Mathematical Lower Bounds for City Logistics Distribution Network: Comparison
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Please note this is a comparison between Version 1 by zhiqiang niu and Version 2 by Dean Liu.

By considering the intra-echelon connection of facilities within the same layer of echelon, rwesearchers propose a new distribution network design model by reformulating the classical quadratic assignment problem (QAP). To minimize the overall transportation costs, the proposed model jointly optimizes two types of decisions to enable agile distribution with dynamic “shortcuts”

  • multi-echelon distribution network
  • lateral-transshipment
  • quadratic assignment problem

1. Basic Types of City Logistics Distribution Networks

A simple distribution network usually consists of three types of facilities/nodes, i.e., warehouses/plants, DCs, and delivery stations/customers[1]. City logistics, as warehouses and DCs occupy relatively large areas, are usually located in suburban areas. Meanwhile, delivery stations are widely distributed across the city to guarantee the fast delivery of orders.
According to the volumes of goods, fulfillment services, and types of goods, different distribution network structures should be designed and continuously improved. Generally, there are three basic types of network structures, as shown in Figure 1. For the distribution network with regional DCs in Figure 1a[2][3], warehouses and DCs are fully connected and each delivery station can only be served by one DC. In the case of the distribution network shown in Figure 1b with fully covered DCs, warehouses are allowed to connect specific DCs, while each DC can connect to all delivery stations. The last type of structure shown in Figure 1c implements the idea of the consolidation of goods to the maximum extent. In this situation, warehouses and DCs do not need to supply all their downstream facilities, but they are required to ensure that demand can always be met by transfer transportation between DCs. It should be noted that there are no extra connections (in comparison to the basic skeleton model) in most analytical studies, while multiple coverages are allowed in practice[4][5][6].
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Figure 1. Different types of distribution networks.

2. Distribution Network Design Problem and Related Models

The distribution network design (DND) problem is concerned with the decisions regarding the number of facilities and their optimal locations and facility capacity allocation[7][8][9][10][11][12]. It can also be viewed as a variation of the facility location-allocation or the production-distribution problem, which are reviewed in this subsection.
The coordination of distribution network design and planning has been addressed by some studies, which aim to integrate a number of different layers of supply chain (SC). For example, ref. [13]designed an integrated model considering production and distribution functions in a two-echelon system on a just-in-time (JIT) basis. Focusing on the production process in supply chain, ref. [14]constructed a continuous flexible process network model to maximize the operating profit. Ref. [15] set up a bi-objective model to minimize costs and the delay for the JIT delivery in a three-echelon supply chain. To provide a system-level optimized supply network, [4] attempted to solve the problems jointly in the entire supply network, including network design, production quota assignment, production planning, capacity planning for various facilities, and distribution planning. Ref. [16]investigated the effectiveness of production and distribution integration by a computational study under different logistic environments. In [17], the configuration of a production and distribution network needs to be optimized; there are two types of constraints (namely operational and financial constraints) in the model. Considering process uncertainty and robustness, ref. [18]studied the design and planning of supply chain networks involving multi-echelon, multi-product, and multi-period situations. Ref. [2]presented mathematical models to optimize inventory control and facility locations for a four-echelon supply chain network. With environmental considerations, some studies designed more realistic production-inventory models [19][20]. Ref. [6]developed a nonlinear mixed-integer programming model to optimize multi-echelon sustainable production-distribution supply networks under carbon emission policies. Aiming to increase the total value of a company by configuring and controlling all parts of a supply chain, [21]proposed a three-echelon, multi-commodity, and multi-period model for tactical and strategic decision-making. To ensure a responsive and resilient supply chain network, [22]addresses a multi-period supply chain (SC) network design problem where customer demands depend on the delivery lead-times of the facilities serving them. Interested readers are further referred to a number of excellent review papers on integrating production and distribution in supply chain management [8][23][24][25][26][27].

3. Solution Algorithms

In multi-echelon distribution network design and planning problems, the combined multi-layer decisions could dramatically increase the size of the search space, especially when additional real-world factors (the capacity, costs, uncertainty, etc.) are included. As a result, the design of algorithms for real-life instances has been challenging for practical applications and theoretical research. Given its inherent multi-layer structure, it is beneficial to adopt decomposition methods that can simplify the original problems and solve them efficiently. The Lagrangian relaxation-based method[28] is one of the most frequently used decomposition techniques. For instance, [29] used Lagrangian relaxation to decompose a location-inventory problem and provide a lower bound rule, then the branch-and-bound algorithm was used to obtain feasible solutions. To solve a distribution planning problem, [30]presented a Lagrangian substitution-based solution approach to transform the original nonlinear model into a mixed-integer linear programming model with univariate (solvable) concave models. Ref. [31]also developed a Lagrangian relaxation solution framework to decompose the network design problem into closely related knapsack and time-dependent least cost path problems. Although there are other decomposition-oriented applications in the field of supply chain management[3][4][13][32], theoretically rigorous and computationally reliable decomposition techniques are much needed for different complex scenarios[6]. Comparatively, commonly used heuristic methods aim to find a close-to-optimal solution rapidly in integrated production-distribution problems. To name a few: adapted imperialist competitive algorithm (AICA), variable neighborhood search (VNS) algorithm[7], genetic algorithm (GA)[13][14], and ant colony (AC) algorithm[9]. Ref. [20]utilized the VNS, Tabu Search (TS), Keshtel Algorithm (KA), Water Wave Optimization (WWO), and Particle Swarm Optimization (PSO) to solve a tri-level location-allocation model for forwarding/reverse supply chain systems. Furthermore, different commercial software packages are also developed to solve production-distribution planning problems[33][2][17][34].
Table 1 compares key modeling components in some closely related literature. Most multi-echelon DND models in the existing literature follow a linear structure in which lateral-transshipments within the same echelon is not considered[7][8] or there is no single sourcing strategy[35]. In the work of [2], they used a general linearization technique in quadratic assignment problems to address the quadratic term in the objective function. TheOur proposed approach is developed from the perspective of quadratic assignment problems, with more emphasis on the problem decomposition scheme and branch-and-bound algorithms with domain-specific lower bound rules.
Table 1. Comparison of key modeling components in some closely related literature.
Publication Number of Echelons Model Problem Decomposition Schemes Solution

Algorithms
[13] 2 MIP, linear Lagrangian relaxation LR
[4] 7 MIP, linear Sequential; Lagrangian relaxation LR; GA
[16] 2 MIP, linear Two-phase heuristic LS
[14] 3 MIP, linear - GA
[17] 3 MIP, linear - CPLEX solver
[3] 3 0–1 IP, linear Lagrangian relaxation LR
[2] 4 CQMIP, nonlinear - CPLEX solver
[7] 9 MIP, linear - AICA; VNS
[18] 3 MIP, linear - CPLEX solver; Simulation
[20] 4 MIP, nonlinear Nested approach VNS; TS; PSO; KA; WWO
[5] 3 MIP, nonlinear Sequential Heuristic
[35] 4 MIP, linear Sequential Heuristic; GAMS
This respapearch 3 0–1 IP, nonlinear Two-stage decomposition via cost estimation BB; ALNS
Model: CQMIP—conic quadratic mixed-integer programming; IP—integer programming; MIP—mixed-integer programming. Solution algorithms: LR—lagrangian relaxation; GA—genetic algorithm; AICA—adapted imperialist competitive algorithm; VNS—variable neighborhood search; LS—local search; TS—tabu search; PSO—particle swarm optimization; KA—keshtel algorithm; WWO—water wave optimization; BB—branch-and-bound.

References

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