3.1. Modified Kerre’s Method for Comparison of Two Bipolar Fuzzy Numbers To compute the fuzzy maximum of two bipolar LR fuzzy numbers, we need to compute fuzzy maximum for each polar as given by (8). Here, we first give a result that leads to a direct and efficient formula to compute the fuzzy maximum of two arbitrary bipolar LR fuzzy numbers. Then, applying the direct formula for max~, we modify Kerre’s method to compare two bipolar LR fuzzy numbers. Next, using our modified Kerre’s method for comparing of two bipolar LR fuzzy numbers, we establish some simple formulas for comparison of bipolar triangular fuzzy numbers.
Define (9)wz=w∣w≤z,Ez=x,y∣x,y∈wz, z=maxx,y.
Lemma 9. Suppose M~=(mL,mP,mN,mR)LR and N~=(nL,nP,nN,nR)LR are two bipolar LR fuzzy numbers. Then, for positive polar and for all z∈R, one has(10)maxminM~x,N~y≤maxM~z,N~z, ∀x,y∈Ez,and, for negative polar and for all z∈R, one has(11)minmaxM~x,N~y≥minM~z,N~z, ∀x,y∈Ez.
Proof. First, we establish (10). Consider z∈R. Without loss of generality, suppose M(z)≥N(z). Then, we need to show (12)maxminM~x,N~y≤M~z, ∀x,y∈Ez. There are two cases as described below.
(1) Case x<z. Since (x,y)∈Ez, y=z and so(13)minM~x,N~z≤N~z≤M~z⟹maxx≤zminM~x,N~z≤M~z.(2) Case y<z. Since (x,y)∈Ez, x=z and so(14)minM~z,N~y≤M~z⟹maxy≤zminM~z,N~y≤M~z.Therefore, from (13) and (14), we have (15)maxminM~x,N~y≤M~z, ∀x,y∈Ez, and the proof is complete. In a similar manner, we can establish (11).
Next, in Theorem 10, we give a direct formula to compute the maximum of two arbitrary bipolar LR fuzzy numbers.
Theorem 10. Suppose M~=(mL,mP,mN,mR)LR and N~=(nL,nP,nN,nR)LR are two arbitrary bipolar LR fuzzy numbers and mP≤nP and mN≤nN. For z∈R, one has(16)max~PM~,N~z=N~z,mP<z<nPmaxM~z,N~z,z≥nPminM~z,N~z,z≤mP,max~NM~,N~z=N~z,mN<z<nNminM~z,N~z,z≥nNmaxM~z,N~z,z≤mN.
Proof. First, we prove max~PM~,N~(z). Let z∈R. According to (6) and mP≤nP, we consider three cases.
(1) Case mP<z<nP. Since the left side of the membership function of N~ is an increasing function, we have (17)∀x,y∈Ez⟹y≤z⟹N~y≤N~z.Then, we conclude from M~(x)≤1=M~(mP) and N~(y)≤N~(z) that (18)minM~x,N~y≤min1,N~z, ∀x,y∈Ez. Therefore, (19)minM~x,N~y≤min1,N~y≤min1,N~z, ∀x,y∈Ez. But, min(1,N~(z))=N~(z), and so we have (20)supx,y∈EzminM~x,N~y=N~z, mP<z<nP, to complete the proof for the case.
(2) Case z≥nP. Without loss of generality, suppose that M~(z)>N~(z). Then, according to Lemma 9, it is clear that(21)maxminM~x,N~y≤M~z.Now, we show(22)maxminM~x,N~y≥M~z.We know that (z,nP)∈Ez. Then, we have (23)minM~z,N~nP≤maxx,y∈EzminM~x,N~y. But (24)minM~z,N~nP=minM~z,1=M~z, and thus (25)maxx,y∈EzminM~x,N~y≥M~z. Therefore, (22) is established and from (21) and (22), and the proof of the case is complete.
(3) Case z≤mP (≤nP). Since M~(x) and N~(y) are increasing functions on [min(mL,nL),mP], we have(26)x,y∈Ez⟹x≤z⟹M~x≤M~z,x,y∈Ez⟹y≤z⟹N~y≤N~z,minM~x,N~y≤minM~z,N~z, ∀x,y∈Ez. Thus, we have (27)maxx,y∈EzminM~x,N~y≤minM~z,N~z, and the proof is complete. Similarly, we can prove max~N(M~,N~)(z).
Definition 11. Suppose M~=(mL,mP,mN,mR) and N~=(nL,nP,nN,nR) are two arbitrary bipolar LR fuzzy numbers and let(28)rbM~,N~=dM~,O~P-dN~,O~P+dM~,O~N-dN~,O~N.If rb(M~,N~)≥0 then M~≤N~; else M~≥N~.
Theorem 12. Let M~=(mL,mP,mN,mR)LR and N~=(nL,nP,nN,nR) be two LR fuzzy numbers. If mP≤nP and mN≤nN, then(29)rbM~,N~=∫minmL,nLmPM~z-N~zdz+∫mPnPM~z-N~zdz+∫nPmaxmR,nRN~z-M~zdz+∫minmL,nLmNN~z-M~zdz+∫mNnNM~z-N~zdz+∫nNmaxmR,nRM~z-N~zdz,where rb(M~,N~) is defined as in (28).
Proof. From Theorem 10 and Definition 7, we have (30)dM~,O~P=∫minmL,nLmPM~z-minM~z,N~zdz+∫mPnPM~z-N~zdz+∫nPmaxmR,nRmaxM~z,N~z-M~zdz,dN~,O~P=∫minmL,nLmPN~z-minM~z,N~zdz+∫mPnPN~z-N~zdz+∫nPmaxmR,nRmaxM~z,N~z-N~zdz. Thus, we have (31)dM~,O~P-dN~,O~P=∫minmL,nLmPM~z-N~zdz+∫mPnPM~z-N~zdz+∫nPmaxmR,nRN~z-M~zdz. Also, for negative polar we have (32)dM~,O~N=∫minmL,nLmNmaxM~z,N~z-M~zdz+∫mNnNM~z-N~zdz+∫nNmaxmR,nRM~z-minM~z,N~zdz,dN~,O~N=∫minmL,nLmNmaxM~z,N~z-N~zdz+∫mNnNN~z-N~zdz+∫nNmaxmR,nRN~z-minM~z,N~zdz.
Thus, we have (33)dM~,O~N-dN~,O~N=∫minmL,nLmNM~z-N~zdz+∫mNnNM~z-N~zdz+∫nNmaxmR,nRM~z-N~zdz.
And therefore (34)rbM~,N~=∫minmL,nLmPM~z-N~zdz+∫mPnPM~z-N~zdz+∫nPmaxmR,nRN~z-M~zdz+∫minmL,nLmNN~z-M~zdz+∫mNnNM~z-N~zdz+∫nNmaxmR,nRM~z-N~zdz.
We can rewrite (29) as(35)rbM~,N~=∫minmL,nLmPM~Lz-N~Lzdz+∫mPnPM~Rz-N~Lzdz+∫nPmaxmR,nRN~Rz-M~Rzdz+∫minmL,nLmNN~Lz-M~Lzdz+∫mNnNM~Rz-N~Lzdz+∫nNmaxmR,nRM~Rz-N~Rzdz.Note that when M~=(mL,mP,mN,mR) and N~=(nL,nP,nN,nR) are bipolar triangular fuzzy numbers, we can simplify (35), and then the computation of ∫mPnPM~R(z)-N~L(z)dz can be simplified, if we can compute the intersection of MR and NL. Since each polar of M~ and N~ is a triangular fuzzy number, x¯P below,(36)x¯P=mRnP-nLmPnP-nL-mP+mR,is the length of the intersection point of MR and NL for the positive polar, and x¯N below,(37)x¯N=mRnN-nLmNnN-nL-mN+mR,is the length of the intersection point of MR and NL for the negative polar. We have the following proposition giving a reformulation of (29).
Proposition 13. Let M~=(mL,mP,mN,mR) and N~=(nL,nP,nN,nR) be two bipolar triangular fuzzy numbers with mP<nP, mN<nN, x¯P∈[mP,nP], and x¯N∈[mN,nN], where x¯P and x¯N are defined by (36) and (37). Then, one has(38)rbM~,N~=∫minmL,nLmPM~Lz-N~Lzdz+∫mPx¯PM~Rz-N~Lzdz+∫x¯PnPN~Lz-M~Rzdz+∫nPmaxmR,nRN~Rz-M~Rzdz+∫minmL,nLmNN~Lz-M~Lzdz+∫mNx¯NN~Lz-M~Rzdz+∫x¯NnNM~Rz-N~Lzdz+∫nNmaxmR,nRM~Rz-N~Rzdz.
Note that the sign of rb(·,·) is adequate to determine M~≤N~ or M~≥N~. But, for bipolar LR fuzzy number linear programming problems, in some situations we need to compute the exact value of rb(·,·).
Example 14. Let M~=(2,4,6,10) and N~=(2,5,8,12) be two bipolar triangular fuzzy numbers. Then, M~LP, M~RP, N~LP, N~RP, M~LN, M~RN, N~LN, and N~RN are (39)M~LPz=0,z<2,12x-2,2≤z≤4,M~RPz=0,z>10,-16x-10,4<z≤10,N~LPz=0,z<5,13x-2,2<z≤5,N~RPz=0,z>12,-17x-12,5<z≤12,M~LNz=0,z<2,-14x-2,2<z≤6,M~RNz=0,z>10,14x-10,6<z≤10,N~LNz=0,z<2,-16x-2,2<z≤8,N~RNz=0,z>12,14x-12,8<z≤12. According to (38), we have rb(M~,N~)=4.4889 and this means M~<N~.
Next, we give some corollaries, the proofs of which are straightforward.
Corollary 15. Let M~=(mL,mP,mN,mR) and N~=(nL,nP,nN,nR) be two bipolar triangular fuzzy numbers. If mR≤nL, then(40)rbM~,N~=2mR-mL2+nR-nL2.
Corollary 16. Let M~=(mL,mP,mN,mR) and N~=(nL,nP,nN,nR) be two bipolar triangular fuzzy numbers. If mP<nP and mN=nN, where y¯P=M~RP(x¯P)=N~LP(x¯P), with x¯P as defined by (36), then(41)rbM~,N~=nR-nL2+mR-mL2-y¯PmR-nL+nR+nL2-mR+mL2.
Corollary 17. Let M~=(mL,mP,mN,mR) and N~=(nL,nP,nN,nR) be two bipolar triangular fuzzy numbers. If mP<nP and mN<nN, where y¯P=M~RP(x¯P)=N~LP(x¯P), with x¯P as defined by (36), and y¯N=M~RN(x¯N)=N~LN(x¯N), with x¯N as defined by (37), then(42)rbM~,N~=nR-nL2+mR-mL2-y¯PmR-nL+nR-nL2+mR-mL2+y¯NmR-nL.
A property of (42) is that for two bipolar triangular fuzzy numbers such as M~ and N~ we have rb(λM~,λN~)=λrb(M~,N~), where λ≥0.