1 Introduction
A supply chain is a network of suppliers, manufacturers, storage houses, and distribution centers organized to acquire raw materials, convert these raw materials to finished products, and distribute these products to customers Liu and Nagurney (2013); Zhang and Xu (2014); Yu and Nagurney (2013). With the globalization of market economies, for many companies, especially for the high tech companies, such as Samsung, Apple, and IBM, their customers are all over the world and the components are also distributed in many places ranging from Taiwan to South Africa. To design an efficient supply chain network the enterprises must identify optimal capacities associated with various supply activities and the optimal production quantities, storage volumes as well as the shipments.They also must take into consideration the cost related with each activity. The cost, including the shipment, the shrinking resources of manufacturing bases, varies from day to day. From the practical standpoint, it is meaningful for us to consider these factors so that the sum of strategic, tactical, and operational costs can be minimized. In this way, the design of the supply chain network should be conducted in a rigorous way so that we can provide an insight into this problem from the systemwide view.
In the past decades the issue of designing the supply chain network got a great deal of attention, see e.g. Ma and Suo (2006); Santoso et al. (2005); Zhou et al. (2002); Trkman and McCormack (2009); Altiparmak et al. (2009); Ahmadi Javid and Azad (2010); Nagurney (2010a); Bilgen (2010). In 1998 Beamon Beamon (1998) presented an integrated supply chain network design model formulated as a multicommodity mixed integer program and treated the capacity associated with each link as a known parameter. Two years later, Sabri and Beamon developed another approach to optimize the strategic and operational planning in the supply chains design problem using a multiobjective function. However, in their model, the cost associated with each link was a linear function and thus the model did nit capture the reality of dynamical networks, which are prone to congestions. Handfield and Nichols Handfield and Nichols (2002) also employed discrete variables in the formulation of supply chain network model and this model was faced with the same problem mentioned above. Recently, Nagurney Nagurney (2010b) presented a framework for supply chain network design and redesign at minimal total cost subjecting to the demand satisfaction from a systemoptimization perspective. They employed Lagrange multiplier to deal with the constraint associated with the link which made the model complicated with many variables.
Computer scientists and engineers are often looking into behaviour, mechanics, physiology of living systems to uncover novel principles of distributed sensing, information processing and decision making that could be adopted in development of future and emergent computing paradigms, architectures and implementations. One of the most popular nowadays living computing substrates is a slime mould Physarum Polycephalum.
Plasmodium is a vegetative stage of acellular slime mould P. polycephalum, a single cell with many nuclei, which feeds on microscopic particles Stephenson et al. (1994). When foraging for its food the plasmodium propagates towards sources of food, surrounds them, secretes enzymes and digests the food; it may form a congregation of protoplasm covering the food source. When several sources of nutrients are scattered in the plasmodium’s range, the plasmodium forms a network of protoplasmic tubes connecting the masses of protoplasm at the food sources.
In laboratory experiments and theoretical studies it is shown that the slime mould can solve many graph theoretical problems, including finding the shortest path Nakagaki et al. (2000); Zhang et al. (2013a, 2014); Tero et al. (2006); Zhang et al. (2013b, c), connecting different arrays of food sources in an efficient manner Nakagaki et al. (2007); Adamatzky (2013); Tero et al. (2008), network design Tero et al. (2010); Adamatzky (2012); Adamatzky et al. (2011).
Physarum can be considered as a parallel computing substrate with distributed sensing, parallel information processing and concurrent decision making. When the slime mould colonizes several sources of nutrients it dynamically updates thickness of its protoplasmic tubes, depending on how much nutrients is left in any particular sources and proximity of source of repellents, gradients of humidity and illumination Adamatzky (2010); Adamatzky and Schubert (2014). This dynamical updating of the protoplasmic networks inspired us to employ principle of Physarum foraging behaviour to solve the supply chain network design problem aiming to minimize total costs of transportation and redistribution of goods and services. In the Physaruminspired algorithm proposed we consider links’ capacities as design variables and use continuous functions to represent costs of the links. Based on the systemoptimization technique developed for supply chain network integration Nagurney (2009); Nagurney et al. (2010), we abstract the economic activities associated with a firm as a network. We make full use of two features of Physarum: a continuity of the flux during the iterative process and the protoplasmic network adaptivity, or reconfiguration.
The paper is structured as follows. In Section 2, we introduce the supply chain network design model and our latest researches related to P. polycephalum. In Section 3, we propose an approach to the supply chain network design problem based on Physarum model. In Section 4, numerical examples are used to illustrate the flow of the proposed method and the methods’s efficiency. We present our conclusions and provide suggestions for further studies in Section 5.
2 Preliminaries
In this section, the supply chain network design model and the Physarum model are introduced.
2.1 The Supply Chain Network Design Model Nagurney (2010b)
Consider the supply chain network shown in Fig. 1: a firm corresponding to node 1 aims at delivering the goods or products to the bottom level corresponding to the retail outlets. The links connecting the source node with the destination nodes represent the activities of production, storage and transportation of good or services. Different network topologies corresponds to different supply chain network problems. In this paper, we assume that there exists only one path linking node 1 with each destination node, which can ensure that the demand at each retail outlet can be satisfied.
As shown in Fig. 1, the firm takes into consideration manufacturers, distribution centers when retailers with demands must be served. The node in the first layer is linked with the possible manufacturers, which are represented as . These edges in the manufacturing level are associated with the possible distribution center nodes, which are expressed by . These links mean the possible shipment between the manufacturers and the distribution centers. The links connecting with reflect the possible storage links. The links between and denote the possible shipment links connecting the storage centers with the retail outlets.
Let a supply chain network be represented by a graph , where is a set of nodes and is a set of links. Each links in the network is associated with a cost function and the cost reflects the total cost of all the specific activities in the supply chain network, such as the transport of the product, the delivery of the product, etc. The cost related with link is expressed by . A path connecting node 1 with a retail node shown in Fig. 1 denotes the whole activities related with manufacturing the products, storing them and transporting them, etc. Assume denotes the set of source and destination nodes and represents the set of alternative associated possible supply chain network processes joining . Then means the set of all paths joining while denotes the flow of the product on path , then the following Eq. (1) must be satisfied:
(1) 
Let represent the flow on link , then the following conservation flow must be met:
(2) 
Eq. (2) means that the inflow must be equal to the outflow on link .
These flows can be grouped into the vector
. The flow on each link must be a nonnegative number, i.e. the following Eq. (5) must be satisfied:(3) 
Suppose the maximum capacity on link is expressed by . It is required that the actual flow on link cannot exceed the maximum capacity on this link:
(4) 
The total cost on each link, for the simplicity, they can be represented as a function of the flow of the product on all the links Nagurney (2006, 2009); Nagurney et al. (2002); Nagurney and Woolley (2010):
(5) 
The total investment cost of adding capacity on link can be expressed as follows:
(6) 
2.2 Physarum polycephalum
Physarum Polycephalum is a large, singlecelled amoeboid organism forming a dynamic tubular network connecting the discovered food sources during foraging. The mechanism of tube formation can be described as follows. Tubes thicken in a given direction when shuttle streaming of the protoplasm persists in that direction for a certain time. There is a positive feedback between flux and tube thickness, as the conductance of the sol is greater in a thicker channel. With this mechanism in mind, a mathematical model illustrating the shortest path finding has been constructed Tero et al. (2007).
Suppose the shape of the network formed by the Physarum is represented by a graph, in which a plasmodial tube refers to an edge of the graph and a junction between tubes refers to a node. Two special nodes labeled as , act as the starting node and ending node respectively. The other nodes are labeled as etc. The edge between nodes and is . The parameter denotes the flux through tube from node to . Assume the flow along the tube is approximated by Poiseuille flow. Then the flux can be expressed as:
(8) 
where is a pressure at a node , is a conductivity of a tube , and is its length.
By assuming that the inflow and outflow must be balanced, we have:
(9) 
For the source node and the sink node the following holds:
(10) 
(11) 
where is the flux flowing from the source node and is a constant value here.
In order to describe such an adaptation of tubular thickness we assume that the conductivity changes over time according to the flux . An evolution of can be described by the following equation:
(12) 
where is a decay rate of the tube. The equation implies that a conductivity becomes nil if there is no flux along the edge. The conductivity increases with the flux. The is monotonically increasing continuous function satisfying .
Then the network Poisson equation for the pressure can be obtained from the Eq. (811) as follows:
(13) 
By setting =0 as a basic pressure level, all can be determined by solving Eq. (13) and can also be obtained.
3 Proposed Method
In this section, we employ the Physarum model to solve the supply chain network design problem. Generally speaking, there are two subproblems to address:

In the shortest path finding model, there is only one source and one destination in the network while there are more than one retails in the supply chain network design problem.

The Physarum model should be modified to satisfy the capacity constraint on each link.
3.1 OneSource MultiSink’s Physarum Model
In the original Physarum model Tero et al. (2007), there is only one source node and one ending node. In the supply chain network design problem, as shown in Fig. 1, there are retail outlets. From left to right, from top to bottom, we can number the nodes shown in Fig. 1. As a result, the following Eq. (15) is formulated to replace the Eq. (13).
(15) 
where means that units of goods are distributed from the firm to the other manufacturing facilities, denotes retail outlet needs units of goods.
In the original Physarum model, the length associated with each link is fixed. In the supply chain network design problem, the cost on each link, be it a production link, a shipment link, or a storage link, is comprised of the cost of the flow on each link and the investment cost . Practically, the flow on each link should be equal to its capacity. Assume there is flow passing through the link , we take the following measure to convert the two costs into one.
(16) 
As a result, Eq. (16) will be updated as below:
(17) 
where represents the flow on the link starting from node to node .
In the Physarum model, it is necessary for us to initialize the related parameters, including the link length , the conductivity , and the pressure at each node. In the supply chain network design model, the capacity and the flow are the unknown design variables. If we don’t know the specific flow on each link, we cannot determine its cost, which further leads to the initialization failure of Physarum algorithm. To prevent this failure we initialize the length on each link as a very small value ranging from 0.01 to 0.0001.
3.2 PhysarumInspired Model Satisfying the Imposed Capacity on Each Link
To satisfy the imposed capacity on each link, we bring in a new parameter called Capacity Ratio (CR). Let be a flow on link is then the imposed capacity on link is represented as . The CR associated with this link is defined as follows:
(18) 
By defining this parameter, we aim at updating the cost on link as below:
(19) 
In this process, the parameter CR has an important role from two aspects. On the one hand, if the flow exceeds its capacity , then will be bigger than 1, which leads to the increase of . Otherwise, will be less than 1, which further results in the decrease of .
According to the mechanism lying in the Physarum model, every time the links’ length changes, the flow along each link is reallocated adaptively. Bonifaci et al. Bonifaci et al. (2012) has proved that the mass of the mould eventually converges to the shortest path. Based on this rule, after a series of reallocation, we can reach the equilibrium state in the supply chain network design problem and the optimal supply chain network can be obtained.
3.3 General Flow of Physarum Model
The main flow of Physarum model is presented in Algorithm 1. The firm and the retail outlets are treated as the starting node and the ending node, respectively.
First of all, the conductivity of each tube (Dij) is initiated with random value between and and other variables are assigned with , including flux through each tube (), pressure at each node ().
Secondly, we can obtain the pressure associated with each node using Eq. (17). Besides, the flux passing through each link and the conductivity in the next iteration can be recorded. Thirdly, we will update the cost on each link using Eq. (16). In order to satisfy the capacity constraint, Eq. (19) is imposed.
There are several possible solutions to decide when to stop execution of Algorithm 1, such as the maximum number of iterations is arrived, conductivity of each tube converges to 0 or 1, flux through each tube remains unchanged, etc. The algorithm described in present paper halts when , where is a threshold value.
4 Numerical Examples
In this section we demonstrate efficiency of the algorithm in numerical examples. The supply chain network topology for all the examples is shown in Fig. 4.1, and is . In addition, we initialize the link length as .
Link  

1  29.08  29.08  
2  24.29  24.29  
3  31.63  31.63  
4  16.68  16.68  
5  12.40  12.40  
6  8.65  8.65  
7  15.64  15.64  
8  18.94  18.94  
9  12.69  12.69  
10  44.28  44.28  
11  40.72  40.72  
12  25.34  25.34  
13  18.94  18.94  
14  0.00  0.00  
15  19.66  19.66  
16  16.06  16.06  
17  5.00  5.00 
Example 4.1
In this example, the demands for each retail outlet is , respectively. The cost of the flow on each link and the investment cost are shown in Table 1; the costs are continuousvalue functions.
Based on the proposed method, Fig. 3 illustrates the flux variation during the iterative process. The flux on each link gets stable with the increase of iterations. The flux of link corresponding to the shipment link connecting the first distribution center with the retailer gradually decreases to and it should be removed from the supply chain network design. Besides, the link ’s flux increases to step by step. Table 1 represents the solution. This solution corresponds to the optimal supply chain network design topology as shown in Fig. 4.
Example 4.2
The data in Example 4.2 has the same total cost as in Example 4.1 except that we use linear term to express the total cost associated with the first distribution center (For specific details, please see data for link 10 in Table 2).
Link a  

1  29.28  29.28  
2  23.78  23.78  
3  31.93  31.93  
4  19.01  19.01  
5  10.28  10.28  
6  13.73  13.73  
7  10.05  10.05  
8  21.77  21.77  
9  10.17  10.17  
10  54.50  54.50  
11  30.50  30.50  
12  29.58  29.58  
13  23.18  23.18  
14  1.74  1.74  
15  15.42  15.42  
16  11.82  11.82  
17  3.26  3.26 
Figure 5 shows a changing trend of the flux associated with each link during the iterative process. The complete solution for this problem is shown in Table 2. Tthe flow on link 14 now has positive capacity and positive product flow in contrast to the data shown in Table 1. As a matter of fact, the flow of all the links from the first distribution center to the retail outlets has increased, comparing to the values in Example 4.1. Namely, the capacity and product flow on links 12, 13, and 14 is bigger than that of Example 4.1. On the contrary, the product flow on the links connecting the second distribution center with the retail outlets decrease. For example, the solution values on links 15, 16, 17 in Example 4.2 are less when compared with that of Example 4.1. Based on the solution shown in Table 2, the final optimal supply chain network for this example can be formulated as shown in Fig. 2.
Example 4.3
Example 4.3 has the same data as Example 4.2 except that we use linear terms to replace nonlinear functions representing the total costs associated with the capacity investments on the first and second manufacturing plants. For instance, the cost associated with the capacity investments on the first link is represented by instead of .
Link a  

1  20.91  20.91  
2  45.18  45.18  
3  18.91  18.91  
4  14.74  14.74  
5  6.16  6.16  
6  23.79  23.79  
7  21.39  21.39  
8  14.79  14.79  
9  4.21  4.21  
10  53.23  53.23  
11  31.77  31.77  
12  29.10  29.10  
13  22.70  22.70  
14  1.44  1.44  
15  15.90  15.90  
16  12.30  12.30  
17  3.56  3.56 
As can be seen from Fig. 6, it shows us the flux variation associated with each link during the iterative process. The solution for this problem is given in Table 4.3. As for the objective function, our result is 10726.48, which is in accordance with that of Ref. Nagurney (2010b). It can be noted that once the cost on each link changes, the proposed method can adaptively allocate the flow and the capacity investments.
5 Conclusions
We solve the supply chain network design problem using bioinspired algorithm. We propose a model for the supply chain network design allowing for the determination of the optimal levels of capacity and product flows in the supply activities, including manufacturing, distribution, and storage and subject to the satisfaction of retail outlets. By employing principles of protoplasmic network growth and dynamical reconfiguration used by slime mould Physarum polycephalum model, we solve the supply chain network design problem, no matter the cost associated with the capacity investments and the product flows is represented by linear or continuous functions.
Further research can focus on the following directions. First, we will adapt the method to the design of supply chain network under complicated environment. For example, the supply chain network design problem with uncertain customer demands, the supply chain network redesign problem, and so on. Second, we will try to apply this method into other fields, such as the transportation network, mobile networks, and telecommunication networks etc.
Acknowledgement
The work is partially supported Chongqing Natural Science Foundation, Grant No. CSCT, 2010BA2003, National Natural Science Foundation of China, Grant No. 61174022, National High Technology Research and Development Program of China (863 Program) (No.2013AA013801), Doctor Funding of Southwest University Grant No. SWU110021.
References
 Liu and Nagurney (2013) Z. Liu, A. Nagurney, Supply chain networks with global outsourcing and quickresponse production under demand and cost uncertainty, Annals of Operations Research 208 (1) (2013) 251–289.
 Zhang and Xu (2014) W. Zhang, D. Xu, Integrating the logistics network design with order quantity determination under uncertain customer demands, Expert Systems with Applications 41 (1) (2014) 168–175.
 Yu and Nagurney (2013) M. Yu, A. Nagurney, Competitive food supply chain networks with application to fresh produce, European Journal of Operational Research 224 (2) (2013) 273–282.
 Ma and Suo (2006) H. Ma, C. Suo, A model for designing multiple products logistics networks, International Journal of Physical Distribution & Logistics Management 36 (2) (2006) 127–135.
 Santoso et al. (2005) T. Santoso, S. Ahmed, M. Goetschalckx, A. Shapiro, A stochastic programming approach for supply chain network design under uncertainty, European Journal of Operational Research 167 (1) (2005) 96–115.

Zhou et al. (2002)
G. Zhou, H. Min, M. Gen, The balanced allocation of customers to multiple distribution centers in the supply chain network: a genetic algorithm approach, Computers & Industrial Engineering 43 (1) (2002) 251–261.
 Trkman and McCormack (2009) P. Trkman, K. McCormack, Supply chain risk in turbulent environments A conceptual model for managing supply chain network risk, International Journal of Production Economics 119 (2) (2009) 247–258.
 Altiparmak et al. (2009) F. Altiparmak, M. Gen, L. Lin, I. Karaoglan, A steadystate genetic algorithm for multiproduct supply chain network design, Computers & Industrial Engineering 56 (2) (2009) 521–537.
 Ahmadi Javid and Azad (2010) A. Ahmadi Javid, N. Azad, Incorporating location, routing and inventory decisions in supply chain network design, Transportation Research Part E: Logistics and Transportation Review 46 (5) (2010) 582–597.
 Nagurney (2010a) A. Nagurney, Supply chain network design under profit maximization and oligopolistic competition, Transportation Research Part E: Logistics and Transportation Review 46 (3) (2010a) 281–294.
 Bilgen (2010) B. Bilgen, Application of fuzzy mathematical programming approach to the production allocation and distribution supply chain network problem, Expert Systems with Applications 37 (6) (2010) 4488–4495.
 Beamon (1998) B. M. Beamon, Supply chain design and analysis::: Models and methods, International journal of production economics 55 (3) (1998) 281–294.
 Handfield and Nichols (2002) R. B. Handfield, E. L. Nichols, Supply chain redesign: Transforming supply chains into integrated value systems, FT Press, 2002.
 Nagurney (2010b) A. Nagurney, Optimal supply chain network design and redesign at minimal total cost and with demand satisfaction, International Journal of Production Economics 128 (1) (2010b) 200–208.
 Stephenson et al. (1994) S. L. Stephenson, H. Stempen, I. Hall, Myxomycetes: a handbook of slime molds, Timber Press Portland, Oregon, 1994.
 Nakagaki et al. (2000) T. Nakagaki, H. Yamada, Á. Tóth, Intelligence: Mazesolving by an amoeboid organism, Nature 407 (6803) (2000) 470–470.
 Zhang et al. (2013a) X. Zhang, Z. Zhang, Y. Zhang, D. Wei, Y. Deng, Route selection for emergency logistics management: A bioinspired algorithm, Safety Science 54 (2013a) 87–91.
 Zhang et al. (2014) X. Zhang, Q. Wang, F. T. S. Chan, S. Mahadevan, Y. Deng, A Physarum Polycephalum Optimization Algorithm for the Biobjective Shortest Path Problem., International Journal of Unconventional Computing 10 (12) (2014) 143–162.
 Tero et al. (2006) A. Tero, R. Kobayashi, T. Nakagaki, Physarum solver: A biologically inspired method of roadnetwork navigation, Physica A: Statistical Mechanics and its Applications 363 (1) (2006) 115–119.
 Zhang et al. (2013b) X. Zhang, S. Huang, Y. Hu, Y. Zhang, S. Mahadevan, Y. Deng, Solving 01 knapsack problems based on amoeboid organism algorithm, Applied Mathematics and Computation 219 (19) (2013b) 9959–9970.
 Zhang et al. (2013c) Y. Zhang, Z. Zhang, Y. Deng, S. Mahadevan, A biologically inspired solution for fuzzy shortest path problems, Applied Soft Computing 13 (5) (2013c) 2356–2363.
 Nakagaki et al. (2007) T. Nakagaki, M. Iima, T. Ueda, Y. Nishiura, T. Saigusa, A. Tero, R. Kobayashi, K. Showalter, Minimumrisk path finding by an adaptive amoebal network, Physical Review Letters 99 (6) (2007) 068104.
 Adamatzky (2013) A. I. Adamatzky, Route 20, Autobahn 7, and Slime Mold: Approximating the Longest Roads in USA and Germany With Slime Mold on 3D Terrains., IEEE Transactions on Cybernetics .
 Tero et al. (2008) A. Tero, K. Yumiki, R. Kobayashi, T. Saigusa, T. Nakagaki, Flownetwork adaptation in Physarum amoebae, Theory in Biosciences 127 (2) (2008) 89–94.
 Tero et al. (2010) A. Tero, S. Takagi, T. Saigusa, K. Ito, D. P. Bebber, M. D. Fricker, K. Yumiki, R. Kobayashi, T. Nakagaki, Rules for biologically inspired adaptive network design, Science 327 (5964) (2010) 439–442.
 Adamatzky (2012) A. Adamatzky, Bioevaluation of World Transport Networks, World Scientific, 2012.
 Adamatzky et al. (2011) A. Adamatzky, G. J. Martínez, S. V. ChapaVergara, R. AsomozaPalacio, C. R. Stephens, Approximating Mexican highways with slime mould, Natural Computing 10 (3) (2011) 1195–1214.
 Adamatzky (2010) A. Adamatzky, Physarum machines: computers from slime mould, vol. 74, World Scientific, 2010.
 Adamatzky and Schubert (2014) A. Adamatzky, T. Schubert, Slime mold microfluidic logic gates, Materials Today 17 (2) (2014) 86–91.
 Nagurney (2009) A. Nagurney, A systemoptimization perspective for supply chain network integration: The horizontal merger case, Transportation Research Part E: Logistics and Transportation Review 45 (1) (2009) 1–15.
 Nagurney et al. (2010) A. Nagurney, T. Woolley, Q. Qiang, Multiproduct supply chain horizontal network integration: models, theory, and computational results, International Transactions in Operational Research 17 (3) (2010) 333–349.
 Nagurney (2006) A. Nagurney, Supply chain network economics: dynamics of prices, flows and profits, Edward Elgar Publishing, 2006.
 Nagurney et al. (2002) A. Nagurney, J. Dong, D. Zhang, A supply chain network equilibrium model, Transportation Research Part E: Logistics and Transportation Review 38 (5) (2002) 281–303.
 Nagurney and Woolley (2010) A. Nagurney, T. Woolley, Environmental and cost synergy in supply chain network integration in mergers and acquisitions, in: Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems, Springer, 57–78, 2010.
 Tero et al. (2007) A. Tero, R. Kobayashi, T. Nakagaki, A mathematical model for adaptive transport network in path finding by true slime mold, Journal of Theoretical Biology 244 (4) (2007) 553–564.
 Bonifaci et al. (2012) V. Bonifaci, K. Mehlhorn, G. Varma, Physarum can compute shortest paths, Journal of Theoretical Biology 309 (2012) 121–133.
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