Abstract
A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis.
This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We concentrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, subject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN optimization problem is much more difficult; the few existing methods cannot handle large-scale problems.
In this paper we introduce a heuristic method for approximately optimizing a SAN, by forming a related DAN optimization problem which includes extra margins in each of the activity durations to account for the variation. Since the method is based on optimizing a DAN, it readily handles large-scale problems. To assess the quality of the resulting suboptimal designs, we describe two widely applicable lower bounds on achievable performance in optimal SAN design.
We demonstrate the method on a simplified statistical digital circuit sizing problem, in which the device widths affect both the mean and variance of the gate delays. Numerical experiments show that the resulting design is often substantially better than one in which the variation in delay is ignored, and is often quite close to the global optimum (as verified by the lower bounds).
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References
Abdel-Malek H, Bandler J (1980a) Yield optimization for arbitrary statistical distributions: part I—theory. IEEE Trans Circuits Syst 27(4):245–253
Abdel-Malek H, Bandler J (1980b) Yield optimization for arbitrary statistical distributions: part II—implementation. IEEE Trans Circuits Syst 27(4):253–262
Abramowitz M, Stegun I (eds) (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Wiley, New York
Agarwal A, Zolotov V, Blaauw D (2003) Statistical timing analysis using bounds and selective enumeration. IEEE Trans Comput Aided Des Integr Circuits Syst 22(9):1243–1260
Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805
Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization. Analysis, algorithms, and engineering applications. SIAM, Philadelphia
Bertsekas D (1999) Nonlinear programming, 2nd edn. Athena Scientific
Bertsekas D, Nedic A, Ozdaglar A (2003) Convex analysis and optimization. Athena Scientific
Bertsimas D, Thiele A (2006) A robust optimization approach to supply chain management. Math Program Ser B 54(1):150–168
Bertsimas D, Natarajan K, Teo C-P (2004) Probabilistic combinatorial optimization: Moments, semidefinite programming and asymptotic bounds. SIAM J Optim 15(1):185–209
Bhardwaj S, Vrudhula S, Blaauw D (2003) TAU: timing analysis under uncertainty. In: International conference on computer-aided design, San Jose, pp 615–620, November 2003
Birge J, Maddox M (1995) Bounds on expected project tardiness. Oper Res 43:838–850
Bowman R (1995) Efficient estimation of arc criticalities in stochastic activity networks. Manag Sci 41(1):58–67
Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge
Boyd S, Kim S-J, Patil D, Horowitz M (2005) Digital circuit optimization via geometric programming. Oper Res 53(6):899–932
Boyd S, Kim S-J, Vandenberghe L, Hassibi A (2007) A tutorial on geometric programming. Optim Eng (to appear). Available from www.stanford.edu/boyd/~gp_tutorial.html
Bozorgzadeh E, Ghiasi S, Takahashi A, Sarrafzadeh M (2004) Optimal integer delay budget assignment on directed acyclic graphs. IEEE Trans Comput Aided Des Integr Circuits Syst 23(8):1184–1199
Burt J, Garman M (1971) Conditional Monte Carlo: A simulation technique for stochastic network analysis. Manag Sci 18(1):207–217
Charnes A, Cooper W, Thompson G (1964) Critical path analyses via chance constrained and stochastic programming. Oper Res 12:460–470
Conn A, Gould N, Toint PhL (1992) LANCELOT: a Fortran package for large-scale nonlinear optimization (release A). Springer series in computational mathematics, vol 17. Springer, Berlin
Conn A, Coulman P, Haring R, Morrill G, Visweswariah C, Wu C (1998) JiffyTune: Circuit optimization using time-domain sensitivities. IEEE Trans Comput Aided Des Integr Circuits Syst 17(12):1292–1309
Davis E (1966) Resource allocation in project network models—a survey. J Ind Eng 17(4):177–187
Devroye L (1979) Inequalities for the completion times of stochastic PERT networks. Math Oper Res 4(4):441–447
Dodin B (1984) Determining the k most critical paths in PERT networks. Oper Res 32:859–877
Dodin B (1985) Bounding the project completion time distribution in PERT networks. Oper Res 33:862–881
Dodin B, Elmaghraby S (1985) Approximating the criticality indices of the activities in PERT networks. Manag Sci 31(2):207–223
Elmaghraby S (1977) Project planning and control by network models. Wiley, New York
Eppstein D (1998) Finding the k shortest paths. SIAM J Comput 28(2):652–673
Esary J, Proschan F, Walkup D (1967) Association of random variables with applications. Ann Math Stat 38:71466–1474
Fishburn J, Dunlop A (1985) TILOS: A posynomial programming approach to transistor sizing. In: IEEE international conference on computer-aided design: ICCAD-85. Digest of technical papers, IEEE Computer Society Press, pp 326–328
Fox B (1994) Integrating and accelerating tabu search. Ann Oper Res 41:47–67
El Ghaoui L, Lebret H (1997) Robust solutions to least-squares problems with uncertain data. SIAM J Matrix Anal Appl 18(4):1035–1064
Glasserman P (2003) Monte Carlo methods in financial engineering. Springer, Berlin
Hagstrom J, Jane N (1990) Computing the probability distribution of project duration in a PERT network. Networks 20:231–244
Heller U (1981) On the shortest overall duration in stochastic acyclic network. Methods Oper Res 42:85–104
Hongliang C, Sapatnekar S (2003) Statistical timing analysis considering spatial correlations using a single PERT-like traversal. In: International conference on computer-aided design, San Jose, pp 621–625, November 2003
Hsiung K-L, Kim S-J, Boyd S (2005) Robust geometric programming via piecewise linear approximation. Manuscript, September 2005
Jess J, Kalafala K, Naidu S, Otten R, Visweswariah C (2003) Statistical timing for parametric yield prediction of digital integrated circuits. In: Proc. of 40th proc. IEEE/ACM design automation conference, Anaheim, pp 343–347, June 2003
Knowles S (1999) A family of adders. In: Proceedings of 14th IEEE symposium on computer arithmetic, IEEE Computer Society Press, pp 30–34
Liou J-J, Krstić A, Jiang Y-M, Cheng K-T (2003) Modeling, testing, and analysis for delay defects and noise effects in deep submicron devices. IEEE Trans Comput Aided Des Integr Circuits Syst 22(6):756–769
Ludwig A, Möhring R, Stork F (2001) A computational study on bounding the makespan distribution in stochastic project networks. Ann Oper Res 102:49–64
Ma S, Keshavarzi A, De V, Brews R (2004) A statistical model for extracting geometric sources of transistor performance variation. IEEE Trans Electron Dev 51(1):36–41
Mejilson I, Nadas A (1979) Convex majorization with an application to the length of critical path. J Appl Probab 16:671–677
Mingozzi A, Maniezzo V, Ricciardelli S, Bianco L (1998) An exact algorithm for the resource constrained project scheduling problem based on a new mathematical formulation. Manag Sci 44:714–729
MOSEK ApS (2002). The MOSEK optimization tools version 2.5. User’s manual and reference. Available from www.mosek.com
Mutapcic A, Koh K, Kim S-J, Vandenberghe L, Boyd S (2005) GGPLAB: a simple matlab toolbox for geometric programming. Available from www.stanford.edu/~boyd/ggplab/
Nesterov Y (2003) Introductory lectures on convex optimization: a basic course. Kluwer Academic, Boston
Nesterov Y, Nemirovsky A (1994) Interior-point polynomial methods in convex programming. Studies in applied mathematics, vol 13. SIAM, Philadelphia
Neumaier A (1998) Solving ill-conditioned and singular linear systems: A tutorial on regularization. SIAM Rev 40(3):636–666
Nocedal J, Wright S (1999) Numerical optimization. Springer series in operations research. Springer, New York
Okada K, Yamaoka K, Onodera H (2003) A statistical gate delay model for intra-chip and inter-chip variabilities. In: Proc. of 40th IEEE/ACM proc. design automation conference, Anaheim, pp 31–36, June 2003
Orshansky M, Keutzer K (2002) A general probabilistic framework for worst case timing analysis. In: Proc. of 39th IEEE/ACM design automation conference, New Orleans, pp 556–561, June 2002
Orshansky M, Chen J, Hu C (1999) Direct sampling methodology for statistical analysis of scaled CMOS technologies. IEEE Trans Semicond Manuf 12(4):403–408
Orshansky M, Milor L, Chen P, Keutzer K, Hu C (2002) Impact of spatial intrachip gate length variability on the performance of high-speed digital circuits. IEEE Trans Comput Aided Des Integr Circuits Syst 21(5):544–553
Orshansky M, Milor L, Hu C (2004) Characterization of spatial intrafield gate CD variability, its impact on circuit performance and spatial mask-level correction. IEEE Trans Semicond Manuf 17(1):2–11
Patil D, Yun Y, Kim S-J, Boyd S, Horowitz M (2004) A new method for robust design of digital circuits. In: Proceedings of the sixth international symposium on quality electronic design (ISQED) 2005, IEEE Computer Society Press, pp 676–681
Pelgrom M (1989) Matching properties of MOS transistors. IEEE J Solid State Circuits 24(5):1433–1439
Pich M, Loch C, De Meyer A (2002) On uncertainty, ambiguity, and complexity in project management. Manag Sci 48(8):1008–1023
Plambeck E, Fu B-R, Robinson S, Suri R (1996) Sample-path optimization of convex stochastic performance functions. Math Program 75(2):137–176
Pollalis S (1993) Computer-aided project management: a visual scheduling and control system. Vieweg Verlag, Wiesbaden
Prekopa A (1983) Stochastic programming. Kluwer Academic, Dordrecht
Robillard P, Trahan M (1976) The completion times of PERT networks. Oper Res 25:15–29
Robinson J (1979) Some analysis techniques for asynchronous multiprocessor algorithms. IEEE Trans Softw Eng 5(1):24–31
Rockafellar R, Uryasev S (2000) Optimization of conditional value-at-risk criterion. J Risk 2(3):21–41
Sapatnekar S (1996) Wire sizing as a convex optimization problem: exploring the area-delay tradeoff. IEEE Trans Comput Aided Des 15:1001–1011
Sapatnekar S (2000) Power-delay optimization in gate sizing. ACM Trans Des Autom Electron Syst 5(1):98–114
Sapatnekar S, Rao V, Vaidya P, Kang S (1993) An exact solution to the transistor sizing problem for CMOS circuits using convex optimization. IEEE Trans Comput Aided Des Integr Circuits Syst 12(11):1621–1634
Serfling R (1980) Approximation theorems of mathematical statistics. Wiley, New York
Shogan A (1977) Bounding distributions for a stochastic PERT network. Networks 7:359–381
Slowinski R (1980) Two approaches to problems of resource allocation among project activities: a comparative study. J Oper Res Soc 31:711–723
Van Slyke R (1963) Monte Carlo methods and the PERT problem. Oper Res 11:839–860
Van Slyke R, Wets R (1969) L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J Appl Math 17:638–663
Styblinski M, Opalski L (1986) Algorithms and software tools for IC yield optimization based on fundamental fabrication parameters. IEEE Trans Comput Aided Des 5(1):79–89
Sullivan R, Hayya J (1980) A comparison of the method of bounding distributions (MBD) and Monte Carlo simulation for analyzing stochastic acyclic networks. Oper Res 28(3):614–617
Sutherland I, Sproul B, Harris D (1999) Logical effort: designing fast CMOS circuits. Kaufmann, San Francisco
van der Vaart A (1998) Asymptotics statistics. Cambridge University Press, Cambridge
Vanderbei R (1992) LOQO user’s manual. Technical Report SOL 92–05, Dept. of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA
Visweswariah C (2003) Death, taxes and failing chips. In: Proc. of 40th IEEE/ACM design automation conference, Anaheim, pp 343–347, June 2003
Wallace S (1989) Bounding the expected time-cost curve for a stochastic PERT network from below. Oper Res 8:89–94
Weiss G (1986) Stochastic bounds on distributions of optimal value functions with applications to PERT, network flows and reliability. Oper Res 34(4):595–605
White K, Trybula W, Athay R (1997) Design for semiconductor manufacturing-perspective. IEEE Trans Compon Packag Manuf Technol Part C 20(1):58–72
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Kim, SJ., Boyd, S.P., Yun, S. et al. A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing. Optim Eng 8, 397–430 (2007). https://doi.org/10.1007/s11081-007-9011-5
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DOI: https://doi.org/10.1007/s11081-007-9011-5