
In today's competitive world, entrepreneurs cannot argue for transporting a single product. It does not provide much profit to the entrepreneur. Due to this reason, multiple products need to be transported from various origins to destinations through various types of conveyances. Real-world decision-making problems are typically phrased as multi-objective optimization problems because they may be effectively described with numerous competing objectives. Many real-life problems have uncertain objective functions and constraints due to incomplete or uncertain information. Such uncertainties are dealt with in fuzzy/interval/stochastic programming. This study explored a novel integrated model bi-objective bi-item solid transportation problem with fuzzy stochastic inequality constraints following a normal distribution. The entrepreneur's objectives are minimizing the transportation cost and duration of transit while maximizing the profit subject to constraints. The chance-constrained technique is applied to transform the uncertainty problem into its equivalent deterministic problem. The deterministic problem is then solved with the proposed method, namely, the global weighted sum method (GWSM), to find the optimal compromise solution. A numerical example is provided to test the efficacy of the method and then is solved using the Lingo 18.0 software. To highlight the proposed method, comparisons of the solution with the existing solution methods are performed. Finally, to understand the sensitivity of parameters in the proposed model, sensitivity analysis (SA) is conducted.
Citation: T. K. Buvaneshwari, D. Anuradha. Solving bi-objective bi-item solid transportation problem with fuzzy stochastic constraints involving normal distribution[J]. AIMS Mathematics, 2023, 8(9): 21700-21731. doi: 10.3934/math.20231107
[1] | Junhong Li, Ning Cui . A hyperchaos generated from Rabinovich system. AIMS Mathematics, 2023, 8(1): 1410-1426. doi: 10.3934/math.2023071 |
[2] | A. E. Matouk . Chaos and hidden chaos in a 4D dynamical system using the fractal-fractional operators. AIMS Mathematics, 2025, 10(3): 6233-6257. doi: 10.3934/math.2025284 |
[3] | Hany A. Hosham, Thoraya N. Alharthi . Bifurcation and chaos in simple discontinuous systems separated by a hypersurface. AIMS Mathematics, 2024, 9(7): 17025-17038. doi: 10.3934/math.2024826 |
[4] | Ning Cui, Junhong Li . A new 4D hyperchaotic system and its control. AIMS Mathematics, 2023, 8(1): 905-923. doi: 10.3934/math.2023044 |
[5] | Rahat Zarin, Abdur Raouf, Amir Khan, Aeshah A. Raezah, Usa Wannasingha Humphries . Computational modeling of financial crime population dynamics under different fractional operators. AIMS Mathematics, 2023, 8(9): 20755-20789. doi: 10.3934/math.20231058 |
[6] | Michael Precious Ineh, Umar Ishtiaq, Jackson Efiong Ante, Mubariz Garayev, Ioan-Lucian Popa . A robust uniform practical stability approach for Caputo fractional hybrid systems. AIMS Mathematics, 2025, 10(3): 7001-7021. doi: 10.3934/math.2025320 |
[7] | Kareem Alanazi, Omar Naifar, Raouf Fakhfakh, Abdellatif Ben Makhlouf . Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function. AIMS Mathematics, 2024, 9(12): 35533-35550. doi: 10.3934/math.20241686 |
[8] | Chatthai Thaiprayoon, Jutarat Kongson, Weerawat Sudsutad . Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches. AIMS Mathematics, 2025, 10(4): 8827-8872. doi: 10.3934/math.2025405 |
[9] | A. M. Alqahtani, Shivani Sharma, Arun Chaudhary, Aditya Sharma . Application of Caputo-Fabrizio derivative in circuit realization. AIMS Mathematics, 2025, 10(2): 2415-2443. doi: 10.3934/math.2025113 |
[10] | Michael Precious Ineh, Edet Peter Akpan, Hossam A. Nabwey . A novel approach to Lyapunov stability of Caputo fractional dynamic equations on time scale using a new generalized derivative. AIMS Mathematics, 2024, 9(12): 34406-34434. doi: 10.3934/math.20241639 |
In today's competitive world, entrepreneurs cannot argue for transporting a single product. It does not provide much profit to the entrepreneur. Due to this reason, multiple products need to be transported from various origins to destinations through various types of conveyances. Real-world decision-making problems are typically phrased as multi-objective optimization problems because they may be effectively described with numerous competing objectives. Many real-life problems have uncertain objective functions and constraints due to incomplete or uncertain information. Such uncertainties are dealt with in fuzzy/interval/stochastic programming. This study explored a novel integrated model bi-objective bi-item solid transportation problem with fuzzy stochastic inequality constraints following a normal distribution. The entrepreneur's objectives are minimizing the transportation cost and duration of transit while maximizing the profit subject to constraints. The chance-constrained technique is applied to transform the uncertainty problem into its equivalent deterministic problem. The deterministic problem is then solved with the proposed method, namely, the global weighted sum method (GWSM), to find the optimal compromise solution. A numerical example is provided to test the efficacy of the method and then is solved using the Lingo 18.0 software. To highlight the proposed method, comparisons of the solution with the existing solution methods are performed. Finally, to understand the sensitivity of parameters in the proposed model, sensitivity analysis (SA) is conducted.
A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Gallian [3] has written a dynamic survey of graph labeling. MacDougall et al. [5] introduced the notion of a vertex magic total labeling of graphs. Let G be a graph of order n and size m. A vertex magic total labeling of G is defined as a one-to-one function
f:V(G)∪E(G)→{1,2,⋯,n+m} |
with the property that for each vertex u of G,
f(u)+∑v∈N(u)f(uv)=k |
for some constant k where N(u) is the neighborhood of u. The constant k is called the magic constant for f. The vertex-magic total labelings of wheels and related graphs were studied in [6], and later in [11]. The properties of the general graphs such as cycles, paths, complete graphs, wheels, bipartite graphs and trees, which satisfy the vertex magic total labelings, were studied in [10]. MacDougall et al. [4] introduced the concept of a super vertex magic total labeling. They defined a vertex magic total labeling to be super if
f[V(G)]={1,2,⋯,n}. |
In 2017, Nagaraj et al. [7] introduced the concept of an even vertex magic total labeling. They called a vertex magic total labeling as even if
f[V(G)]={2,4,⋯,2n}. |
A graph G is called an even vertex magic if there exists an even vertex magic total labeling of G. We note that if G is an even vertex magic, then n≤m. The following results, which appeared in [7], are useful to us.
Theorem 1.1. [7] Let G be a nontrivial graph of order n and size m. If G is an even vertex magic, then magic constant k is given by the following:
k=m2+2mn+mn. |
A wheel Wn, n≥3, is a graph of order n+1 that contains a cycle Cn, for which every vertex in the cycle Cn is connected to one other vertex known as the hub. The edges of the wheel which are incident to the hub are called spokes. The vertices and edges of the cycle Cn in Wn are called rim vertices and rim edges, respectively. It was shown in [7] that a wheel Wn has no even vertex magic total labeling, as we state next.
Theorem 1.2. [7] A wheel Wn is not even vertex magic.
In this paper, the labeling problem is related to the work in [1]. In addition to the aforementioned vertex labeling by even numbers 2,4,⋯,2n, they studied vertex labelings by using three consecutive numbers 0,1,2 with some specific properties. These labelings were referred to as a weak Roman dominating function and a perfect Roman dominating function.
From the studies in [8,9,12], there exist graphs with the same order and size that are even vertex magics. Moreover, the wheel related graphs, namely fans, cycles and suns, having the even vertex magic total labelings were established in [7]. However, since these graphs have the same order and size, it is interesting and challenging to study wheel related graphs when the size is greater than the order, which have an even vertex magic total labeling.
The t-fold wheel Wn,t, n≥3, t≥1, is a wheel related graph derived from a wheel Wn by duplicating the t hubs, each adjacent to all rim vertices, and not adjacent to each other. It is observed that the t-fold wheel Wn,t has a size nt+n that exceeds its order n+t. The goal of this paper is to study conditions for an even vertex magic Wn,t in terms of n and t. Furthermore, we also determine an even vertex magic total labeling of some t-fold wheel Wn,t.
Since the 1-fold wheel Wn,1 is isomorphic to the wheel Wn and by Theorem 1.1, Wn is not an even vertex magic. In this section, we consider the t-fold wheel Wn,t, where n and t are integers with n≥3 and t≥2.
In order to present the conditions for an even vertex magic Wn,t, we initially explore the magic constant of the t-fold wheel Wn,t of order n+t and size nt+n by employing Theorem 1.1.
Proposition 2.1. Let n and t be integers with n≥3 and t≥2. If the t-fold wheel Wn,t is an even vertex magic, then the magic constant is defined as follows:
k=2nt+3n+n2t2+2n2t+nn+t. |
We are able to show the bound of an integer t for the t-fold wheel having an even vertex magic total labeling as follows.
Proposition 2.2. Let n and t be integers with n≥3 and t≥2. If the t-fold wheel Wn,t is an even vertex magic, then 2≤t≤n.
Proof. Suppose that the t-fold wheel Wn,t is an even vertex magic with magic constant k. By Proposition 2.1, we obtain the following:
k=2nt+3n+n2t2+2n2t+nn+t. |
On the contrary, assume that t>n. Let t=n+r, for some r≥1. Then,
n2t2+2n2t+n=n4+2n3r+n2r2+2n3+2n2r+n |
and
n+t=n+(n+r)=2n+r. |
Let
P(n)=n4+2n3r+n2r2+2n3+2n2r+n. |
By using the remainder theorem, the remainder when P(n) is divided by 2n+r is as follows:
P(−r2)=r4+4r3−8r16. |
If
P(−r2)=0, |
then r=−2, which is a contradiction. Thus,
P(−r2)≠0. |
Specifically, n2t2+2n2t+n is not divisible by n+t. Thus, k is not an integer, which is a contradiction. Therefore, 2≤t≤n.
According to Proposition 2.2, the t-fold wheel W3,t is not an even vertex magic, where t≥4. Figure 1 shows the even vertex magics W3,2 and W3,3 with magic constants k=36 and k=50, respectively, where their vertices and edges are labeled by the even vertex magic total labelings. We present an even vertex magic total labeling of the t-fold wheel Wn,t by considering only the integer n as the following results.
Proposition 2.3. For every integer n≥3, if the n-fold wheel Wn,n is an even vertex magic, then n is odd.
Proof. Let n be an integer with n≥3. Suppose that the n-fold wheel Wn,n is an even vertex magic with a magic constant k. On the contrary, assume that n is even. There exists an integer q such that n=2q. By Proposition 2.1,
k=2n2+3n+n3+2n2+12. |
Since
n3+2n2+1=8q3+8q2+1 |
is odd, n3+2n2+1 is not divisible by 2. Thus, k is not an integer, which is a contradiction. Therefore, n is odd.
As we have seen in Figure 1, the 3-fold wheel W3,3 is an even vertex magic, as indicated by Proposition 2.3. By an argument similar to the one used in the proof of Proposition 2.3, we obtain the condition for an even vertex magic Wn,n−2, as we now show.
Proposition 2.4. For every integer n≥4, if the (n−2)-fold wheel Wn,n−2 is an even vertex magic, then n is even.
The even vertex magic total labeling of the 2-fold wheel W4,2 with a magic constant k=50 is shown in Figure 2.
In order to deduce an even vertex magic total labeling of the t-fold wheel for achieving the main result, we need some additional notation for the t-fold wheel Wn,t. For every pair of integers n≥3 and t≥2, let
V(Wn,t)={u1,u2,…,un,v1,v2,…,vt} |
and
E(Wn,t)={uiui+1|1≤i≤n−1}∪{unu1}∪{uivj|1≤i≤n,1≤j≤t}. |
Suppose the t-fold wheel Wn,t is an even vertex magic. Then, for any even vertex magic total labeling f of Wn,t, let
Srv=n∑i=1f(ui), Sre=n−1∑i=1f(uiui+1)+f(unu1) |
and
Sh=t∑j=1f(vj), Ss=t∑j=1n∑i=1f(uivj). |
Next, we present the following lemma to show the necessary condition for an even vertex magic Wn,t with the following magic constant:
k=2nt+3n+n2t2+2n2t+nn+t. |
Note that
Srv+2Sre−Sh=(n−t)k. |
Lemma 2.5. Let n and t be integers where n≥3 and t≥2. If the t-fold wheel Wn,t is an even vertex magic, then
Srv+2Sre−Sh=(t2+4t+3)n2+(−2t3−6t2−3t+1)n+(t3+2t2−1)(2nt)n+t. |
With the aid of Lemma 2.5 and Proposition 2.2, the necessary condition for an even vertex magic total labeling of the t-fold wheel Wn,t can also be given in terms of n and t.
Proposition 2.6. Let n and t be integers where n≥3 and t≥2. If the t-fold wheel Wn,t is an even vertex magic, then
(−t2−2t+1)n2+(2t3+6t2+7t+1)n−(t2+t)−(t3+2t2−1)(2nt)n+t≥0. |
Proof. Suppose that the t-fold wheel Wn,t is an even vertex magic. By Lemma 2.5,
Srv+2Sre−Sh=(t2+4t+3)n2+(−2t3−6t2−3t+1)n+(t3+2t2−1)(2nt)n+t. |
Next, we consider the maximum of (Srv+2Sre−Sh).
By Proposition 2.2, 2≤t≤n, and then 2n+2t<nt+n+t+1. The maximum of
(Srv+2Sre−Sh)=∑2t+2≤i≤2n+2tiis eveni+2nt+2n+t∑i=nt+n+t+1i−∑2≤i≤2tiis eveni=(∑2≤i≤2n+2tiis eveni−∑2≤i≤2tiis eveni)+2(nt+2n+t∑i=1i−nt+n+t∑i=1i)−∑2≤i≤2tiis eveni=((2n+2t)(2n+2t+2)4−(2t)(2t+2)4)+2((nt+2n+t)(nt+2n+t+1)2−(nt+n+t)(nt+n+t+1)2)−(2t)(2t+2)4=2n2t+4n2+4nt−t2+2n−t. |
Since Srv+2Sre−Sh does not exceed the maximum of (Srv+2Sre−Sh), the maximum of
(Srv+2Sre−Sh)−(Srv+2Sre−Sh)≥0. |
Therefore,
(−t2−2t+1)n2+(2t3+6t2+7t+1)n−(t2+t)−(t3+2t2−1)(2nt)n+t≥0. |
Now, we investigate the sufficient condition for a labeling f that can be an even vertex magic total labeling of Wn,n when n is odd.
Theorem 2.7. Let n be an odd integer where n≥3. For every n-fold wheel Wn,n, let
f:V(Wn,n)∪E(Wn,n)→{1,2,…,n2+3n} |
be defined by the following:
f(ui) = 2i, if 1≤i≤n,f(vj) = 2n+2j, if 1≤j≤n,f(uiui+1) = 2n+1−2i, if 1≤i≤n−1,f(unu1) = 1, f(un+1−jvj) = n2+3n+1−2j, if 1≤j≤n,f[EH]−f[{un+1−jvj|1≤j≤n}] = {2n+1,2n+3,…,n2+n−1} ∪{4n+2,4n+4,…,n2+3n}, if EH={uivj|1≤i,j≤n}. |
If
n−1∑j=1f(u1vj)=n3+4n2−52, |
then f can be an even vertex magic total labeling of Wn,n.
Proof. Assume that
n−1∑j=1f(u1vj)=n3+4n2−52. |
We have that
Ss−n∑j=1f(un+1−jvj)=∑2n+1≤i≤n2+n−1iis oddi+∑4n+2≤i≤n2+3niis eveni=(∑1≤i≤n2+n−1iis oddi−∑1≤i≤2n−1iis oddi)+(∑2≤i≤n2+3niis eveni−∑2≤i≤4niis eveni)=((n2+n)24−(2n)24)+((n2+3n)(n2+3n+2)4−4n(4n+2)4)=n4+4n3−4n2−n2, |
and then,
(Ss−n∑j=1f(un+1−jvj))−n−1∑j=1f(u1vj)=n4+3n3−8n2−n+52. |
Next, we consider the sum of the label of each vertex and the labels of all edges incident to this vertex. By the assumption, for 1≤j≤n,
f(vj)+n∑i=1f(uivj)=f(vj)+f(un+1−jvj)+∑1≤i≤ni≠n+1−jf(uivj)=f(vj)+f(un+1−jvj)+Ss−n∑j=1f(un+1−jvj)n=(2n+2j)+(n2+3n+1−2j)+n3+4n2−4n−12=n3+6n2+6n+12, |
f(u1)+f(u1u2)+f(unu1)+n∑j=1f(u1vj)=f(u1)+f(u1u2)+f(unu1)+f(u1vn)+n−1∑j=1f(u1vj)=2+(2n+1−2)+1+(n2+n−1+2)+n3+4n2−52=n3+6n2+6n+12. |
For 2≤i≤n,
f(ui)+f(uiui+1)+f(ui−1ui)+n∑j=1f(uivj)=f(ui)+f(uiui+1)+f(ui−1ui)+f(uivn+1−i)+∑1≤j≤nj≠n+1−if(uivj)=f(ui)+f(uiui+1)+f(ui−1ui)+f(uivn+1−i)+(Ss−n∑j=1f(un+1−jvj))−n−1∑j=1f(u1vj)n−1=2i+(2n+1−2i)+(2n+1−2i+2)+(n2+n−1+2i)+n3+4n2−4n−52=n3+6n2+6n+12. |
Therefore, f can be an even vertex magic total labeling of Wn,n with a magic constant
k=n3+6n2+6n+12. |
Now, we investigate the sufficient condition for a labeling f that can be an even vertex magic total labeling of Wn,n−2 when n is even.
Theorem 2.8. Let n be an even integer with n≥4. For every (n−2)-fold wheel Wn,n−2, let
f:V(Wn,n−2)∪E(Wn,n−2)→{1,2,…,n2+n−2} |
be defined by the following:
f(ui)=2i,if 1≤i≤n,f(vj)=2n+2j,if 1≤j≤n−2,f[EC]={a1,a2,⋯,an},if EC={uiui+1,unu1|1≤i≤n−1},f[EH]={1,2,⋯,n2+n−2}−{2,4,⋯,2n+4,a1,a2,⋯,an},if EH={uivj|1≤i≤n,1≤j≤n−2}. |
If
Ss=n4+n3−15n2+20n−42, |
then f can be an even vertex magic total labeling of Wn,n−2.
Proof. Assume that
Ss=n4+n3−15n2+20n−42. |
It suffices to show that for each vertex u of Wn,n−2,
f(u)+∑v∈N(u)f(uv)=k, |
where
k=n3+3n2−3n2. |
To do this, we consider the relevant sums, as follows.
Since the sum of the labels of all rim edges is equal to the sum of the labels of all vertices and the labels of all edges subtracted by the sum of the labels of all vertices and the labels of all spokes, it follows that
Sre=n2+n−2∑i=1i−∑2≤i≤4n−4iis eveni−Ss=(n2+n−2)(n2+n−1)2−(4n−4)(4n−2)4−n4+n3−15n2+20n−42=n3+5n2−11n+22. |
Since the sum of the labels of all hubs is equal to the sum of even integers from 2n+2 to 4n−4,
Sh+Ss=∑2n+2≤i≤4n−4iis eveni+Ss=(∑2≤i≤4n−4iis eveni−∑2≤i≤2niis eveni)+Ss=(4n−4)(4n−2)4−(2n)(2n+2)4+n4+n3−15n2+20n−42=n4+n3−9n2+6n2. |
Since the sum of the labels of all rim vertices is equal to the sum of even integers from 2 to 2n,
Srv+2Sre+Ss=∑2≤i≤2niis eveni+2Sre+Ss=(2n)(2n+2)4+2(n3+5n2−11n+22)+n4+n3−15n2+20n−42=n4+3n3−3n22. |
Next, we consider the sum of the label of each vertex and the labels of all edges incident to this vertex. We have the sum of the label of each hub and the labels of all edges incident to this hub as follows.
For 1≤j≤n−2,
f(vj)+n∑i=1f(uivj)=Sh+Ssn−2=n3+3n2−3n2. |
We obtain the sum of the label of each rim vertex and the labels of all edges incident to this rim vertex as follows.
For 2≤i≤n−1,
f(ui)+f(uiui+1)+f(ui−1ui)+n−2∑j=1f(uivj)=Srv+2Sre+Ssn=n3+3n2−3n2. |
Similarly,
f(un)+f(unu1)+f(un−1un)+n−2∑j=1f(unvj)=n3+3n2−3n2 |
and
f(u1)+f(u1u2)+f(unu1)+n−2∑j=1f(u1vj)=n3+3n2−3n2. |
Therefore, f can be an even vertex magic total labeling of Wn,n−2 with the following magic constant:
k=n3+3n2−3n2. |
In this section, we establish a characterization of an even vertex magic Wn,t for an integer 3≤n≤9. First, we present an n-fold wheel Wn,n which has an even vertex magic total labeling for every odd integer 3≤n≤9 as follows.
Theorem 3.1. For every odd integer 3≤n≤9, the n-fold wheel Wn,n is an even vertex magic.
Proof. Let n be an odd integer where 3≤n≤9. We define
f:V(Wn,n)∪E(Wn,n)→{1,2,⋯,n2+3n}, |
as the sufficient condition of Theorem 2.7, by
f(ui)=2i,if 1≤i≤n,f(vj)=2n+2j,if 1≤j≤n,f(uiui+1)=2n+1−2i,if 1≤i≤n−1,f(unu1)=1, |
and for 1≤i,j≤n, f(uivj) are shown in Tables 1–4,
f(uivj) | v1 | v2 | v3 |
u1 | 11 | 18 | 13 |
u2 | 14 | 15 | 9 |
u3 | 17 | 7 | 16 |
f(uivj) | v1 | v2 | v3 | v4 | v5 |
u1 | 15 | 21 | 36 | 38 | 31 |
u2 | 13 | 30 | 29 | 33 | 28 |
u3 | 40 | 24 | 35 | 19 | 17 |
u4 | 34 | 37 | 11 | 23 | 32 |
u5 | 39 | 27 | 26 | 22 | 25 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 | v7 |
u1 | 15 | 51 | 64 | 40 | 29 | 68 | 57 |
u2 | 66 | 43 | 17 | 38 | 19 | 59 | 70 |
u3 | 25 | 45 | 56 | 48 | 61 | 32 | 47 |
u4 | 52 | 31 | 33 | 63 | 54 | 53 | 30 |
u5 | 42 | 35 | 65 | 34 | 60 | 36 | 46 |
u6 | 55 | 67 | 23 | 58 | 49 | 27 | 41 |
u7 | 69 | 50 | 62 | 37 | 44 | 39 | 21 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | v9 |
u1 | 73 | 33 | 102 | 69 | 19 | 79 | 63 | 86 | 91 |
u2 | 43 | 56 | 35 | 62 | 49 | 89 | 85 | 93 | 87 |
u3 | 75 | 61 | 48 | 53 | 104 | 92 | 95 | 23 | 50 |
u4 | 80 | 58 | 72 | 25 | 108 | 97 | 77 | 47 | 39 |
u5 | 51 | 96 | 59 | 81 | 99 | 37 | 55 | 57 | 70 |
u6 | 60 | 100 | 44 | 101 | 67 | 40 | 41 | 90 | 64 |
u7 | 38 | 83 | 103 | 82 | 74 | 78 | 29 | 76 | 46 |
u8 | 88 | 105 | 94 | 65 | 42 | 66 | 52 | 31 | 68 |
u9 | 107 | 21 | 54 | 71 | 45 | 27 | 106 | 98 | 84 |
For every odd integer 3≤n≤9, the labeling f, as defined above, is an even vertex magic total labeling of the n-fold wheel Wn,n with magic constants k=50,153,340 and 635, respectively. Therefore, Wn,n is an even vertex magic.
As a consequence of an even vertex magic W3,2, Proposition 2.2 and Theorem 3.1, in any t-fold wheel W3,t, we are able to show that both W3,t and W3,t are only even vertex magics.
Theorem 3.2. For every integer t≥2, the t-fold wheel W3,t is an even vertex magic if and only if t=2,3.
The following result gives the necessary and sufficient condition for the t-fold wheel Wn,t to be an even vertex magic for every odd integer 5≤n≤9.
Theorem 3.3. For every odd integer 5≤n≤9 and an integer t≥2, the t-fold wheel Wn,t is an even vertex magic if and only if t=n.
Proof. Let n be an odd integer where 5≤n≤9 and t is an integer where t≥2. Assume that the t-fold wheel Wn,t is an even vertex magic. By Proposition 2.2, 2≤t≤n.
Case 1. n=5,7. If 2≤t≤n−1, then n2t2+2n2t+n is not divisible by n+t, and hence k is not an integer, which is a contradiction. Therefore, t=n.
Case 2. n=9. If either t=2 or 4≤t≤n−1, then n2t2+2n2t+n is not divisible by n+t, and hence k is not an integer, which is a contradiction. If t=3, then,
2nt3−n2t2−2n2t+6nt2+7nt+n2−t2+n−t−2nt4+4nt3−2ntn+t=−174<0, |
which is a contradiction with Proposition 2.6. Therefore, t=n.
Conversely, assume t=n. By Theorem 3.1, Wn,t is an even vertex magic.
We show an even vertex magic total labeling of Wn,n−2 for every even integer 4≤n≤8 as follows.
Theorem 3.4. For every even integer 4≤n≤8, the (n−2)-fold wheel Wn,n−2 is an even vertex magic.
Proof. Let n be an even integer with 4≤n≤8. We define
f:V(Wn,n−2)∪E(Wn,n−2)→{1,2,…,n2+n−2} |
as the sufficient condition of Theorem 2.8, by
f(ui)=2i,if 1≤i≤n,f(vj)=2n+2j,if 1≤j≤n−2, |
for 1≤i≤n−1, f(uiui+1) and f(unu1) are shown in Tables 5–7.
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u1) |
18 | 9 | 11 | 13 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u5) | f(u5u6) | f(u6u1) |
40 | 39 | 38 | 13 | 17 | 19 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u5) | f(u5u6) | f(u6u7) | f(u7u8) | f(u8u1) |
70 | 68 | 66 | 64 | 62 | 19 | 13 | 11 |
And for 1≤i≤n and 1≤j≤n−2, f(uivj) are shown in Tables 8–10.
f(uivj) | v1 | v2 |
u1 | 16 | 1 |
u2 | 14 | 5 |
u3 | 7 | 17 |
u4 | 3 | 15 |
f(uivj) | v1 | v2 | v3 | v4 |
u1 | 36 | 24 | 3 | 29 |
u2 | 11 | 25 | 33 | 1 |
u3 | 26 | 9 | 5 | 30 |
u4 | 22 | 23 | 34 | 15 |
u5 | 37 | 21 | 28 | 27 |
u6 | 7 | 35 | 32 | 31 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 |
u1 | 69 | 3 | 40 | 56 | 39 | 50 |
u2 | 1 | 67 | 5 | 59 | 31 | 35 |
u3 | 21 | 17 | 65 | 7 | 44 | 46 |
u4 | 42 | 38 | 23 | 33 | 9 | 57 |
u5 | 61 | 27 | 37 | 34 | 30 | 15 |
u6 | 41 | 53 | 49 | 32 | 47 | 25 |
u7 | 29 | 60 | 54 | 52 | 63 | 36 |
u8 | 58 | 55 | 45 | 43 | 51 | 48 |
For every even integer 4≤n≤8, the labeling f, as defined above, is an even vertex magic total labeling of the (n−2)-fold wheel Wn,n−2 with magic constants k=50,153 and 340, respectively. Therefore, Wn,n−2 is an even vertex magic.
There is a similar methodology of the proof of Theorem 3.4, which is also used in the study of graph operations (see [2]). Next, we determine a characterization of the t-fold wheel Wn,t to be an even vertex magic for every even integer 4≤n≤8. In order to we need to present the following lemma involving a 3-fold wheel W8,3.
Lemma 3.5. The 3-fold wheel W8,3 is not an even vertex magic.
Proof. On the contrary, assume that the 3-fold wheel W8,3 is an even vertex magic with a magic constant k. Since W8,3 has an order 11 and a size 32 and by Proposition 2.1, k=160. We have that
Srv=132, 2Sre=1,628−2Ss |
and
Srv+2Sre+Ss=8k=1,280. |
Thus, Ss=480. However, Sh+Ss=3k=480. This is a contradiction because Sh>0. Therefore, W8,3 is not an even vertex magic.
We are able to show that the necessary and sufficient condition for the t-fold wheel Wn,t is an even vertex magic for every even integer 4≤n≤8.
Theorem 3.6. For every even integer 4≤n≤8 and integer t≥2, the t-fold wheel Wn,t is an even vertex magic if and only if t=n−2.
Proof. Let n be an even integer where 4≤n≤8 and t is an integer where t≥2. Assume that the t-fold wheel Wn,t is an even vertex magic. By Proposition 2.2, 2≤t≤n.
Case 1. n=4,6. If either 2≤t≤n−3 or n−1≤t≤n, then n2t2+2n2t+n is not divisible by n+t, and hence k is not an integer, which is a contradiction. Therefore, t=n−2.
Case 2. n=8. If either 4≤t≤n−3 or n−1≤t≤n, then n2t2+2n2t+n is not divisible by n+t, and hence k is not an integer, which is a contradiction. If t=3, then, by Lemma 3.5, Wn,t is not an even vertex magic, which is a contradiction. If t=2, then,
2nt3−n2t2−2n2t+6nt2+7nt+n2−t2+n−t−2nt4+4nt3−2ntn+t=−62<0, |
which is a contradiction with Proposition 2.6. Therefore, t=n−2.
Conversely, assume t=n−2. By Theorem 3.4, Wn,t is an even vertex magic.
In this paper, we have not only established the bound of an integer t for the even vertex magic total labeling of the t-fold wheel, but have also presented the necessary condition for such labeling in terms of n and t. Furthermore, we have conducted an investigation into the sufficient conditions for labelings that can serve as even vertex magic total labelings for Wn,n when n is odd, and Wn,n−2 when n is even.
Our research has led us to the following significant conclusions:
● For every integer t≥2, the t-fold wheel W3,t is an even vertex magic total labeling if and only if t=2,3.
● For every odd integer 5≤n≤9 and an integer t≥2, the t-fold wheel Wn,t is an even vertex magic total labeling if and only if t=n.
● For every even integer 4≤n≤8 and an integer t≥2, the t-fold wheel Wn,t is an even vertex magic total labeling if and only if t=n−2.
In essence, our work has discussed the characterizations of t-fold wheel Wn,t to possess an even vertex magic total labeling for an integer 3≤n≤9. It would be interesting to apply the results of this paper to further study under what conditions for Wn,t will be an even vertex magic, especially for a larger n.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare that they have no conflicts of interest.
[1] | E. Shell, Distribution of a product by several properties, Proceedings of the second symposium in linear programming, 2 (1955), 615–642. |
[2] |
K. B. Haley, New methods in mathematical programming—The solid transportation problem, Oper. Res., 10 (1962), 448–463. https://doi.org/10.1287/opre.10.4.448 doi: 10.1287/opre.10.4.448
![]() |
[3] | P. Pandian, D. Anuradha, A new approach for solving solid transportation problems, Appl. Math. Sci., 4 (2010), 3603–3610. |
[4] |
V. Vidhya, K. Ganesan, An alternate method for finding optimal solution to solid transportation problem under fuzzy environment, IOP Conf. Ser. Mater. Sci. Eng., 912 (2020), 062047. https://doi.org/10.1088/1757-899X/912/6/062047 doi: 10.1088/1757-899X/912/6/062047
![]() |
[5] |
H. A. E. W. Khalifa, P. Kumar, M. G. Alharbi, On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment, J. Intell. Syst., 30 (2021), 620–635. https://doi.org/10.1515/jisys-2020-0095 doi: 10.1515/jisys-2020-0095
![]() |
[6] |
D. Chhibber, D. C. S. Bisht, P. K. Srivastava, Pareto-optimal solution for fixed-charge solid transportation problem under intuitionistic fuzzy environment, Appl. Soft Comput., 107 (2021), 107368. https://doi.org/10.1016/j.asoc.2021.107368 doi: 10.1016/j.asoc.2021.107368
![]() |
[7] |
T. Anithakumari, B. Venkateswarlu, A. Akilbasha, Optimizing a fully rough interval integer solid transportation problems, J. Intell. Fuzzy Syst., 41 (2021), 2429–2439. https://doi.org/10.3233/JIFS-202373 doi: 10.3233/JIFS-202373
![]() |
[8] |
A. Baidya, U. K. Bera, Solid transportation problem under fully fuzzy environment, Int. J. Math. Oper. Res., 15 (2019), 498–539. https://doi.org/10.1504/IJMOR.2019.102997 doi: 10.1504/IJMOR.2019.102997
![]() |
[9] |
S. Ghosh, K. H. Küfer, S. K. Roy, G. W. Weber, Type-2 zigzag uncertain multi-objective fixed-charge solid transportation problem: Time window vs. preservation technology, Cent. Eur. J. Oper. Res., 31 (2023), 337–362. https://doi.org/10.1007/s10100-022-00811-7 doi: 10.1007/s10100-022-00811-7
![]() |
[10] |
S. Pramanik, D. Kumar, J. M. Maiti, Multi-objective solid transportation problem in imprecise environments, J. Transp. Secur., 6 (2013), 131–150. https://doi.org/10.1007/s12198-013-0108-0 doi: 10.1007/s12198-013-0108-0
![]() |
[11] |
M. B. Kar, P. Kundu, S. Kar, T. Pal, A multi-objective multi-item solid transportation problem with vehicle cost, volume and weight capacity under fuzzy environment, J. Intell. Fuzzy Syst., 35 (2018), 1991–1999. https://doi.org/10.3233/JIFS-171717 doi: 10.3233/JIFS-171717
![]() |
[12] |
D. Rani, T. R. Gulati, Uncertain multi-objective multi-product solid transportation problems, Sādhanā, 41 (2016), 531–539. https://doi.org/10.1007/s12046-016-0491-x doi: 10.1007/s12046-016-0491-x
![]() |
[13] |
D. Rani, T. R. Gulati, A. Kumar, On Fuzzy Multiobjective Multi-Item Solid Transportation Problem, J. Optim., 2015 (2015), 787050. https://doi.org/10.1155/2015/787050 doi: 10.1155/2015/787050
![]() |
[14] |
P. Kundu, S. Kar, M. Maiti, Multi-objective multi-item solid transportation problem in fuzzy environment, Appl. Math. Model., 37 (2013), 2028–2038. https://doi.org/10.1016/j.apm.2012.04.026 doi: 10.1016/j.apm.2012.04.026
![]() |
[15] | S. Kataoka, A stochastic programing model, Econometrica, 31 (1963), 181–196. |
[16] | W. Szwarc, The transportation problem with stochastic demand, Manage. Sci., 11 (1964), 33–50. |
[17] |
S. Singh, A. Pradhan, M. P. Biswal, Multi-objective solid transportation problem under stochastic environment, Sādhanā, 44 (2019), 105. https://doi.org/10.1007/s12046-019-1094-0 doi: 10.1007/s12046-019-1094-0
![]() |
[18] | A. C. Williams, A stochastic transportation problem, Oper. Res., 11 (1963), 759–770. |
[19] |
K. Holmberg, H. Tuy, A production-transportation problem with stochastic demand and concave production costs, Math. Program. Ser. B, 85 (1999), 157–179. https://doi.org/10.1007/s101070050050 doi: 10.1007/s101070050050
![]() |
[20] | A. Charnes, W. W. Cooper, Chance-constrained programming, Manage. Sci., 6 (1959), 73–79. |
[21] | S. K. Roy, D. R. Mahapatra, Multi-objective interval-valued transportation probabilistic problem involving log-normal, Int. J. Math. Sci. Comput., 1 (2011), 14–21. |
[22] |
D. R. Mahapatra, S. K. Roy, M. P. Biswal, Multi-choice stochastic transportation problem involving extreme value distribution, Appl. Math. Model., 37 (2013), 2230–2240. https://doi.org/10.1016/j.apm.2012.04.024 doi: 10.1016/j.apm.2012.04.024
![]() |
[23] | S. K. Roy, D. R. Mahapatra, M. P. Biswal, Multi-choice stochastic transportation problem with exponential distribution, J. Uncertain Syst., 6 (2012), 200–213. |
[24] | D. R. Mahapatra, S. K. Roy, M. P. Biswal, Multi-objective stochastic transportation problem involving log-normal, J. Phys. Sci., 14 (2010), 63–76. |
[25] |
P. Agrawal, K. Alnowibet, A. W. Mohamed, Gaining-sharing knowledge based algorithm for solving stochastic programming problems, Comput. Mater. Contin., 71 (2022), 2847–2868. https://doi.org/10.32604/cmc.2022.023126 doi: 10.32604/cmc.2022.023126
![]() |
[26] |
H. Kwakernaak, Fuzzy random variables-Ⅱ. Algorithms and examples for the discrete case, Inf. Sci., 17 (1979), 253–278. https://doi.org/10.1016/0020-0255(79)90020-3 doi: 10.1016/0020-0255(79)90020-3
![]() |
[27] |
L. Zhao, N. Cao, Fuzzy random chance-constrained programming model for the vehicle routing problem of hazardous materials transportation, Symmetry, 12 (2020), 1208. https://doi.org/10.3390/SYM12081208 doi: 10.3390/SYM12081208
![]() |
[28] |
G. Maity, V. F. Yu, S. K. Roy, Optimum intervention in transportation networks using multimodal system under fuzzy stochastic environment, J. Adv. Transp., 2022 (2022), 3997396. https://doi.org/10.1155/2022/3997396 doi: 10.1155/2022/3997396
![]() |
[29] |
S. H. Nasseri, S. Bavandi, Fuzzy stochastic linear fractional programming based on fuzzy mathematical programming, Fuzzy Inf. Eng., 10 (2018), 324–338. https://doi.org/10.1080/16168658.2019.1612605 doi: 10.1080/16168658.2019.1612605
![]() |
[30] |
S. Acharya, N. Ranarahu, J. K. Dash, M. M. Acharya, Computation of a multi-objective fuzzy stochastic transportation problem, Int. J. Fuzzy Comput. Model., 1 (2014), 212–233. https://doi.org/10.1504/ijfcm.2014.067129 doi: 10.1504/ijfcm.2014.067129
![]() |
[31] |
S. Dutta, S. Acharya, R. Mishra, Genetic algorithm based fuzzy stochastic transportation programming problem with continuous random variables, Opsearch, 53 (2016), 835–872. https://doi.org/10.1007/s12597-016-0264-7 doi: 10.1007/s12597-016-0264-7
![]() |
[32] |
P. Agrawal, T. Ganesh, Fuzzy fractional stochastic transportation problem involving exponential distribution, Opsearch, 57 (2020), 1093–1114. https://doi.org/10.1007/s12597-020-00458-5 doi: 10.1007/s12597-020-00458-5
![]() |
[33] |
T. Latunde, J. O. Richard, O. O. Esan, O. O. Dare, Sensitivity Analysis of Road Freight Transportation of a Mega Non-Alcoholic Beverage Industry, J. Appl. Sci. Environ. Manage., 24 (2020), 449–454. https://doi.org/10.4314/jasem.v24i3.8 doi: 10.4314/jasem.v24i3.8
![]() |
[34] |
Y. Sun, M. Lang, Bi-objective optimization for multi-modal transportation routing planning problem based on pareto optimality, J. Ind. Eng. Manage., 8 (2015), 1195–1217. https://doi.org/10.3926/jiem.1562 doi: 10.3926/jiem.1562
![]() |
[35] |
V. Kakran, J. Dhodiya, A belief-degree based multi-objective transportation problem with multi-choice demand and supply, Int. J. Optim. Control Theor. Appl., 12 (2022), 99–112. https://doi.org/10.11121/ijocta.2022.1166 doi: 10.11121/ijocta.2022.1166
![]() |
[36] |
A. A. Gessesse, R. Mishra, M. M. Acharya, Solving multi-objective linear fractional stochastic transportation problems involving normal distribution using simulation-based genetic algorithm, Int. J. Eng. Adv. Technol., 9 (2019), 9–17. https://doi.org/10.35940/ijeat.b3054.129219 doi: 10.35940/ijeat.b3054.129219
![]() |
[37] |
S. K. Roy, Multi-choice stochastic transportation problem involving Weibull distribution, Int. J. Oper. Res., 21 (2014), 38–58. https://doi.org/10.1504/IJOR.2014.064021 doi: 10.1504/IJOR.2014.064021
![]() |
[38] |
D. R. Mahapatra, Multi-choice stochastic transportation problem involving weibull distribution, An Int. J. Optim. Control Theor. Appl., 4 (2013), 45–55. https://doi.org/10.11121/ijocta.01.2014.00154 doi: 10.11121/ijocta.01.2014.00154
![]() |
[39] |
A. Das, G. M. Lee, A multi-objective stochastic solid transportation problem with the supply, demand, and conveyance capacity following the weibull distribution, Mathematics, 9 (2021), 1757. https://doi.org/10.3390/math9151757 doi: 10.3390/math9151757
![]() |
[40] |
M. S. Osman, O. E. Emam, M. A. El Sayed, Stochastic Fuzzy Multi-level Multi-objective Fractional Programming Problem: A FGP Approach, Opsearch, 54 (2017), 816–840. https://doi.org/10.1007/s12597-017-0307-8 doi: 10.1007/s12597-017-0307-8
![]() |
[41] |
P. K. Giri, M. K. Maiti, M. Maiti, Fuzzy stochastic solid transportation problem using fuzzy goal programming approach, Comput. Ind. Eng., 72 (2014), 160–168. https://doi.org/10.1016/j.cie.2014.03.001 doi: 10.1016/j.cie.2014.03.001
![]() |
[42] |
L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
![]() |
[43] |
S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129–132. https://doi.org/10.1016/0165-0114(92)90256-4 doi: 10.1016/0165-0114(92)90256-4
![]() |
[44] |
J. J. Buckley, Uncertain probabilities Ⅲ: The continuous case, Soft Comput., 8 (2004), 200–206. https://doi.org/10.1504/ijfcm.2014.067129 doi: 10.1504/ijfcm.2014.067129
![]() |
[45] | S. Nanda, G. Panda, J. Dash, A new methodology for crisp equivalent of fuzzy chance constrained programming problem, Fuzzy Optim. Decis. Mak., 7 (2008), 59–74. |
[46] |
M. Zelany, A concept of compromise solutions and the method of the displaced ideal, Comput. Oper. Res., 1 (1974), 479–496. https://doi.org/10.1016/0305-0548(74)90064-1 doi: 10.1016/0305-0548(74)90064-1
![]() |
[47] |
H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets Syst., 1 (1978), 45–55. https://doi.org/10.1016/0165-0114(78)90031-3 doi: 10.1016/0165-0114(78)90031-3
![]() |
f(uivj) | v1 | v2 | v3 |
u1 | 11 | 18 | 13 |
u2 | 14 | 15 | 9 |
u3 | 17 | 7 | 16 |
f(uivj) | v1 | v2 | v3 | v4 | v5 |
u1 | 15 | 21 | 36 | 38 | 31 |
u2 | 13 | 30 | 29 | 33 | 28 |
u3 | 40 | 24 | 35 | 19 | 17 |
u4 | 34 | 37 | 11 | 23 | 32 |
u5 | 39 | 27 | 26 | 22 | 25 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 | v7 |
u1 | 15 | 51 | 64 | 40 | 29 | 68 | 57 |
u2 | 66 | 43 | 17 | 38 | 19 | 59 | 70 |
u3 | 25 | 45 | 56 | 48 | 61 | 32 | 47 |
u4 | 52 | 31 | 33 | 63 | 54 | 53 | 30 |
u5 | 42 | 35 | 65 | 34 | 60 | 36 | 46 |
u6 | 55 | 67 | 23 | 58 | 49 | 27 | 41 |
u7 | 69 | 50 | 62 | 37 | 44 | 39 | 21 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | v9 |
u1 | 73 | 33 | 102 | 69 | 19 | 79 | 63 | 86 | 91 |
u2 | 43 | 56 | 35 | 62 | 49 | 89 | 85 | 93 | 87 |
u3 | 75 | 61 | 48 | 53 | 104 | 92 | 95 | 23 | 50 |
u4 | 80 | 58 | 72 | 25 | 108 | 97 | 77 | 47 | 39 |
u5 | 51 | 96 | 59 | 81 | 99 | 37 | 55 | 57 | 70 |
u6 | 60 | 100 | 44 | 101 | 67 | 40 | 41 | 90 | 64 |
u7 | 38 | 83 | 103 | 82 | 74 | 78 | 29 | 76 | 46 |
u8 | 88 | 105 | 94 | 65 | 42 | 66 | 52 | 31 | 68 |
u9 | 107 | 21 | 54 | 71 | 45 | 27 | 106 | 98 | 84 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u1) |
18 | 9 | 11 | 13 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u5) | f(u5u6) | f(u6u1) |
40 | 39 | 38 | 13 | 17 | 19 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u5) | f(u5u6) | f(u6u7) | f(u7u8) | f(u8u1) |
70 | 68 | 66 | 64 | 62 | 19 | 13 | 11 |
f(uivj) | v1 | v2 |
u1 | 16 | 1 |
u2 | 14 | 5 |
u3 | 7 | 17 |
u4 | 3 | 15 |
f(uivj) | v1 | v2 | v3 | v4 |
u1 | 36 | 24 | 3 | 29 |
u2 | 11 | 25 | 33 | 1 |
u3 | 26 | 9 | 5 | 30 |
u4 | 22 | 23 | 34 | 15 |
u5 | 37 | 21 | 28 | 27 |
u6 | 7 | 35 | 32 | 31 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 |
u1 | 69 | 3 | 40 | 56 | 39 | 50 |
u2 | 1 | 67 | 5 | 59 | 31 | 35 |
u3 | 21 | 17 | 65 | 7 | 44 | 46 |
u4 | 42 | 38 | 23 | 33 | 9 | 57 |
u5 | 61 | 27 | 37 | 34 | 30 | 15 |
u6 | 41 | 53 | 49 | 32 | 47 | 25 |
u7 | 29 | 60 | 54 | 52 | 63 | 36 |
u8 | 58 | 55 | 45 | 43 | 51 | 48 |
f(uivj) | v1 | v2 | v3 |
u1 | 11 | 18 | 13 |
u2 | 14 | 15 | 9 |
u3 | 17 | 7 | 16 |
f(uivj) | v1 | v2 | v3 | v4 | v5 |
u1 | 15 | 21 | 36 | 38 | 31 |
u2 | 13 | 30 | 29 | 33 | 28 |
u3 | 40 | 24 | 35 | 19 | 17 |
u4 | 34 | 37 | 11 | 23 | 32 |
u5 | 39 | 27 | 26 | 22 | 25 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 | v7 |
u1 | 15 | 51 | 64 | 40 | 29 | 68 | 57 |
u2 | 66 | 43 | 17 | 38 | 19 | 59 | 70 |
u3 | 25 | 45 | 56 | 48 | 61 | 32 | 47 |
u4 | 52 | 31 | 33 | 63 | 54 | 53 | 30 |
u5 | 42 | 35 | 65 | 34 | 60 | 36 | 46 |
u6 | 55 | 67 | 23 | 58 | 49 | 27 | 41 |
u7 | 69 | 50 | 62 | 37 | 44 | 39 | 21 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 | v7 | v8 | v9 |
u1 | 73 | 33 | 102 | 69 | 19 | 79 | 63 | 86 | 91 |
u2 | 43 | 56 | 35 | 62 | 49 | 89 | 85 | 93 | 87 |
u3 | 75 | 61 | 48 | 53 | 104 | 92 | 95 | 23 | 50 |
u4 | 80 | 58 | 72 | 25 | 108 | 97 | 77 | 47 | 39 |
u5 | 51 | 96 | 59 | 81 | 99 | 37 | 55 | 57 | 70 |
u6 | 60 | 100 | 44 | 101 | 67 | 40 | 41 | 90 | 64 |
u7 | 38 | 83 | 103 | 82 | 74 | 78 | 29 | 76 | 46 |
u8 | 88 | 105 | 94 | 65 | 42 | 66 | 52 | 31 | 68 |
u9 | 107 | 21 | 54 | 71 | 45 | 27 | 106 | 98 | 84 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u1) |
18 | 9 | 11 | 13 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u5) | f(u5u6) | f(u6u1) |
40 | 39 | 38 | 13 | 17 | 19 |
f(u1u2) | f(u2u3) | f(u3u4) | f(u4u5) | f(u5u6) | f(u6u7) | f(u7u8) | f(u8u1) |
70 | 68 | 66 | 64 | 62 | 19 | 13 | 11 |
f(uivj) | v1 | v2 |
u1 | 16 | 1 |
u2 | 14 | 5 |
u3 | 7 | 17 |
u4 | 3 | 15 |
f(uivj) | v1 | v2 | v3 | v4 |
u1 | 36 | 24 | 3 | 29 |
u2 | 11 | 25 | 33 | 1 |
u3 | 26 | 9 | 5 | 30 |
u4 | 22 | 23 | 34 | 15 |
u5 | 37 | 21 | 28 | 27 |
u6 | 7 | 35 | 32 | 31 |
f(uivj) | v1 | v2 | v3 | v4 | v5 | v6 |
u1 | 69 | 3 | 40 | 56 | 39 | 50 |
u2 | 1 | 67 | 5 | 59 | 31 | 35 |
u3 | 21 | 17 | 65 | 7 | 44 | 46 |
u4 | 42 | 38 | 23 | 33 | 9 | 57 |
u5 | 61 | 27 | 37 | 34 | 30 | 15 |
u6 | 41 | 53 | 49 | 32 | 47 | 25 |
u7 | 29 | 60 | 54 | 52 | 63 | 36 |
u8 | 58 | 55 | 45 | 43 | 51 | 48 |