𝑘 ≤   𝑀𝑏𝑖𝑡                  ∀ 𝑖, 𝑡                  (17) 

𝑛𝑗𝑙𝑡
𝑎 ≤ 𝑀𝑔𝑗𝑙𝑡

𝑎                                            ∀ 𝑗, 𝑙, 𝑡                (18) 

𝑛𝑗𝑚𝑡
𝑏 ≤ 𝑀𝑔𝑗𝑚𝑡

𝑏                                                ∀ 𝑗, 𝑚, 𝑡                (19) 

Eq(20) computes the consumption of raw materials or components by remanufacturing, refurbishing and 

production of new products. This must be less than the target consumption of virgin raw materials plus the 

amount which is recovered from cannibalizing. Eq(21) ensures the investments in capacity and in opening 

distribution facilities are within the investment budget for each time period. 

∑  
𝑘𝑚𝑎𝑥
𝑘=𝑟𝑒𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒 𝑝𝑘𝑡 𝑅𝐶𝑘 ≤ 𝑅𝑇𝑟𝑡

 +  𝑝𝑘=𝑐𝑎𝑛𝑛𝑖𝑏𝑎𝑙𝑖𝑧𝑒,   𝑡  𝑅𝐶𝑘=𝑐𝑎𝑛𝑛𝑖𝑏𝑎𝑙𝑖𝑧𝑒            ∀ 𝑡                     (20) 

∑ 𝐸𝐶𝑘  𝑒𝑘𝑡 +𝑘  ∑ 𝐹𝐶  
𝑖𝑖  𝑏𝑖𝑡  ≤ 𝐼𝐵𝑡           ∀ 𝑡                (21) 

The objective of the CLSC optimization model in this study is to maximize profit which is defined as the difference 

between the sales revenue (from the primary and secondary consumers) and fixed and variable costs from both 

upstream and downstream movement of products. Eq(22) to Eq(25) define the different components of the 

aforementioned profit function. Sales revenue is dependent on the amount of products that ship out from the d/r 

to the customers. The total fixed cost is composed of the fixed costs obtained from undertaking the different 

product take back activities, and the fixed costs of operating the d/r and s/m facilities. Variable costs include the 

transportation costs from the s/m and d/r facilities’, the recovery costs for each option, expansion cost for the 

recovery options, shortage penalties for failure to fulfil all customer orders, holding costs for the inventory of 

recovered and returned products, and the incentive costs brought about by each product take back activities. 

𝑀𝑎𝑥 ∑ 𝑝𝑟𝑜𝑓𝑖𝑡𝑡𝑡 = 𝑆𝑎𝑙𝑒𝑠 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑡 − 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑡 −  𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡𝑡               (22) 

𝑆𝑎𝑙𝑒𝑠 𝑅𝑒𝑣𝑒𝑛𝑢𝑒𝑡 = ∑ 𝑆𝑃𝑘 ∑ ∑ 𝑦𝑖𝑗𝑘𝑡        ∀ 𝑡𝑗𝑖𝑘                  (23) 

𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑡 = 𝐴𝐶𝑎 ∑ ∑ 𝑔𝑗𝑙𝑡
𝑎

𝑙𝑗  +  𝐴𝐶𝑏 ∑ ∑ 𝑔𝑗𝑚𝑡
𝑏

𝑚𝑗 + 𝐴𝐶𝑐 ∑ ∑ 𝑔𝑗𝑜𝑡
𝑐

𝑜𝑗 +  𝐹𝐶1𝑧𝑡 + ∑ 𝐹𝐶1
𝑖𝑖  𝑏𝑖𝑡   ∀ 𝑡                    (24) 

𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡𝑡 =  ∑ ∑ 𝑇𝐶𝑖𝑘
1

𝑘𝑖 𝑥𝑖𝑘𝑡 + ∑ ∑ ∑ 𝑇𝐶𝑖𝑗𝑘
2

𝑘𝑗𝑖 𝑦𝑖𝑗𝑘𝑡 + ∑ 𝑅𝐶𝑘  𝑝𝑘𝑡 +𝑘 ∑ 𝐸𝐶𝑘  𝑒𝑘𝑡 +𝑘 ∑ 𝑆𝐶𝑗 𝑠𝑗𝑡𝑗  

+∑ 𝐻𝐶𝑘 ∑ (𝐼𝑖𝑘𝑡 + 𝑉𝑖𝑘𝑡𝑖 ) + ∑ 𝐼𝐶𝑘
𝑎  ∑ ∑ 𝑄𝑗𝑙𝑡

𝑎
𝑙𝑗 +𝑘  ∑ 𝐼𝐶𝑘

𝑏  ∑ ∑ 𝑄𝑗𝑚𝑡
𝑏

𝑚𝑗𝑘𝑘                           ∀ 𝑡                   (25) 

3.  Model Validation 

The model was validated through COUENNE, a nonlinear solver in General Algebraic Modeling System 

(GAMS). The model was relaxed such that quantities of products and product returns are not required to have 

integer values. The model was validated using a logarithmic function for quantity of product returns and an 

exponential distribution for quality of product returns. Hypothetical values were used for the validation. 

3.1.  Quantity and quality function parameters 

The relationship between the incentive levels offered and the quantity of product returns is modeled by an 

arbitrary function. For the model validation, a logarithmic function is assigned for quantity of product returns as 

in Eq(26). The left side consisted of the quantity of product returns. A constant was placed in the equation to 

ensure that an activity should be selected such that a base amount of product returns is obtained.  

𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 +
𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
∗ log𝑏𝑎𝑠𝑒(𝑖𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝑙𝑒𝑣𝑒𝑙 + 𝑠ℎ𝑖𝑓𝑡) (26) 

A logarithmic function allows the incremental quantity received to diminish as incentive levels increase, such 

that additional incentives at high incentive levels are not as effective as in low incentive levels. This represents 

a scenario wherein there is a limit to the quantity that could be collected, regardless of any further increase of 

incentive offer. This could be associated with the actual number of products being used by consumers or the 

number of consumers willing to return their products. The “constant” term denotes the amount of product returns 

generated should no effort on the part of the firm to generate product returns be present. The function also 

52



models distance to affect this incremental quantity, as a greater distance between the distributor/retailer center 

and the consumer would lessen the incremental quantity received for a higher incentive level. Therefore, the 

“distance” term in the equation models the inversely proportional nature of the proximity of the collection center 

to the consumer. The “incentive” variable is put inside the logarithm term to make sure that as the incentive 

levels per activity increase, there is also an increment added to the total quantity of product returns. Finally, a 

“shift” term is added to the incentive level to ensure that the value inside the logarithm term would not be zero, 

which would render the equation invalid. Meanwhile, the quality of product returns is treated as a variable, which 

follows a distribution function. An exponential distribution is used. This is a conservative distribution, which 

assumes that more low-quality product returns would always be collected than high quality product returns. The 

cumulative exponential distribution is used by the model to compute probabilities of collecting product returns 

in each quality range. This follows the form shown in Eq(27). 

𝐹(𝑞) = 1 − 𝑒
−

𝑖𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
𝑖𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝑙𝑒𝑣𝑒𝑙∗ 𝛽

 𝑞
 (27) 

Wherein 𝜷 is the mean quality level and q is the quality level for which the cumulative probability is being taken. 

In order to increase quality of product returns as incentive increases, the mean quality is multiplied by the 

incentive level divided and by an incentive denominator which controls the effect of the incentive on quality. In 

this way, the mean quality of product returns increases as incentive level increases while maintaining the 

properties of an exponential distribution. The plots for the quantity functions are shown in Figure 1. The quantity 

eventually plateaus, making offering higher incentives less effective. When the distance between consumer and 

distributor/retailer center is greater, lesser quantity of product returns is obtained, across product takeback 

activities. The figure on the right shows the base quality function, which occurs at an incentive level of 0, and 

the curve at an incentive level of 80. The movement of the curve shows a higher probability of collecting high 

quality product returns at an increased incentive level. On the other hand, there is a higher probability of 

collecting low quality product returns at lower incentive levels. 

 

Figure 1: Quantity function plots 

3.2 Analysis of results 
The model is run for four time periods, the first period being an initialization period. The results are summarized 

in Figure 2. Combined sales revenue for new products sold to primary consumers consisted of products coming 

from new product production and remanufacturing recovery option, and refurbished products, which were sold 

to secondary consumers. Aside from the Sales Revenue obtained, it was also observed that the production and 

recovery costs constitute the largest portion of costs followed by cash rebate cost. Since the source of revenue 

comes from the amount of products sold to both primary and secondary consumers, the model would therefore 

prioritize allocating cost in the production of new products and recovery via refurbishing of products to be sold 

to secondary consumers. Also, it follows that cash rebate should also constitute a considerable amount of cost 

percentage because this recovery option produces a higher quantity of product returns. Another insight that can 

be gained is that the total cost exhibited a gradual increase, which reflected the effect of having an investment 

budget per period.  Since Time-period 1 is an initialization period, it does not contain any item flows between 

operational distributor/retailer facilities. In time-period 2, forward and reverse item flows begin to take place. 

One cash rebate activity is selected and one discount activity. Forward product flows are only directed through 

one distributor/retailer facility, although the other distributor/retailer facility is also opened in order to carry out 

product take-back activities. Refurbishing and remanufacturing recovery options are utilized. The third-time 

period produced similar results to the second-time period, except that two cash rebate activities are carried out 

 

 

 

 

0

20

40

60

80

100

120

140

0

1
6

3
2

4
8

6
4

8
0

9
6

1
1
2

1
2
8

1
4
4

1
6
0

1
7
6

1
9
2

2
0
8

2
2
4

2
4
0

2
5
6

2
7
2

2
8
8

Q
u
a
n
tit

y 
o
f 
P

ro
d
u
c
t R

e
tu

rn
s

Incentive Level ($)

0

0.2

0.4

0.6

0.8

1

1.2

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Cu
m
ul
at
iv
e	
Pr
ob

ab
ili
ty

Quality	Level	(%)

Base F(q) F(q) at incentive level = 80 Discount incentive (30) 
 
Rebate Incentive (30) 
 

Discount incentive (80) 
 
Rebate Incentive (80) 
 

#
 o

f 
u
n
it
s
 

53



instead of a mix of both types. Both refurbishing and remanufacturing recovery options are utilized. The fourth 

period produced similarly selected two cash rebate activities, but chose not to satisfy secondary consumer 

demand. Instead, remanufacturing was carried out on product returns. 

 

 

Figure 2: Model results  

4.  Conclusions 

A multi-period mathematical model that incorporates decisions on incentive levels, take-back program type and 

location for closed-loop supply chains was proposed in the study. The presented model can properly decide on 

standard CLSC decisions. Furthermore, Monte Carlo simulation was performed and the results revealed that 

the model produced significantly lower (p-value < 0.0001) objective function values under uncertain demand 

than in the equivalent deterministic problem. This implies that the solution delivered by the model is not able to 

achieve the objective function value promised under uncertain demand. This observation was addressed using 

the TORO procedure, which produced solutions that dominated the deterministic solution in terms of mean and 

standard deviation of objective values under uncertain demand. The procedure allows the decision maker to 

select among non-dominated solutions according to risk-appetite. 

References 

Aras N., Aksen D., 2007, Locating Collection Centers for Distance and Incentive-dependent Returns, 

International Journal of Production Economics, 111, 316-333. 

Balisado S., 2013, Modeling Mobile Phone User Preferences on Incentives and Convenience of Mobile Phone 

Take-Back Program in the Philippines: A Choice Experiment Approach, MSc Dissertation, De La Salle 

University-Manila, Philippines. 

Donmez I., Turkay M., 2013, Design of Reverse Logistics Network for Waste Batteries with an Application in 

Turkey, Chemical Engineering Transactions, 35, 1393-1398. 

Govindan K., Soleimani H., Kannan D., 2015, Reverse logistics and Closed-loop Supply Chain: A 

Comprehensive Review to Explore the Future, European Journal of Operational Research, 240, 603-626. 

Lundin J.F., 2012, Redesigning a Closed-loop Supply Chain Exposed to Risks, International Journal of 

Production Economics, 140, 596–603. 

Matter A., Ahsan M., Marback M., Zurbrugg C., 2015, Impacts of Policy and Market Incentives for Solid Waste 

Recycling in Dhaka, Bangladesh, Waste Management, 39, 321-328. 

Yamzon A., Ventura V., Guico P., Sy C., 2016, Optimal Planning of Incentive-Based Quality in Closed-Loop 

Supply Chains, Clean Technologies and Environmental Policy, 18, 1415-1431. 

	

 1 2 3 4 

Rebate activity incentive level ($) 0 0 110 132 

Discount activity incentive level ($) 0 93 0 0 

 Quantity (Units) 

Time Period 1 2 3 4 

Production 0 473 650 457 

Refurbishing 0 75 59 0 

Remanufacturing 0 45 0 254 

Cannibalizing 0 0 0 0 

Controlled Disposal 0 152 13 0 

 Capacity (Units) 

Time Period 1 2 3 4 

Production 0 650 650 650 

Refurbishing 0 75 75 75 

Remanufacturing 0 254 254 254 

Cannibalizing 0 0 0 0 

Controlled Disposal N/A N/A N/A N/A 

Units 1 2 3 4 

Refurbished 
Products 
Forward 

0 25 29 0 

Units 1 2 3 4 

Refurbished Products Forward 0 50 30 0 (Units) 1 2 3 4 

New Products Forward 0 0 0 0 

Refurbished Products Forward 0 0 0 0 

Product Returns Reverse 0 0 72 133 

Units 1 2 3 4 

New 
Products 

Forward 

0 239 400 350 

Units 1 2 3 4 

New Products Forward 0 518 600 712 

Refurbished Products Forward 0 75 60 0 

Product Returns Reverse 0 165 114 121 

Units 1 2 3 4 

New Products 
Forward 

0 279 250 362 

$ 1 2 3 4 

Rebate Incentive 0 147 172 152 

Discount Incentive 0 0 0 0 

54