Integrated operations scheduling with delivery deadlines

Computers & Industrial Engineering 85 (2015) 177–185

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Integrated operations scheduling with delivery deadlines q Gang Wang a,⇑, Lei Lei b a b

Department of Decision and Information Sciences, Charlton College of Business, University of Massachusetts Dartmouth, 285 Old Westport Rd, North Dartmouth, MA 02747, USA Department of Supply Chain Management and Marketing Sciences, Rutgers Business School, Rutgers University, Newark, NJ 07102, USA

a r t i c l e

i n f o

Article history: Received 19 May 2014 Received in revised form 19 March 2015 Accepted 24 March 2015 Available online 30 March 2015 Keywords: Operations scheduling Capacitated supply and distribution operations Delivery deadlines Partial relaxation Median threshold

a b s t r a c t An integrated scheduling problem of supply and distribution operations in a make-to-order supply chain is considered. The supply chain consists of contracted suppliers, capacitated processing centers, and many demand points. Processing centers customize and/or configure semi-finished products supplied by suppliers to finished products in terms of the requirements of demand points: order quantity and delivery deadline. The problem is to find the assignment of both suppliers and demand points to processing centers and the schedules for supply and distribution operations such that the total costs of shipping and penalty that unfulfilled customer orders incur are minimized, subject to network capacity and deadline constraints. In this paper, a heuristic algorithm is developed, based on partial relaxation and median threshold, which makes it possible to avoid dealing with the generalized assignment problem, a strongly NP-hard problem, in the solution process. Computational tests are performed over test cases randomly generated to assess the performance of the proposed algorithm. As compared to the use of commercial optimization package, CPLEX, computational results show that the proposed algorithm is able to find a good near-optimal solution to this problem with remarkably less computational time. Ó 2015 Published by Elsevier Ltd.

1. Introduction The trend of globalization and rising fuel price has brought more challenges to supply chain management. Due to the rise on oil price, there has been an increase of more than 50 percent in transportation costs over the past five years; inventory, in turn, has risen by more than 60 percent from 2002 to 2006 as a result of increasing transportation costs (Simchi-Levi, Peruvankal, Mulani, Read, & Ferreira, 2012). As globalization increases, lead time is also becoming longer and longer so that many companies are exposed to different types of risks. Intensified competition and heightened requirements of customers are forcing companies to significantly reduce inventory costs on one hand and to improve responsiveness on the other. Cost and customer service have become even more critical in global supply chain operations in today’s fast-changing, complex environment. To reduce inventory costs, many companies attempt to hold very little inventory so that inventory holding costs are negligible. To be more responsive, companies have been trying to achieve shorter lead times. Shorter lead times, however, incur higher distribution costs because more delivery shipments and faster transportation modes

q

This manuscript was processed by Area Editor T.C. Edwin Cheng.

⇑ Corresponding author.

E-mail address: [email protected] (G. Wang). http://dx.doi.org/10.1016/j.cie.2015.03.015 0360-8352/Ó 2015 Published by Elsevier Ltd.

may have to be used. In such cases, a tighter integration of supply and distribution operations is required to achieve a desired service level at minimum total cost, with little or no idle time of supply and distribution operations, and with little or no inventory of intermediate and finished products. Motivated by the applications in lumber industry, a scheduling problem of integrated supply and distribution operations (ISDO) was presented by Wang and Lei (2012), in an effort to improve operations efficiency and effectively manage lead time. The supply chain associated with the scheduling problem is three-stage supply and distribution network with contracted suppliers, processing centers and demand points. Suppliers ship semi-finished products to a given set of capacitated processing centers (PCs) for further processing. PCs comply with the requirements that demand points specify to customize and/or finally configure semi-finished products to finished lumber products. Then the finished lumber products are distributed to demand points dispersed geographically by or before the deadlines that demand points request. Due to constraints of delivery deadline and network capacity, orders of certain demand points may be unfulfilled, which leads to penalty costs. Additionally, partial delivery is unacceptable as each demand point adopts single sourcing strategy to reduce purchase costs. The problem is to determine the assignment of both suppliers and demand points to PCs and find the operations schedules so

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as to minimize the total shipping and penalty costs while satisfying service level requirements. Since a detailed discussion on ISDO problem was provided by Wang and Lei (2012), the focus of this paper is on solution approaches determining a close-to-optimal operational schedule for the scheduling problem. In fact, ISDO problem is difficult to solve because of its combinatorial nature, so most related work in the literature tends to employ heuristics to find a near-optimal solution to ISDO problem. In general, the heuristics for solving ISDO problem in the literature can be classified into two categories. The research in the first category aims to decompose ISDO problem into several sub-problems which are in general solvable in polynomial time. A feasible solution to original problem can then be found and improved by combining the optimal solutions to sub-problems. Wang and Lei (2012) investigated a capacitated supply and distribution network scheduling problem with delivery deadlines. The three efficiently solvable cases of the operations scheduling problem were taken into account: (a) identical order quantities; (b) designated suppliers; and (c) divisible customer order sizes. For each case, they designed a decomposition-based algorithm and proved their computational complexity, respectively. Lei, Liu, Ruszczynski, and Park (2006) developed a twophase algorithm to solve an integration problem of production, inventory, and distribution with non-instantaneous traveling times. The first phase found a feasible solution to the original problem by solving a direct shipment problem between manufacturing facilities and customers. The second phase used heuristics to improve the solution obtained in the first phase. Dawandel, Geismar, Hall, and Sriskandarajah (2006) solved a transportation scheduling problem in a supply chain with a short shelf-life product by a two-phase heuristic, where the first phase applied a genetic algorithm to permutate customers and the second phase used Gilmore Momory algorithm to obtain an integrated schedule by combining the subsequences of customers. Kaminsky and Kaya (2009) considered a combined scheduling problem of make-toorder and make-to-stock supply chains. They presented different heuristic algorithms to find appropriate inventory levels for make-to-stock products and for scheduling lead time quotations, respectively. Manoj, Sriskandarajah, and Wagneur (2012) considered the problem of whether coordinating the two stages that are adjacent in a production system was beneficial. They analyzed conflicting costs when the optimal schedule for the other stage was employed, evaluated the computational tractability of the individual problems at each stage, and showed that individual problems were able to be solved in polynomial time. Finally a genetic algorithm was designed for solving the problem. Compared to the first category, works in the second category intend to solve ISDO problem in an integrated fashion by first reducing original problem to an easily solvable problem through relaxing some constraints, and then improving the solution obtained by relaxed problem to a near-optimal one. There are several early studies of this category in the literature (e.g., Cohen & Lee, 1988, Chandra & Fisher, 1994, Fumero & Vercellis, 1999, Hall, Lesaoana, & Potts, 2001, Kreipl and Pinedo, 2004). Recent research on the operations scheduling problem of ISDO is also extensive. Chen and Vairaktarakis (2005) studied a joint scheduling problem of minimizing a convex combination of distribution cost and customer service level, where some efficient algorithms were provided to highlight the advantages of integrated scheduling over the sequential model. Lapierrea and Ruizb (2007) studied an coordination problem of procurement and distribution operations in hospital logistics and proposed a tabu search metaheuristic to solve this problem with two modeling approaches including many decision regarding operations decision. Li and Vairaktarakis (2007) developed efficient heuristics and approximation schemes for an integrated scheduling problem with two machines and bundling

operations to minimize the total costs of transportation and customers’ waiting. Sawik (2009) analyzed a multi-objective integrated scheduling problem in a customer-driven supply chain by applying both monolithic and hierarchical approaches, and showed that monolithic approach was advantageous over hierarchical method through computational studies. Zegordi and Nia (2009) considered a two-stage supply chain where order assignment and the integration of production and distribution are both involved. They designed a genetic algorithm based algorithm and proved that the algorithm performed well. Zhong, Chen, and Chen (2010) proposed and proved a tight, polynomial-time heuristic algorithm with the performance ratio of 2 in order to solve an integrated scheduling problem of production and distribution operations with committed delivery dates. Gaudreault, Frayret, Rousseau, and D’Amours (2011) studied the coordination of process planning and operations scheduling through using a single model so that simultaneous implementation of process planning and scheduling is achieved. Rasti-Barzoki and Hejazi (2013) studied an operations scheduling problem with due date assignment, production, and batch delivery in a make-to-order supply chain. They presented a three-stage heuristic algorithm based on the structure of optimal solution and carried out computational experiments to show the efficiency of their algorithm. Mokhtari and Abadi (2013) dealt with a scheduling problem with a single stage considering scheduling both in-house production or outsourcing. The objective of this study is to jointly schedule in-house and outsourcing production at the same time so as to minimize the sum of the total weighted completion time and outsourcing cost. Liu, Zhang, Zhu, and Rao (2014) presented a multi-objective scheduling model to schedule logistics tasks and resources for fourth party logistics and developed an improved nondominated sorting genetic algorithm to achieve a joint optimization of cost and time of logistics activities between tow adjacent activities and two sequential activities. Although the scheduling problem of ISDO has been extensively studied in the literature, none of the existing algorithms can be directly applied to our model. Compared to those in the literature, ISDO studied is a more complex problem and has some fundamental differences. First, many relevant works consider minimizing either lead times or total costs in supply, production, and distribution operations. However, our objective is to not only minimize the total costs of scheduling supply and distribution operations and penalizing unsatisfied demand points, but also consider the constraints of network capacity and delivery deadlines. Second, most related studies focus on allocating supplies from production facilities, inventory, and distribution centers to demand points, while our model also needs to assign capacitated suppliers to serve the needs of the distribution centers (i.e., PCs in our case). Third, the interaction between assigning suppliers and demand points to PCs and delivering on time results in additional complexities for solving this problem. The integration scheduling of supply and distribution operations requires simultaneous optimization of assigning suppliers and demand points to PCs and scheduling supply and distribution, with a focus on minimizing the total costs of shipping and penalty. This is difficult since the simultaneous assignment of suppliers and demand points to PCs is also subject to network capacities, processing times, shipping times, and delivery deadlines. To solve ISDO problem under study, an approximation algorithm belonging to the second category is developed by using partial relaxation and median threshold. When certain constraints are relaxed, ISDO problem can be reduced to a combination of two NP-hard problems: generalized assignment problem and supplier selection problem, both of which are computationally difficult. To avoid solving NP-hard problems, at the beginning of the proposed algorithm, ISDO problem with multiple sourcing is first solved to

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find an initial operational schedule to ISDO problem. Solution properties to ISDO problem with multiple sourcing are proved that a substantially large number of demand points adopting single sourcing strategy are entirely assigned and thus the number of customers using multiple sourcing is small and less than the number of PCs. According to solution properties obtained, the proposed algorithm first entirely assigns the customers with single sourcing strategy in the ISDO problem with multiple sourcing, and then introduces a threshold to assign the customers with multiple sourcing by evaluating their profit contributions. The threshold is defined by the maximum median of unit profit contributions associated with multiple sourcing oriented customers over all the PCs. With the threshold, multiple sourcing oriented customers with high unit profit contributions are chosen to use single sourcing strategy because single sourcing helps companies realize significant cost savings. In the meantime, the proposed algorithm still allows the rest of customers with small unit profit contributions to be multiple sourcing oriented. The iteration process continues to seek single sourcing oriented customers until all such customers are identified. Three different computational tests over 800 randomly generated cases are conducted to justify the performance of the proposed algorithm in terms of network size, the level of variability in customer demand, and the relative penalty cost. The remainder of the paper is organized as follows. Section 2 introduces an integrated scheduling problem of supply and distribution operations with delivery deadlines. In Section 3, a partial relaxation and threshold based heuristic algorithm is proposed. Section 4 reports the empirical observations of the algorithm performance upon randomly generated test cases under various parameter setting. Finally, Section 5 concludes the study and discusses future extensions. 2. Operations scheduling problem with delivery deadlines For the convenience of further analysis, we restate the integrated operations scheduling problem over capacitated supply and distribution network in the previous work (Wang & Lei, 2012). The shipping network consists of three stages: contracted suppliers, processing centers or PCs, and demand points. Each supplier with a limited production capacity ships semi-finished products to PCs and charges a shipping cost (via chartered vessels) proportional to shipping quantity and reciprocal to shipping time. Each PC has a limited capacity for processing products and starts processing only after all the shipments from assigned suppliers are received. Each customer receives (customized) final products from no more than one PC (single sourcing) because of volume discounts and the shipment arrival time at a customer site must be no later than its specified delivery deadline. If shipments cannot arrive at demand points by specified due dates (due to the delay in shipping and/or the limitation of network capacity), penalty cost proportional to order sizes are imposed. No partial delivery is acceptable, that is, delivered orders must be fully fulfilled. The notations of the parameters and decision variables used are summarized below. Model parameters S set of contracted suppliers, S ¼ f1; 2; . . . ; Sg; J set of demand points, J ¼ f1; 2; . . . ; Jg; set of processing centers (PCs), N ¼ f1; 2; . . . ; Ng; N Fs capacity of supplier s; s 2 S bsn shipping rate ($/unit) from supplier s to PC n ; s 2 S; n 2 N t sn shipping time from supplier s to PC n ; s 2 S; n 2 N C n processing capacity of PC n ; n 2 N

sn

unit processing time at PC n ; n 2 N shipping rate from PC n to demand point j, n 2 N ; j 2 J pnj fixed shipment cost from PC n to demand point j, n 2 N; j 2 J tnj shipping time from PC n to demand point j, n 2 N ; j 2 J kj order quantity of demand point j, j 2 J pj unit penalty cost for unsatisfied demand point j, j 2 J Tj deadline of delivering the order to demand point j, j 2 J big positive number M Decision variables xsn shipping quantity from supplier s to PC n ; s 2 S; n 2 N  zsn 1; if supplier s ships to PC n ; s 2 S; n 2 N 0; otherwise:  ynj 1; if PC n supplies demand point j; n 2 N ; j 2 J 0; otherwise: anj

The objective is to determine the assignment of both suppliers and demand points to the capacitated PCs and the operations schedules for supply and distribution such that the total shipping and penalty cost is minimized. Thus the model is written as an mixed integer program (MIP) below. XX XX X X ISDO : minG ¼ bsn xsn þ ðanj kj þ pnj Þynj þ pj kj ð1  ynj Þ: s2S n2N

n2N j2J

j2J

n2N

The first term in the objective function of ISDO defines the shipping cost from suppliers to PCs, where bsn xsn estimates the cost of shipping a quantity of xsn from supplier s to PC n . The second term represents the variable and fixed shipping cost from PCs to demand points. The third term denotes the total penalty costs incurred by unsatisfied demands. Subject to. (1) Capacity constraints on suppliers:

X

xsn 6F s ; 8s 2 S;

ð1aÞ

xsn 6zsn F s ; 8s 2 S; 8n 2 N :

ð1bÞ

n2N

Constraints (1a) ensure that all the quantities shipped out of supplier s do not exceed supplier’s capacity, while constraints (1b) defines the relationship between the shipping quantities from supplier s to PC n and the flow indicator variables, zsn , which equals to one if xsn > 0. (2) Flow balance constraints for each PC:

X

xsn ¼

s2S

X kj ynj ;

8n 2 N :

ð2Þ

j2J

(3) Single-sourcing constraints:

X

ynj 6 1;

8j 2 J :

ð3Þ

n2N

Constraints (3) ensure that each demand point is supplied by at most one PC in accordance with the single sourcing strategy. (4) Constraints on delivery deadlines through each PC n ; n 2 N :

maxftsn zsn g þ s2S

X xsn sn þ tnj ynj  Mð1  ynj Þ 6T j ;

8j 2 J :

s2S

Constraints (4) impose the requirement that the arrival time of shipment at demand point j must be no later than its specified due date. (5) Capacity constraints on PCs:

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X kj ynj 6 C n ;

8n 2 N :

j2J

Constraints (5) ensure that PC capacities are not violated. (6) Integrality and nonnegativity:

xsn P 0; zsn ; ynj 2 f0;1g; 8s 2 S; 8n 2 N ; 8j 2 J :

ð6Þ

Throughout the remaining discussion in this paper, we shall denote this problem as ISDO. ISDO can be shown to be NP-hard in strong sense, as it can be reduced to a dynamic generalized assignment problem (Kogan & Shtub, 1997) even if the deadlines of all demand points are relaxed and only a single supplier with an infinite capacity is considered. 3. A partial relaxation and threshold based heuristic

supplier, and S n :¼ fs 2 Sjr sn < r n g; n 2 N . Since shipping rates between suppliers and PCs are reciprocal to shipping times, the suppliers with lower values of r sn usually have longer shipping time tsn , and thus such suppliers will not be chosen to serve PCs. This is necessary because longer shipping times result in the violations of more delivery deadline constraints, thus leading to the increase in the number of unassigned demand points. The suppliers with greater values of rsn , on the other hand, will not possibly be selected because of higher shipping rates, which may increase the total cost. Hence a good choice for the latest arrival time at each PC, i.e., max tsn zsn , is the arrival time of shipment from the supplier whose value of rsn is the median, rn . Let us consider the following ISDO problem with multiple sourcing, denoted by ISDOM , where dnj are linear variables between 0 and 1 for all n 2 N and j 2 J .

ISDOM : min

XX XX bsn xsn þ ðanj knj þ pnj Þdnj s2S n2N

An efficient heuristic algorithm in this section is developed by using partial relaxation and median thresholds. Due to the inverse relationship of shipping cost and time from suppliers to PCs, ISDO is converted into an approximation problem. Since ISDO is difficult to solve due to single sourcing constraint (3), an ISDO with multiple sourcing strategy is investigated based on the approximation problem. A detailed analysis on the ISDO is provided as theoretical basis for the design of solution approach. Using this analysis, a partial relaxation and threshold based algorithm is developed, together with the analysis of algorithm complexity. LP relaxation based algorithms have been widely studied in the literature for solving multiple knapsack problems (MKP) or generalized assignment problems (GAP). Dawande, Keskinocak, and Ravi (2000) considered the MKP with assignment restrictions and presented LP rounding approximation algorithm. Dahl and Foldnes (2006) also studied a LP based heuristics for solving the MKP with assignment restrictions. Trick (1992) investigated the basis structure of the LP relaxation of the GAP and presented an improvement heuristic in terms of problem properties and violated inequalities. Romeijin and Morales (2000) studied a LP relaxation of the GAP and proved the relationship derived from the partial solution of its LP relaxation. However, the existing results on MKP and GAP can not be directly exploited to solving the operations scheduling problem. In addition to additional stage (contracted suppliers), the interplay of delivery deadlines, supplier selection, and customer assignment makes ISDO much harder to solve than MKP and GAP. The solution approach proposed here extends the existing results on GAP and use partial relaxation and threshold to find a near optimal operational schedule to ISDO in polynomial time. 3.1. ISDO with multiple sourcing An operational schedule to ISDO consists of a supply schedule that determines when PCs start processing, where DPs’ orders are processed and shipping quantities, and a distribution schedule that defines the departure time of each order and when DPs’ orders are delivered. The single sourcing constraints make ISDO more difficult, so a simplified version of ISDO is taken into account i.e., ISDO problem with multiple sourcing. The multiple sourcing problem still needs to decide on which appropriate set of suppliers will be chosen, which is NP-hard problem. However, the inverse relationship between shipping cost and time from suppliers to PCs provides insight into easily solving this multiple sourcing problem. To begin with, let rsn :¼ bsn =tsn denote the cost per unit of time. Sort rsn in decreasing order, and for any given n 2 N , let rn denote the median of r sn over all suppliers, sm be the corresponding

þ

X

n2N j2J

! X 1 dnj pj kj

j2J

n2N

s.t.

X xsn 6 F s ;

8s 2 S;

ð7Þ

n2N

X X xsn ¼ kj dnj ; s2S

8n 2 N :

ð8Þ

j2J

X dnj 6 1;

8j 2 J :

ð9Þ

n2N

X t sm n þ xsn sn þ t nj dnj  Mð1  dnj Þ 6 T j ;

8j 2 J :

ð10Þ

s2S

X kj dnj 6 C n ;

8n 2 N :

ð11Þ

j2J

xsn ¼ 0;

8s 2 S n ;

0 6 dnj 6 1;

8n 2 N : 8s 2 S;

xsn P 0;

ð12Þ

8n 2 N ;

8j 2 J :

ð13Þ

In each iteration, since the number of customers, J, is usually much greater than the number of processing centers, N, and/or the number of suppliers, S, such a partial relaxed model has potential to achieve a significant reduction in the number of integer variables. 3.2. Analysis One remaining issue here is then how this relaxation may affect the solution quality. The analysis below serves for this purpose. Construct a directed graph G ¼ ðV; EÞ as follows. The node set V of the graph G consists of supplier, PC and customer demand points, i.e., V ¼ S [ N [ J . The edges ðs; nÞ and ðn; jÞ of the graph G are directed from supplier nodes, s, to PC nodes, n, with the capacity of F s , and from PC nodes, n, to customer nodes, j, with the assigned capacity of kj , respectively, i.e., E ¼ fðs; nÞ; ðn; jÞjs 2 S; n 2 N ; j 2 J g. Let x ¼ ðx; dÞ be an operational schedule to ISDOM , where d denotes the vector of fractional variables (0 < dnj < 1), and GR be the subgraph of G induced by the fractional edges dnj . Let W denote a pseudo-tree in GR . A pseudo-tree is a connected subgraph of G on vertex set V if it contains at most jVj edges. P A demand point is partially assigned if n2N dnj < 1, for all j 2 J . In the following theorems, the solution properties of ISDOM are provided to facilitate the design of the heuristic algorithm by extending the theoretical results for the basis structures of GAP (Trick, 1992). In fact, the existing results for GAP cannot be directly applied to our problem since the supply chain under study is three-stage network. With the independence of customer orders on PCs, i.e., knj ¼ kj , however, a feasible solution to ISDOM can be

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constructed to help obtain the solution properties of ISDOM , based on the theoretical results of GAP. Lemma 1. GR consists of pseudo-trees W 1 ; W 2 ; . . . ; W r , where r is the number of pseudo-trees in GR . Proof. By definition of pseudo-tree, it is sufficient to show that each connected component, W k ; 1 6 k 6 r, is either a tree or a tree plus an edge. Case (1) W k is a tree. Suppose that W k contains a cycle with a node sequence from i to j, where i; j 2 V R . Let l be the perturbation of a feasible solution to ISDOM and find a path from i to j, i.e., l ¼ ðþe; ðki =kiþ1 Þe; ðþki =kiþ1 Þe; . . . ; ðki =kj Þe; ðþki =kj ÞeÞ with zero for other arcs (Dawande et al., 2000). Then choose e > 0 to be small enough and construct two feasible solutions v 1 ¼ x þ l and v 2 ¼ x  l, thereby following that x ¼ ðv 1 þ v 2 Þ=2, which contradicts the fact that x is the basic feasible solution. Case (2) W k is a tree plus an edge. It suffices to show that W k includes only one cycle. According to the Case (1), it follows that W k excluding supplier nodes is a tree. When a supplier node is linked with two PC nodes, cycles exist in W k . Suppose that there exist multiple cycles. Similar to the pf of Case 1, an extreme point to W k can be represented as a convex combination of two feasible solutions to ISDOM on W k , leading to a contradiction. We now conclude that the statement holds. h

Theorem 2. Any operational schedule to ISDOM problem, x, satisfies the following properties: (a) at most one PC n is partially utilized to capacity C n in W k ; 1 6 k 6 r; n 2 N ; (b) at most one demand point is partially fulfilled in W k ; 1 6 k 6 r; (c) at most one demand point in W k is partially fulfilled by only one PC, 1 6 k 6 r. (d) a cycle in W k including the nodes of supplier, PC and customer, where PCs are entirely utilized.

Proof (a) and (b) We shall prove (a) and (b) together because their proofs are similar. Let V k denote the vertex set of the pseudo-tree W k ; 1 6 k 6 r. By Lemma 1, W k is a pseudo-tree of at most jV k j edges. This implies that there exist at least jV k j  jV k \ Sj  1 fractional edges in W k , i.e., dnj > 0, where V k \ S denotes the set of supplier nodes in W k . Since x is the basic feasible solution to ISDOM , the linearly independent constraints of at least jV k j  jV k \ Sj  1 in (9) and (11) must be set to equality. Therefore, at most one PC is partially used or at most one demand point is not fully fulfilled in W k ; 1 6 k 6 r. (c) We shall show that if there exist two partially fulfilled demand points by only one PC at the same time, then this will result in a contradiction. Suppose that two

demand points are partially fulfilled by only one PC, that is, they are leaf nodes of the pseudo-tree W k ; 1 6 k 6 r. Let i and j denote these two demand points, respectively, and e be positive and small enough. Fixing the values of all other edges in GR to 0, we perturb any feasible solution x to ISDOM along the path from i to j in W k by l, i.e., l ¼ ðþe; ðki =kiþ1 Þe; ðþki =kiþ1 Þe; . . . ; ðki =kj Þe; ðþki =kj ÞeÞ. Further, two feasible solutions are obtained, i.e., v 1 ¼ x þ l and v 2 ¼ x  l, whereas the convex combination of v 1 and v 2 is x, i.e., x ¼ ðv 1 þ v 2 Þ=2, which contradicts the fact that x is a basic feasible solution to ISDOM . (d) We shall first prove that if there exists a cycle in W k ; 1 6 k 6 r, then the cycle must contain supplier nodes. It is equivalent to show that any subgraph of W k ; 1 6 k 6 r, on vertex set N [ J is a tree. Let H denote the subgraph of GR induced by fractional edges, dnj P 0, on vertex set N [ J . The pf relies on the perturbation technique from (Dawande et al., 2000) in a way similar to the pf of (c). We shall argue that H is actually a forest. Suppose that there exists a cycle, C, on H. We perturb the variables on this cycle, C, while keeping the values of other variables on the edges of H. The perturbation vector f is defined as ðþe; ðki =kiþ1 Þe; ðþki =kiþ1 Þe; . . . ; ðki =kj Þe; ðþki =kj ÞeÞ, where e > 0 is small enough. Adding f and f to any basic feasible solution x to ISDOM produces two feasible solutions to ISDOM , i.e., x1 ¼ x  f and x2 ¼ x þ f . Since x ¼ x1 þ x2 , which contradicts x, there exists no cycle on H, i.e., H is a forest. We complete the first claim. It is obvious that PCs on a cycle in W k must be entirely utilized to their maximum capacity. This can be proved from the fact that each demand point on the cycle of W k always attempts to be entirely fulfilled by the PC with lowest shipping cost and delivery timeliness guarantee before assigned to another PC on the cycle. h

Theorem 3. The number of demand points using multiple sourcing, i.e., 0 < dnj < 1, is above bounded by the number of fully utilized PCs. Proof. Let W k be any pseudo-tree in GR ; 1 6 k 6 r. Let rk be the number of supplier nodes, pk be the number of PC nodes, and qk be the number of customer nodes in W k . The number of edges in W k is rk þ pk þ qk  1. Let h1 and h2 , be the number of suppliers in W k \ S, and the number of demand points in W k \ J , respectively. It follows that h2 6 1 from part (b) of Lemma 1. Let di denote the degree of node i in W k . Then the following inequality holds:

rk þ pk þ qk  1 ¼

X i2W k

X

di ¼ h1 þ h2 þ

i2W k \S:di P2

X

di þ

di

i2W k \J :di P2

P h1 þ h2 þ 2ðr k  h1 Þ þ 2ðqk  h2 Þ P 2ðrk þ qk Þ  h1  h2 :

It follows that pk  ðrk  h1 Þ P qk ) pk P qk , which leads to the statement by summing up over all the trees, W k , in GR .

h

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Theorem 4. There exist at least J  N demand points that are fully fulfilled by single sourcing strategy in the optimal operational schedule

in Step 2 until no improvement can be made or maximum running steps are reached. 3(a) In terms of the saving factor enj calculated using (14) for each link ðn; jÞ 2 X, find the threshold D and partition X into X1 and X2 , where X1 ¼ fðn; jÞjenj P Dg and X2 ¼ fðn; jÞjenj < Dg, for all n 2 N ; j 2 J . 3(b) Solve the following linear program on X, and update X and X0 as follows: X0 ( X0 [ fðn; jÞjdnj ¼ 1g and X ( X n fðn; jÞjdnj ¼ 1g, for all n 2 N ; j 2 J .

to ISDOM . Proof. Let x be an optimal operational schedule to ISDOM . If GF is empty, then there exists no demand point that is partially fulfilled by x. In other words, demand points are either fully fulfilled by a single PC or entirely unassigned. Let p be the number of PCs that serve partially fulfilled demand points in x; a be the number of integrally fulfilled demand points in x; g be the number of partially fulfilled demand points in x. Hence a þ g ¼ J and we have that g 6 p 6 N by Theorem 3. Since each demand point that is partially fulfilled is assigned to at least two PCs, we derive the following inequality, a þ 2g 6 N þ J, based on the statement of (c) in Lemma 1. This means that a P J  N. h

min

XX XX bsn xsn þ ðanj kj þ pnj Þdnj s2S n2N

þ

X

n2N j2J

! X 1 dnj pj kj

j2J

n2N

s.t. 3.3. Solution approach

X xsn 6 F s ;

According to the single-sourcing constraints (3), all such partially fulfilled demand point j with 0 6 dnj 6 1 must be reevaluated to be either dnj ¼ 0 or dnj ¼ 1 in order to obtain the final feasible solution. Nevertheless, Theorem 3 indicates that the total number of bounded linear variables with fractional values

X X xsn ¼ kj dnj ; s2S

under the optimal solution to ISDO ; 0 6 dnj 6 1, is likely to be small as compared to J. This observation leads to the partial relaxation and threshold based heuristic algorithm, called PT, for solving ISDO, where Step 3 is an iterative process. In each iteration, Step 3 fixes a subset of fractional variables with lower saving values to 0 through a prescribed threshold, but still allows the remaining part of binary variables to be relaxed. To do so, let us first define the profit contribution of demand point j if j is fulfilled by PC n

8ðn; jÞ 2 X:

n2N

X xsn sn þ t nj dnj  Mð1  dnj Þ 6 T j ; t sm n þ

8j 2 J :

s2S

X kj dnj 6 C n ;

8n 2 N :

j2J

0 6 dnj 6 1;

8ðn; jÞ 2 X1 ;

xsn P 0; 8s 2 S; 8n 2 N ; dnj ¼ 0; 8ðn; jÞ 2 X2 ; xsn ¼ 0;

pj kj  anj kj  pnj for all n 2 N and j 2 J ; kj

8n 2 N :

j2J

X dnj 6 1;

M

enj ¼

8s 2 S;

n2N

dnj ¼ 1;

ð14Þ

Let D be a pre-specified threshold, defined by maxn2N mn , where mn denotes the median of enj for each given PC n , and thus be called median threshold. Letting dnj be the optimal assignment of demand

8s 2 S n ; 8n 2 N ; 8ðn; jÞ 2 X0 :

Theorem 5. Algorithm PT terminates in finite steps.

points to PC in ISDOM , the proposed heuristic is summarized below. Algorithm PT:

Proof. In Algorithm PT, Step 2 solves ISDOM to obtain an initial assignment of customer demand points to PCs by fixing ynj ¼ 1 if the optimal solution to ISDOM ; dnj , equals to 1. However, demand points associated with dnj ¼ 0 or 0 < dnj < 1 may be fully fulfilled in the optimal solution to ISDO. Hence, in Step 3, the set X is divided into two disjoint subsets, X1 and X2 , based on the maximum median value of saving factors, maxj2N mn , derived by (14). Since customer demand points with a more significant saving if served are more likely to be fully fulfilled, we set variables, dnj , with higher saving factors to binary variables. According to Theorem 3 that the number of variables with fractional values, dnj , is small, we can solve

Step 1: (Initialization) For any n 2 N , sort r sn in decreasing order and use any algorithm for solving weighted median problem to find the median of rsn ; r n ; tsm n , and S n . Step 2: Solve ISDOM , and fix ynj ¼ 1 if dnj ¼ 1 to determine an initial assignment of demand points to PCs. For any link ðn; jÞ over the shipping network, dnj ¼ 1 defines a feasible assignment of demand points to PCs. Divide the values of assignment variables in ISDOM into two disjoint sets: and X ¼ fðn; jÞjdnj ¼ 0 and 0 < dnj < 1g, for all n 2 N ; j 2 J . Step 3: Repeat the following processes by partial relaxation and median threshold to improve the initial solution obtained

X0 ¼ fðn; jÞjdnj ¼ 1g

ISDOM quickly. As Theorem 4 guarantees that Algorithm PT will fix some variables, ynj , to be 1 in each iteration and the total number of binary valuables ynj is bounded by a constant, the heuristic algorithm PT terminates within polynomial steps. h

Table 1 Measurement criteria in the experiment. Criterion

Value

J R

50 100% 10%

r=l

Other parameters 100 120% 20%

200 140% 30%

400 160% 40%

600 180% 50%

800 200% 60%

1000 70%

S=8 S=4 S=4

N=5 N = 5, 10 N = 5,10

J = 50 J = 50

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G. Wang, L. Lei / Computers & Industrial Engineering 85 (2015) 177–185 Table 2 Auxiliary parameters used in the experiments. Parameters

Cs

bsn

t sn

sn

anj

pnj

tnj

kj

Tj

Value

½100; 1000

½0:3; 0:8

½15; 40

½15; 25

½0:1; 0:5

½20; 100

½5; 10

½80; 800

½10; 100

4. Computational testing

Table 4 PT vs. CPLEX on

The purpose of computational experiment performed in this section is to measure the effectiveness and performance of the proposed algorithm. The three criteria for computational analysis as well as test cases in experiment design are first described and then computational results are reported accordingly.

4.1. Experiment design The computational performance of the proposed algorithm will be evaluated from different perspectives. To examine the robustness of the algorithm, the level of variability in customer demand is taken into account, i.e., r=l, where r and l denotes standard deviation and mean of the order sizes kj , respectively. Since penalty costs for unsatisfied demand points play an extremely important role in assigning demand points, the relative penalty cost, R, is considered to assess the impact of penalty costs on the computational performance of the proposed algorithm, defined by

P j2J pj kj R¼P P : n2N j2J ðanj kj þ pnj Þ

ð15Þ

Furthermore, the assessment on the impact of the size of the distribution network, J, is performed. In Table 2, the values of all parameters used in this study are taken from uniform distribution over each corresponding interval. For example, C s is any random number taken from a uniform distribution over the interval ½100; 1000. For each given set of the values ðJ; R; r=lÞ in Table 1, 30 random test cases are generated in the empirical study, based on the values of parameters obtained in Table 2. The computational performance of the proposed search algorithm is measured by its required CPU time (in second) on a Dell 600 (Pentium-M 1.4 GHz with 1 GB RAM) and the error gap defined below

Gap ¼

GPT  G  100% G

Table 3 Impact of network size, J, on the algorithm performance.

50 100 200 400 600 800 1000 a

Performance in total cost

CPLEX

PT

CPLEX(G )

PTðGPT Þ

Gap (%)

Std. dev.

3044.93 3263.97 3651.77 4158.32 4477.83 4992.28 5120.17

0.13 0.24 0.25 0.31 0.36 0.38 0.40

868760.3 59555.73 59671.34 62322.72 59868.15 60544.49 41605.5

881201.8 60511.59 60793.98 63560.39 61114.19 61935.89 42640.01

1.43 1.60 1.88 1.99 2.08 2.30 2.49

0.01 0.01 0.01 0.02 0.01 0.02 0.01

Table 5 PT vs. CPLEX on

r=l (%)

10 20 30 40 50 60 70

r=l (S ¼ 4; N ¼ 10; J ¼ 50).

CPU time (s)

Performance in total cost

CPLEX

PT

CPLEX(G )

PTðGPT Þ

Gap (%)

Std. Dev.

6904.16 7103.98 7490.33 7619.43 7810.18 7900.81 8260.94

0.22 0.28 0.34 0.42 0.48 0.50 0.58

2,393,402 1,971,913 343,295 1,943,578 1,931,199 2,318,260 1,741,057

2,432,013 2,009,988 351039.8 1,988,117 1,976,402 2,375,737 1,784,030

1.61 1.93 2.26 2.29 2.34 2.48 2.47

0.01 0.02 0.01 0.02 0.02 0.01 0.02

Table 6 PT vs. CPLEX on R (S ¼ 4; N ¼ 5; J ¼ 50). R (%)

100 120 140 160 180 200

CPU time (s)

Performance in total cost

CPLEX

PT

CPLEX(G )

PTðGPT Þ

Gap (%)

Std. dev.

2976.37 3013.57 3034.00 3066.97 3070.17 3094.67

0.18 0.16 0.14 0.14 0.13 0.12

77346.87 84759.02 67180.79 63614.78 55917.41 58698.15

79737.32 87394.06 68851.04 65358.51 57016.72 59810.35

3.09 3.11 2.49 2.74 1.97 1.89

0.02 0.01 0.01 0.01 0.01 0.01

Table 7 PT vs. CPLEX on R (S ¼ 4; N ¼ 10; J ¼ 50).

defined by (1)–(7), and GPT represents the total operation cost by our proposed search algorithm PT.

CPU time (s)

10 20 30 40 50 60 70

CPU time (s)

ð16Þ

where G denotes the minimum operation cost obtained by using the commercial CPLEX solver to solve the respective ISDO problems

J

r=l (%)

r=l (S ¼ 4; N ¼ 5; J ¼ 50).

Performance in total cost

CPLEX

PT

CPLEX(G )

PTðGPT Þ

Gap (%)

Std. dev.

3622.99 7329.32 18466.75 30639.89 –a – –

0.26 0.78 1.19 1.90 2.10 2.21 2.55

79915.98 60608.96 77,604 57435.2 – – –

81702.53 62046.94 79482.26 58840.46 88428.83 88224.45 38911.97

2.35 2.37 2.42 2.45 – – –

0.01 0.01 0.01 0.01 – – –

The dash means that CPLEX fails to find solution within 10 h.

R (%)

100 120 140 160 180 200

CPU Time (seconds)

Performance in total cost

CPLEX

PT

CPLEX(G )

PTðGPT Þ

Gap(%)

Std. dev.

7392.7 8103.2 8226.8 8432.1 8600.6 9199.1

0.31 0.28 0.24 0.22 0.21 0.21

64739.35 57703.48 77454.95 76506.89 62744.66 57379.4

66535.97 59217.53 79168.09 78187.34 63870.97 58283.94

2.78 2.62 2.21 2.20 1.80 1.58

0.02 0.02 0.02 0.01 0.01 0.01

4.2. Computational results In Tables 3–7, Columns 1–4 contain the following information in order of network size (J), average CPU time required by CPLEX 11, average running time of algorithm PT, average objective function value of CPLEX, and average objective function value obtained by the PT. Column 6 gives the average performance gap measured

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G. Wang, L. Lei / Computers & Industrial Engineering 85 (2015) 177–185

by (16). Each data point reported in these tables represents the average of 30 observations from the randomly generated test cases. Table 3 reveals the impact of problem size in terms of the network size, J, on the algorithm performance. Given that S ¼ 8 and N ¼ 5, the size of the network ranges from J ¼ 50 to J ¼ 1000. As we can see, when the number of demand points or network size is relatively small (J ¼ 50; 100), using CPLEX solver directly to solve ISDO is a practical option. It requires only a minimal amount of CPU time while guarantees the optimality. However, as the problem size increases, CPLEX solver starts to lose its computational advantage to the proposed heuristic. Especially, when the problem size goes beyond J ¼ 200 in our experiments, the required computation time by the CPLEX solver becomes fairly excessive, while that required by the proposed algorithm PT is constantly within 3 CPU seconds. The resulting error gaps are constantly within 1%. Moreover, for all test cases with J > 400, it is observed that CPLEX is unable to obtain the optimal solution to ISDO within the given maximum CPU time (10 h). One main reason behind this observation is that the ISDO under study contains both the generalized assignment problem and supplier selection as subproblems, which introduces combinatorial nature into search process and therefore the time needed to verify the optimality of a solution becomes excessive when the network size becomes large. With the proposed PT heuristic, however, a linear transportation problem with time constraints, a relatively easier problem, is solved rather than a more computationally difficult problem. Tables 4 and 5 presents the results of an experiment in which the performance of Algorithm PT is assessed against the levels of demand variability (r=l) over the following two instances: S ¼ 4; N ¼ 5; J ¼ 50 and S ¼ 4; N ¼ 10; J ¼ 50, respectively. As the results show, the proposed algorithm PT works very nicely in seeking the optimal solution under fairly homogeneous demand. However, as the variability in demand increases, the level of error gaps also increases slightly. Indeed, observe that the average gap increases from 1% to 2% as the variability level in demand increases from r=l = 10% to r=l = 70%. Our second observation is that the proposed algorithm PT is capable to handle the cases where the level of demand variation is large. As shown in Tables 4 and 5, when the value of r=l exceeds 20%, the average error gap has the tendency to increase at a fairly reasonable rate. For instance, when r=l > 20% for the case with S ¼ 4; N ¼ 5; J ¼ 50, the average error gap is about 2% and as for the cases with S ¼ 4; N ¼ 10; J ¼ 50, the average error gap is about 2%. From Tables 4 and 5, we can see that when the level of r=l is relatively low, the resulting empirical error gaps are fairly small. This indicates that Algorithm PT has the potential to find near optimal solutions when the customer demands in a supply chain network are homogeneous. This observation is consistent with the fact that a knapsack problem can be easily solved optimally if all the items (ordered) sizes are equal (i.e., r=l). However, after the level of r=l goes beyond 20%, the empirical error gaps tend to increase. Nevertheless, about 72% of empirical average error gaps were within 1% from the optimal values obtained in our experiment, with the largest average error gap 2%. In addition, the observation implies that the proposed algorithm can better deal with the fluctuating demand in centralized supply chain system (N = 5) than in decentralized system (N = 10). Tables 6 and 7 show the impact of the relative penalty cost (R) defined by (15) on empirical error gaps for the two instances with S ¼ 4; N ¼ 5; J ¼ 50 and S ¼ 4; N ¼ 10; J ¼ 50, respectively. As the value of R increases, CPLEX takes more time to find an optimal solution to ISDO. This is because the total penalty cost dominates the shipping cost. According to the improvement criterion, CPLEX will allocate more demand points to proper processing centers to lessen penalty costs as much as possible, which requires CPLEX to evaluate more constraints of delivery deadline and capacity.

On the other hand, the proposed algorithm PT uses median threshold to directly assign those demand points with high efficiency values, i.e., enj , and then applies partial relaxation to resolve the relaxed problem in order to avoid the demand points possibly assigned. Therefore, it appears that the average CPU time that Algorithm PT takes steadily decreases when the value of R increases from 100% to 200% as shown in Tables 6 and 7. In the meanwhile, the proposed algorithm PT guarantees a good near optimal solution to ISDO. 5. Conclusions and future extensions In this study, we proposed and analyzed a partial relaxation and threshold based search algorithm for the operations scheduling problem of capacitated supply and distribution network with customer delivery deadlines. Empirical observations on the algorithm performance obtained from randomly generated test cases are reported. It is observed that the partial relaxation and threshold based approach consistently obtained near optimal solutions (within 2%) in more than 600 instances out of 800 test cases, which showed that relaxation and threshold based solutions could be a potentially promising approach for the operations scheduling problems of capacitated network. There are three major extensions for this research. One is the location selection for capacitated multi-echelon shipping networks. While the locations of processing centers are assumed to be known in this study, the result reported here can be extended to the network design problem, where the locations of the processing centers must be optimized simultaneously with the network flows from the suppliers to the customers. The other research direction is to extend the analysis to the multi-modal transshipment/distribution networks with convex cost functions. For such networks, we shall allow multiple transportation modes along each link from the source/supplier locations to processing centers to customer demand points, and aim at choosing the optimal shipping mode for each link and the transshipment locations, each of which can also be a demand point, under convex cost functions. Solving such an optimization problem as a mixed integer program with non-linear cost functions is not much harder than its linear objective function version. The third is to find a subset of supplier candidates in the neighborhood of median supplier for each PC in the design of the algorithm PT. Thus one of their shipping times may be used as the latest arrival times for shipments from suppliers to PCs in Constraints (4), i.e., maxs2S ftsn zsn g. Acknowledgments The authors thank the editor(s) and two anonymous referees for their valuable comments, which led to the improvements on the paper. References Chandra, P., & Fisher, M. L. (1994). Coordination of production and distribution planning. European Journal of Operational Research, 72(3), 503–517. Chen, Z. L., & Vairaktarakis, G. (2005). Integrated scheduling of production and distribution operations. Operations Research, 51(4), 614–628. Cohen, M. A., & Lee, H. L. (1988). Strategic analysis of integrated productiondistribution systems: Models and methods. Operations Research, 36(2), 216–228. Dahl, G., & Foldnes, N. (2006). LP based heuristics for the multiple knapsack problem with assignment restrictions. Annuals of Operations Research, 146(1), 91–104. Dawande, M., Keskinocak, P., & Ravi, R. (2000). Approximation algorithms for the multiple knapsack problems with assignment restrictions. Journal of Combinatorial Optimization, 4(2), 171–186. Dawandel, M., Geismar, H. N., Hall, N. G., & Sriskandarajah, C. (2006). Supply chain scheduling: Distribution systems. Production and Operations Management, 15(2), 243–261.

G. Wang, L. Lei / Computers & Industrial Engineering 85 (2015) 177–185 Fumero, F., & Vercellis, C. (1999). Synchronized development of production, inventory and distribution schedules. Transportation Science, 33(3), 330–340. Gaudreault, J., Frayret, J.-M., Rousseau, A., & D’Amours, S. (2011). Combined planning and scheduling in a divergent production system with co-production: A case study in the lumber industry. Computer & Operations Research, 38(9), 1238–1250. Hall, N. G., Lesaoana, M., & Potts, C. N. (2001). Scheduling with fixed delivery dates. Operations Research, 49(1), 134–144. Kaminsky, P., & Kaya, O. (2009). Combined make-to-order/make-to-stock supply chains. IIE Transactions, 41, 103–119. Kogan, K., & Shtub, A. (1997). Dynamic generalized assignment problem. Annuals of Operations Research, 69, 227–239. Kreipl, S., & Pinedo, M. (2004). Planning and scheduling in supply chains: An overview of issues in practice. Production and Operations Management, 51(1), 77–92. Lapierrea, S. D., & Ruizb, A. B. (2007). Scheduling logistic activities to improve hospital supply systems. Computer & Operations Research, 34(2), 624–641. Lei, L., Liu, S., Ruszczynski, A., & Park, S. (2006). On the integrated production, inventory, and distribution routing problem. IIE Transactions, 38(3), 955–970. Liu, Q., Zhang, C. Y., Zhu, K. R., & Rao, Y. Q. (2014). Novel multi-objective resource allocation and activity scheduling for fourth party logistics. Computer & Operations Research, 44(4), 42–51. Li, C. L., & Vairaktarakis, G. (2007). Coordinating production and distribution of jobs with bundling operations. IIE Transactions, 39(2), 203–215. Manoj, U. V., Sriskandarajah, C., & Wagneur, E. (2012). Coordination in a two-stage production system: Complexity, conflict and cooperation. Computer & Operations Research, 39(6), 1245–1256.

185

Mokhtari, H., & Abadi, I. N. (2013). Scheduling with an outsourcing option on both manufacturer and subcontractors. Computer & Operations Research, 40(5), 1234–1242. Rasti-Barzoki, M., & Hejazi, R. S. (2013). Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries for multiple customers in supply chains. European Journal of Operational Research, 228(2), 345–357. Romeijin, H. E., & Morales, D. R. (2000). A class of greedy Algorithms for the generalized assignment problem. Discrete Applied Mathematics, 103(2), 209–235. Sawik, T. (2009). Monolithic versus hierarchical approach to integrated scheduling in a supply chain. International Journal of Production Research, 47(21), 5881–5910. Simchi-Levi, D., Peruvankal, J. P., Mulani, N., Read, B., & Ferreira, J. (2012). Is it time to rethink your manufacturing strategy? MIT Sloan Management Review, 53(2), 20–22 (Winter). Trick, M. A. (1992). A linear relaxation heuristic for the generalized assignment problem. Naval Research Logistics, 39(2), 137–151. Wang, G., & Lei, L. (2012). Polynomial-time solvable cases of the capacitated multiechelon shipping network scheduling problem with delivery deadlines. International Journal of Production Economics, 2(137), 263–271. Zegordi, S. H., & Nia, M. A. (2009). Integrating production and transportation scheduling in a two-stage supply chain considering order assignment. International Journal of Advanced Manufacturing Technology, 44(9–10), 928–939. Zhong, W. Y., Chen, Z. L., & Chen, M. (2010). Integrated production and distribution scheduling with committed delivery dates. Operations Research Letters, 38(2), 133–138.