A new methodology for solving bi-criterion fractional stochastic programming

Solving a bi-criterion fractional stochastic programming using an existing multi criteria decision making tool demands sufficient efforts and it is time consuming. There are many cases in financial situations that a nonlinear fractional programming, generated as a result of studying fractional stochastic programming, must be solved. Often management is not in needs of an optimal solution for the problem but rather an approximate solution can give him/her a good starting for the decision making or running a new model to find an intermediate or final solution. To this end, this author introduces a new linear approximation technique for solving a fractional stochastic programming (CCP) problem. After introducing the problem, the equivalent deterministic form of the fractional nonlinear programming problem is developed. To solve the problem, a fuzzy goal programming model of the equivalent deterministic form of the fractional stochastic programming is provided and then, the process of defuzzification and linearization of the problem is presented. A sample test problem is solved for presentation purposes. There are some limitations to the proposed approach: (1) solution obtains from this type of modeling is an approximate solution and, (2) preparation of approximation model of the problem may take some times for the beginners.


1.
Introduction. Under some circumstances the measure to be used by researcher is the division of one function of variables to another functions where one or both of these functions can be linear or nonlinear. Data Envelopment Analysis (DEA) considering fractional situations with hundreds applications have been used by researcher many times. This is why we can say that fractional programming has attracted the attention of many researchers during past four decades. In this regard Saad (2007) indicated that: "the main reason for interest in fractional programming stems from the fact that linear fractional objective functions occur frequently as measures of performance in a variety of circumstances. Lara and Stancu-Minasian (1997) have reviewed fractional programming as a tool for studying the sustainability of agricultural systems where the essentials of technique in both single and multi-objective cases are outlined. Authors pointed to this reality that algorithms embedded in the programming packages for solving the models are not friendly and this shortcoming needs to be overcome. Two procedures for avoiding this shortcoming in the multiple objective cases are discussed. Publication of five bibliographies complied by Stancu-Minasian (1999) reflect this reality that a large number of theoretical as well as algorithmic work have been done by many researchers over the years. As Lara and Stancu-Minasian (1999) mentioned in their work, although output/input ratios arise naturally in many economic problems very few real applications of fractional programming have been reported, particularly in the field of agriculture. Perhaps, the lack of friendly procedures for solving the models is one of the main reasons. However, a number of fractional programming applications can be seen in the work of (Stancu-Minasian's, 1997). Lara, P. (1993) reported an application of fractional programming in the field of livestock while Zhu, and Hung (2011) developed a stochastic linear fractional programming approach for sustainable waste management. Pena, et al. (2007) have shown that how the technique of multiple objective fractional programming can enhance the process of animal diet formulation. In another research, Lara (2007) linked production theory and multi-objective fractional programming together to propose a supporting tool for animal diet formulation. Fu et al. (2018) have proposed a simulation based linear fractional programming for water allocation and planning for the Songhua river in China.
More often, an appropriate solution, instead of an optimal solution, to a complex problem can satisfy managements need for making a quick decision. Multi objective fractional stochastic type problem can be regarded as a complex problem provided that a large number of variables are involving in modeling. Although, there are various approaches to tackle the generalized form of problem, there is no approaches available for solving a particular case of fractional stochastic problem discussed here. In this article, the basic idea is to use fuzzy goal programming (FGP) as a tool for solving the special type of fractional stochastic problem. To do that, first we define a fuzzy goal programming problem for the equivalent deterministic form (EDF) of the stochastic programming problem and then we apply the concept of defuzzification for converting the fuzzy model into a model that is not fuzzy. The rest of solution procedure is detailed in the respectful sections of the article as it the process progresses.
The remainder of this article is organized as follow: literature review is the topic of section 2. Model development is discussed in section 3. Linearization technique is the topic of section 4. Compromise goal constraints is discussed in section 5 while Tailors series is the topic of section 6. Computational algorithms are discussed in section 7. An example problem with details in solution is discussed in section 8. Authors conclusion and analysis is given in section 9.
2. Literature Review. There are many situations in which businesses deal with the linear fractional programming problem. In this regard, we can hint to the work of Steuer (1986) saying that the mathematical optimization problems with a goal function being a ratio of a linear numerator and a linear denominator have many applications. In waste management area, Zare Mehrjerdi and Faregh (2017) employed a fractional function modeling of such type. This sort of modeling is employed in finance (corporate planning, bank balance sheet management), in Marine transportation, in water resources, and health care to mention a few. Considering linear functions for F (x) and G(x), then an optimization problem as such as (1) can be proposed where S is assumed to be a nonempty bounded polyhedron.
To solve this problem, many authors as such as , Martos (1975), and Wolf (1985) have conducted research on this problem and proposed different algorithms for different forms and shapes of the problems. Comparative investigations of such algorithms can be found in Arsham and Kahn (1990), and Bhatt (1989). In their book, Nonlinear Programming, Theory and Algorithms, Bazaraa and Shetty (1979) have shown that the fractional type objective function shown above has several important properties -it is (simultaneously): pseudo convex, pseudo concave, quasi-convex, quasi-concave, strict quasi-convex and strict quasi-concave. This means that the point that satisfies the Kuhn-Tucker conditions for the maximization problem gives the global maximum on the feasible set. In addition, each local maximum is also a global maximum. This maximum is obtained at an extreme point of S (Metev and Gueorguieva (1995). Algorithms for generalized fractional programming is the topic studied by Crouzeix and Ferland in 1991. In this research, authors have generalized the work of Dinkelbach (1967) where it is tried to find the root of function F (λ) = 0 where F (λ) is the optimal value of the parametric program of A fractional programming problem with absolute value functions of the type formulated below was proposed by Chadha [4].
Researchers have managed to use exact approach and heuristic approach for solving ratio optimization type problem. A traditional approach is the use of parametric method as are discussed by Wolf (1985). Cooper (1962, 1973) converted fractional programming (FP) into equivalent linear programming and then solved the resulting problem. Other researchers have conducted research on this problem by treating solving the fractional programming problem as the primal and dual simplex algorithm. This type of treatment can be seen in the work of Farag proposed fuzzy goal programming approach for solving fractional programming. Metev and Gueorguieva (1995) have discussed about a simple method for obtaining weakly efficient points in MOLFP problem. Authors shown that the property of strict quasi-convexity allows to use successfully the reference point method for the analysis of MOLFP problems. Omar M . Saad (2007) proposed a solution algorithm for fuzzy MOLFP where fuzzy parameters are considered in the right-hand side of the constraints. Furthermore, the concept ofα-level set of a fuzzy number has been employed by the authors for the purpose of difuzzification. A solution approach was proposed by Saad and Abd-Rabo (1997) for solving integer linear fractional programming where right-hand side constraints are considered to be random variables. Saad and Sharif (2001) developed a solution method for solving integer linear fractional programming problems with chance constraints, assuming the independency of involved parameters in their model building. Zhou, et al. (2019) proposed a type-2 fuzzy chance constrained fractional integrated modeling method for energy system management under uncertainties and risks.
Goal programming (GP) has been employed by many researchers for solving many managerial problems as well as fuzzy type programming problems. Chang (2005) proposed a fuzzy goal programming approach for solving fractional programming problem with absolute-value functions. Masatoshi Sakawa and Kosuke Kato (1998) conducted a research on the interactive decision-making approach for MOLFP problems with block angular structure involving fuzzy numbers. A multi objective linear fractional programming problem with the block angular structure can be formulated as below: Biswas and Bose (2012) studied a fuzzy goal programming approach for solving quadratic fractional bi-level programming. The study considers the general solution approach to bi-level programming taking quadratic functions in the form of equation (7).
Fractional programming also known as ratios  3. Model Development. The basic idea is to use fuzzy goal programming (GP) as a tool for solving fractional stochastic problem. To do that, first we define a fuzzy goal programming for the stochastic programming problem and then we apply the concept of defuzzification to convert the fuzzy model into a model that is not fuzzy. Now, we are in need of developing an equivalent crisp model of the proposed fuzzy system. Linear goal programming was originally introduced by Abraham Charnes and William Cooper in 1961. One can solve a GP model either regularly or interactively. Zare Mehrjerdi applied goal programming and interactive goal programming to various type problems (1986,1993,2009,2011,2019). The main difference between fuzzy goal programming (FGP) and goal programming (GP) is in that GP requires that decision makers to set definite aspiration values for each goal while in the FGP these are specified in an imprecise manner.

3.1.
Notations. In the model developed by this author, the following notations are used:

Decision Variables.
x i = The j th decision variable.

Lower Bounds Values.
LF 1 = Lower bound for probabilistic constraint 2 LF 2 = Lower bound for probabilistic constraint 3 LF 3 = Lower bound for probabilistic constraint 4 LF 4 = Lower bound for probabilistic constraint 5 3.5. Parameters. a ij = Technological coefficients b i = The level of the ith resource α=The probability that the probabilistic constraint (related to numerator) would not hold β=The probability that the probabilistic constraint (related to denominator) would not hold α 2 =The probability that the probabilistic constraint (related to numerator) would not hold β 2 =The probability that the probabilistic constraint (related to denominator) would not hold δ i =The probability that the probabilistic constraint (related to constraints) would not hold

Problem 1.
Maximize A major difficulty in using CCP when input-output coefficients and/or cost vectors are random variables having known distribution functions is the need for a nonlinear computer program. The equivalent deterministic form of chance constraints of (10) through (13) is as shown below. More details on this type of modeling can be seen in the works of Kataok (1963), and Zare Mehrjerdi (1986Mehrjerdi ( , 2009Mehrjerdi ( , 2011Mehrjerdi ( , 2019 to mention a few. By setting F i (x) = LF i for all i = 1, 2, 3, 4 and assuming that technological coefficients are independently normally distributed random variables then, the EDF of problem 1 can be written as Problem 2.
3.7. Solution Methodology. To solve this multi criterion nonlinear fractional programming an approximate linearization technique is proposed below. For this purpose, problem (2) is rewritten as follows: Now, define the membership functions for Z 1 (x) and Z 2 (x) as shown in the section below.
3.9. Fuzzy Goal Programming Modeling. We can write the following goal programming model for the membership function as below: Let us assume that After combining goal constraints (22) and (23) and substitution we have Or, we can write (27) as Now, we have Now, we substitute for Z 1 and Z 2 to get the following model Or Where C 3j , C 4j , D 3j , and D 4j are as defined below: Formula (31) will be converted into the following: we can use formula 36 to develop the upper bound function as shown below: Now, we can use formula 36 to develop the lower bound function as shown below: The proposed nonlinear goal programming can be expanded to include four compromise constraint one for each of the functions F 1 (x), F 2 (x), F 3 (x), and F 4 (x). The process of development of the compromise functions are shown below.

Linearization Technique. Let us assume that
2j , ζ 2 j ) and the variance-covariance matrices of V 1 , V 2 , V 3 , and V 4 , when coefficients are independent normally distributed random variables. Since, we know that considering the first fractional objective function, we have the following inequalities intact: Where u 1j = σ 2 j and u 2j = ϕ 2 j for all j = 1, 2, · · · , n. Therefore, we have Since, we have we can define the following new functions: 5. Compromise Goal Constraints. A goal constraint incorporating the optimum value of the upper and lower bound functions of the numerator and denominator of F 1 (X) and F 2 (X) respectively are also of tremendous value for problem solving. and where following inequalities hold: where f * 1 and f * 2 represent the optimum values of P 4 and P 5 linear programming problems, respectively. It should be noted that F * 1 is the optimal value of F 1 (X) over the defined feasible region of S. However, P 4 and P 5 are defined as below: In a similar fashion, we can introduce problems P 6 and P 7 as they are defined below: Note that in a similar way we can develop compromise constraints for the numerator and denominator of the second objective function. The development of these two constraints is left to the readers. These constraints shown by H 3 (x) and H 4 (x) are used in the model given below. Now, the upper bound nonlinear goal programming can be written as: Now, the upper bound nonlinear goal programming can be written as: Please note that the materials provided in the remainder of this section is for knowing how to calculate the values of upper and lower bound values of nonlinear functions F 3 (x) and F 4 (x). Where, 6. Taylors Series Linearization Technique. A technique that can convert a quadratic function into a linear function is Taylors series. Using this approach, and following specific rules, we can convert any nonlinear constraint into a linear constraint, about a specified solution point X 0 . Following Taylor series rules, the upper bound nonlinear constraint can be written as, The same linearization technique can be used for the nonlinear goal constraint appeared in lower bound goal programming problem (LBGPP) as shown below: Once, such linear approximations are completed then we can solve the upper bound and lower bound goal programming problems using multiple objective goal programming technique.
7. Computational Algorithms. The computation method developed in the previous sections of this article can be organized into the steps described below.
Step 1: using information provided, develop the fractional model of the problem.
Step 2: using the concepts discussed in the previous sections, develop upper bound goal programming model of problem (UBGPP).
Step 3: using the concepts discussed in the previous sections, develop lower bound goal programming model of the problem (LBGPP).
Step 4: ignore the nonlinear goal constraint and solve the remaining GP problem. Call this temporary solution point X •

1
Step 6: Same steps can be followed to solve the LBGPP as well.
Step 1: Convert priority goal (s) as a problem constraint. This can be done using following formula: Step 2: Ask the decision makers for the values of weights Wi, and priority levels P j .
Step 3: apply the steps of Simulated Annealing technique (see Figure 1) to solve the problem. Once an optimum solution is obtained then, identify the Goal Level and the value of positive and negative deviational variables to determine the levels of achievements of priority levels.
Step 4: Show the results to decision maker and continue solving the problem with new weights if decision maker is unsatisfied with the solution obtained. Max Let us assume that the values of Z 1 (x) and Z 2 (x) are requested to be as follows: Now, the lower bound nonlinear goal programming problem can be written as follows: . Results and Analysis. This section presents the computer results obtained for the upper bound and lower bound problems.
9.1. Upper Bound problem. The relaxed upper bound GP problem was solved and solution point satisfying all constraints except the nonlinear goal constraint was obtained. The Taylor series was used and the nonlinear goal constraint was linearized around this solution point. Then, the linearized function was added to the problem instead of the nonlinear goal constrained and problem was resolved. The optimal solution of the goal programming is: x 1 = 0.593 The goal priority level is at 3.585 9.2. Lower Bound Problem. The relaxed lower bound GP problem was solved and a solution point satisfying all constraints except the nonlinear goal constraint was obtained. The Taylor series was used and the nonlinear goal constraint was linearized around this solution point. Then, the linearized function was added to the problem instead of the nonlinear goal constrained and problem was resolved. The optimal solution of the goal programming is: x 1 = 0.0 The goal priority level is at 2.44 Conclusion. The problem of fractional stochastic programming has not been studied with the structure defined by this author in this article. Due to the fact that solving a fractional stochastic programming of this type is not a simple task author has proposed a linearization goal programming technique to solve the problem. The final linear goal programming problem resulted due to linearization can be solved by existing commercial computer packages or customized software developed by the analyst. Since this linearization technique is unique in the sense of development and application it makes a significant contribution to the field of fractional stochastic programming per se. There are some limitations to this type of modeling: (1) the solution obtains from this type of modeling is an approximate one and hence optimal solution of the problem is not achievable at all. There are ways to expand the scope of this research. Using joint chance constrained programming for modeling of the problem is highly recommended, as it is applied in agriculture and engineering areas. This type of modeling is highly regarded for problems related to environment, pollution control, water management, city waste management, financial modeling, and land planning and management. When one needs to deal with systems reliability then chance constrained programming (CCP) is recommended. Due to the facts that CCP is widely used for risk modeling and systems reliability analysis along with fuzzy type modeling and goal programming approach, our proposed approximation technique can make significant contribution to the field of fractional stochastic programming for solving complex problems.