Optimal computing budget allocation for constrained optimization

OPTIMAL COMPUTING BUDGET ALLOCATION FOR CONSTRAINED OPTIMIZATION NUGROHO ARTADI PUJOWIDIANTO NATIONAL UNIVERSITY OF SINGAPORE 2012 OPTIMAL COMPUTING BUDGET ALLOCATION FOR CONSTRAINED OPTIMIZATION NUGROHO ARTADI PUJOWIDIANTO (B. Eng. (Hons.), NTU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. ______________________ Nugroho Artadi Pujowidianto February 2013 Acknowledgments I would like to thank my supervisors, Dr. Yap Chee Meng and Associate Professor Lee Loo Hay for their patient guidance throughout the research journey. In addition, I am grateful for the advices given by Professor Chen Chun-Hung and Associate Professor Raghu Pasupathy. I would also like to thank my Oral Qualifying Examiners, Associate Professor Chew Ek Peng and Associate Professor Ng Szu Hui for their valuable comments and suggestions during the proposal of the thesis. I am grateful to my Oral Defence committee members, Dr. Michel-Alexandre Cardin, Associate Professor Ng Szu Hui, Associate Professor Ng Kien Ming, and the External Examiner. I also appreciate the support given by Professor Tang Loon Ching and Associate Professor Poh Kim Leng and other faculty members of the Department of Industrial and Systems Engineering (ISE). My gratitude also goes towards my co-authors, Dr. Susan R. Hunter and Mr. Li Lingwei. The journey of the research has been made manageable by the Maritime Logistics and Supply Chain Research groups where I learned a lot from fellow students especially from those working on simulation optimization, Li Juxin, Zhang Si, Xiao Hui, and Hu Xiang. I would also thank my Indonesian seniors, Dr. Budi Hartono, Dr. Nur Aini Masruroh, Dr. Markus Hartono, Dr. Hendry Rahardjo, Dr. Aldy Gunawan for their guidance when they were working their doctorates. I am heavily indebted by the assistance from the Registrar’s Office, the Engineering Faculty, and ISE Department particularly Ms Ow Lai Chun, Mdm. Tan Swee Lan, Mr. Lau Pak Kai, and Ms Celine Neo who have supported me in throughout my study in ISE. The seminars provided by both ISE Department and the Department of Decision Sciences i have significantly opened my eyes for high quality research around the world. It has been a great opportunity to experience the hospital settings, thanks to Singapore General Hospital’s staffs particularly Mr. Phua Tien Beng and Dr. Oh Hong Choon, Telogorejo Hospital’s staffs, National Healthcare Group’s researchers such as Mr. Teow Kiok Liang, and Dr. Samuel Ng who provide the opportunity to observe the radiology process in ParkwayHealth. I have been blessed by the great support from many friends such as Pr. Budianto Lim, Dr. Linda Bubod, Mr. Eugene Chong, Yopie Adrianto, Lim Lung Sen, Jefry Tedjokusumo, Dr. Jin Dayu, Nguyen Viet Anh, Freddy Wilyanto Suwandi, Dr. Albertus H. Adiwahono, Stephanie Budiman, Lee Jiun Horng, Felixen M. Wirahadisurya, and many more. I am also thankful for the Ministry of Education staffs and Pioneer Secondary School staffs for their support while I was finishing the thesis on part-time basis. I would like to thank my family for their continuous support. The education given by my parents, Dr. Handojo Pudjowidyanto and Dr. Lanny Indriastuti, and my sister’s, Irma Pujowidiyanto and her husband, Surya Lesmana, together with their love are invaluable. I am also deeply grateful to my girlfriend, Feliz Adrianne and her family for their understanding and unwavering encouragement. There are many times when I felt like quitting the study. It was my girlfriend’s listening ears, her cheerful support, inspiring encouragement, and her love that kept me going. Finally, I would like to thank God who has given me the strength to complete my thesis. The presence of so many supporting people in my life is the evidence of His continuous grace which makes all things possible. ii Table of Contents Acknowledgments i Table of Contents . iii Summary . vii List of Tables . viii List of Figures . x List of Symbols xi List of Abbreviations . xiii Chapter 1. Introduction . 1.1. Background 1.2. Motivation 1.3. Objective 1.4. Scope 1.5. Contribution . 1.6. Organization of the Thesis . Chapter 2. Literature Review 2.1. Simulation Optimization 2.2. Stochastic Constrained Optimization via Simulation . 12 2.3. Ranking and Selection 13 2.4. Constrained R&S 15 2.5. Optimal Computing Budget Allocation (OCBA) . 16 iii 2.6. Summary of the Research Gaps . 17 Chapter 3. Asymptotic Simulation Budget Allocation . 18 3.1. Overview 18 3.2. Stochastic Constrained Optimization via Simulation . 19 3.3. Computing Budget Allocation . 20 3.4. Assumptions . 21 3.5. The Problem Formulation using Bonferroni-bound . 23 3.6. Proposed Allocation . 25 3.6.2. Exact Solution . 27 3.6.3. Insights from the Allocation . 28 3.6.4. Closed-form Solution 29 3.7. Sequential Allocation Procedure 30 3.8. Numerical Experiments 32 3.8.1. Computing Budget Allocation Procedures . 32 3.8.2. Simulation Settings . 33 3.8.3. Experimental Results 34 3.9. The Effect of Correlation in Allocating Simulation Budget 38 Chapter 4. Explicit Consideration of Correlation Between Performance Measures in Simulation Budget Allocation . 39 4.1. The Problem Formulation using Large-Deviations Theory . 39 4.2. Exact Solution 43 iv 4.3. Closed-Form Expressions 45 4.3.1. Properties of the Rate Functions . 45 4.3.2. Properties of the Optimal Allocation 48 4.3.3. Closed-Form Approximation 49 4.3.4. Closed-Form Allocation to the Non-best Designs 54 4.4. Score Functions for Multivariate Normal Distribution 56 4.4.1. Score Functions when the Performance Measures are Independent to Each Other . 57 4.4.2. Score Functions in the Case of Correlated Performance Measures 60 4.5. Allocation to the Best Feasible Design 63 4.6. Numerical Examples 67 4.6.1. Convergence Rate Analysis 68 4.6.2. Finite-Time Performance 71 Chapter 5. Bed Allocation Problem 73 5.1. Motivation 73 5.2. System Description and Modeling . 77 5.3. Problem Description . 79 5.4. Efficient Procedure for Selecting the Best Feasible Bed Alternative 81 5.5. Computational Results and Analysis 83 5.5.1. Selection from a small number of alternatives . 83 5.5.2. Selection from a large number of alternatives 89 v Chapter 6. Conclusions and Future Research . 92 References . 96 Appendix A. Proof of Lemma 3.2 110 Appendix B. Proof of Theorem 3.1 112 Appendix C. The KKT conditions for problem (4.11) . 115 Appendix D. Finding The Solutions for (4.11) 116 Appendix E. Proof of Lemma 4.1 118 Appendix F. The KKT conditions for problem (4.48) . 120 Appendix G. Proof of Theorem 4.4 121 Appendix H. Proof of Theorem 4.5 122 vi Summary We consider the constrained optimization problem from a finite set of designs where their main objective and the constraint measures must be estimated via stochastic simulation. As simulation is time-consuming, the simulation budget needs to be efficiently allocated. This thesis proposes two procedures for determining the number of simulation replications for each design to maximize the probability of correct selection given a fixed computing budget. The first procedure asymptotically maximizes the lower bound of the probability of correct selection. The approximation is based on Bonferroni bounds which are applicable for the cases with independent and correlated performance measures. The second proposed procedure utilizes large deviations theory to derive an asymptotically optimal allocation which is able to explicitly account for the impact of the correlation among the multiple performance measures. As the number of the designs becomes large, the optimal allocation can be approximated by closed-form expressions which are simple and easy-to-implement. The numerical results show that the proposed procedures can enhance the simulation efficiency. An application example of the proposed procedure to a hospital bed allocation problem is also provided. 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Proceedings of the 2008 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 273–280. Teng, S., L. H. Lee, E. P. Chew. 2010. Integration of indifference-zone with multiobjective computing budget allocation. European Journal of Operational Research. 203 419-429. 108 Teow, K.L. and Tan, W.S. 2007. Hospital Beds Reallocation Using Mathematical Programming, in International Conference on Industrial Engineering and Systems Management, Beijing, China. Vieira Junior, H., Kienitz, K.H., Belderrain, M.C.N. 2011. Discrete-valued, stochasticconstrained simulation optimization with COMPASS. Proceedings of the 2011 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 4191-4200. Waeber, R., P. I. Frazier, and S. G. Henderson. 2010. Performance measures for ranking and selection procedures. Proceedings of the 2010 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, 1235-1245. World Health Organization. 2009. Health workforce, infrastructure, essential medicines, in World Health Statistics. Retrieved from http://www.who.int/whosis/whostat/EN_WHS09_Table6.pdf. Yan, D. and H. Mukai. 1992. Stochastic discrete optimization. SIAM Journal of Control and Optimization 30, 594-612. 109 Appendix A. Proof of Lemma 3.2 A function is concave if its Hessian matrix possible values of equation ̂ })] ( ( For {̂ ∑ ∑ } ). is thus concave if {̂ } )) for ( where (3.5), , ( i.e. is negative semi-definite for all ( ( ⁄√ [ {̂ ) . According to } )) for {̂ ( ( } ⁄√ {̂ ) for {̂ , and ̂} are all concave. ,( ( )), ( ( ( (A.1) )) ( ) ) (( )( ) )) The function so that ( ( ( ) for ⁄√ ⁄√ )) is concave because ( ) ⁄ ( ) ( ⁄√ ) . 110 The function ( ( ( ( ⁄√ ) ⁄ ) . Otherwise, required amount of ( ( so that ) ( , the condition become negative as function ( is concave as ( )) ⁄√ As ( )) for ( ( ⁄√ )) ) . is satisfied if should be increased and the term ( ) will ( ). The is a decreasing function of , i.e. to make ( )) is concave as ) is ( ) . Thus, the . This completes the proof that is asymptotically concave and the problem in (3.9) is an asymptotically convex optimization problem. ■ 111 Appendix B. Proof of Theorem 3.1 Based on (3.10) and (3.11), there are three possible cases in comparing two nonbest designs: and ; ; and . The derivation for the first case is provided while that of the other cases are similar. From (3.10) and (3.11): ( ) ( ) ( ⁄ Taking the log of (B.1) and dividing it by ) ⁄√ (B.1) ⁄ ⁄ on both sides: (B.2) ( ) ( ( ) ⁄ ( ⁄ As ) ⁄ ⁄ ) , the following is obtained which completes the proof of equation (3.14). ( ⁄ ⁄ (B.3) )⁄ ⁄ Based on the KKT conditions, the relationship between and for all is investigated. Equation (3.12) means that the number of replications to be allocated to the best feasible design should ideally consider its feasibility terms and its optimality 112 term from the comparison with the non-best designs in . However, we make a simplification due to the complexity of the analysis by using the term that dominates the others. Term dominates term and if when ( . As , the feasibility term related to ) is asymptotically much larger than the other feasibility terms. It is also much larger than the optimality term if (B.4) ( ⁄ ⁄ ) If the condition in (B.5) is fulfilled, only the feasibility term related to and the allocation to the best design based on the feasibility, ⁄ ⁄ ⁄ ( , if )⁄( ⁄ ) , is used can be found. Let )⁄ √( if or . Based on (3.10), (3.11), and the feasibility term in (3.2), the following can be obtained: (B.5) ⁄ ( ) Otherwise, only the optimality term is used. At least one term in ∑ ( ) ( ) ⁄ dominates ( to the optimality term in (3.12) yields relationship with the non-best designs in ⁄√ ) ⁄ ⁄ . Substituting (3.11) , the allocation to the best design based on its , 113 √∑ Substituting (B.5) and (B.6) into (B.4), the condition becomes (B.6) and so . This completes the proof of Theorem 3.1. 114 Appendix C. The KKT conditions for problem (4.11) ( ) (C.1) ) (C.2) ( [ ( ) ] (C.3) (C.4) ( ( ) ( ( ) (C.6) ) ) (C.5) (C.7) (C.8) 115 Appendix D. Finding The Solutions for (4.11) Let denote the -th row and -th column of matrix A. If the column vector of -th column from matrix A while , it represents is for the row vector of - th row from matrix A. ( ) , the value of For and system of linear equations with can be computed by solving the variables from ( ) as follows [ ( ) ] (( ( (D.1) )) ) from equations (C.1) and (C.2), ( ) [ For (D.2) ( ( ), ( ). ) as ( ) ( ( )) ) ) and ( ) ( ) while ( ) can be computed by solving the system and of linear equations with ( ] (( | ( )| ) variables from ( ) and ( ) ( ) as follows 116 [ ( [ ) ( ] (( ) ( ] (( (D.3) )) ) ( (D.4) )) ) ( ) ( ), For ( ). The value of can be computed by solving the system of linear equations with ( variables from ( ) [ ( ( ) and | ( )| ) as follows ) ] (( ( )) ) (D.5) ( ) from equations (C.1) and (C.2), ( ) [ (D.6) ( ) ] (( ( )) ) 117 Appendix E. Proof of Lemma 4.1 Given the optimal values of ( , ) and ( ) We are interested in finding the value of ( , equation (4.11) becomes ) ( ( )⁄ ( )⁄ . Based on (C.1) and (C.2), ( ) ( ) (E.2) [ ( ( ) (E.1) ) ] ) ( ) ( [ ( (E.3) ) ] ) Based on (D.1) to (D.6) and (E.1) to (E.3), Lemma 4.1 can be proven. For [ ( ) , ( ) ( ) [ ( ) and ( ) so that ( ( ) . As ] ) ] , and 118 ( ) ( ( ) ( that for and ) ( ] . Thus, ( ), ( ) ( ) . ( ) ( ) and ( ) ( ) . As ( ) ( ( ). For [ ( ( ) and ( ) ( ) ( ( ( ) ) ). Note , this implies that ( ) and ] ) where ] , ( ( ) ] ) ) ( . Consequently, ) ) [ ( ) as ( ( ) and ) ( ( ) . As ( ), For ) ( ). In addition, for all . Consequently, ) ( and ( ) , and as [ and ( . Thus, [ ( ) ( ), ) ). ( ), For ( ( ) . In addition, ( ) ( ) and . Thus, ) . This completes the proof of Lemma 4.1. 119 Appendix F. The KKT conditions for problem (4.48) ( ) [ ( ) , (F.1) ] ( ) (F.2) , (F.3) , ( ( (F.4) , ) ( ( ) , ) ) (F.5) (F.6) , . (F.7) 120 Appendix G. Proof of Theorem 4.4 The infimum in For , is achieved as follows: and . Based on (4.51) and (F.4), (G.1) For , , , and . Based on (4.51) and (F.5), , For , , , and , and . (G.2) . Based on (4.51), (F.4) and (F.5), (G.3) Thus, we can obtain (4.57) to (4.59) by substituting (G.1) to (G.3) into (4.48). ■ 121 Appendix H. Proof of Theorem 4.5 ( ), For ( ) from (F.4) as . Based on (4.60), we can obtain which are the solutions to the system of linear equations with ( ) variables resulting from ( ), For . ( ) from (F.5) as on (4.60), we can obtain ( ) ( ). Based ( ) which are the solutions to the system of and linear equations with ( ( ) | ( )| ) variables resulting from ( ) and ( ). ( ), For ( ) (F.5) as ( ) from (F.4) as ( ) from and ( ). Based on (4.60), we can obtain which are the solutions to the system of linear equations with | ( ) resulting from ( ) ( )| variables ( ). Based on (4.6) and (4.60), ( ( ), Therefore, for ( ), For ) ( ( ) . Similarly, ( ) for ( ) ( ( )[ ( ) ) ) ( ) as ) ( ( ) as ( ]. (H.1) ( ) ( ) ( ) ) as . ( ) and ( ) ( ). This completes the proof of (4.66) to (4.68). ■ 122 [...]... design , = the rate of approaching zero for a given set of xii List of Abbreviations AK+ = The procedure by Andradóttir and Kim (2010), CS = Correct Selection, EA = Equal Allocation, FS = False Selection, IZ = Indifference Zone, NP = Nested Partitions, OCBA = Optimal Computing Budget Allocation, OCBA-CO = Optimal Computing Budget Allocation for Constrained Optimization, = Probability of Correct Selection,... branch of R&S which aims to maximize given a fixed computing budget called as Optimal Computing Budget Allocation (OCBA) Although OCBA has been shown to be effective for the unconstrained optimization (Branke et al 2007), none of the research has incorporated the notion of OCBA for constrained optimization This motivates research in extending OCBA for handling the presence of stochastic constraints... index for designs, = number of stochastic constraints, = index for the performance measures, , for the main objective while for the constraint measures, = = = sample mean of the main objective for design , variance of the main objective for design , = random variable for the th constraint measure for design , = ̂ mean of the main objective for design , = ̂ random variable for the main objective for design... of correlation towards the budget allocation In addition, a significant additional computational burden for obtaining the allocation is not desirable when the computing budget is limited Thus, it is also needed to derive a closed-form allocation which is easy to implement 17 Chapter 3 Asymptotic Simulation Budget Allocation In this chapter, we present the simulation budget allocation which asymptotically... to the complement of As mentioned before, ̂ is the sample mean of the main objective value while the sample mean of the constraint measures is represented by ̂ Our goal is to intelligently control the number of simulation replications for each design N i so that is maximized given a total computing budget The optimal computing budget allocation for constrained optimization (OCBA-CO) problem is (3.3)... Kleywegtet al., 2001) Fu (2002) called this approach as simulation for optimization as for a given realization of the randomness using a scenario generator, the tools for deterministic optimization can be used On the other hand, simulation optimization can be considered as optimization for simulation This is because for each design, the performance measures are not available analytically and have to be... stochastic constrained optimization via simulation (OvS), considers a huge number of designs while the second problem, called constrained ranking and selection (R&S), considers a finite set of designs Similar to the simulation optimization for the unconstrained problems, the challenge in stochastic constrained OvS is to balance the efforts for searching and sampling from the design space For the case... information in the first stage Kim and Nelson (2001) and Nelson et al (2001) propose the fullysequential indifference zone procedures for unconstrained optimization In a fullysequential procedure, one simulation replication is collected from each alternative until it is eliminated from the consideration The second group aims to maximize given a computing budget such as the Optimal Computing Budget Allocation. .. to the computing budget allocation and provide the framework for extending the result for the general distribution case  From the practitioners’ point of view, we derive a closed-form allocation rule that is easy to implement and insightful This is essential to the practitioners who may not want to spend significantly more time or efforts to compute the allocation for each design  For users who are... more concerned on the optimality, the procedures can also return optimal allocation using a solver  We generalize the OCBA for selecting the best design The proposed procedures show that when all of the designs are feasible, the allocation rules are the same as the OCBA for unconstrained optimization  We provide the proof that as the number of designs becomes large, the optimal allocation can be approximated . OPTIMAL COMPUTING BUDGET ALLOCATION FOR CONSTRAINED OPTIMIZATION NUGROHO ARTADI PUJOWIDIANTO NATIONAL UNIVERSITY OF SINGAPORE 2012 OPTIMAL COMPUTING BUDGET ALLOCATION. Indifference Zone, NP = Nested Partitions, OCBA = Optimal Computing Budget Allocation, OCBA-CO = Optimal Computing Budget Allocation for Constrained Optimization,  = Probability of Correct. maximize  given a fixed computing budget called as Optimal Computing Budget Allocation (OCBA). Although OCBA has been shown to be effective for the unconstrained optimization (Branke et al.