Optimal Pareto front for manufacturing tolerance allocation model

Abstract

This article presents a multi-objective formulation of a tolerance allocation model for interchangeable assembly of pin and hole to complement the need of small-scale industries where there exists two categories of machines, one for pin production and other for hole production. The two objectives considered in this article are minimum total cost of assembly and minimum clearance variation. The problem is formally defined as the determination of optimal Pareto set of tolerances for pin and hole for minimum total cost and minimum clearance variation of a hole and pin assembly, given the design clearance, the process capability of the machines defined with their standard deviations and the mean diameter of either pin or hole. An iterative search algorithm is proposed to explore the entire decision space and evaluate and obtain the Pareto optimal front. This article also presents how the optimal Pareto front of the model can be utilized by the manufacturer to set the optimal tolerances according to demands of the customers. Besides, the effect of process capabilities of pin and hole manufacturing machines on the optimality is discussed to understand their criticality in decision-making.

Keywords

Interchangeable assembly manufacturing tolerance allocation multi-objective clearance variation assembly cost

Introduction

Assembly is a process of joining a number of components to make an integrated and complete product in which relative motion is observed. The function of an assembly is to transform one kind of motion into another kind of required motion of its interconnected parts. The relative motion between a shaft and a hole is either longitudinal or circumferential, or a combination of both. The sliding motion is purely longitudinal if the cross section is non-circular. The sliding motion is both circumferential and longitudinal if the cross section of the part is circular. Machine elements, namely, a piston in a cylinder, a shaft in a bearing, a pin of a fixture, an axle in housing, a cylindrical pin used to connect the piston and connecting rod, and a spindle of a lathe are a few examples for assemblies with circular cross section and is referred generally as pin–hole assembly. The relative motion of a pin–hole assembly is influenced by the degree of tightness between the parts. For example, in a radial ball bearing, the shaft fits into the inner race maintaining an interference fit. The type of fit maintained in an impeller shaft and sleeve of an oil pump is preferably clearance fit. In a pin–hole assembly, various fits, namely, running fit, location fit, force and shrink fit or an interference fit are obtained by appropriate selection of the hole size that correlates with diameter of the pin. The clearance or interference is important as it decides the two important quality attributes, namely, the performance and the life of the assembled product. In practice, the selection of hole size for a given pin diameter is achieved either by interchangeable assembly process or selective assembly process.

In an assembly, the clearance or interference depends solely on the manufacturing tolerances of the hole and the pin. Tolerance is an allowable deviation from a specified value, reasonably non-harmful to the functioning of a product over its life cycle.¹ Tolerance allocation is an approach that judiciously distributes the assembly tolerance among the individual component dimensions considering the functionality requirements.² According to Chase et al.,³ the assembly tolerance is known from design requirements, and the tolerance is distributed or allocated among the components in some rational way. But it affects the fit and function of the final product. For instance, high tolerances lead to poor performance and dissatisfaction among customers. In short, tolerance affects every aspect of the product life cycle.⁴ As tolerance has enormous impact on quality and cost, it is important to design the tolerance so as to achieve high quality at low cost. The successful operation and performance of the mating parts of an assembly depend upon the tolerance allocation on individual components. Tolerance allocation is an important step in the initial period of product development that directly affects the total cost and quality level of the product. Thus, the functional performance and the manufacturing cost of an assembled product are directly influenced by the individual component tolerance.⁵ The product component tolerances must be set to assure that the resulting assembly meets the design specification. When the tolerances of the mating parts are closer to match design (nominal) value, the clearance (or interference) variation is minimal and the motion is transmitted effectively. This leads to improved performance of the assembly and increased life of the assembly. These improvements in the attributes improve the overall quality of the product. Tolerance allocation is one of the main considerations for deciding the quality. In order to achieve the desired level of quality as defined by clearance variation, it becomes essential either to select a high-precision process, where manufacturing cost is high or to fix the limits narrowly in the low-precision processes that would lead to more number of rejections. Hence, tolerance allocation influences the total cost of an assembly which mainly includes the manufacturing cost, rejection cost and quality loss cost (QLC). On the above considerations, the tolerance allocation approaches are attempted focusing on manufacturing and quality costs.

Methods focusing on manufacturing provide a means to select suitable manufacturing processes or machines based on the process capabilities. Literature shows a number of articles that discuss the selection of a suitable manufacturing process which yields optimal tolerance with minimum total cost. Chase et al.³ allocated the tolerance to components of a mechanical assembly considering different processes to minimize total cost. Different cost models are discussed, namely, linear, reciprocal and reciprocal square. Dong et al.⁶ discussed the need for using the various types of cost-tolerance models and introduced the new production cost – tolerance models for tolerance synthesis, such as hybrid model and polynomial model. They demonstrated the curve fitting technique for the existing cost-tolerance model as well as the new production cost-tolerance model. They considered a design of rotor assembly. Singh et al.⁷ dealt with tolerance synthesis problem with alternative manufacturing processes. They considered alternative machining processes to get minimum manufacturing cost under setup reduction constraints. They allocated the optimal tolerance for piston and cylinder assembly. Kopurdekar and Anand⁸ briefly discussed various tolerance allocation methods such as simple division, proportional scaling and tolerance allocation using optimization techniques and the limitations of commonly used assembly model including the statistical model. They demonstrated the tolerance allocation to a three component assembly with three available machines. Singh et al.⁹ proposed the manufacturing design tolerance with availability of alternative machines. They considered the machine selection as a discrete variable and tolerance as a continuous variable. Few researchers have used rejection cost in place of manufacturing cost. Rejection cost is the cost incurred due to the fraction of the number of components that get rejected, as they do not fall within the design tolerance limit. The fraction of rejection is determined by considering the statistical normal distribution of component dimension and using the $\pm 3 σ$ concept.¹⁰

Methods focusing on quality mostly use quality loss function defined by Taguchi, the cost associated with the deviation of a dimension from its target. The method emphasizes the loss incurred due to deviation of dimension from its target, although the parts lie within the limits. Huang and Shiau¹¹ obtained the optimal tolerance allocation for a sliding vane compressor. Asymmetric quadratic quality loss model is used to calculate quality loss caused by the deviation and the mean shift distributions. Such methods are based on the overall ‘cost incurred by the society’ rather than solely on the cost incurred by manufacturer. Asha et al.¹² discussed QLC and loss per assembly; Kannan et al.¹³ discussed a total cost which includes a fixed cost and a variable cost per assembly. However, tolerance allocation with quality cost alone is rarely addressed.

Methods focusing simultaneously on manufacturing and quality have been gaining importance in the recent years. Evans¹⁰ is one of the earliest works which discussed the tolerance allocation considering the rejection rate and its salvage (cost) value. He emphasized the advantage of using statistical approach to assigning component tolerances over the worst case approach. He used the normal distribution to represent the rejection rate. He showed graphically the decreasing trend of raw cost with increasing standard deviation. He demonstrated the presence of a minimum real cost for an optimal tolerance limit, which includes the raw cost and salvage. Wei¹⁴ developed a model to combine manufacturing cost and scrap cost. To find scrap cost, he used normal distribution and compared his model with traditional model. He took quality loss as the manufacturing cost times the scrap rate. Weiss¹⁵ gave a detailed account of various scientific methods for allocating manufacturing tolerances. He discussed some of the basic methods such as worst case method, proportional scaling method and constant precision scaling method. Also, he discussed the statistical method and optimization tools to allocate the manufacturing tolerances to individual parts of an assembly. He showed graphically that the scraping cost drastically increases with decreasing tolerance, which adds to manufacturing cost. He also discussed the use of QLC. Ye and Salustri¹⁶ developed a new method called as simultaneous tolerance synthesis method. Design function is considered as one-dimensional in their method. The products are normally distributed within six sigma limit. Their objective is to reduce the total cost of the product, which is the summation of manufacturing cost and QLC. As manufacturing cost decreases with increasing tolerance, QLC increases with increasing tolerance. They obtained a minimum total cost and increased service life of the product. Design tolerance and process tolerance were taken as the two variables. Here, manufacturing cost is a function of process tolerance and quality loss is a function of design tolerance. Tolerance allocation is considered by taking into account both process and machining allowance. They compared the new method with Zhang’s method, wherein the manufacturing cost is low, but QLC is high. Singh et al.¹⁷ developed a new methodology to allocate the tolerance with stock removal allowance. They allocated the tolerance based on the assembly dimension. As there are different processes for a single product, they determined the better process. Their aim is to minimize the manufacturing cost and QLC. They have taken the quality loss as a quadratic function. Sivakumar et al.¹⁸ considered the following objective functions for optimization, namely, tolerance stack up, total manufacturing cost and QLC for clutch assembly and knuckle joint with three arms.

Ruiz and Forero¹⁹ described multi-objective optimization problem with the following two concurrent objective functions: loss of quality and manufacturing cost. The concurrent engineering approach seeks a balanced single set of tolerances for the two functions by simultaneous analysis of the effect of tolerances in both design and manufacturing. Andolfatto et al.²⁰ described the method to select the appropriate assembly technique based on quality and cost. They proposed and solved a multi-objective optimization problem to minimize cost and maximize quality.

Quite a few literatures included a variety of parameters that are indirectly related to the assembly besides manufacturing and QLCs. Jeang²¹ developed a tolerance allocation model with the total cost expressed as the summation of setup cost, inventory cost, QLC and tolerance (i.e. manufacturing) cost. Moroni et al.²² proposed an approach for inspection cost evaluation assuming process capabilities and costs are known. The inspection cost will be influenced by the inspection system and tolerance value. They considered measurement cost and inspection error cost to find the inspection cost.

There are methods which ignore both the manufacturing cost and QLC but consider other parameters such as process capability. Kumar et al.²³ discussed the use of asymmetric quality loss function approach with a rotor key assembly as an example. Jeang²⁴ developed a new method to find the best process means and process tolerance. He considered the quality loss due to the differences those exist between process mean and design target. They considered process deterioration to find optimal tolerance. Swift et al.²⁵ developed a new methodology to allocate the tolerance considering manufacturing risk and assembly risk. In their model, the tolerance allocation is based on the process selection and process capability index for centred distribution and shifted distribution. They aimed at achieving low standard deviation for the largest possible component tolerances. They did not consider manufacturing cost and QLC. Terry² described the method of allocating the tolerance based on process capability. However, a wide literature is found on selective assembly which is the most preferable method for high-precision assemblies, wherein the optimization is used to obtain the minimum clearance variation coupled with minimum number of classes.

Krogstie et al.²⁶ designed and conducted survey to map the organizational perception on its tolerance engineering (TE) practice. The results demonstrate that employees recognize the importance of TE in order to develop products with appropriate limits on specification. Hernandez and Tutsch²⁷ proposed a statistical dynamic specifications method for allocating tolerances. Observation points are introduced in the middle of assembly line to retrieve measured value of first component to make corrections in the parameters to obtain better tolerances of remaining component.

The salient points observed with various tolerance allocation models of interchangeable assembly are as follows:

The major concern in the literature on tolerance allocation is the determination of optimal tolerance that minimizes the cost;

Manufacturing cost has been primarily considered as the tolerance cost. The concern for tolerance allocation decisions in earlier days is based on this tolerance cost. In this approach, tolerance cost for various processes is arrived using tolerance cost versus tolerance data. Tolerance is used to select appropriate manufacturing processes or machines from alternative processes;

Later on, the focus has shifted towards the simultaneous consideration of manufacturing and quality in which Taguchi’s quality loss function is used extensively to assess the quality cost; this method gives importance for minimizing variation in quality (i.e. fit), in which total cost function involving manufacturing and QLCs is primarily adopted to fix processes and their tolerance bands;

Rejection cost has not been given due consideration, and it was neglected on the assumption that the tolerance limits of the matching parts are fixed at $\pm 3 σ$ limits to all alternative processes, but this is not the case when the limits are different from $\pm 3 σ$ limits;

It is validated that the influence of inspection and other costs is marginal in tolerance allocation decisions in interchangeable assembly; however, they are identified as significant in selective assembly;

The other quality aspect in assembly ‘clearance variation’ has been given little consideration in interchangeable assembly;

Most of the works attempted to integrate quality and manufacturing cost. In these works, a combined total cost function is used for the evaluation and fixing the tolerance.

The review on interchangeable assembly reveals that the rejection cost is ignored and is not included with other tolerance costs such as manufacturing cost, quality cost and inspection cost. Also, consideration for variation in quality due to rejection cost is ignored. This necessitates a general tolerance allocation model connecting cost and quality to assist the manufacturer for fixing the manufacturing tolerances accordingly to varying needs of customers. Besides, tolerance allocation is a key decision in small-scale industries that operate with few manufacturing facilities and need to meet different customer tolerance specifications. Considering the above two points, this article presents a multi-objective formulation of a tolerance allocation model for interchangeable assembly of pin and hole to complement the need of small-scale industries where there exists two categories of machines, one for pin production and other for hole production. The two objectives considered in this article are minimum total cost of assembly and minimum clearance variation. The first objective of minimization of per unit cost of assembly takes care of rejection cost, QLC and inspection cost in addition to the cost of manufacturing. As the manufacturing scenario considered here is fixed manufacturing facility, the cost of manufacturing becomes constant. Besides, the environment warrants cent percent inspection, and hence, the first objective is formulated with rejection and quality costs. The second objective is minimization of clearance variation. The attribute that indicates the variation in quality of assembly is written as a function of maximum variation of clearance from the design fit. The constraints that govern the solution are the $\pm 3 σ$ limits of the manufacturing processes. The bi-objective model belongs to non-linear programming problem. Contrary to traditional optimization, a multi-objective formulation involves the simultaneous minimization of multiple objectives. There does not exist an optimal solution to most of the multi-objective formulations that serves to minimize all the objectives that are conflicting in nature. Such multi-objective optimization formulations can be solved by converting the objectives to a single function value assigning weights to the objectives or prioritizing the objectives. The solution of these methods depends on the estimation of weights or prioritization of objectives and limits their scope to all applied under varied requirements. Under such circumstances, the concept of Pareto optimality is used. A feasible solution is Pareto optimal if and only if does not exist a solution that dominates the other solutions in all objectives. The Pareto frontier is the set of all optimal Pareto solutions and can be used to select the optimal solution depending upon the compromise between the objectives. This article proposes an iterative search algorithm (ISA) to obtain a Pareto optimal tolerance set for the objectives of non-dominated minimum total cost and minimum clearance variation. The rest of the article is organized as follows: the ‘Model formulation’ section describes the goal programming formulation of the tolerance allocation model to pin–hole assembly system; the ‘Solution methodology’ section describes ISA solution methodology; the ‘Model utility’ section describes the model utility; the ‘Model sensitivity studies’ section describes model sensitivity studies on the various process capabilities of hole and pin manufacturing machines and its corresponding precision levels; the ‘Conclusion’ section describes the conclusion and further scope of this ISA model.

Model formulation

The problem under consideration is a manufacturing tolerance allocation for an interchangeable assembly of pin–hole system to the given mean diameter of the pin or hole and the design specification fit $(C_{d})$ . Equation (1) portrays the relationship between the mean diameters of pin and hole, and $C_{d}$

μ_{h} = μ_{p} + C_{d}

(1)

The tolerance model considers a manufacturing environment of small-scale industries where there is a need to fix tolerances in accordance with the process capabilities of their available manufacturing facilities. The model takes into account the quality loss, rejection, inspection and manufacturing costs as the parameters influencing the tolerance allocation decisions. This section delineates the mathematical formulation of the problem describing its environment, objectives and constraints.

Manufacturing environment

This research work considers a manufacturing environment in which production facility possess the following features: the production of both pin and hole follows normal distribution defined with their mean $(μ_{p} / μ_{h})$ and standard deviation $(σ_{p} / σ_{h})$ ; the process capabilities of pin $(C_{p}^{P})$ and hole $(C_{p}^{h})$ are known and the standard deviation of pin $(σ_{p})$ and hole $(σ_{h})$ production process can be estimated reasonably; the manufacturing extremes at the $\pm 3 σ$ limits determine the upper and lower limits of the pin and hole; equations (2)–(5) provide the manufacturing limits for the pin and hole

U^{p} = μ_{p} + 3 σ_{p}

(2)

L^{p} = μ_{p} - 3 σ_{p}

(3)

U^{h} = μ_{h} + 3 σ_{h}

(4)

L^{h} = μ_{h} - 3 σ_{h}

(5)

Cost elements of a pin–hole assembly

QLC

The design of any class of fit (clearance fit has a positive clearance and interference fit has a negative clearance) should meet the required performance (reliability) for the desired life and this is often referred as design clearance. However, maintaining the clearance or interference exactly to the design value of mating parts is highly difficult to be achieved or costly due to manufacturing constraints. The actual value of the fit (C) depends on the manufacturing processes and the resultant dimensions of the pin and hole. Hence, the variation of C from $C_{d}$ is inevitable. This difference is commonly referred as clearance variation $(C_{v})$ and it may be positive or negative. Positive $C_{v}$ leads to reduction in estimated life of the assembly and negative $C_{v}$ results in poor performance of the assembly. Both of them lead to weaken the quality of assembly and the associated cost is addressed in literature as QLC. Equation (6) describes the general Taguchi’s quality loss function to the nominal value of the best product/system²⁸

L (Y) = K {(Y - m)}^{2}

(6)

where $L (Y)$ is the quality loss for the performance Y; Y is the performance output value; m is the target performance of Y and K is the proportionality constant, which depends on financial criticality of Y (relating the performance deviation and cost/price).

With respect to the pin–hole assembly, QLC for the pin–hole assembly can be expressed as the function of actual clearance C and design clearance $C_{d}$ . The proposed QLC function for the pin–hole assembly is given in equation (7)

{QLC}_{p - h} = K {(C - C_{d})}^{2}

(7)

However, the pins and holes are manufactured in bulk and accepted within certain specific tolerance limits in interchangeable assembly. Under this circumstance, the value of C is not a constant and varies from pair to pair. Here, C can be assumed as an average value $C_{a}$ , which is defined as the difference between the mid values of hole and pin and is given in equation (7a)

C_{a} = ((\frac{t_{h}^{u} + t_{h}^{l}}{2}) - (\frac{t_{p}^{u} + t_{p}^{l}}{2}))

(7a)

where $t_{h}^{u}$ is the size of the hole at upper tolerance limit; $t_{h}^{l}$ is the size of the hole at lower tolerance limit; $t_{p}^{u}$ is the size of the pin at upper tolerance limit and $t_{p}^{l}$ is the size of the pin at lower tolerance limit.

The ${QLC}_{p - h}$ assembly can be rewritten as given in equation (8)

{QLC}_{p - h} = K {(C_{a} - C_{d})}^{2}

(8)

The proportionality constant K depends on the factors influencing performance, and it can be defined as a ratio of cost of assembly (A) and maximum permissible clearance variation $(Δ)$ as described in equations (9a) and (9b). The cost of assembly can be equated with the sum of the processing costs of manufacturing a hole $(m_{h})$ and a pin $(m_{p})$ . Equation (10) on substitutions provides the final expression for QLC

K = \frac{A}{Δ^{2}}

(9)

A = m_{h} + m_{p}

(9a)

Δ = (U^{h} - L^{p}) - (L^{h} - U^{p}) or (U^{h} - L^{h}) + (U^{p} - L^{p})

(9b)

\begin{matrix} {QLC}_{p - h} = \frac{m_{h} + m_{p}}{{((U^{h} - L^{h}) + (U^{p} - L^{p}))}^{2}} \\ {((\frac{t_{h}^{u} + t_{h}^{l}}{2}) - (\frac{t_{p}^{u} + t_{p}^{l}}{2}) - C_{d})}^{2} \end{matrix}

(10)

Manufacturing cost

The manufacturing cost of assembly is the total cost of manufacturing the components of the assembly. It is the summation of manufacturing cost of the hole $(m_{h})$ and the manufacturing cost of the pin $(m_{p})$ and is given in equation (11). It is a variable only when different manufacturing processes are taken into consideration. As the tolerance allocation model under consideration is based on fixed manufacturing processes, this cost does not influence the tolerance allocation decisions

{MC}_{p - h} = m_{h} + m_{p}

(11)

Cost of rejections

Whatever be the production method, the dimension/size of the manufactured component follows normal distribution with the mean ‘μ’ and standard deviation ‘σ’. When the components are accepted in $μ \pm 3 σ$ limits, the acceptance percentage of the components is 99.992. Let U and L be the upper and lower limits at $3 σ$ deviation from mean ‘μ’. These values determine the process capability. However, when the components are accepted in the dimensional limits of $t^{u}$ and $t^{l}$ , where $t^{u} < U$ and $t^{l} > L$ , then the percentage acceptance of the components reduces and it depends on the values of $t^{u}$ and $t^{l}$ . The percentage of rejection of the components thus becomes percentage of acceptance subtracted from 100. Figure 1 illustrates the rejection percentage for the tolerance band of $t^{u}$ and $t^{l}$ .

Figure 1.

The rejection percentage – limit on tolerances within $\pm 3 σ$ limits.

The general expression for the percentage of rejection (%R) of components that are manufactured in the process that follow normal distribution is given in equation (12)

% R = 100 - (\frac{1}{σ \sqrt{2 π}} \int_{t^{l}}^{t^{u}} e^{- (\frac{{(x - μ)}^{2}}{2 σ^{2}})} dX) \times 100

(12)

where X is the dimensional value in the range of $t^{l}$ and $t^{u}$ (i.e. $X = t^{l}, [t^{l} + (t^{u} - t^{l}) / 1000] \dots t^{u}$ ).

The integral in equation (12) is calculated numerically using trapezoidal method by dividing the range $t^{l}$ to $t^{u}$ into 1000 divisions. The fraction of rejection (FR) per component is expressed in equation (13)

FR = 1 - \frac{1}{σ \sqrt{2 π}} \int_{t^{l}}^{t^{u}} e^{- (\frac{{(x - μ)}^{2}}{2 σ^{2}})} dx

(13)

For the pin-hole assembly under consideration, the fractional defectives of pin $({FR}_{p})$ and that of hole $({FR}_{h})$ are expressed in terms of the mean diameter $(μ_{p} / μ_{h})$ and the standard deviation of machine $(σ_{p} / σ_{h})$ , and are given by equations 14 and 15 respectively for pin and hole production processes.

{FR}_{p} = 1 - \frac{1}{σ_{p} \sqrt{2 π}} \int_{t_{p}^{l}}^{t_{p}^{u}} e^{- (\frac{{(x_{p} - μ_{p})}^{2}}{2 σ_{p}^{2}})} {dx}_{p}

(14)

{FR}_{h} = 1 - \frac{1}{σ_{h} \sqrt{2 π}} \int_{t_{h}^{l}}^{t_{h}^{u}} e^{- (\frac{{(x_{h} - μ_{h})}^{2}}{2 σ_{h}^{2}})} {dx}_{h}

(15)

where $x_{p} = t_{p}^{l}, [t_{p}^{l} + (t_{p}^{u} - t_{p}^{l}) / 1000] \dots t_{p}^{u}$ ; $x_{h} = t_{h}^{l}, [t_{h}^{l} + (t_{h}^{u} - t_{h}^{l}) / 1000] \dots t_{h}^{u}$ ; ${FR}_{p}$ is the fraction of pins rejected and ${FR}_{h}$ is the fraction of holes rejected.

The cost of the rejection is calculated by multiplying the fraction of rejection with the cost of production. Equations (16) and (17) provide the costs due to the rejection of pin and hole.

Rejection cost of the pins $({RC}_{p})$

{RC}_{p} = m_{p} \times [1 - \frac{1}{σ_{p} \sqrt{2 π}} \int_{t_{p}^{l}}^{t_{p}^{u}} e^{- (\frac{{(x_{p} - μ_{p})}^{2}}{2 σ_{p}^{2}})} {dx}_{p}]

(16)

Rejection cost of the holes $({RC}_{h})$

{RC}_{h} = m_{h} \times [1 - \frac{1}{σ_{h} \sqrt{2 π}} \int_{t_{h}^{l}}^{t_{h}^{u}} e^{- (\frac{{(x_{h} - μ_{h})}^{2}}{2 σ_{h}^{2}})} {dx}_{h}]

(17)

The cost rejection for an assembly is the summation of the rejection cost of the hole and rejection cost of the pin, which is expressed in equation (18)

Rejection cost of an assembly $({RC}_{p - h})$

{RC}_{p - h} = {RC}_{p} + {RC}_{h}

(18)

Cost of inspection

The cost of eliminating the rejections depends on type of inspection, that is, 100% inspection or sampling inspection. Sampling inspection of components may lead to some of the assemblies that deviate from the limits. On the other hand, cent percent inspection can assure cent percent fit assembly. The problem under consideration does not permit more than the limit given for the application as the quality loss would be very high above that limit. Hence, cent percent inspection with gauges is considered here. Let per unit cost of inspection for pin and hole be ${IC}_{p}$ and ${IC}_{h}$ , respectively. The costs ${IC}_{p}$ and ${IC}_{h}$ depend on the method of inspection. The inspection cost of an assembly is the summation of unit cost of inspection for pin and hole

{IC}_{p - h} = {IC}_{p} + {IC}_{h}

(19)

Total cost of a pin–hole assembly

Equation (20) expresses the total cost (TC) function for the cost-tolerance model, which is the summation of quality loss, manufacturing, rejection and inspection costs

TC = {QLC}_{p - h} + {MC}_{p - h} + {RC}_{p - h} + {IC}_{p - h}

(20)

Clearance variation in a pin–hole assembly

The clearance variation $(C_{v})$ is defined as the maximum deviation of the fit (clearance or interference) that would be resulted with the allocated tolerances on the pin and hole from the design specification fit $(C_{d})$ . The maximum clearance/interference of the fit is the difference between the hole diameter at upper tolerance limit and pin diameter at lower tolerance limit (i.e. $t_{h}^{u} - t_{p}^{l}$ ), and the minimum clearance/interference of it corresponds to the difference in the values of hole diameter at lower tolerance limit and pin diameter at upper tolerance limit (i.e. $t_{h}^{l} - t_{p}^{u}$ ). The maximum deviation occurs either at maximum or minimum clearance/interference, and hence, $C_{v}$ is expressed as given in equation (21)

C_{v} = \max {\mod (t_{h}^{u} - t_{p}^{l} - C_{d}), \mod (t_{h}^{l} - t_{p}^{u} - C_{d})}

(21)

Problem definition

The problem is formally defined as the determination of Pareto solution set of tolerances for pin $(t_{p}^{u}, t_{p}^{l})$ and hole $(t_{h}^{u}, t_{h}^{l})$ for the minimum total cost and minimum clearance variation of a hole and pin assembly, given the design clearance $C_{d}$ , the process capability of the machines defined with their standard deviations ( $σ_{p}$ and $σ_{h}$ ) and mean diameter of either pin $(μ_{p})$ or hole $(μ_{h})$ .

Mathematical model

As the model considers predetermined manufacturing processes and cent percent gauging method of inspection, their cost elements do not influence optimal tolerance decisions, and hence, the total cost minimization objective defined in equation (20) can be reduced by neglecting manufacturing and inspection cost elements. Under this line of reasoning, equation (22) becomes the first objective function of this model. The second objective function is same as given in equation (21) and is stated here as a minimization criterion as equation (23). Equation (24) represents that the lower dimensional limit of the pin is always above the minimum size of the pin at $- 3 σ_{p}$ limits. Equation (25) represents that the upper dimensional limit of the pin is always below the maximum size of the pin at $+ 3 σ_{p}$ limits. The constraint equation (26) ensures that the lower dimensional limit of the pin is always below the upper dimensional limit of the pin. Similar manufacturing constraints are used for the hole which are expressed through equations (27)–(29). The objectives and constraints are dependent on the dimensional limits of the pin $(t_{u}^{p}, t_{l}^{p})$ and hole $(t_{u}^{h}, t_{l}^{h})$ , and hence, they are the decision variables to the problem

\begin{matrix} \begin{matrix} Minimise TC \\ = \frac{m_{h} + m_{p}}{{((U^{h} - L^{h}) + (U^{p} - L^{p}))}^{2}} {((\frac{t_{h}^{u} + t_{h}^{l}}{2}) - (\frac{t_{p}^{u} + t_{p}^{l}}{2}) - C_{d})}^{2} \\ + m_{p} \times [1 - \frac{1}{σ_{p} \sqrt{2 π}} \int_{t_{p}^{l}}^{t_{p}^{u}} e^{- (\frac{{((x_{p} - μ_{p}))}^{2}}{2 σ_{p}^{2}})} {dx}_{pi}] \\ + m_{h} \times [1 - \frac{1}{σ_{h} \sqrt{2 π}} \int_{t_{h}^{l}}^{t_{h}^{u}} e^{- (\frac{{(x_{h} - μ_{h})}^{2}}{2 σ_{h}^{2}})} {dx}_{hi}] \end{matrix} \end{matrix}

(22)

\begin{matrix} Minimise C_{v} \\ = \max {\mod (t_{h}^{u} - t_{p}^{l} - C_{d}), \mod (t_{h}^{l} - t_{p}^{u} - C_{d})} \end{matrix}

(23)

Subjected to the manufacturing constraints

t_{p}^{l} > L^{p}

(24)

t_{p}^{u} < U^{p}

(25)

t_{p}^{l} < t_{p}^{u}

(26)

t_{h}^{l} > L^{h}

(27)

t_{h}^{u} < U^{h}

(28)

t_{h}^{l} < t_{h}^{u}

(29)

Solution methodology

The tolerance allocation model under study belongs to constrained non-linear multi-objective problem involving integrals in its objective functions. The decision variables are continuous in its solution space constrained with upper and lower limits. Ryu et al.²⁹ proposed a method for approximating the Pareto front of a multi-objective simulation optimization problem. In their article, the weight on each single objective function is determined by accessing newly introduced points at the current iteration and the non-dominated points so far. Legriel et al.³⁰ proposed a general methodology for approximating the Pareto front of multi-criteria optimization problems. This article presents an ISA to explore the entire decision space. The decision variables are defined in this space using the specified levels of precision (Δx and Δy). The decision space is then transferred to the objective space so as to evaluate and obtain the Pareto optimal front. This method, although time consuming, is capable of capturing Pareto front with reasonable accuracy. Figure 2 outlines the procedural steps of the proposed ISA in the form of pseudo code. This section delineates, with a numerical illustration, the different modules of ISA, namely, data input module, initial size allocation module, evaluation module, design space exploration module and Pareto front determination module.

Figure 2.

ISA pseudo code.

Data input module

The data that govern the problem given in Table 1 are the input to the ISA and used for illustration.

Table 1.

Sample input data.

Parameter	$μ_{p}$	$C_{d}$	$σ_{h}$	$σ_{p}$	$m_{h}$	$m_{p}$	Δx	Δy
Unit	mm	mm	mm	Mm	INR	INR	mm	mm
Value	29.9	0.05	0.02	0.015	15	10	0.005	0.005

INR: Indian Rupee.

Initial size allocation module

The total cost of the assembly TC defined in equation (22) and its corresponding objective function $C_{v}$ stated in equation (23) depend on the input data. The mean diameter of the hole $(μ_{h})$ is determined by equation (1). The minimum and maximum sizes of the pin $(L^{P}, U^{P})$ and hole $(L^{h}, U^{h})$ which are related to standard deviation of the hole and pin are determined by equations (2)–(5). The dimensional limits of the pin $(t_{u}^{p}, t_{l}^{p})$ and hole $(t_{u}^{h}, t_{l}^{h})$ are determined as initial values of the iteration by the equations (30)–(33)

t_{h}^{l} = L^{h} (represents a minimum hole diameter)

(30)

t_{p}^{u} = U^{p} (represents a maximum pin diameter)

(31)

t_{h}^{u} = t_{h}^{l} + Δ x (minimum hole diameter incremented with Δ x)

(32)

t_{p}^{l} = t_{p}^{u} - Δ y (maximum pin diameter decremented with Δ y)

(33)

The corresponding values for the illustrative example are shown in Table 2.

Table 2.

Initial parameters of the problem illustrated.

Parameter	Value (mm)
$μ_{h}$	$29.95$
$t_{h}^{l}$	$29.89$
$t_{p}^{u}$	$29.945$
$L^{p}$	$29.885$
$U^{h}$	$30.01$
$t_{h}^{u}$	$29.91$
$t_{p}^{l}$	$29.940$

Iterative search and evaluation module

The entire design space is explored by incrementing the hole dimension and decrementing the pin dimension. The pin dimension is decremented by ‘Δy’ which is the precision level of the pin manufacturing machine. The hole dimension is decremented by ‘Δx’ which is the precision level of the hole manufacturing machine. The manufacturing constraints are ensured through equations (24)–(29). This module calculates the total cost TC (using equation (22)) and clearance variation $C_{v}$ (using equation (23)) by substituting the input data given in Table 2 and the current pin and hole sizes. Calculated TC, C_v and the corresponding $t_{h}^{u}, t_{h}^{l}$ , $t_{p}^{u}, t_{p}^{l}$ are stored and then proceeded further by updating the pin and hole sizes. Table 3 shows $t_{h}^{u}, t_{h}^{l}$ , $t_{p}^{u}, t_{p}^{l}$ , C_v and TC at different iterations.

Table 3.

Iteration steps and results.

$t_{p}^{u}$	$t_{h}^{l}$	$t_{h}^{u}$	$t_{p}^{l}$	$C_{v}$	$TC$
29.945	29.950	29.955	29.94	0.045	24.40164
			29.935	0.045	24.23195
			⋮	⋮	⋮
			29.855	0.045	13.54995
		29.960	29.94	0.045	22.90046
			29.935	0.045	22.73786
			⋮	⋮	⋮
			29.855
		⋮	⋮	⋮	⋮
		30.01	29.94	0.045	17.58403
			29.935	0.045	17.49223
			⋮	⋮	⋮
			29.855	0.105	8.05745
29.945	29.955	29.960	29.94	0.04	24.27830
			29.935	0.04	24.12279
			⋮	⋮	⋮
			29.855	0.055	13.66755
		⋮
		30.01	29.94	0.04	19.03273
			29.935	0.04	18.94808
			⋮	⋮	⋮
			29.855	0.105	9.62662
	⋮
	30.005	30.01	29.94	0.02	25.00783
			29.935	0.025	25.00645
			⋮	⋮	⋮
			29.855	0.105	16.87684
29.940	29.945	29.95	29.935	0.045	24.36659
⋮	⋮	⋮	⋮	⋮	⋮
29.860	30.005	30.01	29.855	0.105	29.15323

Pareto front determination module

The obtained $C_{v}$ are sorted and stored. For each unique $C_{v}$ , ${TC}_{\min}^{C_{v}}$ is initialized and compared with other TC values. If any TC is found to be lower than the initialized ${TC}_{\min}^{C_{v}}$ then it is updated with the lower TC. The ${TC}_{\min}^{C_{v}}$ obtained at the end of each iterations is stored along with corresponding pin and hole sizes ( $t_{p}^{u}, t_{p}^{l}$ , $t_{h}^{u}, t_{h}^{l}$ ). The above procedure is continued for all other unique $C_{v}$ . At the end, ${TC}_{\min}^{C_{v}}$ is stored as TC and is presented along with $C_{v}$ as the Pareto front optimal solution. Table 4 shows the non-dominated Pareto front solution obtained through ISA to the illustrative problem.

Table 4.

Optimal Pareto front of the illustrative problem.

Pareto front point	$t_{h}^{u}$	$t_{h}^{l}$	$t_{p}^{u}$	$t_{p}^{l}$	$C_{v}$	$TC$
A	29.950	29.945	29.90	29.895	0.005	22.21382
B	29.955	29.940	29.90	29.895	0.01	19.34188
C	29.960	29.940	29.905	29.895	0.015	16.64496
D	29.965	29.935	29.905	29.895	0.02	14.18765
E	29.965	29.935	29.910	29.890	0.025	11.84868
F	29.970	29.930	29.910	29.890	0.03	9.80951
G	29.970	29.930	29.915	29.885	0.035	7.93278
H	29.975	29.925	29.915	29.885	0.04	6.34261
I	29.975	29.925	29.920	29.880	0.045	4.99373
J	29.980	29.925	29.920	29.880	0.05	4.41463
K	29.980	29.925	29.920	29.875	0.055	3.99105
L	29.985	29.925	29.920	29.875	0.060	3.60754
M	29.990	29.925	29.920	29.875	0.065	3.37271
N	29.990	29.925	29.920	29.870	0.070	3.15420
O	29.995	29.925	29.920	29.870	0.075	3.03530
P	29.995	29.930	29.925	29.865	0.080	2.95201
Q	30.000	29.930	29.925	29.865	0.085	2.91493

Model utility

Customer’s preferences on the objectives vary depending upon their capabilities and requirements. The manufacturer is bound to find manufacturing solution according to the needs of the customer. This section presents how the optimal Pareto front of the model can be used by the manufacturer to determine the optimal tolerances to the following demands of the customers: (1) minimum clearance variation with cost constraint, (2) minimum cost with limitation on clearance variation and (3) restriction on both cost and clearance variation. Figure 3 demonstrates all the above cases using the optimal Pareto front of the example problem discussed in the ‘Solution methodology’ section. Tolerances corresponding to point D is the choice of the manufacturer when he is constrained to produce under the TC of INR15 (shown as line 11), the cost evaluated to satisfy the demand of customer cost. Under this cost restriction, the manufacturer can commit clearance variation to the extent of 0.02. Whereas tolerances corresponding to point N is the solution for the customer who imposes higher limit on $C_{v}$ as 0.07 (shown as line 22) and the manufacturer can negotiate for fixing the cost above INR3.15. When the customer poses limits on TC as INR10 (shown as line 33) and $C_{v}$ as 0.06 (shown as line 44), then the G, H, I, J and K become the set of choices for the manufacturer and negotiate accordingly. Besides, the optimal Pareto front indicates the non-existence of solution to certain customers (i.e. not satisfying the requirements of customer demands). Demands indicated by lines 55 and 66 cannot be met with the available manufacturing facility, and under such circumstances, it helps the manufacturer to decline the order. In addition to take decisions with respect to customer demands, this model can provide an estimate of cost for desired quality.

Figure 3.

Pareto front and model utility demonstration.

Model sensitivity studies

The model developed and illustrated is confined to a specific manufacturing scenario defined with capabilities of the processes ( $C_{p}^{h}$ and $C_{p}^{p}$ ). However, the practice experiences variations in process capabilities due to difference in the skill of the operators, ageing of the machines and improper handling of tools. In order to make use of the model, it becomes essential to understand the change in the Pareto front under the different process capabilities of the processes. This would be useful for the manufacturer to take decisions on new machinery purchase and expansion and the needed accuracy of Pareto front. This section studies the effect of process capability on Pareto front by (1) varying the process capability of pin manufacturing machine while keeping the process capability of hole manufacturing machine as same and (2) varying the process capability of hole manufacturing machine while keeping the process capability of pin manufacturing machine as constant. Besides, the solution space and optimal Pareto front are obtained with discrete points under defined precision levels (i.e. Δx and Δy), although the decision variables are continuous in nature. This section further explores the sensitiveness of the model with respect to change in the different levels of accuracy.

Effects of process capability

The effects of process capability of manufacturing facility (pin and hole) are discussed in this section. Figure 4 shows the Pareto front obtained by varying $σ_{p}$ in the range from 0.01 to 0.03 on the illustrative problem data. Figure 5 shows the Pareto front obtained by varying $σ_{h}$ in the range from 0.01 to 0.03 on the illustrative problem data. The observations reveal that

Number of Pareto solutions increases with increase in $σ_{p}$ and $σ_{p}$ ;

Difference in TC is marginal at lower values of $C_{v}$ and increases gradually as $C_{v}$ increases;

As $σ_{p}$ / $σ_{h}$ increases, the Pareto front is shifted towards higher TC.

Figure 4.

Pareto front plot of $C_{v}$ versus TC for various process capabilities of pin.

Figure 5.

Pareto front plot of $C_{v}$ versus TC for various process capabilities of hole.

Effects of precision level

The machining accuracy plays slender role in allocating tolerance of the pin and hole. The effect is studied by varying uniformly Δx and Δy in the range from 0.004 to 0.02 while keeping other parameters same as that used in illustration. Figure 6 shows the Pareto front obtained for different precision level. It is observed from Figure 6 that higher the precision level, smaller the number of Pareto solution, and there is no much significant variation in their Pareto front.

Figure 6.

Plot of $C_{v}$ versus TC for various precision levels.

Conclusion

In this article, a bi-objective (‘minimum cost and minimum clearance variation’) manufacturing tolerance allocation model for an interchangeable assembly of pin and hole is formulated to complement the need of small-scale industries where there exists two categories of machines, one for pin production and other for hole production. An ISA is proposed to obtain a Pareto optimal tolerance set for the objectives of non-dominated minimum cost and minimum clearance variation, and illustrated with a sample data set. The usefulness of the Pareto front in manufacturing tolerance allocation decisions is demonstrated with different customer requirement scenarios that reveal its application domain. Besides, the Pareto front may assist the customer to know the clearance variation that is possible to get for the cost he or she could invest. Besides, the discussions on the effect of process capabilities of pin and hole manufacturing machines on the optimality pave for understanding their criticality. However, the model is limited to a set of manufacturing facility, one for pin and one for hole, a typical case of small-scale industries. However, the model can be extended to medium-scale industries, where there will be variety of facilities (different capabilities, models, types) for both pin and hole. Under the diverse manufacturing scenario, the decision space would comprise the combination of manufacturing facilities and decisions for allocating the tolerance. The tolerance allocation is furthermore challenging, and hence, the intelligent search heuristics such as genetic algorithm, simulated annealing algorithm, bee colony algorithm, ant colony algorithm and particle swarm optimization may be considered as the solution methodology alternative to the ISA.

Footnotes

Appendix 1 Acknowledgements

The authors thank the managements of Thiagarajar College of Engineering, Madurai and SACS M.A.V.M.M Engineering College, Madurai for providing necessary facilities to carry out this work.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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