Operations scheduling in reverse supply chains: Identical demand and delivery deadlines

Operations scheduling in reverse supply chains: Identical demand and delivery deadlines

Author’s Accepted Manuscript Operations Scheduling in Reverse Supply Chains: Identical Demand and Delivery Deadlines Gang Wanga, Angappa Gunasekarana ...

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Author’s Accepted Manuscript Operations Scheduling in Reverse Supply Chains: Identical Demand and Delivery Deadlines Gang Wanga, Angappa Gunasekarana

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S0925-5273(16)30189-X http://dx.doi.org/10.1016/j.ijpe.2016.08.010 PROECO6493

To appear in: Intern. Journal of Production Economics Received date: 6 October 2015 Revised date: 2 August 2016 Accepted date: 7 August 2016 Cite this article as: Gang Wanga and Angappa Gunasekarana, Operations Scheduling in Reverse Supply Chains: Identical Demand and Delivery Deadlines, Intern. Journal of Production Economics, http://dx.doi.org/10.1016/j.ijpe.2016.08.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Operations Scheduling in Reverse Supply Chains: Identical Demand and Delivery Deadlines Gang Wanga,∗, Angappa Gunasekarana a Department

of Decision and Information Sciences, Charlton College of Business, University of Massachusetts Dartmouth, 285 Old Westport Rd, North Dartmouth, MA 02747, USA

Abstract This study addresses an integrated operations scheduling problem in reverse supply chains, where delivery deadlines and identical demand are involved. The supply chains consist of contracted collectors, a manufacturer, and many secondary markets. Both collectors and manufacturer are capacitated. The manufacturer remanufactures returned/used products shipped from collectors and then ships finished products directly to demand points geographically dispersed in secondary markets, following order quantity and delivery due date that each demand point requests. Each demand point orders the same quantity, which can be true in supply chain practices (i.e., grouping demand points into customer zones). Furthermore, the manufacturer is imposed penalties for late deliveries. The problem is to determine shipping quantities from collectors to the manufacturer and the assignment of collectors and demand points to the manufacturer, subject to the capacity constrains on both collectors and the manufacturer. This paper formulates the scheduling problem as a bi-criteria mixed integer program with the objective of minimizing both total shipping and penalty costs and delivery lateness. For the problem with the order sizes of one unit, the total unimodularity of its constraint matrix allows for the development of a polynomial time algorithm. The problem where order sizes are the same is solved by a dynamic programming based algorithm. The respective numerical examples are provided to verify two problems and their corresponding solution approaches. Keywords: Reverse supply chains; integrated operations scheduling; delivery deadlines; bi-criteria decision making; mixed integer programming.

1. Introduction Reverse supply chains have gained considerable attention in industry and academia due to mounting regulatory pressure, growing environmental concerns, and increasing benefits (i.e., material conservation, reduced energy consumption and waste, and lower prices) (Guide, 2002). In reverse supply chains, returned/used products are shipped to inspection and disposition locations, where disposition actions are determined including reuse, remanufacturing, and recycling. The new and never used products are restocked to the forward distribution channels, while other products are sold for remanufacturing and recycling. Then remanufactured products are sold in secondary markets for additional revenue. In certain cases, companies are mandated to recycle used products because of hazardous materials (e.g., in the case of refrigerators in the U.S.). In such a case, reverse supply chains need to be not only well managed, but also tightly integrated (Blackburn et al , 2004). However, the planning and scheduling of reverse supply chain operations is a challenge due to highly variable return flows. The purpose of this paper is to develop optimization tools for integrated operations scheduling in reverse supply chains with remanufacturing. The study was motivated by reverse supply chain practices in computer and electronic industry. Computers and electronic devices are returned or collected at the stores of retailers or wholesalers and then transported to collectors’ facility, where returned products are inspected in terms of products’ conditions. The new or unused products will ∗ Corresponding

author Email address: [email protected] (Gang Wang)

Preprint submitted to International Journal of Productions Economics

April 7, 2016

Operations Scheduling in Reverse Supply Chains: Identical Demand and Delivery Deadlines

Abstract This study addresses an integrated operations scheduling problem in reverse supply chains, where delivery deadlines and identical demand are involved. The supply chains consist of contracted collectors, a manufacturer, and many secondary markets. Both collectors and manufacturer are capacitated. The manufacturer remanufactures returned/used products shipped from collectors and then ships finished products directly to demand points geographically dispersed in secondary markets, following order quantity and delivery due date that each demand point requests. Each demand point orders the same quantity, which can be true in supply chain practices (i.e., grouping demand points into customer zones). Furthermore, the manufacturer is imposed penalties for late deliveries. The problem is to determine shipping quantities from collectors to the manufacturer and the assignment of collectors and demand points to the manufacturer, subject to the capacity constrains on both collectors and the manufacturer. This paper formulates the scheduling problem as a bi-criteria mixed integer program with the objective of minimizing both total shipping and penalty costs and delivery lateness. For the problem with the order sizes of one unit, the total unimodularity of its constraint matrix allows for the development of a polynomial time algorithm. The problem where order sizes are the same is solved by a dynamic programming based algorithm. The respective numerical examples are provided to verify two problems and their corresponding solution approaches. Keywords: Reverse supply chains; integrated operations scheduling; delivery deadlines; bi-criteria decision making; mixed integer programming.

1. Introduction Reverse supply chains have gained considerable attention in industry and academia due to mounting regulatory pressure, growing environmental concerns, and increasing benefits (i.e., material conservation, reduced energy consumption and waste, and lower prices) (Guide, 2002). In reverse supply chains, returned/used products are shipped to inspection and disposition locations, where disposition actions are determined including reuse, remanufacturing, and recycling. The new and never used products are restocked to the forward distribution channels, while other products are sold for remanufacturing and recycling. Then remanufactured products are sold in secondary markets for additional revenue. In certain cases, companies are mandated to recycle used products because of hazardous materials (e.g., in the case of refrigerators in the U.S.). In such a case, reverse supply chains need to be not only well managed, but also tightly integrated (Blackburn et al , 2004). However, the planning and scheduling of reverse supply chain operations is a challenge due to highly variable return flows. The purpose of this paper is to develop optimization tools for integrated operations scheduling in reverse supply chains with remanufacturing. The study was motivated by reverse supply chain practices in computer and electronic industry. Computers and electronic devices are returned or collected at the stores of retailers or wholesalers and then transported to collectors’ facility, where returned products are inspected in terms of products’ conditions. The new or unused products will be sent back to forward distribution channels. Some will be remanufactured or refurbished and shipped to secondary markets. Since computers and/or electronic products are perishable, the key characteristics of reverse supply chains for computers and electronic products is time value of remanufactured products. Hence demand points request delivery deadline for the manufacturer; otherwise late deliveries would lead to contract penalties, lost sales, and customer dissatisfaction. In addition to timely delivery, the manufacturer faces additional challenge of assigning demand points due to its capacity constraints. It is therefore challenging to plan and schedule reverse supply chain operations in computer and/or electronic industries. In this paper, we consider a three-stage, capacitated, reverse supply chain as Preprint submitted to International Journal of Productions Economics

August 8, 2016

above described, which consists of contracted collectors, capacitated manufacturer, and demand points geographically dispersed in secondary markets as depicted in Figure 1. Contracted collectors supply returned/used products to the manufacturer, and the manufacturer remanufactures good-as-new products, and delivers to demand points. Each demand point specifies order quantity and delivery deadline. The manufacturer is imposed penalties if it can not deliver finished products to demand points before or on requested deadlines.

Figure 1: Reverse supply chains with remanufacturing

Figure 1: Reverse supply chains with remanufacturing

The problem under study is formulated as bi-criteria mixed integer linear program. The goal of the problem is to find optimal schedules for supply and distribution operations, and determine a feasible assignment of collectors and demand points to the manufacturer, while minimizing the total shipping and penalty costs as well as the delivery lateness, subject to the capacity restrictions on suppliers and the manufacturer, and the delivery deadlines. The two problems are presented. One is regarding the integrated scheduling problem with unit order size, where a linear programming based algorithm is proposed due to its totally unimodular constraint matrix; the other is the scheduling problem with identical order sizes for which a dynamic programming based algorithm is developed. For both problems, numerical examples demonstrate two considered problems and how the proposed solution approaches can solve the respective problems. The rest of the paper is organized as follows. In Section 2, a literature review is provided. Section 3 presents an operations scheduling problem with product remanufacturing is proposed, where identical order sizes and delivery deadlines are both taken into account. Section 4 is concerned with the case with unit order sizes. In Section 5, the case with identical order sizes are studied. In Section 6, we conclude the study and discuss its future extensions. 2. Literature Review Remanufacturing operations planning in reverse supply chains focuses on important tactical and operational decisions on production schedules and flow quantities through supply chain networks. The research in the literature with regard to reverse supply chains planning can be classified into two categories. The first category concerns the planning of production, inventory and distribution operations, while the second category deals with the operations scheduling and assignment at the operational level. Much research has been done in the first category in the literature. Murayama et al (2006) discussed the problem in reverse supply chain where timing and quantities of returned products and reusable components are not known through the use of production planning method. With this method, they can make productions plans at each time period with respect to the quantities of disassembled products, and new, reusable components. Yuan and Gao (2010) investigated a closed-loop supply chain system, which consists of a supplier, a manufacturer, a retailer, and a collector. They applied one manufacturing cycle and one remanufacturing cycle policy to determine the optimal system policy 2

for both decentralized and centralized decision making through elimination theory and difference function. Kaya et al (2014) investigated the operations problem of disassembling, refurbishing and producing modular products in a reverse supply chain by developing a large-scale mixed integer programming model. They utilized two-stage stochastic and robust optimization approaches to solve this problem in which the first stag focused on capacity decisions at disassembly and refurbishing sites, while the second stage dealt with such operational decisions as production, inventory and disposal rates. Ramos et al (2014) considered a real case of a recyclable waste collection system, where tactical and operational decisions are investigated for establishing service areas for each depot and scheduling collection routes for each vehicles. They presented a multi-objective, multi-depot periodic vehicle routing problem with inter-depot routes. Niknejad and Petrovic (2014) studied optimization problem of inventory control and production planning for integrated reverse logistics network. They developed two phase fuzzy mixed integer optimization algorithm to solve this problem with numerical experiments and sensitivity analysis. Work in the second category has also been completed in the literature (Morgan et al (2013); Wang and Lei (2012, 2014, 2015)). Guide (1997) examined disassembly release mechanisms and priority dispatching rules in the planning and controlling of remanufacturing systems. They found that simple due-date-based priority dispatching procedures generally performed well, while disassembly release mechanisms had very little impact. Teunter et al (2008) derived a mixed-integer program to solve the economic lot-scheduling problem when manufacturing and remanufacturing operations were performed on separate dedicated lines. Given the computational complexity of this approach, they continued the study of the economic lot-scheduling problem by developing fast, but simple, heuristics that could provide near-optimal solutions (Teunter et al, 2009). Ozceylan and Paksoy (2013) developed a non-linear mixedinteger programming model to analyze disassembly line balancing problem in reverse supply chains, which consists of customers, collection/disassembly centers, and plants in order to determine the optimal distribution between facilities, the best number of opened disassembly workstations, and the optimal assignment of tasks to workstations. Kim et al (2014) considered a scheduling problem for remanufacturing systems in which end-of-life products were separated into their major components at a disassembly workstation, each of them was reprocessed at its dedicated flow-shop-type reprocessing line with serial workstations, and finally, the reprocessed components, together with new components if required, were reassembled into remanufactured products at a reassembly workstation. They focused on the one with parallel flow-shop-type reprocessing lines since it was a typical remanufacturing configuration. Li et al (2014) presented a remanufacturing process planning and resource scheduling method by color Petri net used to model the dynamics in remanufacturing. A hybrid meta-heuristic coupled with simulated annealing and tabu search was also proposed to derive the optimal process plan and schedule with the objective of minimizing the total processing cost and tardiness penalty. Lina et al (2014) analyzed a multi-plant remanufacturing system and decided on which plant to perform the remanufacturing. In view of the interdependencies between plant selection, remanufacturing option and job scheduling, They presented a solution method based on the linear physical programming and the multi-level encoding genetic algorithm. Yin (2015) considered two-agent just-in-time scheduling with unrelated parallel machines, provided a bicriterion analysis for the Pareto-optimal solutions, and developed polynomial algorithms for each problem. Yin (2016) studied integrated production and batch delivery scheduling in a make-to-order production system with two competing agents. For each of the problems, they either provided a polynomial-time algorithm to solve it or showed that it is NP-hard, and then conducted numerical experiment to test algorithmic performance. The problem considered is the combination of both categories described above because it incorporates both planning and scheduling processes in reverse supply chains. Therefore, all solution approaches presented in the previous work cannot be directly applied to the problem under study. The new algorithms are presented to solve the operations scheduling problem in this paper, based on linear programming and dynamic programming techniques. 3. Problem Formulation Consider a capacitated reverse supply chain with remanufacturing in which returned products are collected from a set I = {1, 2, · · · , I} of contracted collectors. After inspection, collected products are shipped to the processing center (PC) of the manufacturer for remanufacturing. Remanufactured products are distributed to a number J = {1, 2, · · · , J} of demand points (DPs). Each collector has the capacity of si , and the shipping charge of ai , i ∈ I. A quantity xi of product is shipped out of collector i to the PC so that the total shipping costs from collectors to the PC are  then i∈I ai xi . The PC, with a capacity of c and the processing time τ per unit, processes returned products and delivers remanufactured products to demand points, which specify the order quantity q j , and the delivery deadline d j , 3

j ∈ J. Unassigned orders incur penalty costs of p j per unit. Each shipment between the PC and DPs incurs the fixed shipping cost f j and shipping rate v j , j ∈ J. The problem aims to find optimal shipping quantities and the assignment of collectors and demand points to the manufacturer, which minimize the total shipping and penalty costs, and the delivery lateness simultaneously. Let us first summarize the notations to be used in our problem formulation. • Index Sets: I

Set of contracted collectors I = {i|1, 2, · · · , I}

J

Set of demand points J = { j|1, 2, · · · , J}

• Parameters: si

Capacity of collector i

ai

Shipping rate from collector i to the PC

c

Capacity of the PC

τ

Processing time at the PC per unit

q

Order quantity of demand point j

dj

Deadline of delivering the order to demand point j

tj

Shipping time from PC to demand point j

vj

Variable cost for shipping one unit of product from the PC to demand point j

fj

Fixed cost for the shipment from the PC to demand point j

pj

Unit cost for unfulfilled demand point j

• Decision variables: xi yj

number of units to be shipped from collector i to the PC  1, if demand point j is served by the PC = 0, otherwise.

We now formulate the scheduling problem as the following bi-criteria mixed integer program. • Objective Function  Let LT j (y) = τ j∈J q j y j + t j y j denote the lead time for fulfilling DP j , j ∈ J, and b j = v j q + f j . The problem is to find a feasible assignment of demand points and collectors to the PC such that the total shipping and penalty costs plus order fulfillment lead time are minimized simultaneously. ⎧    ⎪ ⎪ ⎨ f1 (x, y) = i∈I ai xi + j∈J b j y j + j∈J p j q(1 − y j ), (1) min ⎪ ⎪ ⎩ f2 (y) = j∈J LT j (y) − d j . In the first objective function f1 (x, y), the first term defines the shipping cost from suppliers to the PC, where ai xi estimates the cost of shipping a quantity of xi from supplier i to the PC. The second term represents the variable and fixed shipping cost from the PC to demand points. The third term denotes the total penalty costs incurred by unassigned demands. The second objective function f2 (y), defines the bottleneck lead time. 4

• Constraints Capacity constraints on collectors Constraints (2) ensure that all the quantities shipped out of collector i do not exceed its capacity. ≤

xi

si , ∀i ∈ I,

(2)

Flow balance constraint through the PC Constraint (3) is flow conservation constraint for the PC, i.e., inbound and outbound shipping quantities through the PC must remain the same.   xi = qy j , (3) j∈J

i∈I

Capacity constraint on the PC Constraint (4) ensures that all demands of assigned DPs can not exceed the PC’s capacity.  qy j ≤ c, (4) j∈J

Integrality and nonnegativity y j ∈ {0, 1}, xi ≥ 0, ∀i ∈ I, j ∈ J.

(5)

To solve the original problem (1), we introduce a weight α ∈ [0, 1] and convert (1) into a single objective problem, weighted sum problem, expressed in the following. min α f1 (x, y) + (1 − α) f2 (y)

(6) (7)

s.t. (2) − (5), 0 ≤ α ≤ 1.

(8)

Since the above weighted sum problem (6) can be reduced to subset sum problem, it is also NP-hard. Throughout the remaining discussion in this paper, we shall denote the above weighted sum problem (6) by P. 4. Operations Scheduling with Unit Order Size This section considers a special case with unit order size that is polynomially solvable. This is true in supply chain practice when only a full truck load or a container for each demand point is involved (Wang and Lei, 2012), i.e., q = 1. Hence, P can be written as follow. P1 min α f1 (x, y) + (1 − α) f2 (y) s.t. xi ≤ si , ∀i ∈ I,   xi = y j, 

i∈I

j∈J

y j ≤ c, ∀ j ∈ J,

j∈J

xi ≥ 0, y j ∈ {0, 1}, 0 ≤ α ≤ 1, ∀i ∈ I, j ∈ J.

5

4.1. Linear programming based algorithm To solve P1 , consider its linear relaxation problem in the following LP1 min α f1 (x, y) + (1 − α) f2 (y) s.t. xi ≤ si , ∀i ∈ I,   xi =



y j,

j∈J

i∈I

y j ≤ c, ∀ j ∈ J,

j∈J

xi ≥ 0, 0 ≤ y j ≤ 1, 0 ≤ α ≤ 1, ∀i ∈ I, j ∈ J. Thus we will show that solving P1 is equivalent to solving LP1 in the following theorem. Theorem 1. If si and c are integer for all i ∈ I, P1 , can be solved in strongly polynomial time for any α ∈ [0, 1]. Proof. for any α ∈ [0, 1], we will show that the constraint matrix of P1 is totally unimodular and then it can be solved in strongly polynomial time. Consider that the constraint matrix of P1 has N × J columns and N + J rows and takes the form ⎛ ¯ ⎞ ⎜⎜⎜ 1 0 · · · 0 ⎟⎟⎟ ⎜⎜⎜ 0 1¯ · · · 0 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ ⎜⎜⎜ .. .. . . .. ⎟⎟⎟⎟ ⎜ . . ⎟⎟ A = ⎜⎜⎜ . . ⎜⎜⎜ 0 0 . . . 1¯ ⎟⎟⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎝ ⎟⎠ .. . I I I where the 1¯ is a row vector with 1’s and I is an identity matrix. Each column of A contains two entries equal to 1. It suffices to prove that there exists a partition of entries for each column, which consists of two disjoint subsets of entries of each column, such that the difference between the sum of entries in each subset is equal to −1, 0, or 1 in terms of Theorem 5.23 (Korte et al, 2006). Thus it shows that A is totally unimodular, thereby proving the abovementioned equivalence. To start with, let H = {i1 , . . . , il } be the subset of {1, 2, . . . , N + J}, and a partition of H be the two disjoint subset H1 and H2 such that H = H1 ∪ H2 . Furthermore, define H1 = {i1 , . . . , il1 }, H2 = {i1 , . . . , il2 }  and l1 + l2 = l. Since A only contains two nonzero entries in each column, we obtain that k∈H1 ak j ∈ {0, 1} and    k∈H2 ak j ∈ {0, 1} for all j ∈ J. Hence, k∈H1 ak j − k∈H2 ak j ∈ {−1, 0, 1}. Under the condition that all the nonzero right-hand sides are integers, the total unimodularity of the constraint matrix indicates that the optimal solution to LP1 is an optimal integral solution, which is also the optimal solution to P1 (Korte et al, 2006). Moreover, LP1 is known to be solvable in polynomial time. Thus P1 is polynomially solvable in strong sense.  According to the proof of Theorem 1, we develop the following algorithm to solve this special case. The algorithm attempts to find an optimal solution to P1 by performing linear search for α and solving its linear relaxation, LP1 . In doing so, it starts with the initialization of model parameters, α, and step length h and then solves LP1 for any given α. In iteration, the algorithm enumerates all the values of α, and chooses the solution and the value of α with the least cost. For given α ∈ [0, 1], let us define the following subproblem of LP1 , LP1 (α). min α f1 (x, y) + (1 − α) f2 (y) s.t. xi ≤ si , ∀i ∈ I,   xi = y j, 

i∈I

j∈J

y j ≤ c, ∀ j ∈ J,

j∈|

xi ≥ 0, y j ∈ [0, 1], ∀i ∈ I, ∀ j ∈ J. 6

Let G∗ denote the optimal value of LP1 with the optimal solution x∗ , y∗ and α∗ , and G∗α be the optimal value of LP1 (α). Algorithm PUD Step 1 Initialize model parameters and set α = 0, z∗ = ∞, and h = 0.05, where h denotes the step size. Step 2 Solve the subproblem of LP1 , LP1 (α). If G∗α < G∗ , then G∗ = G∗α . Step 3 Set α = α + h and repeat Step 2. Step 4 Output x∗ , y∗ , α∗ , and G∗ . Theorem 2. Algorithm PUD runs in polynomial time. Proof. In terms of Theorem 1, in each iteration, Step 2 in Algorithm PUD can be completed in polynomial time. However, with the value of step size, h, Algorithm PUD terminates after a finite number of steps bounded by αh . Therefore, it runs in polynomial time.  4.2. Numerical example To illustrate the above case, we provide a numerical example where four collectors and eight DPs are considered, and the manufacturer has the capacity of 5 and the processing time of 0.26. The values of other parameters associated with collectors and DPs are shown in Tables 1 and 2, respectively. Table 1: The parameter values with collectors

Collector Shipping rate Capacity

1 2 5

2 2 6

3 3 5

4 1 4

Table 2: The parameter values with demand points

DP Shipping time Shipping rate Order size Deadline Fixed cost Penalty cost

1 3 3 1 6 6 6

2 2 3 1 6 6 6

3 3 3 1 5 5 6

4 2 3 1 5 5 6

5 2 2 1 7 6 7

6 2 3 1 6 5 6

7 2 3 1 6 6 6

8 3 3 1 7 5 7

Using Algorithm PUD, the optimal solution to LP1 is obtained as follows: x1∗ = x2∗ = x3∗ = 0, and x4∗ = 4; = y∗2 = y∗5 = y∗7 = 1 and y∗3 = y∗4 = y∗6 = y∗8 = 0. Based on the result of Theorem 1, we can further obtain the optimal solution to P1 , where collector 4 is selected and DPs 1, 2, 5, and 7 are served. Figure 2 illustrates the optimal operation schedules below, where red solid lines denotes shipments between suppliers and the PC, and blue solid lines show the assignment of demand points to the PC, while dashed lines represent neither shipments nor assignment. y∗1

5. Operations Scheduling with Identical Order Sizes This section presents an exact algorithm to solve P based on dynamic programming, proves its time complexity, and provides a numerical example.

7

Collectors 1

2

4

3

PC

1

8

2 3

6

4

7

5

Demand Points

Figure 2: Optimal operations schedules with unit order size

Figure 2: Optimal operations schedules with unit order size

5.1. Dynamic programming based algorithm Let d be the available capacity of the manufacturer, d = 1, 2, . . . , c, r = qc , and f1 (x, y) = h1 (x) + h2 (y), where    h1 (x) = i∈I ai xi and h2 (y) = j∈J (b j q + f j )y j + j∈J p j q(1 − y j ). Hence, weighted sum problem, P, can be decomposed into two subproblems: transportation problem and assignment problem as shown in the following. min h1 (x) s.t.

min αh1 (y) + (1 − α) f2 (y) s.t. 

xi ≤  si , ∀i ∈ I, xi = d,

y j = r, ∀ j ∈ J,

j∈J

y j ∈ {0, 1}, ∀ j ∈ J.

i∈I

xi ≥ 0, ∀i ∈ I.

Let V( j, d) denote the optimal value of P with j demand points and the capacity of d, and g j be the cost that demand point j incurs if it is assigned to the manufacturer. Hence the recursive function is defined as follows:  V( j − 1, d − q) + g j , q ≤ d V( j, d) = V( j − 1, d), otherwise. To calculate g j , note that g j includes three costs: 1) shipping cost from suppliers to manufacturers due to the assignment of demand point j, denoted by Δh1 ; 2) total shipping and penalty costs from the manufacturer to demand points, denoted by Δh2 ; 3) delay cost for missing timely delivery, denoted by Δ f2 . Let us first derive the first cost, which is incurred by satisfying demand point j. Consider the transportation problems with the capacity of d and d − q and denote their optimal values by h1 (x∗ (d)) and h1 (x∗ (d − q)) where x∗ (·) represents the optimal solution, respectively, so Δh1 = h1 (x∗ (d)) − h1 (x∗ (d − q)). The second cost can be obtained by calculating shipping cost and profit for avoiding penalty, that is, Δh2 = (b j − p j )q + f j . The last cost is seen as the potential penalty cost for the manufacturer because of late delivery to demand point j, which can be calculated in the following: Δ f2 = =

j

k=1 (LT k (y)

− dk ) −

 j−1

k=1 (LT k (y)

Jτq + (t j − d j ).

Hence, it follows that g j = α(Δh1 + Δh2 ) + (1 − α)Δ f2 . 8

− dk )

In what follows, we design a dynamic programming based algorithm in terms of the above-mentioned V( j, d). At each iteration, for any given α, it solves transportation and assignment problems to find the optimal solution based on dynamic programming while using linear search to enumerate the value of α. Finally it terminates with the optimal solution by tracking back after a finite number of steps. Let h be the step size for linearly searching for the optimal value of α, z∗ denote the optimal value of P, and α∗ be the optimal value of α. The algorithm is shown below. Algorithm RDP Step 1 Initiate model parameters and set d = 0, j = 0, α = 0, h = 0.05, V(0, 0) = 0, V(0, −q) = 0, and V(−1, 0) = 0. Step 2 Find the optimal solution to P. For α ∈ [0, 1], for d = 0 : c do for j = 0 : J do solve h1 if q j ≤ d then V( j, d) = max{V( j − 1, d − q j ), V( j − 1, d − q j ) + g j } else V( j, d) = V( j − 1, d) end if z∗ = V( j, d) end for end for

Step 2a)

Step 2b) Track back to find the optimal solution and set α∗ = α and α = α + h. Repeat Step 2a). Step 3 Output x∗ , y∗ , α∗ , and z∗ . Theorem 3. The time complexity of Algorithm RDP is O(Jc αh ). Proof. To prove the time complexity of Algorithm RDP, let us exam the time complexity of each step. Step 1 is with regard to the algorithmic initialization, which requires O(1). Step 2 is an iterative process used for searching for the optimal solution for any given α ∈ [0, 1] and takes O(cJ) time to find the optimal solution to P. Specifically, for any given capacity c of PC, the algorithm takes O(J) to find the optimal solution. Enumerating all values of c will require O(c) time. Given the value of step size, h, in Algorithm RDP, it terminates after a finite number steps, which is bounded by αh . Hence Algorithm RDP runs in O(Jc αh ) as Theorem 3 claims, which completes the proof.  5.2. Numerical example Since the case considers identical order sizes, we develop a numerical example with four collectors and eight DPs, each DP with the order size of 3. The data used in this example is the same as in Section 4.2. The rest of data associated with collectors and DPs are shown in Tables 3 and 4, respectively. Table 3: The parameter values with collectors

Collector Shipping rate Capacity

1 1 5

2 2 4

3 2 5

4 1 7

After applying Algorithm RDP to the above-mentioned data, the solution process can be obtained in Table 5. Thus, the optimal solution to P1 can be obtained as follows: x1∗ = 5, x4∗ = 1, and x2∗ = x3∗ = 0; y∗1 = y∗2 = y∗3 = y∗4 = 0 and y∗5 = y∗6 = 1. Figure 3 shows the optimal operations schedules when order sizes of demand points are the same, where solid and dashed lines have the same indication as in the previous section. It is observed from the optimal solution that collectors 1 and 4 are selected to supply the manufacturer and DPs 5 and 6 are fully served. 9

Table 4: The parameter values with DPs

DP Shipping rate Shipping time Demand Deadline Fixed cost Penalty

1 3 4 3 7 1 28

2 2 2 3 8 5 28

3 3 4 3 2 1 22

4 2 2 3 7 1 26

5 2 2 3 6 2 23

6 2 2 3 6 2 28

Table 5: The solution process obtained by applying Algorithm RDP

item/d 1 2 3 4 5 6

0 -125.25 -125.25 -141 -248.25 -263.25 -267

1 -125.25 -125.25 -141 -248.25 -263.25 -267

2 -125.25 -125.25 -141 -248.25 -263.25 -267

3 -71 -195.25 -195.25 -209 -306.25 -336.25

4 -71 -195.25 -195.25 -209 -306.25 -336.25

5 -71 -195.25 -195.25 -209 -306.25 -336.25

Collectors 1

2

4

3

PC

1

2

3

4

5

6

Demand Points

Figure 3: Optimal operations schedules with identical order sizes

Figure 3: Optimal operations schedules with identical order sizes

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6 -71 -141 -248.25 -263.25 -267 -379.25

6. Conclusions and future extensions Reverse supply chain planning and scheduling has received much attention due to increasingly environmental concerns and regulations. Hence, it is crucial to achieve an efficient and effective management in reverse supply chains through integration. This paper studied an integrated operations scheduling problem in capacitated reverse supply chains with remanufacturing, where identical order sizes and delivery deadlines are taken into account. First, bi-criteria mixed integer linear program was proposed to formulate this operations scheduling problem. Due to NPhard nature, two cases of this problem have been investigated, respectively. One is concerned with unit order size, while the other deals with identical order sizes. Then, the case with unit order size has been analyzed in which it was proved that for any given weight α, the problem can be solved in polynomial time due to the total unimodularity of its constraint matrix, and a linear programming base algorithm was developed. Furthermore, the problem with identical order sizes was discussed, followed by the dynamic programming based algorithm, which is able to solve the problem optimally. Numerical examples associated with two cases are reported to validate two problems and their solution approaches. There are two major extensions for this research. One is the integrated operations scheduling problem in remanufacturing with different order sizes and delivery deadlines. Although the operations scheduling problem with identical order sizes is studied, the results from this study can be extended to solve the problem with different order sizes by using parameterized sensitivity analysis. The other extension is to take into account multiple processing centers at the manufacturer’ plants. Such problems can still be modeled as bi-criteria mixed integer nonlinear problem, but solving such an optimization problem is difficult due to its combinatorial nature. This paper is of interest to both researchers and practitioners. It provides an underlying framework for further research on operations scheduling in multi-echelon reverse supply chains and even in closed-loop supply chains. In addition, two cases studied in this paper are practical as they can help firms develop efficient decision-making tools for evaluating and measuring the reverse (or closed-loop) supply chain performance. Acknowledgments The authors thank the editor(s) and anonymous referees for their valuable comments. Blackburn, J., Guide, V., Souza, G., and Wassenhove, L. (2004). Revese supply chains for commercial returns. California Management Review, 46(2), 6-22. Guide, A., Krausb, M., and Srivastavac, R. 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