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Gamma distribution approach in chance-constrained stochastic programming model
Journal of Inequalities and Applications volume 2011, Article number: 108 (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.
The classical linear programming problem, which is a specific class of mathematical programming problem, is formulated as follows
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.
Stochastic linear programming problem with chance constraints is defined as follows
where c j , a ij and b i are random variables and u i 's are chosen probabilities. k th chance constraint given in model (2.1) is obtained as
If b k is the random variable in the model, and its distribution function is F b then the deterministic equivalent of chance constraint can be calculated as
Assume that a kj is a random variable having normal distribution with the mean E(a kj ) and the variance Var(a kj ). Furthermore, covariance between the random variables a kj and a kl is zero. Then, random variable d k is defined as follows
where a k1 ,..., a kn 's are random variables with normal distribution and x1,..., x n 's are unknowns, chance constraint given with inequality (2.2) is defined as follows
where denotes the value of standard normal variable and . Therefore, deterministic equivalent of inequality (2.4) is stated as
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.
Theorem 3.1 Let X1, X2,..., X n be independent random variables with given
where if it is as follows
is defined. Here, S is an absolute positive constant.
Proof to Theorem 3.1 can be found in [, pp. 109-111]. In case of equality, as a result of Essen inequality we can give the following equation, for large values of n
After defining the Essen inequality given in Theorem 3.1, now we explain Gamma distribution approach for CCSP model. In linear programming, the constraints are constructed as follows:
Here, the matrix A indicates a coefficients matrix. Let d k = a k 'x k = 1,..., m then k th row in (3.3) rewritten as
If a kj 's which are k th row of coefficients matrix A are independent gamma random variables, chance constraints given in model (2.1) are as follows
Assume that each random variable a kj has Gamma distribution with (α kj , β kj ) parameters in (3.4). For the purpose of using Essen inequality given in Theorem 3.1, the random variable r j = a kj x j - E(a kj x j ), j = 1,..., n is taken into account. Expected value and variance of each random variable a kj as follows:
Therefore, the expected value of random variable r j will be as follows:
and its variance will be as follows:
Absolute third moment of random variable d j is found in the following equality
The expected value in equality (3.6) can be written as follows:
Then, I kj is rewritten as follows
If it is taken as, in integral then I kj can be written as follows
Here, by making variable change ,
is obtained. Incomplete gamma function is defined as follows
Therefore, ω1 can be rearranged as follows:
Similarly, it can be written as follows
The second part of the integral can be written as follows
If it is taken as in integral then II kj can be written as follows
In the same way it will be
Therefore, for any finite α kj and β kj , it can easily be seen that Ed j = 0 and E|d j |3 < ∞. Therefore, the conditions in Theorem 3.1 are satisfied, then and B n is obtained as
The third absolute moment of random variable, r j , in terms of integrals I kj and II kj is written as follows
Then, L n is obtained as follows
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.
Lemma 3.1 Maximum value of L n in Equation 3.8 is given by
Proof Maximum value of L n given in Equation 3.8 is obtained by maximizing nominator while minimizing the denominator, i.e.
In Theorem 3.1, using L n given in (3.8), following inequality is obtained
If the suggested constant S = 0.7975  in inequality (3.10) and if the value max L n given with (3.9) is used following inequality is obtained
Here, F n (x) is Gamma distribution function, Φ(x) is that of standard normal distribution. Thus, for d k
is defined. Therefore, constraint (3.5) can be written as follows
Here, the following inequality is written
There are decision variables x j (j = 1,..., n) in L n which is on the left side of the inequality (3.12). Since these decision variables are the results of the problem solved after model (2.1) is made deterministic, they are unknown here. Therefore, L n is not a numeric and it cannot be solved using Φ-1(1-(u k )+SL n ). Therefore, using the approach suggested  right side of inequality (3.12) can be written as follows
and deterministic constraint belonging to inequality (3.12) is then written as fallows
Using Equation (3.2) we can construct the following inequality
4. Numerical experiments
Consider the CCSP model as follows
Here, assume that a kj j = 1,2,3 are independent random variables distributed as Gamma distribution with the following parameters (α kj , β kj )
b2 is normal random variable with the following expected value and variance
In the solving stage of the problem, for using of Essen inequality given in Theorem 3.1 can be defined as
is found and for k = 1, B n is obtained as follows
Then, L n is written as follows
As a result of the solution of the integrals, I kj (k = 1, j = 1,2,3) and II kj (k = 1, j = 1,2,3) in L n can be obtained as
Then, L n is found as
Therefore, in the case where a kj is a random variable with Gamma distribution, deterministic equality of the first chance constraint in model (4.1), using inequality (3.14) is obtained as follows
Using inequality (3.15) we can write as:
Using inequality (2.3) for the second chance constraint, deterministic inequality is obtained as
Then, deterministic equality of CCSP model given in (4.1), using inequality (4.3), can be found as follows
The second constraint is given for controlling of non-negativity on the right side of first constraint. The nonlinear problem given in (4.5) has been solved with condition
using software Lingo 9.0 and the results are shown in Table 1.
Deterministic equality of CCSP model given in (4.1), using inequality (4.4), can be found as follows
As a second case, let us assume that a kj coefficients in the first chance constraint in model (4.1) are independent normal random variables with the following expected value E(a kj ) and variance Var(a kj )
Then, deterministic equality of chance constraint can be arranged as follows
Therefore, deterministic equality of CCSP model given in (4.1) can be found as follows:
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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The authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, read and approved the final manuscript.
Kumru D Atalay and Aysen Apaydin contributed equally to this work.
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Atalay, K.D., Apaydin, A. Gamma distribution approach in chance-constrained stochastic programming model. J Inequal Appl 2011, 108 (2011). https://doi.org/10.1186/1029-242X-2011-108
- chance-constrained programming
- Essen inequality
- Gamma distribution