You do not have permission to edit this page, for the following reason:

You are not allowed to execute the action you have requested.


You can view and copy the source of this page.

x
 
1
<!-- metadata commented in wiki content
2
3
4
<span id='OLE_LINK228'></span><span id='OLE_LINK8'></span><span id='OLE_LINK34'></span><big>Spare Parts Supply Network Optimization with Uncertain Distributed Lead Times and Demands</big>
5
6
<span style="text-align: center; font-size: 75%;">Yadong Wang<sup>1</sup>, Quan Shi<sup>1</sup><span id="fnc-1">[[#fn-1|<sup>*</sup>]]</span>, Wei Xia<sup>1</sup>, Feng Li<sup>12</sup></span>
7
8
<span id='OLE_LINK231'></span><span id='OLE_LINK232'></span><span style="text-align: center; font-size: 75%;">1.Department of Equipment Command and Management, Army Engineering University of PLA, Shijiazhuang, China</span>
9
10
2.UNIT 32031 of PLA, Kaifeng, China
11
12
<span id='_GoBack'></span>
13
14
<span id='OLE_LINK190'></span><span id='OLE_LINK191'></span>
15
--><span id='OLE_LINK179'>'''Abstract: '''In the spare parts supply network, there are many uncertain factories, such as unpredictable demands and changeable lead times. The spare parts shortage caused by those uncertainties may lead to severe losses. To solve the uncertainty of supply network, a determined optimization model is developed and then reformulated as a robust counterpart. In the robust model, it is only necessary to know the moment information of the uncertain parameters rather than the true probability distribution. The solution obtained by the robust model can satisfy the constraints in the worst-case, that is, feasible for any probability distribution within the moment based ambiguity set. Two moment based robust models are studied in this work. The result of the experiment indicates that the robustness of the robust model is stronger than that of the determined model and chance constraint model, and the effect of safety tolerance on the robustness is revealed by sensitivity analysis. Finally, the second order moment model is verified be superior to the first order moment model in spare parts supply network optimization.</span>
16
17
'''Keywords: '''spare parts supply network, uncertainty, moment based ambiguity set, chance constraint optimization, distributed robust optimization
18
19
'''1 Introduction'''
20
21
<span id='OLE_LINK180'></span><span id='OLE_LINK181'>Spare parts supply network optimization is one of the hot issues of spare parts management. Many research works have been done on supply network optimization in certain circumstances. In recent years, many researchers took attention to uncertainty optimization. In the spare parts supply network, many factories such as demands, lead times and risks are uncertain since the difficulty of data collection or complex dynamic scenarios. Although there are many advanced methods, including reliability based</span><span id='cite-_Ref38381732'></span>[[#_Ref38381732|<span style="text-align: center; font-size: 75%;">[1]</span>]]<span style="text-align: center; font-size: 75%;"> and data based methods</span><span id='cite-_Ref38381747'></span>[[#_Ref38381747|<span style="text-align: center; font-size: 75%;">[2]</span>]]<span style="text-align: center; font-size: 75%;">, have been used to predict the demands or lead times, however, most of them are based on over-ideal assumptions and not accurate enough. In fact, in many cases, it almost cannot forecast the demand precisely at all, such as in reverse logistics or in emergency supply. </span>
22
23
<span id='OLE_LINK140'></span><span id='OLE_LINK141'></span><span id='OLE_LINK144'></span><span id='OLE_LINK145'></span><span style="text-align: center; font-size: 75%;">To simplify the problems, many researchers assume that the demands are determined </span><span id='cite-_Ref38381756'></span>[[#_Ref38381756|<span style="text-align: center; font-size: 75%;">[3]</span>]]<span style="text-align: center; font-size: 75%;"> or obeys specified known distribution, such as Normal distribution </span><span id='cite-_Ref38381761'></span>[[#_Ref38381761|<span style="text-align: center; font-size: 75%;">[4]</span>]]<span style="text-align: center; font-size: 75%;">, Poisson distribution </span><span id='cite-_Ref38381766'></span>[[#_Ref38381766|<span style="text-align: center; font-size: 75%;">[5]</span>]]<span style="text-align: center; font-size: 75%;">, Stuttering Poisson distribution </span><span id='cite-_Ref38381799'></span>[[#_Ref38381799|<span style="text-align: center; font-size: 75%;">[6]</span>]]<span style="text-align: center; font-size: 75%;">, etc.. There are also many works assume the lead times are fixed </span><span id='cite-_Ref38381806'></span>[[#_Ref38381806|<span style="text-align: center; font-size: 75%;">[7]</span>]]<span style="text-align: center; font-size: 75%;"> or obey distributions known in advance </span><span id='cite-_Ref38381812'></span>[[#_Ref38381812|<span style="text-align: center; font-size: 75%;">[8]</span>]]<span style="text-align: center; font-size: 75%;">. These assumptions may be effective sometimes, but once not in line with the reality, it may cause out of stock or more serious consequences. To handle the uncertainty in spare parts supply network, the most conmen used methods are stochastic program </span><span id='cite-_Ref38381823'></span>[[#_Ref38381823|<span style="text-align: center; font-size: 75%;">[9]</span>]]<span style="text-align: center; font-size: 75%;">, fuzzy optimization </span><span id='cite-_Ref38381827'></span>[[#_Ref38381827|<span style="text-align: center; font-size: 75%;">[10]</span>]]<span style="text-align: center; font-size: 75%;">, and robust optimization </span><span id='cite-_Ref38381835'></span>[[#_Ref38381835|<span style="text-align: center; font-size: 75%;">[11]</span>]]<span style="text-align: center; font-size: 75%;">. However, in stochastic optimization, it is usually necessary to know the distribution of uncertain parameters in advance or very time-consuming by using Monte Carlo simulation. The fuzzy optimization cannot guarantee the feasibility in all time. Thus, the robust optimization was adopted in this paper because it doesn’t rely on the distribution of uncertain parameters, and can guarantee the feasibility of the solution in the worst case.</span>
24
25
<span id='OLE_LINK188'></span><span id='OLE_LINK164'></span><span id='OLE_LINK178'>This paper optimizes the supply network without thus assumption to avoid the serious losses caused by small probability events. The demands and lead times are uncertain with unknown distribution, the only things can we get from the historical data is certain known moments or known structural properties. A moment based ambiguity set was established to descript all the possible distribution, which must contain the real distribution of uncertain parameters. The robust model only needs to ensure the worst case in the ambiguity set is satisfied.</span>
26
27
<span id='OLE_LINK198'></span><span style="text-align: center; font-size: 75%;">The remaining sections of this article are organized as follows. In section 2, the determined model of spare parts supply network optimization is developed and then reformulated as its moment based robust counterpart. In section 3, the numerical experiment and sensitivity analysis are carried out. In section 4, the results of the robust model are compared with the determined model and chance constraint model, respectively. Finally, we summarize the paper in section 6.</span>
28
29
'''2 Modeling'''
30
31
The classical three echelon spare parts supply network consists of supply centers, distribution centers and customers. The ordered spare parts are supported by supply centers and then distributed through the distribution centers to meet the spare parts demands of the customers. In the process of supplying, in order to save the costs, it is often unnecessary to open all of the alternative distribution centers. The aim of this research is to decide which distribution center should be opening and also the number of spare parts flow among the nodes of the spare parts supply network.
32
33
<span id='OLE_LINK211'></span><span id='OLE_LINK212'>Different from the traditional supply network, there are many uncertainty parameters in our spare parts supply network. Lead times and demands are the two most conmen considered uncertainty factories in the supply network. In terms of spare parts demands, the predicted values usually do not agree with the real data especially in some complex circumstances. Although, there are lots of state-of-the-art forecast methods are studied and used by researchers, there are still big gaps between the mathematic models and the real-world practice. Similarly, it’s also difficult to ensure the lead times of supply remain unchanged every time, even there may be some accidents occur in transit. One of the general strategies to handle these uncertainties is assuming these uncertain parameters obey specified distribution. This assumption will not accurate because of the limitation of the historical data. So, in our supply network optimization model, the probability distribution of lead times and demands are even typically unknown. There are only some useful information can be obtained from the samples, such as certain known moments or known structural properties.</span>
34
35
<span id='OLE_LINK199'></span><span id='OLE_LINK210'></span><span id='OLE_LINK45'></span><span id='OLE_LINK46'>The proposed model focus on the problems that optimize the spare parts network with uncertain lead times and demands to meet the customers’ demands within the given hard time window and with the minimum total cost. The uncertainty makes the problem more complicated to be handled. Firstly, the determined model is developed, and then the corresponding robust counterpart problem will be reformulated. The objective function of the model is minimize the total supply costs, and the constraints of the model include demand fill rate, lead time, captain limitation, flow equilibrium, and so on. The parameters and models are described as follows.</span>
36
37
''2.1 Parameter Description''
38
39
<span id='OLE_LINK47'></span><span id='OLE_LINK48'></span><span id='OLE_LINK59'></span><span id='OLE_LINK60'></span><span style="text-align: center; font-size: 75%;"> <math>I</math> : the amount of supply centers,  <math>i=1,2,\cdots ,I</math> ;</span>
40
41
<span id='OLE_LINK61'></span><span id='OLE_LINK62'></span><span id='OLE_LINK5'></span><span id='OLE_LINK6'></span><span style="text-align: center; font-size: 75%;"> <math>J</math> : the amount of distribution centers,  <math>j=1,2,\cdots ,J</math> ;</span>
42
43
<span id='OLE_LINK80'></span><span id='OLE_LINK81'></span><span style="text-align: center; font-size: 75%;"> <math>K</math> : the amount of customers,  <math>k=1,2,\cdots ,K</math> ;</span>
44
45
<span id='OLE_LINK63'></span><span id='OLE_LINK64'></span><span id='OLE_LINK65'></span><span id='OLE_LINK78'></span><span id='OLE_LINK3'></span><span id='OLE_LINK4'></span><span id='OLE_LINK86'></span><span id='OLE_LINK87'></span><span id='OLE_LINK88'></span><span style="text-align: center; font-size: 75%;"> <math>T_{ij}^{}</math> : the lead time between supply center  <math>i</math> and distribution center <math>j</math> , which is uncertain;</span>
46
47
<span id='OLE_LINK26'></span><span style="text-align: center; font-size: 75%;"> <math>T_{jk}^{}</math> : the lead time between distribution center <math>j</math> and customer <math>k</math> , which is uncertain;</span>
48
49
<span id='OLE_LINK182'></span><span id='OLE_LINK183'></span><span style="text-align: center; font-size: 75%;"> <math>T</math> : the required maximum lead time.</span>
50
51
<span id='OLE_LINK22'></span><span id='OLE_LINK23'></span><span id='OLE_LINK91'></span><span id='OLE_LINK92'></span><span id='OLE_LINK93'></span><span id='OLE_LINK94'></span><span style="text-align: center; font-size: 75%;"> <math>C_{ij}^{trans}</math> : unit spare parts transportation costs between supply center  <math>i</math> and distribution center <math>j</math> ;</span>
52
53
<span id='OLE_LINK37'></span><span id='OLE_LINK35'></span><span id='OLE_LINK36'></span><span id='OLE_LINK97'></span><span id='OLE_LINK98'></span><span id='OLE_LINK79'></span><span id='OLE_LINK82'></span><span style="text-align: center; font-size: 75%;"> <math>C_{jk}^{trans}</math> : unit spare parts transportation costs between distribution center <math>j</math> and customer <math>k</math> ;</span>
54
55
<span id='OLE_LINK38'></span><span id='OLE_LINK39'></span><span id='OLE_LINK40'></span><span style="text-align: center; font-size: 75%;"> <math>C_j^{open}</math> : fixed opening costs of distribution center <math>j</math> ;</span>
56
57
<span id='OLE_LINK11'></span><span id='OLE_LINK12'></span><span id='OLE_LINK41'></span><span id='OLE_LINK42'></span><span id='OLE_LINK89'></span><span id='OLE_LINK90'></span><span style="text-align: center; font-size: 75%;"> <math>C_j^{invent}</math> : unit spare parts inventory cost of distribution center <math>j</math> ;</span>
58
59
<span id='OLE_LINK83'></span><span id='OLE_LINK84'></span><span id='OLE_LINK85'></span><span style="text-align: center; font-size: 75%;"> <math>C_k^{invent}</math> : unit spare parts inventory cost of customer <math>k</math> ;</span>
60
61
<math>C_k^{short}</math> : unit shortage loss of customer <math>k</math> ;
62
63
<math>d_k</math> : spare parts demand of customer, which is uncertain  <math>k</math> ;
64
65
<math>U_j</math> : the maximum spare parts capacity of distribution center <math>j</math> .
66
67
Decision variables:
68
69
<span id='OLE_LINK95'></span><span id='OLE_LINK96'></span><span style="text-align: center; font-size: 75%;"> <math>x_{ij}^{}</math> : the amount of spare parts supply from supply center  <math>i</math> to distribution center <math>j</math> ;</span>
70
71
<span id='OLE_LINK101'></span><span id='OLE_LINK102'></span><span style="text-align: center; font-size: 75%;"> <math>x_{jk}^{}</math> : the amount of spare parts supply from distribution center <math>j</math> to customer <math>k</math> ;</span>
72
73
<span id='OLE_LINK103'></span><span id='OLE_LINK104'></span><span id='OLE_LINK105'></span><span id='OLE_LINK99'></span><span id='OLE_LINK100'></span><span style="text-align: center; font-size: 75%;"> <math>y_j^{}</math> : binary variable, used to label the opening of distribution center <math>j</math> . If distribution center <math>j</math> is opening,  <math>y_j^{}=1</math> , otherwise,  <math>y_j^{}=0</math> .</span>
74
75
''2.2 Determined Model''
76
77
<span id='OLE_LINK218'></span><span id='OLE_LINK106'></span><span id='OLE_LINK107'></span><span id='OLE_LINK108'></span><span id='OLE_LINK109'></span><span style="text-align: center; font-size: 75%;">Firstly, the determined model is developed. In this model, all parameters seem as determined parameters. The objective function is minimizing the total supply costs, which consists of opening cost, transportation costs, inventory costs, and shortage loss. The objective function is formulated as follows: </span>
78
79
<div style="text-align: right; direction: ltr; margin-left: 1em;">
80
<span style="text-align: center; font-size: 75%;"> <math>min\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }C=</math><math>C_{}^{open}+C_{}^{trans}+C_{}^{invent}+C_{}^{short}</math> (1)</span></div>
81
82
<span id='OLE_LINK110'></span><span style="text-align: center; font-size: 75%;">where,  <math>C_{}^{open}</math> represents the total opening costs,  <math>C_{}^{trans}</math> represents the total transportation costs,  <math>C_{}^{invent}</math> represents the total inventory costs, and  <math>C_{}^{short}</math> represents the total shortage loss. Each costs are calculated as follows: </span>
83
84
<div style="text-align: right; direction: ltr; margin-left: 1em;">
85
<span style="text-align: center; font-size: 75%;"> <math>C_{}^{open}=\sum_{j\in J}C_j^{open}\cdot y_j^{}</math> (2)</span></div>
86
87
<div style="text-align: right; direction: ltr; margin-left: 1em;">
88
<span style="text-align: center; font-size: 75%;"> <math>C_{{}_{}}^{trans}=\sum_{i\in I}\sum_{j\in J}C_{ij}^{trans}\cdot x_{ij}^{}+</math><math>\sum_{j\in J}\sum_{k\in K}C_{jk}^{trans}\cdot x_{jk}^{}</math> (3)</span></div>
89
90
<div style="text-align: right; direction: ltr; margin-left: 1em;">
91
<span style="text-align: center; font-size: 75%;"> <math>C_{}^{invent}=\sum_{j\in J}C_j^{invent}\cdot (\sum_{i\in I}x_{ij}^{}-</math><math>\sum_{k\in K}x_{jk}^{})\mbox{ }+\sum_{k\in K}C_k^{invent}\Xi \cdot (\sum_{j\in J}x_{jk}^{}-</math><math>d_k^{})\mbox{ }</math> (4)</span></div>
92
93
<div style="text-align: right; direction: ltr; margin-left: 1em;">
94
<span style="text-align: center; font-size: 75%;"> <math>C_{}^{short}=\sum_{k\in K}C_k^{short}\cdot \Xi \cdot (d_k^{}-</math><math>\sum_{j\in J}x_{jk}^{})</math> (5)</span></div>
95
96
<span id='OLE_LINK15'></span><span id='OLE_LINK16'></span><span style="text-align: center; font-size: 75%;">where,  <math>\Xi \lbrace .\rbrace \mbox{ }</math> represents  <math>max(0,.)</math> .</span>
97
98
The following constraints should be satisfied.
99
100
''s.t.''
101
102
<div style="text-align: right; direction: ltr; margin-left: 1em;">
103
<span style="text-align: center; font-size: 75%;"> <math>\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}^{})-</math><math>T\leq \mbox{ }0</math> (6)</span></div>
104
105
<div style="text-align: right; direction: ltr; margin-left: 1em;">
106
<span style="text-align: center; font-size: 75%;"> <math>\sum_{i\in I}x_{ij}^{}-\mbox{ }\mbox{ }y_{}_j^{}\cdot U_j\leq \mbox{ }0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }j=</math><math>1,2,\cdots ,J</math> (7)</span></div>
107
108
<div style="text-align: right; direction: ltr; margin-left: 1em;">
109
<span style="text-align: center; font-size: 75%;"> <math>\sum_{k\in K}x_{jk}^{}-y_{}_j^{}\cdot U_j\leq \mbox{ }\mbox{ }0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }j=</math><math>1,2,\cdots ,J</math> (8)</span></div>
110
111
<div style="text-align: right; direction: ltr; margin-left: 1em;">
112
<span style="text-align: center; font-size: 75%;"> <math>\mbox{ }d_{}_k^{}-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k=</math><math>1,2,\cdots ,K</math> (9)</span></div>
113
114
<div style="text-align: right; direction: ltr; margin-left: 1em;">
115
<span style="text-align: center; font-size: 75%;"> <math>\sum_{k\in K}x_{jk}^{}-\sum_{i\in I}x_{ij}^{}\leq 0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }j=</math><math>1,2,\cdots ,J</math> (10)</span></div>
116
117
<div id="OLE_LINK19" style="text-align: right; direction: ltr; margin-left: 1em;">
118
<span style="text-align: center; font-size: 75%;"> 
119
{|
120
|-
121
| <math>x_{ij}^{}\in N^+</math>
122
| <math>x_{jk}^{}\in N^+</math>
123
|}
124
 <math>y_{}_j^{}=\left\{0,1\right\}</math> (11)</span></div>
125
126
where, constraint (6) is a hard time window constraint, which specifies that the spare parts must arrive at the customers before the deadline. The constraints (7) and (8) state that the closed distribution centers should not participate in the supply of spare parts. They also specify that the input of spare parts should not exceed the maximum capacity of distribution centers. Constraint (9) specifies that the demands of spare parts of all the customers must be satisfied. Constraint (10) ensures that the output of spare parts should not exceed the spare parts flow into each distribution center. Constraint (11) states the character of decision variables.
127
128
<span id='OLE_LINK113'></span><span id='OLE_LINK114'></span><span style="text-align: center; font-size: 75%;">''2.3 Robust Counterpart Problem Reformulation''</span>
129
130
<span id='OLE_LINK219'></span><span id='OLE_LINK115'></span><span id='OLE_LINK116'>In the determined supply optimization model, there will be two situations of spare parts shortage. One of them is the spare parts cannot arrive in time, and another is the amount of supplied spare parts cannot meet the demand of customers. Both the two kinds of shortage can lead to serious consequences. In the determined model, it is only necessary to find the feasible solutions that satisfy all the constraints of the model. However, in the uncertain model, it is hardly to find feasible solutions because of the uncertainty of lead times and demands. We need to reformulate the mathematical model in the uncertain environment, and find robust solutions to ensure that the spare parts demands are satisfied in any case.This section reformulates the determined model as an uncertain model, and derives a tractable and efficient deterministic robust counterpart for the uncertain model.</span>
131
132
<span id='OLE_LINK172'></span><span id='OLE_LINK173'></span><span id='OLE_LINK49'></span><span id='OLE_LINK50'></span><span id='OLE_LINK174'></span><span id='OLE_LINK175'>For most supply practices, the supply environment is complex and changeable, and the sample data is limited. It is usually difficult to get the true probability distribution of these parameters. What only can we do is that extract useful information from these limited samples and use the statistical features to modeling and analysis. Despite the unknown distribution of parameters, the moment, such as, mean, variance, covariances, etc., can be obtained easily. The known moment of the sample contains a lot of valuable information about the distribution, so we use a moment-based ambiguity sets describe the uncertain parameters. This ambiguity set consists of all possible probability distributions satisfied the moment condition, and deservedly contains the true distribution of our parameters. If we can find the robust solutions that satisfy the constraints in the worst-case, then the constraints with real distribution can also be met by the robust solutions.</span>
133
134
This section focuses on the robust counterpart reformulation in the case of known first order moments and the case of known both the first order and the second order moments. The ambiguity set of these two cases are expressed as follows, respectively:
135
136
<div style="text-align: right; direction: ltr; margin-left: 1em;">
137
<span style="text-align: center; font-size: 75%;"> <math>D=\left\{f(\xi ):\begin{array}{c}
138
{\int }_{\xi \in R^K}f(\xi )d\xi =1\\
139
E[\xi ]-{\mu }_0=0
140
\end{array}\right\}</math> (12)</span></div>
141
142
<div style="text-align: right; direction: ltr; margin-left: 1em;">
143
<span style="text-align: center; font-size: 75%;"> <math>D=\left\{f(\xi ):\begin{array}{c}
144
{\int }_{\xi \in R^K}f(\xi )d\xi =1\\
145
E[\xi ]-{\mu }_0=0\\
146
var(\xi )=E[{\left(\xi -{\mu }_0\right)}^2]={\sigma }_0^2
147
\end{array}\right\}</math> (13)</span></div>
148
149
<span id='OLE_LINK123'></span><span id='OLE_LINK124'>where, Eqs. (12) is the ambiguity set with known first order moment, and Eqs. (13) is the ambiguity set with the known first and second  order moment. The parameter  <math>{\mu }_0</math> and  <math>{\sigma }_0^2</math> represent the mean and variance obtained from historical data.  <math>D</math> denotes the support of the family of all probability distributions with known moment information. <math>f(\xi )</math> is the probability density function (PDF) of the random variable  <math>\xi </math> .</span>
150
151
<span id='OLE_LINK136'></span><span id='OLE_LINK137'>If the function objective consists uncertain parameters, it can be use the expectation of objective to reformulate, or transfer the objective function to constraints </span><span id='cite-_Ref38381847'></span>[[#_Ref38381847|<span style="text-align: center; font-size: 75%;">[12]</span>]]<span style="text-align: center; font-size: 75%;">. In our spare parts optimization model, the uncertain parameters just exist in constraints. In constraint (6) and (9), the lead time  <math>T_{ij}^{}</math> and  <math>T_{jk}^{}</math> and the spare parts demands  <math>d_k</math> are uncertainty with some unknown distribution. It can be formulate as chance constraints to ensure that the constraint must be satisfied under a probability which greater than the specified threshold. </span>
152
153
<span id='OLE_LINK125'></span><span id='OLE_LINK126'></span><span id='OLE_LINK127'></span><span id='OLE_LINK128'></span><span style="text-align: center; font-size: 75%;">The single chance constraint of Eqs. (6) is as follows:</span>
154
155
<div style="text-align: right; direction: ltr; margin-left: 1em;">
156
<span style="text-align: center; font-size: 75%;"> <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}^{})-</math><math>T\leq \mbox{ }\mbox{ }0)\mbox{ }\geq 1-\epsilon \mbox{ }</math> (14)</span></div>
157
158
<span id='OLE_LINK54'></span><span id='OLE_LINK55'></span><span id='OLE_LINK56'></span><span id='OLE_LINK57'></span><span style="text-align: center; font-size: 75%;">The single chance constraint of Eqs. (9) is as follows:</span>
159
160
<div style="text-align: right; direction: ltr; margin-left: 1em;">
161
<span style="text-align: center; font-size: 75%;"> <math>\mbox{ }Pr(d_k-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ })\geq 1-</math><math>\epsilon \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k=</math><math>1,2,\cdots ,K</math> (15)</span></div>
162
163
<span id='OLE_LINK58'></span><span id='OLE_LINK129'></span><span id='OLE_LINK130'></span><span id='OLE_LINK131'></span><span id='OLE_LINK184'></span><span id='OLE_LINK185'></span><span style="text-align: center; font-size: 75%;">where,  <math>\epsilon \in (0,1)</math> </span> represents a specified safety tolerance or violation probability, and all constraints take the same <span style="text-align: center; font-size: 75%;"> <math>\epsilon </math> </span>.
164
165
<span id='OLE_LINK133'></span><span id='OLE_LINK189'></span><span id='OLE_LINK162'></span><span id='OLE_LINK163'></span><span id='OLE_LINK73'></span><span id='OLE_LINK134'></span><span id='OLE_LINK142'></span><span id='OLE_LINK143'></span><span style="text-align: center; font-size: 75%;">In terms of random variable <math>\xi </math> , if the probability density function  <math>\xi </math> is known, it is easy to solve the inequality  <math>Pr(\xi \leq b)\leq \alpha </math> by  <math>b\leq F_{{}_{\xi }}^{-1}(\alpha )</math> </span><span id='cite-_Ref38381866'></span>[[#_Ref38381866|<span style="text-align: center; font-size: 75%;">[13]</span>]]<span style="text-align: center; font-size: 75%;">. However, in our spare parts network, the distribution of lead times and demands are unknown. Even the PDF of the lead times is known, it still be difficult and time-consuming to calculate the PDF of  <math>\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}^{})</math> . Therefore, further derivation of Eqs. (14) and Eqs. (15) is needed. We use the robust chance constraints to formulate inequality (14) and (15) as follows:</span>
166
167
<div style="text-align: right; direction: ltr; margin-left: 1em;">
168
<span style="text-align: center; font-size: 75%;"> <math>\underset{P\in D_T}{inf}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}^{})-</math><math>T\leq \mbox{ }\mbox{ }0)\mbox{ }\geq 1-\epsilon \mbox{ }</math> (16)</span></div>
169
170
<div style="text-align: right; direction: ltr; margin-left: 1em;">
171
<span style="text-align: center; font-size: 75%;"> <math>\mbox{ }\underset{P\in D_d}{inf}Pr(d_k-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ })\geq 1-</math><math>\epsilon \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k=</math><math>1,2,\cdots ,K</math> (17)</span></div>
172
173
<span id='OLE_LINK160'></span><span id='OLE_LINK161'></span><span id='OLE_LINK208'></span><span id='OLE_LINK209'></span><span id='OLE_LINK154'></span><span style="text-align: center; font-size: 75%;">In the case of only the first order moments are known, for an arbitrary distributional random variable  <math>\mbox{ }\mbox{ }\mbox{ }\xi </math> . According to the Markov's inequality, there is  <math>Pr\left\{\xi \geq k\right\}\leq \frac{E(\xi )}{k}\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }X\geq 0,k\geq 0</math> . Therefore, we can get the robust counterparts of constraint (16) and (17) as follows:</span>
174
175
<div style="text-align: right; direction: ltr; margin-left: 1em;">
176
<span style="text-align: center; font-size: 75%;"> <math>E\left\{\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\right. </math><math>\left. \sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}\leq T\epsilon </math> (18)</span></div>
177
178
<div style="text-align: right; direction: ltr; margin-left: 1em;">
179
<span style="text-align: center; font-size: 75%;"> <math>\mbox{ }\frac{E(d_{}_k^{})}{\sum_{j\in J}x_{jk}^{}}\leq \epsilon \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k=</math><math>1,2,\cdots ,K</math> (19)</span></div>
180
181
<span id='OLE_LINK165'></span><span id='OLE_LINK166'></span><span id='OLE_LINK167'></span><span id='OLE_LINK155'></span><span id='OLE_LINK156'></span><span style="text-align: center; font-size: 75%;">'''Proof. '''For inequality (16), there is:</span>
182
183
184
{|
185
|-
186
| <math>\underset{P\in D_T}{inf}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}^{})-</math><math>T\leq \mbox{ }\mbox{ }0)\mbox{ }\mbox{ }</math>
187
| <math>=\underset{P\in D_T}{inf}\left\{1-Pr(\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\right. </math><math>\left. \sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\geq T\mbox{ }\mbox{ })\mbox{ }\right\}\mbox{ }</math>
188
|}
189
190
191
<span style="text-align: center; font-size: 75%;"> 
192
{|
193
|-
194
| <math>\geq 1-\frac{E\left\{\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}}{T}</math>
195
| <math>\mbox{ }\geq </math>
196
|}
197
 <math>1-\epsilon </math> </span>
198
199
<math>\Rightarrow E\left\{\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\right. </math><math>\left. \sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}\leq T\epsilon </math>
200
201
<span style="text-align: center; font-size: 75%;">For inequality (17), there is:</span>
202
203
204
{|
205
|-
206
| <math>\mbox{ }\underset{P\in D_d}{inf}Pr(d_k-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ })</math>
207
| <math>\mbox{ }=1-\underset{P\in D_d}{sup}Pr(d_k-\sum_{j\in J}x_{jk}^{}\geq 0\mbox{ })</math>
208
|}
209
210
211
<span style="text-align: center; font-size: 75%;"> 
212
{|
213
|-
214
| <math>\mbox{ }\geq 1-\frac{E(d_k)}{\sum_{j\in J}x_{jk}^{}}</math>
215
| <math>\mbox{ }\geq 1-\epsilon \mbox{ }\mbox{ }</math>
216
|}
217
</span>
218
219
220
{|
221
|-
222
| <math>\mbox{ }\Rightarrow </math>
223
| <math>\mbox{ }\frac{E(d_{}_k^{})}{\sum_{j\in J}x_{jk}^{}}\leq \epsilon \mbox{ }</math>
224
|}
225
226
227
<span style="text-align: center; font-size: 75%;">In the case of both the first and second order moments are known, for an arbitrary distributional random variable <math>\mbox{ }\mbox{ }\mbox{ }\xi </math> . According to the Cantelli's inequality, there is,  <math>Pr\left\{\xi -\mu \geq k\right\}\leq \frac{{\sigma }^2}{{\sigma }^2+k^2}\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k\geq 0</math> . Therefore, we can get the robust counterparts of constraint (16) and (17) as follows:</span>
228
229
<div style="text-align: right; direction: ltr; margin-left: 1em;">
230
<span style="text-align: center; font-size: 75%;"> [[Image:Draft_Wang_655393420-image83.png|486px]] (20)</span></div>
231
232
<div style="text-align: right; direction: ltr; margin-left: 1em;">
233
<span style="text-align: center; font-size: 75%;"> <math>\mbox{ }\mbox{ }\frac{var(d_k)}{var(d_k)+{\left(\sum_{j\in J}x_{jk}^{}-E(d_k)\right)}^2}\leq \epsilon \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k=</math><math>1,2,\cdots ,K</math> (21)</span></div>
234
235
'''Proof.''' For inequality (16), there is:
236
237
<math>\underset{P\in D_T}{inf}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}^{})-</math><math>T\leq 0\mbox{ }\mbox{ })\mbox{ }\mbox{ }</math>
238
239
<span style="text-align: center; font-size: 75%;"> <math>=\underset{P\in D_T}{inf}\left\{1-Pr(\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\right. </math><math>\left. \sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\geq T\mbox{ }\mbox{ })\mbox{ }\right\}\mbox{ }</math> </span>
240
241
<math>\geq 1-\frac{var\left\{\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}}{var\left\{\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}+{\left(E\left\{\sum_{i\in I}\sum_{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum_{j\in J}\sum_{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}-T\right)}^2}</math>  <math>\mbox{ }\geq </math>  <math>1-\epsilon </math>
242
243
<span id='OLE_LINK170'></span><span id='OLE_LINK171'></span><span id='OLE_LINK229'></span><span id='OLE_LINK230'></span><span style="text-align: center; font-size: 75%;"> <math>\mbox{ }\Rightarrow </math>  [[Image:Draft_Wang_655393420-image83.png|600px]] </span>
244
245
For inequality (17), there is:
246
247
248
{|
249
|-
250
| <math>\mbox{ }\underset{P\in D_d}{inf}Pr(d_k-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ })</math>
251
| <math>=1-\underset{P\in D_d}{sup}Pr(d_k-\sum_{j\in J}x_{jk}^{}\geq 0\mbox{ })</math>
252
|}
253
 
254
{|
255
|-
256
| <math>\geq \mbox{ }1-\frac{var(d_k)}{var(d_k)+{\left(\sum_{j\in J}x_{jk}^{}-E(d_k)\right)}^2}</math>
257
| <math>\geq </math>
258
|}
259
 <math>1-\epsilon </math>
260
261
<span style="text-align: center; font-size: 75%;"> 
262
{|
263
|-
264
| <math>\mbox{ }\Rightarrow </math>
265
| <math>\mbox{ }\frac{var(d_k)}{var(d_k)+{\left(\sum_{j\in J}x_{jk}^{}-E(d_k)\right)}^2}\leq \epsilon </math>
266
|}
267
</span>
268
269
<span id='OLE_LINK220'></span><span id='OLE_LINK168'></span><span id='OLE_LINK169'></span><span style="text-align: center; font-size: 75%;">In summary, the robust counterpart is derivated by replace the inequality (6) and (9) of the determined model by the inequality (18) and (19) or inequality (20) and (21).</span>
270
271
'''3 Case Study'''
272
273
''3.1 Case Description''
274
275
<span id='OLE_LINK221'></span><span id='OLE_LINK186'></span><span id='OLE_LINK187'></span><span id='OLE_LINK192'></span><span style="text-align: center; font-size: 75%;">In this case, the three echelon spare parts supply network consists of 2 supply centers, 5 distribution centers, and 4 customers. The spare parts are supported to meet the demands of customers. The first and second moments of the lead times and demands are counted from 50个samples. The relative data are shown in Table 1-4. The required maximum lead time is 300 hours, and the</span> safety tolerance<span style="text-align: center; font-size: 75%;"> <math>\epsilon </math> is set from 0.1 to 0.9, respectively.</span>
276
277
The problems were solved by CPLEX and differential evolutionary algorithm in a MATLAB 2014b platform on an ASUS laptop with 5-6300HQ 2.3GBz CPU, 4GB RAM in Windows 7.0(64-bit) environment.
278
279
''3.2 Results and Sensitive Analysis''
280
281
The optimal solutions of the robust optimization model with known first order moment (referred to as the first order moment model) and with known first and second moment (referred to as the second order moment model) are obtained, their corresponding objective functions and constraints are also calculated and shown in Table 5. In order to analyze the effect of safety tolerance  <math>\epsilon </math> on the robustness of the model, we take multiple values of  <math>\epsilon </math> in the first order moment model and the second order moment model, respectively. Constraint 1-20 in Table 5 is the value of the left side of the corresponding inequality constraints, that is, if the value is big than zero, the corresponding constraints are violated, otherwise, the constraint is satisfied. The following conclusion can be summed from the table.
282
283
<span id='OLE_LINK193'>(1) As the safety tolerance increases, the robustness of the model is gradually slack. In other words, the smaller the safety tolerance, the stricter the robustness of the constraints is. In the first order moment model, the feasible solution is obtained only when the tolerance is set as 0.9, and the constraint value is greater than zero in other cases (bold in the table), which indicates unfeasible. Comparing the value of these unfeasible solutions, it can be found that the increase of safety tolerance leads to a decrease of constraint violation degree</span>
284
285
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
286
<span style="text-align: center; font-size: 75%;">'''Table 1. '''The opening costs, inventory costs and inventory capacity of distribution centers</span></div>
287
288
{| style="width: 74%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
289
|-
290
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Hlk38295451'></span>
291
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Distribution centers 1</span>
292
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Distribution centers 2</span>
293
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Distribution centers 3</span>
294
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Distribution centers 4</span>
295
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Distribution centers 5</span>
296
|-
297
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Opening costs</span>
298
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1000</span>
299
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1500</span>
300
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2500</span>
301
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1800</span>
302
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2000</span>
303
|-
304
|  style="text-align: center;"|<span id='OLE_LINK176'></span><span id='OLE_LINK177'>Inventory costs</span>
305
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
306
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">8</span>
307
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
308
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
309
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
310
|-
311
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Inventory capacity</span>
312
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">50</span>
313
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">55</span>
314
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">80</span>
315
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">60</span>
316
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">65</span>
317
|}
318
319
320
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
321
<span style="text-align: center; font-size: 75%;">'''Table 2.''' The shortage loss, inventory costs, and demands of customers </span></div>
322
323
{| style="width: 73%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
324
|-
325
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
326
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">customer1</span>
327
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">customer2</span>
328
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">customer3</span>
329
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">customer4</span>
330
|-
331
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Shortage loss</span>
332
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">100</span>
333
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">120</span>
334
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">80</span>
335
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">150</span>
336
|-
337
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Inventory costs</span>
338
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
339
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">15</span>
340
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">13</span>
341
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
342
|-
343
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Expectation of demands</span>
344
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">68</span>
345
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">61</span>
346
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">57</span>
347
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">88</span>
348
|-
349
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Variance of demands</span>
350
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
351
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
352
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
353
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">8</span>
354
|}
355
356
357
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
358
<span style="text-align: center; font-size: 75%;">'''Table 3. '''The transportation costs between nodes of supply network </span></div>
359
360
{| style="width: 87%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
361
|-
362
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
363
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 1</span>
364
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 2</span>
365
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 3</span>
366
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 4</span>
367
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 5</span>
368
|-
369
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Supply center1</span>
370
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">105</span>
371
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">104</span>
372
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">102</span>
373
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">104</span>
374
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">110</span>
375
|-
376
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Supply center2</span>
377
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">94</span>
378
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">108</span>
379
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">105</span>
380
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">98</span>
381
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">96</span>
382
|-
383
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 1</span>
384
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">46</span>
385
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">42</span>
386
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">59</span>
387
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">54</span>
388
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">46</span>
389
|-
390
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 2</span>
391
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">41</span>
392
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">67</span>
393
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">61</span>
394
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">67</span>
395
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">41</span>
396
|-
397
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 3</span>
398
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">49</span>
399
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">53</span>
400
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">51</span>
401
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">49</span>
402
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">49</span>
403
|-
404
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 4</span>
405
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">55</span>
406
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">49</span>
407
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">60</span>
408
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">51</span>
409
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">55</span>
410
|}
411
412
413
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
414
<span style="text-align: center; font-size: 75%;">'''Table 4. '''The lead times between nodes of the supply network </span></div>
415
416
{| style="width: 87%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
417
|-
418
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Hlk38295509'></span>
419
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 1</span>
420
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 2</span>
421
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 3</span>
422
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 4</span>
423
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Distribution centers 5</span>
424
|-
425
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Supply center1</span>
426
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">12/1.6</span>
427
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">11/2.5</span>
428
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">9/1.8</span>
429
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">11/1.1</span>
430
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">10/2.3</span>
431
|-
432
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Supply center2</span>
433
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">7/0.7</span>
434
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">15/1.3</span>
435
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">15/1.6</span>
436
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">7.5/1.8</span>
437
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">8/1.1</span>
438
|-
439
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 1</span>
440
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5/0.2</span>
441
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5/0.3</span>
442
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">7/0.5</span>
443
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3.7/1.2</span>
444
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4/0.8</span>
445
|-
446
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 2</span>
447
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4/0.8</span>
448
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5/1.1</span>
449
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3/0.2</span>
450
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4.5/0.6</span>
451
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4/0.3</span>
452
|-
453
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 3</span>
454
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3.5/0.3</span>
455
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.5/0.3</span>
456
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.9/0.7</span>
457
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.5/0.4</span>
458
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5/0.8</span>
459
|-
460
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Customer 4</span>
461
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2.6/0.2</span>
462
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">4/0.4</span>
463
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1.6/0.2</span>
464
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2/0.5</span>
465
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3.5/0.7</span>
466
|}
467
468
469
<span id='OLE_LINK196'></span><span id='OLE_LINK197'></span><span id='OLE_LINK204'></span><span id='OLE_LINK205'></span><span id='OLE_LINK200'></span><span id='OLE_LINK201'></span><span id='OLE_LINK202'></span><span id='OLE_LINK203'>(2) The value of the objective function decreases gradually as the robustness of the model enhanced. This is because when the robustness is very strict, the solutions try to meet the constraints as much as possible for the price of at the expense of objective function. For example, in order to meet the demand constraint, decision makers have to increase the amount of supplied spare parts, which makes the transportation and inventory costs increase greatly. Therefore, it’s necessary to judge and weigh the smaller objective function and the higher robust constraints to find a balance point.</span>
470
471
<span id='OLE_LINK206'></span><span id='OLE_LINK207'>(3) The robustness of the first order moment model is stronger than that of the second order moment model. However, the robustness of the first order moment model is too strict, and the feasible solutions may not be able to be found. That is because when using the Markov's inequality, the probability of constraints satisfied is reduced too small, and a lot of information is missed during this process. However, the second-order moment model is not as strict as the first-order moment model, because there is more distribution information in the second order moment. In the actual spare parts supply, if the robustness is emphasized too much, the feasibility of the solution cannot be guaranteed, and the cost of supply will be increased at the same time. It can see that, the result of second order moment model is better than the first order moment model.</span>
472
473
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
474
<span style="text-align: center; font-size: 75%;">'''Table 5.''' Objective function and constraints of first order and second-order moment model</span></div>
475
476
{| style="width: 100%;border-collapse: collapse;" 
477
|-
478
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Model</span>
479
|  colspan='5'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">First order moment model</span>
480
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
481
|  colspan='5'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Second order moment model</span>
482
|-
483
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Safety tolerance</span>
484
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.1</span>
485
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
486
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
487
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
488
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9</span>
489
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
490
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.1</span>
491
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
492
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
493
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
494
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9</span>
495
|-
496
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Costs</span>
497
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">42029</span>
498
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">41679</span>
499
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">41168</span>
500
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">40958</span>
501
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">39480</span>
502
|  style="border-top: 1pt solid black;text-align: center;"|
503
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">42634</span>
504
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">41172</span>
505
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">41120</span>
506
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">39459</span>
507
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">39392</span>
508
|-
509
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Constraint 1</span>
510
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''130.3'''</span>
511
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''86.3'''</span>
512
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''23.3'''</span>
513
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-30.2</span>
514
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-199.8</span>
515
|  style="text-align: center;vertical-align: top;"|
516
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1421.134</span>
517
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5796.052</span>
518
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-10530.2</span>
519
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-10105.74</span>
520
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-13000.61</span>
521
|-
522
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
523
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
524
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
525
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
526
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
527
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
528
|  style="text-align: center;vertical-align: top;"|
529
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
530
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-3</span>
531
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
532
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
533
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5</span>
534
|-
535
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
536
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
537
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
538
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
539
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
540
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
541
|  style="text-align: center;vertical-align: top;"|
542
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
543
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
544
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
545
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
546
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
547
|-
548
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
549
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
550
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
551
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
552
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
553
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
554
|  style="text-align: center;vertical-align: top;"|
555
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
556
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
557
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5</span>
558
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
559
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
560
|-
561
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
562
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
563
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
564
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
565
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
566
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
567
|  style="text-align: center;vertical-align: top;"|
568
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
569
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
570
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
571
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
572
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
573
|-
574
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
575
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
576
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-11</span>
577
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
578
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
579
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
580
|  style="text-align: center;vertical-align: top;"|
581
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-3</span>
582
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
583
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
584
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
585
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
586
|-
587
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
588
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
589
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
590
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
591
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
592
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
593
|  style="text-align: center;vertical-align: top;"|
594
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
595
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5</span>
596
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-9</span>
597
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-7</span>
598
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
599
|-
600
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">8</span>
601
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
602
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5</span>
603
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
604
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
605
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
606
|  style="text-align: center;vertical-align: top;"|
607
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
608
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
609
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
610
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
611
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
612
|-
613
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
614
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
615
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
616
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
617
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
618
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
619
|  style="text-align: center;vertical-align: top;"|
620
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
621
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
622
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5</span>
623
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
624
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
625
|-
626
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
627
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
628
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
629
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
630
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
631
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
632
|  style="text-align: center;vertical-align: top;"|
633
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-8</span>
634
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
635
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
636
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-3</span>
637
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
638
|-
639
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
640
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-7</span>
641
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-11</span>
642
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
643
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
644
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
645
|  style="text-align: center;vertical-align: top;"|
646
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-7</span>
647
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5</span>
648
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
649
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
650
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
651
|-
652
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
653
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''59.2'''</span>
654
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''41.6'''</span>
655
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''26.5'''</span>
656
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''15.5'''</span>
657
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-0.4</span>
658
|  style="text-align: center;vertical-align: top;"|
659
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-17.5</span>
660
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-126</span>
661
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-195.5</span>
662
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-82</span>
663
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-56.7</span>
664
|-
665
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">13</span>
666
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''55.4'''</span>
667
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''44.5'''</span>
668
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''31'''</span>
669
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''16.2'''</span>
670
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-0.2</span>
671
|  style="text-align: center;vertical-align: top;"|
672
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2.2</span>
673
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-3.1</span>
674
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-7</span>
675
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-3</span>
676
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2.5</span>
677
|-
678
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">14</span>
679
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''51.8'''</span>
680
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''44.4'''</span>
681
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''26.5'''</span>
682
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''19.2'''</span>
683
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-0.6</span>
684
|  style="text-align: center;vertical-align: top;"|
685
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-0.1</span>
686
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-5.9</span>
687
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-9</span>
688
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-0.7</span>
689
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2.9</span>
690
|-
691
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">15</span>
692
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''77.8'''</span>
693
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''56.2'''</span>
694
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''39'''</span>
695
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''7.5'''</span>
696
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-0.2</span>
697
|  style="text-align: center;vertical-align: top;"|
698
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-12.4</span>
699
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-30.7</span>
700
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-8.5</span>
701
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-98.4</span>
702
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-128.8</span>
703
|-
704
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">16</span>
705
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
706
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
707
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
708
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
709
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
710
|  style="text-align: center;vertical-align: top;"|
711
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
712
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
713
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-7</span>
714
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
715
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
716
|-
717
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">17</span>
718
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
719
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
720
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
721
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
722
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
723
|  style="text-align: center;vertical-align: top;"|
724
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
725
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-3</span>
726
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
727
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
728
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
729
|-
730
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">18</span>
731
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
732
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
733
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
734
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
735
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
736
|  style="text-align: center;vertical-align: top;"|
737
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
738
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
739
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
740
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
741
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
742
|-
743
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">19</span>
744
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-2</span>
745
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
746
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
747
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
748
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
749
|  style="text-align: center;vertical-align: top;"|
750
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-7</span>
751
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
752
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
753
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
754
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
755
|-
756
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">20</span>
757
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-6</span>
758
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
759
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
760
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
761
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
762
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
763
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
764
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-1</span>
765
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-4</span>
766
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
767
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0</span>
768
|}
769
770
771
<span style="text-align: center; font-size: 75%;">'''4 Robustness analysis'''</span>
772
773
<span id='OLE_LINK213'>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.</span>
774
775
<span id='OLE_LINK214'></span><span id='OLE_LINK215'></span><span style="text-align: center; font-size: 75%;">''4.1 Comparison with Determined Model''</span>
776
777
<span id='OLE_LINK43'></span><span id='OLE_LINK44'>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.</span>
778
779
<span id='OLE_LINK216'></span><span id='OLE_LINK217'>Brought the robust solutions under different safety tolerances into the determined models, and the values of all constraints are calculated respectively. The constraint value indicates constraint violation, and if the constraint value is less than or equal to 0, it means that the constraint is satisfied. Figure 1-10 is the histograms of the constraints values in these ten different cases, respectively (two different order moment models with five safety tolerance). Taking Figure 1 as an example, Figure 1 shows the constraints values that bring the robust solutions of the first order moment model with a safety tolerance of 0.1 into the determined model. The horizontal axis represents the index of the constraint, and there are a total of 20 constraints in our case. The vertical axis is a constraint value, the yellow bars (labeled as robust model) are the constraint values of the robust model, and the blue bars (labeled as certain model) are the constraint values of determined the model.</span>
780
781
<span id='OLE_LINK222'>In Figure 1, the constraints 1,12,13,14 and 15 of the robust model are not satisfied. In the corresponding determination model, there are only two violated constraints, that is, constraint 13 and constraint 14. It can be seen that although the robust solution is still unfeasible in the deterministic model, it greatly reduces the constraints violate degree. It can be seen that the first order moment model has good robustness, and the same conclusion can be drawn from Figure 2-10.</span>
782
783
<span id='OLE_LINK223'></span><span id='OLE_LINK24'></span><span id='OLE_LINK25'></span><span id='OLE_LINK1'></span><span id='OLE_LINK2'></span><span id='OLE_LINK7'>Comparing the results in Figure 1-5 (or figure 6-10), it can also be found that the constraint violations of both the determined model and the robust model decrease as the safety tolerance increases, which is consistent with conclusion (2) in section 5.2. Comparing the results of first order moment model with those in second order moment model, we can find that the robust solution of the second-order moment model makes the constraint violations in determined model much smaller than the robust solution of the first-order moment model. This conclusion is consistent with conclusion (3) in section 5.2.</span>
784
785
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
786
 [[Image:Draft_Wang_655393420-image95.png|318px]] </div>
787
788
<span id='OLE_LINK76'></span><div id="OLE_LINK77" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
789
<span style="text-align: center; font-size: 75%;">'''Fig 1.'''</span> <span style="text-align: center; font-size: 75%;">Constraint violation comparison between first order moment model and determined model as safety tolerance is 0.1</span></div>
790
791
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
792
 [[Image:Draft_Wang_655393420-image96.png|324px]] </div>
793
794
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
795
<span style="text-align: center; font-size: 75%;">'''Fig 2.''' Constraint violation comparison between first order moment model and determined model as safety tolerance is 0.3</span></div>
796
797
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
798
 [[Image:Draft_Wang_655393420-image97.png|330px]] </div>
799
800
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
801
<span style="text-align: center; font-size: 75%;">'''Fig 3. '''Constraint violation comparison between first order moment model and determined model as safety tolerance is 0.5</span></div>
802
803
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
804
 [[Image:Draft_Wang_655393420-image98.png|330px]] </div>
805
806
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
807
<span style="text-align: center; font-size: 75%;">'''Fig 4. '''Constraint violation comparison between first order moment model and determined model as safety tolerance is 0.7</span></div>
808
809
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
810
 [[Image:Draft_Wang_655393420-image99.png|336px]] </div>
811
812
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
813
<span style="text-align: center; font-size: 75%;">'''Fig 5.''' Constraint violation comparison between first order moment model and determined model as safety tolerance is 0.9</span></div>
814
815
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
816
 [[Image:Draft_Wang_655393420-image100.png|336px]] </div>
817
818
<span id='OLE_LINK111'></span><div id="OLE_LINK112" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
819
<span style="text-align: center; font-size: 75%;">'''Fig 6. '''Constraint violation comparison between second order moment model and determined model as safety tolerance is 0.1</span></div>
820
821
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
822
 [[Image:Draft_Wang_655393420-image101.png|312px]] </div>
823
824
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
825
<span style="text-align: center; font-size: 75%;">'''Fig 7.''' Constraint violation comparison between second order moment model and determined model as safety tolerance is 0.3</span></div>
826
827
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
828
 [[Image:Draft_Wang_655393420-image102.png|318px]] </div>
829
830
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
831
<span style="text-align: center; font-size: 75%;">'''Fig 8.''' Constraint violation comparison between second order moment model and determined model as safety tolerance is 0.5</span></div>
832
833
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
834
 [[Image:Draft_Wang_655393420-image103.png|318px]] </div>
835
836
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
837
<span style="text-align: center; font-size: 75%;">'''Fig 9. '''Constraint violation comparison between second order moment model and determined model as safety tolerance is 0.7</span></div>
838
839
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
840
 [[Image:Draft_Wang_655393420-image104.png|318px]] </div>
841
842
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
843
<span style="text-align: center; font-size: 75%;">'''Fig 10.''' Constraint violation comparison between second order moment model and determined model as safety tolerance is 0.9</span></div>
844
845
<span id='OLE_LINK224'></span><span id='OLE_LINK225'>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 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.</span>
846
847
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
848
 [[Image:Draft_Wang_655393420-image105.png|600px]] </div>
849
850
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
851
<span style="text-align: center; font-size: 75%;">'''Fig 11.''' Constraint violation comparison between robust model and chance constraint model</span></div>
852
853
''4.2 Comparison with Chance Constraint Model''
854
855
In this section, the feasibility of robust solutions in the chance constraint model is verified. Take the robust solutions of first order moment model and second order moment model into the chance constraint model to test the feasibility. Since the uncertain parameters only exist in the constraints (6) and (9) of the spare parts supply model, only these two inequality constraints need to be tested. The corresponding opportunity constraint models are:
856
857
<div style="text-align: right; direction: ltr; margin-left: 1em;">
858
<span style="text-align: center; font-size: 75%;"> <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{ })\mbox{ }\geq 1-</math><math>\epsilon \mbox{ }</math> (22)</span></div>
859
860
<div style="text-align: right; direction: ltr; margin-left: 1em;">
861
<span style="text-align: center; font-size: 75%;"> <math>\mbox{ }Pr(d_k-\sum_{j\in J}x_{jk}^{}\leq 0\mbox{ })\geq 1-</math><math>\epsilon \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ },\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }k=</math><math>1,2,\cdots ,K</math> (23)</span></div>
862
863
<span style="text-align: center; font-size: 75%;">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 </span><span id='cite-_Ref38381890'></span>[[#_Ref38381890|<span style="text-align: center; font-size: 75%;">[14]</span>]]<span style="text-align: center; font-size: 75%;">. Because the first order moment robust optimization model is not feasible, only the second order moment optimization model is compared and analyzed.</span>
864
865
50 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.
866
867
<div style="text-align: right; direction: ltr; margin-left: 1em;">
868
<span style="text-align: center; font-size: 75%;"> <math>N_T{}'=\sum_{n=1}^N\Delta (\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> (24)</span></div>
869
870
<div style="text-align: right; direction: ltr; margin-left: 1em;">
871
<span style="text-align: center; font-size: 75%;"> <math>N_d{}'=\sum_{n=1}^N\Delta (d_k-\sum_{j\in J}x_{jk}^{}\leq 0)</math> (25)</span></div>
872
873
<span id='OLE_LINK120'></span><span id='OLE_LINK121'></span><span id='OLE_LINK122'></span><span id='OLE_LINK132'></span><span id='OLE_LINK66'></span><span id='OLE_LINK67'></span><span id='OLE_LINK135'></span><span id='OLE_LINK68'></span><span id='OLE_LINK69'>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.</span>
874
875
<span id='OLE_LINK70'>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.</span>
876
877
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
878
<span style="text-align: center; font-size: 75%;">'''Table 6. '''The constraint satisfied probability with second order moment model robust solution</span></div>
879
880
{| style="width: 53%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
881
|-
882
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;"> <math>\epsilon </math> </span>
883
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.1</span>
884
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
885
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
886
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
887
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9</span>
888
|-
889
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;"> <math>N_T{}'/N</math> </span>
890
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9692</span>
891
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8997</span>
892
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7938</span>
893
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7288</span>
894
|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.6992</span>
895
|-
896
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;"> <math>N_d{}'/N</math> , <math>k=1</math> </span>
897
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9269</span>
898
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8194</span>
899
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8187</span>
900
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7938</span>
901
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7994</span>
902
|-
903
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;"> <math>N_d{}'/N</math> , <math>k=2</math> </span>
904
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9524</span>
905
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8219</span>
906
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8170</span>
907
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7922</span>
908
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7223</span>
909
|-
910
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;"> <math>N_d{}'/N</math> , <math>k=3</math> </span>
911
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9461</span>
912
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9403</span>
913
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9396</span>
914
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8959</span>
915
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8843</span>
916
|-
917
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;"> <math>N_d{}'/N</math> , <math>k=4</math> </span>
918
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.9289</span>
919
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8351</span>
920
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7378</span>
921
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7298</span>
922
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.6537</span>
923
|}
924
925
926
<span style="text-align: center; font-size: 75%;">'''5 Conclusion'''</span>
927
928
<span id='OLE_LINK226'></span><span id='OLE_LINK227'></span><span id='OLE_LINK74'></span><span id='OLE_LINK75'></span><span id='OLE_LINK71'></span><span id='OLE_LINK72'>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.</span>
929
930
'''Reference '''
931
932
<span id='_Ref38381732'></span><span style="text-align: center; font-size: 75%;">[[#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>
933
934
<span id='_Ref38381747'></span><span style="text-align: center; font-size: 75%;">[[#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>
935
936
<span id='_Ref38381756'></span><span style="text-align: center; font-size: 75%;">[[#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>
937
938
<span id='_Ref38381761'></span><span style="text-align: center; font-size: 75%;">[[#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>
939
940
<span id='_Ref38381766'></span><span style="text-align: center; font-size: 75%;">[[#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>
941
942
<span id='_Ref38381799'></span><span style="text-align: center; font-size: 75%;">[[#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>
943
944
<span id='_Ref38381806'></span><span style="text-align: center; font-size: 75%;">[[#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>
945
946
<span id='_Ref38381812'></span><span style="text-align: center; font-size: 75%;">[[#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>
947
948
<span id='_Ref38381823'></span><span id='_Ref527702845'></span><span style="text-align: center; font-size: 75%;">[[#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>
949
950
<span id='_Ref38381827'></span><span style="text-align: center; font-size: 75%;">[[#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>
951
952
<span id='_Ref38381835'></span><span style="text-align: center; font-size: 75%;">[[#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>
953
954
<span id='_Ref38381847'></span><span style="text-align: center; font-size: 75%;">[[#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>
955
956
<span id='_Ref38381866'></span><span style="text-align: center; font-size: 75%;">[[#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>
957
958
<span id='_Ref38381890'></span><span style="text-align: center; font-size: 75%;">[[#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>
959
960
----
961
962
* Corresponding author. Tel.: +86 18795428481; E-mail: xwzj0003[mailto:@ @]gmail.com
963

Return to Wang et al 2020f.

Back to Top

Document information

Published on 12/01/21
Accepted on 12/11/20
Submitted on 13/07/20

Volume 37, Issue 1, 2021
DOI: 10.23967/j.rimni.2020.11.002
Licence: CC BY-NC-SA license

Document Score

0

Views 144
Recommendations 0

Share this document

claim authorship

Are you one of the authors of this document?