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A Hybrid Fuzzy Evolutionary Algorithm for A Multi-Objective Resource Allocation Problem Lily Rachmawati Department of Electrical and Computer Engineering National University of Singapore [email protected]

Dipti Srinivasan Department of Electrical and Computer Engineering National University of Singapore [email protected]

Abstract

In this paper we propose a hybrid Fuzzy EA algorithm employing fuzzy representation and reasoning for multi-objective resource allocation problem involving fuzzy objectives. In particular, the objectives are represented as fuzzy variables, which act as inputs to a fuzzy inference system evaluating the fitness of the associated candidate solution. The motivation of incorporating fuzzy logic in this manner is twofold. Firstly, fuzzy logic offers an effective representation and reasoning framework in evaluating the quality of candidate solutions. Secondly, based on the pre-specified preference of the user, the fitness function guides the search to interesting regions of the fitness space. Evolutionary Multi-objective Optimization (EMO) has been dominated recently by the concepts of Pareto-ranking based on domination criteria. This a posteriori approach essentially seeks a set of nondominated solutions, the Pareto-optimal set, which exhibits various trade-off properties. The convergence and extensive coverage of the true Pareto-optimal front of associated multi-objective problems are the main preoccupation of the approach. Having attained the set of Pareto-optimal solutions, it is the task of the decision maker (DM) to select one solution from the set that suits his/her requirements the best. A number of efficient algorithms have been proposed based on these principles [7-14]. However, the definition of domination when many objectives are involved is inadequate to represent that of a human decision maker’s perception. Further, Pareto-ranking based approaches cannot incorporate the DM’s preferences in trade-offs between objectives. The number of Pareto-optimal solutions may well overwhelm a decision maker. These limitations have been highlighted and addressed to a certain degree in several papers [15-20].

In this paper a hybrid fuzzy evolutionary algorithm for a multi-objective resource allocation problem, the student project allocation (SPA) problem, is presented. Student Project Allocation must satisfy a number of soft objectives stemming from multiple points of view. The proposed algorithm employs a fuzzy inference system to model and aggregate the objectives, assuming the role of the fitness function in the evolutionary algorithm. The fuzzy system captures preferences of the decision maker in the compromise between various objectives, thereby guiding the search to interesting regions in the objective space. The results demonstrate the effectiveness of this hybrid approach for a large data set

1. Introduction Resource allocation problems are widely encountered in industrial as well as academic practices. Owing to its combinatorial complexity, large-scale resource allocation often proves mathematically intractable. The search capability of evolutionary algorithm (EA) warrants its extensive application to such resource allocation problems. The allocation of resources often needs to satisfy multiple criteria, some of which may be soft and perception-based. Some examples of such problems are university timetabling [1], nurse scheduling [2], and network topology design [3]. For relevant variables in these problems, fuzzy representation has been suitably chosen [3-6] in the EA. However, the observed trend is that fuzzified values of perception-based objectives are treated with the conventional crisp mathematics instead of the approximate reasoning facility provided by fuzzy logic.

Proceedings of the Fifth International Conference on Hybrid Intelligent Systems (HIS’05) 0-7695-2457-5/05 $20.00 © 2005

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The hybridization of fuzzy logic and evolutionary algorithm to address multiple objectives has been proposed in [19] and [20]. Abido and Bakhashwain [19] proposed a clustering and fuzzy evaluation of the whole set of Pareto-optimal solutions as a post processing step to produce a manageable set of alternatives and a recommendation to the user. Pirjanian [20] employed a set of fuzzy rules to compute a priori the appropriate weights for the weighted approach. The approach proposed in this paper treats the multi-objectivity in a novel manner. The fuzzy framework used articulates the DM’s necessarily fuzzy knowledge of acceptable trade-offs. Fuzzy inference is embedded in the search process, instead of being a post- or pre- processing step, and thereby probabilistically guides the search via the mechanism of selection to an interesting region in the fitness space, i.e. the region with acceptable objective trade-offs. A problem with multiple fuzzy criteria and known desired trade-offs, a final year project allocation for undergraduates at a university in Singapore, is used in this paper as a context for the discussion of the proposed approach. A similar Student Project Allocation (SPA) problem has been considered in [21] with an interesting two-stage heuristics, which yielded a feasible project allocation with around 80% success rate after a 24-hour run. Anwar and Bahaj [22] proposed an integer programming approach which resolved this SPA problem in a much shorter time. An integer programming for a bi-objective SPA problem was also presented. However, the multiple objectives were handled with two successive optimization steps, biasing the approach against the objective optimized in the second step. The hybrid Fuzzy EA proposed here optimizes simultaneously the multiple objectives involved, achieving a success rate in excess of 96% and high scores for the other objectives. The paper is organized as follows. In section 2 the problem definition is given. In section 3 the proposed EA is described. In section 4 the implementation of the EA to a dataset is described and the results are presented. Conclusion and suggestion for future works follow in Section 5.

2. The Student Project Allocation Problem The student project allocation (SPA) problem is a multi-criteria resource allocation problem, where students in their final year of study are required to submit several choices in the order of preference from a list of projects offered by the faculty. As the number

of projects and the number of students is typically very large in big university departments, this problem poses special challenges to the administrators. The allocation of projects to students is typically accomplished manually, using heuristics similar to those presented in [15]. For a large department, the amount of time required can be enormous and the degree of match between student preferences and allocated project somewhat unsatisfactory. In this paper Fuzzy-Evolutionary Algorithm (FEA) is presented to solve this problem. Actual data from a large university department was used for development and testing. The core of the project allocation system is a database containing details on all projects offered. This database is both maintained and viewed using the World Wide Web. Every year the Department offers a large number of senior year projects in four areas, corresponding to four sub-departments. Each of the students in the Department submits a list of eight projects which he/she would like to be assigned, in order of preference. Cumulative Average Point (CAP), defined between zero and five, serves as a measure of academic performance. The SPA problem is to assign a suitable project to each student, subject to several criteria stemming from the students’ perspective, the Department’s perspective and the academic staffs’ perspective. The criteria comprise: 1. The overall student satisfaction. This corresponds to how compatible the students’ preferences are with the projects allocated. 2. Balanced loading across the sub-departments. Loading is defined as the number of students who are assigned projects offered by a sub-department. 3. Balanced loading across staff advisors. 4. Balanced spread of good students across subdepartments. 5. Balanced spread of good students across advisors. In the allocation, a project may be assigned to only one student. Students not allocated their choices are given the remaining unallocated projects randomly. Criteria 1 to 5 are perception-based and hence best expressed as fuzzy quantities. Further, they are in conflict with each other. Solutions involve trade-offs in the satisfaction of the five criteria. Two things need to be noted here: 1. the trade-offs possible are unique to each cohort’s profile, 2. there is a priori knowledge of acceptable trade-offs, albeit with some uncertainty. Because of these, the five criteria are formulated as objectives (f1, f2, f3, f4, and f5 corresponding to criteria 1 to 5). In summary, the problem is formulated as follows:


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Let D = {d1, d2,…, dK}be the set of subdepartments, and S = {s1, s2,…, sL}be the set of advisors. Each advisor offers a total of J projects, making up M projects in total where M = L X J. Let P= {p1, p2,…, pM}be the set of projects offered. Let X = {(x1, c1), (x2, c2), …, (xN, cN)} be the set of N students xn and the students’ corresponding Cumulative Average Point (CAP) cn with M > N. Each student xn is allowed to ballot for Ω projects in the order of preference. The goal is to find for all n from 1 to N the pairing (xn, pm) such that f1, f2, f3, f4, and f5 are minimized. An evolutionary algorithm has been proposed to solve this optimization problem.

3. Fuzzy Evolutionary algorithm In this paper Fuzzy-Evolutionary Algorithm (FEA) is presented to solve this challenging resource allocation problem. Complexity due to the sheer size of the search space (M!/(M-N)!) motivates the incorporation of a priori knowledge to make the search more efficient. In particular, a rough specification of desired trade-offs is used in the fitness function to guide the search process. The objective values (f1, f2, f3, f4, and f5) are represented as fuzzy variables and fuzzy inference is used to compute fitness. Two of the criteria (f4, and f5) involve the concept of a ‘good student’, which in this paper is defined by a fuzzy set in the universe CAP ranging from 0 to 5. A trapezoidal membership function with the minimum at CAP = 2.5 and the maximum at CAP = 4.2 delineates the set of good student. The definition is motivated by the honors classification scheme employed in the Department. Further details of the EA are as below.

Medium (M) and Large (L) in the experiment. The range for the first objective function f1 is defined between 0 to 10, the range for f2 is between 0 to 20, f3 between 0 to 4, f4 between 0 to 10, and the range for f5 between 0 to 2. The membership functions were designed and tuned according to the DM’s preferences. A total of hundred and seventy eight fuzzy rules of the form: Rule r q : If f 1 is λ1q and σ 1D is λ q2 and σ 1S is λ q3 and

σ 2D is λ q4 and σ 2S is λ q5 then fitness = c q

were used with product operator as the t-norm [23], and averaging as the implication operator. The fuzzy rules are designed according to two principles: high fitness reflects how near to optimal the objective values are as well as how good the trade-off property of the objectives is.

3.1. Chromosome representation and population initialization

Let n = r1 , a random number in [1,N] Let Γ be an empty list Yes Length (Γ) = 8? No Choose randomly r2 in [1,B]. Let pm be the r2-th preference of student xn Yes r2 in Γ

Insert n to Λ No

Insert r2 into Γ Yes pm allocated? No

A chromosome consists of N genes. Gene n encodes the project ID assigned to student xn. A population of size NPOP is initialized as follows. In the first stage (Fig. 1), as far as possible, students are given projects which rank high on their list of preferred projects. Students with no projects are allocated randomly picked projects in the second stage (Fig. 2).

3.2. Fitness function The Fuzzy Inference System is based on the relative importance of the objectives, which is as follows: (1). f1, (2) f2 and f3, (3) f4, and f5. Four triangular membership functions were designed for each of the five objectives, corresponding to Zero (Z), Small (S),


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Let n = remainder (n+1,N) No n = r1 ? Yes End

Fig.1. Population initiation: stage 1

Choose r3 randomly in [1,M]. Let pm be the r3-th project in the list. Yes pm allocated? No Allocate pm to xj , the j-th element in the list Λ. Increment j No Length (Λ) = j? Yes End

Fig.2. Population initiation: stage 2

3.3. Selection, Crossover and Mutation A binary tournament with replacement is used to select each parent for crossover. Single point crossover is applied with a probability of px. Let ρ1and ρ2 be the chosen chromosomes to reproduce. With crossover point r3 randomly chosen from [1,N], children (α1 and α2) copy genes 1 to r3 of ρ1 and ρ2 respectively. The rest of α1 is determined as follows: starting from the next gene, if the corresponding gene in ρ2 has an allele corresponding to a project unallocated to other genes in α1, the allele is copied. Otherwise, this gene is placed in a waiting list and the process continues with the next gene until all N genes have been considered. The algorithm in Fig. 2 is applied to the genes in the waiting list. The same is done to determine values of α2. The crossover ensures children inherit the allocation scheme of respective parents as much as possible. Offspring chromosomes are mutated with a probability of pm. Two random numbers r4 in [0,N] and r5 in [1,Ω] are selected. The project ranked r5 is assigned in student r4’s preference to student r4. If the project is already allocated to another student, a mutual swap is performed. A screening after recombination and mutation ensures no duplicates are found in the offspring population. The NPOP parent population and NPOP offspring population are combined and ranked in terms of their fitness. NPOP individuals with the highest fitness form the population at the next iteration.

4. Results and discussion The fuzzy Evolutionary algorithm developed in this paper was tested on actual data set with 148 students, 200 projects, 5 sub-departments, and 40 advisors. The experiments were conducted on a Pentium III 600 MHz machine, each running for 2 minutes. Different sets of membership functions and fuzzy rules reflecting different desired trade-off characteristics were investigated. For each set of parameters 10 runs were conducted and the best result taken. Two groups of representative results are presented here. Experiments #1, #2, #3 correspond to the membership functions and fuzzy rules given in section 3. Experiments #4, #5, #6 used fuzzy rules constructed with the following ranking: (1) f1, f2, f3 and (2)f4 and f5. Table 1 shows the objective function values for the experiments. The results were obtained with NPOP = 50, MAXITER = 200, px =0.8, pm= 0.1. Table 1. f1,f2,f3,f4,f5 values for results #1,#2,#3 Result #1 #2 #3 #4 #5 #6 f1 4.18 3.76 3.70 4.58 4.60 4.89 f2 0.49 0.49 0.49 0.48 0.42 0.41 f3 0.51 0.51 0.56 0.46 0.46 0.45 f4 0.54 0.48 0.44 3.76 3.39 3.22 f5 0.26 0.49 0.66 0.80 0.80 0.67 It is observed that the set of solutions obtained using the same set of fuzzy rules score within the same fuzzy sets for the objectives involved, demonstrating the effect of fuzzy granular “weighting” in this approach. Histograms in Fig. 3 and Fig. 4 below show the number of students allocated projects of priority one to nine (nine corresponding to un-balloted project) for the six sets of results. Results #1, #2 and #3: Allocated Projects Priority 30


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Result #1 Result #2 Result #3

25 No of Students

Let j = 1

20 15 10 5 0

1

2

3 4 5 6 7 8 Priority of Allocated Project

9

Fig.3. Allocated projects priorities (results #1, #2 and #3)

Results #4, #5, #6: Allocated Projects Priority 30 Result #4 Result #5 Result #6

No of Students

25 20 15 10 5 0

1

2

3 4 5 6 7 Priority of Allocated Project

8

9

Fig.4. Allocated projects’ priorities (results (results #4, #5 and #6) Results #1, #2, and #3 exhibit the higher importance attached to student satisfaction. The hybrid approach yielded a success rate of 96% in allocating one of the eight chosen projects to each student, as opposed to the 80% obtained with the heuristic-based method in [15]. The spread of loading across departments and advisors are observed to be very good. The multi-objective approach presented in this paper is superior compared to the integer programming approach in [16] which was biased towards the firstoptimized objective, and produced unfavorable results for the objective considered next. Results #4, #5, and #6 show the lower relative importance of student satisfaction. The degradation in the value of f1 is accompanied by an improvement in the values of f2 and f3, as expected. The effect of is also observed in a slight drop in the objectives f4 and f5. In terms of computation time taken, the proposed approach takes only a few minutes, as opposed to the long computing time reported in [15]. The hybrid approach has performed well for the SPA problem. With some modification the Fuzzy-EA could be tailored to other complex resource allocation problems with multiple soft, perception-based objectives where the user or DM’s notion of preferred tradeoff is available. Based on the notion, the fuzzy inference function is designed to serve as the fitness function.

5. Conclusion This paper has presented a fuzzy evolutionary approach which can be applied to practical resource allocation problems that involve searching through a finite but very large set of possible combinations. Relative importance and acceptable trade-off of

objectives were captured in the fuzzy inference system design. The hybrid approach was applied to the allocation of final year projects in an undergraduate course. Testing was done on actual data obtained from a university department. Through the analysis of results, it is concluded that the algorithm developed is capable of solving this constrained multi-objective problem with a good success rate. The manual design of the fuzzy rule base is the main drawback of the approach for applications of larger scale. Future works will include developing support for an automated design of the rule base.

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