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Current Research & Reviews

Research Article Volume 1 Issue 3

Neutrosophic number goal programming for multi-objective linear programming problem in neutrosophic number environment

Surapati Pramanik,1 Durga Banerjee2

1Department of Mathematics, Nandalal Ghosh B T College, India
2Department of Mathematics, Ranaghat Yusuf Institution, India

Correspondence: Surapati Pramanaik, Department of Mathematics, Nandalal Ghosh B T College, Panpur, P.O. Narayanpur, District. North 24 Parganas, PIN- 743126, West Bengal, India, Tel +919477035544

Received: May 29, 2018 | Published: June 22, 2018

Citation: Pramanik S, Banerjee D. Neutrosophic number goal programming for multi-objective linear programming problem in neutrosophic number environment. MOJ Curr Res & Rev. 2018;1(3):135-141. DOI: 10.15406/mojcrr.2018.01.00021

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Abstract

Purpose: The purpose of the paper is to propose goal programming strategy to multi-objective linear programming problem with neutrosophic numbers which we call NN-GP. The coefficients of objective functions and the constraints are considered as neutrosophic numbers of the form (m+nI), where m, n are real numbers and I denotes indeterminacy.

Design: For this study, the neutrosophic numbers are converted into interval numbers. Then, the problem reduces to multi-objective linear interval programming problem. Employing interval programming technique, the target interval of the objective function is determined. For the sake of achieving the target goals, the goal achievement functions are constructed. Three new neutrosophic goal programming models are developed using deviational variables to solve the reduced problem.

Findings: Realistic optimization problem involves multiple objectives. Crisp multi-objective optimization problems involve deterministic objective functions and/or constrained functions. However, uncertainty involves in real problems. Hence, several strategies dealing with uncertain multi-objective programming problems have been proposed in the literature. Multi-objective linear programming has evolved along with different paradigms and in different environment. Goal programming and fuzzy goal programming have been widely used to solve the multi-objective linear programming problems. In this paper goal programming in neutrosophic number environment has been developed. It deals with effectively multi-objective linear programming problem with neutrosophic numbers. We solve a numerical example to illustrate the proposed NN-GP strategy.

Originality: There are different Schools in optimization field and each has their own distinct strategy. In neutrosophic number environment goal programming for multi-objective programming problem is proposed here at first.

Keywords:Neutrosophic goal programming, fuzzy goal programming, Multi-objective programming, neutrosophic numbers

Introduction

In multi-criteria decision making (MCDM) process, multi-objective programming evolves in many directions. In multi-objective programming, several conflicting objective functions are simultaneously considered. When the objective functions and constraints both are linear, the multi-objective programming problem is considered as a linear multi-objective programming problem. If any objective function and/or constraint is nonlinear, then the problem is considered as a nonlinear multi-objective programming problem. Goal programming is a widely used strong mathematical tool to deal multi-objective mathematical programming problems. The idea of goal programming lies in the work of Chames, Cooper & Ferguson.1 Charnes & Cooper2 first coined the term goal programming to deal with infeasible linear programming in 1961. GP underlies a realistic satisficing philosophy. Charnes & Cooper,2 Ijiri,3 Lee,4 Ignizio,5 Romero,6 Schniederjans,7 Chang,8 Dey & Pramanik9 and many pioneer researchers established different approaches to goal programming in crisp environment. Inuguchi & Kume10 investigated interval goal programming. Narasimhan11 grounded the goal programming using deviational variables in fuzzy environment. Fuzzy goal programming (FGP) has been enriched by several authors such as Hannan,12 Ignizio,13 Tiwari, Dharma & Rao,14,15 Mohamed,16 Pramanik,17,18 Pramanik & Roy,19‒21 Pramanik & Dey,22 Pramanik et al.,23 Tabrizi, Shahanaghi & Jabalameli.24 Pramanik & Roy25‒27 studied fuzzy goal programming strategy for transportation problems. Pramanik & Roy28 presented goal programming in intuitionistic fuzzy environment, which is called intuitionistic FGP (IFGP). Pramanik & Roy29 studied IFGP approach in transportation problems. Pramanik & Roy30 employed IFGP to quality control problem. Pramanik, Dey & Roy31 studied bi-level programming problem in intuitionistic fuzzy environment. Razmi et al.,32 studied Pareto-optimal solutions for intuitionistic multi-objective programming problems. Smarandache33 developed neutrosophic set based on neutrosophy. Neutrosophic set33 accommodates inconsistency, incompleteness, indeterminacy in a new angle by introducing indeterminacy as independent component. Wang, Smarandache, Zhang, et al.,34 made neutrosophic theory popular by defining single valued neutrosophic set (SVNS) to deal with realistic problems. SVNS has been vigorously applied in different areas such as multi criteria/ attribute decision making problems35‒53, conflict resolution,54 educational problem,55‒56 data mining,57 social problem,58‒59 etc. Smarandache60‒61 defined neutrosophic number (NN) using indeterminacy as component and established its basic properties. The NN is expressed in the form m+nI, where m, n are real numbers and I represents indeterminacy. Several authors62‒66 applied NNs to decision making problems. Pramanik & Roy67 applied NNs to teacher selection problem. Ye68 developed linear programming strategy with NNs and discussed production planning problem. Ye69 developed nonlinear programming strategy in NN environment.

Banerjee & Pramanik70 first studied goal programming strategy for single objective linear programming problem and developed three neutrosophic goals programming with NNs. Multi-objective linear programming problem (MOLPP) with NNs is yet to appear in the literature. To fill the gap, we present goal programming strategy for multi-objective linear programming problem with neutrosophic numbers. The coefficients of objective functions and constraints are considered as NNs of the form (m+nI), where m, n are real numbers and I represents indeterminacy. The NNs are converted into interval numbers. The entire programming problem reduces to multi-objective linear interval programming problem. The target interval of the neutrosophic number function is formulated based on the technique of interval programming. Three new neutrosophic goal programming models are formulated. A numerical example is solved to illustrate the proposed NN-GP strategy. The remainder of the paper is presented as follows: Next section presents some basic discussion regarding neutrosophic set, NNs, interval numbers. Then the following section recalls interval linear programming. Then the next section devotes to formulate neutrosophic number goal programming for multi-objective linear goal programming with NNs. Then the next section presents a numerical example. Then the next section presents the conclusion and future scope of research.

Some basic discussions

Here we present some basic definitions and properties of neutrosophic numbers, interval numbers.

Neutrosophic number

An NN60-61 is denoted by α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ =m+nI, where m, n are real numbers and I is indeterminacy.

α=m+nIwhereI[ I L , I U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaamyBaiabgUcaRiaad6gacaWGjbGaam4DaiaadIgacaWG LbGaamOCaiaadwgacaWGjbGaeyicI4Ccfa4aamWaaeaajugibiaadM eajuaGdaahaaqabeaajugWaiaadYeaaaqcLbsacaGGSaGaamysaKqb aoaaCaaabeqaaKqzadGaamyvaaaaaKqbakaawUfacaGLDbaaaaa@4F99@

α=[ m+n I L ,m+b I U ]=[ α L , α U ]( say ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0tcfa4aamWaaeaajugibiaad2gacqGHRaWkcaWGUbGaamys aKqbaoaaCaaabeqaaKqzadGaamitaaaajugibiaacYcacaWGTbGaey 4kaSIaamOyaiaadMeajuaGdaahaaqabeaajugWaiaadwfaaaaajuaG caGLBbGaayzxaaqcLbsacqGH9aqpjuaGdaWadaqaaKqzGeGaeqySde wcfa4aaWbaaeqabaqcLbmacaWGmbaaaKqzGeGaaiilaiabeg7aHLqb aoaaCaaabeqaaKqzadGaamyvaaaaaKqbakaawUfacaGLDbaadaqada qaaKqzGeGaam4CaiaadggacaWG5baajuaGcaGLOaGaayzkaaaaaa@5F18@

Example:

Consider the NN α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ = 5+3I, where 5 is the determinate part and 3I is the indeterminate part. Suppose I [ 0.1,0.2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyicI4 Ccfa4aamWaaOqaaKqzGeGaaGimaiaac6cacaaIXaGaaiilaiaaicda caGGUaGaaGOmaaGccaGLBbGaayzxaaaaaa@4020@ , then α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ becomes an interval α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ = [5.3, 5.6]. Thus for a given interval of the part I, NNs are converted into interval numbers.

Some basic properties of interval number

Here some basic properties of interval analysis71 are presented as follows:

An interval is defined by an order pair α=[ α L , α U ]={ β: α L β α U ,βR } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0tcfa4aamWaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabeaa jugWaiaadYeaaaqcLbsacaGGSaGaeqySdewcfa4aaWbaaSqabeaaju gWaiaadwfaaaaakiaawUfacaGLDbaajugibiabg2da9Kqbaoaacmaa keaajugibiabek7aIjaacQdacqaHXoqyjuaGdaahaaWcbeqaaKqzad GaamitaaaajugibiabgsMiJkabek7aIjabgsMiJkabeg7aHLqbaoaa CaaaleqabaqcLbmacaWGvbaaaKqzGeGaaiilaiabek7aIjabgIGiol aadkfaaOGaay5Eaiaaw2haaaaa@6122@ , where α L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde wcfa4aaWbaaSqabeaajugWaiaadYeaaaaaaa@3AD3@ and α U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde wcfa4aaWbaaSqabeaajugWaiaadwfaaaaaaa@3ADC@ denote the left and right limit of the interval α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ on the real line R.

Assume that m ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ ) and w ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ ) be the midpoint and the width respectively of an interval α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ .

Then, m(α)=(1/2)( α L + α U ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBai aacIcacqaHXoqycaGGPaGaeyypa0JaaiikaiaaigdacaGGVaGaaGOm aiaacMcacaGGOaGaeqySdewcfa4aaWbaaSqabeaajugWaiaadYeaaa qcLbsacqGHRaWkcqaHXoqyjuaGdaahaaWcbeqaaKqzadGaamyvaaaa jugibiaacMcaaaa@4B01@ and w(α)=( α U α L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaai4Dai aacIcacqaHXoqycaGGPaGaeyypa0Jaaiikaiabeg7aHLqbaoaaCaaa leqabaqcLbmacaWGvbaaaKqzGeGaeyOeI0IaeqySdewcfa4aaWbaaS qabeaajugWaiaadYeaaaqcLbsacaGGPaaaaa@4792@    (1)

The different operations on (Moore, 1966) are defined as follows:

The scalar multiplication of  is defined as:

λα={ [λ α L ,λ α U ],λ0 [λ α U ,λ α L ],λ0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaeqySdeMaeyypa0tcfa4aaiWaaKqzGeabaeqakeaajugibiaacUfa cqaH7oaBcqaHXoqyjuaGdaahaaWcbeqaaKqzadGaamitaaaajugibi aacYcacaaMc8Uaeq4UdWMaeqySdewcfa4aaWbaaSqabeaajugWaiaa dwfaaaqcLbsacaGGDbGaaiilaiaaykW7cqaH7oaBcqGHLjYScaaIWa aakeaajugibiaacUfacqaH7oaBcqaHXoqyjuaGdaahaaWcbeqaaKqz adGaamyvaaaajugibiaacYcacaaMc8Uaeq4UdWMaeqySdewcfa4aaW baaSqabeaajugWaiaadYeaaaqcLbsacaGGDbGaaiilaiaaykW7cqaH 7oaBcqGHKjYOcaaIWaaaaOGaay5Eaiaaw2haaaaa@6EA9@    (2)

Absolute value of α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3796@ is defined as |α|={ [ α L , α U ], α L 0 [0, max(- α L , α U )],    α L <0< α U [ α U , α L ],     α U 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiiFai abeg7aHjaacYhacqGH9aqpjuaGdaGabaqcLbsaeaqabOqaaKqzGeGa aGjbVlaacUfacqaHXoqyjuaGdaahaaWcbeqaaKqzadGaamitaaaaju gibiaacYcacqaHXoqyjuaGdaahaaWcbeqaaKqzadGaamyvaaaajugi biaac2facaGGSaGaaGzbVlabeg7aHLqbaoaaCaaaleqabaqcLbmaca WGmbaaaKqzGeGaeyyzImRaaGimaaGcbaqcLbsacaaMe8Uaai4waiaa icdacaGGSaGaaeiiaiaab2gacaqGHbGaaeiEaiaabIcacaqGTaGaeq ySdewcfa4aaWbaaSqabeaajugWaiaabYeaaaqcLbsacaGGSaGaeqyS dewcfa4aaWbaaSqabeaajugWaiaadwfaaaqcLbsacaGGPaGaaiyxai aacYcacaqGGaGaaeiiaiaabccacqaHXoqyjuaGdaahaaWcbeqaaKqz adGaamitaaaajugibiabgYda8iaaicdacqGH8aapcqaHXoqyjuaGda ahaaWcbeqaaKqzadGaamyvaaaaaOqaaKqzGeGaaGjbVlaacUfacqGH sislcqaHXoqyjuaGdaahaaWcbeqaaKqzadGaamyvaaaajugibiaacY cacqGHsislcqaHXoqyjuaGdaahaaWcbeqaaKqzadGaamitaaaajugi biaac2facaGGSaGaaeiiaiaabccacaqGGaGaaeiiaiabeg7aHLqbao aaCaaaleqabaqcLbmacaqGvbaaaKqzGeGaeyizImQaaGimaaaakiaa wUhaaaaa@9456@ (3) (iii) The binary operation ‘*’ is defined between two interval numbers α=[ α L , α U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0Jaai4waiabeg7aHLqbaoaaCaaaleqabaqcLbmacaWGmbaa aKqzGeGaaiilaiaaykW7cqaHXoqyjuaGdaahaaWcbeqaaKqzadGaam yvaaaajugibiaac2facaaMc8oaaa@487E@ and β=[ β L , β U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOSdi Maeyypa0Jaai4waiabek7aILqbaoaaCaaaleqabaqcLbmacaWGmbaa aKqzGeGaaiilaiaaykW7cqaHYoGyjuaGdaahaaWcbeqaaKqzadGaam yvaaaajugibiaac2faaaa@46F9@ as αβ={ab:aα,bβ} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGPaVl abeg7aHjabgEHiQiabek7aIjaaykW7cqGH9aqpcaaMc8Uaai4Eaiaa dggacqGHxiIkcaWGIbGaaiOoaiaaykW7caWGHbGaeyicI4SaeqySde MaaiilaiaaykW7caWGIbGaeyicI4SaeqOSdiMaaiyFaaaa@51A5@ where α L a α U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde wcfa4aaWbaaSqabeaajugWaiaadYeaaaqcLbsacqGHKjYOcaWGHbGa eyizImQaeqySdewcfa4aaWbaaSqabeaajugWaiaadwfaaaaaaa@4414@ , β L b β U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOSdi wcfa4aaWbaaSqabeaajugWaiaadYeaaaqcLbsacqGHKjYOcaWGIbGa eyizImQaeqOSdiwcfa4aaWbaaSqabeaajugWaiaadwfaaaaaaa@4419@ .

‘*’ is designated as any of the operation of four conventional arithmetic operations.

 Some basic properties of NNs

Here we present some properties of NNs60-61.

Let α 1 = a 1 + b 1 I 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaHXo qykmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9aqpcaWGHbGc daWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaamOyaOWaaS baaSqaaKqzadGaaGymaaWcbeaajugibiaadMeakmaaBaaaleaajugW aiaaigdaaSqabaaaaa@4747@ and α 2 = a 2 + b 2 I 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaaMc8 UaeqySdeMcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0Ja amyyaOWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaadk gakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaWGjbGcdaWgaaWc baqcLbmacaaIYaaaleqaaaaa@48D6@ where I 1 [ I 1 L , I 1 U ], I 2 [ I 2 L , I 2 U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGjb GcdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaeyicI4SaaGPaVlaa cUfacaWGjbGcdaqhaaWcbaqcLbmacaaIXaaaleaajugWaiaadYeaaa qcLbsacaGGSaGaaGPaVlaadMeakmaaDaaaleaajugWaiaaigdaaSqa aKqzadGaamyvaaaajugibiaac2facaGGSaGaaGPaVlaadMeakmaaBa aaleaajugWaiaaikdaaSqabaqcLbsacqGHiiIZcaaMc8Uaai4waiaa dMeakmaaDaaaleaajugWaiaaikdaaSqaaKqzadGaamitaaaajugibi aacYcacaaMc8UaamysaOWaa0baaSqaaKqzadGaaGOmaaWcbaqcLbma caWGvbaaaKqzGeGaaiyxaaaa@645E@ then

α 1 =[ a 1 + b 1 I 1 L , a 1 + b 1 I 1 U ]=[ α 1 L , α 1 U ](say) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqGH0i cxcqaHXoqykmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9aqp caGGBbGaamyyaOWaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgU caRiaadkgakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacaWGjbGc daqhaaWcbaqcLbmacaaIXaaaleaajugWaiaadYeaaaqcLbsacaGGSa GaaGPaVlaadggakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH RaWkcaWGIbGcdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaamysaO Waa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGvbaaaKqzGeGaaiyx aiaaykW7cqGH9aqpcaaMc8Uaai4waiabeg7aHPWaa0baaSqaaKqzad GaaGymaaWcbaqcLbmacaWGmbaaaKqzGeGaaiilaiaaykW7cqaHXoqy kmaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamyvaaaajugibiaac2 facaGGOaGaam4CaiaadggacaWG5bGaaiykaaaa@75C5@ and α 2 =[ a 2 + b 2 I 2 L , a 2 + b 2 I 2 U ]=[ α 2 L , α 2 U ](say). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaHXo qykmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqpcaGGBbGa amyyaOWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaadk gakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaWGjbGcdaqhaaWc baqcLbmacaaIYaaaleaajugWaiaadYeaaaqcLbsacaGGSaGaaGPaVl aadggakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRaWkcaWG IbGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaamysaOWaa0baaS qaaKqzadGaaGOmaaWcbaqcLbmacaWGvbaaaKqzGeGaaiyxaiaaykW7 caaMc8Uaeyypa0JaaGPaVlaacUfacqaHXoqykmaaDaaaleaajugWai aaikdaaSqaaKqzadGaamitaaaajugibiaacYcacaaMc8UaeqySdeMc daqhaaWcbaqcLbmacaaIYaaaleaajugWaiaadwfaaaqcLbsacaGGDb GaaGPaVlaacIcacaWGZbGaamyyaiaadMhacaGGPaGaaiOlaaaa@7858@

α 1 + α 2 =[ α 1 L + α 2 L , α 1 U + α 2 U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaHXo qykmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcqaHXoqy kmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqpcaaMc8Uaai 4waiabeg7aHPWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGmbaa aKqzGeGaey4kaSIaeqySdeMcdaqhaaWcbaqcLbmacaaIYaaaleaaju gWaiaadYeaaaqcLbsacaGGSaGaaGPaVlabeg7aHPWaa0baaSqaaKqz adGaaGymaaWcbaqcLbmacaWGvbaaaKqzGeGaey4kaSIaeqySdeMcda qhaaWcbaqcLbmacaaIYaaaleaajugWaiaadwfaaaqcLbsacaGGDbaa aa@6227@

α 1 α 2 =[ α 1 L α 2 U , α 1 U α 2 L ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaHXo qykmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHsislcqaHXoqy kmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqpcaaMc8Uaai 4waiabeg7aHPWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGmbaa aKqzGeGaeyOeI0IaeqySdeMcdaqhaaWcbaqcLbmacaaIYaaaleaaju gWaiaadwfaaaqcLbsacaGGSaGaaGPaVlabeg7aHPWaa0baaSqaaKqz adGaaGymaaWcbaqcLbmacaWGvbaaaKqzGeGaeyOeI0IaeqySdeMcda qhaaWcbaqcLbmacaaIYaaaleaajugWaiaadYeaaaqcLbsacaGGDbaa aa@6248@

α 1 α 2 =[min( α 1 L * α 2 L , α 1 L * α 2 U , α 1 U * α 2 L , α 1 U * α 2 U ),max( α 1 L * α 2 L , α 1 L * α 2 U , α 1 U * α 2 L , α 1 U * α 2 U )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaHXo qykmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHxiIkcqaHXoqy kmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqpcaaMc8Uaai 4waiGac2gacaGGPbGaaiOBaiaacIcacqaHXoqykmaaDaaaleaajugW aiaaigdaaSqaaKqzadGaamitaaaajugibiaacQcacqaHXoqykmaaDa aaleaajugWaiaaikdaaSqaaKqzadGaamitaaaajugibiaacYcacaaM c8UaeqySdeMcdaqhaaWcbaqcLbmacaaIXaaaleaajugWaiaadYeaaa qcLbsacaGGQaGaeqySdeMcdaqhaaWcbaqcLbmacaaIYaaaleaajugW aiaadwfaaaqcLbsacaGGSaGaeqySdeMcdaqhaaWcbaqcLbmacaaIXa aaleaajugWaiaadwfaaaqcLbsacaGGQaGaeqySdeMcdaqhaaWcbaqc LbmacaaIYaaaleaajugWaiaadYeaaaqcLbsacaGGSaGaaGPaVlabeg 7aHPWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGvbaaaKqzGeGa aiOkaiabeg7aHPWaa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGvb aaaKqzGeGaaiykaiaacYcacaaMc8UaciyBaiaacggacaGG4bGaaiik aiabeg7aHPWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGmbaaaK qzGeGaaiOkaiabeg7aHPWaa0baaSqaaKqzadGaaGOmaaWcbaqcLbma caWGmbaaaKqzGeGaaiilaiaaykW7cqaHXoqykmaaDaaaleaajugWai aaigdaaSqaaKqzadGaamitaaaajugibiaacQcacqaHXoqykmaaDaaa leaajugWaiaaikdaaSqaaKqzadGaamyvaaaajugibiaacYcacqaHXo qykmaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamyvaaaajugibiaa cQcacqaHXoqykmaaDaaaleaajugWaiaaikdaaSqaaKqzadGaamitaa aajugibiaacYcacaaMc8UaeqySdeMcdaqhaaWcbaqcLbmacaaIXaaa leaajugWaiaadwfaaaqcLbsacaGGQaGaeqySdeMcdaqhaaWcbaqcLb macaaIYaaaleaajugWaiaadwfaaaqcLbsacaGGPaGaaiyxaaaa@C4E0@

(iv) α 1 ÷ α 2 ={ [ α 1 L , α 1 U ][ 1 α 2 U , 1 α 2 L ]or [min( α 1 L / α 2 L , α 1 L / α 2 U , α 1 U / α 2 L , α 1 U / α 2 U ),max( α 1 L / α 2 L , α 1 L / α 2 U , α 1 U / α 2 L , α 1 U / α 2 U )]if0 α 2 Undefinedif0 α 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaHXo qykmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH3daUcqaHXoqy kmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqpcaaMc8Ucda GabaqcLbsaeaqabOqaaKqzGeGaai4waiabeg7aHPWaa0baaSqaaKqz adGaaGymaaWcbaqcLbmacaWGmbaaaKqzGeGaaiilaiabeg7aHPWaa0 baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGvbaaaKqzGeGaaiyxaiab gEHiQiaacUfakmaalaaabaqcLbsacaaIXaaakeaajugibiabeg7aHP Waa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGvbaaaaaajugibiaa cYcacaaMc8UcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacqaHXoqykm aaDaaaleaajugWaiaaikdaaSqaaKqzadGaamitaaaaaaqcLbsacaGG DbGaaGPaVlaad+gacaWGYbaakeaajugibiaacUfaciGGTbGaaiyAai aac6gacaGGOaGaeqySdeMcdaqhaaWcbaqcLbmacaaIXaaaleaajugW aiaadYeaaaqcLbsacaGGVaGaeqySdeMcdaqhaaWcbaqcLbmacaaIYa aaleaajugWaiaadYeaaaqcLbsacaGGSaGaaGPaVlabeg7aHPWaa0ba aSqaaKqzadGaaGymaaWcbaqcLbmacaWGmbaaaKqzGeGaai4laiabeg 7aHPWaa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGvbaaaKqzGeGa aiilaiabeg7aHPWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGvb aaaKqzGeGaai4laiabeg7aHPWaa0baaSqaaKqzadGaaGOmaaWcbaqc LbmacaWGmbaaaKqzGeGaaiilaiaaykW7cqaHXoqykmaaDaaaleaaju gWaiaaigdaaSqaaKqzadGaamyvaaaajugibiaac+cacqaHXoqykmaa DaaaleaajugWaiaaikdaaSqaaKqzadGaamyvaaaajugibiaacMcaca GGSaGaaGPaVlGac2gacaGGHbGaaiiEaiaacIcacqaHXoqykmaaDaaa leaajugWaiaaigdaaSqaaKqzadGaamitaaaajugibiaac+cacqaHXo qykmaaDaaaleaajugWaiaaikdaaSqaaKqzadGaamitaaaajugibiaa cYcacaaMc8UaeqySdeMcdaqhaaWcbaqcLbmacaaIXaaaleaajugWai aadYeaaaqcLbsacaGGVaGaeqySdeMcdaqhaaWcbaqcLbmacaaIYaaa leaajugWaiaadwfaaaqcLbsacaGGSaGaeqySdeMcdaqhaaWcbaqcLb macaaIXaaaleaajugWaiaadwfaaaqcLbsacaGGVaGaeqySdeMcdaqh aaWcbaqcLbmacaaIYaaaleaajugWaiaadYeaaaqcLbsacaGGSaGaaG PaVlabeg7aHPWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGvbaa aKqzGeGaai4laiabeg7aHPWaa0baaSqaaKqzadGaaGOmaaWcbaqcLb macaWGvbaaaKqzGeGaaiykaiaaykW7caGGDbGaaGPaVlaadMgacaWG MbGaaGPaVlaaicdacqGHjiYZcaaMc8UaeqySdeMcdaWgaaWcbaqcLb macaaIYaaaleqaaaGcbaqcLbsacaWGvbGaamOBaiaadsgacaWGLbGa amOzaiaadMgacaWGUbGaamyzaiaadsgacaaMc8UaamyAaiaadAgaca aMc8UaaGimaiabgIGiolabeg7aHPWaaSbaaSqaaKqzadGaaGOmaaWc beaaaaGccaGL7baajugibiaac6cacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8oaaa@31F9@  

Interval valued linear programming

In this section, first we recall the general model of interval linear programming.

Optimize C p ( Y ¯ )= j=1 n [ c pj L , c pj U ] y j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaqGdb GcdaWgaaWcbaqcLbmacaqGWbaaleqaaKqzGeGaaiikaiqadMfagaqe aiaacMcacqGH9aqpkmaaqahabaqcLbsacaGGBbGaam4yaOWaa0baaS qaaKqzadGaamiCaiaadQgaaSqaaKqzadGaamitaaaajugibiaacYca caWGJbGcdaqhaaWcbaqcLbmacaWGWbGaamOAaaWcbaqcLbmacaWGvb aaaKqzGeGaaiyxaiaadMhajuaGdaWgaaWcbaqcLbmacaWGQbaaleqa aKqzGeGaaiilaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGWbGaeyypa0JaaeymaiaabYcacaaMc8UaaeOmaiaabYca caqGUaGaaeOlaiaab6cacaqGSaGaaGPaVlaabcfaaSqaaKqzadGaam OAaiabg2da9iaaigdaaSqaaKqzadGaamOBaaqcLbsacqGHris5aaaa @6B57@      (4)

subject to

A ¯ Y ¯  ( = )  b ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsaceWGbb GbaebacaaMc8UabmywayaaraGaaeiiaOWaaeWaaKqzGeabaeqakeaa jugibiabgwMiZcGcbaqcLbsacqGH9aqpaOqaaKqzGeGaeyizImkaaO GaayjkaiaawMcaaKqzGeGaaeiiaiqadkgagaqeaaaa@4583@    (5)

Y ¯ =( y 1 , y 2 ,..., y n )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sqaqpu0xh9q8qiW7rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsaceWGzb GbaebacqGH9aqpcaaMc8UaaiikaiaadMhakmaaBaaaleaajugWaiaa igdaaSqabaqcLbsacaGGSaGaaGPaVlaadMhakmaaBaaaleaajugWai aaikdaaSqabaqcLbsacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaa ykW7caWG5bGcdaWgaaWcbaqcLbmacaWGUbaaleqaaKqzGeGaaiykai abgwMiZkaaicdaaaa@50CA@    (6)

where Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara aaaa@36ED@ is a decision vector of order n×1, [ c pj L , c pj U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaGGBb Gaam4yaOWaa0baaSqaaKqzadGaamiCaiaadQgaaSqaaKqzadGaamit aaaajugibiaacYcacaWGJbGcdaqhaaWcbaqcLbmacaWGWbGaamOAaa WcbaqcLbmacaWGvbaaaKqzGeGaaiyxaaaa@45EA@ (j = 1, 2, ..., n; p = 1,2,...,P) is interval coefficient of p-th objective function, A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmyqay aaraaaaa@3758@ b ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOyay aaraaaaa@3779@ is q×n matrix, is q×1 vector and c pj L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4yaK qbaoaaDaaaleaajugWaiaadchacaWGQbaaleaajugWaiaadYeaaaaa aa@3D39@ and c pj U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4yaK qbaoaaDaaaleaajugWaiaadchacaWGQbaaleaajugWaiaadwfaaaaa aa@3D42@ represent lower and upper bounds of the coefficients respectively.

Again, the multi objective linear programming with interval coefficients in objective functions as well as constraints can be presented as:

Optimize C p ( Y ¯ )= j=1 n [ c pj L , c pj U ] y j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaqGdb GcdaWgaaWcbaqcLbmacaqGWbaaleqaaKqzGeGaaiikaiqadMfagaqe aiaacMcacqGH9aqpkmaaqahabaqcLbsacaGGBbGaam4yaOWaa0baaS qaaKqzadGaamiCaiaadQgaaSqaaKqzadGaamitaaaajugibiaacYca caWGJbGcdaqhaaWcbaqcLbmacaWGWbGaamOAaaWcbaqcLbmacaWGvb aaaKqzGeGaaiyxaiaadMhakmaaBaaaleaajugWaiaadQgaaSqabaqc LbsacaGGSaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabchacqGH9aqpcaqGXaGaaeilaiaaykW7caqGYaGaaeilaiaa b6cacaqGUaGaaeOlaiaabYcacaaMc8UaaeiuaaWcbaqcLbmacaWGQb Gaeyypa0JaaGymaaWcbaqcLbmacaWGUbaajugibiabggHiLdaaaa@6AB8@

subject to j=1 n [ a kj L , a kj U ] y j [ b k L , b k U ],    k=1,2,...,q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaabCaeaaju gibiaacUfacaWGHbGcdaqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqc LbmacaWGmbaaaKqzGeGaaiilaiaadggakmaaDaaaleaajugWaiaadU gacaWGQbaaleaajugWaiaadwfaaaqcLbsacaGGDbGaamyEaOWaaSba aSqaaKqzadGaamOAaaWcbeaajugibiabgsMiJkaacUfacaWGIbGcda qhaaWcbaqcLbmacaWGRbaaleaajugWaiaadYeaaaqcLbsacaGGSaGa amOyaOWaa0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGvbaaaKqzGe GaaiyxaiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaae4Aaiabg2da 9iaabgdacaqGSaGaaeOmaiaabYcacaqGUaGaaeOlaiaab6cacaqGSa GaaeyCaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWG UbaajugibiabggHiLdaaaa@6E1A@      (7)

Here Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmyway aaraaaaa@3770@ is a decision vector of order nx1, , [ c pj L , c pj U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaGGBb Gaam4yaOWaa0baaSqaaKqzadGaamiCaiaadQgaaSqaaKqzadGaamit aaaajugibiaacYcacaWGJbGcdaqhaaWcbaqcLbmacaWGWbGaamOAaa WcbaqcLbmacaWGvbaaaKqzGeGaaiyxaaaa@45E5@ [ b k L , b k U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaGGBb GaamOyaOWaa0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGmbaaaKqz GeGaaiilaiaadkgakmaaDaaaleaajugWaiaadUgaaSqaaKqzadGaam yvaaaajugibiaac2faaaa@43FB@ (j = 1, 2,..., n; k = 1, 2, ..., q; p = 1, 2,..., P) are closed intervals.

According to Shaocheng72 & Ramadan73, the interval inequality of the form

j=1 n [ a kj L , a kj U ] y j [ b k L , b k U ],    k=1,2,...,q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaabCaeaaju gibiaacUfacaWGHbGcdaqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqc LbmacaWGmbaaaKqzGeGaaiilaiaadggakmaaDaaaleaajugWaiaadU gacaWGQbaaleaajugWaiaadwfaaaqcLbsacaGGDbGaamyEaOWaaSba aSqaaKqzadGaamOAaaWcbeaajugibiabgwMiZkaacUfacaWGIbGcda qhaaWcbaqcLbmacaWGRbaaleaajugWaiaadYeaaaqcLbsacaGGSaGa amOyaOWaa0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGvbaaaKqzGe GaaiyxaiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaae4Aaiabg2da 9iaabgdacaqGSaGaaeOmaiaabYcacaqGUaGaaeOlaiaab6cacaqGSa GaaeyCaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWG UbaajugibiabggHiLdaaaa@6E2B@    

j=1 n [ a j L y , j a j U y j ] [ b L , b U ] y j 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaabCaeaaju gibiaacUfacaWGHbGcdaqhaaWcbaqcLbmacaWGQbaaleaajugWaiaa dYeaaaqcLbsacaWG5bGcdaWgbaWcbaqcLbmacaWGQbaaleqaaKqzGe GaaiilaiaaykW7caWGHbGcdaqhaaWcbaqcLbmacaWGQbaaleaajugW aiaadwfaaaqcLbsacaWG5bGcdaWgaaWcbaqcLbmacaWGQbaaleqaaK qzGeGaaiyxaiabgwMiZkaaykW7aSqaaKqzadGaamOAaiabg2da9iaa igdaaSqaaKqzadGaamOBaaqcLbsacqGHris5aiaacUfacaWGIbGcda ahaaWcbeqaaKqzadGaamitaaaajugibiaacYcacaaMc8UaamOyaOWa aWbaaSqabeaajugWaiaadwfaaaqcLbsacaGGDbGaaGPaVlaaykW7cq GHaiIicaWG5bGcdaWgaaWcbaqcLbmacaWGQbaaleqaaKqzGeGaeyyz ImRaaGimaaaa@6F2A@ can be written as the two inequalities

j=1 n a j L y j b U j=1 n a j U y j b L y j 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaabCaeaaju gibiaadggakmaaDaaaleaajugWaiaadQgaaSqaaKqzadGaamitaaaa jugibiaadMhakmaaBeaaleaajugWaiaadQgaaSqabaqcLbsacqGHLj YSaSqaaKqzadGaamOAaiabg2da9iaaigdaaSqaaKqzadGaamOBaaqc LbsacqGHris5aiaaykW7caWGIbGcdaahaaWcbeqaaKqzadGaamyvaa aajugibiaaykW7kmaaqahabaqcLbsacaWGHbGcdaqhaaWcbaqcLbma caWGQbaaleaajugWaiaadwfaaaqcLbsacaWG5bGcdaWgbaWcbaqcLb macaWGQbaaleqaaKqzGeGaeyyzImRaaGPaVdWcbaqcLbmacaWGQbGa eyypa0JaaGymaaWcbaqcLbmacaWGUbaajugibiabggHiLdGaamOyaO WaaWbaaSqabeaajugWaiaadYeaaaqcLbsacaaMc8UaeyiaIiIaamyE aOWaaSbaaSqaaKqzadGaamOAaaWcbeaajugibiabgwMiZkaaicdaaa a@73CD@    (8)

Minimization problem73 is stated as:

Minimize C p ( Y ¯ )= j=1 n [ c pj L , c pj U ] y j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVGI8gkVeY=4rFfeuY=Hhbba9q8qqaqFr0de9ps0dbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaae4qaK qbaoaaBaaaleaajugWaiaabchaaSqabaqcLbsacaGGOaGabmywayaa raGaaiykaiabg2da9KqbaoaaqahakeaajugibiaacUfacaWGJbqcfa 4aa0baaSqaaKqzadGaamiCaiaadQgaaSqaaKqzadGaamitaaaajugi biaacYcacaWGJbqcfa4aa0baaSqaaKqzadGaamiCaiaadQgaaSqaaK qzadGaamyvaaaajugibiaac2facaWG5bqcfa4aaSbaaSqaaKqzadGa amOAaaWcbeaajugibiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiCaiabg2da9iaabgdacaqGSaGaaGPaVlaa bkdacaqGSaGaaeOlaiaab6cacaqGUaGaaeilaiaaykW7caqGqbaale aajugWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaad6gaaKqzGeGa eyyeIuoaaaa@6F33@

subject to j=1 n [ a kj L , a kj U ] y j [ b k L , b k U ],    k=1,2,...,q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVGI8gkVeY=4rFfeuY=Hhbba9q8qqaqFr0de9ps0dbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqbaoaaqahake aajugibiaacUfacaWGHbqcfa4aa0baaSqaaKqzadGaam4AaiaadQga aSqaaKqzadGaamitaaaajugibiaacYcacaWGHbqcfa4aa0baaSqaaK qzadGaam4AaiaadQgaaSqaaKqzadGaamyvaaaajugibiaac2facaWG 5bqcfa4aaSbaaSqaaKqzadGaamOAaaWcbeaajugibiabgwMiZkaacU facaWGIbqcfa4aa0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGmbaa aKqzGeGaaiilaiaadkgajuaGdaqhaaWcbaqcLbmacaWGRbaaleaaju gWaiaadwfaaaqcLbsacaGGDbGaaiilaiaabccacaqGGaGaaeiiaiaa bccacaqGRbGaeyypa0JaaeymaiaabYcacaqGYaGaaeilaiaab6caca qGUaGaaeOlaiaabYcacaqGXbaaleaajugWaiaadQgacqGH9aqpcaaI XaaaleaajugWaiaad6gaaKqzGeGaeyyeIuoaaaa@7314@

For the best optimal solution, we solve the problem

Minimize C p ( Y ¯ )= j=1 n c pj L y j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rFfpec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaae4qaO WaaSbaaSqaaKqzadGaaeiCaaWcbeaajugibiaacIcaceWGzbGbaeba caGGPaGaeyypa0JcdaaeWbqaaKqzGeGaam4yaOWaa0baaSqaaKqzad GaamiCaiaadQgaaSqaaKqzadGaamitaaaajugibiaadMhakmaaBaaa leaajugWaiaadQgaaSqabaqcLbsacaGGSaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabchacqGH9aqpcaqGXaGaaeil aiaaykW7caqGYaGaaeilaiaab6cacaqGUaGaaeOlaiaabYcacaaMc8 UaaeiuaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWG UbaajugibiabggHiLdaaaa@6140@    (9)

subject to

j=1 n a kj U y j b k L ,    k=1,2,...,q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaqahabaqcLb sacaWGHbGcdaqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqcLbmacaWG vbaaaKqzGeGaamyEaOWaaSbaaSqaaKqzadGaamOAaaWcbeaajugibi abgwMiZkaadkgakmaaDaaaleaajugWaiaadUgaaSqaaKqzadGaamit aaaajugibiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaae4Aaiabg2 da9iaabgdacaqGSaGaaeOmaiaabYcacaqGUaGaaeOlaiaab6cacaqG SaGaaeyCaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmaca WGUbaajugibiabggHiLdaaaa@5C50@

For the worst solution, we solve the problem

Minimize C p ( Y ¯ )= j=1 n c pj U y j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaae4qaO WaaSbaaSqaaKqzadGaaeiCaaWcbeaajugibiaacIcaceWGzbGbaeba caGGPaGaeyypa0JcdaaeWbqaaKqzGeGaam4yaOWaa0baaSqaaKqzad GaamiCaiaadQgaaSqaaKqzadGaamyvaaaajugibiaadMhakmaaBaaa leaajugWaiaadQgaaSqabaqcLbsacaGGSaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabchacqGH9aqpcaqGXaGaaeil aiaaykW7caqGYaGaaeilaiaab6cacaqGUaGaaeOlaiaabYcacaaMc8 UaaeiuaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWG UbaajugibiabggHiLdaaaa@6144@    10)

subject to

j=1 n a kj L y j b k U ,    k=1,2,...,q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaqahabaqcLb sacaWGHbGcdaqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqcLbmacaWG mbaaaKqzGeGaamyEaOWaaSbaaSqaaKqzadGaamOAaaWcbeaajugibi abgwMiZkaadkgakmaaDaaaleaajugWaiaadUgaaSqaaKqzadGaamyv aaaajugibiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaae4Aaiabg2 da9iaabgdacaqGSaGaaeOmaiaabYcacaqGUaGaaeOlaiaab6cacaqG SaGaaeyCaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmaca WGUbaajugibiabggHiLdaaaa@5C50@

Suppose, the best solution point by solving (9) is

Y ¯ B =( y B 1 , y B 2 ,..., y B n )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmyway aaraqcfa4aaWbaaSqabeaajugWaiaadkeaaaqcLbsacqGH9aqpcaaM c8UaaiikaiaadMhajuaGdaahaaWcbeqaaKqzadGaamOqaaaajuaGda WgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaaiilaiaaykW7caWG5bqc fa4aaWbaaSqabeaajugWaiaadkeaaaqcfa4aaSbaaSqaaKqzadGaaG OmaaWcbeaajugibiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaaGPa VlaadMhajuaGdaahaaWcbeqaaKqzadGaamOqaaaajuaGdaWgaaWcba qcLbmacaWGUbaaleqaaKqzGeGaaiykaiabgwMiZkaaicdaaaa@5D4F@    (11)

With the best objective value C B p ( Y ¯ B )= j=1 n c pj L y B j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaae4qaO WaaWbaaSqabeaajugWaiaabkeaaaGcdaWgaaWcbaqcLbmacaqGWbaa leqaaKqzGeGaaiikaiqadMfagaqeaOWaaWbaaSqabeaajugWaiaadk eaaaqcLbsacaGGPaGaeyypa0JcdaaeWbqaaKqzGeGaam4yaOWaa0ba aSqaaKqzadGaamiCaiaadQgaaSqaaKqzadGaamitaaaajugibiaadM hakmaaCaaaleqabaqcLbmacaWGcbaaaOWaaSbaaSqaaKqzadGaamOA aaWcbeaajugibiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiCaiabg2da9iaabgdacaqGSaGaaGPaVlaabkda caqGSaGaaeOlaiaab6cacaqGUaGaaeilaiaaykW7caqGqbaaleaaju gWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaad6gaaKqzGeGaeyye Iuoaaaa@684C@  (12)

Suppose, the worst solution point by solving (10) is Y ¯ W =( y W 1 , y W 2 ,..., y W n )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGabmyway aaraGcdaahaaWcbeqaaKqzadGaam4vaaaajugibiabg2da9iaaykW7 caGGOaGaamyEaOWaaWbaaSqabeaajugWaiaadEfaaaGcdaWgaaWcba qcLbsacaaIXaaaleqaaKqzGeGaaiilaiaaykW7caWG5bGcdaahaaWc beqaaKqzadGaam4vaaaakmaaBaaaleaajugibiaaikdaaSqabaqcLb sacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaaykW7caWG5bGcdaah aaWcbeqaaKqzadGaam4vaaaakmaaBaaaleaajugWaiaad6gaaSqaba qcLbsacaGGPaGaeyyzImRaaGimaaaa@57D2@  (13)

With the worst objective value C W p ( Y ¯ W )= j=1 n c pj L y W j ,       p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aaboeakmaaCaaaleqabaqcLbmacaqGxbaaaOWaaSbaaSqaaKqzadGa aeiCaaWcbeaajugibiaacIcaceWGzbGbaebakmaaCaaaleqabaqcLb macaWGxbaaaKqzGeGaaiykaiabg2da9OWaaabCaeaajugibiaadoga kmaaDaaaleaajugWaiaadchacaWGQbaaleaajugWaiaadYeaaaqcLb sacaWG5bGcdaahaaWcbeqaaKqzadGaam4vaaaakmaaBaaaleaajugW aiaadQgaaSqabaqcLbsacaGGSaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabchacqGH9aqpcaqGXaGaaeilaiaaykW7 caqGYaGaaeilaiaab6cacaqGUaGaaeOlaiaabYcacaaMc8Uaaeiuaa WcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWGUbaajugi biabggHiLdaaaa@6A16@  (14)

Then the optimal value of the p-th objective function is [C B p ( Y ¯ B ), C W p ( Y ¯ W )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaae4wai aaboeakmaaCaaaleqabaqcLbmacaqGcbaaaOWaaSbaaSqaaKqzadGa aeiCaaWcbeaajugibiaacIcaceWGzbGbaebakmaaCaaaleqabaqcLb macaWGcbaaaKqzGeGaaiykaiaacYcacaaMc8Uaae4qaOWaaWbaaSqa beaajugWaiaabEfaaaGcdaWgaaWcbaqcLbmacaqGWbaaleqaaKqzGe GaaiikaiqadMfagaqeaOWaaWbaaSqabeaajugWaiaadEfaaaqcLbsa caGGPaGaaiyxaaaa@4F7D@ . (15)

Now using the technique of goal programming we would get the optimal solution of the problem.

Neutrosophic number goal programming for multi-objective linear programming problem in neutrosophic number environment

Consider the minimization problem stated as follows:

Minimize C p ( Y ¯ )= j=1 n (a + pj I pj b pj ) y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaamyAaiaad2gacaWGPbGaamOEaiaadwgacaaMc8Ua aGPaVlaadoeakmaaBaaaleaajugWaiaadchaaSqabaqcLbsacaGGOa GabmywayaaraGaaiykaiabg2da9iaaykW7kmaaqahabaqcLbsacaGG OaGaamyyaOWaaSraaSqaaKqzadGaamiCaiaadQgaaSqabaqcLbsacq GHRaWkcaWGjbGcdaWgaaWcbaqcLbmacaWGWbGaamOAaaWcbeaajugi biaadkgakmaaBaaaleaajugWaiaadchacaWGQbaaleqaaKqzGeGaai ykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWGUbaa jugibiabggHiLdGaamyEaOWaaSbaaSqaaKqzadGaamOAaaWcbeaaju gibiaaykW7aaa@67F8@ p=1,2,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaeiCai abg2da9iaabgdacaqGSaGaaGPaVlaabkdacaqGSaGaaeOlaiaab6ca caqGUaGaaeilaiaaykW7caqGqbaaaa@40EE@  (16)

Subjected to j=1 n (c + kj I kj d kj ) y j α K + I k β k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaqahabaqcLb sacaGGOaGaam4yaOWaaSraaSqaaKqzadGaam4AaiaadQgaaSqabaqc LbsacqGHRaWkcaWGjbGcdaWgaaWcbaqcLbmacaWGRbGaamOAaaWcbe aajugibiaadsgakmaaBaaaleaajugWaiaadUgacaWGQbaaleqaaKqz GeGaaiykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmaca WGUbaajugibiabggHiLdGaamyEaOWaaSbaaSqaaKqzadGaamOAaaWc beaajugibiaaykW7cqGHKjYOcqaHXoqykmaaBaaaleaajugWaiaadU eaaSqabaqcLbsacqGHRaWkcaWGjbGcdaWgaaWcbaqcLbmacaWGRbaa leqaaKqzGeGaeqOSdiMcdaWgaaWcbaqcLbmacaWGRbaaleqaaaaa@62F9@ ,

Where I pj [ I pj L , I pj U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamysaO WaaSbaaSqaaKqzadGaamiCaiaadQgaaSqabaqcLbsacqGHiiIZcaGG BbGaamysaOWaa0baaSqaaKqzadGaamiCaiaadQgaaSqaaKqzadGaam itaaaajugibiaacYcacaaMc8UaamysaOWaa0baaSqaaKqzadGaamiC aiaadQgaaSqaaKqzadGaamyvaaaajugibiaac2faaaa@4D1B@ and I kj [ I kj L , I kj U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamysaO WaaSbaaSqaaKqzadGaam4AaiaadQgaaSqabaqcLbsacqGHiiIZcaGG BbGaamysaOWaa0baaSqaaKqzadGaam4AaiaadQgaaSqaaKqzadGaam itaaaajugibiaacYcacaaMc8UaamysaOWaa0baaSqaaKqzadGaam4A aiaadQgaaSqaaKqzadGaamyvaaaajugibiaac2faaaa@4D0C@ I k [ I k L , I k U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamysaO WaaSbaaSqaaKqzadGaam4AaaWcbeaajugibiabgIGiolaacUfacaWG jbGcdaqhaaWcbaqcLbmacaWGRbaaleaajugWaiaadYeaaaqcLbsaca GGSaGaaGPaVlaadMeakmaaDaaaleaajugWaiaadUgaaSqaaKqzadGa amyvaaaajugibiaac2faaaa@4A3F@ j=1, 2,…….., n and k=1, 2, ……… q (17)

Now,

C p ( Y ¯ )= j=1 n (a + pj I pj b pj ) y j = j=1 n [ ( a pj + I pj L b pj ) y j ,( a pj + I pj U b pj ) y j ]=[ j=1 n ( a pj + I pj L b pj ) y j , j=1 n ( a pj + I pj U b pj ) y j ]=[ C p L , C p U ] (say) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aaykW7caWGdbGcdaWgaaWcbaqcLbmacaWGWbaaleqaaKqzGeGaaiik aiqadMfagaqeaiaacMcacqGH9aqpcaaMc8UcdaaeWbqaaKqzGeGaai ikaiaadggakmaaBeaaleaajugWaiaadchacaWGQbaaleqaaKqzGeGa ey4kaSIaamysaOWaaSbaaSqaaKqzadGaamiCaiaadQgaaSqabaqcLb sacaWGIbGcdaWgaaWcbaqcLbmacaWGWbGaamOAaaWcbeaajugibiaa cMcaaSqaaKqzadGaamOAaiabg2da9iaaigdaaSqaaKqzadGaamOBaa qcLbsacqGHris5aiaadMhakmaaBaaaleaajugWaiaadQgaaSqabaqc LbsacaaMc8Uaeyypa0JcdaaeWbqaaKqzGeGaai4waaWcbaqcLbmaca WGQbGaeyypa0JaaGymaaWcbaqcLbmacaWGUbaajugibiabggHiLdGa aiikaiaadggakmaaBaaaleaajugWaiaadchacaWGQbaaleqaaKqzGe Gaey4kaSIaamysaOWaa0baaSqaaKqzadGaamiCaiaadQgaaSqaaKqz adGaamitaaaajugibiaadkgakmaaBaaaleaajugWaiaadchacaWGQb aaleqaaKqzGeGaaiykaiaadMhakmaaBaaaleaajugWaiaadQgaaSqa baqcLbsacaGGSaGaaGPaVlaacIcacaWGHbGcdaWgaaWcbaqcLbmaca WGWbGaamOAaaWcbeaajugibiabgUcaRiaadMeakmaaDaaaleaajugW aiaadchacaWGQbaaleaajugWaiaadwfaaaqcLbsacaWGIbGcdaWgaa WcbaqcLbmacaWGWbGaamOAaaWcbeaajugibiaacMcacaWG5bGcdaWg aaWcbaqcLbmacaWGQbaaleqaaKqzGeGaaiyxaiabg2da9iaacUfakm aaqahabaqcLbsacaGGOaGaamyyaOWaaSbaaSqaaKqzadGaamiCaiaa dQgaaSqabaqcLbsacqGHRaWkcaWGjbGcdaqhaaWcbaqcLbmacaWGWb GaamOAaaWcbaqcLbmacaWGmbaaaKqzGeGaamOyaOWaaSbaaSqaaKqz adGaamiCaiaadQgaaSqabaqcLbsacaGGPaGaamyEaOWaaSbaaSqaaK qzadGaamOAaaWcbeaaaeaajugWaiaadQgacqGH9aqpcaaIXaaaleaa jugWaiaad6gaaKqzGeGaeyyeIuoacaGGSaGaaGPaVRWaaabCaeaaju gibiaacIcacaWGHbGcdaWgaaWcbaqcLbmacaWGWbGaamOAaaWcbeaa jugibiabgUcaRiaadMeakmaaDaaaleaajugWaiaadchacaWGQbaale aajugWaiaadwfaaaqcLbsacaWGIbGcdaWgaaWcbaqcLbmacaWGWbGa amOAaaWcbeaajugibiaacMcacaWG5bGcdaWgaaWcbaqcLbmacaWGQb aaleqaaKqzGeGaaiyxaiabg2da9iaacUfacaWGdbGcdaqhaaWcbaqc LbmacaWGWbaaleaajugWaiaadYeaaaqcLbsacaGGSaGaaGPaVlaado eakmaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaajugibiaa c2faaSqaaKqzadGaamOAaiabg2da9iaaigdaaSqaaKqzadGaamOBaa qcLbsacqGHris5aiaaykW7caGGOaGaam4CaiaadggacaWG5bGaaiyk aaaa@F619@

where, j=1 n ( a pj + I pj L b pj ) y j = C p L and j=1 n ( a pj + I pj U b pj ) y j = C p U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaadI gacaWGLbGaamOCaiaadwgacaGGSaGaaGPaVpaaqahabaGaaiikaiaa dggadaWgaaWcbaGaamiCaiaadQgaaeqaaOGaey4kaSIaamysamaaDa aaleaacaWGWbGaamOAaaqaaiaadYeaaaGccaWGIbWaaSbaaSqaaiaa dchacaWGQbaabeaakiaacMcacaWG5bWaaSbaaSqaaiaadQgaaeqaaa qaaiaadQgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyyp a0Jaam4qamaaDaaaleaacaWGWbaabaGaamitaaaakiaaykW7caWGHb GaamOBaiaadsgacaaMc8+aaabCaeaacaGGOaGaamyyamaaBaaaleaa caWGWbGaamOAaaqabaGccqGHRaWkcaWGjbWaa0baaSqaaiaadchaca WGQbaabaGaamyvaaaakiaadkgadaWgaaWcbaGaamiCaiaadQgaaeqa aOGaaiykaiaadMhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaWGdb Waa0baaSqaaiaadchaaeaacaWGvbaaaaqaaiaadQgacqGH9aqpcaaI XaaabaGaamOBaaqdcqGHris5aaaa@7242@

where, j=1 n ( a pj + I pj L b pj ) y j = C p L and j=1 n ( a pj + I pj U b pj ) y j = C p U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaadI gacaWGLbGaamOCaiaadwgacaGGSaGaaGPaVpaaqahabaGaaiikaiaa dggadaWgaaWcbaGaamiCaiaadQgaaeqaaOGaey4kaSIaamysamaaDa aaleaacaWGWbGaamOAaaqaaiaadYeaaaGccaWGIbWaaSbaaSqaaiaa dchacaWGQbaabeaakiaacMcacaWG5bWaaSbaaSqaaiaadQgaaeqaaa qaaiaadQgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyyp a0Jaam4qamaaDaaaleaacaWGWbaabaGaamitaaaakiaaykW7caWGHb GaamOBaiaadsgacaaMc8+aaabCaeaacaGGOaGaamyyamaaBaaaleaa caWGWbGaamOAaaqabaGccqGHRaWkcaWGjbWaa0baaSqaaiaadchaca WGQbaabaGaamyvaaaakiaadkgadaWgaaWcbaGaamiCaiaadQgaaeqa aOGaaiykaiaadMhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaWGdb Waa0baaSqaaiaadchaaeaacaWGvbaaaaqaaiaadQgacqGH9aqpcaaI XaaabaGaamOBaaqdcqGHris5aaaa@7242@    . (18)

The constraints reduce to

j=1 n (c + kj I kj d kj ) y j α k + I k β k [ j=1 n (c + kj I kj L d kj ) y j , j=1 n (c + kj I kj U d kj ) y j ][ α k + I k L β k , α k + I k U β k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaWaaabCae aajugibiaacIcacaWGJbGcdaWgbaWcbaqcLbmacaWGRbGaamOAaaWc beaajugibiabgUcaRiaadMeakmaaBaaaleaajugWaiaadUgacaWGQb aaleqaaKqzGeGaamizaOWaaSbaaSqaaKqzadGaam4AaiaadQgaaSqa baqcLbsacaGGPaaaleaajugWaiaadQgacqGH9aqpcaaIXaaaleaaju gWaiaad6gaaKqzGeGaeyyeIuoacaWG5bGcdaWgaaWcbaqcLbmacaWG QbaaleqaaKqzGeGaaGPaVlabgsMiJkabeg7aHPWaaSbaaSqaaKqzad Gaam4AaaWcbeaajugibiabgUcaRiaadMeakmaaBaaaleaajugWaiaa dUgaaSqabaqcLbsacqaHYoGykmaaBaaaleaajugWaiaadUgaaSqaba aakeaajugibiabgkDiElaacUfakmaaqahabaqcLbsacaGGOaGaam4y aOWaaSraaSqaaKqzadGaam4AaiaadQgaaSqabaqcLbsacqGHRaWkca WGjbGcdaqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqcLbmacaWGmbaa aKqzGeGaamizaOWaaSbaaSqaaKqzadGaam4AaiaadQgaaSqabaqcLb sacaGGPaaaleaajugWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaa d6gaaKqzGeGaeyyeIuoacaWG5bGcdaWgaaWcbaqcLbmacaWGQbaale qaaKqzGeGaaiilaiaaykW7kmaaqahabaqcLbsacaGGOaGaam4yaOWa aSraaSqaaKqzadGaam4AaiaadQgaaSqabaqcLbsacqGHRaWkcaWGjb GcdaqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqcLbmacaWGvbaaaKqz GeGaamizaOWaaSbaaSqaaKqzadGaam4AaiaadQgaaSqabaqcLbsaca GGPaaaleaajugWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaad6ga aKqzGeGaeyyeIuoacaWG5bGcdaWgaaWcbaqcLbmacaWGQbaaleqaaK qzGeGaaiyxaiabgsMiJkaacUfacqaHXoqykmaaBaaaleaajugWaiaa dUgaaSqabaqcLbsacqGHRaWkcaWGjbGcdaqhaaWcbaqcLbmacaWGRb aaleaajugWaiaadYeaaaqcLbsacqaHYoGykmaaBaaaleaajugWaiaa dUgaaSqabaqcLbsacaGGSaGaaGPaVlabeg7aHPWaaSbaaSqaaKqzad Gaam4AaaWcbeaajugibiabgUcaRiaadMeakmaaDaaaleaajugWaiaa dUgaaSqaaKqzadGaamyvaaaajugibiabek7aIPWaaSbaaSqaaKqzad Gaam4AaaWcbeaajugibiaac2faaaaa@CECF@  

Let α k + I k L β k = b k L , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamitai aadwgacaWG0bGaaGPaVlabeg7aHPWaaSbaaSqaaKqzadGaam4AaaWc beaajugibiabgUcaRiaadMeakmaaDaaaleaajugWaiaadUgaaSqaaK qzadGaamitaaaajugibiabek7aIPWaaSbaaSqaaKqzadGaam4AaaWc beaajugibiabg2da9iaadkgakmaaDaaaleaajugWaiaadUgaaSqaaK qzadGaamitaaaajugibiaacYcaaaa@5107@ , α k + I k U β k = b k U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeqySde McdaWgaaWcbaqcLbmacaWGRbaaleqaaKqzGeGaey4kaSIaamysaOWa a0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGvbaaaKqzGeGaeqOSdi McdaWgaaWcbaqcLbmacaWGRbaaleqaaKqzGeGaeyypa0JaamOyaOWa a0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGvbaaaaaa@4B9B@  

Then[ j=1 n (c + kj I kj L d kj ) y j , j=1 n (c + kj I kj U d kj ) y j ][ b k L , b k U ],k=1,2,...,q. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamivai aadIgacaWGLbGaamOBaiaaykW7caGGBbGcdaaeWbqaaKqzGeGaaiik aiaadogakmaaBeaaleaajugWaiaadUgacaWGQbaaleqaaKqzGeGaey 4kaSIaamysaOWaa0baaSqaaKqzadGaam4AaiaadQgaaSqaaKqzadGa amitaaaajugibiaadsgakmaaBaaaleaajugWaiaadUgacaWGQbaale qaaKqzGeGaaiykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqc LbmacaWGUbaajugibiabggHiLdGaamyEaOWaaSbaaSqaaKqzadGaam OAaaWcbeaajugibiaacYcacaaMc8UcdaaeWbqaaKqzGeGaaiikaiaa dogakmaaBeaaleaajugWaiaadUgacaWGQbaaleqaaKqzGeGaey4kaS IaamysaOWaa0baaSqaaKqzadGaam4AaiaadQgaaSqaaKqzadGaamyv aaaajugibiaadsgakmaaBaaaleaajugWaiaadUgacaWGQbaaleqaaK qzGeGaaiykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbma caWGUbaajugibiabggHiLdGaamyEaOWaaSbaaSqaaKqzadGaamOAaa Wcbeaajugibiaac2facqGHKjYOcaGGBbGaamOyaOWaa0baaSqaaKqz adGaam4AaaWcbaqcLbmacaWGmbaaaKqzGeGaaiilaiaaykW7caWGIb GcdaqhaaWcbaqcLbmacaWGRbaaleaajugWaiaadwfaaaqcLbsacaGG DbGaaiilaiaaykW7caWGRbGaaGPaVlabg2da9iaaykW7caaIXaGaai ilaiaaykW7caaIYaGaaiilaiaaykW7caGGUaGaaiOlaiaac6cacaGG SaGaaGPaVlaadghacaGGUaaaaa@A2CD@    (19)

Assume that the decision maker fixes [ C * p L , C * p U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaaykW7jugibi aacUfacaWGdbqcfa4aaWbaaSqabeaajugWaiaacQcaaaqcfa4aa0ba aSqaaKqzadGaamiCaaWcbaqcLbmacaWGmbaaaKqzGeGaaiilaiaayk W7caWGdbqcfa4aaWbaaSqabeaajugWaiaacQcaaaqcfa4aa0baaSqa aKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGaaiyxaaaa@4CBE@ as the target interval of the p-th objective function.

Applying the procedure discussed in the section 3, we find out the target level of each objective function. The p–th objective function with target is written as:

C p U C p L and C p L C p U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaam4qaK qbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaajugibiab gwMiZkaadoeajuaGdaqhaaWcbaqcLbmacaWGWbaaleaajugWaiabgE HiQiaadYeaaaqcLbsacaaMc8Uaamyyaiaad6gacaWGKbGaaGPaVlaa doeajuaGdaqhaaWcbaqcLbmacaWGWbaaleaajugWaiaadYeaaaqcLb sacqGHKjYOcaWGdbqcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbma cqGHxiIkcaWGvbaaaaaa@5933@  (20)

The goal achievement functions are written as:

C p U + d p U = C p L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 Iaam4qaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaa jugibiabgUcaRiaadsgajuaGdaqhaaWcbaqcLbmacaWGWbaaleaaju gWaiaadwfaaaqcLbsacqGH9aqpcqGHsislcaWGdbqcfa4aa0baaSqa aKqzadGaamiCaaWcbaqcLbmacqGHxiIkcaWGmbaaaaaa@4C95@ and C p L + d p L = C p U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyyai aad6gacaWGKbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaWGWbaa leaajugWaiaadYeaaaqcLbsacqGHRaWkcaWGKbqcfa4aa0baaSqaaK qzadGaamiCaaWcbaqcLbmacaWGmbaaaKqzGeGaeyypa0Jaam4qaKqb aoaaDaaaleaajugWaiaadchaaSqaaKqzadGaey4fIOIaamyvaaaaaa a@4EFF@    (21)

Here d p L 0,and d p U 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVGI8gkVeY=4rFfeuY=Hhbba9q8qqaqFr0de9ps0dbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadsgadaqhaa WcbaGaamiCaaqaaiaadYeaaaGccqGHLjYScaaIWaGaaiilaiaaykW7 caWGHbGaamOBaiaadsgacaaMc8UaamizamaaDaaaleaacaWGWbaaba GaamyvaaaakiabgwMiZkaaicdaaaa@4868@ are negative deviational variables.

Goal programming model I (22)

Min p=1 P ( d p U + d p L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVRWaaabCaeaajugibiaacIcacaWGKbqcfa4a a0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGaey4kaS IaamizaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamitaaaa jugibiaacMcaaSqaaKqzadGaamiCaiabg2da9iaaigdaaSqaaKqzad GaamiuaaqcLbsacqGHris5aaaa@5214@

subject to

C p U + d p U = C p L , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 Iaam4qaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaa jugibiabgUcaRiaadsgajuaGdaqhaaWcbaqcLbmacaWGWbaaleaaju gWaiaadwfaaaqcLbsacqGH9aqpcqGHsislcaWGdbqcfa4aa0baaSqa aKqzadGaamiCaaWcbaqcLbmacqGHxiIkcaWGmbaaaKqzGeGaaiilaa aa@4DD4@ C p L + d p L = C p U , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aadoeajuaGdaqhaaWcbaqcLbmacaWGWbaaleaajugWaiaadYeaaaqc LbsacqGHRaWkcaWGKbqcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLb macaWGmbaaaKqzGeGaeyypa0Jaam4qaKqbaoaaDaaaleaajugWaiaa dchaaSqaaKqzadGaey4fIOIaamyvaaaajugibiaacYcaaaa@4D7C@

j=1 n (c + kj I kj L d kj ) y j b k U , j=1 n (c + kj I kj U d kj ) y j b k L , d p L 0,, d p U 0, y j 0,j=1,2,...,n,andk=1,2,...,q,p=1,2,...,P. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaWaaabCae aajugibiaacIcacaWGJbqcfa4aaSraaSqaaKqzadGaam4AaiaadQga aSqabaqcLbsacqGHRaWkcaWGjbqcfa4aa0baaSqaaKqzadGaam4Aai aadQgaaSqaaKqzadGaamitaaaajugibiaadsgajuaGdaWgaaWcbaqc LbmacaWGRbGaamOAaaWcbeaajugibiaacMcaaSqaaKqzadGaamOAai abg2da9iaaigdaaSqaaKqzadGaamOBaaqcLbsacqGHris5aiaadMha juaGdaWgaaWcbaqcLbmacaWGQbaaleqaaKqzGeGaeyizImQaamOyaK qbaoaaDaaaleaajugWaiaadUgaaSqaaKqzadGaamyvaaaajugibiaa cYcacaaMc8oakeaadaaeWbqaaKqzGeGaaiikaiaadogajuaGdaWgba WcbaqcLbmacaWGRbGaamOAaaWcbeaajugibiabgUcaRiaadMeajuaG daqhaaWcbaqcLbmacaWGRbGaamOAaaWcbaqcLbmacaWGvbaaaKqzGe GaamizaKqbaoaaBaaaleaajugWaiaadUgacaWGQbaaleqaaKqzGeGa aiykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWGUb aajugibiabggHiLdGaamyEaKqbaoaaBaaaleaajugWaiaadQgaaSqa baqcLbsacqGHKjYOcaWGIbqcfa4aa0baaSqaaKqzadGaam4AaaWcba qcLbmacaWGmbaaaKqzGeGaaiilaaGcbaqcLbsacaWGKbqcfa4aa0ba aSqaaKqzadGaamiCaaWcbaqcLbmacaWGmbaaaKqzGeGaeyyzImRaaG imaiaacYcacaGGSaGaamizaKqbaoaaDaaaleaajugWaiaadchaaSqa aKqzadGaamyvaaaajugibiabgwMiZkaaicdacaGGSaGaaGPaVlaadM hajuaGdaWgaaWcbaqcLbmacaWGQbaaleqaaKqzGeGaeyyzImRaaGim aiaacYcacaaMc8UaamOAaiabg2da9iaaigdacaGGSaGaaGPaVlaaik dacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaad6gacaGGSaGaaGPa VlaadggacaWGUbGaamizaiaaykW7caWGRbGaeyypa0JaaGymaiaacY cacaaMc8UaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamyC aiaacYcacaWGWbGaaGPaVlabg2da9iaaigdacaGGSaGaaGPaVlaaik dacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaaccfacaGGUaaaaaa@D26A@

Goal programming model II (23)

Min p=1 P ( ω p U d p U + ω p L d p L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVRWaaabCaeaajugibiaacIcacqaHjpWDjuaG daqhaaWcbaqcLbmacaWGWbaaleaajugWaiaadwfaaaqcLbsacaWGKb qcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGa ey4kaSIaeqyYdCxcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmaca WGmbaaaKqzGeGaamizaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqz adGaamitaaaajugibiaacMcaaSqaaKqzadGaamiCaiabg2da9iaaig daaSqaaKqzadGaamiuaaqcLbsacqGHris5aaaa@60A5@

subject to

C p U + d p U = C p L , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 Iaam4qaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaa jugibiabgUcaRiaadsgajuaGdaqhaaWcbaqcLbmacaWGWbaaleaaju gWaiaadwfaaaqcLbsacqGH9aqpcqGHsislcaWGdbqcfa4aa0baaSqa aKqzadGaamiCaaWcbaqcLbmacqGHxiIkcaWGmbaaaKqzGeGaaiilaa aa@4DD4@

C p L + d p L = C p U , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 Iaam4qaKqbaoaaDaaaleaajugWaiaadchaaSqaaiaadYeaaaqcLbsa cqGHRaWkcaWGKbqcfa4aa0baaSqaaKqzadGaamiCaaWcbaGaamitaa aajugibiabg2da9iabgkHiTiaadoeajuaGdaqhaaWcbaqcLbmacaWG WbaaleaajugWaiabgEHiQiaadwfaaaqcLbsacaGGSaaaaa@4B6F@

j=1 n (c + kj I kj L d kj ) y j b k U , j=1 n (c + kj I kj U d kj ) y j b k L , d p L 0, d p U 0, ω p U 0, ω p L 0, y j 0andj=1,2,...,n;k=1,2,...,q,p=1,2,,...,P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaGaaGPaVp aaqahabaqcLbsacaGGOaGaam4yaKqbaoaaBeaaleaajugWaiaadUga caWGQbaaleqaaKqzGeGaey4kaSIaamysaKqbaoaaDaaaleaajugWai aadUgacaWGQbaaleaajugWaiaadYeaaaqcLbsacaWGKbqcfa4aaSba aSqaaKqzadGaam4AaiaadQgaaSqabaqcLbsacaGGPaaaleaajugWai aadQgacqGH9aqpcaaIXaaaleaajugWaiaad6gaaKqzGeGaeyyeIuoa caWG5bqcfa4aaSbaaSqaaKqzadGaamOAaaWcbeaajugibiabgsMiJk aadkgajuaGdaqhaaWcbaqcLbmacaWGRbaaleaajugWaiaadwfaaaqc LbsacaGGSaGaaGPaVdGcbaWaaabCaeaajugibiaacIcacaWGJbqcfa 4aaSraaSqaaKqzadGaam4AaiaadQgaaSqabaqcLbsacqGHRaWkcaWG jbqcfa4aa0baaSqaaKqzadGaam4AaiaadQgaaSqaaKqzadGaamyvaa aajugibiaadsgajuaGdaWgaaWcbaqcLbmacaWGRbGaamOAaaWcbeaa jugibiaacMcaaSqaaKqzadGaamOAaiabg2da9iaaigdaaSqaaKqzad GaamOBaaqcLbsacqGHris5aiaadMhajuaGdaWgaaWcbaqcLbmacaWG QbaaleqaaKqzGeGaeyizImQaamOyaKqbaoaaDaaaleaajugWaiaadU gaaSqaaKqzadGaamitaaaajugibiaacYcaaOqaaKqzGeGaamizaKqb aoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamitaaaajugibiabgw MiZkaaicdacaGGSaGaaGPaVlaadsgajuaGdaqhaaWcbaqcLbmacaWG WbaaleaajugWaiaadwfaaaqcLbsacqGHLjYScaaIWaGaaiilaiabeM 8a3LqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaajugi biabgwMiZkaaicdacaGGSaGaaGPaVlabeM8a3LqbaoaaDaaaleaaju gWaiaadchaaSqaaKqzadGaamitaaaajugibiabgwMiZkaaicdacaGG SaGaamyEaKqbaoaaBaaaleaajugWaiaadQgaaSqabaqcLbsacqGHLj YScaaIWaGaaGPaVlaadggacaWGUbGaamizaiaaykW7caWGQbGaeyyp a0JaaGymaiaacYcacaaMc8UaaGOmaiaacYcacaGGUaGaaiOlaiaac6 cacaGGSaGaamOBaiaaykW7caGG7aGaam4Aaiabg2da9iaaigdacaGG SaGaaGPaVlaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadg hacaGGSaGaamiCaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGG SaGaaiOlaiaac6cacaGGUaGaaiilaiaadcfaaaaa@E609@

Here ω p U , ω p L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeqyYdC xcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGa aiilaiaaykW7cqaHjpWDjuaGdaqhaaWcbaqcLbmacaWGWbaaleaaju gWaiaadYeaaaaaaa@45C0@ are the numerical weights of corresponding negative deviational variables suggested by decision makers.

Goal programming model III (24)

Minλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GaamyAaiaad6gacaaMc8Uaeq4UdWgaaa@3C77@  

subject to

C p U + d p U = C p L , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 Iaam4qaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaa jugibiabgUcaRiaadsgajuaGdaqhaaWcbaqcLbmacaWGWbaaleaaju gWaiaadwfaaaqcLbsacqGH9aqpcqGHsislcaWGdbqcfa4aa0baaSqa aKqzadGaamiCaaWcbaqcLbmacqGHxiIkcaWGmbaaaKqzGeGaaiilaa aa@4DD4@

C p L + d p L = C p U , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0lK8sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 Iaam4qaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamitaaaa jugibiabgUcaRiaadsgajuaGdaqhaaWcbaqcLbmacaWGWbaaleaaca WGmbaaaKqzGeGaeyypa0JaeyOeI0Iaam4qaKqbaoaaDaaaleaajugW aiaadchaaSqaaKqzadGaey4fIOIaamyvaaaajugibiaacYcaaaa@4C9D@

j=1 n (c + kj I kj L d kj ) y j b k U , j=1 n (c + kj I kj U d kj ) y j b k L , λ d p U , λ d p L , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rFfpec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGceaqabeaadaaeWb qaaKqzGeGaaiikaiaadogajuaGdaWgbaWcbaqcLbmacaWGRbGaamOA aaWcbeaajugibiabgUcaRiaadMeajuaGdaqhaaWcbaqcLbmacaWGRb GaamOAaaWcbaqcLbmacaWGmbaaaKqzGeGaamizaKqbaoaaBaaaleaa jugWaiaadUgacaWGQbaaleqaaKqzGeGaaiykaaWcbaqcLbmacaWGQb Gaeyypa0JaaGymaaWcbaqcLbmacaWGUbaajugibiabggHiLdGaamyE aKqbaoaaBaaaleaajugWaiaadQgaaSqabaqcLbsacqGHKjYOcaWGIb qcfa4aa0baaSqaaKqzadGaam4AaaWcbaqcLbmacaWGvbaaaKqzGeGa aiilaiaaykW7aOqaamaaqahabaqcLbsacaGGOaGaam4yaKqbaoaaBe aaleaajugWaiaadUgacaWGQbaaleqaaKqzGeGaey4kaSIaamysaKqb aoaaDaaaleaajugWaiaadUgacaWGQbaaleaajugWaiaadwfaaaqcLb sacaWGKbGcdaWgaaWcbaqcLbmacaWGRbGaamOAaaWcbeaajugibiaa cMcaaSqaaKqzadGaamOAaiabg2da9iaaigdaaSqaaKqzadGaamOBaa qcLbsacqGHris5aiaadMhajuaGdaWgaaWcbaqcLbmacaWGQbaaleqa aKqzGeGaeyizImQaamOyaKqbaoaaDaaaleaajugWaiaadUgaaSqaaK qzadGaamitaaaajugibiaacYcaaOqaaKqzGeGaeq4UdWMaeyyzImRa amizaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamyvaaaaju gibiaacYcaaOqaaKqzGeGaeq4UdWMaeyyzImRaamizaKqbaoaaDaaa leaajugWaiaadchaaSqaaKqzadGaamitaaaajugibiaacYcaaaaa@A1D9@

d p L 0,, d p U 0, y j 0,j=1,2,...,n,andk=1,2,...,q,p=1,2,...,P. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGKb qcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGmbaaaKqzGeGa eyyzImRaaGimaiaacYcacaGGSaGaamizaKqbaoaaDaaaleaajugWai aadchaaSqaaKqzadGaamyvaaaajugibiabgwMiZkaaicdacaGGSaGa aGPaVlaadMhajuaGdaWgaaWcbaqcLbmacaWGQbaaleqaaKqzGeGaey yzImRaaGimaiaacYcacaaMc8UaamOAaiabg2da9iaaigdacaGGSaGa aGPaVlaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaad6gaca GGSaGaaGPaVlaadggacaWGUbGaamizaiaaykW7caWGRbGaeyypa0Ja aGymaiaacYcacaaMc8UaaGOmaiaacYcacaGGUaGaaiOlaiaac6caca GGSaGaamyCaiaacYcacaWGWbGaaGPaVlabg2da9iaaigdacaGGSaGa aGPaVlaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaaccfaca GGUaaaaa@7BF6@

Numerical example

Consider the following MOLPP with NNs with IÎ[0 , 1]. 

Min C 1 =(2+I) y 1 +(4+I) y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaWgaaWcbaqcLbmacaaIXaaa leqaaKqzGeGaeyypa0JaaiikaiaaikdacqGHRaWkcaWGjbGaaiykai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aiikaiaaisdacqGHRaWkcaWGjbGaaiykaiaadMhajuaGdaWgaaWcba qcLbmacaaIYaaaleqaaaaa@4F42@  

Min C 2 =(3+I) y 1 +(2+I) y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaWgaaWcbaqcLbmacaaIYaaa leqaaKqzGeGaeyypa0JaaiikaiaaiodacqGHRaWkcaWGjbGaaiykai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aiikaiaaikdacqGHRaWkcaWGjbGaaiykaiaadMhajuaGdaWgaaWcba qcLbmacaaIYaaaleqaaaaa@4F42@    

Subject to

(3+I) y 1 +(2+4I) y 2 (4+30I), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaiikai aaiodacqGHRaWkcaWGjbGaaiykaiaadMhajuaGdaWgaaWcbaqcLbma caaIXaaaleqaaKqzGeGaey4kaSIaaiikaiaaikdacqGHRaWkcaaI0a GaamysaiaacMcacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaa jugibiabgwMiZkaacIcacaaI0aGaey4kaSIaaG4maiaaicdacaWGjb GaaiykaiaacYcaaaa@4EF9@  

(4+I) y 1 +(16+I) y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaiikai aaisdacqGHRaWkcaWGjbGaaiykaiaadMhajuaGdaWgaaWcbaqcLbma caaIXaaaleqaaKqzGeGaey4kaSIaaiikaiaaigdacaaI2aGaey4kaS IaamysaiaacMcacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqz GeGaeyyzImRaaGymaiaaiAdacaGGSaaaaa@4AB4@  

y 1 0; y 2 0,I[0,1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaO WaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgwMiZkaaicdacaGG 7aGaaGPaVlaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacq GHLjYScaaIWaGaaiilaiaaykW7caGGjbGaeyicI4Saai4waiaaicda caGGSaGaaGPaVlaaigdacaGGDbaaaa@4E58@ .

The objective functions and the constraints reduce to the following structures:

Min C 1 =[2 y 1 +4 y 2 ,3 y 1 +5 y 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaWgaaWcbaqcLbmacaaIXaaa leqaaKqzGeGaeyypa0Jaai4waiaaikdacaWG5bqcfa4aaSbaaSqaaK qzadGaaGymaaWcbeaajugibiabgUcaRiaaisdacaWG5bqcfa4aaSba aSqaaKqzadGaaGOmaaWcbeaajugibiaacYcacaaMc8UaaG4maiaadM hajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaaGyn aiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaGGDbaaaa@580B@  

Min C 2 =[3 y 1 +2 y 2 ,4 y 1 +3 y 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaWgaaWcbaqcLbmacaaIYaaa leqaaKqzGeGaeyypa0Jaai4waiaaiodacaWG5bqcfa4aaSbaaSqaaK qzadGaaGymaaWcbeaajugibiabgUcaRiaaikdacaWG5bqcfa4aaSba aSqaaKqzadGaaGOmaaWcbeaajugibiaacYcacaaMc8UaaGinaiaadM hakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaaIZaGa amyEaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaGGDbaaaa@580A@

[3 y 1 +2 y 2 ,4 y 1 +6 y 2 ][4,34], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaai4wai aaiodacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiab gUcaRiaaikdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiaacYcacaaMc8UaaGinaiaadMhajuaGdaWgaaWcbaqcLbmacaaI XaaaleqaaKqzGeGaey4kaSIaaGOnaiaadMhakmaaBaaaleaajugWai aaikdaaSqabaqcLbsacaGGDbGaeyyzImRaai4waiaaisdacaGGSaGa aGPaVlaaiodacaaI0aGaaiyxaiaacYcaaaa@576D@

[4 y 1 +16 y 2 ,5 y 1 +17 y 2 ]16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr `0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaai4wai aaisdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiab gUcaRiaaigdacaaI2aGaamyEaOWaaSbaaSqaaKqzadGaaGOmaaWcbe aajugibiaacYcacaaMc8UaaGynaiaadMhajuaGdaWgaaWcbaqcLbma caaIXaaaleqaaKqzGeGaey4kaSIaaGymaiaaiEdacaWG5bGcdaWgaa WcbaqcLbmacaaIYaaaleqaaKqzGeGaaiyxaiabgwMiZkaaigdacaaI 2aGaaiilaaaa@53AD@

y 1 0; y 2 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaK qbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHLjYScaaIWaGa ai4oaiaaykW7caWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaicdacaGGUaaaaa@4616@

The reduced problems are shown in Table 1.

The best and worst solutions are presented in Table 2.

The objective functions with targets can be written as:

2 y 1 +4 y 2 34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGOmai aadMhakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaaI 0aGaamyEaOWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabgsMiJk aaiodacaaI0aGaaiilaaaa@434E@ 3 y 1 +2 y 2 46, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaaI YaGaamyEaOWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabgsMiJk aaisdacaaI2aGaaiilaaaa@4350@ 4 y 1 +3 y 2 2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaaI ZaGaamyEaOWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabgwMiZk aaikdacaGGUaaaaa@42A3@

The goal functions with targets can be written as:

2 y 1 +4 y 2 + d 1 L =34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaqcLbsacaaIYa GaamyEaOWaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaa isdacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey4kaS IaamizaKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamitaaaa jugibiabg2da9iaaiodacaaI0aGaaiilaaaa@49DC@

3 y 1 5 y 2 + d 1 U =4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaG4maiaadMhakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH sislcaaI1aGaamyEaOWaaSbaaSqaaKqzadGaaGOmaaWcbeaajugibi abgUcaRiaadsgajuaGdaqhaaWcbaqcLbmacaaIXaaaleaajugWaiaa dwfaaaqcLbsacqGH9aqpcqGHsislcaaI0aGaaiilaaaa@4ADA@

3 y 1 +2 y 2 + d 2 L =46, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRa WkcaWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGmbaa aKqzGeGaeyypa0JaaGinaiaaiAdacaGGSaaaaa@4A2E@

4 y 1 3 y 2 + d 2 U =2, d 1 U 0, d 1 L 0, d 2 U 0, d 2 L 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frpq0=irpe ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGceaqabeaajugibi abgkHiTiaaisdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaa jugibiabgkHiTiaaiodacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaale qaaKqzGeGaey4kaSIaamizaKqbaoaaDaaaleaajugWaiaaikdaaSqa aKqzadGaamyvaaaajugibiabg2da9iabgkHiTiaaikdacaGGSaGaaG PaVdGcbaqcLbsacaWGKbqcfa4aa0baaSqaaKqzadGaaGymaaWcbaqc LbmacaWGvbaaaKqzGeGaeyyzImRaaGimaiaacYcacaaMc8UaamizaK qbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamitaaaajugibiab gwMiZkaaicdacaGGSaGaaGPaVlaadsgajuaGdaqhaaWcbaqcLbmaca aIYaaaleaajugWaiaadwfaaaqcLbsacqGHLjYScaaIWaGaaiilaiaa ykW7caWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGmb aaaKqzGeGaeyyzImRaaGimaiaac6caaaaa@77CB@

Using the goal programming model (22), the goal programming model I is presented as follows:

GP Model I

Min p=1 2 ( d p U + d p L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVRWaaabCaeaajugibiaacIcacaWGKbqcfa4a a0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGaey4kaS IaamizaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqzadGaamitaaaa jugibiaacMcaaSqaaKqzadGaamiCaiabg2da9iaaigdaaSqaaKqzad GaaGOmaaqcLbsacqGHris5aaaa@521B@

2 y 1 +4 y 2 + d 1 L =34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGOmai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGinaiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRa WkcaWGKbqcfa4aa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaWGmbaa aKqzGeGaeyypa0JaaG4maiaaisdacaGGSaaaaa@4A2B@

3 y 1 5 y 2 + d 1 U =4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaG4maiaadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa eyOeI0IaaGynaiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLb sacqGHRaWkcaWGKbqcfa4aa0baaSqaaKqzadGaaGymaaWcbaqcLbma caWGvbaaaKqzGeGaeyypa0JaeyOeI0IaaGinaiaacYcaaaa@4B5E@

3 y 1 +2 y 2 + d 2 L =46, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey 4kaSIaamizaKqbaoaaDaaaleaajugWaiaaikdaaSqaaKqzadGaamit aaaajugibiabg2da9iaaisdacaaI2aGaaiilaaaa@4AB2@

4 y 1 3 y 2 + d 2 U =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaGinaiaadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa eyOeI0IaaG4maiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaK qzGeGaey4kaSIaamizaKqbaoaaDaaaleaajugWaiaaikdaaSqaaKqz adGaamyvaaaajugibiabg2da9iabgkHiTiaaikdacaGGSaaaaa@4BE0@

4 y 1 +6 y 2 4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOnaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey yzImRaaGinaiaacYcaaaa@43AE@

5 y 1 +17 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGynai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiEdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4528@

3 y 1 +2 y 2 34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey yzImRaaG4maiaaisdacaGGSaaaaa@4466@

4 y 1 +16 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4526@  

4 y 1 +16 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4526@  

d 1 U 0, d 1 L 0, d 2 U 0, d 2 L 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamizaK qbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamyvaaaajugibiab gwMiZkaaicdacaGGSaGaaGPaVlaadsgajuaGdaqhaaWcbaqcLbmaca aIXaaaleaajugWaiaadYeaaaqcLbsacqGHLjYScaaIWaGaaiilaiaa ykW7caWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGvb aaaKqzGeGaeyyzImRaaGimaiaacYcacaaMc8UaamizaKqbaoaaDaaa leaajugWaiaaikdaaSqaaKqzadGaamitaaaajugibiabgwMiZkaaic dacaGGSaaaaa@5FB0@

y 1 0; y 2 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaK qbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHLjYScaaIWaGa ai4oaiaaykW7caWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaicdacaGGUaaaaa@4616@

Using the goal programming model (23), the goal programming model II is presented as follows:

 GP Model II

Min p=1 2 ( ω p U d p U + ω p L d p L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVRWaaabCaeaajugibiaacIcacqaHjpWDjuaG daqhaaWcbaqcLbmacaWGWbaaleaajugWaiaadwfaaaqcLbsacaWGKb qcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGa ey4kaSIaeqyYdCxcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmaca WGmbaaaKqzGeGaamizaKqbaoaaDaaaleaajugWaiaadchaaSqaaKqz adGaamitaaaajugibiaacMcaaSqaaKqzadGaamiCaiabg2da9iaaig daaSqaaKqzadGaaGOmaaqcLbsacqGHris5aaaa@60AC@

2 y 1 +4 y 2 + d 1 L =34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGOmai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGinaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey 4kaSIaamizaKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamit aaaajugibiabg2da9iaaiodacaaI0aGaaiilaaaa@4AAF@

3 y 1 5 y 2 + d 1 U =4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaG4maiaadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa eyOeI0IaaGynaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaK qzGeGaey4kaSIaamizaKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqz adGaamyvaaaajugibiabg2da9iabgkHiTiaaisdacaGGSaaaaa@4BE2@

3 y 1 +2 y 2 + d 2 L =46, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRa WkcaWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGmbaa aKqzGeGaeyypa0JaaGinaiaaiAdacaGGSaaaaa@4A2E@

4 y 1 3 y 2 + d 2 U =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabgkHiTiaais dacaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maiaadMha daWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGKbWaa0baaSqaaiaaik daaeaacaWGvbaaaOGaeyypa0JaeyOeI0IaaGOmaiaacYcaaaa@433F@  

4 y 1 +6 y 2 4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aaisdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiab gUcaRiaaiAdacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGe GaeyyzImRaaGinaiaacYcaaaa@44B5@

5 y 1 +17 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGynai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiEdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4528@

3 y 1 +2 y 2 34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey yzImRaaG4maiaaisdacaGGSaaaaa@4466@

4 y 1 +16 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4526@

d 1 U 0, d 1 L 0, d 2 U 0, d 2 L 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamizaK qbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamyvaaaajugibiab gwMiZkaaicdacaGGSaGaaGPaVlaadsgajuaGdaqhaaWcbaqcLbmaca aIXaaaleaajugWaiaadYeaaaqcLbsacqGHLjYScaaIWaGaaiilaiaa ykW7caWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGvb aaaKqzGeGaeyyzImRaaGimaiaacYcacaaMc8UaamizaKqbaoaaDaaa leaajugWaiaaikdaaSqaaKqzadGaamitaaaajugibiabgwMiZkaaic dacaGGSaaaaa@5FB0@

y 1 0, y 2 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaO WaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgwMiZkaaicdacaGG SaGaaGPaVlaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGe GaeyyzImRaaGimaiaacYcaaaa@4581@

ω p U , ω p L 0,p=1,2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeqyYdC xcfa4aa0baaSqaaKqzadGaamiCaaWcbaqcLbmacaWGvbaaaKqzGeGa aiilaiaaykW7cqaHjpWDjuaGdaqhaaWcbaqcLbmacaWGWbaaleaaju gWaiaadYeaaaqcLbsacqGHLjYScaaIWaGaaiilaiaadchacqGH9aqp caaIXaGaaiilaiaaykW7caaIYaGaaiOlaaaa@4FFE@

Using the goal programming model (24), the goal programming model III is presented as follows:

 

GP Model III

Minλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlabeU7aSbaa@3B95@

2 y 1 +4 y 2 + d 1 L =34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGOmai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGinaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey 4kaSIaamizaKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamit aaaajugibiabg2da9iaaiodacaaI0aGaaiilaaaa@4AAF@

3 y 1 5 y 2 + d 1 U =4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaG4maiaadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa eyOeI0IaaGynaiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaK qzGeGaey4kaSIaamizaKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqz adGaamyvaaaajugibiabg2da9iabgkHiTiaaisdacaGGSaaaaa@4BE2@

3 y 1 +2 y 2 + d 2 L =46, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaaI YaGaamyEaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRa WkcaWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGmbaa aKqzGeGaeyypa0JaaGinaiaaiAdacaGGSaaaaa@4A2E@

4 y 1 3 y 2 + d 2 U =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaGinaiaadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa eyOeI0IaaG4maiaadMhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaK qzGeGaey4kaSIaamizaKqbaoaaDaaaleaajugWaiaaikdaaSqaaKqz adGaamyvaaaajugibiabg2da9iabgkHiTiaaikdacaGGSaaaaa@4BE0@

4 y 1 +6 y 2 4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aaisdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiab gUcaRiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaisdacaGGSaaaaa@4539@

5 y 1 +17 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGynai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiEdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4528@

3 y 1 +2 y 2 34, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhakmaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaaI YaGaamyEaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHLj YScaaIZaGaaGinaiaacYcaaaa@43E2@

4 y 1 +16 y 2 16, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaaiilaaaa@4526@

d 1 U 0, d 1 L 0, d 2 U 0, d 2 L 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamizaK qbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamyvaaaajugibiab gwMiZkaaicdacaGGSaGaaGPaVlaadsgajuaGdaqhaaWcbaqcLbmaca aIXaaaleaajugWaiaadYeaaaqcLbsacqGHLjYScaaIWaGaaiilaiaa ykW7caWGKbqcfa4aa0baaSqaaKqzadGaaGOmaaWcbaqcLbmacaWGvb aaaKqzGeGaeyyzImRaaGimaiaacYcacaaMc8UaamizaKqbaoaaDaaa leaajugWaiaaikdaaSqaaKqzadGaamitaaaajugibiabgwMiZkaaic dacaGGSaaaaa@5FB0@

y 1 0, y 2 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaK qbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHLjYScaaIWaGa aiilaiaaykW7caWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaicdacaGGSaaaaa@4605@

λ d 1 U ,λ d 1 L , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaeyyzImRaamizaKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGa amyvaaaajugibiaacYcacaaMc8Uaeq4UdWMaeyyzImRaamizaKqbao aaDaaaleaajugWaiaaigdaaSqaaKqzadGaamitaaaajugibiaacYca aaa@4BD7@

λ d 2 U ,λ d 2 L . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaeyyzImRaamizaKqbaoaaDaaaleaajugWaiaaikdaaSqaaKqzadGa amyvaaaajugibiaacYcacaaMc8UaaGPaVlabeU7aSjabgwMiZkaads gajuaGdaqhaaqcgayaaKqzadGaaGOmaaqcgayaaKqzadGaamitaaaa jugibiaac6caaaa@4E6E@

The optimal solutions are presented in Table 3.

Objective function

Problem for the best solution

Problem for the worst solution

C1

Min C 1 L =2 y 1 +4 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIXaaa leaajugWaiaadYeaaaqcLbsacqGH9aqpcaaIYaGaamyEaOWaaSbaaS qaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaaisdacaWG5bqcfa4a aSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@4AAC@ 4 y 1 +6 y 2 4; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aaisdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiab gUcaRiaaiAdacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGe GaeyyzImRaaGinaiaacUdaaaa@44C4@ 5 y 1 +17 y 2 16; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGynai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiEdacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGe GaeyyzImRaaGymaiaaiAdacaGG7aaaaa@44B3@ y 1 0; y 2 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaO WaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgwMiZkaaicdacaGG 7aGaaGPaVlaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacq GHLjYScaaIWaGaaiOlaaaa@450E@

Min C 1 U =3 y 1 +5 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIXaaa leaajugWaiaadwfaaaqcLbsacqGH9aqpcaaIZaGaamyEaOWaaSbaaS qaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaaiwdacaWG5bqcfa4a aSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@4AB7@ 3 y 1 +2 y 2 34; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHLj YScaaIZaGaaGinaiaacUdaaaa@43F1@ 4 y 1 +16 y 2 16; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaai4oaaaa@4535@ y 1 0; y 2 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaO WaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgwMiZkaaicdacaGG 7aGaaGPaVlaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacq GHLjYScaaIWaGaaiOlaaaa@450E@ strong>

C2

Min C 2 L =3 y 1 +2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIYaaa leaajugWaiaadYeaaaqcLbsacqGH9aqpcaaIZaGaamyEaOWaaSbaaS qaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaaikdacaWG5bqcfa4a aSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@4AAC@ 4 y 1 +6 y 2 4; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGPaVl aaisdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiab gUcaRiaaiAdacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGe GaeyyzImRaaGinaiaacUdaaaa@44C4@ 5 y 1 +17 y 2 16; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGynai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiEdacaWG5bGcdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGe GaeyyzImRaaGymaiaaiAdacaGG7aaaaa@44B3@ y 1 0; y 2 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaO WaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgwMiZkaaicdacaGG 7aGaaGPaVlaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacq GHLjYScaaIWaGaaiOlaaaa@450E@

Min C 2 U =4 y 1 +3 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIYaaa leaajugWaiaadwfaaaqcLbsacqGH9aqpcaaI0aGaamyEaOWaaSbaaS qaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaaiodacaWG5bqcfa4a aSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@4AB7@ 3 y 1 +2 y 2 34; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaG4mai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGOmaiaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHLj YScaaIZaGaaGinaiaacUdaaaa@43F1@ 4 y 1 +16 y 2 16; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaaGinai aadMhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIa aGymaiaaiAdacaWG5bqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaju gibiabgwMiZkaaigdacaaI2aGaai4oaaaa@4535@ y 1 0; y 2 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamyEaO WaaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgwMiZkaaicdacaGG 7aGaaGPaVlaadMhakmaaBaaaleaajugWaiaaikdaaSqabaqcLbsacq GHLjYScaaIWaGaaiOlaaaa@450E@

Table 1 Reduced problem

Objective function

Best Solution with solution point

Worst solution with solution point

C1

Min C 1 L* =3.765 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIXaaa leaajugWaiaadYeacaGGQaaaaKqzGeGaeyypa0JaaG4maiaac6caca aI3aGaaGOnaiaaiwdaaaa@4549@ at (0, 0.941)

Min C 1 U* =34 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIXaaa leaajugWaiaadwfacaGGQaaaaKqzGeGaeyypa0JaaG4maiaaisdaaa a@431E@ at (11.333, 0)

C2

Min C 2 L* =1.882 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIYaaa leaajugWaiaadYeacaGGQaaaaKqzGeGaeyypa0JaaGymaiaac6caca aI4aGaaGioaiaaikdaaaa@4548@ at (0, 0.941)

Min C 2 U* =45.333 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGaamytai aadMgacaWGUbGaaGPaVlaadoeajuaGdaqhaaWcbaqcLbmacaaIYaaa leaajugWaiaadwfacaGGQaaaaKqzGeGaeyypa0JaaGinaiaaiwdaca GGUaGaaG4maiaaiodacaaIZaaaaa@460A@
at (11.333, 0)

Table 2 Best and Worst solutions

Programming model

C1

C2

Y ¯ * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipq0le9sipGc91rpepec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaKqzGeGabmyway aaraGcdaahaaWcbeqaaKqzadGaaiOkaaaaaaa@38AC@

Goal programming Model I

[22.67, 34]

[34, 45.33]

(11.33, 0)

Goal programming Model II

[22.67, 34]

[34, 45.33]

(11.33, 0)

Goal programming Model III

[22.67, 34]

[34, 45.33]

(11.33, 0)

Table 3 Optimal solution

Conclusion

This paper has presented the solution strategy of multi-objective linear goal programming problem with neutrosophic coefficients of both objective functions and constraints. The neutrosophic coefficients of the form m + nI is converted into interval coefficient with the prescribed range of I. Adopting the concept of solving linear interval programming problem, three new neutrosophic goal programming models have been developed and solved by considering a numerical example. We hope that the proposed method for solving multi-objective linear goal programming with neutrosophic coefficients will lighten up a new way for the future research work. The proposed NN-GP strategy can be extended to multi-objective priority based goal programming with NNs. In future, we shall apply the proposed NN-GP strategies to production planning in brickfield,74 bi-level programming problem75 and health care management.76

Acknowledgements

None.

Conflict of interest

The author declares that there is no conflict of interest.

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