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Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. (English) Zbl 1427.65094
Summary: In this paper, two-step method (TSM), alternative solution method (SOM-2) and best-worst case (BWC) method are introduced to solve a type of interval linear programming (ILP) problem. To compare the performance of the methods, Monte Carlo simulation is also used to solve the same ILP problem, whose solutions are assumed to be real solutions. In the comparison, two scenarios corresponding with two assumptions of distribution functions are considered: (i) all the input parameters obey normal distribution; (ii) all the input parameters obey uniform distribution. Based on the simulation results, coverage rate (CR) and validity rate (VR) are proposed as new indicators to measure the quality of the numerical solutions obtained from the methods. Results from a numerical case study indicate that the TSM and SOM-2 solutions can cover the majority of valid values ($$\mathrm{CR}>50\%$$, $$\mathrm{VR}>50\%$$), compared to the conventional BWC method. In addition, from the point of CR, TSM is more applicable since the solutions of TSM can identify more feasible solutions. However, from the point of VR, SOM-2 is preferred since it can exclude more baseless solutions (this means more feasible solutions are contained in the SOM-2 solutions). In general, TSM would be preferred when only the range of the system objective needs to be determined, while SOM-2 would be much useful in identifying the effective values of the objective.

##### MSC:
 65K05 Numerical mathematical programming methods 65G30 Interval and finite arithmetic 90C05 Linear programming
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##### References:
 [1] Alefeld, G.; Herzberger, J., Introduction to Interval Computations (1983), Academic Press: Academic Press New York [2] Guo, P.; Huang, G. H.; He, L.; Li, H. L., Interval-parameter fuzzy-stochastic semi-infinite mixed-integer linear programming for waste management under uncertainty, Environ. Model. Assess., 14, 521-537 (2009) [4] Carlin, B. P.; Polson, N. G.; Stoffer, D. S., A Monte Carlo approach to nonnormal and nonlinear state-space modeling, J. Am. Stat. Assoc., 87, 493-500 (1992) [5] Chib, S.; Nardari, F.; Shephard, N., Markov chain Monte Carlo methods for stochastic volatility models, J. Econometr., 108, 281-316 (2002) · Zbl 1099.62539 [6] Fantazzini, D., The effects of misspecified marginals and copulas on computing the value at risk: a Monte Carlo study, Comput. Stat. Data Anal., 53, 2168-2188 (2009) · Zbl 05687918 [7] Hansen, E. R., Bounding the solution of interval linear equations, SIAM J. Numer. Anal., 29, 1493-1503 (1992) · Zbl 0756.65035 [8] He, L.; Huang, G. H.; Zeng, G. M.; Lu, H. W., Identifying optimal regional solid waste management strategies through an inexact integer programming model containing infinite objectives and constraints, Waste Manage., 29, 21-31 (2009) [9] He, L.; Huang, G.; Zeng, G.; Lu, H., Fuzzy inexact mixed-integer semiinfinite programming for municipal solid waste management planning, J. Environ. Eng., 134, 572-581 (2008) [10] Huang, G. H.; Baetz, B. W.; Patry, G. G., A grey linear programming approach for municipal solid waste management planning under uncertainty, Civ. Eng. Environ. Syst., 9, 319-335 (1992) [11] Huang, G. H.; Baetz, B. W.; Patry, G. G., Grey integer programming: an application to waste management planning under uncertainty, Eur. J. Oper. Res., 83, 594-620 (1995) · Zbl 0899.90131 [12] Huang, G. H.; Baetz, B. W.; Patry, G. G., Trash-flow allocation: planning under uncertainty, Interfaces, 28, 36-55 (1998) [13] Lu, H. W.; Huang, G. H.; Liu, L.; He, L., An interval-parameter fuzzy-stochastic programming approach for air quality management under uncertainty, Environ. Eng. Sci., 25, 895-909 (2008) [14] Neumaier, A., Interval Methods for Systems of Equations (1990), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0706.15009 [15] Rohn, J., Systems of linear interval equations, Linear Algebra Appl., 126, 39-78 (1989) · Zbl 0712.65029 [16] Lu, H. W.; Huang, G. H.; Liu, Z. F.; He, L., Greenhouse gas mitigation-induced rough-interval programming for municipal solid waste management, J. Air. Waste. Manage. Assoc., 58, 1546-1559 (2008) [17] Shary, S. P., On optimal solution of interval linear equations, SIAM J. Numer. Anal., 32, 610-630 (1995) · Zbl 0838.65023 [18] Lu, H. W.; Huang, G. H.; He, L., A semi-infinite analysis-based inexact two-stage stochastic fuzzy linear programming approach for water resources management, Eng. Opt., 41, 7385 (2009) [19] Tierney, L.; Mira, A., Some adaptive Monte Carlo methods for Bayesian inference, Stat. Probab. Lett., 18, 2507-2515 (1999)
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