Optimization of logistic systems using fuzzy weighted aggregation

Fuzzy Sets and Systems 158 (2007) 1947–1960www.elsevier.com/locate/fss

Optimization of logistic systems using fuzzy weighted aggregation

C.A. Silvaa, J.M.C. Sousaa,∗,1, T.A. Runklerb

aTechnical University of Lisbon, Instituto Superior Técnico, Department of Mechanical Engineering, CIS/IDMEC, 1049-001 Lisbon, PortugalbSiemens AG, Corporate Technology, Information and Communications, Learning Systems Department, 81730 Munich, Germany

Available online 19 April 2007

Abstract

Logistic scheduling problems are often multi-criteria optimization problems, with many contradictory objectives and constraints,which cannot be properly described by conventional cost functions. The use of fuzzy decision making may improve the performanceof this type of systems, since it allows an easier and suitable description of the confluence of the different criteria of the schedulingprocess. This paper introduces the application of fuzzy weighted aggregation to formulate the logistic system optimization problem.Further, this paper also extends the application of this framework to different types of optimization methodologies: dispatchingrules, if it is used as a performance index; or meta-heuristics, such as genetic algorithms (GA) or ant colony optimization (ACO),if it is used as an objective function. Simulation results show that the fuzzy combination of criteria improves the scheduling resultswhatever optimization methodology is used.© 2007 Elsevier B.V. All rights reserved.

Keywords: Logistic systems; Fuzzy weighted aggregation; Dispatching rules; Meta-heuristics

1. Introduction

Recently, the analysis and control of logistic chains has been playing an increasing role in industrial and distributionsystems. Logistics can be defined as a process that deals with planning, handling and control of the storage of goodsbetween the manufacturing point and the consumption point. In the past, goods were produced, stored and then deliveredon demand. Nowadays, many companies do not work with stocks, using instead cross-docking centers [23], which areplaces with limited store capacity, as e.g. airports. The goods are transported from the suppliers to these cross-dockingcenters, stored, and then shipped to the customers. The lack of storage may increase the delivery time, but it considerablyreduces the volume of invested capital and increases the flexibility of the process. The key issue is to deliver the goodson time, in order to assure customer satisfaction and, at the same time, minimizing the stocks. Therefore, the controlof a logistic process is a scheduling problem.

Over the last decades, a wide range of methodologies has been developed to solve different scheduling problems,especially NP-hard ones [14]. Heuristic methods, such as dispatching rules [2] are still often used in industry, becausethey are easy, simple and fast implementation techniques. However, these heuristics cannot efficiently cope withtoday’s increasingly complex optimization environments. Nowadays, meta-heuristics are considered to be very powerful

∗ Corresponding author. Tel.: +351 21 8417471; fax: +351 21 8498097.E-mail address: [email protected] (J.M.C. Sousa).

1 This work is supported by the project POCI/EME/59191/2004, co-sponsored by FEDER, Programa Operacional Ciência e Inovação 2010, FCT,Ministério do Ensino Superior, da Ciência e Tecnologia, Portugal.

0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2007.04.008

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1948 C.A. Silva et al. / Fuzzy Sets and Systems 158 (2007) 1947–1960

scheduling techniques [9], mainly the ones using an evolutionary metaphor, like genetic algorithms [3] or ant colonyoptimization [4].

The choice of the cost function to optimize is as important as the choice of the optimization procedure.An optimizationalgorithm can provide optimal solutions, but if the cost function does not describe properly the objectives, the finalsolution can never be a good solution from the practical point of view. For the logistic system, the scheduling problemis a multi-objective optimization problem. Apart from the desired delivery times, one has also to take into account otherimportant characteristics, such as the priority of the orders. Although a combination of criteria is highly desirable, thiscombination is very difficult to implement in practice.

This paper uses the aggregation of fuzzy criteria to describe the logistic system performance, as introduced in [22].This paper expands the use of this technique, by showing that the use of fuzzy aggregation can improve the logisticsystem performance whether the used optimization strategy is a simple dispatching rule or a more sophisticated method,such as genetic algorithms or ant colony optimization (ACO). As far as the authors know, it is the first time that ACOis used to optimize a fuzzy cost function. In terms of genetic algorithms, there has been some applications in termsof jobshop scheduling problems, such as [7,16]. However, in these works the genetic algorithms are specialized forthe use of specific fuzzy costs functions, while in this paper the GA implementation is independent of the used costfunction.

The paper is organized as follows: the logistic systems are described in Section 2. Fuzzy weighted aggregation isbriefly presented in Section 3. Section 4 explains the different scheduling methods that can be used for the logisticsystem optimization. The simulation results are presented and discussed in Section 5. Section 6 concludes the paper,presenting also the guidelines for future work.

2. The logistic scheduling problem

The logistic optimization problem described in this paper is based on a real-world logistic sub-process at Fujitsu-Siemens Computers (FSC) [17]. In this section, a generic model that is representative of many different types of logisticsystems is presented, and the different optimization criteria that the system has to follow are described. This sectiondiscusses further how the different criteria can be described by a cost function.

2.1. System description

In general terms, a logistic process deals with the problem of collecting orders and assigning them to the clients at thecorrect delivery date. At each day, the logistic system has an order list O of n orders waiting to be delivered. An orderoj ∈ O with j = 1, . . . , n, is a set of m different types of items called the components ci with i = 1, . . . , m, in certainquantities qij . Therefore, an order can be defined as an m-tuple oj = (q1j , . . . , qmj ). When a new order oj arrives, itreceives two labels: the arrival date or release date rj and the desired delivery date or due date dj , which is the datewhen the client wishes to receive the order. The components ci are purchased from external suppliers and collectedat the cross-docking center after some time, called the suppliers service, which is the time that the suppliers take todeliver the components. At this point, the decision process starts. Based on the components stock at the cross-dockingcenter and the order list, the logistic system has to assign the components to the orders. Finally, the orders are deliveredat the completion date Cj . Two disturbances may influence the system: the fact that the suppliers service may not berespected and the fact that some clients ask for desired delivery dates not compatible with the suppliers service.

The difference between the completion date and the due date is called the lateness Lj = Cj − dj . If an order oj isdelivered before the desired delivery date, the lateness Lj is negative; otherwise, the lateness Lj is positive. Notice thatreceiving an order before the desired delivery date may be in many cases as unpleasant for the customer as receiving theorder after the delivery date. As an example, consider the case where a large company purchases new office furniture—the furniture arrival date has to be arranged not only with the workers (so that they could be prepared to proceed withthe exchange), but also with a distribution company that removes the old furniture (otherwise, the company has to storethe new and the old office furniture). Therefore, we consider in this paper that one of the objectives of the logisticsystem is to match both desired and delivery dates, i.e., to have for all orders Lj = 0.

In practice, clients have different importance, which means that the orders have different priority indexes Pj . Anexample of an order that is more important than others is one with a large number of components, not only becausethe order is big and its profit is large, but also because that particular client should be very satisfied in order to assure

C.A. Silva et al. / Fuzzy Sets and Systems 158 (2007) 1947–1960 1949

Fig. 1. Logistic system representation, with seven incoming orders.

possible future orders. In this case, the logistic system may delay less important orders slightly in order to satisfy moreimportant ones. Therefore, the orders priority is another important criterion to be followed by the logistic system.

Fig. 1 presents a schematic representation of a logistic process. In this example, the logistic system collects n = 7orders, composed by different combinations of four types of components. These components are purchased from thesuppliers, who are able to deliver only part of the requested quantity of components. The optimization algorithm hasto decide which of the orders waiting on the order list will receive the components, based on the desired delivery datedj and on the priority index Pj . With the available stock of components, there are several different possible solutions,like delivering orders o6 and o7 or delivering orders o3 and o5. In terms of lateness, both solutions would be similar,but in terms of priority, the first solution is more advantageous because the priorities are higher. Therefore, the finallogistic system solution is to deliver orders o6 and o7.

2.2. Optimization problem

A classical objective function for this type of problems is to assume the minimization of the absolute lateness weightedwith the priority index of all orders, called Total Weighted Lateness, and given by [14]

n∑j=1

‖Pj × Lj‖. (1)

However, with this function a low global lateness value could mask the fact that no orders are delivered at the correctdate, although all of the orders are delivered with small lateness; or an order with a high priority and a small delay canbe masked by an order with low priority but large delay. Moreover, it has to be assured that the maximum number oforders is delivered at the correct date. Therefore, we propose here a new objective function called high priority-nulllateness (HP-NL). This cost function imposes as the most important objective the fact that the highest number of ordersare delivered at the correct date, although it considers also important to have a small lateness variance of the remainingorders according to the priorities. Given the set O of orders in the system waiting to be delivered, the subset of orders


that are going to be delivered is defined as OD ⊆ O. The complementary subset of orders that are not delivered andremain in the system is defined as OND ⊆ O, such that OD ∪ OND = O. Consider further that the subset of orders thatare delivered at the correct date is defined as O0

D ⊆ OD and the subset of orders that are not delivered and are alreadydelayed is defined as Od

ND ⊆ OND. The optimization objective is to minimize the cost function given by

fL =∑

j∈O Lj + |OdND|

|O0D| + �

, (2)

where∑

j∈O Lj accounts for the minimization of the lateness of the total set of orders in the system O; |OdND| is the

cardinality of the subset OdND and refers to the minimization of the number of orders that are not delivered and are

already delayed; and finally |O0D| + � is the cardinality of the subset O0

D and accounts for the maximization of thenumber of orders delivered at the correct date. The � is a small constant that avoids the infinity value when no ordersare delivered at the correct date. This decision step is done once per day, but different solutions for the same dailyproblem originate different next-day scheduling problems. The supply chain management is a dynamic successionof daily optimization problems, that are treated independently, even though they are not. Therefore, to evaluate theperformance of the supply chain, larger periods of time, such as weeks or months, should be considered.

The cost function defined in (2) describes a trade-off amongst two different objectives that can lead to under orover-achievement of at least one of them. Therefore, some flexibility may be present for specifying the constraints ofthe problem. Moreover, the objectives are known only approximately. The objectives can be expressed approximatelyin linguistic terms [22], and a precise mathematical formula is not available. Also, the decision constraints may berelaxed, as long as the decision objectives can be improved.

These types of problems require an extension of the classical optimization and constraint framework in order to dealwith the flexibility of the constraints and with the approximate specification of the objectives. Next section explainshow this can be achieved through fuzzy weighted aggregation.

3. Fuzzy weighted aggregation

Fuzzy optimization is the name given to the techniques that formulate optimization problems with flexible, approx-imate or uncertain criteria by using fuzzy sets [25]. Fuzzy sets can be used to represent flexibility in the goals and inthe constraints. Hence, the criteria normally assumed to be crisp, are now assumed to have some flexibility, which canbe exploited for improving the optimization objective.

An important issue in fuzzy decision making is the aggregation of the different fuzzy criteria. Dubois and Prade[6] provide an extensive survey on fuzzy set-theoretic operations, mainly concentrating on the properties of thoseoperations. Several different types of aggregation operators, such as t-norms, t-conorms or mean operators have beendefined in the last decade [27]. One of the most relevant works is the one by Yager [24], that discusses some of theissues involved in the selection of appropriate operators mainly for implementing the union and intersection of fuzzysubsets, based on the properties satisfied by those operators. Another important aspect, as important as the aggregationitself, is the relative importance of each of the criteria and how this can be incorporated [24]. This is the subject offuzzy weighted aggregation. Inclusion is normally achieved by retaining the aggregation operators as they are definedand then associating the weights with the membership functions as product, power, max-min, t-conorms or t-norms.Kaymak and Sousa present in [10] an extensive analysis on different aggregation operators and weighting factors.

This section describes the general framework of fuzzy weighted aggregation and its application to the logisticscheduling problem.

3.1. General framework

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

fuzzy maximizex∈X

[f1(x), f2(x), . . . , fp(x)]subject to gi(x)�̃0, i ∈ {1, 2, . . . , q}, (3)

where the ∼ sign denotes a fuzzy satisfaction of the constraints and �̃ denotes that gi(x)�0 can be satisfied to a degreesmaller than one [26]. The fuzzy maximization corresponds to achieving the highest possible aspiration level for the


goals f1(x) to fp(x), given the fuzzy constraints of the problem. This optimization problem can be solved by using theapproach of Bellman and Zadeh to fuzzy decision making [1].

Consider a decision making problem, like the logistic problem in this paper, where the decision alternatives arex ∈ X. In the logistic process, x corresponds to a certain assignment of components to the orders [22]. A fuzzy goalFi , i = 1, . . . , p, is a fuzzy subset of X. Similarly, fuzzy constraints Gi , i = 1, . . . , q, can be defined as fuzzy subsetsof X. Goals and constraints can be treated as criteria Ci , with i = 1, . . . , r , and r = p + q is the total number ofcriteria. The membership function of a criterion Ci is defined as ui(x), with ui : X → [0, 1], and it indicates the degreeof satisfaction of the decision criterion (goal or constraint) by the decision alternative x ∈ X. The fuzzy decision D isdefined as the confluence of the criteria, i.e.

D(x) = u1(x) ◦ · · · ◦ up(x) ◦ up+1(x) ◦ · · · ◦ ur(x), (4)

where ◦ denotes an aggregation operator for fuzzy sets. Originally, as introduced in [1], the aggregation operator wasjust the min operator, while here the aggregation operator is general, as proposed in [10].

The optimal decision alternative x∗ is then the argument that maximizes the fuzzy decision, i.e.

x∗ = arg maxx∈X

D(x). (5)

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

Weighted aggregation has been used quite extensively especially in fuzzy decision making, where the weights areused to represent the relative importance that the decision maker attaches to different decision criteria. Usually, anaveraging operator has been used for the weighted aggregation, such as the generalized means, which can be naturallyextended to weighted equivalents. The weighted generalized mean operator has been used in many fields, and it hasbeen studied in the context of fuzzy set aggregation [10].

This paper applies the generalized mean to aggregate the several criteria in the logistic process. The confluence ofthe r criteria using the weighted generalized mean is defined as

Dw� (x) =

(r∑

i=1

wi · ui(x)�

)1/�

(6)

for any � ∈ R \ {0}. The weight vector is defined as w = [wi, w2, . . . , wr ] and satisfies∑r

i=1 wi = 1. The sum of theweights wi is one in order to define a relative importance between the weights.

In multi-objective optimization, one common approach is not to define a specific weight to each criteria, but ratherto present the Pareto surface found by an evolutionary approach [15] and then choose the solution that best fits theproblem. In the logistic scheduling problem, this cannot be done because the optimization problem is dynamic. Thus,the weights in (6) are chosen a priori.

3.2. Fuzzy optimization of the logistic system

The logistic system cost function defined in (2) is a linear combination of orders priority and lateness weighted bythe number of orders delivered at the correct date. With this function, orders with high priority and low lateness mayhave the same weight as orders with low priority and high lateness. However, in real world systems this relation isnever linear: for perishable goods, even if the goods have low priority, they have to be delivered on time or they must bethrown away; important clients, that represent most of the companies profit, have to receive always the goods on time,even if it implies large lateness to other clients. Moreover, the introduction of the term describing the total number oforders delivered at the correct date does not clearly describe the relative importance of delivering orders at the correctdate. This subjective evaluation of priority versus lateness can be easily described by fuzzy criteria. The use of fuzzysets to describe each criterion allows to define mathematically the optimization concept of the logistic system managersin an easier way.

Consider the two criteria for the optimization of the logistic system: the priority C1; and the lateness C2. The priorityindexes can be given in the interval [0, 1], which are already fuzzy values. For the criterion lateness C2, the proposed


Fig. 2. Membership function u2 of the lateness.

membership function u2 is the one presented in Fig. 2. This membership function, for the order lateness Lj ∈ N, isanalytically described by

u1(Lj ) =

⎧⎪⎪⎨⎪⎪⎩

eLj

2if Lj < 0,

1 if Lj = 0,

tanh(0.2 × Lj ) if Lj > 0.

(7)

The membership function u2 is very small for orders delivered before the desired date (following an exponentialfunction). When the delivery date is the correct date (Lj = 0), the membership function is 1, in order to mimic theimportance of delivering orders with zero lateness, as described in function (2). Finally, for lateness values greateror equal than 1, the function grows to 1 following an hyperbolic tangent. This means that an order that is alreadydelayed should be delivered immediately, although it can be further postponed if there are orders that can be deliveredat the correct date. These criteria are combined through the weighted generalized mean given in (6). As described byKaymak and Sousa in [10], a compensation between goals should be allowed since the aspiration level of some goalsis unreachable, e.g. null lateness for all the orders. This can be achieved by using � = 1, which is the arithmetic mean[11].

Since the sum of the weights wi is one, the weights are defined using a parameter � = w1 and consequently(1 − �) = w2. Thus, the aggregation of the criteria is given by

D(x) = �u1 + (1 − �)u2. (8)

The parameter � is chosen based on a sensitivity analysis presented in Section 5. This fuzzy optimization function can beused either to define a simple dispatching rule sorting index [22] or a cost function to be optimized by meta-heuristics,such as GA or ACO [20].

3.2.1. FWA as a dispatching ruleIf the optimization method used by the logistic system is a dispatching rule, the fuzzy weighted aggregation can

be used to compute a fuzzy index for each order and the logistic system solution results from the index sorting indescending order.

3.2.2. FWA as a cost function for meta-heuristicsHere the meta-heuristics cost function defined in (2) evolves to the function defined in (9), since the fuzzy decision

function in (8) is defined for a maximization problem and the meta-heuristics are designed for minimization problems.Therefore, the cost function used by the meta-heuristics is in fact given by

fFWA = 1∑OD

D(x), (9)


where∑

ODD(x) is the sum of fuzzy indexes for all the delivered orders. The use of (9) is justified because the cost

function takes into account the relative importance of orders priority and lateness, and the fact that the lateness is anonlinear function, where orders at the correct date are as important as orders that are very delayed. This implicitlydescribes the fact that delivering orders at the correct date is very important, as described in function (2).

The possible different scheduling strategies to optimize the cost functions are discussed in the next section.

4. Scheduling optimization methods

To schedule the logistic process, two types of strategies can be used: dispatching rules or meta-heuristics. Heuristicsare still often implemented in scheduling systems, since they are easy and cheap to implement. However, in the lastdecades it has been acknowledged that these techniques are insufficient to ensure the accomplishment of the more andmore complex objectives of modern scheduling systems [9,14]. Therefore, to face the increasing complexity of thesystems maintaining high performance results, meta-heuristic methods have to be used.

4.1. Dispatching rules

A dispatching rule consists of assigning a priority index to an item, based on an optimality criterion, and then schedulethe activities according to some index sorting. This typically gives a simple scheduling algorithm that runs in O(n log n)

time. The index of a job is determined without reference to other jobs.In our problem, the most interesting dispatching rules that can be used are the following:

• Earliest due date (EDD)—This method uses the desired delivery times for each order and sorts them in ascendingorder. The order with the earliest desired delivery time is the first in the list. The components in the stock aresubsequently assigned to the orders in this list, starting from the first list entry [14]. In practice, it corresponds tosorting the orders by descending lateness indexes Lj .

• Ordering by priority (P)—This method uses a list sorted by priority indexes Pj . The priority index of the orders canbe derived from the profit, the size or client importance.

• Weighted Lateness (WL)—This method sorts the list by a combined priority-lateness index Pj × Lj . Two orderswith the same lateness will be sorted by priority, and two orders with the same priority will be sorted by lateness.

These rules use simple optimization criteria to sort the activities, which does not guarantee a final optimal solution,which is in this case to have null value on expression (2) and corresponds to deliver all orders at the correct date withnull lateness.

Nevertheless, they are easy, simple and safe scheduling techniques which makes them in general attractive strategiesto implement when the cost function being minimized is as simple as the one defined in (1). However, as previouslydescribed, these heuristics cannot efficiently cope with complex optimization environments, with multiple criteria suchas the one defined by the objective function (2). In that case, it is necessary to use more sophisticated methods, such asmeta-heuristics [14].

4.2. Meta-heuristics

Meta-heuristics is the name given to the optimization methods like genetic algorithms, tabu search, simulatedannealing or ACO [13]. More than heuristics, which are simple rules that provide educated guesses, they can bedescribed as heuristics on top of other heuristics, i.e. algorithms that use other heuristics to search for optimal solutions.Nowadays, genetic algorithms (or in a broader sense evolutionary algorithms) are one of the most used meta-heuristic[12]. However, ant colonies have become more and more a credible option to evolutionary algorithms, particularly inoptimization problems that can be described by graphs [5]. GA and ACO meta-heuristics have been recently appliedto optimize logistic systems [21]. This paper extends the application of these meta-heuristics to logistic problemsformulated using weighted fuzzy optimization. Note that the objective is not to compare the performances (for thismatter, considering different cost functions, see [19]). Here, we only show that the fuzzy decision making framework canbe implemented to describe a multi-criteria cost function for the logistic system, independently from the meta-heuristicthat is used to solve the problem.


Fig. 3. GA implementation to solve the logistic process: (a) Encoding of two different solutions; (b) One-point crossover between solutions A andB, resulting in two new members A′ and B ′; (c) Mutation suffered by offspring A′.

4.2.1. Logistic optimization using genetic algorithmsGenetic algorithms (GA) were introduced by Holland in [8] and have been widely used in the scheduling optimization

field. To implement the algorithm on an optimization problem, there are three main aspects to take into account: theencoding of the solution, the definition of the fitness function and the implementation of the basic genetic operations(selection, crossover and mutation) within the problem framework.

In this application to logistic systems, a chromosome with size n, is a binary vector representing the list of orderswaiting to be delivered, with a 0 value if the order is not delivered, and the value 1 if the order is delivered. Thepopulation is initialized as random binary strings.

The selection of the individuals is done according to the roulette wheel method [12]. The eliminated populationis replaced by the new offsprings. The new offsprings are generated using the traditional one-point crossover, wheretwo parents originate two different offsprings. Finally, the mutation is applied to a small subset of offsprings afterthe crossover step. The GA implementation in this paper is elitist, since the best individuals within the population aremaintained. Fig. 3 represents the operations between two solutions for the example described in Fig. 1, with n = 7orders. The final solution A′′ is to deliver the set of orders {1, 4, 6, 7}.

The fitness function of a solution is given by the objective function of the scheduling problem. GA are applied bothto classical and fuzzy objective functions, which are proposed in (2) and (9), respectively. Due to the constraints presentin our problem, not all the solutions are feasible. In the present implementation, if an infeasible solution is created afterthe application of the genetic operations, the infeasible solution is repaired into a feasible one before the algorithmproceeds to the next iteration. The transformation of an infeasible solution into a feasible one consists of checking ifthere are enough components in the stock to deliver this order or not. If yes, the gene remains 1, the stock is updatedand the algorithm proceeds to the next gene. If there are not enough components in the stock, the gene is changed from1 to 0. The solution depends on the sequence by which the genes are individually checked. Since orders should not bedelivered with delay, we use the EDD heuristic to define the sequence by which the genes are checked, introducingin this way a local heuristic. In the example solution A′′ in Fig. 3, the feasible solution would be to deliver the set oforders {6, 7} (see Fig. 1 to verify the desired delivery dates dj ).

The algorithm runs for O(g × Nmax) ≈ O(N2max) time, where Nmax is the maximum number of iterations allowed

and g is the size of the population.

4.2.2. Logistic optimization using ACOThis section describes the ACO implementation to solve the optimization of logistic processes. The logistic problem

is modeled by a graph, where the nodes represent the orders waiting to be delivered. The role of the ants is to find theminimum cost path connecting the orders that should be delivered. The objective function to be minimized by eachant k is denoted by f k . This function can be either the classical or the fuzzy objective functions defined in (2) and (9),respectively.

One important aspect of the problem is the fact that the number of visited nodes may not be the same from one ant toanother. This is different from other ACO implementations, as e.g. in traveling salesman problem, where the numberof nodes to visit is fixed and equal to the number of cities to visit [4]. We consider that each ant is traveling with a bagcontaining the available stocks. This ant distributes the stocks between the orders that it is visiting. It only visits orderswhich is possible to deliver. In the example presented in Fig. 1, order o2 needs 2 components c2. If the ant only has one(q2 = 1), it will not visit that order. In this way, the ACO only builds feasible solutions. When the stock bag is emptyor the remaining components are not enough to deliver any missing order, the search for this ant is finished. Therefore,


Fig. 4. Graph representing the scheduling problem with seven orders solved by the ACO. The pheromone trails have different intensities: strong (−),medium (..).

the algorithm will not have a constant number of iterations, since the number of orders visited by each ant is different.Since the path is not closed, the initial starting point for each ant assumes an important role.

Fig. 4 represents schematically the optimization graph. The ants have found a good schedule, delivering orders (6,7)and alternative schedules, such as (6,5) or (3,7). Notice that the notation changed from set notation {, } in the GA, tovectorial notation (, ) because ACO solutions also indicate the sequence in which the orders should be scheduled. Inthis way, for an ant k in the graph, the probability of choosing the next order to deliver is given by

pkij (t) =

⎧⎪⎪⎨⎪⎪⎩

��ij × ��

ij∑nj /∈� ��

ij × ��ij

if j /∈ �,

0 otherwise

(10)

where �ij is the pheromone concentration in the path (i, j), �ij is a heuristic function and � is a tabu list.The pheromone trails �ij are restricted to the interval [�min, �max], with �min = 0 and �max = 1. Initially, all the

pheromones track are initiated with the value �max/2 [21].The heuristic function � in this case is the lateness of the order: if an order has already a delay, the ant will feel

a stronger attraction to visit it. The heuristic function depends only on the order and therefore it concerns the nodesonly, therefore �ij = �j = j . Notice that this is again the EDD heuristic, which was also used in the GA. However,we define the heuristic function here as an exponential function in the interval [0, 1] where the value 0 is for the orderthat has the minimum lateness Lmin and 1 is for the most delayed order Lmax. The objective is that the orders alreadydelayed attract ants much more than the orders not yet delayed:

� = e(Lj −Lmin)/(Lmax−Lmin) − 1

e − 1. (11)

As introduced in [21], the values are restrained to this interval to make a more comprehensive weighting of the weights� and �.

The tabu list � is the list of orders already delivered by the ant and also the orders which are not possible to visit dueto lack of stocks. The parameters � and � measure the relative importance of trail pheromone (experience) and localheuristic (knowledge), respectively.

Let a tour be the route made by one ant until it empties the stock bag and an iteration be the set of the tours performedby all the g ants. The update of the pheromone concentration in the trails is done at the end of each iteration, and isgiven by

�ij (t + 1) = �ij (t) × (1 − �) +g∑

k=1

��kij , (12)


where � ∈ [0, 1] expresses the pheromone evaporation phenomenon. The term∑g

k=1 ��kij are pheromones deposited

in the trails (i, j) followed by all the g ants after a complete tour, which are defined as

��kij =

⎧⎨⎩

1

f kif arc (i, j) was used by the k ant,

0 otherwise(13)

The value of f k is given by the evaluation function for each k ant in a minimization problem, which can be eitherthe cost function in (2) or in (9). However, as described in [5], it improves the algorithm convergence if only the bestant of each iteration is allowed to update the trail. In this way, the pheromone matrix is only biased by the solutionfound by the best ant of each iteration. Notice that the time interval taken by the g ants to complete all tours is at mostn × g. In every N th of the Nmax maximum number of tours, a new ant colony is released. The algorithm runs forO(n × g × Nmax) ≈ O(N3

max) time.

5. Experimental results and discussion

The approach proposed in this paper considers a logistic process model as presented in Section 2, that mimics a real-world logistic system at Fujitsu-Siemens Computers [17]. This section describes the simulation environment in detail.Further, it presents the experimental results using dispatching rules and meta-heuristics, where the several differenttechniques described in Section 4 are compared.

5.1. Simulation environment

We consider a process with an average number of n = 20 orders arriving each day, following a Poisson distribution.Each order is a set of 1 up to m = 10 different types of components, and each type of components within an order canhave a maximum quantity of qi = 20. The suppliers service and the desired delivery date have some noise, to simulatedisturbances. The orders have also a priority index, which can be {0.25, 0.5, 1} where 0.25 means low importanceorder, 0.5 medium importance and 1 a very important order. The majority of the orders have low priority (0.25) or highpriority label (1), and some are at the intermediate level (0.5).

The logistic system decision step is done once per day, but different solutions for the same daily problem originatedifferent next-day scheduling problems. The logistic system performance is a dynamic succession of daily optimizationproblems, that are treated independently, even though they are not. Therefore, to evaluate the performance of the logisticsystem, larger periods of time, such as weeks or months, should be considered. The simulation is done for an intervalof 30 days. At each day the system adds the components arriving at that day to the components stock list and findssolution OD of orders to be delivered.

5.2. Logistic optimization using dispatching rules

This section describes the optimization results following the different dispatching rules approaches described inSection 4.1. Recall that dispatching rules are simple algorithms to find solutions for optimization problems. Theyare not optimization methods and therefore the provided solutions are not optimal (although there are cases wheredispatching rules can provide optimal solutions [14]).

The proposed heuristics use different rules to determine the scheduling solution. In order to compare their results, itis necessary to evaluate their performance in light of a performance index. The performance of the different heuristicsis evaluated based on their performance using the function fL defined in (2).

5.2.1. Optimization resultsThe optimization results are presented as follows: the number of orders that are delivered before the desired date

L < 0, the orders that are delivered at the correct date L = 0 and the orders that are delivered after the correct dateL > 0. The tables indicate also the minimum and maximum lateness min(Lj ) and max(Lj ) as an indicator of thevariance. Finally, the table presents the average performance for the 30 days using the objective function defined in (2).The orders are divided into three groups, referring to priority indexes. The objective is not only to deliver the maximumnumber of orders with L = 0, but also to deliver orders with priority 1 as much as possible.


Table 1Dispatching rules

Priority (fL) L < 0 L = 0 L > 0 min(L) max(L)

0.25 134 25 81 −14 8EDD 0.5 62 7 30 −14 4

1 78 21 53 −17 7Total 141.14 274 53 164 −17 8

0.25 100 12 57 −18 28P 0.5 102 2 24 −18 10

1 181 8 13 −18 7Total 145.09 383 22 94 −18 28

0.25 181 9 36 −18 26WL 0.5 106 1 4 −18 29

1 169 3 7 −18 25Total 157.41 456 13 47 −18 29

0.25 102 80 50 −14 6FWA 0.5 57 37 14 −14 3

1 98 51 14 −18 7Total 43.75 257 168 78 −18 8

As described in Section 4.1, the EDD does not take into consideration the priority indexes and even in terms oflateness, the method is blind to the fact that the orders might be ready before the desired date. In this way, many ordersare delivered in advance without taking into account the priority indexes. This can be confirmed by the fact that mostof the delivered orders have priority 0.25. The performance can be considered very poor, as Table 1 shows.

When the orders are sorted using priority only, the P method, a higher number of orders with high priority (P = 1)

is delivered, but again failing the null lateness objective: the number of orders delivered at the correct date is smallerthan with the EDD method and the lateness variance (maximum value) is high. The combined priority-lateness indexused by the WL method does not work much better. The number of orders delivered at the correct date is small, thevariance of the lateness is high, and the priority indexes are not followed. This proves that an index combination usinga simple operation like the product does not guarantee that any of the criteria is being optimized.

When weighted fuzzy optimization is used, considering a weight � = 0.2, there is a large improvement: many ordersare delivered at the correct date and most with high priority. The variance of the lateness is now very small, speciallyfor L > 0. It is clear that the fuzzy aggregation combines the goals much better than the simple product provided bythe WL method. Therefore, we conclude that if a dispatching rule is used, weighted fuzzy optimization assures the bestperformance in terms of the objective function (2).

Next section describes how the weight � = 0.2 was determined through a sensitivity analysis.

5.2.2. Sensitivity analysis for parameter �The weight � defines the relative weight of both criteria priority and lateness in the aggregation function defined

in (9): � = 0 means that only the lateness criteria u2 is considered and � = 1 means that only the priority criteria isconsidered. The sensitivity analysis uses the cost function fL as performance index for different values of � ∈ [0, 1],in 0.1 steps. The results are presented in Fig. 5.

The results show that if � = 1, the dispatching rule using the fuzzy weighted aggregation provides the same solutionas the P dispatching rule.

On the other hand, when � = 0, the fuzzy weighted criteria is not the same as applying the EDD dispatching rule. Thereason is that the membership function defined in (7) is not an increasingly monotonous function of lateness, otherwisethe result of FWA with � = 0 would be the same of applying EDD.

When � = 0.5 is applied, both criteria have the same weight. Also here, it is different from applying the WT heuristic.In this case, even if the membership function was monotonously increasing, the solution would be different except ifthe membership function u2 was a linear function of lateness.


Fig. 5. Performance index of the FWA for different � values.

Table 2Genetic algorithms (GA)

Priority L < 0 L = 0 L > 0 min(L) max(L)

0.25 127 67 35 −12 22GA (fL) 0.5 39 47 8 −12 13

1 50 70 23 −7 21Total 216 184 66 −12 22

0.25 86 116 23 −10 27GA (fFWA) 0.5 14 70 6 −6 20

1 19 107 10 −7 31Total 119 293 39 −12 31

As expected, the best results are obtained when both criteria are considered. The results show further that when��0.5, the lateness criterion is more important than the priority one. For ��0.5 the performance decreases drastically.The best result is obtained when � = 0.2, and therefore, this is the chosen value.

5.3. Logistic optimization using meta-heuristics

When applying the meta-heuristics described in Section 4.2, there are two possibilities: the first one is using thecost function described in (2), designated by fL; the other is to use the fuzzy cost function defined in (9). This sectionpresents a comparison of the optimization results for GA and ACO algorithms when both objective functions are used.

5.3.1. Genetic algorithmsThe parameters for the GA chosen for this implementation are: selection rate is 0.5, crossover rate is 1, mutation

rate is 0.1 and the population size is g = 100. The results are presented in Table 2.Comparing the dispatching rules, it can be generally stated that the GA are a much more powerful scheduling

method. When the genetic algorithms are optimizing the fL function, there are many orders delivered at the cor-rect date and most of them with high priority (P = 1). However, in terms of lateness variance, the results areworse.


Table 3Ant colony optimization (ACO)

Priority L < 0 L = 0 L > 0 min(L) max(L)

0.25 150 51 40 −15 11ACO (fL) 0.5 59 23 15 −12 10

1 75 52 21 −12 11Total 284 126 76 −15 11

0.25 121 86 33 −18 11ACO (fFWA) 0.5 54 38 7 −14 17

1 77 56 18 −14 15Total 252 196 89 −18 17

When we implement the GA to optimize the FWA cost function, the results are even better: it is possible to delivereven more orders with low priority (0.25) without compromising the other types with higher priority, improving theglobal lateness objective; there are also many more orders delivered at the correct day (L = 0). However, the variancearound L = 0 is still high. Notice however that this only happens because the membership function defined in (7) andrepresented in Fig. 2, only assigns a value near one when the lateness is near 30 days. Therefore, this lateness variancecould be decreased if the membership function considered high values for smaller lateness values. Nevertheless, theuse of the FWA improves the scheduling performance.

5.3.2. Ant colony optimization (ACO)The set of parameters for the ACO algorithm is tuned using a trial-and-error approach. The pheromone trails are

initialized with the value of 0.5 and the values � = 0.5 and � = 5 are used.Table 3 confirms that in general, for every type of priorities, more orders are delivered at the correct date (L = 0)

than with dispatching rules, but less than with the GA. In terms of lateness, negative or positive, the spread is not sowide as in the GA. Notice that in this paper we do not pretend to compare the ACO method to the GA in terms ofperformance in this specific problem. The objective here is to show that the use of the FWA function can also improvethe results for ACO meta-heuristic.

When the ACO method is applied to optimize the FWA function, it improved the results in terms of number of ordersdelivered at the correct date, since it was able to deliver more orders of lower priority than before. This effect was alsonoticeable with the GA. Further, with the ACO, the lateness of the delivered orders improves and it is much better thanfor the GA case.

6. Conclusions

This paper presents the use of weighted fuzzy decision functions to optimize logistic systems. The system hascontradictory criteria that have to be respected and fuzzy weighted aggregation proved to be an efficient combinationmethod. The presented results describe a simulation example, where the use of dispatching rules is compared to meta-heuristics with and without the use of the weighted fuzzy optimization. The results are clear: the use of a fuzzy weightedfunction improves the system, whatever the form of the optimization strategy is: dispatching rules or meta-heuristics.

Weighted fuzzy optimization can be applied with any optimization framework in scheduling problems. As thenext step, we plan to implement this technique to other scheduling problems in supply-chains, as e.g. job-shop floorscheduling problems or vehicle routing problems, preferably in a distributed way, as proposed recently in [18].

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Optimization of logistic systems using fuzzy weighted aggregation