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Journal of Cleaner Production 198 (2018) 1494e1502

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Journal of Cleaner Production

journal homepage: www.elsevier .com/locate/ jc lepro

Fuzzy multicycle manufacturing / remanufacturing production decisions considering inflation and the time value of money

Weida Chen, Li Wei*, Yaguang Li School of Economics and Management, Southeast University, Nanjing, 211189, China

a r t i c l e i n f o

Article history: Received 14 December 2017 Received in revised form 8 June 2018 Accepted 1 July 2018 Available online 17 July 2018

Keywords: Inflation Time value of money Manufacturing/remanufacturing Production decisions

* Corresponding author. E-mail addresses: [email protected] (W. Chen),

[email protected] (Y. Li).

https://doi.org/10.1016/j.jclepro.2018.07.004 0959-6526/© 2018 Elsevier Ltd. All rights reserved.

a b s t r a c t

Under the background of global low-carbon production, this paper constructed a multicycle dynamic programming model to study the fuzzy manufacturing/remanufacturing production decisions by taking into account inflation and the time value of money. Furthermore, we analysed the impact of inflation and the time value of money on manufacturing/remanufacturing production decisions by dynamic pro- gramming with the signed distance method. The results indicated that (i) the optimal production cycle is positively related to the adjusting interest rate, and when the optimal production cycle is larger than the minimum feasible cycle number, the maximum profits first increase and then decrease; and (ii) when the demand is fixed, the total production quantity of manufactured products decreases, and the total pro- duction quantity of remanufactured products increases with the increase in the adjusted interest rate.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

With the development of the global low-carbon economy, governments have introduced a series of related policies and reg- ulations, and manufacturing enterprises have begun to change their production and management concepts. Therefore, increas- ingly more manufacturing enterprises have paid attention to the remanufacturing field to protect the environment. Awide variety of products, including mobile phones, photocopiers and computers, have been covered by remanufacturing. For example, IBM resold or reused 2.4 million assets sent for refurbishment from 2010 to 2013, and it established a new remanufacturing centre in China. In addition, Xerox collected a total of 2,057,690 lbs. of broader scope products through customer equipment return programmes in the U.S. during 2015. In view of this, the efficient decisions to manage the remanufacturing system are most important, especially in manufacturing/remanufacturing systems.

In the remanufacturing system, the uncertainty of returning used products has become an element that cannot be neglected when maximizing profits (e.g., Kaya. 2010; Wang et al., 2017; Guchhait et al., 2013). Consequently, the remanufacturing system may face a mismatch between supply and demand. The uncertainty of returned products is expressed as a random parameter in prior

[email protected] (L. Wei),

academic literature. Different distributions and fuzzy sets are usually applied to indicate the stochastic returned products (e.g., Han et al., 2016; Alimoradi et al., 2015). One of the main advantages that fuzzy sets provide is the possibility of describing parameters as linguistic variables (Niknejad and Petrovic, 2014). Therefore, it is crucial to take into account uncertain returns using the fuzzy sets from both the practical and research viewpoints. In this paper, triangular fuzzy numbers are used to represent the statistical un- certainty of returned products.

Apart from the abovementioned uncertainty of the system, the costs of capital are also one of the important features of manufacturing/remanufacturing systems. Inflation and the time value of money are the concepts closely related to capital and time. In practice, the variations in the carrying costs and the selling prices of raw materials and energy, which are affected by different infla- tion rates, are important cost factors in the whole productive pro- cess (Dey et al., 2008). Therefore, inflation and the time value of money are crucial attributes of today’s unpredictable market, especially when the planning horizon is long. These attributes have profound influences on operational decisions. Many key industrial decision-makers have taken inflation into considerationwhen they make production decisions. Additionally, in most of the previous articles, inflation and the time value of money are considered in the inventory management strategy or the supply chain management strategy. However, there is little guidance about how inflation and the time value of money affect the production decision of a rema- nufacturing system, especially in amanufacturing/remanufacturing system. In reality, production and processing are the main stages

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e1502 1495

that consume resources and energy, as well as the key nodes with the greatest potential for savings and recycling waste. Hence, this phenomenon raises some questions. Under inflation and the time value of money, how should a manufacturing/remanufacturing system dynamically allocate the returned products? How much should manufacturing and remanufactured enterprises allocate in every production cycle to maximize profits?

With these motivations, we consider a manufacturing/rema- nufacturing system with inflation and the time value of money. A multi-period is set here. In each period, the quantity of the returned item is set as a triangular fuzzy number, the demand is satisfied with the serviceable inventory fulfilled by manufacturing new products and remanufactured products, and the leftover in- ventories are carried over to the next period. A dynamic pro- gramming model of fuzzy multi-cycle manufacturing/ remanufacturing production decisions, taking into account infla- tion and the time value of money, is developed to formulate the uncertainty of return quantity and the production decision prob- lem. The decision-maker jointly decides the quantity of new and remanufactured products in every period. We show that the optimal production policy is characterized by four parameters, which are the adjusted interest rate, manufacturing costs, holding costs of serviceable inventory and remanufacturing costs. Via a numerical analysis, we study the behaviour of the optimal pro- duction policy and the trend of the production cycle with respect to the inter-temporal changes in the adjusted interest rate.

The remainder of this paper is organized as follows. Section 2 describes the assumptions and notations. Section 3 presents the mathematical model and the corresponding analysis solution. In section 4, the numerical examples are presented to illustrate the model. Finally, we conclude this study and provide future research directions in Section 5.

2. Literature review

By focusing on the impact of inflation and the time value of money on the production decisions of a hybrid manufacturing/ remanufacturing system, our paper draws on two research streams: the operational policy of a hybrid manufacturing/rema- nufacturing system and the influence of inflation and the time value of money on production decisions.

Since Simpson (1978) first studied hybrid systems, there have been numerous studies that contribute to the research topic on the production planning of a hybrid manufacturing/remanufacturing system due in part to the profits and cost savings of remanufactur- ing. Based on the understanding of the characteristics of a hybrid system, these studies concentrate on the optimum policies for acquiring and classifying products subject to variability in the quality, quantity and return times on the costs, random demand, product substitution, product categorization, and so on. Papers in this research stream include Mukhopadhyay and Ma (2009), Behret and Korugan (2009), Inderfurth (2004), Teunter (2004), Feng et al. (2011), Xu et al. (2012), Kenne et al. (2012), Kouedeu et al. (2014), and Cai et al. (2014). In addition, some researchers have studied the uncertainty of a hybrid system in a fuzzy environment. Guchhait et al. (2013) and Roy et al. (2009) considered a production in- ventory problem incorporating fuzzy production. Jing et al. (2014), Alimoradi et al. (2015) and Su and Lin (2015), Su (2017) con- structed the fuzzy optimizationmodel for a remanufacturing system with different constraints. The above studies showed that fuzzy as- sumptions can address the uncertainty of remanufacturing in an efficient way. Some literature take into account the integration of resource capacity, stochastic demand, returns, setup costs, and pricing and production decisions for a hybrid system with a single- period setting. For example, seeMitra (2016) and Kim et al. (2013). In

particular, Kwak and Kim (2017) presented a model that integrated pricing and production planning for a line of new and remanufac- tured products in a competitive market. In addition, the multicycle production decision of manufacturing/remanufacturing has been studied in many literature. Francie et al. (2015) investigated the production rate controls of a hybrid system. Han et al. (2016), Fang et al. (2017) and Polotski et al. (2017) investigated a production planning problem of a hybrid manufacturing/remanufacturing sys- tem. Similarly, Hilger et al. (2016) considered a stochastic dynamic multi-product capacitated lot sizing problem with manufacturing and remanufacturing under demand and return uncertainty. Most of the above papers focused on the integrated decision-making of a hybridmanufacturing/remanufacturing systemwith the uncertainty or the variability in a hybrid system, which are the characteristics of a hybrid system. These systems include methodologies such as dy- namic programming. Similar to these papers, we also assume a homogeneous product base and use a multicycle dynamic pro- gramming model to describe the problem. In contrast, we consider an additional impact factor of the economic environment, which includes inflation and the time value of money. Inflation is a concept closely related to time, and there is no analytical solution to the proposed model. Therefore, the problem is solved using a dynamic programming method combined with a one-dimensional search method. To some degree, our model examines the issue of when and how the manufacturing and remanufactured enterprises should produce remanufactured items during every production cycle under the constraints of inflation and the time value of money. To our knowledge, the influences of inflation and the time value of money on the production decisions of a hybrid system under a multicycle setting have not been investigated.

In practice, many manufacturing enterprises have realized the influence of inflation and the time value of money on production decisions and have started to integrate production and capital op- erations. Furthermore, many researchers have taken into account financial factors in conducting optimization research. Misra (1979) divided inflation into interior inflation and exterior inflation and analysed the influence of the interest rate and inflation on replenishment. Furthermore, Taheri-Tolgari el al. (2012), Gilding (2014), Pal el al. (2014), and Bhunia et al. (2015) investigated the optimal inventory replenishment policy under inflation and other constraints. Zhang et al. (2007) extended the Fisher Equation under the risk-neutral measure and used the martingale method to derive the optimal allocations. In addition, Guill�en et al. (2006) and Alikar et al. (2017) considered a multi-component multi-period mathe- matical model with inflation. Outside the inventory management domain, the effects of inflation on supply chain management under a fuzzy environment have been studied in a number of articles. Nagaraju et al. (2016) demonstrated the optimality of the cycle time and inventory decisions under the phenomena of different inflation rates at the supply chain management points. Braglia et al. (2016) proposed a novel approach to stock management in a coordi- nated supply chain using present value. Mondal el al. (2013) and Yadav et al. (2015) examined the total profits under inflation in fuzzy environments. Several prior studies also analysed inflation and the time value of money for different purposes other than the operation strategy design. Darwish (2006) and Pal el al. (2015) focused on the optimal production and inventory decision using a production inventory model that integrated inflation. Sarkar el al. (2011), Sarkar and Moon (2011) examined the retailer’s optimal replenishment policy, production policy and inventory policy under inflation. Although the studies on inflation and the time value of money are vast, all the studies above consider the influence of inflation and the time value of money on inventory management and supply chain management. Different from the aforementioned

Parameters r interest rate, which represents the time value of money i inflation rate t adjusted interest rate, t ¼ r� i

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e15021496

research, our paper first focuses on a hybrid manufacturing/rema- nufacturing system, which has more uncertainty than a classic manufacturing system. In particular, a fuzzy set is used to describe the uncertainty of the returned products. Second, we explore the production decision of a hybrid system with integrated inflation and the time value of money. Our model investigates how firms can more effectively utilize remanufacturing to gain more profits under inflation and under the time value of money. To the best of our knowledge, the existing literature does not discuss the fuzzy mul- ticycle manufacturing/remanufacturing production decision considering inflation or the time value of money. Our paper com- plements the hybrid manufacturing/remanufacturing system literature by focusing on the effects of inflation and the time value of money on the operational decision, highlighting that inflation and the time value of money play important roles in the optimal manufacturing/remanufacturing product portfolio and in profits.

u sale price H the length of the planning horizon T the length of the production cycle N number of production cycles, N ¼ H=Tand N � NL NL the lower bound of N Pm production rate in the manufacturing process, unit/year Pmk production rate in the manufacturing process in the production cycle k PM the upper bound of Pm

Pr production rate in the remanufacturing process, unit/year Prk production rate in the remanufacturing process in the production cycle k Dk demand rate in the production cycle k, unit/unit time IkðtÞ serviceable inventory (SI) of manufactured and remanufactured products

in the cycle k at time t Mk recoverable inventory (RI) in cycle k at time t TPk total quantity of manufactured and remanufactured products in the

production cycle k, TPk ¼ mk þ rk Tk the total time lapsed in the beginning of production cycle k Im the upper bound of the serviceable inventory Is the lower bound of the serviceable inventory Mm the upper bound of the recoverable inventory cm manufacturing costs, dollar/unit cr remanufacturing costs, dollar/unit c1 holding costs of serviceable inventory, dollar/unit/unit time c2 holding costs of recoverable inventory, dollar/unit/unit time Sm set-up costs of the manufacturing process Sr set-up costs of the remanufacturing process Decision Variables mk quantity to be manufactured in the production cycle k rk quantity to be remanufactured in the production cycle k Rk return items, triangular fuzzy number Rk ¼ ðerk � D1; erk; erk þ D2Þ;

3. Problem description, assumptions and notations

3.1. Problem description

In this paper, we consider that a remanufacturer produces new products and remanufactured products simultaneously in a finite planning horizon. The planning horizon H is divided into N pro- duction cycles. At t ¼ 0, the serviceable inventory Ikð0Þ is zero. In every cycle time k, demand Dk is satisfied by the serviceable in- ventory IkðtÞ, which is periodically replenished by new products and remanufactured products, and the production rates in the manufacturing/remanufacturing processes are Pmk and P

r k, respec-

tively. In the production processes of each lot size of goods in the production process, fixed setup costs of manufacturing and rema- nufacturing ðSm; SrÞ are incurred for each set-up. The optimal policy is decided as follows. The remanufacturer views the serviceable inventory (SI) and the recoverable inventory (RI), and then simul- taneously determines the manufacturing quantity mk, the rema- nufacturing quantity rk and the return item quantity Rk. In addition, the process of manufacturing and remanufacturing decisions takes into account inflation and the time value of money to meet the market demand. The SI and RI inventory levels during the planning period are shown in Fig. 1.

0<D1 < erk; D2 >0;D1;D2 decided by decision makers

3.2. Assumptions

(1). Producing a new product only needs a returned product.

Fig. 1. SI and RI inventory levels.

(2). Manufacturing costs are higher than remanufacturing costs. (3). Manufactured products and remanufactured products are

homogenous, and the sale prices are equal. (4). Shortages are not permitted. (5). The quantity of returns is a triangular fuzzy number. (6). The setup costs of the manufacturing process are higher than

the remanufacturing process.

3.3. Notations

4. Model formulations and solution

4.1. Mathematical model

As mentioned above, we can obtain that Tk ¼ kT . When t ¼ 0, then IðtÞ ¼ 0. When t ¼ TN , then IðtÞ ¼ 0. In this fuzzy multi-cycle dynamic programming, the state variable is a vector of two ele- ments, which contains the recoverable inventory and the service- able inventory. Therefore, the state variable can be written as follows:

sk ¼ ðMk; IkðTkÞÞ (1) The decision variable is a vector of two elements, which contains

the quantity to be manufactured and the quantity to be remanu- factured. Therefore, the decision variable can be written as follows:

xk ¼ ðmk; rkÞ (2) In the production cycle k, the serviceable inventory at time t can

be written as follows:

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e1502 1497

IkðtÞ¼ IkðTkÞþ Zt 0

� dmk p

m k þdrkprk

� da�Dk;0<t<T;ðk¼1;2;/N�1Þ

¼ IkðTkÞþdmk pmk $tþdrkprk$t�Dk ¼ IkðTkÞþ

� dmk p

m k þdrkprk

� $t�Dk

(3)

Where

pmk ¼ pm

N ; prk ¼

pr

N

Iðt ¼ 0Þ ¼ Iðt ¼ HÞ ¼ 0

dmk ¼ � 1 when mk >0 0 when mk ¼ 0

drk ¼ � 1 when rk >0 0 when rk ¼ 0

In the production cycle k, the present value of the holding costs of the serviceable inventory can be written as follows:

Cmh ¼ c1 ZT 0

IðtÞe�ttdt;ðk¼1;2;/N�1Þ

¼ c1$ dmk p

m k þdrkprk t

1�e�tT

t �Te�tT

! � IkðTkÞ�Dk

t $ � 1�e�tT

�! (4)

In the production cycle k, the present value of the holding costs of the recoverable inventory can be written as follows:

Crh ¼ c2 ZT 0

Mke �ttdt; ðk ¼ 1;2;/N � 1Þ

¼ c2$Mk t

$ � 1� e�tT

� (5)

where

Mkþ1 ¼ Mk þ Rk � rk; ðk ¼ 1;2;/N � 1Þ

rk ¼ ZT 0

� drk$p

r k

� dt; ðk ¼ 1;2;/N � 1Þ

¼ drk$prk$T To simulate the returned items, we now consider the triangular

fuzzy number

Rk ¼ ð~rk � D1;~rk;~rk þ D2Þ (6)

where 0<D1 <~rk and D2 >0. The membership grade of Rk is 1 at point ~rk. The grade is decreasing as the point moves away from ~rk, and it reaches 0 at the end points of ~rk � D1 and ~rk þ D2.

By utilizing the signed method to defuzzify Rk, we obtain

d � Rk; ~01

� ¼ 1

2

Z1 0

�ð~rkÞLðaÞ þ ð~rkÞUðaÞ�da ¼ 1

4 ½ð~rk � D1Þ þ 2$~rk þ ð~rk þ D2Þ�

¼ ~rk þ 1 4 ðD2 � D1Þ

(7)

In the production cycle k, the present value of production costs can be written as follows:

Cp ¼ � cmd

m k p

m k þ crdrkprk

� e�tT (8)

In the production cycle k, the present value of revenue can be written as follows:

P ¼ u$Dke�tT (9) In the production cycle k, the present value of the setup costs of

the manufacturing and remanufacturing processes can be written as follows:

Cs ¼ � dmj Sm þ drj Sr

� e�tT (10)

In the production cycle k, the present value of total profits can be written as follows:

pkðsk; xkÞ ¼ pkðMk; IkðTkÞ;mk; rkÞ ¼ P � Cmh � Crh � Cp � Cs

(11)

The objective is to determine the optimal production decision mk ; rkðk ¼ 1;2;/NÞ to maximize the total profits during N periods. Therefore, the objective function is defined as follows:

f *k ðskÞ ¼ max XN i¼k

� e�tði�kÞT$piðsi; xiÞ

� (12)

In the dynamic programming, there is a recursive relationship as follows:

f *k ðskÞ ¼ max h pkðsk; xkÞ þ e�tT$f *kþ1ðskþ1Þ

i (13)

f *k ðskÞ ¼ max h pkðMk; IkðTkÞ;mk; rkÞ þ e�tT$f *kþ1ðMk; IkðTkÞÞ

i (14)

where f *NðsNÞ ¼ 0 Hence, the above model can be formulated as follows:

max h pkðMk; IkðTkÞ;mk; rkÞ þ e�tT$f *kþ1ðMk; IkðTkÞÞ

i (15)

s:t

pmk ðtÞ � pM (16)

Mkþ1 ¼ Mk þ Rk � rk ðk ¼ 1;2;/N � 1Þ (17)

0 � Mk � Mm ðk ¼ 1;2;/NÞ (18)

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e15021498

Is � IkðtÞ � Im; t2½Tk; Tkþ1� ðk ¼ 1;2;/N � 1Þ (19)

IkðTkÞ þmk þ rk � Dk; t2½Tk; Tkþ1� ðk ¼ 1;2;/N � 1Þ (20)

Ikþ1ðTkþ1Þ ¼ IkðTkÞ þmk þ rk � Dk ðk ¼ 1;2;/N � 1Þ (21)

vpkðsk; xkÞ vprk

¼ c1$ drk t

1� e�tT

t � Te�tT

! � T$d

r k

t

� 1� e�tT

� þ c2$d

r k

t $ � 1� e�tT

� � crd

r k

t

� 1� e�tT

� ¼ 0

(28)

rk � Mk; t2½Tk; Tkþ1� ðk ¼ 1;2;/N � 1Þ (22)

0< cr < cm (23)

0< Sr < Sm (24)

Constraint (16) represents the upper bound of the production rate in the manufacturing process. Constraint (17) assures the balance of the recoverable inventory between returns and pro- duction. Constraint (18) is the recoverable bounds in production cycle k. Constraint (19) represents the serviceable inventory bounds in production cycle k. Constraint (20) assures that there is no stock- out. Constraint (21) assures the serviceable inventory balance. Constraint (22) is the upper bound of the quantity of the remanu- factured product. Constraint (23) assures that the remanufacturing costs are less than the manufacturing costs. Constraint (24) assures that the remanufacturing setup costs are less than the manufacturing costs.

4.2. The model solution

To obtain the optimal value of the objective function when N is deterministic, first, pmk ; p

r k in each production cycle are determined.

The process of the solution is as follows:

vpkðsk; xkÞ vpmk

¼ c1$ dmk t

1� e�tT

t � Te�tT

! � T$d

m k

t

� 1� e�tT

� � cmd

m k

t

� 1� e�tT

� ¼ 0

(25)

Then,

c1$ dmk t

1� e�tT

t � Te�tT

! � T$d

m k

t

� 1� e�tT

� � cmd

m k

t

� 1

� e�tT �

¼ 0 (26)

When dmk s0,

c1$ 1 t

1� e�tT

t � Te�tT

! � T

t

� 1� e�tT

� � cm

t

� 1� e�tT

� ¼ 0

(27)

In the above formulation, it is obvious that there is no analytical solutionwith respect to T since it is a high-power expression of the exponential function. However, T ¼ HN, and the production cycleN is discrete. Therefore, we can utilize one-dimensional search theory to obtain the maximum profits.

Similarly, there is no analytical solutionwith respect to T as well. Therefore, we can solve the problem with the dynamic program- ming method combined with the one-dimensional search method.

5. Numerical example implementation

In this section, some numerical experiments are presented to obtain the same managerial insights, which cannot be derived by the theoretic analysis. By combining the datawith investigating the remanufacturers in China and the actual situation in practice, we suppose the base parameters as follows: H ¼ 2 years; cm ¼ 5$=unit=year; cr ¼ 2$=unit=year; c1 ¼ 40$=unit; c2 ¼ 30$=unit; sm ¼ 120$; sr ¼ 80$; t ¼ 0:02; Im ¼ 50unit; Is ¼ 15unit; u ¼ 120$; pm ¼ 135unit=year; the total demandD ¼ 150 unit=year, ~rk � Nð40;2Þ, D3 � Nð32;2Þ, and D4 � Nð42;2Þ. Then, we can obtain the maximum profits as 22528$ and the optimal production cycle as N ¼ 5.

The impact of the production cycle N is shown in Table 1 and Fig. 2.

As shown in Table 1 and Fig. 2, the maximum profits increase with the increase of production cycles; these profits attain a maximum limit and then decrease as the production cycles in- creases. We can obtain the maximum profits of 22528$ when the planning horizon is divided into 5 cycles, and each cycle is 0.4 years. This outcome occurs because the increase of the production cycle initially increases the serviceable inventory level as the demand is fixed, which increases the profits. When the serviceable inventory level increases too far beyond the market demand, the inventory costs will increase. Therefore, the firm can appropriately lower the production cycle to capture the benefits from the manufacturing and remanufacturing processes.

Then, we can get the optimal manufacturing/remanufacturing production decisions shown in Table 2.

Table 2 shows the values of the SI and RI, as well as the manufacturing quantities, remanufacturing quantities and profits of each production cycle, when the optimal production cycle is 5. At the end of production cycle 1, SI is 0, the manufacturing quantity is 45, the remanufacturing quantity is 30, and the market demand is 60. After meeting the market demand, SI is 15 at the beginning of production cycle 2. We use a similar approach to deduce the way to calculate production cycles 2, 3 and 4, and the obtained solutions conform to different binding conditions. In production cycle 5, the

Table 1 The impact of production cycle on the maximum profits.

t ¼ 0:02 Production cycles N The length of each cycle T/years Demand in each cycle Maximum profit

3 0.67 100.00 22454 4 0.50 75.00 22486 5* 0.40 60.00 22528*

6 0.33 50.00 22473 7 0.29 42.86 22400 8 0.25 37.50 22393 9 0.22 33.33 22207 10 0.20 30.00 22180 11 0.18 27.27 22090 12 0.17 25.00 21677

3 4 5 6 7 8 9 10 11 12 Production cycles/year

2.16

2.17

2.18

2.19

2.2

2.21

2.22

2.23

2.24

2.25

2.26

M ax im um

pr of it/ $

104

Fig. 2. The impact of production cycle on the maximum profits.

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e1502 1499

total number of SI, manufacturing and remanufacturing is 60, which exactly equals the market demand. Thus, at the end of pro- duction cycle 5, SI is 0. The solution satisfies the constants of the proposed model.

In the next part, wewill analyse the variation trend of maximum profits by varying the adjusted interest rate from 0.005 to 0.3. The variation trend of the maximum profits is shown in Table 3.

As shown in Table 3, we can obtain the maximum profits when the planning horizon is divided into 3 production cycles and when t ¼ 0:005. Similarly, the maximum profits are obtained under different adjusted interest rates. Obviously, the maximum profits are negatively related to the adjusted interest rate. Moreover, the optimal production cycle increases from 3 to 8 as the adjusted in- terest rate increases. The optimal production cycle and variation trend of the maximum profits are shown in Fig. 3a and Fig. 3b,

Table 2 The optimal manufacturing/remanufacturing production decisions.

t ¼ 0:02 Production cycle SI RI Manufactured products

1 0 30 45 2 15 19 41 3 15 21 39 4 15 19 41 5 15 16 29 In total 60 105 195

when the adjusted interest rate varies from 0.005 to 0.3. As shown in Fig. 3a, when the production cycle is in accordance

with the constraints of the equipment and stuff, the optimal pro- duction cycle increases as the adjusted interest rate increases. Since the production cycle N is discrete, some optimal production cycles may be the same under different adjusted interest rates, but the overall trend is that the optimal production cycle increases as the adjusted interest rate increases. Since, as the adjusted interest rate increases, the total costs of the system increase, this is why the optimal production cycle of the system increases. Thus, the enter- prise can undertake more production cycles to protect itself against capital losses when the adjusted interest rate increases.

As shown in Fig. 3b, the maximum profits decline when the adjusted interest rate varies from 0.005 to 0.3. This result is attributed to the fact that the interest rate has a deep relationship with the production costs. Therefore, an increase in the rate of inflation causes the total profits of the hybrid system to decrease. This change in the system is very appreciable. Thus, the change in the adjusted interest rate will lead to positive changes in the pro- duction cycle and profits. Furthermore, the production cycle and profits are all very sensitive to changes in the parameters of the adjusted interest rates.

Based on the above data, we can obtain the optimal manufacturing/remanufacturing production quantities when the adjusted interest rate varies from 0.005 to 0.3, which are shown in Table 4 and Fig. 4.

As shown in Table 4 and Fig. 4, the total quantity of the manu- factured product decreases, and the total quantity of the remanu- factured product increases, as the adjusted interest rate increases. It is obvious that inflation and the time value of money have considerable impacts on the optimal manufacturing/remanu- facturing production decisions. This result is because the manufacturing costs are more than remanufacturing costs. There- fore, the rate of increase in manufacturing costs is more than the rate of increase in remanufacturing costs, which is the reason for the change of the remanufacturing quantities. Additionally, it is evident that SI and RI increase with the adjusted interest rate. The reason for this outcome is that the enterprise is under pressure

Remanufactured products Maximum Profit in each cycle

30 4171.4 19 4650.2 21 4671.4 19 4654.1 16 5240.8 105

Table 3 The maximum profits versus production cycles under different AIR.

Adjusting interest rate Production cycle N length of each cycle T/years Demand in each cycle Maximum profit

0.005 3* 0.67 100 23661*

4 0.5 75 23592 5 0.4 60 23448 6 0.33 50 23331 7 0.29 43 23325 8 0.25 38 23288

0.01 3* 0.67 100 23272*

4 0.5 75 23244 5 0.4 60 23054 6 0.33 50 22921 7 0.29 43 22878 8 0.25 38 22808

0.015 3 0.67 100 22887 4* 0.5 75 22929*

5 0.4 60 22806 6 0.33 50 22716 7 0.29 43 22681 8 0.25 38 22571 9 0.22 33 22473 10 0.2 30 22024 11 0.18 27 21472 12 0.17 25 21143

0.025 3 0.67 100 22157 4 0.5 75 22179 5* 0.4 60 22199*

6 0.33 50 22134 7 0.29 43 22072 8 0.25 38 22066 9 0.22 33 22043 10 0.2 30 21887 11 0.18 27 21830 12 0.17 25 21311

0.1 3 0.67 100 17685 4 0.5 75 17878 5 0.4 60 17964 6* 0.33 50 18003*

7 0.29 43 17924 8 0.25 38 17865

0.3 3 0.67 100 11035 4 0.5 75 11229 5 0.4 60 11317 6 0.33 50 11375 7 0.29 43 11421 8* 0.25 38 11518*

9 0.22 33 11395 10 0.2 30 11149 11 0.18 27 11084

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Adjusting interest rate

1

2

3

4

5

6

7

8

9

O pt im al pr od uc tio n cy cl es /y ea r

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Adjusting interest rate

0.5

1

1.5

2

2.5

M ax im um

pr of it/ $

104

Fig. 3. a The optimal production cycle when the adjusted rate varies from 0.005 to 0.3. b The maximum profits when the adjusted interest rate varies from 0.005 to 0.3.

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e15021500

Table 4 Optimal manufacturing/remanufacturing production decisions.

Adjusting interest rate Production cycle SI RI Manufactured products Remanufactured products Cycle maximum Profit

0.005 1 0 30 85 30 7354 2 15 26 74 26 7909.4 3 15 26 59 26 8550.1 In total 30 82 218 82

0.01 1 0 30 85 30 7284.6 2 15 30 70 30 7876.1 3 15 30 55 30 8515.4 In total 30 90 210 90

0.015 1 0 30 60 30 5336.4 2 15 26 49 26 5891 3 15 21 54 21 5840 4 15 23 37 23 6487.4 In total 45 100 200 100

0.025 1 0 30 45 30 4153.4 2 15 16 44 16 4599.8 3 15 19 50 19 4259.5 4 24 20 54 20 4045.3 5 38 22 0 22 6289.2 In total 82 107 193 107

0.1 1 0 30 35 30 3196 2 15 19 31 19 3650.8 3 15 15 35 15 3614.5 4 15 14 36 14 3604.6 5 15 15 35 15 3613.2 6 15 16 19 16 4200 In total 75 109 191 109

0.3 1 0 30 22 30 2051.2 2 15 11 26 11 2396 3 15 12 25 12 2405.3 4 15 10 27 10 2387.2 5 15 13 24 13 2414.1 6 15 9 28 9 2377.9 7 15 15 22 15 2430.9 8 15 13 13 13 2900.6 in total 105 113 187 113

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Adjusting interest rate

50

100

150

200

250

M an uf ac tu re d/ re m an uf ac tu re d qu an tit ie s

Total manufactured products Total remanufactured products

Fig. 4. Manufacturing/remanufacturing production quantities when the adjusted in- terest rate varies from 0.005 to 0.3.

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e1502 1501

from capital, with the adjusted interest rate increasing. Hence, the enterprise can earn more profits by remanufacturing to meet the market demand. Thus, the enterprise will take effective measures to improve the return rate. From a managerial point of view, if the adjusted interest rate is high, the enterprise can lower their manufacturing quantity to capture the benefits from remanufacturing.

6. Conclusions

This paper focused on the enterprise’s multi-cycle manufacturing/remanufacturing production decisions by taking into account inflation and the time value of money, and a dynamic planningmodel was constructed tomaximize the remanufacturer’s expected profits. Then, the signed distance method combined with the dynamic method was developed to obtain the optimal pro- duction cycle and the total production quantity. In this paper, some numerical experiments were employed to explore the impacts of inflation and the time value of money on manufacturing/remanu- facturing production decisions. In addition, the variation trend of the maximum profits and the optimal production cycle under different adjusted interest rates were analysed.

Our key findings are summarized as follows. First, the change in the optimal production cycle will lead to the positive changes in the adjusted interest rate. Therefore, the increase of the adjusted in- terest rate can force the enterprise to plan for more production cycles, to some degree. Moreover, when the optimal production cycle is larger than the minimum feasible cycle number, the maximum profits first increase and then decrease. Second, when the demand is fixed, the enterprise may have less incentives to produce the manufactured products but high enthusiasm to pro- duce remanufactured goods, with the increase in the adjusted in- terest rate. Thus, the enterprise should adjust their manufacturing/ remanufacturing production decision according to the changes in the adjusted interest rate.

Our work, however, has some limitations. First, we assumed that the quantity of returns is a triangular fuzzy number. Future research should investigate the case of more generalized return quantities.

W. Chen et al. / Journal of Cleaner Production 198 (2018) 1494e15021502

Second, we assumed that shortages are not permitted, while the case of the shortage being backordered can be considered further, which may be more practical. Finally, we also assumed that the quality of manufacturing production is the same as that of rema- nufacturing production. Exploring the customer’s acceptance of new and remanufacturing products will be of interest for academic research.

Acknowledgements

We gratefully acknowledge the support of the National Natural Science Foundation of China, Research Fund [grant numbers 71571042, 71271054, and 71501046], and the Scientific Research Innovation Project for College Graduates in Jiangsu Province [grant number CXZZ12_0133].

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