Gamma distribution approach in chance-constrained stochastic programming model
© Atalay and Apaydin; licensee Springer. 2011
Received: 7 June 2011
Accepted: 8 November 2011
Published: 8 November 2011
In this article, a method is developed to transform the chance-constrained programming problem into a deterministic problem. We have considered a chance-constrained programming problem under the assumption that the random variables a ij are independent with Gamma distributions. This new method uses estimation of the distance between distribution of sum of these independent random variables having Gamma distribution and normal distribution, probabilistic constraint obtained via Essen inequality has been made deterministic using the approach suggested by Polya. The model studied on in practice stage has been solved under the assumption of both Gamma and normal distributions and the obtained results have been compared.
A chance-constrained stochastic programming (CCSP) models is one of the major approaches for dealing with random parameters in the optimization problems. Charnes and Cooper  have first modelled CCSP. Here, they have developed a new conceptual and analytic method which contains temporary planning of optimal stochastic decision rules under uncertainty. Symonds  has presented deterministic solutions for the class of chance-constraint programming problem. Kolbin  has examined the risk and indefiniteness in planning and managing problems and presented chance-constraint programming models. Stancu-Minasian  has suggested a minimum-risk approach to multi-objective stochastic linear programming problems. Hulsurkar et al.  have studied on a practice of fuzzy programming approach of multi-objective stochastic linear programming problems. They have used fuzzy programming approach for finding a solution after changing the suggested stochastic programming problem into a linear or a nonlinear deterministic problem. Liu and Iwamura  have studied on chance-constraint programming with fuzzy parameters. Chance-constraint programming in stochastic is expanded to fuzzy concept by their studies. They have presented certain equations with chance constraint in some fuzzy concept identical to stochastic programming. Furthermore, they have suggested a fuzzy simulation method for chance constraints for which it is usually difficult to be changed into certain equations. Finally, these fuzzy simulations which became basis for genetic algorithm have been suggested for solving problems of this type and discussing numeric examples. Mohammed  has studied on chance-constraint fuzzy goal programming containing right-hand side values with uniform random variable coefficients. He presented the main idea related with the stochastic goal programming and chance-constraint linear goal programming. Kampas and White  have suggested the programming based on probability for the control of nitrate pollution in their studies and compared this with the approaches of various probabilistic constraints. Yang and Wen  presented a chance-constrained programming model for transmission system planning in the competitive electricity market environment. Huang  provided two types of credibility-based chance-constrained models for portfolio selection with fuzzy returns. Ağpak and Gökçen  developed new mathematical models for stochastic traditional and U-type assembly lines with a chance-constrained 0-1 integer programming technique. Henrion and Strugarek  investigated the convexity of chance constraints with independent random variables. Parpas and Rüstem  proposed a stochastic algorithm for the global optimization of chance-constrained problems. They assumed that the probability measure used to evaluate the constraints is known only through its moments. Xu et al.  developed a robust hybrid stochastic chance-constraint programming model for supporting municipal solid waste management under uncertainty. Abdelaziz and Masri  proposed a chance-constrained approach and a compromise programming approach to transform the multi-objective stochastic linear program with partial linear information on the probability distribution into its equivalent uni-objective problem. Goyal and Ravi  presented a polynomial time approximation scheme for the chance-constrained knapsack problem when item sizes are normally distributed and independent of other items.
where all coefficients (technologic coefficients a ij , right-hand side values b i and objective function coefficients c j (j = 1,..., n i = 1,..., m)) are deterministic. However, when at least one coefficient is a random variable, the problem becomes a stochastic programming problem.
In this article, we have assumed that the a ij , (i = 1,..., m, j = 1,... n) which are the elements of, m × n type technologic matrix A, are random variables having Gamma distribution. In case that these coefficients having Gamma distribution are independent, the estimation of the distance between the distribution of sum of them and normal distribution has been obtained. Essen inequality has been used for these and deterministic equality of chance constraints has been found. The model with random variable coefficients has been solved via the suggested method and it has been implemented on a numeric example. The model has been examined again for the case to have coefficients with normal distribution. It has been observed that the case a ij coefficients have Gamma distribution or normal distribution has given similar results for large values of n with regard to objective function.
2. Chance-constrained stochastic programming
Stochastic programming deals with the case that input data (prices, right hand side vector, technologic coefficients) are random variables. As parameters are random variables, a probability distribution should be determined. Two frequently used approaches for transforming stochastic programming problem into a deterministic programming problem are chance constraint programming and two-staged programming.
"Chance-constrained programming" which is a stochastic programming method contains fixing the certain appropriate levels for random constraints. Therefore, it is generally used for modelling technical or economic systems. The practices include economic planning, input control, structural design, inventory, air and water quality management problems. In chance constraints, each constraint can be realized with a certain probability.
Solution methods for models constituted by dual and triple combinations of c j , a kj and b k coefficients and also for the case that c j 's are random variable are different. In this article, these are not mentioned [5, 19–21].
3. Gamma distribution approach for CCSP
Let, X1, X2,..., X n be independent random variables with a distribution function F n (x). Let Φ(x) be a standard normal distribution function. Then, supremum of absolute distance between F n (x) and Φ(x) can be found. The theorem related to this, which is known as Essen Inequality, is as follows.
is defined. Here, S is an absolute positive constant.
Even if L n defined in Theorem 3.1 is maximum it can be a useful upper bound for left side of (3.1). Following lemma is related to this situation.
4. Numerical experiments
Solutions results of models (4.5), (4.6), (4.9)
Model (4.9) has been solved by software Lingo 9.0 and the results are listed in Table 1.
In this study, a new method is suggested for the solution of the deterministic equivalence of the CCSP. The main purpose of this article is to transform the chance-constrained model into a deterministic model based on the Essen inequality. According to the Essen inequality, the estimation of the distance between the distribution of a sum of independent random variables and the normal distribution is less than or equal to SL n . This study considers a stochastic optimization model with random technology matrix in which the random variables are independent and follow a Gamma distribution. Deterministic equality of these kinds of problems has been obtained via the suggested method. Furthermore, by adding a second constraint having normal distribution in the right-hand side value, a problem with two chance constraints has been obtained. In this problem, both cases that a kj coefficients have gamma and normal distributions have been examined and for the solution of deterministic models Lingo 9.0 has been used.
As a result, the upper bounds of the chance constrained are derived by the Essen inequality and developed approximate deterministic equivalent of the model.
The solutions obtained by including the supremum distance defined by the Essen inequality in the model are shown clearly in the solutions results (4.5) and (4.6) in Table 1.
For large values of n, the solution results of the models having Gamma and normal distributions are closed to each other. This can be observed in Table 1 by examining the solution results (4.6) and (4.9). Here, it can be seen that coefficients of the objective function and decision variables are very similar.
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