# Application of Fuzzy Logic in Operation Management Research

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2 International Journal of Scientific and Research Publications, Volume 4, Issue 10, October Where A and B, are linguistic values defined by fuzzy set on the ranges (universes of discourse) X and Y, respectively. The if- part of the rule " x is A " is antecedent or premise, while the then-part of the rule " y is B" is consequent or conclusion. An example such a rule might be If service is good then price of food is average In this rule, service and price are two fuzzy variables. The fuzzy values in this rule are good and average descriptions. These values are defined on universes of discourse and determine the degree of element x which belongs to the membership functions. In general, the input to an if-then rule is the current value for the input variable (in this case, service) and the output is an entire fuzzy set (in this case, average). III. METHOD AND ALGORITHM The framework of fuzzy logic new product portfolio selection model is composed of three main stages. The first stage is the new-product pre-screening. In this stage, on the basis of business strategy and new product strategy, the managers will set up a set of critical characteristics for the new product must meet to determine a new product is pass or kill. The second stage is the individual new product scoring. In this stage, due to the change in business environment, managerial goals and company s competency, the managers develop a set of criteria that new product should meet to rate the attractiveness of a new product. The third stage prioritizes projects and allocates resources into projects for portfolio selection. A stepwise description is given below: 1. Form a committee of decision-makers and collect the project related data. 2. Select a criteria for scoring project s value, strategy fitting and development risk. 3. Define linguistic variables as well as associated membership functions for measuring the project s value, strategy fitting and development risk. 4. Assess the criteria using linguistic terms and translate them into fuzzy numbers. 5. Aggregate fuzzy numbers to obtain FSFI, FVI and FRI of a new product development project. 6. Alignment of portfolio strategies. 7. Resource allocation and project selection The fuzzy logic new product portfolio selection model [FLNPPSM] can efficiently aid managers dealing with ambiguity and complexity in achieving relatively realistic and informative results in the evaluation process. IV. APPLICATION OF FLNPPSM Select criteria for scoring project s strategy fitting, value and risk. The next step in the product selection process was to decide the criteria to evaluate the proposed products. A new product selection decision depends not only on the value of the product but also on strategy fitting and development risk. Based Table 1: Characteristics of high-performance new product arenas: Major criteria Sub criteria Element criteria Strategy fit (A) Business strategy fit (A 1 ) Degree of fitting the strategy for the product line and/or business (A 11 ) Synergy with other product/business within company (A 12 ) Strategic leverage (A 2 ) Proprietary position (A 21 ) New product value (B) Competitive Marketing (B 1 ) advantages Platform for growth (A 22 ) Matches desired entry timing needed by target segments (B 11 ) Has unique or special functions to meet and/or special features to attract target segments (B 12 ) Market (B 2 ) Conforms to our sales force, channels of distribution and logistical strengths (B 13 ) attractiveness Size of the markets (B 21 ) Long-term potential of markets (B 22 ) Technological (B 3 ) Growth rates of markets (B 23 ) suitability Allows the company to use very best suppliers (B 31 ) Degree of fitting R&D skills/resources (B 32 ) Degree of fitting engineering/design skills/resources (B 33 )

3 International Journal of Scientific and Research Publications, Volume 4, Issue 10, October Potential for gaining product advantage (B 4 ) Magnitude of effect for customers (B 41 ) New products will meet customer needs (B 42 ) New product differentiated from competitive products (B 43 ) New product development risk (C) Organizational Risk (C 1 ) Lack of resource commitment (C 11 ) Lack of implementation capability (C 12 ) Technical risk (C 2 ) Organizational and/or financial impact (C 13 ) uncertainty Technical gap (C 21 ) Program complexity (C 22 ) The Project time frame (C 23 ) Competitive risk (C 3 ) Market competitiveness (C 31 ) Solutions of fully fuzzy linear system by ranking function Definition: The n x n linear system (a 11 x 1 ) (a 12 x 2 ). (a 1n x n )=b 1 (a 21 x 1 ) (a 22 x 2 ). (a 2n x n )=b 2... (a n1 x 1 ) (a n2 x 2 ). (a nn x n )=b n or in its matrix form, A x = b, is called a fully fuzzy linear system of equations (FFLSE) where the coefficient matrix A = [aij] n Ij=1 is a fuzzy matrix and b=[b 1,..., b n ] T is a fuzzy number vector and the fuzzy number vector x is the unknown to be found. Proposition Suppose that the matrices B and M = A+ A- are invertible, C- C+ and (x 1,..., x n ) T given by X j = (x J, y J, z J ), j = 1,..., n, be the solution of equation Then this solution is a nonnegative fuzzy exact solution of (3.4) if it satisfies 0 < x i < y i < z i, i = 1,..., n. {A + x + A z, By, C x + C + z) = b =(d 1, d 2, d 3 ). Numerical example Consider the following system: (1, 2, 5) (x 1, y 1, z 1 ) (3, 4, 4) (x 2, y 2, z 2 ) (0,1,2) (x 3,y 3,z 3 ) = (19,68,115) (2, 3, 5) (x 1, y 1, z 1 ) (0, 1, 11) (x 2, y 2, z 2 ) (4,5,6) (x 3,y 3,z 3 ) = (30,77,261) (2, 5, 7) (x, 1 y 1, Z1 ) (4, 6, 6) (x 2,y2,z2) (5,7,10) (x 3,y3 z3) = (61,167,253) So we must solve two following systems x1 = x2 = x3 = z1 = z2 = 261

4 International Journal of Scientific and Research Publications, Volume 4, Issue 10, October z3 = y1 = y2 = y3 = 167 Using theorem to solve fuzzy linear system: X1 y1 z1 = X2 y2 z2 = X3 y3 z3 = Linguistic variables as well as associated membership functions for measuring the project s strategy fitting, value and risk are defined. Finally, the rating scale R= {Worst [W], Very Poor [VP], Poor [P], Fair [F], Good [G], Very Good [VG], Excellent [E]} was chosen for evaluating the rating effect of the different criteria of the project s strategy fitting and value; the rating scale R' = {Extremely High [EH], Very High [VH], High [H], Fairly High [FH], Medium [M], Fairly Low [FL], Low [L]} was used for estimating the possibility of project development risk; the weighting scale W = {Very Low [VL], Low [L], Fairly Low [FL], Fairly High [FH], High [H], Very High [VH]} were used for evaluating the relative importance of the various criteria. All scales and their associated membership functions are listed in Table 2: Linguistic variables and the corresponding fuzzy numbers Performance rate Risk possibility Importance weight Linguistic variables Fuzzy number Linguistic variables Fuzzy number Linguistic variables Fuzzy number Worst (W) (0, 0, 0.2) Low (L) (0, 0, 0.2) Very Low (VL) (0, 0, 0.2) Very poor (VP) (0, 0.2, 0.4) Fairly Low (FL) (0, 0.2, 0.4) Low (L) (0, 0.2, 0.4) Poor (P) 0.2, 0.35, 0.5 Medium (M) 0.2, 0.35, 0.5 Fairly Low (FL) 0.2, 0.35, 0.5 Fairly (F) (0.3, 0.5, 0.7) Fairly High (FH) (0.3, 0.5, 0.7) Fairly (F) (0.3, 0.5, 0.7) Good (G) (0.5, 0.65, 0.8) High (H) (0.5, 0.65, 0.8) Fairly High (FH) (0.5, 0.65, 0.8) Very Good (VG) (0.6, 0.8, 1.0) Very High (VH) (0.6, 0.8, 1.0) High (H) (0.6, 0.8, 1.0) Excellent (E) (0.8, 1.0, 1.0) Extremely High (EH) (0.8, 1.0, 1.0) Very High (VH) (0.8, 1.0, 1.0) Assess the criteria using linguistic terms and translate them into fuzzy numbers: Once the linguistic variables and associated membership functions for evaluating are defined, the experts use the linguistic terms to directly assess the rating which characterizes the degree of the effect/impact of various factors on the attractiveness of the new product development project as in Table 3. Furthermore, On the basis of Table 2, fuzzy numbers parameterized by quadruples, Table 4 is the linguistic terms approximated by the fuzzy numbers of new product P1 assessed by senior manager of marketing. Aggregate fuzzy numbers to obtain fuzzy value index (FVI), fuzzy risk index (FRI) and fuzzy strategy fitting index (FSFI) of the new product development project. According to the fuzzy weighted-average definition, the FVI, FRI and FSFI can be obtained by a standard fuzzy operation. Applying the same processes, the new project P 1 was assessed by the other four seniors managers. Finally, mean operation is used for integrating the FVIs, FRIs and FSFIs under the same project assessed by different senior managers. Furthermore, the senior managers assess the other eight new product projects.

5 International Journal of Scientific and Research Publications, Volume 4, Issue 10, October Table 3: Linguistic assessment of new product P 1 given by the senior manager of marketing Sub criteria Element criteria Fuzzy rating Fuzzy weight of sub criteria A 1 A11 VG H H A12 E VH A2 A21 G VH H A22 VG VH B 1 B11 G H VH B12 VG FH B13 E H B 2 B21 VG VH VH B22 G VH B23 G H B3 B31 E FH FH B32 VG H B33 VG H B 4 B41 G H H B42 VG VH B43 G H C 1 C11 H H FH C12 VH VH C13 FL F C 2 C21 VH VH VH C22 H H C23 EH VH C3 C31 VH H H C32 H FH Fuzzy weight of sub criteria Alignment of portfolio strategies: To keep a balance between project s strategy fitting, value and development risk, under the consideration of business environments, company s business strategy and marketing direction, the steering committee of company sets a directive of the weights of project s strategy fitting, value and development risk as Very High, High and High, respectively. Table 4: Linguistic terms approximated by fuzzy numbers of new product P 1 given by a senior manager of marketing Sub criteria Element criteria Fuzzy rating Fuzzy weight of sub criteria Fuzzy weight criteria A1 A11 (0.6,0.8,1.0) (0.6,0.8,1.0) (0.6,0.8,1.0) A12 (0.8,1.0,1.0) (0.8,1.0,1.0) A2 A21 (0.5,0.65,0.8) (0.8,1.0,1.0) (0.6,0.8,1.0) A22 (0.6,0.8,1.0) (0.8,1.0,1.0) of sub B1 B11 (0.5,0.65,0.8) (0.6,0.8,1.0) (0.8,1.0,1.0) B12 (0.6,0.8,1.0) (0.5,0.65,0.8) B13 (0.8,1.0,1.0) (0.6,0.8,1.0) B2 B21 (0.6,0.8,1.0) (0.8,1.0,1.0) (0.8,1.0,1.0) B22 (0.5,0.65,0.8) (0.8,1.0,1.0) B23 (0.5,0.65,0.8) (0.6,0.8,1.0) B3 B31 (0.8,1.0,1.0) (0.5,0.65,0.8) (0.5,0.65,0.8) B32 (0.6,0.8,1.0) (0.6,0.8,1.0) B33 (0.6,0.8,1.0) (0.6,0.8,1.0)

6 International Journal of Scientific and Research Publications, Volume 4, Issue 10, October B 4 B41 (0.5,0.65,0.8) (0.6,0.8,1.0) (0.6,0.8,1.0) B42 (0.6,0.8,1.0) (0.8,1.0,1.0) B43 (0.5,0.65,0.8) (0.6,0.8,1.0) C1 C11 (0.5,0.65,0.8) (0.6,0.8,1.0) (0.5,0.65,0.8) C12 (0.6,0.8,1.0) (0.8,1.0,1.0) C13 (0.3,0.5,0.7) (0.3,0.5,0.7) C2 C21 (0.6,0.8,1.0) (0.8,1.0,1.0) (0.8,1.0,1.0) C22 (0.5,0.65,0.8) (0.6,0.8,1.0) C23 (0.8,1.0,1.0) (0.8,1.0,1.0) C3 C31) (0.6,0.8,1.0) (0.6,0.8,1.0) (0.6,0.8,1.0) C32 (0.5,0.65,0.8) (0.5,0.65,0.8) 5. Applying the fuzzy mean and spread method, the mean and variance of each project are calculated. The results are shown in Table Product Table 5: The FPAIs of the nine new product projects and their ranking Cost estimate \$ Million Fuzzy project attractive index (FPAI) µ(m) σ(m) Ranking P1 85 (0.38, 0.63, 0.83) P2 90 (0.44, 0.69, 0.88) P3 93 (0.39, 0.64, 0.85) P4 84 (0.40, 0.64, 0.84) P (0.38, 0.63, 0.84) P6 98 (0.43, 0.69, 0.87) P7 86 (0.41, 0.66, 0.86) P8 83 (0.39, 0.62, 0.83) P9 97 (0.44, 0.62, 0.83) V. CONCLUSIONS This research has highlighted the decision support system for selection of new product portfolio. Because of complexity, incomplete information and ambiguity in the portfolio selection context, a fuzzy logic-based portfolio selection model, which applies linguistic approximation and fuzzy arithmetic operation, has been developed to address the new product portfolio selection. The method incorporates the multiplicity in meaning and ambiguity of factor measurement while considering important interactions among decision levels and criteria. The company and managers involved in the case study illustrated in this study were generally pleased with the approach. [3] Martino, J. P. (1995) R&D Project Selection. Wiley, New York, NY. [4] Bard, J. F., Balachandra, R., and Kaufmann, P.E. (1988) An interactive approach to R&D project selection and termination, IEEE Transactions on Engineering Management 35, pp [5] Kao, C. and Liu, S. T. (1999) Competitiveness of manufacturing firms: an application of fuzzy weighted average, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 29(6), pp [6] Chen, L. H. and Chiou T. W. (1999) A fuzzy credit-rating approach for commercial loans: a Taiwan case, Omega 27(4), pp [7] Lin, C. T. (2000) A knowledge-based method for bid/no-bid decision making in project management, Proceedings of PMI Research Conference 2000 Paris, France: pp [8] Cooper, R. G., Edgett, S. J., and Kleinschmidt, E. J. (2002) Optimizing the stage-gate process: What best-practice companies do-ii. Research Technology Management 45(6), pp. 43. [9] Chen, S. J. and Hwang, C. L. (1992) Fuzzy Multiple Attribute Decision Making Methods and Applications. Berlin Heidelberg: Springer-Verlag. REFERENCES [1] Griff Abbie (1997) PDMA research on new product development practices: Updating trends and benchmarking best practices, Journal of Product Innovation Management, 14, pp , [2] Cooper, R.G., Edgent, S. J. and Kleinschmidi, E. J. (1999) New product portfolio management: Practices and performance, Journal of Product Innovation Management, 16, pp AUTHORS First Author Preeti Kaushik, Assistant Professor, Inderprastha Engineering College, Ghaziabad (U.P), Mobile No.:

7 International Journal of Scientific and Research Publications, Volume 4, Issue 10, October

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