Makara Journal of Technology

Volume 16 Number 1 Article 2

4-1-2012

Concurrent of Tolerance Synthesis and Process Selection for Products with Multiple Quality Characteristcs Considering

Mohamad Imron Mustajib Systems Laboratory, Department of Industrial Engineering, University of Trunojoyo Madura, Kamal, Bangkalan 69162, Indonesia, [email protected]

Follow this and additional works at: https://scholarhub.ui.ac.id/mjt

Part of the Chemical Engineering Commons, Civil Engineering Commons, Computer Engineering Commons, Electrical and Electronics Commons, Metallurgy Commons, Ocean Engineering Commons, and the Structural Engineering Commons

Recommended Citation Mustajib, Mohamad Imron (2012) "Concurrent Engineering of Tolerance Synthesis and Process Selection for Products with Multiple Quality Characteristcs Considering Process Capability," Makara Journal of Technology: Vol. 16 : No. 1 , Article 2. DOI: 10.7454/mst.v16i1.1040 Available at: https://scholarhub.ui.ac.id/mjt/vol16/iss1/2

This Article is brought to you for free and open access by the Universitas Indonesia at UI Scholars Hub. It has been accepted for inclusion in Makara Journal of Technology by an authorized editor of UI Scholars Hub. MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14

CONCURRENT ENGINEERING OF TOLERANCE SYNTHESIS AND PROCESS SELECTION FOR PRODUCTS WITH MULTIPLE QUALITY CHARACTERISTCS CONSIDERING PROCESS CAPABILITY

Mohamad Imron Mustajib

Manufacturing Systems Laboratory, Department of Industrial Engineering, University of Trunojoyo Madura, Kamal, Bangkalan 69162, Indonesia

E-mail: [email protected]

Abstract

The existences of variances that are very difficult to be removed from manufacturing processes provide significance of tolerance to the product quality characteristics target of customer functional requirement. Furthermore, quality loss incurred due to deviation of quality characteristics of the target with a specified tolerance. This article discusses the development of concurrent engineering optimization model of tolerance design and manufacturing process selection on product with multiple quality characteristics by minimizing total costs in the system, namely total manufacturing cost and quality loss cost as functions of tolerance, also rework and scrap costs. The considered multiple quality characteristics have interrelated tolerance chain. The formulation of proposed model is using mixed integer non linear programming as the method of solution finding. In order to validate of the model, this study presents a numerical example. It was found that optimal solution are achieved from proposed model in the numerical example.

Abstrak

Rekayasa Simultan Sintesis Toleransi dan Pemilihan Proses untuk Produk dengan Multi-karakteristik Kualitas yang Mempertimbangkan Kapabilitas Proses. Keberadaan variansi yang sangat sulit untuk dihilangkan dalam proses manufaktur memberikan peran penting adanya toleransi terhadap target karakteristik kualitas produk yang menjadi kebutuhan fungsional bagi konsumen. Selanjutnya timbul kerugian kualitas yang disebabkan oleh penyimpangan karakteristik kualitas dari target dengan toleransi yang ditetapkan. Makalah ini membahas pengembangan model optimisasi rekayasa simultan desain toleransi dan pemilihan proses manufaktur pada produk dengan multi karakteristik kualitas untuk meminimasi total ongkos dalam sistem, yaitu total ongkos manufaktur dan ongkos kerugian kualitas yang merupakan fungsi dari toleransi serta ongkos rework dan ongkos scrap. Karakteristik kualitas produk yang diperhatikan dalam penelitian ini memiliki rantai toleransi yang saling berkaitan (interrelated chain). Formulasi model yang dikembangkan menggunakan mixed integer non linear programming sebagai metode pencarian solusi.

Keywords: optimization model, quality loss, rework, scrap, tolerance

1. Introduction design function to define minimum and maximum values allowable for the product to work properly. In Tolerance synthesis is a critical issue in design and quality engineering, tolerance allocation reflects a trade- manufacturing stages, which affects both of product and off between customer requirements and the ability of process design because the tolerance is the producers to satisfy them. Thus, the loss of both between product requirements and manufacturing cost customers and producers can be balanced. [1]. Tolerance synthesis, which is also called tolerance allocation, is carried out in a direction opposite to Tolerance synthesis in the product design phase focuses tolerance analysis; from the tolerance of the function of on efforts to meet products functional requirements with interest to the individual tolerances [2]. According to the tolerance values as tight as possible. Due to tight Gryna et al. [3] tolerance is set by the engineering tolerance desired in the design phase, it is usually less

7 8 MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14

considering the process capability of production. While tolerances but also lower manufacturing cost. On the tolerance synthesis at the process design/planning phase other hand, when the precision limit of the process and is more focused on the ease in performing the tolerance limits of final product are based on worst-case production process. Thus, in this stage the desired criteria it tends to require tighter component tolerances tolerances allocation is set as loose as possible. with relatively expensive manufacturing cost, compared Meanwhile, during a production process, variance leads with statistical approaches (root sum square criteria). to imperfections of the product. Therefore, tolerance synthesis must be considered in both product and Altough statistical approaches improved performance, process design. many researchers have not addresed the problem of process selection by considering process capability. The aim of tolerance synthesis is to determine the Process capability indices, including Cp, Cpk, and Cpm, maximum and minimum deviation limits of the product have been proposed in the manufacturing industry to quality characteristics due to imperfections of the provide numerical measures on whether a process is production process. Costs consideration and the capable of reproducing items meeting the manufacturing capability of production process to tolerance allocation quality requirement preset in the factory [19]. In have made the problem become more complex. The addition, tolerance allocation could minimize rework tolerance values will affect the ability of components to and scrap costs, but most of the researches in tolerance be assembled into a final product (assembly), process synthesis did not consider the scrap and rework rates of selection, tooling, set-up costs, operator skills, the process resulted from allocated tolerance. inspection and measurement, scrap and rework. Loose tolerance facilitates implementation of the production Based on those shortcomings of existing model of process resulting in lower production costs. tolerance synthesis of multiple quality characteristics in Consequently, if tolerance is too loose it will reduces previous studies and the need for reducing the quality performance of the product. On the other manufacturing cost, we propose the development of hand, strict tolerance will improve product performance, concurrent engineering optimization model of tolerance but scrap and rework costs are higher. A lower process synthesis and process selection by considering process capability causes more deviation of quality capability and non conformance rate; scrap and rework characteristics from maximum and minimum limits. rates. The overall objective of the model is minimizing Thus, allocation of optimal tolerance values should take total cost of the system, i.e. manufacturing and quality into account to quality loss and manufacturing cost by costs. The development was carried out by enhancing considering design and tolerance constraints and also on the model of previous study [8,9,11-14] with respect other constraints [4]. Quality loss function has been to process capability and statistical approaches, also introduced by Taguchi [5] in order to make it easy for taking into account the quality loss cost is a function of designer to make trade-off between manufacturing and tolerance as well as rework and scrap cost. quality loss cost. Moreover, process (machine) selection problem in the planning process can be done by 2. Methods minimizing total cost of manufacturing and quality loss cost. Taguchi in Taguchi et al. [5] defined quality of a product as the minimum product loss imparted by the A major focus in tolerance design model was for a society from the time product shipped. Loss due to the single quality characteristic by incorporating quality characteristics of the product quality cannot meet loss [4,6-9]. Recently, Mustajib and Irianto [10] customer needs and satisfaction. Taguchi argues that propossed an integrated model for tolerance allocation there is a loss (in the form of cost and quality) when the and selection process in multi-stage processes. quality characteristic deviating from the target product, Meanwhile, the importance of determining tolerances although the quality characteristic deviating from the for multiple quality characteristics products has been target is still within the specifications or the specified demonstrated by Lee and Tang [11], Chou and Chen tolerances. The losses arise because of the waste, loss of [12], Huang et al. [13], and Peng et al. [14]. opportunity (opportunity cost) and cost when the Furthermore, many researches [15-17] extended this product fails to meet the target value specified by work on assembly products with multiple quality quality characteristics. From the producers’ view, these characteristics based on statistical approach. losses can be quantified in the form of costs incurred by producers of quality; the cost of rework and scrap, cost Statistical approach estimates the accumulation of of inspection. From customers’s view, these losses are tolerance (stack-up condition) on the assembled form of disatisfaction of the product variability product, which is based on the fact that the probability received. Based on the variability of the product quality, of the component is at an extreme lapse very low it can be quantified in monetary terms. Moreover, the tolerance [18]. The impact of this is not only on tighter quantification of the quality loss cost can be done by tolerance assembled product with looser component using Taguchi quadratic loss function approach, Eq. 1: MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14 9

2 The mean of nominal dimension of the ith component L(y) = r(y − m) (1) (Ki) is µ1,...µi,...µI. Expanding of right-hand side Eq. 4 with C r = in Taylor series around µ1,...µi,...µI, by ignoring the 2 (2) t higher order we obtained [21]: ⎛ ⎞ I ⎜ ∂Yq ⎟ In Eq.1 r is a constant that converts the characteristics Yq = M q + ∑ ⎜ ΔKi ⎟ (5) become the characteristics of engineering costs which i=1⎜ ∂Ki ⎟ ⎝ μi ⎠ is the loss coefficient of the final product quality. This This study consider that each of components of the ith costant is estimated based on the cost of rework (C) component (K ) can be produced by alternative where is required when quality characteristics of the i processes 1,...j,...J, so that quality characteristic final product y deviating from the target m, but still Y =f (K ,...K ,...K ) in Eq. 5 become: within an acceptable tolerance limits customers (t). q q 11 ij IJ Further determination of the optimal value of tolerance ⎛ ⎞ I J ⎜ ∂Y ⎟ should pay attention to aspects of the quality loss cost q Yq = M q + ∑ ∑ ⎜ ΔKij ⎟ (6) L(t) and the manufacturing cost P(t) with respect to i=11j= ⎜ ∂K ⎟ ⎜ ij μ ⎟ design constraints and tolerance of other constraints that ⎝ i ⎠ are relevant. In Fig. 1, L(t) states the cost of quality loss as a function of tolerance, whereas P (t) states the cost Furthermore, deviation of of the qth quality of manufacturing as a function of tolerance. characteristic (Dq) can be estimated based on the Furthermore, P(t) also states the overall cost of difference between the qth quality characteristic (Yq) manufacturing on the quality characteristics with with the qth quality characteristic target (Mq): tolerance t. ⎛ ⎞ I ⎜ ∂Yq ⎟ Dq = Yq − M q = ∑ ⎜ ΔKi ⎟ (7) Quality Characteristics. The problem, however, in Eq. i=1⎜ ∂Ki ⎟ 1 above is applied only for products with single quality ⎝ μi ⎠ characteristic, while for products with multiple quality Thus, the vector of multiple quality characteristics T characteristics the value of quality loss can be expanded deviation is D=[D1,...Dq,...DQ] . Based on the vector of [9]: multiple quality characteristics deviation, Eq. 3 can be L()()()Y = Y − M T R Y − M (3) written with Eq. 8 to evaluate the multivariate quality loss function with: T Eq. 2 is general; it means that each of quality L()D = D RD (8) characteristics are not considered whether it is or T correlated or not. In Eq. 3 above, Y=[Y1,...Yq,...YQ] C = DT RD (9) states the vector for the qth quality characteristic and the vector of the qth quality characteristic target is denoted T Elements of R matrix can be calculated based on a basic by M=[M1,...Mq,...MQ] , and R is r x r matrix for constant of the qth quality characteristic. correlation in Eq. 2 that are extended to be loss of multiple quality characteristics. This expansion is The value of the qth quality characteristic (Y ) can be formulated with [9]: q p p estimated from the dimension of ith component. This (r) (r) ∑ ∑ Rqi Dq D = C r , relationship can be stated in Eq. [4]: q=11i= i (10) Yq = f q (K1,KK i KK I ). (4) with r = 1, 2,K, p(p +1)/ 2 Rqi represents the qth row and the ith column in matrix elements of R. When both set of product quality (r) (r) deviation of D (denoted by D q ) and vector of D i and quality loss cost (Cr) are known, then the matrix elements of R (denoted by Rqi) can be finded. In case p=2, it is required three data sets for matrix elements.

Let we consider again that Eq. 7 can be expressed as the 2 variance of the qth quality characteristic (σq ) with: 2 ⎛ ⎞ ⎜ ⎟ I J ∂Yq 2 ⎜ ⎟ 2 σ q = ∑ ∑ ΔKij (11) Figure 1. The Solution Space for an Unconstrained i=1j=1⎜ ∂K ⎟ ⎜ ij μ ⎟ Space [4] ⎝ ij ⎠

10 MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14

2 ΔKi in Eq. 11 is nothing but the variance of the ith 2 2 ⎛ ⎞ component dimension that produced by the jth process I J ⎜ ∂Y ⎟ ⎛ t ⎞ 2 2 2 q ∂Ys ⎜ ij ⎟ (18) (σij ) which is statistically can be written with: Cov()σ r ,σ s = ∑ ∑ ⎜ ⎟ i=1 j=1⎜ ∂K ∂K ⎟ ⎜ 5.51Cpm ⎟ 2 ⎜ ij μ ij μ ⎟ ⎝ ij ⎠ ⎛ ⎞ ⎝ i ij ⎠ ⎜ ⎟ I J ∂Yq σ 2 = ⎜ ⎟ σ 2 (12) q ∑ ∑ ij Meanwhile, the variance covariance matrix can be i=1j=1⎜ ∂Kij ⎟ ⎜ μij ⎟ 2 ⎝ ⎠ expressed by σ q vector of parameter [11]:

⎡σ 2 Cov σ 2,σ 2 Cov σ 2,σ 2 ⎤ Eq. 12 is in accordance with the statistical criteria ⎢ 1 ( 1 2 ) L ( 1 p )⎥ ⎢ 2 2 2 ⎥ approach (root sum square criteria) which states that the 2 Cov(σ ,σ ) σ L M Vσq ()t = ⎢ 1 2 2 ⎥ (19) dimension of assembly product consists of variance ⎢ ⎥ ⎢M M O M ⎥ dimensional of its components. Furthermore, if we ⎢ 2 2 2 ⎥ consider that the qth quality characteristic derived from ⎣Cov(σ1 ,σ p ) L L σ p ⎦ one or more components of interrelated dimensions chain, then the covariance of the rth and rth quality Quality loss cost. Expected cost of quality loss due to characteristic can be finded by Eq. 13: deviation of quality characteristic from its target can be 2 estimated by tracking the variance covariance matrix ⎛ ⎞ [11]. Thus, the quality loss expectations in Eq. 3 can be I J ⎜ ∂Y ⎟ 2 2 ⎜ q ∂Ys ⎟ 2 calculated by tracing the variance covariance matrix Cov(σ r ,σ s )= ∑ ∑ σ ij (13) i=1j=1⎜ ∂K ∂K ⎟ based on Eq. 17 which can be formulated in Eq. 20. ⎜ ij μ ij μ ⎟ ⎝ ij ij ⎠ 2 E[L(tij )]= Trace[RVσ q (tij )] (20)

Variance for the ith component produced by the jth 2 Cost of non conformance. The process alternatives and process (σij ) in Eq. 12 and 13 can be estimated by considering process capability index (Cp): tolerance allocation have possibility to produce non fonformance if the process variances are out from USLi − LSLi Cpi = (14) minimum and maximum values allowed. Costs of non 6σ i conformance includes rework and scrap cost. By Where are USLi and LSLi state upper and lower assuming that all variance for the ith component specification limits of the ith component. By produced by the jth process are normally distributed, considering the tolerance of one side only, Eq. 14 can be then the probability of non concormance (ḡij) is simplified to obtain variances and relationships to the determined by probability meets tolerance limits (gij): jth component tolerance (t ) in Eq 15: j g = 1 − g (21) 2 ij ij ⎛ tij ⎞ σ 2 ()t = ⎜ ⎟ (15) ij ij ⎜ 3Cp ⎟ ⎝ ij ⎠

This research consider that on average process have a mean shift of 1.5σ. It is consistent with Motorola’s original Six Sigma program which stipulate that a process was said to be six sigma quality level when Cp=2 and Cpk=1.5. By assuming 1.5σ process shift, Taguchi process capability index can be obtained [20]: Cp Cpm = = 0.5447Cp 2 ⎛1.5σ ⎞ (16) 1− ⎜ ⎟ ⎝ σ ⎠

Moreover, the variance of the qth quality characteristic in the Eq. 12 and 13 can be rewritten into Eq. 17 and 18 by substituting Eq. 15 into Eq. 12 and 13 as follows. 2 ⎛ ⎞ 2 I I ⎜ ∂Y ⎟ ⎛ t ⎞ 2 ⎜ q ⎟ ⎜ ij ⎟ σq ()tij = ∑ ∑ ,∀q ∈ Q (17) i=1i=1⎜ ∂K ⎟ ⎜ 5.51Cpm ⎟ ⎜ ij τ ⎟ ⎝ ij ⎠ ⎝ ij ⎠ Figure 2. Standarized Process Tolerances MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14 11

If the product is not to meets tolerance (ḡij) is classified Notations into two types, namely: (1) ḡ1ij, for probability where i. Indexs: tolerance is not met owing to undersize, and (2) ḡ2ij, for q = qth quality characteristic probability where tolerance is not met due to oversize. i = ith component j = jth process alternative Thus, for symmetric bilateral tolerance (see Fig 2), both Q = set of quality characteristics of rework rate and scrap rate can be obtained by I = set of components following equation: J = set of process alternatives 1 ii. Decision variables: g1ij = g 2ij = .g ij (22) 2 Xij = 1, if the ith component is processed by using jth process alternative. And 0, if Model formulation. The objective function the model not. is minimizing total costs (TC), which is the sum of tij = tolerance of the ith component which is manufacturing cost and quality costs. Quality costs produced by using jth process includes quality loss cost and failure costs. The cost of alternative failure arises when the dimensional tolerance cannot be iii. Performance: met and results in component dimensions are undersized TC = total costs or oversized. In mechanical product, if the dimension is iv. Parameters: undersize at the features likes a hole, a step, a groove, A , B ,C = coefficients of cost-process tolerance and a slot, it is need to be reworked with rework cost, ij ij ij r function for the ith component by using c ij. On the other hand, if the dimension is oversize, it is s jth process alternative to generate need to be scrapped with certain of scrap cost, c ij. If the dimensional tolerance is bilateral, then probability tolerance tij process cannot meets the dimensional tolerance (ḡij) R = p x p positive definte matrix for quality due to oversized is equal to probability cannot meets the loss constant the qth quality dimensional tolerance due to undersized component characteristic [22]. Therefore, complete model of the objective 2 Vσ 2 = vector parameter σq of the variance function can be expressed by Eq. 23. Meanwhile, all of q covariance matrix the model constraints are formulated by Eq. 24 to 27 2 = variance for the ith component produced respectively. σ ij by the jth process

Yq = qth quality characteristic ⎡ IJ⎛ −B t ⎞ ⎤ ij ij ∂Y = partial derivative of qth quality ⎢ ∑ ∑ ⎜ Aij e + Cij ⎟X ij + ⎥ q i=1j=1⎝ ⎠ characteristic with respect to ith ⎢ ⎥ ∂Kij ⎢ 2 ⎥ component produced by the jth process Min TC = Trace[]RVσ q ()tij X ij + (23) ⎢ ⎥ Cpmij = Taguchi process capability index for the ith component produced by the jth ⎢ IJ r r IJ s s ⎥ ⎢ ∑ ∑ Cij g ij X ij + ∑ ∑ Cij g ij X ij ⎥ process ⎢i=1j=1 i=1j=1 ⎥ ⎣ ⎦ CpmYq = Taguchi process capability index for the subject to: qth quality characteristic 1) Quality spesification of design r = rework cost for the ith component cij 2 2 2 produced by the jth process I J ⎛ ∂Y ⎞ t t ⎜ q ⎟ ij Yq s = scrap cost for for the ith component ∑ ∑ x ≤ , ∀q ∈Q (24) c ⎜ ∂K ⎟ Cpm ij Cpm ij produced by the jth process i=11j= ⎝ ij ⎠ ij Yq tYq = tolerance design of qth quality 2) Process precision limits characteristic min max t max = upper tolerance limit for the ith tij ≤ tij ≤ tij , ∀i ∈ I ∀ j ∈ J (25) ij component produced by the jth process t min = lower tolerance limit for the ith 3) Alternative process ij component produced by the jth process J s = scrap rate of non conforming to ∑ X ij = 1, ∀i ∈ I (26) g ij J =1 specification for the ith component produced by the jth process.

4) Binary decision variables r = rework rate of non conforming to g ij X ∈[]0,1 , ∀ ∈ I, ∀ ∈ J specification for the ith component ij i j (27) produced by the jth process. 12 MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14

3. Results and Discussion is called spring constant (k) or spring stiffness. Spring constant can be calculated by dividing the changing This section presents numerical example which is force and deflection. related to quality characteristic of one mechanical F k = (28) product, helical spring. This product is a type of ΔL compression spring made of round wire (diameter, dw), wrapped in a straight line (length, L), cylindrical form Because the loading is transmitted through wire spring, with outer diameter Do and the number of coil springs is it will cause torsion. Therefore, the stress arising in the n with constant distance between one coil with the wire is the torsion shear stress (τ) which can be derived others coils (Fig. 3). The use of helical spring is often from the classical equation [23]: found in a variety of equipments. In motor , the Tc τ = (29) springs are usually used in suspension systems, engine J valve springs and the clutch plate. Meanwhile, in the p manufacturing process it is often used for the striper and the control valve of hydraulic and pneumatic systems. Meanwhile, the angular deflection (θ) on the wire can The springs are also widely used in small appliances be calculated by using Eq. 30 such as on electrical switches, pen ballpoint, etc. ⎛ Do ⎞ ⎜ F ⎟πDon 2 TL ⎝ 2 ⎠ 16FDo n The main performance of a helical spring is an aspect θ = = = (30) GJ p π 4 d 4 G that must be met within the design specifications. The dwG w purpose of helical spring design is to determine the 32 dimensions of spring that can be operated at the limit load (F) and certain axial deflection ( ΔL ). Allowable Where T is the applied torque, G is the modulus of stress depends on the material and dimensions of helical elasticity of spring material, and Jp states moment of spring. Thus, the purpose of a helical spring design is polar inertia of wire material. Furthermore, axial quality specifications that are expected by the designers deflection ( ΔL ) can be obtained by Eq. 31. as customer needs. D ΔL = θ o (31) 2 Compressive force imposed axially on the helical spring (F) will cause axial deflection ( ΔL ). The relationship Note that the precision of spring outer diameter (Do) is between the resulted force and deflection influenced by the precision of spring internal diameter (Di) and the spring wire diameter (dw). We consider that the quality characteristic of outer diameter helical spring (Do) can be formulated by Eq. 32. Do = Di + 2dw (32)

By looking back to Eq. 28 as the basic equation represents spring performance, then the next quality characteristic helical spring which states the spring constant (k) or the spring stiffness can be formulated by Eq. 33 through substitution the value of θ in Eq. 28 and 32 into Eq. 31 Gd 4 k = w 3 (33) 8()Di + dw n Thus, both Eq. 32 and 33 above are the formulations for two quality characteristics of the helical spring. Furthermore, D0 is called the quality characteristic of Y1 and k is called the quality characteristic of Y2.

Partial derivative quality characteristic of Y1 and Y2 to the component dimensions Di, dw, and n which is resulted by process alternative (see Table 1) can be calculated as follows.

∂Y1 ∂D0 Figure 3. Helical Spring = = 1 (34) ∂K1 ∂Di MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14 13

Table 1. Cost Parameters, Tolerance Limits, and Non Conformance Rates

Process Cost Parameter Lower Upper Rework Dimension Scrap Rate Alternative A B C Tolerance Tolerance Rate dw 1 1.581 78.735 1.44 0.018 0.80 0.003 0.001 2 14.132 7.262 1.44 0.020 0.82 0.002 0.002 Di 1 17.364 39.333 0.50 0.218 1.20 0.007 0.003 2 78.735 3.124 0.55 0.230 1.26 0.006 0.002 n 1 65.634 214.097 1.50 0.220 0.26 0.002 0.004 2 61.138 20.682 1.50 0.220 0.25 0.004 0.003

Table 2. The Optimal Solutions of Three Process ∂Y1 ∂D0 = = 2 (35) Capability Scenarios ∂K 2 ∂d w Cpm In similar manner, the partial derivatives for Y2 are as 0.888 1 1.109 follows: t1 0.031 0.032 0.032 4 t2 0.218 0.230 0.218 ∂Y2 ∂k 37500dw = = − = −0.256 (36) t3 0.220 0.220 0.220 ∂K ∂D 4 1 i n()Di + dw X1 1 2 1 3 4 X2 1 2 1 ∂Y2 ∂k 50000dw 37500dw X 1 1 1 = = − = 3.5 (37) 3 ∂K ∂d 3 4 TC 19.40203 18.4911 16.38516 2 w n()Di + dw n()Di + dw

4 ∂Y2 ∂k 12500dw = = − = −0.238 (38) ∂K ∂n 4 3 R12 = R21 3 n ()Di + dw ⎡ 2 2 2 2⎤ ⎛ 3 ⎞ ⎛ 1⎞ ⎛ 3 ⎞ ⎛ 2 ⎞ ⎢C3 − C1⎜ D1 ⎟ ⎜ D1⎟ − C1⎜ D2 ⎟ ⎜ D2 ⎟ ⎥ ⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎥ = ⎣ ⎦ 3 (3) The data obtained from previous study [12], for ⎜⎛2D D ⎟⎞ (44) ⎝ 1 2 ⎠ example, are the number of coils (n ≈ 10), modulus of ⎡60 − 60 0,58 2 0,88 2 − 60 0,47 2 0,772⎤ elasticity of helical spring material, G = 100.000 ⎢ ()() () ⎥ = ⎣ ⎦ = 21.24 kg/mm, and quality tolerance of product design (Table ()2 × 0.58 × 0.47 1). Suppose that the values of product design tolerances are D0=1.107 and k=1.106. Furthermore, the quality Meanwhile, elements of the variance covariance matrix characteristic of Y1 and Y2 deviate from their target can be obtained through Eq. 17 and 18. In case Cpm=1, vector by values: the expected of quality lost cost can be generated by using of MathCAD 14.0 software as follows: T (1) D(1) = ⎡D , 0⎤ = []0.88, 0 T (39) ⎣⎢ 1 ⎦⎥ 2 2 2 E[L(tij )]= 71.36t11 + 73.93t12 +144.66t21 + T (2) ⎡ (2) ⎤ T (45) D = 0, D = []0, 0.77 (40) 147.23t 2 +123.72t 2 +123.72t 2 ⎣⎢ 2 ⎦⎥ 22 31 32 T (3) ⎡ (3) (3) ⎤ T D = D , D = []0.58, 0.47 (41) Finally, decision variables of the optimization model of ⎣⎢ 1 2 ⎦⎥ tolerance design and manufacturing process selection on will result in product failure and cause a loss of 60$. helical spring can be obtained by solved in the proposed

model by using the LINGO software package. The Furthermore, elements of the quality loss constant optimal tolerances and process with another two process matrix can be obtained through Eq. 10. capability scenarios are given in Table 2. C 60 R = 1 = = 77.48 11 2 2 The optimization result data in Table 2 indicates a more ⎛ (1) ⎞ 0.88 (42) ⎜ D ⎟ reasonable relationship between process capability and ⎝ 1 ⎠ total costs in the system. As can be seen, the impact of C 60 R = 2 = = 101.20 process capability increases has significantly reduced 22 2 2 total costs. On the other hand, the changes of process ⎛ (2) ⎞ 0.77 (43) ⎜ D ⎟ capability have changed tolerances allocation and ⎝ 2 ⎠ selected processes. 14 MAKARA, TEKNOLOGI, VOL. 16, NO. 1, APRIL 2012: 7-14

4. Conclusion [6] S. Shin, B.R. Cho, Eng. Optimiz. 40/11 (2008) 989. [7] P. Muthu, V. Dhanalakshmi, K. Sankaranara- This study proposed the development of concurrent yanasamy, Int. J. Adv. Manuf. Tech. 44 (2010) engineering optimization model of tolerance synthesis 1154. and process selection by considering process capability [8] K. Sivakumar, C. Balamurugan, S. Ramabalan, Int. and costs of non conformance. The objective of the J. Adv. Manuf. Technol. 53 (2011) 711. doi: model is minimizing total cost of the system, i.e. 10.1007/s00170-010-2871-4. manufacturing and quality costs. Formulation of the [9] K. Sivakumar, C. Balamurugan, S. Ramabalan, model developed using mixed integer non linear Comput. Aided Design, 43 (2011) p.207. programming as the method of solution finding. In [10] M.I. Mustajib, D. Irianto, J. Adv. Manuf. Sys. 9/1 order to validate of the developed model, this study (2010) 31. doi: 10.1142/S02196867000-1788. presents a numerical example. It was found that optimal [11] C.L. Lee, G.R. Tang, Mech. Mach. Theory. 35 solution are resulted from proposed model in the (2000) 1658. numerical example. The optimization results data in [12] C.Y. Chou, C.H. Chen, Int. J. Prod. Econ. 70 indicate that there were relationship between process (2001) 279. capability and total costs in the system. Meanwhile, the [13] M.F. Huang, Y.R. Zhong, Z.G. Xu, Int. J. Adv. impact of process capability increasing have Manuf. Technol. 25 (2005) 714. significantly reduced total cost. Moreover, the changes [14] H.P. Peng, X.Q. Jiang, X.J. Liu, Int. J. Adv. Manuf. of process capability have changed tolerance allocation Technol. 39/1 (2007) 1. doi:10.1007/s00170-007- and selected process. In summarize, this finding is 1205-7. promising and should be explored with other methods [15] P.K. Singh, S.C. Jain, P.K. Jain, Comput. Ind. 56 for finding better optimal solutions. It is possible to use (2005) 179. computational intelligence to enhance the method of [16] H.P. Peng, X.Q. Jiang, X. J. Liu, Int. J. Prod. Res. solution finding for proposed model. 46/24 (2008) 6963. [17] I. González, I. Sánchez, Mech. Mach. Theory. 44 (2009) 1097. References [18] S.S. Lin, H.P. Wang, C. Zhang, In: H.C. Zhang, Advanced Tolerancing Techniques, Willey Series [1] H.C. Zhang, Advanced Tolerancing Techniques, in Engineering Design and Automation, John Willey Series in Engineering Design and Willey & Sons, Inc., New York, USA, 1997, p.261. Automation, John Willey & Sons, New York, USA, [19] W.L. Pearn, M.H. Shu, Int. J. Adv. Manuf. 1997, p. 207 Technol. 23 (2005) 116. [2] Y.S. Hong, T.C. Chang, Int. J. Prod. Res. 40/11 [20] C.X. Feng, R. Balusu, Con. Innov. Design Manuf. (2002) 2425. 103 (1999) 1. [3] M.F. Gryna, R.C.H. Chua, J.A. DeFeo, Juran’s [21] D.C. Montgomery, Introduction to Statistical Quality Planning and Analysis: For Enterprises th th Quality Control, 4 ed., John Willey & Sons, Inc., Quality, 5 ed., McGraw-Hill, Inc., New York Hoboken, New Jersey, USA, 2001, p.392. 2007, p.655. [22] Y.H. Lee, C.C. Wei, C.B. Chen, C.H. Tsai, Eng. [4] C.X. Feng, J. Wang, J.S. Wang, Comput. Ind. Eng. Optimiz. 32 (2000) 619. 40 (2001) 15. [23] R.L. Mott, Machine Elements in Mechanical [5] G. Taguchi, S. Chowdhury, Y. Wu, Taguchi’s Design, 4th ed., Person Education, Inc., Upper Quality Engineering Handbook, John Willey & Saddle River, New Jersey, USA, 2004. p.80. Sons. Inc., Hoboken, New Jersey, USA, 2005, p.134.