An Optimization Model of Reverse Logistics Network Design

: With the increasing development of China’s economy, product and materials of reverse logistics is also increasing. Reverse logistics network design is of great significance to reduce cost of reverse logistics enterprises and society. In this paper, 0-1 mixed integer programming is used to establish a reverse logistics network optimization model which takes into account establishment of fixed facilities and flow between different logistics nodes. The objective function is profit maximization, and the constraints contain the processing capacity of facilities vehicle loading capacity. Lingo 14.0 is used to solve the model. The results show that the application of the model can improve the profits of reverse logistics enterprises on the basis of optimizing logistics facility allocation and material flow.


Introduction
China's output of industrial products ranks first in the world. While producing a large number of industrial products, more and more waste products are produced. Therefore, design and planning of waste product logistics network is important in current logistics research. In addition, due to equirements of "carbon neutralization", reverse logistics enterprises should not only reduce logistics costs, but also reduce carbon emissions. In reverse logistics network design, Waste fire extinguisher (WFE) plays important role.
Many scholars study reverse logistics network design. Mutha A and pokharel [1] established a mixed integer programming model for dealing with the recovery and return of products with different structures. Das K and Chowdhury A H [2] studied the modular structure of the product to simplify the recycling process. Qiang [3] built a production planning model to evaluate the benefits of re-manufacturing in closed-loop supply chain. Liao T Y [4] established a reverse logistics network model with mixed integer programming considering the return quality in the model, and solved the model with hybrid genetic algorithm. Kara and Onut [5] studied the logistics network planning combining forward and reverse logistics in the field of paper industry. Alshamsi A and Diabat A [6] established a reverse supply chain network considering different return quality. Figure 1 depicts the networks of WFE supply chain.  So far, the research on reverse logistics mainly aims at cost minimization and less focuses on profit maximization. In addition, vehicle scheduling problem is rarely considered in the design of reverse logistics network. This paper aims to make up for the shortcomings of the current research and establish reverse logistics network model with the goal of profit maximization, and vehicle scheduling problems between nodes are embedded in the model. The example analysis shows that this model can effectively improve the efficiency of reverse logistics enterprises through reasonable network node setting and vehicle scheduling between nodes. CO2j: Grams of carbon dioxide emitted by vehicle j driving one kilometer.

Parameters
g FIXC : Cost of establishing and putting processing center g into operation, that is, fixed cost of processing center g. g VAC : Cost of handling a WFE in processing center g, that is, variable cost of processing center g. Constraint(12) indicates flow of WFE into processing center g and out to secondary market i.
Constraint (13) indicates flow of WFE/its components into processing center g and out to a recycling center k.
Constraint (14) indicates flow of WFE/its components into processing center g and out to a destruction center h.
Constraint (15) indicates that only vehicle j used in the whole logistics process, it can be used in the transportation process from processing center g to secondary market i/recycling center Constraint (16) indicates that only vehicle j used in the whole logistics process, it can be used in the transportation process from processing center g to destruction center h.
Constraint (17) indicates that only vehicle j used in the whole logistics process, it can be used in the transportation process from processing center g to recycling center k.
Constraint (18) indicates that only vehicle j used in the whole logistics process, it can be used in the transportation process from processing center g to secondary market i. , , , 0 Constraint (19) indicates non-negative conditions.

Solution
Multi-objective programming is changed into single objective programming.

Numerical Examples
Zhejiang Qianjin logistics company is business with reverse logistics services of WFE. This paper analyzes a practical case of the company. Up to now, the company has 10 suppliers, 6 collecting points, 2 processing centers and 2 secondary markets, 2 destruction centers and 2 recycling centers are served by it. Cost of buying one vehicle is 23 yuan and cost of driving a kilometer per vehicle is 5 yuan. Cost of building a collecting center and putting it into operation is 150 yuan. Cost per ton of WFE processed by ta collecting point is 15 yuan. Cost of building a processing center and put it into operation is 125 yuan and cost of handling WFE per ton is 22 yuan. Loading capacity of a vehicle is 8 tons. Amounts of WFE given by a supplier is 70 tons, handling capacity of a collecting point/processing center is 400/500 tons. Selling price of a WFE in secondary market is 180 yuan. Distance from a collecting point to a processing center is illustrated in Table 1  ). Objective function value z1=59678 and z2=21795. Because the objective function is to minimize value of the objective function, we always selects the route with the lowest transportation cost to reduce transportation cost in the transportation plan. For example, for WFEs' transportation from collection point 2 to processing centers, transportation to processing center 2 is given priority, and for WFEs' transportation from processing center1 to secondary markets, transportation to secondary market2 is given priority, and for WFEs' transportation from processing center1 to secondary markets. Objective function value is minimized by selecting the route with lower transportation cost. If 1  and 2  are given different values, logistics cost ) 1 z ( and carbon emission )