Genetic optimisation of a fuzzy distribution model

Transcription

1 The current issue and full text archive of this journal is available at Genetic optimisation of a fuzzy distribution model Enrique LoÂpez GonzaÂlez and Miguel A RodrõÂguez FernaÂndez Department of Economy and Business Management, University of LeoÂn, LeoÂn, Spain Genetic optimization 681 Keywords Transport, Distribution, Fuzzy sets, Supply chain, Decision making Abstract Distribution problems deal with distribution from a number of sources to a number of destinations Each source offers amounts of goods, while each destination demands quantities of these goods The object is to find the cheapest transporting schedule that satisfies demand without violating supply restraints In this paper we propose the use of fuzzy sets to represent the provisional information related to costs, demands and other variables Moreover, we suggest including the problem of shortest route for the distribution vehicles Finally, to solve this complex problem we propose using a genetic algorithm with a fuzzy fitness function 1 Introduction Many different optimisation problems can be formulated on networks For instance, we might be interested in finding the shortest path from one node to another, finding the cheapest way to connect all the nodes, or finding the best way to move objects through a network Network optimisation represents one of the most important areas of management science for several reasons (Camm and Evans, 1996; Heizer and Render, 1996) First, networks can be used to model many diverse applications such as transportation systems, communication systems, vehicle routing problems, production planning and cash flow analysis Second, managers accept networks more readily because they provide a visual picture of the problem under study Finally, networks have certain mathematical properties that allow management scientists to develop special algorithms that are able to solve much larger problems than other optimisation methods In this paper we aim to show a new approach to a classical network optimisation problem, the transportation problem, first by allowing the vehicles that can establish routes, and also permitting management with imprecise or vague information We propose using fuzzy sets theory (Zadeh, 1965; Kaufmann, 1975; Zimmermann, 1985), with the aim of being able to handle the uncertainty, which is a characteristic of decision-making processes in distribution problems Moreover, to optimise the distribution network we suggest using a genetic algorithm (GA) (Goldberg, 1989; Biethahn and Nissen, 1995; LoÂpez-GonzaÂlez et al, 1998; Herrera et al, 1999) The main reason for this is that GAs are heuristic optimisation methods which do not impose restrictions on posing the problem In this study, the algorithm is characterised by its use of a fuzzy fitness function that allows the evaluation of imprecise information International Journal of Physical Distribution & Logistics Management, Vol 30 No 7/8, 2000, pp # MCB University Press,

2 IJPDLM 30,7/8 682 In the light of the above, the next section shows a description of the problem to be solved The third section presents the fuzzy approach to this problem In the fourth section we introduce the genetic algorithm used to solve the aforementioned approach The fifth section offers a practical example of the distribution problem and, finally, we include some concluding remarks 2 The distribution problem We consider the problem of finding the cheapest means of shipping supplies from several origins to several destinations Origin points or sources can be factories, warehouses, or any other points from which goods are shipped Destinations are any points that receive goods To solve this problem we must determine how many goods are supplied from each source to each destination and what route the distribution vehicle must follow in order to minimize the total distribution costs Traditionally this problem has been analysed by two different decisions: first, obtaining the quantities to be shipped from each origin to each destination, minimising the costs; and then satisfying the demand, or transportation problem (Figure 1) On the other hand, finding the shortest route that one vehicle must follow to reach all the destinations, or the shortest route problem, is known as travelling salesman problem (Figure 2) In this paper we suggest combining both problems in order to obtain an optimal solution that minimises the distribution costs of the quantities supplied and the routes followed A distribution problem has a very large, sometimes infinite, number of feasible solutions Traditionally, this problem has partially reduced not permitting shipments among destinations This kind of problem can be solved with linear programming and other methods such as Stepping Stone, MODI, Houthtaker Method or Hammer Balas Method But the global approach, considering the vehicles distribution routes, is poorly studied (Turban and Meredith, 1991; Heizer and Render, 1996) Figure 1 Transportation problem

3 Moreover, some of the information considered is imprecise Costs, supplies and demands may be not known in a crisp numerical way (Camm and Evans, 1996) Therefore, in this paper we suggest using fuzzy sets to represent vague knowledge 3 Fuzzy distribution model With this method the companies must know how many commodities supply from each origin to each destination when the information held is not precise and a fuzzy representation in more accurate As we show below, the information necessary is related to: Destination or client demands The company must satisfy the client demands Sometimes this information is vague; therefore we consider a more realistic representation using fuzzy sets So, for n destinations we could have: ~C ˆ f ~ C 1 ; ~ C 2 ; ; ~ C n g Genetic optimization 683 Number of sources and supply capacity We suppose that each source has a supply capacity, normally known in a strict way So for m sources we could have: F ˆ ff 1 ; F 2 ; ; F m g Vehicle capacity We consider that in each source the company has a vehicle used to distribute the commodities Therefore, we must take into account the vehicle capacity, because the number of journeys will depend on it For the aforementioned m sources, the vehicle capacities could be: V ˆ fv 1 ; V 2 ; ; V m g Transporting costs of empty vehicles Sometimes, the vehicles must return to their respective sources due to each one having distributed its capacity Moreover, the last journey is made between the last client supplied and the source of the vehicle Thus, we must take into account Figure 2 Travelling salesman problem

4 IJPDLM 30,7/8 684 what the shipping cost is when the vehicle is empty This normally depends of the distance (routed kilometres) So, for m sources, the transporting cost of empty vehicles could be: cv ˆ fcv 1 ; cv 2 ; ; cv m g Increasing cost for each unit of commodities shipped When the vehicles distribute the commodities the transporting costs increase with each unit shipped As well, this increasing cost depends of the distance travelled Moreover, the knowledge of this cost could not be precise, and representing it using fuzzy sets could improve the results obtained Thus, for m vehicles the increasing cost could be: ~4cv ˆ f 4cv ~ 1 ; 4cv2 ~ ; ; 4cvm ~ g Distance between sources and destinations The distribution is made from the sources to the destinations or clients So, it is necessary to know the distance between them 8 9 D 11 ; D 12 ; ; D 1n >< D 21 ; D 22 ; ; D 2n >= D ˆ >: D m1 ; D m2 ; ; D mn >; Distance between destinations As we mentioned before, the vehicles can establish routes from the source to the several destinations that are supplied Thus, it is necessary to know the distances between the destinations 8 ; D12 0 ; ; 9 D0 1n >< D21 0 ; ; ; D0 2n >= D 0 ˆ >: D 0 n1 ; D0 n2 ; ; >; According to this approach, we are trying to reach the real distribution problem Thereafter, we need a tool capable of solving this complex problem or able to give a reasonable solution In this paper we propose using a genetic algorithm with a fuzzy fitness function Although we describe the use of fuzzy sets, we suggest representing all the fuzzy information as trapezoidal fuzzy numbers (TFN), where the possibility distribution is shown in Figure 3

5 Genetic optimization 685 Figure 3 The use of trapezoidal fuzzy numbers to solve distribution problems The TFN are characterised by four values: a 1 ; a 2 ; a 3 ; a 4 a 1 indicates minor value that the variable can take; a 2 ; a 3 Š indicates an interval where the variable has the maximum level of belief; and a 4 indicates the upper value that the variable can take The reason is that these fuzzy numbers are easily understood and have a good suitability to different real situations, particularly to economic and entrepreneurial estimates (Kaufmann and Gil-Aluja, 1986) 4 Genetic algorithm and fuzzy distribution problems Some authors have applied genetic algorithms to transportation problems (Vignaux and Michalewicz, 1991; Mizunuma and Watada, 1995) In many cases they solve the problem by reducing their complexity or using precise knowledge for the variables In our approach, we suggest a fuzzy representation (TFN) of the information on held and a less restrictive model Due to this, the GA must determine the quantities supplied from each source to each destination and the routes travelled for the vehicles For both objectives the criterion is the same: minimising the total distribution costs 41 Genetic algorithms GAs are search algorithms which use principles inspired by natural genetics to evolve solutions to problems (Holland, 1975) The basic idea is to maintain a population of chromosomes, which represents candidate solutions to the concrete problem being solved, which evolves over time through a process of competition and controlled variation GAs have got a great measure of success in search and optimisation problems A GA starts off with a population of randomly generated chromosomes (solutions), and advances toward better chromosomes by applying genetic

6 IJPDLM 30,7/8 686 operators modelled on the genetic processes occurring in nature During successive iterations, called generations, chromosomes in the population are rated for their adaptation as solutions and, on the basis of these evaluations, a new population of chromosomes is formed using a selection mechanism and specific genetic operators such as crossover and mutation An evaluation or fitness function must be devised for each problem to be solved Given a particular chromosome, a possible solution, the fitness function returns a single numerical fitness, which is supposed to be proportional to the utility or adaptation of the solution represented by that chromosome GAs may deal successfully with a wide range of problem areas, particularly in management applications The main reasons for this success are: GAs can solve hard problems quickly and reliably; GAs are easy to interface to existing simulations and models; GAs are extendible; and GAs are easy to hybridise All these reasons may be summed up in only one: GAs are robust GAs are powerful in difficult environments where the space is usually large, discontinuous, complex and poorly understood They are not guaranteed to find the global optimum solution to a problem, but they are generally good at finding acceptably good solutions to problems quickly These reasons have been behind the fact that, during the last few years, GA applications have grown enormously in many fields The basic principles of GAs were first laid down rigorously by Holland (Holland, 1975), and are well described in many books (eg Goldberg, 1989; Michalewicz, 1996) 42 A genetic algorithm for fuzzy distribution problems To solve the fuzzy distribution problem we propose to use a GA with the following components Genetic representation The solutions to the problems are a set of distribution quantities to be shipped from each source to each destination and the routes followed by the vehicles of each source To codify this solution we propose to use two matrices: one that contains the quantities distributed from each origin to each destination, and the other that represents the path followed for the various vehicles With regard to the codification of quantities matrix, an example of five sources that must supply four possible destinations is shown in Table I, that indicates the quantities supplied from each source to each destination On the other hand, the codification of vehicles routed could be used as shown in Table II This indicates the position of each destination in the vehicle routes Moreover, in order to generate useful solutions, the company must decide the covering level of each client's fuzzy demand The TFN can be identified as

7 possibility distributions So, the company establish the risk of not covering the demand of each client According to this, the quantity matrix represents available solutions for the decision Fuzzy fitness function The fitness function must assign more value to these solutions that are good solutions to the distribution problem For this we propose to calculate the total distribution cost that the two matrices suppose The steps are as follows At the beginning, each vehicle departs from its source full of commodities The first destination, established by the routes matrix, determines the cost of this first journey Once the first destination is satisfied (with one or more journeys) the second destination is attempted If the vehicle is empty during the route, it returns empty to the source to fill its capacity Also, when all the destinations are satisfied then the vehicle goes back to its origin, completing the route The sum of all these transporting costs (departure from origin fully laden attempt first destination, attempt second, goes back to origin when empty, etc) would be a part of the solution To obtain the total cost we must add the cost of the other distribution routes contained in the matrices and established for other vehicles Due to the fact that some of the information can be represented by fuzzy sets, we suggest using the operations designed for them (Dubois and Prade, 1980) Moreover, in cases when the operations could not produce a TFN, we approximate the result as one of this (Kaufmann and Gil-Aluja, 1986), assuming some error To set up a hierarchy among the solutions, the proposal is to use the fuzzy distance (Kaufmann and Gil-Aluja, 1987), based on Hamming Distance We Genetic optimization 687 S 1 1 Source 1 Source 2 Source 3 Source 4 Source 5 Total Destination Destination Destination Destination Total Table I An example of five sources that must supply four destinations in a codification of quantities matrix S 2 1 Source 1 Source 2 Source 3 Source 4 Source 5 Destination Destination Destination Destination Table II An example of five sources that must supply four destinations in a codification of vehicles matrix

8 IJPDLM 30,7/8 688 calculate the distance from the origin B ~ (singleton 0) to each solution fuzzy cost, which is defined as follows: d ~ A; ~ B ˆ 1 aˆ0 ja 1 a B1 a j ja2 a B2 a j da where A 1 a ; A2 a Š is confidence interval of A ~ at the significance level a The more accurate solutions will have a lower distance Thus for obtaining the fitness value we propose calculating the inverse of the distance of each solution Selection process We propose to use roulette wheel ranking (Goldberg, 1989) to select the individuals that will be the ``parents'' of the next generation Crossover operator Traditional crossovers cannot be used, because the solutions are two matrices that have to accomplish some constraints Due to this we propose two different crossovers for each matrix, as follows: (1) Distribution quantities matrices To combine the information of these matrices obtained from two ``parents'' we have used the crossover proposed by Vignaux and Michalewicz (1991) for the transportation problem The process is as follows: Continuing with the aforementioned example of five sources that must supply four destinations, imagine that after the selection process we have two quantity matrices to be crossed, as in Table III Using these ``parents'', we obtain two temporal matrices: one containing the integer average of both ``parents'' (MED), and another that includes a 1 in those cells that have been rounded to obtain the average, and a 0 in the remaining (RED) Thus we obtained information as shown in Table IV for the example The RED matrix has some interesting properties One of them indicates that the number of ones in each row or column is a par one In other words, the source capacity values and the destination Source 1 Source 2 Source 3 Source 4 Source 5 Total S1 1 Destination Destination Destination Destination Total Table III Examples of two quantity matrices which need to be crossed S2 1 Destination Destination Destination Destination Total

9 demand values are par integers Thus, we can divide the RED matrix into two matrices (RED1 and RED2), which accomplish the following condition: RED ˆ RED1 RED2; being equal to the supplying capacity of RED1 and RED2 for each source, and, also, the demands of RED1 and RED2 for each destination For the example, we could obtain the following matrices RED1 and RED2 as shown in Table V Finally, the offspring is obtained as follows: S1 0 S2 0 ˆ MED RED1 ˆ MED RED2 Genetic optimization 689 Then, for the example, the ``sons'' could be as in Table VI Source 1 Source 2 Source 3 Source 4 Source 5 Total MED Destination Destination Destination Destination Total RED Destination Destination Destination Destination Total Table IV Two temporal matrices demonstrating MED and RED Source 1 Source 2 Source 3 Source 4 Source 5 Total RED1 Destination Destination Destination Destination Total RED2 Destination Destination Destination Destination Total Table V The division of the RED matrix into RED1 and RED2

10 IJPDLM 30,7/8 690 Table VI An example of the ``sons'' outcome Source 1 Source 2 Source 3 Source 4 Source 5 Total S1 10 Destination Destination Destination Destination Total S2 10 Destination Destination Destination Destination Total (2) Vehicle routes matrices In addition, we must cross the route matrices of both ``parents'' To do this, we select one source and interchange between the ``parents'' the routes that the vehicle of this origin must follow If we have to cross the following route matrices of the aforementioned problem, first we need to randomly select one source (source 2) as shown in Table VII Second, we need to interchange the list of routes followed (see Table VIII) Mutation operator The intention of this operator is to introduce diversity into the solutions In trying to do this we must distinguish between the two types of matrices (1) Distribution quantities mutation We propose using a special mutation method designed to keep the solutions resulting as feasible ones The steps are as follow: Step 1 We select one source (source A) and one destination (destination A) from the matrix This destination must receive a non-zero quantity from the source selected For the example, if we Source 1! Source 2 Source 3 Source 4 Source 5 Table VII Examples of two vehicle route matrices which need to be crossed S1 2 Destination Destination Destination Destination S2 2 Destination Destination Destination Destination

11 Source 1 Source 2 Source 3 Source 4 Source 5 S1 20 Destination Destination Destination Destination S2 20 Destination Destination Destination Destination Genetic optimization 691 Table VIII The interchange of the list of routes followed mutate the first matrix obtained after the crossing process, the source and destination could be as shown in Table IX Step 2 We generate a random number between 0 and the quantity assigned to this destination For the example it could be 20 Step 3 We search in the remaining sources one quantity assigned to a different destination which is bigger than the random number generated before (source B and destination B) (see Table X) Step 4 We discount the random number to the assigned amounts from source A to destination A and from the source B to destination B After that, we add the random number to the assigned amounts from source A to destination B and from source B to destination A Source 1 Source 2 Source 3 Source 4 Source 5 Total S1 1 Destination Destination Destination Destination Total Table IX Distribution quantities mutation (step 1) Source 1 Source 2 Source 3 Source 4 Source 5 Total S1 1 Destination Destination Destination Destination Total Table X Distribution quantities mutation (step 3)

12 IJPDLM 30,7/8 692 According to this, for the example, the mutated matrix could be as shown in Table XI Finally, with the aforementioned process we can mutate the quantity matrices, maintaining them as feasible ones (2) Vehicle routes mutation On the other hand, for mutating the route matrix we propose an interchange method First, we select randomly one source After that, we select two destinations and interchange their position in the route followed for the source vehicle Halt criteria for the best solution search The proposal is for the algorithm to go through a number of generations specified by the user until the best solution is found Moreover, in order not to lose good solutions, the characteristic termed ``eâlitism'' (Goldberg, 1989) has been introduced This procedure consists of keeping the best individual from a population in successive generations unless and until some other individual succeeds in doing better in respect of suitability In this way, the best solution for a previous population is not lost until outclassed by a more suitable solution As explained, application of the model proposed here allows solving the distribution problem under uncertain environmental conditions 5 Experiment: an example of a practical application In this section we present an example that deals with the distribution problem for a production furniture company To do that we divide this section into subsections: one which poses the problem and other containing the proposed solution using the GA 51 Introduction to the problem Let it be imagined that a furniture company supplies tables to several clients To obtain the tables they have four factories and each one has its own distribution vehicle In Table XII we show the production capacity of each factory and the capacity, transporting cost empty and increase for transporting unit of their respective vehicles Moreover, the company supplies eight clients (furniture shops) They are located in different places The estimated demand of each one is shown in Table XIII Source 1 Source 2 Source 3 Source 4 Source 5 Total Table XI Distribution quantities mutation (step 4) S1 10 Destination Destination Destination Destination Total

13 In addition, we know the distance between the factories and the clients in kilometres and the distance between these clients, shown in Tables XIV and XV respectively According to the aforementioned information, the company wants to know the cheapest solution To do this, it is necessary to answer two questions: (1) How many tables are supplied from each source to each destination? (2) What are the routes that each vehicle must follow? Genetic optimization Designed GA application The problem explained can be solved using the GA introduced in Section 4 So, for the purposes of operational model application, the parameters used in finding the solution were: Factory Production capacity Vehicle capacity Transporting cost empty ($/Km) Transporting cost increasing for product unit ($/Km) A 1, (2; 4; 4; 5) B 2, (4; 5; 5; 6) C 3, (1; 2; 2; 3) D 1, (3; 3; 3; 3) Table XII Statistical information of the company in the experimental application of the proposed model Clients Table estimated demand Client one (450; 500; 500; 600) Client two (1,450; 1,500; 1,500; 1,650) Client three (600; 800; 800; 950) Client four (1,500; 1,600; 1,600; 1,650) Client five (600; 600; 600; 600) Client six (450; 500; 500; 550) Client seven (950; 1,000; 1,000; 1,050) Client eight (450; 500; 500; 600) Table XIII Estimated demand of experimental company Distance (kilometres) Source A Source B Source C Source D One Two Three Four Five Six Seven Eight Table XIV Distance between factions and clients

14 IJPDLM 30,7/8 Beginning crossover probability: 50 percent Ending crossover probability: 50 percent Beginning mutation probability: 30 percent Ending mutation probability: 20 percent 694 Number of generations: 100 Number of individuals: 10 It should be pointed out that the use of a high mutation probability was motivated by the need to bring diversity into the few chains, since if this were not so all that would be obtained would be the best combination of those initially considered who passed the first selections In the practical example analysed, the final solutions are shown in Tables XVI and XVII The distribution cost that we expect for this solution is: Total fuzzy distribution cost (TFN) = ( ; ; ; ) The above TFN means that the lower expectation cost incurred will be between and , the most probable value being Finally, the graph that shows the evolution of the cost of the best solution (y axis) on each generation (x axis) is displayed in Figure 4 This explains the good performance of the algorithm in the best solution searching Distance (kilometres) One Two Three Four Five Six Seven Eight Table XV Distance between clients One ± Two 40 ± Three ± Four ± Five ± Six ± Seven ± 45 Eight ± Distribution quantities Source A Source B Source C Source D Total Table XVI Solution through distribution quantities matrix One Two ,500 Three Four ,600 Five Six Seven Eight Total 1,000 2,000 3,000 1,000 7,000

15 Vehicle routes Source A (8) Source B (8) Source C (8) Source D (8) Genetic optimization One Two Three Four Five Six Seven Eight Table XVII Solution through vehicle routes matrix 5106, , , , , ,3 3810, ,2 3378, , , , , , , ,3 3810, ,2 3378, ,1 Figure 4 Best solution cost 6 Concluding remarks There are two main results obtained from this paper One is the formulation of an approach to the distribution problem that could be adapted to real situations The other is showing the GA capacity to solve complex problems with imprecise or vague information Fuzzy sets allow us to represent the imprecise information that characterised the decision making related to physical distribution Moreover, the genetic algorithms could supply solutions to the distribution problems when they are considered in depth However, trying to adjust properly the posing of the problem, as a proposal of future work, could be interesting, taking into account the accomplishing of production and demand cadences, multiple product distribution, the intermediate warehouses, etc References Biethahn, V and Nissen, V (1995), Evolutionary Algorithms in Management Application, Springer-Verlag, Dordrecht Camm, JD and Evans, JR (1996), Management Science: Modelling, Analysis and Interpretation, South-Western College Publishing, Cincinati, OH Dubois, D and Prade, H (1980), Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, NY

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