extension of wolfe method for solving quadratic...

Research ArticleExtension of Wolfe Method for Solving QuadraticProgramming with Interval Coefficients

Syaripuddin,1 Herry Suprajitno,2 and Fatmawati2

1Department of Mathematics, Faculty of Mathematics and Natural Sciences, Mulawarman University,Kampus Gunung Kelua, Jl. Barong Tongkok, Samarinda 1068, Indonesia2Department of Mathematics, Faculty of Science and Technology, Airlangga University, Kampus C Unair,Jl. Mulyorejo, Surabaya 60115, Indonesia

Correspondence should be addressed to Fatmawati; [email protected]

Received 10 April 2017; Accepted 1 August 2017; Published 14 September 2017

Academic Editor: Frank Werner

Copyright © 2017 Syaripuddin et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quadratic programming with interval coefficients developed to overcome cases in classic quadratic programming where thecoefficient value is unknown and must be estimated. This paper discusses the extension of Wolfe method. The extended Wolfemethod can be used to solve quadratic programming with interval coefficients.The extension process ofWolfe method involves thetransformation of the quadratic programming with interval coefficients model into linear programming with interval coefficientsmodel. The next step is transforming linear programming with interval coefficients model into two classic linear programmingmodels with special characteristics, namely, the optimum best and the worst optimum problem.

1. Introduction

Quadratic programming is a special form of nonlinear pro-grammingwhich has special characteristics; that is, the objec-tive function is in quadratic forms and constraint functionsare linear form [1]. Although the quadratic programming ispart of nonlinear programming, the completion is still adopt-ing some linear programming problem solving methods, oneof which is the Wolfe method. This method transforms thequadratic programming problem into a linear programmingproblem. Wolfe [2] modified the simplex method to solvequadratic programming problem by adding conditions ofthe Karush-Kuhn-Tucker (KKT) and changing the objectivefunction of quadratic forms into a linear form.

Interval quadratic programming is a development of theclassic quadratic programming that utilizes interval analysistheory developed by Moore [3]. This development aim is toaccommodate cases which contain the uncertainty, that is,when the data value is unknown for certain, but the datalies within an interval where the values of the upper limitand lower limit are known. The special characteristics of theinterval quadratic programming problem are the coefficients

of the objective function and constraint functions are in theinterval form.

Research on quadratic programming with interval coef-ficients has been conducted by Liu and Wang [4]. However,the coefficients of quadratic forms in the objective functionon the developed model are not in the interval form yet.Furthermore, Li and Tian [5] generalized the model in [4]by assuming that the quadratic coefficients of the objectivefunction are in the interval form. References [4, 5] used theduality theory to create a method of solving the quadraticprogramming with interval coefficients. Quadratic program-ming model with interval coefficients is transformed intotwo classic quadratic programming models with the specialcharacteristics, called the best optimum and worst optimumproblem. A completion method was developed based onthe method of solving the linear programming with intervalcoefficients that have been discussed by some researchers [6–10].

This paper will discuss the extension of Wolfe methodto solve quadratic programming with interval coefficients.Therefore, this article will focus on how to transform thequadratic programming with interval coefficients into linear

HindawiJournal of Applied MathematicsVolume 2017, Article ID 9037857, 6 pageshttps://doi.org/10.1155/2017/9037857

https://doi.org/10.1155/2017/9037857

2 Journal of Applied Mathematics

programming with interval coefficients. Furthermore, thelinear programmingwith interval coefficients which has beenobtained from the transformation will be solved by using themethod in [8].

This paper is organized as follows. Section 2 discussesinterval arithmetic operations. In Section 3, a general formof linear programming with interval coefficients is stated. InSection 4, a general form of quadratic programming withinterval coefficients is stated. Extension of Wolfe methodas one method of solving the quadratic programming withinterval coefficients is discussed in Section 5, whereas Sec-tion 6 discusses numerical examples, and Section 7 providessome concluding remarks.

2. Interval Arithmetic

The basic definition and properties of interval number andinterval arithmetic can be seen at Moore [3], Alefeld andHerzberg [11], and Hansen [12].

Definition 1. A closed real interval 𝑥 = [𝑥𝐼, 𝑥𝑆] denoted by 𝑥is a real interval number which can be defined completely by

𝑥 = [𝑥𝐼, 𝑥𝑆] = {𝑥 ∈ R | 𝑥𝐼 ≤ 𝑥 ≤ 𝑥𝑆; 𝑥𝐼, 𝑥𝑆 ∈ R} , (1)

where 𝑥𝐼 and 𝑥𝑆 are called infimum and supremum, respec-tively.

Definition 2. A real interval number 𝑥 = [𝑥𝐼, 𝑥𝑆] is calleddegenerate, if 𝑥𝐼 = 𝑥𝑆.Definition 3. Let 𝑥 = [𝑥𝐼, 𝑥𝑆] and 𝑦 = [𝑦𝐼, 𝑦𝑆]; then

1. 𝑥 + 𝑦 = [𝑥𝐼 + 𝑦𝐼, 𝑥𝑆 + 𝑦𝑆] (addition),2. 𝑥 − 𝑦 = [𝑥𝐼, 𝑥𝑆] − [𝑦𝐼, 𝑦𝑆] = [𝑥𝐼, 𝑥𝑆] + [−𝑦𝑆, −𝑦𝐼] =[𝑥𝐼 − 𝑦𝑆, 𝑥𝑆 − 𝑦𝐼] (subtraction),3. 𝑥 ⋅ 𝑦 = [min{𝑥𝐼𝑦𝐼, 𝑥𝐼𝑦𝑆, 𝑥𝑆𝑦𝐼, 𝑥𝑆𝑦𝑆}, max{𝑥𝐼𝑦𝐼, 𝑥𝐼𝑦𝑆,𝑥𝑆𝑦𝐼, 𝑥𝑆𝑦𝑆}] (multiplication),4. 𝑥/𝑦 = 𝑥(1/𝑦) = [𝑥𝐼, 𝑥𝑆][1/𝑦𝑆, 1/𝑦𝐼], 0 ∉ 𝑦 (division).

3. Linear Programming withInterval Coefficients

The general form of linear programming with interval coeffi-cients is defined as follows:

Maximize 𝑧 = 𝑛∑𝑗=1

[𝑐𝑗𝐼, 𝑐𝑗𝑆] 𝑥𝑗 (2a)

subject to

𝑛∑𝑗=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝑥𝑗 ≤ [𝑏𝑖𝐼, 𝑏𝑖𝑆] , 𝑖 = 1, 2, . . . , 𝑚 (2b)

𝑥𝑗 ≥ 0, 𝑗 = 1, 2, . . . , 𝑛, (2c)

where 𝑥𝑗 ∈ R, [𝑐𝑗𝐼, 𝑐𝑗𝑆], [𝑏𝑖𝐼, 𝑏𝑖𝑆], and [𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] ∈ 𝐼(R).

The model in (2a)–(2c) is solved by means of transform-ing the linear programmingwith interval coefficients into twoclassic linear programming models with the special charac-teristics, namely, the best optimum and the worst optimumproblems. The best optimum problem has properties of bestversion on the objective function andmaximum feasible areaon the constraint function. On the other hand, the worstoptimum problem has a characteristic that it is the worstversion of the objective function and the minimum feasiblearea of the constraint function.

Chinneck and Ramadan [8] provide a rule to determinethe best and worst optimum problem in a linear program-ming problem with interval coefficients. The constraints oflinear programming with interval coefficients which havean inequality sign (≤) in (2b) have characteristics of themaximum feasible area and theminimum feasible area whichis given by the following theorem.

Theorem 4 (Chinneck and Ramadan [8]). Suppose that wehave an interval inequality given ∑𝑛𝑗=1[𝑎𝑗𝐼, 𝑎𝑗𝑆]𝑥𝑗 ≤ [𝑏𝐼, 𝑏𝑆],where 𝑥𝑗 ≥ 0. Then, ∑𝑛𝑗=1 𝑎𝑗𝐼𝑥𝑗 ≤ 𝑏𝑆 is maximum feasible areaand ∑𝑛𝑗=1 𝑎𝑗𝑆𝑥𝑗 ≤ 𝑏𝐼 is minimum feasible area.

The objective function of linear programming withinterval coefficients for the case of maximizing (2a) hascharacteristics of the best version and the worst version of theobjective function is expressed in the following theorem.

Theorem 5 (Chinneck and Ramadan [8]). If 𝑧 = ∑𝑛𝑗=1[𝑐𝑗𝐼,𝑐𝑗𝑆]𝑥𝑗 is the objective function for 𝑥𝑗 ≥ 0, then ∑𝑛𝑗=1 𝑐𝑗𝑆𝑥𝑗 ≥∑𝑛𝑗=1 𝑐𝑗𝐼𝑥𝑗, where ∑𝑛𝑗=1 𝑐𝑗𝑆𝑥𝑗 is the best version of the objectivefunction and ∑𝑛𝑗=1 𝑐𝑗𝐼𝑥𝑗 is the worst version of the objectivefunction.

4. Quadratic Programming withInterval Coefficients

The general form of quadratic programming with intervalcoefficients introduced by Li and Tian [5] is defined asfollows:

Maximize 𝑧= 𝑛∑𝑗=1

[𝑐𝑗𝐼, 𝑐𝑗𝑆] 𝑥𝑗+ 12𝑛∑𝑗=1

𝑛∑𝑘=1

[𝑞𝑗𝑘𝐼, 𝑞𝑗𝑘𝑆] 𝑥𝑗𝑥𝑘(3a)

subject to

𝑛∑𝑗=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝑥𝑗 ≤ [𝑏𝑖𝐼, 𝑏𝑖𝑆] , 𝑖 = 1, 2, . . . , 𝑚 (3b)

𝑥𝑗 ≥ 0, 𝑗 = 1, 2, . . . , 𝑛, (3c)

Journal of Applied Mathematics 3

where 𝑥𝑗 ∈ R, [𝑐𝑗𝐼, 𝑐𝑗𝑆], [𝑞𝑖𝑗𝐼, 𝑞𝑖𝑗𝑆], [𝑏𝑖𝐼, 𝑏𝑖𝑆], [𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] ∈ 𝐼(R),∑𝑛𝑗=1∑𝑛𝑘=1[𝑞𝑗𝑘𝐼, 𝑞𝑗𝑘𝑆]𝑥𝑗𝑥𝑘 is negative semidefinite, and 𝐼(R)are the set of all interval numbers inR.

The model as shown in (3a)–(3c) is a generalization ofthe model in [4]. The coefficient of the objective functionand constraints of the quadratic programming with intervalcoefficients model in (3a)–(3c) have an interval form. Theidea to solve the model is the extension of Wolfe method.This method focuses on how to transform the quadraticprogramming with interval coefficients in (3a)–(3c) intolinear programming with interval coefficients in (2a)–(2c).Furthermore, linear programming with interval coefficientsobtained from the transformation is done using the methodin [8].

Extension of Wolfe method is the main result in thispaper. The fundamental difference between the extensions ofWolfe method and the method in [5] is, on the extension ofWolfe method, quadratic programming model with intervalcoefficients is transformed into linear programming withinterval coefficients, while, in [5], the model of quadraticprogramming with interval coefficients is maintained.

5. Extension of Wolfe Method

Wolfe method is one method for solving quadratic pro-gramming problems by means of transforming the quadraticprogramming problems into a linear programming problem.Wolfe [2] modified the simplex method to solve quadraticprogramming problems by adding a requirement Karush-Kuhn-Tucker (KKT) and changing the quadratic objectivefunction into a linear objective function.

The extension of Wolfe method is used to solve quadraticprogramming problem with interval coefficients. Steps ofextension of Wolfe method are declared as follows.

Form of Lagrange function for the problem in (3a)–(3c)is

𝐿 (𝑥, 𝑦, 𝑟, 𝜆, 𝜇) = 𝑛∑𝑗=1

[𝑐𝑗𝐼, 𝑐𝑗𝑆] 𝑥𝑗 + 12⋅ 𝑛∑𝑗=1

𝑛∑𝑘=1

[𝑞𝑗𝑘𝐼, 𝑞𝑗𝑘𝑆] 𝑥𝑗𝑥𝑘

− 𝑚∑𝑖=1

𝜆𝑖( 𝑛∑𝑗=1

([𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝑥𝑗 − [𝑏𝑖𝐼, 𝑏𝑖𝑆] + 𝑦𝑖2))

− 𝑛∑𝑗=1

𝜇𝑗 (−𝑥𝑗 + 𝑟𝑗2) ,

(4)

where 𝜆𝑖, 𝑖 = 1, 2, . . . , 𝑚, 𝜇𝑗, 𝑗 = 1, 2, . . . , 𝑛, are Lagrangemultipliers and 𝐿(𝑥, 𝑦, 𝑟, 𝜆, 𝜇) is Lagrange function withinterval coefficients.

Local minimum points of the function 𝐿 were obtainedby the first partial derivatives of the function 𝐿 with respect

to the variables and equating to zero (KKT necessary condi-tions) (see [13, 14]).

𝜕𝐿𝜕𝑥𝑖 = [𝑐𝑗𝐼, 𝑐𝑗𝑆] + 𝑛∑𝑘=1

[𝑞𝑗𝑘𝐼, 𝑞𝑗𝑘𝑆] 𝑥𝑘− 𝑚∑𝑖=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝜆𝑖 + 𝜇𝑗 = 0,𝑗 = 1, 2, . . . , 𝑛,

(5a)

𝜕𝐿𝜕𝑦𝑖 = −2𝜆𝑖𝑦𝑖 = 0, 𝑖 = 1, 2, . . . , 𝑚, (5b)

𝜕𝐿𝜕𝑟𝑖 = −2𝜇𝑗𝑟𝑗 = 0, 𝑗 = 1, 2, . . . , 𝑛, (5c)

𝜕𝐿𝜕𝜆𝑖 =𝑛∑𝑗=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑠] 𝑥𝑗 + 𝑦𝑖2 − [𝑏𝑖𝐼, 𝑏𝑖𝑆]= 0, 𝑖 = 1, 2, . . . , 𝑚,

(5d)

𝜕𝐿𝜕𝜇𝑖 = 𝑥𝑗 − 𝑟2𝑗 = 0, 𝑗 = 1, 2, . . . , 𝑛, (5e)

𝑥𝑗, 𝜆𝑖, 𝜇𝑗, 𝑦𝑖, 𝑟𝑗 ≥ 0, 𝑖 = 1, 2, . . . , 𝑚, 𝑗 = 1, 2, . . . , 𝑛. (5f)

Results simplification of (5a)–(5f) is

− 𝑛∑𝑘=1

[𝑞𝑗𝑘𝐼, 𝑞𝑗𝑘𝑆] 𝑥𝑘 + 𝑚∑𝑖=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝜆𝑖 − 𝜇𝑗= [𝑐𝑗𝐼, 𝑐𝑗𝑆] , 𝑗 = 1, 2, . . . , 𝑛,

(6a)

𝑛∑𝑗=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑠] 𝑥𝑗 + 𝑠𝑖 = [𝑏𝑖𝐼, 𝑏𝑖𝑆] , 𝑖 = 1, 2, . . . , 𝑚, (6b)

𝑥𝑗, 𝜆𝑖, 𝜇𝑗, 𝑠𝑖 ≥ 0, 𝑗 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑚 (6c)

and satisfies the complementary conditions,

𝜇𝑗𝑥𝑗 = 0, 𝑗 = 1, 2, . . . , 𝑛, 𝜆𝑖𝑠𝑖 = 0, 𝑖 = 1, 2, . . . , 𝑚. (6d)

Add artificial variables V𝑗, 𝑗 = 1, 2, . . . , 𝑛, in (6a) for an initialbasis, as follows:

− 𝑛∑𝑘=1


[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝜆𝑖 − 𝜇𝑗 + V𝑗

= [𝑐𝑗𝐼, 𝑐𝑗𝑆] .(7)

Furthermore, create a linear programming with intervalcoefficients, where the objective function is to minimize thenumber of artificial variables V𝑗, 𝑗 = 1, 2, . . . , 𝑛, and constraintis (7), (6b), (6c), and (6d) obtained fromnecessary conditionsof KKT.

Minimize 𝑧 = V1 + V2 + ⋅ ⋅ ⋅ + V𝑛 (8a)


subject to,

− 𝑛∑𝑘=1


[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑆] 𝜆𝑖 − 𝜇𝑗 + V𝑗

= [𝑐𝑗𝐼, 𝑐𝑗𝑆] , 𝑗 = 1, 2, . . . , 𝑛(8b)

𝑛∑𝑗=1

[𝑎𝑖𝑗𝐼, 𝑎𝑖𝑗𝑠] 𝑥𝑗 + 𝑠𝑖 = [𝑏𝑖𝐼, 𝑏𝑖𝑆] , 𝑖 = 1, 2, . . . , 𝑚 (8c)

𝑥𝑗, 𝜆𝑖, 𝜇𝑗, 𝑠𝑖, V𝑗 ≥ 0, 𝑗 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑚 (8d)

satisfying the complementary conditions,

𝜇𝑗𝑥𝑗 = 0, 𝑗 = 1, 2, . . . , 𝑛,𝜆𝑖𝑠𝑖 = 0, 𝑖 = 1, 2, . . . , 𝑚, (8e)

where V𝑗 ≥ 0 is artificial variable.The model as shown in (8a)–(8e) is linear programming

with interval coefficients which is added by complementaryconditions. This model is the result of the transformationfrom the quadratic programming with interval coefficientsmodel by extension of Wolfe method.

The next step, linear programming with interval coeffi-cients model in (8a)–(8e), was solved by transforming intotwo linear programming cases with the special characteris-tics, namely, the best and the worst optimum problem. Thetransformation process can be written in Algorithm 6 asfollows.

Algorithm 6.

(1) Given a quadratic programming problem with inter-val coefficients in (3a)–(3c), Extension of Wolfemethod is based on (3a)–(3c) equivalent to the linearprogramming with interval coefficients in (8a)–(8e).

(2) Use Theorems 4 and 5 for transforming the linearprogramming with interval coefficients in (8a)–(8e)into two classic linear programming models withspecial characteristics; namely,

(a) the best optimum problem is

Minimize 𝑧𝑆 = V1 + V2 + ⋅ ⋅ ⋅ + V𝑛 (9a)

subject to,

− 𝑛∑𝑘=1

𝑞𝑗𝑘𝑆𝑥𝑘 + 𝑚∑𝑖=1

𝑎𝑖𝑗𝐼𝜆𝑖 − 𝜇𝑗 + V𝑗 = 𝑐𝑗𝑆,𝑗 = 1, 2, . . . , 𝑛

(9b)

𝑛∑𝑗=1

𝑎𝑖𝑗𝐼𝑥𝑗 + 𝑠𝑖 = 𝑏𝑖𝑆,𝑖 = 1, 2, . . . , 𝑚

(9c)

𝑥𝑗, 𝜆𝑖, 𝜇𝑗, 𝑠𝑖, V𝑗 ≥ 0,𝑗 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑚 (9d)


𝜇𝑗𝑥𝑗 = 0, 𝑗 = 1, 2, . . . , 𝑛,𝜆𝑖𝑠𝑖 = 0, 𝑖 = 1, 2, . . . , 𝑚, (9e)

(b) the worst optimum problem is

Minimize 𝑧𝐼 = V1 + V2 + ⋅ ⋅ ⋅ + V𝑛 (10a)

subject to,

− 𝑛∑𝑘=1

𝑞𝑗𝑘𝐼𝑥𝑘 + 𝑚∑𝑖=1

𝑎𝑖𝑗𝑆𝜆𝑖 − 𝜇𝑗 + V𝑗 = 𝑐𝑗𝐼,𝑗 = 1, 2, . . . , 𝑛

(10b)

𝑛∑𝑗=1

𝑎𝑖𝑗𝑆𝑥𝑗 + 𝑠𝑖 = 𝑏𝑖𝐼,𝑖 = 1, 2, . . . , 𝑚

(10c)

𝑥𝑗, 𝜆𝑖, 𝜇𝑗, 𝑠𝑖, V𝑗 ≥ 0,𝑗 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑚 (10d)


𝜇𝑗𝑥𝑗 = 0, 𝑗 = 1, 2, . . . , 𝑛,𝜆𝑖𝑠𝑖 = 0, 𝑖 = 1, 2, . . . , 𝑚. (10e)

(3) The optimum value of the quadratic programmingwith interval coefficients is obtained by combining theoptimum value from the worst and the best optimumproblem; that is, 𝑧 = [𝑧𝐼, 𝑧𝑆].

Algorithm 6 shows that the best and the worst optimumproblem are linear programming models added by comple-mentary conditions. Thus, both problems can be solved bysimplex method.

6. Numerical Example

Consider the following example of quadratic programmingwith interval coefficients in the journal Li and Tian [5].

Minimize 𝑧= [−10, −6] 𝑥1 + [2, 3] 𝑥2 + [−1, 1] 𝑥1𝑥2

+ [4, 10] 𝑥21 + [10, 20] 𝑥22(11a)

subject to

[1, 2] 𝑥1 + 3𝑥2 ≤ [1, 10] (11b)

[−2, 8] 𝑥1 + [4, 6] 𝑥2 ≤ [4, 6] (11c)

𝑥1, 𝑥2 ≥ 0. (11d)

Journal of Applied Mathematics 5

Table 1: Two classic linear programming models with special characteristics.

The best optimum problem The worst optimum problem(1) Classic linear programming model (2) Classic linear programming modelMinimize 𝑧𝑆 = V1 + V2 Minimize 𝑧𝐼 = V1 + V2subject to subject to8𝑥1 − 𝑥2 + 𝜆1 − 2𝜆2 − 𝜇1 + V1 = 10 20𝑥1 + 𝑥2 + 2𝜆1 + 8𝜆2 − 𝜇1 + V1 = 6𝑥1 − 20𝑥2 − 3𝜆1 − 4𝜆2 + 𝜇2 + V2 = 2 −𝑥1 − 40𝑥2 − 3𝜆1 − 6𝜆2 + 𝜇2 + V2 = 3𝑥1 + 3𝑥2 + 𝑠1 = 10 2𝑥1 + 3𝑥2 + 𝑠1 = 1−2𝑥1 + 4𝑥2 + 𝑠2 = 6 8𝑥1 + 6𝑥2 + 𝑠2 = 4𝑥𝑖, 𝜆𝑖, 𝜇𝑖, 𝑠𝑖, V𝑖 ≥ 0, 𝑖 = 1, 2 𝑥𝑖, 𝜆𝑖, 𝜇𝑖, 𝑠𝑖, V𝑖 ≥ 0, 𝑖 = 1, 2satisfying complementary conditions: satisfying complementary conditions:𝜆1𝑠1 = 0, 𝜆2𝑠2 = 0, and 𝜇1𝑥1 = 0, 𝜇2𝑥2 = 0 𝜆1𝑠1 = 0, 𝜆2𝑠2 = 0, and 𝜇1𝑥1 = 0, 𝜇2𝑥2 = 0Solution: 𝑧𝑆 = 6.25, 𝑥1 = 1, 25, and 𝑥2 = 0. Solution: 𝑧𝐼 = 0.9, 𝑥1 = 0.3, and 𝑥2 = 0

According to Li and Tian [5], for the solution of the model in(11a)–(11c), the best optimum problem is 𝑧𝑆 = −0.9, 𝑥1 = 0.3,and 𝑥2 = 0, the worst optimum problem is 𝑧𝐼 = −6.25, 𝑥1 =1, 25, and 𝑥2 = 0, and the optimum value is 𝑧 = [𝑧𝐼, 𝑧𝑆] =[−6.25, −0.9].

This paper presents only the maximization problemso that any minimization problem will be converted intomaximization problem, the simple procedure to convert aminimization problem to a maximization problem and viceversa. Simply multiply the objective function of a minimiza-tion problemby−1 converting it into amaximization problemand vice versa.

Maximize 𝑧= [6, 10] 𝑥1 + [−3, −2] 𝑥2

+ [−1, 1] 𝑥1𝑥2 + [−10, −4] 𝑥21+ [−20, −10] 𝑥22

(12a)

subject to

[1, 2] 𝑥1 + 3𝑥2 ≤ [1, 10] (12b)

[−2, 8] 𝑥1 + [4, 6] 𝑥2 ≤ [4, 6] (12c)

𝑥1, 𝑥2 ≥ 0. (12d)

We apply the extension of Wolfe method for transformingquadratic programming with interval coefficients model in((12a)–(12d)) into linear programming with interval coeffi-cients model. We have

Minimize 𝑧 = V1 + V2 (13a)

subject to

[8, 20] 𝑥1 + [−1, 1] 𝑥2 + [1, 2] 𝜆1 + [−2, 8] 𝜆2 − 𝜇1+ V1 = [6, 10] (13b)

[−1, 1] 𝑥1 + [20, 40] 𝑥2 + [3, 3] 𝜆1 + [4, 6] 𝜆2 − 𝜇2+ V2 = [−3, −2] (13c)

[1, 2] 𝑥1 + 3𝑥2 ≤ [1, 10] (13d)

[−2, 8] 𝑥1 + [4, 6] 𝑥2 ≤ [4, 6] (13e)

𝑥1, 𝑥2 ≥ 0. (13f)

We apply Algorithm 6 for transforming linear programmingwith interval coefficients model in ((13a)–(13f)) into two clas-sic linear programming models with special characteristics,namely, the best optimum and the worst optimum problem.The result of the transformation is shown in Table 1.

So, the optimum value of the quadratic programmingwith interval coefficients is obtained by combining the opti-mum value from the worst and the best optimum problem;that is, 𝑧 = [𝑧𝐼, 𝑧𝑆] = [0.9, 6.25]. This solution gives the samevalue as obtained by Li and Tian [5].

7. Conclusion

This paper presents an extension of Wolfe method. Theextension of Wolfe method performed by transforming thequadratic programming with interval coefficients modelinto linear programming with interval coefficients model.Furthermore, linear programming with interval coefficientsmodel is transformed into two classic linear programmingmodels using Algorithm 6. The extension of Wolfe methodhas a particular benefit: the final model is linear program-ming. Hence, it can be solved by the simplex method.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

References

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[10] H. Suprajitno and I. B.Mohd, “Interval linear programming,” inProceedings of ICOMS-3, Bogor, Indonesia, 2008.

[11] G. Alefeld and J. Herzberger, Introduction to Interval Computa-tions, Academic Press, New York, NY, USA, 1983.

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[13] J. Zhang, “Optimality condition and wolfe duality for invexinterval-valued nonlinear programming problems,” Journal ofApplied Mathematics, vol. 2013, Article ID 641345, 2013.

[14] H.-C. Wu, “The Karush-KUHn-Tucker optimality conditionsin an optimization problem with interval-valued objectivefunction,” European Journal of Operational Research, vol. 176,no. 1, pp. 46–59, 2007.

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