Applied Mathematical Sciences, Vol. 4, 2010, no. 63, 3121 - 3132
Route Selection Based on Soft MODM Framework in Transportation of Hazardous Materials J. Jassbi Department of Industrial Management, Faculty of Economy and Management Science and Research Branch, Islamic Azad University, Tehran, Iran
[email protected] P. Makvandi1 Department of Industrial Management, Faculty of Economy and Management Science and Research Branch, Islamic Azad University, Tehran, Iran
[email protected]
Abstract Route selection could be a part of most transportation planning scenarios from transportation planning for metropolitan areas to transportation of goods and materials. Meanwhile, as the importance of the subject of transportation increase, the sensitivity of the route selection procedure increase meaningfully. Due to the risk that is tied with transportation of hazardous material (Hazmat Shipment), developing algorithms to find safest route of transportation of such these goods could be very vital. There are two basic concerns exist in hazmat shipment: first is to define the risk of transportation and second one is to develop proper algorithm for route selection that minimize the risk of transportation. In fact, hazmat shipment is a Multi Objective Decision Making (MODM) problem. In this paper a soft framework is proposed to define and solve the problem. Genetic Algorithm is used as a tool to solve the problem and different scenarios are tested to evaluate the proposed framework. Keywords: Route Selection, Hazardous Materials, Transportation Modeling, Genetic Algorithm, MODM, Soft Framework
1 . corresponding author e-Mail:
[email protected]
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1. Introduction Transportation of hazardous material (Hazmat Shipment) is a new emerging problem of the modern societies and specially industrialized countries. The main topic of such problems is to minimize the risk of the shipment. Although it is obvious that we want to minimize the risk of shipment but there is no comprehensive definition of risk while this definition robustly tied to our objectives. For example if the object country is in peace the risk of hazmat could be defined dependent on the accidents consequences but if the probability of terrorist attacks exists in the object country, then the robbery of hazmat products will be placed in a high potential risk category. In a hazmat shipment problem the main objective is to find the best path to transport hazardous materials such as explosives, gases, flammable liquids and solids, oxidizing substances, poisonous and infection substances, radioactive materials, corrosive substances and hazardous wastes. Because of the nature of these kinds of transportations and due to high impact of probable incidents that could be occurred in hazmat shipments, it is a sensitive area for decision making and it is very import for producers of hazmat products, consumers, transporters, local governments, insurance companies and specially for the people that live alongside of the selected path and also in the region of these transportations. The modeling of transportation problems is a popular application in Operation Research (OR). In most transportation planning models, the objective is to move products from origins to destinations at minimal cost. However, for hazmat shipment, a cost minimizing objective usually is not suitable. The risk associated with hazmat products make these problems more complicated (and also more interesting) than many other transportation problems. Reflection of the risk in hazmat transport models is not merely an academic exercise and in real world it has first priority for many other official entities. For example, the Hazardous Material Transportation Uniform Act (HMTUSA) of 1990 requires the US department of transportation to determine standards for designating the routes to be used for hazmat shipment [7]. Some researchers believe that research in this area focuses on two basic issues, the first one is related to assessing the risk induced on the population by hazmat vehicles traveling on various segments of the road network, and the second one involves the selection of safest routes to take [3]. Different works have been done in this area based on different criteria and various definition of risk. Abkovitz, for example, defined the risk based on transported substance and transport modality [1]. Environmental conditions is also had entered in definition of risk and formed the risk model by evaluation of environmental condition of hazmat shipment [12]. Erkut and Varter used incident probabilities on unit segments of a network and applied different risk models for a given route to determine the route risk. In this paper some similarity indices have been modified and base on these indices, the similarity between routes that are selected by different criteria are computed. It is also highlighted that one
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of the most popular risk model used by researchers and practitioners is societal risk, that’s being the product between the incident probability per unit length and the incident consequence, which is evaluated as the population area [7]. Designating hazardous materials routes in and through a major population center has been also considered by researchers of hazmat shipment to minimize the total risk of transportation [6]. Meanwhile, the role of information systems is vital to optimize the hazmat shipment problem. Using information systems such as GIS and DSS also has been considered as an alternative to find optimal path for hazmat shipment [14]. For repeated shipments where the arc incident in a route between a pairs of points probabilities are unknown, different strategies has been proposed such as use a mix of routes. A simple heuristic solution based on a shortest path algorithm and the method of successive averages is proposed. Connections to game theory provide useful insights into the nature of the solution [2]. Multi-objective optimization is also considered in hazmat shipment researches such as to minimize maximum population exposure and to minimize total travel time [5]. As it described before, the main problem is to find the route with minimum risk while the definition of the risk differs from case to case. For example we could minimize the accident probability but in this case the population exposure would be increased. In recent works, some models have been proposed for determining path of minimum total risk while guaranteeing equitable risk spreading [4,9,10]. In fact we face to a MODM (Multi Objective Decision Making) model with inconsistence objective, since the mentioned works did not use a soft MODM approach to form and solve the risk model. Novelties of proposed method Although hazmat shipment is a multi objective optimization problem in nature, but Due to our best knowledge, a soft decision framework based on risk definition has never been reported in literature before. In this paper we have tried to eliminate the absence of a flexible MODM approach and developed a soft MODM model to satisfy different and even inconsistence objectives based on priorities and weights. Although the weights of decision criteria in this problem could be determined based on weight assigning such as entropy, but enjoying dynamic nature of the proposed model, the weights could be changed depending on priorities of decision maker. There are different methods exist to solve MODM models such as goal programming, penalty method and also soft computing techniques. In this paper we have applied Genetic Algorithm as a branch of soft computing techniques to solve MODM model in hazmat shipment. Problem definition, methodology and Genetic Algorithm are presented in section 2. Section 3 is presented model development and also the result of developed model. At the end, the results are aggregated in conclusion section.
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P D Definition n and Metthodologyy 2. Problem blem definnition and methodologgy that hass been useed to solvee the probllem are Prob intro oduced in thhis section. Hazmat shhipment, riskk definitionn and Genettic Algorithm (GA) are addressed in this sectioon.
2.1 Hazmat Sh hipment Theere is no gloobal definitioon of risk but b in generaal, risk dealls with two bbasic subjeccts: 1- Probabiility of an undesired u evvent 2- Conseqquence of ann undesired event ng the traditional risk definition, d the risk of trransporting hazmat B oover a unit segment s Usin (succh as one kiilometer) caan be writtenn: R ABB = PAB C AB Where: ncident on thhe unit roadd segment A for hazm mat B and PAB is probabillity of an in n the neighbborhood asssociated C ABB is populattion along the unit road segmentt A within withh hazmat B . In this t field the impact arrea should be defined.. The impacct area is a circle withh impact radiius that the transporter of hazmat product p is llocated in itts center. Baased on the hazmat typee, this circlle and conssequently itts impact w would be deefined. Figuure1 illustraates this concept.
Figgure1- Impacct Area
Althhough the rrisk of incid dent in hazzmat shipmeent is very important bbut there arre other kindds of risks exist in thiis area suchh as robberry or terrorrist attacks. Due to theese new emeerging threaats, the objeective of rouute selectioons could bee changed iinconsistenttly. For exam mple, if we want to traansport radiooactive garbbage, what criteria c shouuld be used to
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select the best path? Minimum accident probability, minimum incident, safest path or the lowest populated path? The answer to this question leads us to select objectives to form the logical and mathematical model. In this paper we have selected the 6 following objectives and tried to combine them in unique MODM model. The selected objectives and the proper definition of each objective are presented in table1. It is obvious that any other criteria could be inserted to model by decision maker. Table1. The Objectives of path Selection Objective Shortest Travel Distance Minimum population Exposure Minimum Societal Risk Minimum Accident Probability Safest Road (minimum Robbery Rate) Minimum Terrorist Attacks
Calculation Criteria Path length The average population in impact area
L × AR × CR × Pd × π × r 2 (expected number of people to be impacted in one trip of hazmat truck) Average accident probability for the path Average Robbery probability for the path Average Terrorist Attacks probability for the path
Where:
L : Length of the path AR : Accident Rate (probability) on the path (per kilometer) CR : Conditional Release given an accident Pd : Population density in the neighborhood of the path (person per kilometer-sq) r : Impact Radius 2.2 Genetic algorithm Genetic Algorithm (GA) is a kind of search and optimized algorithm that have been produced from simulating biologic heredities and long evolutionary processes of creatures. It stimulates the mechanism of “survival competitions; the superior survive while the inferior are eliminated, the fittest survive.” The mechanism searches after the optimal subject by means of a successive iterative algorithm. From the late 80s, GA, as a new cross discipline which has drawn people’s attention, has already shown its increasing vitality in many fields [13]. GA has demonstrated substantial improvement over a variety of random and local search methods. The GA is based on the laws of natural selection in genetics. The principal idea is to search for optimal solution in a large population. It uses a fixed length binary string called a chromosome to represent a possible solution or individual for a given problem domain. Usually a simple GA consists of three operations: • Selection • Crossover • Mutation
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The population comprises a group of chromosomes from which candidates can be selected as a solution of a given problem. The initial population (set of possible solutions) consists of member solutions. Each one has a fixed member of randomly selected features from a feature pool. The fitness (a measure of appropriateness of the solution of each chromosome) is evaluated by fitness function. The selection operator selects chromosomes in the population based on fitness. Individuals with higher fitness have more chance for being selected as parents. A couple of selected chromosomes or parents are then crossovered and their information is exchanged to generate new chromosomes or offsprings. Mutation fillips the value of all or some bits of randomly selected chromosome. This operation increases exploration of search space and prevents converging to a local optimum point. The process runs cyclically until a stopping criterion is reached. Each cycle is called a generation. The key assumption of parent selection is to give preference to fitter individuals. Here we use the Roulette Wheel selection scheme [8]: 1. Computing cumulative fitness of individuals. 2. Generating a random number n between 0 and total fitness. 3. Returning the first individual that its cumulative fitness is greater than or equal to n. In this paper the Uniform crossover scheme is used. That is, each bit position 1 to L, is randomly picked from either of the two parents’ strings. This means that each bit is inherited independently from other bit and that there is, in fact, no linkage between bits. After crossover and mutation, there are two groups of candidates: the parents and the offsprings. Each of these groups contributes a fraction of new generation. We have used the elitist selection strategy to transfer a copy of a few fittest individuals in present generation to next generation to ensure crucial convergence of algorithm [11].While any successful application of genetic algorithm for solving a problem is greatly dependent on finding a suitable method for encoding the possible solutions to chromosomes, the creation for the appropriate fitness function is important to have a successful training process.
3. Model Development and Results In this paper we have formed a MODM model base on 6 different objectives that are presented in table1. Each of 6 different objectives could be determine by associated weights as w1 , w2 , w3 , w4 , w5 , w6 respectively. The presented model is dynamic and the proper codes have been written in MATLAB™ language that can get the priorities from decision maker, calculate the results and then select the best path base on input weights. Though, the decision maker could change the weights due to the situation and based on his/her priorities. The only constraint is:
Rou ute selection n based on soft MODM M frameworrk
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Where; is the weighhts of 6 diffferent objectives respecctively. Thee conceptuall framework k of the mo odel is illusttrated in figuure 2. As itt shown in figure f 2, the output of thhe model is the best path based onn decision maker’s m prioorities. The soft and work of thee model allows decisioon maker too revise thee priorities and the flexxible framew resuulted weightts based on output feeddback.
Figure2. Coonceptual moodel framew work
Herre, Genetic Algorithm is used as a tool to ssolve the MODM M moddel based on o input weights (prioriities). As itt is shown in table 2, eight pathhs between a proposedd Origin del. Desstination (O-D) pair witth determinned calculatiion criteria are used to feed to mod
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Table2 – Eight different paths between a proposed origin-destination pair
Path
No. of Edges
Length
population exposure
societal risk
1 2 3 4 5 6 7 8
2 2 2 3 3 3 4 4
1000 950 1050 1150 1050 1200 1500 1400
0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2
30000 30000 30000 35000 35000 35000 40000 40000
Accident Probability(Rate) in every 1000 km 0.00219 0.00219 0.00219 0.0026 0.0026 0.0026 0.00449 0.00449
Robbery Rate 0.095 0.095 0.095 0.07 0.07 0.07 0.05 0.05
Terrorism Attacks Rate 0.001 0.001 0.001 0.0005 0.0005 0.0005 0.00025 0.00025
Due to the dynamic nature and implementation of the model, the input weights could be subject of change. The input data have been preprocessed by normalizing each set. Then, the GA parameters have been set. Since GA is a heuristic algorithm, its parameters should be set based on the case problem. The parameters of the GA that used to solve MODM model are presented in table 3.
Table3- GA Parameters Parameters Population Type Population Size Selection Function Crossover Function Mutation Function Mutation Rate
Value Bit String 100 Roulette Single point Uniform 0.01
Now we are ready to check the model and select the best path for transportation of hazardous materials. Due to this fact that weights are determined based on decision maker’s priorities, different scenarios are considered here to run the model. The results are presented in table 4.
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Table 4- Results of Scenario test
Scenario
Weights Combination
Selected Path
Convergence Diagram Best: 0.35631 Mean: 0.35722 0.48
0.46
1
w1 = w2 = w3 = w4 = w5 = w6
5
Fitness value
0.44
0.42
0.4
0.38
0.36
0.34
5
10
15
20
25
30 35 Generation
40
45
50
55
60
Best: 0.41379 Mean: 0.41406
0.48
0.47
w4 = 0.1, w5 = 0.1, w6 = 0.1
0.46
5
Fitness value
2
w1 = 0.1, w2 = 0.3 , w3 = 0.3,
0.45
0.44
0.43
0.42
0.41
5
10
15
20
25
30 35 Generation
40
45
50
55
60
Best: 0.28182 Mean: 0.28418 0.46 0.44 0.42
w4 = 0.1, w5 = 0.3 , w6 = 0.3
0.4
8
Fitness value
3
w1 = 0.1, w2 = 0.1 w3 = 0.1,
0.38 0.36 0.34 0.32 0.3
5
10
15
20
25
30 35 Generation
40
45
50
55
60
Best: 0.28509 Mean: 0.28588 0.35
0.34
4
w1 = 0.1, w2 = 0.1, w3 = 0.1, w4 = 0.5 , w5 = 0.1, w6 = 0.1
5
Fitness v alue
0.33
0.32
0.31
0.3
0.29
5
10
15
20
25
30 35 Generation
40
45
50
45
50
55
60
Best: 0.2858 Mean: 0.28705 0.37 0.36 0.35
w4 = 0.3 , w5 = 0.1, w6 = 0.1
0.34
5
Fitness value
5
w1 = 0.3 , w2 = 0.1, w3 = 0.1,
0.33 0.32 0.31 0.3 0.29
5
10
15
20
25
30 35 Generation
40
55
60
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Best: 0.36495 Mean: 0.36495 0.45 0.44 0.43
w4 = 0.2 , w5 = 0.1, w6 = 0.2
0.42
5
Fitness value
6
w1 = 0.1, w2 = 0.2 , w3 = 0.2 ,
0.41 0.4 0.39 0.38 0.37 0.36
5
10
15
20
25
30 35 Generation
40
45
50
55
60
We have tested 6 different scenarios to select the path. It means that 6 different combinations of weights are used for path selection.
Conclusions Hazmat shipment is a problem of industrialized societies and due to sensitive nature of these kinds of problems many works have been done in this area. As it was mentioned in literature review, most of these works did not look at this problem as a flexible MODM model and their formulation and model formation, strongly tied with their definition of risk in hazmat problems. In this paper we had tried to propose a soft and flexible MODM approach and suitable dynamic GA based model to making proper decisions about path selection in hazmat shipment. Flexible definition of the problem and also the soft GA capabilities give us a new vision to hazmat shipment problem that let us to make decisions quicker in complex cases.
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[4] Curren J, Ratick S., (1995). A model to assess risk, equity and efficiency in facility location and transportation of hazardous materials, Location Science; Vol.3. pp.187201 [5] Erkut E., Glickman Theodore. (2007). Minimax Population Exposure in Routing Highway Shipments of Hazardous Materials. Journal Transportation Research Record: Journal of the Transportation Research Board, Issue Volume 1602 / 1997, pp 93-100. [6] Erkut E., Osman Alp. (2007). Designing a road network for hazardous materials shipments, Computers & Operations Research, Volume 34, Issue 5, Hazardous Materials Transportation. pp. 1389-1405 [7] Erkut E., Verter V, (1998), Modeling of transport risk of hazardous materials, Operation Research; Vol. 46, pp. 625-642 [8] Goldberg DE., Genetic algorithm in search, optimization and machine learning. (1989). Addison Wesley publishing Co. [9] Gopalan R, Patta R., Karwn MH,. (1990). The equity constrained shortest path problem. Computers and Operation Research, Vol. 17, pp. 297-307 [10] Konstantinos N. Androutsopoulos, Konstantinos G. Zografos, (2010). Solving the bicriterion routing and scheduling problem for hazardous materials distribution, Transportation Research Part C: Emerging Technologies, Volume 18, Issue 5, Applications of Advanced Technologies in Transportation: Selected papers from the 10th AATT Conference. pp 713-726 [11] Makvandi, P., Jassbi, J., Khanmohammadi, S., (2005). Application of Genetic Algorithm and Neural Network in Forecasting with Good Data, WSEAS Transactions on systems, April, Vol.4, pp. 337-343 [12] Patel MH, Horowitz AJ,.(1994). Optimal routing of hazardous materials considering risk of spill. Transportation Research.; vol.28A, pp. 119-132 [13] Sun Yanfeng, Wang Zhongtuo, (1995). Parallel Genetic Algorithm [J]. Systematic Engineering; 13(2):146. [14] Verter V, Kara BY. (2001). A GIS-based framework for hazardous materials transport risk assessment. Risk Anal. Dec; 21(6):1109-20.
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[15] William C. Frank, Jean-Claude Thill, Rajan Batta. (2000). Spatial decision support system for hazardous material truck routing, Transportation Research Part C: Emerging Technologies, Volume 8, Issues 1-6, February-December. pp. 337-359.
Received: August, 2010
