An Equivalent Linear Programming Form of General Linear Fractional Programming: A Duality Approach

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.


Introduction
Fractional programming has been attracting the attention of a fair number of researchers all over the world (see [1]). The author of [2] presented the last bibliography with 520 entries, mainly from the period 1997-2019, which emphasizes the amount of effort that has been made in the field. This bibliography is an extension of eight other bibliographies previously published by him. Some researchers have been studying the role of duality in fractional programming. The authors of [3] dealt with programming with linear fractional functionals and utilized a transformation of variables to obtain an equivalent linear programming form of general linear fractional programming. The author of [4] studied the duality for a special class of linear fractional functionals programming problem where its dual is a linear programming problem. The authors of [5] showed that the dual of a linear fractional program, under some assumptions, is itself a fractional linear program. The author of [6] investigated the primal-dual relation in linear fractional programming when the constraints are in an equation form. It was concluded that some primal-dual relations in such problems may not hold. The author of [7] extended a dual linear fractional program of a given linear fractional program which resulted in necessary and sufficient conditions for the optimality of a given feasible solution. The authors of [8] extended the duality in linear fractional programming and developed a dual linear programming model for a general maximization linear fractional functionals programming problem. They proved duality theorems in linear fractional functionals programming. The authors of [9] and [10] extended fuzzy duality in linear fractional programming problems under uncertainty. Recently, the author of [11] presented a linear fractional programming problem and its dual problem under a fuzzy environment using hyperbolic membership functions.
In this paper, we first show that the dual linear programming model of the dual model proposed by [8] is the same linear programming model that was formulated by [3] as an equivalent model of linear fractional functionals programming problems. Next, we utilize some interesting properties obtained in this study and also the duality properties to prove (in an easy manner) that the linear programming and linear fractional functional programming models are equivalent. There is an involutory property of duality in linear programming which states that the dual of the dual is the original (primal) model (see [12]). We show that in linear fractional functional programming the dual of the dual is also a model that is equivalent to the original model. We use a numerical example to show the applicability of the proposed linear programming model.
As a special case of the general maximization linear fractional functional programming problem, we consider the well-known data envelopment analysis (DEA) approach. DEA, originated by [13], is an operations research method to evaluate the efficiency score of a set of similar decision-making units (DMUs) in which each DMU uses multiple inputs to produce multiple outputs. DEA assigns a weight (multiplier) to each input and output and to evaluate the efficiency score of a specific DMU, finds the maximum ratio of the weighted sum of outputs to the weighted sum of inputs subject to the condition that the same ratio for all DMUs must be less than or equal to unity.

Related Work
Consider the following general linear fractional programming:  is the right-hand side column vector, c = (c 1 , . . . , c n ) and d = (d 1 , . . . , d n ) are row vectors, α and β are arbitrarily scalar constants, and 0 n is the origin in R n . It is assumed that dx + β > 0 for all x ∈ X F , the objective function is continuously differentiable, and that the feasible region X F is regular (nonempty and bounded). The authors of [8] formulated the following linear programming and proved that it is the dual of general linear fractional programming (1): minimize g(y, z) = z subject to (y, z) ∈ X D where X D = (y, z) : Here, the symbol T denotes transposition, y =    and z ∈ R is a decision variable. The authors of [8] stated and proved the well-known weak duality and complementary slackness theorems for models (1) and (2).

Dual of the Dual in Linear Fractional Programming
This section shows that in contrast to linear programming where the dual linear program is itself a linear program, the dual linear fractional programming problem is not necessarily a linear fractional programming problem. The linear programming of model (2) is the dual of the general linear fractional programming (1) (see [8]). We first formulate the dual linear programming of model (2) to obtain the following linear programming problem and then show that the following model is equivalent (not necessarily equal) to the general linear fractional programming (1): As a result, there are two various models which are the dual of the dual model (2), i.e., linear fractional programming (1) and linear programming (3) problems. In other words, we can conclude that a linear programming model and a fractional programming model might have a common dual problem. We utilize the dual model proposed by [8] to prove that models (1) and (3) are equivalent. Toward this end, we first prove (x, 0) / ∈ X L and then state some interesting propositions which show the relationship between the feasible regions and objective functions of the linear fractional programming problem (1) and linear programming (3) problems.
Subsequently, x ∈ D is a recession direction for X F (for more details about recession direction, we refer the reader to [12]) which contradicts the given assumption that X F is bounded.
Referencing Lemma 1 helps us to validate the following corollaries:

Corollary 2. x ∈ R n is a feasible solution for X F if and only if
x dx+β , 1 dx+β ∈ R n+1 is a feasible solution of X L .

Corollary 3. For a given
According to Lemma 1 and the given assumptions for model (1), all denominators in Corollaries 1-4 are nonzero. These corollaries help us to prove the following theorem. Proof. Let x * be an optimal solution for the general linear fractional programming (1). We show x * dx * +β , 1 dx * +β is an optimal solution for the linear programming model (3). From the optimality conditions for dual linear programming, there is an optimal solution for model (2), say (y * , z * ), in which f (x * ) = g(y * , z). From Corollary 2, Hence, x * dx * +β , 1 dx * +β is an optimal solution for model (3) and h x * dx * +β , 1 dx * +β = f (x * ). Conversely, let (x * , λ * ) be the optimal solution of the linear programming model (3). We prove that 1 λ * x * is an optimal solution for the linear fractional programming (1). From Corollary 1, 1 λ * x * ∈ X F , which leads to On the contrary, suppose w * is an optimal solution for model (1) which shows that there is a feasible solution with a higher objective value than the optimal solution, which is impossible.
The following corollaries can be directly obtained from Theorem 1: Corollary 5. If x * is an optimal solution of the general fractional programming model (1), then 1 λ * x * is an optimal solution of the equivalent linear programming model (3).
is an optimal solution of the linear programming model (3), then 1 λ * x * is an optimal solution of the equivalent linear fractional programming model (1).

Theorem 2 (Weak Duality Theorem).
For all x in X F and for all (y, z) in X D , we have f (x) ≤ g(y, z).

A Special Case (DEA Method)
In this section, we consider the fractional functional programming problem in DEA as a special case of the general linear fractional functional programming problem (1) where According to the proposed approach in this study, the following linear programming model is equivalent to the above linear fractional model: As a result, to obtain an equivalent linear programming form of the linear fractional model (7), we easily set the denominator of the model equal to 1, include it in the feasible region, and maximize the numerator, as is done in the equivalent linear programming model (8).
It should be highlighted here that the following model is the dual of both models (7) and (8): Assume there are n DMUs (DMU j : j = 1, . . . , n) of which DMU j consumes m inputs x j = x 1j , . . . , x mj to produce s outputs y j = y 1j , . . . , y sj . The authors of [13] formulated the following fractional functional programming to evaluate the efficiency score of DMU o∈{1,...,n} , the unit under consideration, which is known as the Charnes, Cooper, and Rhodes (CCR) model under the constant returns-to-scale (CRS) assumption: Here, (decision variables) weights (multipliers) for inputs and outputs, respectively. In fact, since the weights in model (10) are derived from the data in favor of the unit under evaluation instead of being fixed in advance for all the units, the model is called the multiplier model (for more details, see [14]). According to [8], the following linear programming model is the dual of fractional functional programming (10): Here, λ =    λ 1 . . .
If the optimal θ, denoted by θ * , is less than one, then the obtained linear combination vector outperforms DMU o .
Since the feasible region of the linear programming (11) envelops all data by the efficient frontier, it is called the envelopment model (for more details, see [15]).
Considering the proposed approach in this paper, the following linear programming model which is the dual of model (11) is equivalent to the fractional functional programming (10): Here, µ =    Let (u * F , v * F ), (λ * , θ * ), and (u * L , v * L ) be the optimal solutions for models (10)-(12), respectively. As inspection makes clear, e D (λ * , θ * ) = e L (u * L , v * L ) and 0 T s , for more details, see [16]). As a result, there is an identical dual problem for both fractional functional programming (10) and linear programming (12) which points out that these models are equivalent.
Numerical Case Study (Healthcare Systems) Table 1 exhibits data of twelve hospitals with two inputs (Doctors and Nurses) and two outputs (Inpatients and Outpatients), as adopted from [17]: (13) The optimal solution and the optimal objective value of the fractional CCR model are u * 1 , u * 2 , v * 1 , v * 2 = (265.29, 950.15, 0, 539.99) and e * F u * 1 , u * 2 , v * 1 , v * 2 = 0.693, respectively, which points out that the first hospital is inefficient and its efficiency score is 0.693.
The following model is also another alternative dual linear programming model of the model (14) which is not identical to the fractional CCR model (13): This example illustrates that the dual of the dual model (14) is not necessarily the original model. As a matter of fact, these models are equivalent. In other words, the linear model (15) is equivalent to the fractional linear programming problem (13) (see Theorem 1) and both are the dual model. It should be highlighted here that the optimal solution for the linear model (15) is η * 1 , η * 2 , µ * 1 , µ * 2 = 1.80 × 10 −3 , 6.45 × 10 −3 , 0, 3.67 × 10 −3 with e * L = e L µ * 1 , µ * 2 = 0.693. In summary, the optimal objective values of models (13-15) are identical, i.e., e F u * 1 , u * 2 , v * 1 , v * 2 = e D (θ * ) = e L µ * 1 , µ * 2 = 0.693. The main aim of this paper is to point out that, in contrast to linear programming, in fractional linear programming, the dual of the "dual" is not necessarily the original model. As a matter of fact, the dual of the dual is a linear programming problem that is equivalent to the fractional linear programming problem. Toward this end, we employed the approach of [8] to formulating a linear programming problem from a general fractional linear programming problem. Although both problems have an identical dual problem, they are equivalent, not equal. In other words, we demonstrated that for a linear programming problem, under some assumptions, two different dual problems exist that are indeed equivalent. We provided a numerical example along with one of the most important DEA models (α = β = 0), to illustrate the provided theorems. We applied the method of [8] to directly find the dual of the linear fractional DEA model and then utilized the suggested approach in this study to directly obtain its equivalent linear DEA model. Extending our approach for α > 0 (β > 0) leads to another DEA model with an input (output) orientation form under the variable returns-to-scale assumption (for a deeper discussion about different orientations and returns-to-scale assumptions, we refer the reader to [18]). An interesting further research direction is investigating duality in linear fractional programming with multiple objective functions. Finally, our approach can be applied to a wide range of practical applications, including energy [19], the banking industry [20], manufacturing systems [21], finance [22], and sport [23], among others.

Conflicts of Interest:
The author declares no conflict of interest.