Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach
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Lachhwani, Kailash; Poonia, Mahaveer Prasad Article Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Lachhwani, Kailash; Poonia, Mahaveer Prasad (2012) : Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 8, pp. 1-11, https://doi.org/10.1186/2251-712X-8-16 This Version is available at: https://hdl.handle.net/10419/78570 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/
ORIGINAL RESEARCH Open Access Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach Kailash Lachhwani 1* and Mahaveer Prasad Poonia 2 Abstract In this paper, we show a procedure for solving multilevel fractional programming problems in a large hierarchical decentralized organization using fuzzy goal programming approach. In the proposed method, the tolerance membership functions for the fuzzily described numerator and denominator part of the objective functions of all levels as well as the control vectors of the higher level decision makers are respectively defined by determining individual optimal solutions of each of the level decision makers. A possible relaxation of the higher level decision is considered for avoiding decision deadlock due to the conflicting nature of objective functions. Then, fuzzy goal programming approach is used for achieving the highest degree of each of the membership goal by minimizing negative deviational variables. We also provide sensitivity analysis with variation of tolerance values on decision vectors to show how the solution is sensitive to the change of tolerance values with the help of a numerical example. Keywords: Multilevel fractional programming, Fuzzy goal programming, Membership function, Tolerance values Background Hierarchical optimization or multilevel programming problems (MLPPs) have the following common characteristics: interactive decision making units exist within predominantly hierarchical structures; the execution of decision is sequential from higher level to lower level; each decisionmaking unit independently controls a set of decision variables and is interested in maximizing its own objective but is affected by the reaction of lower level decision makers (DMs). Due to their dissatisfaction with the decision of the higher level DMs, decision deadlock arises frequently in the decision-making situation. Multilevel fractional programming problems (MLFPPs) involve objective functions in fractional form, i.e., fX ðÞ ¼NXðÞ DXðÞat each level with the assumption that the denominator of objectives remains positive at each level in the feasible region. Some important existing solution approaches such as the extreme point search, the procedure based on the Karush-Kuhn Tucker condition, and the decent method (Anandilingam 1988; Anandilingam and Apprey 1991; Biswas and Pal 2005; Bellmann 1957; Charnes and Cooper 1962; Craven and Mond 1975; Lai 1996) are effective only for solving simple types of multilevel programming problems. Initially, fuzzy approach was used to handle multiobjective optimization problems (Chakraborty and Gupta 2002; Jimenez and Bilbas 2009). Lai and Hwang (1993) at first developed an effective fuzzy approach using the concept of tolerance membership functions for solving MLPPs in 1996. Shih et al. (1996) extended Lai’s concept using a non-compensatory maximum-minimum aggregation operator for solving MLPPs. Shih and Lee (2000) further extended Lai’s concept by introducing the compensatory fuzzy operator for solving MLPPs. Sinha (2003a,b) studied alternative MLP techniques based on fuzzy mathematical programming (FMP). The basic concept of these fuzzy approaches is the same, and evaluation of the problem again and again by redefining the elicited membership values is essentially needed in the solution search process to obtain a satisfactory solution. So, computational load is also inherently involved in the fuzzy approaches developed so far. In the FMP techniques of Sinha (2003a,b), the last (lower) level is the most important, and the decision of the lowest level remains either unchanged or closest to individual best decisions, which leads * Correspondence: [email protected]om 1 Department of Mathematics, Government Engineering College, Bikaner 334004, India Full list of author information is available at the end of the article © 2012 Lachhwani and Poonia; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 http://www.jiei-tsb.com/content/8/1/16
to the paradox that the decision power of the lowest level DM dominates the higher level DM. To overcome such difficulties, the fuzzy goal programming (FGP) approach to multidecision-making problems was introduced by Mohamed (1997) which is extended by Pramanik and Roy (2007) to solve MLPPs. Baky (2009) used fuzzy goal programming to solve decentralized bilevel multiobjective programming problems. Chang (2009) suggested goal programming approach for fuzzy multiobjective fractional programming problems. Recently, Pal and Gupta (2009) studied a genetic algorithm to fuzzy goal programming formulation of fractional multiobjective decision-making problems. In real-world decision-making situations, decision makers sometimes may be faced with the decision to optimize inventory/sales, actual cost/standard cost, output/employee, etc. with respect to some constraints. Such type of problems in a large hierarchical organization from higher level to lower level and their sequential decisions on complex and conflicting objectives formulate the MLFPPs. Practical optimization situations involving multilevel with fractional objectives have been rare, but such problems can be encountered in the most complex design, pattern recognition, control theory, and resource allocation situations (Mohamed 1997). Motivated by the concept of interactive fuzzy goal programming and fractional programming, an effort has been made to examine the possibility of unifying the level-wise (hierarchical) and stage-wise operations with the assumption of a positive denominator of objective functions at each level. The aim of this paper is to present a procedure to solve multilevel fractional programming problems. Our proposed methodology involves the fuzzy goal levels of the numerator and denominator part of each objective as well as decision vectors controlled by the higher level DMs, which are determined by individual optimal solutions. Then, the fuzzy goals are characterized by the associated membership functions which are transformed into fuzzy flexible membership goals by means of introducing negative and positive deviational variables and assigning a higher membership value (unity) as aspiration level to each of them. Since overdeviation from any fuzzy goal implies the full achievement of the membership values, we assign only negative deviational variables to the achievement function and minimize negative deviational variables to get a compromise optimal solution. To illustrate our proposed method, we solve a numerical example and compare the results with the change in tolerance limits. The paper is organized as follows: In the ‘Formulation of MLPP’section, we discuss formulation of MLFPP and the related terminology. In the ‘Fuzzy programming formulation of MLFPP’section, we characterize the linear membership functions for the numerator and denominator of objective functions at each level as well as decision vectors controlled by the higher level DMs. In the next section, we discuss the proposed FGP approach to tackle MLFPPs and formulate different mathematical models related to it. In the ‘Selection of compromise solution’section, selection criteria of compromise optimal solution are described. To illustrate the proposed methodology, a numerical example is considered and sensitivity analysis is performed with the change in tolerance limits in the ‘Numerical example’section. Concluding remarks are given in the last sections. Results and discussion Formulation of MLFPP We consider a T-level fractional programming problem of maximization-type objectives at each level. Mathematically, we can state it as follows: Max X1 Z1 XðÞ¼ C11 X1þ C12 X2þ:::: þ C1T XTþα1 D11 X1þ D12 X2þ:::: þ D1T XTþβ1 Max X2 Z2 XðÞ¼ C21 X1þ C22 X2þ:::: þ C2T XTþα2 D21 X1þ D22 X2þ:::: þ D2T XTþβ2 Max XT ZT XðÞ¼ CT1 X1þ CT2 X2þ:::: þ CTT XTþαT DT1 X1þ DT2 X2þ:::: þ DTT XTþβT subject to Ai1 X1þ Ai2 X2þ::::::::: þ AiT XT≤;¼;≥ ðÞ bi 8i¼1;2; ::::; m and X1≥0; X2≥0; ::::; XT≥0 ð1Þ X1¼X1 1;X2 1; ::::::::; XN1 1 0decision variables are under the control of the first level DM; XT¼X1 T;X2 T; ::::::::; XNT T 0 decision variables are under the control of the t-level DM. Where 0denotes transposition, Aij i¼1;2; ::::; m;and j¼1;2;...;Tare mrow vectors, each with dimension 1Nj . Ait Xt;t¼1;2; ::::; Tis a column vector of dimension M1ðÞ. C11; C21; ::::; CT1are row vectors of dimension 1 N1 ðÞ . Similarly, C1T; C2T; ::::; CTT and D1T; D2T; ::::; DTT are row vectors of dimension 1 NT ðÞ. We take X¼ X1∪ X2∪...∪ XTand N¼N1þN2þ::::::: þ NT. Here, one DM is located on each level. Decision vector Xt;t¼1;2;...;Tis the control of the t-th level DM having Ntnumber of decision variables. Here, it is assumed that the denominator of objective functions is positive at each level for all the values of decision variables in the constraint region. Fuzzy programming formulation of MLFPP To formulate the fuzzy programming model of MLFPP, the objective numerator fiN XðÞþαi;8t¼1;2;...;Tand objective denominator fiD XðÞþαi;8t¼1;2;...;Tat Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 2 of 11 http://www.jiei-tsb.com/content/8/1/16
each level and the decision vector Xt;t¼1;2; ::; T1ðÞ would be transformed into fuzzy goals by means of assigning an aspiration level to each of them. Then, they are to be characterized by the associated membership functions by defining tolerance limits for the achievement of the aspired levels of the corresponding fuzzy goals. Characterization of membership function of MLFPP In the decision-making context, each DM is interested in maximizing his or her own objective function; the optimal solution of each DM when calculated in isolation would be considered as the best solution, and the associated objective values can be considered as the aspiration level of the corresponding fuzzy goal. Let XB tbe the best solution of the t-th level DM. It is quite natural that objective values which are equal to or larger than ZB t¼Zt XB t ¼Max X2X Zt XðÞ;t¼1; 2;...;Tshould be absolutely satisfactory to the t-th level DM. If the individual best solutions XB t;t¼1;2;...;Tare the same, then a satisfactory optimal solution of the system is reached. However, this rarely happens due to the conflicting nature of the objectives. To obtain a satisfactory solution, the higher DM should give some tolerance (relaxation), and the relaxation of the decision of the higher level DM depends on the needs, desires, and practical situations in the decision-making situation. Then, the fuzzy goals take the form Zt XðÞ≥Zt XB t ;t¼1;2;...;T and Xtffi XB t;t¼1;2;...;T1ðÞ. To build membership functions, fuzzy goals and tolerance should be determined first. However, they could hardly be determined without meaningful supporting data. Using the individual best solution, we find the values of all the numerator objective functions and denominator objective functions at each best solution and construct a payoff matrix as follows: XB 1N1N2:: NT XB 1N1 XB 1 N2 XB 1 ::NT XB 1 XB 2N1 XB 2 N2 XB 2 ::NT XB 2 :: :::: :: :::: XB TN1 XB T N2 XB T ::NT XB T 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 and XB 1D1D2:: DT XB 1D1 XB 1 D2 XB 1 ::DT XB 1 XB 2D1 XB 2 D2 XB 2 ::DT XB 2 :: :::: :: :::: XB TD1 XB T D2 XB T ::DT XB T 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð2Þ Here, Xtt¼1;2;...;TðÞare assumed to be the main decision vectors. The maximum value of each column Nt XB t and Dt XB t give upper tolerance limit or aspired level of achievement for the t-th numerator objective function and denominator objective function, respectively, where NB t¼Nt XB t ¼Max X2X Nt XðÞ;t¼1;2;...;T.The minimum value of each column gives the lower tolerance limit or lowest acceptable level of achievement for the t-th numerator objective function and denominator objective function, respectively, where NL t¼Min X2X Nt XB 1 ; Nt XB 2 ;...;Nt XB T g;.t¼1;2;...;T.Then,thelinear membership functions for the defined fuzzy goals are as follows (see also Figures 1 and 2): μZtNt XðÞðÞ¼ 1if Nt XðÞ≥NB t Nt XðÞNL t NB tNL t if NL t≤Nt XðÞ≤NB t 0if Nt XðÞ≤NL t 8t¼1;2;...;T 8 > > > > > > > < > > > > > > > : ð3Þ (()) t Zt DX L t DB t D 1 (()) t Zt NX () t NX L t NB t N 1 () t DX (a) (b) Figure 1 (a): Membership function for μZtDi XðÞðÞ(b): Membership function for μZtNi XðÞðÞ. Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 3 of 11 http://www.jiei-tsb.com/content/8/1/16
μZtDt XðÞðÞ¼ 0if Dt XðÞ≥DB t DB tDt XðÞ DB tDL t if DL t≤Dt XðÞ≤DB t 1if Dt XðÞ≤DL t 8t¼1;2;...;T 8 > > > > > > > < > > > > > > > : ð4Þ Here, linear membership functions are more suitable than nonlinear functions as less computational difficulties arise in models due to it. Let p t; pþ tt¼1;2;...;T1ðÞ be the negative and positive tolerance values on decision vectors Xtconsidered by the t-th level DM. This is a triangular fuzzy number. Then, the linear membership functions for decision vectors Xtcan be formulated as follows: μ Xt Xt ðÞ¼ Xt XB t p t p t if XB t p t ≤ Xt≤ XB t XB tþ pþ t Xt pþ t if XB t≤ Xt≤ XB tþ pþ t 0otherwise 8t¼1;2;...;T 8 > > > > > > > > > > < > > > > > > > > > > : ð5Þ Here, p tand pþ tare the negative and positive tolerance vectors; p tand pþ tare not necessarily same. Generally, Xtlies between XB t p tand XB tþ pþ t. DMs may prefer to shift the range of XB twhich may be the left of XB tor the right of XB t, only depending on the needs and desires of the higher level DMs in the decision-making situation. Then, the membership function becomes one-sided. For example, if X¼ 0, then Xshould lie on the right of 0. Then, the DM should assign p t≤ 0; pþ t≥ 0 and p t ≤ pþ t . If the DM wants the shift towards the left of XB t, then p tshould be assigned a positive value while pþ t should be assigned a negative value, i.e., p t≥ 0, pþ t≤ 0, and p t ≥ pþ t . Similarly, if the shift is required to the right of XB t, then the DM should assign p t≤ 0, pþ t≥0, and p t ≤ pþ t . We may treat the tolerance as variables with the restrictions that p t≤ XB t(so that the value of the variables remain non-negative). FGP solution approach FGP is an extension of conventional goal programming (GP) introduced by Charnes and Cooper (1962). GP has been extensively studied and widely circulated in literature (Arora and Gupta 2009; Pramanik and Roy 2007). In this paper, GP approach to fuzzy multiobjective decision-making problems introduced by Mohamed (1997) is extended to solve MLFPP problems. In a decision-making situation, the aim of each DM is to achieve the highest membership value (unity) of the associated fuzzy goal in order to obtain the absolute satisfactory solution. However, in real practice, achievement of all membership values to the highest degree (unity) is not possible due to conflicting objectives. Therefore, the decision policy for minimizing the regrets of the DMs for all the levels should be taken into consideration. Then, each DM should try to maximize his or her membership function by making them as close as possible to unity by minimizing its negative deviational variables. Therefore, in effect, we are simultaneously optimizing all the objective functions. So, for the defined membership functions in Equations 3, 4, and 5, the flexible membership goals having the aspired level unity can be represented as follows: μZtNt XðÞðÞþD t1Dþ t1¼1; t¼1;2;...;Tð6Þ μZtDt XðÞðÞþD t2Dþ t2¼1; t¼1;2;...;Tð7Þ μ Xt Xt ðÞþ D t3 Dþ t3¼ I;t¼1;2;...;T1ðÞ ð8Þ Here, D t1;D t2are negative deviational variables, and Dþ t1;Dþ t2are positive deviational variables; Dþ t3; D t3 represent the vector of negative deviational and positive deviational variables. It is to be noted that any overdeviation from a fuzzy goal implies the full achievement value. Then, Equations 6, 7, and 8 can be written as follows: μZtNt XðÞðÞþD t1≥1;t¼1;2;...;Tð9Þ μZtDt XðÞðÞþD t2≥1;t¼1;2;...;Tð10Þ μ Xt Xt ðÞþ D t3≥ I;t¼1;2;...;T1ðÞ ð11Þ () tt XX t X B tt Xp B t X 1 B tt Xp Figure 2 Membership functions of decision vector Xtt¼1;2; ::;ð T1Þ. Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 4 of 11 http://www.jiei-tsb.com/content/8/1/16
FGP formulation can be presented as follows: Model I: Find Xso as to minimize λsubject to μZtNt XðÞðÞþ D t1≥1; t¼1;2;...;T μZtDt XðÞðÞþD t2≥1; t¼1;2;...;T μ Xt Xt ðÞþ D t3≥ I;t¼1;2;...;T1ðÞ λ≥D t1;t¼1;2;...;T λ≥D t2;t¼1;2;...;T λ≥ D t3 I;t¼1;2;...;T1ðÞ D t1≥0;t¼1;2;...;T D t2≥0;t¼1;2;...;T D t3≥ 0;t¼1;2;...;T1ðÞ Ai1 X1þ Ai2 X2þ:::::::::::::: þ AiT XT≤;¼;≥ðÞbi 8i¼1;2; ::::; m and X1≥0; X2≥0; :::::::::::; XT≥0 ð12Þ The above problem can be rewritten as follows: Minimize λsubject to Nt XðÞNL t NB tNL t þD t1≥1; t¼1;2;...;T DB tDt X ðÞ DB tDL t þD t2≥1; t¼1;2;...;T Xt XB t p t p tþ D t31≥ I;t¼1;2;...;T1ðÞ XB tþ pþ t Xt pþ tþ D t32≥ I;t¼1;2;...;T1ðÞ λ≥D t1;t¼1;2;...;T λ≥D t2;t¼1;2;...;T λ≥ D t31 I;t¼1;2;...;T1ðÞ λ≥ D t32 I;t¼1;2;...;T1ðÞ D t1≥0; t¼1;2;...;T D t2≥0; t¼1;2;...;T D t31≥ 0; t¼1;2;...;T1ðÞ D t32≥ 0; t¼1;2;...;T1 ðÞ Ai1 X1þ Ai2 X2þ:::::::::::::: þ AiT XT≤;¼;≥ðÞbi 8i¼1;2; ::::; m and X1≥0; X2≥0; :::::::::::; XT≥0 ð13Þ Here, D t1;D t2are negative deviational variables. D t31; D t32 represent the vector of underdeviational variables. Iis the column vector having all components equal to 1, and its dimension depends on X. Model IIa: Find Xso as to minimize λ¼X T t¼1 W t1D t1þX T1 t¼1 W t2D t2þX T1 t¼1 W t3 D t3 Model IIb: Find Xso as to minimize λ¼X T t¼1 D t1þX T1 t¼1 D t2þX T1 t¼1 D t3 subject to μZtNt XðÞðÞþD t1≥1; t¼1;2;...;T μZtDt XðÞðÞþD t2≥1; t¼1;2;...;T μ Xt Xt ðÞþ D t3≥ I;t¼1;2;...;T1ðÞ D t1≥0; t¼1;2;...;T D t2≥0; t¼1;2;...;T D t3≥ 0; t¼1;2;...;T1ðÞ Ai1 X1þ Ai2 X2þ:::::::::::::: þ AiT XT≤;¼;≥ðÞbi 8i¼1;2; ::::; m and X1≥0; X2≥0; :::::::::::; XT≥0 ð14Þ Here,D t1;D t2are negative deviational variables. D t3 represents the vector of underdeviational variables. The numerical weights are taken as W t1¼1 NB tNL t ðÞ ., W t2¼1=DB tDL t ðÞ ,and W t3¼1 p t ðÞ ;1 pþ t ðÞ .i.h . Iis the column vector having all components equal to 1, and its dimension depends on X.Theaboveproblem in models IIa and IIb can be rewritten as follows: Minimize λ¼X T t¼1 W t1D t1þX T1 t¼1 W t2D t2þX T1 t¼1 W t31 D t31 þX T1 t¼1 W t32 D t32 Minimize λ¼X T t¼1 D t1þX T1 t¼1 D t2þX T1 t¼1 D t31 þX T1 t¼1 D t32 subject to Nt X ðÞ NL t NB tNL t þD t1≥1;t¼1;2;...;T DB tDt XðÞ DB tDL t þD t2≥1;t¼1;2;...;T Xt XB t p t p tþ D t31≥ I;t¼1;2;...;T1ðÞ XB tþ pþ t Xt pþ tþ D t32≥ I;t¼1;2;...;T1ðÞ D t1≥0;t¼1;2;...;T D t2≥0;t¼1;2;...;T D t3≥ 0;t¼1;2;...;T1ðÞ Ai1 X1þ Ai2 X2þ:::::::::::::: þ AiT XT≤;¼;≥ðÞbi 8i¼1;2; ::::; m and X1≥0; X2≥0; :::::::::::; XT≥0 ð15Þ Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 5 of 11 http://www.jiei-tsb.com/content/8/1/16
Here, D t1;D t2are negative deviational variables. D t31 and D t32 represent the vector of underdeviational variables. W t1¼1 NB tNL t ðÞ .,W t2¼1=DB tDL t ðÞ and W t31 ¼1 p t ðÞ ; . W t31 ¼1 pþ t ðÞ .. Iis the column vector having all components equal to 1, and its dimension depends on X. By solving Equation 15, if the DMs are satisfied with this solution, then a satisfying solution is reached. Otherwise, higher level DMs should provide new tolerance limits for the control variable until a satisfying solution is reached. In general, considering a set of positive relaxation offered by the higher level DMs, the solution of Equation 15 becomes satisfying for all the level DMs. Numerical example Let us consider the following tri-level linear fractional programming problem as Max x1;x2 Z1¼7x1þ3x24x3þ2x4 x1þx2þx3þ1 Max x3 Z2¼x2þ3x3þ4x4 x1þx2þx3þ2 Max x4 Z3¼2x1þx2þx3þx4 x1þx2þx3þ3 subject to x1þx2þx3þx4≤5 x1þx2x3x4≤2 x1þx2þx3≥1 x1x2þx3þ2x4≤4 x1þ2x3þ2x4≤3 x4≤2 and x1≥0;x2≥0;x3≥0;x4≥0. We find the best optimal solution NL 1¼6at 0;0;1:5;0ðÞ; NB 1¼17 at 2:3333;0;0;0:3333ðÞ, NB 2¼9:5atð0;3:5;0;1:5Þ, NL 2¼0at ð1;0;0;0Þ, NL 3¼1atð0;1;0;0Þ, and NB 3¼5at 2:3333;0;0:3333;0ðÞ. Similarly, DB 1¼6at0;3:5;1:5;0ðÞ; DL 1¼2at 1;0;0;0 ðÞ ; DB 2¼7at 0;3:5;ð1:5;0Þ; DL 2¼3atð1;0;0;0Þ, DB 3¼8atð0;3:5;1:5;0Þ, and DL 3¼4atð1;0;0;0Þ. Let the first level DM decide that x1¼2:3333 with −2 (negative) and +2 (positive) tolerance limits and x2¼0 with −6.43 (negative) and +6.43 (positive) tolerance limits. Let the second level DM decide that x3¼0with−1 (negative) and +1 (positive) tolerance limits. Then, following the procedure, FGP model I gives the problem as follows: Minimize λsubject to Nt XðÞNL t NB tNL t þD t1≥1; t¼1;2;...;T ⇒7x1þ3x24x3þ2x4þ23D 11≥17 ⇒x2þ3x3þ4x4þ9:5D 21≥9:5 ⇒2x1þx2þx3þx4þ4D 31≥5 DB tDt X ðÞ DB tDL t þD t2≥1;t¼1;2;...;T ⇒x1þx2þx34D 12≥1 ⇒x1þx2þx34D 22≤1 ⇒x1þx2þx34D 32≤1 Xt XB t p t p tþ D t31≥ I; XB tþ pþ t Xt pþ tþ D t32≥ I;t¼1;2;...;T1 ðÞ ⇒0:5x1þD 1311≥1:1666 ⇒0:5x1þD 1321≥1:1666 ⇒0:1555x2þD 1312≥0 ⇒0:1555x2þD 1322≥0 ⇒x3þD 2311≥0 ⇒x3þD 2321≥0 Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 6 of 11 http://www.jiei-tsb.com/content/8/1/16
Figure 3 Description (a) and solution (b) of example 1 with model I in LINDO 10.0 (trial version). Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 7 of 11 http://www.jiei-tsb.com/content/8/1/16
Figure 4 Description (a) and solution (b) of example 1 with model IIb in LINDO 10.0 (trial version). Lachhwani and Poonia Journal of Industrial Engineering International 2012, 8:16 Page 8 of 11 http://www.jiei-tsb.com/content/8/1/16