A hybrid solution to collaborative decision-making in a decentralized supply-chain

J. Eng. Technol. Manage. 29 (2012) 95–111

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Journal of Engineering and Technology Management journal homepage: www.elsevier.com/locate/jengtecman

A hybrid solution to collaborative decision-making in a decentralized supply-chain Steven Y.P. Lu a, Henry Y.K. Lau a, Cedric K.F. Yiu b a b

Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Decentralized supply chain Cross-facility capacity management Lagrangian relaxation Artificial immune systems

This paper considers a decentralized supply chain, where multiple independent manufacturing facilities manage some capital-intensive equipments or resources shared among them. In particular, these manufacturing facilities operate somewhat in isolation to serve their own customers, but coordinate closely with each other to ensure the shared resources are effectively utilized. Such crossfacility capacity management problems are common in high-tech industries, they are typical examples of collaborative decisionmaking in supply chain integration, and are critical to create a competitive edge in a more interconnected business environment. In this paper, a hybrid algorithm that integrates Lagrangian relaxation and immunity-inspired coordination scheme, known as LR-ICI, is proposed and investigated by extensive numerical experiments, and is shown to be competitive compared to similar algorithms. ß 2011 Elsevier B.V. All rights reserved.

Introduction A supply chain can be defined as a network of autonomous or semiautonomous business entities collectively responsible for customer satisfaction with procurement, manufacturing and distribution activities. Optimal supply chain performance requires the execution of a set of precise actions. Often, the actual decision (action) authority and processes are distributed among the members in a supply chain, who are primarily concerned with optimizing their own objectives. As a result, making appropriate decisions to attain global optimal performance in a decentralized supply chain is a very challenging problem. Traditionally, contractual agreements and complex accounting schemes are used to ensure that the supply-chain works effectively during daily operations (Giannoccaro and Pontrandolfo, 2004). However, the centralized or hierarchical decision-making process in these supply chains results in losses of efficiency in a competitive and dynamic market environment. Members in a 0923-4748/$ – see front matter ß 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jengtecman.2011.09.008

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supply chain have to coordinate their individual decision-making closely and cooperatively to achieve optimal supply chain performance with balanced services level and inventory level. As stated before, a supply chain is a network of facilities that performs the functions of procurement, production and distribution. Often, organizational barriers between these facilities exist, and information flows can be restricted such that centralized control of material, cash and information flows in a supply chain may not be feasible or desirable. Lee and Billington (1993) described the needs of model support in managing material flows in the supply chains of Hewlett– Packard (HP) and developed a model for supply chains that is not under a complete centralized control scheme. Incentive problems that may arise in multi-echelon supply chains when decisions are delegated to corresponding site managers were discussed by Lee and Whang (1999), where alternative performance mechanisms that are used to align the incentives of the different managers in a supply chain are investigated. The decentralized supply chain formation process which determines the structure and terms of exchange relationships among different entities in the chain was discussed by Walsh and Wellman (2003) where they introduced a market price system to resolve the resource contention and achieve competitive equilibrium. Krishnan et al. (2004) proposed a sales-rebate contract to coordinate a newsvendor supply chain with a fixed price, where the supplier charges the retailer a per-unit wholesale price but gives the retailer a rebate per unit sold above a fixed threshold and the retailer continues to salvage leftover units. Bernstein and Federgruen (2005) designed contractual arrangements between the parties that allow the decentralized chain to perform as well as a centralized one in the context of two-echelon supply chains. Liu et al. (2007) constructed a Stackelberg game to analyze the pricing and lead time decisions by the suppliers as the leader and the retailer as the follower in a decentralized supply chain. Van De Panne (1991) dealt with the decentralization of decision making for multi-division enterprises and proposed an organization structure with one division setting resource prices and the remaining ones determining quantities. A manufacturing system populated by heterogeneous agents and distributed structures of control using contract net was introduced by Maturana and Norrie (1997). Chen (1999) considered a supply chain whose members are divisions of the same firm where the divisions are managed by different individuals with only local inventory information. The optimal decision rules for the divisions under the assumption that the division managers share a common goal to optimize the overall performance are characterized in his work. Jeong and Leon (2002) developed a methodology for decision-making in organizationally distributed systems where decision authorities and information are dispersed in multiple divisions. The proposed methodology is based on Lagrangian relaxation techniques. Guo et al. (2007) provided a market-based decomposition method for decentralized problem solving and information processing in large decentralized organizations. Over the past ten years, many businesses have realized that they need to form alliances in the form of different forms of supply chain networks in order to maintain their viability in their businesses. The common vision is that these enterprises strive to link together and collaborate to ensure that real customer needs and actual business requirements would drive them toward a set of optimized operating conditions. Collaborative manufacturing networks in the commercial aerospace industry has been investigated by Johansen et al. (2005), who discussed a number of organizational, structural, and cultural issues in projects coordination between business partners. Danilovic and Winroth (Steensma, 1996) identifies barriers and developed a tentative fourdimensional analytical framework of inter-organizational collaboration in network settings. Steensma (Sicotte and Langley, 2000) examined the relationship between an organization’s learning capability and inter-organizational collaboration in acquiring technological competencies. Sicotte and Langley (Antonio et al., 2009) discussed different types of integration mechanisms in R&D project management, and examined the use of these mechanisms in a sample of 121 R&D projects in a large research laboratory. Antonio et al. (Langner and Seidel, 2009) empirically explored the individual effects as well as interaction effects of product modularity and internal integration on competitive capabilities. Langner and Seidel (Farmer et al., 1986) examined the details of collaborative concept development through matched cases of novel convertible roof projects in the European automotive industry. An alternative tool to model the interactive nature of multiple decision-makers is market-based methods. There is a significant amount of literature covering applications of the auction mechanism to

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various problems. One of the early works is the auction algorithm proposed by Bertsekas (Fisher, 2004) for finding shortest paths in a directed graph. In this work, parallel asynchronous implementations of the auction algorithm show that the algorithm is well suited for parallel computation and suggests a significant speedup potential. Later on, this algorithm was extended to applications in transportation problems and other linear network optimization problems (Bertsekas, 1993; Bertsekas and Tsitsiklis, 1989). There are many other auction algorithms proposed for a variety of applications. Examples include production planning for multiple facilities by a tailored auction mechanism (Bertsekas, 1990), an auction-base decomposition method for solving large-scale linear programs using distributed agents in a market setting (Ertogral and Wu, 2000), and so on. In essence, contract net protocol also belongs to the family of game-theoretical or market-based algorithms. Originally perceived by Smith (1980), contract net specifies the interaction between agents through the use of contracts and allows tasks to be distributed among a group of agents. It was first applied to a simulated distributed acoustic sensor network (Guttman et al., 2001), and then extended to commercial applications for simulating marketplaces (Sandholm and Lesser, 1995) and others (Saad et al., 1997; Ouelhadj et al., 2003). This paper considers a supply chain model where multiple manufacturing facilities operate in a distributed manner to serve their own customers, and common resources (e.g. warehouses) are shared between these independent manufacturing facilities. Specifically, we deal with the crossfacility capacity management problem in such a decentralized supply chain. Cross-facility capacity management problem is very common in high-tech industries such as semiconductor, pharmaceutical manufacturing and telecommunications service providers (Wu and Golbasi, 2005). In these cases, capital-intensive equipments are shared by multiple production facilities, with each production unit is headed by a manufacturing manager (MM). Each MM makes its own production plan to satisfy local demands and maximize local profits. Quite often, it is very expensive to gather detailed information about the production facilities in a form usable by top-level coordinators or other production facilities. For these reasons, it is often best for each facility to operate somewhat in isolation, passing only those information required to coordinate its activities such as decisionmaking properly so that the capacity of shared capital-intensive equipments is effectively utilized. The major concern of this study is to determine means to maximize the total profits of the independent production facility networked with shared equipments without creating any conflict among the local production plans. This cross-facility capacity management problem is a realistic scenario of collaborative decision-making in supply chain management. Most companies are tightly integrated with their suppliers, customers and business partners, manufacturers can no longer afford to treat suppliers like vendors from whom every lost ounce of cost-savings can be wrong. Nor can they treat customers simply like a market for products and services at the best possible prices. Instead, they need to treat suppliers, customers and business partners like collaborators-together looking for ways to improve efficiency and value across the entire spectrum of the value chain, not just in their respective businesses or operations. A number of forces are driving these companies to synchronize their operations and decision processes with customers, suppliers and business partners, especially globalization, strategic partnerships, the Internet, regulatory changes, and increasing specialization and outsourcing of key business processes. Companies with high level of collaboration have cut costs and cycle times significantly, increased revenue and new product and service introductions, and achieved other major business benefits. But most companies are not highly integrated with suppliers, customers, and business partners- in fact, on average most of their interactions are at the level of exchanging information regularly, but with limited ability to collaboratively make operational decisions. Research focus is unfolding the facts: how are companies with their business partners? Do high levels of external integration measurably improve the way these companies operate? How should companies make their operational decisions collaboratively to create a competitive advantage? The technology-enabled collaboration of multiple companies is only beginning. The biggest obstacles to collaboration include overcoming distrust and the sizable organizational and technological barriers that stand in the way. The hype about collaborative commerce-fanned by technology suppliers and consulting firms-has been deafening. Collaborative commerce in operations management is also one of the

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potential topics selected for this special issue. With the aim of advancing knowledge on the fact of collaborative commerce-how integrated are companies with their partners to create a competitive edge in operations management, this paper has employed a variety of mathematical modelling and optimization methodologies in developing the hybrid solution scheme for solving the crossfacility capacity management problem. Specifically, the cross-facility capacity management problem is formulated as an optimization problem with a decomposable structure. By exploiting this specific structure, we adopt a modified Lagrangian relaxation method to decompose the original optimization problem into a set of subproblems. Traditionally, there will be a master problem to coordinate the solving of these subproblems, and communication only happens between each sub-problem and the master problem. In contrary, the proposed decompose approach does not introduce a master problem to serve as the toplevel coordinator in the decomposition of the original problem. Instead, the resulted sub-problems are equally ranked, and a novel self-coordination scheme is developed which enables the solving of subproblems is coordinated through peer-to-peer communication, rather than communication between each sub-problem and the master problem. The innovative self-coordination scheme is developed with inspiration from the human immune system. Specifically, the inspiration is obtained from the immune network theory, which assumes that a network of stimulatory and suppressive interactions exists between antibodies in our human body. In the immunity-inspired self-coordination scheme, when a solution to the sub-problem is founded, the subsystem exchanges this solution with its neighbouring subsystems in the network. The individual subsystem then locally evaluates the stimulatory or suppressive effect from each of its neighbours, and updates its associated Lagrange multipliers locally according to the consolidated effect from all of its neighbours. These coordination processes continue iteratively until predefined termination conditions are satisfied. In the experiments, randomly generated problem instances are solved and the proposed distributed algorithm was shown to be competitive in terms of computation costs. This advantage was demonstrated by comparing the results with that of the Lagrangian relaxation algorithm with subgradient method (LR-SM) (Fisher, 2004) and the Cooperative Interaction via Coupling Agent (CICA) (Jeong and Leon, 2002); both of these algorithms are similar in nature to the proposed algorithm. Distributed optimization formulation This section presents a model of the cross-facility capacity management problem. Our model involves multiple production facilities, with each facility being headed by a MM. In order to be responsive to an increasingly complex and rapid changing supply chain environment, there are usually a wide variety of products produced in each of these production facilities. Since these products may belong to different supply channels that operate under different demand and delivery characteristics, actual production decisions of the products are made by individual manufacturing manager. On the other hand, these production facilities are linked together by sharing some common resources, such as a warehouse, some capital intensive equipment and so on. Therefore, manufacturing managers must coordinate their individual decision processes to ensure the capacity of shared common resources is effectively utilized. Fig. 1 shows such a decentralized supply chain with five manufacturers and three shared warehouses between them. In modelling the simple cross-facility capacity allocation problem, we use the following notation: Indices i = 1, 2, . . ., n: manufacturing facilities in the decentralized supply chain network; j 2 Si, for i = 1, 2, . . ., n: set of products produced by manufacturer i; k = 1, 2, . . ., m: common resources shared by manufacturing facilities in the network; Rk, for k = 1, 2, . . ., m: set of manufacturing facilities connected via common resource k;

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Manufacturer 1

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Manufacturer 4 Common Resource (e.g. a warehouse)

Common Resource (e.g. a warehouse)

Common Resource (e.g. a warehouse)

Manufacturer 3

Manufacturer 2

Manufacturer 5

Fig. 1. Five manufacturers connected via the coupling operational constraints: shared warehouses.

Decision variables xij: decision vector (production quantity of product j by manufacturer i) controlled by MM i.

Data and parameters pij: unit profit margin for product j of manufacturer i; bi: capacity of the local resources of manufacturer i; aij: consumption rate of local resources at manufacturer i to produce product j; ck: capacity of common resources k; qijk: consumption rate of the common resources k to produce product j at manufacturer i; From a centralized perspective, the cross-facility capacity allocation problem can be formulated as follows: Problem (P)

max

n X X

pi j x i j

i¼1 j 2 Si

subject to XX

X

ai j xi j  bi 8 i ¼ 1; 2; . . . ; n

j 2 Si

qi jk xi jck 8 k ¼ 1; 2; . . . ; m

i 2 Rk j 2 Si

xi j  0 8 j 2 Si 8 i ¼ 1; 2; . . . ; n In the above centralized model, the global system objective is to maximize the sum of the Pn P P manufacturing facilities’ profits, i¼1 j 2 Si pi j xi j . Constraint set j 2 Si ai j xi j  bi 8 i ¼ 1; 2; . . . ; n describes the local resource constraints of each manufacturer i, while the coupling constraints caused by the limited capacity of each shared resource k are represented by the constraint set P P i 2 Rk j 2 Si qi jk xi j  c k 8 k ¼ 1; 2; . . . ; m. Solving this centralized cross-facility capacity allocation problem requires a central computation facility with access to (i) all manufacturers’ local information, and (ii) the global capacity constraints between them. However, in a decentralized supply chain, global communication and complete information sharing among manufacturers are not allowed. As stated above, we are interested in solving the problem through local computations by each manufacturer and their cooperative local interactions. The centralized problem (P) can be rewritten as a combination of the manufacturers’ sub-problems, with the sub-problem for manufacturer i define as follows:

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Sub-problem i (SPi)

max

X

pi j xi j

j 2 Si

subject to X ai j xi j  bi j 2 Si

X

j 2 Si

X

X

qi jk xi j þ

qg jk xg j  ck 8 k 2 Z i

g 2 Rk ;g 6¼ i j 2 Si

xi j  0 8 j 2 Si where Zi is the set of common resources shared by manufacturing i and others. An integrated algorithm Generally, the centralized cross-facility capacity allocation problem is a multi-agent coordination problem in the following form:

maximize

m X

f i ðX i Þ

i¼1

subject to All local constraints : Ai ðX i Þ  bi ; i 2 V All coupling constraints : g i j ðX i ; X j Þ  ci j ; ði; jÞ 2 E

(1)

Under a centralized model given by Eq. (1), there are two types of constraints: (1) ‘‘easy’’ constraints where only local variables of one agent appear; (2) ‘‘complicating’’ constraints where local variables of two (or more) coupled agents appear together. This problem exhibits a special block angular structure (this structure is defined as the pattern of having zero and nonzero coefficients in the constraints), as depicted in Fig. 2. To solve this problem, the problem is decomposed by removing the complicating constraints and to split the multi-agent coordination problem into a set of independent sub-problems, one for each agent. After decomposition, each sub-problem is optimized locally by an agent in an independent and simultaneous manner. At the same time, information of the sub-problem is gathered, analyzed and stored by each agent separately. In this way, the proposed solution approach then fits in with the actual decision-making and information processing structure of a networked system.

Fig. 2. Block angular structural pattern of the constraints.

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The multi-agent coordination problem defined by Eq. (1) can be rewritten as a combination of multiple sub-problems. For each agent i, the sub-problem is equivalent to the problem defined by Eq. (1). P:

! m X max f i ðX i Þ; s:t: Ai ðX i Þ  bi ; i 2 V and g i j ðX i ; X j Þ  ci j ; ði; jÞ 2 E i¼1

¼ fSP 1 : ðmax f 1 ðX 1 Þ; s:t: A1 ðX 1 Þ  b1 and g 1 j ðX 1 ; X j Þ  c1 j ; j 2 N1 Þ; ... SP i : ðmax f i ðX i Þ; s:t: Ai ðX i Þ  bi and g i j ðX i ; X j Þ  ci j ; j 2 Ni Þ; ... SP m : ðmax f m ðX m Þ; s:t: Am ðX m Þ  bm and g m j ðX m ; X j Þ  cm j ; j 2 Nm Þg

 Lemma 1. If Xi is an optimal solution of agent i’s sub-problem SPi, for all i = 1, 2, . . ., m, then (X1 , X2 , . . ., Xm ) constitutes an optimal solution to the centralized multi-agent coordination problem P and the sum of the optimal objective values of SPi, for all i = 1, 2, . . ., m is the optimal objective value of P. As such, the solving of the agents’ sub-problems is equivalent to solving the centralized multi-agent coordination problem (for the proof this lemma, please refer to (Hirayama, 2006)).

Decomposition We apply Lagrange duality to the coupling constraints of each agent i, such that gij(Xi, Xj)  cij, j 2 Ni. These coupling constraints are relaxed with respect to the objective function fi(Xi) by adding another function hi(Xi, Xj, 8 j 2 Ni), that must reflects the preference of agent i’s neighbours. Then, the subproblem for agent i with augmented objective function can be expressed as: LR-SP i ðli j Þ : fmaxð f i ðX i Þ þ hi ðX i ; X j ; 8 j 2 Ni ÞÞ; s:t: Ai ðX i Þ  bi g; and hi ðX i ; X j ; 8 j 2 N i Þ X X ¼ hi ðX i ; X j Þ ¼ li j ðci j  g i j ðX i ; X j ÞÞ j 2 Ni

(2)

j 2 Ni

where lij is the nonnegative Lagrangian multiplier associated with coupling constraint gij(Xi, Xj)  cij and hi(Xi, Xj) is the corresponding penalty function. Lemma 2

(.).

 For any value of lij, suppose Xi is optimal to LR-SPi(lij), for all i = 1, 2, . . ., m. If (1) (X1 , X2 , . . ., Xm ) is feasible to P, and (2) for each (i, j) 2 E, (Xi , X j ) satisfies the complementary slackness condition hi(Xi, Xj) = lij(cij  gij(Xi,  Xj)) = 0, then (X1 , X2 , . . ., Xm ) is optimal to P (for the proof of this lemma, please refer to Appendix A).

Therefore, in order to achieve the global system objective (optimal solution to P) through local optimizations of sub-problems by each agent, agent i needs to find an optimal solution to LR-SPi(lij) given the values of lij, 8 j 2 Ni, as well as satisfying the following conditions: (1) gij(Xi, Xj)  cij, if lij = 0 (2) gij(Xi, Xj) = cij, if lij > 0 Agent i is free to determine the value of Xi. From Eq. (2), it can be seen that the objective function of agent i’s sub-problem also depends on a few variables Xj, for j 2 Ni which are under the control of other agents within its neighbourhood Ni. In order to solve its local optimization problem as given by Eq. (2), agent i needs to know: (1) the values of lij, 8 j 2 Ni, and (2) the values of Xj, for j 2 Ni. While the decision variables of agent i’s neighbours can be obtained through communication, the key issue is to determine the values of the corresponding Lagrange multipliers lij, 8 j 2 Ni. As previously stated, each Lagrange multiplier lij is associated with a relaxed coupling constraint gij(Xi, Xj)  cij, for(i, j) 2 E. Ideally, the corresponding penalty functions should enter each associated agent’s sub-problem

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in an identical manner. That means that the values of lij in the sub-problems of coupled agents should be identical for each pair of (i, j) 2 E. It is well-known that the best choice for the values of lij would be an optimal solution to the dual problem that contains the local objective functions and local constraints of the agents coupled by gij(Xi, Xj)  cij (Krishnan et al., 2004). However, in the adopted definition of the networked system, local objective fi(Xi) and local constraints Ai(Xi)  bi are private information to agent i, where such private information is not shared with others in its neighbourhood via communication. In the proposed algorithm, the values of lij, 8 j 2 Ni are locally computed by agent i after receiving updated information from its neighbours via communication. Therefore, local optimization of sub-problem LR-SPi(lij) by agent i is equivalent to the following iterative (optimization) process: 1. Compute an optimal solution Xi to LR-SPi(lij). 2. Exchange non-private information with neighbouring agents. 3. Update lij, 8 j 2 Ni based on local information and received information from neighbours Given the values of lij and the values of Xj, for j 2 Ni, solving sub-problem LR-SPi(lij) is computationally less intensive compared with that of original problem P. As a result, the key issues of developing the proposed distributed algorithm are: (1) to determine the appropriate (minimum) information to be exchanged between neighbouring agents, and (2) to develop an effective multiplier update method. Sub-problem coordination For each agent i, to compute an optimal solution for LR-SPi(lij) can be achieved independently given the values of lij and the values of Xj, for j 2 Ni. According to Lemma 2, it can be seen that satisfaction of the feasibility condition and the complementary slackness condition for each (i, j) 2 E must be realized through coordination between agents coupled by this (i, j) 2 E. As stated above, there does not exist a ‘‘top-level’’ coordinating unit to accomplish this task in the definition of the networked system. Instead, the coordination is realized through a cooperative interaction scheme between coupled agents. Since the values of Xj, forj 2 Ni are under the authority of the neighbours of agent i, information about the values of Xj, forj 2 Ni has to be exchanged via communication with neighbours. Once agent i decides the values of lij, for j 2 Ni, it is then able to search for an optimal solution to LR-SPi(lij). Therefore, in order to satisfy the feasibility condition and the complementary slackness condition, the locally computed values of lij, for j 2 Ni act as a tool to guide the iterative search for an optimal solution to LR-SPi(lij) in a sense that reduces the violation of the conditions (in other words, more cooperation). Based on the analysis given in the above section, the satisfaction of the feasibility condition and the complementary slackness condition given the values of lij, forj 2 Ni is equivalent to satisfying: (1) gij(Xi, Xj)  cij, if lij = 0, (2) gij(Xi, Xj) = cij, if lij > 0. For the case of lij = 0, the sub-problem of agent i, i.e., LR-SPi(lij) will be an optimization problem that is independent of multiplier lij. That is to say, multiplier lij cannot impose any effect in the optimization of LR - SPi(lij), let alone guiding the search toward a reduction of violating gij(Xi, Xj)  cij. Therefore, in this thesis, the case of lij > 0 is considered. Our focus in the development of the cooperative interaction scheme is to determine the values of lij, for j 2 Ni such that they can guide the local search toward a reduction in the violation of gij(Xi, Xj) = cij with lij > 0 in a distributed and simultaneous manner by the agents. Traditionally, the values of lij, for j 2 Ni are calculated by a centralized computational facility controlling agent i and agent j for each (i, j) 2 E. And the role of lij is to enable the penalty function hi(Xi, Xj) to suppress the search for an optimal solution to sub-problem LRSPi(lij) if the relaxed constraint gij(Xi, Xj)  cij is not satisfied. The difference between the proposed algorithm and traditional Lagrangian relaxation based subgradient optimization algorithms are: (1) the values of lij, for j 2 Ni are calculated locally by each agent i based on local information and received information from neighbours via communication, (2) the role of lij is to enable the penalty function hi(Xi, Xj) to suppress local optimization of agent i when gij(Xi, Xj) > cij or to stimulate it when gij(Xi, Xj) < cij in the search for an optimal solution to

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Fig. 3. An idiotypic network.

sub-problem LR-SPi(lij), so that the condition gij(Xi, Xj) = cij is satisfied or nearly satisfied progressively in the iterative local optimization process. Such characteristics have motivated the development of the cooperative interaction scheme in this research based on the biological immune system, which has similar mechanisms for maintaining the concentration of immune effectors in the human body. These important mechanisms of the human immune system are further discussed in the following paragraphs. In the human immune system, it can be readily observed that part of an antibody known as paratope will bind to part of an antigen known as epitope in the immunity process. Immune network theory assumes that antibodies also have epitope, which can be bounded by other antibodies’ paratopes. These antibodies both stimulate and suppress each other in certain ways that lead to the stabilization of the network of interconnected antibodies. With this assumption, a network of stimulatory and suppressive interactions exists between antibodies that affect the concentrations of each type of antibody. The idiotypic network can be illustrated pictorially by Fig. 3, where immunization of an antigen (Ag) may lead to the generation of a chain of antibodies (Ab). By assuming that there is a single epitope-binding region on each antibody and antigen, and a single paratope-binding region on each antibody, Farmer et al. (Farmer et al., 1986) suggested an abstracted mathematical model of the immune network theory as follows: 2 3 N N n X X X dxi 4 ¼c m ji xi x j  k1 mi j xi x j þ m ji xi y j 5  k2 xi dt j¼1 j¼1 j¼1

(3)

Eq. (3) was introduced to model the changes in the concentration of particular types of antibody. Let there be N antibodies with concentrations {x1, x2, . . ., xN}, and n antigens with concentrations {y1, y2, . . ., yn}. The first term mjixixj represents the stimulation of the paratope of an antibody of type i by the epitope of an antibody of type j. The second term mijxixj represents the suppression of antibody of type i when its epitope is recognized by the paratope of type j. The form of these terms is based on the fact that the probability of a collision (a paratope and an epitope are close enough to attempt to bind) between an antibody of type i and an antibody of type j is proportional to xixj. The third term mjixiyj represents the stimulation of an antibody type by binding with the epitope of an antigen. The constant k1 represents a possible inequality between stimulation and suppression. The parameter c is a rate constant that depends on the number of bindings per unit time and the rate of antibody production stimulated by a binding. The final term k2xi models the tendency of immune cells to die in the absence of any interactions, at a rate determined by k2. The value of k2 is adjusted until the total concentration of the system is kept at a constant value. According to Eq. (3), the change in concentration of an

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Table 1 A mapping between the networked system and the biological immune system. Interactions between agents in a networked system Values of agent i’s local variables Xi Stimulation from local objective value fi(Xi) Stimulation from positive values of cij  gij(Xi, Xj), for j 2 Ni Suppression from negative values of cij  gij(Xi, Xj), for j 2 Ni

Interactions between antibodies in a biological immune system Concentration of an antibody of type i Stimulation by binding with antigens Stimulation by binding with other antibodies Suppression by be recognized by other antibodies

antibody of type i depends on both the stimulation of antigens bounded by antibody i and the aggregated stimulation (or suppression) from antibodies recognized by antibody i. With inspiration from the immune network theory that a network of stimulatory and suppressive interactions exists between antibodies that affect the concentrations of each type of antibody, in the proposed immunity-inspired cooperation scheme, the selection of the values of lij, for j 2 Ni by agent i reflects the stimulatory or suppressive effect of both the local objective value fi(Xi) and the slackness of the relaxed coupling constraint associated with lij, |cij  gij(Xi, Xj)|. Before describing the complete immunity-inspired cooperative interactions scheme, a comparison between the multi-agent coordination problem and the interactions between antibodies in biological immune system is presented, which is given in Table 1. As stated above, each agent i use the updated values of lij, for j 2 Ni as a tool to determine the stimulatory interactions (when gij(Xi, Xj) < cij, for j 2 Ni) and the suppressive interactions with its neighbours (when gij(Xi, Xj) > cij, for j 2 Ni) in performing local optimization of sub-problem LR-SPi(lij). The updated values of lij, for j 2 Ni which enables a network of stimulatory or suppressive interactions between agents continues until reaching a stable condition where gij(Xi, Xj) = cij is satisfied (or nearly satisfied). According to Lemma 2 and the associated analysis given, the satisfaction of the condition gij(Xi, Xj) = cij is equivalent to attaining a global optimal solution to the original multi-agent coordination problem. Prior to the design of the updating policy of lij, for j 2 Ni, an additional property of the parameter lij is described as follows: Lemma 3. If lij, for j 2 Ni is very big so that (@fi(Xi)/@Xi)  lij  (@gij(Xi, Xj)/@Xi) < 0, the interaction between agent i and agent j becomes dominated by agent j. That is to say, the interaction between them will enforce a suppressive effect on the values of Xi in the next iteration of the optimization process performed by agent i. In the contrary, if lij, for j 2 Ni is very small so that (@fi(Xi)/@Xi)  lij  (@gij(Xi, Xj)/@Xi) > 0, the interaction between agent i and agent j becomes dominated by agent i. In addition, the larger the value of |(@fi(Xi)/ @Xi)  lij  (@gij(Xi, Xj)/@Xi)|, the bigger the suppressive effect on the value of Xi will be (for the proof of this lemma, please refer to Appendix B). According to Lemma 3, in order to ensure that the selection of the values of lij, for j 2 Ni by agent i reflects the appropriate stimulatory or suppressive effect from neighbouring agents in local interactions, the updating policy for the values of lij needs to satisfy the following conditions:  @g ðX ;X Þ < 0; if g i j ðX i ; X j Þ > ci j (1) @ f@iXðX i Þ  li j  i j@Xi j > 0; if g i j ðX i ; X j Þ < ci j i i @g ðX ;X Þ (2) j @ f@iXðX i Þ  li j  i j@Xi j j is proportional to |cij  gij(Xi, Xj)| i i

Therefore, the following formula is introduced to locally compute the values of lij, forj 2 Ni by agent i:

@g i j ðX i ; X j Þ @ f i ðX i Þ  li j  ¼ u  ðci j  g i j ðX i ; X j ÞÞ @X i @X i

(4)

where u is a positive constant that is given at the beginning of the optimization process and is known to all agents as their a prior knowledge.

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After transforming Eq. (4), the following expression for lij is obtained:

li j ¼

ð@ f i ðX i Þ=@X i Þ  u  ðci j  g i j ðX i ; X j ÞÞ ð@g i j ðX i ; X j Þ=@X i Þ

(5)

Once agent i finds an optimal solution Xj to its sub-problem and receives the current optimal solutions Xj of its neighbouring agent j, for j 2 Ni, it may update lij using Eq. (5). An integrated solution to the cross-facility capacity management problem According to the locally multiplier updating formula (Eq. (5)), we have:

ð@ f i ðX i Þ=@X i Þ  u  ðci j  g i j ðX i ðnÞ; X j ðnÞÞ ð@g i j ðX i ; X j ðnÞÞ=@X i Þ h   i  P P P ck  g 2 Rk ;g 6¼ i j 2 Sr qg jk xg j ðnÞ =Si  qi jk xi j ðnÞ j 2 Si ð@ pi j xi j =@xi j Þ  u  h P   i ¼ P P j 2 Si @ g 2 Rk ;g 6¼ i j 2 Sr qg jk xg j ðnÞ=Si  qi jk xi j =@xi j h   i  P P P ck  g 2 Rk ;g 6¼ i j 2 Sr qg jk xg j ðnÞ =Si  qi jk xi j ðnÞ j 2 Si pi j  u  P ¼ j 2 Si qi jk

li j ðn þ 1Þ ¼

Thus, the procedures for solving the cross-facility capacity allocation problem using the proposed distributed algorithm are given below: Initialization (performed locally by each manufacturer i) set the initial iteration n = 0, the maximum number of iteration = N, lk(0) = 0, 8 k 2 Zi set u = a positive constant, where 0 < u < (1/2), data received = 1, e = acceptable capacity violation for all common resources Main loop (performed locally by each manufacturer i) while n < N do manufacturer i locally finds an optimal production plan for its products nP  o P P P P xi j ðnÞ ¼ arg maxP : j 2 Si pi j xi j þ k 2 Z i lk ðnÞ c k  g 2 Rk ;g 6¼ i j 2 Sr qg jk xg j ðn  1Þ  j 2 Si qi jk xi j a x bi ;xi j  0 8 j 2 Si j 2 Si i j i j P  q x ðnÞ, to the manufacturers that are For each k 2 Z i , manufacturer i communicates its required common resource k, i jk j 2 Si ij sharing common resource k with manufacturer i. Simultaneously, it receives the required common resources by those P  manufacturers, j 2 Sr qr jk xr j ðnÞ, for g 2 Rk ; g 6¼ i. if data received = 0, wait; else P P P for each k 2 Z i , calculates jck  g 2 R ;g 6¼ i j 2 Sr qg jk xg j ðnÞ  j 2 S qi jk xi j ðnÞj kP i P P   if jck  g 2 R ;g 6¼ i j 2 Sr qg jk xg j ðnÞ  j 2 S qi jk xi j ðnÞj  e for all k 2 Z i , stop; else i k sets n = n + 1; manufacturer i locally updates the values of lk ðnÞ, for k 2 Z i by h  P   i P P pi j u ck  qg jk xg j ðn1Þ =Si qi jk xi j ðn1Þ g 2 Rk ;g 6¼ i P j 2 Sr lk ðnÞ ¼ j 2 Si sets data received = 0; end end Repeat the main loop

q j 2 Si i jk

In the above solution process, the information exchanged between interacting manufacturers is limited to their proposals for the consumption of each common resource k. This is tailored to fit in with the actual operating condition of the multi-manufacturer network, where manufacturers may be sensitive to reveal their production plans for the products manufactured by them to others. This algorithm is a hybrid distributed optimization methodology that integrates a mathematical decomposition and an immunity-inspired coordination scheme, and is applicable to collaborative decision-making scenarios where multiple companies are integrated via sharing some common resources. The mathematical programming formulation with a decomposable structure adopted in this paper is an ideal model of the realistic scenario of collaborative decision-making: each

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manufacturer has its own operational decision problem to solve (this local problem is represented by a sub-problem in the formulation), an individual manufacturer cannot solve its local problem in isolation, and it has to determine the capacity allocation of the shared warehouses together with its neighbouring manufacturers to maximize their overall benefits (this connection is represented by a coupling constraint in the formulation). A workable and efficient interaction rule is required for these manufacturers to resolve potential conflicts and to achieve their common objective to maximize the overall benefits in this collaborative decision-making scenario. The innovative self-coordination scheme developed in this paper is essentially a cooperative interaction rule that specifies the information exchanged and the self-coordination procedures based on locally available information throughout the collaborative decision-making process. This cooperative interaction rule is based on a peer-to-peer communication method and a synchronous iterative distributed decision-making process. It is developed with inspiration from human immune system which is a natural and efficient cooperative system with numerous collaborative behaviors. It is very innovative compared with traditional coordination scheme:  The sub-problems of all manufacturers are equally ranked and no one has priority over another in their interactions, rather than the two-level master/slaves scheme widely adopted in existing distributed algorithms.  Individual manufacturer is allowed to exchange partial information only in its neighbourhood, instead of all manufacturers passing information to a top-level coordinator or manufacturers sharing complete information with their neighbours.  Individual manufacturer updates the values of associated Lagrange multipliers independently according to the stimulative or suppressive effect its neighbouring manufacturers’ decisions made in the last iterate have on it.

Experimental set-up The operation and performance of the proposed distributed optimization method is investigated by simulation studies. All simulation experiments were performed on a parallel multi-PC distributed computing platform (PMDCP) (Fig. 4). In this platform, there are 24 interconnected PCs. Each PC is equipped with an Intel1 Core (TM) 2 CPU 6600 @ 2.40 GHz 1.58 GHz processor with 4 GB of RAM, and

Fig. 4. The parallel multi-PC distributed computing platform.

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Table 2 Basic problem characteristics. Group # Group Group Group Group Group Group Group Group Group Group

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10

Number of facilities (n)

Number of products (identical for all facilities)

Number of common resources and the associated connection topology (Rk)

2 3 4 5 6 2 3 4 5 6

10 10 10 10 10 20 20 20 20 20

k = 1, k = 2, k = 4, k = 4, k = 5, k = 1, k = 2, k = 4, k = 4, k = 5,

R1 = (1, R1 = (1, R1 = (1, R1 = (1, R2 = (1, R1 = (1, R1 = (1, R1 = (1, R1 = (1, R1 = (1,

2) 2), 2), 2), 2), 2) 2), 2), 2), 2),

R2 = (2, R2 = (1, R2 = (2, R2 = (2,

3) 3), R3 = (2, 3), R4 = (4, 4) 3), R3 = (3, 4), R4 = (3, 5) 3), R3 = (2, 4), R4 = (4, 5), R5 = (4, 6)

R2 = (2, R2 = (1, R2 = (2, R2 = (2,

3) 3), R3 = (2, 3), R4 = (4, 4) 3), R3 = (3, 4), R4 = (3, 5) 3), R3 = (2, 4), R4 = (4, 5), R5 = (4, 6)

they communicate with each other under a peer-to-peer model. The simulation experiments were implemented with this parallel distributed computing platform, supported by the message passing interface (MatlabMPI) of MATLAB. To instantiate the insights gained from the analytic solution procedures and investigate the performance of the proposed algorithm, we conduct extensive empirical tests on the distributed computing platform as described. The test problems are randomly generated by varying the number of manufacturing facilities, the number of products manufactured at each facility, the number of common resources shared by the manufacturing facilities (together with the network connection topology), the unit profit of each product, capacity of local resources at each facility, capacity of each common resource, the consumption rate of local resources by each product, and the consumption rate of common resources by each product. By varying the number of manufacturing facilities, the number of products manufactured at each facility, and the number of common resources shared by the manufacturing facilities (together with the network connection topology), 10 groups of test problems are generated as defined in Table 2. For each of the above 10 groups, 10 test problems are generated by varying the following parameters: (1) The unit profit margins, pij, that are randomly generated from a Uniform (Chen, 1999; Bertsekas, 1993) distribution. (2) The capacity of local resources, bi, that are randomly generated from a Uniform [50,100] distribution. (3) The consumption rate of local resources, aij, that are randomly generated from a Uniform (Giannoccaro and Pontrandolfo, 2004; Chen, 1999) distribution (4) The capacity of common resources, ck, that are randomly generated from a Uniform [100,200] distribution. (5) The consumption rate of common resources, qijk, that are randomly generated from a Uniform (Giannoccaro and Pontrandolfo, 2004; Chen, 1999) distribution.

Thus, 100 test problems were generated in total in the experiment. Matlab was used to implement the algorithm. The manufacturing managers solved their corresponding sub-problems, LR-SPi (lk), using ILOG CPLEX, and they were able to exchange messages using the Message Passing Interface (MatlabMPI) supported by Matlab. In the experiments, the parameters of the algorithm were fixed as follows: the maximum number of iteration = 100n, u = 0.5, e = 1.0. Results and analysis The results of the proposed algorithm (LR-ICI) were compared with those obtained by the Lagrangian relaxation algorithm with subgradient method (LR-SM) (Fisher, 2004) and the Cooperative

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Interaction via Coupling Agent (CICA) (Jeong and Leon, 2002). The main differences between the three methods are:  LR-SM is a centralized algorithm where the master problem coordinates the sub-problem solutions by updating Lagrangian multipliers with global system information without restriction at any given time.  CICA is a distributed algorithm where coupling agents are introduced as the third parties to guide sub-problems interrelated by coupling constraints in finding compromised solutions.  LR-ICI is a distributed algorithm where subsystems solve their sub-problems through local optimization, exchange partial information (depends on specific problem domains) with only neighbouring subsystems, and then self-coordinate their sub-problems by computing Lagrangian multipliers locally with renewed information received by communicating with its neighbours. A common feature of the three methods is that they do not guarantee the feasibility and convergence to a global optimal solution, which is also a common property of Lagrangian relaxation based algorithms. To evaluate the quality of the solutions and the speed of convergence, the following performance indexes are measured: (1) Percentage deviation from optimal: d fo ¼ 100 

Global objective value  Optimal global objective value % Optimal global objective value

(2) Average computation time: P10 CPU minutes of solving prolem instance i C 1 ¼ i¼1 10 (3) Average communication cost: P10 the number of rounds in solving problem instance i C 2 ¼ i¼1 10 The first performance measure provides benchmarks for the quality of the solution, the second performance measure indicates the average total computation time required to solve a test problem, while the third performance measure indicates the communication cost (which is assumed to be proportional to the number of messages exchanged). The main loop between solving its sub-problem and updating the Lagrange multipliers is called a round. The number of messages exchanged between neighbouring manufacturers in a given network topology will increases with the number of rounds. Similarly, the number of calling a local solver by a manufacturer will also increase with the number of rounds. Table 3 Percentage deviation from optimal dfo. Problem group (manufacturers #  products #)

Group Group Group Group Group Group Group Group Group Group

#1 (2  10) #2 (3  10) #3 (4  10) #4 (5  10) #5 (6  10) #6 (2  20) #7 (3  20) #8 (4  20) #9 (5  20) #10 (6  20)

CICA

LR-SM

LR-ICI

Min

Avg

Max

Min

Avg

Max

Min

Avg

Max

0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.00 0.10 0.10

1.80 2.18 3.29 3.47 3.93 1.98 2.33 3.50 3.99 4.36

3.97 6.21 8.19 8.86 9.15 4.03 6.77 8.65 9.74 10.60

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.92 2.27 3.33 3.69 4.08 2.12 2.44 3.69 3.81 4.78

4.16 6.79 8.34 9.05 9.91 4.36 7.69 9.18 10.46 11.34

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

2.96 2.01 3.98 3.74 3.12 2.31 2.64 3.35 5.16 4.44

8.14 7.19 7.54 9.16 8.59 10.14 9.24 9.57 10.67 10.45

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Table 4 Average computation time C1. Problem group (manufacturers #  products #) Group Group Group Group Group Group Group Group Group Group

#1 (2  10) #2 (3  10) #3 (4  10) #4 (5  10) #5 (6  10) #6 (2  20) #7 (3  20) #8 (4  20) #9 (5  20) #10 (6  20)

LR-SM

CICA

LR-ICI

5 14 27 46 135 7 16 34 65 158

13 22 31 55 115 17 29 45 79 137

22 38 52 68 82 29 45 59 75 89

Fig. 5. Comparison of the communication costs C2 (measured by the average number of rounds) in solving the problem groups with number of products = 20 by LR-SM, CICA and LR-ICI respective.

A summary of the experimental results is given in Tables 3 and 4, and Fig. 5. The findings of the experiments are as follows:  In terms of deviation from optimal dfo, LR-SM, CICA and LR-ICI show similar performance. In addition, LR-SM and CICA performs better for problem groups with fewer manufacturers (and also fewer cross-facility common resources). The reason for this phenomenon is that there are fewer cross-facility common resources to coordinate their capacity allocation. The dfo of LR-ICI does not seem to be dependent on the problem structure even through there are more cross-facility capacity for problem groups with a larger number of manufacturers (as shown in Table 3).  In terms of total computation time C1, the computation time increases in a near linear fashion as the number of facilities increases for LR-ICI. On the other hand, when the test problem instances were solved by LR-SM and CICA, the computation time increases much more dramatically and an exponential growth can be observed for both the problems with 10 products manufactured at each facility and the problems with 20 products manufactured at each facility (as shown in Table 4).  In terms of average communication costs C2 that are measured by the average number of rounds, LRICI obviously spends more rounds to find a global solution than LR-SM and CICA. In addition, the number of facilities does not seem to affect the number of rounds required by LR-SM, the number of rounds required by CICA increases in a near linear fashion as the number of facilities, while LR-ICI seems to have an exponential growth in the number of rounds as the number of facilities increases (as shown in Fig. 4).

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Conclusion In this paper, the cross-facility capacity allocation problem in a distributed environment is studied to investigate the performance of the proposed distributed solution technique LR-ICI in different applications. Cross-facility capacity management problem is a realistic scenario of collaborative decision-making in supply chain integration, and managers face great challenges in the integration in order to create a competitive advantage in today’s more and more interconnected business environment. Therefore, the development of an efficient solution technique for solving the cross-facility capacity management problem contributes a lot to the theory-driven conceptual and empirical research in the domain of collaborative decision-making and supply chain integration. The experimental results in this paper show that the performance of LR-ICI is similar to that of LR-SM and CICA in terms of solution quality measured by the percentage deviation from optimal solution. In terms of computation cost measured by total CPU minutes and communication cost measured by average number of rounds to find a global solution, LR-ICI outperforms LR-SM and CICA for total computation time, but underperforms LR-SM and CICA for average number of rounds required to find a global solution. However, we consider that the extra number of rounds required by LR-SM is an inevitable cost for finding a global solution with reduced global information. One should select an algorithm that provides an optimal balance between computational cost and communication cost depending on one’s specific requirement. Acknowledgement The work described in this paper was partly supported by the Research Grant Council of the Hong Kong Special Administrative Region, PRC under the GRF Project Nos. HKU7142/06E and HKU7137/07E. The third author would like to thank the Research Committee of the Hong Kong Polytechnic University for support. Appendix A. Proof of Lemma 2 According to the assumption, the dual problem P  LR-SP i ðli j Þ ¼ f i ðXi Þ þ j 2 N li j ðci j  g i j ðXi ; X j ÞÞ ¼ f i ðXi Þ, since (X1 , X2 , . . ., Xm ) is feasible to P, then i  Xi is optimal to SPi, for all i = 1, 2, . . ., m. According to Lemma 1, (X1 , X2 , . . ., Xm ) is optimal to P. Appendix B. Proof of Lemma 3 The objective function of agent i for optimizing its sub-problem, LR-SPi(lij), can be rewritten as: X max : X i  ½ð@ f i ðX i Þ=@X i Þ  li j  ð@g i j ðX i ; X j Þ=@X i Þ þ li j  ci j þ l ðc  g ir ðX i ; X r ÞÞg r 2 N ;r 6¼ j ir ir i

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