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Optimal policy for a closed-loop supply chain inventory system with remanufacturing
Mathematical and Computer Modelling 48 (2008) 867–881 www.elsevier.com/locate/mcm
Optimal policy for a closed-loop supply chain inventory system with remanufacturing Shen-Lian Chung a , Hui-Ming Wee b,∗ , Po-Chung Yang c a Department of Information Management, St. John’s University, Tamsui, Taipei 25135, Taiwan b Department of Industrial Engineering, Chung Yuan Christian University, Chungli 32023 Taoyuan, Taiwan c Industrial Engineering and Management Department, St. John’s University, Tamsui, Taipei 25135, Taiwan
Received 6 June 2007; received in revised form 23 October 2007; accepted 2 November 2007
Abstract Previous researchers have developed ways of managing forward-oriented supply chains, and gave insight to solve single-stage inventory systems. In this study, we analyze an inventory system with traditional forward-oriented material flow as well as a reverse material flow supply chain. In the reverse material flow, the used products are returned, remanufactured and shipped to the retailer for resale. A multi-echelon inventory system with remanufacturing capability is proposed. We then develop a closed-loop supply chain inventory model and maximize the joint profits of the supplier, the manufacturer, the third-party recycle dealer and the retailer under contractual design. The analytical results of this study show a significant increase in the joint profit when the integrated policy is adopted. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Supply chain management; Closed-loop supply chain; Multi-echelon inventory; Reverse logistics
1. Introduction In order to meet environmental concerns/regulations, manufacturers often attempt to recover the residual value of their used products through remanufacturing. Product remanufacturing such as transforming used items into marketable products through refurbishment, repair and upgrading can also yield substantial cost benefits. Most of the former researchers have developed methods of managing forward-oriented supply chains to optimize single-stage inventory systems. Clark and Scarf [2] were among the first authors to propose a two-echelon inventory model. They developed a system for optimizing base stock in serial inventory systems and achieved an efficient method for establishing an ordering policy for an optimal base inventory. Shu [17] and Goyal [6,7] presented an optimal solution to determine the economic packaging frequencies of items jointly replenished. Many researchers have investigated an integrated approach to price discounting [5,13]. Lu [11] developed an optimal one-vendor multi-buyer integrated model. Hill [8] offered a more general policy for a single-vendor, single-buyer production-inventory model based ∗ Corresponding author. Tel.: +886 3 456 6142; fax: +886 3 456 3171/4499.
E-mail address: [email protected] (H.-M. Wee). c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.11.014
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on successive shipments of a single production batch size. Several researchers (Viswanathan, 1998; Goyal and Nebebe [3]; Goyal [3]) integrated the proposed strategies of Lu [11], Goyal [4] and Hill [8] to obtain an improved relevant inventory cost model. Yang and Wee [19] developed an arborescent model that integrated considerations of the producer, distributor and retailer. Yang and Wee [20,18] and Yang [21] developed an integrated vendor–buyer model for optimizing the delivery number and lot size of deteriorating items. Schrady [16] was the earliest author to propose a deterministic model with infinite production rates for manufacturing and remanufacturing. Schrady argued that optimal lot sizes for manufacturers and remanufacturers can be determined by the classical EOQ formula. The following authors extended Schrady’s analysis: Nahmias and Revera [14] examined finite remanufacturing rates; Mabini et al. [12] considered stockout service level constraints and a multi-product model; Koh et al. [10] analyzed finite manufacturing/remanufacturing rates. Further, Inderfurth, Lindner and Rachaniotis [9] investigated lot-sizing decisions in a hybrid production/rework system characterized by defective products. Several other studies have addressed collection issues. Savaskan et al. [15] determined the optimal collection channel configuration of a monopolist manufacturer. Bautista and Pereira [1] proposed a method of identifying these collection area problems and established the relationship between the set covering problem and the MAX-SAT problem. The present study analyzes a closed-loop supply chain inventory system. In addition to traditional forward material flows, the model examines used products returned to a reconditioning facility where they are stored, remanufactured then shipped back to retailers for retail sale. Fig. 1 illustrates the proposed process. The proposed optimal policy for a multi-echelon inventory system with remanufacturing is developed by integrating the concerns of the supplier, the manufacturer, the retailer and the third-party recycle dealer. An example of the closed-loop supply chain with a remanufacturing system illustrates the system. The solutions are obtained for both decentralized and integrated centralized decision making. The findings of this study demonstrate that the proposed integrated centralized decisionmaking approach can substantially improve efficiency. 2. Assumptions and notations The mathematical models in this analysis have the following assumptions: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
An infinite planning horizon. No deterioration. No stock shortages. The manufacturing and remanufacturing rates and constant lead-times. The multi-echelon inventory system contains a single item. The product annual demand rate and the annual return rate are constant, and the annual return rate is less then the annual demand rate. No space, capacity or capital constraints. No quantity discount. The number of deliveries within the manufacturing cycle is an integer. The setup cost per run and the annual holding cost fraction are known and constant. To meet the retailer demand, the remanufactured products are available before the manufactured products. The remanufactured products are comparable to newly manufactured products. Single suppliers, single manufacturers, single retailers and single third parties in the closed-loop supply chain multi-echelon inventory system are considered.
The following denotes the retailer’s parameters: D Ar Fr Pc Pr Tr TCr TPr
annual demand rate ordering cost to the retailer inventory holding cost to retailer in annual percentage per dollar retail price wholesale price retailer ordering cycle time (decision variable) total relevant cost to the retailer per unit time total retailer profit per unit time
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Fig. 1. The integrated closed-loop supply chain inventory system.
The following denotes the manufacturer’s parameters: setup cost per manufacturing run setup cost per remanufacturing run ordering cost for the manufacturer material warehouse ordering cost for the manufacturer used product warehouse product inventory holding cost percentage per year per dollar to the manufacturer material inventory holding cost percentage per year per dollar for the manufacturer warehouse used product inventory holding cost percentage per year per dollar for the manufacturer warehouse manufacturer unit purchase price from the supplier manufacturer unit purchase price from the third party manufacturer reproduction period in each remanufacturing cycle manufacturer non-production period in each remanufacturing cycle manufacturer production period in each manufacturing cycle manufacturer non-production period in each manufacturing cycle number of deliveries per manufacturing cycle time from the manufacturer to the retailer (decision variable) number of deliveries per remanufacturing cycle time from the manufacturer to the retailer (decision variable) number of deliveries per remanufacturing/manufacturing cycle time from the manufacturer to the retailer, I = m + n, where I is a positive integer M fixed cost to manufacturer for processing buyer order of any size P annual manufacturer production rate (P > D) R annual manufacturer reproduction rate (R > D) TCm total relevant cost per unit time to the manufacturer TC Mw total relevant cost per unit time for the material warehouse TC Rw total relevant cost per unit time for the used product warehouse TPm total manufacturer profit per unit time AM AR A Mw A Rw FM FMw FRw PM PR TR1 TR2 TM1 TM2 m n I
The following denotes the third party’s parameters: C A3 F3 P3 k K TC3 TP3
annual return rate setup cost per run for the third party used product inventory holding cost percentage per year per dollar to the third party third party collecting unit cost from the consumer number of deliveries per TR1 from the third party to the manufacturer (decision variable) fixed third-party cost to process manufacturer orders of any size; total relevant cost to third party per unit time total third-party profit per unit time
The following denotes the supplier’s parameters: As fixed cost to supplier per order Fs material inventory holding cost to supplier, in percentage per year per dollar
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Fig. 2. The integrated manufacturer and retailer multi-delivery inventory model with remanufacturing.
Ps l L TCs TPs
supplier purchase unit price number of deliveries per TM1 from the supplier to the manufacturer (decision variable) fixed supplier cost to process manufacturer orders of any size total relevant supplier cost per unit time total supplier profit per unit time
The following denotes the objective function of the optimization problem: JP(l, k, n, m, Tr ) joint profit per unit time for the whole system 3. Model development For a continuous review system, Fig. 2 represents the manufacturer–retailer inventory system. The average inventory of the retailer is DT r /2, and the total relevant inventory cost for the retailer can be expressed as follows: TCr (Tr ) =
Fr Pr DTr Ar + + Pr D. Tr 2
(1)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Total retailer profit is the sales revenue minus total retailer cost. It is given as follows: Fr Pr DTr Ar + Pr D . (2) TPr (Tr ) = Pc D − + Tr 2 When coordination is considered, the total relevant cost per unit time for the remanufacturing and manufacturing, TCm (n, m, Tr ), as shown in Appendix A consists of the setup cost and the holding cost: A M + A R + (n + m) M n PR + m PM TCm (n, m, Tr ) = + FM n+m (n + m) Tr Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D × + − . (3) 2R (n + m) 2P (n + m) 2 Fig. 3 illustrates the integrated third-party, supplier and manufacturer multi-supply inventory model. The total relevant cost per unit time for the used product and material warehouse, TC Rw (k, n, m, Tr ) and TC Mw (l, n, m, Tr ), derived in Appendix B consisted of the setup cost, the holding cost and the purchase cost: TC Rw (k, n, m, Tr ) =
FRw PR D 2 n 2 Tr k A Rw n PR D + + (m + n)Tr 2k(n + m)R n+m
(4)
TC Mw (l, n, m, Tr ) =
l A Mw FMw PM D 2 m 2 Tr m PM D + + (m + n)Tr 2l(n + m)P n+m
(5)
and
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Fig. 3. The integrated third-party, supplier and manufacturer multi-supply inventory model with remanufacturing and manufacturing.
respectively. The total manufacturer profit is the sales revenue minus the total manufacturer cost of remanufacturing, manufacturing and relevant warehouse inventory. It is shown as follows: TPm (l, k, n, m, Tr ) = Pr D − TCm − TC Rw − TC Mw A R + A M + (n + m)M + k A Rw + l A Mw n PR + m PM = Pr D − − FM (n + m)Tr n+m Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D × + − 2R (n + m) 2P (n + m) 2 −
FRw PR D 2 n 2 Tr FMw PM D 2 m 2 Tr (n PR + m PM )D − − . 2k(n + m)R 2l(n + m)P n+m
(6)
The total relevant used product inventory cost to the third party, TC3 (k, n, m, Tr ), as shown in Appendix C, includes the setup cost, the holding cost and the purchase cost:
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F3 P3 Tr A3 + k K + TC3 (k, n, m, Tr ) = (n + m)Tr
D 2 n 2 − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 P3 Dn + . (7) 2 (n + m) Rk n+m
The total profit of the third party is sales revenue minus the relevant inventory cost to the third party. It is shown as follows: (PR − P3 )Dn A3 + k K TP3 (k, n, m, Tr ) = − n+m (n + m)Tr 2 2 F3 P3 Tr D n − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 . (8) − 2 (n + m) Rk The total relevant inventory cost to the supplier, TCs (l, n, m, Tr ), as derived in Appendix D, consists of the setup cost, the holding cost and the purchase cost: As + l L Fs Ps D 2 m 2 Tr 1 Ps Dm TCs (l, n, m, Tr ) = + . (9) 1− + (n + m)Tr 2P(n + m) l n+m The total supplier profit is the sales revenue minus total relevant inventory cost to the supplier. It is shown as follows: Fs Ps D 2 m 2 Tr (PM − Ps )Dm As + l L 1 TPs (l, n, m, Tr ) = − − 1− . (10) n+m (n + m)Tr 2P(n + m) l The joint annual profit is the total annual profits of the retailer, the manufacturer, the supplier and the third party. One has J P (l, k, n, m, Tr ) = TPr + TPm + TP3 + TPs .
(11)
The optimization problem is stated as max J P (l, k, n, m, Tr )
(12)
s.t. Tr ≥ 0 m (D − C) = n C m+n = I l, k, I = 1, 2, 3, . . . . 4. Solution procedure The optimization problem is to determine the values of k, l, n, m and Tr that maximize JP(k, l, n, m, Tr ). Two cases are discussed. The first case does not consider integration (sequential decision making by retailer and manufacturer). The second case considers system integration (centralized decision making). Since the problem is a constrained nonlinear mixed programming problem, the values of k, l, n, m and Tr can be derived by the following procedure: Case I. Integration of the supply chain members is not considered, and optimal policy is derived by sequencing optimization from the retailer to the manufacturer. The optimal ordering cycle time of the retailer can be found by taking the derivative of (2) with respect to Tr , and setting the results to zero. The optimal variables of Tr denoted by Tr# are derived as follows: s 2Ar # Tr = . (13) Fr Pr D The maximized retailer total annual profit with individual decision making is p TPr Tr# = Pc D − 2Ar Fr Pr D.
(14)
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Substituting (13) into (3), one has A M + A R + (n + m) M n PR + m PM TCm n, m, Tr# = + F M n+m (n + m) Tr# # # Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr# D . × + − 2R (n + m) 2P (n + m) 2
873
(15)
The remanufacturing and manufacturing periods of the manufacturer are set at nT r# and mT r# ; and the annual remanufacturing quantity equals the annual used product return rate C as follows 1 nTr# D = C. (16) (n + m)Tr# Solving Eq. (16) for I = n + m, one has n=
IC D
(17)
m=
I (D − C) D
(18)
and
respectively. Substituting (17) and (18) into (15), one has AM + AR + I M C PR + (D − C)PM TCm I, Tr# = + F M I Tr# D " # C Tr# 2D 2 + I C R − I C D (D − C)Tr# D 2D 2 + I (D − C)P − I (D − C)D Tr# D × + − . 2R D 2P D 2
(19)
The remanufacture/manufacture period is (n + m)Tr# = IT r# where I is a positive integer. The first derivatives of Eq. (19) with respect to I to zero are equated to solve the equations. dTCm I, Tr# C PR + (D − C)PM M AM + AR + I M = − + F M dI I Tr# D I 2 Tr# C Tr# (C R − C D) (D − C)Tr# D ((D − C)P − (D − C)D) × + = 0. (20) 2R D 2P D The value of I 0 that minimizes total annual manufacturer production cost is s AM + AR 0 I = GTr#
(21)
where G = FM
C PR + (D − C)PM D
C Tr# (C R − C D) (D − C)Tr# D ((D − C)P − (D − C)D) + . 2R D 2P D
Since the value of I is a positive integer, the optimal solution I # approximates the values of I 0 that satisfy TCm I # − 1, Tr# ≥ TCm I # , Tr# ≤ TCm I # + 1, Tr# .
(22)
(23)
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For a known value of I # , the optimal values n # and m # can be obtained by (17) and (18); the total relevant cost per unit time of the used product and material warehouse are TC Rw k, n # , m # , Tr# =
k A Rw FRw PR D 2 n 2 Tr n PR D + + (m # + n # )Tr 2k(n + m)R n+m
TC Mw l, n # , m # , Tr# =
l A Mw FMw PM D 2 m # Tr m # PM D + + # # # # # (m + n )Tr 2l(n + m )P n + m#
(24)
and 2
(25)
respectively. The time period of deliveries from the third party and supplier to the used product and material warehouse of the manufacturer are TR1 /k and TM1 /l; where k and l are positive integers. The first derivatives of Eqs. (24) and (25) with respect to k and l and equating to zero are solved. The values for k 0 and l 0 that minimize the manufacturer cost of the used products and the material warehouse total annual inventory are s FRw PR 0 # # k = n Tr D (26) 2R A Rw and s 0
l =m
#
Tr# D
FMw PM 2P A Mw
(27)
respectively. Since the values of k and l are positive integers, the optimal k # and l # approximate the values of k 0 and l 0 that satisfy (28) TC Rw k # − 1, n # , m # , Tr# ≥ TC Rw k # , n # , m # , Tr# ≤ TC Rw k # + 1, n # , m # , Tr# and TC Mw l # − 1, n # , m # , Tr# ≥ TC Mw l # , n # , m # , Tr# ≤ TC Mw l # + 1, n # , m # , Tr# . The maximal joint annual profit with decentralized decision making is: J P l # , k # , n # , m # , Tr# = TPr (Tr# ) + TPm l # , k # , n # , m # , Tr# + TP3 k # , n # , m # , Tr# + TPs l # , n # , m # , Tr# .
(29)
(30)
Case II. A centralized decision-making procedure with integration is assumed in this case. The problem is to determine the values of k, l, n, m and Tr that maximize JP(k, l, n, m, Tr ). Since the numbers k, l and I are discrete variables (where I = n + m, and n = IC/D and m = I (D − C)/D), the values of k, l and I can be derived by the following procedure: (a) Set the values of l, k and I around l # , k # and I # , and determine the derivative of JP(Tr ) with respect to Tr . The optimal value of Tr for l, k and I are denoted by Tr+ . (b) Derive the optimal value of l, k and I , denoted by l ∗ , k ∗ and I ∗ , such that J P l ∗ − 1, k ∗ , I ∗ , Tr+ ≤ J P l ∗ , k ∗ , I ∗ , Tr∗ ≥ J P l ∗ + 1, k ∗ , I ∗ , Tr+ (31) ∗ ∗ ∗ + ∗ ∗ ∗ ∗ ∗ ∗ ∗ + J P l , k − 1, I , Tr ≤ J P l , k , I , Tr ≥ J P l , k + 1, I , Tr (32) and J P l ∗ , k ∗ , I ∗ − 1, Tr+ ≤ J P l ∗ , k ∗ , I ∗ , Tr∗ ≥ J P l ∗ , k ∗ , I ∗ + 1, Tr+ respectively.
(33)
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Given a known I ∗ , the optimal values of n ∗ and m ∗ can be obtained by (17) and (18). Hence the optimal values of m ∗ and Tr∗ are derived. The following numerical examples illustrate the above procedure.
l ∗, k ∗, n∗,
5. Numerical example Example. To illustrate the result of the above theory, the parameters to illustrate the concept are as follows: The customer annual demand rate, D = 2000 units per year; the third-party annual return rate, C = 770 unit; the retailer, the manufacturer used product warehouse, the manufacturer material warehouse and the supplier fixed costs to place an order are Ar = $100, A Rw = $350, A Mw = $350 and As = $200 respectively; the manufacturer manufacturing setup cost, the remanufacturing setup cost and the third-party setup cost are A M = $2000, A R = $2500 and A3 = 250 respectively; the manufacturer fixed cost to process retailer order of any size, M = $350; the thirdparty and the supplier fixed costs to process manufacturer order of any size are K = $150 and L = $150 respectively; the retailer, the manufacturer, the manufacturer warehouse, the third-party and the supplier inventory holding cost rates are, Fr = 0.30 per unit price per year, FM = 0.20 per unit price per year, FRw = 0.25 per unit price per year, FMw = 0.25 per unit price per year, F3 = 0.35 per unit price per year and Fs = 0.3 per unit price per year respectively; the consumer, the retailer, the manufacturer, the third-party and the supplier purchase unit prices are Pc = $175, Pr = $150, PR = $110, PM = $115, P3 = $70 and Ps = $90 respectively; the annual production and the reproduction rates of the manufacturer are P = 5000 unit per year and R = 4000 unit per year respectively. What are the values of Tr# , n # , m # , k # , l # and the maximal joint profit with decentralized decision making, JP(l # , k # , n # , m # , Tr# )? What are the values of Tr∗ , n ∗ , m ∗ , k ∗ , l ∗ and the maximal joint profit with centralized decision making, JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ )? And what is the percentage of the extra joint profit (PEJP) in this model? Substitute the above parameters into (12). The optimization problem is stated as: 400 350n + 27m 2 Tr + 450m 100 − 45 000Tr − max J P (l, k, n, m, Tr ) = 350 000 − Tr n+m 13 750n 2 Tr 700m 2 Tr 50(99 + 7n + 7m + 10k + 10l) − − − k(n + m) l (n + m) (n + m)Tr 2 100(22n + 23m)Tr (5n − 2m + 6m 2 ) − (n + m)2 2 122.5Tr (100n k − 23n 2 + 77knm + 77m 2 k + 77mn) − (34) k(n + m) s.t. Tr ≥ 0 n = I × 77/200 m = I × 123/200 l, k, I = 1, 2, 3, . . . . Case I. For decentralized decision making with step-by-step optimization from the retailer to the manufacturer, substitute the above parameters into (13) and (21). The optimal values of Tr# and I 0 satisfying (23) are 0.04714 and 17.24 respectively. A value of 17 is obtained for I # . The optimal values of n # and m # , derived from (17) and (18) are 6.545 and 10.455 respectively. From (26) and (27), the optimal values of k # and l # satisfying (28) and (29) are 2 and 3 respectively. With known optimal values of Tr# , n # , m # , k # and l # , the maximal joint annual profit with decentralized decision making is $148,119. Case II. For centralized decision making with integration, the optimal solution can be derived by the following procedure: (a) For l = 2, k = 2, I = 8, n = 3.08 and m = 4.92, from (34), the optimization problem can be stated as: MAX J P (Tr ) = 185 400 − s.t. Tr ≥ 0.
1318.75 − 206 770.89 Tr
(35)
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Table 1 The results of l, k and I values for a range approximating l # , k # and I # l
k
I
n
m
Tr+
JP+
PEJP (%)
3 3 3 2 3 3 3 2 2 2 1 1∗ 1 1 2 2 1
2 2 2 2 2 2 1 1 2 3 3 2∗ 2 2 2 2 1
17 12 8 8 5 3 5 5 5 5 5 5∗ 4 6 6 4 5
6.545 4.62 3.08 3.08 1.925 1.155 1.925 1.925 1.925 1.925 1.925 1.925∗ 1.54 2.31 2.31 1.54 1.925
10.455 7.38 4.92 4.92 3.075 1.845 3.075 3.075 3.075 3.075 3.075 3.075∗ 2.46 3.69 3.69 2.46 3.075
0.0477 0.0609 0.0818 0.0799 0.1157 0.1681 0.1091 0.1061 0.1126 0.1169 0.1136 0.1093∗ 0.1284 0.0957 0.0985 0.1326 0.1027
148 121.24 150 245.81 151 629.32 152 374 151 874.16 150 502.44 151 681.23 152 586.89 152 724.8 152 222.11 153 013.22 153 552.71∗ 153 283.82 153 537.72 152 756.92 152 395.03 153 475.67
0.00 1.44 2.37 2.87 2.54 1.61 2.40 3.02 3.11 2.77 3.30 3.67 3.49 3.66 3.13 2.89 3.62
Table 2 Comparing the results of integrated decision with those achieved by the decentralized decision making Relative variable of model
l
k
Integrated policy 1 2 Decentralized policy 3 2 Integrated extra profit Percentage integrated extra joint profit (%)
I
n
m
Tr
TPr
TPm
TP3
TPs
JP
5 17
1.925 6.54
3.075 10.46
0.1093 0.0471
44 168 45 757 −1590 −3.47
54 140 51 323 +2817 +5.49
25 136 23 282 +1854 +7.96
30 109 27 757 +2353 +8.48
153 553 148 119 +5434 +3.67
Profit decrease; +: profit increase.
Taking the derivative of (35) with respect to Tr , set the results to zero, the optimal values of Tr and JP denoted by Tr+ and JP+ are 0.07986 year and $152,374, respectively. (b) Table 1 presents the results of l, k and I values for a range of values approximating l # , k # and I # . The optimal values of l ∗ , k ∗ , I ∗ , n ∗ , m ∗ and Tr∗ satisfying (31)–(33) are 1, 2, 5, 1.925, 3.075 and 0.1093 years, respectively. The maximal joint profit JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ ) is $153,552.71; the percentage of the extra joint profit (PEJP) is 3.67%. Table 2 compares the results of the integrated decision making with those by independent decision making. 6. Sensitivity analysis In the numerical example, the optimal value of joint profit JP(l, k, n, m, Tr ) for a fixed set of parameters, Φ = {Ai , A R , A Mw , A Rw , Fi , FMw , FRw , Pv , Pi , Pc , PR , D, C, R, P, M, K , L}, where i = r, M, 3, s are denoted by JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ ). The change in JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ ) is observed when the parameters in Φ change. The following analysis identifies the percentage joint profit change (PJPC) when only one of the parameters in the subset of Φ increases or decreases by 5%, 10% and 15%, and all the other parameters remain unchanged. Figs. 4–7 show the results of the sensitivity analysis. The key conclusions of the sensitivity analysis are as follows: (a) The optimal values of the joint profit (JP) are significantly influenced by the retailer price to the consumer (Pc ) and/or the annual demand rate (D). (b) The optimal values of the joint profit (JP) are significantly influenced by the third-party recycle dealer collecting unit cost from the consumer (P3 ), the supplier purchase unit price (Ps ) or the annual return rate (C).
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Fig. 4. The effect of Ar , A M , A R , A Mw , A Rw , A3 and As on PJPC.
Fig. 5. The effect of Fr , FM , FMw , FRw , F3 and Fs on PJPC.
(c) The joint profit (JP) increases when the annual return rate (C), the retailer selling price to the consumer (Pc ) or the annual demand rate (D) increases whereas the third party collecting unit cost from the consumer (P3 ), the supplier purchase unit price (Ps ), the setup cost parameter (Ai ) or the holding cost parameter (Fi ) decreases. (d) The percentage of joint profit change, PJPC, ranges from −34% to 34% with 15% increase or decrease in parameter values. 7. Conclusion This study derives an optimal strategy for a closed-loop supply chain system with remanufacturing. From the perspectives of the supplier; the manufacturer; the retailer and the third-party recycling dealer, an optimal production and replenishment policy was formulated to maximize the joint profit. An example comparing the decentralized decision-making system with the centralized decision-making system is illustrated. The findings in this study revealed a substantial profit increase using the integrated approach. The percentage extra joint profit (PEJP) is 3.67%. Sensitivity analysis showed that the optimal value of the joint profit is highly sensitive to the retail selling price
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Fig. 6. The effect of Pc , Pr , PM , PR , P3 and Ps on PJPC.
Fig. 7. The effect of D, C, R, P, M, K and L on PJPC.
and/or the annual demand. To achieve a win–win system, profit sharing should be considered in the supply chain collaborations. Future research should examine price sensitive demand and risk management. Acknowledgments The authors would like to express their appreciation to the referees for their contributions in improving the quality of the paper, and the National Science Council of the Republic of China, Taiwan for supporting this research under Contract No. NSC 95-2221-E-129 -007. Appendix A As Fig. 2 shows, when g equals Tr D 2 /R, the amount of stock requited by the retailer during reproduction of first shipment equals Tr D. The total stock increases at a rate of (R − D) during the manufacture of batch quantity of nT r D. The maximum inventory level at the completion of a reproduction batch equals Tr D 2 nTr D + (R − D) . R R
(A.1)
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Accordingly, the average total stock in the remanufacturing period of manufacturer–retailer inventory system (AIL Rr ) is given by n Tr D 2 (R − D) nTr D AIL Rr = + . (A.2) R 2R n+m Similarly, the average total stock in the manufacturing period of manufacturer–retailer inventory system (AIL Mr ) is given by Tr D 2 m (P − D) mTr D AIL Mr = . (A.3) + P 2P n+m The average total stock in the manufacture–retailer inventory system (AILmr ) is given by AILmr = AIL Mr + AIL Rr =
Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m + . 2R (n + m) 2P (n + m)
(A.4)
Consequently, average manufacturer inventory level can be determined by subtracting the average retailer inventory level from the average total inventory of the manufacturer–retailer inventory system. The average manufacturer inventory level (AILm ) is given by AILm =
Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D + − . 2R (n + m) 2P (n + m) 2
Additionally, the total relevant production cost of the manufacturer is given by n PR + m PM A M + A R + (n + m) M TCm (n, m, Tr ) = + FM n+m (n + m) Tr Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D × + − . 2R (n + m) 2P (n + m) 2
(A.5)
(A.6)
The first term is the setup cost and the second term is the holding cost. Appendix B As Fig. 3 shows, the average used product inventory in the manufacturer warehouse (AIL Rw ) is R TR1 TR1 D 2 n 2 Tr DnTr AIL Rw = = since TR1 = . 2 k (n + m)Tr 2k(n + m)R R
(B.1)
The total relevant inventory cost of the manufacturer used product warehouse is: TC Rw (k, n, m, Tr ) =
k A Rw FRw PR D 2 n 2 Tr n PR D + + . (m + n)Tr 2k(n + m)R n+m
(B.2)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Similarly, as Fig. 3 shows, the total relevant inventory cost of the manufacturer material warehouse is: TC Mw (l, n, m, Tr ) =
FMw PM D 2 m 2 Tr l A Mw m PM D + + . (m + n)Tr 2l(n + m)P n+m
(B.3)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Appendix C As Fig. 3 shows, the average used product inventory of the third-party warehouse (AIL3 ) is 1 TR1 TR1 1 RTR1 − C(k − 1) (k − 1) AIL3 = (n + m)Tr 2 k k
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" 2 # DnTr 1 1 TR1 (k − 1) since TR1 = + C (n + m)Tr − (n + m)Tr 2 k R 2 2 Tr D n − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 = . 2 (n + m) Rk
(C.1)
The total relevant used product inventory cost to the third party is: A3 + k K P3 Dn + F3 P3 × AIL3 + (n + m)Tr n+m 2 2 F P T D n − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 A3 + k K 3 3 r = + (n + m)Tr 2 (n + m) Rk P3 Dn + . n+m
TC3 (k, n, m, Tr ) =
(C.2)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Appendix D As Fig. 3 shows, the lot size of the supplier is DmT r , and the net average material inventory level of the supplier in the integrated system (AILs ) can be determined by subtracting the average material inventory level of the manufacturer from the average total inventory in the supplier–manufacturer inventory system. TM1 D 2 m 2 Tr DmTr × − 2 (n + m)Tr 2l(n + m)P 2 2 D m Tr 1 = 1− . 2P(n + m) l
AILs =
since TM1 =
DmTr P (D.1)
The total relevant inventory cost to the supplier is: Fs Ps D 2 m 2 Tr As + l L + TCs (l, n, m, Tr ) = (n + m)Tr 2P(n + m)
1 1− l
+
Ps Dm . n+m
(D.2)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. References [1] J. Bautista, J. Pereira, Closed-loop supply chain models with product remanufacturing modeling the problem of locating collection areas for urban waste management, An application to the metropolitan area of Barcelona, Omega 34 (2006) 617–629. [2] A.J. Clark, H. Scarf, Optimal policies for a multi-echelon inventory problem, Management Science 6 (4) (1960) 475–490. [3] S.K. Goyal, F. Nebebe, Determination of economic production-shipment policy for a single-vendor-single-buyer system, European Journal of Operational Research 121 (2000) 175–178. [4] S.K. Goyal, A one-vendor multi-buyer integrated inventory model: A comment, European Journal of Operational Research 82 (1995) 209–210. [5] S.K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal Production Research 15 (1977) 107–111. [6] S.K. Goyal, Determination of economic packaging frequency for items jointly replenished, Management Science 20 (2) (1973) 232–235. [7] S.K. Goyal, Determination of optimum packaging frequency of items jointly replenished, Management Science 21 (4) (1974) 436–443. [8] M.R. Hill, The single-vendor single-buyer integrated production-inventory model with a generalized policy, European Journal of Operational Research 97 (1997) 493–499. [9] K. Inderfurth, G. Lindner, N.P. Rachaniotis, Lot sizing in a production system with rework and product deterioration, International Journal of Production Research 43 (7) (2005) 1355–1374. [10] S.G. Koh, H. Hwang, K.I. Sohn, C.S. Ko, An optimal ordering and recovery policy for reusable items, Computers and Industrial Engineering 43 (2002) 59–73. [11] L. Lu, A one-vendor multi-buyer integrated inventory model, European Journal of Operational Research 81 (1995) 312–323. [12] M.C. Mabini, L.M. Pintelon, L.F. Gelders, EOQ type formulation for controlling repairable inventories, International Journal of Production Economics 28 (1992) 21–33. [13] J.P. Monahan, A quantity discount pricing model to increase vendor profits, Management Science 30 (1984) 720–726.
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[14] S. Nahmias, H. Rivera, A deterministic model for a repairable item inventory system with a finite repair rate, International Journal of Production Research 17 (3) (1979) 215–221. [15] R.C. Savaskan, S. Bhattacharya, L.N. Van Wassenhove, Closed-loop supply chain models with product remanufacturing, Management Science 50 (2) (2004) 239–252. [16] D.A. Schrady, A deterministic inventory model for repairable items, Naval Research Logistics Quarterly 14 (1967) 391–398. [17] F.T. Shu, Economic ordering frequency for two items jointly replenished, Management Science 17 (6) (1971) B406–B410. [18] P.C. Yang, H.M. Wee, A single-vendor and multiple-buyers, production-inventory policy for a deteriorating item, European Journal of Operational Research 143 (2002) 570–581. [19] P.C. Yang, H.M. Wee, An arborescent inventory model in a supply chain system, Production Planning & Control 12 (8) (2001) 728–735. [20] P.C. Yang, H.M. Wee, Economic ordering policy of deteriorated item for vendor and buyer: An integrated approach, Production Planning & Control 11 (5) (2000) 474–480. [21] P.C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research 157 (2004) 389–397.
Optimal policy for a closed-loop supply chain inventory system with remanufacturing Shen-Lian Chung a , Hui-Ming Wee b,∗ , Po-Chung Yang c a Department of Information Management, St. John’s University, Tamsui, Taipei 25135, Taiwan b Department of Industrial Engineering, Chung Yuan Christian University, Chungli 32023 Taoyuan, Taiwan c Industrial Engineering and Management Department, St. John’s University, Tamsui, Taipei 25135, Taiwan
Received 6 June 2007; received in revised form 23 October 2007; accepted 2 November 2007
Abstract Previous researchers have developed ways of managing forward-oriented supply chains, and gave insight to solve single-stage inventory systems. In this study, we analyze an inventory system with traditional forward-oriented material flow as well as a reverse material flow supply chain. In the reverse material flow, the used products are returned, remanufactured and shipped to the retailer for resale. A multi-echelon inventory system with remanufacturing capability is proposed. We then develop a closed-loop supply chain inventory model and maximize the joint profits of the supplier, the manufacturer, the third-party recycle dealer and the retailer under contractual design. The analytical results of this study show a significant increase in the joint profit when the integrated policy is adopted. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Supply chain management; Closed-loop supply chain; Multi-echelon inventory; Reverse logistics
1. Introduction In order to meet environmental concerns/regulations, manufacturers often attempt to recover the residual value of their used products through remanufacturing. Product remanufacturing such as transforming used items into marketable products through refurbishment, repair and upgrading can also yield substantial cost benefits. Most of the former researchers have developed methods of managing forward-oriented supply chains to optimize single-stage inventory systems. Clark and Scarf [2] were among the first authors to propose a two-echelon inventory model. They developed a system for optimizing base stock in serial inventory systems and achieved an efficient method for establishing an ordering policy for an optimal base inventory. Shu [17] and Goyal [6,7] presented an optimal solution to determine the economic packaging frequencies of items jointly replenished. Many researchers have investigated an integrated approach to price discounting [5,13]. Lu [11] developed an optimal one-vendor multi-buyer integrated model. Hill [8] offered a more general policy for a single-vendor, single-buyer production-inventory model based ∗ Corresponding author. Tel.: +886 3 456 6142; fax: +886 3 456 3171/4499.
E-mail address: [email protected] (H.-M. Wee). c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.11.014
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on successive shipments of a single production batch size. Several researchers (Viswanathan, 1998; Goyal and Nebebe [3]; Goyal [3]) integrated the proposed strategies of Lu [11], Goyal [4] and Hill [8] to obtain an improved relevant inventory cost model. Yang and Wee [19] developed an arborescent model that integrated considerations of the producer, distributor and retailer. Yang and Wee [20,18] and Yang [21] developed an integrated vendor–buyer model for optimizing the delivery number and lot size of deteriorating items. Schrady [16] was the earliest author to propose a deterministic model with infinite production rates for manufacturing and remanufacturing. Schrady argued that optimal lot sizes for manufacturers and remanufacturers can be determined by the classical EOQ formula. The following authors extended Schrady’s analysis: Nahmias and Revera [14] examined finite remanufacturing rates; Mabini et al. [12] considered stockout service level constraints and a multi-product model; Koh et al. [10] analyzed finite manufacturing/remanufacturing rates. Further, Inderfurth, Lindner and Rachaniotis [9] investigated lot-sizing decisions in a hybrid production/rework system characterized by defective products. Several other studies have addressed collection issues. Savaskan et al. [15] determined the optimal collection channel configuration of a monopolist manufacturer. Bautista and Pereira [1] proposed a method of identifying these collection area problems and established the relationship between the set covering problem and the MAX-SAT problem. The present study analyzes a closed-loop supply chain inventory system. In addition to traditional forward material flows, the model examines used products returned to a reconditioning facility where they are stored, remanufactured then shipped back to retailers for retail sale. Fig. 1 illustrates the proposed process. The proposed optimal policy for a multi-echelon inventory system with remanufacturing is developed by integrating the concerns of the supplier, the manufacturer, the retailer and the third-party recycle dealer. An example of the closed-loop supply chain with a remanufacturing system illustrates the system. The solutions are obtained for both decentralized and integrated centralized decision making. The findings of this study demonstrate that the proposed integrated centralized decisionmaking approach can substantially improve efficiency. 2. Assumptions and notations The mathematical models in this analysis have the following assumptions: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
An infinite planning horizon. No deterioration. No stock shortages. The manufacturing and remanufacturing rates and constant lead-times. The multi-echelon inventory system contains a single item. The product annual demand rate and the annual return rate are constant, and the annual return rate is less then the annual demand rate. No space, capacity or capital constraints. No quantity discount. The number of deliveries within the manufacturing cycle is an integer. The setup cost per run and the annual holding cost fraction are known and constant. To meet the retailer demand, the remanufactured products are available before the manufactured products. The remanufactured products are comparable to newly manufactured products. Single suppliers, single manufacturers, single retailers and single third parties in the closed-loop supply chain multi-echelon inventory system are considered.
The following denotes the retailer’s parameters: D Ar Fr Pc Pr Tr TCr TPr
annual demand rate ordering cost to the retailer inventory holding cost to retailer in annual percentage per dollar retail price wholesale price retailer ordering cycle time (decision variable) total relevant cost to the retailer per unit time total retailer profit per unit time
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Fig. 1. The integrated closed-loop supply chain inventory system.
The following denotes the manufacturer’s parameters: setup cost per manufacturing run setup cost per remanufacturing run ordering cost for the manufacturer material warehouse ordering cost for the manufacturer used product warehouse product inventory holding cost percentage per year per dollar to the manufacturer material inventory holding cost percentage per year per dollar for the manufacturer warehouse used product inventory holding cost percentage per year per dollar for the manufacturer warehouse manufacturer unit purchase price from the supplier manufacturer unit purchase price from the third party manufacturer reproduction period in each remanufacturing cycle manufacturer non-production period in each remanufacturing cycle manufacturer production period in each manufacturing cycle manufacturer non-production period in each manufacturing cycle number of deliveries per manufacturing cycle time from the manufacturer to the retailer (decision variable) number of deliveries per remanufacturing cycle time from the manufacturer to the retailer (decision variable) number of deliveries per remanufacturing/manufacturing cycle time from the manufacturer to the retailer, I = m + n, where I is a positive integer M fixed cost to manufacturer for processing buyer order of any size P annual manufacturer production rate (P > D) R annual manufacturer reproduction rate (R > D) TCm total relevant cost per unit time to the manufacturer TC Mw total relevant cost per unit time for the material warehouse TC Rw total relevant cost per unit time for the used product warehouse TPm total manufacturer profit per unit time AM AR A Mw A Rw FM FMw FRw PM PR TR1 TR2 TM1 TM2 m n I
The following denotes the third party’s parameters: C A3 F3 P3 k K TC3 TP3
annual return rate setup cost per run for the third party used product inventory holding cost percentage per year per dollar to the third party third party collecting unit cost from the consumer number of deliveries per TR1 from the third party to the manufacturer (decision variable) fixed third-party cost to process manufacturer orders of any size; total relevant cost to third party per unit time total third-party profit per unit time
The following denotes the supplier’s parameters: As fixed cost to supplier per order Fs material inventory holding cost to supplier, in percentage per year per dollar
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Fig. 2. The integrated manufacturer and retailer multi-delivery inventory model with remanufacturing.
Ps l L TCs TPs
supplier purchase unit price number of deliveries per TM1 from the supplier to the manufacturer (decision variable) fixed supplier cost to process manufacturer orders of any size total relevant supplier cost per unit time total supplier profit per unit time
The following denotes the objective function of the optimization problem: JP(l, k, n, m, Tr ) joint profit per unit time for the whole system 3. Model development For a continuous review system, Fig. 2 represents the manufacturer–retailer inventory system. The average inventory of the retailer is DT r /2, and the total relevant inventory cost for the retailer can be expressed as follows: TCr (Tr ) =
Fr Pr DTr Ar + + Pr D. Tr 2
(1)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Total retailer profit is the sales revenue minus total retailer cost. It is given as follows: Fr Pr DTr Ar + Pr D . (2) TPr (Tr ) = Pc D − + Tr 2 When coordination is considered, the total relevant cost per unit time for the remanufacturing and manufacturing, TCm (n, m, Tr ), as shown in Appendix A consists of the setup cost and the holding cost: A M + A R + (n + m) M n PR + m PM TCm (n, m, Tr ) = + FM n+m (n + m) Tr Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D × + − . (3) 2R (n + m) 2P (n + m) 2 Fig. 3 illustrates the integrated third-party, supplier and manufacturer multi-supply inventory model. The total relevant cost per unit time for the used product and material warehouse, TC Rw (k, n, m, Tr ) and TC Mw (l, n, m, Tr ), derived in Appendix B consisted of the setup cost, the holding cost and the purchase cost: TC Rw (k, n, m, Tr ) =
FRw PR D 2 n 2 Tr k A Rw n PR D + + (m + n)Tr 2k(n + m)R n+m
(4)
TC Mw (l, n, m, Tr ) =
l A Mw FMw PM D 2 m 2 Tr m PM D + + (m + n)Tr 2l(n + m)P n+m
(5)
and
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Fig. 3. The integrated third-party, supplier and manufacturer multi-supply inventory model with remanufacturing and manufacturing.
respectively. The total manufacturer profit is the sales revenue minus the total manufacturer cost of remanufacturing, manufacturing and relevant warehouse inventory. It is shown as follows: TPm (l, k, n, m, Tr ) = Pr D − TCm − TC Rw − TC Mw A R + A M + (n + m)M + k A Rw + l A Mw n PR + m PM = Pr D − − FM (n + m)Tr n+m Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D × + − 2R (n + m) 2P (n + m) 2 −
FRw PR D 2 n 2 Tr FMw PM D 2 m 2 Tr (n PR + m PM )D − − . 2k(n + m)R 2l(n + m)P n+m
(6)
The total relevant used product inventory cost to the third party, TC3 (k, n, m, Tr ), as shown in Appendix C, includes the setup cost, the holding cost and the purchase cost:
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F3 P3 Tr A3 + k K + TC3 (k, n, m, Tr ) = (n + m)Tr
D 2 n 2 − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 P3 Dn + . (7) 2 (n + m) Rk n+m
The total profit of the third party is sales revenue minus the relevant inventory cost to the third party. It is shown as follows: (PR − P3 )Dn A3 + k K TP3 (k, n, m, Tr ) = − n+m (n + m)Tr 2 2 F3 P3 Tr D n − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 . (8) − 2 (n + m) Rk The total relevant inventory cost to the supplier, TCs (l, n, m, Tr ), as derived in Appendix D, consists of the setup cost, the holding cost and the purchase cost: As + l L Fs Ps D 2 m 2 Tr 1 Ps Dm TCs (l, n, m, Tr ) = + . (9) 1− + (n + m)Tr 2P(n + m) l n+m The total supplier profit is the sales revenue minus total relevant inventory cost to the supplier. It is shown as follows: Fs Ps D 2 m 2 Tr (PM − Ps )Dm As + l L 1 TPs (l, n, m, Tr ) = − − 1− . (10) n+m (n + m)Tr 2P(n + m) l The joint annual profit is the total annual profits of the retailer, the manufacturer, the supplier and the third party. One has J P (l, k, n, m, Tr ) = TPr + TPm + TP3 + TPs .
(11)
The optimization problem is stated as max J P (l, k, n, m, Tr )
(12)
s.t. Tr ≥ 0 m (D − C) = n C m+n = I l, k, I = 1, 2, 3, . . . . 4. Solution procedure The optimization problem is to determine the values of k, l, n, m and Tr that maximize JP(k, l, n, m, Tr ). Two cases are discussed. The first case does not consider integration (sequential decision making by retailer and manufacturer). The second case considers system integration (centralized decision making). Since the problem is a constrained nonlinear mixed programming problem, the values of k, l, n, m and Tr can be derived by the following procedure: Case I. Integration of the supply chain members is not considered, and optimal policy is derived by sequencing optimization from the retailer to the manufacturer. The optimal ordering cycle time of the retailer can be found by taking the derivative of (2) with respect to Tr , and setting the results to zero. The optimal variables of Tr denoted by Tr# are derived as follows: s 2Ar # Tr = . (13) Fr Pr D The maximized retailer total annual profit with individual decision making is p TPr Tr# = Pc D − 2Ar Fr Pr D.
(14)
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Substituting (13) into (3), one has A M + A R + (n + m) M n PR + m PM TCm n, m, Tr# = + F M n+m (n + m) Tr# # # Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr# D . × + − 2R (n + m) 2P (n + m) 2
873
(15)
The remanufacturing and manufacturing periods of the manufacturer are set at nT r# and mT r# ; and the annual remanufacturing quantity equals the annual used product return rate C as follows 1 nTr# D = C. (16) (n + m)Tr# Solving Eq. (16) for I = n + m, one has n=
IC D
(17)
m=
I (D − C) D
(18)
and
respectively. Substituting (17) and (18) into (15), one has AM + AR + I M C PR + (D − C)PM TCm I, Tr# = + F M I Tr# D " # C Tr# 2D 2 + I C R − I C D (D − C)Tr# D 2D 2 + I (D − C)P − I (D − C)D Tr# D × + − . 2R D 2P D 2
(19)
The remanufacture/manufacture period is (n + m)Tr# = IT r# where I is a positive integer. The first derivatives of Eq. (19) with respect to I to zero are equated to solve the equations. dTCm I, Tr# C PR + (D − C)PM M AM + AR + I M = − + F M dI I Tr# D I 2 Tr# C Tr# (C R − C D) (D − C)Tr# D ((D − C)P − (D − C)D) × + = 0. (20) 2R D 2P D The value of I 0 that minimizes total annual manufacturer production cost is s AM + AR 0 I = GTr#
(21)
where G = FM
C PR + (D − C)PM D
C Tr# (C R − C D) (D − C)Tr# D ((D − C)P − (D − C)D) + . 2R D 2P D
Since the value of I is a positive integer, the optimal solution I # approximates the values of I 0 that satisfy TCm I # − 1, Tr# ≥ TCm I # , Tr# ≤ TCm I # + 1, Tr# .
(22)
(23)
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For a known value of I # , the optimal values n # and m # can be obtained by (17) and (18); the total relevant cost per unit time of the used product and material warehouse are TC Rw k, n # , m # , Tr# =
k A Rw FRw PR D 2 n 2 Tr n PR D + + (m # + n # )Tr 2k(n + m)R n+m
TC Mw l, n # , m # , Tr# =
l A Mw FMw PM D 2 m # Tr m # PM D + + # # # # # (m + n )Tr 2l(n + m )P n + m#
(24)
and 2
(25)
respectively. The time period of deliveries from the third party and supplier to the used product and material warehouse of the manufacturer are TR1 /k and TM1 /l; where k and l are positive integers. The first derivatives of Eqs. (24) and (25) with respect to k and l and equating to zero are solved. The values for k 0 and l 0 that minimize the manufacturer cost of the used products and the material warehouse total annual inventory are s FRw PR 0 # # k = n Tr D (26) 2R A Rw and s 0
l =m
#
Tr# D
FMw PM 2P A Mw
(27)
respectively. Since the values of k and l are positive integers, the optimal k # and l # approximate the values of k 0 and l 0 that satisfy (28) TC Rw k # − 1, n # , m # , Tr# ≥ TC Rw k # , n # , m # , Tr# ≤ TC Rw k # + 1, n # , m # , Tr# and TC Mw l # − 1, n # , m # , Tr# ≥ TC Mw l # , n # , m # , Tr# ≤ TC Mw l # + 1, n # , m # , Tr# . The maximal joint annual profit with decentralized decision making is: J P l # , k # , n # , m # , Tr# = TPr (Tr# ) + TPm l # , k # , n # , m # , Tr# + TP3 k # , n # , m # , Tr# + TPs l # , n # , m # , Tr# .
(29)
(30)
Case II. A centralized decision-making procedure with integration is assumed in this case. The problem is to determine the values of k, l, n, m and Tr that maximize JP(k, l, n, m, Tr ). Since the numbers k, l and I are discrete variables (where I = n + m, and n = IC/D and m = I (D − C)/D), the values of k, l and I can be derived by the following procedure: (a) Set the values of l, k and I around l # , k # and I # , and determine the derivative of JP(Tr ) with respect to Tr . The optimal value of Tr for l, k and I are denoted by Tr+ . (b) Derive the optimal value of l, k and I , denoted by l ∗ , k ∗ and I ∗ , such that J P l ∗ − 1, k ∗ , I ∗ , Tr+ ≤ J P l ∗ , k ∗ , I ∗ , Tr∗ ≥ J P l ∗ + 1, k ∗ , I ∗ , Tr+ (31) ∗ ∗ ∗ + ∗ ∗ ∗ ∗ ∗ ∗ ∗ + J P l , k − 1, I , Tr ≤ J P l , k , I , Tr ≥ J P l , k + 1, I , Tr (32) and J P l ∗ , k ∗ , I ∗ − 1, Tr+ ≤ J P l ∗ , k ∗ , I ∗ , Tr∗ ≥ J P l ∗ , k ∗ , I ∗ + 1, Tr+ respectively.
(33)
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Given a known I ∗ , the optimal values of n ∗ and m ∗ can be obtained by (17) and (18). Hence the optimal values of m ∗ and Tr∗ are derived. The following numerical examples illustrate the above procedure.
l ∗, k ∗, n∗,
5. Numerical example Example. To illustrate the result of the above theory, the parameters to illustrate the concept are as follows: The customer annual demand rate, D = 2000 units per year; the third-party annual return rate, C = 770 unit; the retailer, the manufacturer used product warehouse, the manufacturer material warehouse and the supplier fixed costs to place an order are Ar = $100, A Rw = $350, A Mw = $350 and As = $200 respectively; the manufacturer manufacturing setup cost, the remanufacturing setup cost and the third-party setup cost are A M = $2000, A R = $2500 and A3 = 250 respectively; the manufacturer fixed cost to process retailer order of any size, M = $350; the thirdparty and the supplier fixed costs to process manufacturer order of any size are K = $150 and L = $150 respectively; the retailer, the manufacturer, the manufacturer warehouse, the third-party and the supplier inventory holding cost rates are, Fr = 0.30 per unit price per year, FM = 0.20 per unit price per year, FRw = 0.25 per unit price per year, FMw = 0.25 per unit price per year, F3 = 0.35 per unit price per year and Fs = 0.3 per unit price per year respectively; the consumer, the retailer, the manufacturer, the third-party and the supplier purchase unit prices are Pc = $175, Pr = $150, PR = $110, PM = $115, P3 = $70 and Ps = $90 respectively; the annual production and the reproduction rates of the manufacturer are P = 5000 unit per year and R = 4000 unit per year respectively. What are the values of Tr# , n # , m # , k # , l # and the maximal joint profit with decentralized decision making, JP(l # , k # , n # , m # , Tr# )? What are the values of Tr∗ , n ∗ , m ∗ , k ∗ , l ∗ and the maximal joint profit with centralized decision making, JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ )? And what is the percentage of the extra joint profit (PEJP) in this model? Substitute the above parameters into (12). The optimization problem is stated as: 400 350n + 27m 2 Tr + 450m 100 − 45 000Tr − max J P (l, k, n, m, Tr ) = 350 000 − Tr n+m 13 750n 2 Tr 700m 2 Tr 50(99 + 7n + 7m + 10k + 10l) − − − k(n + m) l (n + m) (n + m)Tr 2 100(22n + 23m)Tr (5n − 2m + 6m 2 ) − (n + m)2 2 122.5Tr (100n k − 23n 2 + 77knm + 77m 2 k + 77mn) − (34) k(n + m) s.t. Tr ≥ 0 n = I × 77/200 m = I × 123/200 l, k, I = 1, 2, 3, . . . . Case I. For decentralized decision making with step-by-step optimization from the retailer to the manufacturer, substitute the above parameters into (13) and (21). The optimal values of Tr# and I 0 satisfying (23) are 0.04714 and 17.24 respectively. A value of 17 is obtained for I # . The optimal values of n # and m # , derived from (17) and (18) are 6.545 and 10.455 respectively. From (26) and (27), the optimal values of k # and l # satisfying (28) and (29) are 2 and 3 respectively. With known optimal values of Tr# , n # , m # , k # and l # , the maximal joint annual profit with decentralized decision making is $148,119. Case II. For centralized decision making with integration, the optimal solution can be derived by the following procedure: (a) For l = 2, k = 2, I = 8, n = 3.08 and m = 4.92, from (34), the optimization problem can be stated as: MAX J P (Tr ) = 185 400 − s.t. Tr ≥ 0.
1318.75 − 206 770.89 Tr
(35)
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Table 1 The results of l, k and I values for a range approximating l # , k # and I # l
k
I
n
m
Tr+
JP+
PEJP (%)
3 3 3 2 3 3 3 2 2 2 1 1∗ 1 1 2 2 1
2 2 2 2 2 2 1 1 2 3 3 2∗ 2 2 2 2 1
17 12 8 8 5 3 5 5 5 5 5 5∗ 4 6 6 4 5
6.545 4.62 3.08 3.08 1.925 1.155 1.925 1.925 1.925 1.925 1.925 1.925∗ 1.54 2.31 2.31 1.54 1.925
10.455 7.38 4.92 4.92 3.075 1.845 3.075 3.075 3.075 3.075 3.075 3.075∗ 2.46 3.69 3.69 2.46 3.075
0.0477 0.0609 0.0818 0.0799 0.1157 0.1681 0.1091 0.1061 0.1126 0.1169 0.1136 0.1093∗ 0.1284 0.0957 0.0985 0.1326 0.1027
148 121.24 150 245.81 151 629.32 152 374 151 874.16 150 502.44 151 681.23 152 586.89 152 724.8 152 222.11 153 013.22 153 552.71∗ 153 283.82 153 537.72 152 756.92 152 395.03 153 475.67
0.00 1.44 2.37 2.87 2.54 1.61 2.40 3.02 3.11 2.77 3.30 3.67 3.49 3.66 3.13 2.89 3.62
Table 2 Comparing the results of integrated decision with those achieved by the decentralized decision making Relative variable of model
l
k
Integrated policy 1 2 Decentralized policy 3 2 Integrated extra profit Percentage integrated extra joint profit (%)
I
n
m
Tr
TPr
TPm
TP3
TPs
JP
5 17
1.925 6.54
3.075 10.46
0.1093 0.0471
44 168 45 757 −1590 −3.47
54 140 51 323 +2817 +5.49
25 136 23 282 +1854 +7.96
30 109 27 757 +2353 +8.48
153 553 148 119 +5434 +3.67
Profit decrease; +: profit increase.
Taking the derivative of (35) with respect to Tr , set the results to zero, the optimal values of Tr and JP denoted by Tr+ and JP+ are 0.07986 year and $152,374, respectively. (b) Table 1 presents the results of l, k and I values for a range of values approximating l # , k # and I # . The optimal values of l ∗ , k ∗ , I ∗ , n ∗ , m ∗ and Tr∗ satisfying (31)–(33) are 1, 2, 5, 1.925, 3.075 and 0.1093 years, respectively. The maximal joint profit JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ ) is $153,552.71; the percentage of the extra joint profit (PEJP) is 3.67%. Table 2 compares the results of the integrated decision making with those by independent decision making. 6. Sensitivity analysis In the numerical example, the optimal value of joint profit JP(l, k, n, m, Tr ) for a fixed set of parameters, Φ = {Ai , A R , A Mw , A Rw , Fi , FMw , FRw , Pv , Pi , Pc , PR , D, C, R, P, M, K , L}, where i = r, M, 3, s are denoted by JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ ). The change in JP(l ∗ , k ∗ , n ∗ , m ∗ , Tr∗ ) is observed when the parameters in Φ change. The following analysis identifies the percentage joint profit change (PJPC) when only one of the parameters in the subset of Φ increases or decreases by 5%, 10% and 15%, and all the other parameters remain unchanged. Figs. 4–7 show the results of the sensitivity analysis. The key conclusions of the sensitivity analysis are as follows: (a) The optimal values of the joint profit (JP) are significantly influenced by the retailer price to the consumer (Pc ) and/or the annual demand rate (D). (b) The optimal values of the joint profit (JP) are significantly influenced by the third-party recycle dealer collecting unit cost from the consumer (P3 ), the supplier purchase unit price (Ps ) or the annual return rate (C).
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Fig. 4. The effect of Ar , A M , A R , A Mw , A Rw , A3 and As on PJPC.
Fig. 5. The effect of Fr , FM , FMw , FRw , F3 and Fs on PJPC.
(c) The joint profit (JP) increases when the annual return rate (C), the retailer selling price to the consumer (Pc ) or the annual demand rate (D) increases whereas the third party collecting unit cost from the consumer (P3 ), the supplier purchase unit price (Ps ), the setup cost parameter (Ai ) or the holding cost parameter (Fi ) decreases. (d) The percentage of joint profit change, PJPC, ranges from −34% to 34% with 15% increase or decrease in parameter values. 7. Conclusion This study derives an optimal strategy for a closed-loop supply chain system with remanufacturing. From the perspectives of the supplier; the manufacturer; the retailer and the third-party recycling dealer, an optimal production and replenishment policy was formulated to maximize the joint profit. An example comparing the decentralized decision-making system with the centralized decision-making system is illustrated. The findings in this study revealed a substantial profit increase using the integrated approach. The percentage extra joint profit (PEJP) is 3.67%. Sensitivity analysis showed that the optimal value of the joint profit is highly sensitive to the retail selling price
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Fig. 6. The effect of Pc , Pr , PM , PR , P3 and Ps on PJPC.
Fig. 7. The effect of D, C, R, P, M, K and L on PJPC.
and/or the annual demand. To achieve a win–win system, profit sharing should be considered in the supply chain collaborations. Future research should examine price sensitive demand and risk management. Acknowledgments The authors would like to express their appreciation to the referees for their contributions in improving the quality of the paper, and the National Science Council of the Republic of China, Taiwan for supporting this research under Contract No. NSC 95-2221-E-129 -007. Appendix A As Fig. 2 shows, when g equals Tr D 2 /R, the amount of stock requited by the retailer during reproduction of first shipment equals Tr D. The total stock increases at a rate of (R − D) during the manufacture of batch quantity of nT r D. The maximum inventory level at the completion of a reproduction batch equals Tr D 2 nTr D + (R − D) . R R
(A.1)
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Accordingly, the average total stock in the remanufacturing period of manufacturer–retailer inventory system (AIL Rr ) is given by n Tr D 2 (R − D) nTr D AIL Rr = + . (A.2) R 2R n+m Similarly, the average total stock in the manufacturing period of manufacturer–retailer inventory system (AIL Mr ) is given by Tr D 2 m (P − D) mTr D AIL Mr = . (A.3) + P 2P n+m The average total stock in the manufacture–retailer inventory system (AILmr ) is given by AILmr = AIL Mr + AIL Rr =
Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m + . 2R (n + m) 2P (n + m)
(A.4)
Consequently, average manufacturer inventory level can be determined by subtracting the average retailer inventory level from the average total inventory of the manufacturer–retailer inventory system. The average manufacturer inventory level (AILm ) is given by AILm =
Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D + − . 2R (n + m) 2P (n + m) 2
Additionally, the total relevant production cost of the manufacturer is given by n PR + m PM A M + A R + (n + m) M TCm (n, m, Tr ) = + FM n+m (n + m) Tr Tr D (2D + n R − n D) n Tr D (2D + m P − m D) m Tr D × + − . 2R (n + m) 2P (n + m) 2
(A.5)
(A.6)
The first term is the setup cost and the second term is the holding cost. Appendix B As Fig. 3 shows, the average used product inventory in the manufacturer warehouse (AIL Rw ) is R TR1 TR1 D 2 n 2 Tr DnTr AIL Rw = = since TR1 = . 2 k (n + m)Tr 2k(n + m)R R
(B.1)
The total relevant inventory cost of the manufacturer used product warehouse is: TC Rw (k, n, m, Tr ) =
k A Rw FRw PR D 2 n 2 Tr n PR D + + . (m + n)Tr 2k(n + m)R n+m
(B.2)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Similarly, as Fig. 3 shows, the total relevant inventory cost of the manufacturer material warehouse is: TC Mw (l, n, m, Tr ) =
FMw PM D 2 m 2 Tr l A Mw m PM D + + . (m + n)Tr 2l(n + m)P n+m
(B.3)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Appendix C As Fig. 3 shows, the average used product inventory of the third-party warehouse (AIL3 ) is 1 TR1 TR1 1 RTR1 − C(k − 1) (k − 1) AIL3 = (n + m)Tr 2 k k
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" 2 # DnTr 1 1 TR1 (k − 1) since TR1 = + C (n + m)Tr − (n + m)Tr 2 k R 2 2 Tr D n − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 = . 2 (n + m) Rk
(C.1)
The total relevant used product inventory cost to the third party is: A3 + k K P3 Dn + F3 P3 × AIL3 + (n + m)Tr n+m 2 2 F P T D n − 2Cn D(n + m) (k − 1) + C Rk(n + m)2 A3 + k K 3 3 r = + (n + m)Tr 2 (n + m) Rk P3 Dn + . n+m
TC3 (k, n, m, Tr ) =
(C.2)
The first term is the ordering cost, the second term is the holding cost and the third term is the purchasing cost. Appendix D As Fig. 3 shows, the lot size of the supplier is DmT r , and the net average material inventory level of the supplier in the integrated system (AILs ) can be determined by subtracting the average material inventory level of the manufacturer from the average total inventory in the supplier–manufacturer inventory system. TM1 D 2 m 2 Tr DmTr × − 2 (n + m)Tr 2l(n + m)P 2 2 D m Tr 1 = 1− . 2P(n + m) l
AILs =
since TM1 =
DmTr P (D.1)
The total relevant inventory cost to the supplier is: Fs Ps D 2 m 2 Tr As + l L + TCs (l, n, m, Tr ) = (n + m)Tr 2P(n + m)
1 1− l
+
Ps Dm . n+m
(D.2)
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