MULTI-OBJECTIVE PROBABILISTIC FRACTIONAL PROGRAMMING PROBLEM INVOLVING TWO PARAMETERS CAUCHY DISTRIBUTION

The paper presents the solution methodology of a multi-objective probabilistic fractional programming problem, where the parameters of the right hand side constraints follow Cauchy distribution. The proposed mathematical model can not be solved directly. The solution procedure is completed in three steps. In ﬁrst step, multi-objective probabilistic fractional programming problem is converted to deterministic multi-objective fractional mathematical programming problem. In the second step, it is converted to its equivalent multi-objective mathematical programming problem. Finally, ε -constraint method is applied to ﬁnd the best compromise solution. A numerical example and application are presented to demonstrate the procedure of proposed mathematical model.

objective functions are deterministic in nature. But many real world decision making problems have uncertain objective functions and constraints due to incomplete or vague or uncertain information. Such type of uncertainties are addressed via fuzzy programming problem or stochastic programming problem. The mathematical models are said to be stochastic linear fractional programming problem, if some or all parameters of linear fractional programming are considered as random variables. There are two methods to deal with stochastic programming problem, namely two stage programming problem and chance constrained (probabilistic) programming problem. Optimizing two or more fractional objective functions which are conflicting in nature is called multi-objective fractional programming problem. Multiobjective fractional programming problem is a beneficial means of modeling real life decision making problems with objective functions in the form of fractions such as minimizing actual cost/standard cost, maximizing profit/cost, minimizing inventory/sales, etc. Multi-objective LFP problem is also applicable in various disciplines like production planning problem, planning of agricultural problem, inventory problem, health care and hospital planning, financial planning, etc.
Multi-objective probabilistic linear fractional programming (MOPLFP) problem is nothing but it is optimizing the multi-objective linear fractional functions subject to some constraints which follows known random distribution. MOPLFP problem can not be solved directly. In order to solve it, the proposed model is transformed to known standard mathematical model in three steps. In the first step of transformation, MOPLFP problem is converted to its equivalent deterministic multi-objective linear fractional mathematical programming problem. In second step, multi-objective fractional mathematical programming problem is transformed to its equivalent multi-objective linear programming problem. Finally, multi-objective mathematical programming problem is solved using -constrained method. This paper has been organized into nine sections. The Section 1 presents the brief introduction of fractional programming problem. Section 2 presents motivation of the study. Section 3 presents review of literatures that are related to proposed work. Section 4 states the general model of MOPLFP problem. Section 5 presents the procedure to obtain deterministic equivalence of probabilistic programming problem. Section 6 presents the conversion of multiobjective linear fractional programming into equivalent multi-objective linear programming problem. Section 7 presents the solution procedure of multiobjective probabilistic linear fractional programming problem. One numerical and one practical example are given in Section 8 to illustrate the procedure. Finally, conclusions and references are provided in Section 9.

Motivation
Uncertain programming problem is an integral part of real life optimization problems. In most of the situations decision makers solve optimization problems that involves specific parameters which are mostly unknown. This leads to uncertainties in the input model parameters. Therefore parameters are han-dled by using known probability distribution functions. Stochastic programming is formulated due to presence of randomness in optimization problems. Most real life mathematical models involve more than one objective function which are conflicting in nature. Unlike single objective programming problems which give unique optimal solution, there may be many compromise solutions in multi-objective programming problem. Solving multi-objective programming problem is not an easy task. Sometimes decision makers need to determine the allocation (ratio) of resources to achieve certain specification that maximize or minimize the ratios. This motivates us to conduct and consider the proposed mathematical model as multi-objective probabilistic fractional programming problem.

Literature survey
To solve fractional programming problem, researchers presented the transformation of fractional programming problem into its equivalent mathematical programming problem. Chakraborty and Gupta [3] presented fuzzy programming method to obtain solution of multi-objective fractional mathematical programming problem after converting multi-objective fractional programming to multi-objective linear mathematical programming problem. An iterative parametric method is suggested by Valipour et al. [26] to obtain the solution of multi-objective linear fractional mathematical programming problem. A computer based method is developed by Hasan and Acharjee [1] to solve single objective linear fractional mathematical programming after converting into equivalent linear mathematical programming. Kornbluth and Steuer [12] presented a simplex-based method for the solution of multi-objective fractional mathematical programming problem. Latter Ponnaiah and Mohan [16] proposed a simplex method to find the solution of linear fractional mathematical programming problems by restricting the denominator of objective function. A fortran computer programme is developed by Saha et al. [20] to find the solution of LFP by transforming to a linear programming problem where the constant term of the denominator and numerator are negative. Odior [15] adopted a solution method for linear fractional mathematical programming problems using the concept of duality and partial fractions. Taylor series method was proposed by Güzel and Sivri [11] to find the solution of multi-objective linear fractional programming problem. Tantawy [24] developed iterative technique to find solution of linear fractional mathematical programming problems based on conjugate gradient projection. A method to convert stochastic sum of probabilistic fractional program problems in to stochastic constraints is presented by Charles and Dutta [6]. Wen and Wu [27] proposed a parametric method to solve linear fractional mathematical programming problems. Gradient method presented by Enkhbat et al. [10] to find the solution of convex concave fractional minimization problems. Roy [19] proposed genetic algorithm for solving single objective fractional programming problems. It is possible to convert linear fractional mathematical programming to equivalent linear mathematical programming by adding one extra [9]constraint and variable. Marchi [13] extended the transformation of Charnes and Cooper to other problems, where the functional involves quotient of linear functions. As Borza et al. [2] presented, the methodology for solution of LFP problems having interval coefficients in the objective function by using variable transformation. Bibliography on fractional mathematical programming has been presented by Stancu [22,23]. Udhayakumar et al. [25] solved probabilistic fractional programming problems using stochastic simulation based genetic algorithm, where random variables follow any continuous distribution. Charles and Dutta [5] obtained the efficient solution of multi-objective stochastic fractional mathematical programming problems when the continuous random variables involved in the objective functions and constraints. Charles and Dutta [8] presented a method to recognize redundant objective functions for a given multi-objective probabilistic fractional mathematical programming.
Some applications of linear fractional mathematical programming problems are studied by many researchers. Charles and Dutta [4] applied linear probabilistic fractional mathematical programming by using branch and bounded method for machine manufacturing problem. Ren et al. [17] developed a model for the uncertainty of water resources allocation for industry having probabilistic mathematical programming problem and fractional programming problem. Ren et al. [18] developed a model for optimal allocation of uncertain water resources for industries by mixing probabilistic mathematical programming and fractional programming. Charles and Dutta [7] presented application of multiobjective stochastic fractional programming for assembled printed circuit board problem. Zhu and Huang [28] presented the application of probabilistic fractional programming problem for maintaining municipal solid waste controlling under uncertainty. Latter Zhu and Huang [29] presented dynamic stochastic fractional programming approach for sustainable management of electric power under uncertainty.
In this paper, we study MOPLFP problem where parameters in the right hand side of the constraints follow Cauchy distribution. To solve the problem, we convert the fractional mathematical programming problem having multiobjective functions into equivalent multi-objective mathematical programming problem and probabilistic programming problem into equivalent deterministic programming problem. Finally, -constraint technique is used to obtain the solution.

Formulation of multi-objective probabilistic linear fractional programming problem
A MOPLFP problem is expressed as: The functions N q (x j ) and D q (x j ) are linear functions of x j , c qj , d qj ∈ R n , a ij ∈ R m×n , α q and β q are scalars, "P" indicates probability, η i is the given probability at which the i th constraint violations are admitted.

Deterministic equivalence of probabilistic constraints
We assumed that the random variables b i follow Cauchy distribution with two parameters γ i and δ i . The parameters δ i and γ i denote locations and scales of the random variables b i . The probability density function (pdf) of Cauchy random variable b i is expressed by: There are different methods of estimating parameters of Cauchy distribution.
To select the location and scale parameters of random sample of Cauchy distribution, one can use maximum likelihood estimation. The procedure of selecting parameters of Cauchy distribution is described as follows. Let x 1 , x 2 , ..., x n be random sample of Cauchy distribution. The likelihood function L(δ, γ|x 1 , x 2 , ..., x n ) is a function that selects the value of the parameter values δ and γ where the data is most probable. Maximum likelihood is a technique that helps to find the value of the parameters that maximize the likelihood function. The log-likelihood function of the Cauchy distribution is given by: Putting ∂L ∂δ = 0 and ∂L ∂γ = 0, it is possible to find the maximum value of the likelihood function.
Solving (5.1) and (5.2) simultaneously, one can find the values of the two parameters γ and δ. From the probabilistic constraint (4.2), we have After integration, we have Substituting the limit of integration, we obtain the following result which is simplified as Taking tangent both sides of (5.4), we obtain solving for y i , we have 6 Conversion of multi-objective linear fractional programming problem into equivalent multi-objective linear programming problem Consider a single objective linear fractional mathematical programming problem max : For simplicity, assume that d 1j Consider the concave-convex programming problem max : Using the transformation y j = x j t and t = 1 D1(xj ) , the programming problem (6.3)-(6.4) is written as 3)-(6.4) is called concave-convex fractional programming if the following conditions are satisfied: ii ) D 1 (x j ) is convex function and positive.
Theorem 2 [Schaible [21]]. If (6.3)-(6.4) is concave-convex LFP problem having global maximum at point x * , then (6.5)-(6.6) attains maximum value at the point (t * , y * j ), x * j = y * j /t * . But if N 1 (x j ) is concave and negative and D 1 (x j ) concave and positive, then −N 1 (x j ) is convex and positive. Hence (6.3)-(6.4) is changed to the following linear mathematical programming problem max : Now, let us extend the single objective LFP problem into multi-objective fractional programming problem which is expressed in Section 5 by: For some x j in the feasible set, consider the two index sets where I I c = {1, 2, ..., Q}.

Solution procedure
It is difficult to solve the mathematical programming problem (4.1)-(4.3) directly. In this paper, we use -constraint approach after the problem is changed to deterministic multi-objective programming problem. The basic steps of -constraint method for multi-objective linear programming problem are described as follows.
Step 1 : Find the first ideal solution X (1) by solving the first objective function disregarding the other objective functions.
Similarly, ideal solutions for second, third and so on are obtained by solving individual mathematical programming problem.
Step 2 : With the help of ideal solutions found in Step 1, construct a pay-off matrix which is shown in Table 1. Estimate the upper and lower bounds of the objective functions from the pay-off matrix as L q ≤ Z q ≤ U q , q = 1, 2, ..., Q.
Step 3 : Determine the bounds of q , q = 1, 2, ..., Q which is the point in the range of Z q .
iii ) : Proceed the procedure Q times for Q objective functions.
Step 5 : Solve the single objective mathematical model by using suitable mathematical method or software to obtain optimal compromise solution for different values of q , q = 1, 2, ..., Q.

Examples
In this section, we see one numerical example and one practical example on nurse scheduling problem to illustrate the proposed model.

Numerical example
Consider a MOPLFP problem where the parameters of the right hand side constraints follow Cauchy distribution.
y (1) 0.8026938 1.108459544 y (2) 1.17647064 1.470588 From the pay-off matrix given by Table 2, the lower and the upper bounds of the two objective functions are given by: Therefore the bounds of q , q = 1, 2 is the point in the range of the objective function f q (y), i.e., 0.8026938 ≤ 1 ≤ 1.17647064, 1.108459544 ≤ 2 ≤ 1.470588.
Using q (q = 1, 2), we define two different single objective programming problems by taking one objective function as constraint as follows: The first single objective programming problem is given by considering the second objective function f 2 as constraint.
From Table 3, we obtained different compromise solutions for the given pro- gramming problem. The optimal value of Z 2 is similar to 2 . The decision maker can select best compromise solutions by selecting appropriate value of 2 . The second single objective programming problem is given by considering the first objective function f 1 as a constraint.

Application on nurses schedule problem
Nurses scheduling problem helps for any health organization to have sufficient number of nurses in each shift. In this example, we focus on nurse scheduling problem where the nurses manager wants to create a schedule for two shifts namely day and night shifts for 1 week. The schedule is done by optimizing two objectives. The primary objective of any health organization is minimizing the total cost assigned for nurses. In this problem, the manager wants to minimize the the ratio of total over time cost and regular time cost assigned for nurses at each shift. The second objective is maximizing the ratio of senior nurses and junior nurses in each shift. The total number of over time hours assigned for each nurse in each shit must satisfy the maximum stay period of patients in the health care at each shift. Since the stay period of patients in the health care are unknown at each shift, the maximum stay period of patients in the health care is uncertain which is expressed by random variable following Cauchy distribution having location and scale parameters. The mathematical programming problem is expressed by: where n is index of nurse in health care, n = 1 represents senior nurse and n = 2 represents junior nurse. s = 1, 2 is index of shift, t = 1, 2, ..., 7 is index of date in a week, x nst is number of nurses needed in each shift on each day. o ns is over time cost assigned for each nurse at each shift, r ns is regular time cost assigned for each nurse at each shift, h nst are over time hours assigned for nurse n for each shift and date. M st is the maximum period of patients staying in the health care at each shift in each time, η st is the probability at which the constraint violations are admitted. Assume that the total over time cost for senior and junior nurses is $30 and $20 per hour respectively, where as the total over time needed at each shift and date is given by Table 5. Since the maximum over time in each shift and date is random variable following Cauchy distribution, the values of the two parameters δ and γ are given by Table 6. The probabilities at which the constraint violations admitted are given by Table 7. The regular time cost assigned for senior and junior nurses is Table 7. Values of probabilities. $185 and $120 per day respectively. The total regular time per day is 8 hours.
Since the formulated nurses scheduling problem is multi-objective probabilistic fractional programming problem, it is impossible to solve the problem directly. After substituting all given values, first change the probabilistic programming problem into deterministic equivalent programming problem using (5.6)-(5.7).
In the next step, we transform the multi-objective fractional programming problem into equivalent multi-objective programming problem by the transformation given in (7.1)-(7.2) using y nst = tx nst . Finally, we apply -constraint method to solve the multi-objective programming problem. Solving each objective functions separately subject to the given constraints, we obtain ideal solutions. By contracting pay-off matrix using each individual solution, we obtain the bounds of 1 and 2 which are in the bounds of the first objective function and second objective function respectively. 0.2657 ≤ 1 ≤ 0.456851 and 0.001 ≤ 2 ≤ 0.005256. By taking q (q = 1, 2), we formulate the following two different single objective programming problem by taking one objective function as constraint. The first single objective programming problem is given by considering the second objective function as constraint.
min : Z 1 = h st y nst ≤ t(δ + γ tan(π/2 − η st π)), y nst ≥ 0, t > 0.  s = 1, 2 and t = 1, 2, ..., 7, the solution of the nurses scheduling programming problem is given by the Table 8. The results in Table 8 shows that for different values of 2 , we obtain the compromise solution of the nurse scheduling problem which minimize the ratio of over time cost to regular time cost and maximize the ratio of number of senior nurses to number of junior nurses. If we take 2 ≥ 0.0032, the number of junior nurses is much more less than number of senior nurses. The nurses manager has a great chance to select how many nurses must be assigned for each shift so as to a chive his objective. The second single objective programming problem is maximized by considering the first objective function as constraint. h st y nst ≤ t(δ + γ tan(π/2 − η st π)), y nst ≥ 0, t > 0. (8.8) Again solving (8.7)-(8.8) by taking different values of 1 in the interval [0.2657, 0.4568510] by LINGO software and applying the transformation x nst = y nst /t, n = 1, 2, s = 1, 2 and t = 1, 2, ..., 7, the solution of the nurses scheduling programming problem is given by the Table 9. The results in Table 9 shows that for different values of 1 , we obtain the compromise solutions of the nurses scheduling problem which minimizes the ratio of over time cost to regular time cost and maximizes the ratio of number of senior nurses to number of junior nurses. If we take 1 > 0.26752, the number of junior nurses is much more less than number of senior nurses. From the two tables we see that a single point cannot optimize the two objective functions simultaneously. By choosing different values of 1 in [0.2657, 0.4568510] and 2 in [0.001, 0.005256], we obtain the compromise solutions of the nurse scheduling problem.

Conclusions
In this study, we have considered MOPLFP problem, where the parameters of the right hand side constraints are random variables following Cauchy distribution. We have assumed other parameters of the mathematical model as deterministic variables. In the methodology, MOPLFP problem is converted to deterministic equivalent multi-objective linear fractional mathematical programming and the deterministic multi-objective linear fractional mathematical programming is transformed to multi-objective linear mathematical programming problem using the transformation y j = tx j , t > 0, j = 1, 2, . . . , n. The resulting multi-objective linear programming problem is solved using -constraint method. Numerical example and a real life application are provided. Depending on different situations independent random variables following other distributions may be considered. This problem may be solved by using genetic algorithm approach.