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Abstract

Optimal component tolerances can reduce product cost and improve enterprise competitiveness. In order to obtain optimal tolerances accurately and efficiently, analytical methods are applied to solve tolerance optimization model. In this article, both manufacturing cost and quality loss are included in the total cost, and two kinds of constraints, including assembly tolerance constraint and process accuracy constraints, are considered in the tolerance optimization model. In order to allocate the optimal tolerances among the related components, the following procedure is presented. First, the two kinds of constraints are ignored, and analytical method is applied to find the initial values of component tolerances. If the initial component tolerances cannot satisfy assembly tolerance constraint, the Lagrange multiplier method is executed and the Lambert W function is applied to solve the corresponding equations to obtain the new values of component tolerances. Then, process accuracy constraints are satisfied through the adjustment of component tolerances. Finally, an example is used to demonstrate the effectiveness of the method proposed in this article.

Keywords

Tolerance optimization quality loss Lagrange multiplier method

Introduction

Because the component tolerances have significant influence on production cost, tolerance design becomes an important part of product development. Tighter tolerance would result in high manufacturing cost, while looser tolerance may lead to poor performance. Therefore, component tolerance should be optimized, and this issue has received the attention of many researchers in the last decades.

In these works about tolerance design, an important issue is how to establish the proper relationship between the manufacturing cost and tolerance. In order to establish cost–tolerance function, first, the discrete manufacturing cost data should be obtained. Then, curve-fitting techniques are adopted to fit the discrete data, and proper mathematical model is obtained. Over the past decades, many cost–tolerance functions have been proposed. These cost functions proposed prior to 1990 are referred to as traditional cost functions, while those proposed after 1990 are named as nontraditional cost functions.¹ Wu et al.² and Chase et al.³ introduced several traditional cost functions, such as reciprocal function, reciprocal square function, reciprocal power function, exponential function, reciprocal power and exponential hybrid function, linear function, and piecewise linear function. Dong et al.⁴ proposed several nontraditional cost functions, including combined reciprocal powers and exponential function, combined linear and exponential function, B-spline curve function, cubic polynomial function, fourth-order polynomial function, and fifth-order polynomial function.

In the early studies about tolerance optimization, only manufacturing cost was considered and tolerance was optimized to minimize manufacturing cost.^2–5 In these years, quality loss has attracted more attention.^6–16 In order to estimate monetary loss incurred by poor product’s performance, Taguchi et al.⁶ proposed quality loss function, which is a quadratic function. At present, the sum of manufacturing cost and quality loss is usually treated as the objective function in the studies about tolerance optimization.^7–16

When tolerance optimization models are established, suitable methods should be employed to obtain optimal tolerances. Over the past decades, a number of numerical optimization methods were adopted to calculate optimal tolerances, including nonlinear programming,⁹ integer programming,¹⁰ genetic algorithm,¹¹ continuous ant colony algorithm,¹² particle swarm optimization,^13,14 hybrid of simplex method and particle swarm optimization algorithm,¹⁵ self-organizing migration algorithm,¹⁶ and so on.

Although numerical methods have been widely used in tolerance optimization, analytical method should be the first choice because of its high efficiency and accuracy. In fact, the Lagrange multiplier method seems to be the oldest analytical method for constrained nonlinear programming problems.¹ Because the Lagrange multiplier method can generate closed-form optimal solution, it has been studied by a few of researchers.^{1–3,11,17,18} Chase et al.³ modeled the manufacturing cost–tolerance characteristics using reciprocal power function and assembly tolerance constraint using root-sum-square method. In the model proposed by Spotts,¹⁷ the manufacturing cost functions were represented using reciprocal square function, and both the worst-case method and root-sum-square method were applied to establish assembly tolerance constraints. Singh et al.¹¹ and Kumar and Stalin¹⁸ used exponential cost function and worst-case model. Although Lagrange multiplier method has obtained the attention of these researchers, some limitations exist in these studies: (1) Quality loss was not included in the objective function and (2) process accuracy constraints were not studied.

Jeang¹⁹ and Shin et al.²⁰ considered quality loss in their objective function and applied calculus-based analytical method to obtain the optimal tolerances. However, assembly tolerance constraint was not studied in their research.

In this article, analytical method is executed to obtain the optimal tolerance to minimize the sum of manufacturing cost and quality loss. Both assembly constraint and process accuracy constraints are included in the model proposed in this article, and closed-form solution of optimal tolerance is derived.

Model formulation

Manufacturing cost

In this article, the exponential function is used to model the manufacturing cost, and the total manufacturing cost associated with a mechanical assembly can be addressed as

C_{M} = \sum_{i = 1}^{n} (a_{i} + b_{i} e^{- c_{i} t_{i}})

(1)

where n is the number of components in the assembly, coefficients a_i, b_i, and c_i are unique for each component, and t_i is the tolerance of component i.

Quality loss cost

Based on Taguchi’s standpoints, the expected quality loss cost can be computed by

Q_{L} = \sum_{i = 1}^{n} \frac{A}{T^{2}} σ_{i}^{2}

(2)

where A is quality loss coefficient, T is the single-side functional tolerance stack up limit for the assembly dimension chain, and $σ_{i}$ is the standard deviation of component i. If the component dimension has a normal distribution, the standard deviation $σ_{i}$ can be related to the tolerance of component i

σ_{i} = \frac{t_{i}}{3}

(3)

Then, the expected quality loss can be rewritten as follows

Q_{L} = \sum_{i = 1}^{n} B_{i} t_{i}^{2}

(4)

where

B_{i} = \frac{A}{9 T^{2}}

(5)

Total cost

In this article, the objective function is the total cost, which includes manufacturing cost and quality loss

C_{T} = C_{M} + Q_{L} = \sum_{i = 1}^{n} (a_{i} + b_{i} e^{- c_{i} t_{i}}) + \sum_{i = 1}^{n} B_{i} t_{i}^{2}

(6)

The total cost, manufacturing cost, and quality loss cost are shown in Figure 1. It can be found that the minimum total cost can be obtained at the optimal tolerance $t^{*}$ in Figure 1.

Figure 1.

Cost–tolerance curve.

Design constraints

Assembly tolerance constraint

Over the past decades, a few of tolerance analysis methods have been applied to establish assembly tolerance constraint, and worst-case method and root-sum-square method are the two most widely used methods. Based on the worst-case method, the assembly tolerance constraint can be written as follows

\sum_{i = 1}^{n} ξ_{i} t_{i} \leq t_{0}

(7)

where $ξ_{i} = | \partial f / \partial x_{i} |$ (where f is the design function of the assembly dimension chain), t₀ is the functional tolerance specified by the designers, and n is the number of components in the dimension chain. If the dimension chain includes n_k standard/purchased parts, tolerances of these parts are predetermined, and the assembly tolerance constraint can be rewritten based on the worst-case method

\sum_{i = 1}^{n_{v}} ξ_{i} t_{i} \leq t_{0} - \sum_{j = 1}^{n_{k}} ξ_{j} t_{j}

(8)

where n_v is the number of component tolerances that need to be calculated. Based on the root-sum-square method, assembly tolerance constraint can be established as follows

\sum_{i = 1}^{n_{v}} ξ_{i}^{2} t_{i}^{2} \leq t_{0}^{2} - \sum_{j = 1}^{n_{k}} ξ_{j}^{2} t_{j}^{2}

(9)

Equation (9) is based on the assumption that the distribution is normal with its mean centered at the nominal value.

Process accuracy constraints

Each component tolerance has its own limits

t_{i}^{min} \leq t_{i} \leq t_{i}^{max}

(10)

where $t_{i}^{min}$ and $t_{i}^{max}$ are lower and upper constraints on component tolerance t_i, respectively. Optimal component tolerances should satisfy process accuracy constraints.

Analytical solution for optimal tolerance

In order to satisfy assembly tolerance constraint and process accuracy constraints, this article proposes the following procedure to calculate optimal tolerance.

Unconstrained tolerance optimization

First, both assembly tolerance constraint and process accuracy constraints are not considered, and analytical method is applied to find the minimum total cost by setting the first derivative of equation (6) to 0

- b_{i} c_{i} e^{- c_{i} t_{i}} + 2 B_{i} t_{i} = 0 (i = 1, 2, \dots, n_{v})

(11)

The solution of equation (11) is

t_{i}^{*} = \frac{1}{c_{i}} Lambertw (\frac{b_{i} c_{i}^{2}}{2 B_{i}}) (i = 1, 2, \dots, n_{v})

(12)

where Lambertw is Lambert W function,²¹ and $t_{i}^{*}$ is the optimal tolerance of component i. Since the Lambert W function is included in many mathematical software packages, the optimal tolerances can be obtained quickly.

Using equation (12), optimal tolerances $t_{i}^{*}$ ( $i = 1, 2, \dots, n_{v}$ ) are obtained, but it is not sure that $t_{i}^{*}$ satisfies both the assembly tolerance constraint and process accuracy constraints. In the following, these two constraints are researched.

Optimal tolerance satisfying assembly tolerance constraint

In this article, both worst-case method and root-sum-square method are considered.

Worst-case method

Combining equations (6) and (8), the augmented system of equations is obtained

\begin{matrix} \sum_{i = 1}^{n} (a_{i} + b_{i} e^{- c_{i} t_{i}}) + \sum_{i = 1}^{n} B_{i} t_{i}^{2} \\ + λ [\sum_{i = 1}^{n_{v}} ξ_{i} t_{i} + \sum_{j = 1}^{n_{k}} ξ_{j} t_{j} - t_{0}] \end{matrix}

(13)

where $λ$ is the Lagrange multiplier. By setting the first derivatives of equation (13) with respect to t_i equal to 0

- b_{i} c_{i} e^{- c_{i} t_{i}} + 2 B_{i} t_{i} + ξ_{i} λ = 0 (i = 1, 2, \dots, n_{v})

(14)

$λ$ is calculated as follows

λ = \frac{b_{i} c_{i} e^{- c_{i} t_{i}} - 2 B_{i} t_{i}}{ξ_{i}}

(15)

Eliminating $λ$ by establishing the relationship between each component tolerance with t₁, the following equation is obtained

\begin{matrix} e^{- c_{i} t_{i}} & = \frac{2 B_{i}}{b_{i} c_{i}} (t_{i} - \frac{2 ξ_{i} B_{1} t_{1} - ξ_{i} b_{1} c_{1} e^{- c_{1} t_{1}}}{2 ξ_{1} B_{i}}) \\ (i = 1, 2, \dots, n_{v}) \end{matrix}

(16)

The solution of equation (16) is

t_{i} = r_{i} + \frac{1}{c_{i}} lambertw (\frac{c_{i} e^{- c_{i} r_{i}}}{l_{i}}) (i = 1, 2, \dots, n_{v})

(17)

where

r_{i} = \frac{2 ξ_{i} B_{1} t_{1} - ξ_{i} b_{1} c_{1} e^{- c_{1} t_{1}}}{2 ξ_{1} B_{i}} and l_{i} = \frac{2 B_{i}}{b_{i} c_{i}}

Substituting equation (17) into the corresponding assembly tolerance constraint, the following equation is established

ξ_{1} t_{1} + \sum_{i = 2}^{n_{v}} ξ_{i} (r_{i} + \frac{1}{c_{i}} lambertw (\frac{c_{i} e^{- c_{i} r_{i}}}{l_{i}})) + \sum_{j = 1}^{n_{k}} ξ_{j} t_{j} = t_{0}

(18)

Equation (18) can be solved iteratively to determine t₁, and other component tolerances t_i can also be calculated using equation (17). Although the worst-case tolerances t_i, which are obtained using equations (17) and (18), satisfy assembly constraint, it is not sure that they satisfy process accuracy constraints.

Root-sum-square method

Equations (6) and (9) are combined into an augmented system of equations

\begin{matrix} \sum_{i = 1}^{n} (a_{i} + b_{i} e^{- c_{i} t_{i}}) + \sum_{i = 1}^{n} B_{i} t_{i}^{2} \\ + λ [\sum_{i = 1}^{n_{v}} ξ_{i}^{2} t_{i}^{2} + \sum_{j = 1}^{n_{k}} ξ_{j}^{2} t_{j}^{2} - t_{0}^{2}] \end{matrix}

(19)

We set the first derivatives of equation (19) equal to 0

- b_{i} c_{i} e^{- c_{i} t_{i}} + 2 B_{i} t_{i} + 2 λ ξ_{i}^{2} t_{i} = 0 (i = 1, 2, \dots, n_{v})

(20)

According to equation (20), the following relationships are established

c_{i} t_{i} e^{c_{i} t_{i}} = \frac{b_{i} c_{i}^{2} ξ_{1}^{2} t_{1}}{b_{1} c_{1} ξ_{i}^{2} e^{- c_{1} t_{1}} - 2 B_{1} t_{1} ξ_{i}^{2} + 2 B_{i} t_{1} ξ_{1}^{2}}

(21)

The solution of equation (21) is

\begin{matrix} t_{i} & = \frac{1}{c_{i}} lambertw (\frac{b_{i} c_{i}^{2} ξ_{1}^{2} t_{1}}{b_{1} c_{1} ξ_{i}^{2} e^{- c_{1} t_{1}} - 2 B_{1} t_{1} ξ_{i}^{2} + 2 B_{i} t_{1} ξ_{1}^{2}}) \\ (i = 1, 2, \dots, n_{v}) \end{matrix}

(22)

Equation (22) can be substituted into the corresponding assembly tolerance constraint

\begin{matrix} ξ_{1}^{2} t_{1}^{2} & + \sum_{i = 2}^{n_{v}} {[\frac{ξ_{i}}{c_{i}} lambertw (\frac{b_{i} c_{i}^{2} ξ_{1}^{2} t_{1}}{b_{1} c_{1} ξ_{i}^{2} e^{- c_{1} t_{1}} - 2 B_{1} t_{1} ξ_{i}^{2} + 2 B_{i} t_{1} ξ_{1}^{2}})]}^{2} \\ + \sum_{j = 1}^{n_{k}} ξ_{j}^{2} t_{j}^{2} = t_{0}^{2} \end{matrix}

(23)

Equation (23) can be solved iteratively to determine t₁, which can be substituted into equation (22) to obtain all component tolerance t_i ( $i = 1, 2, \dots, n_{v}$ ). The statistical tolerances t_i, which are obtained using equations (22) and (23), satisfy assembly constraint, but they do not necessarily satisfy process accuracy constraints.

Optimal tolerance satisfying both assembly tolerance constraint and process accuracy constraints

In order to obtain the optimal tolerances satisfying both assembly tolerance constraint and process accuracy constraints, the following optimization procedure is proposed:

First, the initial values of optimal tolerances are calculated using equation (12), which are denoted by $t_{i}^{(0)}$ ( $i = 1, 2, \dots, n_{v}$ ). If $t_{i}^{(0)}$ satisfies both the assembly tolerance constraint and process accuracy constraints, then the final values of optimal tolerances are $t_{i}^{(0)}$ .

If $t_{i}^{(0)}$ cannot satisfy the assembly tolerance constraint, the Lagrange multiplier method is adopted, and equations (17) and (18) (or equations (22) and (23)) are used to obtain $t_{i}^{(1)}$ ( $i = 1, 2, \dots, n_{v}$ ) as the new values of optimal tolerances. If $t_{i}^{(0)}$ satisfies the assembly tolerance constraint, then the new values of optimal tolerances are rewritten as $t_{i}^{(1)} = t_{i}^{(0)}$ ( $i = 1, 2, \dots, n_{v}$ ). If $t_{i}^{(1)}$ satisfies process accuracy constraints, the final values of optimal tolerances are $t_{i}^{(1)}$ .

If $t_{i}^{(1)}$ cannot satisfy process accuracy constraints, all component tolerances would be divided into three subsets: in the first subset, $t_{i}^{(1)} < t_{i}^{min}$ ; in the second subset, $t_{i}^{min} \leq t_{i}^{(1)} \leq t_{i}^{max}$ ; in the third subset, $t_{i}^{max} < t_{i}^{(1)}$ . If the number of component tolerances in the first subset is denoted by n_l and those in the third subset is denoted by n_u, then there are $n_{v} - n_{l} - n_{u}$ component tolerances in the second subset. In the first and third subsets, component tolerances do not satisfy process accuracy constraints and should be adjusted. In this article, the following rules are proposed to adjust the value of component tolerances to satisfy process accuracy constraints:

If $t_{i}^{(1)} < t_{i}^{min}$ , the final value of optimal tolerance is $t_{i} = t_{i}^{min}$ .

If $t_{i}^{max} < t_{i}^{(1)}$ , the final value of optimal tolerance is $t_{i} = t_{i}^{max}$ .

Based on these two rules, component tolerances are determined in the first and third subsets.

Steps 1–3 are repeated sequentially until optimal tolerances of all components are obtained. Because the component tolerances have been determined in the first and third subsets, the component tolerances in the second subset should be optimized based on the following assembly tolerance constraint

\begin{matrix} \sum_{i = 1}^{n_{v} - n_{l} - n_{u}} ξ_{i} t_{i} = t_{0} - \sum_{i = 1}^{n_{l}} ξ_{i} t_{i}^{min} - \sum_{i = 1}^{n_{u}} ξ_{i} t_{i}^{max} \\ - \sum_{j = 1}^{n_{k}} ξ_{j} t_{j} (worst - case) \end{matrix}

(24)

\begin{matrix} \sum_{i = 1}^{n_{v} - n_{l} - n_{u}} ξ_{i}^{2} t_{i}^{2} = t_{0}^{2} - \sum_{i = 1}^{n_{l}} ξ_{i}^{2} {(t_{i}^{min})}^{2} \\ - \sum_{i = 1}^{n_{u}} ξ_{i}^{2} {(t_{i}^{max})}^{2} - \sum_{j = 1}^{n_{k}} ξ_{j}^{2} t_{j}^{2} \\ (root - sum - square) \end{matrix}

(25)

Using equations (17) and (24), the following equation is established based on worst-case method

\begin{matrix} t_{0} & = \sum_{i = 1}^{n_{l}} ξ_{i} t_{i}^{min} + \sum_{i = 1}^{n_{u}} ξ_{i} t_{i}^{max} + \sum_{j = 1}^{n_{k}} ξ_{j} t_{j} + ξ_{1} t_{1} \\ + \sum_{i = 2}^{n_{v} - n_{l} - n_{u}} ξ_{i} (r_{i} + \frac{1}{c_{i}} lambertw (\frac{c_{i} e^{- c_{i} r_{i}}}{l_{i}})) \end{matrix}

(26)

Using equations (22) and (25), the following equation is established based on the root-sum-square method

\begin{matrix} t_{0}^{2} = ξ_{1}^{2} t_{1}^{2} + \sum_{i = 2}^{n_{v} - n_{l} - n_{u}} \\ {[\frac{ξ_{i}}{c_{i}} lambertw (\frac{b_{i} c_{i}^{2} ξ_{1}^{2} t_{1}}{b_{1} c_{1} ξ_{i}^{2} e^{- c_{1} t_{1}} - 2 B_{1} t_{1} ξ_{i}^{2} + 2 B_{i} t_{1} ξ_{1}^{2}})]}^{2} \\ + \sum_{i = 1}^{n_{l}} ξ_{i}^{2} {(t_{i}^{min})}^{2} + \sum_{i = 1}^{n_{u}} ξ_{i}^{2} {(t_{i}^{max})}^{2} + \sum_{j = 1}^{n_{k}} ξ_{j}^{2} t_{j}^{2} (27) \end{matrix}

(27)

Solving equation (26) (or equation (27)), t₁ is obtained, and other component tolerances can be calculated using equation (17) (or equation (22)).

According to the optimization procedure proposed in this article, the closed-form solution of optimal tolerance is obtained, which satisfy both assembly tolerance constraint and process accuracy constraints.

Illustrative example

The overrunning clutch presented by Feng and Kusiak¹⁰ is applied to demonstrate the effectiveness of the method proposed in this article. The clutch, which is shown in Figure 2, consists of three kinds of components: hub, roller, and cage, denoted by X₁, X₂, and X₃, respectively. The contact angle Y is the assembly functional dimension

Y = f (X_{1}, X_{2}, X_{3}) = \cos^{- 1} (\frac{X_{1} + X_{2}}{X_{3} - X_{2}})

Figure 2.

Overrunning clutch.

The normal value and the tolerance of contact angle Y are $0.122 \pm 0.035 rad$ . The nominal values of X₁, X₂, and X₃ are 2.17706, 0.9000, and 4.0000 in, respectively. The derivatives of Y with respect to X_i (i =1, 2, 3) are calculated as

| \frac{\partial f}{\partial X_{1}} | = \frac{1}{X_{3} - X_{2}} {(1 - \frac{X_{1} + X_{2}}{X_{3} - X_{2}})}^{- 1 / 2} = 3.7499 rad / in

| \frac{\partial f}{\partial X_{2}} | = \frac{X_{3} + X_{1}}{{(X_{3} - X_{2})}^{2}} {(1 - \frac{X_{1} + X_{2}}{X_{3} - X_{2}})}^{- 1 / 2} = 7.4721 rad / in

| \frac{\partial f}{\partial X_{3}} | = \frac{X_{1} + X_{2}}{{(X_{3} - X_{2})}^{2}} {(1 - \frac{X_{1} + X_{2}}{X_{3} - X_{2}})}^{- 1 / 2} = 3.7222 rad / in

Process accuracy constraints for these components are $0.0001 \leq t_{1} \leq 0.0120 in$ , $0.0001 \leq t_{2} \leq 0.0005 in$ , and $0.0001 \leq t_{3} \leq 0.0120 in$ .

The cost–tolerance data proposed by Feng and Kusiak¹⁰ are used in this article, which are illustrated in Table 1. Exponential function is applied to model manufacturing cost functions, which are obtained through curve fitting

Table 1.

Cost–tolerance data for the clutch (tolerance in 10⁻⁴ in and cost in dollars).

Hub tolerance	Cost	Roller tolerance	Cost	Cage tolerance	Cost
2	19.38	1	3.513	1	18.637
4	13.22	2	2.48	2	12.025
8	5.99	4	1.24	4	5.732
16	4.505	8	1.24	8	2.686
30	2.065	16	1.20	16	1.984
60	1.24	30	0.413	30	1.447
120	0.825	60	0.413	60	1.200
–	–	120	0.372	120	1.033

M (t_{1}) = 1.0998 + \frac{22.6709}{e^{1.4366 \times 1 0^{3} t_{1}}}

M (t_{2}) = 0.1273 + \frac{2.5396}{e^{0.6030 \times 10^{3} t_{2}}}

M (t_{3}) = 1.4152 + \frac{26.2090}{e^{4.3704 \times 10^{3} t_{3}}}

Quality loss cost is calculated as follows

Q_{L} = \frac{A}{{(3 \times 0.035)}^{2}} (t_{1}^{2} + 4 t_{2}^{2} + t_{3}^{2})

The objective function is the sum of manufacturing cost and quality loss

\begin{matrix} Min C_{T} = C_{M} + Q_{L} = 3.0242 + \frac{22.6709}{e^{1.4366 \times 1 0^{3} t_{1}}} \\ + 4 \times \frac{2.5396}{e^{0.6030 \times 10^{3} t_{2}}} + \frac{26.2090}{e^{4.3704 \times 10^{3} t_{3}}} \\ + \frac{A}{{(3 \times 0.035)}^{2}} (t_{1}^{2} + 4 t_{2}^{2} + t_{3}^{2}) \end{matrix}

subject to

\begin{matrix} 3.7499 t_{1} + 2 \times 7.4721 t_{2} + 3.7222 t_{3} \leq 0.0350 \\ (worst - case) \end{matrix}

\begin{matrix} {(14.0618 t_{1}^{2} + 111.6646 t_{2}^{2} + 13.8548 t_{3}^{2})}^{1 / 2} \leq 0.0350 \\ (root - sum - square) \end{matrix}

0.0001 \leq t_{1} \leq 0.0120

0.0001 \leq t_{2} \leq 0.0005

0.0001 \leq t_{3} \leq 0.0120

Assuming quality loss coefficient A = 50, the optimal worst-case tolerances can be calculated as follows:

Using equation (12), the initial values of optimal tolerances are calculated as follows: $t_{1}^{(0)} = 0.00463127$ , $t_{2}^{(0)} = 0.00563758$ , and $t_{3}^{(0)} = 0.00200200$ . It is obvious that initial optimal tolerances do not satisfy constraints.

In order to satisfy assembly tolerance constraint, equations (17) and (18) are used to obtain the new values of optimal tolerances: $t_{1}^{(1)} = 0.00268985$ , $t_{2}^{(1)} = 0.00137365$ , and $t_{3}^{(1)} = 0.00117826$ . Although $t_{i}^{(1)}$ (i = 1, 2, 3) satisfies assembly tolerance constraint, $t_{2}^{(1)}$ , the tolerance of the second component does not satisfy the process accuracy constraints.

The tolerance of second component is adjusted to satisfy process accuracy constraints, and the following optimal tolerance is determined: $t_{2} = t_{2}^{max} = 0.0005$ .

Because the optimal tolerance of the second component is determined, the tolerance of the first and third components should be calculated again to minimize the objective function. First, Step 1 is repeated to obtain the new value of component tolerances $t_{i}^{(2)}$ (i = 1, 3): $t_{1}^{(2)} = 0.00463127$ and $t_{3}^{(3)} = 0.00200200$ . $t_{i}^{(2)}$ (i = 1, 3) and t₂ not only satisfy both assembly tolerance constraint and process accuracy constraints, but also minimize the objective function.

Based on the method proposed in this article, the optimal tolerances are obtained and shown in Table 2. In order to demonstrate the effectiveness of the method proposed in this article, numerical optimization method is also used to resolve this problem, and the optimal tolerances are obtained and listed in Table 3.

Table 2.

Results obtained by the method proposed in this article (tolerance in inches and cost in dollars).

A	t ₁	t ₂	t ₃	Manufacturing cost	Quality loss	Total cost	Assembly constraint
50	0.00463127	0.00050000	0.00200200	10.57184348	0.11998491	10.69182839	0.03229026
100	0.00421443	0.00050000	0.00186021	10.59938681	0.20155837	10.80094518	0.03019940
500	0.00327061	0.00050000	0.00153580	10.77681888	0.63743982	11.41425870	0.02545270
1000	0.00287730	0.00050000	0.00139861	10.95983075	1.01904448	11.97887523	0.02346722
5000	0.00200754	0.00050000	0.00108785	12.03172441	2.81797352	14.84969793	0.01904905

Table 3.

Results obtained by numerical method (tolerance in inches and cost in dollars).

A	t ₁	t ₂	t ₃	Manufacturing cost	Quality loss	Total cost	Assembly constraint
50	0.00463126	0.00050000	0.00200233	10.57183792	0.11999048	10.69182840	0.03229202
100	0.00421442	0.00050000	0.00186081	10.59936705	0.20157816	10.80094521	0.03020216
500	0.00327056	0.00050000	0.00153574	10.77684168	0.63741701	11.41425870	0.02545275
1000	0.00287723	0.00050000	0.00139856	10.95987879	1.01899645	11.97887524	0.02346720
5000	0.00200764	0.00050000	0.00108799	12.03140378	2.81829422	14.84969800	0.01905030

The comparison of Tables 2 and 3 clearly indicates that the analytical method, which is proposed in this article, is more accurate than the numerical method, and improvement in the objective function (0.00000001, 0.00000003, 0.00000001, and 0.00000007) is found for different values of A (50, 100, 1000, and 5000). In fact, when A = 500, the objective functions obtained by the analytical method and the numerical method are equal to 11.4142586953 and 11.4142586972, respectively. Although the difference is very small and cannot be illustrated in Tables 2 and 3, the analytical method actually obtains smaller objective function. Tables 2 and 3 indicate that the assembly tolerance constraint does not influence the value of optimal worst-case tolerances, and only process accuracy constraints influence it. Therefore, the optimal statistical tolerances should be equal to the optimal worst-case tolerances.

Tables 2 and 3 also indicate that the process accuracy constraint for the roller is overqualified. In order to reduce manufacturing cost, this article adjusts the process accuracy constraints for the roller, and the new process accuracy constraints for these components are $0.0001 \leq t_{1} \leq 0.0120 in$ , $0.0001 \leq t_{2} \leq 0.0050 in$ , and $0.0001 \leq t_{3} \leq 0.0120 in$ . Based the new process accuracy constraints, the optimal worst-case tolerances are calculated and shown in Tables 4 and 5, and the optimal statistical tolerances are shown in Tables 6 and 7. The comparison of Tables 4 and 5 indicates the improvement in the objective function (0.00009795, 0.00009607, 0.00008064, 0.00006079, and 0.00000004) for different values of A (50, 100, 500, 1000, and 5000). The comparison of Tables 6 and 7 shows apparent improvement in the objective function (0.10188436, 0.17486377, 0.00000164, 0.00000145, and 0.00000239) for different values of A (50, 100, 500, 1000, and 5000). The comparisons demonstrate that the proposed analytical method can obtain better results.

Table 4.

Optimal worst-case tolerances obtained by the method proposed in this article according to new constraints (tolerance in inches and cost in dollars).

A	t ₁	t ₂	t ₃	Manufacturing cost	Quality loss	Total cost	Assembly constraint
50	0.00268985	0.00137365	0.00117825	8.08893511	0.07333903	8.16227415	0.0350
100	0.00267907	0.00137612	0.00117920	8.08912825	0.14641993	8.23554818	0.0350
500	0.00260402	0.00139307	0.00118677	8.09414589	0.72344274	8.81758865	0.0350
1000	0.00253013	0.00140926	0.00119619	8.10587242	1.43097675	9.53684917	0.0350
5000	0.00200754	0.00095134	0.00108785	10.24138677	4.00621579	14.24760252	0.02579364

Table 5.

Optimal worst-case tolerances obtained by numerical method according to new constraints (tolerance in inches and cost in dollars).

A	t ₁	t ₂	t ₃	Manufacturing cost	Quality loss	Total cost	Assembly constraint
50	0.00268984	0.00137362	0.00117820	8.08903501	0.07333710	8.16237210	0.0350
100	0.00267908	0.00137608	0.00117920	8.08922779	0.14641645	8.23564425	0.0350
500	0.00260393	0.00139305	0.00118675	8.09425464	0.72341464	8.81766929	0.0350
1000	0.00252997	0.00140924	0.00119627	8.10600565	1.43090431	9.53690996	0.03500000
5000	0.00200751	0.00095131	0.00108784	10.24146678	4.00613577	14.24760256	0.02579385

Table 6.

Optimal statistical tolerances obtained by the method proposed in this article according to new constraints (tolerance in inches and cost in dollars).

A	t ₁	t ₂	t ₃	Manufacturing cost	Quality loss	Total cost	Assembly constraint
50	0.00365173	0.00299088	0.00166990	4.83467547	0.23539761	5.07007308	0.0350
100	0.00360225	0.00299963	0.00165238	4.83607958	0.46891350	5.30499308	0.0350
500	0.00327060	0.00291365	0.00153580	5.01564749	2.13210360	7.14775109	0.03363071
1000	0.00287730	0.00221717	0.00139861	6.11362913	2.71186976	8.82549889	0.02631397
5000	0.00200754	0.00095132	0.00095132	10.24138677	4.00621579	14.24760256	0.01319552

Table 7.

Optimal statistical tolerances obtained by numerical method according to new constraints (tolerance in inches and cost in dollars).

A	t ₁	t ₂	t ₃	Manufacturing cost	Quality loss	Total cost	Assembly constraint
50	0.00448655	0.00282110	0.00196044	4.91886447	0.25309297	5.17195744	0.0350
100	0.00421507	0.00275730	0.00185927	5.01151650	0.46834035	5.47985685	0.03386264
500	0.00327240	0.00291488	0.00153545	5.01385815	2.13389457	7.14775273	0.03364533
1000	0.00287714	0.00221621	0.00139761	6.11552036	2.70997998	8.82550034	0.02630425
5000	0.00200825	0.00095204	0.00108774	10.23772267	4.00988228	14.24760495	0.01320288

Conclusion

In order to minimize the sum of manufacturing cost and quality loss, the article proposes analytical methods to obtain closed-form solutions for the optimal tolerances, and all constraints, including assembly tolerance constraint and process accuracy constraints, are satisfied. Comparing with numerical method, both the computational efforts and the total cost are reduced. This demonstrates that the proposed method is indeed efficient and effective. It should be noted that the analytical method proposed in this article is only applicable to a few of manufacturing cost functions. In the future, the proposed method should be investigated to solve as many as tolerance optimization problems.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Singh

Jain

. Important issues in tolerance design of mechanical assemblies. Part 2: tolerance synthesis. Proc IMechE, Part B: J Engineering Manufacture 2009; 223(10): 1249–1287.

ElMaraghy

. Evaluation of cost-tolerance algorithms for design tolerance analysis and synthesis. Manuf Rev 1988; 1(3): 168–179.

Chase

Greenwood

Loosli

. Least cost tolerance allocation for mechanical assemblies with automated process selection. Manuf Rev 1990; 3(1): 49–59.

Dong

Xue

. New production cost-tolerance models for tolerance synthesis. J Eng Ind: T ASME 1994; 116: 119–206.

Zhang

. Simultaneous tolerancing for design and manufacturing. Int J Prod Res 1996; 34(12): 3361–3382.

Taguchi

Elsayed

Hsiang

. Quality engineering in production systems. New York: McGraw-Hill, 1989.

Hsieh

. The study of cost-tolerance model by incorporating process capability index into product lifecycle cost. Int J Adv Manuf Tech 2006; 28(5–6): 638–642.

Cao

Mao

Ching

. A robust tolerance optimization method based on fuzzy quality loss. Proc IMechE, Part C: J Mechanical Engineering Science 2009; 223(11): 2647–2653.

Cheng

Maghsoodloo

. Optimization of mechanical assembly tolerances by incorporating Taguchi’s quality loss function. J Manuf Syst 1995; 14(4): 264–276.

10.

Feng

Kusiak

. Robust tolerance design with the integer programming approach. J Manuf Sci E: T ASME 1997; 119: 603–610.

11.

Singh

Jain

. A genetic algorithm based solution to optimum tolerance synthesis of mechanical assemblies with alternate manufacturing processes—benchmarking with the exhaustive search method using the Lagrange multiplier. Proc IMechE, Part B: J Engineering Manufacture 2004; 218(7): 765–778.

12.

Prabhaharan

Asokan

Rajendran

. Sensitivity-based conceptual design and tolerance allocation using the continuous ants colony algorithm (CACO). Int J Adv Manuf Tech 2005; 25(5–6): 516–526.

13.

Haq

Sivakumar

Saravanan

. Particle swarm optimization (PSO) algorithm for optimal machining allocation of clutch assembly. Int J Adv Manuf Tech 2006; 27(9–10): 865–869.

14.

Forouraghi

. Optimal tolerance allocation using a multiobjective particle swarm optimizer. Int J Adv Manuf Tech 2009; 44(7–8): 710–724.

15.

Zahara

Kao

. A hybridized approach to optimal tolerance synthesis of clutch assembly. Int J Adv Manuf Tech 2009; 40(11–12): 1118–1124.

16.

Coelho

. Self-organizing migration algorithm applied to machining allocation of clutch assembly. Math Comput Simulat 2009; 80: 427–435.

17.

Spotts

. Allocation of tolerances to minimize cost of assembly. J Eng Ind: T ASME 1973; 95: 762–764.

18.

Kumar

Stalin

. Optimum tolerance synthesis for complex assembly with alternative process selection using Lagrange multiplier method. Int J Adv Manuf Tech 2009; 44(3–4): 405–411.

19.

Jeang

. Optimal tolerance design for product life cycle. Int J Prod Res 1996; 34(8): 2187–2209.

20.

Shin

Govindaluri

Cho

. Integrating the Lambert W function to a tolerance optimization problem. Qual Reliab Eng Int 2005; 21(8): 795–808.

21.

Corless

Gonnet

Hare

DEG

. On the Lambert W function. Adv Comput Math 1996; 5(4): 329–359.