Probabilistic Quadratic Programming Problems with Some Fuzzy Parameters

Abstract

We present a solution procedure for a quadratic programming problem with some probabilistic constraints where the model parameters are either triangular fuzzy number or trapezoidal fuzzy number. Randomness and fuzziness are present in some real-life situations, so it makes perfect sense to address decision making problem by using some specified random variables and fuzzy numbers. In the present paper, randomness is characterized by Weibull random variables and fuzziness is characterized by triangular and trapezoidal fuzzy number. A defuzzification method has been introduced for finding the crisp values of the fuzzy numbers using the proportional probability density function associated with the membership functions of these fuzzy numbers. An equivalent deterministic crisp model has been established in order to solve the proposed model. Finally, a numerical example is presented to illustrate the solution procedure.

1. Introduction

In real-world applications, it is very much difficult to know all the information about the input parameters of the mathematical programming model because relevant data are inexistent or scarce, difficult to obtain or to estimate, the system is subject to changes, and so forth, that is, input parameters are uncertain. One of the best ways of modeling these uncertainties in the form of mathematical programming is known as stochastic programming. It is widely used in several research areas such as agriculture, capacity planning, finance, forestry, military, production control and scheduling, sport, telecommunications, transportation, environmental management planning.

It should also be noted, that in many practical situations, knowledge about the data (i.e., the coefficients/parameters of the model) is not purely probabilistic or possibilistic but rather a mixture of both kinds. For an example consider a firm that desires to maximize its profit by meeting all the customers demands which fluctuate due to random change in price. By the stochasticity of the demand and the fact that the production may not fulfill all the possible demands, this point cannot be satisfactorily answered by the true or false statement. This optimization problem is related to two types of uncertainties, namely, randomness and fuzziness, which motivates the proposed study in this direction.

Quadratic programming (QP) is an optimization technique where we minimize/maximize a quadratic objective function of several variables subject to a set of linear constraints. QP has a wide range of applications, such as portfolio selection, electrical energy production, agriculture, and crop selection. Probabilistic quadratic programming is applicable for financial and risk management. In this direction several important-related literatures are found, which are cited below.

McCarl et al. [1] presented some of the methodologies where QP is applicable. Liu and Wang [2] proposed an interval quadratic programming problem where the cost coefficients, constraint coefficients, and right-hand sides are represented by interval parameters. Silva et al. [3] developed an original and novel fuzzy sets based method that solves a class of quadratic programming problems with vagueness. Liu [4] presented a fuzzy quadratic programming problem where the cost coefficients, constraint coefficients, and right-hand sides values are represented by fuzzy data. Ammar [5] proposed a multiobjective quadratic programming problem having fuzzy random coefficient matrix in the objectives and constraints, where the decision vector is treated as a fuzzy vector. Liu [6] discussed the fuzzy quadratic programming problem where the cost coefficients, constraint coefficients, and right-hand side parameters of the constraints are represented by convex fuzzy numbers. He also described a solution procedure [7] for a class of fuzzy quadratic programming problems, where the cost coefficients of the objective function, constraint coefficients, and right-hand side parameters values are fuzzy numbers. Qin and Huang [8] presented an inexact chance constrained quadratic programming model for stream water quality management. Nasseri [9] defined a quadratic programming problem with fuzzy numbers where the cost coefficients, constraint coefficients, and right-hand side parameters values are represented by trapezoidal and/or triangular fuzzy numbers. An inexact fuzzy-stochastic quadratic programming method is developed by Guo and Huang [10] for effectively allocating waste to a municipal solid waste management system by considering the nonlinear objective function and multiple uncertainties on parameters in the constraints.

In this paper, we develop a new method for solving probabilistic quadratic programming problems involving some of the coefficients that are triangular fuzzy numbers and/or trapezoidal fuzzy numbers. Only the right- hand side parameters of the constraints are considered as Weibull random variables with known probability distributions. Both randomness and fuzziness are considered together within the QP framework. A defuzzification method is introduced for finding the crisp values of the fuzzy numbers using the Mellin transformation [11].

1.1. Probabilistic Fuzzy Quadratic Programming Problem

A probabilistic fuzzy quadratic programming problem is a modified QP having a quadratic objective function and some linear constraints involving fuzziness and randomness in some situations. When some of the input parameters of the QP are characterized by stochastic and fuzzy parameters, the problem is treated as a probabilistic fuzzy quadratic programming problem. A general probabilistic fuzzy quadratic programming problem can be presented asmin∶𝑧=^𝑛_𝑗=1𝑐_𝑗̃𝑥_𝑗+12^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝑗𝑥_𝑖𝑥_𝑗 (1.1) subject toPr^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑏_𝑖≥1−𝛼_𝑖𝑥,𝑖=1,2,…,𝑚,_𝑗≥0,𝑗=1,2,…,𝑛 (1.2)(1.3) where 0<𝛼_𝑖<1, 𝑖=1,2,…,𝑚, are the specified probabilities. The coefficients 𝑐_𝑗̃, 𝑗=1,2,…,𝑛, and 𝑞_𝑖𝑗, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑛, are considered as triangular fuzzy numbers but 𝑎_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, are consider as trapezoidal fuzzy number. Only 𝑏_𝑖, 𝑖=1,2,…,𝑚, are considered as random variables with known distributions. The decision variables 𝑥_𝑗, 𝑗=1,2,…,𝑛, are treated as deterministic in the problem. In the following section, we discuss some useful preliminaries related to fuzzy numbers, then the method of defuzzification is introduced.

2. Some Preliminaries

In this section, we present triangular and trapezoidal membership functions that are used in the model formulation. Also we introduce the Mellin transform to find the expected value of the function of a random variable using proportional probability density function associated with the membership functions of the fuzzy numbers.

Definition 2.1 (triangular fuzzy number). A fuzzy number denoted by the triplet 𝐴=(𝑎₁,𝑎₂,𝑎₃) is called a triangular fuzzy number with a piecewise linear membership function 𝜇_𝐴(𝑥) defined by 𝜇_𝐴(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,𝑎₃−𝑥𝑎₃−𝑎₂,𝑎₂≤𝑥≤𝑎₃,0,otherwise. (2.1)

Definition 2.2 (trapezoidal fuzzy number). A fuzzy number denoted by the quadruplet 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄) is called a trapezoidal fuzzy number with a piecewise linear membership function 𝜇_𝐴(𝑥) defined by 𝜇_𝐴(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,1,𝑎₂≤𝑥≤𝑎₃,𝑎₄−𝑥𝑎₄−𝑎₃,𝑎₃≤𝑥≤𝑎₄,0,otherwise. (2.2)

2.1. Defuzzification with Probability Density Function and Membership Function

Let 𝐹(ℝ) be the set of all fuzzy numbers. Let (𝑎₁,𝑎₂,𝑎₃) and (𝑎₁,𝑎₂,𝑎₃,𝑎₄) be the triangular and trapezoidal fuzzy numbers, respectively, in 𝐹(ℝ). Now we define the method associated with a probability density function for the membership function of 𝐴 as follows [12, 13].

Proportional probability distribution: define a probability density function 𝑓₁=𝑐𝜇_𝐴(𝑥) associated with 𝐴, where 𝑐 is a constant obtained by using the property of probability density function, that is, ∫^∞_−∞𝑓₁(𝑥)𝑑𝑥=1, that is, ∫^∞_−∞𝑐𝜇_𝐴(𝑥)𝑑𝑥=1.

2.2. Mellin Transform

As we know that any probability density function with finite support is associated with an expected value, the Mellin transform [12, 13] is used to find this expected value.

Definition 2.3. The Mellin transform 𝑀_𝑋(𝑡) of a probability density function 𝑓(𝑥), where 𝑥 is a positive, is defined as 𝑀_𝑋(𝑡)=^∞₀𝑥^𝑡−1𝑓(𝑥)𝑑𝑥, (2.3) where the integral exists.

Now we find the Mellin transform in terms of expected values. Recall that the expected value of any function 𝑔(𝑋) of the random variable 𝑋, whose probability density function is 𝑓(𝑥), is given by𝔼𝑔(𝑋)=^∞_−∞𝑔(𝑥)𝑓(𝑥)𝑑𝑥. (2.4) Therefore, it follows that 𝑀_𝑋(𝑡)=𝔼[𝑋^𝑡−1]=∫^∞₀𝑥^𝑡−1𝑓(𝑥)𝑑𝑥.

Hence, 𝔼[𝑋^𝑡]=𝑀_𝑋(𝑡+1). Thus, the expected value of random variable 𝑋 is 𝔼[𝑋]=𝑀_𝑋(2).

For an example, if 𝐴₁=(𝑎₁,𝑎₂,𝑎₃) and 𝐴₂=(𝑎₁,𝑎₂,𝑎₃,𝑎₄) are the triangular and trapezoidal fuzzy numbers, respectively, then their crisp values are calculated by finding expected values using the probability density function corresponding to the membership functions of the given fuzzy number.

Now, the probability density function corresponding to triangular fuzzy number 𝐴₁=(𝑎₁,𝑎₂,𝑎₃) is given as𝑓𝐴₁(𝑥)=𝑐₁𝜇𝐴₁(𝑥), (2.5) where 𝜇𝐴₁(𝑥) is defined as𝜇𝐴₁(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,𝑎₃−𝑥𝑎₃−𝑎₂,𝑎₂≤𝑥≤𝑎₃,0,otherwise. (2.6)

Now 𝑐₁ is calculated as^∞_−∞𝑓𝐴₁(𝑥)𝑑𝑥=1, (2.7) that is,^∞_−∞𝑐₁𝜇𝐴₁(𝑥)𝑑𝑥=1, (2.8) that is,𝑐₁^𝑎2_𝑎1𝑥−𝑎₁𝑎₂−𝑎₁𝑑𝑥+𝑐₁^𝑎3_𝑎2𝑎₃−𝑥𝑎₃−𝑎₂𝑑𝑥=1. (2.9) On integration, we get𝑐₁=2𝑎₃−𝑎₁. (2.10) The proportional probability density function corresponding to triangular fuzzy number 𝐴₁ is given by𝑓_𝑋𝐴₁(𝑥)=2𝑥−𝑎₁𝑎₂−𝑎₁𝑎₃−𝑎₁,𝑎₁≤𝑥≤𝑎₂,2𝑎₃−𝑥𝑎₃−𝑎₂𝑎₃−𝑎₁,𝑎₂≤𝑥≤𝑎₃,0,otherwise. (2.11) Graphically it is shown in Figure 1.

Figure 1 

Proportional probability density function of triangular fuzzy number.

Further, using the Mellin transform, we obtain𝑀_𝑋(𝑡)=^∞₀𝑥^𝑡−1𝑓_𝑋𝐴₁(𝑥)𝑑𝑥=^𝑎2_𝑎1𝑥^𝑡−12𝑥−𝑎₁𝑎₂−𝑎₁𝑎₃−𝑎₁𝑑𝑥+^𝑎3_𝑎2𝑥^𝑡−12𝑎₃−𝑥𝑎₃−𝑎₂𝑎₃−𝑎₁𝑑𝑥. (2.12) On integration, we obtain𝑀_𝑋𝐴₁(𝑡)=2𝑎₃−𝑎₁𝑡()𝑎𝑡+1³𝑎^𝑡₃−𝑎^𝑡₂𝑎₃−𝑎₂−𝑎₁𝑎^𝑡₂−𝑎^𝑡₁𝑎₂−𝑎₁. (2.13) Thus the mean (𝜇_𝑋𝐴₁) and variance (𝜎²_𝑋𝐴₁) of the random variable 𝑋𝐴₁ can be obtained as𝜇_𝑋𝐴₁𝑋𝐴=𝔼₁=𝑀_𝑋𝐴₁(2)=𝑎₁+𝑎₂+𝑎₃3𝜎_2𝑋,𝐴₁=𝑀_𝑋𝐴₁(3)−𝑀_𝑋𝐴₁(2)²=𝑎²₁+𝑎²₂+𝑎²₃−𝑎₁𝑎₂−𝑎₂𝑎₃−𝑎₃𝑎₁.18 (2.14)(2.15) Further, the probability density function corresponding to trapezoidal fuzzy number 𝐴₂=(𝑎₁,𝑎₂,𝑎₃,𝑎₄) is given as 𝑓𝐴₂(𝑥)=𝑐₂𝜇𝐴₂(𝑥), where 𝜇𝐴₂(𝑥) is defined as𝜇𝐴₂(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,1,𝑎₂≤𝑥≤𝑎₃,𝑎₄−𝑥𝑎₄−𝑎₃,𝑎₃≤𝑥≤𝑎₄,0,otherwise. (2.16) Now 𝑐₂ is calculated as^∞_−∞𝑓𝐴₂(𝑥)𝑑𝑥=1, (2.17) that is,^∞_−∞𝑐₂𝜇𝐴₁(𝑥)𝑑𝑥=1, (2.18) that is,𝑐₂^𝑎2_𝑎1𝑥−𝑎₁𝑎₂−𝑎₁𝑑𝑥+𝑐₂^𝑎3_𝑎2𝑑𝑥+𝑐₂^𝑎4_𝑎3𝑎₄−𝑥𝑎₄−𝑎₃𝑑𝑥=1. (2.19) On integration, we get𝑐₂=2𝑎₄+𝑎₃−𝑎₁−𝑎₂. (2.20) The proportional probability density function corresponding to triangular fuzzy number 𝐴₁ is given by𝑓_𝑋𝐴₂(𝑥)=2𝑥−𝑎₁𝑎₂−𝑎₁𝑎₄+𝑎₃−𝑎₁−𝑎₂,𝑎₁≤𝑥≤𝑎₂,2𝑎₄+𝑎₃−𝑎₁−𝑎₂,𝑎₂≤𝑥≤𝑎₃,2𝑎₄−𝑥𝑎₄−𝑎₃𝑎₄+𝑎₃−𝑎₁−𝑎₂,𝑎₃≤𝑥≤𝑎₄,0,otherwise. (2.21) Graphically it is shown in Figure 2.

Figure 2 

Proportional probability density function of trapezoidal fuzzy number.

Using the Mellin transform, we get𝑀𝐴₂(𝑡)=^∞₀𝑥^𝑡−1𝑓_𝑋𝐴₂(𝑥)𝑑𝑥=^𝑎2_𝑎1𝑥^𝑡−12𝑥−𝑎₁𝑎₂−𝑎₁𝑎₄+𝑎₃−𝑎₁−𝑎₂+𝑑𝑥_𝑎3𝑎2𝑥^𝑡−12𝑎₄+𝑎₃−𝑎₁−𝑎₂𝑑𝑥+^𝑎4_𝑎3𝑥^𝑡−12𝑎₄−𝑥𝑎₄−𝑎₃𝑎₄+𝑎₃−𝑎₁−𝑎₂𝑑𝑥. (2.22) On integration, we obtain𝑀𝐴₂(𝑡)=2𝑎₄+𝑎₃−𝑎₁−𝑎₂𝑡()𝑎𝑡+1^4𝑡+1−𝑎₃^𝑡+1𝑎₄−𝑎₃−𝑎₂^𝑡+1−𝑎₁^𝑡+1𝑎₂−𝑎₁. (2.23) Thus, the mean (𝜇_𝑋𝐴₂) and variance (𝜎²_𝑋𝐴₂) of the random variable 𝑋𝐴₂ can be obtained as𝜇_𝑋𝐴₂𝑋𝐴=𝔼₂=𝑀_𝑋𝐴₂(2)=13𝑎₁+𝑎₂+𝑎₃+𝑎₄+𝑎₁𝑎₂−𝑎₃𝑎₄𝑎₄+𝑎₃−𝑎₂−𝑎₁𝜎_2𝑋,𝐴₂=𝑀_𝑋𝐴₂(3)−𝑀_𝑋𝐴₂(2)²=16𝑎²₁+𝑎²₂+𝑎²₃+𝑎²₄+𝑎₁+𝑎₂𝑎²₃+𝑎²₄−𝑎₃+𝑎₄𝑎²₁+𝑎²₂𝑎₄+𝑎₃−𝑎₂−𝑎₁−𝜇_𝑋𝐴₂². (2.24)(2.25)

2.3. Probabilistic Fuzzy Quadratic Programming Problem and Its Crisp Model

Let 𝑐_𝑗̃=(𝑐¹_𝑗,𝑐²_𝑗,𝑐³_𝑗), 𝑗=1,2,…,𝑛, 𝑎_𝑖𝑗=(𝑎¹_𝑖𝑗,𝑎²_𝑖𝑗,𝑎³_𝑖𝑗), 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, and 𝑞_𝑖𝑗=(𝑞¹_𝑖𝑗,𝑞²_𝑖𝑗,𝑞³_𝑖𝑗), 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑛 be triangular fuzzy numbers. The crisp values of these fuzzy numbers obtained by using the method of defuzzification with probability density function of given membership function as described in Section 2 are given as𝑐_𝑗̂=𝑐¹_𝑗+𝑐²_𝑗+𝑐³_𝑗3,𝑗=1,2,…,𝑛,𝑎_𝑖𝑗=𝑎¹_𝑖𝑗+𝑎²_𝑖𝑗+𝑎³_𝑖𝑗3,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,𝑞_𝑖𝑗𝑞¹_𝑖𝑗+𝑞²_𝑖𝑗+𝑞³_𝑖𝑗3,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑛, (2.26) where the symbol 𝑐_𝑗̂ represents the crisp value of the given fuzzy number 𝑐_𝑗̃ and so on.

Similarly, if all the coefficients are trapezoidal fuzzy numbers such as, 𝑐_𝑗̃=(𝑐¹_𝑗,𝑐²_𝑗,𝑐³_𝑗,𝑐⁴_𝑗), 𝑗=1,2,…,𝑛, 𝑎_𝑖𝑗=(𝑎¹_𝑖𝑗,𝑎²_𝑖𝑗,𝑎³_𝑖𝑗,𝑎⁴_𝑖𝑗), 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, and 𝑞_𝑖𝑗=(𝑞¹_𝑖𝑗,𝑞²_𝑖𝑗,𝑞³_𝑖𝑗,𝑞⁴_𝑖𝑗), 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑛, then the crisp values are given as𝑐_𝑗̂=13𝑐¹_𝑗+𝑐²_𝑗+𝑐³_𝑗+𝑐⁴_𝑗+𝑐¹_𝑗𝑐²_𝑗−𝑐³_𝑗𝑐⁴_𝑗𝑐³_𝑗+𝑐⁴_𝑗−𝑐¹_𝑗−𝑐²_𝑗,𝑗=1,2,…,𝑛,𝑎_𝑖𝑗=13𝑎¹_𝑖𝑗+𝑎²_𝑖𝑗+𝑎³_𝑖𝑗+𝑎⁴_𝑖𝑗+𝑎¹_𝑖𝑗𝑎²_𝑖𝑗−𝑎³_𝑖𝑗𝑎⁴_𝑖𝑗𝑎³_𝑖𝑗+𝑎⁴_𝑖𝑗−𝑎¹_𝑖𝑗−𝑎²_𝑖𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛𝑞_𝑖𝑗=13𝑞¹_𝑖𝑗+𝑞²_𝑖𝑗+𝑞³_𝑖𝑗+𝑞⁴_𝑖𝑗+𝑞¹_𝑖𝑗𝑞²_𝑖𝑗−𝑞³_𝑖𝑗𝑞⁴_𝑖𝑗𝑞³_𝑖𝑗+𝑞⁴_𝑖𝑗−𝑞¹_𝑖𝑗−𝑞²_𝑖𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑛., (2.27)

Thus, the probabilistic quadratic programming problem having crisp objective function can be stated asmin∶𝑧=^𝑛_𝑗=1𝑐_𝑗̂𝑥_𝑗+12^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝑗𝑥_𝑖𝑥_𝑗 (2.28) subject toPr^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑏_𝑖≥1−𝛼_𝑖𝑥,𝑖=1,2,…,𝑚,_𝑗≥0,𝑗=1,2,…,𝑛, (2.29)(2.30) where 0<𝛼_𝑖<1, 𝑖=1,2,…,𝑚.

2.4. Deterministic Model of the Probabilistic Quadratic Programming Problem

It is assumed that the 𝑖th random variables 𝑏_𝑖, 𝑖=1,2,…,𝑚 follow the Weibull distribution [14]. The probability density function (pdf) of the random variables 𝑏_𝑖 is given by𝑓𝑏_𝑖=𝑝_𝑖𝑠_𝑖𝑏_𝑖𝑠_𝑖^𝑝𝑖−1𝑒^{−(𝑏𝑖/𝑠𝑖)𝑝𝑖}if𝑏_𝑖≥0,𝑝_𝑖>0,𝑠_𝑖>0,0,elsewhere, (2.31) with𝐸𝑏_𝑖=𝑠_𝑖Γ11+𝑝_𝑖,𝑝_𝑖>0,𝑠_𝑖𝑏>0,𝑖=1,2,…,𝑚,Var_𝑖=𝑠²_𝑖Γ21+𝑝_𝑖−𝑠_𝑖Γ11+𝑝_𝑖²,𝑝_𝑖>0,𝑠_𝑖>0,𝑖=1,2,…,𝑚, (2.32)(2.33) where 𝑝_𝑖, 𝑖=1,2,…,𝑚, and 𝑠_𝑖, 𝑖=1,2,…,𝑚, are called shape parameters and scale parameters, respectively.

Now, using pdf of the Weibull distribution, the 𝑖th probability constraint can be written as^∞_𝑦𝑖𝑓𝑏_𝑖𝑑𝑏_𝑖≥1−𝛼_𝑖,𝑖=1,2,…,𝑚, (2.34) where 𝑦_𝑖=∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗 and 𝑦_𝑖≥0.

It can be further written as^∞_𝑦𝑖𝑝_𝑖𝑠_𝑖𝑏_𝑖𝑠_𝑖^𝑝𝑖−1𝑒^{−(𝑏𝑖/𝑠𝑖)𝑝𝑖}𝑑𝑏_𝑖≥1−𝛼_𝑖,𝑖=1,2,…,𝑚, (2.35) On integration, we obtain𝑒^{−(𝑦𝑖/𝑠𝑖)𝑝𝑖}≥1−𝛼_𝑖,𝑖=1,2,…,𝑚. (2.36) It can be further simplified as𝑦_𝑖≤𝑠_𝑖ln1−𝛼_𝑖^1/𝑝𝑖,𝑖=1,2,…,𝑚, (2.37) that is, ^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑠_𝑖ln1−𝛼_𝑖^1/𝑝𝑖,𝑖=1,2,…,𝑚, (2.38) Thus, the equivalent deterministic model of the probabilistic quadratic programming problem (2.28)–(2.30) can be stated asmin∶𝑧=^𝑛_𝑗=1𝑐_𝑗̂𝑥_𝑗+12^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝑗𝑥_𝑖𝑥_𝑗 (2.39) subject to^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑠_𝑖ln1−𝛼_𝑖^1/p𝑖𝑥,𝑖=1,2,…,𝑚,_𝑗≥0,𝑗=1,2,…,𝑛. (2.40)

3. Numerical Example

Let us consider the following probabilistic fuzzy quadratic programming problem:min∶𝑧=26𝑥₁+35𝑥₂+128𝑥²₁−6𝑥₁𝑥₂+8𝑥²₂ (3.1) subject toPr13𝑥₁+25𝑥₂≤𝑏₁≥0.95,Pr17𝑥₁+23𝑥₂≤𝑏₁≥0.85,13𝑥₁+15𝑥₂≥𝑥25,₁,𝑥₂≥0, (3.2) where 𝛼₁=0.05 and 𝛼₁=0.15 are specified probability levels. The fuzzy coefficients 26=(24,26,27), 35=(32,35,37), and 8=(5,8,10), 6=(4,6,7), 8=(6,8,11) are defined by triangular fuzzy numbers. But the fuzzy coefficients 13=(10,12,14,17), 25=(22,24,25,27), 17=(14,16,18,22), 23=(21,23,24,26), 13=(10,12,15,17), 15=(12,14,16,20), and 25=(23,25,26,28) are defined by trapezoidal fuzzy numbers. The right-hand side parameters 𝑏₁ and 𝑏₂ follow the Weibull distribution with known parameters. The parameters are given as𝑝₁=12,𝑝₂=14,𝑞₁=0.12,𝑞₂=0.03. (3.3) The crisp values of the above triangular and trapezoidal fuzzy numbers are calculated by using (2.14) and (2.24). That is, 26=(24+26+27)/3=25.67, 35=(32+35+37)/3=34.67, 8=(5+8+10)/3=7.67, 6=(4+6+7)/3=5.67, 8=(6+8+11)/3=8.33, 13=(1/3)[(10+12+14+17)+(10×12−14×17)/(14+17−10−12)]=13.36, 25=(1/3)[(22+24+25+27)+(22×24−25×27)/(25+27−22−24)]=24.5, 17=(1/3)[(14+16+18+22)+(14×16−18×22)/(18+22−14−16)]=17.6, 23=(1/3)[(21+23+24+26)+(21×23−24×26)/(44+26−21−23)]=23.5, 13=(1/3)[(10+12+15+17)+(10×12−15×17)/(15+17−10−12)]=13.5, 15=(1/3)[(12+14+17+20)+(12×14−16×20)/(16+20−14−12)]=15.6, 25=(1/3)[(23+25+26+28)+(23×25−26×28)/(26+28−23−25)]=25.5.

Now, using the parameter values of the random variables, crisp values of the triangular and trapezoidal fuzzy numbers, and the deterministic constraint (2.38), we formulate an equivalent deterministic crisp model of the given probabilistic fuzzy quadratic programming model problem:min∶𝑧=25.67𝑥₁+34.67𝑥₂+128.33𝑥²₁−5.67𝑥₁𝑥₂+7.67𝑥²₂ (3.4) subject to13.36𝑥₁+24.5𝑥₂≤45.610034,17.6𝑥₁+23.5𝑥₂≤42.979943,13.5𝑥₁+15.6𝑥₂𝑥≥25.5,₁,𝑥₂≥0. (3.5) The above deterministic crisp quadratic programming problem is solved using LINGO 11.0 [15] software. The optimal solution is obtained as 𝑥₁=0.9623144, 𝑥₂=0.9394838, and 𝑧=62.91564.

4. Conclusions

Quadratic programming is one of the important techniques for improving the efficiency and increasing the productivity of business companies and public organizations. It is also applied to some of the managerial decision making problems such as demand-supply response and enterprise selection. In this paper, a single-objective probabilistic fuzzy quadratic programming problem is presented where both fuzziness and randomness are involved within the quadratic programming framework. The present work can be extended in the multiobjective framework considering the parameters as different random variables and other fuzzy numbers depending on the problems.