April  2019, 15(2): 633-645. 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, 2019, 15 (2) : 633-645. doi: 10.3934/jimo.2018062
References:
[1]

M. BartuschR. H. Mohring and F. J. Randermacher, Scheduling project networks with resource constraints and time windows, Annals of Operations Research, 16 (1988), 201-240.   Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization, MOS-SIAM Series on Optimization, 2001.  Google Scholar

[3]

J. BlazewiczJ. 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.  Google Scholar

[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.  Google Scholar

[5]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, first edition, 2004.  Google Scholar

[6]

P. Brucker, Scheduling and constraint propagation, Discrete Applied Mathematics, 123 (2002), 227-256.  doi: 10.1016/S0166-218X(01)00342-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

U. MalikI. M. JaimoukhaG. 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.  Google Scholar

[12]

I. Pólik, Addendum to the SeDuMi user guide version 1.1, http://sedumi.ie.lehigh.edu/?page_id=58, 2005. Google Scholar

[13]

A. A. B. PritskerL. 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.  Google Scholar

[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.  Google Scholar

[15]

N. Z. Shor, Quadratic optimization problems, Soviet. J. Comput. Systems Sci., 25 (1987), 1-11.   Google Scholar

[16]

J. F. Sturm, Using SeDuMi 1.02, a matlab toolbox for optimizativer over smmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653.   Google Scholar

[17]

X. Sun and R. Li, New progress in integer programming, Operations Research Transactions, 18 (2014), 39-67.   Google Scholar

[18]

L. Vandenerghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95.  doi: 10.1137/1038003.  Google Scholar

[19]

H. WakiS. KimM. 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.  Google Scholar

[20]

H. Wolkowicz, R. Saigak and L. Vandenerghe, Handbook of Semidefinite Programming, Kluwer's Publisher, 2000.  Google Scholar

[21]

F. Zhang, The Schur Complement and Its Applications: Numerical Methods and Algorithms, Springer Science Business Media, 2005.  Google Scholar

[22]

X. ZhengX. SunD. 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.  Google Scholar

show all references

References:
[1]

M. BartuschR. H. Mohring and F. J. Randermacher, Scheduling project networks with resource constraints and time windows, Annals of Operations Research, 16 (1988), 201-240.   Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization, MOS-SIAM Series on Optimization, 2001.  Google Scholar

[3]

J. BlazewiczJ. 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.  Google Scholar

[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.  Google Scholar

[5]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, first edition, 2004.  Google Scholar

[6]

P. Brucker, Scheduling and constraint propagation, Discrete Applied Mathematics, 123 (2002), 227-256.  doi: 10.1016/S0166-218X(01)00342-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

U. MalikI. M. JaimoukhaG. 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.  Google Scholar

[12]

I. Pólik, Addendum to the SeDuMi user guide version 1.1, http://sedumi.ie.lehigh.edu/?page_id=58, 2005. Google Scholar

[13]

A. A. B. PritskerL. 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.  Google Scholar

[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.  Google Scholar

[15]

N. Z. Shor, Quadratic optimization problems, Soviet. J. Comput. Systems Sci., 25 (1987), 1-11.   Google Scholar

[16]

J. F. Sturm, Using SeDuMi 1.02, a matlab toolbox for optimizativer over smmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653.   Google Scholar

[17]

X. Sun and R. Li, New progress in integer programming, Operations Research Transactions, 18 (2014), 39-67.   Google Scholar

[18]

L. Vandenerghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95.  doi: 10.1137/1038003.  Google Scholar

[19]

H. WakiS. KimM. 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.  Google Scholar

[20]

H. Wolkowicz, R. Saigak and L. Vandenerghe, Handbook of Semidefinite Programming, Kluwer's Publisher, 2000.  Google Scholar

[21]

F. Zhang, The Schur Complement and Its Applications: Numerical Methods and Algorithms, Springer Science Business Media, 2005.  Google Scholar

[22]

X. ZhengX. SunD. 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.  Google Scholar

Figure 1.  Project Network Diagram in Example 1
Figure 2.  Project Network Diagram in Example 2
Table 1.  Basic Notations
SymbolDefinition
$Pred(i)$ 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
SymbolDefinition
$Pred(i)$ 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
Table 2.  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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