2007 International Conference on Management Science & Engineering (14th)August 20-22, 2007 Harbin, P.R.China

Goal Programming in Quality Function DeploymentUsing Genetic Algorithm

TIAN Na, CHE A-daSchool of Management, Northwestern Polytechnical University, P.R.China, 710072

Abstract: Quality Function Deployment (QFD) is asystematic approach that captures customer requirementsand translates them, through House of Quality (HOQ),into technical characteristics of the product. Animportant activity in constructing a HOQ is to determineimprovement ratios for the technical characteristics,based on the collected customer requirements, with aview to achieving a high level of overall customersatisfaction. Traditional methods for this planningprocess are mainly subjective, and often result in anon-optimal or sub-optimal solution, especially withmany customer requirements and technicalcharacteristics. This paper presents a goal programmingapproach to QFD planning. We first present a genericgoal programming model for QFD. We then propose agenetic algorithm for linear and nonlinear goalprogramming models in QFD. Computationalexperiments show that the proposed approach iseffective.

Keywords: genetic algorithm, goal programming,optimization, quality function deployment

1 Introduction

Today, manufacturing enterprises are facingincreasing challenge as a result of increased marketcompetition and changing customer demands. The mainobjective of any manufacturing enterprise is to bring newproducts to market sooner than the competitors withlower cost and improved quality. Changing demandsrequire enterprises to focus more on the customerrequirements at the earlier stage of the productdevelopment. The key to success is customer drivendesign and manufacturing['], and all decisions must relateto customer satisfactory by listening to the voice of thecustomer [2-4]

In order to develop and produce products which willachieve high profits and satisfy the customer, it isnecessary and essential for the manufacturing enterprisesto have up-to-date approaches and means to understandcustomer requirements, and launch a competitive product

This work was supported by the Program for New CenturyExcellent Talents in Universities of Ministry of Education,China, under grant No. NCET-06-0875.

into market based on them with lower cost, shorter time,and high quality.

Quality Function Deployment[5-9] (QFD) is anapproach to customer driven product development,which adopts scientific and systematic methods forsurveying and analyzing customer requirements, andtranslating them into the appropriate technicalrequirements for each stage of the product developmentcycle using a structured framework known as House ofQuality (HOQ) [1O]. The QFD approach has beensuccessfully used for defining new products, as well asfor diagnosing and improving existing products.

One of the most important activities in constructinga HOQ is to determine improvement ratios for thetechnical characteristics of the product, based upon thecollected customer requirements and other information inthe HOQ, with a view to achieving a high level of overallcustomer satisfaction. This is a complex planningprocess with multiple decision variables and goals"[1113],requiring compromise and optimize all kinds of conflictsin HOQ. In practice, the process of setting theimprovement ratios is often accomplished in a subjectivemanner, upon experience and expertise of members ofthe product development team. Traditional methods oftenresult in a non-optimal or sub-optimal solution,especially with many customer requirements andtechnical characteristics in a HOQ. As a result,optimization approaches such as linear programming andgoal programming are required in such a complexplanning process['4].

Wasserman[15] proposed a linear programmingmodel for determining the optimal improvement ratios oftechnical characteristics subject to a given cost. Kim et al.[16] proposed a fuzzy theoretic modeling approach toQFD by developing fuzzy, multi-criteria models to aid adesign team in choosing target level for technicalcharacteristics. Moskowitz et al.[171 and Lee['8] presentedan integrated mathematical programming formulationand solution approach for determining optimal targetlevels of technical characteristics in order to maximizecustomer satisfaction. But all the authors assumed alinear relationship between customer requirements andtechnical characteristics. However, in practice, therelationships between customer requirements andtechnical characteristics are often nonlinear. Traditional

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optimization approach like simplex method cannot dealwith nonlinear optimization models. This paper presentsa genetic algorithm for linear and nonlinear goalprogramming in QFD.

The rest of this paper is organized as follows.Section 2 introduces the basic approach to QFD. Section3 gives a generic goal programming model for QFDplanning. In Section 4, we propose a genetic algorithmfor linear and nonlinear goal programming models inQFD. Section 5 presents an illustrative example to verifythe proposed algorithm. Section 6 concludes the paper.

2 Basic approach to quality functiondeployment

Quality Function Deployment (QFD) provides ameans of defining customer requirements, and translatingthem into the appropriate technical requirements andactions for each stage of the product development cycle,and a customer-satisfactory product can be achieved bycoordinating every activity in an organization. GenerallyQFD process contains two stages: capturing customerrequirements and the waterfall decomposition process ofthe customer requirements. In the first phase, themarketing members use all means available to gather orcollect the customer requirements, i.e. what the customerreally wants and expects for the product being designed,and then analyze them. This phase is the most critical aswell as the most difficult part of the QFD process. In thesecond phase, the QFD team members use agraphic-based tool known as House of Quality (HOQ) totranslate the customer's requirements into productcharacteristics, parts characteristics, key processcharacteristics and production requirements through fourphases, i.e. product planning, parts deployment, processplanning and production planning, as shown in Fig. 1. Itthus ensures that the processes of product developmentare directly related to the customer requirements.

As a tool for QFD, the HOQ facilitates decisionmaking in product or service development by providing astructured framework and an organized approach toimproving the product/service quality and satisfying thecustomer's expectations. Taking the first HOQ as anexample, the process of building a HOQ includes thefollowing tasks:

(l)Identifying the customer needs ("Whats"), theirrelative importance, their improvement ratios and targetlevels;

(2)Identifying the product or technicalcharacteristics ("Hows");

(3)Identifying the relationship between "Whats" and"Hows";

(4)Identifying the correlation between "Hows";(5)Determining the importance, improvement ratios

and target values for product characteristics.

3 Goal programming model for QFD

Goal Programming (GP) is a modification andextension of Linear Programming (LP). GP is anoptimization technique that is capable of handlingdecision problems that deal with a single goal withmultiple subgoals, as well as problems with multiplegoals with multiple subgoals['920]. Often, the multiplegoals in a GP model are in conflict and incommensurable.In GP, deviations between goals and what can beachieved within the given set of constraints are to beminimized, rather than try to maximize or minimize theobjective criterion directly as in LP. Therefore, theobjective function of a GP model contains primarily thedeviational variables which represent each type of goalor subgoal accordingly. There are two kinds ofdeviational variables from a goal: the positive and thenegative. The former represents the overachieved level ofa goal while the latter denotes the underachieved level.

The GP model for QFD planning could beformulated as follows:

Problem GP1productracteri E

product L

planning

partsaracteri s process

.racteri sti

parts

\depl oyment processplanning producti on

planning

.orrel ati onmatri x'HOW'

relationshipmatrix

teclii cal importanicetarget values

techliiical competitivebenchimiark

Fig.1 QFD approach

m

Minimize ( iYy)i=l

subject to:

f(x1,x2.x)y1 ++y i=y1i =1,2,...mxj = gj(Xly'X2y .. Xi-I Y Xj+l Y . Xn )yj= 1,2, . .. n

U(x1,x2..xn)-CV(x1,X2..Xn) < T

Yi > °, Yi+ > O, i = 1,2, . .. m, j = 1,2, . .. n

whereWi weight of the ith customer requirementyi ratio of improvement of the ith customer

requirementyi-: under achievement of the ith customer

requirementyi+ : over achievement of the ith customer

requirement

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21

x : ratio of improvement for the jth technicalcharacteristic(decision variable), j=1,2,..., n

f: functional relationship between customerrequirement i and technical characteristics

gj : functional relationship between technicalcharacteristic j and other technical characteristics, j= 1,2

U(x],X2, ..., X9: total cost of implementation oftechnical characteristics, a function which associateswith ratios of improvements of technical characteristics

V(x],X2, ..., X9: total time of implementation oftechnical characteristics, a function which associateswith ratios of improvements of technical characteristics

C: predetermined upper limit of the total cost ofimplementation of technical characteristics

T: predetermined upper limit of the total time ofimplementation of technical characteristics

L;: lower limit of the ratio of improvement for jthtechnical characteristic, j=1,2,..., n

MjA: upper limit of the ratio of improvement for jthtechnical characteristic, j=1,2,..., n

The objective function of the GP model ensures thatthe under-achievement of the goals, degree of customerdissatisfaction, is minimized. The solution of the aboveoptimization model first requires quantification ofrelationships f, i=1,2,..., m and gj j=1,2,..., n. If anexisting data set (for example, customer and technicalcompetitive analysis data ) is available, the quantificationof relationships may be done using multivariatetechniques such as statistical regression analysis, fuzzyregression analysis, artificial neural networks. In thisstudy, we capture the functional relationship f and g1from the assessed relationship matrix between customerrequirements and technical characteristics as well as thecorrelation matrix between technical characteristics,respectively.

For simplicity, we first assume f and gj are linearfunctions. Therefore, the constraint equation showing therelationship between customer requirements andtechnical characteristics can be written in the followinggeneral format[18]:

UiiXi + Ui2X2 +* UinXn 2 Yi

where uij denotes the degree of relationship between theith customer requirement and the jth technicalcharacteristic.

Similarly, the correlation equation betweentechnical characteristics can be written as follows[18]:

x1 - pljxl + p21X2 + * - - + PnjXnwhere pij denotes the degree of correlation between theith technical characteristic and the jth technicalcharacteristic, andpij =1, if i=j.

Considering the correlation between technicalcharacteristics, the relationship equation betweencustomer requirements and technical characteristics canbe written as:

n n n

( Uik Plk)X + ( Uik P2k )X2 +* + ( Uik Pnk )Xn . Yik=l k=l k=l

After normalization, the final relationship equation can

be formulated as follows:rilxl + r12X2 + ... + rin Xn . Yi

wheren

y ikP jkk=1r k=rIJ n n

E y ikPjkj=l k=1

Based on the above relation, Problem GP1 can berewritten as Problem GP2 as follows:

Problem GP2 Minimize (yWiyi )i=1

subject to:

ri1xi + r12X2 + ...+ rInXn - Y1+ + Yi Yir21X1 + r22X2 + *-+ r2nXn - Y2+ + Y2- Y2

rmlxl + rm2X2 +. + rmnXn -Ym + Ym = Ym

cx +C x + +--c+x + -vCI11 2 2 n n Y

IX + t2 2 n n* _nYt + t<TLj < xi <m

Yi- >°0, Yi+ >°0, i =1,2, ---,m, j=12--nYc >°X Y+ >°, Yt >°, Yt >°

where cj and tj are the cost and time coefficients forimplementation of technical characteristics j,respectively.

In the above model, it is assumed that the functionalrelationship between customer requirement i andtechnical characteristics (i.e. f), as well as the functionalrelationship between technical characteristic j and othertechnical characteristics (i.e. gj) are linear. But this is notthe case in most real-world applications. Therefore, theoptimization problems we deal with in practice arenonlinear in nature. It should be noted that Problem GP1becomes a nonlinear programming model iff and/or gjare nonlinear. Traditional optimization methods likesimplex method cannot deal with such a nonlinearprogramming model.

4 Genetic Algorithms (GAs) for goalprogramming in QFD

Genetic Algorithms (GAs) are used to solve manyreal-world problems and have received a great deal ofattention regarding their potential as optimizationtechniques for multi-objective optimization problems.Genetic algorithms seek to mimic the biologicalprocesses of reproduction and natural selection. Naturalselection determines which members of a populationsurvive to reproduce, and reproduction ensures that thespecies will continue. The flow chart of GA is depictedin Fig. 2

4.1 Representation and initializationFor our problem, it is convenient to represent a

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chromosome directly as a solution vector. Let V denotethe chromosome in a population as VI[x12, ..., xn]. Theelements of V are the decision variables in theoptimization model. For our problem, they represent theimprovement ratios of technical characteristics. In thefollowing, we denote pop size as the number ofchromosomes in the population.

Termination test

End |

Fig.2 The flow chart of GA

4.2 Evaluation and selectionGenerally, the fitness value of a chromosome is

obtained by the evaluation function. For our problem, theevaluation function is given by the following equation:

Kk1eval(Vp ) =Ewiy + -E, dki=1 K k=1

where dk is the penalty factor for the kth constraints inthe optimization model. dk is defined as follows:

d {; V1 satisfies the constraint kk M; otherwise

where M is a large positive number. The fitness value ofa chromosome is given by:

fit(V" ) = , p = 1,2,..., pop _ sizeeval(V)

The best chromosome V* with the largest fitness valuecan be kept at each generation from the followingequation:

V* = arg maxf fit(V, )Ip = 1,2,..., pop _ size]where argmax means argument maximum.

The selection process is based on spinning theroulette wheel pop size times, each time a singlechromosome is selected using the following steps.

Step 1 Calculate a cumulative probability ap foreach chromosome Vp,(p=1,2, ...,pop size)

Step 2 Generate a random real number r in [0,1]

Step 3 If r<a1, then select the first chromosome VI;Otherwise select the pth chromosome Vp(2.p.pop_size)such that ap-1<r<ap.

Step 4 Repeat steps 2 and 3 pop_size times andobtainpop size copies of chromosomes.

4.3 Crossover operationThe probability of crossover pR gives us the

expected number pR pop size of chromosomes whichundergo the crossover operation. The steps of crossoveroperation is depicted as follows:

Step 1: Generate a random real number r in [0,1 ]Step 2: Select the given chromosome for crossover

if r< pcStep3: repeat steps 1 and 2 pop-size times and

produce Pc pop size parents.Step4: For each pair of parents (Vj and Vk), the

crossover operator on Vj and Vk will produce twochildren Vj and Vk as follows:

VL =LV1 + (1-b)VkVk =IVk + (1-A)/Vj

where 04X<1.

4.4 Mutation operationThe probability of mutation Pm gives us the

expected number Pm pop size of chromosomes whichundergo the mutation operation. The steps of mutationoperation is depicted as follows:

Step 1: Generate a random real number r in [0,1 ]Step 2: Select the given chromosome for crossover

if r< pmStep3: repeat steps 1 and 2 pop-size times and

produce Pm pop_size parents.Step4: For a given parent V, if the element xj of it is

randomly selected for mutation, the resulted offspring isV =[xI,.x,...x.], where x is randomly selectedfrom the following two choice:

fxi + random(O, M1 - xj)xi - random(O, x - Lj)

where random(0, y) returns a value in the range [0, y].

5 Numerical examples

Fifteen customer requirements were captured in theprocess of using QFD in the development of an airconditioner. Seventeen technical characteristics wereaccordingly determined based on these customerrequirements. A relationship matrix between thecustomer requirements and technical characteristics anda correlation matrix between the technical characteristicswere then established. The two matrices are omitted inthis paper. Based on the definition of Problem GP2, thefollowing linear GP model was derived from theconstructed two matrices.

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Minimize(10.48y1 + 5.7ly2 + 5.7ly3 + 8.57y4 + 5.7ly5

+10.48y6 + 8.57y7 + 10.48y8 + 5.7ly- + 2.86y10

+5.7ly1- + 10.48y1- + 0.95y1 + 2.86y14 + 5.71y15)

subject to:0. 17x1 +0.36x2 +0. 17x13 +0.13x14 +0. 17x16 y1 +y1 1.45

0.58x1 +0.52x2 +0.1x6 -0.14x13 -0.1x16 +0.03x17

-Y2 +y2 =1.250.51x3+0.49x4 y3 +y3 1.0

-0.05x2 +0.41x3 +0.49x4 +0.05x6 +0. 1x17 y4 +y4 1.25

0.17x1 +0.14x2 +0.86x5 -0.09x13 -0.09x16 y5 +y5 1.0

0. 31x1 + 0. 23x2 + 0. 37x6 + 0. 06x7 -0. 03x12 -0.09x13

-0.09x16 +0.23x17 y6 +y6 1.5

0.17x1 +0.1x2 +0.1x6 +0.24x7 +0.07x8 +0.03x9+0. 14x10 + 0. 03x12 + 0. 1x17 y7 +y7 1.17

0.38x8 +0.32x9 +0.29xo -y8 +y8 =1.4

-0.03x7 +0.30x8 +0.35x9 +0.33x10 +0.05x12 -y +y9 =1.0

0.07x7 + 0. 19X8 + 0.26x9 + 0.36x10 +0.26x12 -y1 +y1= 1.1

1.0Ox1l -y11 +y11= 1.3

0. 14x7 +0. 17x8 +0.22x9 +0.38x10 +0.36x12 -Y12 +Y12 =1.4

0.08x1 + 0. 19x +0.39x +0. 19X14 +0.31x16 -y13 +y13 = 1.25

0. 15x1 +0.46x13 +0.5x14 +0.19x16 -Y14 +y14 =1.0

0.08x13 +0. 15x14 +0.77x15 -y15 +y15 =1.25

1 171x <1.317 j=1

1.0<xi. < 1.5,j =1,2,...17

y- <0.1,i =1,2,...15y- > 0,y+ > 0,i = 1,2,...15, j = 1,2,...17

The above model is a linear goal programmingmodel, and can be solved using Simplex Method (SM).To test the effectiveness of the genetic algorithm (GA),we solve this model using SM and GA. Tab. 1 depicts theoptimal improvement ratios of technical characteristicsusing SM and GA. It can be seen from Tab. 1 that there isalmost no divergence between the optimal improvementratios of technical characteristics by simplex method andGA, which indicates that GA is an effective optimizationtechnique for goal programming in QFD..

Tab.1 The optimal improvement ratios of technicalcharacteristics by sim>lex method and GA

No. 1 2 3 4 5 6 7 8 9~jbySM.5 1.5 .0 1.44 1.0 1.5 1.0 1.24 1.5

xjby GA 1.5 1.5 1.0 1.43 1.0 1.5 1.0 1.23 1.5No. 10 11 12 13 14 15 16 17

xjby SM 1.5 1.3 1.15 1.42 1.5 1.18 1.0 1.37x1byGA 1.5 1.3 1.21 1.44 1.47 1.19 1.0 1.37

It is assumed that the functional relationshipbetween customer requirement i and technicalcharacteristics (i.e. f), as well as the functional

relationship between technical characteristic j and othertechnical characteristics (i.e. gj) are linear in the abovemodel. If this assumption is relaxed, then a nonlineargoal programming model can be derived. An example ofa nonlinear programming model is given as follows,where we assume a nonlinear relation between technicalcharacteristics 1 and its related customer requirements,as well as a nonlinear relation between technicalcharacteristics 3 and its related customer requirements.

Minimize(10.48y1 + 5.7ly2 + 5.7ly3 + 8.57y4 + 5.7ly5

+10.48y6 + 8.57y7 + 10.48y8 + 5.7ly- + 2.86y10

+5.7ly1- + 10.48y12 + .95y13 + 2.86y14 + 5.7ly15)subject to:0. 17(0.7x, + 0.3x2) + 0.36X2 + 0. 17X13 + 0. 13x14

+0. 17x16 - y1 + y1 = 1.45

0.58(0.7x1 + 0.3x2) + 0.52x2 + 0. x6 -0. 14x13

0. 1x16 + 0. 03x17 y2 +y2 =1.25

0.51(0.7x3 + 0.3x3 )+ 0.49x4 - y3 +y3 =1.0

-0.05x2 + 0.41(0.7x3 + 0.3x2) + 0.49x4 + 0.05x6

+0.1x17 y4 +y4 =1.25

0. 17(0.7x1 + 0. 3x2) + 0. 14x2 + 0.86x5 - 0.09x130.09X16 y5 +y5 =1.0

0.3 1(0.7x1 + 0.3x2) + 0.23x2 + 0.37x6 + 0.06x7 0.03x12

0.09x13 - 0.09x16 + 0.23x17 - y6 + y6 = 1.5

0. 17(0.7x1 + 0.3x )+0.1x2 +0.1x + 0.24x7 + 0.07x8

+0.03x9 + 0. 14x1 + 0.03x12 + 0. x17 - y7 + y7 = 1. 17

0.38x8 + 0.32x + 0.29x1O - y8 +y8 = 1.4

-0.03x7 +0.30x8 +0.35x9 +0.33x10 +0.05x12 -y +y9 1.0

0.07x7 + 0. 19X8 + 0.26xQ + 0.36x1o +0.26x12 -Y1 +Y1= 1.1

1.00x1l -y11 +y11 = 1.3

0. 14x7 + 0. 17x8 + 0.22x9 + 0.38x10 + 0.36x12 -Y12 +y12 =1.4- 0.08(0.7x1 + 0.3x2 ) + 0. 19X2 + 0.39x13 + 0. 19x14

+0.31x16 -y13 +y13 =1.25

- 0. 15(0.7x, + 0.3x2) + 0.46x13 + 0.5x14 + 0. 9x16

-Y14 +Y14 =1.0

0. 08x13 + 0. 15x14 +0.77x15 -y15 +y15 = 1.25

1 171x <1.317 j=1

1.0<xi. <1.5,j= 1,2,....17

y- <0.1,i=1,2,...15

y- >O,y+ >0, =1,2,...15, j = 1,2 ... 17

The above optimization model is a nonlinearprogramming model, and could not be solved usingtraditional simplex method. However, it could be solvedusing GA. Table 2 depicts the optimal improvementratios of technical characteristics using GA.

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No. 1xj 1.5No. 10x 1.5

Tab.2 The optimal improvement ratiosof technical characteristics usinog GA2 3 4 5 6 7 81.5 1.5 1.0 1.0 1.5 1.0 1.3411 12 13 14 15 16 17

1.31 1.11 1.2 1.5 1.23 1.29 1.08

91.5

6 Conclusion

An important activity in constructing a HOQ is todetermine improvement ratios for the technicalcharacteristics, based on the collected customerrequirements, with a view to achieving a high level ofoverall customer satisfaction. This paper proposed a goalprogramming approach for QFD planning using geneticalgorithm. Computational experiments have shown thatGAs are effective optimization techniques for linear andnonlinear programming in QFD.

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