Multi-objective linear programming

Multi-objective linear programming problem (MOLPP) emerges naturally in decision making when several rates are to be optimized simultaneously and a compromise is sought which optimizes a weighted sum of these rates. In the light of the applications of single objective linear programming problem, objective function may represent output, input, profit, cost, capital, return on investment, productivity, liquidity, risk or time (Kornbluth and Steuer 1981) . A multitude of applications of the MOLPP can be envisioned in this way. The important applications of MOLPP can be seen in several areas such as production, transportation (marine), finance (corporate planning, bank balance sheet), engineering, statistics, water resources, health care and so forth (Steuer 1986 & Korhonen, Salo and Steuer 1997).

Abstract:- This paper presents FERCIPA solver for linear programming problems. The solver which can handle both single objective and multi-objective linear programming problems of large scales generates a sequence of interior feasible points that converge at the optimal solution for single objective linear programming problems and an optimal compromise solution for multi-objective linear programming problems. The solver is validated by its application to handle single objective linear programming problems and multi-objective linear programming problems involving up to six bounded variables and functional constraints. The solution obtained by FERCIPA solver is seen to compare favourably with those of other software like the Feasible Region Contraction Algorithm (FRCA) and MATLAB.

Abstract—This study presents a method to determine weights of objectives in multi objective linear programming without decision maker/s preference. The method is developed by modifying Belenson and Kapur’s approach under fuzziness. It is used two-person zero-sum game with mixed strategies. Degree of linear membership functions of objectives are used in pay-off matrix. The proposed method is shown with a numerical example and several fuzzy solution approaches are used to get a solution by using obtained weights. Also the results of problems that are obtained from literature are presented.

In this study we have introduced a new algorithm for solving fully fuzzy Multi objective linear programming problem with triangular fuzzy number without converting to equivalent classical problem. The triangular fuzzy numbers are represented in terms of location index number, left fuzziness index function and right fuzziness index function respectively. Multi objective linear programming with imprecise parameters makes the problem complicated and hence the traditional approaches fails to give a solution to those problems. Based on the fuzzy ideal and fuzzy negative ideal solution of each single fuzzy objective function of the given fuzzy MOLP, the proposed algorithm provides a fuzzy Pareto- optimal solution for the given fully fuzzy multi objective linear programming problem in an improved way. This idea can be extended for solving multi-level multi objective linear programming problem with fuzzy coefficients. A numerical example discussed by Buckley [3] is solved using the proposed method without converting the given problem to crisp equivalent problem. It is to be noted that by applying the proposed method, the Decision Maker have the flexibility of choosing r  [0,1] depending upon the situation and he can obtain the fuzzy Pareto optimal solution for the given problem.

In most cases, method of converting multi-objective linear programming problem into a single objective linear programming problem is often used because the solution procedure is already known. Subjective selection of weights in method of combining objective functions may favour some objective functions and thus suppressing the impact of others in the overall analysis of the system. It may not be possible to generate all possible Pareto optimal solution as required in some cases.

In this paper we discuss about infeasibility diagnosis and infeasibility resolution, when the constraint method is used for solving multi objective linear programming problems. We propose an algorithm for resolution of infeasibility, which is a combination of interactive, weighting and constraint methods. Numerical examples are provided to illustrate the tech- niques developed.

This paper uses parametric study for providing essential information about the problem's behavior to the decision maker. Two novel algorithms are presented in this work. The first algorithm is obtained to find the complete stability set of the first kind for parametric multi- objective linear programming problems. It is based on the weighting method for scalarizing the multi- objective linear programming problems and Kuhn- Tucker conditions for mathematical programming in general.

Multi objective programming in crisp and fuzzy environment have been well developed in order to deal realistic problems. Multi-objective programming in intuitionistic fuzzy environment is still in its infancy. MOLP in fuzzy and intuitionistic fuzzy environment are not capable of dealing with indeterminacy which exists in realistic multi-objective programming problem. So to deal MOLP involving indeterminacy, neutrosophic set studied by F. Smarandache [94, 95, 96, 97] and single valued neutrosophic set [98] are suitable tools. In 2015, Roy and Das [99] proposed neutrosophic optimization approach to solve multi-objective linear programming problem that can be considered as an extension of fuzzy programming [11] and intuitionistic fuzzy optimization [70]. In 2015, Das and Roy [100] proposed multi-objective non-linear programming problem based on neutrosophic optimization technique. Hezam et al. [101] presented Taylor series approximation to solve neutrosophic multi-objective programming problem. Kar et al. [102] applied single valued neutrosophic set theory to generalized assignment problem. Kar et al. [103] also presented neutrosophicmulti-criteria assignment problem. In 2015, Kour and Basu [104] presented neutrosophicreal life transportation problem. In 2016, Thamaraiselvi and Santhi [105] presented neutrosophgic transportation model. In the optimum solution, Thamaraiselvi and Santhi [105] considered that the degrees of indeterminacy and falsity are the same in the optimum level. In 2016, Abdel-Baset et al. [106] presented two models of neutrosophic goal programming. Roy and Das [107] applied neutrosophic goal programming model of Abdel-Baset et al. [106] to bank investment problem. In 2016, S. Pramanik [108] critically studied the results of neutrosophic optimization models presented in [99, 100,101, 106] and presented new direction of research and proposed new framework of neutrosophic linear goal programming. In Pramanik’s model [107] falsity membership function and indeterminacy membership functions are minimized while truth membership functions are maximized.

A multi-objective linear programming problem is introduced by Chandra Sen [1] and suggests an approach to construct the multi-objective function under the limitation that the optimum value of individual problem was greater than zero. [2] studied the multi-objective function by solving the multi-objective programming problem, using mean and mean value. [3] solved the multi objective fractional programming problem by Chandra Sen’s technique. In order to extend this work, we have defined a multi-objective linear programming problem and in-

Consider the modeling of complex system where several objectives are to be optimized at a time. For example, agricultural production system is a real life example of such complex systems comprising of multiple objectives that too of conflicting nature. In general modeling of an agricultural management system requires optimizing the profit cutting the cost of cultivation. Thus the core objective of the problem is to maximize the profit subject to minimization of resource requirements. But since situations are not ideal and have limited scope in production system. With changing scenario of capital intensive agricultural with mechanization and proper availability of resources, the problem changes to develop a model giving optimal profit in accordance of fulfilling resource goals. Thus such problems of multi objective linear programming can be better dealt with goal programming approach. Here we deal such problem as fractional programming problem in which fractional function (profit/ cash-input) is to be optimized and other objectives are to be dealt as constraints for getting optimal solution.

Traveling salesman problem (TSP) is one of the challenging real-life problems, attracting researchers of many fields including Artificial Intelligence, Operations Research, and Algorithm Design and Analysis. The problem has been well studied till now under different headings and has been solved with different approaches including genetic algorithms and linear programming. Conventional linear programming is designed to deal with crisp parameters, but information about real life systems is often available in the form of vague descriptions. Fuzzy methods are designed to handle vague terms, and are most suited to finding optimal solutions to problems with vague parameters. Fuzzy multi-objective linear programming, an amalgamation of fuzzy logic and multi- objective linear programming, deals with flexible aspiration levels or goals and fuzzy constraints with acceptable deviations.

This multiple and heterogeneous nature of objectives in such situations has called for an approach that can take this nature into consideration. Multi-objective decision making is an area of multiple criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of economics, science, finance, engineering, and logistics where optimal decisions need to be taken in the presence of trade-offs between conflicting objectives. For a multi-objective optimization problem, there is no single solution exists that simultaneously optimizes each objective. In this case, the objective functions are said to be conflicting and there exist a number of Pareto optimal solutions. A solution is called non- dominated; Pareto optimal, Pareto- efficient or non-inferior, if none of the objective functions can be improved in value without degrading some of the other objective values. A multi-objective optimization problem can be formulated as

A novel hybrid model for supplier selection integrated factor analysis is introduced by He and Zhang, 2018. Based on hesitant fuzzy sets, Zhou et al. 2018 investigated a preference model to select the suppliers. Pan, 1989 proposed a linear programming model used to determine the number of suppliers to utilize and purchase quantity allocations among suppliers. Fuzzy set theory is useful for solving multi-objective supplier selection problems to enhance and improved the suggested solution techniques. Fuzzy linear constraints with fuzzy numbers were studied by Dubois and Prade, 1980. Zimmermann, 1978, developed fuzzy programming approach for solving multi- objective linear programming problem. Sakawa, 1993 introduced basics of interactive fuzzy multiple objective optimization. Agakishiyev, 2016 suggested a new method for solving SSP using Z- numbers. Polat et al. 2017 proposed an integrated fuzzy MCGDM approach for the SSP to select the most appropriate rail suppler. Chan and Kumar, 2007 applied fuzzy extended analytic hierarchy process to SSP with different criteria such as cost and service performance. Kumar et al. 2004 studied VSP via fuzzy programming approach. Khalifa, 2017 studied fuzzy vendor selection problem. Kumar et al., 2006 treated VSPs via fuzzy goal programming. Arikan, 2013 proposed an interactive approach for solving fuzzy multiple sourcing SSP. Diaz- Madronero et al. 2010 investigated an interactive approach for solving multiobjective VSP with fuzzy data represent by S- curve membership functions.

Our practical experience shows that the solution X ch * by transforming the multi objective convex programming to the multi objective linear programming problem and using the Chebyshev’s approach for its solution, provides us a satisfactory point in the sense that the values of the various objective functions at this point remain very close to the optimal values obtained by individually solving the convex programming problems (2.5) for various j  1 , 2 , ..., p .

Abstract—The uncertainty in real-world decision making originates from several sources, i.e., fuzziness, randomness, ambiguous. These uncertainties should be included while translating real-world problem into mathematical programming model though handling such uncertainties in the decision making model increases the complexities of the problem and make the solution of the problem hard. In this paper, a linear fractional programming is used to solve multi-objective fuzzy random based possibilistic programming problems to address the vague decision maker’s preference (aspiration) and ambiguous data (coefficient), in a fuzzy random environment. The developed model plays a vital role in the construction of fuzzy multi- objective linear programming model, which is exposed to various types of uncertainties that should be treated properly. An illustrative example explains the developed model and highlights it’s effectiveness.

Sophia Porchelvi and Rukmani [7] solved Multi-objective intuitionistic fuzzy linear programming problem (MOIFLPP) using a ranking procedure. This paper is used to transform the MOIFLP problem into a Multi-objective linear programming problem (MOLPP) and can be solved accordingly.

In the recent years, some other approaches have been reported for solving MOLFP problems. Guzel and Sivri [3] worked together to propose a method for finding an efficient solution of MOLFP problem using goal programming. Later Guzel [4] presented a simplex-based algorithm to find an efficient solution of MOLFP problem based on a theorem studied in a work by Dinkelbach [5], where he converted the main problem into a single linear programming problem. Jain [6] proposed a method using Gauss elimination technique to derive numer- ical solution of multi-objective linear programming (MOLP) problem. Then Jain [7] in 2014 extended his work for MOLFP problem. Porchelvi et al. [8] presented procedures for solving multi-objective linear fractional programming problems for both crisp and fuzzy cases using the complementary development method [9], where the fractional linear programming is transformed into linear pro- gramming problem. All of these methods provide only one efficient solution of MOLFP problem. S.F. Tantway [10] proposed a feasible direction method to find all efficient solutions of MOLFP problem. But his proposed method is applicable only for a special class of MOLFP problem, where all denominators of the frac- tional objectives are equal.

Abstract – In this paper, a method of solving the fuzzy fractional assignment problems, where the cost of the objective function is expressed as triangular fuzzy number, is proposed. In the proposed method, the linear programming of fuzzy fractional assignment problem (FFAP) is transformed to a multi objective linear programming (MOLP) of assignment problem and resultant problem is converted to a linear programming problem, using Taylor’s series method. An illustrative numerical example is given to justify the proposed theory.

In this paper, we presents simple Neutrosophic optimization approach to solve Multi- objective linear programming problem.it can be considered as an extension of fuzzy and intuitionistic fuzzy optimization .Also lower and upper bounds for the indeterminacy membership functions are defined. The empirical tests show that optimal solutions of Neutrosophic optimization approach can satisfy the objective function with higher degree than the solutions of fuzzy and intuitionistic fuzzy programming approach. The results thus obtained also reveal that neutrosophic optimization by proposed algorithm- 2 using non-linear Truth, Indeterminacy, Falsity membership functions give a better result than neutrosophic optimization by proposed algorithm- 1 using linear Truth, Indeterminacy, Falsity membership functions.

This paper proposes the method to the solution of multi objective linear programming problems in fuzzy environment. Here attention has been paid to the study of optimal compromise solution for multi objective fuzzy linear programming problems. Two algorithms have been presented to gives efficient solutions as well as an optimal compromise solution. An illustrative example is given to describe our proposed two algorithms.