# Upgrading Strategy, Warranty Policy and Pricing Decisions for Remanufactured Products Sold with Two-Dimensional Warranty

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Assumptions and Notation

#### Assumptions

- The base-warranty of an end-of-life product has expired when it is sold by a user to a dealer.
- The purchasing price of an end-of-life product depends on the reliability of the product, which is considered to be dependent on its age and usage.
- Each end-of-life product is disassembled into parts, and after an appropriate upgrading process, the parts will be reused.
- The warranty coverage offered by the dealer for remanufactured products is two-dimensional (2D), i.e., both the product’s age and usage will be used to characterise the warranty.
- Customers will have the option to choose the upgrading strategy as well as the warranty policy for the products that they purchase.
- The number of product failures during a given operation period is assumed to follow a non-homogeneous Poisson process (NHPP).
- All the product failures during warranty are repaired minimally by the dealer with no charge to the consumer.
- The time to repair a faulty item is considerably smaller than the mean inter-failure time, so the repair time is assumed to be negligible.
- Customers are considered to be the market leader, aiming to minimise their expected costs during the remaining life of the product by choosing the most affordable upgrading strategy and warranty policy.
- The dealer is considered to be the market follower, aiming to maximise his/her profit margin based on customer preferences about the upgrading and warranty.
- The demand for the remanufactured product is a nonlinear function of its selling price, as well as the expected warranty servicing costs.

## 3. Components of the Model

#### 3.1. The Purchasing Price of an End-of-Life Product from an End-User

_{p}) is a decreasing function of the product’s age (x) and/or past usage (y); this is given by:

_{0}is the sale price of the new product, c

_{sv}is the salvage value of the product at the end of its life (i.e., either at age L or usage U, whichever comes first), and b has a Beta distribution with parameters a

_{1}and a

_{2}, that are dependent on x and y. The Beta distribution parameters a

_{1}and a

_{2}are modelled by the following equations [7]:

_{1}, α

_{2}, β

_{1}, and β

_{2}are constant parameters that must be chosen such that the value of b declines with an increase of the product’s age (x) and/or usage (y). In practice, these parameters are determined using an appropriate regression model. As parameter b has a Beta distribution, the purchase price of a product with age x and usage y from an end-user can be expressed by:

_{sv}, i.e., $b=0$.

#### 3.2. The Upgrading Cost

_{0}= [0, x] × [0, y]. Therefore, the failure/repair process of the product follows a non-homogeneous Poisson process (NHPP) with a rate of occurrence of failures (ROCOF) of r(t,s) = f(t,s)/[1 − F(t,s)]. At age x and usage y, the product is subjected to an upgrading action with the upgrade level u = (u

_{t}, u

_{s}), where 0 ≤ u

_{t}≤ x and 0 ≤ u

_{s}≤ y. We assume the failure rate function after upgrading is expressed by r(t,s;u) = r (t − γu

_{t}, s − τu

_{s}), where γ and τ ∈ [0, 1] represent the effectiveness of the upgrading process on the age and usage, respectively.

_{t}, u

_{s}) is modelled by a nonlinear function, given by Equation (5):

_{s}is the fixed set-up cost of the upgrading process, and parameters c

_{u}, ζ, ψ and φ are non–negative constant values that can be estimated by means of an appropriate regression model. As the parameter b is assumed to have a Beta distribution, the expected upgrading cost will be modelled by:

#### 3.3. Dealer’s Expected Costs during the Warranty Period

_{R}(.) and G

_{R}(.), respectively.

_{w}(u

_{t}, u

_{s}, w

_{1}, v

_{1})] represent the expected number of repairs carried out by the dealer within the 2D warranty coverage region, i.e., Ω

_{w}= [x − u

_{t}, x − u

_{t}+ w

_{1}] × [y − u

_{s}, y − u

_{s}+ v

_{1}]. The warranty coverage regions for two cases of (i) r < r

_{0}, and (ii) r ≥ r

_{0}, where r

_{0}= v

_{1}/w

_{1}, are shown in Figure 1. While 1D warranties have a certain expiration time, in 2D cases, the warranty coverage begins from the point that the product is upgraded and expires at an unknown time depending on the customers’ usage rate. The usage rate r varies from one customer to another, but will remain constant for a given customer. When the warranty coverage expires, the usage of the product during the warranty coverage period changes from y − u

_{s}to s = rt (see Figure 1a). On the other hand, if the warranty coverage expires due to usage, then the age of the product during the warranty coverage period will change from x − u

_{t}to (y − u

_{s}+ v

_{1})/r (see Figure 1b). Therefore,

_{R}(.), the expected number of repairs carried out by the dealer within the 2D warranty coverage region is estimated by:

_{r}represent the average cost of a repair action carried out by the dealer during warranty coverage. Then, the total expected cost of the dealer associated with repair services during warranty coverage, E [c

_{w}(u

_{t}, u

_{s}, w

_{1}, v

_{1})] is given by:

#### 3.4. The Expected Price of Upgrading and Warranty for Customers

_{u,w}(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{u,w})], is expressed by:

_{u,w}represents the percentage of the dealer’s profit margin from offering the warranty and upgrading services to customers, c

_{u}(x,y) is the expected upgrade cost which is given by Equation (6), and E [c

_{w}(u

_{t}, u

_{s}, w

_{1}, v

_{1})] is the total expected cost of the dealer due to repair services during warranty coverage, which is given by Equation (10). Therefore, the expected selling price of the remanufactured product to customers is given by:

_{p}and E [p

_{u}

_{,w}(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{u,w})] are given by Equations (3) and (11), respectively.

#### 3.5. The Dealer’s Expected Cost for Post-Warranty Services

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1})] represent the expected number of repairs carried out by the dealer during the post-warranty coverage period. The post-warranty regions, Ω

_{pw}, for four cases are shown in Figure 2, including two cases for r < r

_{1}, where r

_{1}= [U − (y − u

_{s}+ v

_{1})]/[L − (x − u

_{t}+ w

_{1})] (see Figure 2a,b) and two cases for r ≥ r

_{1}(see Figure 2c,d). Therefore,

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1})], is given by:

#### 3.6. The Expected Price of Post-Warranty Service for Customers

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{pw})], is expressed by:

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1})] is the total expected cost of the dealer associated with repair services during the post-warranty period, which is given by Equation (16).

## 4. The Proposed Model and Solution Approach

#### 4.1. The Customer–Dealer Stackelberg Game Model

_{1}, v

_{1}and upgrade level u = (u

_{t}, u

_{s}). The customer uses the product until the maximum age limit L or usage limit U is reached. The expected cost of customers during the product use is obtained by summing the expected selling price of the product in the region ${\Omega}_{w}$ and the expected price of post-warranty coverage ${\Omega}_{pw}$. Therefore, we have:

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{u,w})] and E [p

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{pw})] are given by Equations (12) and (17), respectively.

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{u,w})] and E [p

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{pw})] are given by Equations (12) and (17), respectively; k

_{1}> 0 is a constant amplitude factor; k

_{2}> 0 is a constant for the post-warranty price, which allows for non-zero demand when there is no post-warranty coverage; $\omega >1$ is a constant parameter of the selling price elasticity; and $\mu >1$ is a constant parameter of the post-warranty price elasticity.

_{u}(x,y), E[c

_{w}(u

_{t}, u

_{s}, w

_{1}, v

_{1})], E[c

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1})] and d(E[P(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{u,w})], E[p

_{pw}(u

_{t}, u

_{s}, w

_{1}, v

_{1}, δ

_{pw})]) are given by Equations (6), (10), (16) and (19), respectively.

#### 4.2. The Solution Approach

## 5. Numerical Example

^{5}) miles, whichever comes first. The product is subject to random failures with the time to first failure following the Weibull distribution with $r\left(t,s\right)=\frac{{\beta}_{3}}{{{\alpha}_{3}}^{{\beta}_{3}}}\frac{{\beta}_{4}}{{{\alpha}_{4}}^{{\beta}_{4}}}{\left(t\right)}^{{\beta}_{3}-1}{\left(s\right)}^{{\beta}_{4}-1}$. The effectiveness of the upgrading process in terms of age and usage reduction, γ and τ respectively, are considered as two uniformly distributed random variables. The usage rate is assumed to be distributed according to a Gamma distribution with $g\left(r\right)=\frac{{\left(\lambda r\right)}^{\left(\rho -1\right)}\lambda exp\left(-\lambda r\right)}{\mathsf{\Gamma}\left(\rho \right)}$, where $\mathsf{\Gamma}\left(\rho \right)$ is the gamma function which is given by $\mathsf{\Gamma}\left(\rho \right)={{\displaystyle \int}}_{0}^{\infty}{z}^{\rho -1}{e}^{-z}dz$. We assume that u

_{t}, u

_{s}, w

_{1}and v

_{1}are decision variables to be determined such that the customer’s expected cost is minimised, and δ

_{u}

_{,w}and δ

_{pw}are decision variables to be determined such that the dealer’s expected profit is maximised. Table 1 presents a summary of the input parameter values used in the analysis.

^{5}km) miles. It was found that the minimum expected cost for customers during product use, i.e., E* (C

_{customer}) = 3679, will be achieved when the customer chooses the upgrading strategy of (u

_{t}* = 0.89, u*

_{s}= 2.49) and warranty coverage of (w

_{1}* = 2.14 years, v

_{1}* = 4.06 × 10

^{5}miles). In such a case, if the dealer chooses the percentage of the profit margin from offering the warranty coverage and upgrade services as δ

_{u,w}* = 1.49, and from the post-warranty services as δ

_{pw}* = 1.11, the demand for the remanufactured product will be estimated to be 3098, the optimal selling price of the product to be 1839, and the optimal expected profit of the dealer to be 4,956,658. The customer’s optimal strategy, the dealer’s optimal strategy, the upgrade cost, the expected number of repairs during the warranty period as well as the expected number of failures during the post-warranty period are all presented in Table 2.

- (1)
- As the past usage of the remanufactured product increases, the customer’s expected cost decreases and, as a result, demand for the product increases. This also results in an increase in the dealer’s expected profit. At lower usage levels, the rate of cost decrease and the rate of demand increase are lower than those at higher usage levels. This means that between two products having same age, customers will prefer the one with larger/longer past usage because it is less expensive. Moreover, the upgrading cost is lower compared to the purchasing price from an end user.
- (2)
- Considering the product’s reliability at the time of purchase, it is found that the reliability decreases with an increase in the product’s usage. Therefore, the purchasing price from an end user also decreases.
- (3)
- The upgrading cost increases with an increase in the upgrade level and decreases with an increase in the product’s reliability.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

L | Maximum life of a product in terms of ‘age’ |

U | Maximum life of a product in terms of ‘usage’ |

p_{0} | Sale price of a new product |

c_{sv} | Disposal or salvage value of a product |

x | Past age of the remanufactured product, where 0 < x < L |

y | Past usage of the remanufactured product, where 0 < y < U |

Ω_{0} | Past 2D life region of the remanufactured product = [0, x] × [0, y] |

c_{p} | Purchasing price of a product with age x and usage y |

u = (u_{t}, u_{s}) | Upgrading strategy, where 0 ≤ u_{t} ≤ x and 0 ≤ u_{s} ≤ y |

c_{u}(.) | Upgrading cost function |

c_{s}, c_{u}, ζ, ψ, φ | Parameters of the upgrading cost function |

w_{1}, v_{1} | Age and usage limits of the warranty coverage for the remanufactured products, where 0 < w_{1} ≤ L − x, 0 < v_{1} ≤ U − y |

r | Costumer’s usage rate, where r > 0 |

Ω_{w} | Warranty region for the remanufactured product = [x − u_{t}, x − u_{t} + w_{1}] × [y − u_{s}, y − u_{s} + v_{1}] |

Ω_{pw} | Post-warranty region for the remanufactured product = [x − u_{t} + w_{1}, L] × [r(x − u_{t} + w_{1}), U] ∪ [(y − u_{s} + v_{1})/r, L] × [y − u_{s} + v_{1}, U] |

g_{R}(.), G_{R}(.) | Probability density function (PDF) and cumulative distribution function (CDF) of variable R |

T | A random variable describing the time to failure of the product |

S | A random variable describing the usage to failure of the product |

f(t,s;u), F(t,s;u) | PDF and CDF of the variables T and S after applying the upgrading strategy u |

r(t,s;u) | Failure rate function of the variables T and S after applying the upgrading strategy u |

γ [τ] | Effectiveness of the update action in ‘age’ [‘usage’] reduction |

B(.) | PDF of Beta distribution |

c_{r} | Expected cost of a repair action at any point of time |

δ_{u,w} | Dealer’s profit margin percentage for offering the warranty and upgrading services to customers, where δ_{u,w} ≥ 0 |

δ_{pw} | Dealer’s profit margin percentage for offering the post-warranty services to customers, where δ_{pw} ≥ 0 |

c_{w}(u_{t}, u_{s}, w_{1}, v_{1}) | Dealer’s expected cost within the warranty coverage region |

c_{pw}(u_{t}, u_{s}, w_{1}, v_{1}) | Dealer’s expected cost within the post-warranty region |

## References

- Kwak, M. Optimal line design of new and remanufactured products: A model for maximum profit and market share with environmental consideration. Sustainability
**2018**, 10, 4283. [Google Scholar] [CrossRef] [Green Version] - Neto, J.Q.F.; Bloemhof, J.; Corbett, C. Market prices of remanufactured, used and new items: Evidence from eBay. Int. J. Prod. Econ.
**2016**, 171, 371–380. [Google Scholar] [CrossRef] - Wahjudi, D.; Gan, S.S.; Anggono, J.; Tanoto, Y.Y. Factors affecting purchase intention of remanufactured short life-cycle products. Int. J. Bus. Soc.
**2018**, 19, 415–428. [Google Scholar] - Shafiee, M.; Saidi-Mehrabad, M.; Naini, S.G.J. Warranty and sustainable improvement of used products through remanufacturing. Int. J. Prod. Lifecycle Manag.
**2009**, 4, 68–83. [Google Scholar] [CrossRef] - Saidi-Mehrabad, M.; Noorossana, R.; Shafiee, M. Modeling and analysis of effective ways for improving the reliability of second-hand products sold with warranty. Int. J. Adv. Manuf. Technol.
**2010**, 46, 253–265. [Google Scholar] [CrossRef] - Jalali-Naini, S.G.; Shafiee, M. Joint determination of price and upgrade level for a warranted second-hand product. Int. J. Adv. Manuf. Technol.
**2011**, 54, 1187–1198. [Google Scholar] [CrossRef] - Shafiee, M.; Chukova, S.; Yun, W.Y.; Niaki, S.T.A. On the investment in a reliability improvement program for warranted second-hand items. IIE Trans.
**2011**, 43, 525–534. [Google Scholar] [CrossRef] - Shafiee, M.; Finkelstein, M.; Chukova, S. On optimal upgrade level for used products under given cost structures. Reliab. Eng. Syst. Saf.
**2011**, 96, 286–291. [Google Scholar] [CrossRef] - Shafiee, M.; Chukova, S. Optimal upgrade strategy, warranty policy and sale price for second-hand products. Appl. Stoch. Models Bus. Ind.
**2013**, 29, 157–169. [Google Scholar] [CrossRef] - Su, C.; Wang, X. Optimizing upgrade level and preventive maintenance policy for second-hand products sold with warranty. J. Risk Reliab.
**2014**, 228, 518–528. [Google Scholar] [CrossRef] - Kim, D.-K.; Lim, J.-H.; Park, D.H. Optimal maintenance level for second-hand product with periodic inspection schedule. Appl. Stoch. Models Bus. Ind.
**2015**, 31, 349–359. [Google Scholar] [CrossRef] - Liao, B.; Li, B.; Cheng, J. A warranty model for remanufactured products. J. Ind. Prod. Eng.
**2015**, 32, 551–558. [Google Scholar] [CrossRef] - Otieno, W.; Liu, Y. Warranty analysis of remanufactured electrical products. In Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management, Detroit, MI, USA, 23–25 September 2016; pp. 734–743. [Google Scholar]
- Alqahtani, A.Y.; Gupta, S.M. Warranty as a marketing strategy for remanufactured products. J. Clean. Prod.
**2017**, 161, 1294–1307. [Google Scholar] [CrossRef] - Kim, D.-K.; Lim, J.-H.; Park, D.-H. Optimization of post-warranty sequential inspection for second-hand products. J. Syst. Eng. Electron.
**2017**, 28, 793–800. [Google Scholar] - Darghouth, M.N.; Chelbi, A.; Ait-kadi, D. Investigating reliability improvement of second-hand production equipment considering warranty and preventive maintenance strategies. Int. J. Prod. Res.
**2017**, 55, 4643–4661. [Google Scholar] [CrossRef] - Darghouth, M.N.; Chelbi, A. A decision model for warranted second-hand products considering upgrade level, past age, preventive maintenance and sales volume. J. Qual. Maint. Eng.
**2018**, 24, 544–558. [Google Scholar] [CrossRef] - Tang, J.; Li, B.-Y.; Li, K.W.; Liu, Z.; Huang, J. Pricing and warranty decisions in a two-period closed-loop supply chain. Int. J. Prod. Res.
**2019**, 58, 1688–1704. [Google Scholar] [CrossRef] - Zhu, X.; Yu, L. The impact of warranty efficiency of remanufactured products on production decisions and green growth performance in closed-loop supply chain: Perspective of consumer behavior. Sustainability
**2019**, 11, 1420. [Google Scholar] [CrossRef] [Green Version] - Cao, K.; Xu, B.; Wang, J. Optimal trade-in and warranty period strategies for new and remanufactured products under carbon tax policy. Int. J. Prod. Res.
**2020**, 58, 180–199. [Google Scholar] [CrossRef] - Shafiee, M.; Chukova, S.; Saidi-Mehrabad, M.; Niaki, S.T.A. Two-dimensional warranty cost analysis for second-hand products. Commun. Stat. Theory Methods
**2011**, 40, 684–701. [Google Scholar] [CrossRef] - Su, C.; Wang, X. Optimal upgrade policy for used products sold with two-dimensional warranty. Qual. Reliab. Eng. Int.
**2016**, 32, 2889–2899. [Google Scholar] [CrossRef] - Su, C.; Wang, X. Modeling flexible two-dimensional warranty contracts for used products considering reliability improvement actions. J. Risk Reliab.
**2016**, 230, 237–247. [Google Scholar] [CrossRef] - Alqahtani, A.Y.; Gupta, S.M. Evaluating two-dimensional warranty policies for remanufactured products. J. Remanufacturing
**2017**, 7, 19–47. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Liu, Y.; Liu, Z.; Li, X. On reliability improvement program for second-hand products sold with a two-dimensional warranty. Reliab. Eng. Syst. Saf.
**2017**, 167, 452–463. [Google Scholar] [CrossRef] - Kijima, M.; Morimura, H.; Suzuki, Y. Periodical replacement problem without assuming minimal repair. Eur. J. Oper. Res.
**1988**, 37, 194–203. [Google Scholar] [CrossRef] - Malik, M.A.K. Reliable preventive maintenance scheduling. AIIE Trans.
**1979**, 11, 221–228. [Google Scholar] [CrossRef] - Brown, M.; Proschan, F. Imperfect Repair. J. Appl. Probab.
**1983**, 20, 851–859. [Google Scholar] [CrossRef] - Shafiee, M.; Chukova, S.; Yun, W.Y. Optimal burn-in and warranty for a product with post-warranty failure penalty. Int. J. Adv. Manuf. Technol.
**2014**, 70, 297–307. [Google Scholar] [CrossRef] - Luptácik, M. Mathematical Optimization and Economic Analysis; Springer: New York, NY, USA, 2010. [Google Scholar]
- Lv, Y.; Hu, T.; Wang, G.; Wan, Z. A penalty function method based on Kuhn–Tucker condition for solving linear bilevel programming. Appl. Math. Comput.
**2007**, 188, 808–813. [Google Scholar] [CrossRef]

Product’s Maximum Lifetime | L = 10 Years, U = 10 (×10^{5}) Miles |
---|---|

Sale price of a new product | p_{0} = 2500 |

Disposal or salvage value of a product | C_{sv} = 100 |

Parameters of the Beta distribution | a_{1}= x, a_{2} = 1/y, α_{1} = 1, α_{2} = 1, β_{1} = 1, β_{2} = 1 |

Parameters of the Weibull distribution | α_{3} = 1/0.345, α_{4} = 1/0.354, β_{3} = 2, β_{4} = 2 |

Parameters of the upgrade action cost | c_{s} = 10, c_{u} = 500, ζ = 0.2, ψ = 0.25, φ = 0.25 |

Parameters of the Uniform distribution | [0, 1] |

Parameters of the Gamma function | λ = 1, ρ = 4 |

Parameters of the demand function | k_{1} = 10^{11}, ω = 1.15, µ = 1.15, k_{2} = 0.01 |

Expected cost of a repair action at any point of time | c_{r} = 100 |

**Table 2.**Customer’s optimal strategy (u

_{t}*, u

_{s}*, w

_{1}*, v

_{1}*), dealer’s optimal strategy (δ

_{u,w}*, δ

_{pw}*), and their optimal objective functions for a product with past age of x = 1 and different past usages.

x = 1 | |||||||
---|---|---|---|---|---|---|---|

y = 0.5 | y = 1 | y = 2 | y = 2.5 | y = 3 | y = 3.5 | y = 4 | |

u_{t}* | 0 | 0.78 | 0.68 | 0.89 | 0.98 | 0.28 | 0.14 |

u_{s}* | 0 | 0.03 | 1.93 | 2.49 | 2.96 | 3.49 | 3.89 |

w_{1}* | 0.80 | 2.86 | 1.90 | 2.14 | 2.18 | 1.38 | 0.95 |

v_{1}* | 9.47 | 8.15 | 6.06 | 4.06 | 3.17 | 3.00 | 2.10 |

δ_{u,w}* | 4.05 | 2.08 | 1.38 | 1.49 | 1.40 | 1.87 | 0.83 |

δ_{pw}* | 1.59 | 1.16 | 1.28 | 1.11 | 1.08 | 0.97 | 2.35 |

c_{p} | 1700 | 1300 | 900 | 785 | 700 | 633 | 580 |

c_{u}(x,y)
| 10.03 | 178 | 393 | 417 | 418 | 303 | 250 |

E[N_{w} (u_{t}*, u_{s}*, w_{1}*, v_{1}*)] | 1.47 | 2.28 | 0.32 | 0.05 | 0.01 | −0.02 | −0.02 |

E[c_{w} (u_{t}*, u_{s}*, w_{1}*, v_{1}*)] | 147 | 228 | 32 | 5.1 | 1.88 | −2.40 | −2.42 |

E[p_{u,w} (u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{u,w}*)] | 795 | 1257 | 1015 | 1053 | 1011 | 865 | 831 |

E[P(u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{u,w}*)] | 2495 | 2557 | 1915 | 1839 | 1711 | 1498 | 1411 |

E[N_{pw} (u_{t}*, u_{s}*, w_{1}*, v_{1}*)] | 9.63 | 11.8 | 8.39 | 8.71 | 8.40 | 7.60 | 7.67 |

E[c_{pw} (u_{t}*, u_{s}*, w_{1}*, v_{1}*)] | 963 | 1183 | 839 | 871 | 840 | 760 | 767 |

E[p_{pw} (u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{pw}*)] | 2495 | 2557 | 1915 | 1839 | 1748 | 1498 | 1411 |

d(E[P(u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{u,w}*)], E[p_{pw} (u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{pw}*)]) | 1537 | 1452 | 2822 | 3098 | 3568 | 4962 | 5695 |

E[Profit_{dealer} (u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{u,w*}, δ_{pw}*)] | 3,334,949 | 3,230,301 | 4,702,092 | 4,956,658 | 5,356,424 | 6,468,434 | 6,992,316 |

E[C_{customer} (u_{t}*, u_{s}*, w_{1}*, v_{1}*, δ_{u,w*}, δ_{pw}*)] | 4990 | 5114 | 3831 | 3679 | 3460 | 2997 | 2823 |

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## Share and Cite

**MDPI and ACS Style**

Baghdadi, E.; Shafiee, M.; Alkali, B.
Upgrading Strategy, Warranty Policy and Pricing Decisions for Remanufactured Products Sold with Two-Dimensional Warranty. *Sustainability* **2022**, *14*, 7232.
https://doi.org/10.3390/su14127232

**AMA Style**

Baghdadi E, Shafiee M, Alkali B.
Upgrading Strategy, Warranty Policy and Pricing Decisions for Remanufactured Products Sold with Two-Dimensional Warranty. *Sustainability*. 2022; 14(12):7232.
https://doi.org/10.3390/su14127232

**Chicago/Turabian Style**

Baghdadi, Esmat, Mahmood Shafiee, and Babakalli Alkali.
2022. "Upgrading Strategy, Warranty Policy and Pricing Decisions for Remanufactured Products Sold with Two-Dimensional Warranty" *Sustainability* 14, no. 12: 7232.
https://doi.org/10.3390/su14127232