The multiproduct two-layer supply chain is very common in various industries. In this paper, we introduce a possible modeling and algorithms to solve a multiproduct two-layer supply chain network design problem. The decisions involved are the DCs location and capacity design decision and the initial distribution planning decision. First we describe the problem and give a mixed integer programming (MIP) model; such problem is NP-hard and it is not easy to reduce the complexity. Inspired by it, we develop a transformation mechanism of relaxing the fixed cost and adding some virtual nodes and arcs to the original network. Thus, a network flow problem (NFP) corresponding to the original problem has been formulated. Given that we could solve the NFP as a minimal cost flow problem. The solution procedures and network simplex algorithm (INS) are discussed. To verify the effectiveness and efficiency of the model and algorithms, the performance measure experimental has been conducted. The experiments and result showed that comparing with MIP model solved by genetic algorithm (GA) and Benders, decomposition algorithm (BD) the NFP model and INS are also effective and even more efficient for both small-scale and large-scale problems.

The supply chain network system is the soul and backbone throughout the overall supply chain management (SCM). Many decisions have to be made over the SCM. For instance, to determine the structure of the supply chain; to determine number, size, and location of facilities in supply chain; and to make the distribution or transportation plan, sometimes, the decision of product design could also seem as an issue in SCM. Not going that far, in this paper, we focus on the supply chain network design (SCND) problem. The SCND makes the decision of the structure of a chain and affects its costs and performance, which is not only of significant importance to supply chain management but also a classic case in operation research [

Although in the last decades thousands of research literatures could be brought into SCND scope, it is not hard to classify the proposed works into several parts: (i) from the type of problem, SCND can be divided into two parts, the design in strategy phase and the operation phase [

A realistic two-layer supply chain network.

To specify our research problem in this paper, we notice that there is a very common SC network (SCN) structure in the real world. Some of manufacturers usually pay more attention to optimization of their multicategory products supply and marketing chain rather than the certain raw material supply sources, food manufacturing, or other products with single or simple source of materials for instance [

In the multiproduct two-layer SCN, decision makers often faced such a problem of, when the location of plants sites and the design capacity of various sites have been fixed, how to locate the DCs and design the DCs’ capacity in the retail market (or demand area) and how to make the preliminary distribution plan so that the total cost of location-operation-distribution is minimal and customers’ demands are met at the same time. In fact, customers think that the more DCs the better; in this condition, it takes less time and costs less for distribution in the SCN, but obviously it takes much to construct and operate the DCs. On the contrary, the less DCs have been set, the more time and costs will be taken for distribution in the SCN. Moreover, the costs for constructing and operating the DCs will be reduced but it may be difficult to meet the customers’ demand, and the delivery time from the plants via DCs to customers becomes longer. Notice that there are many tradeoffs in the multiproduct two-layer SCND problem and we are trying to formulate an integrated optimal model and solve it in an effective way.

As mentioned before, the multiproduct two-layer SCN structure is very common in various industries. Hence, there already existed numerous literatures dealing with this kind of SCND in both application and research area. However, most of the literatures are related to modeling the problem with various characteristics or modeling the problem in different planning phases [

As a matter of fact, based on existing literatures, many NP-hard problems or nonlinear optimization problems can also be formulated as a network flow problem [

The rest of this paper is structured as follows. The second part is literature review of relevant problems; the third part provides the transformation mechanisms and network flow models in designing multiproduct multilayer supply chain network. We propose a solution program and apply the network simplex algorithm to solve the model and discuss the algorithm’s performances compared with genetic algorithm, and Benders’ decomposition algorithm through numerical examples. The last part gives the conclusion and future research direction.

Broadly speaking, our research could be related to a number of literature streams. First, it pertains to the facility location research, in particular, the inventory-location problem and location-allocation problem. Secondly, it relates to SCND problem. When we just consider the details of the costs involving in the problem, the research is relevant to fixed cost transportation problem. The transformation mechanisms and algorithms are related to network flow algorithms domain. In this paper, we just review the most closely related literatures.

The aim of SCND is to design an efficient network structure for new chain’s entities or to reengineer an existing network to increase its total value. Various traditional SCNDs aim at obtaining the optimal distribution plan and facility location in the supply chain and optimizing the original supply chain network. Thus, it can be said that SCND subordinates to the joint optimization of product distribution and facility location [

Study of this problem can generally be expressed as mixed integer programming model whose goal is to minimize the costs of various expenses. Hanssmann [

A general network flow problem could be shown in Figure

A general network flow diagram.

The multiproduct multilayer SCND in this paper can be stated asfollows. As shown in Figure

The manufacturers supply a variety of products; the plants location and production capacity are fixed; final market customer demand can be also predicted.

We only take the forward logistics flow into account.

The candidate location of the DCs can be obtained, and the fixed costs in constructing and the basic operation costs in operating the DCs can be estimated.

The costs for plants to produce each product are known; the basic costs of delivery products through the supply chain are known, regardless of the carbon tax and other additional costs accompanied with logistics.

Original two-layer supply chain network.

Decisions to be made in this problem are (i) determining the location of distribution center and design capacity and (ii) determining the initial distribution plan.

The following parameters and labels are introduced.

Sets and constancies

Variables

It is not hard to model the problem as a mixed integer programming model (MIP):

The objective function (

Constraints (

Such model has already been classified to a NP-hard problem. In the next section, we will show the transformation mechanisms of how to formulate a lower bound network flow model of the proposed MIP model; therefore, we can solve the MIP model through solving the lower bound problem in polynomial time.

Observation shows that the costs involved in objective function and constraints almost linear over the flow in supply chain network (production, operation, and distribution amount), and the only fixed cost is the facility’s construction costs. In this section we give the transformation mechanisms for costs, network structures, and flows and how we formulate the proposed problem as a network flow model.

Palekar et al. and others developed a new relaxation algorithm that can be used to solve the transportation problem with fixed costs, and they had proved this relaxation method a more effective one [

The transform function could be written as

After the relaxation, the lower bound model can be formulated as minimal cost flow problem so that it is easy to find the solution.

Although the relaxation functions ((

First, we build a network

Add virtual decision nodes for distribution centers location for each distribution center. Thus, as illustrated in Figure

Transformation process of the DCs nodes.

Add virtual source node and its corresponding virtual arcs. As shown in Figure

Transformation process of the plants nodes.

Add virtual sink node and its corresponding virtual arcs. Figure

So far, the network flow supply chain network structure has been obtained. We provide the framework in Figure

Transformation process of the customers nodes.

Framework of transformed supply chain network structure.

We first introduce some additional notations:

Capacity analysis of each arc is as follows.

As investigated in transformation techniques in Steps

Since the flow associated with the arc

Now we investigate the upper bound and lower bound of the flows. To verify if there is always a flow entering the customer demand node, we set that the upper bound and lower bound of

Cost analysis of each arc is as follows.

The cost associated with virtual source is the unit manufacturing cost of product; then

It is clear that cost associated with virtual sink is 0; then

The cost associated with virtual arc

The cost associated with the arc

It is hard to define the cost associated with the arc

Substituting the cost function into (

However, it is easy to notice that the

Now we give the derivation of (

Since the total customer’s demand is the upper bound of each DCs capacity, we can obtain

If

then,

It is obvious that

When

So far, we deduce that (

Based on the above analysis and derivation, we formulate the network flow problem of multiproduct multilayer SCND problem (NFP) as

The objective function (

Constraints (

The structure of algorithms to solve the model we proposed in Section

Solution technique.

After the application of transformation techniques to the original problem, the multiproduct two-layer SCND problem has been transformed into a network flow problem with some virtual nodes and arcs. In this condition, the network simplex algorithm could be employed to solving the NFP model. The steps of solve algorithms are as follows.

Initializing viable forest structure, order

Adding a corresponding supernode

Define

Provide the rule of enter arc, which is to find a variable which does not meet the optimal conditions; that is, if there exist the following conditions, the solution to the problem is not optimal. If

Generally speaking, any arc fulfilling the conditions above can be taken as an enter arc to go through the iteration in network map. In order to increase the efficiency of the algorithm, this paper compared the reduced cost

Endow the enter arc with flow until the flow of a basic arc reaches the upper or lower bound of the capacity. Assuming the enter arc in Step

Thus the available mechanisms for leaving arc are

Renew the augmented forest structure through selecting the leaving arc and enter arc in viable base solutions. It is known that this kind of iterative method iterates from a relatively augmented forest to another one, until the optimal solution is reached.

The function

Function

For

End

End

When all arcs meet the optimum conditions, the minimum cost flow network problems get the optimal solution, and the optimal solution is

For small-scale problem, we could directly program “

In this section, we will solve some numerical problems by MIP and corresponding algorithm, and by NFP and network simplex algorithm we have proposed in this paper to verify the effective and efficiency of NFP models.

In recent years, the related algorithms to MIP problem are heuristic algorithm Benders’ decomposition algorithm, and so forth. We choose the genetic algorithm (GA) and Benders’ decomposition algorithm (BD), such two-type well-known effective algorithm for MIP problem, to measure the MIP model of this paper and algorithm performance. We then apply the NFP model and network simplex algorithm (INS) with the same data to compare the optimal solution and algorithm running time with MIP model solved by GA and BD, respectively. The scale of experimental data is given in Table

The limitation of data scale in experiments.

Experiment | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Category of products | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

Manufacturer | 1 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

Distribution centers | 1 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

Customers | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

Number of actual network arcs | 18 | 144 | 486 | 1152 | 2250 | 3888 | 6174 | 9216 | 13122 | 18000 |

Number of solving network arcs | 27 | 178 | 561 | 1284 | 2455 | 4182 | 6573 | 9736 | 13779 | 18810 |

Range of randomly generated constant data for experiment.

Experiment data | Name of constant | Range of data generated | Unit of measurement |
---|---|---|---|

1 | The scale of customers’ demand | 100–10,000 | Piece |

2 | The capacity of plants | 100–10,000 | Piece |

3 | Manufacturing costs per unit of product | 5–18 | RMB/piece |

4 | Operation costs per unit of product in DCs | 6–12 | RMB/piece |

5 | Transportation costs per unit of product | 0.8–2.2 | RMB/piece |

6 | Fixed costs of facilities construction | 3,000–10,000 | RMB/period |

The comparison of result of experimenting and performance is shown in Table

Comparison between running time and the optimal solution.

Problem | GA | BD | INS | |||
---|---|---|---|---|---|---|

Minimum cost | Running time | Minimum cost | Running time | Minimum cost | Running time | |

1 | 1025,640 | 3 | 1025,640 | 2 | 1025,580 | 2 |

2 | 3,710,342 | 6 | 3,710,342 | 5 | 3,709,573 | 4 |

3 | 10,855,375 | 189 | 10,855,375 | 141 | 10,835,485 | 92 |

4 | 24,779,022 | 463 | 24,779,022 | 399 | 24,778,267 | 351 |

5 | 46,520,863 | 655 | 47,526,986 | 592 | 47,513,097 | 521 |

6 | 678,362,757 | 1255 | 678,362,922 | 1202 | 678,354,052 | 978 |

7 | 138,225,390 | 2231 | 138,226,758 | 1931 | 138,210,352 | 1429 |

8 | 177,462,355 | 3732 | 177,462,355 | 3211 | 176,677,827 | 2534 |

9 | 402,100,473 | 5002 | 402,100,473 | 4887 | 402,080,659 | 3997 |

10 | 482,395,459 | 6974 | 482,395,459 | 6499 | 482,211,954 | 6240 |

Deviation comparison of INS with GA and BD.

Problem number | Percentage deviation from GA | Percentage deviation from BD | Minimal and maximal deviation from GA | Minimal and maximal deviation from BD |
---|---|---|---|---|

1 | −0.00058 | −0.00058 | ||

2 | −0.00207 | −0.00207 | ||

3 | −0.18000 | −0.18000 | ||

4 | −0.00030 | −0.00030 | −0.00030 | −0.00030 |

5 | −0.00169 | −0.00292 | ||

6 | −0.00128 | −0.00014 | ||

7 | −0.00109 | −0.00118 | ||

8 | −0.3900 | −0.3900 | −0.3900 | −0.3900 |

9 | −0.00049 | −0.00049 | ||

10 | −0.00380 | −0.00380 |

The deviation of the INS from GA and BD is tiny, which indicates that the NFP model is almost equivalent to the MIP model. From the point of time, as for small-scale experiments, all the performance of algorithm is shown in Figure

Algorithms performance comparison for small-scale network problem.

Algorithms performance comparison for large-scale network problem.

Comparing the algorithms performance for small-scale problem, it is interesting that, for the problem with the arc scale less than 1200, the INS performs much better than GA and BD. For the problem with the arc scale more than 1500, although the INS also performs better than others, its running time yield is greater than others with the expanding of network scales. More numerical studies should be conducted to analyze the efficiency of NFP and INS for small-scale SCND problem.

While for the algorithms performance for large-scale problem, it is noteworthy that the benefits of the INS appear to be somewhat remarkable and stable. The experimental result indicates that the NFP and INS are particularly suitable for solving large-scale SCND problem.

For a manufacturer-center multiproduct two-layer supply chain, the SCND problem is involved with the DCs location and capacity design decision and the initial distribution planning decision. In this paper, we have first proposed a classic MIP (mixed integer programming) model to solve such problem, and then we developed a transformation mechanism for modeling such problem as a network flow problem. After that, the NFP (network flow problem) model has been formulated. Since the NFP model could be solved as a minimal cost flow problem. The solution procedures and corresponding network simplex algorithm (INS) are designed. To verify the effectiveness and efficiency of the NFP model and algorithms, the performance measure experimental has been conducted for 10 various-scale problems, in which the plant’s capacity, the customer demand, and the relevant costs data are randomly generated in a reasonable region. The experiments and result showed that, comparing with MIP model solved by GA and and BD, the NFP model and INS are even more efficient for both small-scale and large-scale problem; furthermore, the advantage of NFP model and INS is stable for solving large-scale SCND problem.

However, there are still many works that have to be accomplished in the future. In this paper, we have just initiated work on the former. Although this paper affords a new solving approach to SCND problem. It only dealt with certain customer demand and a two-layer supply chain. The uncertainty and random demand scenario and more complicated competitive SCND should be considered in the future. Besides, as mentioned previously, a tight relaxation mechanism for the DCs construction cost should be developed and more numerical experimental should be investigated in the future study.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors gratefully thanks for the editors’ work and anonymous reviewers for their helpful comments on an earlier draft of this paper. The authors also thank the financial support by the Fundamental Research Funds for the Central Universities (Grant no. SWJTU12CX114) and by the Soft science research funds of science and technology department of Sichuan Province (Grant no. 2014ZR0019).