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Online a : h ps:// esea ch endsjou nal.com ISSN No: 2584-282X
Indexed Jou nal Pee Re iewed Jou nal
INTERNATIONAL JOURNAL OF TRENDS IN EMERGING RESEARCH AND DEVELOPMENT
Volume 3; Issue 5; 2025; Page No. 80-85
Recei ed: 14-07-2025
Accep ed: 20-08-2025
Published: 06-10-2025
Solu ion o Fuzzy Ma ix Game P oblem by Me hod o Oddmen s using
Oc adecagonal Fuzzy Numbe s
1Anuj Sha ma and 2D . Dha mend a Badal
1Resea ch Schola , Depa men o Ma hema ical Sciences and Compu e Applica ions, Bundelkhand Uni e si y, Jhansi, U a
P adesh, India
2Assis an P o esso , Depa men o Ma hema ical Sciences and Compu e Applica ions, Bundelkhand Uni e si y, Jhansi,
U a P adesh, India
DOI: h ps://doi.o g/10.5281/zenodo.17279088
Co esponding Au ho : Anuj Sha ma
Abs ac
In his pape we ha e conside ed uzzy game ma ix. The inexac alues in he uzzy game ma ix a e Oc adecagonal uzzy numbe s. We
con e he uzzy alued game p oblem o a c isp alued game p oblem by using anking which we ha e ied o sol e using me hod o
oddmen s.
Keywo ds: Fuzzy se s, Oc adecagonal Fuzzy Numbe s (ODFN), Ranking o Fuzzy Numbe s, Fuzzy Game P oblem
In oduc ion
The heo y o games s a ed in he 20 h cen u y. Bu he ma hema ical ea men o games ook i e in 1944 when Neumann,
J.V. and Mo gens e n, O. [15] published hei enowned a icle on “Theo y o games and economic beha io ”. The Neumann’s
app oach u ilizes he mini-max p inciple which in ol es he elemen a y idea o minimiza ion o maximum loss.
All he pa ame e s in he uzzy game p oblem a e uzzy numbe s. The uzzy numbe s can be iangula , apezoidal,
hexagonal, oc agonal, hendecagonal, oc adecagonal e c. The anking me hod using α-cu s, o anking o Fuzzy numbe s was
p oposed by Basi zadeh [11] in which he has anked iangula and apezoidal uzzy numbe s. Fo Oc adecagonal uzzy
numbe s, he a i hme ic ope a ions, alpha cu , and anking echnique a e in oduced by Ba ya V, Sha ma A and Badal D [1]. By
using his anking, he Fuzzy Game p oblem is con e ed o a c isp alue p oblem, which can be sol ed using he me hod o
oddmen s.
P elimina ies
The aim o his sec ion is o h ow some ligh on some no a ions, no ions and esul s which a e used u he .
Fuzzy Se : Le X be a non-emp y se . A uzzy se “A” in X is cha ac e ized by i s membe ship unc ion A: X [0, 1]
and A(x) is in e p e ed as he deg ee o membe ship o elemen x in uzzy A o each x Ɛ X. Comple e non-membe ship is
ep esen ed by he alue ze o; comple e pa icipa ion is ep esen ed by he alue one and in e media e deg ees o membe ship
a e ep esen ed by alues in be ween. The membe ship unc ion o uzzy se A is also known as he mapping A.
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Fuzzy Numbe
A uzzy numbe “A” is a con ex no malized uzzy se on he eal line R, such ha :
a. The e exis s a leas one x˳ Ɛ R wi h µᴀ (x)= 1
b. µᴀ (x) is piecewise con inuous.
Oc adecagonal Fuzzy Numbe
A Oc adecagoanal uzzy numbe is deno ed as ᾹODFN = (a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16,
a17, a18) and i s membe ship unc ion is gi en by:
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Pa ame ic Fo m o Oc adecagonal Fuzzy Numbe
The pa ame ic o m o Oc adecagonal uzzy numbe is de ined as U = (P1( ), Q1(s), R1( ), S1(u), T1( ), U1(w), V1(x), W1(y),
P2( ), Q2(s), R2( ), S2(u), T2( ), U2(w), V2(x), W2(y)) o ϵ [0,1/8], s ϵ [1/8,2/8], ϵ [2/8,3/8], u ϵ [3/8,4/8], ϵ [4/8,5/8], w ϵ
[5/8,6/8], x ϵ [6/8,7/8], y ϵ [7/8,1] whe e,
a. P1( ), Q1(s), R1( ), S1(u), T1( ), U1(w), V1(x) and W1(y) a e bounded le con inuous non-dec easing unc ion o e [0,1/8],
[1/8,2/8], [2/8,3/8], [3/8,4/8], [4/8,5/8], [5/8,6/8], [6/8,7/8], and [7/8,1].
b. P2( ), Q2(s), R2( ), S2(u), T2( ), U2(w), V2(x) and W2(y) a e bounded le con inuous non-inc easing unc ion o e [0,1/8],
[1/8,2/8], [2/8,3/8], [3/8,4/8], [4/8,5/8], [5/8,6/8], [6/8,7/8], and [7/8,1].
Fig 1: G aphical Rep esen a ion o Oc adecagonal Fuzzy numbe
Ranking o Oc adecagonal Fuzzy Numbe
The anking unc ion :F(R) ⟶ R whe e F(R) is a se o uzzy numbe s de ined on se o eal numbe s, which maps each uzzy
numbe in o he eal line, whe e he na u al o de exis s (Yage 1986), i.e.
a. A > B i (A) > (B)
b. A < B i (A) < (B)
c. A = B i (A) = (B)
Le A = (a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16, a17, a18) and B = (b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12,
b13, b14, b15, b16, b17, b18) be wo Oc adecagonal uzzy numbe s hen
𝑟(𝐴) = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 + 𝑎11 + 𝑎12 + 𝑎13 + 𝑎14 + 𝑎15 + 𝑎16 + 𝑎17 + 𝑎18
18
And
𝑟(𝐵)=𝑏1 + 𝑏2 + 𝑏3 + 𝑏4 + 𝑏5 + 𝑏6 + 𝑏7 + 𝑏8 + 𝑏9 + 𝑏10 + 𝑏11 + 𝑏12 + 𝑏13 + 𝑏14 + 𝑏15 + 𝑏16 + 𝑏17 + 𝑏18
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Solu ion o Fuzzy Ma ix Game by Me hod o Oddmen s
Conside he gene al (3x3) game ma ix, whose elemen s a e Oc adecagonal uzzy numbe s.
Algo i hm
▪ Fi s we shall con e he gi en uzzy game p oblem in o a c isp alue p oblem
▪ Check o he saddle poin in he payo ma ix.
▪ I he e is no saddle poin no i is educible by dominance, sol e using me hod o oddmen s.
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S ep: 1 Sub ac each ow om he ow abo e i.e. sub ac 2nd ow om 1s and 3 d ow om he 2nd and w i e he di e ences
in he o m o wo successi e ows benea h he ma ix's ows.
S ep: 2 Now Sub ac each column o m he column o i s le i.e. sub ac 2nd column om 1s and 3 d column om 2nd and
w i e he di e ences in he o m o wo successi e columns o he igh o he ma ix.
S ep: 3 Now calcula e he oddmen s o he A’s 1, 2, 3 s a egies and B’s I, II, III s a egies.
S ep: 4 W i e hese oddmen s, neglec ing he signs.
S ep: 5 Now check i he sum o oddmen s o bo h he playe s is same hen bo h he playe s uses hei all pu e s a egies and
hence game is con o mable o ma ix me hod. Bu i he sum o oddmen s o bo h he playe s is di e en , hen bo h he
playe s do no uses hei all pu e s a egies.
S ep: 6 Now di ide he oddmen s by he sum o he oddmen s o ge he op imal s a egies o bo h he playe s.
S ep: 7 Finally calcula e he alue o game by using he o mula gi en below,
O
Example: Conside he ollowing uzzy game p oblem,
Solu ion: Fi s o all we shall use anking o Oc adecagonal uzzy numbe s o con e he gi en uzzy game p oblem o a c isp
alue p oblem,
a11 = (-14, -12, -10, -8, -6, -4, -2, 0, 2,
4, 6, 8, 10, 12, 14, 16, 18, 20)
a12 = (-16, -14, -12, -10, -8, -6, -4, -2,
2, 3, 4, 5, 6, 7, 8, 9, 20, 26)
a13 = (-16, -14, -12, -10, -8, -6, -4, -2,
2, 3, 4, 5, 6, 7, 8, 9, 20, 26)
a21 = (-16, -14, -12, -10, -8, -6, -4, -2,
2, 3, 4, 5, 6, 7, 8, 9, 20, 26)
a22 = (-16, -14, -12, -10, -8, -6, -4, -2,
2, 3, 4, 5, 6, 7, 8, 9, 20, 26)
a23 = (-16, -14, -12, -10, -8, -6, 2, 4, 6,
8, 10, 12, 14, 16, 18, 20, 22, 24)
a31 = (-16, -14, -12, -10, -8, -6, -4, -2,
2, 3, 4, 5, 6, 7, 8, 9, 20, 26)
a32 = (-9, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13)
a33 = (-16, -14, -12, -10, -8, -6, -4, -2,
2, 3, 4, 5, 6, 7, 8, 9, 20, 26)
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Reduced C isp alue p oblem is gi en by he ollowing ma ix,
Now Sub ac ing 2nd ow o m he 1s ow and 3 d ow om he 2nd ow and w i e he di e ences in he o m o wo
successi e ows below he ma ix,
Playe B
3
1
1
Playe A
1
1
5
1
4
1
A1 - A2
2
0
-4
A2 - A3
0
-3
4
Simila ly, sub ac ing 2nd column o m he 1s column and 3 d column om he 2nd column and w i e he di e ences in he
o m o wo successi e columns o he igh o he ma ix,
Playe B
B1 - B2
B2 - B3
3
1
1
2
0
Playe A
1
1
5
0
-4
1
4
1
-3
3
A1 - A2
2
0
-4
A2 - A3
0
-3
4
Now we shall calcula e he oddmen s o A’s 1,2,3 s a egies and B’s I, II, III s a egies,
Oddmen o A’s 1s S a egy = de |0 −4
−3 3 | = -12
Oddmen o A’s 2nd S a egy = de |2 0
−3 3| = 6
Oddmen o A’s 3 d S a egy = de |2 0
0 −4| = -8
Oddmen o B’s Is S a egy = de |0 −4
−3 4 | = -12
Oddmen o B’s IInd S a egy = de |2 −4
0 4 | = 8
Oddmen o B’s III dS a egy = de |2 0
0 −3| = -6
Now we shall w i e hese oddmen s, neglec ing he signs as shown in he able below,
Now check he sum o oddmen s o bo h he playe s A and B which is same i.e. 26 as shown in he able abo e. So we can say
ha bo h he playe s A and B uses hei all pu e s a egies and hence he game is sol able by ma ix me hod bu i he sum o
oddmen s o bo h he playe s is no equal ha means bo h he playe s do no use hei all pu e s a egies and hence he game is
no sol able by ma ix me hod.
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Now we shall calcula e he op imal s a egies o playe s A and B, by di iding he oddmen s by he sum o oddmen s, he e o e
Conclusion
▪ In his pape , Me hod o Oddmen s is used o sol ing he uzzy ma ix game p oblem.
▪ In he abo e example we ha e conside ed a uzzy ma ix game p oblem whose elemen s a e Oc adecagonal uzzy
numbe s and sol ed using me hod o oddmen s.
Re e ences
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