Multi-objective fuzzy linear programming and its application in transportation model.Abstract In this study, the solution procedure of Multi-objective Fuzzy fuzz·y adj. fuzz·i·er, fuzz·i·est 1. Covered with fuzz. 2. Of or resembling fuzz. 3. Not clear; indistinct: a fuzzy recollection of past events. 4. Linear Programming Problem (MOFLPP) with mixed constraints CONSTRAINTS - A language for solving constraints using value inference. ["CONSTRAINTS: A Language for Expressing Almost-Hierarchical Descriptions", G.J. Sussman et al, Artif Intell 14(1):1-39 (Aug 1980)]. and its application in solid transportation problem, is going to be presented. There are two parts in this paper. In the first part, a Multi-objective Linear Programming Problem MOLPP) with fuzzy coefficients occurring in constraints and objective functions and fuzzy constraint Constraint A restriction on the natural degrees of freedom of a system. If n and m are the numbers of the natural and actual degrees of freedom, the difference n - m is the number of constraints. goals, has been considered. Here fuzzy constraint goals and coefficients of objective and constraint functions are characterised by Triangular Fuzzy Numbers (TFNs). Using Bellman and Zadeh's (1970) multicriteria fuzzy decision-making decision-making, n the process of coming to a conclusion or making a judgment. decision-making, evidence-based, n a type of informal decision-making that combines clinical expertise, patient concerns, and evidence gathered from process, the very problem has been converted to a crisp non-linear programming problem. Then it has been solved using fuzzy decisive set method. In other part, a linear multi-objective solid transportation problem with mixed constraint as well as additional restriction restriction - A bug or design error that limits a program's capabilities, and which is sufficiently egregious that nobody can quite work up enough nerve to describe it as a feature. in fuzzy environment is considered. In this transportation problem, cost coefficients of objective functions and additional restriction function, the supply, demand and conveyance The transfer of ownership or interest in real property from one person to another by a document, such as a deed, lease, or mortgage. conveyance n. capacity have been expressed as TFNs. This MOFLPP is solved by fuzzy decisive set method as before. Numerical numerical expressed in numbers, i.e. Arabic numerals of 0 to 9 inclusive. numerical nomenclature a numerical code is used to indicate the words, or other alphabetical signals, intended. examples have been provided for two parts to illustrate the solution procedure. Keywords Keywords are the words that are used to reveal the internal structure of an author's reasoning. While they are used primarily for rhetoric, they are also used in a strictly grammatical sense for structural composition, reasoning, and comprehension. and Phrases: Fuzzy multi-objective linear programming, Solid transportation problem, Triangular fuzzy number. 1. Introduction Generally, in a Multi-Objective Linear Programming Problem (MOLPP), coefficients (of objective and constraint functions) as well as constraint goals are assumed to be fixed in value. But there are many practical situations where this assumtions are not valid. These coefficients as well as constraint goals may not be well defined due to lack of information of data and/or and/or conj. Used to indicate that either or both of the items connected by it are involved. Usage Note: And/or is widely used in legal and business writing. uncertain market situations. For this reasons, the different coefficients and constraint goals may be chacterised by fuzzy numbers. The idea of fuzzy set Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent was first proposed by Zadeh [11], as a mean of handling uncertainty that is due to imprecision im·pre·cise adj. Not precise. im  pre·cise ly adv. rather than to
randomness. After that Bellman and Zadeh [11] proposed that a fuzzy
decision might be defined as the fuzzy set, defined by the intersection intersection /in·ter·sec·tion/ (-sek´shun) a site at which one structure crosses another. intersection a site at which one structure crosses another. of fuzzy objective and constraint goals. From this view point , Tanakka and Asai [6], Zimmermann Zimmermann (German for carpenter) may refer to the following people:
tr.v. tonged, tong·ing, tongs To seize, hold, or manipulate with tongs. [Back-formation from tongs. [14], Gasimov and Yenilmez [12] among others, considered single objective mathematical programming mathematical programming Application of mathematical and computer programming techniques to the construction of deterministic models, principally for business and economics. with all fuzzy parameters. Tong considered the fuzzy linear programming problem with fuzzy constraints. After defuzzification Defuzzification is the process of producing a quantifiable result in fuzzy logic. Typically, a fuzzy system will have a number of rules that transform a number of variables into a "fuzzy" result, that is, the result is described in terms of membership in fuzzy sets. he solved the so-obtained crisp problem by fuzzy decisive set method proposed by Sakawa and Yano [10]. Gasimov and Yenilmez considered fuzzy linear programming (FLP FLP Family Limited Partnership FLP Follow Up FLP Fiji Labor Party FLP Flashpoint FLP Fast Link Pulse FLP Flameproof FLP Flippase (genetics) FLP Front de Libération de la Palestine FLP Fasting Lipid Profile ) problem with less than type constraints. In their paper Coefficients of constraints were taken as fuzzy numbers. They solved it by fuzzy decisive set method and modified mod·i·fy v. mod·i·fied, mod·i·fy·ing, mod·i·fies v.tr. 1. To change in form or character; alter. 2. sub-gradient method. Lai-Hawng [15] considered MOLPP with all parameters, having a triangular possibility distribution. They used an auxiliary auxiliary In grammar, a verb that is subordinate to the main lexical verb in a clause. Auxiliaries can convey distinctions of tense, aspect, mood, person, and number. model and it was solved by multi-objective linear programming methods. Chanas Chanas may refer to:
There are so many Transportation models where FLP have been applied. Bit et al [1] considered fuzzy programming approach to multicriteria decision making transportation problem in which the constraints are of equality equality Generally, an ideal of uniformity in treatment or status by those in a position to affect either. Acknowledgment of the right to equality often must be coerced from the advantaged by the disadvantaged. Equality of opportunity was the founding creed of U.S. types. Latter Bit et al [2] also considered a fuzzy programming approach to multi-objective solid transportation problem in which the supply, demand and capacity constraints are of equality and inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. types. They solved it by fuzzy programming technique. Das See direct attached storage and FDDI. DAS - Digital Analog Simulator. Represents analog computer design. et al [13] considered the multi-objective transportation problem with interval interval, in music, the difference in pitch between two tones. Intervals may be measured acoustically in terms of their vibration numbers. They are more generally named according to the number of steps they contain in the diatonic scale of the piano; e.g. cost, source and destination parameters. They converted the interval cost, source and destination parameters into deterministic 1. (probability) deterministic - Describes a system whose time evolution can be predicted exactly. Contrast probabilistic. 2. (algorithm) deterministic - Describes an algorithm in which the correct next step depends only on the current state. one and finally it was solved by fuzzy programming technique. In this paper, we have proposed a MOFLPP with mixed constraints in which right hand side of constraints are fuzzy numbers. Using Bellman and Zadeh's fuzzy decision-making process, the MOFLPP is converted into an equivalent crisp LPP LPP Legitimate Peripheral Participation (community of practice) LPP Liberian People's Party (Liberia) LPP Leak Point Pressure LPP Land Partnership Plan LPP Lean Premixed Prevaporized . Then it is solved by simplex method simplex method Standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. . Next we have also considered MOFLPP with coefficients of objective as well as constraint functions and right hand sides of constraints are TFNs. Converting it into an equivalent crisp non-linear programming problem, it is also solved by fuzzy decisive set method. We have also considered an application of MOFLPP on a transportation model .We have considered a multi-objective solid transportation model with an additional restriction and mixed constraints in which coefficients of objective functions, additional restriction function, demand, supply and conveyance capacities are expressed as TFNs. It is then solved by fuzzy decisive set method as shown before. 2. Triangular Fuzzy Number (TFN TFN Tax File Number (Australia) TFN TheForce.Net (Star Wars Fan Site) TFN Taiwan Fixed Network TFN Texas Freedom Network TFN Tribe Flood Network ) Let F([??]) be a set of all triangular fuzzy number in a real line [??]. A triangular fuzzy number [??] ([member of] F([??])) is a normal and convex Convex Curved, as in the shape of the outside of a circle. Usually referring to the price/required yield relationship for option-free bonds. fuzzy set with the following membership function [[mu].sub.[??]] : [??] [right arrow] [1,0] (which satisfies both normality normality, in chemistry: see concentration. i.e [[mu].sub.[??]] ([??]) = 1 for at least one [??] [member of] R and convexity Convexity A measure of the curvature in the relationship between bond prices and bond yields. Notes: Positive convexity corresponds to curvature that opens upward. Negative convexity corresponds to curvature that opens downward. i.e [[mu].sub.[??]]([[??].sup./]) [greater than or equal to] min.([[mu].sub.[??]]([[??].sub.1]), [[mu].sub.[??]]([[??].sub.2])) where [[mu].sub.[??]]([??]) [member of] [1,0] and [for all] [[??].sup./] [member of] [[[??].sub.1],[[??].sub.2]]). [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .] [FIGURE 1 OMITTED] It is parameterized by a triplet triplet /trip·let/ (trip´let) 1. one of three offspring produced at one birth. 2. a combination of three objects or entities acting together, as three lenses or three nucleotides. 3. ([a.sub.1],[a.sub.2],[a.sub.3]) where [a.sub.1],[a.sub.3] are the lower and upper limits of support of [??] and [a.sub.2] is the pick value of [??] (fig fig, name for members of the genus Ficus of the family Moraceae (mulberry family). This large genus contains some 800 species of widely varied tropical vines (some of which are epiphytic); shrubs; and trees, including the banyan, the peepul, or bo tree, and .-1). Triangular fuzzy numbers are very often used in different application (e.g. fuzzy controls, managerial decision making, business and finance, social sciences etc.). More generally, the left [??] and right [??] are branches of the TFN. They can be denoted by left [??] = ([a.sub.1], [a.sub.2], [a.sub.2]) and right [??] = ([a.sub.2], [a.sub.2], [a.sub.3]). i) The left TFN [??] =([a.sub.1], [a.sub.2], [a.sub.2]) (fig.-2) is suitable to represent positive large or words with similar meaning (e.g. old age, big profit , high risk, etc.) provided that [a.sub.2] > [a.sub.1] . It is represented by the following membership functions: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], = [[x - [a.sub.1]]/[[a.sub.2] - [a.sub.1]]] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = 1 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [FIGURE 2 OMITTED] ii) The right TFN [??] = ([a.sub.2] ,[a.sub.2] ,[a.sub.3]) (fig.-3) [??] is suitable to represent positive small or words with similar meaning (e.g. young age, small cost, small risk etc.) provided that [a.sub.3] > [a.sub.2]. It is represented by the following membership functions: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], = [[[a.sub.3] - x]/[[a.sub.3] - [a.sub.2]]] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], = 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [FIGURE 3 OMITTED] Note: A TFN = ([a.sub.1],[a.sub.2],[a.sub.3]) is positive (negative) if [a.sub.1] [greater than or equal to] 0 ([a.sub.3] < 0). 3. Multi-Objective Linear Programming Problem (MOLPP) with Fuzzy Resources The General Multi-Objective Linear Programming Problem (GMOLPP) with mixed constraints may be written as follows: Minimize In a graphical environment, to hide an application that is currently displayed on screen. For example, in Windows and Mac, the application's window is removed from the screen and represented by an icon on the Windows Taskbar. In the Mac, the icon is placed in the Dock. See Win Minimize windows. Z = [[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (3.1) subject to [n.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] = [b.sub.i] for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [Z.sup.k] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j], k = 1, 2, 3, ......, K 3.1 MOLPP with fuzzy resources When constraint goals are TFNs, (3.1) becomes Minimize Z =[[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (3.2) subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = 1,2,3, ......, [m.sub.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.2]+1 , [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [Z.sup.k] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j], k = 1, 2, 3, ......, K We will accept some assumptions. Assumption1: [[??].sub.i] are considered as the following positive TFNs: Left TFN [[??].sub.i] = ([b.sub.i] - [b.sup.0.sub.i], [b.sub.i], [b.sub.i]) with tolerance tolerance /tol·er·ance/ (tol´er-ans) 1. diminution of response to a stimulus after prolonged exposure. 2. the ability to endure unusually large doses of a poison or toxin. 3. drug t. 4. [b.sup.0.sub.i] (<[b.sub.i]) for [n.summation over (j=1)] [a.sub.ij][x.sub.j] [greater than or equal to] [[??].sub.i]; 1,2,3, ......, [m.sub.1]; RightTFN [[??].sub.i] = ([b.sub.i], i, [b.sub.i] + [b.sup.0.sub.i]) with tolerance [b.sup.0.sub.i] (>0) for [n.summation over (j=1)] [a.sub.ij][x.sub.j] [less than or equal to] [[??].sub.i]; = [m.sub.1]+1,[m.sub.1]+2, ......, [m.sub.2]; and TFN [[??].sub.i] = ([b.sub.i]- [b.sup.l.sub.i],[b.sub.i],[b.sub.i]+ [b.sup.r.sub.i]) with tolerances [b.sup.l.sub.i](< [b.sub.i]), [b.sup.r.sub.i] (> 0) for [n.summation over (j=1)] [a.sub.ij][x.sub.j] = [[??].sub.i], i = [m.sub.2]+1 , [m.sub.2]+2, ......, m. The problem (3.2) becomes with extreme tolerances as Minimize Z =[[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (3.3) subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.0.sub.i] i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.0.sub.i] i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [Z.sup.k] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j], k = 1, 2, 3, ......, K 3.2 Fuzzy Programming Technique for the Solution of MOLPP with Fuzzy Resources The MOFLPP can be considered as a Vector Minimum Problem (VMP VMP Vampire VMP Validation Master Plan VMP Value of Marginal Product VMP Veterinary Medicinal Product VMP Veterans Memorial Park VMP Variable Message Panel VMP Value Management Program VMP Vector Map Product VMP Vacuum Metallised Pigment ). Let [L.sub.k] and [U.sub.k] be the lower and upper bound for the k-th objective, where [L.sub.k] = aspired level of achivement for the k-th objective function, and [U.sub.k] = highest acceptable level of achivement for the k-th objective function. When the aspiration aspiration /as·pi·ra·tion/ (as?pi-ra´shun) 1. the drawing of a foreign substance, such as the gastric contents, into the respiratory tract during inhalation. 2. levels for each objective have been specified spec·i·fy tr.v. spec·i·fied, spec·i·fy·ing, spec·i·fies 1. To state explicitly or in detail: specified the amount needed. 2. To include in a specification. 3. , we formed a fuzzy model. Our next step is to transform the fuzzy model into a crisp model (i.e a coventional LPP). The details of the foregoing steps may be presented as follows: Algorithm algorithm (ăl`gərĭth'əm) or algorism (–rĭz'əm) [for Al-Khowarizmi], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical. Step- step- pref. Related by means of a remarriage rather than by blood: stepparent. [Middle English, from Old English st 1. Solve the MOLPPs (3.1) and (3.3) as a single objective LPP using each time only one objective and ignore all others. Step-2. From the results of step-1, determine the corresponding value for every objective functions at each solutions. Step-3. Find upper and lower bounds This article is about order theory and lattice theory. For analysis of algorithms in computational complexity, see Big O notation. In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (P (i.e [U.sub.k] and [L.sub.k]) for kth objective from the 2k objective values derived de·rive v. de·rived, de·riv·ing, de·rives v.tr. 1. To obtain or receive from a source. 2. in step-2. Step-4. The initial fuzzy model is equivalent to following: Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4) so as to satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = 1, 2, 3, ......, K. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = 1,2,3, ......, [m.sub.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] [congruent con·gru·ent adj. 1. Corresponding; congruous. 2. Mathematics a. Coinciding exactly when superimposed: congruent triangles. b. to] [b.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Here membership functions for fuzzy constraints of (3.4) are defined as: (for kth constraints [[??].sub.k] (k = 1, 2, 3, ......, K) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] [less than or equal to] [L.sub.k], k = 1, 2, 3, ......, K. = [[[U.sub.k] - [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j]]/[[U.sub.k] - [L.sub.k]]] for [L.sub.k] [less than or equal to] [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] [less than or equal to] [U.sub.k], = 0 for [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] > [U.sub.k], (for the ith constraints [[??].sub.i] (i = 1,2,3, ......, [m.sub.1])) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.0.sub.i] = [[[n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sub.i]]/[b.sup.0.sub.i]] for [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.0.sub.i] [less than or equal to] [b.subi] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] = 0 for [b.sub.i] > [n.summation over (j=1)][a.sub.ij][x.sub.j] (for the ith constraints [[??].sub.i] (i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2])) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] < [n.summation over (j=1)][a.sub.ij][x.sub.j] = [[[b.sub.i] - [n.summation over (j=1)][a.sub.ij][x.sub.j]]/[b.sup.0.sub.i]] for [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] + [b.sup.0.sub.i] = 1 for [b.sub.i] [greater than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] + [b.sup.0.sub.i] (for the ith constraints [[??].sub.i] (i = [m.sub.2]+1, [m.sub.2]+2, ......, m)) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] > [n.summation over (j=1)][a.sub.ij][x.sub.j] + [b.sup.l.sub.i] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.summation over (j=1)] [a.sub.ij][x.sub.j] < [b.sub.i] [less than or equal to] [n.summation over (j=1)] [a.sub.ij][x.sub.j] + [b.sup.l.sub.i] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.r.sub.i] [less than or equal to] [b.sub.i] < [n.summation over (j=1)][a.sub.ij][x.sub.j] = 0 for [b.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.r.sub.i] Step-5. Using the max-min operator (as Zimmermann [7]) crisp LPP for (3.2) is formulated for·mu·late tr.v. for·mu·lat·ed, for·mu·lat·ing, for·mu·lates 1. a. To state as or reduce to a formula. b. To express in systematic terms or concepts. c. as follows: Max [lambda] (3.5) subject to, [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] + [lambda]([U.sub.k] - [L.sub.k]) [less than or equal to] [U.sub.k] for k = 1, 2, 3, ......, K. [n.summation over (j=1)][a.sub.ij][x.sub.j] - [lambda] [b.sup.0.sub.i] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)][a.sub.ij][x.sub.j] + [lambda] [b.sup.0.sub.i] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] - [lambda] [b.sup.l.sub.i] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [n.summation over (j=1)][a.sub.ij][x.sub.j] + [lambda] [b.sup.r.sub.i] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m 0 [less than or equal to] [lambda] [less than or equal to] 1, [x.sub.j] [greater than or equal to] 0 . j=1,2,3, ......, n It can be solved by any simplex method. Numerical Example1: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1) subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i, j, k = 1, 2 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; [[??].sub.2] = [??] = (9, 10, 10) respectively constraint goals. To solve this problem, we first solve the following four Sub-Problems (SPs): Minimize [Z.sup.11] = 5[[??].sub.1]+3[[??].sub.2] ... (1.1) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sup.12] = 5[[??].sub.1]+3[[??].sub.2] ..........(1.2) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sup.21] = 2[[??].sub.1]+7[[??].sub.2] ......(1.3) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sup.22] = 2[[??].sub.1]+7[[??].sub.2] ..........(1.4) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; So the optimal solutions of (1.1), (1.2), (1.3) and (1.4) are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], respectively. So [L.sub.1] = min [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]= min{30, 27, 50, 45} = 27 and [U.sub.1] = max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = max{30, 27, 50, 45} = 50 [L.sub.2] = min [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = min{70, 63, 20, 18} = 18 and [U.sub.2] = max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = max{70, 63, 20, 18} = 70 Following the step-4, the problem (1) is equivalent to Find {[x.sub.j] , j = 1, 2.} (1.5) so as to satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here membership functions for fuzzy constraints of (1.5) are defined as: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = [50 - [(5[x.sub.1] + 3[x.sub.2])]/23] for 27 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = [70 - [(2[x.sub.1] + 7[x.sub.2])]/52] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = [2[x.sub.2] + 4[x.sub.2] - 20]/2] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], = 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Using the max-min operator (as Zimmermann [7]) crisp LPP for (1) is formulated as follows: Max [lambda] (1.6) 5[[??].sub.1]+3[[??].sub.2]+23[lambda] [less than or equal to] 50, 2[[??].sub.1]+7[[??].sub.2]+52[lambda] [less than or equal to] 70, 2[[??].sub.1]+4[[??].sub.2] -2[lambda] [greater than or equal to] = 20, [[??].sub.1]+[[??].sub.2] -[lambda] [greater than or equal to] 10, 0 [less than or equal to] [lambda] [less than or equal to] 1, [[??].sub.1],[[??].sub.2] [greater than or equal to] 0. So, optimal solution of MOFLPP (1) are [[??].sup.*.sub.1] =4.758065; [[??].sup.*.sub.2] = 5.645161; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with aspiration level [[lambda].sup.*] = 0.403226. 4. MOLPP with Fuzzy Coefficients and Fuzzy Resources When the objective function's coefficients, technological coefficients and also right hand side of constraints are fuzzy numbers then (3.1) becomes Minimize Z =[[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (4.1) subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = 1,2,3, ......, [m.sub.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [x.sub.j] [greater than or equal to] 0 . j=1, 2, 3, ......, n where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], k = 1, 2, 3, ......, K Assumption1: Fuzzy objective and constraints coefficients are considered as the following positive TFN's: Right TFN [c.sup.k.sub.j] = ([c.sup.k.sub.j], [c.sup.k.sub.j], [c.sup.k.sub.j] + [p.sup.k.sub.j]) with tolerance [p.sup.k.sub.j] (> 0) for the objective function [n.summation over (j=1)][[??].sup.k.sub.j][x.sub.j] for k = 1, 2, 3, ......, K. Left TFNs [[??].sub.ij] = ([a.sub.ij] - [d.sup.0.sub.ij], [a.sub.ij], [a.sub.ij]) with tolerance [d.sup.0.sub.ij] (< [a.sub.ij]) and [[??].sub.i] = ([b.sub.i]-[b.sup.0.sub.i], [b.sub.i], [b.sub.i]) with tolerance [b.sup.0.sub.i](< [b.sub.i]) for [n.summation over (j=1)][[??].sub.ij] [x.sub.j] [greater than or equal to] [[??].sub.i]; i = 1,2,3, ......, [m.sub.1]. Right TFNs [[??].sub.ij] = ([a.sub.ij] , [a.sub.ij] , [a.sub.ij] + [d.sup.0.sub.ij]) with tolerance [d.sup.0.sub.ij] (> 0) and [[??].sub.i] = ([b.sub.i], [b.sub.i], [b.sub.i] + [b.sup.0.sub.i] with tolerance [b.sup.0.sub.i] (> 0) for [n.summation over (j=1)][[??].sub.ij] [x.sub.j] [less than or equal to] [[??].sub.i]; i =[m.sub.1]+ 1, [m.sub.1+2], ......, [m.sub.2]. and TFNs [[??].sub.ij] = ([a.sub.ij] - [d.sup.l.sub.ij], [a.sub.ij], [a.sub.ij] + [d.sup.r.sub.ij]) with tolerances [d.sup.l.sub.ij](< [a.sub.ij]), [d.sup.r.sub.ij] (> 0) and [[??].sub.i] = ([b.sub.i] - [b.sup.l.sub.i], [b.sub.i], [b.sub.i] + [b.sup.r.sub.i]) with tolerances [b.sup.l.sub.ij] (< [b.sub.i]), [b.sup.r.sub.ij] (> 0) for [n.summation over (j=1)][[??].sub.ij][x.sub.j] = [[??].sub.i]; i = [m.sub.2+1], [m.sub.2]+2, ......, m. For the calculation of upper ([U.sub.k]) and lower ([L.sub.k]) bounds of the k-th(k = 1, 2, 3, ......, K) objective function, we first construct the following eight sub-problems(4.2 - 4.9): Minimize [Z.sup.k1] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.2) subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] = [b.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Minimize [Z.sup.k2] = [n.summation over (j=1)]([c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.3) subject to same constraints of (4.2) Minimize[Z.sup.k3] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.4) subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i]-[b.sup.0.sub.i], for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.0.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Minimize [Z.sup.k4] = [n.summation over (j=1)][c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.5) subject to same constraints of (4.4) Minimize [Z.sup.k5]= [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.6) subject to [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)]([a.sub.ij] - [d.sup.l.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [n.summation over (j=1)]([a.sub.ij] + [d.sup.r.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Minimize [Z.sup.k6] = [n.summation over (j=1)][c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.7) subject to same constraints of (4.6). Minimize [Z.sup.k7] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.8) subject to [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.0.sub.i], for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)]([a.sub.ij] + [d.sup.0.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.0.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)]([a.sub.ij] - [d.sup.l.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [n.summation over (j=1)]([a.sub.ij] + [d.sup.r.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Minimize [Z.sup.k8] = [n.summation over (j=1)]([c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.9) subject to same constraints of (4.8). 4.1 Fuzzy Programming Technique for the Solution of MOLPP with Fuzzy Coefficients and Fuzzy Resources The MOFLPP can be considered as a vector minimum problem. Let [L.sub.k] and [U.sub.k] be the lower and upper bound for the k-th objective, where [L.sub.k] = aspired level of achivement for the k-th objective function, and [U.sub.k] = highest acceptable level of achivement for the k-th objective function. When the aspiration levels for each objectives have been specified, we formed a fuzzy model. Our next step is to transform the fuzzy model into a crisp model The foregoing steps may be presented as follows: Step-1. Solve the MOLPPs (4.2), (4.3), (4.4), (4.5), (4.6), (4.7), (4.8) and (4.9) for each kth objectives (k = 1, 2, 3, ...., K). Step-2. From the results of step-1, determine the corresponding value for every objective function at each solution. Step-3. Find upper and lower bounds (i.e [U.sub.k] and [L.sub.k]) for kth objective from the 8k objective values derived in step-2, as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] k= 1, 2, 3, ......, K 1 [less than or equal to] r [less than or equal to] K 1 [less than or equal to] s [less than or equal to] 8 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] k= 1, 2, 3, ......, K 1 [less than or equal to] r [less than or equal to] K 1 [less than or equal to] s [less than or equal to] 8 Step-4. The initial fuzzy model becomes (in terms of aspiration levels with each objectives) Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.10) so as to satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = 1, 2, 3, ......, K. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = 1,2,3, ......, [m.sub.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)][a.sub.ij][x.sub.j] [congruent to] [b.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m Here the membership functions for the fuzzy constraints of (4.10) are defined as: (for kth constraints [[??].sub.k] (k = 1, 2, 3, ......, K) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) for [U.sub.k] < [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] = [[[U.sub.k] - [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j]]/[[n.summation over (j=1)][p.sup.k.sub.j] [x.sub.j] + M]] for [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] [less than or equal to] [U.sub.k] [less than or equal to] [n.summation over (j=1)](c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] + M = 1 for [U.sub.k] [greater than or equal to] [n.summation over (j=1)]([c.sup.k.sub.j] + [p.sup.k.sub.j]) [x.sub.j] + M where M = [U.sub.k] - [L.sub.k], k = 1, 2, 3, ......, K. (for the ith constraints [[??].sub.i] (i = 1,2,3, ......, [m.sub.1])) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] [less than or equal to] [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] - [b.sup.0.sub.i] = [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sub.i]/[n.summation over (j=1)] [d.sup.0.sub.ij][x.sub.j] + [b.sup.0.sub.i] for [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij]) [x.sub.j] - [b.sup.0.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] = 0, for [b.sub.i] > [n.summation over (j=1)][a.sub.ij][x.sub.j] (for the ith constraints [[??].sub.i] (i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2])) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [b.sub.i] < [n.summation over (j=1)][a.sub.ij][x.sub.j] = [[[b.sub.i] - [n.summation over (j=1)][a.sub.ij][x.sub.j]]/[[[n.summation over (j=1)][d.sup.0.sub.ij][x.sub.j]]+[b.sup.0.sub.i]]], for [n.summation over (j=1)][a.sub.ij][x.sub.j]] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)] ([a.sub.ij] + [d.sup.0.sub.ij])[x.sub.j] + [b.sup.0.sub.i] = 1, for [b.sub.i] [greater than or equal to] [n.summation over (j=1)]([a.sub.ij] + [d.sup.0.sub.ij])[x.sub.j] + [b.sup.0.sub.i] (for the ith constraints [[??].sub.i] (i = [m.sub.2]+1, [m.sub.2]+2, ......, m)) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [b.sub.i] < [n.summation over (j=1)] ([a.sub.ij] - [d.sup.l.sub.ij]) [x.sub.j] - [b.sup.r.sub.i] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [n.summation over (j=1)] ([a.sub.ij] - [d.sup.l.sub.ij]) [x.sub.j] - [b.sup.r.sub.i] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)] [a.sub.ij][x.sub.j] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [n.summation over (j=1)] [a.sub.ij] [x.sub.j] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)] ([a.sub.ij] + [d.sup.r.sub.ij]) [x.sub.j] + [b.sup.l.sub.i] = 0, for [b.sub.i] > [n.summation over (j=1)] ([a.sub.ij] + [d.sup.0.sub.ij]) [x.sub.j] + [b.sup.l.sub.i] Step-5. Using the max-min operator (as Zimmermann [15]) crisp LPP for (4.1) is formulated as follows: Max [lambda] (4.11) subject to, [n.summation over (j=1)] ([c.sup.k.sub.j] + [lambda][p.sup.k0.sub.j])[x.sub.j] + [lambda] ([Z.sup.k.sub.U] - [Z.sup.k.sub.L]) [less than or equal to] [Z.sup.k.sub.L] for k = 1, 2, 3, ..., K. [n.summation over (j=1)] ([a.sub.ij] - [lambda][d.sup.0.sub.ij])[x.sub.j] - [lambda] [b.sup.0.sub.i] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1] [n.summation over (j=1)] ([a.sub.ij] + [lambda][d.sup.0.sub.ij])[x.sub.j] + [lambda][b.sup.0.sub.i] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2] [n.summation over (j=1)] ([a.sub.ij] - (1 - [lambda]) [d.sup.l.sub.ij]) [x.sub.j] + [lambda][b.sup.r.sub.i] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m [n.summation over (j=1)] ([a.sub.ij] + (1 - [lambda])[d.sup.r.sub.ij])[x.sub.j] - [lambda][b.sup.l.sub.i] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m 0 [less than or equal to] [lambda] [less than or equal to] 1, [x.sub.j] [greater than or equal to] 0 j=1,2,3, ......, n Note: The constraints in problem (4.11) containing cross product terms [lambda][x.sub.j](j = 1, 2, 3, ..., n) which are not convex. Therefore the solution of this problem requires the special approach adopted for solving general non-convex application problems. It may be solved by fuzzy decisive set method [12]. 4.2 The algorithm of the fuzzy decisive set method (Sakawa and Yano [12]): This method is based on the idea that for a fixed value of [lambda], the problem (4.11) are linear programming problem. Obtains the optimal solution [[lambda].sup.*] to the problem (4.11) is equivalent to determining the maximum value of [lambda] so that the feasible (algorithm) feasible - A description of an algorithm that takes polynomial time (that is, for a problem set of size N, the resources required to solve the problem can be expressed as some polynomial involving N). set is nonempty. The algorithm to this method for the problem (4.11) is presented below. Algorithm: Step1- Set [lambda] = 1 and test whether a feasible set satisfying the constraints of the problem (31) exist or not, using phase one of the Simplex method. If a feasible set exist, set [lambda] = 1. Otherwise, set [[lambda].sup.L] = 0 and [[lambda].sup.R] = 1 and go to the next step. Step2- For the value of [lambda] = [[[lambda].sup.L] + [[lambda].sup.R]]/2 up date the value of [[lambda].sup.L] and [[lambda].sup.R] using the bisection method In mathematics, the bisection method is a root-finding algorithm which works by repeatedly dividing an interval in half and then selecting the subinterval in which the root exists. Suppose we want to solve the equation f(x) = 0. as follows: [[lambda].sup.L] = [lambda] if feasible set is non-empty for [lambda], [[lambda].sup.R] = [lambda] if feasible set is empty for [lambda]. Consequently for each [lambda], test whether a feasible set of the problem (4.11) exists or not exists , using phase one of the simplex method and determine the maximum value of [[lambda].sup.*] satisfying the constraints of the problem (4.11). Numerical Example 2: Minimize [Z.sup.1] = [[??].sup.1.sub.1] [[??].sub.1] + [[??].sup.1.sub.2] [[??].sub.2] (2) Minimize [Z.sup.2] = [[??].sup.2.sub.1] [[??].sub.1] + [[??].sup.2.sub.2] [[??].sub.2] subject to [[??].sub.11] [[??].sub.1] + [[??].sub.12] [[??].sub.2] [greater than or equal to] [[??].sub.1] [[??].sub.21] [[??].sub.1] + [[??].sub.22] [[??}.sub.2] [greater than or equal to] [[??].sub.2] [x.sub.1], [x.sub.2] [greater than or equal to] 0 where [[??].sup.1.sub.1] = [??] = (5, 5, 6); [[??].sup.1.sub.2] = [??] = (3, 3, 4.5) and [[??].sup.2.sub.1] = [??] = (2, 2, 4); [[??].sup.2.sub.2] = [??] = (7, 7, 7.5) respectively for objective coefficients. [[??].sub.11] = [??] = (1.5,2,2); [[??].sub.12] = [??] = (3,4,4) [[??].sub.12] = [??] = (.5,1,1); [[??].sub.22] = [??] = (1,1,1) respectively for technological coefficients. and [[??].sub.1] = [??] = (18, 20, 20); [[??].sub.2] = [??] = (9, 10, 10) respectively for constraint goals. To solve the problem (2), we first solve the following sixteen sub-problems: Minimize [Z.sub.11] = 5[x.sub.1]+3[x.sub.2], -- (2.1) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sup.12] = 6[[??].sub.1]+4.5[[??].sub.2], -- (2.2) subject to same constraints of (2.1). Minimize [Z.sub.13] = 5[[??].sub.1]+3[[??].sub.2], -- (2.3) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sub.14] = 6[[??].sub.1]+4.5[[??].sub.2] -- (2.4) subject to same constraints of (2.3). Minimize [Z.sup.15] = 5[[??].sub.1]+3[[??].sub.2], -- (2.5) subject to 1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 20, 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sub.16] = 6[[??].sub.1]+4.5[[??].sub.2], -- (2.6) subject to same constraints of (2.5). Minimize [Z.sup.17] = 5[[??].sub.1]+3[[??].sub.2], -- (2.7) subject to 1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 18, 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sup.18] = 6[[??].sub.1]+4.5[[??].sub.2], -- (2.8) subject to same constraints of (2.7). Minimize [Z.sup.21] = 2[[??].sub.1] + 7[[??].sub.2], -- (2.9) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] = 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0 ; Minimize [Z.sup.22] = 4[[??].sub.1]+7.5[[??].sub.2], -- (2.10) to same constraints of (2.9). Minimize [Z.sup.23] = 2[[??].sub.1]+7[[??].sub.2], -- (2.11) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0 ; Minimize [Z.sup.24] = 4[[??].sub.1]+7.5[[??].sub.2] -- (2.12) subject to same constraints of (2.11). Minimize [Z.sup.25] = 2[[??].sub.1]+7[[??].sub.2], -- (2.13) subject to 1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 20, 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0 ; Minimize [Z.sup.26] = 4[[??].sub.1]+7.5[[??].sub.2], -- (2.14) subject to same constraints of (2.13). and Minimize [Z.sup.27] = 2[[??].sub.1]+7[[??].sub.2], -- (2.15) subject to 1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 18 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0; Minimize [Z.sup.28] = 4[[??].sub.1]+7.5[[??].sub.2], -- (2.16) subject to same constraints of (2.15). so the optimal solutions of the sub-problems ((2.1) - (2.16)) are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] So, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] According to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. step-4 formulating membership functions and following step-5, crisp LPP of (2) is formulated as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.17) Using gradient gradient In mathematics, a differential operator applied to a three-dimensional vector-valued function to yield a vector whose three components are the partial derivatives of the function with respect to its three variables. The symbol for gradient is ∇. based non- non- word element [L.]not . non- pref. Not: noninvasive. linear programming package the optimal solutions be [[??].sup.*.sub.1] = 10.88221, [[??].sup.*.sub.2] = 2.041447, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and aspiration level [[lambda].sup.*] = 0.4539063. The problem (2.17) may also be solved by the fuzzy decisive set method. For [lambda] = 1 , the problem can be written as 6[[??].sub.1]+4.5[[??].sub.2] [less than or equal to] 27 4[[??].sub.1]+7.5[[??].sub.2] [less than or equal to] 18 1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 22 .5[[??].sub.1]+[[??.sub.2] [greater than or equal to] 11 [[??].sub.1], [[??].sub.2] [greater than or equal to] 0. Since the feasible set is empty, by taking [[lambda].sup.L] =0 & [[lambda].sup.R] = 1, the new value of [lambda] = [[0 + 1]/2] = 1/2 is tired. For [lambda] = 1/2 = .5, the problem (2.17) can be written as 5.5[[??].sub.1]+3.75[[??].sub.2] [less than or equal to] 63.5 3[[??].sub.1]+3.25[[??].sub.2] [less than or equal to] 44 1.75[[??].sub.1]+3.5[[??].sub.2] [greater than or equal to] 21 0.75[[??].sub.1]+[[??].sub.2] [greater than or equal to] 10.5 [[??].sub.1], [[??].sub.2] [greater than or equal to] 0. Since the feasible set is empty, by taking [[lambda].sup.L] =1 & [[lambda].sup.R] = 1/2, the new value of [lambda] = [0 + [1/2]/2] = 1/4 is tired. And so on. The following values of [lambda] are obtained in the next 25th iterations: [lambda] = 0.25; [lambda] = 0.375; [lambda] = 0.4375; [lambda] = 0.46875; [lambda] = 0.453125; [lambda] = 0.4609375; [lambda] = 0.4570312; [lambda] = 0.4550781; [lambda] = 0.4541016; [lambda] = 0.4536133; [lambda] = 0.4538574; [lambda] = 0.4539795; [lambda] = 0.4539185; [lambda] = 0.4538879; [lambda] = 0.4539032; [lambda] = 0.4539109; [lambda] = 0.4539070; [lambda] = 0.4539051; [lambda] = 0.4539060; [lambda] = 0.4539065; [lambda] = 0.4539063; [lambda] = 0.4539064; [lambda] = 0.4539063. Consequently, we obtain the optimal value [[lambda].sup.*] = 0.4539063 at the 25th iteration One repetition of a sequence of instructions or events. For example, in a program loop, one iteration is once through the instructions in the loop. See iterative development. (programming) iteration - Repetition of a sequence of instructions. by using the fuzzy decisive set method and solutions of the problem (2) are [[??].sup.*.sub.1] = 10.88221, [[??].sup.*.sub.2] = 2.041448, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] =60.53539, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 36.05455 and aspiration Level [[lambda].sup.*] = 0.4539063. 5. Application in Transportation Models The following model adopted from Bit, at al [3] is used to show that the above MOFLPP can be employed to solve the multi-objective transportation problems. Consider m origins (or sources) [O.sub.i] (i = 1, 2, ......, m) and n destinations [D.sub.j] (j = 1, 2, ..., n). At each origin [O.sub.i], let [a.sub.i] be the amount of homogeneous The same. Contrast with heterogeneous. homogeneous - (Or "homogenous") Of uniform nature, similar in kind. 1. In the context of distributed systems, middleware makes heterogeneous systems appear as a homogeneous entity. For example see: interoperable network. product which we want to transport to n destinations [D.sub.j] to satify the demands for [b.sub.j] units of the product there. Let [e.sub.k] (k =1, 2, ...., K) be the units of product which can be carried by K different modes of transport called conveyance, such as trucks, air freight air freight n → flete m por avión air freight n → fret aérien air freight air n → Luftfracht f , freight The price or compensation paid for the transportation of goods by a carrier. Freight is also applied to the goods transported by such carriers. The liability of a carrier for freight damaged, lost, or destroyed during shipment is determined by contract, statute, or train, ship, etc. A penalty [p.sup.p.sub.ijk] associated with transportation of a unit of the product from source i to destination j by means of k-th conveyance for the p-th criterion
n. The process or condition of becoming worse. amount, quantity goods delivered, under used capacity, etc. A variable [x.sub.ijk] represents the unknown quantity to be transported from origin [O.sub.i] to destination [D.sub.j] by means of the k-th conveyance. A general Multi-Objective solid transportation model with mixed constraints, written as follows: Minimize Z =[ [Z.sup.1], [Z.sup.2], [Z.sup.3], ......, [Z.sup.P] (5.1) Subject to: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [Z.sup.p] = [summation over (i[member of]I)] [summation over (j[member of]J)] [summation over (k[member of]K)] [c.sup.p.sub.ijk] [x.sub.ijk], p = 1, 2, 3, ......, P. It is noted that the restriction on total delivery time is necessary for transportation of perishable goods PERISHABLE GOODS, Goods which are lessened in value and become worse by being kept. Vide Bona Peritura. , delivery of emergency supplies, etc. We are now adding an additional restriction to the above model that the total delivery time ([summation over (i)] [summation over (j)] [summation over (k)] [t.sub.ijk] [x.sub.ijk]) is not more than T units. Here [t.sub.ijk] represents delivery time of unit item of transportation from i-th zone to j-th zone by means of k-th conveyance for the p-th criterion. In the above model, penalties, supply and demand amount, etc. are assumed to be fixed in value. In general, transportation penalties, delivery time, demand and supply amount are somewhat uncertain (non-stochastic) imprecise im·pre·cise adj. Not precise. im pre·cisely adv. and vague in nature. So in
real life situation, to depict de·pict tr.v. de·pict·ed, de·pict·ing, de·picts 1. To represent in a picture or sculpture. 2. To represent in words; describe. See Synonyms at represent. this nature, all the parameters in the above model may be taken as fuzzy numbers. Then the above model (5.1) in fuzzy environment may be rewritten as Minimize Z =[[Z.sup.1], [Z.sup.2], [Z.sup.3], ......, [Z.sup.P] (5.2) Subject to: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] It is a MOFLPP. It can be solved as before. Numerical Example3: 5.1 Fuzzy Multi-Objective Solid Transportation Model with Restriction on Total delivery Time Minimize [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3) Subject to: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [[??].sub.ijk] = 0, for i [member of] 1, 2, 3; j [member of] 1, 2, 3; k [member of] 1, 2, 3 where three penalties and delivery time are given as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [Here [c.sup.1.sub.111] = (9,9,10), [c.sup.1.sub.112] = (12,12,12.5), [c.sup.1.sub.113] = (9,9,10) and similar representation for other elements.] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively for constraint goals. The optimal solution of the above fuzzy transportation problem is [x.sup.*.sub.121] = 4.805142, [x.sup.*.sub.122] = 0.6301453, [x.sup.*.sub.132] = 1.872076, [x.sup.*.sub.211] = 5.556556, [x.sup.*.sub.212] = 2.136081, [x.sup.*.sub.222] = 0.08009255, [x.sup.*.sub.223] = 1.099346, [x.sup.*.sub.232] = 0.5889684, [Z.sup.1*] ([x.sup.*]) = 97.09676, [Z.sup.2*] ([x.sup.*]) = 58.50913, [Z.sup.3*] ([x.sup.*]) = 58.08856 with aspiration level [[lambda].sup.*]=0.3073629. 6. Conclusion In this paper, we have proposed two types of FLPP FLPP Future Launchers Preparatory Programme (European Space Agency) FLPP Foreign Language Proficiency Pay . One is FLPP with fuzzy number resources and another is a FLPP with fuzzy number coefficients and resources. A procedure is developed for solving said FLPPs. It is also applied to a fuzzy solid transportation problem. This procedure may be very helpful for any fuzzy multi criteria criteria (krītēr´ē  ),n. decision making problem. Received July July: see month. 27, 2004, Accepted December December: see month. 31, 2004 References [1] A. K. Bit, M. P. Biswal and S. S. Alam (language) ALAM - A language for symbolic mathematics, especially General Relativity. See also CLAM. ["ALAM Programmer's Manual", Ray D'Inverno, 1970]. , Fuzzy programming approach to multiobjective solid transportation problem, Fuzzy Sets and Systems Fuzzy sets and systems A fuzzy set is a generalized set to which objects can belong with various degrees (grades) of memberships over the interval [0,1]. Fuzzy systems are processes that are too complex to be modeled by using conventional mathematical methods. 57 (1993) 183-194. [2] A. K. Bit, M. P. Biswal and S. S. Alam, Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy sets and systems 50 (1992), 135-141. [3] D. Chanas, Fuzzy programming in multiobjective linear programming- a parametric approach, Fuzzy Set and system 29 (1989) 303-313 [4] D. 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The sanyasins or disciples of Adi Shankaracharyas are also called "Dash Nam" as the Title Goswami is further divided into ten groups viz. , and S. S. Alam, Multiobjective transportation problem with interval; cost, source and destination parameters., European Journal of Operational Research 117 (1999), 100-112. [14] S. Tong, Interval number and fuzzy number linear programming. , Fuzzy Sets and Systems 66 (1994), 301-306. [15] Y. J. Lai and C. L. Hawng, Fuzzy Mathematical Programming, Lecture notes in Economics and Mathematical systems., Springer-Verlag, (1992). Department of Mathematics, Bengal Bengal (bĕng-gôl`, bĕn–), region, 77,442 sq mi (200,575 sq km), E India and Bangladesh, on the Bay of Bengal. The inland section is mountainous, with peaks up to 12,000 ft (3,660 m) high in the northwest, but most of Bengal is the Engineering college (Deemed University Deemed University is a status of autonomy granted to high performing institutes and departments of various universities in India. It is granted by the University Grants Commission (UGC) of India. ), Howrah Howrah: see Haora, India. , West-Bengal, Pin 711103, India India, officially Republic of India, republic (2005 est pop. 1,080,264,000), 1,261,810 sq mi (3,268,090 sq km), S Asia. The second most populous country in the world, it is also sometimes called Bharat, its ancient name. India's land frontier (c. * E-mail: roy_t_k@yahoo.co.in |
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