Integrated economic design of quality control and maintenance management: Implications for managing manufacturing process

Abstract

This paper develops an integrated economic model for the joint optimization of quality control parameters and a preventive maintenance policy using the cumulative sum (CUSUM) control chart and variable sampling interval fixed time sampling policy. To determine the in-control and out-of-control cost for both mean and variance, a Taguchi quadratic loss function and modified linear loss function are used, respectively. Imperfect preventive maintenance and minimal corrective maintenance policies were considered in developing the model, which determines the optimal values for significant process parameters to minimize the total expected cost per unit of time. A numerical example is used to test the model, which is followed by a sensitivity analysis. The integration of CUSUM mean and variance charts with the maintenance actions are proven successful to detect the slightest shift of the process. The findings reveal that among all the cost components, process failure due to external causes and equipment breakdown has a noteworthy attribute to the total costs of the optimized model. It is expected that top managers can utilize the suggested combined model to minimize the costs related to quality loss and maintenance policy and achieve economical advantages as well.

Appendices

Appendix 1

The initial transition probability matrix for CUSUM-m chart,

$${\mathbf{Q}}_{{{\varvec{\upmu}}}} = \left[ {\begin{array}{*{20}c} {0.47} & {0.47} & {0.03} & 0 & 0 & 0 & 0 & 0 \\ {0.09} & {0.59} & {0.31} & {0.01} & 0 & 0 & 0 & 0 \\ 0 & {0.15} & {0.64} & {0.15} & {0.06} & 0 & 0 & 0 \\ 0 & {0.01} & {0.19} & {0.65} & {0.09} & {0.06} & 0 & 0 \\ 0 & 0 & {0.03} & {0.35} & {0.45} & {0.13} & {0.04} & 0 \\ 0 & 0 & 0 & {0.09} & {0.3} & {0.41} & {0.16} & {0.04} \\ 0 & 0 & 0 & 0 & {0.05} & {0.25} & {0.45} & {0.2} \\ 0 & 0 & 0 & 0 & {0.02} & {0.13} & {0.32} & {0.39} \\ \end{array} } \right]$$

The initial transition probability matrix for CUSUM-S² chart,

$${\varvec{Q}}_{{{\mathbf{s}}^{2 } }} = \left[ {\begin{array}{*{20}c} {0.48} & {0.45} & 0 & 0 & 0 & 0 & 0 & 0 \\ {0.28} & {0.45} & {0.19} & {0.05} & 0 & 0 & 0 & 0 \\ 0 & {0.25} & {0.61} & {0.12} & {0.02} & 0 & 0 & 0 \\ 0 & {0.01} & {0.3} & {0.45} & {0.21} & {0.03} & 0 & 0 \\ 0 & 0 & {0.02} & {0.28} & {0.41} & {0.28} & {0.01} & 0 \\ 0 & 0 & 0 & {0.05} & {0.21} & {0.49} & {0.24} & {0.01} \\ 0 & 0 & 0 & 0 & 0 & {0.45} & {0.47} & {0.08} \\ 0 & 0 & 0 & 0 & 0 & 0 & {0.43} & {0.57} \\ \end{array} } \right]$$

Appendix 2

2.1 Cycle length calculation

According to Eq. (20), expected cycle length

$$E\left( {T_{cycle} } \right) = E\left( {T_{1} } \right) + E\left( {T_{2} } \right) + E\left( {T_{3} } \right) + E\left( {T_{4} } \right).$$

The expected in control process time

$$E\left( {T_{1} } \right) = \frac{1}{\lambda } + \left( {1 - \gamma_{1} } \right)*t_{0} *\frac{s}{{ARL_{j1}}} .$$

(27)

where t₀ is the expected time of searching an assignable cause under a false alarm, and s is the expected sampling frequency while in control; ARL_j1 denotes in-control joint average run length.

$\gamma_{1} = 1$; if production goes on during searches.

0_; if production stops during searches.λ is the process failure rate. Process failure rate due to an external cause is denoted by $\lambda_{1}$ and the process failure rate owing to inferior machine condition is denoted by $\lambda_{2}$.

Here it is assumed that λ₁ and λ₂ are independent of each other and do not occur simultaneously.

So,

$$\lambda = \lambda_{1} + \lambda_{2} .$$

(28)

$$\lambda_{1} = \left( {1/{\text{Meantime between process failures due to external causes}}} \right),$$

(29)

$$\lambda_{2} = \frac{{N_{f} *P_{{FM_{2} }} }}{{T_{eval} }},$$

(30)

where $N_{f}$ is the number of machine failures and T_eval is the evaluation period and t_PM is the preventive maintenance interval.

According to Pandey et al., (2012), the relationship between ${\text{N}}_{{\text{f}}}$ and $t_{pm}$ is expressed by

$$N_{f} = 0.0437*\left( {t_{PM} } \right)^{0.8703} .$$

(31)

where the sampling frequency is calculated based on the concept given by Chou et al., (2008). The first sampling takes place at the time point h during the first interval of [0, h]. When the manufacturing process is in control, aside from the first interval, the possible values of the sampling frequency for any interval with fixed length h (i.e., the intervals [h, 2 h], [2 h, 3 h], [3 h, 4 h] …) are 1, 2,…. η. Thus, the expected in control sampling frequency is,

$$s = 1 + (\frac{{1 - \rho e^{\eta - 1} }}{1 - \rho } + \left( {2\eta - 1} \right) \cdot \rho^{\eta - 1} ) *\mathop \sum \limits_{j = 2}^{\infty } e^{{ - \left( {j - 1} \right)\lambda h}} ,$$

$$= 1 + \left( {\frac{{1 - \rho e^{\eta - 1} }}{1 - \rho } + \left( {2\eta - 1} \right) \cdot \rho^{\eta - 1} } \right) * \left( {\frac{{e^{ - \lambda h} }}{{1 - e^{ - \lambda h} }}} \right),$$

(32)

$$\eta = \frac{h}{d}$$

(33)

where ρ indicates the conditional probability that the sample point shows a warning signal whereas the process is actually in control. Therefore,

$$\rho = \frac{{2\left[ {\phi \left( k \right) - \phi \left( \omega \right)} \right]}}{{\phi \left( k \right) - \phi \left( { - k} \right)}} + \frac{{ \left[ {\phi \left( {k_1} \right) - \phi \left( {\omega_1} \right)} \right]}}{{\phi \left( {k_1} \right) - \phi \left( 0 \right)}} - \left( {\frac{{2\left[ {\phi \left( k \right) - \phi \left( \omega \right)} \right]}}{{\phi \left( k \right) - \phi \left( { - k} \right)}}*\frac{{\phi \left[ {\left( {k_1} \right) - \phi \left( {\omega_1} \right)} \right]}}{{\phi \left( {k_1} \right) - \phi \left( 0 \right)}}} \right).$$

(34)

where $\phi \left( . \right)$ denotes the standard normal cumulative distribution function, and.

$\alpha$ is the probability of false alarm given by

$$\alpha = 1/ARL_{j1} .$$

(35)

The expected out of control time before declaring the process is out of control can be denoted as,

$$E \left( {T_{2} } \right) = ATS_{2} {-}\xi ,$$

(36)

where

$$ATS_{2} = + \left( {ATS_{2} } \right)_{external} ,$$

(37)

ATS₂ is the average time interval from the last in control sampling point to the time to give out of control signal

$$ATS_{2} = [ \rho \mathop \sum \limits_{i = 0}^{\eta - 2} \rho^{i} d + \left( {1 - \rho } \right) \mathop \sum \limits_{i = o}^{\eta - 1} \rho^{i} \left( {h - id} \right)\left] * \right[(ARL_{j2} )_{mc} *\frac{{\lambda_{2} }}{\lambda } + \left( {ARL_{j2} )_{E} *\frac{{\lambda_{1} }}{\lambda }} \right].$$

(38)

where $(ARL_{j2} )_{mc}$ and $(ARL_{j2} )_{E}$ are the joint out of control ARL for machine degradation and external cause respectively. $\rho \mathop \sum \limits_{i = 0}^{\eta - 2} \rho^{i} d + \left( {1 - \rho } \right) \mathop \sum \limits_{i = o}^{\eta - 1} \rho^{i} \left( {h - id} \right)$ is the average sampling interval.$\xi$ indicates the average time gap between the last sampling time point, while the process was in control and the time point at which an assignable cause actually occurs and it can be presented as

$${\upxi } = \mathop \sum \limits_{{{\text{j}} = 0}}^{{{\upeta } - 1}} \left( {{\uprho }_{{1{\text{j }}}} {\uptau }_{{1{\text{j}}}} } \right) + \rho_{2 } \tau_{2} .$$

(39)

Here,

$$p_{\rho_{1j}} = \frac{{{\uprho }^{{\text{j}}} \left( {1 - {\uprho }} \right)\left( {h - {\text{jd}}} \right)}}{{{\uprho }\sum\nolimits_{{{\text{i}} = 0}}^{\eta - 2} {{\uprho }^{{\text{i}}} {\text{d + }}\left( {1 - {\uprho }} \right)} \sum\nolimits_{{{\text{i}} = 0}}^{\eta - 1} {{\uprho }^{{\text{i}}} {\text{}}}{\left( {h - {\text{id}}} \right)} }},$$

(40)

$$\rho_{2} = \frac{{{\uprho }\sum\nolimits_{i = 0}^{\eta - 2} {{\uprho }^{{\text{i}}} {\text{d}}} }}{{{\uprho }\sum\nolimits_{{{\text{i}} = 0}}^{\eta - 2} {{\uprho }^{{\text{i}}} {\text{d + }}\left( {1 - {\uprho }} \right)} \sum\nolimits_{{{\text{i}} = 0}}^{\eta - 1}{{\uprho }^{{\text{i}}} {\text{}}} {\left( {h - {\text{id}}} \right)} }}.$$

(41)

τ_1j is defined as the expected in-control time interval while an assignable cause occurs between the sampling time points $ih + jd\;and \; \left( {i + 1} \right)h$.

$$\tau_{10 } = \frac{{1 - \left( {1 + {\uplambda }h} \right){\text{e}}^{{ - {\uplambda }h}} }}{{{\lambda }\left( {1 - {\text{ e}}^{{ - {\lambda d}}} } \right)}},$$

(42)

$$\tau_{1j } = \frac{1}{\lambda } - \frac{{\left( {\eta - j} \right)de^{{ - \lambda \left( {\eta - j} \right)h}} }}{{1 - e^{{ - \lambda \left( {\eta - j} \right)h}} }},$$

(43)

For $j = 1, 2 \ldots ..\eta - 1$. and τ₂ is defined as the expected in-control time interval while an assignable cause occurs between the sampling time points $i$ and $\left( {i + 1} \right)$ having sampling interval d.

$$\tau 2 = \frac{{1 - \left( {1 + \lambda d} \right)e^{ - \lambda d} }}{{\lambda \left( {1 - e^{ - \lambda d} } \right)}}.$$

(44)

The expected sampling time,

$$E \left( {T_{3} } \right) = n*T_{s} .$$

(45)

where $T_{s}$ is the average sampling, inspecting, evaluating, and plotting time for each sample.

The expected time interval for searching and correcting assignable cause is

$$E \left( {T_{4} } \right) = t_{1} + E \left( {T_{restore} } \right),$$

$$= t_{1} + \left( {T_{resetting} *\frac{{\lambda_{1} }}{\lambda } + MTTR_{cr} *\frac{{\lambda_{2} }}{\lambda }} \right).$$

(46)

where t₁ is the average time to search for the assignable cause and $E \left( {T_{restore} } \right)$ is the expected time to repair or reset the process.

Appendix 3

3.1 Process failure cost calculation

The expected cost of false alarm

$$E\left[ {C_{false} } \right] = C_{false} \cdot \frac{S}{{ARLj{}_{1}}} \cdot t_{0} ,$$

(47)

where $C_{false}^{{}}$ is the cost per unit time for scrutinizing the false alarm.

If ‘a’ and ‘b’ respectively are the fixed and variable cost of sampling for unit sample, then the expected cost per cycle for sampling is,

$$E\left[ {cost\;of\,sampling} \right] = \frac{{\left( {a + bn} \right)\left[ {\frac{1}{\lambda } + \left\{ {\left( {1 - \gamma_1} \right)*t0 *\frac{s}{{ARL_{j1}}}} \right\} + ATS_{2} {-} \xi + \eta Ts + r_{1} t_{1} + r_{2} E\left( {T_{restore} } \right)} \right]}}{{\rho \mathop \sum \nolimits_{i = 0}^{\eta - 2} \rho^{i} d + \left( {1 - \rho } \right) \mathop \sum \nolimits_{i = o}^{\eta - 1} \rho^{i} \left( {h - id} \right)}}.$$

(48)

$$\begin{aligned} \gamma_{2} = & 1;{\text{if production continues while correcting the process}} \\ & {\text{0; if production stops during correcting the process}} \\ \end{aligned}$$

Let,${C}_{resetting}$ be the cost of finding an assignable cause. It also includes downtime costs if production stops functioning while searching and resetting. So, the expected value of ${C}_{resetting}$ can be calculated as:

$$E \left[ {C_{resetting} \left] = \right[C_{resetting} *T_{resetting} } \right]* \frac{{\lambda_{1} }}{\lambda }.$$

(49)

The average cost of corrective actions and repairing machine owing to FM₂ is computed as:

$$E\left[ {\left( {C_{repair} } \right) FM_{2} } \right] = \left\{ {MTTR_{CM} \left[ {PR. C _{lp} + LC} \right] + C_{FCPPM} } \right\}*\frac{{\lambda_{2} }}{\lambda }.$$

(50)

3.2 Consideration of Taguchi loss

Taguchi loss function (TLF) is utilized to find out the in control and out of control quality loss occurred due to the production of defective products (Al-Ghazi et al. 2007). Here, a critical to quality (CTQ) characteristic is considered with bilateral tolerances of equal value ($\Delta$) for CUSUM-m chart and unilateral tolerance of value (${\Delta }_{1}$) for the CUSUM-S² chart. The penalty cost of producing a defective product is A cost/unit, and uniform rejection cost is incurred beyond the control limits.

CUSUM-m chart.

[L_{in control}] determination: At in control state quality loss per unit time is computed as,

$$\left[ {L_{in control} } \right]_{mean} = [PR*\frac{A}{{\Delta^{2} }}\mathop \smallint \limits_{{\mu - \frac{k\sigma }{{\sqrt n }}}}^{{\mu + \frac{k\sigma }{{\sqrt n }}}} \left( {x - \mu } \right)^{2} f\left( x \right)dx] + \left( {PR*R*C_{frej } } \right),$$

(51)

where $PR$ is the production rate $x$ representing sample means of the quality characteristic, and $f(x)$ is its normal density function with a mean of $\mu$ and a standard deviation of n $\frac{\sigma }{\sqrt{n}}$. Now, under this loss function, unlike the classical SPC approach, any deviation from the target value can be counted as a loss. While running the process within control limits the proportion of nonconforming unit, $R = 1 - \left\{ {\phi \left( k \right) - \phi \left( { - k} \right)} \right\}$ and $C_{frej}$ represent the cost of rejection per unit.

After some algebraic manipulations,

$$\left[ {L_{in control} } \right]_{mean} = PR * \frac{{\text{A}}}{{\Delta^{2} }} * \frac{{\sigma^{2} }}{n}\left[ {1 - \frac{2k}{{\sqrt {2\pi } }} e^{{\frac{{ - k^{2} }}{2}}} - \alpha } \right] + \left( {PR*R*C_{frej } } \right).$$

where $\alpha = 2{ }\phi \left( { - {\text{k}}} \right).$

$\left[ {L_{out\; of \;control} } \right]_{}$ determination

$$\left[ {L_{out of control} } \right]_{mean} = \left. {PR*\frac{A}{{\Delta^{2} }}\int_{ - \infty }^{\infty } {(x^{\prime} - \mu )^{2} f(x^{\prime})dx^{\prime}} - \mathop \smallint \limits_{{\mu - \frac{k\sigma }{{\sqrt n }}}}^{{\mu + \frac{k\sigma }{{\sqrt n }}}} \left( {x^{\prime} - \mu } \right)^{2} f\left( {x^{\prime}} \right)dx^{\prime}} \right\},$$

(52)

After some algebraic manipulations,

$$\begin{aligned} \left[ {L_{out of control} } \right]_{mean} = & PR* \frac{A}{{\Delta^{2} }} *\frac{{\sigma^{2} }}{n}[\left( {1 + \delta^{2} n} \right)*\left( {1 - \beta } \right) \\ & + \left( {\frac{{k + \delta \sqrt n { }}}{{\sqrt {2\pi } }}} \right) * e^{{\frac{{ - \left( {k - \delta \sqrt n } \right)^{2} }}{2}}} + \left( {\frac{{k - \delta \sqrt n { }}}{{\sqrt {2\pi } }}} \right) * e^{{\frac{{ - \left( {k + \delta \sqrt n } \right)^{2} }}{2}}} \\ & + \left( {PR*R_{\delta } *C_{frej } } \right). \\ \end{aligned}$$

(53)

$$\beta = \phi \left( {k - \delta * \sqrt n } \right) - \phi \left( {k - \delta * \sqrt n } \right).$$

where ${R}_{\delta }$ is the proportion of non-conforming units while the process runs out of control state.

For an out of control state, the quality loss per unit time owing to a machine failure is computed as:

$$\begin{aligned} \left[ {L_{out of control} } \right]_{mean m/c} = & PR* \frac{A}{{\Delta^{2} }} *\frac{{\sigma^{2} }}{n}[\left( {1 + \delta_{m/c}^{2} n} \right)*\left\{ {1 - \phi \left( {k - \delta_{m/c} *\sqrt n } \right) + \phi \left( { - k - \delta_{m/c} *\sqrt n } \right) } \right\} \\ & + (\frac{{k + \delta_{m/c} \sqrt n }}{{\sqrt {2\pi } }})* + e^{{\left( {\frac{{ - \left( {k - \delta_{m/c} \sqrt n } \right)^{2} }}{2}} \right)}} + (\frac{{k - \delta_{m/c} \sqrt n }}{{\sqrt {2\pi } }}) * \\ + & e^{{\left( {\frac{{ - \left( {k + + \delta_{m/c} \sqrt n } \right)^{2} }}{2}} \right)}} + \left( {PR*R_{\delta m/c} *C_{frej } } \right). \\ \end{aligned}$$

(54)

where

$$~R_{{\delta \;m/c}} = 1 - \left\{ {\phi \left( {k - \delta _{{m/c}} \sigma } \right) - \phi \left( { - k - \delta _{{m/c}} \sigma } \right)} \right\}.$$

Similarly, for an out of control state, the quality loss per unit time owing to external causes is computed as:

$$\left[ {L_{out of control} } \right]_{mean E} = PR* \frac{A}{{\Delta^{2} }} *\frac{{\sigma^{2} }}{n}\left[ {\left( {1 + \delta_{E}^{2} n} \right)*\left\{ {1 - \phi \left( {k - \delta_{E} *\sqrt n } \right) + \phi \left( { - k - \delta_{E} *\sqrt n } \right)} \right\} + \left( {\frac{{k + \delta_{E} \sqrt n }}{{\sqrt {2\pi } }})e^{{\frac{{ - \left( {k - \delta_{E} \sqrt n } \right)^{2} }}{2}}} } \right) + \left( {\frac{{k - \delta_{E} \sqrt n }}{{\sqrt {2\pi } }}e^{{\frac{{ - \left( {k + \delta_{E} \sqrt n } \right)^{2} }}{2}}} } \right)} \right] + \left( {PR*R_{\delta E} *C_{frej } } \right).$$

(55)

where

$$R_{\delta m/c} = 1 - \left\{ {\phi \left( {k - \delta_{E} \sigma } \right) - \phi \left( { - k - \delta_{E} \sigma } \right)} \right\}.$$

(56)

3.3 CUSUM-S² chart

Since it is a “smaller the better” situation. I.e. it is better if the variance is smaller in the CUSUM-S² chart, the desired value for variance is 0. In this case, only the upper control limit is considered to monitor the chart. Trietsch (1999) stated that when the expected cost of exceeding the tolerance limits does not equal to both sides of the target, Taguchi quadratic loss function seems inappropriate in that situation. That's why here in control and out of control loss for the CUSUM-S² chart is determined considering the modified Kapoor and Wang (1994) model stated by C. H. Chen and Chou (2005), which is a linear loss function.

[$L_{in control}$] determination: At in control state quality loss per unit time is computed as$,$

$$\left[ {L_{in control} } \right]_{variance} = PR*\frac{{\text{A}}}{\Delta 1}*\mathop \smallint \limits_{ - \infty }^{{\frac{k1*\sigma }{{\sqrt n }}}} yf\left( y \right)dy + \left( {PR*R^{\prime}*C_{frej } } \right),$$

(57)

$$f\left( y \right) = \frac{1}{{\phi \left( {k1} \right) * \frac{\sigma }{\sqrt n } * \sqrt {2\pi } }}*e^{{ - \frac{{\left( {\frac{y - \mu }{{\frac{\sigma }{\sqrt n }}}} \right)^{2} }}{2}}} .$$

(58)

where $PR$ is the production rate, y represents sample variance of the quality characteristic. Now under this loss function, unlike the classical SPC approach, any deviation from the target value is count as a loss. In this work μ/σ > 5, the probability that y < 0 tends to 0.

After some algebraic manipulations,

$$\left[ {L_{in control} } \right]_{variance} = PR*\frac{{\text{A}}}{\Delta 1}*\frac{1}{{\phi \left( {k1} \right)}}\{ \mu *\phi \left( {k1 \frac{\sigma }{\sqrt n } - \mu } \right) - \frac{\sigma }{\sqrt n }*\varphi \left( {k1 \frac{\sigma }{\sqrt n } - \mu } \right) + \left( {PR*R^{\prime}*C_{frej } } \right)$$

(59)

where $\varphi \left( . \right)$ Signifies the standard normal probability density function.$R'$ denotes the proportion of defective items while the process is in control state.

$$R' = 1 - \phi \left( {k1} \right).$$

(60)

$\left[ {{L_{out~of~control}}} \right]$ Determination:

$${\left[ {{L_{out~of~control}}} \right]_{variance}} = PR*\frac{{\text{A}}}{{\Delta 1}}*\{ \mathop \smallint \limits_{ - \infty }^{\infty ~} y'f\left( {{\text{y'}}} \right)d{\text{y'}}\mathop \smallint \limits_{ - \infty }^{\frac{{k\sigma }}{{\sqrt n }}} y'f\left( {{\text{y'}}} \right)d{\text{y'}} + \left( {PR*R{'_{\delta ~}}*{C_{frej~~}}} \right),$$

$$f~\left( {~y'} \right)~ = ~\frac{1}{{\phi \left( {k1} \right)~*~\frac{\sigma }{{\sqrt n }}~*~\sqrt {2\pi } }}*{e^{ - ~\frac{{{{\left( {\frac{{y - \mu - \delta 1\sigma }}{{\frac{\sigma }{{\sqrt n }}}}} \right)}^2}}}{2}}}.$$

(61)

After some algebraic manipulations,

$${\left[ {{L_{out~of~control}}} \right]_{variance}} = PR*\frac{{\text{A}}}{{\Delta 1}}*\frac{1}{{\phi \left( {k1} \right)}}\left[ {\varphi \left( {k1 - \frac{{\mu \sqrt n }}{\sigma } - \delta 1\sigma } \right) + \left\{ {\{ 1 - \phi \left( {k1 - \frac{{\mu \sqrt n }}{\sigma } - \delta 1\sigma } \right)\} \left( {\mu + \delta 1\sigma } \right)} \right\} + ~\left( {PR*R{'_{\delta ~}}*{C_{frej~~}}} \right)} \right]$$

(62)

where $R{'_\delta }$ is the percentage of the non-conforming unit while the process runs out of control state.

For an out of control state, the quality loss per unit time owing to the machine failure is computed as:

$${\left[ {{L_{out~of~control}}} \right]_{variance~m/c}} = ~PR * \frac{{\text{A}}}{{\Delta 1}}*\frac{1}{{\phi \left( {k1} \right)}}\left[ {\varphi \left( {k1 - \frac{{\mu \sqrt n }}{\sigma } - \delta {1_{m/c}}\sigma } \right) + \left\{ {\{ 1 - \phi \left( {k1 - \frac{{\mu \sqrt n }}{\sigma } - \delta {1_{m/c}}\sigma } \right)\} \left( {\mu + \delta {1_{m/c}}\sigma } \right)} \right\} + ~\left( {PR*{{R'}_{\delta \frac{m}{c}}}*{C_{frej~~}}} \right)} \right].$$

(63)

where

$$R{'_{\delta ~m/c}} = 1 - \phi \left( {k1 - {\delta _{\frac{m}{c}}}\sigma } \right).$$

(64)

Similarly, at out of control state quality loss per unit time owing to external causes is computed as:

$${\left[ {{L_{out~of~control}}} \right]_{variance~E}}~ = PR*\frac{{\text{A}}}{{\Delta 1}}*\frac{1}{{\phi \left( {k1} \right)}}\left[ {\varphi \left( {k1 - \frac{{\mu \sqrt n }}{\sigma } - \delta {1_E}\sigma } \right) + \left\{ {\{ 1 - \phi \left( {k1 - \frac{{\mu \sqrt n }}{\sigma } - \delta {1_E}\sigma } \right)\} \left( {\mu + \delta {1_E}\sigma } \right)} \right\} + ~\left( {PR*{{R'}_{\delta ~E}}*{C_{frej~~}}} \right)} \right].$$

(65)

where

$$R{'_{\delta E}} = 1 - \phi \left( {k1 - {\delta _{~E~}}\sigma } \right).$$

(66)

Thus, the expected process quality loss for a cycle in the in-control state is:

$$E\left[ {{L_{in~control}}} \right] = ~\left\{ {{{\left[ {{L_{in~control}}} \right]}_{mean}}~ + {{\left[ {{L_{in~control}}} \right]}_{variance}}} \right\}~*\frac{1}{\lambda }.$$

(67)

Therefore, for an out of control state, the expected quality loss incurred per cycle owing to the machine failure is:

$$\begin{aligned} E~\left[ {\left( {cost~of~L_{{out~of~control}} } \right)_{{m/c}} ~} \right]~ = ~{\text{~}} & \left[ {L_{{out~of~control}} } \right]_{{~mean~m/c}} + ~ \\ & {\text{~}}\left[ {L_{{out~of~control}} } \right]_{{variance~m/c}} ]~*~\left\{ {ATS_{2} ~{-}~\xi ~ + ~n*T_{S} + ~r_{1} t_{1} ~ + r_{2} *E~\left( {T_{{restore}} } \right)} \right\}*~\frac{{\lambda _{2} }}{\lambda }. \\ \end{aligned}$$

(68)

Thus, for an out of control state, the expected quality loss incurred per cycle owing to external causes is:

$$\begin{aligned} E~\left[ {\left( {cost~of~L_{{out~of~control}} } \right)_{E} ~} \right]~ = & ~~\left[ {L_{{out~of~control}} } \right]_{{~mean~E}} \\ & + ~~\left[ {L_{{out~of~control}} } \right]_{{variance~E}} ]~*~\left\{ {ATS_{2} ~{-}~\xi ~ + ~n*T_{S} + ~r_{1} t_{1} ~ + r_{2} *E~\left( {T_{{restore}} } \right)} \right\}*~\frac{{\lambda _{1} }}{\lambda }. \\ \end{aligned}$$

(69)

Adding Eqs. [48], [49], [50], [51], [67], [68] and [69] gives the expected cost of the manufacturing process failure per cycle as:

$$\begin{aligned} E[C_{{process}} ] = & E[C_{{false}} \left] { + ~E~\left[ {Cost~of~sampling} \right] + ~E~} \right[C_{{resetting}} \left] {~ + ~E~} \right[\left( {C_{{repair}} } \right)_{{FM_{2} }} ] \\ & + E[L_{{in~control}} \left] {~ + E~} \right[\left( {cost~of~L_{{out~of~control}} } \right)_{{\frac{m}{c}}} ~] \\ & + E[\left( {cost~of~L_{{out~of~control}} } \right)_{E} ~. \\ \end{aligned}$$

(70)

Appendix 4

4.1 Parameters of Nelder-Mead simplex algorithm

The following parameters have been set for the Nelder-Mead Simplex algorithm:

Initial simplex parameters:

alpha (α): 1

beta (β): 0.5

lambda (λ): 1

Convergence parameters:

epsilon1 (ε₁): 1e-6.

epsilon2 (ε₂): 1e-6.

Shrinkage parameters:

gamma (γ): 2

delta (δ): 0.5

4.2 Parameters of genetic algorithm

Parameter settings for a meta-heuristic algorithm like the Genetic algorithm depend on the specific problem of interest (Fathollahi-Fard et al., 2020). For our problem, we have considered the following parameters of the Genetic algorithm:

Population Size: 100.

Population type: Double vector.

Creation function: Constraint dependent.

Crossover Fraction: 0.8.

Maximum number of generations: 5000.

Function tolerance: 1e-8.

Nonlinear constraint tolerance: 1e-8.