Predictive maintenance: strategic use of IT in manufacturing organizations

Abstract

A combination of big data and predictive analytics orchestrated through the Internet of Things (IoT) offers many opportunities for researchers in Information Systems, Operations Management and Strategy to look at old problems in new ways, and to identify completely new research areas. While there is much hype, little research has been conducted that informs companies about how to profitably integrate the IoT with strategic or operational processes. This paper views the IoT through the lens of predictive maintenance -- the use of real-time data and predictive analytics algorithms to dynamically manage preventive maintenance policies. These are being used by numerous manufacturing organizations to transition from product-oriented to service-oriented business models. In particular, we analyze optimal preventive maintenance policies in an environment where equipment is subject to a deterioration, which shifts it from its initial, fully-productive state, having a specified, age-dependent failure rate to a less-productive or deteriorated state, having a different, presumably higher, age-dependent failure rate. The deterioration, itself, is a random process.

Appendices

Appendix A. Optimal preventive maintenance schedule when the state shift is observable

Panagiotidou and Tagaras (2007) developed a model to optimize maintenance for equipment with two quality states and general failure time distributions assuming the transition from the in-control state to the out-of-control state is observable. The model developed by Panagiotidou and Tagaras (2007) is a renewal reward process model with two decision variables:

• t_m0 = the scheduled time for preventive maintenance if the equipment remains in the in-control state during that time; and

• t_m1 = the rescheduled time for preventive maintenance if the equipment transitions to the out-of-control state prior to t_m0.

1.1 Preventive Maintenance Policy

• If the shift occurs prior to t_m1, then preventive maintenance is performed at time t_m1.

• If the shift occurs after t_m1 but before t_m0 then preventive maintenance is performed at the time of the shift.

• If no shift occurs prior to t_m0 then preventive maintenance is performed at t_m0.

Of course, if a failure occurs prior to t_m0 (or t_m1) then corrective maintenance is performed.

Specifically the model determines the values for t_m0 and t_m1 that maximize the expected profit per cycle as:

$$ {\operatorname{Max}\ \mathrm{EPT}}_{\mathrm{IoT}}\left({\mathrm{t}}_{\mathrm{m}0},{\mathrm{t}}_{\mathrm{m}1}\right)=\mathrm{EP}\left({\mathrm{t}}_{\mathrm{m}0},{\mathrm{t}}_{\mathrm{m}1}\right)/\mathrm{ET}\left({\mathrm{t}}_{\mathrm{m}0},{\mathrm{t}}_{\mathrm{m}1}\right) $$

where EP(t_m0, t_m1) is the expected profit per cycle and ET(t_m0, t_m1) is the expected cycle time, each given by the following expressions.

$$ \begin{array}{l}\mathrm{EP}\left({\mathrm{t}}_{\mathrm{m}0},{\ \mathrm{t}}_{\mathrm{m}1}\right)={R}_0\kern0.5em {\int}_0^{t_{m0}}\kern0.5em \overline{F}(t)\ {\overline{\Phi}}_0(t)\ dt+{R}_1\kern0.75em {\int}_0^{t_{m1}} f(t){\overline{\Phi}}_0(t)\kern1.25em {\int}_t^{t_{m1}}{\overline{\Phi}}_1\left({t}^{\prime}\right)\kern0.5em /\kern0.75em {\overline{\Phi}}_1(t)\ {dt}^{\prime }\ dt\ \hfill \\ {}\kern7.75em - W-\left({W}_P- W\right)\left[{\overline{\Phi}}_0\left({t}_{m0}\right)\ \overline{F}\left({t}_{m0}\right)+{\int}_0^{t_{m1}} f(t)\ {\overline{\Phi}}_0(t)\ {\overline{\Phi}}_1\left({t}_{m1}\right)\kern0.5em /\kern0.5em {\overline{\Phi}}_1(t)\ dt+{\int}_{t_{m1}}^{t_{m0}} f(t)\ {\overline{\Phi}}_0(t)\ dt\kern0.75em \right]\hfill \end{array} $$

$$ \begin{array}{l}\mathrm{ET}\left({\mathrm{t}}_{\mathrm{m}0},{\ \mathrm{t}}_{\mathrm{m}1}\right)={\int}_0^{t_{m0}}\kern0.5em \overline{F}(t)\ {\overline{\Phi}}_0(t)\ dt+\kern0.5em {\int}_0^{t_{m1}} f(t){\overline{\Phi}}_0(t)\kern1.25em {\int}_t^{t_{m1}}{\overline{\Phi}}_1\left({t}^{\prime}\right)\kern0.5em /\kern0.75em {\overline{\Phi}}_1(t)\ {dt}^{\prime }\ dt\ \hfill \\ {}\kern7.75em + Z+\left({Z}_P-\mathrm{Z}\right)\left[{\overline{\Phi}}_0\left({t}_{m0}\right)\ \overline{F}\left({t}_{m0}\right)+{\int}_0^{t_{m1}} f(t)\ {\overline{\Phi}}_0(t)\ {\overline{\Phi}}_1\left({t}_{m1}\right)\kern0.5em /\kern0.5em {\overline{\Phi}}_1(t)\ dt+{\int}_{t_{m1}}^{t_{m0}} f(t)\ {\overline{\Phi}}_0(t)\ dt\kern0.75em \right]\hfill \end{array} $$

Where

f(t) :

density function of the time to quality shift

F(t):

cumulative distribution function of the time to quality shift

$\overline{\mathrm{F}}\left(\mathrm{t}\right)$ :

1 – F(t)

φ _i (t) :

density function of the time of failure (equipment age) if the process is in state i (i = 0, 1) at t = 0; note that the density function of the time to failure if a quality shift occurs at time t_s is φ₁(t)/Φ₁(t_s) for t > t_s.

Φ _i (t) :

cumulative distribution function of the time of failure in state i

$\overline{\Phi}(t)$ :

1 – Φ_i(t)

h _i (t) :

φ_i(t) / Φ_i(t) (failure rate in state i)

t :

equipment age; t = 0 at the beginning of each cycle

t _m0 :

scheduled preventive maintenance time in the in-control state

t _m1 (t) :

rescheduled preventive maintenance time if a shift to the out-of-control state occurs at t < t_m0

Z :

expected time to repair the equipment after failure

Z _P :

expected time to perform preventive maintenance

R _i :

expected net revenue per unit time of operation in state i (i = 0,1)

W :

cost of repair after failure

W _P :

cost of preventive maintenance

Appendix B. Optimal preventive maintenance schedule when the state shift is not observable

If the transition from the in-control state to the out-of-control state is not observable, then there is only one decision variable, t_m0, which must be determined considering that the unobservable transition can occur prior to that time (and a failure may occur prior to that time).

Modifying the model developed by Panagiotidou and Tagaras (2007), if the transition is not observable then in all expressions, replace t_m1 with t_m. The objective function is then given as:

$$ {\operatorname{Max}\ \mathrm{EPT}}_{\mathrm{N}\mathrm{on}\ \mathrm{IoT}}\left({\mathrm{t}}_{\mathrm{m}}\right)={\mathrm{EP}}_{\mathrm{N}}\left({\mathrm{t}}_{\mathrm{m}}\right)/{\mathrm{ET}}_{\mathrm{N}}\left({\mathrm{t}}_{\mathrm{m}}\right) $$

where EP_N(t_m) is the expected profit per cycle and ET_N(t_m) is the expected cycle time, each given by the following expressions.

$$ \begin{array}{l}{\mathrm{EP}}_{\mathrm{N}}\left({\mathrm{t}}_{\mathrm{m}}\right)\kern1.5em ={R}_0\ {\int}_0^{t m}\kern1em \overline{F}(t)\ {\overline{\Phi}}_0(t)\ dt+\kern0.5em {R}_1\ {\int}_0^{t m} f(t)\ {\overline{\Phi}}_0(t)\kern0.75em {\int}_t^{t m}\ {\overline{\Phi}}_1\left({t}^{\prime}\right)\kern0.5em /\kern0.5em {\overline{\Phi}}_1(t)\ {dt}^{\prime }\ dt\ \hfill \\ {}\kern5.25em - W-\left({W}_P- W\right)\left[{\overline{\Phi}}_0\left({t}_m\right)\ \overline{F}\left({t}_m\right)+{\int}_0^{t m} f(t)\ {\overline{\Phi}}_0(t)\ {\overline{\Phi}}_1\left({t}_m\right)\kern0.5em /\kern0.5em {\overline{\Phi}}_1(t)\ dt\right]\hfill \end{array} $$

$$ \begin{array}{l}{\mathrm{ET}}_{\mathrm{N}}\left({\mathrm{t}}_{\mathrm{m}}\right)\kern1.5em ={\int}_0^{t m}\kern1em \overline{F}(t)\ {\overline{\Phi}}_0(t)\ dt+\kern0.5em {\int}_0^{t m} f(t)\ {\overline{\Phi}}_0(t)\kern0.75em {\int}_t^{t m}\ {\overline{\Phi}}_1\left({t}^{\prime}\right)\kern0.5em /\kern0.5em {\overline{\Phi}}_1(t)\ {dt}^{\prime }\ dt\ \hfill \\ {}\kern5.25em + Z+\left({Z}_P- Z\right)\left[{\overline{\Phi}}_0\left({t}_m\right)\ \overline{F}\left({t}_m\right)+{\int}_0^{t m} f(t)\ {\overline{\Phi}}_0(t)\ {\overline{\Phi}}_1\left({t}_m\right)\kern0.5em /\kern0.5em {\overline{\Phi}}_1(t)\ dt\right]\hfill \end{array} $$

All variables and functions are as defined above.

Appendix C. Example failure and shift distributions

Following Panagiotidou and Tagaras (2007) we consider a numerical example where the failure mechanism in both in-control and out-of-control states is expressed by Weibull distributions of the failure age. The Weibull density function associated with state t (0 = in-control and 1 = out-of-control) is:

$$ {\varphi}_i(t)={\lambda}_i\;{c}_i\;{t}^{c i-1}\;{e}^{-{\lambda}_i\;{t}^{c i}}\kern2.75em t>0,{c}_i>0,{\lambda}_i>0, i=0,1 $$

where λ_i is the scale parameter and c_i is the shape parameter of the distribution. In order to ensure that the failure rate when operating in the out-of-control state with equipment age t is larger than or equal to the failure rate in the in-control state with the same equipment age, it is necessary to have c₀ = c₁ and λ₀ ≤ λ₁. For the graphical illustrations in Appendix Figs. 7 and 8 below c₀ = c₁ = 2, λ₀ = 0.005 and λ₁ = 0.01 (mean times: μ₀ = 12.53, μ₁ = 8.86; standard deviations: σ₀ = 6.55, σ₁ = 4.63).

Fig. 7

Failure distribution for initial (in-control) state (φ₁(t); c₀ = 2, λ₀ = .005)

Fig. 8

Failure distribution for deteriorated (out-of-control) state (φ₂(t); c₁ = 2, λ₁ = .01)

The quality shift mechanism of the process is expressed by a Gamma density function.

$$ f(t)={\lambda}_c/\Gamma (c)\ {t}^{c-1}{e}^{-\lambda t} $$

For the graphical illustration in Fig. 9 below c = 2 and λ = 0.15 (mean time: μ = 13.33; standard deviation: σ = 9.43).

Fig. 9

Time to shift distribution (ƒ(t); c = 2, λ = .15)