Evolved genetic algorithms with fuzzy aggregation applied to priorities in logistic systems

C.A. Silva * *, J.M. Sousa *, T. Runkler 3 and J.M.G. Sb da Costa *

*Technical University of Lisbon, Instituto Superior TCcnico Dept. of Mechanical Engineering/ GCAR - IDMEC

AV. Rovisco Pais, 1049-001 Lisbon, Portugal * Siemens AG, Corporate Technology lnformation and Communications, CT IC 4

81730 Munich, Germany

Absrracr-This paper addresses the problem uf optimizing the s r h d u l r uf lui?i\lic prucessn using genetic algurilhmr and fuzz) deririun making. We cunsider hem the problem uf d)namicall) assign cumpunenLs lo orders and choose the sululiun lhdt is able lo de]i,*r more orden ill the cUrrrCl dale, ~ a m r time the priority degree of the ..\ ronlprumise belHren lhese conflicting goals is achieved h) ujing a genctic algorithm lo optimize a fuzz) neighted functiun. The simulation I~TUIIS rhuw (hat lhe propus& genetic algi>rilhm c d w d *ilh f u w optimization pwsmls good rewlh for this t ~ p r of prohlcmr.

rei ir'u o i solutions to the 13g1s11c s;hrduling problem 35 PJ11

0, the ,~ppl!-chain inanapernsnt problem. can be iound In Shen and NJmlc h j the multl-agent cunlmunit! [X I , 191 From the pure scheJbllng prohlem p i n 1 ut \'IC\\. no \\ark hd\e been dune %I CJr. Therc.furr. ;In !\nt CJluny Opt~nti/.dt~on (.ACOl approsch. (101. ,,hich pr,,,c,i IJ h r a poweriui scheduling methodolog) iur i,,gi,l,c I ) l e of prohlen,s, hJs heen proposed IO thls type ut problems \er! rccenlly I 1 I I

TIth Droblem I\ houever, mare than a "delner in time"

T~~ problem has been tackled

I. INTRODUCTION

ECENTLY. the analysis and control of logistic chains R has been playing an increasing role in industrial and distribution systems. Logistics can be defined as a process that deals with the plaoning, handling. and control of the storage of goods between the manufacturing point and the consumption point. In the past, goods were produced, stored and then delivered on demand. Nowadays, many companies do not work with stocks, using instead cross-docking cenrers I l l : the goods are transported from the suppliers to these cross- docking centers (e.g. airports), stored, and then shipped to the customers. The lack of storage may increase the delivery time, but it considerably reduces the volume of invested capital and increases the flexibility of the process. The key issue is to deliver the goods in time by minimizing the stocks, which is a typical scheduling problem in a new industrial framework.

Over the last decades, a wide range of methodologies has been developed to solve different scheduling problems, usually NP-hard ones. Pinedo (21 presents the state of the art of scheduling techniques, more focused on job-shop problems, but the survey stands for scheduling problems in general. Nowadays, meta-heuristics are considered to be the most powerful scheduling techniques [3], mainly the ones using an evolutionary metaphor, like the genefic nlgorirhms [41. However, many other heuristics have been also successfully applied, like simulated annenling [ 5 ] or tab# search [6]. A

scheduling problem. Apart from the desired delivery times, one has also to take into account other important character- istics, such as the priority of the orders. In general, a multi- criteria scheduling problem has to be solved, and although a combination of criteria is highly desirable, this combination is very difficult to implement in practice. A new methodology to deal with the priorities as is a combination of ordering principles by means of fuzzy decision making in a fuzzy optimization framework has been proposed very recently I12J. This approach can be applied in conjunction with ACO [l31, increasing the performance of the scheduling system.

This paper proposes a genetic algorithm to solve the logistic scheduling problem, evolved with a fuzzy criteria aggregation optimization function to deal with the orders priorities. We pretend to demonstrate that the fuzzy aggregation can easily he incorporated with any scheduling strategy, in particular with genetic algorithms. The purpose of choosing genetic al- gorithms is to motivate the application of fuzzy aggregation in other scheduling applications, since the genetic algorithms are the most used meta-heuristic applied in scheduling problems. The remaining of the paper is organized as follows: the logistic process is described in Section 11. The different scheduling methods that can be used in the process are explained in Section Ill. Section IV presents and discuss the simulation results. Section V concludes the paper, with the guidelines for future wok.

This work i s ~upwned by Ihe German hlinistry of Education and Research (BMBF) under Canlracl no. 13N7906 (project Niveili). the "PRl8Tama de Financiamento Plurianual de Unidades de I&D (POCTI), do Quadro Co. m u n i ~ * ~ de poi^ l~r, and by [he G ~ ~ , no, SmHiBDi6366iZ~1, FC-, Minisrdrio do Enrino Superior da Citncia e da Tecnolugis. Ponugal

I t . THE LOGISTIC PROCESS

Figure 1 presents a schematic representation Of a logistic process, based on a real-world logistic subprocess at Fujitsu- Siemens Computers (FSC) [I I] . The process can be described

0-7803-7937-3/03/%17.00 02003 IEEE 175

The focus of this paper is to optimize this component assignment process. If the delivery of two orders with different priority indices is mutually exclusive, than the order with the highest priority has to be delivered.

5 ) Order delivery. The order is delivered to the client, with a certain tardiness T3. where tardiness stands for the difference between the delivery date and the desired date. The tardiness should always he zero.

e 111. SCHEDULING POLICIES

Fig. 1. General representation of the logistic process

in probabilistic terms. In fact, the birth process of the system (arrival of new orders in a certain period of time) and the death process (delivery of orders per unit of time, or the time it took them to be processed), can be described by the classical theory of queuing processes [14]. For the process being studied, this theory asserts the Poisson distribution for the model of the birth process,

(1)

where x is the random variable number of orders and AT is the birth rare, i.e. the parameter indicating the probability that this event occur on a certain time T . The death process is modelled by the exponential distribution

p(., AT) = __ (AT)5,-AT X!

P(T, d = fie-PT (2) where the time T is the random variable and fl is the death rare, which accounts for the number of days that an order remains in the system. In this case it can be seen also as the service rare of the system. The process can he divided into five sequential steps:

1) Order arrival. The client buys a product, called order, which is a set of one or more different items, the components. An order must contain a desired deliveq date, which is the date required by the client for the order to be delivered. Each order has a priority index, which may reflect the importance of the client, or the profit of the order.

2) Component request. The different components must be requested from the extemal suppliers. Each component is characterized by a certain quantity.

3 ) Component arrival. Each component takes some time to be delivered to the logistic system. This time is called the supplier delay. After this time, the component is delivered to the so-called cross-docking places. A component sfock list is built at these places, which contains the available components and their quantity.

4) Component assignment. Usually the components are not all available at the same time. For this reason, the orders have to wait for all the required components to be available. This waiting list is called the order stock. Each order has its own desired delivery date. The decision process has to decide which orders are going to be delivered, taking into account the availability of their components. This is normally performed once per day.

The componenr assignment step is the key issue in logistic processes. The company can not influence the arrival rates of the orders (birth process), or the suppliers delay. The service or death rates (death process) of the orders are the only conrml variable, by influencing the assignment of components to the orders. The control goal is to generate particular service rates for each order hy some scheduling process using external information like desired times. The next sections describe possible scheduling possibilities that can be used in a logistic system.

A. Pre-assignment (PA) When the components arrive at the cross-docking center

from the external suppliers, they are already assigned to specific orders. The components are stored there until all the missing components arrive and then the order is completed and can be delivered. This strategy can not deal efficiently with disturbances, e.g. a delay in the component arrival. This assignment method is a static scheduling method.

8. Dispatching rules These methods use some sorting between the orders based

on a performance index. They are dynamic strategies, since they allow the exchange of components between orders. In our problem, the most interesting dispatching rules that can be used are: - Firsr Desired Firsr Delivered (FDFS) - This method uses

the desired delivery times for each order and sorts them in ascending order. The order with the earliest desired delivery time is the first in the list. The components in the stock are subsequently assigned to the orders in this list, starting from the first list entry. It is equivalent to the so called Earlier Due Dare dispatching scheduling rule used for manufacturing systems [Z]. In practice, corresponds to a sorting by rardiness indices Ti. Ordering by priority (P) - This methods uses a list sorted by priority indices P3. Some orders can have priority over others. An example of an important order is one with a large number of components required by a very important client. This order should have priority over others, because not only the order is big and the profit associated with it is large, but also the client should be very satisfied in order to assure possible future orders. Note that the company may have to delay less important orders slightly, in order to satisfy more important ones. Weighred rardiness (WT) - This method sorts the list by a combined priority-tardiness index wjTj = Tj /Pj .

776

Two orders with the same tardiness will he sorted by priority, two orders with the same priority will be sorted by tardiness.

C. Fuzzy Weighred Aggregarion (FWA) This method can be seen also as a dispatching rule, since it

is based on a list where the index is defined following a fuzzy optimization method. Fuzzy optimization is the name given to the techniques that formulate optimization problems with flexible, approximate or uncertain criteria by using fuzzy sets. Fuzzy sets cat1 be used to represent flexibility in the goals and in the constraints. Hence, the criteria normally assumed to be crisp, are now assumed to have some flexibility, which can be exploited for improving the optimization objective.

The general formulation for fuzzy optimization in the pres- ence of flexible goals and constraints is given by

fuzzy maximize [h fx ) , f z ( x ) , . . . f , ( x ) ] subjectto gi(x)%O: < E {1,2; . . . , q } ,

where the'sign denotes a fuzzy satisfaction of the constraints and 2 denotes that gi(x) 5 0 can he satisfied to a degree smaller than I . The fuzzy maximization corresponds to achiev- ing the highest possible aspiration level for the goals fl(x) to f,(x), given the fuzzy constraints of the problem. This optimization problem can be solved by using the approach of Bellman and Zadeh to fuzzy decision making [IS].

Consider a decision making problem, like the logistic prob- lem in this paper, where the decision alternatives are x E X. In the logistic process, x corresponds to a certain assignment of components to the orders. A fuzzy goal F,, i = 1, is a fuzzy subset of X. Similarly, fuzzy constraints Gi. i L 1,. . . , g. can he defined as fuzzy subsets of X. Goals and constraints can be treated as criteria C,, with i = 1, and m = p + q is the total number of criteria. The membership function of a criterion Ci is defined as u,(x), with ui : X i 10: 11, and it indicates the degree of satisfaction of the decision criterion (goal or constraint) by the decision alternative x E X. The fuzzy decision D is defined as the confluence of the criteria, i.e.

X t X (3)

D(x) = ut(x) o . . . o u,(x) ouP+l(x) o . . . ouZlm(x), (4)

where o denotes an aggregation operator for fuzzy sets. The optimal decision alternative X' is then the argument that maximizes the fuzzy decision, i.e.

x* = argmaxD(x) X E X ( 5 )

When some trade off amongst the goals is desirable, the aggregation may be modelled by an averaging operation. The aggregation of goals allowing some compensation between goals can be done using averaging operators. When the cri- teria are not equally important they can be combined using weighted fuzzy aggregation.

Weighted aggregation has been used quite extensively es- pecially in fuzzy decision making, where the weights are used to represent the relative impoltance that the decision maker attaches to different decision criteria. Almost always

an averaging operator has been used for the weighted aggre- gation, such as the generalized means, which can be naturally extended to weighted equivalents. The weighted generalized mean operator has been used in many fields, and it has been studied in the context of fuzzy set aggregation.

This paper applies the generalized means to aggregate the several criteria in the logistic process. The confluence of the m criteria using the weighted generalized mean is defined as:

o;(x) = c w i ' 21i(X)V (6)

for any y E R \ {O} . The weight vector is defined as w = [tui, w?,. . . ,U,] and satisfies Czl w, = 1. The sum of the weights w, is one in order to define a relative importance between the weights.

The logistic process can be seen as a fuzzy optimization problem when it is formulated as follows. At each time period the optimization algorithm computes a fuzzy index (membership grade) for each order. This index combining the several criteria is computed for each order using the weighted generalized mean given in (6). This paper uses y = 1, which is the arithmetic mean, in order to allow the compensation between criteria. For the logistic problem we defined two criteria: C1 is the tardiness and Cz is the priority. The weighted aggregation of the criteria membership functions U , and u2 is given by

(*Il )l'?

2

D(x) = c u i i , U{(.). (7) ,=I

The result of this weighted aggregation gives a membership grade for each order. The membership grades are then used to sort the orders as in the other dispatching rules.

D. Generic rrfgorirhms Genetic Algorithms (GA) are an adaptive method to solve

search and optimization problems, based on the principles of natural evolution and they were introduced by Holland in I161 and later developed by Goldberg in [17]. In the scheduling optimization field, we find the work of Lawton [le] as an introductory work that led to a growing research volume of evolutionary techniques in this specific area.

The genetic mechanisms through which species have sur- vived and adapted to the environment are: selection, iecom- bination and mutation. The role of mutation is the random variation of the existing genetic material; recombination, also called crossover, hybridizes different genetic material in order to integrate the advantageous features of parents into off- springs; selection increases the proportion of better adapted individuals in the population. All this properties are based on the classification of the individual's fitness, which is a measure of how well it performs as a member of the species. GA is a methodology that applies these genetic operations to a population of individuals (solutions). These operations originate a new population of individuals that, hopefully, will be more fitted to the environment (optimization problem) than the previous one. This process is carried on throughout several

777

generations (iterations) until a sufficient fitness is achieved. The algorithm runs for O(n, x N,,& time, where 7% is the size of the population and N,,, is the maximum number of iterations allowed.

To implement the algorithm on an optimization problem, there are three main aspects to take into account:

the coding which consists of finding a suitable represen- tation of the solutions as a chromosome (sequence of genes); the fitness function, which determines how well fitted an individual i s to the environment; . the operations, since the way the operations are executed can influence the efficiency of the algorithm significantly. This includes the parameter tuning, such as the size of the population, crossover and mutation rates, etc.

I ) Encoding the possible solutions: Historically, GA rolu- lions have been first represented by binary strings [I61 and they are still frequently used. They are the easiest way to implement the genetic operations, such as mutation, which consists of changing a gene from 0 to I or vice-versa. In our problem, the use of binary encoding seems reasonable. The problem is intrinsically discrete and the (0, 1) pair can simply represent the fact that an order is delivered or not.

In this way, this paper adopts the following representation: a solution is a binary vector with the size of all the orders waiting to be delivered, m, with value 0 (zero) if the order is not delivered today, and value 1 if it is. Figure 2a represents an example of two solutions, A and 8 , in an optimization problem of rn. = 5 orders waiting to be delivered. Solution A represents that orders 1 and 3 will be delivered and solution B represents that orders I , 4 and 5 will be delivered.

Due to the constraints present in our problem, not all the solutions are feasible. A feasible solution is a solution that is pan of the solution space and respects the constraints, while an infeasible one belongs to the solution space but does not respect the constraints. An example of an infeasible solution would be, e.g. one solution that "delivers" order 2, when is missing a type of component in the stock IO deliver it. If an infeasible solution is created after the genetic operations, this solution is transformed into a feasible solution before the algorithm proceeds.

The transformation of an infeasible solution into a feasible one consists of the following procedure: for every gene 1 in the solution, i.e. a delivered order, it is checked if there are enough components in the stock to deliver this order or not. If yes the gene remains 1, the stock is updated and the proceeding checks the next gene. If there are not enough components in the stock, the gene is changed from 1 to 0. When there are no more components available in the stock, the genes that were not checked yet are set to 0. The critical aspect of this algorithm is to decide in which order the check of the genes is performed. This could be done following a random sorting or, as it is done in this paper, sorting the genes by delivery date, introducing in this way a local heuristic in the algorithm. The solution is checked following the sequence of genes that have the earliest delivery date. In this way, the solution is not only repaired but this is done in such a way that indirectly

*

B

(a) Encoding of two different S0l"tiO"S

B

(b) One-poinr cmssover between solutions A and B, resulting in two new m e m k n A' and E'.

(c) iMuration suffered by offspring A.

Fig. 2. GA implemenlation 10 solve the logistic process.

the orders that have to be delivered with more urgency are the first ones to be checked.

2) Fifness function: The fitness function is the objective function of the problem. The objective is to deliver all orders with zero tardiness, or at least with the minimum variance. A classical objective function in scheduling problems f is to assume the minimization of the sum of the tardiness absolute values of all orders. However, with this function, a low global tardiness value could mask the fact that no orders are delivered at the comect dare, although all of the orders are delivered with small tardiness. Therefore, we adopt a new objective function called Null Tardiness Deviation (NTD). Here, we impose as the most imporlant objective the fact that the highest number of orders are delivered at the correct date, although we consider also important to have a small tardiness variance of the remaining orders. In this way, formally, the fitness function is given by:

where IVT stands for #Ti = O,Vo, E 0, i.e. the number of orders delivered at the correct date and E,"=, IT,I is the tardiness for all orders and accounts for the variance of the tardiness. Note that if the absolute value is not used, orders with negative tardiness could nullify orders with positive value. Thus, the fitness function of the individual could be good, but not the solution. Note further that the global tardiness term can

778

be replaced by a weighted tardiness term, e.g. En==, JT,/P,(. as defined previously. In this case the GA & d method is applied.

3) Operutions over rhe populution: The initial population is initialized as random binary strings with the size of the number of orders in the system waiting to be delivered.

The selection consists of evaluating the fitness of each solution and then choosing the ones to create offsprings in the next generation, while the rest of the members of the population will disappear. The rate of individuals allowed to make offsprings is defined as the selection rate. Here we use a selection rate of 50% and eliminate the rest of the population. The eliminated population is replaced by the new offsprings.

The crossover is the most important operation in the algo- rithm and there are several critical points in this stage that have to he analyzed. We chose to have only two parents that generate two offsprings. The crossover method we use is the most simple and traditional way to do it, the one- point crossover. It consists on aligning both parents A and 8, randomly chose a crossover section, and then the parents swap the segments located to the right of the crossover, resulting in two new offsprings A' and B', as shown in Fig. 2b. Note that often. the crossovers are not done between all parents, but only on a subset of the parents. The rest of the parents simply generate a direct copy of themselves. This makes our GA methodology elirisr, since we maintain the best individuals within the population. There are also other possible crossover techniques, such as the two-point crossover or the uniform crossover [ 191.

The mutation is applied to a subset of the offsprings after the crossover step. It randomly changes each gene with a small probability. Figure 2c shows how the offspring A' suffered a mutation. The mutation rates are usually very low. We use here a mutation rate of 1%.

E. Evolved CA with Fuzzy Weighted Aggregation (CA & FWA)

The difference to the previous optimization method is that the objective function to optimize presented in (8). the term E;=, IT,\ which describes the deviation of the tardiness, is replaced by the fuzzy decision function D ( x ) as defined in (7). The fitness function is then given then by

(9)

order. However, we introduce some noise associated with the amval of the components to the system from the suppliers. This noise follows a normal distribution with average i~ = 1 and standard deviation U = 2. This simulates the fact that some of the components will miss the expected supplier delay inducing in this way a delivery date different (and usually later) from the desired delivery date.

The orders have also a priority index, which can be {0.25,0.5; I} where 0.25 means low importance order, 0.5 medium importance and 1 important order. The majority of the orders have low priority (0.25) or high priority label (11, and some are at the intermediate level (0.5). The simulation is done for an interval of 1 month. At each day the system adds the components arriving at that day to the components stock list (except for the PA strategy).

This paper uses tables representing the number of orders classified by type tardiness T to measure the performance of the different methods. If T < 0 it means that the orders where delivered before the desired date, if T = 0 it means that they were delivered at the correct date. and if T > 0 it means that the orders were delivered after the correct date. The tables indicate also the minimum and maximum tardiness as an indicator of the variance. The orders are divided into 3 groups, referring to priority indices, The objective is not only to deliver the maximum number of orders with T = 0, but also to deliver orders with priority 1 as much as possible. The last row of each table represents the total results and also the global service rate p of the system.

In the next sections, we present the results of the simulations divided by scheduling policies: pre-assignment (PA), dispatch- ing rules (FDFS, P, WT), the FWA method and the genetic algorithm approaches (GA, GA & WT and GA & WFA). The simulations consider the period of one month, and start with a stable running system, where we have already some orders to be delivered and some components in the stock.

A. Pre-assignment (PA) As explained in Section 11, in this method, the components

are pre-assigned to the orders, so the orders will he delivered as scan as the last component arrives to the docking center. While it waits for the missing components to arrive, all the other components have to stay there. Even if an order B is missing a component that exists in the stock assigned to an order A, and the order B could be delivered on time if it received this component, the component is not assigned to order B. The result, as it can he seen in Table I, is that very few

IV. EXPERIMENTS Considering the description of the system presented in orders are delivered at the date, In of priorities,

Section 11, we simulated a simplified logistic process based this is also there are orders delivered at the correct date with low priority than with high priority. In this way we conclude that this method is very poor to achieve the T = 0 objective, as well as to respect the priorities.

on a real-world example and representative of a large number of logistic systems. We consider here a logistic process where the number of amving orders on each day follows a Poisson distribution as in (I), with At = 20. Each order is a set of 1 up to 10 different types of components, and each type of component within an order can have a maximum quantity of 20. The suppliers delay follow a uniform distribution between 1 and 10 for each type of component. The suppliers delays are constant throughout the simulation. The desired delivery date is equal to the largest supplier delay of components within the

B. DisParching It is expectable that any dynamic scheduling, like the FDFS.

where the components can be exchanged between the orders, performs better than the PA method. However, as Table 11 shows, if the method only follows a desired delivery date

119

TABLE 1 PRE-ASSlFNMENT (PA)

Prionty T < 0 T=O T > 0 min(T) max(T) p 0.25 25 37 140 -4 I5 0.5 15 20 50 -6 I2 I 29 27 75 -6 12

. Total 69 89 265 -6 IS 15.2

Priority 0.25 0.5 I

Tola1

TABLE II FIRST DESIRED PIRST SERVED (FDFS)

T < 0 T=O T > 0 min(T) max(T) # 134 25 81 -14 8 62 7 30 -14 4 78 21 53 -17 7

274 53 I64 -17 8 12.6 0.5 I

sorting, the method is blind to the fact that the orders might be ready before the desired date. In this way, the majority of the orders will be delivered in advanced decreasing the service rate from 15.2 to 12.6. This shows that the system as the capacity to process orders faster than the PA method does and that the dynamic assignment between orders enlarges the possible solutions for the scheduling. But it also shows that this new freedom has to be properly controlled. In terms of tardiness, the results are quite bad. In terms of priorities, they are not respected anyway, since the majority of the delivered orders is from type 0.25.

When the orders are sorted using priority only, the P method, a higher number of orders of type 1 is delivered, but again failing the null tardiness objective, as Table 111 shows. The number of orders delivered at the correct date is even smaller than with the FDFS method.

Finally, the combined priorily-tardiness index used by the WT method does not work as good as expected, as we can see in Table IV. The number of orders delivered at the correct date is small, the variance of the tardiness is high, and the priority indices are not followed. This proves that an index combination using a simple operation like the product does not guarantee that any of the indices is being followed.

In conclusion, the dispatching rules perform worse than the original PA method mainly because they are not effective in terms of tardiness.

TABLE 111 PRIORITIES (P)

Prionty T < 0 T = O T > 0 min T max T <I 181 -18

Total 383 22 94 -18 28 11.2

TABLE I V WElGHTED TARDINESS (WT)

58 28 12 -14 3 85 60 14 -18 7

0.25 0.5 I

Total

Fig. 3. membership function U, of the tardiness (delayed days)

TABLE V FUZZY WEltiHTED AOFREOATTION (FWA)

max(T) p P 181 9 -I8 26

106 I 4 -18 29 169 3 7 -18 25 456 13 47 -18 29 9.7

0.25 I 130 19 87 -14 8

C. Fuzzy oprimization (FWA) To apply the generic framework defined in Section 111-C,

the weighted aggregation in (7) was defined as follows: since the sum of the weights wi is one, the weights are defined using a parameter lul = e and consequently w9 = (1 - e ) . Thus, the aggregation of the criteria is given by

(IO)

The criterion C1 is the tardiness, so the proposed membership function UI is the one presented in Fig. 3. The membership function U, is very small for orders delivered before the desired date (following an exponential function). When the delivery date is the correct date (T = 0), the membership function is 0.5, which is much higher than the previous values. Finally, for tardiness values greater or equal than 1, the function grows again exponentially to 1. This means that an order that is already delayed should be delivered immediately. The criterion Cz, the priority, is already a fuzzy number. The used value of e = 0.8, derived from the study presented in [12]. Notice that fort = 1, we would have the FDFS rule and for c = 0, we would have the P rule.

With this strategy, where both tardiness and priority are taken into account in a fuzzy index, the results improve in terms of tardiness and also in terms of priorities, as shown in Table V. With this method. it was possible to deliver many orders with the highest priority in the correct date. Moreover, the result for the other indices of priority is also good in terms of tardiness. Notice that the fuzzy rule with L = 0.8 takes into account mainly the tardiness of the orders. Nevertheless, it is visible that follows the priority criteria, since it delivers more orders of type 1 than the FDFS method, as it is desirable.

D. Generic Algorithms Genetic algorithms are a much more powetful scheduling

method than dispatching rules and that influences naturally also the performance in terms of priority index. There are much more orders delivered at the correct date for any type of orders, as Table VI shows. However, we see that the number of orders delivered at the correct date of type 0.25 is higher

D(x) = €211 + (1 - E ) W .

780

TABLE VI GENETK ALOORlTHMS (CA)

Pnorily T C O T=O T > 0 min(Tj max(T) p 0.25 89 101 37 -12 13 0.5 35 48 II -I2 6

1 50 73 25 -13 14 Total 174 222 73 -13 19 13.8 '

~ Pnonty 0.25 0.5 I

Total

T<O T=0 T > 0 min(T1 max(T) p 108 84 33 -12 32 40 42 11 -10 I9 5 8 60 26 -14 12 206 186 70 -14 32 13.7

TABLE VI1 GENETIC ALGORITHMS (GA) WITH WEIGHTED TARDINESS (WT)

Priority 0.25 0.5

1 T o ~ l

T<O T=O T > O min(T) max(Tj p 127 67 35 - I2 22 39 47 8 -I2 13 50 70 23 -7 21

216 184 66 -12 22 13.6

than of type 1 . In terms of priorities, the results are not what we want from the system, hut again, the priority was not taken into account.

If we combine the genetic algorithms with WT, we can observe that in terms of tardiness, the results are similar, but the number of orders of type 1 delivered is higher than the number of orders of type 0.25. The results can he seen in Table VII. Note that using this , the priority index was respected. It is visible that the trade-off done between number of orders delivered with T = 0 and the priority indices, is done at at the expenses of the lowest priority indices. In terms of tardiness, the results improved, since we have smaller variance for every type of order.

E. Generic algorirhms wirk jiuuuy weighted agg regntion Finally, our objective is to prove that using GA to optimize a

fuzzy optimization function that takes into account priority in- dices, will outperform all the previous strategies and achieves the complex objective of T = 0 for the most important orders. Both the CA and the FWA were applied using the parameters defined previously. The results are presented in Table VIII. For every degree of priority, there are more orders delivered at the correct day (T = 0). without changing the variance around T = 0. Moreover, the variance of tardiness, expressed in terms of the min T and niaxT variables, is also much better. Therefore we conclude that the GA &RNA method is the best scheduling method to a logistic process where priority indices have to be respected.

V. CONCLUSIONS This paper presents a new genetic algorithm optimization

methodology using n fuzzy decision making framework to optimize the scheduling of a logistic process, respecting the priority of the orders. This type of process can be modelled probabilisrically as birth and death processes. The dynamic assignment of components to orders is the kep issue in the optimization of logistic processes, and the genetic algorithm is able to do it in a better way. The use of multiple optimization criteria is solved using a weighted fuzzy aggregation in a fuzzy decision making environment, This type of optimization func- tion can he easily incorporated in different optimization strate- gies, like dispatching rules or meta-heuristics. A simple but

illustrative example shows that the proposed evolved method performs very well in a scheduling problem. As future work, we pretend to extend this hybridization of meta-heuristics and fuzzy decision making For other types of optimization problems.

REFERENCES [I] 1. M. Swuminathan. S . E Smith, and N. M. Sadeh. "Modeling supply

chain dynamics: A multiagent approach," Decision Sciences Joumnl.

[Z] M. Pinedo, scheduling: The"?, AlgOrirhms, and Syslenir. 2nd ed. Prentice Hall. 2W2.

[3] A. lain and S. Meeran. "A state-of-the-an review of job-shop scheduling techniques:' 1998. [Odine]. Available: citcseer.nj.nec.com/jain98sralean.himl

[4] C. R, M. Gen, and Y. Tsujimura. "A tot31 survey of job-shop scheduling problems using genetic dgorlchms-i. representation," Computers and

Vol. 29. "0. 3, pp. 607432, 1998.

industrial engineerieg. vol. 30, no. 4, pp. 983-997. 1996. [ 5 ] P. I. M. V. Laarhooven, E. H. L. Aanr. and 1. K. Lenstra, "Job-shop

scheduling by simulaled annealing," Opmmiom restorch. vol. 40, no. I, nn 11?-125. 1992. = / ~~ ~~

[6) M. Laguna and F. Glover. "Integration target analysis and tabu seach for improved scheduling systems:' Expert rysremr with applicolions. vol. 6. pp. 287-297. 1993.

[7] W. Shen and D. Nome, "Agent-based systems for intelligent manufac- tunng: A state-of-the-an survey:' Knowledge and Informarion Systems, on lnlermlloml Journal. vol. 1. no. 2, pp. 129-156. 1999.

[8] M. Barbuceanu and M. Fox, "Coordinating multiple agents in the supply chain," Barbuceanu. M. and Fox, M.S. (1996). Coordinating Multiple Agents in (he Supply Chain. In: Pmceedings of the Fifth Workshops on Enabling Technology for Collaborative Enterprises, WET 1CE96. IEEE Computer Society Press. pp. I34-14L 19%. [Online]. Available: ~iieseer.nj.nec.comibarbuceanu96coordinari~g.hl~l

(91 H. V. D. Parunak. "Practical and industrial applications of agent-based systems:' P a n " (1998). Practical and Industrial Applicarions of Agent-Based Syslcmr. hrrp:llwwwcr.umbc.edu/agentsi, 1998. [Onlinel. Available: c i r e r e e r n j . n c c . c o m i p ~ ~ ~ 9 8 p ~ ~ ~ t I ~ ~ l . h t

[IO] M. Dorigo, V. Manierro. and A. Colomi, 'The Ant System: Optimization by a colony of cooperating agents." IEEE Trrinsaclions on Syscrms, Men. and Cybernerirs Pan B : Cybemeric$, vol. 26, no. I , pp. 2941, 1996. [Online]. Available: citereernj.nec.comido"go96ant.html

( I I ] C. Silva, T. Runkler. 1. M. Sausa, and R. Palm. "Ant colonies LF logistic processe "ptimi~er~:' in Pmceedings of AntrZWZ, Brussels. Belgium. Brussels. Belgium. Seprember 2W2, pp. 7 6 8 8 .

I121 1. M. Sousa. R.Pdm, C. Silva, and T. RunkleLer, "Optimizing lagiatic processes using a fuzzy decision making approach." 2003, IO appear in IEEE Tranraclions on Systems. Man. and Cybernetics Pan U: Cybernetics.

[ 131 C. Silva. T. Runkler, and J. M. Soura, "Evolved ant colonies optimization in the scheduling of logistic prwesro," in Join1 GMA ond GI Workhop Fuuy Syramr. Donmund, November 2002, pp. 145-158.

[I41 R. Wolff, Sforhosiic Modeling ond the Theory u/Qwucr. Prentice-Hall, Landan, 1989.

[IS] U. E. Bellman and L. A. Zadeh. "Decision-making in B fuzzy envimn- ment," Management Science. vol. 17, no. 4, pp. 141-164. Dec. 1970.

[I61 1. H. Holland, AdDprorion in " M o l and AWiciaI Swems. The University of Michigan Press. 1975.

(I71 D. E. Goldberg. Cenrric Algorilhmr in Search. Oprimiralion and Mn- chine Laming . Addison-Wesley, 1989.

vol. May Isrue. pp. 23-21, 1992.

Springer-Verlag. 1999.

1181 0. Lawton, "Genetic algorithms for schedule oplimizat!on," AI Expen,

1191 Michalewicz, GA + Dora Srrucrurer = Evolctlionar). Computing.

78 1