An Adapted Firefly Algorithm for Product Development Project Scheduling with Fuzzy Activity Duration

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 973291, 11 pages http://dx.doi.org/10.1155/2015/973291 Research Article An Adapted Firefly Algorithm for Product Development Project Scheduling with Fuzzy Activity Duration Minmei Huang,1 Jijun Yuan,2 and Jing Xiao3 1 School of Public Administration, South China Normal University, Guangzhou 510006, China School of Finance, Guangdong University of Finance & Economics, Guangzhou 510320, China 3 School of Computer Science, South China Normal University, Guangzhou 510631, China 2 Correspondence should be addressed to Minmei Huang; Received 3 June 2014; Revised 11 August 2014; Accepted 27 August 2014 Academic Editor: Fang Zong Copyright © 2015 Minmei Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Efficient scheduling plays an important role in product development project management, especially for the product development project with fuzzy activity times. In this research a trapezoidal fuzzy number is used to represent fuzzy activity duration, and an improved magnitude of the trapezoidal fuzzy number is adopted for fuzzy time comparison. Firstly, a mathematical model for the scheduling problem with minimizing the project completion time for the product development project is established. Then, an adapted fuzzy firefly algorithm is developed to solve the model. The priority value based coding method is used; the fuzzy parallel schedule generation scheme is adopted to generate feasible solutions, and the brightness comparisons are made before updating fireflies’ locations in the proposed algorithm. Finally, the performance of the proposed algorithm is presented by computational experiments based on PSPLIB benchmarks. An example of resource allocation of an electronic product development project is also used to illustrate the effectiveness and efficiency of the proposed algorithm. 1. Introduction Product development has become a source of competitive advantage for many industries in recent years [1]. In order to bring the new product to the market as early as possible, it is essential to schedule product development project efficiently with the constraints of time, resources, and cost. Because of the uniqueness of new product development project and the influence of uncertainty, the precise prediction of activity duration is difficult. Thus the activity duration is always estimated by experts’ experiences and is usually imprecise. In resource constrained project scheduling problems (RCPSP), time parameters are considered to be deterministic. Thus traditional approaches for RCPSP cannot accommodate product development project in uncertain environment. Some research has been performed on fuzzy resource constrained project scheduling problems. The methods in the literature can be grouped into two categories: the heuristic methods and the artificial intelligence based methods. (1) For heuristic methods, traditional critical path method (CPM), plan evaluation, and review technique (PERT) were adopted to solve project scheduling problems with uncertain times without resource constraints [2, 3]. Hapke and Slowinski [4] extended the known priority heuristic method for solving RCPSP with fuzzy time parameters. Bhaskar et al. [5] proposed a heuristic method for resource constrained project scheduling problem with fuzzy activity times. The heuristic method was based on priority rule for parallel schedule generation scheme, and the proposed priority rule was called Schedule Performance Index (SPI). (2) For artificial intelligence based methods, Wang [1] proposed the fuzzy beam search algorithm to determine a schedule with the minimum schedule risk, and the start time of each activity was selected to maximize the minimum satisfaction degrees of all temporal constraints. Wang [6] developed a genetic algorithm based on fuzzy set theory for uncertain product development projects. Ke and Liu [7] built three types of fuzzy models to solve the project scheduling problem with fuzzy activity duration and developed a hybrid intelligent algorithm to solve the fuzzy models. 2 Mathematical Problems in Engineering The metaheuristics firefly algorithm (FA) inspired from intelligent social behavior of fireflies was recently presented by Yang [8]. Firefly algorithm is one of the biology-derived algorithms and it was proved by Yang [9] that FA is more efficient than particle swarm algorithms when dealing with multimodal functions. The FA has also been applied successfully to nonlinear design problems [10], constrained continuous optimization tasks [11], permutation flowshop scheduling problems [12], and resource constrained project scheduling problems [13, 14]. Yuan [15] proposed a modified firefly algorithm to solve multiobjective constraint optimization problem, and the proposed method was applied to the optimization design of motor product family. Yang [16] developed a multiobjective firefly algorithm (MOFA) for continuous optimization, and Luna et al. [17] applied MOFA to the software project scheduling problem. The objective of this research is to develop an effective method by adapting firefly algorithm to handle product development project scheduling problem with fuzzy activity duration times. The paper is organized as follows. In Section 2 fuzzy set theory is used to represent uncertain activity duration; the magnitude of trapezoidal fuzzy number is employed to rank the fuzzy time parameters, and then the mathematical model for the problem is described. Section 3 proposes the fuzzy firefly algorithm, which is based on priority value coding and parallel schedule generation scheme, and an example is used to illustrate the algorithm. Computational experiments on 30 benchmark datasets and an electronic product development project are conducted in Section 4. Finally, Section 5 concludes the paper, and the future research directions are proposed. u(x) ̃ A 1 0 b a c d x Figure 1: Activity duration represented by trapezoidal fuzzy number. and the membership function 𝑢 can be expressed as 𝑥−𝑎 , { { 𝑏−𝑎 { { { {1, 𝑢 (𝑥) = { 𝑑 − 𝑥 { { , { { {𝑑−𝑐 {0, 𝑎 ≤ 𝑥 < 𝑏, 𝑏 ≤ 𝑥 < 𝑐, 𝑐 ≤ 𝑥 < 𝑑, (1) otherwise. A fuzzy activity duration represented by four real paramẽ = (𝑎, 𝑏, 𝑐, 𝑑) and denoted ters 𝑎, 𝑏, 𝑐, and 𝑑 can be written as 𝐴 by a trapezoidal (𝑎, 𝑏, 𝑐, 𝑑) (see Figure 1). ̃ = If two fuzzy activity durations are defined as 𝐴 ̃ (𝑎1 , 𝑏1 , 𝑐1 , 𝑑1 ) and 𝐵 = (𝑎2 , 𝑏2 , 𝑐2 , 𝑑2 ), respectively, then the following operations can be expressed as Addition: ̃ ⊕ 𝐵̃ = (𝑎1 + 𝑎2 , 𝑏1 + 𝑏2 , 𝑐1 + 𝑐2 , 𝑑1 + 𝑑2 ) . 𝐴 (2) Subtraction: 2. Problem Description 2.1. Definition on Fuzzy Activity Duration. For most product development projects, it is difficult to precisely give activity durations. Activity durations are often estimated by human (...truncated)