Skip to main content
SpringerLink
Log in

A mathematical programming approach to sample coefficient of variation with interval-valued observations

  • Original Paper
  • Published:
TOP Aims and scope Submit manuscript

Abstract

The coefficient of variation is a useful statistical measure, which has long been widely used in many areas. In real-world applications, there are situations where the observations are inexact and imprecise in nature and they have to be estimated. This paper investigates the sample coefficient of variation (CV) with uncertain observations, which are represented by interval values. Since the observations are interval-valued, the derived CV should be interval-valued as well. A pair of mathematical programs is formulated to calculate the lower bound and upper bound of the CV. Originally, the pair of mathematical programs is nonlinear fractional programming problems, which do not guarantee to have global optimum solutions. By model reduction and variable substitutions, the mathematical programs are transformed into a pair of quadratic programs. Solving the pair of quadratic programs produces the global optimum solutions and constructs the interval of the CV. The given example shows that the proposed model is indeed able to help the manufacturer select the most suitable manufacturing process with interval-valued observations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Almeida CWD, Souza RMCR, Candeias ALB (2013) Fuzzy Kohonen clustering networks for interval data. Neurocomputing 99:65–75

    Article  Google Scholar 

  • Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming- theory and algorithm, 2nd edn. Wiley, New York

    Google Scholar 

  • Benson HP (2006) Fractional programming with convex quadratic forms and functions. Eur J Oper Res 173:351–369

    Article  Google Scholar 

  • Ben-Tal A, Boyd S, Nemirovski A (2006) Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math Program 107:63–89

    Article  Google Scholar 

  • Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88:411–424

    Article  Google Scholar 

  • Ben-Tal A, Nemirovski A (2008) Selected topics in robust convex optimization. Math Program 112B:125–158

    Google Scholar 

  • Cambini R, Sodini C (2010) A unifying approach to solve some classes of rank-three multiplicative and fractional programs involving linear functions. Eur J Oper Res 207:25–29

    Article  Google Scholar 

  • Černý M, Hladík M (2014) The complexity of computation and approximation of the t-ratio over one-dimensional interval data. Comput Stat Data Anal 80:26–43

    Article  Google Scholar 

  • Chang CT (2002) On the posynomial fractional programming problem. Eur J Oper Res 143:42–52

    Article  Google Scholar 

  • Charnes A, Cooper WW (1962) Programming with linear fractional functional. Nav Res Logist Q 9:181–186

    Article  Google Scholar 

  • Christodoulou C, Bugmann G (2001) Coefficient of variation vs. mean interspike interval curves: what do they tell us about the brain? Neurocomputing 38–40:1141–1149

    Article  Google Scholar 

  • Cooper WW, Park KS, Yu G (1999) IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manag Sci 45:597–607

    Article  Google Scholar 

  • Dai J, Wang W, Mi JS (2013) Uncertainty measurement for interval-valued information systems. Inf Sci 251:63–78

    Article  Google Scholar 

  • Dymova L, Sevastjanov P, Tikhonenko A (2013) A direct interval extension of TOPSIS method. Expert Syst Appl 40:4841–4847

    Article  Google Scholar 

  • Dymova L, Sevastjanov P, Tikhonenko A (2015) An interval type-2 fuzzy extension of the TOPSIS method using alpha cuts. Knowl Based Syst 83:116–127

    Article  Google Scholar 

  • Goberna MA, Jeyakumar V, Li G, Vicente-Pérez J (2014) Robust solution of multiple-objective linear semi-infinite programs under constraint data uncertainty. SIAM J Optim 24:1402–1419

    Article  Google Scholar 

  • Goberna MA, Jeyakumar V, Li G, Vicente-Pérez J (2015) Robust solutions to multiple-objective linear programs with uncertain data. Eur J Oper Res 242:730–743

    Article  Google Scholar 

  • Guo J, Li W, Li C, Gao S (2012) Standardization of interval symbolic data based on the empirical descriptive statistics. Comput Stat Data Anal 56:602–610

    Article  Google Scholar 

  • He X, Oyadiji SO (2001) Application of coefficient of variation in reliability-based mechanical design and manufacture. J Mater Process Technol 119:374–378

    Article  Google Scholar 

  • Hladík M (2010) Generalized linear fractional programming under interval efficiency. Eur J Oper Res 205:42–46

    Article  Google Scholar 

  • Hladík M (2011) Optimal value bounds in nonlinear programming with interval data. TOP 19:93–106

    Article  Google Scholar 

  • Hladík M, Černý M (2012) Interval regression by tolerance analysis approach. Fuzzy Sets Syst 193:85–107

    Article  Google Scholar 

  • Iancu DA, Trichakis N (2014) Pareto efficiency in robust optimization. Manag Sci 60:130–147

    Article  Google Scholar 

  • Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48:219–225

    Article  Google Scholar 

  • Jiang C, Zhang ZG, Zhang QF, Han X, Xie HC, Liu J (2014) A new nonlinear interval programming method for uncertain problems with dependent interval variables. Eur J Oper Res 238:245–253

    Article  Google Scholar 

  • Kalczynski PJ, Kamburowski J (2006) A heuristic for minimizing the expected makespan in two-machine flow shops with consistent coefficients of variation. Eur J Oper Res 169:742–750

    Article  Google Scholar 

  • Kreinovich V, Nguyen HT, Wu B (2007) On-line algorithms for computing mean and variance of interval data. Inf Sci 177:3228–3238

    Article  Google Scholar 

  • Liu ST (2006) Posynomial geometric programming with parametric uncertainty. Eur J Oper Res 162:345–353

    Article  Google Scholar 

  • Liu ST (2008) Posynomial geometric programming with interval exponents and coefficients. Eur J Oper Res 186:17–27

    Article  Google Scholar 

  • Liu ST (2011) Mean-absolute deviation portfolio selection problem with interval-valued returns. J Comput Appl Math 235:4149–4157

    Article  Google Scholar 

  • Mahapatra GS, Mandal TK (2012) Posynomial parametric geometric programming with interval valued coefficient. J Optim Theory Appl 154:120–132

    Article  Google Scholar 

  • Mahapatra GS, Roy TK (2012) Reliability evaluation of complex system with fuzzy reliability of components using interval nonlinear programming. Electron J Appl Stat Anal 5:151–163

    Google Scholar 

  • Neto EAL, Carvalho FAT (2010) Constrained linear regression models for symbolic interval-valued variables. Comput Stat Data Anal 54:333–347

    Article  Google Scholar 

  • Oliveira C, Antunes CH, Barrico C (2014) An enumerative algorithm for computing all possibly optimal solutions to an interval LP. TOP 22:530–542

    Article  Google Scholar 

  • Pal D, Mahapatra GS, Samanta GP (2013) Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model. Math Biosci 241:181–187

    Article  Google Scholar 

  • Pang WK, Yu BWT, Troutt MD, Hou SH (2008) A simulation-based approach to the study of coefficient of variation of dividend yields. Eur J Oper Res 189:559–569

    Article  Google Scholar 

  • Pardalos PM (1986) An algorithm for a class of nonlinear fractional problems using ranking of the vertices. BIT 26:392–395

    Article  Google Scholar 

  • Reed GF, Lynn F, Meade BD (2002) Use of coefficient of variation in assessing variability of quantitative assays. Clin Diagn Lab Immunol 9:1235–1239

    Google Scholar 

  • Roy P, Mahapatra BS, Mahapatra GS, Roy PK (2014) Entropy based region reducing genetic algorithm for reliability redundancy allocation in interval environment. Expert Syst Appl 41:6147–6160

    Article  Google Scholar 

  • See CT, Sim M (2009) Robust approximation to multiperiod inventory management. Oper Res 58:583–594

    Article  Google Scholar 

  • Sengupta A, Pal TK (2000) On comparing interval numbers. Eur J Oper Res 127:28–43

    Article  Google Scholar 

  • Sevastjanov P (2007) Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory. Inf Sci 177:4645–4661

    Article  Google Scholar 

  • Spadoni M, Stefanini L (2011) Computing the variance of interval and fuzzy data. Fuzzy Sets Syst 165:24–36

    Article  Google Scholar 

  • Vavasis SA (1991) Nonlinear optimization: complexity issues. Oxford University Press, New York

    Google Scholar 

  • Wang YM, Yang JB, Xu DL (2005) A preference aggregation method through the estimation of utility intervals. Comput Oper Res 32:2027–2049

    Article  Google Scholar 

  • Wang CF, Shen PP (2008) A global optimization algorithm for linear fractional programming. Appl Math Comput 204:281–287

    Article  Google Scholar 

  • Wu HC (2010) Duality theory for optimization problems with interval-valued objective functions. J Optim Theory Appl 144:615–628

    Article  Google Scholar 

Download references

Acknowledgments

Research was supported by the National Science Council of Republic of China under Contract No. NSC102-2410-H-238-005. The author is indebted to editor and the reviewers for their constructive comments that significantly improved the quality of this paper.

Author information

Authors and Affiliations

  1. Graduate School of Business and Management, Vanung University, Chung-Li, Tao-Yuan, 320, Taiwan, ROC

    Shiang-Tai Liu

Authors
  1. Shiang-Tai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiang-Tai Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, ST. A mathematical programming approach to sample coefficient of variation with interval-valued observations. TOP 24, 1–18 (2016). https://doi.org/10.1007/s11750-015-0391-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11750-015-0391-y

Keywords

Mathematics Subject Classification

Search

Navigation