Multidimensional solution of fuzzy linear programming

Multidimensional solution of FLPP

In this section, we propose a primal Simplex-based strategy for solving FLPP.

Assumptions of the study are as follows:

• The study assumes that existing methods for solving FLPP, which rely on standard interval and fuzzy arithmetic, have limitations and shortcomings.

• It assumes a new approach based on horizontal membership functions and multidimensional relative-distance-measure fuzzy interval arithmetic can provide more comprehensive and practical solutions to FLPP.

• The study assumes that the proposed multidimensional solution can satisfy any equivalent form of FLPP.

• It assumes that horizontal membership functions can effectively incorporate uncertainty into mathematical equations without relying on Zadeh’s extension principle.

Consider the following fuzzy linear programming with m constraints and n variables; (1)${\displaystyle \text{Minimize/Maximize}\left(z\text{\%}\right)\approx C^T\otimes \tilde{x}\text{Subjected to constraints:}A\otimes \tilde{x}\le \tilde{b},\text{and}\tilde{x}\text{is a non-negative fuzzy vector},}$

where the vector matrix $C^T={\left[c_j\right]}_{1\times n}$, the coefficient matrix A = [a_ij]_m×n are crisp matrices with c_j and a_ij are the elements of the set of real numbers, x% = [x%_j]_n×1, and b% = [b%_i]_m×1 are fuzzy vectors where ${\tilde{x}}_j,{\tilde{b}}_i$ are triangular/trapezoidal fuzzy numbers ∀1 ≤ j ≤ n and 1 ≤ i ≤ m.

Next, we propose the Multidimensional Primal Simplex Based Method (MPSBM) algorithm.

Algorithm. MPSBM algorithm

Step 1. Using remarks 3–4 and Definition 7, obtain $b\left(r,{\alpha }_{\tilde{b}}\right),$ and $R\left(\tilde{b}\right).$

Step 2. Using the conventional primal Simplex method, solve the following crisp LP and get the optimal basis inverse matrix (B⁻¹) for it: (2)${\displaystyle Minimize/Maximize\left(z\right)\left(z\right)=C^{T}xS.t:Ax\le R\left(b\right),x\ge 0.}$

It is worth mentioning that B⁻¹ can be extracted from the optimal tableau as the submatrix in rows 1 through m under the original identity columns; Bazaraa, Jarvis & Sherali (2008).

Step 3. Find the multidimensional optimal solution and value as follows: ${\displaystyle {\tilde{x}}^{\mathrm{\ast }}=B^{-1}b\left(r,{\alpha }_{\tilde{b}}\right),{\tilde{z}}^{\mathrm{\ast }}=C^T{\tilde{x}}^{\mathrm{\ast }}.}$

By the fact that if B isthe optimal basis of the equivalent crisp problem, then it will be the optimal basis of the corresponding FLPP, we presented the MPSBM algorithm to reduce the complexity of the problem.

Remark 6. We can use the main steps of the fuzzy primal Simplex method directly. However, this work leads to substantial computational effort.

It is worth mentioning that based on Definition 4 and Remark 5, the consequences of the MPSBM algorithm to solve the FLPP followed as regular extensions of primal results for linear programming (LP) problems with crisp data (such as optimality, unboundedness, and degeneracy conditions, finite convergence, etc.); see Bazaraa, Jarvis & Sherali (2008).

Numerical experiments

Here, we give some examples to illustrate the results obtained in the previous section.

Example 4.1 (Nasseri, 2008). Consider the following FLPP: (3)${\displaystyle \text{Maximize}\left(z\text{\%}\right)\approx 40{\tilde{x}}_1\oplus 30{\tilde{x}}_2\text{Subjected to constraints:}\left\{\begin{array}{cc}\hfill & {\tilde{x}}_1\oplus {\tilde{x}}_2\le \left(395,400,405\right),\hfill \\ \hfill & 2{\tilde{x}}_1\oplus {\tilde{x}}_2\le \left(493,500,507\right),\hfill \end{array}\right.\text{and}x{\text{\%}}_j\ge 0,\forall j=1,2.}$

If we use the existing methods, such as Nasseri (2008), or the existing ranking methods, such as Najafi & Edalatpanah (2013), Kaur & Kumar (2013), Das, Mandal, & Edalatpanah, (2017), Pérez-Cañedo & Concepción-Morales (2019), the optimal fuzzy solution is obtained as ${\tilde{x}}_1=\left(98,100,102\right)$ and ${\tilde{x}}_2=\left(297,300,303\right)$ which, when plugged into the objective function, gives the maximum fuzzy value of $\tilde{z}=\left(12830,13000,13170\right)$.

Now, we have started to use the MPSBM algorithm.

Step 1. Using Remark 3 and Definition 7, we have:

(4)${\displaystyle {\tilde{b}}_1=\left(395,400,405\right),b_1\left(r,{\alpha }_{{\tilde{b}}_1}\right)=\left(395+5r\right)+{\alpha }_1\left(10-10r\right),R\left({\tilde{b}}_1\right)=400,{\tilde{b}}_2=\left(493,500,507\right),b_2\left(r,{\alpha }_{{\tilde{b}}_2}\right)=\left(493+7r\right)+{\alpha }_2\left(14-14r\right),R\left({\tilde{b}}_2\right)=500.}$

Step 2. Using Step 1, we solve the following crisp LP using the conventional primal Simplex method: (5)${\displaystyle \text{Maximize}\left(z\right)\approx 40x_1+30x_2\text{Subjected to constraints:}\left\{\begin{array}{cc}\hfill & x_1+x_2\le 400,\hfill \\ \hfill & 2x_1+x_2\le 500,\hfill \end{array}\right.}$

x_j ≥ 0, ∀j = 1, 2.

After completing the solution, the optimal basic variables are x_B = [x₂, x₁], the non-basic variables are x_N = [x₃, x₄] and $B^{-1}=\left[\begin{array}{cc}\hfill 2\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right].$ Therefore, we obtain: (6)${\displaystyle {\tilde{x}}^{\ast }=\left[\begin{array}{c}\hfill {\tilde{x}}_2\hfill \\ \hfill {\tilde{x}}_1\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 2\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{c}\hfill \left(395+5r\right)+{\alpha }_1\left(10-10r\right)\hfill \\ \hfill \left(493+7r\right)+{\alpha }_2\left(14-14r\right)\hfill \end{array}\right],}$

Step 3. Finally, using Definition 4, we find the multidimensional optimal solution and value as follows: (7)${\displaystyle {\tilde{x}}^{\ast }=\left[\begin{array}{c}\hfill {\tilde{x}}_2\hfill \\ \hfill {\tilde{x}}_1\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(297+3r\right)+2{\alpha }_1\left(10-10r\right)-{\alpha }_2\left(14-14r\right)\hfill \\ \hfill \left(98+2r\right)-{\alpha }_1\left(10-10r\right)+{\alpha }_2\left(14-14r\right)\hfill \end{array}\right]}$ (8)${\displaystyle ,{\tilde{z}}^{\ast }=\left[\begin{array}{c}\hfill 4030\hfill \end{array}\right]\left[\begin{array}{c}\hfill \left(98+2r\right)-{\alpha }_1\left(10-10r\right)+{\alpha }_2\left(14-14r\right)\hfill \\ \hfill \left(297+3r\right)+2{\alpha }_1\left(10-10r\right)-{\alpha }_2\left(14-14r\right)\hfill \end{array}\right]=\left(12830+170r\right)+20{\alpha }_1\left(10-10r\right)+10{\alpha }_2\left(14-14r\right).}$

Figures 1–3 show the optimal solution and value of FLPP (Eq. 3) in three-dimensional space.

It should also be observed that solution (Eq. 7) is not commonly fuzzy numbers defined in 2D space. The full solutions of FLPPs can only be made from multidimensional granules. These granules are not visualizable in 2D space. Nevertheless, Definition 5 can obtain their 2D-space indicators as span. Therefore, we have:

${\displaystyle Span\left({\tilde{x}}_1\right)=\left[88+12r,112-12r\right],}$ ${\displaystyle Span\left({\tilde{x}}_2\right)=\left[283+17r,317-17r\right].}$

These spans can also be accessible in the forms of triangular fuzzy numbers as:

${\displaystyle Span\left({\tilde{x}}_1\right)=\left(88,100,112\right),}$ ${\displaystyle Span\left({\tilde{x}}_2\right)=\left(283,300,317\right).}$

Optimal solutions and triangular fuzzy objective values obtained by the existing and present methods are given in Table 1 for comparison.

For the comparison of the new solution with the existing solution, we use the testing point strategy (TP strategy) (Landowski, 2015; Landowski, 2019). Using the TP strategy, the points in an inappropriate solution that does not fulfill the FLPP or those that are solutions that are not included in the obtained results can be found. For example, consider α₁ = 1, α₂ = 0,  and r = 0, so by Eq. (7), the FLPP Eq. (3) converted into the following LP: (9)${\displaystyle \text{Maximize}\left(z\right)\approx 40x_1+30x_2\text{Subjected to constraints:}\left\{\begin{array}{cc}\hfill & x_1+x_2\le 405,\hfill \\ \hfill & 2x_1+x_2\le 493,\hfill \end{array}\right.}$

x_j ≥ 0, ∀j = 1, 2.

So, by Simplex or Geometric methods such as Fig. 4, it is easy to see that the optimal solution of linear programming (Eq. 9) is x₁ = 88,  and x₂ = 317,  but they do not belong to the existing solutions; see Table 1; however, by inserting α₁ = 1, α₂ = 1,  and r = 0 in Eqs. (7)–(8), we can see that the solution of the new method coincides with the exact solution. Moreover, the obtained solution of Eqs. (7)–(8) satisfies any equivalent form of FLPP (3) and the universality conditions.

We emphasize that even if the obtained solution of the span of the new algorithm is equal to the existing solutions, only multidimensional granules can be the complete solutions of FLPPs. In the following example, we describe this fact.

Example 4.2. Consider the following FLP problem (10)${\displaystyle \text{Maximize}\left(z\text{\%}\right)\approx 14\otimes {\tilde{x}}_1\oplus 13\otimes {\tilde{x}}_2\oplus 16\otimes {\tilde{x}}_3\text{Subjected to}:\left\{\begin{array}{c}12\otimes {\tilde{x}}_1\oplus 13\otimes {\tilde{x}}_2\oplus 12\otimes {\tilde{x}}_3\le \left(469,475,505,511\right),\hfill \\ 14\otimes {\tilde{x}}_1\oplus 13\otimes {\tilde{x}}_3\le \left(452,460,480,488\right),\hfill \\ 12\otimes {\tilde{x}}_1\oplus 15\otimes {\tilde{x}}_2\le \left(460,465,495,500\right),\hfill \end{array}\right.}$

and x%_j ≥ 0, ∀j = 1, 2, 3.

If we use the existing simplex-based methods such as Maleki, Tata & Mashinchi (2000), Nasseri (2008), Ebrahimnejad & Tavana (2014), Khan, Ahmad & Maan (2017), Stanojević & Stanojević (2020) or the existing non-simplex-based methods such as Najafi & Edalatpanah (2013), Kaur & Kumar (2013), Das, Edalatpanah & Mandal (2018), Pérez-Cañedo & Concepción-Morales (2019), Voskoglou (2020), Mohan, Kannusamy & Sidhu (2021), the optimal fuzzy solution is obtained as follows: (11)${\displaystyle \left\{\begin{array}{cc}\hfill & {\tilde{x}}_2=\left(\frac{241}{169},\frac{415}{169},\frac{1045}{169},\frac{1219}{169}\right),\hfill \\ \hfill & {\tilde{x}}_3=\left(\frac{452}{13},\frac{460}{13},\frac{480}{13},\frac{488}{13}\right),\hfill \\ \hfill & {\tilde{x}}_6=\left(\frac{59455}{169},\frac{62910}{169},\frac{77430}{169},\frac{80885}{169}\right),\hfill \\ \hfill & {\tilde{x}}_1={\tilde{x}}_4={\tilde{x}}_5=\tilde{0}.\hfill \end{array}\right.}$

Now, by the MPSBM algorithm, we have:

Using Remarks 4 and Definition 7, we get: (12)${\displaystyle b_1\left(r,{\alpha }_{{\tilde{b}}_1}\right)=\left(469+6r\right)+{\alpha }_1\left(42-12r\right),R\left({\tilde{b}}_1\right)=490,b_2\left(r,{\alpha }_{{\tilde{b}}_2}\right)=\left(452+8r\right)+{\alpha }_2\left(36-16r\right),R\left({\tilde{b}}_2\right)=470,b_3\left(r,{\alpha }_{{\tilde{b}}_3}\right)=\left(460+5r\right)+{\alpha }_3\left(40-10r\right),R\left({\tilde{b}}_3\right)=480.}$

By solution of the related crisp LP, the optimal basic variables are x_B = [x₂, x₃, x₆], the non-basic variables are x_N = [x₁, x₄, x₅], and $B^{-1}=\left[\begin{array}{ccc}\hfill \frac{1}{13}\hfill & \hfill \frac{-12}{169}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{13}\hfill & \hfill 0\hfill \\ \hfill \frac{-15}{13}\hfill & \hfill \frac{180}{169}\hfill & \hfill 1\hfill \end{array}\right].$

So: (13)${\displaystyle {\tilde{x}}^{\ast }=\left[\begin{array}{c}\hfill {\tilde{x}}_2\hfill \\ \hfill {\tilde{x}}_3\hfill \\ \hfill {\tilde{x}}_6\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill \frac{1}{13}\hfill & \hfill \frac{-12}{169}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{13}\hfill & \hfill 0\hfill \\ \hfill \frac{-15}{13}\hfill & \hfill \frac{180}{169}\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{c}\hfill \left(469+6r\right)+{\alpha }_1\left(42-12r\right)\hfill \\ \hfill \left(452+8r\right)+{\alpha }_2\left(36-16r\right)\hfill \\ \hfill \left(460+5r\right)+{\alpha }_3\left(40-10r\right)\hfill \end{array}\right],}$

Therefore: (14)${\displaystyle {\tilde{x}}^{\ast }=\left[\begin{array}{c}\hfill {\tilde{x}}_2\hfill \\ \hfill {\tilde{x}}_3\hfill \\ \hfill {\tilde{x}}_6\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(\frac{673}{169}-\frac{18}{169}r\right)+\frac{1}{13}{\alpha }_1\left(42-12r\right)-\frac{12}{169}{\alpha }_2\left(36-16r\right)\hfill \\ \hfill \left(\frac{452}{13}-\frac{8}{13}r\right)+\frac{1}{13}{\alpha }_2\left(36-16r\right)\hfill \\ \hfill \left(\frac{69664}{169}-\frac{1061}{169}r\right)-\frac{12}{13}{\alpha }_1\left(42-12r\right)+\frac{144}{169}{\alpha }_2\left(36-16r\right)+{\alpha }_3\left(40-10r\right)\hfill \end{array}\right],}$ (15)${\displaystyle {\tilde{z}}^{\ast }=\left(\frac{7905}{13}+\frac{110}{13}r\right)+{\alpha }_1\left(42-12r\right)+\frac{4}{13}{\alpha }_2\left(36-16r\right).}$

Figures 5–7 also show the optimal solution and value of FLPP (Eq. 10) in three-dimensional space.

It is worth mentioning that the span of Eq. (14) is equal to Eq. (11). However, the consequences considered in Eq. (11) do not fulfill the FLPP of Eq. (10). For example, from the first constraint of Eq. (10), we have: (16)${\displaystyle 12\otimes {\tilde{x}}_1\oplus 13\otimes {\tilde{x}}_2\oplus 12\otimes {\tilde{x}}_3\oplus {\tilde{x}}_4=\left(469,475,505,511\right),}$

But replacing the results of Eq. (11) into the left-hand side of Eq. (16) with the standard fuzzy arithmetic leads to $\left[435.7692,456.5385,523.4615,544.2308\right]$. Hence, the left-hand side is not equal to the right-hand side of Eq. (16). Therefore, Eq. (11) is not the complete solution of the mentioned FLPP. Nevertheless, it is easy to check that the obtained solution of Eq. (14) satisfies all constraints of FLPP (Eq. 10). Therefore, the multidimensional granules of Eq. (14) are the only complete solutions of FLPP (Eq. 10).