# American Institute of Mathematical Sciences

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Multi-stage distributionally robust optimization with risk aversion
doi: 10.3934/jimo.2019117

## Multi-period hazardous waste collection planning with consideration of risk stability

 School of Economics and Management, Beijing University of Chemical Technology, Beijing 100029, China

* Corresponding author: Xiang Li

Received  January 2019 Revised  March 2019 Published  September 2019

Hazardous wastes are likely to cause danger to humans and the environment. In this paper, a new mathematical optimization model is developed for the multi-period hazardous waste collection planning problem. The hazardous wastes generated by each source are time-varying in weight and allow incomplete and delayed collection. The aim of the model is to help decision makers determine the weight of hazardous wastes to collect from each source and the transportation routes of vehicles in each period. In the developed model, three objectives are considered simultaneously: (1) minimisation of total cost over all periods, which includes start-up fee of vehicles, transportation cost of hazardous wastes, and penalty fee for the delayed collection; (2) minimisation of total transportation risk posing to the surrounding of routes over all periods; and (3) even distribution of transportation risk among all periods, also called risk stability. The developed multi-objective model is transformed into a single-objective one based on the weighted sums method, which is finally equated to a mixed 0-1 linear programming by introducing a set of auxiliary variables and constraints. Numerical experiments are computed with CPLEX software to find the optimal solutions. The computational results and parameters analysis demonstrate the applicability and validity of the developed model. It is found that the consideration of the risk stability can reduce the total transportation risk, the uneven distribution of the transportation risk among all periods, and the maximum number of vehicles used, though increasing the total cost to some extent.

Citation: Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019117
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##### References:
The results of period 1
The influence of $\lambda_3$
The influence of $W$
The weight of hazardous wastes generated by each factory in each period (Tons)
 Period Factory 1 2 3 4 5 6 7 1 28 19 21 24 32 20 22 2 19 27 23 16 25 11 14 3 12 20 15 10 14 13 17
 Period Factory 1 2 3 4 5 6 7 1 28 19 21 24 32 20 22 2 19 27 23 16 25 11 14 3 12 20 15 10 14 13 17
The transportation cost (＄/Ton) and the number of population exposed (Pop/Ton) of transporting a unit weight of hazardous wastes from one node to another
 Node 0 1 2 3 4 5 6 7 0 0/0 8/7 10/10 11/3 6/11 11/13 8/4 10/14 1 0/0 17/9 17/6 7/8 14/11 3/11 11/17 2 0/0 2/7 16/2 9/4 16/12 12/10 3 0/0 17/8 7/11 15/6 10/13 4 0/0 17/3 9/13 15/12 5 0/0 11/16 4/12 6 0/0 8/13 7 0/0
 Node 0 1 2 3 4 5 6 7 0 0/0 8/7 10/10 11/3 6/11 11/13 8/4 10/14 1 0/0 17/9 17/6 7/8 14/11 3/11 11/17 2 0/0 2/7 16/2 9/4 16/12 12/10 3 0/0 17/8 7/11 15/6 10/13 4 0/0 17/3 9/13 15/12 5 0/0 11/16 4/12 6 0/0 8/13 7 0/0
The computational results
 $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 0$ $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 5$ Period 1 Period 2 Period 3 Period 1 Period 2 Period 3 No. of vehicles 6 4 4 5 5 5 Transportation cost(＄) 6373 5278 4704 5399.8 5134.7 5445 Penalty fee(＄) 0 0 0 1415.7 1250 0 Transportation risk (Pop) 2442 2279 1726 1971.7 1971.7 1955 Total cost(＄) 16355 18645.2 Total transportation risk (Pop) 6447 5898.4
 $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 0$ $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 5$ Period 1 Period 2 Period 3 Period 1 Period 2 Period 3 No. of vehicles 6 4 4 5 5 5 Transportation cost(＄) 6373 5278 4704 5399.8 5134.7 5445 Penalty fee(＄) 0 0 0 1415.7 1250 0 Transportation risk (Pop) 2442 2279 1726 1971.7 1971.7 1955 Total cost(＄) 16355 18645.2 Total transportation risk (Pop) 6447 5898.4

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