Proof. For inequality (16), there is:

For inequality (17), there is:</span>

</span>

In the case of both the first and second order moments are known, for an arbitrary distributional random variable ${\displaystyle {\mbox{ }}{\mbox{ }}{\mbox{ }}\xi }$ . According to the Cantelli's inequality, there is, ${\displaystyle Pr\left\{\xi -\mu \geq k\right\}\leq {\frac {{\sigma }^{2}}{{\sigma }^{2}+k^{2}}}{\mbox{ }},{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}k\geq 0}$ . Therefore, we can get the robust counterparts of constraint (16) and (17) as follows:</span>

Proof. For inequality (16), there is:

${\displaystyle {\underset {P\in D_{T}}{inf}}Pr(\sum _{i\in I}\sum _{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+}$${\displaystyle \sum _{j\in J}\sum _{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})-}$${\displaystyle T\leq 0{\mbox{ }}{\mbox{ }}){\mbox{ }}{\mbox{ }}}$

${\displaystyle ={\underset {P\in D_{T}}{inf}}\left\{1-Pr(\sum _{i\in I}\sum _{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\right.}$${\displaystyle \left.\sum _{j\in J}\sum _{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\geq T{\mbox{ }}{\mbox{ }}){\mbox{ }}\right\}{\mbox{ }}}$ </span>

${\displaystyle \geq 1-{\frac {var\left\{\sum _{i\in I}\sum _{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum _{j\in J}\sum _{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}}{var\left\{\sum _{i\in I}\sum _{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum _{j\in J}\sum _{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}+{\left(E\left\{\sum _{i\in I}\sum _{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+\sum _{j\in J}\sum _{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\right\}-T\right)}^{2}}}}$ ${\displaystyle {\mbox{ }}\geq }$ ${\displaystyle 1-\epsilon }$

<span id='OLE_LINK170'<span id='OLE_LINK171'<span id='OLE_LINK229'<span id='OLE_LINK230' ${\displaystyle {\mbox{ }}\Rightarrow }$ File:Review 789578267838-image83.png </span>

For inequality (17), there is:

${\displaystyle 1-\epsilon }$

</span>

<span id='OLE_LINK220'<span id='OLE_LINK168'<span id='OLE_LINK169'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).

3 Case Study

3.1 Case Description

<span id='OLE_LINK221'<span id='OLE_LINK186'<span id='OLE_LINK187'<span id='OLE_LINK192'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 samples. The relative data are shown in Table 1-4. The required maximum lead time is 300 hours, and the</span> safety tolerance ${\displaystyle \epsilon }$ is set from 0.1 to 0.9, respectively. The problems were solved by CPLEX and differential evolutionary algorithm.

3.2 Results and Sensitive Analysis

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 ${\displaystyle \epsilon }$ on the robustness of the model, we take multiple values of ${\displaystyle \epsilon }$ 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.

(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 id='OLE_LINK196'<span id='OLE_LINK197'<span id='OLE_LINK204'<span id='OLE_LINK205'<span id='OLE_LINK200'<span id='OLE_LINK201'<span id='OLE_LINK202'(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 id='OLE_LINK206'(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.

4 Robustness analysis</span>

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 id='OLE_LINK214'<span id='OLE_LINK215'4.1 Comparison with Determined Model

<span id='OLE_LINK43'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 id='OLE_LINK216'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.

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 id='OLE_LINK223'<span id='OLE_LINK24'<span id='OLE_LINK25'<span id='OLE_LINK1'<span id='OLE_LINK2'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 id='OLE_LINK272'<span id='OLE_LINK273'<span id='OLE_LINK224'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 id='OLE_LINK242'<span id='OLE_LINK243'<span id='OLE_LINK249'<span id='OLE_LINK258'4.2 Comparison with Chance Constraint Model

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:

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 id='cite-_Ref38381890'[14]. Because the first order moment robust optimization model is not feasible, only the second order moment optimization model is compared and analyzed.

50 samples of uncertain parameters are selected, take the robust solution of the second-order moment model into these samples, and calculate the times ${\displaystyle N_{T}{}'}$ and ${\displaystyle N_{d}{}'}$ that inequality (22) and (23) are satisfied.

<span id='OLE_LINK120'<span id='OLE_LINK121'<span id='OLE_LINK122'<span id='OLE_LINK132'<span id='OLE_LINK66'<span id='OLE_LINK67'<span id='OLE_LINK135'<span id='OLE_LINK68'where, ${\displaystyle \Delta (.)=1}$ , if ${\displaystyle .}$ is satisfied. else if, ${\displaystyle \Delta (.)=0}$ . Then, ${\displaystyle N_{T}{}'/N}$ and ${\displaystyle N_{d}{}'/N}$ are used to represent the value of ${\displaystyle Pr(\sum _{i\in I}\sum _{j\in J}T_{ij}^{}\cdot sgn(x_{ij}^{})+}$${\displaystyle \sum _{j\in J}\sum _{k\in K}T_{jk}^{}\cdot sgn(x_{jk}^{})\leq T{\mbox{ }}{\mbox{ }})}$ and ${\displaystyle {\mbox{ }}Pr(d_{k}-\sum _{j\in J}x_{jk}^{}\leq 0{\mbox{ }}){\mbox{ }}{\mbox{ }}}$ . If ${\displaystyle N_{T}{}'/N}$ or ${\displaystyle N_{d}{}'/N}$ were greater than or equal to ${\displaystyle 1-\epsilon }$ , the constraint (22) and (23) are satisfied.

The constraint satisfactions of the chance constraint model are shown in table 6. It can be seen that the constraints (22) and (23) are satisfied in all scenarios. As the safety tolerance decrease, the value of ${\displaystyle N_{T}{}'/N}$ and ${\displaystyle N_{d}{}'/N}$ 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.

5 Conclusion

<span id='OLE_LINK226'<span id='OLE_LINK227'<span id='OLE_LINK74'<span id='OLE_LINK75'<span id='OLE_LINK71'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.

Reference

<span id='_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 id='_Ref38381747'[2] Mao, H. L., Gao, J. W., Chen, X. J., & Gao, J. D. (2014). Demand Prediction of the Rarely Used Spare Parts Based on the BP Neural Network. Applied Mechanics and Materials, 1513–1519.

<span id='_Ref38381756'[3] Sadeghi, J., Sadeghi, S., & Niaki, S. T. A. (2014). A hybrid vendor managed inventory and redundancy allocation optimization problem in supply chain management: An NSGA-II with tuned parameters. Computers & Operations Research, 41, 53–64.

<span id='_Ref38381761'[4] Qiu Zhiping, Ni Zao.(2004). Probabilistic interval approach for determining the demand of aviation spares. Acta Aeronautica Et Astronautica Sinica, 30(5), 861-866.

<span id='_Ref38381766'[5] Axsäter, S. (1993). Optimization of order-up-to-s policies in two-echelon inventory systems with periodic review. Naval Research Logistics, 40(2), 245–253. </span>

<span id='_Ref38381799'[6] Kouki, C. , Babai, M. Z. , Jemai, Z. , & Minner, S. . (2018). Solution procedures for lost sales base-stock inventory systems with compound poisson demand. International Journal of Production Economics, S0925527318300513.

<span id='_Ref38381806'[7] Al-Rifai, M. H., & Rossetti, M. D. (2007). An efficient heuristic optimization algorithm for a two-echelon (R, Q) inventory system. International Journal of Production Economics, 109(1-2), 195–213.

<span id='_Ref38381812'[8] Hnaien, F., Delorme, X., & Dolgui, A. (2010). Multi-objective optimization for inventory control in two-level assembly systems under uncertainty of lead times. Computers & Operations Research, 37(11), 1835–1843.

<span id='_Ref38381823'<span id='_Ref527702845'[9] Gicquel, C. , & Cheng, J. . (2018). A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2), 123-155.

<span id='_Ref38381827'[10] M. Talaei, M. B. Farhang, M. S.Pishvaee, et.al. “A robust fuzzy optimization model for carbon-efficient closed-loop supply chain network design problem: a numerical illustration in electronics industry,” Journal of Cleaner Production, vol. 113, pp. 662–673, 2016.

<span id='_Ref38381835'[11] Shang, C., You, F. . (2018). Distributionally robust optimization for planning and scheduling under uncertainty. Computers & Chemical Engineering, 110(FEB.2), 53-68.

<span id='_Ref38381847'[12] Gicquel, C., & Cheng, J. (2017). A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Annals of Operations Research, 264(1-2), 123–155</span>

<span id='_Ref38381866'[13] Shi, Y. , Boudouh, T. , & Grunder, O. . (2017). A Fuzzy Chance-constraint Programming Model for a Home Health Care Routing Problem with Fuzzy Demand. International Conference on Operations Research & Enterprise Systems.

<span id='_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.

• Corresponding author. Tel.: +86 18795428481; E-mail: xwzj0003@gmail.com