Mathematical Problems in Engineering

Volume 2016, Article ID 2601561, 13 pages

http://dx.doi.org/10.1155/2016/2601561

## An Efficient Imperialist Competitive Algorithm for Solving the QFD Decision Problem

^{1}School of Mechanical Engineering, Shandong University, Jinan, China^{2}Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Shandong University, Ministry of Education, Jinan, China

Received 11 May 2016; Accepted 5 October 2016

Academic Editor: Marco Mussetta

Copyright © 2016 Xue Ji 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.

#### Abstract

It is an important QFD decision problem to determine the engineering characteristics and their corresponding actual fulfillment levels. With the increasing complexity of actual engineering problems, the corresponding QFD matrixes become much huger, and the time spent on analyzing these matrixes and making decisions will be unacceptable. In this paper, a solution for efficiently solving the QFD decision problem is proposed. The QFD decision problem is reformulated as a mixed integer nonlinear programming (MINLP) model, which aims to maximize overall customer satisfaction with the consideration of the enterprises’ capability, cost, and resource constraints. And then an improved algorithm G-ICA, a combination of Imperialist Competitive Algorithm (ICA) and genetic algorithm (GA), is proposed to tackle this model. The G-ICA is compared with other mature algorithms by solving 7 numerical MINLP problems and 4 adapted QFD decision problems with different scales. The results verify a satisfied global optimization performance and time performance of the G-ICA. Meanwhile, the proposed algorithm’s better capabilities to guarantee decision-making accuracy and efficiency are also proved.

#### 1. Introduction

Quality Function Deployment (QFD), as a well-known planning methodology originated in the late 1960s in Japan by Akao and Mazur [1], focuses on translating customer requirements (CRs) into a series of relevant design requirements. During this planning process, a lot of decisions should be made. One of them is to determine the engineering characteristics (ECs) to be implemented among numerous possible selections and their corresponding actual fulfillment levels, which not only directly influences the deployment of other design requirements and the final quality of the product but also influences overall customer satisfaction and the distribution of resources.

In practice, it is a complex process to analyze and organize a lot of information related to the QFD decision problem, including multifacets of customer needs, market driven factors, price/revenue streams of enterprises, and various regulation compliance. Multiple variables and complex relationships bring many difficulties for decision makers to make decisions. Mathematical programming, an optimization technique, such as integer programming (IP), linear programming (LP), nonlinear programming (NLP), and mixed integer nonlinear programming (MINLP), has been widely used in helping people solve these difficulties. Firstly, in order to clarify those complex relationships, the problem is abstracted as a mathematical model. Then the model is calculated by an algorithm to offer a result for decision makers as a valuable reference [2–13].

However, a rapidly changing decision environment makes people not only satisfied with obtaining a decision outcome but also want to seek for a higher decision-making efficiency, in other words, a shorter time spent on making decisions. Especially in such a society with highly developed economy, technology, and culture, customer requirements tend to be more and more diverse, product systems become gradually complex, and science technologies are continuously developing. All of these may lead to a huge and complex QFD matrix. As an important step in tackling the above QFD decision problem, the selection of algorithms influences not only the accuracy of decisions but also total decision time. And if the calculating time of the algorithm is too long, the decision efficiency and time cannot be guaranteed.

The QFD decision problem in this paper is reformulated as a MINLP model aiming to maximize overall customer satisfaction with the consideration of the enterprises’ capability, cost, and resource constraints. Actually, many decision problems in various application fields, such as chemical engineering [14, 15], gas/water distribution networks optimal design [16, 17], and engineering design [18], can be formulated as MINLP models. MINLP is a NP-hard problem and the calculating algorithms to MINLPs have been developed since 1980s, which can be broadly divided into two main classes: deterministic algorithms and stochastic algorithms. Deterministic algorithms, such as branch and bound method, outer approximation method, and extended cutting plane method, can provide accurate solutions for problems in a continuous space but require an additional effort of analysis (such as the gradient analysis of objective function and constraints), which may considerably add to the work [19, 20]. Considering the existence of integer variables in QFD decision problems and easy-operation of engineers, stochastic algorithms, which are more capable of handling NP-hard problems and need not much professional mathematical knowledge, will be focused on in this paper. A lot of stochastic algorithms have been proposed. An algorithm called M-SIMPSA suitable for the optimization of MINLP is presented by Cardoso et al. [19]. Mohan and Nguyen [21] developed an improved computational algorithm called RST2ANU, which was based on the original RST2U [22]. Yiqing et al. [23] improved the particle swarm optimization (PSO) algorithm for solving nonconvex NLP/MINLP problem with equality and/or inequality constraints. Schlüter et al. [24] introduced two novel extensions for ant colony optimization (ACO) framework. Deep et al. [25] proposed a real coded genetic algorithm named MI-LXPM.

In this paper, Imperialist Competitive Algorithm (ICA), one of the stochastic algorithms proposed by Atashpaz-Gargari and Lucas [26], which simulates the social-political process of imperialism and imperialist competition, is applied to solve MINLP models. The key features of ICA are its fast convergent rate to reach global optimum, which has been proved in dealing with various optimization problems, such as structure design [27, 28], distribution network optimization [29, 30], and process planning and scheduling [31]. The advantages of ICA are beneficial to improvement of decision efficiency. In this QFD decision problem, considering the existence of integer variables in this problem, an improvement by combining genetic algorithm (GA) is carried out because the original ICA is only applied to real variables. The improved ICA, called G-IGA in the following, is compared with other mature algorithms to MINLPs, including RST2U and RST2ANU mentioned above, by calculating 7 simple numerical MINLP problems and 4 different scale QFD decision problems, and the results verify the advantages of G-ICA.

The remainder of the paper is organized as follows: the QFD decision problem is defined in Section 2. The G-ICA to resolve the problem is introduced in Section 3. Some numerical experiments and case studies are shown in Section 4, and Section 5 summarizes the paper.

#### 2. Problem Definition

The QFD matrix is just as in Figure 1. Before the decision problem based on QFD is defined, some assumptions should be made in advance:(i)The QFD matrix for a certain product, including CRs, possible ECs, weight vectors, relationship, and correlationship matrixes, is already established.(ii)Data about capability (lower and upper bounds of enterprise fulfillment levels for every EC), cost (maximum total budget and upper bound of investment cost for every EC), and resource (maximum total resources and required resource to implement every EC) is offered by the enterprise.(iii)The fulfillment level is a percentage from 0 to 100%. For every EC, actual fulfillment level is only influenced by the corresponding investment cost, whose value is from 0 to 10 with no units. And the enterprise can determine the influence relationship based on long-term practical experience all by itself.(iv)There are no natural units or maximum values for customer satisfaction. It is the result of synthesizing all the actual fulfillment levels of engineering characteristics to be implemented.