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doi: 10.3934/jimo.2018062

## Immediate schedule adjustment and semidefinite relaxation

 1 School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China 2 Business School, Nankai University, 94 Weijin Road, Nankai District, Tianjin 300071, China

* Corresponding author: Su Zhang

Received  November 2017 Revised  January 2018 Published  April 2018

Fund Project: The first author is supported by National Natural Science Foundation of China No. 11101028,11271206, and the Fundamental Research Funds for the Central Universities. The third author is supported by National Natural Science Foundation of China No. 11401322

This paper considers the problem of temporary shortage of some resources within a project execution period. Mathematical models for two different cases of this problem are established. Semidefinite relaxation technique is applied to get immediate solvent of these models. Relationship between the models and their semidefinite relaxations is studied, and some numerical experiments are implemented, which show that these mathematical models are reasonable and feasible for practice, and semidefinite relaxation can efficiently solve the problem.

Citation: Jinling Zhao, Wei Chen, Su Zhang. Immediate schedule adjustment and semidefinite relaxation. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018062
##### References:
 [1] M. Bartusch, R. H. Mohring and F. J. Randermacher, Scheduling project networks with resource constraints and time windows, Annals of Operations Research, 16 (1988), 201-240. [2] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization, MOS-SIAM Series on Optimization, 2001. [3] J. Blazewicz, J. K. Lenstra and A. H. G. Kan Rinnooy, Scheduling subject to resource constraints: Classification and complexity, Discrete Applied Mathematics, 5 (1983), 11-24. doi: 10.1016/0166-218X(83)90012-4. [4] S. Boyd and L. Vandenberghe, Semidefinite programming relaxations of non-convex problems in control and combinatorial optimization, Communications, Computation, Control, and Signal Processing, Springer, 1997,279-287 doi: 10.1007/978-1-4615-6281-8_15. [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, first edition, 2004. [6] P. Brucker, Scheduling and constraint propagation, Discrete Applied Mathematics, 123 (2002), 227-256. doi: 10.1016/S0166-218X(01)00342-0. [7] M. Goemans and D. Williamson, Imporved approximation algorihtms for maximum cut and satisfiablity problems using semidefinite programming, J. Assoc. Comput. Mach., 42 (1995), 1115-1145. doi: 10.1145/227683.227684. [8] S. Hartmann and D. Briskorn, A survey of variants and extensions of the resource-constrained project scheduling problem, European Journal of Operational Research, 207 (2010), 1-14. doi: 10.1016/j.ejor.2009.11.005. [9] D. Henrion, J. Lasserre and J. Loefberg, GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw., 24 (2009), 761-779, http://homepages.laas.fr/henrion/software/gloptipoly3. doi: 10.1080/10556780802699201. [10] H. Li and N. K. Womer, Solving stochastic resource-constrained project scheduling problems by closed-loop approximate dynamic programming, European Journal of Operational Research, 246 (2015), 20-33. doi: 10.1016/j.ejor.2015.04.015. [11] U. Malik, I. M. Jaimoukha, G. D. Halikias and S. K. Gungah, On the gap between the quadratic integer programming problem and its semidefinite relaxation, Math. Program., 107 (2006), 505-515. doi: 10.1007/s10107-005-0692-2. [12] I. Pólik, Addendum to the SeDuMi user guide version 1.1, http://sedumi.ie.lehigh.edu/?page_id=58, 2005. [13] A. A. B. Pritsker, L. J. Watters and P. M. Wolfe, Multiproject scheduling with limited resources: A zero-one programming approach, Management Science, 16 (1969), 93-108. doi: 10.1287/mnsc.16.1.93. [14] F. Rendl, Semidefinite relaxations for integer programming, 50 Years of Integer Programming 1958-2008, (2009), 687-726. doi: 10.1007/978-3-540-68279-0_18. [15] N. Z. Shor, Quadratic optimization problems, Soviet. J. Comput. Systems Sci., 25 (1987), 1-11. [16] J. F. Sturm, Using SeDuMi 1.02, a matlab toolbox for optimizativer over smmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653. [17] X. Sun and R. Li, New progress in integer programming, Operations Research Transactions, 18 (2014), 39-67. [18] L. Vandenerghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95. doi: 10.1137/1038003. [19] H. Waki, S. Kim, M. Kojima and M. Muramatsu, Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity, SIAM Journal on Optimization, 17 (2006), 218-242. doi: 10.1137/050623802. [20] H. Wolkowicz, R. Saigak and L. Vandenerghe, Handbook of Semidefinite Programming, Kluwer's Publisher, 2000. [21] F. Zhang, The Schur Complement and Its Applications: Numerical Methods and Algorithms, Springer Science Business Media, 2005. [22] X. Zheng, X. Sun, D. Li and Y. Xia, Duality gap estimation of linear equality constraintd binary quadratic programming, Mathematics of Operations Research, 35 (2010), 864-880. doi: 10.1287/moor.1100.0472.

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##### References:
 [1] M. Bartusch, R. H. Mohring and F. J. Randermacher, Scheduling project networks with resource constraints and time windows, Annals of Operations Research, 16 (1988), 201-240. [2] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization, MOS-SIAM Series on Optimization, 2001. [3] J. Blazewicz, J. K. Lenstra and A. H. G. Kan Rinnooy, Scheduling subject to resource constraints: Classification and complexity, Discrete Applied Mathematics, 5 (1983), 11-24. doi: 10.1016/0166-218X(83)90012-4. [4] S. Boyd and L. Vandenberghe, Semidefinite programming relaxations of non-convex problems in control and combinatorial optimization, Communications, Computation, Control, and Signal Processing, Springer, 1997,279-287 doi: 10.1007/978-1-4615-6281-8_15. [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, first edition, 2004. [6] P. Brucker, Scheduling and constraint propagation, Discrete Applied Mathematics, 123 (2002), 227-256. doi: 10.1016/S0166-218X(01)00342-0. [7] M. Goemans and D. Williamson, Imporved approximation algorihtms for maximum cut and satisfiablity problems using semidefinite programming, J. Assoc. Comput. Mach., 42 (1995), 1115-1145. doi: 10.1145/227683.227684. [8] S. Hartmann and D. Briskorn, A survey of variants and extensions of the resource-constrained project scheduling problem, European Journal of Operational Research, 207 (2010), 1-14. doi: 10.1016/j.ejor.2009.11.005. [9] D. Henrion, J. Lasserre and J. Loefberg, GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw., 24 (2009), 761-779, http://homepages.laas.fr/henrion/software/gloptipoly3. doi: 10.1080/10556780802699201. [10] H. Li and N. K. Womer, Solving stochastic resource-constrained project scheduling problems by closed-loop approximate dynamic programming, European Journal of Operational Research, 246 (2015), 20-33. doi: 10.1016/j.ejor.2015.04.015. [11] U. Malik, I. M. Jaimoukha, G. D. Halikias and S. K. Gungah, On the gap between the quadratic integer programming problem and its semidefinite relaxation, Math. Program., 107 (2006), 505-515. doi: 10.1007/s10107-005-0692-2. [12] I. Pólik, Addendum to the SeDuMi user guide version 1.1, http://sedumi.ie.lehigh.edu/?page_id=58, 2005. [13] A. A. B. Pritsker, L. J. Watters and P. M. Wolfe, Multiproject scheduling with limited resources: A zero-one programming approach, Management Science, 16 (1969), 93-108. doi: 10.1287/mnsc.16.1.93. [14] F. Rendl, Semidefinite relaxations for integer programming, 50 Years of Integer Programming 1958-2008, (2009), 687-726. doi: 10.1007/978-3-540-68279-0_18. [15] N. Z. Shor, Quadratic optimization problems, Soviet. J. Comput. Systems Sci., 25 (1987), 1-11. [16] J. F. Sturm, Using SeDuMi 1.02, a matlab toolbox for optimizativer over smmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653. [17] X. Sun and R. Li, New progress in integer programming, Operations Research Transactions, 18 (2014), 39-67. [18] L. Vandenerghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95. doi: 10.1137/1038003. [19] H. Waki, S. Kim, M. Kojima and M. Muramatsu, Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity, SIAM Journal on Optimization, 17 (2006), 218-242. doi: 10.1137/050623802. [20] H. Wolkowicz, R. Saigak and L. Vandenerghe, Handbook of Semidefinite Programming, Kluwer's Publisher, 2000. [21] F. Zhang, The Schur Complement and Its Applications: Numerical Methods and Algorithms, Springer Science Business Media, 2005. [22] X. Zheng, X. Sun, D. Li and Y. Xia, Duality gap estimation of linear equality constraintd binary quadratic programming, Mathematics of Operations Research, 35 (2010), 864-880. doi: 10.1287/moor.1100.0472.
Project Network Diagram in Example 1
Project Network Diagram in Example 2
Basic Notations
 Symbol Definition $Pred(i)$ set of direct predecessors of Activity $i$ $Succ(i)$ set of direct successors of Activity $i$ $d_i$ processing time (or duration) of Activity $i$ $s_i$ start time of Activity $i$ according to the existing schedule $f_i$ completion time of Activity $i$ according to the existing schedule $R_k$ amount of originally available units of renewable resource $k$ in unit time $r_{ik}$ usage of Activity $i$ of renewable resource $k$ in unit time $t_0$ start time of the temporary shortage of resources $T$ lasting time of the resources shortage $\Delta R_k$ amount of decrement of resource $k$ in unit time
 Symbol Definition $Pred(i)$ set of direct predecessors of Activity $i$ $Succ(i)$ set of direct successors of Activity $i$ $d_i$ processing time (or duration) of Activity $i$ $s_i$ start time of Activity $i$ according to the existing schedule $f_i$ completion time of Activity $i$ according to the existing schedule $R_k$ amount of originally available units of renewable resource $k$ in unit time $r_{ik}$ usage of Activity $i$ of renewable resource $k$ in unit time $t_0$ start time of the temporary shortage of resources $T$ lasting time of the resources shortage $\Delta R_k$ amount of decrement of resource $k$ in unit time
Comparing the 1/3-Method with the Rounding Method
 $\Delta R_k$ 1/3-Method Rounding Method SDR Opt.Val $t^*$ Opt.Val $t^*$ Delay Time $2$ 15 infeasible 15.0000 15 1 $5$ 15 15 15.0000 15 1 $8$ 18 infeasible 15.4583 18 4 $11$ 19 19 16.7083 19 5
 $\Delta R_k$ 1/3-Method Rounding Method SDR Opt.Val $t^*$ Opt.Val $t^*$ Delay Time $2$ 15 infeasible 15.0000 15 1 $5$ 15 15 15.0000 15 1 $8$ 18 infeasible 15.4583 18 4 $11$ 19 19 16.7083 19 5
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