Wang et al 2020f - Revision history

Rimni at 14:53, 19 November 2020

2020-11-19T14:53:02Z

Rimni: /* 4.1 Comparison with Determined Model */

2020-11-19T13:54:58Z

‎4.1 Comparison with Determined Model

Rimni at 13:53, 19 November 2020

2020-11-19T13:53:57Z

Rimni: /* References */

2020-11-17T13:45:09Z

‎References

Rimni at 13:44, 17 November 2020

2020-11-17T13:44:39Z

Rimni at 13:14, 17 November 2020

2020-11-17T13:14:37Z

Rimni: /* 4.2 Comparison with chance constraint model */

2020-11-17T13:14:01Z

‎4.2 Comparison with chance constraint model

Rimni: /* 4.2 Comparison with chance constraint model */

2020-11-16T15:21:34Z

‎4.2 Comparison with chance constraint model

Rimni at 15:00, 16 November 2020

2020-11-16T15:00:27Z

Show changes

Rimni at 14:48, 16 November 2020

2020-11-16T14:48:19Z

@@ Line 963: / Line 963: @@
-In the second aspect, take the feasible solutions obtained from the determined model into robust models to test the feasibility of the determined solutions in the robust model, and the results are shown in [[#img-11|Figure 11]] . It can be seen from the figure that all the determined solutions are infeasible in the robust model. The results indicate that the solutions of the determined model can only guarantee the feasibility in a specific case, but not in the models with uncertain distribution parameters. Generally, the constraints violation degree in the second order moment model is significantly smaller than that in the first order moment model, which indicates that the first order moment is stricter than the second order moment model. With the increase of safety tolerance, the constraint violation degree decreases gradually, which indicates that the smaller the safety tolerance, the more strict of the robustness of the model is.
+In the second aspect, take the feasible solutions obtained from the determined model into robust models to test the feasibility of the determined solutions in the robust model, and the results are shown in [[#img-11|Figure 11]]. It can be seen from the figure that all the determined solutions are infeasible in the robust model. The results indicate that the solutions of the determined model can only guarantee the feasibility in a specific case, but not in the models with uncertain distribution parameters. Generally, the constraints violation degree in the second order moment model is significantly smaller than that in the first order moment model, which indicates that the first order moment is stricter than the second order moment model. With the increase of safety tolerance, the constraint violation degree decreases gradually, which indicates that the smaller the safety tolerance, the more strict of the robustness of the model is.
 <div id='img-11'></div>
@@ Line 1,001: / Line 1,001: @@
 Because the first order moment robust optimization model is not feasible, only the second order moment optimization model is compared and analyzed.
-samples of uncertain parameters are selected, take the robust solution of the second-order moment model into these samples, and calculate the times <math>N_T{}'</math> and <math>N_d{}'</math> that inequalities (22) and (23) are satisfied
+samples of uncertain parameters are selected, take the robust solution of the second-order moment model into these samples, and calculate the times <math>N_T'</math> and <math>N_d'</math> that inequalities (22) and (23) are satisfied
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 863: / Line 863: @@
 To verify the robustness of the uncertain model, we compare the results of the robust model with the determined model and the opportunity constraint model, respectively. Robust model refers to the first or second order moment model. The determined model refers to the model with the expected value of the uncertain parameter, which is adopted by most decision makers. The chance constraint model refers to the model with the assumption that the distribution of uncertain parameters is known. Because the distribution is unknown, the Monte Carlo method is used to solve the chance constraint model.
-===4.1 Comparison with Determined Model===
+===4.1 Comparison with determined model===
 In the determined model developed in section 2.3 of this paper, the expectation of lead times时and spare parts demands are adopted as the determined parameters. The comparison with the determined model is taken from two aspects. In the first aspect, the solutions of the first order moment model and the second order moment model (referred to as the robust solution) are brought into the deterministic model to test the feasibility of the robust solution in the determined model.

@@ Line 599: / Line 599: @@
 |  style="text-align: center;"|0.5
 |  style="text-align: center;"|0.7
 |  style="text-align: center;"|0.1
 |  style="text-align: center;"|0.3

← Older revision		Revision as of 13:45, 17 November 2020
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	<span id='_Ref38381835'>[[#cite-_Ref38381835\|[11]]] Shang C., You F. Distributionally robust optimization for planning and scheduling under uncertainty. Computers & Chemical Engineering, 110(FEB.2):53-68, 2018.</span>		<span id='_Ref38381835'>[[#cite-_Ref38381835\|[11]]] Shang C., You F. Distributionally robust optimization for planning and scheduling under uncertainty. Computers & Chemical Engineering, 110(FEB.2):53-68, 2018.</span>
−
	<!--<span id='_Ref38381847'>[[#cite-_Ref38381847\|[12]]] Gicquel C., Cheng J. A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2):123–155, 2017.</span>-->		<!--<span id='_Ref38381847'>[[#cite-_Ref38381847\|[12]]] Gicquel C., Cheng J. A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2):123–155, 2017.</span>-->

@@ Line 1,076: / Line 1,076: @@
 ==References==
-<span id='_Ref38381732'>[[#cite-_Ref38381732|[1]]] Eaves, A H C，Kingsman, B G. (2004). Forecasting for the ordering and stock-holding of spare parts. Journal of the Operational Research Society 55(4):431-437</span>
+<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-<span id='_Ref38381747'>[[#cite-_Ref38381747|[2]]] Mao, H. L., Gao, J. W., Chen, X. J., & Gao, J. D. (2014). Demand Prediction of the Rarely Used Spare Parts Based on the BP Neural Network. Applied Mechanics and Materials, 1513–1519.
+<span id='_Ref38381732'></span>
-<span id='_Ref38381756'>[[#cite-_Ref38381756|[3]]] Sadeghi, J., Sadeghi, S., & Niaki, S. T. A. (2014). A hybrid vendor managed inventory and redundancy allocation optimization problem in supply chain management: An NSGA-II with tuned parameters. Computers & Operations Research, 41, 53–64.
+<span id='_Ref38381747'></span>
-<span id='_Ref38381761'>[[#cite-_Ref38381761|[4]]] Qiu Zhiping, Ni Zao.(2004). Probabilistic interval approach for determining the demand of aviation spares. Acta Aeronautica Et Astronautica Sinica, 30(5), 861-866.
+<span id='_Ref38381756'></span>
-<span id='_Ref38381766'>[[#cite-_Ref38381766|[5]]] Axsäter, S. (1993). Optimization of order-up-to-s policies in two-echelon inventory systems with periodic review. Naval Research Logistics, 40(2), 245–253. </span>
+<span id='_Ref38381761'></span>
-<span id='_Ref38381799'>[[#cite-_Ref38381799|[6]]] Kouki, C. , Babai, M. Z. , Jemai, Z. , & Minner, S. . (2018). Solution procedures for lost sales base-stock inventory systems with compound poisson demand. International Journal of Production Economics, S0925527318300513.
+<span id='_Ref38381766'></span>
-<span id='_Ref38381806'>[[#cite-_Ref38381806|[7]]] Al-Rifai, M. H., & Rossetti, M. D. (2007). An efficient heuristic optimization algorithm for a two-echelon (R, Q) inventory system. International Journal of Production Economics, 109(1-2), 195–213.
+<span id='_Ref38381799'></span>
-<span id='_Ref38381812'>[[#cite-_Ref38381812|[8]]] Hnaien, F., Delorme, X., & Dolgui, A. (2010). Multi-objective optimization for inventory control in two-level assembly systems under uncertainty of lead times. Computers & Operations Research, 37(11), 1835–1843.
+<span id='_Ref38381806'></span>
-<span id='_Ref38381823'><span id='_Ref527702845'>[[#cite-_Ref527702845|[9]]] Gicquel, C. , & Cheng, J. . (2018). A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2), 123-155.
+<span id='_Ref38381812'></span>
-<span id='_Ref38381827'>[[#cite-_Ref38381827|[10]]] M. Talaei, M. B. Farhang, M. S.Pishvaee, et.al. “A robust fuzzy optimization model for carbon-efficient closed-loop supply chain network design problem: a numerical illustration in electronics industry,” ''Journal of Cleaner Production'', vol. 113, pp. 662–673, 2016.
+<span id='_Ref38381823'></span>
-<span id='_Ref38381835'>[[#cite-_Ref38381835|[11]]] Shang, C., You, F. . (2018). Distributionally robust optimization for planning and scheduling under uncertainty. Computers & Chemical Engineering, 110(FEB.2), 53-68.
+<span id='_Ref38381827'></span>
-<span id='_Ref38381847'>[[#cite-_Ref38381847|[12]]] Gicquel, C., & Cheng, J. (2017). A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2), 123–155</span>
+<span id='_Ref38381835'>[[#cite-_Ref38381835|[11]]] Shang C., You F.  Distributionally robust optimization for planning and scheduling under uncertainty. Computers & Chemical Engineering, 110(FEB.2):53-68, 2018.</span>
-<span id='_Ref38381866'>[[#cite-_Ref38381866|[13]]] Shi, Y. , Boudouh, T. , & Grunder, O. . (2017). A Fuzzy Chance-constraint Programming Model for a Home Health Care Routing Problem with Fuzzy Demand. International Conference on Operations Research & Enterprise Systems.
+<!--<span id='_Ref38381847'>[[#cite-_Ref38381847|[12]]] Gicquel C.,  Cheng J.  A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2):123–155, 2017.</span>-->
-<span id='_Ref38381890'>[[#cite-_Ref38381890|[14]]] Rubinstein, R. Y, & Kroese, D. P.(1983). Simulation and the Monte Carlo Method.by Reuven Y. Rubinstein. Journal of the American Statistical Association, 78(382):511-512.
+<span id='_Ref38381866'>[[#cite-_Ref38381866|[12]]] Shi Y., Boudouh T., Grunder O.  A fuzzy chance-constraint programming model for a home health care routing problem with fuzzy demand. International Conference on Operations Research & Enterprise Systems, 2017.</span>
 ----
 *Corresponding author. Tel.: +86 18795428481; E-mail: xwzj0003[mailto:@ @]gmail.com

@@ Line 1,070: / Line 1,070: @@
 |}
-==5 Conclusion==
+==5. Conclusion==
 In this paper, the spare parts network optimization with uncertainty lead times and demands is studied. The determined supply model and its robust counterpart are developed, respectively. In the uncertainty model, the probability of the uncertainties are unknown, we focus on the reformulation of this model with the known first and second order moment. The robust solutions of uncertain models are obtained and the sensitivity analysis with different safety tolerance is curry out. Then the robustness is compared between the robust model and determined model and chance constraint model. The main contributions of this paper are as follows. Firstly, the ambiguity set is used to handle the uncertain probability distributions, and the robust counterpart of the supply model is reformulated to make the model tractable and computable. Secondly, the robust solutions that ensure the feasibility in the worst-case are obtained. These solutions ensure that the uncertain constraints are satisfied under any probability distribution within the moment based ambiguity set. The results of the case study show that the robustness of these solutions are stronger than those in the determined model and chance constraint model. Finally, we verified that the second order moment model is superior to the first moment order model because the second order moment model can make a balance in robustness and flexibility.

@@ Line 1,024: / Line 1,024: @@
 where,  <math>\Delta (.)=1</math>, if  <math>.</math> is satisfied. else if, <math>\Delta (.)=0</math>. Then,  <math>N_T{}'/N</math> and <math>N_d'/N</math> are used to represent the value of  <math>Pr(\sum_{i\in I}\sum_{j\in J}T_{ij}\cdot sgn(x_{ij})+</math><math>\sum_{j\in J}\sum_{k\in K}T_{jk}\cdot sgn(x_{jk})\leq T)</math> and  <math>Pr(d_k-\sum_{j\in J}x_{jk}\leq 0)</math>. If  <math>N_T'/N</math> or  <math>N_d'/N</math> were greater than or equal to  <math>1-\epsilon </math>, the constraints (22) and (23) are satisfied.
-The constraint satisfactions of the chance constraint model are shown in Table 6. It can be seen that the constraints (22) and (23) are satisfied in all scenarios. As the safety tolerance decrease, the value of  <math>N_T'/N</math> and  <math>N_d'/N</math> are larger. This is because when the safety tolerance is small, the robust model is more strict, and the robustness of the robust solution is stronger, which lead to the greater the probability that the constraint is satisfied in chance constraint model.
+The constraint satisfactions of the chance constraint model are shown in [[#tab-6|Table 6]]. It can be seen that the constraints (22) and (23) are satisfied in all scenarios. As the safety tolerance decrease, the value of  <math>N_T'/N</math> and  <math>N_d'/N</math> are larger. This is because when the safety tolerance is small, the robust model is more strict, and the robustness of the robust solution is stronger, which lead to the greater the probability that the constraint is satisfied in chance constraint model.
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="font-size: 75%;">
-'''Table 6. '''The constraint satisfied probability with second order moment model robust solution</span></div>
+'''Table 6'''. The constraint satisfied probability with second order moment model robust solution</div>
-{| style="width: 53%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
+<div id='tab-6'></div>
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"| <math>\epsilon </math> </span>
+|  style="text-align: center;"| <math>N_T'/N</math>
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.1</span>
+|  style="text-align: center;"|0.9692
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.3</span>
+|  style="text-align: center;"|0.8997
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.5</span>
+|  style="text-align: center;"|0.7938
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.7</span>
+|  style="text-align: center;"|0.7288
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|0.9</span>
+|  style="text-align: center;"|0.6992
 |-
-|  style="border-top: 1pt solid black;text-align: center;"| <math>N_T{}'/N</math> </span>
+|  style="text-align: center;"| <math>N_d'/N</math>, <math>k=1</math>
-|  style="border-top: 1pt solid black;text-align: center;"|0.9692</span>
+|  style="text-align: center;"|0.9269
-|  style="border-top: 1pt solid black;text-align: center;"|0.8997</span>
+|  style="text-align: center;"|0.8194
-|  style="border-top: 1pt solid black;text-align: center;"|0.7938</span>
+|  style="text-align: center;"|0.8187
-|  style="border-top: 1pt solid black;text-align: center;"|0.7288</span>
+|  style="text-align: center;"|0.7938
-|  style="border-top: 1pt solid black;text-align: center;"|0.6992</span>
+|  style="text-align: center;"|0.7994
 |-
-|  style="text-align: center;"| <math>N_d{}'/N</math> , <math>k=1</math> </span>
+|  style="text-align: center;"| <math>N_d'/N</math>, <math>k=2</math>
-|  style="text-align: center;"|0.9269</span>
+|  style="text-align: center;"|0.9524
-|  style="text-align: center;"|0.8194</span>
+|  style="text-align: center;"|0.8219
-|  style="text-align: center;"|0.8187</span>
+|  style="text-align: center;"|0.8170
-|  style="text-align: center;"|0.7938</span>
+|  style="text-align: center;"|0.7922
-|  style="text-align: center;"|0.7994</span>
+|  style="text-align: center;"|0.7223
 |-
-|  style="text-align: center;"| <math>N_d{}'/N</math> , <math>k=2</math> </span>
+|  style="text-align: center;"| <math>N_d'/N</math>, <math>k=3</math>
-|  style="text-align: center;"|0.9524</span>
+|  style="text-align: center;"|0.9461
-|  style="text-align: center;"|0.8219</span>
+|  style="text-align: center;"|0.9403
-|  style="text-align: center;"|0.8170</span>
+|  style="text-align: center;"|0.9396
-|  style="text-align: center;"|0.7922</span>
+|  style="text-align: center;"|0.8959
-|  style="text-align: center;"|0.7223</span>
+|  style="text-align: center;"|0.8843
 |-
-|  style="text-align: center;"| <math>N_d{}'/N</math> , <math>k=3</math> </span>
+|  style="text-align: center;"| <math>N_d'/N</math>, <math>k=4</math>
-|  style="text-align: center;"|0.9461</span>
+|  style="text-align: center;"|0.9289
-|  style="text-align: center;"|0.9403</span>
+|  style="text-align: center;"|0.8351
-|  style="text-align: center;"|0.9396</span>
+|  style="text-align: center;"|0.7378
-|  style="text-align: center;"|0.8959</span>
+|  style="text-align: center;"|0.7298
-|  style="text-align: center;"|0.8843</span>
+|  style="text-align: center;"|0.6537
-|-
-|  style="border-bottom: 1pt solid black;text-align: center;"| <math>N_d{}'/N</math> , <math>k=4</math> </span>
-|  style="border-bottom: 1pt solid black;text-align: center;"|0.9289</span>
-|  style="border-bottom: 1pt solid black;text-align: center;"|0.8351</span>
-|  style="border-bottom: 1pt solid black;text-align: center;"|0.7378</span>
-|  style="border-bottom: 1pt solid black;text-align: center;"|0.7298</span>
-|  style="border-bottom: 1pt solid black;text-align: center;"|0.6537</span>
 |}

@@ Line 996: / Line 996: @@
 |}
-Since unknown the probability distribution of lead times and demands, it cannot be solved by probability density function. Even if the distribution of uncertain parameters is known, the solution of the joint probability density function in formula (23) is very time-consuming. So the Monte Carlo method is used to verify the feasibility of the solution
+Since unknown the probability distribution of lead times and demands, it cannot be solved by probability density function. Even if the distribution of uncertain parameters is known, the solution of the joint probability density function in Equation (23) is very time-consuming. So the Monte Carlo method is used to verify the feasibility of the solution
 Because the first order moment robust optimization model is not feasible, only the second order moment optimization model is compared and analyzed.
-samples of uncertain parameters are selected, take the robust solution of the second-order moment model into these samples, and calculate the times <math>N_T{}'</math> and <math>N_d{}'</math> that inequality (22) and (23) are satisfied
+samples of uncertain parameters are selected, take the robust solution of the second-order moment model into these samples, and calculate the times <math>N_T{}'</math> and <math>N_d{}'</math> that inequalities (22) and (23) are satisfied
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 1,022: / Line 1,022: @@
 |}
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+The constraint satisfactions of the chance constraint model are shown in Table 6. It can be seen that the constraints (22) and (23) are satisfied in all scenarios. As the safety tolerance decrease, the value of  <math>N_T'/N</math> and  <math>N_d'/N</math> are larger. This is because when the safety tolerance is small, the robust model is more strict, and the robustness of the robust solution is stronger, which lead to the greater the probability that the constraint is satisfied in chance constraint model.
-<math>N_d{}'=\sum_{n=1}^N\Delta (d_k-\sum_{j\in J}x_{jk}^{}\leq 0)</math> (25)</div>
-−
-where,  <math>\Delta (.)=1</math> , if  <math>.</math> is satisfied. else if, <math>\Delta (.)=0</math> . Then,  <math>N_T{}'/N</math> and <math>N_d{}'/N</math> are used to represent the value of  <math>Pr(\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+</math><math>\sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\leq T\mbox{ }\mbox{ })</math> and  <math>\mbox{ }Pr(d_k-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ })\mbox{ }\mbox{ }</math> . If  <math>N_T{}'/N</math> or  <math>N_d{}'/N</math> were greater than or equal to  <math>1-\epsilon </math> , the constraint (22) and (23) are satisfied.
-−
-The constraint satisfactions of the chance constraint model are shown in table 6. It can be seen that the constraints (22) and (23) are satisfied in all scenarios. As the safety tolerance decrease, the value of  <math>N_T{}'/N</math> and  <math>N_d{}'/N</math> are larger. This is because when the safety tolerance is small, the robust model is more strict, and the robustness of the robust solution is stronger, which lead to the greater the probability that the constraint is satisfied in chance constraint model.
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

@@ Line 873: / Line 873: @@
 <div id='img-1'></div>
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
-|style="padding:10px;"| [[Image:Review_789578267838-image95.png|318px]]
+|style="padding:10px;"| [[Image:Review_789578267838-image95.png|418px]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" style="padding:10px;"| '''Figure 1'''. Constraint violation comparison between first order moment model and determined model as safety tolerance is 0.1