DENG et al 2021b - Revision history

Rimni at 12:02, 14 February 2022

2022-02-14T12:02:45Z

Rimni: /* 2.1 Performance degradation process based on Wiener process */

2022-02-14T12:01:30Z

‎2.1 Performance degradation process based on Wiener process

Rimni at 14:45, 3 November 2021

2021-11-03T14:45:04Z

Rimni at 14:43, 3 November 2021

2021-11-03T14:43:53Z

Rimni at 14:36, 3 November 2021

2021-11-03T14:36:04Z

Rimni at 15:42, 2 November 2021

2021-11-02T15:42:27Z

Rimni at 15:30, 2 November 2021

2021-11-02T15:30:43Z

Rimni at 15:28, 2 November 2021

2021-11-02T15:28:08Z

Rimni at 15:26, 2 November 2021

2021-11-02T15:26:37Z

Rimni: /* 3.1 Notations */

2021-11-02T15:24:20Z

‎3.1 Notations

@@ Line 34: / Line 34: @@
 In the current research, this kind of model has gradually developed to general and nonlinear problems related to time or state [13]. Baraldi, Mangili, Zio ''et al.'' sort out the diversity of model research, including time diversity, unit diversity, measurement diversity, and structural diversity [14]. Another point of the study is to consider the impact of the working environment on the performance degradation process. Bian, Gebraeel, Bian, Gebraeel ''et al.'' add covariates in the Wiener process as the environmental factors in the formula [15].
-The model based on the Wiener process has also been applied to the field of spare parts. Elwany, Gebraeel ''et al''. firstly combine the maintenance decision based on the Wiener process with the spare parts ordering to evaluate the optimal replacement and ordering time. They finally put forward the general framework of the joint policy as well [16]. After that, Zhang and Chen ''et al.'' tried to introduce the Bayesian model into the joint decision model, and update the model parameters with real-time performance data [17]. On the other hand, Zhang ''et al.'' extend the decision-making process to propose a random order lead time for inventory [18].
+The model based on the Wiener process has also been applied to the field of spare parts. Elwany, Gebraeel ''et al''. firstly combine the maintenance decision based on the Wiener process with the spare parts ordering to evaluate the optimal replacement and ordering time. They finally put forward the general framework of the joint policy as well [16]. After that, Zhang and Chen tried to introduce the Bayesian model into the joint decision model, and update the model parameters with real-time performance data [17]. On the other hand, Zhang and Chen extend the decision-making process to propose a random order lead time for inventory [18].
 ===2.2 Inventory policy===

@@ Line 34: / Line 34: @@
 In the current research, this kind of model has gradually developed to general and nonlinear problems related to time or state [13]. Baraldi, Mangili, Zio ''et al.'' sort out the diversity of model research, including time diversity, unit diversity, measurement diversity, and structural diversity [14]. Another point of the study is to consider the impact of the working environment on the performance degradation process. Bian, Gebraeel, Bian, Gebraeel ''et al.'' add covariates in the Wiener process as the environmental factors in the formula [15].
-The model based on the Wiener process has also been applied to the field of spare parts. Elwany, Gebraeel ''et al''. firstly combine the maintenance decision based on the Wiener process with the spare parts ordering to evaluate the optimal replacement and ordering time. They finally put forward the general framework of the joint policy as well [16]. After that, Zhang, Chen ''et al.'' tried to introduce the Bayesian model into the joint decision model, and update the model parameters with real-time performance data [17]. On the other hand, Zhang, Si ''et al.'' extend the decision-making process to propose a random order lead time for inventory [18].
+The model based on the Wiener process has also been applied to the field of spare parts. Elwany, Gebraeel ''et al''. firstly combine the maintenance decision based on the Wiener process with the spare parts ordering to evaluate the optimal replacement and ordering time. They finally put forward the general framework of the joint policy as well [16]. After that, Zhang and Chen ''et al.'' tried to introduce the Bayesian model into the joint decision model, and update the model parameters with real-time performance data [17]. On the other hand, Zhang ''et al.'' extend the decision-making process to propose a random order lead time for inventory [18].
 ===2.2 Inventory policy===

@@ Line 1,239: / Line 1,239: @@
 <!--[[Image:Review_510195169132-image70.jpeg|center|1268x1268px]]-->
 |-
-|  style="text-align: center;font-size: 75%;"|(a) The comparison of every aspect of costs between the (<math>s -1</math>, <math>s</math>) policy and (<math>s</math>, <math>S</math>) policy (The part of <math>C</math>, <math>C_{pr}</math>, <math>C_{o}</math>, <math>C_{s}</math>, <math>C_{v}</math>)
+|  style="text-align: center;font-size: 75%;"|(a) The comparison of every aspect of costs between the (<math>s -1</math>, <math>s</math>) policy and (<math>s</math>, <math>S</math>) policy (the part of <math>C</math>, <math>C_{pr}</math>, <math>C_{o}</math>, <math>C_{s}</math>, <math>C_{v}</math>)
 |-
 |style="padding:10px;"|[[File:DENG_et_al_2021b_6718_Fig9b.jpeg]]
 <!--[[Image:Review_510195169132-image71.jpeg|center|1268x1268px]]-->
 |-
-|  style="text-align: center;font-size: 75%;"|(b) The comparison of every aspect of  costs between the (<math>s -1</math>, <math>s</math>) policy and (<math>s</math>, <math>S</math>) policy (The part of <math>C_{c}</math>, <math>C_{pr}</math>)
+|  style="text-align: center;font-size: 75%;"|(b) The comparison of every aspect of  costs between the (<math>s -1</math>, <math>s</math>) policy and (<math>s</math>, <math>S</math>) policy (the part of <math>C_{c}</math>, <math>C_{pr}</math>)
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" style="padding:10px;"| '''Figure 9'''.

@@ Line 1,176: / Line 1,176: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-!''No.'' !!  <math>a</math> !! <math>T_c</math> !! <math>S</math> !! <math>L_p</math> !! <math>E(EC_\infty)</math>
+!No. !!  <math>a</math> !! <math>T_c</math> !! <math>S</math> !! <math>L_p</math> !! <math>E(EC_\infty)</math>
 |-
 |  style="text-align: center;"|1

@@ Line 205: / Line 205: @@
 (1) Assume initial quantity is <math>E(t) = (0,0,0\cdots 0)</math>. When <math>t = 0</math>, <math>E(t)</math> is continuous.
-(2) <math>\Delta E = E(t + \Delta t) - E(t)</math> shows the changing amount between <math>t + \Delta t</math> and <math>t</math>. It obeys <math>p</math>-dimensional Normal distribution
+(2) <math>\Delta E = E(t + \Delta t) - E(t)</math> shows the changing amount between <math>t + \Delta t</math> and <math>t</math>. It obeys <math>p</math>-dimensional normal distribution
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 538: / Line 538: @@
 '''1) Initialization'''
-Firstly, initialize <math>E_{n_1}^0 </math>, <math>E_{n_2}^0 </math>, <math>E_{n_3}^0 \cdots  E_{n_k}^0</math>. Those are the initial state of parts. And then, initialize the parameters by inputting, such as <math> T_c </math>, <math> S </math>, <math> L_p </math> , <math> \mu </math>, <math> \sigma </math>, <math> n_K </math>, <math>\Phi </math>, <math> K </math>, <math> L </math>, <math> \sigma </math>, <math>T^0_{knk}</math>, <math>C_c</math>, <math>C_{pf}</math>, <math>C_{pr}</math>, <math>C_{o}</math>, <math>C_{s}</math>, <math>C_{v}</math>.
+Firstly, initialize <math>E_{n_1}^0 </math>, <math>E_{n_2}^0 </math>, <math>E_{n_3}^0 \cdots  E_{n_k}^0</math>. Those are the initial state of parts. And then, initialize the parameters by inputting, such as <math> T_c </math>, <math> S </math>, <math> L_p </math> , <math> \mu </math>, <math> \sigma </math>, <math> n_K </math>, <math>\Phi </math>, <math> K </math>, <math> L </math>, <math> \sigma </math>, <math>T^0_{kn_k}</math>, <math>C_c</math>, <math>C_{pf}</math>, <math>C_{pr}</math>, <math>C_{o}</math>, <math>C_{s}</math>, <math>C_{v}</math>.
 Before turning to the general degradation process, the system will pay attention to whether there is resupply getting or not. As (21) following:

@@ Line 307: / Line 307: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>T=inf\lbrace t\vert E(t)=L,t\geq 0\rbrace </math>
+| <math>T=\inf\lbrace t\vert E(t)=L,t\geq 0\rbrace </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (9)
@@ Line 467: / Line 467: @@
 For the same kind of parts that adopt data monitoring to estimate the residual life. Too frequent inspection cannot provide a more accurate data basis for the life prediction. What’s more, it may affect the fitting speed of fault function based on the Wiener process because of the large amount. At the same time, this behavior will continue to increase the cost of testing, resulting in excessive cost.
-On the contrary, it may increase the probability of part failure in the next inspection that <math> T_c </math> is too large so that it could lose the significance of preventive maintenance policy. Secondly, if <math>L_{nkp}</math> is too high, it will increase the times and the ordering cost. Apart from the two aspects, increasing maximum inventory could affect the inventory cost as well. What’s more, the decreasing inventory may cause the delay loss so that the availability of equipment will be affected.
+On the contrary, it may increase the probability of part failure in the next inspection that <math> T_c </math> is too large so that it could lose the significance of preventive maintenance policy. Secondly, if <math>L_{n_kp}</math> is too high, it will increase the times and the ordering cost. Apart from the two aspects, increasing maximum inventory could affect the inventory cost as well. What’s more, the decreasing inventory may cause the delay loss so that the availability of equipment will be affected.
-In conclusion, the joint policy of maintenance and inventory ordering focuses on adjusting the decision variables which are <math> T_c </math>,  <math> S </math> and <math>L_{nkp}</math> to control the costs of preventive maintenance, collective maintenance, check, ordering even delay loss. Under the condition of certain equipment reliability, it may reduce the total cost as much as possible to obtain the optional joint combination of corresponding decision variables.
+In conclusion, the joint policy of maintenance and inventory ordering focuses on adjusting the decision variables which are <math> T_c </math>,  <math> S </math> and <math>L_{n_kp}</math> to control the costs of preventive maintenance, collective maintenance, check, ordering even delay loss. Under the condition of certain equipment reliability, it may reduce the total cost as much as possible to obtain the optional joint combination of corresponding decision variables.
 When calculating the joint model, there are mainly two methods. One is establishing an accurate set of formulas after a complete mathematical derivation. But it is difficult to use explicit expressions to clearly express the relationship between objective functions and decision variables. Then, the joint optimization for the same kind of multi-type parts in this article is a logical process triggered randomly at a discrete time by the change of decision variables. Another is the Marko simulation. It can solve this kind of problem well. By adjusting the decision parameters and analyzing a large number of simulations, the value distribution of objective function in this interval of time is obtained to get the optional joint policy.

@@ Line 442: / Line 442: @@
 (<math> s -1 </math>, <math> s </math>) policy is one of the new aspects in modern inventory policy. For some valuable parts, it is often adopted and requires monitoring of inventory. Assuming that in the beginning, the <math>k^{th}</math> inventory is up to the <math> S </math>. The parts couldn’t degenerate in inventory. Once the replacement happens, the resupply applicant will be sent out.
-It takes time for spare parts from application to arrival. During it, the parts on work are still performing. So, <math>Q_k^{\tau_i}</math> is not only determined by <math>\Delta s_k^\tau</math> which is the difference value in period <math>\tau</math> between <math> S </math> and <math>s_k^\tau</math>, but also <math>Q_k^{\tau i}</math>’ which is the consumptions in <math>\tau_i</math>.
+It takes time for spare parts from application to arrival. During it, the parts on work are still performing. So, <math>Q_k^{\tau_i}</math> is not only determined by <math>\Delta s_k^\tau</math> which is the difference value in period <math>\tau</math> between <math> S </math> and <math>s_k^\tau</math>, but also <math>Q_k^{\tau_i}</math>’ which is the consumptions in <math>\tau_i</math>.
 Still take the condition-based maintenance process of the two parts mentioned in [[#img-1|Figure 1]] as an example. It supposes the <math>k^{th}</math> customer owns these two types of parts. The parts play an important role in each machine and consume the same kind of spare parts. So, the threshold of them is the same (<math>L_1 = L_2</math> and <math>L_{1_p} = L_{2_p} </math>). We can make a decision in inventory ordering. That is as [[#img-2|Figure 2]] following.

@@ Line 141: / Line 141: @@
 |  style="text-align: center;"|''yuan''
 |-
-|  style="text-align: center;"|<math>Q_k^{\tau i}</math>
+|  style="text-align: center;"|<math>Q_k^{\tau_i}</math>
 |  style="text-align: center;"|Number of spare parts replenishments of customer <math>k</math> in resupply period <math>\tau_i</math>
 |  style="text-align: center;"|
 |-
-|  style="text-align: center;"|<math>Q_k^{\tau i}</math>
+|  style="text-align: center;"|<math>Q_k^{\tau_i}</math>
 |  style="text-align: center;"|Spare parts consumption of customer <math>k</math> in resupply period <math>\tau_i</math>
 |  style="text-align: center;"|
@@ Line 157: / Line 157: @@
 |  style="text-align: center;"|Use cycle
 |-
-|  style="text-align: center;"|<math>t_k^{r\tau i}</math>
+|  style="text-align: center;"|<math>t_k^{r\tau _i}</math>
 |  style="text-align: center;"|Ending time of resupply in customer <math>k</math> in resupply period <math>\tau_i</math>
 |  style="text-align: center;"|Use cycle