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Optimal crashing of an activity network with disruptions

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Abstract

In this paper, we consider an optimization problem involving crashing an activity network under a single disruption. A disruption is an event whose magnitude and timing are random. When a disruption occurs, the duration of an activity that has yet to start—or alternatively, yet to complete—can change. We formulate a two-stage stochastic mixed-integer program, in which the timing of the stage is random. In our model, the recourse problem is a mixed-integer program. We prove the problem is NP-hard, and using simple examples, we illustrate properties that differ from the problem’s deterministic counterpart. Obtaining a reasonably tight optimality gap can require a large number of samples in a sample average approximation, leading to large-scale instances that are computationally expensive to solve. Therefore, we develop a branch-and-cut decomposition algorithm, in which spatial branching of the first stage continuous variables and linear programming approximations for the recourse problem are sequentially tightened. We test our decomposition algorithm with multiple improvements and show it can significantly reduce solution time over solving the problem directly.

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Abbreviations

I :

Set of activities;

\(J_i\) :

Set of crashing options for activity \(i \in I\);

\(\varOmega \) :

Index set for disruption scenarios (sample space);

\(\mathcal{A}\) :

Set of arcs, which represents precedence relationships;

\(D_{ik}\) :

Nominal duration between possible start times of activities i and k, \((i,k) \in \mathcal{A}\);

\(e_{ij}\) :

Effectiveness of crashing option \(j \in J_i\) for activity \(i \in I\);

B :

Total budget for crashing options;

\(b_{ij}\) :

Cost of crashing option \(j \in J_i\) for activity \(i \in I\);

\(\bar{x}_{ij}\) :

Upper bound on crashing option \(j \in J_i\) for activity \(i \in I\) with \(\bar{x}_{ij} \le 1\);

\(H^\omega \) :

Disruption time under scenario \(\omega \in \varOmega \);

\(d_{ik}^\omega \) :

Increase in duration of \((i,k) \in \mathcal{A}\) under \(\omega \in \varOmega \), if started after the disruption;

\(p^\omega \) :

The probability of scenario \(\omega \in \varOmega \);

\(p^0\) :

The probability of no disruption;

\(t_{i}\) :

Continuous nominal start time of activity \(i \in I\);

\(x_{ij}\) :

Continuous nominal crashing of activity \(i \in I\) by option \(j \in J_i\);

\(t_{i}^\omega \) :

Continuous start time of activity \(i \in I\) under scenario \(\omega \in \varOmega \);

\(x_{ij}^\omega \) :

Continuous crashing of activity \(i \in I\) by option \(j \in J_i\) under scenario \(\omega \in \varOmega \);

\(G_i^\omega \) :

Binary indicator whether activity \(i \in I\) starts after disruption under \(\omega \in \varOmega \);

\(z_{ij}^\omega \) :

Continuous term to linearize bilinear term, \(G_i^\omega x_{ij}^\omega \), \(i \in I, j \in J_{i}, \omega \in \varOmega \).

References

  1. Aghaie, A., Mokhtari, H.: Ant colony optimization algorithm for stochastic project crashing problem in PERT networks using MC simulation. Int. J. Adv. Manuf. Technol. 45(11), 1051–1067 (2009)

    Article  Google Scholar 

  2. Ahipasaoglu, S.D., Natarajan, K., Shi, D.: Distributionally robust project crashing with partial or no correlation information. Optimization-Online (2016). http://www.optimization-online.org/DB_FILE/2016/11/5715.pdf

  3. Belotti, P., Cafieri, S., Lee, J., Liberti, L.: On feasibility based bounds tightening. Optimization-Online (2012). http://www.optimization-online.org/DB_FILE/2012/01/3325.pdf

  4. Birge, J.R., Louveaux, F.V.: A multicut algorithm for two-stage stochastic linear programs. Eur. J. Oper. Res. 34, 384–392 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blazewicz, J., Lenstra, J.K., Kan, A.H.G.R.: Scheduling subject to resource constraints: classification and complexity. Discrete Appl. Math. 5(1), 11–24 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bowman, R.A.: Stochastic gradient-based time-cost tradeoffs in PERT networks using simulation. Ann. Oper. Res. 53(1), 533–551 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burt, J.M., Garman, M.B.: Conditional Monte Carlo: a simulation technique for stochastic network analysis. Manag. Sci. 18(3), 207–217 (1971)

    Article  MATH  Google Scholar 

  8. Camm, J.D., Raturi, A.S., Tsubakitani, S.: Cutting big M down to size. Interfaces 20(5), 61–66 (1990)

    Article  Google Scholar 

  9. Cardoen, B., Demeulemeester, E., Beliën, J.: Operating room planning and scheduling: a literature review. Eur. J. Oper. Res. 201(3), 921–932 (2010)

    Article  MATH  Google Scholar 

  10. Carøe, C., Tind, J.: L-shaped decomposition of two-stage stochastic programs with integer recourse. Math. Program. 83(1–3), 451–464 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, B., Küçükyavuz, S., Sen, S.: A computational study of the cutting plane tree algorithm for general mixed-integer linear programs. Oper. Res. Lett. 40(1), 15–19 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, X., Sim, M., Sun, P., Zhang, J.: A linear decision-based approximation approach to stochastic programming. Oper. Res. 56(2), 344–357 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Coffrin, C., Hijazi, H.L., Van Hentenryck, P.: Strengthening convex relaxations with bound tightening for power network optimization. In: International Conference on Principles and Practice of Constraint Programming, pp. 39–57. Springer, Berlin (2015)

  14. Cohen, I., Golany, B., Shtub, A.: The stochastic time-cost tradeoff problem: a robust optimization approach. Networks 49(2), 175–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Crow, E.L., Shimizu, K.: Lognormal Distributions: Theory and Applications. CRC Press, Amsterdam (1987)

    MATH  Google Scholar 

  16. De, P., Dunne, E.J., Ghosh, J.B., Wells, C.E.: Complexity of the discrete time-cost tradeoff problem for project networks. Oper. Res. 45(2), 302–306 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Demeulemeester, E., Herroelen, W.S.: Project Scheduling: A Research Handbook, vol. 49. Springer, Berlin (2006)

  18. Demeulemeester, E., Vanhoucke, M., Herroelen, W.: RanGen: a random network generator for activity-on-the-node networks. J. Sched. 6(1), 17–38 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dunning, I., Huchette, J., Lubin, M.: JuMP: a modeling language for mathematical optimization. SIAM Rev. 59(2), 295–320 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Elmaghraby, S.E.: Activity Networks: Project Planning and Control by Network Models. Wiley, London (1977)

    MATH  Google Scholar 

  21. Fulkerson, D.R.: A network flow computation for project cost curves. Manag. Sci. 7(2), 167–178 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gade, D., Küçükyavuz, S., Sen, S.: Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs. Math. Program. 144(1–2), 39–64 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co. (1979)

  24. Gurobi Optimization, Inc.: Gurobi Optimizer Reference Manual (2016). http://www.gurobi.com

  25. Hall, N.G., Sriskandarajah, C.: A survey of machine scheduling problems with blocking and no-wait in process. Oper. Res. 44(3), 510–525 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hartmann, S., Kolisch, R.: Experimental evaluation of state-of-the-art heuristics for the resource-constrained project scheduling problem. Eur. J. Oper. Res. 127(2), 394–407 (2000)

    Article  MATH  Google Scholar 

  27. Hellemo, L., Barton, P.I., Tomasgard, A.: Decision-dependent probabilities in stochastic programs with recourse. CMS 15(3–4), 369–395 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jaselskis, E.J., Ashley, D.B.: Optimal allocation of project management resources for achieving success. J. Constr. Eng. Manag. 117(2), 321–340 (1991)

    Article  Google Scholar 

  29. Ke, H.: A genetic algorithm-based optimizing approach for project time-cost trade-off with uncertain measure. J. Uncertain. Anal. Appl. 2(1), 8 (2014)

    Article  Google Scholar 

  30. Kelly, J.E.: Critical-path planning and scheduling: mathematical basis. Oper. Res. 9(3), 296–320 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kim, S., Boyd, S.P., Yun, S., Patil, D.D., Horowitz, M.A.: A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing. Optim. Eng. 8(4), 397–430 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kim, S., Pasupathy, R., Henderson, S.G.: A guide to sample average approximation. In: Handbook of Simulation Optimization, pp. 207–243. Springer (2015)

  33. Klotz, E., Newman, A.M.: Practical guidelines for solving difficult mixed integer linear programs. Surv. Oper. Res. Manag. Sci. 18(1–2), 18–32 (2013)

    MathSciNet  Google Scholar 

  34. Kuhl, M.E., Tolentino-Peña, R.A.: A dynamic crashing method for project management using simulation-based optimization. In: Proceedings of the 40th Conference on Winter Simulation, pp. 2370–2376. Winter Simulation Conference (2008)

  35. Lamas, P., Demeulemeester, E.: A purely proactive scheduling procedure for the resource-constrained project scheduling problem with stochastic activity durations. J. Sched. 19(4), 409–428 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, Z., Ierapetritou, M.: Process scheduling under uncertainty: review and challenges. Comput. Chem. Eng. 32(4–5), 715–727 (2008)

    Article  Google Scholar 

  37. Magnanti, T.L., Wong, R.T.: Accelerating Benders decomposition: algorithmic enhancement and model selection criteria. Oper. Res. 29(3), 464–484 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mak, W.K., Morton, D.P., Wood, R.K.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24(1), 47–56 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Malcolm, D.G., Roseboom, J.H., Clark, C.E., Fazar, W.: Application of a technique for research and development program evaluation. Oper. Res. 7(5), 646–669 (1959)

    Article  MATH  Google Scholar 

  40. Möhring, R.H., Stork, F.: Linear preselective policies for stochastic project scheduling. Math. Methods Oper. Res. 52(3), 501–515 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mullen, R.E.: The lognormal distribution of software failure rates: origin and evidence. In: Proceedings Ninth International Symposium on Software Reliability Engineering (Cat. No. 98TB100257), pp. 124–133 (1998)

  42. Naderi, B., Zandieh, M., Fatemi Ghomi, S.M.T.: Scheduling job shop problems with sequence-dependent setup times. Int. J. Prod. Res. 47(21), 5959–5976 (2009)

  43. Oberlender, G.D.: Project Management for Engineering and Construction, vol. 2. McGraw-Hill, New York (1993)

  44. Philpott, A.B., Wahid, F., Bonnans, J.F.: MIDAS: a mixed integer dynamic approximation scheme. Math. Program. 181(1), 19–50 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. Plambeck, E.L., Fu, B., Robinson, S.M., Suri, R.: Sample-path optimization of convex stochastic performance functions. Math. Program. 75(2), 137–176 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  46. Project Management Institute: A guide to the project management body of knowledge: PMBOK guide. Project Management Institute, Newtown Square, Pennsylvania (2017)

  47. Qi, Y., Sen, S.: The ancestral Benders’ cutting plane algorithm with multi-term disjunctions for mixed-integer recourse decisions in stochastic programming. Math. Program. 161(1–2), 193–235 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ross, S.M.: Stochastic Processes, vol. 2. Wiley, New York (1996)

    MATH  Google Scholar 

  49. Salmerón, J., Wood, R.K., Morton, D.P.: A stochastic program for optimizing military sealift subject to attack. Military Oper. Res. 14(2), 19–39 (2009)

    Article  Google Scholar 

  50. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM (2009)

  51. Söderlund, J.: Building theories of project management: past research, questions for the future. Int. J. Project Manag. 22(3), 183–191 (2004)

    Article  Google Scholar 

  52. Srinivasan, S., Brooks, J.P., Wilson, J.H.: Batching-based approaches for optimized packing of jobs in the spatial scheduling problem, pp. 243–263. Springer, Cham (2015)

  53. Sundar, K., Nagarajan, H., Misra, S., Lu, M., Coffrin, C., Bent, R.: Optimization-based bound tightening using a strengthened QC-relaxation of the optimal power flow problem. arXiv preprint arXiv:1809.04565 (2018)

  54. Tonchia, S.: Industrial Project Management. Springer, Berlin (2018)

  55. van Slyke, R.M.: Letter to the editor: Monte Carlo methods and the PERT problem. Oper. Res. 11(5), 839–860 (1963)

  56. Wiesemann, W., Kuhn, D., Rustem, B.: Robust resource allocations in temporal networks. Math. Program. 135(1), 437–471 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  57. Yang, H., Nagarajan, H.: Optimal power flow in distribution networks under stochastic N-1 disruptions. Electr. Power Syst. Res. 189, 106689 (2020)

  58. Yu, G., Qi, X.: Disruption Management: Framework, Models and Applications. World Scientific (2004)

  59. Yuan, W., Wang, J., Qiu, F., Chen, C., Kang, C., Zeng, B.: Robust optimization-based resilient distribution network planning against natural disasters. IEEE Trans. Smart Grid 7(6), 2817–2826 (2016)

    Article  Google Scholar 

  60. Zou, J., Ahmed, S., Sun, X.A.: Stochastic dual dynamic integer programming. Math. Program. 175(1–2), 461–502 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research is based upon work supported, in part, by the U.S. Department of Homeland Security under Grant Award Number, 2017-ST-061-QA0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Homeland Security. The Center for Nonlinear Studies at Los Alamos National Laboratory supported Haoxiang Yang’s work. The authors thank two anonymous referees and an associate editor, along with the area editor Dr. Alper Atamtürk, for comments and suggestions that improved the paper.

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Appendices

Appendix

Model when disruption affects activities yet to end

Model (2) assumes that activities that have started are not affected by the disruption, even if they have yet to end. That is, the disruption only affects activities that have not started. Here, we formulate a model under the complementary assumption that the disruption affects activities that have yet to end. To do so, we model the end time of activities as the decision variables denoted by \(t\) and \(t^\omega \). In addition, variable \(G_i^\omega \) indicates whether activity i finishes after the disruption in scenario \(\omega \). In this situation, we further assume that the crashing decisions can be adjusted before the activity ends.

To obtain comparable computational results, we assume that durations \(D_i\) and \(d_i^\omega \) only depend on the activity, and not the precedence relationship, which is the same assumption made in the computational experiments in the main text. For simplicity we formulate the model under this assumption.

$$\begin{aligned} z^* = \min \quad&p^0 t_T + \sum _{\omega \in \varOmega } p^\omega t_T^\omega \end{aligned}$$
(35a)
$$\begin{aligned} \text {s.t.} \quad&t_k - t_i \ge D_k \left( 1 - \sum _{j \in J_k} e_{kj} x_{kj} \right)&\forall \, (i,k) \in \mathcal{A}\end{aligned}$$
(35b)
$$\begin{aligned}&\sum _{i \in I} \sum _{j \in J_i} b_{ij}x_{ij} \le B&\end{aligned}$$
(35c)
$$\begin{aligned}&\sum _{j \in J_i} x_{ij} \le 1&\forall \,i \in I\end{aligned}$$
(35d)
$$\begin{aligned}&H^\omega + M G_i^\omega \ge t_i&\forall \,i \in I, \omega \in \varOmega \end{aligned}$$
(35e)
$$\begin{aligned}&H^\omega - M (1 - G_i^\omega ) \le t_i&\forall \,i \in I, \omega \in \varOmega \end{aligned}$$
(35f)
$$\begin{aligned}&t_i^\omega + {M'} G_i^\omega \ge t_i&\forall \,i \in I, \omega \in \varOmega \end{aligned}$$
(35g)
$$\begin{aligned}&t_i^\omega - {M'} G_i^\omega \le t_i&\forall \,i \in I, \omega \in \varOmega \end{aligned}$$
(35h)
$$\begin{aligned}&x_{ij}^\omega + {\bar{x}_{ij}} G_i^\omega \ge x_{ij}&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35i)
$$\begin{aligned}&x_{ij}^\omega - {\bar{x}_{ij}} G_i^\omega \le x_{ij}&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35j)
$$\begin{aligned}&t_k^\omega - t_i^\omega \ge D_{k} + d_{k}^\omega G_k^\omega \nonumber \\&\quad - \sum _{j \in J_k} D_{k} e_{kj} x_{kj}^\omega - \sum _{j \in J_k} d_{k}^\omega e_{kj} z_{kj}^\omega&\forall \,(i,k) \in \mathcal{A}, \omega \in \varOmega \end{aligned}$$
(35k)
$$\begin{aligned}&\sum _{i \in I}\sum _{j \in J_i} b_{ij}x_{ij}^\omega \le B&\forall \,\omega \in \varOmega \end{aligned}$$
(35l)
$$\begin{aligned}&\sum _{j \in J_i} x_{ij}^\omega \le 1&\forall \,i \in I, \omega \in \varOmega \end{aligned}$$
(35m)
$$\begin{aligned}&z_{ij}^\omega \le {\bar{x}_{ij}} G_i^\omega&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35n)
$$\begin{aligned}&z_{ij}^\omega \le x_{ij}^\omega&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35o)
$$\begin{aligned}&z_{ij}^\omega \ge x_{ij}^\omega + {\bar{x}_{ij}} (G_i^\omega - 1)&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35p)
$$\begin{aligned}&t_i \ge 0&\forall \,i \in I \end{aligned}$$
(35q)
$$\begin{aligned}&t_i^\omega \ge H^\omega G_i^\omega&\forall \, i \in I, \omega \in \varOmega \end{aligned}$$
(35r)
$$\begin{aligned}&0 \le x_{ij} \le {\bar{x}_{ij}}&\forall \,i \in I, j \in J_i\end{aligned}$$
(35s)
$$\begin{aligned}&0 \le x_{ij}^\omega \le {\bar{x}_{ij}}&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35t)
$$\begin{aligned}&0 \le z_{ij}^\omega \le 1&\forall \,i \in I, j \in J_i, \omega \in \varOmega \end{aligned}$$
(35u)
$$\begin{aligned}&G_i^\omega \in \{0,1\}.&\forall \,i \in I, \omega \in \varOmega . \end{aligned}$$
(35v)

The bulk of the model mirrors that of model (2), and so we do not repeat the corresponding narrative. Constraints (35b) and (35k) differ due to the altered definition of the end-time decision variables: the duration between the end time of activity \(i\) and \(k\), if \(i\) precedes \(k\), is the duration of activity \(k\), instead of that of activity \(i\) in model (2). Constraint (35r) ensures that if the disruption occurs during an activity, the crashing decision cannot cause the activity to end before the disruption.

Next, we present the effect of this altered assumption on the optimal values, i.e., the expected completion times of the projects, in the first two rows of Table 6 using \(|\varOmega | = 500\). We also test how well the optimal solution to model (2) performs in model (35), with an optimality gap that ranges from 5% to 20%; see the final row of the table. This suggests the importance of selecting the most realistic assumption when formulating the model.

The decomposition algorithm can still be used to solve model (35) with necessary, but minimal, changes in the recourse problem formulation. We observe that changing the assumption does not significantly change the required computational effort.

Table 6 The first two numerical rows show the optimal values of models (2) and (35), with sample size 500

While we do not explicitly write the model here, it is possible to partition activities into two groups that correspond to the respective assumptions of models (2) and (35) regarding whether in-process activities are affected by a disruption. This can be done by defining both start and end time variables for each activity in the same model, and enforcing the nonanticipativity constraints differently for the two types of activities.

Test cases data

We present the data of four test cases here. For all test cases we assume there is only one possible crashing option for each activity. The option consumes 1 unit of resource and has the effectiveness parameter of \(e_{i1} = 0.5\) for all \(i \in I\). The nominal scenario probability is \(p^0 = 0.1\) and \(p^\omega = \frac{1 - p^0}{|\varOmega |}\). The timing of the disruption follows a lognormal distribution with parameters \(\mu \) and \(\sigma \) where the mode is \(e^{\mu - \sigma ^2}\). We also assume that the duration only depends on the predecessor, i.e., \(D_{ik} = D_i\) and \(d_{ik}^\omega = d_i^\omega \). All \(d_i\) follow an exponential distribution with a mean of \(\mu _i\). The values of \(D_i\) and \(\mu _i\) for the four cases are shown in Tables 7, 8, 9, and 10. Figures 10, 11, 12, and 13 show the four activity networks, and the crashing budget, B, and lognormal parameters are enumerated as follows:

  • Case 11: \(B = 3, \mu = \ln 6, \sigma = 0.5\)

  • Case 14: \(B = 4, \mu = \ln 35, \sigma = 0.5\)

  • Case 19: \(B = 4, \mu = \ln 8, \sigma = 0.5\)

  • Case 35: \(B = 8, \mu = \ln 4, \sigma = 0.3\)

Fig. 10
figure 10

Activity network of Case 11

Table 7 Activity duration \(D_i\) and the mean of disruption magnitude \(\mu _i\) for Case 11
Fig. 11
figure 11

Activity network of Case 14

Table 8 Activity duration \(D_i\) and the mean of disruption magnitude \(\mu _i\) for Case 14
Fig. 12
figure 12

Activity network of Case 19

Table 9 Activity duration \(D_i\) and the mean of disruption magnitude \(\mu _i\) for Case 19
Fig. 13
figure 13

Activity network of Case 35

Table 10 Activity duration \(D_i\) and the mean of disruption magnitude \(\mu _i\) for Case 35

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Yang, H., Morton, D.P. Optimal crashing of an activity network with disruptions. Math. Program. 194, 1113–1162 (2022). https://doi.org/10.1007/s10107-021-01670-x

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