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A Lagrangian relaxation based approach for the capacitated lot sizing problem in closed-loop supply chain
Int. J. Production Economics 140 (2012) 249–255
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A Lagrangian relaxation based approach for the capacitated lot sizing problem in closed-loop supply chain Zhi-Hai Zhang n, Hai Jiang, Xunzhang Pan a
Department of Industrial Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e i n f o
abstract
Article history: Received 27 September 2010 Accepted 19 January 2012 Available online 30 January 2012
This paper investigates the capacitated lot sizing problem in closed-loop supply chain considering setup costs, product returns, and remanufacturing. We formulate the problem as a mixed integer program and propose a Lagrangian relaxation-based solution approach. The resulting Lagrangian subproblems are then solved by polynomial time algorithms. Compared to existing solution methods in the literature, our Lagrangian relaxation based approach is advantageous in that it naturally provides a lower bound on the optimal objective function value, which allows us to assess the quality of solutions found. Numerical experiments using synthesized data demonstrate that our approach can find quality solutions efficiently. & 2012 Elsevier B.V. All rights reserved.
Keywords: Capacitated lot sizing Lagrangian relaxation Closed-loop supply chain
1. Introduction In recent years, the lot sizing problem (LSP) in closed-loop supply chain has received growing attention (Jr. and Wassenhove, 2009). Unlike traditional supply chain, where products flow from manufacturers to customers, in closed-loop supply chain, the manufacturers often setup a program to collect used products from customers and further process them to make a profit or reduce their environmental impacts. Such collection programs then incur reverse product flows from customers back to manufacturers and form what is called a closed-loop supply chain. In closed-loop supply chain, the lot sizing problem needs to take into consideration not only time-varying demands for a set of products over certain periods but also the inventory costs and processing costs associated with collected used products. The goal is to produce a cost-minimizing production schedule, that is, the production quantities for each product at each period (Brahimi et al., 2006b). In this paper, we study the capacitated lot sizing problem in closed-loop supply chain where setup costs, product returns and remanufacturing are considered. Our research originated from the collaboration with a paper product manufacturer. Besides making newly manufacturers paper product from virgin pulp, the manufacturer also utilizes deinked pulp processed from collected recyclable paper to make remanufactured product. The two paper products made from virgin pulp and deinked pulp are branded differently to serve different market segments. In its plant, the production using virgin pulp and that using deinked pulp are subject to an overall
n
Corresponding author. Tel.: þ86 010 62772874; fax: þ86 010 62794399. E-mail addresses: [email protected] (Z.-H. Zhang), [email protected] (H. Jiang), [email protected] (X. Pan). 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2012.01.018
production capacity limit. When recyclable paper products can not be processed into deinked pulp fast enough, they incur inventory cost at the warehouse. The manufacturing cost for virgin pulp and the remanufacturing cost for de-inked pulp are different and so are the demands and prices of the final paper products made from the two types of pulps. Since the pioneering work of Wagner and Whitin (1958), there has been a growing interest in lot sizing problem. For a review of lot sizing problems, their extensions and solution approaches, please refer to Karimi et al. (2003), Brahimi et al. (2006b), Degraeve and ¨ Jans (2007), Quadt and Kuhn (2008) and Buschkuhl et al. (2010). Recent research can be referred to Hop and Tabucanon (2005), ¨ Brahimi et al. (2006a), Pan et al. (2009), Sural et al. (2009), Smith ¨ nal and Romeijn (2010), et al. (2009), Ca´rdenas-Barro´n (2010), O ˜eyro and Viera (2010), Helber and Sahling (2010), Sancak and Pin Salman (2011), Kenne´ et al. (2011). Existing literature on capacitated closed-loop lot-sizing problem is limited. Li et al. (2007) examine the capacitated lot sizing problem with substitutions and return products. They first develop a heuristic genetic algorithm (GA) to determine all periods requiring production setup and then take a dynamic programming approach to determine the quantities produced for new products and remanufactured products. The performance of their heuristic genetic algorithm is compared to a branch-and-bound algorithm implemented in Lingo. Although genetic algorithm can produce good feasible solutions, it is difficult to gage its optimality. More recently, Pan et al. (2009) study the capacitated lot sizing problem in closed-loop supply chain where returned products are collected from customers. They look at several variants of the problem and take a dynamic programming (DP) approach to solve the models. In the variant where both production and remanufacturing are capacitated, a pseudo polynomial time algorithm is proposed. Experiments show that their approach can find optimal solutions for small problem instances, however, for
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Z.-H. Zhang et al. / Int. J. Production Economics 140 (2012) 249–255
large-scale problems, dynamic programming based approaches suffer from the renowned problem of ’’curse of dimensionality’’ and it is difficult, if not impossible, to solve the model. In this paper, we study a variant of the capacitated lot sizing problem in closed-loop supply chain. The problem context in our mind is as follows. A factory produces two types of products, one manufactured from raw materials and the other remanufactured from collected used products. The demands for these two products are separate, deterministic, and time varying during a finite planning horizon, and should be satisfied without backlogging. The total production capacity for manufacturing new product and remanufacturing used product are limited. We formulate the problem as a mixed integer programming model and develop a Lagrangian relaxation (LR) based solution approach. The Lagrangian subproblems are solved by polynomial time algorithms. Compared to existing GA or DP based algorithms, our approach is advantageous in that: (a) the LR approach can provide a lower bound to assess the quality of solutions found while GA approach cannot; and (b) although the DP approach can find optimal solutions to small size problems, it often fails for larger instances. Computational experiments demonstrate that our approach can find satisfactory solutions efficiently. The remainder of the paper is organized as follows. In Section 2, we define notations used in this paper and present our model. We discuss the details of our Lagrangian relaxation based approach in Section 3. In Section 4, we perform extensive computational experiments to evaluate the performance of the algorithm. Finally, we conclude this paper and outline future research directions in Section 5.
Table 2 Decision variables.
at
1, if new products are manufactured in period t; 0, otherwise Quantity of new products manufactured in period t Inventory stock of new products at the end of period t 1, if returned products are remanufactured in period t; 0, otherwise Quantity of returned products remanufactured in period t Inventory stock of remanufactured products at the end of period t Inventory stock of returned products at the end of period t
xt It
bt yt it itr
The manufacturing capacity is sufficient to meet the demands in each period, in particular we have: 1. The capacity can satisfy the demands for new products and remanufactured products simultaneously, i.e., t X
ðDi þdi Þ r
i¼1
t X
Ci,
8t ¼ 1,2,. . .,T:
i¼1
2. The quantity of returned products can satisfy the demand for remanufactured products, i.e., t X
di r
i¼1
t X
Ri ,
8t ¼ 1,2,. . .,T:
i¼1
3. The capacity can satisfy the demand of remanufactured products in each period, i.e., dt rC t ,
8t ¼ 1,2,. . .,T:
Initial inventory stocks are zero, i.e., 2. Notation and model formulation We assume that the facility in focus produces one type of product. Simultaneously, it collects returned product and makes remanufacturing production. The amount of returned products is deterministic over the planning horizon. New products and remanufactured products face deterministic but time-varying demands for a finite planning horizon and should be satisfied separately without backlogging. The costs consist of a fixed setup cost incurred whenever production is scheduled, a linear production cost proportional to the production quantity, and a linear inventory holding cost. All of the cost components are considered for both manufacturing and remanufacturing activities. In addition, the inventory holding cost of returned products is taken into account as well. Tables 1 and 2 summarize the notation used in this paper. Without loss of generality, we further make the following underlying assumptions.
The demands for new products and remanufactured products are separate and backlog is not allowed. Table 1 Data and parameters. T St Ut Pt rt Ht ht hr Dt dt Rt Ct M
Number of time periods in the planning horizon Setup cost for manufactured new products in period t Setup cost for remanufactured products in period t Unit production cost for new products in period t Unit production cost for remanufactured products in period t Unit holding cost for new products in period t Unit holding cost for remanufactured products in period t Unit holding cost for returned products in period t Demand of new products in period t Demand of remanufactured product in period t Quantity of returned product in period t Available capacities for manufacturing and remanufacturing activities in period t A sufficiently large positive number
I0 ¼ i0 ¼ ir0 ¼ 0:
Inventory holding cost of returned products is less than that of remanufactured products, i.e., T X
T X
r
hi r
i¼t
hi ,
8t ¼ 1,2,. . .,T:
i¼t
Now we formally present our model as: Min
T X
r r
½ðSt at þ pt xt þ Ht It Þ þ ðU t bt þr t yt þht it Þ þ ht it ,
ð1Þ
t¼1
s:t: It ¼ It1 þ xt Dt ,
8t ¼ 1,2,. . .,T,
ð2Þ
it ¼ it1 þyt dt ,
8t ¼ 1,2,. . .,T,
ð3Þ
irr ¼ irt1 þ Rt yt ,
8t ¼ 1,2,. . .,T,
ð4Þ
xt þ yt r C t ,
8t ¼ 1,2,. . .,T,
ð5Þ
xt rM at ,
8t ¼ 1,2,. . .,T,
ð6Þ
yt r M bt ,
8t ¼ 1,2,. . .,T,
ð7Þ
at , bt A f0,1g, 8t ¼ 1,2,. . .,T,
ð8Þ
xt ,yt Z0,
ð9Þ
8t ¼ 1,2,. . .,T:
The objective function minimizes the sum of setup cost, production cost, and inventory cost for new products and remanufactured products in all periods. Constraints (2)–(4) are inventory balance constraints for new products, remanufactured products, and returned products, respectively. Constraints (5) represent capacity constraints for manufacturing and remanufacturing activities. Constraints (6) and (7) specify the setup costs of manufacturing and remanufacturing. Constraints (8) and (9) are standard integrality and non-negative constraints.
Z.-H. Zhang et al. / Int. J. Production Economics 140 (2012) 249–255
Hence, the lower bound provided by Lagrangian relaxation can be written as:
3. Solution approach Florian et al. (1980) and Bitran and Yanasse (1982) have shown that the lot sizing problem with time-varying capacity constraints are NP-hard. In Li et al. (2007), heuristic genetic algorithms for capacitated production planning problems with batch processing and remanufacturing are proposed. They compare their planning results with those produced by Lingo for problems with relatively small size (Tr21). Experiments show that their approach can generate satisfactory results. However, for large problem instances, it is difficult to assess the quality of the solution. Pan et al. (2009) later take a dynamic programming approach and develop a pseudopolynomial time algorithm. For instances of small size (Tr25), their approach can yield optimal solution, however, for larger instances, dynamic programming based approach often fail due to the renowned ‘‘curse of dimensionality’’ (Powell, 2007). To address challenges faced by existing solution approaches, we propose a Lagrangian relaxation based approach to solve the capacitated closed-loop lot sizing problem. The advantages of our approach include:
Our approach decomposes the full problem into two uncapacitated lot sizing problem to fully exploit existing solution methods for uncapacitated lot sizing problems. Besides producing good quality feasible solutions, we also provide a lower bound to gage its optimality. Lagrangian relaxation based approach has proven to be an effective approach to solve large optimization problems.
LB : ¼ V n þ V r
T X
Sub-problem LSnew is a classical uncapacitated single-item lot sizing problem, which has been well researched. The best algorithm so far is proposed independently by Federgruen and Tzur (1991), Wagelmans et al. (1992) and Aggarwal and Park (1993) with computational complexity of O(TlogT). We implement the algorithm of Wagelmans et al. (1992) in this research to solve sub-problem LSnew. Unfortunately, sub-problem LSreturned is different from the classical uncapacitated single-item lot sizing problem due to constraints (4), which make the sub-problem intractable. To find its optimal solution, we reformulate this sub-problem as is done in Heuvel and Wagelmans (2008). We introduce marginal P r production cost, ct ¼ r t þ lt þ Ti ¼ t ðhi hi Þ and net cumulative Pt demand, st ¼ j ¼ t þ 1 dj it , which represents the amount of replenishment required to satisfy demand from period t þ1 to T at the end of period t. Then, the reformulated sub-problem is presented below: ðNCDÞ :
Min
T X
ðU t bt þ ct yt Þ þO,
t¼1
s:t:
T X
dj
s0 ¼ We decide to relax the capacity constraints, that is, Constraints (5). By introducing non-negative Lagrangian multipliers lt , where t A f1,2, Tg, we obtain the Lagrangian dual:
lt
r
½ðSt at þ pt xt þ Ht It Þ þ ðU t bt þ r t yt þ ht it þ ht irt Þ
t¼1
þ
T X
lt ðxt þ yt C t Þ
t¼1 T X
T X
Rj r st r
j¼1
T X
T X
dj ,
8t ¼ 1,2,. . .,T1,
j ¼ tþ1
8t ¼ 1,2,. . .,T,
dj ,
j¼1
sT ¼ 0, yt r M bt , 8t ¼ 1,2,. . .,T, bt A f0,1g, 8t ¼ 1,2,. . .,T, st ,yt Z 0, 8t ¼ 1,2,. . .,T: PT P r where, O ¼ t ¼ 1 ti ¼ 1 ðht Ri ht di Þs. Note that O is a constant term, and therefore can be omitted. Lemma 1. NCD and LSreturned are equivalent.
½St at þ ðpt þ lt Þxt þ Ht It
Proof. Follows from Heuvel and Wagelmans (2008). &
t¼1
þ
t X
st ¼ st1 yt ,
3.1. Lower bound
T X
lt C t x:
t¼1
j¼1
MaxMin
251
r
½U t bt þ ðr t þ lt Þyt þ ht it þ ht irt
t¼1
T X
lt C t :
ð10Þ
t¼1
Subject to constraints (2)–(4), (6)–(9). After we remove the constant of the last term in Eq. (10) and fix the value of lt , the relaxed problem can be decomposed into two subproblems for manufacturing and remanufacturing activities, respectively. They are given as follows: ðLSnew Þ :
V n ¼ Min
T P
½St at þ ðpt þ lt Þxt þ Ht It ,
t¼1
s:t: ð2Þ,ð6Þ,
at A f0,1g, 8t ¼ 1,2,. . .,T, xt Z 0, ðLSreturned Þ :
V r ¼ Min
T X
The reformulated sub-problem is solved by an efficient backward dynamic programming algorithm proposed by Liu (2008) in O(T2) time. 3.2. Upper bound After obtaining the lower bound, we propose a procedure to find a feasible solution at each iteration of the Lagrangian procedure. Capacity constraints (5) are relaxed. Thus, manufacturing/remanufacturing quantities need to be shifted to fit available capacity at each period. The procedure for finding a feasible solution is summarized as follows: 3.2.1. Procedure for upper bound:
8t ¼ 1,2,. . .,T: r
½U t bt þ ðr t þ lt Þyt þ ht it þ ht irt ,
t¼1
s:t: ð3Þ,ð4Þ,ð7Þ,
bt A f0,1g, 8t ¼ 1,2,. . .,T, yt Z 0,
8t ¼ 1,2,. . .,T:
Step 1: Set t ¼1. Step 2: If t oT and capacity constraints (5) are not violated, goto Step 7; else goto Step 3. If t ¼T and capacity constraints (5) are not violated, stop; else goto Step 5. Step 3: Move the remanufacturing of returned products from period t to t þ1. If capacity constraints (5) are satisfied after the modification, goto Step 7; else goto Step 4. In order to get a
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feasible solution, the modification for period t can not destroy the feasibility of the solutions before period t. Therefore, we denote slt as the amount of new and returned products processed at period t which exceed the capacity and alyt as the amount of returned products at period t can be shifted to be remanufactured at the following periods. And, they are expressed, respectively as follows: slt ¼ xt þyt 2C t , t X ðyi di Þ: alyt ¼ i¼1
Then, the amount of remanufactured products which is shifted to the following periods, denoted by y, is as follow: y ¼ minfslt ,yt ,alyt g: Note that slt, xt and yt are updated at each movement. For example, slt reduces to slt y at the end of this step. Step 4: Move the manufacturing of new products from period t to tþ 1. If capacity constraints (5) are satisfied, goto Step 7; else goto Step 5. In this step, yt does not shift anymore. We just modify xt. Similarly, the moved amount of new products, denoted by x, is determined by x ¼ minfslt ,alxt g, P where alxt ¼ ti ¼ 1 ðxi Di Þ. It refers to the amount of new products can be shifted to be manufactured in the following periods. Since dt rC t and yt r dt after the modification in the previous step, slt r xt . Therefore, xt is excluded from the above formula. Step 5: Move the manufacturing of new products from t to j (1rjrt 1). If capacity constraints (5) are satisfied, goto Step 7; else goto Step 6. So far, xt and yt can not be further shifted to the following periods. In order to satisfy the demand at period t under the capacity constraints, we consider backward movement of the manufacturing of new products in this step. The amount of new product, which is shifted from period t to j (1rj rt 1) that increases the total cost the least, is x ¼ minfslt ,C j 2xj 2yj g, where Cj xj yj is the remaining manufacturing capacity at period j. Once slt ¼0, goto Step 7. Step 6: Move the remanufacturing of returned products of the previous t 1 periods to period tþ1 so as to have the capacity to manufacture new products which are shifted from period t. Start at period t 1 and move remanufacturing of returned products in period j (1rj rt 1) to period t þ1 meanwhile guarantee that the demands of remanufactured products from period j to t must be satisfied. The shifted amount from period j is determined by: y ¼ minfalyj ,yj ,slt g, P where, alyj ¼ ji ¼ 1 ðyi di Þ. Goto Step 7 once slt ¼0. Step 7: If t ¼T, stop; else set t ¼t þ1, goto Step 2.
Lemma 2. The computational complexity of the procedure is OðT 2 Þ . Proof. It is shown due to the number of shifts of the quantities of manufacturing/remanufacturing at each period in the procedure. That is, j (1rj rt 1) times shifts occur at period t. Therefore, the upper bound can be found in O(T2) time. &
At the end of our Lagrangian relaxation procedure, a local search heuristic, which is an extension of the smoothing heuristic proposed by Trigeiro et al. (1989), is proposed to further improve the incumbent solution. Step 1: Backward pass for manufacturing new product. Start at period T and move the production of new product to the earlier periods that increase the total cost the least. The movable quantity of new products from period i (TZiZ2) to j (i 1 ZjZ1) is determined by minfxj ,C j xj yj g: Step 2: Forward pass for manufacturing new product. Start at period 1 and move the production of new product to the next later periods that increase the total cost the least. The movable quantity of new products from period i (1rirT 1) to j (iþ1 rjrT) is determined by ( ) t X min xi , min ðxk Dk Þ,C j xj yj : i r t r j1
k¼1
In contrast to the backward pass, we must guarantee the demands of new products from period i to j-1 must be satisfied P in this step. So, mini r t r j1 tk ¼ 1 ðxk Dk Þ is taken into account. Step 3: Backward pass for remanufacturing returned product. Start at the last period and move the remanufacturing of returned product to the earlier periods that increase the total cost the least. The movable quantity of remanufactured products from i (TZi Z2) to j (i-1Zj Z1) is: 8 9 8 9 j < = < X = i1 t X X min yi , ðRt yt Þ max 0, ðyk Rk Þ ,C j xj yj , : ; : t¼1 ; t ¼ jþ1
k ¼ jþ1
Pj
where stock of returned t ¼ 1 ðRt yt Þ refers to the inventory Pi1 Pt products at the end of period j, t ¼ j þ 1 maxf0, k ¼ jþ1 ðyk Rk Þg is the inventory stock of returned products at the end of period j which is remanufactured between period j þ1 and period i 1. Step 4: Forward pass for remanufacturing returned product. Start at period 1 and move the remanufacturing of returned product to the next later periods that increase the total cost the least. The movable quantity of remanufactured products from period i (1rirT 1) to j (iþ1 rjrT) is: ( ) t X min yi , min ðyk dk Þ,C j xj yj : i r t r j1
k¼1
Step 5: Repeat from Step 1 to Step 4 once again. Lemma 3. The improvement feasible solution can be obtained in O T 2 time. Proof. The proof is similar with that of Lemma 2.
&
3.3. Updating the Lagrangian multipliers The Lagrangian multipliers lt are updated at each iteration using standard sub-gradient optimization (Fisher, 2004). n o lkt þ 1 ¼ max 0, lkt þ directionk ðxkt þ ykt C t Þ , directionk ¼ d PT
Z nUB Z LB
t¼1
ðxkt þykt C t Þ2
,
where the superscript (kþ 1) indicates that this update is for the (kþ 1) iteration. Z nUB is the best upper bound so far. ZLB is the lower
Z.-H. Zhang et al. / Int. J. Production Economics 140 (2012) 249–255
Table 3 Parameters for generating data sets.
253
Table 4 Parameters for the Lagrangian relaxation procedure.
Parameter
Value
Parameter
Value
Ht ht htr pt rt St Ut Dt dt Rt
Uniformly drawn from [5, 10) Uniformly drawn from [3, 8) f t ht , where ft is uniformly drawn from [0, 1) Uniformly drawn from [15, 20) Uniformly drawn from [10, 15) Uniformly drawn from [6000, 8000) Uniformly drawn from [4000, 6000) Uniformly drawn from [300, 500) drawn from [200, 300) P P kt dt þ maxf ti ¼ 1 di ti ¼ 1 Ri g, where kt is uniformly drawn from [0.5, 1.5) TFðDt þ dt Þ , where TF¼ 1.2
Minimum LB–UP gap Maximum iteration count Maximum number of iterations before halving is d Initial value of d Minimum value of d
1% 300 5 2.0
Ct
Table 5 Performance of the Lagrangian relaxation procedure, TF¼ 1.2. T
Avg. CPU time (s)
Avg. GAP (%)
TF is a coefficient and used to adjust the capacity of each period.
bound at current iteration. d is initialized to 2.0 and halved if there are five consecutive iterations in which the lower bound does not increase.
3.4. Stopping criteria The Lagrangian relaxation procedure is terminated once one of the following stopping criteria is satisfied.
Optimality gap: The optimality gap between the lower and
upper bounds is less than a defined value. Maximum iteration count: A defined number of iterations is performed. Step size multiplier: d is less than a defined epsilon.
4. Computational results The algorithm is implemented in ILOG OPL Script language and tested on a computer with an Intel Core 1.67 GHz CPU. The standard optimization software Cplex version 12.0 was used to find optimal solutions for small-scale instances. We randomly generate numerous data sets to test the quality of the solution algorithm described in the previous section. Data generated for all of the test instances are available upon request. Table 3 shows the parameter values which are used to generate test data in this section. Table 4 lists the parameter values for the Lagrangian relaxation procedure.
4.1. Performance evaluation We use three criteria to evaluate the quality of the algorithm.
The gap between the lower bound (LB) and the upper bound (UB), which is calculated as follows (Brahimi et al., 2006a): GAP ¼ 2
UBLB : UB þ LB
The distances between the upper bound (UB) and the optimal solution (OP), are denoted as UGAP, and between the lower bound (LB) and the optimal solution (OP), are denoted as LGAP. We obtain the optimal solution by solving the problem in Cplex for small-scale instances. And the distances are calculated as follows: UGAP ¼ 2
UBOP , UB þ OP
108
3 5 6 12 20 24 35 48 50 65 80
0.70 1.12 1.58 3.65 7.16 9.35 16.96 30.53 35.65 52.96 75.16
6.46 6.86 6.83 7.47 8.33 8.44 8.37 8.99 7.59 8.39 9.14
UGAP (%)
LGAP (%)
Avg.
Max.
Avg.
Max.
0.00 0.47 0.38 2.27 2.63 2.50
0.00 1.52 1.48 3.97 4.70 4.05
6.46 6.39 6.46 5.21 5.70 5.95
8.92 8.42 8.90 6.99 8.34 7.41
Table 6 Average gap and CPU time for instances with different capacity and T¼ 20. TF
Avg. GAP (%)
Avg. CPU time (s)
1.0 1.2 1.4 1.6 1.8 2.0
4.74 8.33 4.22 2.18 1.38 1.03
4.38 7.16 5.53 3.39 2.23 0.71
LGAP ¼ 2
OPLB : OP þ LB
Running time. Ten instances for each T are generated randomly and the computational results are reported in Table 5. Note that the blank space in Table 5 indicates that Cplex can not find the optimal solution in 10 min. We find that the average GAP is less than 10%. Compared with the gaps of some capacitated lot sizing problems (Trigeiro et al., ¨ 1989; Toledo and Armentano, 2006; Sural et al., 2009) solved by Lagrangian relaxation-based methods, it reveals the good performances of the algorithm. Furthermore, we analyze the distance of the upper/lower bounds from the optimal solution for small-scale instances. And UGAP o LGAPindicates that the solution has a pretty good quality and the lower bound makes GAP poor. Table 6 shows the impact of production capacity on the performance of the algorithm. Except the case of TF¼1.0, it is clear that the gaps and CPU times decrease with larger capacity. TF¼1.0 implies the optimal production policy is Lot-for-Lot. UGAP is zero in all of the instances. Hence, GAP is generated by relaxing capacity constraints (5). Therefore, it has a smaller gap and a short CPU time. Table 7 reports GAP, UGAP and LGAP of instances with different capacity. We can obtain the similar conclusion with that in Table 5.
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Table 7 UGAP and LGAP of instances with different capacity. T
TF¼ 1.1
TF ¼1.2
TF¼ 1.3
TF ¼1.4
TF¼ 1.5
TF ¼1.6
6
GAP (%) UGAP (%) LGAP (%)
8.17 0.18 7.82
6.81 0.39 6.43
6.47 0.82 5.65
5.35 1.11 3.59
3.80 1.03 2.77
3.31 1.29 1.49
12
GAP (%) UGAP (%) LGAP (%)
9.80 2.02 7.78
7.44 2.25 5.19
6.22 2.06 4.16
5.15 2.23 2.93
3.32 1.21 2.10
2.47 1.10 1.37
24
GAP (%) UGAP (%) LGAP (%)
8.90 2.03 6.87
8.47 2.57 5.90
6.63 2.79 3.83
4.58 2.19 2.40
4.34 2.31 2.03
2.37 1.17 1.20
hr
Total cost
51,893.4 46,307.8 44,076.4 42,614.8 42,567.6
496,296.7 449,266.8 433,454.5 425,223.9 421,327.8
Table 8 Cost components for the instances with different capacity, T¼ 20. TF
1.0 1.2 1.4 1.6 1.8
Manufacturing
Remanufacturing
S
p
H
U
r
h
139,636.2 98,220.0 84,834.2 76,605.9 68,548.9
142,288.7 141,047.5 140,472.2 139,339.5 138,525.4
0.0 19,346.9 26,365.7 27,969.3 34,802.7
100,778.3 68,329.0 64,845.8 62,598.5 63,554.1
61,700.2 60,893.5 60,089.8 60,115.9 59,770.3
0.0 10,245.3 13,229.5 15,764.7 15,531.1
4.2. Managerial insights We also observe the changes of cost components with the increase of the capacity, which are shown in Table 8. The first column lists TF. The following three columns show the total setup, production and holding costs associated with manufacturing activities, respectively. Then, the costs associated with remanufacturing activities are reported in the fifth, sixth and seventh columns. The column labeled hr lists the total holding cost associated with returned products. And, the last column reports the total costs. The manufacturing and remanufacturing activities are more likely performed at periods with smaller unit production and reproduction costs if the available capacity is large enough. Therefore, setup and production costs associated with manufacturing and remanufacturing activities show decreasing trends. And inventory holding costs show reverse trends. Accordingly, holding costs of returned products decrease. The total costs become smaller. In conclusion, it makes sense to have more production capacity to reduce the total cost. However, managers have to balance larger initial investment and smaller operational costs.
5. Conclusions and future research directions In this paper, we propose a Lagrangian relaxation-based solution algorithm to solve a capacitated lot sizing problem with setup costs, product returns and remanufacturing, which has received a growing attention in recent years. The lower bound is calculated by decoupling the relaxed problem into two subproblems, which are solved by two polynomial-time algorithms, respectively. Two efficient and effective heuristics are proposed to find the good upper bound. Numerous computational experiments show that the proposed algorithm is capable of solving the problem efficiently. The impact of capacity on the performance and the behavior of the model are investigated as well. There are some extensions of this work. First, manufacturing and remanufacturing activities are only considered in the model. There exist more activities in a closed-loop supply chain, for instance, refurbuishing, recycling, etc, all of them have a closed
relation with production planning. It is interesting to consider more general planning scenarios. Second, handling returned products is a strongly random activity. Therefore, it is our future work to study the stochastic counterpart of the model.
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Li, Y., Chen, J., Cai, X., 2007. Heuristic genetic algorithm for capacitated production planning problems with batch processing and remanufacturing. International Journal of Production Economics 105, 301–317. Liu, T., 2008. Economic lot sizing problem with inventory bounds. European Journal of Operational Research 185, 204–215. ¨ nal, M., Romeijn, H.E., 2010. Multi-item capacitated lot-sizing problems with O setup times and pricing decisions. Naval Research Logistics 57 (2), 172–187. Pan, Z., Tang, J., Liu, O., 2009. Capacitated dynamic lot sizing problems in closedloop supply chain. European Journal of Operational Research 198, 810–821. ˜ eyro, P., Viera, O., 2010. The economic lot-sizing problem with remanufacturing Pin and one-way substitution. International Journal of Production Economics 124, 482–488. Powell, W.B., 2007. Approximate Dynamic Programming: Solving the Curses of Dimensionality. Wiley-Blackwell. Quadt, D., Kuhn, H., 2008. Capacitated lot-sizing with extensions: a review. 4OR: A Quarterly Journal of Operations Research 6, 61–83. Sancak, E., Salman, F.S., 2011. Multi-item dynamic lot-sizing with delayed transportation policy. International Journal of Production Economics 131, 595–603.
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Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A Lagrangian relaxation based approach for the capacitated lot sizing problem in closed-loop supply chain Zhi-Hai Zhang n, Hai Jiang, Xunzhang Pan a
Department of Industrial Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e i n f o
abstract
Article history: Received 27 September 2010 Accepted 19 January 2012 Available online 30 January 2012
This paper investigates the capacitated lot sizing problem in closed-loop supply chain considering setup costs, product returns, and remanufacturing. We formulate the problem as a mixed integer program and propose a Lagrangian relaxation-based solution approach. The resulting Lagrangian subproblems are then solved by polynomial time algorithms. Compared to existing solution methods in the literature, our Lagrangian relaxation based approach is advantageous in that it naturally provides a lower bound on the optimal objective function value, which allows us to assess the quality of solutions found. Numerical experiments using synthesized data demonstrate that our approach can find quality solutions efficiently. & 2012 Elsevier B.V. All rights reserved.
Keywords: Capacitated lot sizing Lagrangian relaxation Closed-loop supply chain
1. Introduction In recent years, the lot sizing problem (LSP) in closed-loop supply chain has received growing attention (Jr. and Wassenhove, 2009). Unlike traditional supply chain, where products flow from manufacturers to customers, in closed-loop supply chain, the manufacturers often setup a program to collect used products from customers and further process them to make a profit or reduce their environmental impacts. Such collection programs then incur reverse product flows from customers back to manufacturers and form what is called a closed-loop supply chain. In closed-loop supply chain, the lot sizing problem needs to take into consideration not only time-varying demands for a set of products over certain periods but also the inventory costs and processing costs associated with collected used products. The goal is to produce a cost-minimizing production schedule, that is, the production quantities for each product at each period (Brahimi et al., 2006b). In this paper, we study the capacitated lot sizing problem in closed-loop supply chain where setup costs, product returns and remanufacturing are considered. Our research originated from the collaboration with a paper product manufacturer. Besides making newly manufacturers paper product from virgin pulp, the manufacturer also utilizes deinked pulp processed from collected recyclable paper to make remanufactured product. The two paper products made from virgin pulp and deinked pulp are branded differently to serve different market segments. In its plant, the production using virgin pulp and that using deinked pulp are subject to an overall
n
Corresponding author. Tel.: þ86 010 62772874; fax: þ86 010 62794399. E-mail addresses: [email protected] (Z.-H. Zhang), [email protected] (H. Jiang), [email protected] (X. Pan). 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2012.01.018
production capacity limit. When recyclable paper products can not be processed into deinked pulp fast enough, they incur inventory cost at the warehouse. The manufacturing cost for virgin pulp and the remanufacturing cost for de-inked pulp are different and so are the demands and prices of the final paper products made from the two types of pulps. Since the pioneering work of Wagner and Whitin (1958), there has been a growing interest in lot sizing problem. For a review of lot sizing problems, their extensions and solution approaches, please refer to Karimi et al. (2003), Brahimi et al. (2006b), Degraeve and ¨ Jans (2007), Quadt and Kuhn (2008) and Buschkuhl et al. (2010). Recent research can be referred to Hop and Tabucanon (2005), ¨ Brahimi et al. (2006a), Pan et al. (2009), Sural et al. (2009), Smith ¨ nal and Romeijn (2010), et al. (2009), Ca´rdenas-Barro´n (2010), O ˜eyro and Viera (2010), Helber and Sahling (2010), Sancak and Pin Salman (2011), Kenne´ et al. (2011). Existing literature on capacitated closed-loop lot-sizing problem is limited. Li et al. (2007) examine the capacitated lot sizing problem with substitutions and return products. They first develop a heuristic genetic algorithm (GA) to determine all periods requiring production setup and then take a dynamic programming approach to determine the quantities produced for new products and remanufactured products. The performance of their heuristic genetic algorithm is compared to a branch-and-bound algorithm implemented in Lingo. Although genetic algorithm can produce good feasible solutions, it is difficult to gage its optimality. More recently, Pan et al. (2009) study the capacitated lot sizing problem in closed-loop supply chain where returned products are collected from customers. They look at several variants of the problem and take a dynamic programming (DP) approach to solve the models. In the variant where both production and remanufacturing are capacitated, a pseudo polynomial time algorithm is proposed. Experiments show that their approach can find optimal solutions for small problem instances, however, for
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large-scale problems, dynamic programming based approaches suffer from the renowned problem of ’’curse of dimensionality’’ and it is difficult, if not impossible, to solve the model. In this paper, we study a variant of the capacitated lot sizing problem in closed-loop supply chain. The problem context in our mind is as follows. A factory produces two types of products, one manufactured from raw materials and the other remanufactured from collected used products. The demands for these two products are separate, deterministic, and time varying during a finite planning horizon, and should be satisfied without backlogging. The total production capacity for manufacturing new product and remanufacturing used product are limited. We formulate the problem as a mixed integer programming model and develop a Lagrangian relaxation (LR) based solution approach. The Lagrangian subproblems are solved by polynomial time algorithms. Compared to existing GA or DP based algorithms, our approach is advantageous in that: (a) the LR approach can provide a lower bound to assess the quality of solutions found while GA approach cannot; and (b) although the DP approach can find optimal solutions to small size problems, it often fails for larger instances. Computational experiments demonstrate that our approach can find satisfactory solutions efficiently. The remainder of the paper is organized as follows. In Section 2, we define notations used in this paper and present our model. We discuss the details of our Lagrangian relaxation based approach in Section 3. In Section 4, we perform extensive computational experiments to evaluate the performance of the algorithm. Finally, we conclude this paper and outline future research directions in Section 5.
Table 2 Decision variables.
at
1, if new products are manufactured in period t; 0, otherwise Quantity of new products manufactured in period t Inventory stock of new products at the end of period t 1, if returned products are remanufactured in period t; 0, otherwise Quantity of returned products remanufactured in period t Inventory stock of remanufactured products at the end of period t Inventory stock of returned products at the end of period t
xt It
bt yt it itr
The manufacturing capacity is sufficient to meet the demands in each period, in particular we have: 1. The capacity can satisfy the demands for new products and remanufactured products simultaneously, i.e., t X
ðDi þdi Þ r
i¼1
t X
Ci,
8t ¼ 1,2,. . .,T:
i¼1
2. The quantity of returned products can satisfy the demand for remanufactured products, i.e., t X
di r
i¼1
t X
Ri ,
8t ¼ 1,2,. . .,T:
i¼1
3. The capacity can satisfy the demand of remanufactured products in each period, i.e., dt rC t ,
8t ¼ 1,2,. . .,T:
Initial inventory stocks are zero, i.e., 2. Notation and model formulation We assume that the facility in focus produces one type of product. Simultaneously, it collects returned product and makes remanufacturing production. The amount of returned products is deterministic over the planning horizon. New products and remanufactured products face deterministic but time-varying demands for a finite planning horizon and should be satisfied separately without backlogging. The costs consist of a fixed setup cost incurred whenever production is scheduled, a linear production cost proportional to the production quantity, and a linear inventory holding cost. All of the cost components are considered for both manufacturing and remanufacturing activities. In addition, the inventory holding cost of returned products is taken into account as well. Tables 1 and 2 summarize the notation used in this paper. Without loss of generality, we further make the following underlying assumptions.
The demands for new products and remanufactured products are separate and backlog is not allowed. Table 1 Data and parameters. T St Ut Pt rt Ht ht hr Dt dt Rt Ct M
Number of time periods in the planning horizon Setup cost for manufactured new products in period t Setup cost for remanufactured products in period t Unit production cost for new products in period t Unit production cost for remanufactured products in period t Unit holding cost for new products in period t Unit holding cost for remanufactured products in period t Unit holding cost for returned products in period t Demand of new products in period t Demand of remanufactured product in period t Quantity of returned product in period t Available capacities for manufacturing and remanufacturing activities in period t A sufficiently large positive number
I0 ¼ i0 ¼ ir0 ¼ 0:
Inventory holding cost of returned products is less than that of remanufactured products, i.e., T X
T X
r
hi r
i¼t
hi ,
8t ¼ 1,2,. . .,T:
i¼t
Now we formally present our model as: Min
T X
r r
½ðSt at þ pt xt þ Ht It Þ þ ðU t bt þr t yt þht it Þ þ ht it ,
ð1Þ
t¼1
s:t: It ¼ It1 þ xt Dt ,
8t ¼ 1,2,. . .,T,
ð2Þ
it ¼ it1 þyt dt ,
8t ¼ 1,2,. . .,T,
ð3Þ
irr ¼ irt1 þ Rt yt ,
8t ¼ 1,2,. . .,T,
ð4Þ
xt þ yt r C t ,
8t ¼ 1,2,. . .,T,
ð5Þ
xt rM at ,
8t ¼ 1,2,. . .,T,
ð6Þ
yt r M bt ,
8t ¼ 1,2,. . .,T,
ð7Þ
at , bt A f0,1g, 8t ¼ 1,2,. . .,T,
ð8Þ
xt ,yt Z0,
ð9Þ
8t ¼ 1,2,. . .,T:
The objective function minimizes the sum of setup cost, production cost, and inventory cost for new products and remanufactured products in all periods. Constraints (2)–(4) are inventory balance constraints for new products, remanufactured products, and returned products, respectively. Constraints (5) represent capacity constraints for manufacturing and remanufacturing activities. Constraints (6) and (7) specify the setup costs of manufacturing and remanufacturing. Constraints (8) and (9) are standard integrality and non-negative constraints.
Z.-H. Zhang et al. / Int. J. Production Economics 140 (2012) 249–255
Hence, the lower bound provided by Lagrangian relaxation can be written as:
3. Solution approach Florian et al. (1980) and Bitran and Yanasse (1982) have shown that the lot sizing problem with time-varying capacity constraints are NP-hard. In Li et al. (2007), heuristic genetic algorithms for capacitated production planning problems with batch processing and remanufacturing are proposed. They compare their planning results with those produced by Lingo for problems with relatively small size (Tr21). Experiments show that their approach can generate satisfactory results. However, for large problem instances, it is difficult to assess the quality of the solution. Pan et al. (2009) later take a dynamic programming approach and develop a pseudopolynomial time algorithm. For instances of small size (Tr25), their approach can yield optimal solution, however, for larger instances, dynamic programming based approach often fail due to the renowned ‘‘curse of dimensionality’’ (Powell, 2007). To address challenges faced by existing solution approaches, we propose a Lagrangian relaxation based approach to solve the capacitated closed-loop lot sizing problem. The advantages of our approach include:
Our approach decomposes the full problem into two uncapacitated lot sizing problem to fully exploit existing solution methods for uncapacitated lot sizing problems. Besides producing good quality feasible solutions, we also provide a lower bound to gage its optimality. Lagrangian relaxation based approach has proven to be an effective approach to solve large optimization problems.
LB : ¼ V n þ V r
T X
Sub-problem LSnew is a classical uncapacitated single-item lot sizing problem, which has been well researched. The best algorithm so far is proposed independently by Federgruen and Tzur (1991), Wagelmans et al. (1992) and Aggarwal and Park (1993) with computational complexity of O(TlogT). We implement the algorithm of Wagelmans et al. (1992) in this research to solve sub-problem LSnew. Unfortunately, sub-problem LSreturned is different from the classical uncapacitated single-item lot sizing problem due to constraints (4), which make the sub-problem intractable. To find its optimal solution, we reformulate this sub-problem as is done in Heuvel and Wagelmans (2008). We introduce marginal P r production cost, ct ¼ r t þ lt þ Ti ¼ t ðhi hi Þ and net cumulative Pt demand, st ¼ j ¼ t þ 1 dj it , which represents the amount of replenishment required to satisfy demand from period t þ1 to T at the end of period t. Then, the reformulated sub-problem is presented below: ðNCDÞ :
Min
T X
ðU t bt þ ct yt Þ þO,
t¼1
s:t:
T X
dj
s0 ¼ We decide to relax the capacity constraints, that is, Constraints (5). By introducing non-negative Lagrangian multipliers lt , where t A f1,2, Tg, we obtain the Lagrangian dual:
lt
r
½ðSt at þ pt xt þ Ht It Þ þ ðU t bt þ r t yt þ ht it þ ht irt Þ
t¼1
þ
T X
lt ðxt þ yt C t Þ
t¼1 T X
T X
Rj r st r
j¼1
T X
T X
dj ,
8t ¼ 1,2,. . .,T1,
j ¼ tþ1
8t ¼ 1,2,. . .,T,
dj ,
j¼1
sT ¼ 0, yt r M bt , 8t ¼ 1,2,. . .,T, bt A f0,1g, 8t ¼ 1,2,. . .,T, st ,yt Z 0, 8t ¼ 1,2,. . .,T: PT P r where, O ¼ t ¼ 1 ti ¼ 1 ðht Ri ht di Þs. Note that O is a constant term, and therefore can be omitted. Lemma 1. NCD and LSreturned are equivalent.
½St at þ ðpt þ lt Þxt þ Ht It
Proof. Follows from Heuvel and Wagelmans (2008). &
t¼1
þ
t X
st ¼ st1 yt ,
3.1. Lower bound
T X
lt C t x:
t¼1
j¼1
MaxMin
251
r
½U t bt þ ðr t þ lt Þyt þ ht it þ ht irt
t¼1
T X
lt C t :
ð10Þ
t¼1
Subject to constraints (2)–(4), (6)–(9). After we remove the constant of the last term in Eq. (10) and fix the value of lt , the relaxed problem can be decomposed into two subproblems for manufacturing and remanufacturing activities, respectively. They are given as follows: ðLSnew Þ :
V n ¼ Min
T P
½St at þ ðpt þ lt Þxt þ Ht It ,
t¼1
s:t: ð2Þ,ð6Þ,
at A f0,1g, 8t ¼ 1,2,. . .,T, xt Z 0, ðLSreturned Þ :
V r ¼ Min
T X
The reformulated sub-problem is solved by an efficient backward dynamic programming algorithm proposed by Liu (2008) in O(T2) time. 3.2. Upper bound After obtaining the lower bound, we propose a procedure to find a feasible solution at each iteration of the Lagrangian procedure. Capacity constraints (5) are relaxed. Thus, manufacturing/remanufacturing quantities need to be shifted to fit available capacity at each period. The procedure for finding a feasible solution is summarized as follows: 3.2.1. Procedure for upper bound:
8t ¼ 1,2,. . .,T: r
½U t bt þ ðr t þ lt Þyt þ ht it þ ht irt ,
t¼1
s:t: ð3Þ,ð4Þ,ð7Þ,
bt A f0,1g, 8t ¼ 1,2,. . .,T, yt Z 0,
8t ¼ 1,2,. . .,T:
Step 1: Set t ¼1. Step 2: If t oT and capacity constraints (5) are not violated, goto Step 7; else goto Step 3. If t ¼T and capacity constraints (5) are not violated, stop; else goto Step 5. Step 3: Move the remanufacturing of returned products from period t to t þ1. If capacity constraints (5) are satisfied after the modification, goto Step 7; else goto Step 4. In order to get a
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feasible solution, the modification for period t can not destroy the feasibility of the solutions before period t. Therefore, we denote slt as the amount of new and returned products processed at period t which exceed the capacity and alyt as the amount of returned products at period t can be shifted to be remanufactured at the following periods. And, they are expressed, respectively as follows: slt ¼ xt þyt 2C t , t X ðyi di Þ: alyt ¼ i¼1
Then, the amount of remanufactured products which is shifted to the following periods, denoted by y, is as follow: y ¼ minfslt ,yt ,alyt g: Note that slt, xt and yt are updated at each movement. For example, slt reduces to slt y at the end of this step. Step 4: Move the manufacturing of new products from period t to tþ 1. If capacity constraints (5) are satisfied, goto Step 7; else goto Step 5. In this step, yt does not shift anymore. We just modify xt. Similarly, the moved amount of new products, denoted by x, is determined by x ¼ minfslt ,alxt g, P where alxt ¼ ti ¼ 1 ðxi Di Þ. It refers to the amount of new products can be shifted to be manufactured in the following periods. Since dt rC t and yt r dt after the modification in the previous step, slt r xt . Therefore, xt is excluded from the above formula. Step 5: Move the manufacturing of new products from t to j (1rjrt 1). If capacity constraints (5) are satisfied, goto Step 7; else goto Step 6. So far, xt and yt can not be further shifted to the following periods. In order to satisfy the demand at period t under the capacity constraints, we consider backward movement of the manufacturing of new products in this step. The amount of new product, which is shifted from period t to j (1rj rt 1) that increases the total cost the least, is x ¼ minfslt ,C j 2xj 2yj g, where Cj xj yj is the remaining manufacturing capacity at period j. Once slt ¼0, goto Step 7. Step 6: Move the remanufacturing of returned products of the previous t 1 periods to period tþ1 so as to have the capacity to manufacture new products which are shifted from period t. Start at period t 1 and move remanufacturing of returned products in period j (1rj rt 1) to period t þ1 meanwhile guarantee that the demands of remanufactured products from period j to t must be satisfied. The shifted amount from period j is determined by: y ¼ minfalyj ,yj ,slt g, P where, alyj ¼ ji ¼ 1 ðyi di Þ. Goto Step 7 once slt ¼0. Step 7: If t ¼T, stop; else set t ¼t þ1, goto Step 2.
Lemma 2. The computational complexity of the procedure is OðT 2 Þ . Proof. It is shown due to the number of shifts of the quantities of manufacturing/remanufacturing at each period in the procedure. That is, j (1rj rt 1) times shifts occur at period t. Therefore, the upper bound can be found in O(T2) time. &
At the end of our Lagrangian relaxation procedure, a local search heuristic, which is an extension of the smoothing heuristic proposed by Trigeiro et al. (1989), is proposed to further improve the incumbent solution. Step 1: Backward pass for manufacturing new product. Start at period T and move the production of new product to the earlier periods that increase the total cost the least. The movable quantity of new products from period i (TZiZ2) to j (i 1 ZjZ1) is determined by minfxj ,C j xj yj g: Step 2: Forward pass for manufacturing new product. Start at period 1 and move the production of new product to the next later periods that increase the total cost the least. The movable quantity of new products from period i (1rirT 1) to j (iþ1 rjrT) is determined by ( ) t X min xi , min ðxk Dk Þ,C j xj yj : i r t r j1
k¼1
In contrast to the backward pass, we must guarantee the demands of new products from period i to j-1 must be satisfied P in this step. So, mini r t r j1 tk ¼ 1 ðxk Dk Þ is taken into account. Step 3: Backward pass for remanufacturing returned product. Start at the last period and move the remanufacturing of returned product to the earlier periods that increase the total cost the least. The movable quantity of remanufactured products from i (TZi Z2) to j (i-1Zj Z1) is: 8 9 8 9 j < = < X = i1 t X X min yi , ðRt yt Þ max 0, ðyk Rk Þ ,C j xj yj , : ; : t¼1 ; t ¼ jþ1
k ¼ jþ1
Pj
where stock of returned t ¼ 1 ðRt yt Þ refers to the inventory Pi1 Pt products at the end of period j, t ¼ j þ 1 maxf0, k ¼ jþ1 ðyk Rk Þg is the inventory stock of returned products at the end of period j which is remanufactured between period j þ1 and period i 1. Step 4: Forward pass for remanufacturing returned product. Start at period 1 and move the remanufacturing of returned product to the next later periods that increase the total cost the least. The movable quantity of remanufactured products from period i (1rirT 1) to j (iþ1 rjrT) is: ( ) t X min yi , min ðyk dk Þ,C j xj yj : i r t r j1
k¼1
Step 5: Repeat from Step 1 to Step 4 once again. Lemma 3. The improvement feasible solution can be obtained in O T 2 time. Proof. The proof is similar with that of Lemma 2.
&
3.3. Updating the Lagrangian multipliers The Lagrangian multipliers lt are updated at each iteration using standard sub-gradient optimization (Fisher, 2004). n o lkt þ 1 ¼ max 0, lkt þ directionk ðxkt þ ykt C t Þ , directionk ¼ d PT
Z nUB Z LB
t¼1
ðxkt þykt C t Þ2
,
where the superscript (kþ 1) indicates that this update is for the (kþ 1) iteration. Z nUB is the best upper bound so far. ZLB is the lower
Z.-H. Zhang et al. / Int. J. Production Economics 140 (2012) 249–255
Table 3 Parameters for generating data sets.
253
Table 4 Parameters for the Lagrangian relaxation procedure.
Parameter
Value
Parameter
Value
Ht ht htr pt rt St Ut Dt dt Rt
Uniformly drawn from [5, 10) Uniformly drawn from [3, 8) f t ht , where ft is uniformly drawn from [0, 1) Uniformly drawn from [15, 20) Uniformly drawn from [10, 15) Uniformly drawn from [6000, 8000) Uniformly drawn from [4000, 6000) Uniformly drawn from [300, 500) drawn from [200, 300) P P kt dt þ maxf ti ¼ 1 di ti ¼ 1 Ri g, where kt is uniformly drawn from [0.5, 1.5) TFðDt þ dt Þ , where TF¼ 1.2
Minimum LB–UP gap Maximum iteration count Maximum number of iterations before halving is d Initial value of d Minimum value of d
1% 300 5 2.0
Ct
Table 5 Performance of the Lagrangian relaxation procedure, TF¼ 1.2. T
Avg. CPU time (s)
Avg. GAP (%)
TF is a coefficient and used to adjust the capacity of each period.
bound at current iteration. d is initialized to 2.0 and halved if there are five consecutive iterations in which the lower bound does not increase.
3.4. Stopping criteria The Lagrangian relaxation procedure is terminated once one of the following stopping criteria is satisfied.
Optimality gap: The optimality gap between the lower and
upper bounds is less than a defined value. Maximum iteration count: A defined number of iterations is performed. Step size multiplier: d is less than a defined epsilon.
4. Computational results The algorithm is implemented in ILOG OPL Script language and tested on a computer with an Intel Core 1.67 GHz CPU. The standard optimization software Cplex version 12.0 was used to find optimal solutions for small-scale instances. We randomly generate numerous data sets to test the quality of the solution algorithm described in the previous section. Data generated for all of the test instances are available upon request. Table 3 shows the parameter values which are used to generate test data in this section. Table 4 lists the parameter values for the Lagrangian relaxation procedure.
4.1. Performance evaluation We use three criteria to evaluate the quality of the algorithm.
The gap between the lower bound (LB) and the upper bound (UB), which is calculated as follows (Brahimi et al., 2006a): GAP ¼ 2
UBLB : UB þ LB
The distances between the upper bound (UB) and the optimal solution (OP), are denoted as UGAP, and between the lower bound (LB) and the optimal solution (OP), are denoted as LGAP. We obtain the optimal solution by solving the problem in Cplex for small-scale instances. And the distances are calculated as follows: UGAP ¼ 2
UBOP , UB þ OP
108
3 5 6 12 20 24 35 48 50 65 80
0.70 1.12 1.58 3.65 7.16 9.35 16.96 30.53 35.65 52.96 75.16
6.46 6.86 6.83 7.47 8.33 8.44 8.37 8.99 7.59 8.39 9.14
UGAP (%)
LGAP (%)
Avg.
Max.
Avg.
Max.
0.00 0.47 0.38 2.27 2.63 2.50
0.00 1.52 1.48 3.97 4.70 4.05
6.46 6.39 6.46 5.21 5.70 5.95
8.92 8.42 8.90 6.99 8.34 7.41
Table 6 Average gap and CPU time for instances with different capacity and T¼ 20. TF
Avg. GAP (%)
Avg. CPU time (s)
1.0 1.2 1.4 1.6 1.8 2.0
4.74 8.33 4.22 2.18 1.38 1.03
4.38 7.16 5.53 3.39 2.23 0.71
LGAP ¼ 2
OPLB : OP þ LB
Running time. Ten instances for each T are generated randomly and the computational results are reported in Table 5. Note that the blank space in Table 5 indicates that Cplex can not find the optimal solution in 10 min. We find that the average GAP is less than 10%. Compared with the gaps of some capacitated lot sizing problems (Trigeiro et al., ¨ 1989; Toledo and Armentano, 2006; Sural et al., 2009) solved by Lagrangian relaxation-based methods, it reveals the good performances of the algorithm. Furthermore, we analyze the distance of the upper/lower bounds from the optimal solution for small-scale instances. And UGAP o LGAPindicates that the solution has a pretty good quality and the lower bound makes GAP poor. Table 6 shows the impact of production capacity on the performance of the algorithm. Except the case of TF¼1.0, it is clear that the gaps and CPU times decrease with larger capacity. TF¼1.0 implies the optimal production policy is Lot-for-Lot. UGAP is zero in all of the instances. Hence, GAP is generated by relaxing capacity constraints (5). Therefore, it has a smaller gap and a short CPU time. Table 7 reports GAP, UGAP and LGAP of instances with different capacity. We can obtain the similar conclusion with that in Table 5.
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Table 7 UGAP and LGAP of instances with different capacity. T
TF¼ 1.1
TF ¼1.2
TF¼ 1.3
TF ¼1.4
TF¼ 1.5
TF ¼1.6
6
GAP (%) UGAP (%) LGAP (%)
8.17 0.18 7.82
6.81 0.39 6.43
6.47 0.82 5.65
5.35 1.11 3.59
3.80 1.03 2.77
3.31 1.29 1.49
12
GAP (%) UGAP (%) LGAP (%)
9.80 2.02 7.78
7.44 2.25 5.19
6.22 2.06 4.16
5.15 2.23 2.93
3.32 1.21 2.10
2.47 1.10 1.37
24
GAP (%) UGAP (%) LGAP (%)
8.90 2.03 6.87
8.47 2.57 5.90
6.63 2.79 3.83
4.58 2.19 2.40
4.34 2.31 2.03
2.37 1.17 1.20
hr
Total cost
51,893.4 46,307.8 44,076.4 42,614.8 42,567.6
496,296.7 449,266.8 433,454.5 425,223.9 421,327.8
Table 8 Cost components for the instances with different capacity, T¼ 20. TF
1.0 1.2 1.4 1.6 1.8
Manufacturing
Remanufacturing
S
p
H
U
r
h
139,636.2 98,220.0 84,834.2 76,605.9 68,548.9
142,288.7 141,047.5 140,472.2 139,339.5 138,525.4
0.0 19,346.9 26,365.7 27,969.3 34,802.7
100,778.3 68,329.0 64,845.8 62,598.5 63,554.1
61,700.2 60,893.5 60,089.8 60,115.9 59,770.3
0.0 10,245.3 13,229.5 15,764.7 15,531.1
4.2. Managerial insights We also observe the changes of cost components with the increase of the capacity, which are shown in Table 8. The first column lists TF. The following three columns show the total setup, production and holding costs associated with manufacturing activities, respectively. Then, the costs associated with remanufacturing activities are reported in the fifth, sixth and seventh columns. The column labeled hr lists the total holding cost associated with returned products. And, the last column reports the total costs. The manufacturing and remanufacturing activities are more likely performed at periods with smaller unit production and reproduction costs if the available capacity is large enough. Therefore, setup and production costs associated with manufacturing and remanufacturing activities show decreasing trends. And inventory holding costs show reverse trends. Accordingly, holding costs of returned products decrease. The total costs become smaller. In conclusion, it makes sense to have more production capacity to reduce the total cost. However, managers have to balance larger initial investment and smaller operational costs.
5. Conclusions and future research directions In this paper, we propose a Lagrangian relaxation-based solution algorithm to solve a capacitated lot sizing problem with setup costs, product returns and remanufacturing, which has received a growing attention in recent years. The lower bound is calculated by decoupling the relaxed problem into two subproblems, which are solved by two polynomial-time algorithms, respectively. Two efficient and effective heuristics are proposed to find the good upper bound. Numerous computational experiments show that the proposed algorithm is capable of solving the problem efficiently. The impact of capacity on the performance and the behavior of the model are investigated as well. There are some extensions of this work. First, manufacturing and remanufacturing activities are only considered in the model. There exist more activities in a closed-loop supply chain, for instance, refurbuishing, recycling, etc, all of them have a closed
relation with production planning. It is interesting to consider more general planning scenarios. Second, handling returned products is a strongly random activity. Therefore, it is our future work to study the stochastic counterpart of the model.
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